Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari
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Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo
Luca Benvenuti Alberto De Santis Lorenzo Farina (Eds.)
Positive Systems Proceedings of the First Multidisciplinary International Symposium on Positive Systems: Theory and Applications (POSTA 2003), Rome, Italy, August 28{30, 2003. With 43 Figures
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Series Advisory Board A. Bensoussan · P. Fleming · M.J. Grimble · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · J.N. Tsitsiklis
Editors Luca Benvenuti Alberto De Santis Lorenzo Farina Dipartimento di Informatica e Sistemistica “Antonio Ruberti” Universit`a di Roma “La Sapienza” Via Eudossiana, 18 00184, Roma, Italia {luca.benvenuti, lorenzo.farina}@uniroma1.it
[email protected]
ISSN 0170-8643 ISBN 3-540-40342-6
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Preface
Mathematical modelling is concerned with choosing the relevant variables of the phenomenon at hand and revealing the relationships among those. Positivity of the variables often emerges as the immediate consequence of the nature of the phenomenon itself. A huge number of evidences are just before our eyes: any variable representing any possible type of resource measured by a quantity such as time, money and goods, buffer size and queues, data packets flowing in a network, human, animal and plant populations, concentration of any conceivable substance you may think of and also – if you haven’t conceived this - mRNAs, proteins and molecules, electric charge and light intensity levels. Moreover, also probabilities are positive quantities, so one should also mention in this list any model such as hidden Markov models and phase–type distributions models. Positive Systems are dynamical systems whose state variables are positive (or at least nonnegative) in value at all times. Such systems have the peculiar property that any nonnegative input and nonnegative initial state generate a nonnegative state trajectory and output at all times. Positive systems have a long and rich history with antecedents in the work of Markov, Perron and Frobenius, Leontieff and Leslie, just to mention a few. The unifying approach of system theory was initiated in the 80’s by David Luenberger in his celebrated book Introduction to Dynamic Systems: Theory, Models and Applications. Chapter VI of his text is devoted to the theory and applications of positive systems. Indeed, to quote Luenberger: It is for positive system that dynamic systems theory assumes one of its most potent forms. From that time on, an impressive number of theoretical and applicative contributions to this field has appeared. This volume contains the proceedings of the First Multidisciplinary Symposium on Positive Systems: Theory and Applications (POSTA 2003) held in Rome, Italy, on August 28–30, 2003. The Symposium aimed to join together researchers working in different areas related to positive systems, in
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order to provide a multidisciplinary forum where they could have the opportunity of exchange ideas and compare results in a unifying framework. The contributions actually well served this aim since they addressed key crosscutting issues of relevance to most thematic areas of positive systems theory and applications. We wish to thank the Program Committee for the outstanding work in reviewing the papers thus providing a substantial contribution to the improvement of the quality of the Symposium. Furthermore, we wish to thank the IEEE Control Systems Society and A.N.I.P.L.A. for their technical sponsorship, and especially all the participants to POSTA 2003 for making this meeting a success. In fact, as Ovid said: the rivers are not very broad near their source: it is the little tributaries that make them wide. The final remark is dedicated to Professors David Luenberger and Jan van Schuppen for their availability, support to the initiative and for enriching the Symposium with their inspired lectures.
Roma August 2003
Luca Benvenuti Alberto De Santis Lorenzo Farina
Organization
Program Committee Dirk Aeyels (Universiteit Gent, Belgium) Brian D.O. Anderson (Australian National University, Canberra, Australia) Georges Bastin (CSEM, Louvain, Belgium) Luca Benvenuti (Universit` a di Roma “La Sapienza”, Italy) Franco Blanchini (Universit` a di Udine, Italy) Vincent Blondel (University of Louvain, Belgium) Rafael Bru (Universidad Politecnica de Valencia, Spain) Bart De Moor (Catholic University of Lovain, Belgium) Alberto De Santis (Universit` a di Roma “La Sapienza”, Italy) Elena De Santis (Universit` a dell’Aquila, Italy) Lorenzo Farina (Universit` a di Roma “La Sapienza”, Italy) Stephane Gaubert (INRIA Roquencourt, France) Jean-Luc Gouz´e (INRIA Sophia Antipolis, France) Diederich Hinrichsen (University of Bremen, Germany) Tadeusz Kaczorek (Warsaw Technical University, Poland) Ventsi Rumchev (Curtin University, Perth, Australia) Jan H. van Schuppen (CWI, Amsterdam, The Netherlands) Anton A. Stoorvogel (Eindhoven University of Technology, The Netherlands) Elena Valcher (Universit` a di Padova, Italy)
Organizing Committee Luca Benvenuti (Universit` a di Roma “La Sapienza”, Italy) Alberto De Santis (Universit` a di Roma “La Sapienza”, Italy) Lorenzo Farina (Universit` a di Roma “La Sapienza”, Italy)
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Additional Referees Bego˜ na Cant´ o Colomina Rafael Cant´ o Colomina Paolo Caravani Carmen Coll Aliaga Tobias Damm Greg Gamblev Vladimir Kharitonov Javier Oliver Villarroya Beatriz Ricarte Benedito Sergio Romero Vivo Elena Sanchez Juan
Contents
Abstracts of Plenary Talks Positive Random Systems with Application to Investment David G. Luenberger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Rational Positive Systems for Reaction Networks Jan H. van Schuppen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Invited Session Max-plus Algebra Organizer: L. Hardouin Min-plus and Max-plus System Theory Applied to Communication Networks Jean-Yves Le Boudec, Patrick Thiran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Reachability and Invariance Problems in Max-plus Algebra St´ephane Gaubert, Ricardo Katz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Modelling of Urban Bus Networks in Dioids Algebra S´ebastien Lahaye, Laurent Houssin, Jean-Louis Boimond . . . . . . . . . . . . . 23 Modal Logic and Dioids Christiano P. Pessanha, Rafael Santos-Mendes . . . . . . . . . . . . . . . . . . . . . . 31 Monotone Linear Dynamical Systems over Dioids Laurent Truffet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Optimal Control for (max,+)-linear Systems in the Presence of Disturbances Mehdi Lhommeau, Laurent Hardouin, Bertrand Cottenceau . . . . . . . . . . . . 47
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Invited Session Continuous and Hybrid Petri Nets Organizers: A. Giua and M. Silva Unforced Continuous Petri Nets and Positive Systems Manuel Silva, Laura Recalde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Reachability Graph for Autonomous Continuous Petri Nets Ren´e David, Hassane Alla . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Modeling Hybrid Positive Systems with Hybrid Petri Nets Marco Gribaudo, Andr´ as Horv´ ath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Simulation and Control of a Bottling Plant using First-order Hybrid Petri Nets Roberta Armosini, Alessandro Giua, M. Teresa Pilloni, Carla Seatzu . . . 79 Invited Session Modelling and Identification of Biological Systems Organizer: M. Saccomani Parameter Identifiability of Nonlinear Biological Systems Mariapia Saccomani, Stefania Audoly, Giuseppina Bellu, Leontina D’Angi` o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Towards Whole Cell “in Silico” Models for Cellular Systems: Model Set-up and Model Validation Andreas Kremling, Katja Bettenbrock, Sophia Fischer, Martin Ginkel, Thomas Sauter, Ernst Dieter Gilles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Guaranteed Parameter Estimation for Cooperative Models Michel Kieffer, Eric Walter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Modeling and Simulation of Genetic Regulatory Networks Hidde de Jong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Qualitative Analysis of Regulatory Graphs: A Computational Tool Based on a Discrete Formal Framework Claudine Chaouiya, Elisabeth Remy, Brigitte Moss´e, Denis Thieffry . . . 119 A Reconstruction Algorithm for Gene Regulatory Sparse Networks using Positive Systems Ilaria Mogno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
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Invited Session Positive Modelling and Control of Biological Systems Organizers: G. Bastin and J.L. Gouz´ e The Basic Reproduction Number in a Multi-city Compartmental Epidemic Model Julien Arino, Pauline van den Driessche . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Stability Analysis of a Metabolic Model with Sequential Feedback Inhibition Yacine Chitour, Fr´ed´eric Grognard, Georges Bastin . . . . . . . . . . . . . . . . . . 143 Differential Systems with Positive Variables Jean-Luc Gouz´e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Positivity and Invariance Properties of Nonisothermal Tubular Reactor Nonlinear Models Mohamed Laabissi, Mohamed E. Achhab, Joseph J. Winkin, Denis Dochain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 A Feedback Perspective for Chemostat Models with Crowding Effects Patrick De Leenheer, David Angeli, Eduardo D. Sontag . . . . . . . . . . . . . . . 167 Positive Control for a Class of Nonlinear Positive Systems Ludovic Mailleret . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Competitive and Cooperative Systems: A Mini-review Morris W. Hirsch, Hal L. Smith . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Invited Session Positivity and Stability Organizer: T. Damm Small-gain Theorems for Predator-prey Systems Patrick De Leenheer, David Angeli, Eduardo D. Sontag . . . . . . . . . . . . . . . 191 Positive Particle Interaction Ulrich Krause . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Stability of Linear Systems and Positive Semigroups of Symmetric Matrices Tobias Damm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
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Invited Session Nonnegative Matrices Organizers: R. Bru and V. Rumchev Digraph-based Conditioning for Markov Chains Stephen J. Kirkland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Paths and Cycles in the Totally Positive Completion Problem Cristina Jord´ an, Juan R. Torregrosa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Completion Problems for Positive Matrices with Minimal Rank Rafael Cant´ o, Ana M. Urbano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Invited Session Reachability and Controllability Organizers: V. Rumchev and R. Bru Some Problems about Structural Properties of Positive Descriptor Systems Rafael Bru, Carmen Coll, Sergio Romero-Vivo, Elena S´ anchez . . . . . . . . 233 Positive Linear Systems Reachability Criterion in Digraph Form Ventsi G. Rumchev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 A Characterization of Reachable Positive Periodic Descriptor Systems Bego˜ na Cant´ o, Carmen Coll, Elena S´ anchez . . . . . . . . . . . . . . . . . . . . . . . . 249 A PLDS Model of Pollution in Connected Water Reservoirs Snezhana P. Kostova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Contributed Papers Positivity for Matrix Systems: A Case Study from Quantum Mechanics Claudio Altafini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 A Simple Food Chain Model with Delay Mario Cavani, Sael Romero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Linear Positive Systems and Phase-type Representations Christian Commault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
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Blending Positive Matrix Pencils with Economic Models Teresa P. de Lima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 On the Positive Reachability of 2D Positive Systems Ettore Fornasini, Maria Elena Valcher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 On Nonnegative Realizations Karl-Heinz F¨ orster, Bela ´ Nagy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Estimation and Strong Approximation of Hidden Markov Models L´ aszl´ o Gerencs´er, G´ abor Moln´ ar-S´ aska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 A Paradigm for Derivatives of Positive Systems Bernd Heidergott . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Nonlinear Positive 2D Systems and Optimal Control Dariusz Idczak, Marek Majewski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 State Feedback Set Stabilization for a Class of Nonlinear Systems Lars Imsland, Bjarne A. Foss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Some Recent Developments in Positive 2D Systems Tadeusz Kaczorek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Nonnegative Infinite Hankel Matrices having a Finite Rank Andrea Morettin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 The Character of an Idempotent-analytic Nonlinear Small Gain Theorem Henry G. Potrykus, Frank Allg¨ ower, S. Joe Qin . . . . . . . . . . . . . . . . . . . . . . 361 Positive Systems with Nondecreasing Controls. Existence and Well-posedness Stanis"law Walczak, Dariusz Idczak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 Reachability and Controllability of Positive Linear Discrete-time Systems with Time-delays Guangming Xie, Long Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Countercurrent Double-pipe Heat Exchangers are a Special Type of Positive Systems Arturo Zavala-R´ıo, Ricardo Femat, Ricardo Romero-Mendez ´ . . . . . . . . . . 385 Note on Structural Properties and Sizes of Eigenspaces of Min-max Functions Qianchuan Zhao, Da-Zhong Zheng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
Author Index
Achhab M.E., 159 Alla H., 63 Allg¨ ower F., 361 Altafini C., 265 Angeli D., 167, 191 Arino J., 135 Armosini R., 79 Audoly S., 87 Bastin G., 143 Bellu G., 87 Bettenbrock K., 95 Boimond J.-L., 23 Bru R., 233 Cant´ o B., 249 Cant´ o R., 225 Cavani M., 273 Chaouiya C., 119 Chitour Y., 143 Coll C., 233, 249 Commault C., 281 Cottenceau B., 47
Fornasini E., 297 Foss B.A., 337 F¨ orster K.-H. , 305
Le Boudec J.Y., 7 Lhommeau M., 47 Luenberger D.G., 1
Gaubert S., 15 Gerencs´er L., 313 Gilles E.D., 95 Ginkel M., 95 Giua A., 79 Gouz´e J.-L., 151 Gribaudo M., 71 Grognard F., 143
Mailleret L., 175 Majewski M., 329 Mogno I., 127 Moln´ ar-S´ aska G., 313 Morettin A., 353 Moss´e B., 119
Hardouin L., 47 Heidergott B., 321 Hirsch M.W., 183 Horv´ ath A., 71 Houssin L., 23 Idczak D., 329, 369 Imsland L., 337 Jord´ an C., 217
D’Angi` o L., 87 Damm T., 207 David R., 63 De Leenheer P., 167, 191 de Lima T.P., 289 de Jong H., 111 Dochain D., 159
Kaczorek T., 345 Katz R., 15 Kieffer M., 103 Kirkland S.J., 215 Kostova S.P., 257 Krause U., 199 Kremling A., 95
Femat R., 385 Fischer S., 95
Laabissi M., 159 Lahaye S., 23
Nagy B., 305 Pessanha C.P., 31 Pilloni M. T., 79 Potrykus H.G., 361 Qin S.J., 361 Recalde L., 55 Remy E., 119 Romero S., 273 Romero-M´endez R., 385 Romero-Vivo S., 233 Rumchev V.G., 241 S´ anchez E., 233, 249 Saccomani M., 87 Santos-Mendes R., 31 Sauter T., 95 Seatzu C., 79 Silva M., 55 Smith H.L., 183 Sontag E.D., 167, 191 Thieffry D., 119
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Author Index
Thiran P., 7 Torregrosa J.R., 217 Truffet L., 39 Urbano A.M., 225 Valcher M.E., 297
van den Driessche P., 135 van Schuppen, J.H., 3 Walczak S., 369 Walter E., 103 Wang L., 377 Winkin J.J., 159
Xie G., 377
Zavala-R´ıo A., 385 Zhao Q., 393 Zheng D.-Z., 393
Cui tria sunt octo, tu me servabis, ut opto, ne voret innumerus cui tria sex numerus.
Positive Random Systems with Application to Investment David G. Luenberger Management Science and Engineering Department, Stanford University, Stanford, 94305 CA, [email protected]
Abstract The theory of positive systems can be extended to random positive systems, along lines originally developed by Bellman and extended by Furstenberg and Kesten. This theory, in turn, can be extended to nonlinear random positive systems that are homogeneous of degree one. These results generalize the Frobenius–Perron theory which defines a maximal growth rate for linear positive systems. An interesting application, which also motivates many of the results, is to the construction of an investment portfolio where individual assets behave randomly, where there are constraints on when assets can be traded, and where there are commissions associated with trading. The theory shows that it is possible to define both a long-term growth rate associated with any such portfolio and a standard deviation of that growth rate. One interesting portfolio choice is the portfolio that will grow as rapidly as possible. The theory itself can be regarded as a synthesis of three major theorems of mathematics: the Law of Large Numbers, the Central Limit Theorem, and the nonlinear version of the Frobenius–Perron Theorem. This powerful combined result may have wide applications. In general, it is computationally difficult to determine the growth rate of a complex system of this type. Two main approaches are simulation and special fixed-point methods. An important version of the theory applies to continuous-time systems. In that case it is possible to define a nonlinear eigenvalue problem that yields the growth rate in a manner similar to solving for the Frobenius-Perron eigenvalue, thus making the application of the general theory of random nonlinear systems almost as simple as application of the familiar linear non-random theory.
L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, p. 1, 2003. Springer-Verlag Berlin Heidelberg 2003
Rational Positive Systems for Reaction Networks Jan H. van Schuppen CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands, [email protected]
Abstract The purpose of the lecture associated with this paper is to present problems, concepts, and theorems of control and system theory for a subclass of the rational positive systems of which examples have been published as models of biochemical cell reaction networks. The recent advances in knowledge for the genome of plants, animals, and humans now lead to increased interest in cell biology. Knowledge is needed on how a cell as a functional unit operates biochemically and how the reaction network is influenced by the genome via the enzymes. In principle it is possible to model the complete biochemical reaction network of a cell though this program has so far been carried out only for small compartments of such networks. Mathematical analysis for such reaction networks then leads to a system of ordinary differential equations or of partial differential equations. Often the ordinary differential equations are of polynomial or of rational form. The number of reactions in a cell can be as high as 15.000 (about half the number of estimated genomes) and the number of chemical compounds as high as 20.000. A detailed mathematical analysis of a mathematical model of the complete cell reaction network may therefore not be possible in the short run. Hence there is an interest to develop procedures to obtain from high-order mathematical models approximations in the form of low-order mathematical models. The formulation of approximate models requires understanding of the dynamics of the system, in particular of its algebraic and graph-theoretic structure and of its rate functions. It is the aim of the author to contribute to this research effort. In this lecture attention is restricted to mathematical models for biochemical cell reaction networks in the form of rational positive systems. These systems are called positive because the state vector represents masses or concentrations of chemical compounds and the external input vectors represent inputs into the network of externally available chemical compounds and of L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 3-5, 2003. Springer-Verlag Berlin Heidelberg 2003
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Jan H. van Schuppen
enzymes produced by the nucleus of the cell. The dynamics of the system is often modelled as a polynomial map but in this lecture attention it is restricted to rational maps (each component equals a quotient of two polynomials). Such a dynamics arises for example in the model of Michealis-Menten kinetics due to a singular perturbation of a bilinear system. The mathematical model of the glycolysis of Trypanosoma brucei is phrased almost entirely in terms of a rational positive system and this model is regarded as realistic, see [2]. A book on biochemical reaction networks is that of R. Heinrich and S. Schuster, see [1]. The subclass of rational positive systems considered in this lecture is specific due to the conditions imposed by the modeling of biochemical cell reaction networks. It is precisely because of these physically determined conditions that the subclass merits further study. The properties of such systems differ to a minor extent from those of polynomial systems considered. The graphtheoretic and the algebraic structure of rational positive systems make the analysis interesting. A book on mathematical control and system theory is [3] and a paper on polynomial positive systems is [4]. The main topics of the lecture are: • The mathematical framework of rational positive systems for biochemical reaction networks. • The system theoretic results on the interconnection and decomposition of rational positive systems, on the realization problem, and the dissipation and conservation properties. • The formulation of control problems for biochemical reaction networks and preliminary concepts and results for these problems.
Acknowledgements The author acknowledges the stimulation and advice provided by Hans V. Westerhoff and Barbara M. Bakker both of the Department of Cell Physiology, of the Faculty of Earth and Life Sciences, of the Vrije Universiteit in Amsterdam. He also acknowledges discussions on the topic of this lecture with Mr. Siddhartha Jha during an internship of the latter at CWI in the Summer of 2001. He thanks Dorina Jibetean for symbolic calculations for the example. He also thanks several unnamed researchers for comments on drafts of the paper.
References 1. R. Heinrich and S. Schuster. The regulation of cellular systems. Chapman and Hall, New York, 1996.
Rational Positive Systems for Reaction Networks
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2. Sandre Helfert, Antonio M. Est´evez, Barbara Bakker, Paul Michels, and Christine Clayton. Roles of triosephosphate isomerase and aerobic metabolism in trypanosoma brucei. Biochem. J., 357:117–125, 2001. 3. E.D. Sontag. Mathematical control theory: Deterministic finite dimensional systems (2nd. Ed.). Number 6 in Graduate Text in Applied Mathematics. Springer, New York, 1998. 4. E.D. Sontag. Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction. IEEE Trans. Automatic Control, 46:1028–1047, 2001.
Min-plus and Max-plus System Theory Applied to Communication Networks Jean-Yves Le Boudec and Patrick Thiran LCA-ISC-I&C, EPFL, Lausanne, Switzerland, [email protected], [email protected] Abstract. Network Calculus is a set of recent developments, which provide a deep insight into flow problems encountered in networking. It can be viewed as the system theory that applies to computer networks. Contrary to traditional system theory, it relies on max-plus and min-plus algebra. In this paper, we show how a simple but important fixed-point theorem (residuation theorem) in min-plus or max-plus algebra can be applied to window flow control.
1 Introduction In this paper, we first review the basic concepts of network calculus, namely the way we characterize flows by arrival curves and network element(s) by service curves, in particular rate-latency service curves (Section 2). A flow x(t) is defined as the cumulative amount of data or bits seen on the data flow in time interval [0, t]. It is therefore a non-decreasing function of time t, which can be continuous or discrete. These tools will enable us to derive some deterministic performance bounds on quantities such delays and backlogs (Section 3), which are defined as follows, for a lossless system with input flow x(t) and output flow y(t): The backlog at time t is x(t) − y(t), the virtual delay at time t is d(t) = inf {τ ≥ 0 : x(t) ≤ y(t + τ )} .
(1)
For packet-switched networks, it is convenient to define a flow by the sequence of packet arrival times X(n) and packet lengths ln , where n is the index of the nth packet of the flow. In Section 3, we mention how the rate-latency service curve can be implemented by a Guaranteed Rate Scheduler. The second part of the paper (Sections 4 and 5) is a systematic method for modeling situations arising in communication networks, as sets of inequalities using min-plus and/or max-plus operators. We will find the maximal and/or minimal solution of these systems of inequalities using a central result of minplus and max-plus algebra using the concept of closure of an operator [3] L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 7-14, 2003. Springer-Verlag Berlin Heidelberg 2003
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Jean-Yves Le Boudec and Patrick Thiran
We apply this theorem to the problem of window-flow control, whether the window size is expressed in bits (min-plus formulation) or packets (max-plus formulation). The interested reader is also referred to the pioneering work of Cruz [7], Chang[5], Agrawal and Rajan[1], as well as to [6, 10].
2 Arrival and service curves To provide guarantees to data flows requires some specific support in the network; as a counterpart, the traffic sent by sources needs to be limited. This is done by using the concept of arrival curve, defined below. Definition 1 (Arrival Curve). Given a wide-sense increasing function α defined for t ≥ 0 (α ∈ F , where F is the set of wide-sense increasing functions), we say that a flow x is constrained by α if and only if for all s ≤ t: x(t) − x(s) ≤ α(t − s). Note that this is equivalent to imposing that for all t ≥ 0 x(t) ≤ inf {α(t − s) + x(s)} = (α ⊗ x)(t) 0≤s≤t
where ⊗ is the min-plus convolution operator. One can always replace an arrival curve α by its sub-additive closure, which is defined as α = inf{δ0 , α, α ⊗ α, . . . , α(n) , . . .} where α(n) = α ⊗ . . . ⊗ α (n times) and δ0 is the “impulse” function defined by δ0 (t) = ∞ for t > 0 and δ0 (0) = 0. In order to provide reservations, flows must be constrained by arrival curves, and network nodes in return need to offer some guarantees to flows. This is done by packet schedulers. The details of packet scheduling are abstracted using the concept of service curve, which we introduce in this section. Definition 2 (Service Curve). Consider a system S and a flow through S with input and output function x and y. We say that S offers to the flow a service curve β if and only if for all t ≥ 0, there exists some t0 ≥ 0, with t0 ≤ t, such that y(t) − x(t0 ) ≥ β(t − t0 ). Again, we can recast this definition as y(t) ≥ inf 0≤s≤t {β(t − s) + x(s)} = (β ⊗ x)(t). The IETF assumes that RSVP routers offer a service curve, which we call the rate-latency service curve, of the form
βR,T (t) = R[t − T ]+ =
½
Network Calculus
9
R(t − T ) if t > T 0 otherwise.
Concatenation of nodes enjoys the same property as in traditional system theory. Assume a flow traverses systems S1 and S2 in sequence. Assume that Si offers a service curve of βi , i = 1, 2 to the flow. Then the concatenation of the two systems offers a service curve of β1 ⊗ β2 to the flow. Bounds for lossless systems with service guarantees [1] can now be obtained as a straightforward applications of the definitions of service and arrival curves. The first theorem says that the backlog is bounded by the vertical deviation between the arrival and service curves: Theorem 1 (Backlog Bound). Assume a flow, constrained by arrival curve α, traverses a system that offers a service curve β. The backlog x(t) − y(t) for all t satisfies: x(t) − y(t) ≤ sup{α(s) − β(s)} s≥0
Let ∆(s) = inf {τ ≥ 0 : α(s) ≤ β(s + τ )}. From (1), ∆(s) is the virtual delay for a hypothetical system which would have α as input and β as output, assuming that such a system exists. Let h(α, β) be the supremum of all values of ∆(s). The second theorem gives a bound on delay for the general case. Theorem 2 (Delay Bound). Assume a flow, constrained by arrival curve α, traverses a system that offers a service curve of β. The virtual delay d(t) for all t satisfies: d(t) ≤ h(α, β).
3 Guaranteed rate scheduler Flows made of variable length packets, such as most Internet flows, introduce some additional subtleties [5, 8], as a packet switching device normally outputs entire packets, and not a continuous data stream. In such a flow,Pthe nth n packet has length ln and arrives at time X(n). We call L(n) = m=1 lm , with L(0) = 0, the sequence of cumulative packet lengths. We assume that lmin = inf {L(n + 1) − L(n)} > 0 n∈N
lmax = sup {L(n + 1) − L(n)} < ∞. n∈N
Consider a network node that has a rate-latency service curve βR,T . The bit-by-bit output of this node, denoted by x0 (t), is fed into a L-packetizer, which is a “device” that transforms such a fluid input x0 (t) into a L-packetized flow y(t) = PL (x0 )(t), where PL (x0 ) = inf n∈N {L(n)1{L(n)>x0 } }. In this expression, 1A denotes the indicator function of event A, namely here 1{L(n)>x0 } = 1
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Jean-Yves Le Boudec and Patrick Thiran
if L(n) > x0 and 0 otherwise. We say that a flow y(t) is L-packetized if PL (y)(t) = y(t) for all t. Another approach to cope with packet-switched networks is to use maxplus algebra instead of min-plus algebra. The L-packetized flow x(t) can indeed be described by the sequence {L(n), n ∈ N} and by the sequence of packet arrival times {X(n), n ∈ N} with the convention that X(0) = 0. The sequence of departure times at the output of the packetizer is denoted by {Z(n), n ∈ N}. These two sequences are linked by the following recursion, which defines the guaranteed rate scheduler, with rate R and delay T : V (0) = 0 V (n) = X(n) ∨ V (n − 1) + ln /R Y (n) ≤ V (n) + T where ∨ stands for maximum. The proof of this property is found in [9, 10]. Eliminating V from this recursion, we find that the output sequence is linked to the input sequence by Y (n) ≤ max {X(k) + (L(n) − L(k − 1))/R} + T. 1≤k≤n
(2)
4 Two window flow control problems We now describe two examples of window flow control, which we will model as min-plus system and as max-plus system in the next section. A window flow control limits the amount of data admitted into the network in such a way that the total backlog is less than or equal to W . This window size can be expressed in bits (such as in TCP-IP) or in packets. For a fluid or bit-by-bit model of the flows and a window size expressed in bits, the min-plus framework is the most convenient. For a packet model of the flow and a window size expressed in packets, the dual max-plus framework is better suited. Example 1: Window in bits - min-plus approach This example is found independently in [4] and [2]. A data flow a(t) is fed via a window flow controller to a network offering a service curve of β. The window flow controller limits the amount of data admitted into the network in such a way that the total backlog is less than or equal to W bits or data units, where W (the window size) is a fixed number (Figure 1). Call x(t) the flow admitted to the network, and y(t) the output. The definition of the controller means that x(t) is the maximum solution to ½ x(t) ≤ a(t) (3) x(t) ≤ y(t) + W
Network Calculus
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controller a(t)
x(t)
network y(t)
Fig. 1. Window Flow Control, from [4] or [2]
which implies that x(t) = a(t) ∧ (y(t) + W ). Note that we do not know the mapping x(t) → y(t), but we do know that y(t) ≥ (β ⊗ x)(t). We will use the later property to derive a service curve for the closed loop system using min-plus methods. Example 2: Window in bits - min-plus approach This second example is the same as the previous one, but now the L-packetized input flow a is expressed as a sequence of packets arrival times {A(n), n ∈ N}. The network service curve βR,T is replaced by a guaranteed rate scheduler, with rate R and delay T . The window flow controller limits the amount of data admitted into the network in such a way that the total backlog is less than or equal to w packets. Call {X(n), n ∈ N} the sequence of admitted packets arrival times in the network, and {Y (n), n ∈ N} the sequence of exit times. The definition of the controller means that X(n) is the minimum solution to ½ X(n) ≥ A(n) (4) X(n) ≥ Y (n − w). Note that we do not know the mapping X(n) → Y (n), but we do know that Y (n) verifies (2). We will see that we can use this expression and max-plus methods to compute an upper bound on the exit time Y (n) of the nth packet.
5 Space method The examples above involve particular types of operators Π : F → F, which are denoted as follows • • • •
Min-plus convolution: Cσ (x)(t) = (σ ⊗ x)(t) = inf 0≤s≤t {σ(t − s) + x(s)}, Packetization: PL (x)(t) = inf n∈N {L(n)1L(n)>x(t) }, Shift operator: Sw (X)(n) = X(n − w), o n + X(k) + Max+ linear operator: LR,T (X)(n) = max1≤k≤n L(n)−L(k−1) R T. We also define a set of properties, which are direct applications of [3]:
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Jean-Yves Le Boudec and Patrick Thiran
• Π is upper-semi-continuous (resp. lower-semi-continuous) if for any decreasing (resp. increasing) sequence of trajectories (xi (t)) we have inf i Π(xi ) = Π(inf i xi ) (resp. supi Π(xi ) = Π(supi xi )). Cσ , PL are upper semi-continuous, LR,T is lower semi-continuous, and Sw is both. • Π is min-plus linear (resp. max-plus linear if it is upper-semi-continuous (resp. lower-semi-continuous) and Π(x + K) = Π(x) + K for all constant K. Cσ is min-plus linear, LR,T is max-plus linear, Sw is both and PL is neither one. In this paper we apply Theorem 4.70, item 6 [3] to the problems formulated in the previous section. Theorem 3. Let Π be an operator F → F. If it is upper-semi-continuous, then for any fixed function a ∈ F, the problem x ≤ Π(x) ∧ a
(5)
has one maximum solution, given by n o x = Π(a) = inf a, Π(a), (Π ◦ Π)(a), . . . , Π (n) (a), . . . . where (Π ◦ Π)(a) = Π(Π(a)) and Π (n) = Π ◦ Π ◦ . . . ◦ Π (n times). If Π is lower-semi-continuous, then for any fixed function a ∈ F, the problem x ≥ Π(x) ∨ a
(6)
has one minimum solution, given by n o x = Π(a) = sup a, Π(a), (Π ◦ Π)(a), . . . , Π (n) (a), . . . . The theorem is proven in [3, 10].
6 Application to the examples of window flow control Example 1: Min-plus approach. Define Π as the operator that maps x(t) to y(t). From Equation (3), we derive that x(t) is the maximum solution to x ≤ a ∧ (Π(x) + W )
(7)
The operator Π can be assumed to be upper-semi-continuous, but not necessarily linear. We know from Theorem 3 that (7) has one maximum solution, and that it is given by x(t) = (Π + W )(a)(t). Now we have Π(x) + W ≥ β ⊗ x + K. One easily shows that x ≥ (β + W ) ⊗ a. It means that the complete system offers a service curve βwfc = β ⊗ (β + W ). For example, if β = βR,T then the service curve of the closed-loop system is the function represented on Figure 2. When RT ≤ W , the window does not
Network Calculus
βwfc1(t) = β(t) = R[t-T]+
13
βwfc1(t)
R W t
T
R
W
t
T 2T 3T 4T Case 1: RT ≤ W
Case 2: RT > W
Fig. 2. The service curve βwfc of the closed-loop system with static window flow control, when the service curve of the open loop system is βR,T with RT ≤ W (left) and RT > W (right).
add any restriction on the service guarantee offered by the open-loop system, as in this case βwfc = β. If RT > W on the other hand, the service curve is smaller than the open-loop service curve. Example 2: Max-plus approach. Define Π as the operator that maps X(n) to Y (n). From Equation (4), we derive that X(n) is the minimum solution to X ≥ A ∨ (Sw ◦ Π) (X).
(8)
The operator Π can be assumed to be lower-semi-continuous. We know from Theorem 3 that (8) has one minimum solution, and that it is given by X(n) = (Sw ◦ Π)(A)(n). Now, (2) yields that Π ≤ LR,T and hence that Sw ◦ Π ≤ Sw ◦ LR,T by isotonicity. Therefore X(n) ≤ (Sw ◦ LR,T )(A)(n), which in turn yields that ´ ³ (9) Y (n) ≤ LR,T ◦ (Sw ◦ LR,T ) (A)(n). We can compute that (2)
(Sw ◦ LR,T )
(A)(n) ≤ (Sw ◦ LR,T ) (A)(n) + T −
(w − 1)lmin R
so that if (w − 1)lmin ≥ RT , X(n) ≤ A(n) ∨ (Sw ◦ LR,T ) (A)(n), and (9) becomes Y (n) ≤ LR,T (A)(n). This shows that if (w − 1)lmin ≥ RT , the window does not add any restriction on the service guarantee offered by the open-loop system.
7 Conclusion Network calculus belongs to what is sometimes called topical (or exotic) algebras, a set of mathematical results, often with high description complexity,
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but offering deep insights into man-made systems such as communication networks. This paper has underlined the importance of a simple fixed point residuation theorem in the networking context. We have illustrated its dual application in both the min-plus and max-plus contexts, by taking two versions of a window flow control problem. Acknowledgment This work was supported by Grant DICS 1830 of the Hasler Foundation, Bern, Switzerland.
References 1. R. Agrawal, R. L. Cruz, C. Okino and R. Rajan, ‘Performance Bounds for Flow Control Protocols’, IEEE Trans. on Networking, vol 7(3), pp 310–323, June 1999. 2. R. Aggrawal and R. Rajan. ‘Performance bounds for guaranteed and adaptive services’, Technical report RC 20649, IBM, December 1996. 3. F. Baccelli, G. Cohen, G. J. Olsder and J.-P. Quadrat. Synchronization and Linearity, An Algebra for Discrete Event Systems, John Wiley and Sons, August 1992. 4. C.S. Chang. ‘A filtering theory for deterministic traffic regulation’, in Proceedings Infocom’97, Kobe, Japan, April 1997. 5. C.S. Chang, Performance Guarantees in Communication Networks, SpringerVerlag, New York, 2000. 6. C.S. Chang, R. L. Cruz, J. Y. Le Boudec, P. Thiran ‘A Min-Plus System Theory for Constrained Traffic Regulation and Dynamic Service Guarantees’, IEEE/ACM Transactions on Networking, vol. 10(6), pp. 805–817, 2002. 7. R. L. Cruz, ‘Quality of service guarantees in virtual circuit switched networks’, IEEE Journal on Selected Areas in Communication, pp. 1048–1056, August 1995. 8. J. Y. Le Boudec, ‘Some properties of variable length packet shapers’ Proceedings of ACM Sigmetrics 2001, Boston, June 2001. 9. J. Y. Le Boudec and G. H´ebuterne, ‘Comments on a deterministic approach on the end-to-end analysis of packet flows in connection-oriented networks’, IEEE/ACM Transactions on Networking, vol. 8, 2000. 10. J. Y. Le Boudec and P. Thiran, Network Calculus: A Theory of Deterministic Queuing Systems for the Internet, Springer-Verlag, vol. LCNS 2050, New York, 2001.
Reachability and Invariance Problems in Max-plus Algebra St´ephane Gaubert1 and Ricardo Katz2 1 2
INRIA, Domaine de Voluceau, 78153, Le Chesnay C´edex, France, [email protected] CONICET, Dep. of Mathematics, Universidad Nacional de Rosario, Avenida Pellegrini 250 2000 Rosario, Argentina, [email protected]
Abstract. We present a synthesis of recent results concerning reachability and invariance problems for max-plus linear dynamical systems. Semigroup membership and orbit problems, reachable spaces, and A, B invariant spaces, are discussed.
1 Introduction The max-plus semiring is the set R ∪ {−∞}, equipped with max as addition, and with the usual sum as multiplication. As is now well known, max-plus linear dynamical systems play a fundamental role in the modeling and analysis of discrete event systems (see [11, 3, 19, 21, 8, 7]). Whereas some basic parts of classical control theory (such as the connection between spectral theory and stability questions, or transfer series methods), have known max-plus analogues, leading to efficient algorithms for discrete event systems, the max-plus adaptation of some other classical results lead to new problems. This short paper is a synthesis of positive and negative results, obtained in the current study of three of such problems: noncommutative reachability problems (section 2), commutative reachability problems (section 3), and A, B invariance problems (section 4). The results of sections 2 and 3 are taken from [17] and [16], respectively. The results of section 4 appear here for the first time.
2 Semigroup membership and orbit problems The following problems can be defined over any semiring. Problem 1 (Matrix reachability). Given n × n matrices A1 , . . . , Ar and M with entries in a semiring S , is there a finite sequence 1 ≤ i1 , . . . , ik ≤ r such that Ai1 · · · Aik = M ?
L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 15-22, 2003. Springer-Verlag Berlin Heidelberg 2003
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St´ephane Gaubert and Ricardo Katz
This problem, which asks whether M belongs to the semigroup generated by A1 , . . . , Ar , may be called more classically the semigroup membership problem. We chose our terminology to show the interplay with the two following problems: Problem 2 (Vector reachability). Given n × n matrices A1 , . . . , Ar and two 1 × n matrices α, η, all with entries in a semiring S , is there a finite sequence 1 ≤ i1 , . . . , ik ≤ r such that αAi1 · · · Aik = η? (The latter problem may be called more classically the orbit problem.) Problem 3 (Scalar reachability). Given n × n matrices A1 , . . . , Ar , a 1 × n matrix α, a n×1 matrix β, all with entries in a semiring S , and a scalar γ ∈ S , is there a finite sequence 1 ≤ i1 , . . . , ik ≤ r such that αAi1 · · · Aik β = γ? When S = (Z, +, ×), and M is the zero matrix, the matrix reachability problem is the well studied mortality problem. Paterson [30] proved that the mortality problem is undecidable, even when n = 3 and r = 2nP + 2, where nP is the minimal number of pairs of words for which Post’s correspondence problem is undecidable (we know from [29] that nP ≤ 7). Paterson’s result was subsequently refined in a series of works by Blondel and Tsitsiklis [5], Cassaigne and Karhumaki [10], Halava and Harju [22], and Bournez and Branicky [1, Prop. 1]. Currently, mortality is known to be undecidable when n = 3 and r = nP + 1 [22], which implies [5, 10] that mortality is also undecidable when n = 3(nP + 1) and r = 2. When γ is zero and S = (Z, +, ×), the scalar reachability problem is equivalent to the classical zero corner problem [26, 23, 10], which is undecidable when n = 3 and r = nP [26], and also when r = 2 and n = 3nP +3 [10, Theorem 2 and § 2.3]. When r = 1, the zero-corner problem is the celebrated (open) Pisot problem (which asks whether an recurrent integer linear sequence has a zero). See [23], [6], and [22] for overviews. Let us now consider the case where S is the max-plus semiring, or rather, in order to make more transparent decision issues, the semiring of max-plus integers, Zmax = (Z ∪ {−∞}, max, +). Then, the vector reachability problem arises in the verification of time properties of discrete event systems. Indeed, it has been shown in several works, including [3, 14, 18], that max-plus linear dynamical systems of the form: ξ(k) = ξ(k − 1)A(k) ,
ξ(k) ∈ Z1×n max
(1)
where the matrix A(k) ∈ Zn×n max is taken from a finite set, arise as models of discrete systems with controlled or random parameters (the control can be a scheduling decision, and ξ(k) represents a vector giving the release times of the different resources of the system, at a given logical instant k). A fundamental discrepancy between the ring of integers Z, and the semiring of max-plus integers, Zmax , is that no free semigroup with at least two letters can be embedded in Zn×n max , because the maximal value of the entries of a product of k matrices taken from a finite subset of Zn×n max grows at most
Reachability and Invariance Problems in Max-plus Algebra
17
linearly with k, so that finitely generated subsemigroups of Zn×n max have polynomial growth function (this observation was made by Krob, and by Simon [32]). This implies that the Post correspondence based undecidability proofs `a la Paterson do not work in Zmax . For this reason, the equality problem for max-plus rational series (in at least two letters) remained open, until Krob [25] showed that this problem is undecidable, by reducing Hilbert’s tenth problem to it. Krob’s proof also yields a negative answer to the scalar reachability problem over Zmax . We show: Theorem 1 ([17]). For r = 2, the matrix and vector reachability problems over the max-plus semiring Zmax are undecidable. Theorem 1 is obtained by combining Krob’s undecidability result with the general result of [17] showing that in any semiring, for any r, r0 , r00 , r000 ≥ 2, the scalar reachability problem for r matrices is (Turing) equivalent to the scalar reachability problem for r0 matrices, which reduces to the vector reachability problem for r00 matrices, which is equivalent to the matrix reachability problem for r000 matrices. A (large, probably coarse) fixed dimension n for which the problem is undecidable could be explicitly computed from Krob’s proof, since Hilbert’s tenth problem remains undecidable for instances of bounded degree and bounded number of variables [27]. The r ≥ 2 bound is optimal, since when r = 1, the matrix reachability problem in Zmax is decidable [13, 24]. We also remark in [17] that the scalar, vector, and matrix reachability problems are decidable for a class of semirings, which includes the tropical semiring Nmin = (N∪{+∞}, min, +), the boreal semiring, Nmax = (N∪{−∞}, max, +), ¯ max = (N ∪ {±∞}, max, +) (with (−∞) + (+∞) = −∞), and its completion N as well as classical semirings such as (N, +, ×).
3 Max-plus reachable spaces Let us know consider the max-plus linear system x(k) = Ax(k − 1) ⊕ Bu(k), y(k) = Cx(k),
q×1 x(k) ∈ Zn×1 max , y(k) ∈ Zmax , (2)
where A, B, C are matrices with entries in Zmax , of dimension n × n, n × p, and q × n, respectively. We know from [3] that (2) represents the behavior of a class of discrete event systems called Timed Event Graphs. We call reachable space in time k, and denote by Rk , the set of states x(k) reachable from the initial state x(0) = 0, where 0 denotes the zero vector. We also define the reachable space in arbitrary time, Rω = ∪k≥0 Rk . We shall sometimes write Rk (A, B) or Rω (A, B) to emphasize the dependence in A, B. Introducing the reachability matrices Rk = (B, AB, . . . , Ak−1 B), for k = 1, 2, . . . and Rω = (B, AB, A2 B . . .) , it is readily seen that Rk is the semimodule generated by the columns, or column space, of the matrix Rk , for k ∈ {1, 2, . . . , ω}. (Semimodules over
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St´ephane Gaubert and Ricardo Katz
semirings, generating families, etc., are defined as in the case of modules over rings, see [9].) Identifying matrices with operators, we will write Rk = Im Rk . Dually, we can define observable congruences, see [16]. See also [8, 31] for more background. By comparison with system theory over fields, a difficulty is that reachable spaces need not be finitely generated. To see this, consider the following example from [16]: 1 −∞ −∞ 0 2 −∞ , A= 5 B = −∞ , with (3) −∞ 6 3 −∞
0 1 2 3 4 5 6 ··· Rω = Im Rω , where Rω = −∞ 5 7 9 11 13 15 · · · . −∞ −∞ 11 14 17 20 23 · · ·
(4)
The semimodules R4 and R12 are shown on Figure 1. To interpret the figure, recall the theorem of Moller [28] and Wagneur [33], which shows that a finitely generated subsemimodule of a free semimodule like Znmax , is minimally generated by its set of extremal rays (as for classical polyhedral cones). Extremal generators are shown by bold points in the figure, and the limiting shape of Rω is readily seen from R12 . For any pair of generators, we showed the line between these two points, which is represented by a broken segment. In order to show generators with infinite coordinates, we used an exponential representation, in which a point x ∈ (R ∪ {−∞})3 is represented by the barycenter of the vertices of a fixed triangle, with respect to the weights exp(βx1 ), exp(βx2 ), exp(βx3 ), for some fixed β > 0. We have no space here to represent observable congruences, which have dual shapes [8, 16]. x3
x3
R4
x1
R12
x2 x1
x2
Fig. 1. Exponential representation of the reachable spaces R4 and R12
This raises the question of computing with such infinitely generated semimodules (in many control problems, computing Rω is not enough, we must
Reachability and Invariance Problems in Max-plus Algebra
19
also perform algebraic operations, like intersection, inverse image by morphisms, etc.) An answer we propose is to define the class of rational semimodules. We only consider here subsemimodules of the free semimodule S n . If G ⊂ S n , we will denote by span G the subsemimodule of S n generated by G. We say that a subsemimodule X ⊂ S n is rational if it has a generating family that is a rational subset of S n , S n being thought of as a monoid under entrywise product. The definition of rational sets of monoids is standard: it is useful here to recall that for commutative monoids, rational sets and semilinear sets coincide [12, 20] (a subset of a monoid (M, ·) is semilinear if it can be written as a finite union of sets of the form {x} · B ∗ , where x ∈ M , B is a finite subset of M , and B ∗ = {1M } ∪ B ∪ B 2 ∪ · · · ). As a straightforward consequence of the cyclicity theorem for reducible max-plus matrices and for rational series (see [13, 24]), which tells that max-plus linear sequences are merges of ultimately geometric sequences, we get: n×p Proposition 1 ([16]). For all A ∈ Zn×n max and B ∈ Zmax , the reachable space Rω (A, B) is a rational semimodule.
For instance, the column space of the matrix Rω in (4) is rational, because the set of columns of Rω can be written as U ∪ ({v} + {w}∗ ), with U = {(0, −∞, −∞)T , (1, 5, −∞)T }, v = (2, 7, 11)T , and w = (1, 2, 3)T . It remains to show that when S = Zmax , rational semimodules are closed under the natural algebraic operations. Whereas operations like sum or Cartesian product, are easily seen to preserve rationality, the difficulty of proving the closure under, say, inverse image by morphisms, or intersection, is that, coming back to finitely generated semimodules, there is still relatively little geometric insight on the structure of the solution set of the general system of p equations with n unknowns (see [4, 19, 2, 15] for existing algorithmic results). A way to avoid solving this difficulty is to use Presburger arithmetics: let us recall the theorem of Ginsburg and Spanier, which shows that the class of rational sets of (Nk , +) coincides with the class of sets defined by Presburger formulas over N, that is, by first order formulas of (N, +, ≤). We state in [16] a small extension of this result to algebraic structures like Zmax , and get: Theorem 2 ([17]). If X , Y ⊂ Znmax , Z ⊂ Zpmax , and W ⊂ (Znmax )2 are 1×n rational semimodules, and if A ∈ Zp×n max , and a, b ∈ Zmax , then, the following sets all are rational semimodules: AX = {Ax | x ∈ X } −1
A
Z = {x ∈
X ª Y = {u ∈ W
⊥
= {x ∈
Znmax Znmax Znmax
(5a)
| Ax ∈ Z}
(5b)
| ∃y ∈ Y, u ⊕ y ∈ X }
(5c)
| ax = bx, ∀(a, b) ∈ W},
(5d)
X > = {(a, b) ∈ (Znmax )2 | ax = bx, ∀x ∈ X } .
(5e)
Thus, spaces which can be derived from reachable spaces and observable congruences have “simple” shapes. However, the price to pay for using Presburger logic is complexity: efficient methods remain to be designed.
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St´ephane Gaubert and Ricardo Katz
4 Max-plus A, B-invariant spaces Let us now consider the max-plus analogues of A, B invariant spaces [34]. n×p n Given matrices A ∈ Zn×n max and B ∈ Zmax , we say that a subspace X ⊂ Zmax is (geometrically) A, B-invariant if x ∈ X ⇒ ∃u ∈ Zpmax , Ax ⊕ Bu ∈ X . We address the problem of finding the maximal A, B-invariant space K∗ contained in a given space K ⊂ Znmax . We set B = Im B. Thus, we have a certain specification K for the state space of the system (2), and we want to find the maximal set K∗ of initial conditions for which there is a control sequence which makes the trajectory x(0), x(1), x(2), . . . stay in K forever. Inspired by the classical case, we introduce the self-map ϕ of the set of subsemimodules of Znmax , such that ϕ(X ) = K ∩ A−1 (X ª B), for all subsemimodules X of Znmax (the operation ª is defined in (5c)). The usual algorithm for computing K∗ , consists in building the sequence X1 = K, Xk = ϕ(Xk−1 ) ⊂ Xk−1 . (In the present max-plus case, Xk can be computed by the elimination algorithm of [4, 19] if K is finitely generated, and by Theorem 2 if K is rational.) As in the classical case, it is easy to see that K∗ ⊂ Xω := ∩k≥1 Xk , and that if the sequence Xk is ultimately stationary, its limit is equal to K∗ . The difficulty of the max-plus case, which is reminiscent of difficulties of the theory over rings, is that there are infinite decreasing sequences of subsemimodules of Znmax , so that Xk need not stationarize. However, the sequence Xk does stationarize under some finiteness conditions, as we next show. Define the max-plus parallelism relation ∼ over Znmax by u ∼ v if u = λv, for some λ ∈ Z (so that ui = λ + vi , with the usual notation), and define the volume of a semimodule X , to be the cardinality of X / ∼, i.e., the number of lines in X , including the trivial line generated by the zero vector. Theorem 3. Assume that K ⊂ Znmax has finite volume, which is the case in particular if K = Im K, where K ∈ Zn×s max has only finite entries. Then, for n×p all A ∈ Zn×n and B ∈ Z , the maximal A, B-invariant space, K∗ is finitely max max generated, and K∗ = Xk , for k large enough. Proof (Sketch). From Xk ⊂ Xk−1 ⊂ K, we deduce that vol Xk ≤ vol Xk−1 ≤ vol K < ∞ if K has finite volume, and since vol Xk ∈ N, vol Xk = vol Xk−1 holds for some k. Then, Xk = Xk−1 . The volume of K = Im K can be bounded in terms of the additive version of Hilbert’s projective metric: for all u, v ∈ Zn , define kukH = maxi ui − mini ui , and ∆H (K) = max kukH = max kK·i kH , u∈K\{0}
1≤i≤s
where K·i denotes the i-th column of K. Then vol K ≤ 1 + (∆H (K) + 1)n−1 . Theorem 3 is useful in many practical problems, since finite volume conditions are typically satisfied when the specification K models some stability requirements (bounded inter-event delays). However, we would like to compute K∗ under more general circumstances, for instance when K is a rational semimodule. Another problem is to determine for which semimodules K
Reachability and Invariance Problems in Max-plus Algebra
21
there is a linear feedback u(k) = F x(k − 1) such that the closed loop system x(k) = (A ⊕ BF )x(k − 1) leaves K∗ invariant. Nevertheless, if K∗ is known, and is finitely generated, the existing results on max-plus linear equations [4, 19, 2, 15] allow us to decide whether such a linear feedback exists (and to compute it). Acknowledgement. The authors thank Vincent Blondel, Guy Cohen, Jean-Jacques Loiseau, and Jean-Pierre Quadrat, for helpful comments.
References 1. O. Bournez and M. Branicky. The mortality problem for matrices of low dimensions. Theory of Computing Systems, 35(4):433–448, 2002. 2. P. Butkoviˇc and R. Cuninghame-Green. The equation A ⊗ x = B ⊗ y over (R ∪ {−∞}, max, +). Theor. Comp. Sci., 293, 2003. 3. F. Baccelli, G. Cohen, G. Olsder, and J. Quadrat. Synchronization and Linearity. Wiley, 1992. 4. P. Butkoviˇc and G. Heged¨ us. An elimination method for finding all solutions of the system of linear equations over an extremal algebra. Ekonomickomatematicky Obzor, 20, 1984. 5. V. D. Blondel and J. N. Tsitsiklis. When is a pair of matrices mortal? Information Processing Letters, 63:283–286, 1997. 6. V. D. Blondel and J. N. Tsitsiklis. A survey of computational complexity results in systems and control. Automatica, 36:1249–1274, 2000. 7. J.-Y. L. Boudec and P. Thiran. Network calculus. Number 2050 in LNCS. Springer, 2001. 8. G. Cohen, S. Gaubert, and J. Quadrat. Max-plus algebra and system theory: where we are and where to go now. Annual Reviews in Control, 23:207–219, 1999. 9. G. Cohen, S. Gaubert, and J.-P. Quadrat. Duality and separation theorems in idempotent semimodules. eprint arXiv:math.FA/0212294, 2002. 10. J. Cassaigne and J. Karhum¨ aki. Examples of undecidable problems for 2generator matrix semigroups. Theoret. Comput. Sci., 204(1-2):29–34, 1998. 11. G. Cohen, P. Moller, J. Quadrat, and M. Viot. Algebraic tools for the performance evaluation of discrete event systems. IEEE Proceedings: Special issue on Discrete Event Systems, 77(1), Jan. 1989. 12. S. Eilenberg and M. Sch¨ utzenberger. Rational sets in commutative monoids. J. Algebra, 13:173–191, 1969. 13. S. Gaubert. Rational series over dioids and discrete event systems. In Proc. of the 11th Conf. on Anal. and Opt. of Systems: Discrete Event Systems, number 199 in Lect. Notes. in Control and Inf. Sci, Sophia Antipolis, June 1994. Springer. 14. S. Gaubert. Performance evaluation of (max,+) automata. IEEE Trans. on Automatic Control, 40(12):2014–2025, Dec 1995. 15. S. Gaubert and J. Gunawardena. The duality theorem for min-max functions. C.R. Acad. Sci., 326:43–48, 1998. 16. S. Gaubert and R. Katz. Rational semimodules over the max-plus semiring and geometric approach of discrete event systems. eprint arXiv:math.OC/0208014, 2002.
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17. S. Gaubert and R. Katz. Reachability problems for products of matrices in semirings. preprint, 2002. 18. S. Gaubert and J. Mairesse. Modeling and analysis of timed Petri nets using heaps of pieces. IEEE Trans. Automat. Control, 44(4):683–697, 1999. 19. S. Gaubert and M. Plus. Methods and applications of (max,+) linear algebra. In R. Reischuk and M. Morvan, editors, STACS’97, number 1200 in LNCS, L¨ ubeck, 1997. Springer. 20. S. Ginsburg and E. H. Spanier. Semigroups, Presburger formulas, and languages. Pacific Journal of Mathematics, 16(2), 1966. 21. J. Gunawardena, editor. Idempotency. Publications of the Isaac Newton Institute. Cambridge University Press, 1998. 22. V. Halava and T. Harju. Mortality in matrix semigroups. Amer. Math. Monthly, 108(7):649–653, 2001. 23. T. Harju and J. Karhum¨ aki. In G. Rozenberg and A. Salomaa, editors, Handbook of formal languages, volume 1. Springer, Berlin, 1997. 24. D. Krob and A. B. Rigny. A complete system of identities for one letter rational expressions with multiplicities in the tropical semiring. J. Pure Appl. Algebra, 134:27–50, 1994. 25. D. Krob. The equality problem for rational series with multiplicities in the tropical semiring is undecidable. Int. J. of Algebra and Comput., 3, 1993. 26. Z. Manna. Mathematical Theory of Computations. McGraw-Hill, 1974. 27. Y. V. Matiyasevich. Hilbert’s tenth problem. Foundations of Computing Series. MIT Press, Cambridge, MA, 1993. ´ enements Discrets. Th`ese, Ecole ´ 28. P. Moller. Th´eorie alg´ebrique des Syst`emes ` a Ev´ des Mines de Paris, 1988. 29. Y. Matiyasevich and G. S´enizergues. Decision problems for semi-thue systems with a few rules. In Proceedings LICS’96, pages 523–531. IEEE Computer Society Press, 1996. 30. M. S. Paterson. Unsolvability in 3 × 3 matrices. Studies in Appl. Math., 49:105– 107, 1970. 31. J.-M. Prou and E. Wagneur. Controllability in the max-algebra. Kybernetika (Prague), 35(1):13–24, 1999. 32. I. Simon. Recognizable sets with multiplicities in the tropical semiring. In MFCS’88, number 324 in LNCS. Springer, 1988. 33. E. Wagneur. Moduloids and pseudomodules. 1. dimension theory. Discrete Math., 98:57–73, 1991. 34. W. Wonham. Linear multivariable control: a geometric approach. Springer, 1985.
Modelling of Urban Bus Networks in Dioids Algebra S´ebastien Lahaye, Laurent Houssin, and Jean-Louis Boimond Laboratoire d’Ing´enierie des Syst`emes Automatis´es, 62 Avenue Notre-Dame du Lac, 49000 Angers, France , {lahaye,houssin,boimond}@istia.univ-angers.fr Abstract. We consider the modelling of urban bus networks in dioid algebras. In particular, we show that their dynamic behavior can be modeled by a Min-Max recursive equation.
1 Introduction The evolution of a class of Discrete Event Dynamic System (DEDS), viz those which involve synchronization phenomena, can be described by linear models provided that a particular algebraic structure, called dioid or idempotent semi-ring, is used. A linear system theory has been developed by analogy with conventional theory [1, 3]. Applications of this theory have essentially concerned manufacturing systems [8, 6], communication networks [7] and transportation networks [2, 4]. In the latter, the focus has been on systems such as railway networks, which evolve according to timetables. In these systems, synchronization phenomena follow from planned connections and from respect of timetables. In this paper, we are interested in modelling of urban bus networks whose behaviors differ significantly. In fact, in such systems, synchronizations with timetables occur only at some particular stops (terminus or departure of lines, main stations). In addition, connections between buses are not necessarily planned, but may rather be decided according to various objectives: to absorb peaks of charge in the network, minimize the connection time at intermodal stations, and/or improve the offer of service on strategic itineraries. For those reasons previous models are not appropriate, and we attempt at establishing specific representations for these systems. More precisely, we show that their dynamic behavior can be described by a Min-Max recursive equation. Extending well-known results on fixed-point problems, an ’input/ouput representation’ is also deduced. The outline of the paper is as follows. In §2, we recall elements of dioid theory and principles of DEDS description over dioids. In §3, we study particular L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 23-30, 2003. Springer-Verlag Berlin Heidelberg 2003
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fixed-point equations over complete dioids. Their solutions are useful for the modelling of urban bus networks. More precisely, in §4, we first describe how such networks operate in practice, and we next propose their modelling in dioid algebras.
2 Preliminaries In this section, we give basic notions from the dioid theory and recall succinctly how some DEDS can be modeled in dioid algebras [1, 3]. 2.1 Elements of dioid theory Definition 1. A dioid is a set D endowed with two inner operations denoted ⊕ and ⊗. The sum is associative, commutative, idempotent (∀a ∈ D, a ⊕ a = a) and admits a neutral element denoted ε. The product is associative, distributes over the sum and admits a neutral element denoted e. The element ε is absorbing for the product. A dioid (D, ⊕, ⊗) is complete if it is closed for infinite sums and if multiplication distributes over infinite sums too. Definition 2. A dioid (D, ⊕, ⊗) is endowed with a partial order relation denoted º defined by the following equivalence: a º b ⇔ a = a ⊕ b. A complete dioid has a structure of complete lattice [1, §4.3]. L On this account, the greatest lower bound of two elements exists: a ∧ b = {x¹a,x¹b} x. Note that ∧ generally distributes over ⊕1 , but not over ⊗. We only have a subdistributivity property of ⊗ with respect to ∧: ∀ a, b, c ∈ D, (a ∧ b)c ¹ ac ∧ bc. Finally, the following property, called absorption law, holds true ∀ a, b ∈ D,
a ∧ (a ⊕ b) = a ⊕ (a ∧ b) = a . (1) S Example 1 (Dioid Zmax ). The set Z = Z {+∞, −∞} endowed with the max operator as sum and the classical sum as product is a complete dioid, usually denoted Zmax , with ε = −∞ and e = 0. Example 2 (Dioid Zmax JγK). Let d be a mapping from Z to Zmax . The formal power L serie D(γ) in one variable γ and coefficients in Zmax is defined by: D(γ) = k∈Z d(k)γ k . Let us denote hD(γ), γ k i the coefficient d(k) of D(γ) for γ k . The set of formal power series in variable γ and coefficients in Zmax endowed with operations C(γ)⊕D(γ) : hC(γ)⊕D(γ), γ k i = hC(γ), γ k i⊕hD(γ), γ k i L k and C(γ) ⊗ D(γ) : hC(γ) ⊗ D(γ), γ i = i+j=k hC(γ), γ i i ⊗ hD(γ), γ j i is a dioid denoted Zmax JγK. 1
In all complete dioids considered hereafter, ∧ distributes over ⊕. Nevertheless, complete dioids are not necessarily distributive [1, ex. 4.37]
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2.2 DEDS description over dioids It is now well known that the class of discrete event dynamic systems involving only synchronization phenomena can be seen as linear systems over the particular algebraic structure called dioid. For instance, by dating each event, i.e. by associating with each event indexed x a dater 2 function {x(k)}k∈Z , it is possible to get a linear state representation in Zmax . As in conventional system theory, output {y(k)}k∈Z of a SISO DEDS is then expressed as a convolution of its input {u(k)}k∈Z by its impulse response {h(k)}k∈Z . An analogous transform to Z-transform (used to represent discrete-time trajectories in classical theory) can be introduced for daters. Indeed, one can represent a dater {x(k)}k∈Z by L its γ-transform which is defined Las the following formal power series: X(γ) = k∈Z x(k)γ k . Since γX(γ) = k∈Z x(k)γ k+1 = L k k∈Z x(k − 1)γ , variable γ can be interpreted as the backward shift operator in event domain. Thus, one can express DEDS behavior over the dioid of formal power series in one variable and coefficients in Zmax , denoted Zmax JγK3 (see example 2). In particular, the γ-transform of its impulse response plays the role of transfer matrix.
3 Fixed-point equations over complete dioids In this section, we are interested in solving ”fixed-point” equations f (x) = x, in which f is an isotone (f s.t. a ¹ b ⇒ f (a) ¹ f (b)) mapping from a complete dioid D into D. Well known Tarski’s theorem4 states that f admits a least fixed point which coincides with the least solution of inequation f (x) ¹ x. Formally, we denote µf the least fixed-point of f , then µf = Inf {x | f (x) ¹ x}. NotationL 1 Let f : D → D, we denote f 0 = Id, f n = f ◦ f ◦ . . . ◦ f (n times) and f ∗ = n∈N f n . ThisL’star notation’ applies also for elements a ∈ D: a0 = e, a2 = a ⊗ a and a∗ = n∈N an . Furthermore, we have a∗ = a∗ a∗ = (a∗ )∗ . Let us note that the set of fixed point of f ∗ coincides with the set of prefix point of f (x s.t. f (x) ¹ x) [1, th. 4.70, p. 186] f (x) ¹ x ⇔ f ∗ (x) = x
(2)
Proposition 1. Let D be a complete dioid and h : D → D an isotone mapping. Let w ∈ D, mapping g : D → D is defined by g(x) = h(x) ⊕ w. If condition h(h∗ (w)) ¹ h∗ (w) is satisfied, then µg = h∗ (w). 2 3
4
x(k) denotes the k + 1-th occurence of event x. Actually, since daters are monotone functions, only a sub-dioid of Zmax JγK would be more appropriate to represent γ-transforms of daters (see [1] or [3] for further explanations). Originally stated for mappings defined over complete lattices, this theorem applies over complete dioids due to their ordered structure (see def. 2).
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Proof. According to equivalence (2) we have g(x) = h(x) ⊕ w ¹ x ⇔ h(x) ¹ x and w ¹ x ⇔ h∗ (x) = x and w ¹ x which implies h∗ (w) ¹ w. This means that any any prefix point of g, and a fortiori µg , is greater than h∗ (w). Conversely, if h(h∗ (w)) ¹ h∗ (w) we have g(h∗ (w)) = h(h∗ (w)) ⊕ w ¹ h∗ (w) ⊕ w = h∗ (w) which means that h∗ (w) is a prefix point of g and as a by-product h∗ (w) º µg . Definition 3. A mapping f : D →L D is said L to be lower semi-continuous (l.s.c.) if for every subset C of D, f ( x∈C x) = x∈C f (x). The following corollary is a well known result (see e.g. [1, th. 4.75]). Corollary 1 Let h : D → D be a l.s.c. mapping and g(x) = h(x)⊕w, we have µg = h∗ (w). In particular, the least fixed point of g(x) = ax ⊕ w is µg = a∗ w. Definition 4. An isotone mapping f : D → D is said to be a closure mapping if f º Id and f ◦ f = f . If f is a closure mapping, then f ∗ = f which implies ∀x, f (f ∗ (x)) = f (f (x)) = f (x). With regard to proposition 1, this leads to the following corollary. Corollary 2 Let h : D → D be a closure mapping and g(x) = h(x) ⊕ w, we have µg = h∗ (w). For instance, let g1 (x) = x∗ ⊕ w, we have5 µg1 = w∗ . In the next proposition, we present two ’classes of mappings’ which are neither l.s.c. nor closure mappings, but for which proposition 1 will even so apply. Proposition 2. Let f : D → D be a closure mapping. Mapping h : D → D, h(x) = f (x) ∧ v satisfies h∗ (x) = x ⊕ (f (x) ∧ v) and h(h∗ (x)) ¹ h∗ (x). Proof. If f is a closure mappingLh2 (x) = f (f (x) ∧ v) ∧ v ¹ f (f (x)) ∧ v = f (x) ∧ v, we then have h∗ (x) = i hi (x) = Id ⊕ h(x) = x ⊕ (f (x) ∧ v). Since Id ¹ f and using absorption law (1), we have h(h∗ (x)) = f (x⊕(f (x)∧v))∧v ¹ f (f (x) ⊕ (f (x) ∧ v)) ∧ v = f (f (x)) ∧ v = f (x) ∧ v ¹ h∗ (x). The following corollary directly follows from propositions 1 and 2. Corollary 3 Let f : D → D be a closure mapping. Let v, w ∈ D and g(x) = (f (x) ∧ v) ⊕ w, we have µg = (f (w) ∧ v) ⊕ w. For instance, let g2 (x) = (a∗ x ∧ v) ⊕ w and g3 (x) = (x∗ ∧ v) ⊕ w, we have µg2 = (a∗ w ∧ v) ⊕ w and µg3 = (w∗ ∧ v) ⊕ w.
4 Modelling of public transportation networks In the following, we are interested in the modelling of urban bus networks. In a first part, we will describe how such networks operate. A model in dioids algebra is proposed in a second part. 5
Note that generally (x ⊕ y)∗ 6= x∗ ⊕ y ∗ , thus corollary 1 cannot apply.
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4.1 Exploitation of urban bus networks As presented in [5, 9], traffic exploitation in urban bus networks can be decomposed in the two following stages. Definition of an operating schedule. The ”operating schedule” is established with the aim of optimizing the offer of service according to objectives and exploitation constraints (bus fleet, line layouts, staff hours of work, etc). It is calculated for mean conditions of exploitation. In practical terms, this optimization results in: • the distribution of resources throughout the network: number of buses allocated to each line, drivers distribution, etc. • the synthesis of timetables defining times at which buses should theoretically run at each stop. This operating schedule partially conditions the dynamics of the network. In fact, buses are effectively synchronized with timetables at only some stops such as terminus or departures of lines and/or main stations. Regulation. This stage corresponds to adjustments or adaptations from the operating schedule in reaction to current exploitation conditions. Common conditions leading to such adjustment operations are disturbances: breakdowns of buses, modifications of traffic flows (for instance due to accidents), etc. A supervisor6 may then decide to transfer passengers, stop or reroute buses... Differently, we are here interested in modelling adjustment operations which rather aim at improving the offer of service by attempting: 1. to quickly absorb a planned peak of charge in the network. This operation comes down to postponing buses departures if a sizeable arrival of users is imminent : for instance, near a factory just before closing time, or near a school before home-time... 2. to provide connections at intermodal stations of the networks. Such bus stops are located in or near a station where different modes of transport converge (train, subway, tram etc.). If an arrival of passengers is imminent, then the operation also consists in waiting for and departing as soon as this quota of users has arrived. 3. to improve the travelling time on itineraries having priority. Here, the focus is on itineraries spreading on several bus lines which should be promoted for strategic and/or commercial reasons. With the aim of improving the offer of service on such itineraries, operations then tend to minimize connection times at line changes/switchings. Let us note that, at a given stop, only one of the above objectives is at most satisfied. In fact, the regulation is at the earliest, as specified by the rule below. Rule 1 At a given stop, a bus departs as soon as a quota of users has arrived from one of the origins presented at items 1), 2) and 3). 6
Visualizing evolutions inside the network and communicating with bus drivers.
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4.2 A model for urban bus networks In this section, we propose a model for urban bus networks operating as described in section 4.1. We assume that such a network includes n bus stops denoted S1 , S2 , . . . Sn . We are interested in the departure times of buses from stops. As the description of traffic exploitation, the modelling issues will be decomposed in two stages: nominal dynamics according to the operating schedule, and behaviors induced by operations of regulation. Nominal dynamics imposed by the operating schedule. In the following, let xi (k) denote the departure time for the k + 1-st bus from stop Si . This departure time will be deduced from conditions according to which buses evolve in the network. We assume, without loss of generality, that at the beginning of operation a bus departs from each stop7 . A first and obvious condition is that, before departing, buses arrive at stops. Suppose that stop Sj immediately precedes Si , then this gives rise to xi (k) º aij + xj (k − 1) , k > 1, in which aij denotes the travelling time from Sj to Si . Let x(k) = (x1 (k), x2 (k), . . . xn (k))> , for the whole network this condition can be written in max-algebraic matrix notation x(k) º A ⊗ x(k − 1)
(3)
in which Aij = ε if Sj does not precede Si , otherwise Aij equals to the travelling time from Sj to Si . Another condition is given by the timetable generated for each line. More precisely, at specific stops (see §4.1), buses are synchronized with timetables, that is, they do not depart before the scheduled time. At such a stop Si , we have xi (k) º ui (k), where ui (k) denotes the scheduled departure time for the (k + 1)-st bus from Si . For the whole network, we obtain x(k) º B ⊗ u(k) in which Bij = e if i = j and Si is a specific stop, Bij = ε otherwise. Finally, in addition with (3), we get x(k) º Ax(k − 1) ⊕ Bu(k)
(4)
Behaviors induced by the regulation operations. We assume that peaks of charge described at item 1. are known a priori and can consequently be traduced by a vector of daters ζ(k). Precisely, a coefficient ζi (k) denotes the planned date of arrival at stop Si of the k-st quota of users from these flows. In the same manner, we consider that flows of users from others modes of transports are exogenous to our system (see item 2 of §4.1), and we then assume that their dates of occurrence are known a priori. In practice, we denote ρ(k) the vector of daters representing dates of arrival at bus stops of quotas of users from other modes of transport. We consider that several itineraries having priority (defined at item 3 of §4.1) 7
If no or several bus(es) initially depart from stops, then this results only in indexes modifications. These cases can be dealt exactly as cases of places initially containing no or several token(s) for the modelling of timed event graph [1, §2.5.2]
Modelling of Urban Bus Networks in Dioids Algebra
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have been selected for the considered urban bus network. Each itinerary is indexed by an element α of the alphabet Σ. Let Si be a stop belonging to α, we denote ξiα (k) the date of arrival at Si of the k-st quota of passengers following this itinerary. If Sj precedes Si on itinerary α, but does not belong to the same bus line, users have to walk between these stops. We then have α α is equal to the connection time ⊗ ξjα (k) , k ≥ 1, in which fij ξiα (k) º fij α = ε otherwise. between Sj and Si (e.g. walking time between these stops), fij For the whole network, this inequality writes ξ α (k) º F α ⊗ ξ α (k) , k ≥ 1
(5)
with ξ α = (ξ1α (k), ξ2α (k), . . . , ξnα (k)))> . Differently, if stops Sj and Si follow one another on itinerary α and belong to the same bus line, then we consider that α passengers use bus on this portion. We have ξiα (k) º gij ⊗ xj (k) , k ≥ 1, and α α α globally, ξ (k) º G ⊗ x(k − 1), k ≥ 1, in which Gij = Aij if Sj and Si follow one another on itinerary α and belong to the same bus line, Gα ij = ε otherwise. In association with (5), we deduce for itinerary indexed α the following implicit inequation ξ α (k) º F α ξ α (k) ⊕ Gα x(k − 1) , k ≥ 1 . Since we are interested by the earliest functioning of the network, we select the least solution which is given by (corollary 1) ξ α (k) = F α ∗ Gα x(k − 1) .
(6)
Finally, following rule 1, Eq. (6) as well as vectors ζ and ρ can be gathered in an unique inequality representing the influence of regulation operations: ^ x(k) º F α ∗ Gα x(k − 1) ∧ ζ(k) ∧ ρ(k) . (7) α∈Σ
Aggregate model. Inequalities (4) and (7) model behaviors induced respectively by the operating schedule and by the regulation operations. Taking into account both aspects leads to ³ ^ ´ x(k) = F α ∗ Gα x(k − 1) ∧ ζ(k) ∧ ρ(k) ⊕ Ax(k − 1) ⊕ Bu(k) . α∈Σ
This recurrent equation can be used for the simulation of bus networks. From this ’state equation’, we next deduce an input/output representation which should be more suitable to tackle in future works performance evaluation and control of such systems. With that intention, we establish the γ-transform of previous equation using properties8 ∀α ∈ Σ Gα ¹ A and F α Gα = F α A: ³ ^ ´ x(γ) = F α ∗ Aγx(γ) ∧ ζ(γ) ∧ ξ(γ) ⊕ Aγx(γ) ⊕ Bu(γ) . α∈Σ 8
Deduced from definition of F α and Gα .
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S´ebastien Lahaye, Laurent Houssin, and Jean-Louis Boimond
V Setting h(x) = ( α∈Σ F α ∗ Aγx(γ) ∧ ζ(γ) ∧ ξ(γ)) ⊕ Aγx(γ) corollary 1 applies (since h is l.s.c.) to state the least solution x(γ) = h∗ (Bu(γ)). To make explicit this solution, we furthermore assume that each itinerary α ∈ Σ includes an unique change of bus-line9 . We then have F α 2 = ε and F α Ai F α = ε, i ≥ 1. Calculations using notably proposition 1 and corollary 2 lead finally to: ³ ^ ´ F α ∗ (Aγ)∗ Bu(γ) ∧ ζ(γ) ∧ ξ(γ) ⊕ (Aγ)∗ Bu(γ). x(γ) = (Aγ)∗ α∈Σ
5 Conclusion This work is a first attempt at modelling dynamic behaviors of urban bus networks in dioids algebra. First of all, we have tried to describe their exploitation, i.e., how they operate in practice. Specificities of such systems have then appeared compared to transportation systems which are governed by timetables (e.g. railway networks). We have shown that their dynamic behavior can be described by a Min-Max recurrent equation which can be used for their simulation. An input/ouput representation is also deduced to tackle, in future works, performance evaluation and control of such systems.
References 1. F. Baccelli, G. Cohen, G.J. Olsder, and J.P. Quadrat. Synchronization and Linearity. Wiley, 1992. 2. H. Braker. Algorithms and Applications in Timed Discrete Event Systems. PhD thesis, Delft University of Technology, Dec 1993. 3. G. Cohen, P. Moller, J.P. Quadrat, and M. Viot. Algebraic tools for the performance evaluation of discrete event systems. IEEE Proceedings: Special issue on Discrete Event Systems, 77(1), Jan. 1989. 4. R. de Vries, B. de Schutter, and B. de Moor. On max-algebraic models for transportation networks. In Proceedings of the International Workshop on Discrete Event Systems (WODES’98), Cagliary, Italy, 1998. 5. S. Hayat and S. Maouche. R´egulation du trafic des autobus : am´elioration de la qualit´e des correspondances. Rapport interne LI-TU0192, INRETS, 1997. 6. S. Lahaye, L. Hardouin, and J. L. Boimond. Models combination in (max,+) algebra for the implementation of a simulation and analysis software. Kybernetika, 2003. to appear in special issue on Max-Plus Algebras. 7. Le Boudec J.Y. and Thiran P. Min-plus and max-plus system theory applied to communication networks. In Submitted to POSTA’2003, Roma, 2003. 8. E. Menguy, J.-L. Boimond, L. Hardouin, and J.-L. Ferrier. Just in time control of timed event graphs: Update of reference input, presence of uncontrollable input. IEEE TAC, 45(11):2155–2159, 2000. 9. A. Soulhi. Contribution de l’intelligence artificielle ` a l’aide ` a la d´ecision dans la gestion des syst`emes de transport urbain collectif. Ph. d. thesis, Universit´e des sciences et technologies de Lille, Jan. 2000. 9
This means that each matrix F α has only one coefficient different from ε.
Modal Logic and Dioids Christiano P. Pessanha and Rafael Santos-Mendes DCA/FEEC/Unicamp - C.P.6101, 13083-970 Campinas SP, Brazil, [email protected], [email protected]
Abstract. This paper presents results concerning the relations between a propositional modal logic (NK-logic) and the algebra of dioids. The technique of analytic tableau, a well known proof technique in logic systems, is used in combination with the NK-logic to verify specifications written in dioid algebra. The concept of terminality, herein introduced, allows the establishment of important relations that support the proposition of simple algorithms for the verification of specifications. An example illustrates the application of the proposed algorithm.
1 Introduction This paper presents results concerning the relations between modal logic and the algebra of dioids. A previous paper (Magossi and Santos-Mendes, 1998) shows a strong correspondence between the propositional modal logic called NK and the dioid M = B [[γ, δ]] /(γ ⊕ δ −1 ) where B is the boolean set {0, 1}. Moreover, it proposes a solution for the problem of verification of specifications in dioid M based on the technique of the analytic tableaux in logic NK. The algorithm presented by Magossi and Santos-Mendes (1998) is limited in two senses: there is not an efficient stopping rule and an auxiliary algorithm has to be run in order to detect if an obtained solution is minimal. The present paper further develops the correspondence between NK-logic and dioid M . First it proposes the concept of “termination” of a branch, meaning that a certain situation has to be attained for every logical variable, allowing the end of the algorithm. The main result of the paper is the proof that the above concept is equivalent to the concepts of causality and minimality in dioid M . This result allows the proposition of a new verification algorithm with an efficient stopping rule and capable to discard non-minimal solutions. The paper is organized as follows. Session 2 reviews the logic NK and its relation to dioid theory. Session 3 introduces the concept of terminality and relates it to the concepts of minimality and causality. Session 4 proposes an L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 31-38, 2003. Springer-Verlag Berlin Heidelberg 2003
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Christiano P. Pessanha and Rafael Santos-Mendes
algorithm to the verification of specifications and finally, in session 5, examples illustrate the application of the method.
2 Dioid M and NK-modal logic This session summarizes the results presented in Magossi and Santos-Mendes (1998). Initially the dioid algebra, particularly dioid M , is introduced, then the propositional Modal Logic-NK and its relations to dioid M are presented. A dioid is an algebraic structure hD, ⊕, ⊗i where D is a non-empty set and ⊕ and ⊗ represent closed binary operations in D specifically addition and multiplication; defined such that addition be commutative, addition and multiplication be associative and multiplication be distributive over addition; moreover, null and unitary elements must exist, every element multiplied by the null element produces the null element, and addition is idempotent. A dioid is commutative if multiplication is commutative. Usually, the null element is denoted by ε and the identity element by e. The following relation defines a partial ordering in a generic dioid: a ≥ b if and only if a = a ⊕ b. An element a is equivalent to b modulo z if and only if az ∗ = bz ∗ where a∗ = ⊕i∈N ai . The relation of equivalence modulo z defines equivalence classes over D and the resulting quotient D/z is also a dioid (if operations are properly redefined). L is defined as the set of formal power series in two variables γ and δ, with boolean coefficients and exponents in Z. The set L together with the usual operations for a formal series is¢a dioid. The dioid M is defined ¡ as the quotient L/z, where z = γ ⊕ δ −1 . The dynamics of an event graph can be described through a set of linear algebraic equations written in dioid M . Event graphs are Petri nets such that each place has one and only one input transition, and also one and only one output transition. The parameters of any specific event graph are the initial marking and the delay for each place. In such a graph, each transition can be associated with a variable which assumes values in M . In general, if the transitions are associated with variables xi , i ∈ {1, . . . , n}, then the following n P algebraic equations relate each variable to the others: xi = γ mij δ dij xj , i = j=1
1, · · · , n where mij and dij are, respectively, the initial marking and the delay of the place with an input transition xj and output transition xi . The event graphs herein considered satisfy the properties of structural observability and structural controllability as defined in Baccelli et al. (1992). In modal logic the semantics of classical propositional logic is usually extended by the definition of a setW often called the set of possible worlds, as well as of one or more relations among the members of this set. When interpreting such logic, a formula is said to be true or false at a possible world, and a single formula can be considered true for one world yet false for another in the same interpretation or model. It is the kind of relations admissible between the members of W that is responsible for the distinction between one
Modal Logic and Dioids
33
logic and another. In NK logic, the relations involved are R and S, with the following property: for every w ∈ W , there exist unique and distinct w0 , w00 and w000 such that wRw0 , wSw00 , w00 Rw000 and w0 Sw000 . These relations allow the establishment of a one to one correspondence between any admissible W and the set Z 2 of ordered pairs of integers, so that©each member of W ªwill be associated with an ordered pair of integers: W = w (i, j) | (i, j) ∈ Z2 . The alphabet of NK-logic is extended from the classical propositional logic by the addition of two modal operators: G and D. The semantics of these operators can be summarized as follows. The formula GX is true at a world w (i, j) if and only if X is true at the world w (i − 1, j). Similarly, the formula DX is true at a world w (i, j) if and only if X is true at the world w (i, j − 1). A signed formula is a formula X preceded by the symbols “T ” or “F ”. Intuitively, the expression TX behaves as the formula X and FX as ¬X. Signed formulas allow a classification of all formulas of classical propositional logic in two types: α-formulas and β-formulas. α-formulas are those which behave conjunctively (with components α1 e α2 ) and β-formulas are those which behave disjunctively (with components β1 and β2 ). Some examples of α-formula are F (X → Y ) (whose components are TX and FY), F (X ∨ Y ) and T (X ∧ Y ). Some examples of β-formula are T (X → Y ) (whose components are FX and TY), T (X ∨ Y ) and F (X ∧ Y ). Analytic tableaux are proof procedures which are elaborated in the form of binary trees with a finite number of branches, each constituted by a set of nodes. Each node contains a signed formula and has a world (i, j) associated with it. The objective of the tableau proof is to verify if a given formula X is valid or not. To do this, a tableau is initiated by FX, and one of the branch extension rules is applied. X is shown to be a tautology and can not be disproved if contradictions occur in all branches of the tableau, but if any branch is free of contradictions among its formulas, there is a possibility of disproving the formula and X is proved to be not valid. The following rules are observed when developing a tableau proof. When an α-formula occurs in a tableau, both of its components must be added to the same branch, although when a β-formula occurs, there must be a bifurcation in the tableau, with each new branch containing one of the components of the formula. Formulas TGX and FGX are said to be ν-type formulas containing TX and FX as components, while formulas TDX and FDX are said to be π-type with components TX and FX. These formulas are different from those previously described due to their modal nature. When a ν-formula occurs in a tableau for a world (i, j), its component must be written at world (i − 1, j). Similarly, when a π-formula occurs in a tableau for world (i, j), its component must be written at world (i, j − 1). The set ℘ of propositional variables can be associated with the dioid M by means of an injective function. Given the injective function H : ℘ → M , the set MH is the image of this function. It is thus possible to define a function I : MH → ℘ such that I(H(p)) = p, for any propositional variable p.
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Christiano P. Pessanha and Rafael Santos-Mendes
Given a function H and its corresponding function I, an I-interpretation is defined in NK logic as a truth-value assignment in which for every a ∈ MH and for every i, j ∈ Z: I(a) is true at the world w(i, j) if and only if a ≥ γ i δ j . The following formulas are I-valid (i.e. true in every I-interpretation) and permit the relation of a set of equations in dioid M to a set of I-valid formulas. For any a, b ∈ MH : I(a ⊕ b) ↔ (I(a) ∨ I(b)); Gn Dd I(a) ↔ I(γ n δ d a); GI(a) → I(a); I(a) → DI(a). The reciprocal of the two last formulas can also be stated for, if A is a truth-value assignment of NK logic such that the formulas Gp → p and p → Dp are satisfied for every propositional variable for every world under A, then the truth-value assignment A is an I-interpretation. Moreover, a ≥ b if and only if I(b) → I(a). The above relations allow the statement of the following: Theorem : If xi , i = 1, . . . , n, are elements of dioid M , I : MH → ℘ is a function that defines an I-interpretation and pi are propositional variables ( n P such that pi = I(xi ), then the set Θ of equations: Θ = xi = γ mij δ dij xj , j=1
i = 1, . . . , n} is satisfied if and only if the set Ω of NK-formulas: Ω = {pi ↔ n ∨ Gmij Ddij pj ; Gpi → pi ; pi → Dpi ( i = 1,...,n)}is true in every world of
j=1
the given I-interpretation. A consequence of this theorem is that a performance specification expressed by formula X in NK logic is satisfied by a system which is described by the set Θ of formulas written in dioid M , if and only if formula X is a logical consequence of the set Ω of NK formulas.
3 Terminality, causality and minimality The correspondence established above can not guarantee that an NK-model obtained by the tableau technique correponds to a practically useful solution (i.e. a minimal and causal solution) in dioid M . In Magossi and Santos-Mendes (1998) an auxiliary algorithm is used to test if a solution is minimal. Moreover, since models are unbounded, the original algorithm lacks an efficient stopping rule. The results presented hereafter will show that useless models can be readly discarded making the auxiliary algorithm unnecessary. At the same time, these results will allow the proposition of an efficient stopping rule for the main algorithm. The following definitions relate different possible worlds in an NK-tableau. Definition: Given an open branch in a tableau, a world (i, j) is a vertice of a propositional variable x if the following formulas occur in the given branch: T x(i, j); F x(i − 1, j); F x(i, j + 1).
Modal Logic and Dioids
35
Definition: Given an open branch in a tableau, a vertice (i, j) of a propositional variable xp is said to be connected to a vertice (k, l) of a propositional variable x1 if the following conditions are satisfied: a) There is a subset F ⊂ Ω with the formulas: xp ↔ Gnp−1 Dmp−1 xp−1 ∨ yp−1 ; xp−1 ↔ Gnp−2 Dmp−2 xp−2 ∨ yp−2 ; . . .; x2 ↔ Gn1 Dm1 x1 ∨ y where: yi = ∨ Gαij Dβij xij , i = 1, . . . , p − 1 and xij are propositional variables j
b) The following formulas occur in the indicated worlds: T xp (i, j); T xp−1 (i − np−1 , j − mp−1 ); . . . ; p−1 p−1 P P mp−r ) = (k, l) np−r , j − T x1 (i − r=1
The ordered pair:(
r=1 p−1 P
p−1 P
r=1
r=1
np−r ,
mp−r ) = (i − k, j − l) is called the distance
of the connection. Obviously both elements of this ordered pair are positive and consequently i ≥ k and j ≥ l. The defined relation is transitive but it is not symmetric. Proposition: In an open branch in a tableau, every vertice of every variable (excepted the input variable) is connected to some other vertice of some variable. The following definitions and lemma will relate the above conditions to the concept of causality in dioid M . Definition: Given an open branch in a tableau, a propositional variable is said to be terminated in one of its vertices (i, j) if (i, j) is a vertice and the following formulas occur: F x(i − 1, j − β), ∀β > 0. Definition: In an open branch in a tableau, a terminated propositional variable is said to be impulsively terminated if the following formulas occur: F x(i + α, j + 1∀α > 0. Lemma: In a terminated branch in a tableau, each vertice of termination of a variable is connected to the (unique) vertice of termination of the input variable. Finally, thanks to the fact that the distance of connection is always a pair of positive integers and taking into account the definition of a causal solution (Cottenceau, 1999) the following theorem can be immediatelly stated: Theorem: In a terminated branch in a tableau, every propositional variable corresponds to a causal solution for the equations of the set Θ. Terminated NK-model, are related to solutions in dioid M that, besides being causal, are also unique and minimal. This is established by the following: Lemma: In a terminated branch in a tableau, the relative position of every vertice with respect to the (unique) vertice of the input variable is uniquely determined. Thanks to the completeness of the NK-logic, if there is a model (i.e. a solution for the set of equations Θ) then there is a corresponding branch in the tableau. The following theorem is an immediate consequence of this fact.
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Christiano P. Pessanha and Rafael Santos-Mendes
Theorem: In a terminated branch in a tableau, every propositional variable corresponds to a unique (and therefore minimal) causal solution for the equations of the set Θ. Suppose now that a terminated but non-impulsive model has been obtained. The following theorem guarantees that such a branch can be discarded and that the tableau can look only for impulsively terminated solutions. Theorem: If there exists an NK-model in which the input variable is not impulsive then there exists another NK-model in which the input variable is impulsive.
4 Algorithm The preceding results guarantee that only impulsively terminated branches are useful, allowing to discard all others. Some considerations must be done before the proposition of the main algorithm. The analytic tableau is a tree in which the rule β defines the bifurcations. Therefore the algorithm should mantain one or more stacks to retrieve the search, if a branch is to be discarded (closed, non-terminated or non-impulsive). Besides, since we look for the termination of every variable, one possibility is to use the ”excluded-third law” (actually a β-formula) to chose T x or F x in each world for each variable x. Within the same branch, a sequence of worlds must be visited. In every new world, formulas resulting from the application of π and ν-rules to formulas from the preceding worlds must be introduced. The simplest sequence is formed by worlds in the same column. This sequence can be deviated when, for every variable x, one has T x and the algorithm should proceed with the next column. This procedure must continue until, for every variable x, one has F x in the whole column. This is, potentially, a termination. To abbreviate the search one must first open the branches produced by the use of the ”excluded-third law” and then those produced by other conjunctions in the set Ω. The proposed algorithm is as follows: Algorithm: 1)If initializing then write F z → y; else look for applications of π-rules and ν-rules from preceding worlds; 2)Write all α-formulas and respective components; 3)Write all β-formulas, chose one component and stack bifurcations (stack 2); 4)Apply “ excluded-third law” to all remaining propositional variables and stack bifurcations (stack 1); 5)if branch is not terminated and branch is not discardable and stopping rule is not true then select new world and go to step 1); 6)if branch is terminated then output branch; 7) if stacks are empty then stop; else select new branch from stack 1 (if available) or from stack 2 and go to 1).
Modal Logic and Dioids k7
x3
z
x2
x1
y
x5
37
x4
u
k1
k4
k5
k2
k3
k6
Fig. 1. Event Graph for example
The “stopping rule” establishes the maximal distance (imax, jmax) that a world can be from the initial world (0, 0). It should be chosen by practical motivations or obtained from previous information about the system. Thanks to the “stopping rule”, it is guaranteed that the stacks will eventually be empty and therefore the algorithm will stop.
5 Example Consider the pair of event graphs given in Figure 1 in which clearly one has z = y ∗ : The question is: z = y ∗ ≤ y? (In other words: y = y ∗ ?). The branch presented in Table 1 shows that the answer is no since a model is found corresponding to a solution in which it is not the case that z ≤ y.
6 Conclusion As established in Magossi and Santos-Mendes (1998), a propositional modal logic (NK-logic) exists in complete correspondence with the dioid M . The technique of the analytic tableau, can be used as proof technique for NKlogic, determining if a specification written in dioid M is satisfied. In this paper some properties have been demonstrated establishing the relations between the concepts of terminality, causality and minimality. These properties allow the elimination of branches corresponding to useless solutions, supporting therefore a simple algorithm to verify a given specification. The proposed methodology can be applied to the synthesis of controllers for discrete event dynamic systems. Supposing that it is not the case that (z ≤ y) (i.e. a terminated branch has been found), the question is how to modify one of the given event graphs such that no terminated branch can be found. In a first context, formulas can be added to the set of formulas describing y, in order to close the tableau. These formulas can be associated to places and transitions constituting a feedback controller to system y designed
38
Christiano P. Pessanha and Rafael Santos-Mendes Table 1. Branch for Example
(4,5) Fx4 Fx1 Fu
(4,4) Tk2 Tk1 ∨k7 Tk1 ; Tu
(0,3)
(0,2) (0,1) Fx2 Fx1 Fu
(1,3)
(1,2) (1,1) Fx5 ; FG2 D2 x4 Fx2 ; Fx1 ; Fu FGDx2 ; Tk5 (2,2) (2,1) (2,0) (3,2) (3,1) (3,0)
(2,3) (3,3) Tk4 ; TG2 Dk4 ∨ k1 ; Fx4 Fx1 ; Fu; FG2 Dx4 Tk1 ; T k7 ∨u; Tk7 Tk6 ; Tk3 ∨ GDk5 ; Tk3 (4,3) (4,2) (4,1)
(0,0) Fz →y; Tz; Fy Tk7 ; Tk6 ; Tk3 ∨GDk5 TGDk5 ; FGDx5 ; Fx3 FGx2 ; FGDx2 (1,0)
(4,0)
to satisfy the specification z. Moreover considering the results presented in L¨ uders and Santos Mendes (2002), the inequality x ≤ a∗ is important for the controller design problem.. If a 6= a∗ , what is the ”best” x? Starting from x = a the tableau could indicate how to ”decrease” x (as less as possible) in order to satisfy the given inequality.
References 1. Baccelli, F. L., G. Cohen, G. J. Olsder and J. P Quadrat (1992) - Synchronization and Linearity - An Algebra for Discrete Event Systems - New York, John Wiley and Sons. 2. Cottenceau, B., (1999) – Contribution ` a la Commande de Syst`emes ` a ´ enements Discrets: Synth`ese de Correcteurs pour les Graphes d’Ev´ ´ enements Ev´ Temporis´es dans les Dio¨ıdes. – Doctoral Thesis – LISA/ISTIA/Universit´e d’Angers- 1999 3. L¨ uders, R. and Santos-Mendes, R. (2002) – Generalized Multivariable Control of Discrete Event Systems in Dioids – Wodes’02 - pp. 197 –202 – October 2002 – Zaragoza SP 4. Magossi, J. C. and Santos-Mendes, R. (1998) – Modal Logic based Algorithms for the Verification of Specifications in Discrete Event Systems – Wodes’98 – pp. 508 –513 – August 1998 –Cagliari IT
Monotone Linear Dynamical Systems over Dioids Laurent Truffet1,2 1 2
IRCCyN UMR-CNRS 6597, 1 rue de la No¨e BP 92101, 44321 Nantes Cedex 3, France, [email protected] Ecole des Mines de Nantes, France, [email protected]
Abstract. Linear systems over naturally ordered dioids are other kinds of positive systems than the usual ones over semiring (R+ , +, ·). In this short paper we study some monotonicity concepts of linear systems over dioids inspired by results on monotonicity of Markov chains which are also particular cases of positive systems. We derive a necessary and sufficient condition for monotonicity in a simple case which requires strong assumptions on dioid (lattice distributivity and invertibility of the multiplication law). The result suggests links between monotonicity and positive invariance which plays an important role in control theory and also with aggregation problems of linear systems (i.e. conditions for the existence of aggregated variables and their linear dynamic from which the complete behavior can be retraced).
1 Introduction Comparison methods play an important role when complex dynamical systems have to be simplified keeping the control of the approximation error made. Stochastic orderings are now a well-established topic of research. They lead to powerful bounding methods where realistic stochastic models are too complex for a rigorous treatment. An important literature now exists on the subject (see e.g. the bibliography published by Mosler and Scarsini [8]). The aim of the paper is to present concepts borrowed from stochastic majorization and adapt them to compare iterated linear functions over dioids defined by: x(0) ∈ Sd , x(n + 1) = A ⊗ x(n). And denoted (x(0), A), where (S, ⊕, ⊗) denotes an idempotent semiring such as the (max, +)-semiring (R ∪ {−∞}, max, +), and A ∈ Sd×d . The main ideas we borrow from stochastic majorization results of Markov chains [7] which are particular cases of positive linear systems in the usual algebra are: - the K-comparison between vectors of Sd which is a preorder defined by: def
∀x, y ∈ Sd , x ≤K y ⇔ K ⊗ x ≤ K ⊗ y (componentwise),
where K ∈ S
m×d
.
L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 39-45, 2003. Springer-Verlag Berlin Heidelberg 2003
(1)
40
Laurent Truffet
- And the K-monotonicity of a linear operator A, defined as follows: ∀x, y ∈ Sd , (x ≤K y ⇒ A ⊗ x ≤K A ⊗ y).
(2)
The results of the paper are as follows. We establish a very similar result for the K-comparison of two (⊕, ⊗)-linear systems (x(0), A) and (y(0), B) as the well-known sufficient condition for comparison of Markov chains (see e.g. [10]). We give a necessary and sufficient algebraic characterization of Kmonotonicity for (⊕, ⊗)-linear system. We mention that the proof is based on a linear extension result developed in Cohen et al. [4]. This remark leads to a deeper related question dealing with relations between three concepts: aggregation, monotonicity and positive invariance of linear systems. We present some results on this subject.
2 Preliminaries In this Section we recall basic statements on dioids and residuation theory needed in the paper. More details can be found in e.g. [1], [3] and [5].
2.1 Ordered sets and elements of residuation theory Let (Ω, ≤) be a (partially) ordered set. (Ω, ≤) is W a sup-semilattice, infsemilattice iff any set {ω , ω } ⊂ Ω has a supremum {ω1 , ω2 }, an infimum 1 2 V {ω1 , ω2 }. (Ω, ≤) is a lattice iff (Ω, ≤) is a supand inf-semilattice. (Ω, ≤) is W complete iff any set A ⊂ Ω has a supremum A. A complete ordered set is V def W also a complete lattice because A = {ω ∈ Ω : ∀a ∈ A, ω ≤ a}. A lattice is distributive iff ∧ and ∨ are distributive with respect to (w.r.t) one another. A map f : (Ω, ≤) → (Ω 0 , ¹) where (Ω, ≤) and (Ω 0 , ¹) denote two ordered sets is (≤, ¹)-nondecreasing or monotone if it is a compatible morphism with respect to ≤ and ¹. The map f : (Ω, ≤) → (Ω 0 , ¹) is residuated iff there exists a map f \ : (Ω 0 , ¹) → (Ω, ≤) such that: ∀ω ∈ Ω, ∀ω 0 ∈ Ω 0 , f (ω) ¹ ω 0 ⇔ ω ≤ f \ (ω 0 ). It means that the following definition is sensible: _ def f \ (·) = {ω ∈ Ω : f (ω) ≤ · }. 0 A monotone map f : (Ω, ≤) → (Ω 0 , ¹) where (Ω, are W ≤) and W (Ω , ¹) W complete sets is said to be continuous iff ∀A ⊂ Ω, f ( ≤ A) = ¹ f (A), ≤ W def (resp. ¹ ) denotes supremum w.r.t ≤ (resp. ¹); f (A) = {f (a) : a ∈ A}. The next result provides simple characterization of residuated map over complete ordered sets.
Monotone Linear Systems
41
Result 2.1 Let (Ω, ≤) and (Ω 0 , ¹) be complete sets. A map f : (Ω, ≤) → (Ω 0 , ¹) is residuated iff f is continuous. 2.2 Basic algebraic structures For any set S, (S, ⊕, ⊗, ε, e) is a semiring iff (S, ⊕, ε) is a commutative monoid, (S, ⊗, e) is a monoid, ⊗ distributes over ⊕, and ε is the neutral element for ⊕ which is also absorbing element for ⊗, i.e. ∀a ∈ S, ε ⊗ a = a ⊗ ε = ε, e is the neutral element for ⊗. (S, ⊕, ⊗, ε, e) is an idempotent semiring or a dioid iff (S, ⊕, ⊗, ε, e) is a semiring which internal law ⊕ is idempotent, i.e. ∀a ∈ S, a ⊕ a = a. (S, ⊕, ⊗, ε, e) is a semifield (resp. idempotent semifield) iff (S, ⊕, ⊗, ε, e) is a semiring (resp. idempotent semiring) and (S − {ε}, ⊗, e) is a group, i.e. (S − {ε}, ⊗, e) is a monoid such that all its elements are invertible (∀a ∈ S, ∃a−1 : a ⊗ a−1 = a−1 ⊗ a = e). Let (S, ⊕, ⊗, ε, e) denote any arbitrary semiring. Each element of Sn is a n dimensional column vector. We equip Sn with the laws ⊕ and .: ∀x, y ∈ not. def Sn , (x ⊕ y)i = xi ⊕ yi , ∀s ∈ S, (s.x)i = (s ⊗ x)i = s ⊗ xi , i = 1, . . . , n. n It makes (S , ⊕, .) be a left S-semimodule free finitely generated with basis e(i) = (δ{k=i} )k=1,...,n ; δ{·} = e if assertion {·} is true and ε otherwise. The addition ⊕ and the multiplication ⊗ are naturally extended to matrices with compatible dimension. Any n × p matrix A is associated with a (⊕, ⊗)-linear map A : Sp → Sn . (A)i,j , (A)l,· and (A)·,· denote the (i, j) entry, the lth row (row-vector) and the kth column of matrix A, respectively. Let us consider any dioid (S, ⊕, ⊗, ε, e). We can equip the idempotent commutative monoid (S, ⊕, ε) with the natural order relation ≤ defined by: def
∀a, b ∈ S, a ≤ b ⇔ a ⊕ b = b.
(3)
We say that dioid (S, ⊕, ⊗, ε, e) is complete if it is complete as a naturally ordered set and if the left and right multiplications, λa , ρa : S → S, λa (x) = a ⊗ x, ρa (x) = x ⊗ a are continuous, for all a ∈ S. In such case we adopt the following notations: a\b = λ\a (b), and b/a = ρ\a (b), ∀a, b ∈ S. A typical example of complete dioid is the top completion of an idempotent semifield. Let us note that if a ∈ S is invertible we have: a\b = a−1 ⊗ b and b/a = b ⊗ a−1 . Because ⊕ = ∨, the operations ·/·, ·\· are extended to all matrices with compatible dimensions assuming these matrices are elements of complete Ssemimodule in a natural way: _ def (A\B)i,j = ( {X : A ⊗ X ≤ B (coefficientwise)})i,j = ∧ ak,i \bk,j (4a) k
def
(D/C)i,j = (
_
{X : X ⊗ C ≤ D (coefficientwise)})i,j = ∧ di,l /cj,l l
(4b)
Because (D/C) ⊗ C ≤ D we have the following useful well-known result.
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Laurent Truffet
Proposition 2.1 Let us consider two matrices C and D with entries in a complete dioid S. Then: ∃X, X ⊗ C = D ⇔ D ≤ (D/C) ⊗ C (coefficientwise).
(5)
3 K-comparison of linear systems on dioids Let us consider a complete dioid (S, ⊕, ⊗, ε, e), three (⊕, ⊗)-linear systems (x(0), A), (y(0), B) and (z(0), C), with x(0), y(0), z(0) elements of Sd , A, B and C elements of Sd×d . And the rectangular matrix K ∈ Sm×d . Let ≤K be the preorder defined by (1) in the Introduction. Theorem 3.1 (K-comparison result.) Let us assume that B is K-monotone in the sense of (2, in the Introduction with A = B). If (i). x(0) ≤K y(0) ≤K z(0), (ii). K ⊗ A ≤ K ⊗ B ≤ K ⊗ C (coefficientwise). Then: ∀n, x(n) = A⊗n ⊗ x(0) ≤K C ⊗n ⊗ z(0) = z(n). Proof. The proof is similar to the one of [11]. The fundamental remark is that it is based on the compatibility of the natural order ≤ w.r.t ⊕ and ⊗. 2 The main difficulty is then to give a simpler characterization of the Kmonotonicity.
4 K-monotone linear operator Next results characterize monotonicity of matrices in the sense defined by (2) in the Introduction. Theorem 4.1 (Sufficient condition for K-monotonicity) Let (S, ⊕, ⊗; ≤ ) be an ordered semiring such that ≤ is compatible w.r.t ⊕ on S and ⊗ on set def
S+ = {x ∈ S : ε ≤ x}. A sufficient condition for a matrix A ∈ Sd×d to be K-monotone, with K ∈ Sm×d is : K ⊗ A = Aˆ ⊗ K. ∃Aˆ ∈ Sm×m +
(6)
(6) Proof. . Let us consider x, y ∈ Sd such that x ≤K y. Now, K ⊗ A ⊗ x = Aˆ ⊗ K ⊗ x. Because ≤ is compatible w.r.t ⊕ on S and ⊗ on S+ , then x ≤K y (6) implies Aˆ ⊗ K ⊗ x ≤ Aˆ ⊗ K ⊗ y = K ⊗ A ⊗ y, which ends the proof. 2
Monotone Linear Systems
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Theorem 4.2 (Necessary condition for K-monotonicity) Let S be a distributive idempotent semifield. A necessary condition (which is obviously sufficient) for a matrix A ∈ Sd×d to be K-monotone, with K ∈ Sm×d is K ⊗ A ≤ ((K ⊗ A)/K) ⊗ K (coefficientwise).
(7)
Proof. Let us assume that A is K-monotone. Then, because the componentwise ordering ≤ is antisymmetric we have: ∀x, y, (K ⊗ x = K ⊗ y) ⇒ (K ⊗ A ⊗ x = K ⊗ A ⊗ y). Because of [4], Corollary 6 with G(·) = K ⊗ ·, and F (·) = K ⊗ A ⊗ · there exists a linear map H(·) = H ⊗ ·, with H ∈ Sm×m , such that G = H ◦ F , or equivalently, K ⊗ A = H ⊗ K. Using (5), Proposition 2.1, this is equivalent to K ⊗ A ≤ ((K ⊗ A)/K) ⊗ K (coefficientwise). 2
5 Monotonicity and related fields In this section, giving some results, we only suggest the existence of links between monotonicity, positive invariance of sets, and aggregation of linear systems over algebraic structures. Positive invariance. Let f : Sn → Sn be a map. A set E ⊂ Sn is positively invariant iff f (E) ⊂ E. Inspired by e.g. Hennet [6] and Bitsoris [2], we study a particular set ½ P (K, w) = {x : K ⊗ x ≤ w} (8) w = K ⊗ φ for some φ ∈ Sd . In the usual algebra P (K, w) is called a polyhedral set. Proposition 5.1 Let (S, ⊕, ⊗, ε, e) be a distributive idempotent semifield. Let A be a K-monotone matrix. (P (K, w) is positively invariant by x 7→ A ⊗ x) ⇔ ((K ⊗ A)/K) ⊗ w ≤ w. Proof. Using Theorem 4.2 and (5) the K-monotonicity of A is equivalent to K ⊗ A = ((K ⊗ A)/K) ⊗ K. If P (K, w) is positively invariant, then because φ ∈ P (K, w) it obviously implies ((K ⊗ A)/K) ⊗ w ≤ w. Conversely, let us consider x such that K ⊗ x ≤ w = K ⊗ φ. By K-monotonicity of A, we have K ⊗ A ⊗ x ≤ K ⊗ A ⊗ φ = ((K ⊗ A)/K) ⊗ K ⊗ φ ≤ w. 2 The following proposition is a semimodule version of [2], Proposition 1. Proposition 5.2 Let (S, ⊕, ⊗, ε, e) be a distributive idempotent commutative semifield. The set P (K, w), with wi 6= ε, ∀i = 1, . . . , m is positively invariant by x 7→ A ⊗ x iff v(A ⊗ x) ≤ v(x), ∀x def
def
def
where v : Sd → S, ∀x, v(x) = Kw ⊗ x, with Kw = w−T ⊗ K, w− = −1 T ) and (·)T denotes transpose operator. (w1−1 , . . . , wm
44
Laurent Truffet
Proof. Necessity. If P (K, w) is positively invariant, then (R1): P (K, c ⊗ w), for all c 6= ε is positively invariant. Let us also remark that, by definition of v(·), we have (R2): P (K, c ⊗ w) = {x : v(x) ≤ c}, ∀c 6= ε. Now, for all x: K ⊗ x ≤ v(x) ⊗ w, by definition of v. Then, (R1) implies K ⊗ A ⊗ x ≤ v(x) w, which implies using (R2) that: v(A ⊗ x) ≤ v(x). Sufficiency. x ∈ P (K, w) ⇔ v(x) ≤ e. Thus, v(A ⊗ x) ≤ v(x) ≤ e which is equivalent to A ⊗ x ∈ P (K, w). 2 Now, we just have to remark that A is K-monotone implies A is Kw monotone, thus: (A K-monotone)∧(A⊗x ≤K x) ⇒ ({A⊗n ⊗x} Kw -decreasing) and ({A⊗n ⊗x} Kw -decreasing)
Proposition 5.2
⇔
(P (K, w) is positively invariant).
Aggregation. This part is clearly inspired by [9] and references therein. We consider m < d, the Sd -valued series {x(n)} defined by x(0) ∈ S, x(n + 1) = A⊗x(n), ∀n, and the Sm -valued series {y(n)} defined by y(n) = V ⊗x(n), ∀n, with A ∈ Sd×d , V ∈ Sm×d is defined by (V )i,j = δ{ϕ(j)=i} , 1 ≤ i ≤ m, 1 ≤ j ≤ d, where map ϕ : {1, . . . , d} → {1, . . . , m} is supposed to be nondecreasing surjective. The series {x(n)} is said to be lumpable w.r.t ϕ iff the series {y(n)} satisfies y(n + 1) = Aˆ ⊗ y(n), for some Aˆ ∈ Sm×m . We now focus our attention on two sufficient conditions of lumpability. We say that A is strong lumpable w.r.t ϕ or simply V -lumpable iff ∃Aˆ ∈ Sm×m : V ⊗ A = Aˆ ⊗ V . We say that matrix A is C-coherent w.r.t ϕ iff ˆ V ⊗ C = Im , Im denotes the m × m identity ∃Aˆ ∈ Sm×m : A ⊗ C = C ⊗ A, d×m matrix, and C ∈ S . Proposition 5.3 Let (S, ⊕, ⊗, ε, e) be a distributive idempotent semifield. ( A is V -monotone) ⇔ ( A is V -lumpable) and for any matrix C ∈ Sd×m such that V ⊗ C = Im ( AT is C T -monotone) ⇔ ( A is C-coherent) Proof. It is obvious using (5) and Theorem 4.2.
2
6 Conclusion In this short paper we aim to convince the reader that stochastic orderings research could be adapted in order to simplify linear systems over dioids which modelized important discrete event systems such as manufacturing systems, and networks. To the best of our knowledge this paper is also the first attempt to present links between (linear) monotonicity, positive invariance and aggregation. This part is obviously not complete and we hope that it may suggest some research activities on this subject. Let us mention some related questions about these possible links: What is the importance of the algebraic
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structure? What is the importance of the definition of the (pre)orders on sets? Is it possible to generalize linear monotonicity and make links with other positive invariant sets? Acknowledgements. This work has been supported by the Programme “Optimisation des Processus Industriels“ of the grant between the Govt of France and the region Pays-De-La-Loire.
References 1. F. Baccelli, G. Cohen, G.J. Olsder, and J-P. Quadrat. Synchronization and Linearity. John Wiley and Sons, 1992. 2. G. Bitsoris. On the Positive Invariance of Polyhedral Sets for Discrete-Time Systems. Syst. and Control Letters, 11, 1988. (243-248). 3. T. S. Blyth and M. F. Janowitz. Residuation Theory. Pergamon Press, 1972. 4. S. Cohen, G. Gaubert and J-P. Quadrat. Linear Projectors in the Max-Plus Algebra. In Proceedings of 5th IEEE-Mediterranean Conf. Paphos, Cyprus., July 1997. 5. J.S Golan. The Theory of Semirings With Applications in Mathematics and Theoretical Computer Science, volume 54. Longman Sci. & Tech., 1992. 6. J. C. Hennet. Une Extension du Lemme de Farkas et Son Application au Probl`eme de R´egulation Lin´eaire sous Contraintes . C.R.A.S, t.308, S´erie I, 1989. (415-419). 7. M. Kijima. Markov Processes for Stochastic Modeling. Chapman-Hall, 1997. 8. K. Mosler and M. Scarsini. Stochastic Orders and Applications: A Classified Bibliography. Springer-Verlag, Berlin, 1993. 9. J. P. Quadrat and Max-Plus WG. Min-Plus Linearity and Statistical Mechanics. Markov Processes and Related Fields, 3(4), 1997. (565-597). 10. D. Stoyan. Comparison Methods for Queues and Other Stochastic Models. J. Wiley and Sons, 1983. 11. L. Truffet. Some Ideas to Compare Bellman Chains. Kybernetika-Special issue on max-plus algebra, 39(2), April 2003. (To appear).
Optimal Control for (max,+)-linear Systems in the Presence of Disturbances Mehdi Lhommeau, Laurent Hardouin, and Bertrand Cottenceau Laboratoire d’Ing´enierie des Syst`emes Automatis´es, 62, avenue Notre-Dame du lac 49000 Angers, France, {lhommeau,hardouin,cottence}@istia.univ-angers.fr Abstract. This paper deals with control of (max,+)-linear systems when a disturbance acts on system state. In a first part we synthesize the greatest control which allows to match the disturbance action. Then, we look for an output feedback which makes the disturbance matching. Formally, this problem is very close to the disturbance decoupling problem for continuous linear systems.
1 Introduction The (max,+) working group [1] has developed a linear theory for discrete event systems which are characterized by synchronization phenomena and time-delays. They have also proposed an optimal control law in regards of just in time criterion. Roughly speaking, it consists in computing the latest date of input events (which are controllable) in order to obtain output events before given desired output dates. This control synthesis needs a complete knowledge of the desired output. Since it is an open loop control, it is not robust when disturbances act on the system. In [5] we have proposed a closed loop control approach where the control objective is expressed as a reference model. The controller design is based on the residuation theory applied to particular mappings. Residuation theory makes possible to consider a kind of mapping inversion defined on ordered sets, and then plays naturally a significant role in controller synthesis. This presentation deals with controller design when disturbances act on the system. As in conventional linear systems theory [10], the control is synthesized in order to keep the system state x in the kernel of the output matrix C. Section 2 recalls some algebraic tools and in particular that the kernel of a linear mapping defined on dioids (or lattices) is an equivalence relation. In Section 3 it is shown that our objective is equivalent to match the output due to the disturbance. Then we show that the optimal control is the greatest (in the dioid sense) which keeps the system state in the equivalence class gene-
L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 47-54, 2003. Springer-Verlag Berlin Heidelberg 2003
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Mehdi Lhommeau, Laurent Hardouin, and Bertrand Cottenceau
rated by the disturbance. This means that the inputs are delayed as much as possible in order to match the output due to the disturbance.
2 Elements of dioid and residuation theories 2.1 Dioid theory We first recall in this section some notions from the dioid theory. A general introduction is given in [4], and a detailed introduction can be found1 in [1]. Definition 1 (Dioid). A dioid is a set D endowed with two inner operations denoted ⊕ and ⊗. The sum is associative, commutative, idempotent (∀a ∈ D, a ⊕ a = a) and admits a neutral element denoted ε. The product is associative, distributes over the sum and admits a neutral element denoted e. The element ε is absorbing for the product. Definition 2 (Order relation). An order relation can be associated with a dioid D by the following equivalence : ∀a, b ∈ D, a º b ⇔ a = a ⊕ b. Definition 3 (Complete dioid). A dioid D is complete if it is closed for infinite sums and if the product distributes over infinite sums too. implicit equation a = ax⊕b admits Theorem 1. Over a complete dioid D, theL x = a∗ b as least solution, where a∗ = i∈IN (Kleene star operator) with a0 = e. The Kleene star operator, over a complete dioid D, will be represented by the following mapping K : D → D, x 7→ x∗ . Definition 4 (Kernel [4],[3]). Let C : X → Y be a mapping. We call kernel of C (denoted by ker C), the equivalence relation over X : ker C
x ≡ y ⇔ C(x) = C(y).
(1)
The set of equivalence classes is denoted by X/ ker C and [x]C denotes the equivalence class of x. Remark 1. The usual kernel definition {x ∈ X | C(x) = ε} becomes meaningless in dioid algebra. Each equivalence class contains all the elements which map to the same image, in [4], the term ”fibration” is used. Relation (1) corresponds to the kernel definition of a mapping defined on lattices [6]. 1
An electronic version is available on http://maxplus.org.
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2.2 Residuation theory The residuation theory provides, under some assumptions, optimal solutions to inequalities such as f (x) ¹ b where f is an isotone mapping (f s.t. a ¹ b ⇒ f (a) ¹ f (b)) defined over ordered sets. Some theoretical results are summarized below. Complete presentations are given in [2] and [1]. Definition 5 (Residual and residuated mapping). An isotone mapping f : D → E, where D and E are ordered sets, is a residuated mapping if for all y ∈ E, the least upper bound of the subset {x|f (x) ¹ y} exists and belongs to this subset. It is then denoted f ] (y). Mapping f ] is called the residual of f . When f is residuated, f ] is the unique isotone mapping such that f ◦ f ] ¹ IdE , and f ] ◦ f º IdD where Id is the identity mapping respectively on E and D. Theorem 2 ([1]). Consider the mapping f : E → F where E and F are complete dioids of which the bottom elements are, respectively, denoted L by εE L and εF . Then, f is residuated iff f (εE ) = εF and f ( x∈G x) = x∈G f (x) for each G ⊆ E. Corollary 1. The mappings La : x 7→ ax and Ra : x 7→ xa defined over a complete dioid D are both residuated 2 . Their residuals are usually denoted, respectively, L]a (x) = a\◦ x and Ra] (x) = x/◦ a in (max, +) literature.3 Theorem 3 ([1]). Let D be a complete dioid and A ∈ Dq×m be a matrix with entries in D. Then, A/◦ A is a matrix in Dq×q which verifies A/◦ A = (A/◦ A)∗ .
(2)
2.3 Mapping restriction In this subsection, the problem of mapping restriction and its connection with the residuation theory is addressed. In particular the Kleene star mapping, becomes residuated as soon as its codomain is restricted to its image. Definition 6 (Restricted mapping). Let f : E → F be a mapping and A ⊆ E. We will denote4 f|A : A → F the mapping defined by f|A = f ◦ Id|A where Id|A : A → E, x 7→ x is the canonical injection. Identically, let B ⊆ F with Imf ⊆ B. Mapping B| f : E → B is defined by f = Id|B ◦ B| f , where Id|B : B → F , x 7→ x is the canonical injection. Definition 7 (Closure mapping). An isotone mapping f : E → E defined on an ordered set E is a closure mapping if f º IdE and f ◦ f = f . 2 3 4
This property concerns as well a matrix dioid product, for instance X 7→ AX ◦ B and B / ◦ A. where A, X ∈ Dn×n . See [1] for the computation of A \ ◦ b is the greatest solution of ax ¹ b. a\ These notations are borrowed from classical linear system theory see [10].
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Mehdi Lhommeau, Laurent Hardouin, and Bertrand Cottenceau
Proposition 1 ([5]). Let f : E → E be a closure mapping. A closure mapping restricted to its image Imf | f is a residuated mapping whose residual is the canonical injection Id|Imf : Imf → E, x 7→ x. Corollary 2. The mapping ImK| K is a residuated mapping whose residual is ¡ ¢] = Id|ImK . ImK| K This means that x = a∗ is the greatest solution to inequality x∗ ¹ a∗ . Actually, the greatest solution achieves equality. 2.4 Projectors [4, 3] Lemma 1. Let C : X → Y be a residuated mapping and let Π C = C ] ◦ C.
(3)
Π C is a projector, i.e. Π C ◦ Π C = Π C and C ◦ Π C = C. Lemma 2. Let B : U → X be a residuated mapping and let ΠB = B ◦ B ] .
(4)
ΠB is a projector, i.e. ΠB ◦ ΠB = ΠB and ΠB ◦ B = B. 2.5 Projections on the image of a mapping parallel to the kernel of another mapping We consider B : U → X and C : X → Y, the projection of x ∈ X on ImB parallel to ker C is any x0 which belongs to ImB and is equivalent to x modulo ker C, that is, find x0 ∈ X , s.t. ∃u ∈ U : C(x0 ) = C(x) and B(u) = x0 . From (3)-(4), it comes that z = Π C (x) = C ] ◦ C(x) is the greatest element in the equivalence class of x modulo ker C , and ξ = ΠB (z) = B ◦ B ] (z) is the greatest element in ImB which is less than z. Then z is ’subequivalent’ (see [4]) to x modulo ker C, i.e. C ◦ ΠB ◦ Π C (x) = C(ξ) ¹ C(x). If equality C holds true (i.e. C(ξ) = C(x)), ΠB ◦ Π C will be denoted by ΠB , which is a C C C projector (i.e. ΠB = ΠB ◦ ΠB ). The question of existence and uniqueness of projections for given operators B and C are studied in [4, 3]. We summarize the results C • Existence of projections for all x is equivalent to the condition C = C ◦ΠB C (i.e. ξ = ΠB (x) ∈ [x]C ). C ◦ B (i.e. any x ∈ ImB • Uniqueness is equivalent to the condition B = ΠB C remains invariant by ΠB ).
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3 Control in the presence of disturbances ½
½ x = Ax ⊕ Bu ⊕ Sq x = A∗ Bu ⊕ A∗ Sq ⇒ (5) y = Cx y = Cx ¡ ¢p ¢n ¡ where u ∈ ZZmax [[γ]] is the control vector, x ∈ ZZmax [[γ]] the state vector, ¡ ¡ ¢q ¢m y ∈ ZZmax [[γ]] the output vector, q ∈ ZZmax [[γ]] the disturbance (uncontrollable input) vector. Matrices of proper size A, B, C, S have entries in dioid ZZmax [[γ]] with only non-negative exponents integer values. In the conventional linear system theory [10], the disturbance decoupling problem consists in finding a control u such that the disturbance q has no influence on the controlled output y (i.e. y = 0, ∀q ∈ Q, the control keeps system state x in the kernel of C). Our problem must be stated in a different way since trajectories u, x, y and q are monotonous and no decreasing. The output cancellation is consequently meaningless in this context. Here we seek for a control u which keeps the system state x in the equivalence class of A∗ Sq modulo ker C. We say that such a control u ensures the disturbance matching, if u is such that ker C
ker C
A∗ Sq ≡ x ⇔ A∗ Sq ≡ A∗ Bu ⊕ A∗ Sq ⇔ CA∗ Sq = CA∗ Bu ⊕ CA∗ Sq.
(6)
The right statement shows that the objective will be achieved iff CA∗ Sq º CA∗ Bu. Obviously the set of controls verifying (6) may contain many elements, hence we are interested in computing the greatest one, formally L u. uopt = (7) ker C {u|A∗ Bu⊕A∗ Sq ≡ A∗ Sq}
¡ ¢n The greatest element in ZZmax [[γ]] such that y = CA∗ Sq is by definition the greatest element in [A∗ Sq]C , i.e. Π C (A∗ Sq) = C ] ◦ C(A∗ Sq). We denote with the same symbol the matrix C and the linear mapping x 7→ Cx. However, since this greatest state is not necessarily reachable, we seek for the greatest reachable state x ensuring the disturbance matching. This state is the projection of Π C (A∗ Sq) in ImA∗ B, i.e., ξ = ΠA∗ B ◦ Π C (A∗ Sq) = A∗ B ◦ (A∗ B)] ◦ C ] ◦ C(A∗ Sq).
(8)
It is the greatest element in ImA∗ B which is ’subequivalent’ to Π C (A∗ Sq), i.e. C such that C(ξ) ¹ C(A∗ Sq). If C ◦ ΠA∗ B ◦ Π C = C then ΠA∗ B ◦ Π C = ΠA ∗B ∗ is a projector in ImA B parallel to ker C and ξ is the greatest element in [A∗ Sq]C . Remark 2. System (5) can represent a Timed Event Graph (TEG), where u represents controllable transitions, x internal transitions and q represents uncontrollable transitions which delay the firing of internal transitions. In
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this context, it is useless that tokens enter too soon into the system. Then the control objective is to delay maximally tokens input by taking the disturbance into account. The control uopt achieves optimally the just-in-time criterion when some disturbances q acts on the system. The greatest control uopt allowing to reach this greatest state x (for a given q), is uopt = (A∗ B)] ◦ C ] ◦ C(A∗ Sq) = (CA∗ B) \◦ (CA∗ Sq). Practically, this control computation requires the disturbance 5 knowledge. Our problem is then to find a feedback F which allows to avoid this assumption. 3.1 Output feedback We discuss the existence and the computation of an output feedback controller which leads to a closed-loop system making the disturbance matching. The ¢p×q ¡ ) is to objective of the control (denoted by u = F y with F ∈ ZZmax [[γ]] keep the transfer relation between y and q unchanged. System (5) becomes ½ x = A∗ BF y ⊕ A∗ Sq (9) y = Cx = CA∗ BF y ⊕ CA∗ Sq The output equation (y = CA∗ BF y ⊕ CA∗ Sq) can be solved by considering Theorem 1. We obtain6 y = (CA∗ BF )∗ CA∗ Sq = CA∗ (BF CA∗ )∗ Sq. According to the previous section, the disturbance matching problem with output feedback can be stated as follows : find the greatest output feedback (denoted Fˆ ) such that the transfer function between y and q remains unchanged, i.e. L (10) Fˆ = { F |M (F ) ¹ CA∗ S}, where mapping M : X 7→ (CA∗ BX)∗ CA∗ S is not residuated since M (ε) = CA∗ S 6= ε. Nevertheless the following result shows that it is possible to compute the greatest output feedback Fˆ . Proposition 2. The greatest solution of (10) is Fˆ = CA∗ B\◦ CA∗ S /◦ CA∗ S. Proof. F = ε is a solution of (10) (since M (ε) = CA∗ S), hence the greatest solution, if it exists, also achieves equality. From (10), we seek for the greatest feedback verifying (CA∗ BF )∗ CA∗ S ¹ CA∗ S. Since RCA∗ S : x 7→ xCA∗ S is residuated (cf. Corollary 1), we have (CA∗ BF )∗ CA∗ S ¹ CA∗ S ⇔ (CA∗ BF )∗ ¹ CA∗ S /◦ CA∗ S. According to (2), the last expression shows that 5
6
In a manufacturing system, q may represent the supply of raw material which is a priori known. The problem is then very similar to the problem introduced in [9] which establishes an optimal open-loop control in presence of known uncontrollable inputs. We recall that (ab)∗ a = a(ba)∗ (see [5]).
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CA∗ S /◦ CA∗ S belongs to the image of K. Since ImK| K is residuated (cf. Corollary 2), there is also the following equivalence (CA∗ BF )∗ ¹ CA∗ S /◦ CA∗ S ⇔ CA∗ BF ¹ CA∗ S /◦ CA∗ S. Finally, since mapping LCA∗ B : x 7→ CA∗ Bx is residuated too (see Corollary 1), we verify that Fˆ = CA∗ B\◦ CA∗ S /◦ CA∗ S is the greatest solution of M (Fˆ ) = (CA∗ B Fˆ )∗ CA∗ S ¹ CA∗ S. Property 1. This feedback is the greatest such that x = A∗ (BF CA∗ )∗ Sq ∈ [A∗ Sq]C and obviously the resulting state x is lower than ξ = Π C (A∗ Sq). Furthermore x = A∗ (BF CA∗ )∗ Sq º A∗ Sq. Therefore, if x ∈ [ξ]C it exists a control u = F y such that x = Ax ⊕ Bu ⊕ Sq ∈ [ξ]C . It seems interesting to characterize under which conditions if x ∈ [ξ]C it exists a control u = F y such that x = Ax ⊕ Bu ∈ [ξ]C and to exhibit the links with the (A, B)−invariant definition given in [7].
4 Conclusion The objective is to synthesize a control law keeping state x in the kernel of C. It presents a strong analogy with the disturbance decoupling of the traditional control systems. However it must be noted that the reached objective does not lead to an output cancellation. Indeed the specific kernel definition of a mapping on a lattice and the nature of the considered systems allow to obtain the greatest control such that the output remains unchanged for any disturbance. The problem is obviously linked with the problem of characterization of (A, B)−invariant and future works will discuss this point [8].
References 1. F. Baccelli, G. Cohen, G.-J. Olsder, and J.-P. Quadrat. Synchronization and Linearity : An Algebra for Discrete Event Systems. Wiley and Sons, 1992. 2. T.S. Blyth and M.F. Janowitz. Residuation Theory. Pergamon press, 1972. 3. G. Cohen, S. Gaubert, and J.-P. Quadrat. Linear projectors in the max-plus algebra. In Proceedings of the IEEE-Mediterranean Conference, Cyprus, July. 1997. 4. G. Cohen, S. Gaubert, and J.-P. Quadrat. Max-plus algebra and system theory: Where we are and where to go now. In IFAC Conference on System Structure and Control, Nantes, 1998. 5. B. Cottenceau, L. Hardouin, J.-L. Boimond, and J.-L. Ferrier. Model Reference Control for Timed Event Graphs in Dioid. Automatica, 37:1451–1458, August 2001. 6. B. Davey and H. Priestley. Introduction to Lattices and Order. Cambridge University Press, 1990. 7. S. Gaubert and R. Katz. Reachability and invariance problems in max-plus algebra. In POSTA’2003, Roma, August 2003. 8. M. Lhommeau. Th`ese, LISA - Universit´e d’Angers, 2003. In preparation.
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9. E. Menguy, J.-L. Boimond, L. Hardouin, and J.-L. Ferrier. Just-in-Time Control of Timed Event Graphs Update of Reference Input, Presence of Uncontrollable Input. IEEE Trans. on Automatic Control, 45(11):2155–2158, November 2000. 10. W.M Wonham. Linear multivariable control : A geometric approach, 3rd edition. Springer Verlag, 1985.
Unforced Continuous Petri Nets and Positive Systems ? Manuel Silva and Laura Recalde Departamento de Inform´ atica e Ingenier´ıa de Sistemas, Universidad de Zaragoza, Spain, {silva,lrecalde}@posta.unizar.es Abstract. Petri nets (PNs) are a well-known family of formalisms whose definition immediately sets them, in a broad sense, as positive systems. Although they are originally discrete event models, their relaxation through continuization transforms them in continuous models. In this paper one of the most relevant timing interpretations of continuous PNs, unforced infinite servers semantics continuous PNs, is compared with linear positive systems and compartmental models.
1 Introduction Continuization is one of the possible relaxations applicable to Petri Nets (PNs), a well known family of discrete event dynamic formalisms [6, 7]. This technique is particularly well suited to deal with heavy traffic (or highly populated) systems. Although not all PN systems allow such kind of approximate modelling [8], this relaxation is possible in many practical cases, leading to a continuous-time formalism that generalizes in several aspects other classical models of positive (continuous) systems theory. Among the structural aspects that differentiate PNs from other models, are the possible existence of attributions, choices, forks and joins, and the absence of “strict” (material) conservation rules. Moreover, timed interpretations lead to different firing/flow policies. This paper explores some of the differences and similarities among unforced infinite servers semantics continuous PNs, linear positive systems and compartmental models. The PN models under consideration lead to piecewise linear, state-homothetic behaviours. Not only superposition principles do not hold, but also many kind of non monotonous behaviours may appear. Nevertheless, monotonicity can be observed in some PN subclasses, while linearity asks for even more constrained net subclasses, namely join free systems (JF). The transformation of continuous JF nets into P-graphs allows, by some weights removing operations, to deal with directed graphs (DG), the underlying structure of classical compartmental models. ?
Partially supported by the CICYT-FEDER project TIC2001-1819
L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 55-62, 2003. Springer-Verlag Berlin Heidelberg 2003
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Positive and compartmental systems will be introduced in Section 2. Section 3 presents basic PN concepts. Continuous PNs under infinite servers semantics will be studied in Section 4. This kind of models will be compared with compartmental systems in Section 5. In Section 6 the comparison will concentrate on the subclass of JF nets, which can be considered to be quite close to classical linear positive systems.
2 Positive and compartmental systems In a broad sense, a system is said to be positive if none of its variables takes negative values. In a more restricted sense, according to Luenberger in [4], a positive system automatically preserves the non-negativity of the state variables. In other words, if non-negativity constraints on the state are added, they ˙ are redundant. More formally, let Σ : x(t) = A · x(t) + B · u(t) be a linear system. Definition 1. [3] Σ is said to be positive iff for every nonnegative initial state and for every nonnegative input its state is nonnegative . If B = 0, the system is said to be uncontrolled or unforced (in automatic control it is also called autonomous; since autonomous has a quite different meaning for PNs [7], this term will not be used here). For the particular semantics that apply to positive systems, unforced means input-flow closed. A closed system is both input and output-flow closed. A matrix (vector) is positive if all its elements are nonnegative. A square matrix is Metzler if non-diagonal elements are nonnegative. ˙ Property 1. [3] A linear system x(t) = A · x(t) + B · u(t) is positive, iff A is a Metzler matrix and B ≥ 0. A particular case of positive systems are compartmental systems. Compartmental systems are composed of a finite number of subsystems (compartments), interacting by exchanging material among the compartments and with the environment [2, 11]. A compartmental system can be represented as a graph (with compartments as nodes) with an associated interpretation (compartmental networks). The level of each compartment, xi ,P changes according P to the input and output flows through the arcs, i.e., x˙ i = k fki − j fij . Inside compartmental systems generation of matter is forbidden. This ˙ means, in a system x(t) = A · x(t) + B · u(t), that P for every column of A the sum of its elements is non positive, i.e., aii + j6=i aji ≤ 0, or 1 · A ≤ 0. Hence, all the eigenvalues have a non-positive real part [3], and so the systems are either asymptotically or marginally stable. If it is an unforced system, then ˙ ˙ x(t) = A · x(t). Therefore, 1 · x(t) ≤ 0, that is, 1 · x(t) is bounded. If it is a closed system, then 1 · A = 0 (or equivalently, A is singular), and then 1 · x(t)
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is constant. That is, the system is strictly conservative. Otherwise, there are losses (evaporation, . . . ) in the system. The flows can be defined according to different semantics [11]: (pure) donor controlled when fij depends only on xi (fij = aij · xi in the linear case), (pure) recipient controlled when fij depends only on xj (fij = bij · xj in the linear case), donor and recipient controlled if fij depends on both, xi and xj (for example, fij = cij · xi · xj ), . . . Pure recipient controlled systems are not positive systems according to Definition 1. In general any kind of control can be defined, i.e., the flows may depend (global function) on any set of variables. In a linear donor controlled system, A is a Metzler matrix. According to Property 1, for any non-negative initial state the variables are always non-negative, i.e., x ≥ 0 is a redundant constraint.
3 Basics of continuous Petri nets A (discrete) PN system is a pair hN , m0 i, where N = hP, T, Pre, Posti is a PN net (P and T are disjoint (finite) sets of places and transitions, and Pre and Post are |P | × |T | sized, natural valued, incidence matrices), and m0 is the initial marking (a |P | sized, non-negative, integer valued, vector). The PN structure has also a graphical interpretation as a bipartite graph in which places and transitions are represented as circles and bars, respectively. In continuous PN systems the restriction on the integrality of firings is removed. Hence, the marking becomes a non-negative real number. In the sequel, unless otherwise stated, only continuous net systems will be considered. For v ∈ P ∪ T , the set of its input and output nodes are denoted as • v, and • v , respectively. A transition t is enabled at m iff for every p ∈ • t, m[p] > 0. The enabling degree of a transition measures the maximal amount in which the transition can be fired in one go, i.e. e(m)[t] = minp∈• t {m[p]/Pre[p, t]}. The firing of t in a certain amount α ≤ e(m)[t] leads to a new marking m0 = m + α · C[P, t], where C = Post − Pre, the token-flow matrix. This is αt denoted as m−→ m0 . A certain marking m0 is reachable from m if a fireable sequence exists leading from m to m0 . If m is reachable from m0 through a sequence σ, a state (or fundamental) equation can be written: m = m0 +C·σ, where σ ∈ IR|T | is a vector with the number of times each transition is fired. Right and left natural annullers of the token-flow matrix are called T- and P-semiflows, respectively. When y · C = 0, y > 0 (C · x = 0, x > 0) the net is said to be conservative(consistent). Some basic subclasses of nets are for example: choice free (CF) nets, each place has at most one output transition; join free (JF) nets, each transition has at most one input place; P-graphs, each transition has one input and one output place. The formalism introduced up to this moment is autonomous (in the PN sense!): which, when and how much the enabled transitions will fire is not defined (in other words, the net model is fully non-deterministic). In particular, it has no notion of time. In order to introduce it, and looking for coherence, let
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us take discrete PNs as inspiration. A simple and interesting way to introduce time in discrete PNs is to assume that all the transitions are timed with an exponential probability distribution function [5]. For a basic timing interpretation of continuous PN systems a first order (or deterministic) approximation of the discrete case [8] can be used, assuming that the delays associated to the firing of transitions can be approximated by their mean values. Let hN , m0 , λi denote the timed net, with λ ∈ QT the speeds of the transitions. Now the state equation has an explicit dependence on time m(τ ) = m0 + C · σ(τ ). ˙ ) = C · σ(τ ˙ ) is obtained. Let us denote Deriving with respect to time, m(τ ˙ since it represents the flow through the transitions. Observe that in f = σ, ˙ ) = 0, and so, from the state equation, C · f = 0. Since the steady state m(τ f ≥ 0, the flow in the steady state is a T-semiflow. Extrapolating from discrete markovian PNs [5], the most important semantics for the purpose of this work is infinite servers (variable speed ) [1, 8]. In that case, assuming the net system is unforced, the flow through a transition t is f [t] = λ[t] · e(m)[t] = λ[t] · minp∈• t {m[p]/Pre[p, t]}, that is, the speed of a server times the number of “active servers”. Putting all together: ˙ =C·f m f [t] = λ[t] · minp∈• t {m[p]/Pre[p, t]} m(0) = m0
for every t ∈ T
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or, equivalently, ˙ = C · Λ · e(m) m e(m)[t] = minp∈• t {m[p]/Pre[p, t]} m(0) = m0
where Λ = diag(λ[t]) is a diagonal matrix with the rates of the transitions. Observe that the above system non linear (due to the “min”, its evolution is defined by a set of piecewise linear differential equations).
4 Some properties of unforced continuous Petri nets under infinite servers semantics One of the characteristics of infinite servers semantics continuous PNs is that they are positive systems. There is no need of explicitly imposing it, since it is guaranteed by (1). Another characteristic is that, unlike the discrete models, continuous PNs preserve their properties under scaling of the marking. Property 2. [8] Let hN , m0 , λi be a continuous infinite servers semantics PN. • •
For every τ ≥ 0, it holds f (τ ) ≥ 0 and m(τ ) ≥ 0. For any marking m that can be reached in hN , m0 i, k·m can be reached in hN , k · m0 i. Moreover, for any timing λ, if f (τ ) is the flow in hN , m0 , λi, then k · f (τ ) is the flow in hN , k · m0 , λi.
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Superposition properties do not hold, and quite intricate behaviours may appear. Even more, the flow of a continuous net system does not fulfill in general any significant monotonicity property, neither w.r.t. the initial marking, nor w.r.t. the structure of the net, nor w.r.t. the transitions rates. For example, in the net system in Fig. 1 (a), if the marking of p5 is augmented to 5, it deadlocks, i.e., the throughput goes down to 0. However, the throughput increases from 0.535 to 1.071 if m0 [p5 ] is reduced to 3! Notice that this token (i.e., resource) reduction is equivalent to adding a place “parallel” to p5 (i.e., with an input arc from t2 and an output arc to t1 ), marked with 3 tokens. Hence, adding constraints may increase the throughput. Finally, an increase in a transition rate (for example, due to a replacement by a faster machine) may also lead to a decrease in the global throughput. In other words, local improvements do not necessarily lead to global improvements. For example, Fig. 1 (b) shows how the throughput of the net system in Fig. 1 (c) changes if the rate of t1 varies from 0 to 5, assuming λ[t2 ] = 1. Notice that even a discontinuity (!) appears at λ[t1 ] = 2. Hopefully, monotonicity properties do hold for some subclasses.
5 Continuous Petri nets vs. positive and compartmental systems: a structural point of view An immediate similarity between PNs and compartmental systems is that both allow a representation based on graphs. However, PNs are bipartite graphs, while compartmental models have a single kind of nodes. In discrete PNs there are two kinds of nodes: OR nodes (attributions/choices) and AND nodes (joins/forks). Nevertheless, in continuous PNs with infinite servers semantics the forward OR node is transformed into a “+” operation: choices can be
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Fig. 2. In continuous PNs, under infinity servers semantics, forks can be transformed into equal choices and viceversa.
seen as flow splitters. Analogously, forks are also “+” operations, and can be immediately transformed into choices. Let p1 and p2 be output places of t0 (to simplify, let us assume there are just two output places), see Figure 2. Then, t0 can be replaced by two transitions t01 and t02 , connected as follows: w01 = w02 = w0 /2, w11 = w1 , w22 = w2 , λ[t01 ] = λ[t02 ] = λ[t0 ]/2. Notice that although these arc weights may be rational, they can be transformed into integer numbers multiplying by two all the arc weights and the marking. Hence, a continuous net system can be transformed into an equivalent one without forks. By equivalent it is meant that they have the same markings (i.e., states) and flows, up to scaling factors or duplication of some variables. Similarly, choices (with the same input set) can be transformed into forks. Let t01 and t02 be the output transitions of p0 . These two transitions can be merged as follows: w11 · w02 · λ[t01 ] (λ[t01 ] + λ[t02 ]) · w01 · w02 , w1 = , define anaw0 = λ[t01 ] · w02 + λ[t02 ] · w01 λ[t01 ] · w02 + λ[t02 ] · w01 logously w2 , and λ[t0 ] = λ[t01 ] + λ[t02 ]. Now, multiply the arcs and the initial marking by the adequate constant so that the arcs are integers. Any linear compartmental system based on donor controller can be “naturally” simulated by means of a PN. However, to our knowledge there is no way to simulate a pure recipient controlled system with any of the policies that are defined in continuous PNs. A basic element in the evolution of PNs is that behaviour is of the consumption/production type, but that is not the idea in a recipient controlled system. Some mixed donor and recipient controls may appear in a natural way through decoloration (see [8]). Another difference between PNs and compartmental systems is with respect to matter conservation. In compartmental systems there is a strong “strict” conservation law: matter is not created, although it may “evaporate” and disappear if the system is not (output) closed. In PNs such kind of constraint does not exist. However, conservativeness (i.e., y > 0 exists such that y · C = 0) is a similar property that is often required. From this and the ˙ = 0 can be deduced and so, y · m = y · m0 = constant. state equation, y · m Since the tokens do not exactly represent a flow of matter and may change their meaning along the net, it guarantees not a strict matter conservation, but a weighted conservation. This is a structural property, and is independent of the transitions speeds. Conservativeness in nets is related to the notion of closeness in compartmental systems.
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6 Join free and linear positive systems Joins are associated to “min” functions and are at the base of important nonlinearities of continuous PN systems. The PN subclass more closely related to linear positive systems is the class of (connected) join free (JF) nets. In that class all the transitions have just one input place, and hence the “minimum operator” problem disappears. Under infinite server semantics, the evolution of a JF system can be represented as a linear positive system: g ·m ˙ = C · Λ · Pre m g p] = 1/Pre[p, t] (thus, for ordinary (unweighted) nets, Pre g = where Pre[t, T Pre ). Observe that A (in x˙ = A · x) is a node–to–node matrix (adyacence), while C is node–to–arc (incidence). We may also assume that the nets are also choice free (CF), since choices can be transformed into forks (see Figure 2). Now, if the net is JF and CF, |P | = |T |, and so Pre, Post, C are square matrices. In that case, since Pre is g p] = Pre[p, t]−1 , and so diagonal, Pre[t, ˙ = C · Λ · Pre−1 · m m Obviously, C · Λ · Pre−1 = (Post · Pre−1 − I) · Λ is a Metzler matrix. Nevertheless, this system is not necessarily a compartmental system because flow conservation is not guaranteed. Let us assume that it is a conservative system, which as we said, is “close” to the matter conservation law of compartmental systems. Strongly connected and conservative JF PNs are also consistent, i.e., x > 0 exists such that C · x = 0 [10]. Moreover, in JF and CF nets this annuller is unique but for a multiplying factor. If mss is a steady state marking, i.e., an equilibrium point, ˙ ss = 0, and so C · f = C · Λ · Pre−1 · mss = 0. Then, the following then m property can be deduced. Property 3. In JF and CF nets the equilibrium point is uniquely defined by: the token conservation law, y·mss = y·m0 , and C·f = 0, f = Λ·Pre−1 ·mss . Another difference between PNs and the graphs associated to compartmental systems is the arc weights. However, this is not significant, since for any conservative JF net, an equivalent one exists with arc weights one. Doing as in Figure 2, the JF net can be transformed so that each transition has just one input and one output place, i.e, it is transformed in a P-graph. Then, the following result can be applied. Property 4. [9] Let hN , m0 , λi be a conservative P-graph, and define N 0 = hP 0 , T 0 , Pre0 Post0 i, an ordinary net with the same topological structure (P 0 = P, T 0 = T, Pre0 [p, t] = 1 iff Pre[p, t] > 0 and Post0 [p, t] = 1 iff Post[p, t] > 0). An initial marking m0 0 exists such that hN , m0 , λi and hN 0 , m0 0 , λi express equivalent behaviour under infinite servers semantics. Therefore, strongly connected and conservative JF PNs are equivalent, from the modelling point of view, to closed linear compartmental systems.
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7 Conclusions In a broad sense, PNs are positive systems. In particular, under so called infinite servers semantics, continuous PNs are positive systems in the sense of Luenberger. In this paper continuous PNs are compared with compartmental systems. The structure in PNs is richer than the directed graphs of compartmental systems. This is reflected in two main aspects: the existence of synchronizations, which implies that the system is in general non-linear due to the “min” operators; and the arc weights. It has been seen that PNs without synchronizations, i.e., JF subnets, can be transformed into P-graphs. Moreover, if the net is conservative the P-graph is equivalent to a state machine, i.e., a directed graph without arc weights. In other words, weights in JF models are a modelling convenience (i.e., do not add theoretical expressive power). However, that is not always the case with more general net subclasses, and there exist nets in which weights are not so easily removed. On the other hand, there is a problem to represent recipient controlled compartmental systems in a “natural” way, since the common practice in PNs is to define the flow according to the enabling, i.e., the input places.
References 1. H. Alla and R. David. Continuous and hybrid Petri nets. Journal of Circuits, Systems, and Computers, 8(1):159–188, 1998. 2. L. Benvenuti and L. Farina. Positive and compartmental systems. IEEE Transacions on Automatic Control, 47(2):370–373, Feb 2002. 3. L. Farina and S. Rinaldi. Positive Linear Systems. Theory and Applications. Pure and Applied Mathematics. John Wiley and Sons, New York, 2000. 4. Luenberger. Introductions to Dynamic Systems: Theory, Models and Applications. John Wiley and Sons, New York, 1979. 5. M. K. Molloy. Performance analysis using stochastic Petri nets. IEEE Trans. on Computers, 31(9):913–917, 1982. 6. T. Murata. Petri nets: Properties, analysis and applications. Proceedings of the IEEE, 77(4):541–580, 1989. 7. M. Silva. Introducing Petri nets. In Practice of Petri Nets in Manufacturing, pages 1–62. Chapman & Hall, 1993. 8. M. Silva and L. Recalde. Petri nets and integrality relaxations: A view of continuous Petri nets. IEEE Trans. on Systems, Man, and Cybernetics, 32(4):314– 327, 2002. 9. M. Silva and L. Recalde. Unforced continuous Petri nets and positive systems. Research report, Dep. Inform´ atica e Ingenier´ıa de Sistemas, Universidad de Zaragoza, Mar´ıa de Luna, 13, 50018 Zaragoza, Spain, 2002. 10. E. Teruel, J. M. Colom, and M. Silva. Choice-free Petri nets: A model for deterministic concurrent systems with bulk services and arrivals. IEEE Trans. on Systems, Man, and Cybernetics, 27(1):73–83, 1997. 11. G. Walter and M. Contreras. Compartmental Modeling With Networks. Birkhauser Boston, 1999.
Reachability Graph for Autonomous Continuous Petri Nets Ren´e David and Hassane Alla Laboratoire d’Automatique de Grenoble (INPG-UJF-CNRS) B.P. 46, 38402 Saint-Martin-d’H`eres, France, {Rene.David, Hassane.Alla}@inpg.fr Abstract. An autonomous continuous Petri net is a model in which the time is not involved, the marking is a vector or non-negative real numbers, and a transition firing corresponds to some ”quantity of firing” (positive number) compatible with the current marking. The paper presents the new concepts of OG-firing (standing for ”at one go firing”) and macro-marking. From these concepts, a reachability graph can be built for an autonomous continuous Petri net: the number of markings may be infinite, but the number of macro-markings is always finite. Since an autonomous hybrid Petri net is made up of a discrete part and a continuous part, some results in this paper may by useful for analyzing such an hybrid model.
1 Introduction The nets introduced by C. A. Petri [10] are usually called Petri nets. Basically, a Petri net (PN) is a bipartite graph: each node is either a place, represented by a circle, or a transition, represented by a bar; if the initial extremity of an arc is a place, its terminal extremity is a transition, and vice-versa. Each arc has a weight (positive integer) denoted by P re(Pi , Tj ) for arc Pi −→ Tj and P ost(Pi , Tj ) for arc Tj −→ Pi . If the weight of all existing arcs is 1, the PN is ordinary; otherwise it is generalized (the arc does not exist if the weight is 0). An autonomous Petri net is a model in which the time is not involved; all the models considered in this paper are autonomous, hence this feature may be implicit. If the PN is marked, each place contains a number of tokens (non-negative integer). The marking of the PN is then a vector of nonnegative numbers. The marking (i.e. the state) of the PN evolves by firing of transitions. In an ordinary PN, a transition can be fired if there is at least one token in every input place; firing of a transition consists of withdrawing one token from each of the input places and to adding a token to each output place of the transition. (For a generalized PN, the number of tokens withdrawn or added correspond to the arc weights.) L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 63-70, 2003. Springer-Verlag Berlin Heidelberg 2003
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More recently, continuous PNs were defined [3]. In [6], autonomous PNs are shown to be a limit case of discrete PNs. In a continuous PN, the markings, arc weights, and firing quantities are non-negative, as in a discrete PN, but are not necessarily integers. Other authors have proposed various results concerning these PNs [12, 7, 11]. Given an initial marking m0 , let M(m0 ) denoted the set of reachable markings and L(m0 ) the set of firing sequences which can be performed. Two basic problems are as follows: given a marking m0 , is it in M(m0 )? given a firing sequence S, is it in L(m0 )? For a bounded discrete PN, a way to tackle these problems consists of building the reachability graph (i.e. graph of markings). However, for a continuous PN, the number of markings is infinite (even if it is bounded), because a marking may change continuously. It follows that, up to now, no reachability graph exists for a continuous PN. The paper presents the new concepts of OG-firing (standing for ”at one go firing”) and macro-marking. From these concepts, a reachability graph can be built for an autonomous continuous PN : the number of markings may be infinite, but the number of macro-markings is always finite. The paper is organized as follows. Section 2 presents the definition of autonomous continuous PNs. In Section 3, the concept of macro-marking and the definition of a reachability graph, for a continuous PN, are presented, then Section 4 concludes the paper.
2 Definition of autonomous continuous Petri nets Some part of this section, particularly the presentation of a continuous PN as a limit case of discrete PN (Fig. 1) was first shown in [6] and [4]. 2.1 From discrete Petri net to continuous Petri net Let us consider a (discrete) PN R defined by its graph Q (places, transitions, arcs) and its marking m = (m1 , m2 , . . . ), and let us apply a transformation which consists of dividing each mark into k equal parts (without any other modification of the PN). This new discrete PN and its marking is denoted 0 by R(k) and m0(k) (or simply R0 and m0 ). Normally, token and mark are synonymous. We use the word mark for the marking of the initial PN. Each mark is divided into k and the new unit which is one kth of mark is called a token. This is consistent with the usual meanings, since there is equivalence for k = 1. See Fig. 1 for example. The considered transformation applied to the PN of Fig. b gives the PN of Fig. c in which the markings are expressed in tokens, i.e., m0(k) = (2k, 0). The new PN possesses all the characteristics of 0 an usual discrete PN. For the PN R(k) of Fig. c (as well as for the PN R of Fig. b) the firing of T1 consists of removing a token from place P1 and adding a token in place P2 . The marking of a place can thus be expressed in tokens
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Fig. 1. Transformation of a PN. (a) Unmarked PN Q. (b) Marked PN R = 0 < Q, m >. (c) Transformed PN R(k) = < Q, m >.
(integer) or in marks (rational number if k is finite). Let m0i be the marking 0 of the place Pi expressed in tokens of the PN R(k) . For the same PN, the m0
marking of place Pi expressed in marks is denoted by mi = ki . If we compare the markings expressed in unit marks for the PNs R and 0 R(k) , it is clear that the reachable markings of the PN of Fig. 1b are included in those of Fig. 1c. If a marking m is such that transition Tj can be fired q times (but not more), Tj is said to be q-enabled [6, 5]; in [1], the number q is called enabling degree. Some notations are presented in the sequel: notation 1a is usual; notation 1b was used in [4, 5]; notation 1c is introduced here. Notation 1 a) A sequence of two successsive firings of T1 is denoted by (T1 )2 = T1 T1 . b) Notation [T1 T2 ] corresponds to the simultaneous firing of both T1 and T2 . c) Let [Tj ]α = [(Tj )α ], denote the firing of Tj , α times simultaneously, i.e., at one go (α is a non-negative number ). Then, a double firing of T1 can be denoted by [T1 T1 ] = [(T1 )2 ] = [T1 ]2 . In other words, [Tj ]α represents α firings of Tj at one go, whereas (Tj )α represents α successive firings of Tj . For continuous PNs, non-integer values of α will be considered. Notation 2 S
→ means that the sequence S of firings can be performed from m1 . a) m1 − S b) m1 − → m2 means, in addition, that the marking reached is m2 . Figure 2a shows the set of possible markings and the corresponding transition firings for the PN R in Fig. 1b, in the plane defined by m1 and m2 . T1 In addition to the single transition firings, for example (2, 0) −→ (1, 1), all [T1 ]2
the multiple transition firings are represented, for example (2, 0) −−−→ (0, 2). The possible markings of R0 (k) are shown in Fig. 2b for k = 4. There are very many possible multiple transition firings. Only two of them are illustrated, namely
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[T2 ]6
(5, 3) −−−→ (2, 6) and (2, 6) −−−→ (8, 0), in tokens, i.e., [T1 ]0.75
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(1.25, 0.75) −−−−−→ (0.5, 1.5) and (0.5, 1.5) −−−−→ (2, 0), in marks. When k tends to infinity, the set of reachable markings becomes infinite. It can be represented by a segment of line between (2, 0) and (0, 2) as illustrated in Fig. 2c. The marking can no longer be expressed in tokens (since m01 may become infinite). We use the marking expressed in marks. For marking m = (α, 2 − α), where α is any real number in the range [0, 2], enabling degrees of T1 and T2 are, respectively, α and 2 − α. Figure c illustrates the possible firings of these transitions according to their enabling degrees. In fact, from m, T1 can be fired β times at one go (β is called the firing quantity), such [T1 ]β
that 0 ≤ β ≤ α: (α, 2 − α) −−−→ (α − β, 2 − α + β). Similarly, the firing of [T2 ]γ , 0 ≤ γ ≤ (2 − α) is possible. Finally, the multiple firing of [(T1 )β (T2 )γ ] is possible at one go from m.
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2 m1
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2–α
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T1
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0
0
2 m1
α (c)
Fig. 2. Graphs of reachable markings for PN R in Fig. 1. (a) For Fig. b. (b) For 0 0 R(k) = < Q, m0(k) > (for k = 4). (c) For R(∞) .
2.2 Definition Definition 1. A marked autonomous continuous PN is a 5-uple R = < P, T, P re, P ost, m0 >: P = P1 , P2 , ..., Pn is a finite, not empty, set of places; T = T1 , T2 , ..., Tm is a finite, not empty, set of transitions; P ∩ T = ∅ , i.e. the sets P and T are disjointed; P re : P × T → Q+ is the input incidence application1 ; P ost : P × T → Q+ is the output incidence application; 1
Notations Q+ and R+ correspond respectively to the non-negative rational numbers and real numbers
Reachability Graph for Autonomous Continuous Petri Nets
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m0 : P → R+ is the initial marking. As for a discrete PN, R =< Q, m0 > where Q =< P, T, P re, P ost > represents the unmarked PN. In a continuous PN, places and transitions are represented by a double line (Fig. 3a for example). Let us recall some notations and vocabulary, familar for everybody having some knowledge of Petri nets (see [5], for example). The sets ◦Pi , Pi◦ , ◦Tj , and Tj◦ , represent respectively the sets of input transitions of Pi , output transitions of Pi , input places of Tj , and output places of Tj . Definition 2. In a continuous PN, the enabling degree of transition Tj for marking m, denoted by q or q(Tj , m) is the real number q defined in (1) (if q > 0, Tj is enabled; it is said to be q-enabled). ¶ µ m(Pi ) q = min . (1) Pi ∈ ◦Tj P re(Pi , Tj )
3 Macro-markings and reachability graph We now introduce concepts which will be useful for studying reachability. Since the number of markings in a continuous PN may be infinite, we define macro-markings whose number is finite. 3.1 Macro-marking Definition 3. Let mk be a marking. The set P of places may be divided into two subsets: P + (mk ) the set of places Pi such that mk (Pi ) > 0, and the set of places Pi such that mk (Pi ) = 0. A macro-marking is the union of all markings mk with the same set P + (mk ) of marked places. A macro-marking will be denoted by m∗j (or possibly mj if it contains a single marking). It may be specified by its set of marked places P + (m∗j ). Property 1. The number of reachable macro-markings of a n-place continuous PN is less than or equal to 2n . This property is a direct consequence of Def. 3, since each macro-marking is based on the Boolean state of every place: marked or not marked. The number of macro-markings is necessarily finite because n is finite. The continuous PN in Fig. 3a has three macro-markings (illustrated in Fig. b to d), namely m∗0 , m∗1 , and m∗2 , such that P + (m∗0 ) = {P1 }, P + (m∗1 ) = {P1 , P2 }, and P + (m∗2 ) = {P2 }. The fourth macro-marking, m∗3 = (0, 0) is not reachable. Given a marking, the set of enabled transitions is known; this is true for any PN, discrete or continuous. An interesting feature of any continuous PN
68
þþ þþ ýÿÿ ÿýÿ ü ü
Ren´e David and Hassane Alla
P1 T1 P2 T2
ø ø ø ùùø ùù
2
(a)
ö ö öûû ûû
m2
m2
2
2
2
0 m0
÷ ÷ ÷úú úú
m2
[T1]2
0
(b)
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0
[T1]α
m 1*
0
[T2]2
[T2]2–α 2 m1
(c)
m 2*
0
0
2 m1
(d)
Fig. 3. (a) Continuous Petri net. (b) to (d) Illustration of its macro-markings.
is that knowledge of the set of marked places (i.e. knowledge of the macromarking) is sufficient to know the set of enabled transitions (not true for a generalized discrete PN). This is a direct consequence of Def. 2. Hence, a reachability graph whose vertices are the macro-markings, can be built. Property 2. In a continuous PN, a change of macro-marking, hence a change of set of enabled transitions, can occur only if an event belonging to one of the followings types occurs (C in their names stands for continuous). C1-event: the marking of a marked place becomes zero. C2-event: an unmarked place becomes marked. 3.2 Reachability graph When a continuous PN is obtained from a discrete PN (same Pre and Post functions and same initial marking), the initial PN is called the discrete counterpart of the continuous PN and vice-versa (Fig. 3a is the continuous counterpart of the PN in Fig. 1b). A continuous PN built from scratch may be converted into a continuous PN with a discrete counterpart if its initial marking is a vector of rational numbers. Let us analyze the reachability of the continuous PN in Fig. 4a. The set of reachable markings will be shown to correspond to all the markings in the grey triangle in Fig. 4b. For the discrete counterpart of the considered PN, the reader may verify that the reachable markings correspond to the three vertices of the triangle plus (1, 1, 0). The language generated is the set of prefixes of L = T1 (T2 T3 )∗ T1 . One can observe that the set of markings of the discrete counterpart is included in the set of markings of the continuous PN ; this is a general property. According to Property 1, the number of reachable macro-markings of the 3-place continuous PN in Fig. 4a cannot be more than 23 = 8. In fact, there are only 7 macro-markings because (0, 0, 0) is not reachable. These macromarkings m∗0 to m∗6 , illustrated in Fig. 4b, correspond to: three vertices of the triangle, each one corresponding to one marked place; three sides of the triangle, except the adjacent vertices, each one corresponding to two marked
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places; all the area of the triangle, except the sides, m∗2 for which all the places are marked. The reachability graph is shown in Fig. 4c. The initial macro-marking corresponds to a single marking, m∗0 = m0 = (2, 0, 0); only T1 is enabled and its enabling degree is 2. If T1 is fired according to its enabling degree, i.e. [T1 ]2 , m∗6 = (0, 2, 0) is reached. If T1 is fired with a firing quantity less than its enabling degree, m∗1 = (m1 , m2 , 0) is reached (this notation means that, in m∗1 , m1 and m2 have a positive value and that P3 is not marked). The arrow from m∗0 to m∗1 is labelled by T1 without specification of the firing quantity; this means that the firing quantity is positive but less than the enabling degree.
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] [T 3
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] [T 3
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2] m
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1 =
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3] 1
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Fig. 4. Continuous PN and its reachability graph.
For m∗1 , both T1 and T2 are enabled. Firing of T1 leads to m∗6 (firing quantity equal to enabling degree, i.e. m1 ) or to m∗1 (firing quantity less than enabling degree). Firing of T2 with a firing quantity less than its enabling degree leads to m∗2 . Firing of T2 with a firing quantity equal to its enabling degree leads to m∗3 (if m2 < m1 ), m∗4 (if m1 < m2 ), or m∗5 (if m1 = m2 ). And so on. If several transitions have the same initial macro-marking and the same final macro-marking, they are separated by a comma. For example, firings of T1 , T2 , or T3 lead from m∗2 to itself. Macro-marking m∗6 is a deadlock (as in the discrete counterpart). The set of reachable markings (represented by the grey triangle in Fig. 4b) corresponds to all markings satisfying (2) and such that m1 ≥ 0, m2 ≥ 0, and m3 ≥ 0 (this is called a marking invariant [4]). 3m1 + 3m2 + 2m3 = 6
(2)
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Ren´e David and Hassane Alla
4 Concluding remarks The new concepts of OG-firing and macro-marking, then the building of a reachability graph for an autonomous continuous Petri net, have been introduced. Hybrid Petri nets were studied in [9]. Other authors have worked on more or less similar models. A special issue of Journal of Discrete Event Dynamic Systems: Theory and Applications was devoted to these models [8]. Since an autonomous hybrid Petri net is made up of a discrete part and a continuous part, the results in this paper may by useful for analyzing such an hybrid model. In particular, a reachability graph can be built for an autonomous hybrid PN [2]. Aknowledgement. The authors thank St´ephane Mocanu for his unvaluable help in preparing this paper.
References 1. J. Campos and M. Silva. Structural Techniques and Performance Bounds of Stochastic Petri Net Models. Design Methods Based on Nets (DEMON), pages 352–391. LNCS 607. Springer-Verlag, Berlin, 1992. Adv. in PN ’92. 2. R. David and H. Alla. Discrete, Continuous, and Hybrid Petri Nets. in preparation. 3. R. David and H. Alla. Continuous Petri Nets. In 8th European Workshop on Application and Theory of Petri Nets, pages 275–294, Saragosse (S), June 1987. 4. R. David and H. Alla. Du Grafcet aux r´eseaux de Petri. Herm`es Pub, Paris, 1989. Second Edition 1992. 5. R. David and H. Alla. Petri Nets and Grafcet: Tools for Modelling Discrete Event Systems. Prentice Hall Int., London, 1992. 6. R. David and H. Alla. Autonomous and Timed Continuous Petri Nets, pages 71–90. LNCS. Springer-Verlag, 1993. First version in 11th Int. Conf. on Applic. and Theory of Petri Nets, Paris, pp. 367-386, June 1990. 7. I. Demongodin, M. Mostefaoui, and N. Sauer. Steady state of continuous neutral weighted marked graphs. In Proc. Int. Conf. on Systems, Man and Cybernetics, IEEE / SMC, pages 3186–3194, USA, 2000. 8. A. Di Febbraro, A. Giua, and G. Menga, editors. Special Issue on Hybrid Petri Nets, J. of Discrete Event Dynamic Systems: Theory and Applications, volume 11, 2001. 9. J. Le Bail, H. Alla, and R. David. Hybrid Petri Nets. In Proc. European Control Conference, Grenoble (F), July 1991. 10. C.A. Petri. Communication with Automata, Supplement 1 to Technical Report RADC-TR-65-337. PhD thesis, University of Bonn, 1962. 11. L. Recalde and M. Silva. PN fluidification revisited: Semantics and steady state. In Proc. Int. Conf. on Automation of Mixed Processes: Hybrid Dynamic Systems (ADPM 2000), pages 279–286, Dortmund (D), September 2000. 12. L. Recalde, E. Teruel, and M. Silva. Autonomous continuous P / T systems. In Springer, editor, 20th Int. Conference on Application and Theory of Petri Nets, pages 108–126, Williamsburg, USA, June 1999.
Modeling Hybrid Positive Systems with Hybrid Petri Nets Marco Gribaudo and Andr´as Horv´ath Dipartimento di Informatica, Universit` a di Torino, Corso Svizzera 185, 10149 Torino, Italy, {marcog,horvath}@di.unito.it Abstract. In this paper the possibility of modeling positive systems by means of Hybrid Petri Nets (HPN) is discussed. Hybrid (or Fluid) Petri Nets are Petri net (PN) based model with two classes of places: discrete places that carry a natural number of distinct objects (tokens), and fluid places that hold a positive amount of fluid, represented by a real number. The HPN formalism we present in this work allows for defining the system model both in stochastic and non-stochastic setting.
1 Introduction Hybrid Petri Nets (HPN) are Petri net (PN) based models in which some places may hold a discrete number of tokens and some places a continuous quantity represented by a real number. Places that hold continuous quantities are referred to as fluid or continuous places, and the non-negative real number is said to represent the fluid level in the place. Discrete tokens move along discrete arcs with the enabling and firing rules of standard PN, while the fluid moves along special continuous (or fluid) arcs according to an assigned instantaneous flow rate. On the non-stochastic side HPN were introduced in [2], on the stochastic side Fluid Stochastic Petri Nets were presented in [20]. Both works aimed at providing an approximation to discrete-state systems in which the number of objects (customers, packets, tasks, workpieces etc.) becomes exceedingly large to be treated with the discrete state approach common to PN. Traditionally, non-stochastic PN based formalisms are applied for addressing qualitative questions (like reachability) and model checking, while on the stochastic side the main objective is performance evaluation. In [5] a unified modeling view for PN was presented that allows for both non-stochastic and stochastic analysis of the same model. In this paper, we present a similar unified view for HPN. The stochastic side of this formalism is similar to the one in [12]. The non-stochastic side of the formalism is derived from its stochastic side. The formalism we present hereinafter was already applied without being discussed in detail to the analysis of a temperature control system in [10]. L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 71-78, 2003. Springer-Verlag Berlin Heidelberg 2003
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Marco Gribaudo and Andr´ as Horv´ ath pi
discrete place
pi tokens
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discrete arc inhibitor arc test arc
cl
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Fig. 1. All the primitives of the formalism
For what concerns the relationship between positive systems and HPN, as noted in [4], a HPN can be seen as a formalism for the description of hybrid positive systems, i.e. positive systems with both discrete events and fluid dynamics. The paper is organized as follows. Section 2 introduces the proposed formalism. An example is discussed in Section 3. A brief overview of solution techniques is provided in Section 4. Conclusions are drawn in Section 5.
2 The formalism We define an HPN as a tuple hP, T , A, M0 i. Hereinafter we introduce all the elements of this tuple and the evolution of the model emphasizing the difference between the non-stochastic and the stochastic part of the formalism. placesª Pd = © P is the ªset of places partitioned into a set of discrete © p1 , . . . , p|Pd | , and a set of continuous places Pc = c1 , . . . , c|Pc | (with Pd ∩ Pc = ∅ and Pd ∪ Pc = P). The discrete places may contain tokens (the number of tokens is a natural number), while the marking of a continuous place is a non negative real number. Graphically a discrete place is drawn as a single circle while a continuous place is drawn with two concentric circles. Graphical representation of all the modeling primitives is given in Figure 1. The complete state (marking) of a HPN is described by a pair of vectors M = (m, x). In M the vector m is the marking of the discrete part of the state representing the token distribution in the discrete places with mi ≥ 0 for any i : pi ∈ Pd . The fluid levels of the continuous places is represented by the vector x with xi ≥ 0 for any i : ci ∈ Pc . The marking M = (m, x) evolves in time, which we indicate by τ , so, formally, we can think of it as a process M(τ ) = {(m(τ ), x(τ )), τ ≥ 0}. On the non-stochastic side, in a given instant of time a state is either possible or not. Instead, on the stochastic side, in a given instant of time the likeliness of the state (m0 , x0 ) is reflected by the quantity P r(mi (τ ) = m0i , 1 ≤ i ≤ |Pd |; xj (τ ) ≤ x0j , 1 ≤ j ≤ |Pc |), i.e. by the probability of observing the given discrete marking joined with the joint cumulative distribution of the fluid levels. T is the set of transitions partitioned into a set of timed transitions Tt , a set of immediate transitions Ti , and a set of fluid transitions Tf (with Tu ∩Tk = ∅,
Modeling Hybrid Positive Systems with Hybrid Petri Nets p1
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Fig. 2. All the possible ways of placing arcs in a net
for u, k = t, i, f ; u 6= k and Tt ∪ Ti ∪ Tf = T ). A timed transition is drawn as an empty rectangle, an immediate transition as a thin bar, while a fluid one as a filled rectangle. A is the set of arcs partitioned into five subsets: Ad , Ac , As , Ah and At . The subset Ad contains the discrete arcs which can be seen as a function Ad : ((P × (Tt ∪ Ti )) ∪ ((Tt ∪ Ti ) × P)) → IN , i.e. discrete arcs connect places (either discrete and continuous) to timed and immediate transitions. All the possible ways of placing arcs in a net is depicted in Figure 2. The arcs Ad are drawn as single arrows. The subset Ac defines the continuous arcs along which fluid is moved. These arcs are drawn as double arrows to suggest a pipe. Ac is a subset of ((Pc ×Tf )∪(Tf ×Pc )) → IR, i.e., a continuous arc can connect a fluid place to a fluid transition or it can connect a fluid transition to a fluid place. The subset As contains the set arcs. As is a subset of ((Tt ∪ Ti ) × Pc ) → IR. These arcs connect continuous places to timed or immediate transitions, and describe the capability of a transition to set the fluid level into a continuous place when it fires. The arcs As are drawn as thick single arrows. The subset Ah contains the inhibitor arcs, Ah : (Pd × T ) → IN ∪ (Pc × T ) → IR, and has the usual meaning. These arcs are drawn with a small circle at the end. The subset At contains the test arcs, At : (Pd × T ) → IN ∪ (Pc × T ) → IR. Test arcs are drawn as double arrow. They test if a place contains at least a given number of token (or a quantity of fluid). Based on the result of the test, transitions can be enabled or disabled. When test arcs connect discrete place to timed or immediate transitions, they can be considered equivalent to an input and an output discrete arc, with the same weight, that connect that place to that transition. The subsets Ad and Ah define the input/output and the inhibitor arcs of common notation of Generalized Stochastic Petri Nets (GSPN, see [1] for further details) while the subsets Ac and As define arcs that are related with the continuous places. Test arcs At have been introduced to visually define enabling condition on fluid transitions. Given a transition tj ∈ T , we denote with • tj = {pi ∈ Pd : Ad (pi , tj ) > 0} ∪ {pi ∈ P : At (pi , tj ) > 0} and with t•j = {pi ∈ Pd : Ad (tj , pi ) > 0} the input and the output set of transition tj , and with ◦ tj = {pi ∈ P : Ah (pi , tj )} the inhibition set of transition tj . The definition of t•j involves only discrete places and hence is exactly the one defined for common GSPN. The definitions of • tj and ◦ tj are instead different since they include also fluid places. M0 = (m0 , x0 ) denotes the initial state of the FSPN. In the following, by providing the enabling and firing rules, we describe how the marking process
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evolves in time. Let us denote by mi the i-th component of vector m, i.e., the number of tokens in place pi when the discrete marking is m. We say that a transition tj ∈ Td ∪ Ti has concession in marking M = (m, x) if and only if ∀ pi ∈
•
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ci ∈
•
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If any immediate transition tj has concession in M = (m, x), and its enabling condition does not depend on the continuous component of the marking (i.e. ( • tj ∪ ◦ tj ) ∩ Pc = ∅), it is said to be enabled and the marking is said to be vanishing. If some immediate transition tj has concession in M = (m, x), but its enabling condition depends on the continuous component of the marking (i.e. ( • tj ∪ ◦ tj ) ∩ Pc 6= ∅), then this particular transition may become enabled or disabled due to a change in the fluid part of the marking. In this case the marking is said to be potentially vanishing, and both the immediate and timed transitions that have concession must be considered potentially enabled. Otherwise, the marking is said to be tangible and any timed transition with concession is enabled in it. In other words, a timed transition is not enabled in a vanishing marking even if it has concession. It may however be enabled in a potentially vanishing marking due to the fluid part of marking1 . The firing of a transition Tj ∈ (Tt ∪ Ti ) enabled in marking M = (m, x) Tj
yields a new marking M 0 = (m0 , x0 ), i.e., (m, x) −→ (m0 , x0 ), where m0 = mi + Ad (Tj , pi ) − Ad (pi , Tj ) and ½i As (cl , Tj ) if (cl , Tj ) ∈ As ∀ cl ∈ Pc , x0l = xl + Ad (Tj , cl ) − Ad (cl , Tj ) otherwise.
∀ pi ∈ Pd ,
In other words, the firing of a timed transition Tj immediately set the level of all the continuous places ck that are connected with set arcs to Tj (of weight wk ) to the value associated to the arc, that is xk = wk (ignoring the effect that standard arcs may have). The firing time of a timed transition can be defined either in stochastic or non-stochastic manner. The definition on the stochastic side follows [12]. The firing time is defined through the firing rate that can depend on the actual marking of the net. In this manner non-Markovian behavior can be modeled. The definition on the non-stochastic side follows the idea of [17]. Firing time is assigned as a constant value or as an interval defined by earliest 1
Note that the previous definition is different from the one of standard GSPNs [1], since it must take into account problems that may arise due to the fluid part of the model. The firing rule is also different from the one of GSPNs because the firing of a transition may affect the continuous part of the marking due to set and discrete arcs.
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and latest firing time values. The firing semantics is interleaving with nondeterminism (no weight is assigned to the action of atomic firing inside the allowed interval or for resolving conflicts). Moreover, we follow the extended firing semantics introduced in [7]: time is assigned as intervals, and firing may be forced when the maximum time expires (strong firing semantics) or firing may be not mandatory when the maximum time expires (weak firing semantics). The earliest and the latest firing time can depend on the actual marking of the net. In any time instant when the amount of time since which the transition is enabled is inside the interval defined by the earliest and the latest firing time, the transition can fire. The evolution of the discrete part of the HPN in a tangible marking is governed by a race. In a vanishing marking instead, the choice of which transition should fire is left non-deterministic2 . The evolution of the continuous part depends on the fluid transitions. Fluid transitions can be enabled or disabled in any marking, tangible or vanishing. A fluid transition tj is enabled if and only if ∀ pi ∈
•
∀ ci ∈
tj , pi ∈ Pd , mi ≥ At (pi , tj ), •
tj , ci ∈ Pc , xi ≥ At (ci , tj ),
∀ pi ∈
◦
∀ ci ∈
◦
tj , pi ∈ Pd , mi < Ah (pi , tj ), tj , ci ∈ Pc xi < Ah (ci , tj ).
Each continuous arc that connects a fluid place ck ∈ Pc to an enabled fluid transition Tj ∈ Tf (or an enabled transition Tj to a fluid place ck ), causes a non-deterministic “change” in the fluid level of place ck . On the non-stochastic side, along a fluid arc the rate at which the fluid is moved into a fluid place or away from a fluid place is defined by an interval. The actual fluid rate along the arc can be any value from this interval chosen in non-deterministic but non-stochastic manner. The potential rate of change of fluid level of a given place can be computed by superposing the effect of the connected fluid arcs. On the stochastic side, we need a stochastic process that describe the flow rate. A simple way is to use a Markov-chain that modulate the flow rate by associating a flow rate to each state of the chain.
3 A simple example Figure 3 presents a HPN model of a rail-crossing. The model is divided into two parts: the left part represents the train, and the right part describes the barrier. The train may be away from the rail-crossing, or approaching it. Place away is marked when all the trains are far away, while place approaching is marked when a train is near the crossing. Transition next-train represents the arrival of a new train in the rail-crossing area. The position of the train is represented by fluid place train-position, which is filled by fluid transition 2
Note that priority as defined in [1] could be introduced as well, but it has been avoided to simplify the presentation of the formalism.
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train-moving. That transition is enabled as long as place approaching is marked thanks to a test arc. As soon as the train is gone far away (at a distance xaway ), transition train-away fires, removing the token from place approaching and putting it in place away. During this event the position of the train is reset using a set arc. The barrier may be either open or closed, depending on the marking of places open and closed. The angle of the barrier is represented by fluid place barrier-angle. As soon as the angle reaches 0, the token moves from place open to place close thanks to immediate transition closing. This transition can fire due to the inhibitor arc that connects it to fluid place barrier-angle. When the angle reaches 90, the system jumps from closed to open, thanks to transition opening and the test arc that connects it to place barrier-angle. The movement of the barrier is regulated by fluid transitions up and down. Transition down is enabled when the train is after (test arc) a given point (xnear ) and place open is marked. Transition up is enabled when the train has passed the rail-crossing (inhibitor arc) and is far enough (xf ar ). As discussed in the previous section, the behavior of the model can be defined both in stochastic and non-stochastic manner.
4 Analysis Completely non-stochastic setting Hybrid automata A hybrid automaton (HA) [3] is a finite state machine whose nodes contain real valued variables with definition of their first derivatives and their possible bounds. The edges represent discrete events and are labeled with assignments on the variables. Given a HA and a formula on its variables, model checking is aimed at computing the regions that satisfies the formula and to provide counterexamples. A conversion algorithm from HPN to HA could be envisaged based on [21]. Having the translation, one can analyze the model with HyTech. HyTech [3] is a model checker for Linear HA, i.e. for HA which only use a restricted form of linear differential equations to define the dynamics of the continuous state variables. If the model does not fit this restriction approximations can be necessary. See [10] for an example.
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Model checking of finite state machine An HPN model can be approximated by a finite state machine by discretizing the continuous variables of the HPN model. The properties to be checked can be specified in Computational Tree Logic (CTL) or a Real Time CTL (RTCTL). Translation of a HPN into the finite state machine description of NuSMV [19] is discussed in [10]. Completely stochastic setting Performance evaluation Two kinds of solution approaches exist for S-HPN (HPN in stochastic setting): numerical techniques (see [14, 12, 8, 9]) and simulation (see [6, 11]). Numerical techniques are aimed at finding the solution of the set of partial differential equations that characterize the stochastic process described by the S-HPN. The solution achieved by numerical analysis can be very detailed, at the expense of a very high computational cost. The equations becomes almost impossible to solve for models with more than two fluid places. Simulative techniques instead do not suffer from the limitations imposed by the numerical techniques, but accuracy is often very hard to control. Model checking By discretizing the differential equations that describe evolution of the model, a discrete-time Markov-chain (DTMC) can be constructed that approximates the behavior of the model. Such techniques are described connected to non-Markovian PN models in [9, 15]. Then, the resulting DTMC can be model checked against probabilistic CTL [13] by a probabilistic model checker. Such model checker is, for example, PRISM [16]. Models with mixed stochastic and non-stochastic behavior The proposed formalism can model systems with mixed stochastic and non-stochastic behavior. Recently, model checking of this kind of models is also considered. A tool for this purpose is Prism [18].
5 Conclusions The proposed formalism allows for defining the model of a hybrid positive system either in stochastic or non-stochastic setting. When the model is nonstochastic, the focus is on verifying logical properties of the system through model checking. In the stochastic case, modeling is aimed at performance analysis or stochastic model checking. In both cases, the analysis can be performed applying already known techniques and tools. Applicability of the approach was shown through a case study in [10].
References 1. M. Ajmone Marsan, G. Balbo, G. Conte, S. Donatelli, and G. Franceschinis. Modelling with Generalized Stochastic Petri Nets. John Wiley & Sons, 1995.
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2. H. Alla and R. David. Continuous and Hybrid Petri Nets. Journal of Systems Circuits and Computers, 8(1):159–188, Feb. 1998. 3. R. Alur, T.A. Henzinger, and P. H. Ho. Automatic symbolic verification of embedded systems. IEEE Tr. on Software Engineering, 22, 1996. 4. R. Armosini, A. Giua, M. T. Pilloni, and C. Seatzu. Simulation and control of a bottling plant using first-order hybrid petri nets. In Proc. of POSTA’03, Rome, Italy, Aug. 2003. 5. A. Bobbio and A. Horv´ ath. Petri nets with discrete phase type timing: A bridge between stochastic and functional analysis. In Proc. of MTCS’01, volume 52 No. 3 of ENTCS, Aalborg, Denmark, Aug. 2001. 6. G. Ciardo, D. M. Nicol, and K. S. Trivedi. Discrete-event Simulation of Fluid Stochastic Petri Nets. IEEE Tr. on Software Engineering, 2(25):207–217, 1999. 7. C. Ghezzi, D. Mandrioli, S. Morasca, and M. Pezze`e. A unified high level Petri net formalism for time-critical systems. IEEE Tr. on Software Engineering, 17:160–171, 1991. 8. M. Gribaudo. Hybrid formalism for performance evaluation: Theory and applications. PhD thesis, Dipartimento di Informatica, Universit` a di Torino, 2001. 9. M. Gribaudo and A. Horv´ ath. Fluid stochastic petri nets augmented with flushout arcs: A transient analysis technique. IEEE Tr. On Software Engineering, 28(10):944–955, 2002. 10. M. Gribaudo, A. Horv´ ath, A. Bobbio, E. Tronci, E. Ciancamerla, and M. Minichino. Model-checking based on fluid petri nets for the temperature control system of the icaro co-generative plant. In Proc. of SAFECOMP’02), volume 2434 of LNCS, Catania, Italy, Sept. 2002. To appear in Int. Journal of Reliability Engineering & System Safety. 11. M. Gribaudo and M. Sereno. Simulation of Fluid Stochastic Petri Nets. In Proc. of MASCOTS’00, pages 231–239, San Francisco, CA, Aug. 2000. 12. M. Gribaudo, M. Sereno, A. Horv´ ath, and A. Bobbio. Fluid stochastic Petri nets augmented with flush-out arcs: Modelling and analysis. Discrete Event Dynamic Systems, 11:97–111, 2001. 13. H. Hansson and B. Jonsson. A logic for reasoning about time and reliability. Formal Aspects of Computing, 6(5):512–535, 1994. 14. G. Horton, V. G. Kulkarni, D. M. Nicol, and K. S. Trivedi. Fluid stochastic Petri Nets: Theory, Application, and Solution Techniques. European Journal of Operations Research, 105(1):184–201, Feb. 1998. 15. A. Horv´ ath, A. Puliafito, M. Scarpa, and M. Telek. Analysis and evaluation of non-Markovian stochastic Petri nets. In Proc. of TOOLS’00, volume 1786 of LNCS, pages 171–187, Schaumburg, IL, USA, March 2000. 16. M. Kwiatkowska, G. Norman, and D. Parker. Prism: Probabilistic symbolic model checker. In Proc. of TOOLS’02, volume 2324 of LNCS, April 2002. 17. P. Merlin and D. J. Faber. Recoverability of communication protocols. IEEE Tr. on Communication, 24(9):1036–1043, 1976. 18. PRISM. http://www.cs.bham.ac.uk/∼dxp/prism/ . 19. NuSMV. http://nusmv.irst.itc.it/index.html. 20. K. Trivedi and V. Kulkarni. FSPNs: Fluid Stochastic Petri nets. In Proc. of ICATPN’93, volume 691 of LNCS, pages 24–31, Chicago, USA, June 1993. 21. B. Tuffin, D.S. Chen, and K. Trivedi. Comparison of hybrid systems and fluid stochastic Petri nets. Discrete Event Dynamic Systems, 11 (1/2):77–95, 2001.
Simulation and Control of a Bottling Plant using First-order Hybrid Petri Nets Roberta Armosini1 , Alessandro Giua2 , M. Teresa Pilloni1 , and Carla Seatzu2 1 2
Dip. Ingegneria Meccanica, Universit` a di Cagliari, Italy, [email protected] Dip. Ingegneria Elettrica ed Elettronica, Universit` a di Cagliari, Italy, {giua,seatzu}@diee.unica.it
Abstract. In this paper we show how First–Order Hybrid Petri nets, an hybrid positive model that combines fluid and discrete event dynamics, may be efficiently used to simulate the dynamic concurrent activities of manufacturing systems. In particular we deal with the performance analysis via simulation of a mineral water bottling plant according to the variations of the production controlling input parameters. The model allows a simulation of the productive line behavior through changes in the production capacity of the producing bottles and PET prototype machines, of the filling machines, of the volume and type of the bottles, of the silos dimensions, and so on.
1 Introduction In this paper we show how hybrid Petri nets [5], a model for positive systems [3] that combines fluid and discrete event dynamics, may be efficiently used to simulate the concurrent activities of high-speed manufacturing systems. The considered application. Problems related to production management and optimization become particularly critical in high-speed production plants, a particular example of which are mineral water bottling plants. Difficulties in production management arise, as a matter of fact, from two conflicting requirements: on one side we have the market, usually characterized by a very variable demand as far as formats and quantity outputs are concerned; on the other one we have the production system, whose best performances are obtained in stable conditions characterized by a constant output production. Simulation techniques represent an important and valid support for coping with these problems, as they allow to estimate plant behavior and performances resulting from different market scenarios, in which variations in the number and size of PET units produced or of bottles filled may occur. Simulation is useful both in the design phase, providing important information for the subsequent decision choices, and in the management phase. L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 79-86, 2003. Springer-Verlag Berlin Heidelberg 2003
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In the present work, a production line of an existing plant was simulated. The plant under study is the Sarda Acque Minerali (SAM) unit, a mineral water bottling plant located in southern Sardinia, at about 20 km from the city of Cagliari. The company production [6] achieves about 110 millions of bottles per year; several formats (0.25l, 0.5l, 1l, 1.5l, 2l) of bottles are produced, filled and finally sold, both with still mineral and sparkling water. Moreover four different mineral water brands are produced. Petri nets as positive systems. Discrete Petri nets [7] are a discrete event model whose state space belongs to the set of non-negative integers. This is a major advantage with respect to other formalisms such as automata, where the state space is a symbolic unstructured set, and has been exploited to develop many analysis techniques that do not require to enumerate the state space (structural analysis [4]). Recently, much work has been devoted to the extension of the classical discrete Petri net formalism to continuous Petri nets obtained by fluidification [8]. In fact, in many applications dealing with complex systems it happens that the model of the plant has a discrete event dynamics whose number of reachable states is typically very large. The analysis and optimization of these systems require large amount of computational efforts, and problems of realistic scale quickly become analytically and computationally untractable. To cope with this problem it is often possible to give a fluid (i.e., continuous) approximation of the “fast” discrete event dynamics [9]. Note that the discrete event dynamics that can be represented by a fluid model are usually related to the flow of materials, thus making fluid models essentially a type of compartmental models [3], a sub-class of positive systems. In general different fluid approximations are necessary to describe the same system, depending on its discrete state. Thus, the resulting model can be better described as an hybrid model , where different dynamics are associated to each discrete state. This has recently lead to the definition of a new family of Petri net models that combine discrete and continuous subsystems into a so called hybrid Petri net [1, 5]. Note that the area of hybrid systems has received a lot of attention in the automatic control community, lately: we believe that in the next years much attention will also be devoted to hybrid positive systems, i.e., positive systems combining both discrete event and continuous dynamics, and hybrid Petri nets are a good example of these class of systems. The hybrid Petri net model considered in this paper is called First–Order Hybrid Petri nets (FOHPN) because its continuous behavior is piece-wise constant. FOHPN were originally presented in [2].
2 First–order hybrid Petri nets In this paper we use the Petri net formalism firstly presented in [2]. Net structure. An (untimed) FOHPN is a structure N = (P, T, P re, P ost, D, C). The set of places P = Pd ∪ Pc is partitioned into a set of discrete
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places Pd (represented as circles) and a set of continuous places Pc (represented as double circles). The cardinality of P , Pd and Pc is denoted n, nd and nc . The set of transitions T = Td ∪ Tc is partitioned into a set of discrete transitions Td and a set of continuous transitions Tc (represented as double boxes). The cardinality of T , Td and Tc is denoted q, qd and qc . The pre+ and post-incidence functions that specify the arcs are (here R+ 0 = R ∪{0}): + P re, P ost : Pc × T → R0 , Pd × T → N. We require that ∀t ∈ Tc and ∀p ∈ Pd , P re(p, t) = P ost(p, t), so that the firing of continuous transitions does not change the marking of discrete places. The function D : Td → R+ 0 specifies the timing associated to timed discrete transitions. The function + C : Tc → R+ 0 × R∞ specifies the firing speeds associated to continuous tran+ sitions (here R+ ∞ = R ∪ {∞}). For any continuous transition tj ∈ Tc we let C(tj ) = (Vj0 , Vj ), with Vj0 ≤ Vj . Here Vj0 represents the minimum firing speed (mfs) and Vj represents the maximum firing speed (MFS). The incidence matrix of the net is defined as C(p, t) = P ost(p, t) − P re(p, t). The restriction of C to PX and TY is denoted C XY . A marking is a function that assigns to each discrete place a non-negative number of tokens, represented by black dots and assigns to each continuous place a fluid volume. A continuous place can be seen as a tank that can fill up with fluid (marking). However, we also consider some connecting elements (such as a pipe) with a zero capacity where fluid can flow but not accumulate. Thus we partition the set of continuous places Pc = P0 ∪ P+ into a set of places P0 (represented as full dark circles) whose marking is always equal to zero (connecting elements), and a set of places P+ (represented as double circles) whose marking may assume any nonnegative real number (tanks). Therefore m : P+ → R+ 0 , P0 → 0, Pd → N. The marking of place pi is denoted mi , while the value of the marking at time τ is denoted m(τ ). The restriction of m to Pd and Pc are denoted with md and mc , respectively. An FOHPN system hN, m(τ0 )i is an FOHPN N with an initial marking m(τ0 ). Net dynamics. The enabling of a discrete transition depends on the marking of all its input places, both discrete and continuous. More precisely, a discrete transition t is enabled at m if for all pi ∈ • t, mi ≥ P re(pi , t), where • t denotes the preset of transition t. If a discrete transition tj fires at a certain time instant τ − , then its firing at m(τ − ) yields a new marking m(τ ) such that mc (τ ) = mc (τ − ) + C cd σ, and md (τ ) = md (τ − ) + C dd σ, where σ is the firing count vector associated to the firing of transition tj . To every continuous transition tj is associated an instantaneous firing speed (IFS) vj (τ ). For all τ it should be Vj0 ≤ vj (τ ) ≤ Vj , and the IFS of each continuous transition is piecewise constant between events. An empty continuous place pi can be fed, i.e., supplied, by an input transition, which is enabled. Thus, as a flow can pass through an unmarked continuous place, this place can deliver a flow to its output transitions. Consequently, a continuous transition tj is enabled at time τ if and only if all its input discrete places pk ∈ Pd have a marking mk (τ ) at least equal to P re(pk , tj ),
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and all its input continuous places are either marked or fed. If all input continuous places of tj have a not null marking, then tj is called strongly enabled, else tj is called weakly enabled. Finally, transition tj is not enabled if one of its empty input places is not fed. We can write the equation which governs the evolution in time of the P marking of a place pi ∈ Pc as m ˙ i (τ ) = tj ∈Tc C(pi , tj )vj (τ ) where v(τ ) = [v1 (τ ), . . . , vnc (τ )]T is the IFS vector at time τ . The enabling state of a continuous transition tj defines its admissible IFS vj . If tj is not enabled then vj = 0. If tj is strongly enabled, then it may fire with any firing speed vj ∈ [Vj0 , Vj ]. If tj is weakly enabled, then it may fire with any firing speed vj ∈ [Vj0 , V j ], where V j ≤ Vj since tj cannot remove more fluid from any empty input continuous place p than the quantity entered in p by other transitions. We say that a macro–event occurs when: (a) a discrete transition fires, thus changing the discrete marking and enabling/disabling a continuous transition; (b) a continuous place becomes empty, thus changing the enabling state of a continuous transition from strong to weak. Let τk and τk+1 be the occurrence times of two consecutive macro–events as defined above; we assume that within the interval of time [τk , τk+1 ), denoted as a macro–period, the IFS vector is constant and we denote it v(τk ). Then the continuous behavior of an FOHPN for τ ∈ [τk , τk+1 ) is described by mc (τ ) = mc (τk ) + C cc v(τk )(τ − τk ), md (τ ) = md (τk ).
3 Modeling plant subsystems with FOHPN In this section we briefly describe some components of the considered plant and the corresponding net models. Transportation lines and switches. Transportation lines consist of pipes of appropriate diameter, depending on the bottle sizes, where bottles are conveyed at high speed thanks to the force produced by the compressed air. Due to the high speed, the main feature of these elements is that there is no accumulation of bottles in their inside. Therefore, transportation lines may be seen as connecting elements and the corresponding places in the Petri net model are zero capacity places, i.e., places in P0 . The connections among different lines may vary and this can be modeled by a MIMO (multi input - multi output) switch. In figure 1 a switch is represented in the case of two input and two output lines, where place pc has been denoted as a dark circle because it is a zero capacity place. The discrete marking is such that one possible path at a time is enabled. The delay times associated to discrete transitions determine the the paths that bottles follow at the different time intervals: thus they are design parameters to be optimized.
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Machines. In this plant we have two different types of machines. The first type is involved in bottles production, while the second one is involved in bottles filling and corking. Machines of the first type are equipped so as to produce bottles of different sizes. In the following, we consider the case of a machine that can be used to produce 1.5 lt bottles and 2 lt bottles. The Petri net model for such a machine is shown in figure 2.a. In particular, the firing of tc,1 denotes the production of 1.5 lt bottles, whereas the firing of tc,2 denotes the production of 2 lt bottles. Clearly, the productivity of the machine is not the same in the two cases, thus the weights of the input arcs to pc are different. pon,1
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Fig. 2. A machine that produces bottles (a). A machine that fills bottles (b).
A dual scheme may be used to describe the functioning of those machines that are involved in the bottle filling and corking. An example in the case of bottles of two different sizes is reported in figure 2.b. The delay times associated to discrete transitions determine the machine production cycle and are the design parameters to be optimized.
4 A real bottling plant Plant description. The production cycle considered in this paper consists of several stages [6]. The first stage consists in the creation of the PET bot-
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tles and the last stage consists in self-filling and corking. More precisely, the first operational machine is M1 that produces PET bottles starting from rawmaterial of PET granules (PET chips). Thanks to an appropriate equipment, this machine may be extremely versatile and may produce different bottle sizes, e.g., 1.5 lt and 2 lt. Then, the produced bottles are directed to appropriate lines of different diameter, depending on their size. The flow of bottles through the conveyor lines occurs at a high speed and is induced by a jet of compressed air. Bottles may follow different paths and may be assigned to different buffers. Path assignment may be seen as a decision problem whose solution aims to optimize the production process. In particular, in the case we are dealing with, there are 7 buffers (S1 , S2 , · · · , S7 ) and the partitioning is established so as to compensate as much as possible the delay due to the reduced productivity of the machines that fill bottles of mineral water with respect to those that produce them. Finally, from buffers bottles are conveyed to the zone of self–filling through other appropriate flow lines. Even in this case, bottles may follow different paths so as to better exploit the filling machines. In particular, there are 3 filling machines that can be used to fill bottles of all sizes. The FOHPN model. The FOHPN model of the above production process can be obtained by simply putting together all the elementary modules previously defined. The resulting Petri net model has not been reported here for brevity’s sake but it can be seen by looking at [6].
5 A numerical optimization problem In this section we present the results of several numerical simulations whose main goal is that of determining the operational configuration of the production process that enables us to optimize the efficiency of the bottling plant with respect to a given performance index. All simulations have been carried out using Simulink, a Toolbox of Matlab. The design parameters are the following: the initial configuration of the plant, i.e., the initial marking of the net; the paths that bottles should follow at the different time intervals, i.e., the timing delays associated to discrete transitions in the Petri net model of switches; the time intervals at which machines should produce (fill) bottles of different formats, i.e., the timing delays associated to discrete transitions in the Petri net model of machines producing (filling) bottles. Different numerical simulations have been carried out using the real data of the machines (namely, their productivity) and the buffers (namely, their maximum capacity). In the following we focus our attention to 1.5 and 2 lt bottles. A time period of 48 hours has been considered during simulation (the behavior of the plant is periodic with a period of 48 hours). The main goal of the company is that of maximizing the net profit resulting from selling its end items. We first assume that all the produced bottles are
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sold. In such a case the net profit is P = (SP1.5 − U C1.5 ) · N1.5 + (SP2 − U C2 ) · N2 where SP1.5 (SP2 ) is the selling price of 1.5 (2) lt bottles, while U C1.5 (U C2 ) is the unitary cost associated to 1.5 (2) lt bottles. The selling price is the price at which the end item is sold to the customer. In all numerical simulations we assumed SP1.5 = 18 c and SP2 = 22 c, where c denotes a cent of Euro. The unitary cost is the cost that the company pays for one unit of end item. It includes the cost that the company pays for the PET and the water, plus an additional term taking into account the production costs pertaining to one bottle. In particular, we assumed U C1.5 = 5 c and U C2 = 6 c. The resulting net profit, computed under the assumption that all the produced bottles are sold, is that shown by the thin curve in figure 3. Thus we can conclude that the fifth simulation corresponds to the best configuration of the plant with respect to the considered performance index P . Note that it is possible to prove that the fifth simulation corresponds to the maximal productivity of no format. This means that the maximum profit is guaranteed by appropriately partitioning the production resources among bottles of different sizes. 6
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Fig. 3. The net profit P under the assumption that all bottles are sold and the net profit P taking into account some constraints in the sale.
Finally, we compute the net profit under the following two realistic assumptions. Firstly, we assume that there is an upper bound on the demand of bottles of each format: if the number of produced bottles is greater than such a limit, then there is a certain number of bottles that are not sold, thus producing no profit. Secondly, we assume that if the number of bottles is less than a given lower bound then the whole demand cannot be met. This produces a shortage which usually has many associated costs. Apart from the loss of profit, the effects of shortage include loss of goodwill, loss of future sales, and so on. In particular, in all numerical simulations we assumed that within the considered time period of simulation, the maximum number of bottles of
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each format that can be sold is Nmax = 7 · 105 , while the number of produced bottles under which there is shortage is Nmin = 105 . Finally, we evaluated that shortage cost is equal to SC = 2 c for unit of end item for both formats. In such a case the net profit is equal to P = SP1.5 · max{N1.5 , Nmax } − U C1.5 · N1.5 − SC · max{0, Nmin − N1.5 } +SP2 · max{N2 , Nmax } − U C2 · N2 − SC · max{0, Nmin − N2 }. When the performance index to be maximized it P the resulting curve is the thick one in figure 3. Thus we can conclude that even in this case the best configuration of the plant is the fifth one.
6 Conclusions An analysis of the operating conditions of a mineral water bottling plant was performed by means of a simulation model based on first order hybrid Petri nets and Simulink. The tests accomplished demonstrate the ability of the model to correctly describe the behavior of the single machines and of the global plant; it also allows to foresee the main plant performances for different operating plant conditions, so representing a valid instrument to cope with complex production optimization problems.
References 1. H. Alla, R. David, “Continuous and Hybrid Petri Nets,” Journal of Circuits, Systems, and Computers, Vol. 8, No. 1, 1998. p. 159-88. 2. F. Balduzzi, A. Giua, G. Menga, “First–Order Hybrid Petri Nets: a Model for Optimization and Control,” IEEE Trans. on Robotics and Automation, Vol. 16, No. 4, pp. 382-399, 2000. 3. L. Benvenuti, L. Farina, “Positive and Compartmental Systems,” IEEE Trans. on Automatic Control , Vol. 47, No. 2, pp. 370-373, 2002. 4. J.M. Colom, M. Silva, “Improving the linearly based characterization of P/T nets,” Advances in Petri Nets 1990 , LNCS 483, pp. 113–145, Springer, 1991. 5. A. Di Febbraro, A. Giua, G. Menga, (eds.) “Special Issue on Hybrid Petri Net,” Discrete Event Dynamic Systems Vol. 11, No. 1/2, 2001. 6. A. Giua, A. Meloni, M.T. Pilloni, C. Seatzu, “Modeling of a bottling plant using hybrid Petri nets,” 2002 IEEE Int. Conf. SMC, Hammamet (Tunisia), Oct 2002. 7. T. Murata, “Petri Nets: Properties, Analysis and Applications,” Proceedings IEEE , Vol. 77, No. 4, pp. 541–580, 1989. 8. M. Silva, L. Recalde, “Petri nets and integrality relaxations: A view of continuous Petri nets,” IEEE Trans. Syst., Man, & Cybern., Vol. 32, No. 4, 2002. 9. R. Suri, B.R. Fu, “On Using Continuous Flow Lines to Model Discrete Production Lines,” Discrete Event Dynamic Systems, No. 4, pp. 129–169, 1994.
Parameter Identifiability of Nonlinear Biological Systems Mariapia Saccomani1 , Stefania Audoly2 , Giuseppina Bellu3 , and Leontina D’Angi` o3 1 2 3
Department of Information Engineering, University of Padova, Padova, Italy, [email protected] Department of Structural Engineering, University of Cagliari, Italy, [email protected] Department of Mathematics, University of Cagliari, Italy, [email protected], [email protected]
Abstract. Parameters characterizing the internal behaviour of biological and physiological systems are usually not directly accessible to measurement. Their measurement is usually approached indirectly as a parameter estimation problem. A dynamic model describing the internal structure of the system is formulated and an input-output experiment is designed for model identification. Identifiability is a fundamental prerequisite for model identification; it concerns uniqueness of the model parameters determined from the input-output data, under ideal conditions of noise-free observations and error-free model structure. Recently, differential algebra tools have been applied to study the identifiability of nonlinear dynamic systems described by polynomial equations, however very few results have been obtained. Given that biological/physiological systems are usually characterized by nonlinear dynamics, e.g. threshold processes, and that the identification experiments are often performed on systems started from known (equilibrium) initial conditions, our purpose was to develop a new differential algebra algorithm, which tests a priori identifiability of nonlinear models with given initial conditions. The algorithm is presented together with an example.
1 Introduction A priori global identifiability is a fundamental prerequisite for model identification. It concerns unique solvability for the parametric structure of a dynamic model from ideal, noise-free, input-output experiments. Assuming that measured input-output variables are available in the absence of noise, one would like to recover a unique model (i.e. a unique parametric structure) from an experiment. For nonlinear models very few results have been obtained and no standard algorithm exists for testing a priori global identifiability. The early efforts have not been very successful; in particular, the method based on power series leads to an infinite number of nonlinear algebraic equations, the L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 87-93, 2003. Springer-Verlag Berlin Heidelberg 2003
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similarity transformation method of [3], although leading to a finite number of equations, is very difficult to implement. The introduction of concepts of differential algebra in control and system theory by Fliess [4] has led to a better understanding of the nonlinear identifiability problem. In particular, Ollivier [6] and Ljung and Glad [5] have shown that the concept of characteristic set of a differential ideal introduced by Ritt [7] is a very useful tool in identifiability. Although differential algebra methods has been an important factor for addressing the identifiability problem for nonlinear models, the construction of an efficient algorithm still remains a difficult task [5, 8]. In [1] we have presented a new algorithm to test global identifiability based on differential algebra and on several conceptual improvements on the methods existing in the literature. However, all the algorithms based on differential algebra analyse identifiability of systems assuming generic initial conditions. It has been pointed out that they may give wrong answers in special cases when the initial condition is fixed to some special value, a situation frequently encountered in identification of biological and medical systems. Our recent work has been devoted to extend the applicability of our algorithm to systems started at specific initial conditions [9, 10].
2 Background and definitions 2.1 The system Consider a parameterised nonlinear system described in state space form ½ ˙ x(t) = f [x(t), p] + G[x(t), p]u(t) (1) y(t) = h[x(t), u(t), p] where x is the n-dimensional state variable; u the m-dimensional input vector made of smooth functions; y is the r-dimension output; p ∈ P is the νdimensional parameter vector. If initial conditions are specified, the relevant equation x(t0 ) = x0 is added to the system. Although this is not strictly necessary, we have assumed the system affine in the control variable. The essential assumption here is that the entries of f , G = [g1 , . . . , gm ] and h are polynomial or rational functions of their arguments. 2.2 A priori identifiability Let y = Φx0 (p, u) be the input-output map of the system (1) started at the initial state x0 . The definition below describes identifiability from input-output data, which is the concept of interest when (as it is usually the case) the initial state is not known exactly to the experimenter.
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Definition The system (1) is a priori globally (or uniquely) identifiable from input-output data if, for at least a generic set of points p∗ ∈ P, there exists (at least) one input function u such that the equation Φx0 (p, u) = Φx0 (p∗ , u)
(2)
has only one solution p = p∗ for all initial states x0 ∈ X ⊆ IRn . A weaker notion is that of local identifiability. The system is locally (or nonuniquely) identifiable at p∗ ∈ P if there exists (at least) one input function u and an open neighbourhood Up∗ of p∗ , such that the equation (2) has a unique solution p ∈ Up∗ for all initial states x0 ∈ X ⊆ IRn . According to these definitions, for a system which is not even locally identifiable, equation (2) has generically an infinite number of solutions for all input functions u. This is commonly called nonidentifiability or unidentifiability [1, 3, 5, 12]. As we shall review in detail later in this paper, the use of differential algebra permits to write the input-output relation of the system in implicit form, i.e. as a set of r polynomial differential equation in the variables (y, u) and coefficients depending, in general polynomially, on the parameter p [6]. In order to analyse the a priori identifiability of the model (1) one has just to define a proper “canonical” set of coefficients of the polynomial differential equations, say c(p). One refers to this family of functions of p as the exhaustive summary of the model [6, 12] since the map c embodies the parameter dependence of the input-output model completely. After the exhaustive summary is found, to study a priori global identifiability of the model, one has to check if the map c(p) is injective.
3 Identifiability and characteristic sets For a formal description of the fundamentals of differential algebra and of the characteristic set, the reader is referred to [7]. Here we only recall that the peculiarity of a characteristic set is that it can be used to generate the differential ideal by means of a finite number of polynomials. We can now return to the dynamic model (1). This can be looked upon as a set of n + r differential polynomials which are the generators of a differential ideal I in a differential ring. The characteristic set of the ideal I is a finite set of n + r nonlinear differential equations which describes the same solution set of the original system [5]. Its special structure allows to construct the exhaustive summary of the model used to test identifiability. The problem now is to construct, in an algorithmic way, the characteristic set starting from the model equations. To solve this problem we have chosen [8, 1] the differential ring R(p)[u, y, x,] and the standard ranking which defines the inputs as the smallest components, followed by the outputs, and the state variables. Thus the characteristic set of the polynomials defining the model has the following form:
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A1 (u, y) . . . Ar (u, y) Ar+1 (u, y, x1 ) Ar+2 (u, y, x1 , x2 ) . . .
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(3)
We will refer to the corresponding first r differential polynomial equations A1 (u, y) = 0
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of (3) as the input-output relation. These polynomials are obtained after elimination of the state variables x and hence represent exactly the pairs (u, y) which are described by the original system. Further, we shall introduce a suitable normalization to make the characteristic set unique. It follows that the coefficients cij (p) of the input-output relation (4) constitute the exhaustive summary of the model. In order to test global identifiability of the system (1) the injectivity of the map c from the parameter space P to its range, a subset of the ν-dimensional Euclidean space, has to be checked. This is the same as unique solvability of the equations cij (p) = c∗ij
i = 1, . . . , r
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(5)
for arbitrary right-hand members c∗ij in the range of c. To solve the resulting system of algebraic nonlinear equations (5) we use the Buchberger algorithm [2].
4 The question of initial conditions As we have seen, the construction of the characteristic set ignores the initial conditions. In particular, the input-output relation (4) represents the inputoutput pairs of the system for “generic” initial conditions. Often, however, physical systems have to be started at special initial conditions, e.g. all radiotracer kinetics experiments in humans [1] are necessarily started at the initial state x(0) = 0. Thus, the problem arises if some specific initial conditions can change the input-output relation. 4.1 The role of accessibility In the following we shall refer to the concept of accessibility as defined in [11]. A full understanding of the identifiability problem with specific initial conditions requires to study the role of accessibility in the structure of the characteristic set [9]. For reasons of space here we shall only present the main ideas leaving the details to [10]. In [9] we have shown that, when the system is accessible from x0 , adding the specific initial condition x(0) = x0 as a constraint, cannot change the characteristic set. This is so since the variety where the motion of the system takes place has the same dimension of the initial variety and the order of the system can not drop. Conversely, suppose that the system is
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algebraically observable [1], generically accessible and assume that x0 belongs to the “thin” set from which the system is non accessible. There is an invariant subvariety where the motion of the system takes place when started at the initial condition x0 . This subvariety can be calculated by the construction of the accessibility Lie Algebra, see [11, p.154]. Let φ(x) = 0 be the equation of the invariant subvariety of non-accessible states. This algebraic equation must be added to the characteristic set in order to get a reduced representation of the system.
5 A computer algebra algorithm The starting point of the algorithm is the differential polynomials defining the dynamical system. The principal steps of the algorithm are: 1. The accessibility Lie Algebra of the system is constructed and the set of zero measure, if exists, where the accessibility rank condition does not hold is calculated. Let φ(x) = 0 be its equation; 2. if φ(x) exists and φ(x(0)) = 0, φ(x) is added to the initial polynomials; 3. if one or more polynomials are rationale, they are reduced to the same denominator; 4. a ranking is introduced; 5. the leaders of each polynomial are found; 6. the polynomials are ordered. Each polynomial is compared with the previous ones and, if it is of equal or higher rank, is reduced with respect to them. This step is repeated until the autoreduced set of minimum rank is reached. This is the characteristic set; 7. the input-output relations are made monic and their coefficients, belonging to the field R(p), are extracted; 8. a random numerical point pˆ from the parameter space is chosen and the exhaustive summary of the system is calculated; 9. the Buchberger algorithm is applied to solve the equations and the number of solutions for each parameter is provided.
6 Example Consider the nonlinear model discussed in [3]. It is a two compartment model which describes the kinetics of a drug in the human body. The drug is injected into the blood where it exchanges linearly with the tissues; the drug is irreversibly removed with a nonlinear saturative characteristic from the blood and with a linear one from the tissue. The system is x1 (0) = 0 x˙ 1 = −(k21 + VM /(Km + x1 ))x1 + k12 x2 + b1 u x˙ 2 = k21 x1 − (k02 + k12 )x2 x2 (0) = 0 (6) y = c1 x1
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where x1 , x2 are drug masses in blood and tissues respectively, u is the drug input, y the measured drug output in the blood, k12 , k21 and k02 are the constant rate parameters, VM and Km are the classical Michaelis-Menten parameters, b1 and c1 are the input and output parameters respectively. The question is whether the unknown vector p = [k21 , k12 , VM , Km , k02 , c1 , b1 ] is globally identifiable from the input-output experiment. Let the ranking of the variables be u < y < x1 < x2 . Since the accessibility Lie Algebra rank is equal to 2 the system is accessible from every point. The reduction procedure is started and the characteristic set is calculated. y¨y 2 + k21 k02 y 3 − (k21 c1 b1 + k02 c1 b1 )y 2 u + (k21 + k12 + k02 )yy ˙ 2+ 2 3 2 −c1 b1 y u˙ − Km c1 b1 u˙ + (2k21 Km k02 + k12 VM + k02 VM )c1 y + −2(k12 + k02 )c1 2 b1 Km yu + 2(k21 Km c1 + k12 Km c1 + k02 Km c1 )y y+ ˙ +2Km c1 y y¨ − 2Km c1 2 b1 y u˙ + Km c1 2 (k21 k02 + k12 VM + k02 VM )y+ −(k12 Km 2 b1 c1 3 + k02 Km 2 b1 c1 3 )u + Km 2 c1 2 y¨+ +(k21 Km 2 c1 2 + k12 Km 2 c1 2 + VM Km c1 2 + k02 Km 2 c1 2 )y˙ y − c1 x1 yy ˙ + Km c1 y˙ − c1 b1 y u˙ − Km c1 b1 u + k21 y 2 − k12 c1 x2 y+ +(Km c1 k21 + VM c1 )y − k12 Km c1 2 x2
(7)
Note that only the first differential polynomial of (7) represents the inputoutput relation of the model, in fact it does not contains as variable neither x nor its derivatives. Coefficients of the input-output relation are extracted (these are the exhaustive summary of the model) and evaluated at a numerical point pˆ randomly chosen in the parameter space P. Each coefficient, in its polynomial form, is then set equal to its corresponding numerical value. The Buchberger’s algorithm is applied to calculate the Gr¨obner basis of the polynomial system (7) −13VM + 11Km k21 − 1 k12 − 17 (8) 11b1 − 7VM k02 − 3 VM c1 − 44 It is easy to see that the system of equations obtained by setting to zero the Gr¨ obner basis polynomials (8) has an infinite number of solutions, thus the model is a priori non identifiable. Note that if the input parameter is assumed to be known, i.e. b1 = 1, the model becomes a priori globally identifiable.
7 Conclusions A priori identifiability is a necessary prerequisite for parameter identification. Checking a priori global identifiability, i.e. the uniqueness of the solution, is
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particularly difficult for nonlinear dynamical systems. In this paper, we briefly describe a new algorithm, recently developed by the authors, for testing a priori identifiability of nonlinear systems. The algorithm is based on the concept of characteristic set of the ideal generated by the polynomials defining the model. We propose a new version of the algorithm which allows to successfully deal with systems starting from initial conditions fixed to some specific value. In order to do this, the accessibility property of the system is checked by using the accessibility Lie Algebra and, whenever needed, a new ideal associated to the dynamical system which takes into account initial conditions is calculated. The algorithm has been used successfully to analyse a priori identifiability of several biological system models.
References 1. S. Audoly, G. Bellu, L. D’Angi` o, M.P. Saccomani and C. Cobelli, Global identifiability of nonlinear models of biological systems, IEEE Trans. Biomed. Eng., vol. 48, n. 1, pp.55-65, 2001. 2. B. Buchberger, An algorithmical criterion for the solvability of algebraic system of equation, Aequationes Mathematicae, vol. 4, no. 3, pp. 45-50, 1988. 3. M.J. Chappell and K.R. Godfrey, Structural identifiability of the parameters of a nonlinear batch reactor model, Math. Biosci., vol. 108, pp. 245-251, 1992. 4. M. Fliess and S.T. Glad, An Algebraic Approach to Linear and Nonlinear Control, in Essays on Control: Perspectives in the Theory and its Applications, H.L.Trentelman, J.C. Willems, Eds. Birkh¨ auser, Boston, pp. 223-267, 1993. 5. L. Ljung and S.T. Glad, On global identifiability for arbitrary model parameterizations,Automatica, vol. 30, no. 2, pp. 265-276, 1994. 6. F. Ollivier, Le probl`eme de l’identifiabilit´e structurelle globale: ´etude th´eorique, ´ m´ethodes effectives et bornes de complexit´e, Th`ese de Doctorat en Science, Ecole Polyt´echnique, Paris, France, 1990. 7. J.F. Ritt, Differential Algebra, Providence, RI: American Math. Society, 1950. o and C. Cobelli, Global iden8. M.P. Saccomani, S. Audoly, G. Bellu, L. D’Angi` tifiability of nonlinear model parameters, Proc. SYSID ’97 11th IFAC Symp. System Identification, Kitakyushu, Japan, vol. 3, pp. 219-224, 1997. 9. M.P. Saccomani, S. Audoly, L. D’Angi` o, A new differential algebra algorithm to test identifiability of nonlinear systems with given initial conditions, Proc. 40th IEEE Conference on Decision and Control, Orlando, Florida, USA, pp.31083113, 2001. 10. M.P. Saccomani, S. Audoly, L. D’Angi` o, Parameter identifiability of nonlinear systems: the role of initial conditions, Automatica, to appear. 11. E.D. Sontag, Mathematical Control Theory, 2nd ed., Berlin: Springer, 1998. 12. E. Walter and Y. Lecourtier, Global approaches to identifiability testing for linear and nonlinear state space models, Math. and Comput. in Simul., vol. 24, pp. 472-482, 1982.
Towards Whole Cell “in Silico” Models for Cellular Systems: Model Set-up and Model Validation Andreas Kremling, Katja Bettenbrock, Sophia Fischer, Martin Ginkel, Thomas Sauter, and Ernst Dieter Gilles Max-Planck-Institute Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany, [email protected] Abstract. Based on recent developments for new measurement technologies that enable researches to get quantitative information on intracellular processes, the setup of very detailed models describing metabolism as well as regulatory networks becomes very popular. However, biochemical networks are rather complex including many feed-forward and feedback loops. In this contribution we propose an interdisciplinary approach including the computer based set-up of models and strategies to validate the models with apparent experiments. This approach will offer a new way to meaningful models that can be used to make simulation experiments analogous to real laboratory experiments. The approach is applied to the bacterium Escherichia coli: A mathematical model to describe carbon catabolite repression is developed and in part validated. The model is aggregated from functional units describing carbohydrate transport and degradation. These units are members of the crp modulon and are under control of a global signal transduction system which calculates the signals that turn on or off gene expression for the specific enzymes. Problems of parameter identification for whole cell models are discussed.
1 Introduction Recent efforts for a better understanding of cellular systems have resulted in multidisciplinary research alliances (mainly in the US) where researchers from biology, informatics and systems engineering work together. The aim of these initiatives is to model complex biological systems in such a way that experiments can be performed with the help of a computer analogous to experiments in a real laboratory. Even biological working groups have recognized the need of frameworks for a quantitative description of cellular processes and the importance of integrating experimental and theoretical/computational approaches [3]. Central in the work of biologists is the definition of ’modules’ or ’functional units’ as a critical level of cellular organization. A concept stating that cellular metabolism is structured in functional units which could be used L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 95-102, 2003. Springer-Verlag Berlin Heidelberg 2003
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in modeling has been proposed [8]. The concept is based on the definition of submodels with characteristic features. The submodels are implemented in the computer tool ProMoT [2] that provides a graphical user interface where the submodels can be chosen from the model library and can be connected to other submodels. Based on the concept a mathematical model considering functional units describing uptake and metabolism of a number of carbohydrates in Escherichia coli is set up and analyzed [7, 6]. A novel experimental approach using isogenic mutant strains, i.e. a number of strains derived from one wild-type with a clearly defined background, was used to determine yet unknown or uncertain parameters. The intention of the contribution at hand is twofold: (i) It summarizes the current state of the model and describes the model extensions having taken place recently. (ii) Discuss the problems on the way to whole-cell-models. Up to now only a few models are available that describe parts of the metabolism, and, simultaneously, provide adequate data for model validation.
2 Modeling concept Our approach is based on the analysis and the combination of the available knowledge on metabolism, signal transduction and cellular control with system-theoretical methods. The modeling procedure thus has to be based on the molecular structure of the functional units in such a way that a cellular unit is represented by an equivalent mathematical submodel. This modular approach is a new feature in the mathematical modeling procedure and guarantees a high transparency for biologists and engineers. The basis of the framework is the definition of a complete set of elementary modeling objects. They should be disjunct with respect to the biological knowledge they comprise to prevent overlapping. The modeling process proceeds along two coordinates: a structural and a behavioral coordinate. The structural coordinate represents a progressive combination and linkage of elementary modeling objects to higher aggregated model structures. Higher aggregated model structures are called functional units. Modeling along the behavioral coordinate means that to each of the elementary modeling objects equations have to be assigned. Functional units are defined according to three biological motivated criteria: (i.) A common physiological task. (ii.) A common genetic unit. The genes for all enzymes of a functional unit are organized in genetical units (operons, regulons and modulons) and/ or in a hierarchical structure. (iii.) A common signal transduction network. All elements of a functional unit are interconnected within a common signal transduction system. The signal flow across the unit border (“cross-talk” or “cross- regulation”) is small compared to the information exchange within the unit, such that the coordinated response to a common stimulus (“stimulon”) helps to identify the members of a unit.
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3 Modeling environment ProMoT/Diva ProMoT enables the use of object-oriented modeling techniques including encapsulation, aggregation, and inheritance. In ProMoT, dynamic models are built by aggregating structural and behavioral modeling entities. The modeling entities in ProMoT are organized in an object-oriented class hierarchy with multiple inheritance. This concept from computer science was adopted to allow a better organization of complex modeling libraries and flexible implementation of large scale models. Every entity in this hierarchy inherits all parts and attributes from its respective super-classes. With this method abstraction is possible and more general and reusable entities can be formed. ProMoT provides a special modeling language as well as a graphical user interface (GUI) for interactive modeling. The modeling tool, as well as the simulation environment, are developed under different Unix-derived operating systems, however the main platform is Linux. The kernel of the system is implemented as a modeling server in object-oriented Common Lisp (using the Common Lisp Object System CLOS). The numerical analysis of the models is done with the simulation environment Diva. Within Diva many different numerical computations are possible, based on facilities to calculate the steady state and dynamic behavior of the model using non-linear equation solvers and integrators. For metabolic models 2 methods are of special interest: (i) Parameter analysis with respect to experimental data. (ii) Identification of parameters and model accuracy.
4 Model for carbohydrate uptake in E. coli Figure 1 shows the modeling objects with relevant in- and outputs. The global signal transduction system comprises the phosphoenolpyruvate (PEP)dependent: glucose phosphotransferase system (PTS), the synthesis of cAMP, and the interaction of the cAMP·Crp complex with the specific DNA binding sites. Besides its sensory function, the PTS is the main glucose uptake system. Uptake of glucose by other transport systems and uptake of lactose, galactose, glycerol, and sucrose1 are described in separate functional units. For the bacterial physiology it is well known that the control of transcription initiation is the main control principle. Therefore control of post transcriptional and of translation processes are not modeled in detail here. One main feature of the model is the hierarchical structure of regulatory network. Based on the analysis of molecular interactions of proteins with DNA binding sites a new approach to develop mathematical models describing gene expression is applied. Detection of hierarchical structures in metabolic networks can be used to decompose complex reaction schemes. This is achieved by assigning each regulator protein to one level in the hierarchy. Signals are 1
E. coli is not able to grow on sucrose; therefore an engineered strain is used here.
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then transduced from the top level to the lower level, but not vice versa. The top level in the model is represented by the RNA polymerase, the second level by the global regulator Crp and the lowest level by e.g. the lactose repressor LacI. The overall comprises 63 states (ode’s/ algebraic equations) and 251 parameters (see Table 1).
5 Model validation Model validation is an essential part in modeling. In order to validate a model it is necessary to compare predictions given by the model with results from real experiments. The experiments have to be designed in a way that the measured data contain information about the different functional units included in the model. A strategy to identify parameters in very large models for cellular systems is still missing. Although the functional units are only weakly coupled to each-other, a number of problems arise during parameter identification: • Up to now it is not clear if the available software is capable to solve all parameter fitting problems. In the present study we used maximal 10 experiments in one fit. A further problem is the finding of the most suitable state that should be measured to get the best information for the fit. From our modular approach results the idea using only one representative for every functional unit that can describe the overall dynamics of the whole
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unit. However, the development of measurement methods for interesting metabolites/ proteins is still expensive. Although the units are only weakly connected, it is useful to analyze the interconnection, e.g. with a sensitivity analysis. However, this requires some starting values for the parameters and the states in the model. The chosen values might be far away from the real values and may lead to incorrect conclusions. Most experiments are performed with batch experiments. Here, the specific growth rate is maximal. The identification of Michaelis-Menten KM values however requires low substrate concentration and therefore other growth rates are required. Low growth rates may lead to stress responses of the organism and the model is normally not able to describe this situation. Measurements of extracellular components are normally available and the uptake rate may be calculated. The simplest kinetic expression requires also information on the amount of enzyme. Since the amount may change during the experiment - this information is available seldom - parameters for the transport step can hardly be found. Incorporation of quantitative knowledge. Sometimes knowledge on the range of concentration of metabolites is available or, based on array data, knowledge that a gene have been expressed is measured.
A brief description of the theoretical background for parameter identification used so far is given in the following. 5.1 Parameter identification To solve the equations the simulation environment Diva was used. The integration algorithm DDASAC [1] has been chosen. To identify the model parameters the following approach is used: (i) Starting with parameters from literature, the model is analyzed with the method of Hearne [4], calculating a combination of parameters which have a maximal effect on the interesting states (states for which measured data are available). This sensitivity analysis gives a first impression on the sensitive parameters. (ii) Together with the measured data and the Fisher information matrix it was checked, if the sensitive parameters could be estimated. Applying a method introduced by [9] a set of parameters from the sensitive parameters were determined which could be estimated together with a given minimal variance γ. For the glucose/ lactose diauxic experiment, m = 8 states were measured (extracellular glucose and lactose, biomass and intracellular LacZ activity which is used as measure for LacZ concentration, galactose, acetate, cAMP in the medium, degree of phosphorylation of EIIA) and it can be expected that the parameters which can be estimated are related to the respective transport units. (iii) Parameter estimation: The whole model is given in the form x˙ = f (x, p) ,
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with states x and parameter vector p. For a subset of the states i = 1, m measurement data are available (zik ) at time point tk (k = 1, N ). The aim of the parameter identification is to minimize the objective function Φ(p) Φ(p) =
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To solve the optimization problem the SQP (Sequential Quadratic Programming) algorithm E04UPF from the NAG library was used. 5.2 Experimental approach Published measurements dealing with diauxic experiments are often not well suited for the validation of mathematical models. The strains that have been used are not isogenic and measurements of different groups are difficult to compare. As the genetic background is often only poorly defined it is almost impossible to consider the genetic variations in model validation. In addition, the experimental setup is often not well documented or the design of the experiments is not useful for model validation. A biological system can be characterized in different ways. One possibility is to stimulate the system by (i) changing the external conditions like growth medium, substrate or temperature, (ii) using different culture conditions like batch, continuous fermentations, deflection from steady state by a pulse, and transient conditions, (iii) by introducing a mutation and/or (iv) by alter the intracellular state of the cell e.g. by using different pre-culture conditions. Figure 2 summarizes all types of stimulation used in this study with respect to
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changes of the specific growth rate µ. In batch cultivations only information during growth with the maximal growth rate is obtained. Steady-state conditions are reached only after a long time period that may cause problems due to genetic alterations and due to substrate limiting conditions. If steady-state conditions are reached, pulse experiments can be performed to analyze very fast kinetics. As a wild-type strain LJ110, a well characterized derivative of the
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E. coli K-12 reference strain W3110 was chosen. Mutations were introduced in cyaA, lacI, dgsA and ptsG, i.e. in genes important in signal transduction influencing diauxic behavior. By characterizing the wild-type and these mutants with respect to growth on different carbohydrates and especially by recording time series of states during diauxic growth we were able to get enough measurements to estimate a relatively high number of parameters although few different states were measured. 5.3 Results Based on the available measurements and the experiments performed a number of parameters could be estimated. Table 1 summarizes the findings for all functional units. Figure 3 shows exemplarily an experiment with the wild-type strain LJ110 when glucose and lactose are present in the medium.
6 Conclusion The present study marks a starting point to set up whole cell models. A detailed model for carbohydrate uptake and metabolism with focus on the cellular control was developed and in part validated. Problems for parameter identification are the lack of consistent experimental data and the uncertainness about the choice of the measured quantity with respect of their importance for the fitting. Here, the development and application of new technologies like cDNA-arrays and proteomics will help to come to better solutions. A problem not addressed here is model structure identification. Current work focuses on the development of methods for experimental design if two or more different model are formulated as hypotheses.
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Table 1. Summary of functional units, number of parameters and number of estimated parameters. About 20 different experiments are used for parameter fitting. a Parameters estimated with Metabolic Flux Analysis. module name PTS (general) PTS Glc Cya Crp 2nd Glc transporter Lac transporter Scr transporter Gly transporter Gal transporter Catabolic reactions Monomer synthesis Liquid phase
param. 21 12 9 17 18 16 26 24 43 51 7 7
param. estimated 9 4 2 3 3 7 9 5 4 11 4a +3 5
number of states 9 1 2 1 3 4/2 6 5 11/2 8 1 8
type ODE ODE ODE ODE ODE ODE/ algebraic ODE ODE ODE/ algebraic ODE ODE ODE
References 1. M. Caracotsios and W. E. Stewart. Sensitivity analysis of initial value problems with mixed odes and algebraic equations. Computers and Chemical Engineering, 9(4):350–365, 1985. 2. M. Ginkel, A. Kremling, T. Nutsch, R. Rehner, and E. D. Gilles. Modular modeling of cellular systems with ProMoT/Diva. Bioinformatics, 2003. In press. 3. L. H. Hartwell, J. J. Hopfield, S. Leibler, and A. W. Murray. From molecular to modular cell biology. Nature, 402(Supp.):C47 – C52, 1999. 4. J. W. Hearne. Sensitivity analysis of parameter combinations. Appl. Math. Modelling, 9:106–108, 1985. 5. A. Kreming, K. Bettenbrock, S. Fischer, K. Jahreis, T. Sauter, and E.D. Gilles. Mathematical modeling of carbohydrate uptake systems in Escherichia coli : I. Growth under unlimited conditions. 2003. Submitted. 6. A. Kremling, K. Bettenbrock, B. Laube, K. Jahreis, J.W. Lengeler, and E.D. Gilles. The organization of metabolic reaction networks: III. Application for diauxic growth on glucose and lactose. Metab. Eng., 3(4):362–379, 2001. 7. A. Kremling and E.D. Gilles. The organization of metabolic reaction networks: II. Signal processing in hierarchical structured functional units. Metab. Eng., 3(2):138–150, 2001. 8. A. Kremling, K. Jahreis, J.W. Lengeler, and E.D. Gilles. The organization of metabolic reaction networks: A signal-oriented approach to cellular models. Metab. Eng., 2(3):190–200, 2000. 9. C. Posten and A. Munack. On-line application of parameter estimation accuracy to biotechnical processes. In Proceedings of the American Control Conference, volume 3, pages 2181–2186, 1990.
Guaranteed Parameter Estimation for Cooperative Models Michel Kieffer and Eric Walter Laboratoire de Signaux et Syst`emes – CNRS – Sup´elec – Universit´e Paris-Sud Plateau de Moulon, 91192 Gif-sur-Yvette Cedex, France, {kieffer,walter}@lss.supelec.fr Abstract. The parameters of cooperative models are estimated in a boundederror context, i.e., all uncertain quantities are assumed to be bounded, with known bounds. Guaranteed estimation is then the characterization of the set of all parameter vectors that are consistent with the model and experimental data, given these bounds. Interval techniques provide an approximate but guaranteed enclosure of this set. No parameter vector consistent with the experimental data and model structure can be missed, so this approach bypasses the structural identifiability study required by the usual approaches based on the local optimization of some cost function.
1 Introduction This paper is about guaranteed estimation of the parameters of cooperative systems from experimental measurements. Estimation is performed in a bounded-error context, i.e., all uncertain quantities (measurement noise, parameters to be estimated) are assumed to be bounded, with known bounds. In this context, parameter estimation may be formulated as finding the set of all parameter vectors that are consistent with the parametric model and experimental data, given the error bounds. Interval techniques provide an approximate but guaranteed enclosure of this set between two subpavings, i.e., union of non-overlapping boxes. The approximation is guaranteed, as no consistent parameter vectors can be missed. Moreover, the precision of the approximation can be tuned by the user. With such techniques, no prior identifiability study is required. For example, a solution set consisting of two or more disconnected subsets may correspond to a model that is only locally identifiable. So far, this approach was mainly applied to models for which an analytical expression of the solution as a function of the parameters was available. This was because guaranteed integration of differential equations is very pessimistic when the parameter vector is only known to belong to some potentially large interval vector. The purpose of this paper is to show that the concept of cooperativity makes it possible to extend the methodology to a very large class of L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 103-110, 2003. Springer-Verlag Berlin Heidelberg 2003
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continuous-time differential models of practical interest. Bounded-error parameter estimation via interval analysis will be presented first, as an alternative to the optimization of a cost function. The special case of cooperative systems will then be considered, before describing some examples.
2 Approaches to parameter estimation Consider a system with input u (t) and output y (t) , and assume this system is described by a parametric model M (p) with the same input u (t) and an output ym (p,t), where p is a vector of unknown but constant parameters. b for p is to be obtained such that the output of the model An estimate p M (b p) is an acceptable approximation of the output of the system. Standard b as the argutechniques (see, e.g., [10] and the references therein) compute p ment of the minimum of a given cost function, e.g., T
j (p) = (y − ym (p)) (y − ym (p)) , T
T
where y = (y (t1 ) , . . . , y (tN )) and ym (p) = (ym (p, t1 ) , . . . , ym (p, tN )) are the system and model outputs collected at given time instants ti , i = 1, . . . , N . This minimization can be performed by local-search algorithms such as GaussNewton or Levenberg-Marquardt, but there is no guarantee of convergence to a global minimizer of j (p) and this minimizer may even not be unique. Random search, using, e.g., simulated annealing or genetic algorithms cannot provide any guarantee either that the global minimum has been found after finite computations. Only global guaranteed techniques, such as Hansen’s algorithm [2], based on interval analysis, can obtain such guaranteed results. Parameter bounding represents an attractive approach to optimization. The idea is to look for the set of all parameter vectors that are consistent (in a sense to be specified) with the experimental data, model structure and error bounds. It is assumed that to each experimental datum y (ti ) corresponds a known interval [εi , εi ], i = 1, . . . , N of acceptable errors. A parameter vector p ∈ P0 is deemed acceptable if εi 6 y (ti ) − ym (p, ti ) 6 εi for all i = 1, . . . , N . Parameter estimation then amounts to characterizing the set S of all acceptable p ∈ P0 S = {p ∈ P0 | y (ti ) − ym (p, ti ) ∈ [εi , εi ] , i = 1, . . . , N } .
(1)
How S may be characterized depends mainly on whether ym (p, ti ) is linear in p. If it is, then S is a polytope that may be described exactly [9] or by an outer-approximation for instance using ellipsoids [1], [7]. When ym (p, ti ) is nonlinear in p, S is no longer a polytope and may even be disconnected. One may nevertheless get a guaranteed enclosure of S using interval analysis [4], [5], as explained in the next section.
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3 Parameter bounding via interval analysis Interval analysis (see, e.g., [3]) is a tool for computing with intervals. An interval [x] = [x, x], with x 6 x, is a closed and connected subset of R. As it is a set, operators on sets apply to intervals. Moreover, each arithmetical operation can be extended to intervals, according to [x] ◦ [y] = {x ◦ y |x ∈ [x] , y ∈ [y] }, with ◦ ∈ {+, −, ∗, /}. For example, [3, 6] + [−2, 3] = [1, 9]. Function definitions may also be extended to interval arguments in order to get interval extensions according to f ([x]) = {f (x) |x ∈ [x] }. Interval extensions are easily computed for monotonous functions such as the exponential function exp ([x]) = [exp (x) , exp (x)]. For non-monotonous elementary functions such as the trigonometric functions, simple algorithms may be put at work to obtain tight lower and upper bounds for the images of intervals. For a more general function f (.), it may no longer be possible to compute its interval extension, and one may then use instead an inclusion function [f ] (.), the image of which is an interval that must satisfy the two properties ∀ [x] , f ([x]) ⊂ [f ] ([x])
(2)
∀ [x] , [y] such that [x] ⊂ [y] then [f ] ([x]) ⊂ [f ] ([y]) .
(3)
The easiest way to obtain an inclusion function for f is to replace all occurrences of real-valued variables by interval ones. The result is called a natural inclusion function. This¡ technique may however be sometimes pessimistic, ¢ as the width w ([y]) = y − y /2 of the interval [y] = [f ] ([x]) provided by the inclusion function may be much larger than the width of the smallest interval containing f ([x]). For example, consider f (x) = x − x = 0. Its natural inclusion function is [f ] ([x]) = [x] − [x], which evaluated at [0, 1] gives [f ] ([0, 1]) = [−1, 1] 6= 0. This problem may be partly solved by using more efficient inclusion functions, see [3]. Interval vectors (or boxes) are Cartesian product of intervals. All previously mentioned definitions extend to boxes componentwise, except for the width of a box, which is the maximum of the widths of its components. Using interval analysis, it is possible to provide inner and outer approximations of S as defined by (1) using the algorithm Sivia (for Set Inverter Via Interval Analysis, [4]). Sivia partitions P0 into three subpavings, namely Sin contained in S, Sout such that its intersection with S is empty and Sbound for which no conclusion could be reached. To obtain this partition, first rewrite (1) as S = {p ∈ P0 | ym (p) ∈ [y] , i = 1, . . . , N } , with [y] = [y (t1 ) − ε1 , y (t1 ) − ε1 ]×· · ·×[y (tN ) − εN , y (tN ) − εN ]. Now, consider a box [p] ⊂ P0 . If [ym ] ([p]) ⊂ [y], then according to (2), ym ([p]) ⊂ [y] and [p] is entirely included in S; it is thus stored in Sin . If [ym ] ([p]) ∩ [y] = ∅, then ym ([p]) ∩ [y] = ∅ and [p], proved to have an empty intersection with S, can be stored in Sout . If neither of the previous tests is satisfied, then [p]
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is undetermined. If the width of such an undetermined box is larger than a precision parameter ², then it is bisected into two subboxes [p1 ] and [p2 ] and the same tests are applied to these two subboxes. Undetermined boxes that are too small to be bisected are stored into Sbound . S is thus bracketed (in the sense of inclusion) between Sin and Sin ∪ Sbound . The volume of the uncertainty subpaving Sbound may be reduced, at the cost of increasing computational effort.
4 Parameter bounding for cooperative systems As evidenced by the previous section, the main requirement of Sivia is an efficient inclusion function [ym ] ([p]) for the model output. The model structure that will be considered in the remainder of this paper consists of a dynamical state equation and an observation equation ½ 0 x (t) = f (x (t) , p, w (t) , u (t)) , (4) ym (t) = g (x (t)) , where x ∈ D is the state of the model and x0 its derivative with respect to time, p is a vector of unknown parameters, w is a vector of bounded state perturbation and u is the known input of the model. Moreover, the state perturbation is supposed to remain bounded, with known bounds, so w (t) ∈ [w (t) , w (t)] for all t > 0. It is not possible in general to obtain an explicit expression of ym (t) for models such as those described by (4). However, for models whose dynamical state equation can be bounded between cooperative systems (systems such that the off-diagonal entries of the Jacobian matrix of their dynamical state equation are non-negative), an inclusion function [ym ] ([p]) can still be computed using the following theorem. Theorem 1 (see [8]). If there exists a pair of cooperative systems ¯0 = f (x, t) x0 = f (x, t) and x
(5)
satisfying ¤ x0 6 x (0) 6 x0 and f (x, t) 6 f (x, p, w, u) 6 f (x, t) , for all £ p ∈ p, p , w (t) ∈ [w (t) , w (t)], t > 0 and x ∈ D then the state of the system (4) satisfies x (t) 6 x (t) 6 x (t) , for all t > 0, where x (t) = φ (x0 , t) is the flow associated with {x0 = f (x, t) , x (0) = x0 } © ª and x (t) = φ (x0 , t) is the flow associated with x0 = f (x, t) , x (0) = x0 . £ ¤ For any t > 0, the box-valued function [φ] ([x0 , x0 ] , t) = φ (x0 , t) , φ (x0 , t) is thus an inclusion function for the solution of (4). Usually, no explicit solutions are available for φ (x0 , t) and φ (x0 , t) , but interval analysis provides tools for computing guaranteed outer approximations of the solution of initial
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value problems, see, e.g., [6]. Using these techniques, it becomes possible to compute tight enclosures of φ (x0 , t) and φ (x0 , t) as £
i i ¤ h ¤ h £ φ (x0 , t) = φ (x0 , t), φ (x0 , t) and φ (x0 , t) = φ (x0 , t), φ (x0 , t) .
The function
h i [[φ]] ([x] , t) = φ (x, t), φ (x, t)
(6)
is such that [φ] ([x0 , x0 ] , t) ⊂ [[φ]] ([x0 , x0 ] , t) and is therefore an inclusion function for the solution x (t) of (4), which can be numerically evaluated for any t > 0. Finally, using, e.g., a simple inclusion function for g (.) evaluated at [[φ]] ([x0 ,x0 ] , t), it is possible to get an inclusion function for ym (t) that can also be numerically evaluated for any t > 0. Guaranteed parameter bounding can thus be achieved for models such as (4) using Sivia, as illustrated in the next section.
5 Examples Two examples will be considered; both correspond to compartmental models, which are positive systems widely used in biology. The first one illustrates the capability of parameter bounding to provide guaranteed results even if the model under study is not uniquely identifiable. The second one shows that models with a larger number of unknown parameters can still be treated in a reasonable amount of time. 5.1 Two-compartment model Assume that the evolution of the quantity of material in each compartment of a two-compartment model is given by ½ 0 x1 = − (p1 + p2 ) x1 + p3 x2 + u (7) x02 = p2 x1 − p3 x2 where xi is the (positive) quantity of material in Compartment i. Assume further that only x2 is measured, according to ys = x2 (1 + η1 ), where η1 is a bounded measurement perturbation. Data have been simulated with p∗ = (2, 0.15, 0.25)T , (x1 , x2 ) (0) = (0, 0) and u (t) = δ (t). At 20 regularly-spaced time instants from 0.5s to 10s, a measurement of x2 is taken and corrupted additively by a bounded relative noise η1 ∈ [−0.1, 0.1]. For each measurement time, Figure 1 presents an interval guaranteed to contain the (supposedly unknown) noise-free value of x2 at that time.
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0.05 0.04 0.03 0.02 0.01 0 0
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Fig. 1. Intervals containing the (unknown) noise-free x2
The dynamical model (7) can be bounded by the two models ½ 0 ½ 0 x1 = − (p1 + p2 ) x1 + p3 x2 + u x1 = −(p1 + p2 )x1 + p3 x2 + u and x02 = p2 x1 − p3 x2 x02 = p2 x1 − p3 x2 which are easily proved to be cooperative as pi > 0 for i = 1, 2, 3. An inclusion function for the model output ym = x2 can then be computed with the technique presented in Section 4. Sivia has been used on this problem for various values of the precision parameter ². The initial search box is taken 3 as [p0 ] = [0, 5] . Table 1 presents computing time as a function of ², on an Athlon 1800+. Table 1. Two-compartment model : computing time for various values of the precision parameter Precision parameter ² Computing time
0.1 60s
0.01 11mn
0.005 27mn
Figure 2 displays the projection of the outer approximation of the solution set onto the (p1 , p2 ) and (p1 , p3 ) planes for ² = 0.005. The solution set consists of two disconnected subsets that are guaranteed to contain all parameter vectors consistent with the observed data and assumed noise bounds. It is actually easy to prove that the model under consideration is only locally identifiable and that the parameters p1 and p3 can be exchanged without modifying input-output behavior, but it should be noted that this knowledge was not taken into account during computation. 5.2 Three-compartment model Consider now a three-compartment model corresponding, e.g., to the behavior of a drug such as Glafenine administered orally. The evolution of the quantities of material in each compartment is given by 0 x1 = −(p1 + p2 )x1 + u, x0 = p1 x1 − (p3 + p5 ) x2 , (8) 02 x3 = p2 x1 + p3 x2 − p4 x3 .
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p3 5
0
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Fig. 2. Projection of an outer approximation of the solution set onto the (p1 , p2 ) and (p1 , p3 ) planes (two-compartment model)
Assume that x2 and x3 are measured, according to T
ys = (x2 (1 + η2 ) , x3 (1 + η3 )) . Data have been simulated with p∗ = (0.6, 1, 0.3, 0.2, 0.3)T , (x1 , x2 , x3 ) (0) = (0, 0, 0) and u (t) = δ (t). At 20 regularly-spaced time instants from 0.5s to 10s, the vector (x2 , x3 ) is sampled and corrupted additively by a bounded rel2 ative noise vector (η2 , η3 ) ∈ [−0.1, 0.1] . For each measurement time, Figure 3 presents intervals guaranteed to contain the noise-free x2 and x3 at that time. Again (8) can be bounded between two cooperative systems. An inclusion function for the two-dimensionnal model output is again obtained with the technique presented in Section 4. Sivia is used with the initial search box 5 [p0 ] = [0, 5] . Table 2 presents computing time as a function of ². The obtained approximation of the solution set now consists of a single connected set in a five-dimensionnal space which is included for ² = 0.025 in the box [s] = [0.508, 0.762]×[0.781, 1.25]×[0, 0.665]×[0.136, 0.254]×[0, 0.645].
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
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Fig. 3. Intervals containing the (unknown) noise-free x2 and x3
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Table 2. Three-compartment model : computing time for various values of the precision parameter Precision parameter ² Computing time
0.1 176s
0.05 11mn
0.025 66mn
6 Conclusion This paper presents an alternative approach for the estimation of the parameters of cooperative models. An enclosure of the set of all parameter vectors that are consistent with the model and experimental data, given known error bounds is obtained. No prior identifiability study is required as identifiability problems (if any) are evidenced as a by-product of the estimation process. The only requirement is that the dynamical state equation of the system can be bounded between two cooperative systems. This is the case for linear compartment models, but the method readily extends to non-linear compartmental models and other positive systems.
References 1. E. Fogel and Y. F. Huang. On the value of information in system identification - bounded noise case. Automatica, 18(2):229–238, 1982. 2. E. R. Hansen. Global Optimization Using Interval Analysis. Marcel Dekker, New York, NY, 1992. 3. L. Jaulin, M. Kieffer, O. Didrit, and E. Walter. Applied Interval Analysis. Springer-Verlag, London, 2001. 4. L. Jaulin and E. Walter. Set inversion via interval analysis for nonlinear bounded-error estimation. Automatica, 29(4):1053–1064, 1993. 5. R. E. Moore. Parameter sets for bounded-error data. Mathematics and Computers in Simulation, 34(2):113–119, 1992. 6. N. S. Nedialkov and K. R. Jackson. Methods for initial value problems for ordinary differential equations. In U. Kulisch, R. Lohner, and A. Facius, editors, Perspectives on Enclosure Methods, pages 219–264, Vienna, 2001. SpringerVerlag. 7. F. C. Schweppe. Uncertain Dynamic Systems. Prentice-Hall, Englewood Cliffs, NJ, 1973. 8. H. L. Smith. Monotone Dynamical Systems : An Introduction to the Theory of Competitive and Cooperative Systems, volume 41 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1995. 9. E. Walter and H. Piet-Lahanier. Exact recursive polyhedral description of the feasible parameter set for bounded-error models. IEEE Transactions on Automatic Control, 34(8):911–915, 1989. 10. E. Walter and L. Pronzato. Identification of Parametric Models from Experimental Data. Springer-Verlag, London, 1997.
Modeling and Simulation of Genetic Regulatory Networks Hidde de Jong Institut National de Recherche en Informatique et en Automatique (INRIA), Unit´e de recherche Rhˆ one-Alpes, 655 avenue de l’Europe, Montbonnot, 38334 Saint Ismier Cedex, France, [email protected] Abstract. The analysis of genetic regulatory networks will much benefit from the recent upscaling to the genomic level of experimental methods in molecular biology. In addition to high-throughput experimental methods, mathematical and bioinformatics approaches are indispensable for the analysis of genetic regulatory networks. Given the size and complexity of most networks of biological interest, an intuitive comprehension of their behavior is often difficult, if not impossible to obtain. A variety of methods for the modeling and simulation of genetic regulatory networks have been proposed in the literature. In this tutorial, the two principal approaches that have been used will be reviewed: methods based on differential equation models and stochastic models. In addition, we will indicate some alternative methods that have emerged in response to the difficulties encountered in applying the classical approaches.
1 Introduction It is now commonly accepted that most interesting properties of an organism emerge from the interactions between its genes, proteins, metabolites, and other constituents. This implies that, in order to understand the functioning of an organism, we need to elucidate the networks of interactions involved in gene regulation, metabolism, signal transduction, and other cellular and intercellular processes. Genetic regulatory networks control the spatiotemporal expression of genes in an organism, and thus underlie complex processes like cell differentiation and development in prokaryotic and eukaryotic organisms. Genetic regulatory networks consist of genes, proteins, metabolites, and other small molecules, as well as their mutual interactions. Their study has taken a qualitative leap through the use of modern genomic techniques that allow simultaneous measurement of the expression levels of all genes of an organism. In addition to experimental tools, mathematical methods supported by computer tools are indispensable for the analysis of genetic regulatory networks. As most networks of interest involve many genes connected through interlocking positive L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 111-118, 2003. Springer-Verlag Berlin Heidelberg 2003
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and negative feedback loops, an intuitive understanding of their dynamics is difficult to obtain and may lead to erroneous conclusions. Modeling and simulation tools allow the behavior of large and complex systems to be predicted in a systematic way. A variety of methods for the modeling and simulation of genetic regulatory networks have been proposed in the literature [3, 12, 15, 25]. In this tutorial, the two principal approaches that have been used in the literature will be briefly reviewed: differential equation models and stochastic models (section 2 and 3). The networks described by these models are examples of positive systems [7], in the sense that the state and output variables remain nonnegative on a time-interval T , if the input variables are positive on T . In fact, the variables in the models represent positive quantities, in particular the concentrations or numbers of molecules of proteins, mRNA, metabolites, and other constituents. In section 4, we will discuss the difficulties encountered in applying the classical approaches and point at alternative approaches that have emerged.
2 Differential equation models Being arguably the most widespread formalism to model dynamical systems in science and engineering, ordinary differential equations (ODEs) have been widely used to analyze genetic regulatory systems. The ODE formalism models the concentrations of mRNAs, proteins, and other molecules by timedependent variables having non-negative real values. Regulatory interactions take the form of functional and differential relations between the concentration variables. More specifically, gene regulation is modeled by nonlinear equations expressing the rate of production or degradation of a component of the system as a function of the concentrations of other components. The equations have the mathematical form dxi = fi (x), 1 ≤ i ≤ n, (1) dt where x = [x1 , . . . , xn ]0 ≥ 0 is the vector of concentrations of proteins, mRNAs, or small molecules, and fi : Rn → R a usually nonlinear function. The rate of synthesis of i is seen to be dependent upon the concentrations x, possibly including xi . Figure 1 shows a simple example of a genetic regulatory network. Genes a and b, transcribed from separate promoters, encode proteins A and B, each of which independently controls the expression of both genes. More specifically, proteins A and B repress gene a as well as gene b at different concentrations. Repression of the genes is achieved by binding of the proteins to regulatory sites overlapping with the promoters. Figure 2(a) shows how the regulatory network in figure 1 can be modeled in terms of differential equations. The model consists of four variables denoting
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Fig. 1. Example of a simple regulatory network, consisting of the genes a and b, proteins A and B, and their mutual interactions. The notation follows, in a somewhat simplified form, the graphical conventions proposed by Kohn [14].
the concentration of mRNA and protein for genes a and b. The transcriptional inhibition of these genes is described by means of sigmoidal functions h− : R2 → [0, 1], which is motivated by the usually nonlinear, switch-like character of gene regulation. The translation of mRNA and the degradation of mRNA and proteins are assumed to be non-regulated and proportional to the substrate concentration. Due to the nonlinearity of fi , analytical solution of the rate equations (1) is not normally possible. In special cases, qualitative properties of the solutions, such as the number and the stability of steady states and the occurrence of limit cycles, can be established. Most of the time, however, one has to take recourse to numerical techniques. In figure 2(b) the results of a numerical simulation of the example network are shown. As can be seen, the system reaches a steady state in which protein A is present at a high concentration, whereas protein B is nearly absent. For different initial conditions, but the same parameter values, a steady state may be reached in which the concentrations of A and B are reversed Differential equations of the form (1) do not take into account the spatial dimension of regulatory processes, essential though in multicellular organisms. The equations can be generalized by defining compartments that correspond to cells or nuclei, by introducing concentration variables specific to each compartment, and by allowing diffusion between the compartments to take place. In the limit of the number of compartments, the resulting equations can be approximated by partial differential equations (PDEs). Partial differential equations are even more difficult to solve analytically than ordinary differential equations, and in almost every situation of practical interest their use requires numerical techniques.
3 Stochastic models An implicit assumption underlying (1), and differential equations more generally, is that concentrations of substances vary continuously and deterministically. Both of these assumptions may be questionable in the case of gene regulation, due to the usual small number of molecules of certain components [13, 16]. Instead of taking a continuous and deterministic approach, some authors have proposed to use discrete and stochastic models of gene regulation.
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dxra 1 2 = κra h− (xpb , θpb ) h− (xpa , θpa ) dt − γra xra
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Fig. 2. (a) ODE model of the regulatory network in figure 1. The variables xpa and xpb denote the concentration of protein A and B, the variables xra and xrb the concentration of the corresponding mRNA, the parameters κra , κpa , κrb , and κpb production rates, the parameters γra , γpa , γrb , and γpb degradation rates, and the 1 2 1 2 parameters θpa , θpa , θpb , and θpb threshold concentrations. The variables are nonnegative and the parameters positive. (b) Time-concentration plot resulting from a numerical simulation of the system described in (a), given specified values for the parameters.
Discrete amounts X of molecules are taken as state variables, and a joint probability distribution p(X, t) is introduced to express the probability that at time t the cell contains X1 molecules of the first species, X2 molecules of the second species, etc. The time evolution of the function p(X, t) can then be specified as follows: p(X, t + ∆t) = p(X, t)(1 −
m X j=1
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m X
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(2)
j=1
where m is the number of reactions that can occur in the system, αj ∆t the probability that reaction j will occur in the interval [t, t + ∆t] given that the system is in the state X at t, and βk ∆t the probability that reaction j will bring the system in state X from another state in [t, t+∆t] [8, 9]. Rearranging (2), and taking the limit as ∆t → 0, gives the master equation [30]: m
X ∂ (βj − αj p(X, t)). p(X, t) = ∂t j=1
(3)
Compare this equation with the rate equations (1) above. Whereas the latter determine how the state of the system changes with time, the former describes how the probability of the system being in a certain state changes with time.
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Notice that the state variables in the stochastic formulation can be reformulated as concentrations by dividing the number of molecules Xi by a volume factor. Although the master equation provides an intuitively clear picture of the stochastic processes governing the dynamics of a regulatory system, it is even more difficult to solve by analytical means than the deterministic rate equation. In order to approximate the solution of the master equation, stochastic simulation methods have been developed [8, 21]. Given a set of possible reactions, the temporal evolution of the state X, the number of molecules of each species, is predicted. The evolution of the state is determined by stochastic variables τ and ρ, representing the time interval between two successive reactions and the type of the next reaction, respectively. At each state a value for τ and ρ is randomly chosen from a set of values whose joint probability density function p(τ, ρ) has been derived from the same principles as those underlying the master equation (2).
8
A + A ←→ A2 A2 + DNAb ←→ A2 · DNAb
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+ RNAb B + B ←→ B2 B2 + DNAb ←→ B2 · DNAb B2 · DNAb + A2 ←→ A2 · B2 · DNAb A2 · DNAb + B2 ←→ A2 · B2 · DNAb (a)
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(b)
Fig. 3. (a) Some of the reactions involved in the expression of gene b in the regulatory network of figure 1. The following abbreviations are used: A and B (protein A and B), A2 and B2 (homodimer of A and B), RNAP (RNA polymerase), DNAb (promoter region of gene b), and RNAb (mRNA b). (b) A typical time-concentration plot resulting from stochastic simulation of the reaction system described in (a).
In figure 3(a) a few examples of reactions occurring in the network of figure 1 are shown: dimerization of the repressor A, binding of the repressor complex A·A to the promoter region, fixation of DNA polymerase to the promoter in the absence of the repressor complex, transcription of the gene b, etc. Typical results of a stochastic simulation of the example network are shown in figure 3(b). Notice the noisy aspect of the time evolution of the protein and mRNA concentrations. This effect, reflecting the stochastic nature of the initiation of transcription and the number of protein molecules produced per
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transcript, may have important consequences. More particularly, fluctuations in the rate of gene expression may lead to phenotypic variation in an isogenic population [16, 18]. Indeed, starting from the same initial conditions, two different simulations may lead to qualitatively different outcomes. Whereas in one simulation protein A may be ultimately present at a high concentration and B at a low concentration like in figure 3(b), another simulation could lead to the opposite result.
4 Discussion In summary, differential equation and stochastic models provide detailed descriptions of genetic regulatory networks, down to the molecular level. In addition, they can be used to make precise, numerical predictions of the behavior of regulatory systems. Many excellent examples of the application of these methods to prokaryote and eukaryote networks can be found in the literature. McAdams and Shapiro [17] have simulated the choice between lytic and lysogenic growth in bacteriophage λ using nonlinear differential equations, while Arkin and colleagues have studied the same system by means of a detailed stochastic model [1]. In a series of publications, the groups of Novak and Tyson have developed ODE models of the kinetic mechanisms underlying cell cycle regulation in Xenopus [2] and in yeast [22] (see [29] for a review). Differential equation models for the segmentation of Drosophila have been studied, focusing on the formation on the expression patterns of the gap, the pair-rule, and the segment polarity gene products in the trunk of the embryo [23, 24, 31]. In many situations of biological interest, however, the application of differential equation and stochastic models is seriously hampered. In the first place, the biochemical reaction mechanisms underlying regulatory interactions are usually not or incompletely known. This means that it is difficult to specify the rate functions fi in (1) and the reactions j in (3). In the second place, quantitative information on kinetic parameters and molecular concentrations is only seldom available, even in the case of well-studied model systems. As a consequence, the numerical simulation methods mentioned above are often difficult to apply. The above two constraints call for methods based on coarse-grained models that, while abstracting from the precise molecular mechanisms involved, capture essential aspects of gene regulation. Moreover, these methods should allow a qualitative analysis of the dynamics of the genetic regulatory systems to be carried out. A number of such methods have been proposed, such as the qualitative analysis of genetic regulatory networks described by piecewise-linear (PL) differential equations [4, 6, 10, 11, 20, 26], and the analysis of genetic regulatory networks by means of asynchronous, multivalued logic [19, 27, 28]. Although the methods are based on different formalisms, differential and log-
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ical equations, they share important biological intuitions, in particular the description of gene activation in terms of on/off-switches. The above-mentioned methods have been used to study a variety of prokaryotic and eukaryotic model systems, such as the choice between vegetative growth and sporulation in B. subtilis and the genetic control of the segmentation in the early Drosophila embryo (see [5] for a review). The applications show that, in order to understand the functioning of an organism in terms of the interactions in regulatory networks, it is not always necessary to model the process down to individual biochemical reactions. In fact, when a global understanding of the evolution of spatiotemporal patterns of gene expression is sought, coarse-grained and qualitative models might be profitably employed. However, when a more detailed and quantitative view of the dynamics of a regulatory system is required, the qualitative approaches need to be supplemented by conventional methods of the type discussed in sections 2 and 3.
References 1. A. Arkin, J. Ross, and H.A. McAdams. Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected Escherichia coli cells. Genetics, 149:1633–1648, 1998. 2. M.T. Borisuk and J.J. Tyson. Bifurcation analysis of a model of mitotic control in frog eggs. Journal of Theoretical Biology, 195:69–85, 1998. 3. H. de Jong. Modeling and simulation of genetic regulatory systems: A literature review. Journal of Computational Biology, 9(1):69–105, 2002. 4. H. de Jong, J. Geiselmann, C. Hernandez, and M. Page. Genetic Network Analyzer: Qualitative simulation of genetic regulatory networks. Bioinformatics, 19(3):336–344, 2003. 5. H. de Jong, J. Geiselmann, and D. Thieffry. Qualitative modeling and simulation of developmental regulatory networks. In S. Kumar and P.J. Bentley, editors, On Growth, Form, and Computers. Academic Press, London, 2003. In press. 6. H. de Jong, J.-L. Gouz´e, C. Hernandez, M. Page, T. Sari, and J. Geiselmann. Hybrid modeling and simulation of genetic regulatory networks: A qualitative approach. In A. Pnueli and O. Maler, editors, Hybrid Systems: Computation and Control (HSCC 2003), Lecture Notes in Computer Science. Springer-Verlag, Berlin, 2003. 7. L. Farina and S. Rinaldi. Positive Linear Systems: Theory and Applications. Wiley, New York, 2000. 8. D.T. Gillespie. Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry, 81(25):2340–2361, 1977. 9. D.T. Gillespie. A rigorous derivation of the chemical master equation. Physica D, 188:404–425, 1992. 10. L. Glass and S.A. Kauffman. The logical analysis of continuous non-linear biochemical control networks. Journal of Theoretical Biology, 39:103–129, 1973. 11. J.-L. Gouz´e and T. Sari. A class of piecewise linear differential equations arising in biological models. Dynamical Systems, 2003. To appear.
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12. J. Hasty, D. McMillen, F. Isaacs, and J.J. Collins. Computational studies of gene regulatory networks: in numero molecular biology. Nature Review Genetics, 2(4):268–279, 2001. 13. M.S.H. Ko. Induction mechanism of a single gene molecule: Stochastic or deterministic? BioEssays, 14(5):341–346, 1992. 14. K.W. Kohn. Molecular interaction maps as information organizers and simulation guides. Chaos, 11(1):1–14, 2001. 15. H.H. McAdams and A. Arkin. Simulation of prokaryotic genetic circuits. Annual Review of Biophysics and Biomolecular Structure, 27:199–224, 1998. 16. H.H. McAdams and A. Arkin. It’s a noisy business! Genetic regulation at the nanomolar scale. Trends in Genetics, 15(2):65–69, 1999. 17. H.H. McAdams and L. Shapiro. Circuit simulation of genetic networks. Science, 269:650–656, 1995. 18. H.M. McAdams and A. Arkin. Stochastic mechanisms in gene expression. Proceedings of the National Academy of Sciences of the USA, 94:814–819, 1997. 19. L. Mendoza, D. Thieffry, and E.R. Alvarez-Buylla. Genetic control of flower morphogenesis in Arabidopsis thaliana: A logical analysis. Bioinformatics, 15(78):593–606, 1999. 20. T. Mestl, E. Plahte, and S.W. Omholt. A mathematical framework for describing and analysing gene regulatory networks. Journal of Theoretical Biology, 176:291–300, 1995. 21. C.J. Morton-Firth and D. Bray. Predicting temporal fluctuations in an intracellular signalling pathway. Journal of Theoretical Biology, 192:117–128, 1998. 22. B. Novak, A. Csikasz-Nagy, B. Gyorffy, K.C. Chen, and J.J. Tyson. Mathematical model of the fission yeast cell cycle with checkpoint controls at the G1/S, G2/M and metaphase/anaphase transitions. Biophysical Chemistry, 72:185–200, 1998. 23. J. Reinitz, D. Kosman, C.E. Vanario-Alonso, and D.H. Sharp. Stripe forming architecture of the gap gene system. Developmental Genetics, 23:11–27, 1998. 24. J. Reinitz, E. Mjolsness, and D.H. Sharp. Model for cooperative control of positional information in Drosophila by bicoid and maternal hunchback. Journal of Experimental Zoology, 271:47–56, 1995. 25. P. Smolen, D.A. Baxter, and J.H. Byrne. Modeling transcriptional control in gene networks: Methods, recent results, and future directions. Bulletin of Mathematical Biology, 62:247–292, 2000. 26. E.H. Snoussi. Qualitative dynamics of piecewise-linear differential equations: A discrete mapping approach. Dynamics and Stability of Systems, 4(3-4):189–207, 1989. 27. R. Thomas and R. d’Ari. Biological Feedback. CRC Press, Boca Raton, FL, 1990. 28. R. Thomas, D. Thieffry, and M. Kaufman. Dynamical behaviour of biological regulatory networks: I. Biological role of feedback loops and practical use of the concept of the loop-characteristic state. Bulletin of Mathematical Biology, 57(2):247–276, 1995. 29. J.J. Tyson, K. Chen, and B. Novak. Network dynamics and cell physiology. Nature Reviews Molecular Cell Biology, 2(12):908–916, 2001. 30. N.G. van Kampen. Stochastic Processes in Physics and Chemistry. Elsevier, Amsterdam, revised edition, 1997. 31. G. von Dassow, E. Meir, E.M. Munro, and G.M. Odell. The segment polarity network is a robust developmental module. Nature, 406:188–192, 2000.
Qualitative Analysis of Regulatory Graphs: A Computational Tool Based on a Discrete Formal Framework Claudine Chaouiya1 , Elisabeth Remy2 , Brigitte Moss´e2 , and Denis Thieffry1 1 2
LGPD-IBDM, Campus de Luminy, Case 907, 13288 Marseille Cedex 9, France, {chaouiya, thieffry}@ibdm.univ-mrs.fr IML, Campus de Luminy, Case 907, 13288 Marseille Cedex 9, France, {remy, mosse}@iml.univ-mrs.fr
Abstract. Building upon the logical approach developed by the group of R. Thomas in Brussels, we are defining a rigorous mathematical framework to model genetic regulatory graphs. Referring to discrete mathematics and graph-theoretic notions, our formal approach supports the development of a software suite in Java, GIN-sim, which allows the qualitative simulation and the analysis of the dynamics of regulatory graphs, under either synchronous or asynchronous updating assumptions.
1 Introduction Our formal approach roots in the logical formalism previously developed by R. Thomas and colleagues [5, 6]. Combining graph-theoretic and discrete mathematical notions, we propose a series of definitions enabling a proper mathematical description of genetic regulatory graphs, as well as of the corresponding qualitative dynamical behaviour (Sections 2 and 3) (see [2] for a recent review of this field). This formal framework serves as a basis for the study of formal properties of regulatory graphs (Section 3.4), as well as for the development of a simulation software, GIN-sim (Section 4). 3
2 Regulatory graphs 2.1 Definitions A regulatory graph is a labeled graph where vertices represent genes, whereas edges represent interactions; when oriented (e.g. transcriptional regulation), an interaction is represented by an arc, possibly signed (positively for 3
We thank H. de Jong for his suggestions concerning a previous version of this manuscript. We further acknowledge the financial support of the French Action inter-EPST bioinformatique.
L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 119-126, 2003. Springer-Verlag Berlin Heidelberg 2003
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an activation, negatively for a repression). Note that we mainly refer to interactions between genes, though these interactions may involve various types of molecular mechanisms. On each arc, a label indicates the conditions under which the interaction is functional, together with the sign of the interaction. Finally, we consider the following data: • A finite set G = {g1 , . . . , gn } constituted by n elements, called genes. • A set of positive integers {max1 , . . . , maxn }, where, for each i, maxi is the maximum expression level of gene gi . Therefore, the different expression levels allowed for gi are the integers {0, . . . , maxi }. • A labeled oriented graph R = (G, L), where G is the set of vertices (genes) and L is the set of arcs, which represent interactions between genes. A label (A, q) is associated to each arc, specifying the conditions under which the interaction takes place, and the nature of this interaction: i) A is an integer interval included in {1, . . . , maxi }. If several arcs join gi to gj , then the different intervals are mutually disjoined. ii) q ∈ {−1, 0, 1} is the sign of the interaction, denoting an activation (q = +1), an inhibition (q = −1), or undetermination(q = 0). Interaction from gi (source) to gj (target) is a tuple T = (gi , gj , A(T ), q(T )) where (A(T ), q(T )) is the label of the arc from gi to gj . Interval A(T ) = [sinf (T ), ssup (T )], with sinf (T ) > 0, is the set of consecutive expression levels of gi for which T is functional. Integer q(T ) is the sign of the interaction. Interactions are subjected to the following conditions: For any gi in G, for any l in {1, . . . , maxi }, there exists an interaction T with source gi such that l = sinf (T ); consequently, any non trivial expression level of gene gi corresponds to a threshold from which an interaction (with source gi ) becomes functional (thus for each gene, the maximum level equals at most the number of interactions exerted by this gene). Let Ij be the set of incoming interactions (or inputs set) of gj . For any gene gj , a subset X of Ij is admissible if it does not contain interactions having the same source. When expression levels of the genes are given, we know which interactions are functional, and we would like to describe their action. This is done by means of logical parameters: • for any gene gj , the application Kj , called logical function for gene gj , associates an integer Kj (X) (0 ≤ Kj (X) ≤ maxj ) to any admissible subset X of Ij . This integer is called logical parameter Kj (X) and corresponds to the expression level to which gene gj tends, when the set of functional incoming interactions is equal to X. Remark 1. For a gene gj , absence of inhibition can lead to increase its level of expression of gj , and consequently, parameter Kj (∅) may be greater than zero. Remark 2. When X is not an admissible subset of Ij , Kj (X) = 0.
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2.2 A toy example As a toy example, we consider the regulatory graph defined by G = {a, b, c}, maxa = 2, maxb = maxc = 1, and by the labeled graph R (see Figure 1). Gene a is a dual regulator of gene b (it activates or inhibits b depending on the context); gene b activates itself and gene c; finally, gene c inhibits gene a.
a
({1},+1) ({2},−1)
({1},−1)
({1},+1)
b
({1},+1)
c
Fig. 1. The regulatory graph of the toy example.
There are five interactions: T1a = (a, b, {1}, +1), T2a = (a, b, {2}, −1), = (b, b, {1}, +1), T2b = (b, c, {1}, +1) and T1c = (c, a, {1}, −1). The logical parameters are given by Table 1. Inputs sets are : Ia = {T1c }, Ib = {T1a , T2a , T1b } and Ic = {T2b }.
T1b
Table 1. The logical parameters of the toy example. Ka :
∅ → 7 2 {T1c } → 7 0
Kb :
∅ {T1a } {T2a } {T1b } {T1a , T1b } {T2a , T1b }
7→ 7→ 7→ 7→ 7→ 7→
0 1 0 1 1 0
Kc :
∅ → 7 0 {T2b } → 7 1
3 Dynamical graphs Consider ((G, L), (Kj )1≤j≤n ) a regulatory graph. To characterise the dynamics of the system, we have to address the following question: given an initial state x0 = (x01 , . . . , x0n ) (where x0i is the initial expression level of gene gi ), what are the following consecutive states (possibly) reached by the system? Let us denote by E the set of all possible states: E = {x = (x1 , . . . , xn ); ∀i = 1, . . . , n, 0 ≤ xi ≤ maxi } .
(1)
For any given state x and for a gene gj , we call Ij (x) the set of all incoming interactions which are functional at state x. It is an admissible set defined by:
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Ij (x) = {U = (gi , gj , A(U ), q(U )) ∈ Ij ; xi ∈ A(U )} .
(2)
A tuple I(x) = (I1 (x), . . . , In (x)), called instruction at state x, defines for each gene which incoming interactions are functional. Using applications (Kj )1≤j≤n , we obtain all the values of the parameters which describe the evolution of the system. In order to represent the discrete dynamics of the system, we define a dynamical graph, where vertices represent states, each labeled by a tuple of n integers representing the actual levels of the genes. In our toy example, in state x = (2, 1, 0) gene a is at its maximum level (xa = 2), as well as b (xb = 1), while c has no signifiant expression (xc = 0). In dynamical graphs, arcs represent spontaneous transitions between pairs of states. For instance, an arc between x0 = (0, 0, 0) and x1 = (1, 0, 0) corresponds to a transition from x0 to x1 , as a consequence of the definition of the corresponding parameters Ka (Ia (x0 )), Kb (Ib (x0 )) and Kc (Ic (x0 )). We have still to define an updating method to specify the temporal ordering of the transitions. We successively consider a fully synchronous versus a fully asynchronous assumptions. 3.1 Synchronous dynamical graphs Under the synchronous assumption, at each time step, all update orders (i.e. calls for changes of expression level for a subset of genes at a given state) are executed simultaneously. As a result, each state has exactly one successor. From a biological point of view, this frequently used assumption implies that all macromolecular processes are realised in identical amounts of times (or “delays”), which is clearly unrealistic and often at the origin of simulation artefacts. We denote by ξs = (E, Fs ) the synchronous graph, where E is defined by (1), and Fs is the set of arcs defined as follows. There exists a unique arc from x to y ∈ E defined by y = (y1 , . . . yn ) and for all j ∈ {1, . . . n} : xj if Kj (Ij (x)) = xj , (3) yj = xj − 1 if Kj (Ij (x)) < xj , xj + 1 if Kj (Ij (x)) > xj . In other words, the dynamical synchronous graph corresponds to an application of E on itself, which associates to a state x a unique state y obtained by a simultaneous update of all coordinates of x, following instruction I(x). Remark 3. Our definition forbids jumping over integer values, something which may occur when using the simple definition yj = Kj (Ij (x)) in a multilevel context. Given a set of initial states, a sub-graph corresponding to a particular pathway can be extracted from ξs . We denote by ξs (x0 ) the sub-graph which represents the pathway of the system when initial state is x0 , under a synchronous updating (note that ξs (E) = ξs ).
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3.2 Asynchronous dynamical graphs Under the asynchronous assumption, when multiple update orders occur at a given state, additional information is needed to select a specific transition (i.e. the values of relevant time delays or some ordering relationships). Here, specific time-delays are associated to each reaction (synthesis, degradation, activation, inhibition). As we have no information about these time delays, all possible transitions are generated. As a consequence, each state x has a number of successors equals to the number of update orders in this state. Let us denote by ξa = (E, Fa ) the asynchronous dynamical graph, where the set of vertices is E, and Fa is the set of arcs. Let x be a state; ∀j ∈ {1, . . . , n} such that Kj (Ij (x)) 6= xj , there exists an arc between x and ½ (x1 , . . . , xj−1 , xj − 1, xj+1 , . . . , xn ) if Kj (Ij (x)) < xj , y= (4) (x1 , . . . , xj−1 , xj + 1, xj+1 , . . . , xn ) if Kj (Ij (x)) > xj . Therefore, two linked states x and y differ by at most one coordinate. Moreover, in an asynchronous graph, an arc represents a unique update order. We denote by ξa (x0 ) the sub-graph of ξa which represents all possible pathways when initial state is x0 , under an asynchronous updating. 3.3 Illustration through our toy example The example of Section 2.2 is small enough to enumerate all possible states with the corresponding instructions and parameters values (Table 2). Table 2. States and corresponding instructions States x (0, 0, 0) (0, 0, 1) (0, 1, 0) (0, 1, 1) (1, 0, 0) (1, 0, 1) (1, 1, 0) (1, 1, 1) (2, 0, 0) (2, 0, 1) , 0) , (2, 1, (2 1 1)
Ia (x) ∅ {T1c } ∅ {T1c } ∅ {T1c } ∅ {T1c } ∅ {T1c } ∅ {T1c }
Ib (x) ∅ ∅ {T1b } {T1b } {T1a } {T1a } {T1a , T1b } {T1a , T1b } {T2a } {T2a } {T2a , T, 1b } {T2a T1b }
Ic (x) Ka (Ia (x)) Kb (Ib (x)) Kc (Ic (x)) ∅ 2 0 0 ∅ 0 0 0 {T2b } 2 1 1 {T2b } 0 1 1 ∅ 2 1 0 ∅ 0 1 0 {T2b } 2 1 1 {T2b } 0 1 1 ∅ 2 0 0 ∅ 0 0 0 {T2b } 2 0 1 {T2b } 0 0 1
Using states and corresponding instructions in Table 2, for an initial state, e.g. x0 = (0, 0, 0), we can generate the dynamical pathway(s) of the system. Figure
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2(A) illustrates the synchronous dynamical sub-graph ξs ((0, 0, 0)), leading to a 3-states cycle. Figure 2(B) illustrates ξa ((0, 0, 0)), leading to two alternative stable states. Recall that in asynchronous graphs, all possible updates are represented. Superscripts (+ or -) indicate whether the instruction tends to increase or decrease the level of expression of a gene. Absence of superscript denotes a stationary level. (A)
(B) + 000
− − 201
++ 100
− 001
+ 000
−+− 101
−+ 210
++ 100 + + 110
200 − − 201
−+ 210 −− 21 1
− 111
011
Fig. 2. (A) Synchronous (B) asynchronous dynamical sub-graphs for the toy example. Note that loops are omitted on terminal nodes.
3.4 Dynamical properties A state x = (x1 , . . . , xn ) is stationary if, for all i = 1, . . . , n, xi = Ki (Ii (x)). Using previous definitions and Remark 2, it is possible to show that stationary states satisfy the following equations: for j = 1, . . . , n, X Y Y xj = Kj (X) 1A(U ) (xi ) (1 − 1A(U ) (xi )) , X⊂Ij
U ∈X
U ∈Ij \X
where U stands for interactions (gi , gj , A(U ), q(U )) and 1A denotes the characteristic function of set A, i.e. 1A (x) = 1 if x ∈ A, 1A (x) = 0 otherwise. Stationary (stable) states are easily identified as final vertices. In our toy example, states (2, 0, 0) and (0, 1, 1) are stationary in the asynchronous subgraph ξa ((0, 0, 0)) (Figure 2(B)). The synchronous graph of Figure 2(A) presents no stationary state, but contains a dynamical cycle. Note that the notion of stationary state is independant of the updating method. Nevertheless, the choice of a specific updating method can considerably change the connectivity between the states (compare Figure 2(A) and (B)). Stationary states or dynamical cycles correspond to the notion of attractors in the field of dynamical systems, though dynamical cycles may be followed only transiently. More generally, attractors are related to strongly connected components of dynamical graphs. Using the same analogy, the notion
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of basin of attraction of an attractor is encompassed by the set of vertices having a path to a given strongly connected component (or “attractor”). Having defined a regulatory graph, we focus on the circuits of the graph. When the signs of the interactions are determined, a circuit is said to be positive if the product of these signs is positive, negative otherwise. These circuits can generate differentiative (positive circuits) or homeostatic (negative circuits) properties [5]. Forming strongly connected components of the regulatory graph, intertwined circuits can be related to the biological notion of cross-regulatory modules. With this mathematical framework, we aim at establishing formal links between regulatory graphs and the corresponding dynamical graphs. We have already precisely defined the structure of the dynamical graphs corresponding to elementary regulatory circuits. This structure depends only on the sign and length of the circuits. Furthermore, the complex structure of ξa can be simply described on the basis of the simpler structure of ξs .
4 GIN-sim From a computational perspective, our approach takes the form of a series of Java classes, collectively called GIN-sim. This simulation tool is part of a wider software project, which provides a series of modules covering the integration, the processing, and the modelling of functional regulatory data [1]. In GINsim, both synchronous and asynchronous simulations have been implemented. Graphical interfaces are currently under development, as well as algorithms to exhibit structural properties of both regulatory and dynamical graphs. Given a set of initial states, GIN-sim generates a dynamical graph, qualitatively representing all allowed spontaneous state transitions corresponding to the model encoded in the original regulatory graph. The initial states and the parameter values can be defined by the user or by default (including the number of distinct levels for each regulatory product, and the qualitative weights of the different combinations of interactions on each gene). The user can progressively refine his model, depending on simulation results. Given a regulatory graph, a set of parameters, and a set of initial states, our simulation algorithm is essentially a variant of the standard depth-first traversal algorithm. For each current state, relevant parameters are determined to generate the successor(s) of this vertex.
5 Discussion and conclusion Leaning on the logical method previously developed by R. Thomas [5], we have introduced a rigorous, discrete, dynamical formalisation of genetic regulatory graphs. The originality of our approach lies in : (1) the coverage of multi-arcs in regulatory graphs (labelled by non overlaping intervals); (2) a
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generic representation of all kinds of logical relationships when multiple interactions are exerted on a given gene. Note that the corresponding logical parameter values constraint the signs attached to the interactions involved. In other words, the determination of the sign of an interaction (+ or -) imposes inequalities on relevant parameters to insure consistency. This discrete mathematical framework opens the way to a systematic analytical study of the link between regulatory and dynamical graphs as well as between synchronous and asynchronous dynamical graphs. GIN-sim implements this formal framework and allows the validation of analytic results, as well as biological applications. Up to now, this approach has been applied to the dynamical modelling of the networks involved in the control of the cell cycle, cell differentiation, and pattern formation during Drosophila melanogaster embryonic development (see e.g. [4]). As the number of genes and interactions of regulatory graphs increases, the size of the corresponding dynamical graphs may grow exponentially. However, there are at least three ways to cope with this problem: (1) using features of genetic regulatory networks such as modularity and limited values for in/out degrees of vertices; (2) focusing on relevant part of dynamical graphs (partial exploration); (3) exploiting analytical results, for example concerning the role of feedback circuits [6] or the location of all stationary states [3]. Other analytical tools are available for the modelling of regulatory graphs [2]. Often complementary, these approaches should be combined to cope with the complexity and the variety of biological networks.
References 1. Chaouiya C., Sabatier C., Verheecke-Mauz C, Jacq B. and Thieffry D. (2002): GIN-tools: Towards a software suite for the integration, the analysis, and the simulation of Gene Interaction Networks., Proceedings of JOBIM 2002. SaintMalo, France, June 2002, pp. 17-26. 2. de Jong H.(2001): Modeling and simulation of genetic regulatory systems: A literature review, J. Comp. Biol. 9, pp.69-105. ´ M. (2003): Identification of all steady states 3. Devloo V., Hansen P. and Labbe in large biological systems by logical analysis, Bull. Math. Biol., in revision. 4. S´ anchez, L. and Thieffry D.(2001): A logical analysis of the gap gene system, J. theor. Biol. 211: 115-141. 5. Thomas R. (1991): Regulatory networks seen as asynchronous automata: a logical description, J. theor. Biol. 153:pp. 1-23. 6. Thomas R, Thieffry D, Kaufman M. (1995): Dynamical behaviour of biological regulatory networks, I. Biological role of feedback loops and practical use of the concept of the loop-characteristic state. Bull. Math. Biol. 57, pp.247-276.
A Reconstruction Algorithm for Gene Regulatory Sparse Networks using Positive Systems Ilaria Mogno Dipartimento di Informatica e Sistemistica ”A. Ruberti”, Universit` a degli Studi di Roma ”La Sapienza”, Via Eudossiana 18, 00184 Rome, Italy, [email protected] Abstract. In this paper we propose a new gene network reconstruction (or identification) scheme which takes advantage of the sparseness of a gene network using a decomposition of the given linear dynamical system describing the network, into two positive linear systems. First, we will describe how gene networks can be modelled as linear systems and an ”ideal” situation is considered in order to state an identification problem for gene regulatory networks. Finally, some preliminary results on the algorithm performances obtained using artificially generated data will be presented.
1 Introduction The problem of modelling and identification of gene regulatory networks is a formidable one. In fact, a huge number of attempts have been made, mainly in the last decade (see [7] for a literature review). The modelling difficulties are mainly due to the fact that the biological knowledge is far from complete and new advances in this field appear everyday. However, the advent of cDNA and oligonucleotide microarrays technologies has provided large amounts of data on mRNA expression levels so that different models can be tested, validated and compared. Unfortunately, gene networks are usually very large (e.g. about 6000 genes for the budding yeast) so that the actual situation is that the available data are usually insufficient for a reliable identification of gene regulatory networks. Moreover, microarrays do not directly measure gene expression levels and the relationships with mRNA levels are currently under investigation. Nevertheless, the study of gene networks using mRNA abundance as a direct measure for gene expression levels, has proved to be a fundamental tool for understanding the mechanism that control the expression of genes (see [1] for an overview of approaches and results in this field). Another important issue is that usually gene networks are sparse, since generally each gene interacts with only a small percentage of all the genes in the entire genome (see [10]). L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 127-134, 2003. Springer-Verlag Berlin Heidelberg 2003
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In this paper we will propose a gene networks reconstruction (or identification) scheme which takes advantage of this sparseness assumption using a decomposition of the given system describing the dynamics of the gene network, into two positive systems (see [6] for general results on positive systems). More specifically, in Section 2 we will describe how gene networks can be modelled as linear systems and in Section 3 an ”ideal” situation is considered in order to state an identification problem for gene regulatory networks. Finally, Section 4 contains the description of the proposed identification algorithm and Section 5 provides some results on its performances on artificially generated data. The fundamental issue of finding conditions for algorithm convergence will not be treated in this preliminary paper.
2 Modelling gene networks as LTI continuous–time systems Hereafter, we will refer to the work of Yeung et al. [16] and model a gene regulatory network as a linear time invariant (LTI) system described by the following differential equations: x˙ i (t) = −λi xi (t) +
N X
Wij xj (t) + bi (t) + ξi (t)
(1)
j=1
for i = 1, 2, ..., N , where the state variables xi ’s are the concentration of mRNA measured as a difference from the equilibrium state preceding the impulsive stimulus, the λi ’s are the self–degradation rates, the bi ’s are the external stimuli (depending on the specific experiment performed), and the ξi ’s represent (internal) noise. The elements of the matrix W describe the type and strength of the ”influence” of the j-th gene on the i-th gene with a positive, zero or negative sign indicating activation, no interaction and repression respectively. It is important to note that, even though this simplifying hypothesis may not be very realistic from a biological point of view, nevertheless it is a fundamental tool for studying and gaining insight into the basic regulation mechanism, thus providing a valuable testbed for different gene networks reconstruction algorithms. In fact, starting from the simplest and then moving toward the more and more complex is a typical scenario in the applied sciences. However, it is worth mentioning that linear modelling has proved to be useful also in real biological situation (see for example references [2], [4], [16], [8], [14], [15], [3]). As described in [16], an experiment consists in applying a prescribed (i.e. known) stimulus bi (t) which is a transient random perturbation (ideally a Dirac function δ0 (t) of ”amplitude” bi , i.e. bi (t) = bi δ0 (t)) and then use a microarray to measure the response after a time T . In terms of system theory, this situation corresponds to measuring the impulse response of the system or, equivalently, to consider its free evolution starting from an initial state
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xi (0) = bi . In this paper, for the sake of simplicity, we will assume the time derivative of the data matrix at time T to be known exactly. Morever, we will refer the usual more compact notation: x˙ = Ax, where the matrix A is a N × N matrix and it incorporates both self–degradation rates (on its main diagonal entries) and the strenght of the gene–to–gene interaction (on its off diagonal entries).
3 Problem formulation In order to evaluate on an ideal simple case the gene networks recontruction algorithm, we will neglect both internal and measurement noise, thus concentrating on the following ”ideal” identification problem. The Ideal Identification Problem Formulation for Gene Networks Given a network composed of N genes, M independent measurements xi (T ) and their derivatives x˙ i (T ) taken at a given time T , then we can write the following data matrices: 1 x1 (T ) x21 (T ) · · · xM 1 (T ) x21 (T ) x22 (T ) · · · xM 2 (T ) XN ×M = .. .. . . .. .. . . x1N (T ) x2N (T ) · · · xM N (T )
X˙ N ×M
x˙ 11 (T ) x˙ 21 (T ) · · · x˙ M 1 (T ) x˙ 12 (T ) x˙ 22 (T ) · · · x˙ M 2 (T ) = .. .. . . .. .. . . x˙ 1N (T ) x˙ 2N (T ) · · · x˙ M (T ) N
where superscripts denote experiments (which are then repeated M times) and the subscripts denote individual genes. Then, the ”ideal” identification problem consists in finding the ”best” matrix A such that X˙ N ×M = AXN ×M (2) holds. It is important to note that the matrix A in equation (2) is unique if and only if M = N , i.e. provided that the number of experiments equals the number of genes in the network. In what follows we will assume the typical situation in which M ¿ N so that equation (2) is underdetermined, that is, it has many solutions. Consequently, in order to find the best choice for the matrix A, some a priori information has to be exploited thus incorporating
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some biological knowledge into the model at hand. One possibility, as discussed in [16], is to try to impose the additional biological constraint that usually gene networks are sparse, i.e. that generally each gene interacts with only a small percentage of all the genes in the entire genome (see [10]). This assumption is obviously equivalent to the fact that the matrix A is sparse, that is it has a ”large” number of zero entries. Moreover, we shall also concentrate on the case in which the number k of nonzero entries in each row of A is not greater than the number of available measurements, i.e. k ≤ M . This additional condition is a very mild one, especially in view of the sparseness assumption and the small number of experiments repetitions. Consequently, we have that k ≤ M < N with k ¿ N . In the next section, we will describe a new identification method which takes advantage of this sparseness assumption using a decomposition of the given dynamical linear system describing the dynamics of the gene network, into two appropriate positive linear systems.
4 An iterative identification algorithm via positive systems Hereafter, we will propose a method to impose sparseness in the identification process by forcing to zero some appropriate entries of A. This can be done in many alternative ways (see for example [16] and [3]), but we will follow reference [13] where first a minimal L2 norm solution to equation (2) is found via singular value decomposition (SVD) on the data matrix XN ×M and the PN 2 2 matrix AL2 with the smallest L2 norm, that is with minimal i,j=1 (aL ij ) 2 where the aL ij ’s are the entries of AL2 , is obtained and afterwards, for each of the N rows of AL2 , the smallest in magnitude N − M entries are detected and considered as ”zero”. Finally, taking into account such estimated zero pattern, the solution to (2) is unique and therefore immediately found by inversion of N matrices of dimension M × M . More precisely, following the same lines as in reference [13], the data matrix XN ×M 1 is decomposed as T XN ×M = UN ×N SN ×M WM ×M T 2 where UN ×N and WM ×M are unitary matrices and the entries σij of the matrix SN ×M are such that σij = 0 for all i 6= j and σ11 ≥ σ22 ≥ . . . ≥ σM M > 0. The numbers σii := σi are the nonnegative square roots of the T eigenvalues of XN ×M XN ×M . They are known as the singular values of XN ×M . As previously stated, the SVD provide a simple way to find a solution AL2 to (2) with minimal L2 norm: 1 2
When the initial conditions are properly chosen (i.e. they are linearly independent) one has is rank(XN ×M ) = M . Unitary matrices X are such that XX T = I, the identity matrix.
Gene Networks Reconstruction via Positive Systems
˙ AL2 = XW diag
i=1,...,M
µ
1 σi
¶
131
UT
Here, the basic idea is to exploit the fact that, the computed matrix AL2 , is the one with minimal L2 norm, so that it is reasonable to assume that the smallest magnitude entries correspond to zeros in the ”true” matrix A. The starting point of the algorithm we are proposing in this paper is the observation that the above mentioned algorithm works better as much as the sparseness hypothesis is fulfilled. A possibility is to decompose the original system into two positive systems as follows X˙ = AX =: (A+ − A− ) X where the matrices A+ and A− are nonnegative and contain only the nonnegative and nonpositive entries of A, respectively. Clearly, both matrices A+ and A− are, in general, more sparse than A. Moreover, we can try to set up an identification algorithm able to identify separately the two matrices by imposing at each step of the algorithm also the nonnegativity of their entries. By doing so, one could also easily embed some a priori information on some regulatory effects by forcing to zero the corresponding elements of the matrix A+ or A− , in case of known activation or repression, respectively. The proposed procedure, hereafter formally described, consists of the following steps: An Iterative Identification Algorithm 1. Assuming a gene network with k ≤ M < N and using the full rank data matrices XN ×M and X˙ N ×M , find the minimal L2 norm solution µ ¶ 1 ˙ UT AL2 = XW diag σ i i=1,...,M to equation
X˙ N ×M = AXN ×M
using SVD of the data matrix XN ×M . 2. For each of the N rows of AL2 , determine the smallest in magnitude N −M entries and consider them as ”zero”. 3. Taking into account the zero pattern detected at the previous step, find the unique (sparse) solution to equation X˙ N ×M = A(0) s XN ×M and then let (0)
(0)
(0)
A(0) s =: A+s − A−s (0)
where A+s and A−s contain only the nonnegative and nonpositive entries (0) of As .
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4. Consider the iterative scheme: ( (k−1) (k) A+s X = A−s X + X˙ (k−1) (k) A−s X = A+s X − X˙
k = 1, 2, . . .
(3)
where the solutions at each step k has to be found as follows: a) Find minimal L2 solutions (via SVD decomposition) to equations (3) (k) (k) and let A+ , A− be such solutions. (k) (k) b) For each of the N rows of A+ and A− , determine the smallest in magnitude N − M entries and consider them as ”zero”. Taking into (k) (k) account such zero pattern, find the (sparse) solutions A+s , A−s to (3) with minimum L2 norm and nonnegative elements minimizing the L2 norm of the error3 . 5. Stop the iterations when the total error ° ° ° ° ° ° (k) ° ° (k) (k−1) (k−1) ε(k) = °A+s X − A−s X − X˙ ° + °A−s X − A+s X + X˙ ° 2
2
is below a preassigned threshold. In this paper, we will not discuss the fundamental issue of the algorithm’s convergence, but we will show hereafter some preliminary results on its performances on artificial data.
5 Simulation results on artificial data In order to illustrate the algorithm performance, we generate the artificial data as described in [16], for a network with N = 100 genes thus obtaining the corresponding matrix A. Then, we compute the data matrices XN ×M and X˙ N ×M for M = 10, 20, ..., 60 measurements. Using this data, we apply the proposed reconstruction algorithm and obtain the estimates A¯ for the true matrix A. As in [16], we measure the error by counting the percentage of discrepancies: N N X X eij E = 100 where
½ eij =
3
1 0
i=1 j=1 N2
¯ ¯ if ¯Aij − A¯ij ¯ > δ otherwise
r command NNLS (Nonnegative This computation can be done using the Matlab° Least Square).
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with δ being a small value corresponding to an error tolerance of ±10%. Obviously, this computation is possible only in this case because the data are generated by an artificial problem in which the matrix A is assumed to be known. The overall errors, separately for positive and negative entries, are depicted in the following figure. Number of Errors on Positive Entries 80 70 60 50 40 30 20 10 0
100 genes
10
20
30
40
50
60
Number of Measurements
Number of Errors on Negative Entries 50 40 30
100 genes
20 10 0 10
20
30
40
50
60
Number of Measurements
References 1. P. Baldi and G.W. Hatfield, DNA Microarrays and Gene Expression: from Experiments to Data Analysis and Modeling, Cambridge University Press, Cambridge, 2002
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2. T.Chen, H.L. He and G.M. Church, Modeling gene expression with differential equations, Proceedings of the Pacific Symposium on Biocomputing 4 (1999) 29– 40 3. M.J.L. De Hoon, S. Imoto, K. Kobayashi, N. Ogasawara and S. Miyano, Inferring gene regulatory networks from time-ordered gene expression data of bacillus subtilis using differential equations, Proceedings of the Pacific Symposium on Biocomputing 8 (2003) 17-28 4. P. D’haeseleer, X. Wen, S. Fuhrman and R. Somogyi, Linear modeling of mRNA levels during CNS development and injury, Proceedings of the Pacific Symposium on Biocomputing 4 (1999) 41–52 5. J. Hasty, J. Pradines, M. Dolnik and J.J. Collins, Noise–based switches and amplifiers for gene expression, Proceedings of the National Academy of Sciences 97 (2000) 2075–2080 6. L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications, Wiley–Interscience, New York, 2000 7. H. De Jong, Modeling and simulation of genetic regulatory systems: A literature review, Journal of Computational Biology 9 (2002) 67–103 8. N.S. Holter, A. Maritan, M. Cieplak, N.V. Fedoroff and J.R. Banavar, Dynamic modeling of gene expression data, Proceedings of the National Academy of Sciences 98 (2001) 1693–1698 9. R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985 10. H. Jeong, B. Tomber, R. Albert, Z.N. Oltvai and A.L. Barabasi, The large scale organization of metabolic networks, Nature 407 (2000) 651–654 11. T. Kailath, Linear Systems, Prentice Hall, 1980 12. D.G. Luenberger, Introduction to Dynamic Systems: Theory, Models, and Applications, Wiley, New York, 1979 13. I. Mogno, L. Farina and S. Monaco, A Reconstruction Algorithm from Expression Data for Sparse Noninteracting Gene Networks, preprint 14. E.P. van Someren, L.F.A.Wessels and M.J.T. Reinders, Linear modeling of genetic networks from experimental data, Proceedings of the International Conference on Intelligent Systems for Molecular Biology (2000) 355–366 15. D.C. Weaver, C.T. Workman and G.D. Stormo, Modeling regulatory networks with weight matrices, Proceedings of the Pacific Symposium on Biocomputing 4 (1999) 112–123 16. M.K.S. Yeung, J. Tegner and J.J. Collins, Reverse engineering gene networks using singular value decomposition and robust regression, Proceedings of the National Academy of Sciences 99 (2002) 6163–6168
The Basic Reproduction Number in a Multi-city Compartmental Epidemic Model Julien Arino1 and Pauline van den Driessche2 1 2
Department of Mathematics, McMaster University, Canada, [email protected] Department of Mathematics, University of Victoria, Canada, [email protected]
Abstract. A directed graph with cities as vertices and arcs determined by outgoing (or return) travel represents the mobility component in a population of individuals who travel between n cities. A model with 4 epidemiological compartments in each city that describes the propagation of a disease in this population is formulated as a system of 4n2 ordinary differential equations. Terms in the system account for disease transmission, latency, recovery, temporary immunity, birth, death, and travel between cities. The basic reproduction number R0 is determined as the spectral radius of a nonnegative matrix product, and easily computable bounds on R0 are obtained.
1 Introduction Modeling the spatial spread of infectious diseases is a complex task. One possible approach is to consider the travel of individuals between discrete geographical regions (cities), considering that the transmission does not take place during travel. The situation is then that of a directed graph, with the vertices representing the cities and the arcs representing the links between these cities. Disease transmission is assumed to occur between individuals present in a given city. Sattenspiel and Dietz [7] introduced such a model with travel between cities, and a similar type of model was considered in [8]. More recently Fulford et al [4] and Wang and Zhao [10] have formulated and discussed other models for the spread of a disease among discrete geographical regions. We consider the time evolution of a disease with 4 epidemiological compartments in each city for residents of n cities who may travel between them. The model formulated here is an extension of that of [1], which is adapted from [7]. We give a rigorous derivation of the basic reproduction number R0 , which represents the average number of new infections produced in a totally susceptible population by the introduction of an infective individual (see [3, 5, 9]). L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 135-142, 2003. Springer-Verlag Berlin Heidelberg 2003
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Easily computable bounds on R0 are derived. Local analysis and numerical simulations indicate that R0 , is a sharp threshold with the disease dying out or becoming endemic according as R0 < 1 or R0 > 1.
2 The SEIRS epidemic model The total number of cities considered is n. The number of residents of (i.e., individuals who normally live Pnin) city i who are present in city j at time t is denoted by Nij , and Nir = j=1 Nij denotes the resident population of city Pn i at time t. Also, Nip = j=1 Nji denotes the population of city i at time t, i.e., the number of individuals who are physically present in city i. As in [1, 7], residents of city i leave this city at a per capita rate gi ≥ 0 per unit time with aPfraction mji ≥ 0 of these outgoing individuals going to city j. n If gi > 0, then j=1 mji = 1, with mii = 0, and gi mji is the travel rate from city i to city j. Residents of city i who are in city j return to i with a per capita rate of rij ≥ 0, with rii = 0. With these assumptions, an individual resident in a given city who is present in another city, must first return to their city of residence before travelling to a third city. The outgoing matrix [gi mji ] and the return matrix [rij ], which represent the outgoing travel from i to j and the return to i from j, respectively, are assumed to have the same zero/nonzero pattern. Thus the directed graph with vertices representing cities and arcs representing travel between these cities can be determined by either matrix. The terms mji and rij implicitly take into account the distance between cities i and j. In each of the n cities, an epidemic model is superimposed; see Hethcote [5] for a recent review of mathematical models of infectious diseases. In [7], an SIR epidemic model with 3 compartments (susceptible, infective, recovered) is formulated in each city (called region), with two types of mobility (infants and adults) in each region. In [8], each region has an SIR model and, as in [7], there is no birth or natural death of individuals. Here we construct an SEIRS model with 4 compartments (susceptible, exposed, infective, recovered) and include birth in the city of residence and natural death in any city. Our general SEIRS model is applicable for diseases with a latent period that confers immunity upon recovery (e.g., pertussis), and can be reduced to simpler models by formally setting parameter(s) (or inverse(s)) to zero. For example, tuberculosis has a long latent period and treated infectives move back into the susceptible class; thus an SEIS model is appropriate. Some childhood diseases (e.g., scarlet fever) have short latent periods and confer permanent immunity upon recovery, thus an SIR model is appropriate; others (e.g., measles) have a longer latent period, thus an SEIR model is preferred. For a disease with no latent period and that confers no immunity (e.g., gonnorhea) an SIS model, as formulated and analyzed in [1], is adequate. Let Sij , Eij , Iij and Rij denote respectively the number of susceptible, exposed, infective and recovered individuals resident in city i who are present
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in city j at time t; thus Nij = Sij + Eij + Iij + Rij for all i, j = 1, . . . , n. Disease transmission is modelled using standard incidence, namely n n X X
κj βikj
j=1 k=1
Sij Ikj Njp
(1)
where the disease transmission coefficient βikj > 0 is the proportion of adequate contacts in city j between a susceptible from city i and an infective from city k that actually results in transmission of the disease and κj > 0 is the average number of such contacts in city j per unit time. Let 1/d, 1/ε, 1/γ and 1/ν denote the average lifetime, exposed period, infective period and period of temporary immunity, respectively. Note that d, ε, γ and ν are assumed to be positive and the same for all cities. Birth and death are assumed to occur with the same rate constant, thus the total population remains a fixed constant. For residents of city i present in city i (with i = 1, . . . , n), the following 4 differential equations describe the dynamics of the susceptible, exposed, infective and recovered individuals, n n X X Sii Iki dSii κi βiki rik Sik − gi Sii − = + d(Nir − Sii ) + νRii (2a) dt Nip
dEii = dt dIii = dt dRii = dt
k=1 n X
k=1 n X k=1 n X
k=1 n X
rik Eik − gi Eii +
κi βiki
k=1
Sii Iki − (ε + d)Eii Nip
(2b)
rik Iik − gi Iii + εEii − (γ + d)Iii
(2c)
rik Rik − gi Rii + γIii − (ν + d)Rii
(2d)
k=1
and, for j 6= i, the following equations describe the dynamics of residents of city i present in city j, n
X Sij Ikj dSij − dSij + νRij = gi mji Sii − rij Sij − κj βikj dt Njp k=1 n X
dEij = gi mji Eii − rij Eij + dt
k=1
κj βikj
Sij Ikj − (ε + d)Eij Njp
dIij = gi mji Iii − rij Iij + εEij − (γ + d)Iij dt dRij = gi mji Rii − rij Rij + γIij − (ν + d)Rij dt
(2e) (2f) (2g) (2h)
As there are n cities, there are 4n2 equations. These equations, together with nonnegative initial conditions and fixed Nir , constitute the SEIRS epidemic
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model. The following result is easily shown and assures that the system is well posed. 2
Proposition 1. The nonnegative orthant R4n is positively invariant under + the flow of (2), and for all t > 0, Sii > 0 and Sij > 0 provided that gi mji > 0. Furthermore, solutions of (2) are bounded. 2.1 The underlying travel model Summing (2a) to (2d) gives the evolution of the number of residents of city i present in city i, n
X dNii = d(Nir − Nii ) + rik Nik − gi Nii dt
(3a)
k=1
Similarly, summing (2e) to (2h) gives the evolution of the number of residents of city i who are present in city j 6= i, dNij = gi mji Nii − rij Nij − dNij dt
(3b)
From (3), it can be shown that the resident population Nir of city i is constant, the current population Nip need not be. The total population Pn whereas P n p r i=1 Ni = i=1 Ni in the system is constant. Equations (3) subject to the initial values Nij ≥ 0 at t = 0 with fixed Nir > 0 constitute the travel model, which is identical to that in the SIS model [1], where the following is proved. Theorem 1. The travel model (3) has the (globally) asymptotically stable equilibrium ¶ µ 1 ˆ Nir Nii = (4) 1 + gi Ci and, for j 6= i
where Ci =
Pn
ˆij = gi mji N d + rij
mki k=1 d+rik
µ
1 1 + gi Ci
¶
Nir
(5)
for i = 1, . . . , n.
2.2 The basic reproduction number The system is at an equilibrium if the time derivatives in (2) are zero. City i is at the disease free equilibrium (DFE) if Iji = 0 for all j = 1, . . . , n, giving ˆji from (4) and (5). The n-city model given by Eji = Rji = 0 and Sji = N (2) is at the DFE if every city is at the DFE. The DFE of (2) always exists, and in the case in which the disease is absent in all cities, (2) reduces to the underlying travel model (3).
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To discuss local stability of the DFE in the n-city model given by (2), we use the method of [3, 9], and R0 , the basic reproduction number for the whole system, is the spectral radius of the next generation matrix. Ordering the infected variables (exposed and infectives) as E11 , . . . , E1n , E21 , . . . E2n , . . . Enn , I11 , . . . , I1n , I21 , . . . I2n , . . . , Inn gives the lower triangular block matrix n M Ak ¸ · k=1 A 0 = V = n C B M −diag(ε) Bk
0
k=1
where each block A, B and C is n2 × n2 . For k = 1, . . . n, Ak is an n × n matrix with rk1 + ε + d 0 · · · 0 −gk m1k 0 · · · 0 0 rk2 + ε + d −gk m2k 0 · · · 0 Ak = −r −r g + ε + d −r k1 k2 k kn 0 ··· −gk mnk 0 rkn + ε + d For a fixed k, and j 6= k, the (k, j) entry of Ak is −rkj , the (j, k) entry is −gk mjk , the j th diagonal entry is rkj + ε + d, the (k, k) entry is gk + ε + d, and other entries are zero. Matrices Bk have the same entries as Ak but with ε replaced by γ. Since Ak and Bk have the Z-sign pattern and have all positive column sums, Ak and Bk are nonsingular M-matrices [2, p. 136]. The inverse of V is the nonnegative matrix n M −1 (Ak ) k=1 −1 ! n ! Ã n V = Ã n M M M −1 −1 −1 (Bk ) (Ak ) (Bk ) diag(ε)
0
k=1
k=1
Matrix F is a block matrix
· F =
0 G 0 0
k=1
¸
where G is an n2 × n2 matrix having n2 blocks, with each block Gij an n × n ˆiq /N ˆ p , for diagonal matrix of the form Gij = diag(gijq ), where gijq = κq βijq N q q = 1, . . . , n.
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Since V −1 is lower triangular by blocks, F V −1 can be given by blocks. By [9, Theorem 2], the basic reproduction number for system (2) is, factoring ε out of the expression, !) ( Ã n M −1 (Ak Bk ) (6) R0 = ε · ρ G k=1
where ρ{·} is the spectral radius, and the following result holds. Theorem 2. Let R0 be defined as in (6). If R0 < 1, then the DFE of (2) is locally asymptotically stable. If R0 > 1, then the DFE of (2) is unstable. From (6), to compute R0 , it is sufficient to invert an n × n matrix. The following bounds hold for R0 . Theorem 3. For system (2), κk βijk ε κk βijk ε ≤ R0 ≤ max i,j,k=1,...,n (γ + d)(ε + d) i,j,k=1,...,n (γ + d)(ε + d) min
¡ ¢ Proof. The i, j block of G ⊕(Ak Bk )−1 is Gij (Aj Bj )−1 for all i, j. As Gij is diagonal, left multiplication with (Aj Bj )−1 amounts to multiplyˆiq /N ˆqp for q = 1, . . . , n. Let v −1 (j) deing row q of (Aj Bj )−1 by κq βijq N kl note the (k, l) entry of (Aj Bj )−1 , for k, l = 1, . . . , n. Consider the first column of Gi1 (A1 B1 )−1 , and denote the sum of entries in the first column of Gi1 (A1 B1 )−1 by [1lT Gi1 (A1 B1 )−1 ]1 , with 1lT = (1, . . . , 1). Then [1lT Gi1 (A1 B1 )−1 ]1 ˆ12 ˆ1n ˆ11 N N N −1 −1 −1 (1) (1) + κ2 β112 p v21 (1) + · · · + κn β11n p vn1 =κ1 β111 p v11 ˆ ˆ ˆn N N N 1 2 ˆ21 ˆ22 ˆ2n N N N −1 −1 −1 + κ1 β211 p v11 (1) + · · · (1) + κ2 β212 p v21 (1) + · · · + κn β21n p vn1 ˆ ˆ ˆ N1 N2 Nn ˆn1 ˆn2 ˆnn N N N −1 −1 −1 (1) + κ1 βn11 p v11 (1) + κ2 βn12 p v21 (1) + · · · + κn βn1n p vn1 ˆ ˆ ˆn N N N 1 2 ´ κ1 ³ ˆ11 + β211 N ˆ21 + · · · + βn11 N ˆn1 v −1 (1) + · · · = p β111 N 11 ˆ N 1 (7) ´ κn ³ ˆ1n + β21n N ˆ2n + · · · + βn1n N ˆnn v −1 (1) + p β11n N n1 ˆn N Suppose that min
i,j,k=1,...,n
and
κk βijk ε κkm βim jm km ε = (γ + d)(ε + d) (γ + d)(ε + d)
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κk βi j k ε κk βijk ε = M M M M i,j,k=1,...,n (γ + d)(ε + d) (γ + d)(ε + d) max
Then
κkm βim jm km ε ≤ . . . ≤ κk βijk ε ≤ . . . ≤ κkM βiM jM kM ε
Using these inequalities in (7) and the definition of Nip , ¢ ¡ −1 −1 (1) ≤ [1lT Gi1 (A1 B1 )−1 ]1 κkm βim jm km ε v11 (1) + · · · + vn1 ¡ −1 ¢ −1 ≤ κkM βiM jM kM ε v11 (1) + · · · + vn1 (1) Note that 1lT Aj = (ε + d)1lT and 1lT Bj = (γ + d)1lT for all j. This implies that 1lT (Aj Bj )−1 = 1/[(γ + d)(ε + d)]1lT , i.e., each column sum of (Aj Bj )−1 is equal to 1/[(γ + d)(ε + d)]. Therefore, κk βi j k ε κkm βim jm km ε ≤ [1lT Gi1 (A1 B1 )−1 ]1 ≤ M M M M (γ + d)(ε + d) (γ + d)(ε + d) The same argument ¡ ¢ shows that this inequality remains true for every column of G ⊕(Ak Bk )−1 . From (6) and using a standard result on the localization of the dominant eigenvalue of a nonnegative matrix (see, e.g., [6, Theorem 1.1]), the result then follows. u t If city i is isolated from the others, then the basic reproduction number in city i is Ri0 = κi βiii ε/[(γ + d)(ε + d)]. This is the product of the average number of contacts, the disease transmission coefficient, the average fraction surviving the latent period ε/(ε+d), and the average time spent in the infective compartment. In the case of disease transmission coefficients equal for all populations present in a city, i.e., βijk = βk for all i, j, giving Ri0 = κi βi ε/[(γ+ d)(ε + d)], the following easily computable bounds hold for R0 . Corollary 1. Suppose that βijk = βk for all i, j = 1, . . . , n. Then min Ri0 ≤ R0 ≤ max Ri0
i=1,...,n
i=1,...,n
Note that in this case, if Ri0 < 1 for all i, then R0 < 1, thus from Theorem 2, the DFE is locally asymptotically stable. Similarly, if Ri0 > 1 for all i, then R0 > 1, thus the DFE is unstable. If κk βk = κβ (i.e., the disease transmission parameters are identical in all cities), then R0 = κβε/[(γ + d)(ε + d)], as in a classical SEIRS model with no mobility.
3 Discussion The SEIRS epidemic model formulated in (2) describes the dynamics of an infectious disease in a population of individuals with travels between discrete
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cities as incorporated in a model by Sattenspiel and Dietz [7]. The disease free equilibrium of the epidemic model (2) has population numbers given by (4) and (5). An explicit formula (6) for the basic reproduction number R0 is derived; the DFE of (2) is locally asymptotically stable if R0 < 1, and unstable if R0 > 1. Numerical simulations indicate that R0 acts as a sharp threshold between the extinction(R0 < 1) and the invasion (R0 > 1) of the disease. They also indicate that the endemic equilibrium is unique with infective numbers tending to this equilibrium whenever R0 > 1. Thus to control the disease, measures should be taken to reduce R0 below 1. However, since R0 depends on the disease transmission parameters, the average lifetime, the exposed and infective periods as well as the outgoing and return travel matrices, such control strategies are not in general easily quantified. However, with parameter values appropriate for a specific disease, R0 can be readily computed from (6) and its variation with respect to some parameters can be estimated.
References 1. J. Arino and P. van den Driessche. A multi-city epidemic model. To appear in Mathematical Population Studies, 2003. 2. A. Berman and R. J. Plemmons. Nonnegative Matrices in the Mathematical Sciences. Academic Press, 1979. 3. O. Diekmann and J. A. P. Heesterbeek. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley, 2000. 4. G. R. Fulford, M. G. Roberts, and J. A. P. Heesterbeek. The metapopulation dynamics of an infectious disease: tuberculosis in possums. Theoretical Population Biology, 61:15–29, 2002. 5. H. W. Hethcote. The mathematics of infectious diseases. SIAM Review, 42(4):599–653, 2000. 6. H. Minc. Nonnegative Matrices. Wiley Interscience, 1988. 7. L. Sattenspiel and K. Dietz. A structured epidemic model incorporating geographic mobility among regions. Math. Biosci., 128:71–91, 1995. 8. L. Sattenspiel and D. A. Herring. Structured epidemic models and the spread of influenza in the central Canadian subartic. Human Biology, 70:91–115, 1998. 9. P. van den Driessche and J. Watmough. Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Math. Biosci., 180:29–48, 2002. 10. W. Wang and X.-Q. Zhao. An epidemic model in a patchy environment. Submitted, 2002.
Stability Analysis of a Metabolic Model with Sequential Feedback Inhibition Yacine Chitour1 , Fr´ed´eric Grognard2 , and Georges Bastin3 1 2 3
D´epartement de Math´ematiques, Universit´e de Paris Sud, 91405 Orsay, France, [email protected] INRIA, BP 93 06902 Sophia-Antipolis Cedex, France, [email protected] CESAME- Universit´e Catholique de Louvain, 1348 Louvain la Neuve, Belgium, [email protected]
Abstract. This paper deals with the stability analysis of a simple metabolic system with feedback inhibition. The system is a sequence of monomolecular enzymatic reactions. The last metabolite acts as a feedback regulator for the first enzyme of the pathway. The enzymatic reactions of the pathway satisfy Michaelis-Menten kinetics. The inhibition is described by an hyperbolic model. Without inhibition, it is clear that the system is cooperative and has a single globally asymptotically stable equilibrium. But, in the common situation where there is inhibition, the system is no longer cooperative and the stability analysis is more intricate. In this paper we exhibit sufficient conditions on the kinetic parameters in order to guarantee that this simple metabolic system with inhibition still has a single globally asymptotically stable equilibrium.
1 Introduction The huge set of biochemical reactions which occur inside living cells is called the Cellular Metabolism. It is usually represented by an intricate network connecting the involved biochemical species (called ”metabolites”). The pathways of the network are called ”metabolic pathways”. In the metabolic engineering literature, it is widely accepted that ”despite their immense complexity, metabolic systems are characterized by their ability to reach stable steady states” (quoted from [7], Chapter 4). It should however be fair to recognize that a mathematical analysis of this fundamental stability property is a difficult question which was not much investigated. Our objective in this paper is to provide a modest contribution to this question. We shall limit ourselves to simple metabolic pathways which are made up of a sequence of mono-molecular enzyme-catalysed reactions as where Xi (i = 1, · · · , n) represent the successive metabolites of the pathway: X1 → X2 → · · · → Xn . L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 143-150, 2003. Springer-Verlag Berlin Heidelberg 2003
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A typical situation is when such a simple pathway is located between two branch points of a complex metabolic network. We shall consider the case of a so-called sequential feedback inhibition (cf. [7]) where the last metabolite Xn acts as an inhibitor of the first reaction X1 −→ X2 . This inhibition is represented by the dotted feedback arrow in Fig.1. The velocity of each enzymatic reaction Xi −→ Xi+1 is represented by a Michaelis-Menten kinetic function : ϕi (xi ) =
ai xi , ki + xi
(1)
where xi denotes the intracellular molar fraction of the metabolite Xi , ai is the maximal velocity and ki the so-called half-saturation constant. It is assumed that the velocity of the first reaction X1 −→ X2 is inhibited by the last metabolite with a multiplicative hyperbolic inhibition function of the form: ψα (xn ) =
1 1 + αxn
(2)
In addition, it is assumed that the cell metabolism is analysed during a period of exponential cell growth with a constant specific growth rate µ. Under these assumptions and notations, a mass balance dynamical model is formulated as: a1 x 1 x˙ 1 = − (k1 +x1 )(1+αxn ) − µx1 + c, x2 a1 x1 − µx2 , x˙ 2 = (k1 +x1 )(1+αxn ) − (ka22+x (Σ) (3) 2) x˙ i = ai−1 xi−1 − ai xi − µxi , 3 ≤ i ≤ n, (ki−1 +xi−1 ) (ki +xi ) where n ≥ 3 is a positive integer, x = (x1 , · · · , xn )T ∈ IRn , and all the ai , ki ’s, c, α, µ are positive constants. In this model c denotes the inflow rate of the first metabolite X1 of the sequence and is assumed to be constant. Without inhibition (i.e. α = 0), the system Σ is clearly compartmental and cooperative which implies that it has a single globally asymptotically stable equilibrium. But if there is inhibition, the system is no longer cooperative and the stability analysis is more difficult. Our contribution in this paper will be to exhibit sufficient conditions on the kinetic parameters that guarantee that the simple metabolic system Σ with feedback inhibition still has a single globally asymptotically stable equilibrium.
2 Notations and statement of the theorem 2.1 Notations Consider the metabolic system (Σ) with feedback inhibition (3). Up to a change of variable (the xi ’s are multiplied by α) and a time reparameterization (the time is multiplied by µ), we may assume that µ = α = 1 and the model is rewritten:
Stability Analysis of a Metabolic Model
x˙ 1 = −ϕ1 (x1 )ψ(xn ) − x1 + c, x˙ 2 = ϕ1 (x1 )ψ(xn ) − ϕ2 (x2 ) − x2 , (Σ) x˙ i = ϕi−1 (xi−1 ) − ϕi (xi ) − xi , 3 ≤ i ≤ n,
145
(4)
where ϕi is defined in (1) and ψ := ψ1 in (2). In this model, c stands for αc µ (with the original c in that last formula) and similarly, for 1 ≤ i ≤ n, ai i stands for αa µ and ki for αki . We introduce some notations: for 1 ≤ i ≤ n, Fi (x) is the real function defining x˙ i . For 2 ≤ i ≤ n, fi (x) = x + ϕi (x) and f1,xn (x) = x + ψ(xn )ϕ1 (x). It is clear that the ϕi ’s, 1 ≤ i ≤ n, are strictly monotone functions on IR+ and realize bijections between IR+ and [0, ai ). We use ϕ−1 to denote the inverse i function. For 2 ≤ i ≤ n, the fi ’s are strictly monotone functions on IR+ and realize bijections from IR+ to IR+ ; fi−1 denotes the inverse function of fi and gi = ϕi ◦ fi−1 . Let M : IR+ → IR+ by M = fn−1 ◦ gn−1 ◦ · · · ◦ g2 . For every x ≥ 0 and 2 ≤ i ≤ n, we have ki 1 ai < 1. , ≤ (fi−1 )0 (x) = 0 ki ai + ki 1 + ϕi (fi−1 (x)) (5) Note that the fi ’s, the ϕi ’s are concave functions on IR+ (negative second derivative) for 2 ≤ i ≤ n. This implies that the gi ’s are also concave. Therefore, we have for 2 ≤ i ≤ n and every x ≥ 0 1 < fi0 (x) = 1 + ϕ0i (x) ≤ 1 +
0 < gi0 (x) =
ϕ0i (fi−1 (x)) ai , ≤ gi0 (0) = ki + ai 1 + ϕ0i (fi−1 (x))
and since 0
M (x) =
(fi−1 )0
(gn−1 ◦ · · · ◦ g2 (x))
"n−1 Y
(6)
# gi0
(gi−1 ◦ · · · ◦ g2 (x)) g20 (x),
i=3
we can then conclude from (5) and (6) that, for every x ≥ 0 0
0 < M (x) <
n−1 Y i=2
ai . ki + ai
(7)
As for f1,xn , for every xn ∈ IR+ , it behaves like any fi , 2 ≤ i ≤ n. Define −1 (c). Later we will study in more details that z : IR+ → [0, c) by z(b) = f1,b application. Let F be the vector field on IRn simply defined by the rightn hand side of (Σ). Let K = IR+ the non-negative orthant and K+ the positive orthant. The positive cone K defines a closed partial order relation ≤ on IRn defined by x ≤ y if and only if y − x ∈ K. It means that xi ≤ yi holds for every 1 ≤ i ≤ n. We write x < y if x ≤ y and x 6= y, and x ¿ y whenever y − x ∈ Int(K) = K+ . This notation extends trivially to subsets of IRn . Moreover, if x ≤ y, then the set Px,y = {z ∈ IRn , x ≤ z ≤ y} is a parallelepiped. Let v ∈ K+ defined by v = (1, · · · , 1).
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If f : IR+ → IR, set limf = lim supt→∞ f and limf = lim inf t→∞ f . This notation is naturally extended to the vectorial case using the partial order defined previously. P We will also consider sometimes the function V : IRn → IR n defined by V (x) = i=1 xi . Let Td , d > 0, be the simplex of K defined as the the set of x ∈ K so that V (x) ≤ d. If x, y ∈ X, then [x, y] denotes the segment with extremities x and y, i.e. the set of points tx+(1−t)y for t ∈ [0, 1]. A set X ⊂ IRn is said to be p-convex if for every x, y ∈ X with x ≤ y then [x, y] ⊂ X. Let m be a positive integer. An m × m matrix A = (aij ) is said to be irreducible if for every nonempty, proper subset I ⊂ {1, · · · , n}, there is an i ∈ I and j ∈ {1, · · · , n}/I such that aij 6= 0. There is a graph-theoretic formulation of irreducibility (cf. [6]): consider the directed graph G whose set of vertices is {1, · · · , n}; two vertices i, j have a directed edge from i to j if aij 6= 0. Then A is irreducible if its directed graph G is connected. A dynamical system (G) given by x˙ = G(x), x ∈ D with D open, G : D → IRn of class C 1 is said to be cooperative (see [6]) if, for every x ∈ D, 1 ≤ i, j ≤ n and i 6= j, ∂Gi ≥ 0. ∂xj If in addition the jacobian matrix DG(x) is irreducible for every x ∈ D, then (G) is said to be irreducible cooperative. Remark 2. 1 It is worth noticing that (Σ) is not cooperative with respect to the partial order defined by K. or by any other partial order defined by an orthant of IRn (cf. [6]). We will consider auxiliary systems (Σ)b , b ≥ 0, given by x˙ 1 = −ϕ1 (x1 )ψ(xn ) − x1 + c, x˙ 2 = ϕ1 (x1 )ψ(b) − ϕ2 (x2 ) − x2 , (Σ)b x˙ k = ϕk−1 (xk−1 ) − ϕk (xk ) − xk , 3 ≤ k ≤ n,
(8)
where the difference with (Σ) lies in the equation defining x˙ 2 : the variable xn is frozen at the constant value b. We use Fb (x) to denote the right-hand side of (Σ)b . Now, the (2, n)-coefficient in DFb (x) is identically equal to zero. If x ∈ K, we use γx , γxb respectively, to denote the trajectory of (Σ), (Σ)b respectively, which starts at x. 2.2 Preliminary results and statement of the theorem Proposition 2. 2
The system (Σ) has the following properties:
(1) (Σ) has a unique equilibrium point x ¯ ∈ K+ ; (2) For every x ∈ K and every t > 0, γx (t) ∈ K+ i.e. K is a positively invariant set for (Σ);
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(3) For every x ∈ K, lim V (γx ) ≤ c; i.e. Tc is a global attractor of all the trajectories starting in K. Remark 2. 3 Even though (Σ) is not cooperative, it has some of the basic features that are required for the investigation of the ω-limit sets of cooperative systems: an invariant cone with a repelling boundary, a bounded attractor and a unique equilibrium point. Remark 2. 4 At the light of Proposition 2. 2, the relevance of the auxiliary systems (Σ)b for understanding the dynamics of (Σ) can be put forward. It is based on the two following remarks: (a) for every b ≥ 0, (Σ)b is an irreducible cooperative system (use the graphtheoretic formulation of irreducibility). This easily implies that (Σ)b verifies (2) (cf. Theorem 1.1 p.56 of [6]) and (3) (with possibly another positive constant instead of c). In fact (Σ)b is a hypercycle for which a Poincar´e-Bendixon theory was developed for the compact ω-limit sets of (Σ)b , regardless of the dimension of the system(cf. [5]). Then, we expect taking advantage of the many deep results relative to that class of irreducible cooperative systems (for an excellent reference, cf [6]). (b) for every x ∈ K and 0 ≤ b0 < b1 , we have b0 ≤ xn ≤ b1 ⇒ Fb1 (x) ≤ F (x) ≤ Fb0 (x),
(9)
and, if x1 > 0, then ≤ can be replaced everywhere by < in the above equation. The monotonicity property expressed in (9) translates to the trajectories of F and Fb as explained next. Assume that we have shown the existence of 0 ≤ b0 < b1 such that for every x ∈ K, there is some tx > 0 for which b0 ≤ xn (t) ≤ b1 , if t ≥ tx . (10) (This is clearly the case by (3) of Proposition 2. 2.) Using (9), we have, for t ≥ tx , Fb1 (γx (t)) ≤ F (γx (t)) ≤ Fb0 (γx (t)). Set yx = γx (tx ). Since Fb is a function of type K, we can apply a standard theorem of comparison for differential inequalities (cf. for instance Theorem 10 p.29 of [3]): for t ≥ tx , γybx1 (t − tx ) ≤ γx (t) ≤ γybx0 (t − tx ).
(11)
(c) The elegant theory recently developed by Angeli and Sontag in [1] about interconnections of controlled monotone systems implies to the system (Σ), as indicated to us by D. Angeli. However, to draw any conclusion with the Angeli-Sontag theory, it is required additional information (with respect to the present procedure) on the asymptotic behaviors of some
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auxiliary systems. Therefore, our approach is more flexible and general, since it can provide results on the bounded attractor of (Σ), regardless on global asymptotic stability of auxiliary systems. One should notice though that the theorem proved in that paper can be recovered by using the Angeli-Sontag theory. Assume now that, according to Part (a), the ω-limit sets of (Σ)b0 and (Σ)b1 are investigated in details and one is able to show e.g. that every trajectory of (Σ)b0 ((Σ)b1 respectively) starting in K converges to xb0 (xb1 respectively). In addition, assume that b0 ¿ xb0 and xb1 ¿ b1 . Then the pair (xbn0 , xbn1 ) can be used in (10) instead of (b0 , b1 ) in the bounding process for γx (t) described above. If that procedure can be reproduced, one may hope to get more and more precise information on the ω-limit sets of (Σ). It is even tempting to conjecture that every trajectory of (Σ) starting in K converges to x ¯. We prove it but for a restricted set of the problem’s parameters. Theorem 2. 5
Under the following condition (C), (C) (a1 + c)
n−1 Y i=2
ai ≤1 ki + ai
(12)
the system (Σ) is globally asymptotically stable in K with respect to x ¯. Remark 2. 6 We may express condition (C) in terms of the original parameters, i.e. with α and µ. Equation (12) becomes n−1 Y ai α (a1 + c) ≤ 1. µ µk i + ai i=2
(13)
It is not surprising that if α = 0 (i.e. no inhibition) or if µ is large enough then the condition expressed in (13) holds true.
3 Proof of Theorem 2. 5 3.1 Technical lemmas Before starting the proof of the theorem, we establish a series of useful lemmas whose proofs are omitted. We first study the application z : IR+ → [0, c) by −1 z(b) = f1,b (c). We have Lemma 3. 1 (i) The application z is strictly increasing from IR+ to [0, c); (ii) The application ϕ1 ◦ z is strictly increasing from IR+ to [0, ϕ1 (c)) and is concave.
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Lemma 3. 2 A point e ∈ K is an element of Eb if and only if its n-th coordinate en is solution in IR+ of the following equation in the unknown y ³ ´ y = M ϕ1 (z(y))ψ(b) . (14) Moreover the previous equation has always solutions and it has exactly one if condition (C) holds. Remark 3. 3 We chose in this paper to investigate the sets Eb ’s in an elementary way rather than using the deep work of [5] and [4]. Doing so leads to obtain results on Eb which are only valid under condition (C) even though they are more general. The more complete characterization of the Eb ’s will appear in the final version of the paper, see [2]. From now on, assume that condition (C) holds. Then, for every b ≥ 0, (Σ)b has a unique equilibrium point e(b) in K+ . All the assumptions of Theorem 3.1 of [6] are satisfied. Therefore, (Σ)b is globally (with respect to initial states in K) asymptotically stable with respect to e(b). The next lemma studies the application e : [0, c] → K+ that associates to b ∈ [0, c], e(b). Set en : [0, c] → IR+ for the application that associates to b the n-th coordinate of e(b). Note that x ¯n is a fixed point of en . Lemma 3. 4
Assume that condition (C) holds. Then
(1) en is a strictly decreasing function and x ¯n is its unique fixed point; (2) if b1 < b2 , then e(b2 ) ¿ e(b1 ); (3) if b < x ¯n , then b < x ¯n < en (b) and x ¯ ¿ e(b); similarly if x ¯n < b, then en (b) < x ¯n < b and e(b) ¿ x ¯. Lemma 3. 5 Consider (I l ) and (S l ) the sequences of points of K defined inductively as follows ½ l+1 I = e(Snl ), (15) l+1 S = e(Inl ), with I 0 = 0 and S 0 so that e(0) ¿ S 0 and, for every x ∈ Tc , x ¿ S 0 . Here e(0) is the equilibrium point of (Σ)0 . Then, for every l ≥ 0, we have I l ¿ x ¯ ¿ Sl and lim I l = lim S l = x ¯, (16) l→∞
l→∞
where x ¯ is the equilibrium point of (Σ). Lemma 3. 6
Let x ∈ K. Then 0 ¿ limγx
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3.2 Final part of the proof of Theorem 2. 5 We are now ready to establish Theorem 2. 5. From what precedes, the conclusion is the consequence of the next statement: for every x ∈ K and for every l≥0 (17) (Ql ) I l ≤ limγx ≤ limγx ≤ S l . Fix x ∈ K. Proposition (Ql ) is proved inductively. For l = 0, this is a consequence of Proposition 2. 2, (3). Applying Lemma 3. 6, there exists ε > 0 and t0 (ε) > 0 such that for every t ≥ t0 (ε), εv ≤ γx (t) ≤ S 0 − εv.
(18)
Then, passing to the limit we have e(Sn0 − ε) ≤ limγx ≤ limγx ≤ e(ε).
(19)
Note that, in equation (18), ε may be replaced by any 0 < η ≤ ε. Since e is globally Lipschitz over IR+ , equation (19) implies (Q1 ) but also the existence of t1 (ε) > 0 such that for every t ≥ t1 (ε), I 1 + C1 εv ≤ γx (t) ≤ S 1 − C1 εv,
(20)
for some 0 < C1 ≤ 1 independent of ε. Notice that equation (20) is of the same type as equation (18) and then leads to equations similar to (19) and again (20). In that way, we obtain, for every l ≥ 2, e(Snl − Cl ε) ≤ limγx ≤ limγx ≤ e(Inl + Cl ε),
(21)
and the existence of tl (ε) such that, for every t ≥ tl (ε), I l + Cl+1 εv ≤ γx (t) ≤ S l − Cl+1 εv, with Cl+1 ≤ Cl ≤ 1 independent of ε. Letting ε tend to zero in (21), we get (Ql ).
References 1. Angeli D. and Sontag E. D., preprint. 2. Chitour Y., Grognard F. and Bastin G., “Stability analysis of a metabolic model with sequential feedback inhibition,” in preparation. 3. Coppel W. A., “Stability and asymptotic behavior of differential equations,” Heath Math. Mono., D. C. Heath Comp., 1965. 4. Li Y. M. and Muldowney J., “Global Stability for the SEIR Model in Epidemiology,” Math. Bio. 125 (1995) pp. 155-164. 5. Mallet-Paret, J. and Smith, Hal L., “The Poincar´e-Bendixson theorem for monotone cyclic feedback systems,” J. Dynam. Differential Equations 2 (1990), no. 4, pp. 367–421. 6. Smith H. L., “Monotone Dynamical Systems, An introduction to the theory of competitive and cooperative systems,” Math. Surveys and Mono., Vol. 41, AMS, Providence, RI, 1995. 7. Stephanopoulos G., Aristidou A. and Nielsen J., “Metabolic Engineering: Principles and Metodologies,” Academic Press (1997).
Differential Systems with Positive Variables Jean-Luc Gouz´e INRIA COMORE, BP 93, 06902 Sophia-Antipolis Cedex, France, [email protected]
Abstract. The variables of biological, ecological, or chemical systems are often positive, because they measure concentrations, numbers,... We study polynomial ndimensional differential systems defined in the positive orthant. We use the tools of the usual linear algebra to exploit the specific structure of such systems, and obtain some indications on their behavior. In some cases, we are able to exhibit functions that decrease along the trajectories and therefore to give sufficient conditions for a regular global behavior: that is, all the trajectories either converge towards the equilibria or are unbounded. Our main example will be the n - dimensional LotkaVolterra system (arising in biological modeling of species interactions). We apply the above results to stabilize the Lotka-Volterra system by controlling either the total growth rates of some species, or, alternatively, the individual growth rates of some species.
1 Introduction Consider the n-dimensional differential system x˙ = f (x)
(1)
We are going here to study this system for positive x only, i.e. the system is defined in a subset of the open positive n-dimensional orthant. Such situations, for example, arise in chemical or biological modeling (where the variables are constrained to be positive because they represent numbers or concentrations). Interestingly, some tools have been developed, that are deeply founded on the sign properties of the variables: let us cite the work on chemical kinetics of Feinberg, Horn and Jackson ([3]), of Clarke ([2]) and others (see the references in [2]). A more biological domain of application is population dynamics, where one models the way species interact together by competition, cooperation ...([13]). Some other tools have also been developed, that use the signs of the Jacobian matrix inside a region of the space. For example, there exist strong results when this matrix is off-diagonal positive, i.e. the species are cooperative L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 151-158, 2003. Springer-Verlag Berlin Heidelberg 2003
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(mutualistic) ([10, 14]). The fact that the elements of the Jacobian matrix are positive is strongly bound to the positivity of the variables. The positivity of the variables is therefore important; in some cases, when ”strong positivity” is demanded, it leads to convergence towards equilibrium ([8]). A famous related example is the generalized Lotka-Volterra (LV) systems in dimension n, that can have a complex behavior for n ≥ 3 ([11]) : x˙ i = xi (bi +
n X
Aij xj ) (i = 1, . . . , n)
(2)
j=1
The (n × n) real matrix A describes the quadratic interactions, and is usually supposed to be bijective (the equilibrium will be unique if it exists). The variables xi , standing for populations, are real non-negative. The positive orthant is invariant. Our aim in this paper is to give some results (related to [3]) concerning positivity, already obtained by the author ([5, 7, 6]). The tools used here are mainly changes of variables, tools from the usual linear algebra, results from positivity and theory of positive matrices ([1]) and auxiliary functions that decrease along the trajectories and Lasalle’s theorem ([12]. The typical result that we obtain under some hypotheses is that all the trajectories either converge towards some attractor set S containing the set of equilibria, or that they cannot remain in any compact set of the positive orthant; they have therefore limit points either at infinity or on the faces of the orthant, or they leave the orthant if it is not invariant. If, moreover, the set S is actually the set of equilibria, we deduce that the system cannot have a complex behavior (such as periodic solutions, recurrent trajectories, chaos, . . . ) in any compact set of the interior of the positive orthant. Notations: For x in Rn , we write x > 0 if xi > 0 (i = 1, . . . , n) and x ≥ 0 if xi ≥ 0 (i = 1, . . . , n). The closed positive orthant is Rn+ = {x ∈ Rn ; x ≥ 0}. We will frequently use the open positive orthant Pn = {x ∈ Rn ; x > 0}. Let us denote by t u the transpose of u, by ex the vector t (ex1 , . . . , exn ), and similarly for ln x. 1 is the vector t (1 . . . 1) and diag (x) the diagonal matrix with diagonal (x1 , x2 , . . . , xn ).
2 Polynomial positive systems The basic system will be in a polynomial form x˙ = P (x), where P is a vector polynomial in the variables xi , i = 1, . . . , n. More precisely: x˙i =
q X
aij vj (x1 , . . . , xn ) (i = 1, . . . , n)
(3)
j=1
where the aij are real and the vj (x1 , . . . , xn ) monomials of the form xβ1 1 . . . xβnn , where the βi are real; q is the number of distinct monomials.
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We study the system when x is positive (the system is defined in Pn ). We remark that the powers βj in the monomials can be real or negative: the system is always C ∞ in Pn . It is clear that this kind of polynomial system is very often met in the models. Let us make here an important remark: nearly all the biological models are in a rational fraction form (after reduction to the same denominator): P (x) where P and Q are two polynomials (Q is a real polynomial). It hapx˙ = Q(x) pens very frequently that, for x > 0, Q(x) > 0 (it is sufficient, for example, that Q have positive coefficients). Then we can multiply the right member by the positive number Q(x), and the new system has the same geometrical phase portrait ([11]). The new system is now in a polynomial form. It is easy to see we can write (3) into the form : x˙ = diag (x)Ae(B ln x)
(4)
where A is a n × p matrix and B a p × n matrix. We can do that by factoring xi in the ith equation. The elements of B are nothing but the powers βj or (βj − 1) of xi in the monomials; p is the number of distinct monomials in the system after factorization of xi . The change of variables (well defined because x > 0) y = ln x gives: y˙ = AeBy
(5)
We now study the equilibria of this system; let s = rank A. The equilibria are such that: y˙ = 0 ⇔ eBy ∈ ker A and therefore belong to the intersection, in a p-space, of ker A (a vector space of dimension (p − s)) and of a manifold (generated by eBy ) of dimension r = rank B (r ≤ (max(n, p))) and located in the positive orthant Pp . If ker A ∩ Pp = ∅, then there is no equilibrium. In particular, if A is injective, there is no equilibrium; this result does not depend on B. In this last case, we have shown that: Theorem 1 If ker A ∩ Pp = ∅, then all positive orbits of (5) have an unbounded closure and every solution eventually leaves every compact of Pn and never comes back. Let us recall the basis of the proof : if ker A ∩ Pp = ∅, then im t A ∩ Rp+ 6= {0}, and we can choose r = t Aq in this intersection. Let V (y) = t qy, then ˙ V (y) = t qAeBy = t reBy , where r is a nonnegative and non-zero vector. So this last expression never vanishes, and Lasalle’s theorem ([12]) gives us the first result. It means that, given an initial condition (and if ker A ∩ Pp = ∅), the orbit cannot remain in any compact; in particular, there is no complicated behavior (like periodic or recurrent solutions, chaos...) inside any compact; for the original system (4), it means that, given a positive initial condition, the orbit is unbounded or has points of the faces of the orthant as limit points,
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or leaves the orthant: that is, the positive orbit definitely leaves any compact set of the interior of the positive orthant. These results heavily depend on the number n and p of variables and distinct monomials; suppose that n ≥ p and that s = p; then A is injective and the condition ker A ∩ Pp = ∅ is trivially verified. We have therefore the simple following corollary: Corollary 1 If n ≥ p, then for almost any choice of the matrix A (if A is injective), all the trajectories of (5) are unbounded. This result does not depend on the matrix B. That is, the unboundedness of the trajectories does not depend on the precise form of the non-linear monomials, nor on the value of the elements of A, but only on the injectivity of A.
3 Lotka-Volterra systems As we have seen before, the most interesting case is n ≤ p. We make this hypothesis now, and we define z ∈ Rp : z = By to obtain z˙ = BAez
(6)
This is a change of variables for z ∈ im B. Let us remark that this system is quadratic homogeneous and analogous to a LV system but without the linear terms. In fact a LV system can be written into this form by taking one more state variable xn+1 with a zero dynamics. We have built tools to study such systems; we just give the results and refer to the papers ([7, 6]). If, given A and B, we can find an off-diagonal nonnegative and singular matrix M and two vectors k > 0 and w, such that t M 1 = 0, M k = 0 and ln k = Bw, and a symmetric matrix P , with P A = t BM , we will summarize these hypotheses by saying the system (A, B) admits a “PM-decomposition”. This property is a linear programming problem, and can be compute with finite algorithms. Theorem 2 If A and B admit a PM-decomposition, the trajectories of (5) are unbounded or converge towards the maximal invariant set included in the set of y such that: l X eBy = λj kj j=1
Here, the graph of M is made of l disconnected classes, associated with matrices Mj (j = 1, . . . , l), kj is a positive vector in ker Mj having the property that the components not corresponding to the vertices of the graph of Mj are zero, and the λj are real nonnegative.
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See [7, 6, 5] for the proof. Corollary 2 If A and B admit a PM-decomposition with ker P ∩ APp = {0} (in particular if P is bijective), then the trajectories of (5) are unbounded or converge towards the (non empty) set of equilibria. Indeed, because of the preceding theorem, the bounded trajectories converge towards the points y such that M eBy = 0 ⇒ P AeBy = 0, and with the hypothesis of the corollary, it implies that AeBy = 0, so y is an equilibrium. If the hypothesis of the corollary are fulfilled, the differential system has a rather simple behavior: a trajectory either goes towards the set of equilibria or is unbounded (it can happen that there is no equilibrium on the linear first integrals of the solution, in such a case the trajectory is unbounded). For the original system (3), it means that a trajectory either converges towards the set of equilibria, or has limit points at infinity or on the faces of the orthant, or leaves the orthant. Intuitively, the trajectory converges to equilibrium or “leaves” the interior of the orthant. Of course, such a trajectory can reenter the orthant later, and infinitely often (this behavior makes no sense when the variables have a biological meaning). For more details, see ([4]). These theorems applied to true Lotka-Volterra systems (2) give the following particular case. We can write a new n-dimensional quadratic homogeneous system: n X Bij xj ) (i = 1, . . . , n) x˙ i = xi ( j=1
with the initial condition x(0) satisfying xn (0) = 1, and ¶ µ Ab B= 0 0 We obtain the new system :
y˙ = Bey
(7)
(8)
and the theorem follows, giving the same results on the behavior as above: Theorem 3 Suppose that, for a given B, there exist a square off-diagonal nonnegative singular matrix M , such that t M 1 = 0, and a symmetric square matrix P such that P B = M . Then, if V (y) =
1 t (y − ln k)P (y − ln k) 2
with k ∈ ker B ∩ Pn , the function V (y) decreases along the trajectories of (8) ˙ (V (y(t)) ≤ 0). Moreover, if the graph of M is made of l disconnected classes, associated with matrices Mj (j = 1, . . . , l), then this derivative vanishes if and only if
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ey =
l X
λj k j
j=1
where kj is a positive vector in ker Mj having the property that the components not corresponding to the vertices of the graph of Mj are zero, and the λj are real nonnegative.
4 Stabilization of LV systems by a positive control Let us suppose now that the LV system (2) is subject to a control on the growth rates (the vector b). We can imagine, for example, that the growth rate of each species depends on one single factor that could be the lighting, the temperature, or the pollution of the ambient space . . . . The new system is now : n X Aij xj ) (i = 1, . . . , n) (9) x˙ i = xi (ubi + j=1
In this system the scalar control u is supposed to be positive ; it means that the external factor cannot change, for example, a death rate into a growth rate, which is quite realistic. The mathematical problem that we want to address here is the following: ” Given a reference equilibrium x∗ , u∗ , which turns out to be unstable, is it possible to stabilize the system around this equilibrium ?” The idea to stabilize the system is simple : we consider u as a new variable, and we choose its dynamic in order to verify the constraint of positivity and to stabilize the new system with the help of a Lyapunov function. The equation for u is chosen of the following form : n X ci xi + du) u˙ = u( i=1
where the coefficients ci , d, and the initial condition u(0) have to be chosen. The equilibrium gives us that n X
ci x∗i + du∗ = 0
i=1
Let us remark that u will stay positive if the initial control is positive, because the hyperplane u = 0 is invariant by this equation. The new state vector is now y = (x, u) and is a solution of the system : (after change of variables z = ln y) µ ¶ Ab 0 z 0 z˙ = A e ; A = t (10) cd
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We can also scale the equilibrium z ∗ = (ln x∗ , ln u∗ ) to 0 by a translation. We keep the same notation for A0 which verifies now A0 1 = 0. We want to choose the parameters c, d verifying t c1 + d = 0 such that the system (10) is globally stable. This system is very similar to the previous one and we could use the same theorem (3). We use here a different Lyapunov function, based on cooperativity. It is not difficult to obtain the result ([9]): Proposition 1 If the system (10) is such that n+1 X j=1
min A0ij > 0 i6=j
then the positive equilibrium is globally asymptotically stable. We apply this result to our control system to obtain a sufficient condition of global stabilization : Theorem 4 If the system (10) is such that n X j=1
min Aij + min bi > 0 i6=j
i
then there exists u positive stabilizing globally (9). Indeed, we can choose cj ≥ mini6=j Aij , and the resulting system will verify the above proposition. In fact, we can even obtain an explicit formulation of the feedback, because the system (10) has a first integral t qz = C, where q belongs to the kernel of t 0 A . Finally C = 0 because when x = 1, then u = 1. We obtain the following explicit form: Corollary 3 A smooth positive stabilizing feedback for the original system (9) is: n Y xi ( ∗ )−qi /qn+1 u = u∗ x i i=1 where q belongs to the kernel of t A0 .
5 Example Let us consider the following three-dimensional LV system : x˙ = x( 34 x − z − 31 u) y˙ = y(3x − 25 y − z + 21 u) z˙ = z(3x − 32 y − 25 z + u)
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For the reference control u = 1, the system has the equilibrium x = y = z = 1, which is unstable: indeed, the Jacobian matrix at the equilibrium is nothing but A, which has a positive eigenvalue. We add the following equation for u : u˙ = u(αx + βy + χz + δu) If we write the whole matrix (10), the criterion will be verified if we can find the coefficients such that: min(α, 3) + min(β, −3/2) + min(χ, −1) − 1/3 > 0 We can choose α = 3, β = χ = 0. The coefficient δ is determined so that u = 1 is an equilibrium. We obtain that u˙ = u(3x − 3u) stabilizes globally the system for positive variables. The kernel of t A0 is (−171/40, −27/20, 9/4, 1), and the explicit form for the positive feedback is therefore : u = x171/40 y 27/20 z −9/4 .
References 1. A. Berman and R.J. Plemmons, Nonnegative matrices in the mathematical sciences, Academic Press, New-York (1979) 2. B.L. Clarke, Stability of complex chemical reaction networks, Advances in Chemical Physics, 43, pp 1-216 (1980) 3. M. Feinberg, Complex balancing in general kinetic systems, Arch. Rational Mech. Anal., 49 (1972) pp 187-194 4. J.L. Gouz´e, Global behaviour of polynomial differential systems in the positive orthant, Research report Inria 1345 (1990) 5. J.L. Gouz´e. Transformation of polynomial systems in the positive orthant. In Kimura and Kodama, editors, Proceedings of the MTNS-91, pages 87–93. Kobe, Japon, MITA Press, 1992. 6. J.L. Gouz´e. Global behaviour of Lotka-Volterra systems. Mathematical Biosciences, 113:231–243, 1993. 7. J.L. Gouz´e. Global behaviour of polynomial differential systems in the positive orthant. In C. Perello, C. Simo, and J. Sola-Morales, editors, International Conference on Differential Equations, pages 561–567. Barcelone, aoˆ ut 91, World Scientific, 1993. 8. J.L. Gouz´e. Positivity, space scale, and convergence towards the equilibrium. Journal of Biological Systems, 3(2):613–620, 1995. 9. J.-L. Gouz´e, Dynamical behaviour of Lotka-Volterra systems. In Proceedings of the MTNS, CD-ROM. Perpignan, France, 2000. 10. M.W. Hirsch, Systems of differential equations which are competitive or cooperative II: Convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), pp 432-439 11. J. Hofbauer and K. Sigmund. The theory of evolution and dynamical systems. Cambridge University Press, 1988. 12. H. K. Khalil, Nonlinear systems. Macmillan Publishing Company, 1992. 13. J.D. Murray. Mathematical Biology. Springer-Verlag, 1990. 14. H. L. Smith. Systems of ordinary differential equations which generates an order preserving flow : A survey of results. SIAM Review, 30:87–113, 1988.
Positivity and Invariance Properties of Nonisothermal Tubular Reactor Nonlinear Models Mohamed Laabissi,1 Mohamed E. Achhab,1 Joseph J. Winkin2 and Denis Dochain3 1
2 3
Universit´e Chouaib Doukkali, Laboratoire d’Ing´enierie Math´ematique, Facult´e des Sciences, BP 20, El Jadida, Morocco, [email protected], [email protected], University of Namur (FUNDP), Department of Mathematics, 8 Rempart de la Vierge, B-5000 Namur, Belgium, [email protected] Universit´e Catholique de Louvain (UCL), CESAME, 4-6 avenue G. Lemaˆıtre, B-1348 Louvain-la-Neuve, Belgium, [email protected]
Abstract. The existence and uniqueness of the state trajectories (temperature and reactant concentration) and the existence and multiplicity of equilibrium profiles are analyzed for a nonisothermal axial dispersion tubular reactor model. It is reported that the trajectories exist on the whole (nonnegative real) time axis and the set of all physically feasible state values is invariant under the dynamics equations. The main nonlinearity in the model originates from the Arrhenius-type kinetics term in the model equations. The analysis uses Lipschitz and dissipativity properties of the nonlinear operator involved in the dynamics and the concept of state trajectory positivity. In addition, the multiplicity of the equilibrium profiles is reported: there is at least one steady state among the physically feasible states for such models, and conditions which ensure the multiplicity of equilibrium profiles are given.
1 Introduction The dynamics of tubular reactors are typically described by nonlinear partial differential equations (PDE), derived from mass and energy balance laws , see e.g. [3]. In the case of isothermal reactions, the dynamical analysis of the linearized tangent model of such systems has been carried out in [13], by using a C0 -semigroup Hilbert state-space formulation. Axial dispersion reactors and plug flow reactors have been studied there. However if the objective is e.g. to control the process, depending on the type of reactions, the nonlinearities may be such that they can not be neglected without a serious deterioration of the desired behavior of the system. It is then important to account for such nonlinearities as much as possible, especially in the case of nonisothermal reactors, where in addition the PDE’s may be highly coupled. This paper is L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 159-166, 2003. Springer-Verlag Berlin Heidelberg 2003
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dedicated to the dynamical analysis of a nonisothermal axial dispersion reactor using a nonlinear model. The state trajectories of the latter are studied by taking the nonlinear terms explicitly into account. The existence and the uniqueness of the temperature and reactant concentration profile trajectories on the whole time axis are reported, [4]. Here the approach is based on nonlinear functional analysis tools, as developed e.g in [6], see [4]. In particular, in the main result reported here, Lipschitz and dissipativity properties of the nonlinear operator involved in the dynamics generator are considered. This result is also based on the invariance of the domain of the nonlinear operator with respect to the C0 -semigroup generated by the linear part of the system. Note that a related nonisothermal reactor model has been considered in [2]. In addition, the multiplicity of the equilibrium profiles is reported for axial dispersion nonisothermal tubular reactors described by Arrhenius type nonlinear models. There is at least one steady state among the physically feasible states for such models. Moreover physically meaningful conditions which ensure the multiplicity of equilibrium profiles are reported, [5].
2 Nonlinear dynamical model The dynamics of an axial dispersion reactor for one nonisothermal reaction are given by the following mass and energy balance equations : ³ ´ µx1 ∂x1 ∂ 2 x1 ∂x1 = D − ν − k x + αδ(1 − x ) exp 1 ∂z 2 0 1 2 ∂t ∂z ´ 1+x1 ³ (1) 2 µx1 ∂x2 ∂x2 ∂ x2 = D + α(1 − x ) exp − ν 2 2 2 ∂t ∂z ∂z 1+x1 with the boundary conditions Di
∂xi (z = 0, t) − νxi (z = 0, t) = 0; i = 1, 2 ∂z ∂xi (z = 1, t) = 0; i = 1, 2 . Di ∂z
(2) (3)
Cin − C T − Tin is the normalized temperature and x2 = is the Tin Cin normalized concentration of the reactant. The index “in” holds for the values in the process inlet. Tin and Cin are constant reference values of the inlet temperature and reactant concentration, respectively. Here, we assume that the inlet temperature is equal to the heat exchanger temperature. It is experimentally observed that, for all z in [0, 1] and for all t ≥ 0, x1 (z, t) ≥ −1 and 0 ≤ x2 (z, t) ≤ 1 (see e.g. [3]). In addition, the real constants D1 , D2 , α, k0 , ν and µ are positive, whereas the constant δ is positive in case of exothermic reactions and negative in case of endothermic reactions. In line with [13], we consider the Hilbert space H = L2 [0, 1]×L2 [0, 1] with the usual inner product. If we define x(t) = (x1 (t), x2 (t))T , the state-space description is given by the following (abstract) differential equation on the Hilbert space H: Here x1 =
Nonisothermal Tubular Reactor Nonlinear Models
x(t) ˙ = Ax(t) + N (x(t)) , x(0) = x0 ∈ H ,
161
(4)
where A is the linear (unbounded) operator defined on its domain D(A) = {x = (x1 , x2 )T ∈ H such that x and dx dz ∈ H are absolutely continuous, dxi dxi d2 x d2 z ∈ H and Di dz (0) − νxi (0) = Di dz (1) = 0 ; i = 1, 2} by ∂ 2 x1 ∂x1 µ ¶ A1 x1 D1 ∂z 2 − ν ∂z − k0 x1 Ax = (5) = A x , ∂ 2 x1 ∂x2 2 2 D2 − ν ∂z 2 ∂z and the nonlinear operator N is defined on D := {(x1 , x2 )T ∈ H such that 0 ≤ x2 (z) ≤ 1 and x1 (z) ≥ −1, for almost all z ∈ [0, 1]}
(6)
by ¶ µ ¶¶T µ µ µx1 µx1 , α(1 − x2 ) exp . N (x) = αδ(1 − x2 ) exp 1 + x1 1 + x1
(7)
As in [13], it can be shown that the operators A1 and A2 are the infinitesimal generators of exponentially stable C0 -semigroups (T1 (t))t≥0 and (T2 (t))t≥0 ; whence by using standard arguments (e.g. [1, Lemma 3.2.2]), the linear operator given by (5) is the infinitesimal generator of the exponentially stable C0 -semigroup (T (t))t≥0 of bounded linear operators on H given by ¶ µ T1 (t) 0 . (8) T (t) = 0 T2 (t) Moreover, as the nonlinear operator N is locally Lipschitz continuous, equa¯ tion (4) has a unique mild solution on some interval [0, tmax ) , (tmax ∈ IR) Rt given by (see e.g. [9, p. 185-186]): x(t) = T (t)x0 + 0 T (t − s)N (x(s))ds , 0 ≤ t < tmax . In order to investigate the asymptotic behavior of the state trajectories, we need the existence of solutions on the whole interval [0, +∞).
3 Preliminary result Let X be a Hilbert space and (T (t))t≥0 a C0 -semigroup of linear operators such that k T (t) k≤ ewt , ∀t ≥ 0, for some w ∈ IR. Let A be the infinitesimal generator of (T (t))t≥0 and D be a closed subset of X. Assume also that N is a continuous function from D into X. Consider the following initial value problem: x(t) ˙ = Ax(t) + N (x(t)) , x(0) = x0 ∈ D . (9) Let I denote the identity operator on X. For y ∈ X, define the distance from y to D by d(y; D) = inf x∈D d(y, x) where d(y, x) denotes the distance induced
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by the norm of the Hilbert space X. The following result, proved in [6, p. 355], gives sufficient conditions for the existence and the uniqueness of the mild solution of system (9) on the whole interval [0, +∞). Theorem 3.1 Assume that (i) D is T (t)-invariant; i.e. T (t)D ⊂ D, for all t ≥ 0; (ii) for all x ∈ D, limh→0+ h1 d(x + hN (x); D) = 0 , and (iii) N is continuous on D and there exists lN ∈ IR+ such that the operator (N − lN I) is dissipative on D. Then (9) has a unique mild solution x(t, x0 ) on [0, +∞[, for all x0 ∈ D. Moreover (S(t))t≥0 defined on D by S(t)x0 = x(t, x0 ) for all t ≥ 0 and x ∈ D, is a nonlinear semigroup on D, with (A+N ) as its generator.
4 Positivity This section is devoted to the positivity of the semigroup corresponding to the linear part of the dynamics. Let us first recall some definitions related to the concept of positive semigroups (see e.g. [7], [11]). Let E be a real Banach lattice and E + be the positive cone that introduces in E a partial order relation defined for all x, y ∈ E by: x ≥ y if and only if x − y ∈ E + . Therefore E + := {x ∈ E : x ≥ 0}. Let Γ be a linear operator on E, then Γ is said to be a positive operator if Γ x ≥ 0 for all x ≥ 0 , i.e. Γ E + ⊂ E + . Definition 4.1 A familly of bounded linear operators (Γ (t))t≥0 on E is said to be a positive C0 -semigroup on E if (Γ (t))t≥0 is a C0 -semigroup on E and Γ (t) is a positive operator for all t ≥ 0. Note that L2 [0, 1] is a real Banach lattice whose positive cone is given by: L2 [0, 1]+ := {h ∈ L2 [0, 1] : h ≥ 0 a.e.} Let us consider the differential operator B on L2 [0, 1] given by : Bh =
d dh 1 (− (p ) + qh) w dz dz
(10)
where w(z), p(z), dp dz (z), q(z) are real continuous functions on the interval [0, 1] and p(z) > 0, w(z) > 0. The operator B is defined on the domain dh d2 h ∈ ∈ L2 [0, 1] are absolutely continuous, dz dz 2 dh dh L2 [0, 1], P1 h := ∆1 (0) − v1 h(0) = 0, P2 h := ∆2 (1) − v2 h(1) = 0}. (11) dz dz D(B) = {h ∈ L2 [0, 1] : h,
Assume that the real constants ∆1 , ∆2 , v1 , v2 verify | ∆1 | + | v1 |> 0 and | ∆2 | + | v2 |> 0. Therefore, if 0 is not in the spectrum of B then: ([1], p. 82) (B −1 h)(x) = where:
Z 0
1
g(x, y)h(y)w(y)dy
(12)
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−h1 (x)h2 (y) 0 ≤ x ≤ y ≤ 1 −h2 (x)h1 (y) 0 ≤ y ≤ x ≤ 1
(13)
½
W (0)p(0)g(x, y) =
and h1 and h2 are linearly independent solutions of : dp dh1 d 2 h1 − qh1 = 0 + dx2 dx dx 2 dp dh2 d h2 p 2 + − qh2 = 0 dx dx dx p
P1 h1 = 0, P2 h1 6= 0
(14)
P1 h2 6= 0, P2 h2 = 0
(15)
W (0) = h1 (0)
dh1 dh2 (0) − h2 (0) (0). dx dx
(16)
Let us now consider the particular case where, for some ∆ > 0 and λ > 0, −v −v p(z) = ∆ exp( −v ∆ z), w(z) = exp( ∆ z), and q(z) = λ exp( ∆ z). Then the operator B given by (10) reads : ˜ , Bh = (λI − A)h
(17)
where A˜ is defined on D(B) by : 2 ˜ = ∆ d h − v dh . Ah dz 2 dz
(18)
It is shown in [13] that the operator A˜ is the generator of an exponentially stable C0 -semigroup on L2 [0, 1], which is denoted by (Λ(t))t≥0 . Lemma 4.1 The semigroup (Λ(t))t≥0 is positive. Proof: For all λ > 0, 0 is not in the spectrum of B. Therefore B −1 h = ˜ ˜ is given by (12). Let us choose h1 (z) = exp(r1 z) − R(λ, A)h. Hence R(λ, A) r2 r2 r1 exp(r2 z) > 0 and h2 (z) = − r1 exp(r2 − r1 ) exp(r1 z) + exp(r2 z) > 0, where r1 =
√ v− v 2 +4∆λ 2∆
< 0 , r2 =
√ v+ v 2 +4∆λ 2∆
> 0 , and |r1 | < r2 . Therefore r2 r1 )[−1
r22 r12
exp(r2 − r1 )]. ˜ ≥ 0 for Now by (13), g(x, y) ≥ 0, which implies by using (12) that R(λ, A)h all h ≥ 0, for all λ > 0 . Finally use the exponential formula: for all t > 0, for ˜ nh ≥ 0 . all h ≥ 0 , Λ(t)h = limn→∞ [ nt R( nt , A)] h1 > 0 and h2 > 0. Observe that W (0) = r1 (1 −
+
In [4], the following inequality is proved, which will be needed below: Lemma 4.2 Let 1I denote the function identically equal to 1. Then, for all ˜ 1I ≤ 1I . λ > 0, λR(λ, A) Clearly the set D given by (6) is a closed convex subset of H and the nonlinear operator N , given by (7), is well defined and continuous on D. In order to be able to apply Theorem 3.1, it is needed to prove that D is T (t)-invariant. This is done in the following section by using the result below, which shows the positivity of the semigroup corresponding to the linear part of the dynamics.
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Lemma 4.3 The C0 -semigroup (T (t))t≥0 given by (8) is positive. Proof: The positivity of the C0 -semigroup (T (t))t≥0 is equivalent to the positivity of (T1 (t))t≥0 and (T2 (t))t≥0 , which can be proved by using Lemma 4.1 for specific values of the parameters in (11) and (17)-(18). See [4].
5 Invariance It is notably reported here that the set D of physically feasible state values is invariant under the dynamics (1)-(3), i.e. (4)-(7). The following result follows ˜ and Lemma 4.2. from the positivity of the resolvent operator R(λ, A) Lemma 5.1 For i = 1, 2 ,
λR(λ, Ai ) 1I ≤ 1I for all λ > 0.
(19)
We are now able to state the following invariance result. Lemma 5.2 Let D be the closed convex set given by (6). Then T (t)D ⊂ D for all t ≥ 0. Proof: This result follows directly from Lemmas 4.3 and 5.1, by using the exponential formula for a semigroup. See [4]. Theorem 5.1 [4] For every x0 ∈ D, equation (4) has a unique mild solution x(t, x0 ) on the interval [0, +∞[. Moreover, if we set S(t)x := x(t, x0 ), then (S(t))t≥0 is a strongly continuous nonlinear semigroup on D, generated by the operator A + N . Proof: The proof consists of checking the conditions of Theorem 3.1. The first one is stated in Lemma 5.2. The proofs of the two other conditions can be found in [4], where it is shown that the third condition holds with respect to an equivalent norm for which (T (t))t≥0 is a semigroup of contractions.
6 Equilibrium profiles By using notably Lemma C and Theorem D in [10], the existence of at least one equilibrium profile is proved in [5] for the model (1)-(3). Theorem 6.1 [5] The axial dispersion nonisothermal tubular reactor with nonlinear model, given by (1)-(3), i.e. (4)-(7), has at least one equilibrium profile in D, i.e. the functional equation Ax + N (x) = 0 ,
x = (x1 , x2 )T ∈ D ∩ D(A) ,
(20)
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which is equivalent to the following equations, admits at least one solution: µx dx d2 x D1 d2 z1 − ν dz1 − k0 x1 + αδ(1 − x2 ) exp( 1+x11 ) = 0, 2 µx1 2 D2 dd2xz2 − ν dx dz + α(1 − x2 ) exp( 1+x1 ) = 0, x = (x1 , x2 )T ∈ D ∩ D(A). Observe that Theorem 6.1 holds independently of the fact that the reactions are endothermic (δ < 0) or exothermic (δ > 0). The analysis of the multiplicity of the equilibrium profiles performed in [5] is mainly based on Lemma 2.1 in [12, p. 442]. The corresponding result can be stated as follows: see Theorem 6.2 below. Let us define the functions φi i = 1, 2, the functional η, and the constant $ as follows: φi = R(0, Ai ) 1I,
i = 1, 2,
(21)
η : X + → [0, +∞) x → inf x1 (z),
(22)
z∈[0,1]
0 < $ = η(φ) = inf φ1 (z) , z∈[0,1]
φ = (φ1 , φ2 ).
(23)
It is clear that η is a continuous concave functional. By a simple variational analysis, one can easily prove that, if µ > 4, the function v(t) = √ at τ1 (µ) = t exp( −µt 1+t ) , t ≥ 0, has a local maximum √ (µ−2)+ µ2 −4µ . local minimum at τ2 (µ) = 2
µ2 −4µ
(µ−2)−
2
and a
Theorem 6.2 [5] For positive parameters α, D1 , D2 , ν, δ, k0 , µ such that µ > 4 and αeµ < 1, the axial dispersion nonisothermal tubular reactor nonlinear model, given by (1)-(3), i.e. (4)-(7), has at least three equilibrium profiles provided that $−1 v(τ2 (µ)) < αδ(1 − αeµ ) < α max(δ, 1)
(24)
where the function v(·) and the real numbers τ1 (µ) and τ2 (µ) are defined as above, the functions φi i = 1, 2 are given by (21) and the constant $ is defined by (22)-(23).
7 Concluding remarks The analysis performed here is a preliminary step of a study dedicated to the design of controllers for axial dispersion nonisothermal reactors based on the nonlinear PDE model of the process, see e.g. [8]. An important open question is the stability of the equilibrium profiles for this model. Acknowledgment. This paper presents research results of the Belgian Programme on Inter-University Poles of Attraction initiated by the Belgian State,
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Prime Minister’s office for Science, Technology and Culture. The scientific responsibility rests with its authors. The work of the two first named authors has been partially carried out within the framework of a collaboration agreement between CESAME (UCL) and LINMA of the Faculty of sciences (UCD), funded by the Secretary of the State for Development Cooperation and by the CIUF (Conseil Interuniversitaire de la Communaut´e Fran¸caise de Belgique). The work of the first and third named authors was partially supported by a Research Grant (2001-2003) from the Facult´es Universitaires Notre-Dame de la Paix at Namur (Fonds Sp´ecial de Recherche de la Communaut´e Fran¸caise de Belgique).
References 1. R.F. Curtain and H.J. Zwart, “An introduction to infinite-dimensional linear systems theory”, Springer Verlag, New York, 1995. 2. K. Deimling, “Nonlinear Functional Analysis”, Springer Verlag, Berlin, 1985. 3. D. Dochain, “Contribution to the Analysis and Control of Distributed Parameter Systems with Application to (Bio)chemical Processes and Robotics”, Th`ese d’Agr´egation de l’Enseignement Sup´erieur, Universit´e Catholique de Louvain, Louvain-la-Neuve, Belgium, 1994. 4. M. Laabissi, M.E. Achhab, J.J. Winkin, D. Dochain, Trajectory Analysis of Nonisothermal Tubular Reactor Nonlinear Models, Systems & Control Letters, Vol. 42, 2001, pp. 169-184. 5. M. Laabissi, M.E. Achhab, J.J. Winkin, D. Dochain, Multiple Equilibrium Profiles for Nonisothermal Tubular Reactor Nonlinear Models, FUNDP Namur, Department of Mathematics, Internal Report No 2002/13; submitted. 6. R.H. Martin, “Nonlinear operators and differential equations in Banach spaces”, John Wiley & Sons, New York, 1976. 7. R. Nagel (ed.) “One-parameter Semigroups of Positive Operators”, Lecture notes in mathematics, 1184, Springer-Verlag, Berlin, 1986. 8. Y. Orlov and D. Dochain, ”Discontinuous feedback stabilization of minimumphase semilinear infinite-dimensional systems with application to chemical tubular reactor”, IEEE Transactions on Automatic Control, vol. 47, No 8, 2002, pp. 1293-1304. 9. A. Pazy, “Semigroups of Linear Operators and Applications to Partial Differential Equations”, Springer Verlag, Berlin, 1983. 10. J. Pr¨ uß, Equilibium Solutions of Age-specific Population Dynamics of Several Species, J. Math. Biology, 11, pp. 65-84, (1981). 11. H. Scheafer, “Banach Lattices and Positive Operators”, Springer-Verlag, Berlin, 1974. 12. K. Taira and K. Umezu, Semilinear elliptic boundary problems in chemical reactor theory, Journal of Differential Equations 142, 434-454, (1998). 13. J. Winkin, D. Dochain, P. Ligarius, “Dynamical analysis of distributed parameter tubular reactors”, Automatica, vol . 36, 2000 ; pp. 349-361.
A Feedback Perspective for Chemostat Models with Crowding Effects Patrick De Leenheer1 , David Angeli2 , and Eduardo D. Sontag3 1 2 3
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA, [email protected] Dip. di Sistemi e Informatica Universit´ a di Firenze, Via di S. Marta 3, 50139 Firenze, Italy, [email protected] Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA, [email protected]
Abstract. This paper deals with an almost global stability result for a chemostat model with including effects. The proof relies on a particular small-gain theorem which has recently been developed for feedback interconnections of monotone systems.
1 Introduction The chemostat is a well-known model used to describe the interaction between microbial species which are competing for a single nutrient, see [12] for a review. One of the prominent results in this area is the so-called ’competitive exclusion principle’ which states roughly that in the long run only one of the species survives. This is in contrast to what is observed in nature where several species seem to coexist. This discrepancy has lead to modifications of the model to try and bring theory and practice in better accordance; see [14, 3, 9, 7]. Recently the chemostat has been made coexistent by means of feedback control of the dilution rate [4]. In this paper we propose another modification of the chemostat model: x˙ i = xi (fi (S) − Di − ai xi ) n X xi fi (S) S˙ = 1 − S −
(1)
i=1
where i = 1, 2, ..., n, xi is the concentration of species i and S is the nutrient concentration. The positive parameters Di are the sum of the (natural) death rates of species i and the dilution rate, while the positive parameters ai give rise to death rates ai xi which are due to crowding effects. Throughout this paper we will assume the following: L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 167-174, 2003. Springer-Verlag Berlin Heidelberg 2003
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fi : R+ → R+ is continuously differentiable Moreover the functions fi are globally Lipschitz continuous on R+ with Lipschitz constants Li . The classical Monod function f (S) = M S/(b + S) with b, M > 0 satisfies these assumptions with global Lipschitz constant M/b. The only difference with the classical chemostat model [12] is that here crowding effects -modeled by the ai - are taken into consideration. Our main result is the following: Theorem 1. If
Li ). max (fi (1)) < 1 (2) i ai then there exists an equilibrium point E ∗ of system (1) such that every solution ξ(t) = (x1 (t), x2 (t), ..., xn (t), S(t))T starting in {(x1 , x2 , ..., xn , S)T ∈ ∗ Rn+1 + | xi > 0, ∀ i = 1, ..., n} converges to E . n . max ( i
Notice that our main result does not guarantee coexistence since the equilibrium point E ∗ could belong to the boundary of Rn+1 and correspond to the + absence of one of the species. However, in the sequel we will provide conditions that do imply coexistence. The proof of our main result is based on the observation that system (1) can be written as a negative feedback interconnection of monotone subsystems and the availability of a particular small-gain theorem for such feedback systems. To see this, let us first introduce some notation. Define x = (x1 , x2 , ..., xn )T , f (S) = (f1 (S), f2 (S), ..., fn (S))T , D = (D1 , D2 , ..., Dn )T and a = (a1 , a2 , ..., an )T . System (1) can then be compactly rewritten as follows: S˙ = 1 − S + f T (S)u1 , y1 = S x˙ = diag(x)(f (u2 ) − D − diag(a)x), y2 = x u1 = −y2 , u2 = y1
(3) (4) (5)
System (3) − (5) is a negative feedback system consisting of two input/output (I/O) subsystems (3) and (4) with inputs u1 , respectively u2 and outputs y1 , respectively y2 . The development of a theory for monotone I/O systems has recently been initiated in [1]. One of its purposes is to extend the rich theory of monotone dynamical systems developed by Hirsch [8], see [11] for a review and [11, 1, 6, 10] for applications in biology. For biological applications of monotone I/O systems see [5] and the use of small-gain theorems in biology see [13].
2 Preliminaries and proofs 2.1 Monotone I/O systems and a small-gain theorem The material in this section can be found in a far more general setting in [1, 2]. We restrict to a framework that serves our purposes, namely I/O systems
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described by differential equations. Consider the following I/O system: x˙ = f (x, u), y = h(x)
(6)
where x ∈ Rn is the state, u ∈ U ⊂ Rm the input and y ∈ Y ⊂ Rp the output. It is assumed that f and g are smooth (say continuously differentiable) and that the input signals u(t) : R → U are Lebesgue measurable functions and locally essentially bounded (i.e. for every compact time interval [Tm , TM ], there is some compact set C such that u(t) ∈ C for almost all t ∈ [Tm , TM ]). This implies that solutions with initial states x0 ∈ Rn are defined for all inputs u(.) and will be denoted by x(t, x0 , u(.)), t ∈ I where I is the maximal interval of existence for this solution. From now on we will assume that a fixed set X ⊂ Rn is given which is forward invariant, i.e. for all inputs u(.) and for every x0 ∈ X it holds that x(t, x0 , u(.)) ∈ X, for all t ∈ I ∩ R+ . Henceforth initial conditions are restricted to this set X. For our purposes X will be Rn+ or R+ and U will be R+ or −Rn+ . We denote the usual partial order on Rn by ¹, i.e. for x, y ∈ Rn , x ¹ y means that xi ≤ yi for i = 1, ..., n. The state space X (input space U , output space Y ) inherits the partial order from Rn (Rm , Rp ) as the former sets are subsets of the latter ones. Similarly, the partial order on Rm carries over to the set of input signals in a natural way (hence we use the same notation for the partial order on this latter set): u(.) ¹ v(.) if u(t) ¹ v(t) for almost all t ≥ 0. The next definition introduces the concept of a monotone I/O system which, loosely speaking means that ordered initial conditions and input signals lead to subsequent ordered solutions. Definition 1. The I/O system (6) is monotone (with respect to the usual partial orders) if the following conditions hold: x1 ¹ x2 , u(.) ¹ v(.) ⇒ x(t, x1 , u(.)) ¹ x(t, x2 , v(.)), ∀ t ∈ (I1 ∩ I2 ) ∩ R+ . (7) and
h is a monotone map, i.e. x1 ¹ x2 ⇒ h(x1 ) ¹ h(x2 ).
(8)
Of particular interest is how an I/O system behaves when it is supplied with a constant input. Next we introduce a notion which implies that this behavior is fairly simple [2]. Definition 2. Assume that X has positive (Lebesgue) measure. The I/O system (6) possesses an Input/State (I/S) quasi-characteristic k : U → X if for every constant input u ∈ U (and using the same notation for the corresponding u(.)), there exists a set of (Lebesgue) measure zero Bu such that: ∀x0 ∈ X \ Bu : lim x(t, x0 , u) = k(u) t→+∞
(9)
If system (6) possesses an I/S quasi-characteristic k then it also possesses an Input/Output (I/O) quasi-characteristic g : U → Y defined as g := h ◦ k.
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Next we recall the main tool (see [2]) for proving our main result. In the following statement we use the concept of an almost globally attractive equilibrium point of an autonomous system, which means that there exists an equilibrium point of this system which attracts all solutions which are not initiated in a certain set of (Lebesgue) measure zero. Theorem 2. Consider the following two I/O systems: x˙ 1 = f1 (x1 , u1 ), x˙ 2 = f2 (x2 , u2 ),
y1 = h1 (x1 ) y2 = h2 (x2 )
(10) (11)
where xi ∈ Xi ⊂ Rni , ui ∈ Ui ⊂ Rmi and yi ∈ Yi ⊂ Rpi for i = 1, 2. Suppose that Y1 = U2 and Y2 = −U1 and that the I/O systems are interconnected through a (negative) feedback loop: u2 = y1 , u1 = −y2
(12)
Assume that: 1. Both I/O systems (10) and (11) are monotone. 2. Both I/O systems (10) and (11) possess continuous I/S quasi-characteristics k1 and k2 respectively (and thus also I/O quasi-characteristics g1 and g2 ). 3. All forward solutions of the feedback system (10) − (12) are bounded. Then the feedback system possesses an almost globally attractive equilibrium point (¯ x1 , x ¯2 ) ∈ X1 × X2 if the following discrete-time system, defined on U2 : uk+1 = (g1 ◦ (−g2 ))(uk )
(13)
possesses a globally attractive fixed point u ¯ ∈ U2 . In that case (¯ x1 , x ¯2 ) = ((k1 ◦ (−k2 ))(¯ u), k2 (¯ u)). In the sequel we will refer to this result as a small-gain theorem and to the last condition as a small-gain condition. 2.2 Properties of the full system and both subsystems is a forward invariant set for system (1) and the solutions Lemma 1. Rn+1 + initiated in this set remain bounded. (Sketch of proof) The first claim follows from e.g. Theorem Pn 3 in [1]. The second claim follows from the fact that for V (x, S) = S + i=1 xi , we have V˙ ≤ ∗ 1 − D∗ V with D∗ = min(1, D1 , ..., Dn ) and hence V (t) ≤ V (0) e−D t +1/D∗ . Next we investigate the I/O-properties of the subsystems (3) and (4) which have the following input, state and output spaces. S˙ = 1 − S + f T (S)u1 y1 = S
(14)
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where S ∈ X1 := R+ denotes the state, u1 ∈ U1 := −Rn+ denotes the input and y1 ∈ Y1 := R+ denotes the output. The input signals u1 (t) : R → U1 are assumed to Lebesgue be measurable and essentially locally bounded, ensuring existence and uniqueness of solutions as discussed in the previous subsection. Similarly consider x˙ = diag(x)(f (u2 ) − D − diag(a)x) y2 = x
(15)
where x ∈ X2 := Rn+ denotes the state, u2 ∈ U2 := R+ denotes the input and y2 ∈ Y2 := Rn+ denotes the output. As before, input signals u2 (t) : R → U2 are Lebesgue measurable and essentially locally bounded. Lemma 2. X1 , respectively X2 , is forward invariant for system (14), respectively system (15). Proof. The proof follows from an application of Theorem 3 in [1]. Lemma 3. Systems (14) and (15) are monotone. Proof. This follows from an application of Proposition 3.3 in [1]. The next result is the key to proving the main theorem and reveals that both subsystems possess I/S quasi-characteristics with certain smoothness properties. Lemma 4. System (14) possesses a continuously differentiable I/S quasicharacteristic k1 : U1 → X1 . Moreover, k1 is globally Lipschitz with Lipschitz constant L∗1 := n . maxi=1,...,n fi (1), i.e. ∀ ua1 , ub1 ∈ U1 : |k1 (ua1 ) − k1 (ub1 )| ≤ L∗1 ||ua1 − ub1 ||max
(16)
where ||.||max denotes the max-norm on Rn , i.e. ||z||max = maxi=1,...,n |zi | when z ∈ Rn . System (15) possesses a globally Lipschitz continuous I/S quasi-characteristic k2 : U2 → X2 with Lipschitz constant L∗2 := maxi=1,...,n Li /ai , i.e. ∀ ua2 , ub2 ∈ U2 : ||k2 (ua2 ) − k2 (ub2 )||max ≤ L∗2 |ua2 − ub2 |
(17)
Proof. Due to space limitations we leave out the proof of this result. It will be included in an extended version of this paper. Remark 1. Notice that the output spaces Y1 , Y2 of systems (14) and (15) are identical to their respective state spaces X1 , X2 and that the output mappings h1 and h2 are just the identity mappings. Therefore the I/O quasicharacteristics g1 and g2 of these systems equal their respective I/S quasicharacteristics and possess the same smoothness properties.
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2.3 Proof of the main result Consider system (1) or its equivalent feedback representation (3) − (5). We will show that the three conditions and the small-gain condition in theorem 2 are satisfied. The first, second and third conditions follow from respectively lemma 3, lemma 4 and lemma 1. To see that small-gain condition is satisfied, recall from lemma 4 and remark 1 that g1 = k1 and g2 = k2 are globally Lipschitz with Lipschitz constants L∗1 , respectively L∗2 . This implies that for all ua , ub ∈ U2 , the composition g := g1 ◦ (−g2 ) satisfies the following: |g(ua ) − g(ub )| ≤ L∗1 ||(−g2 )(ua ) − (−g2 )(ub )||max ≤ L∗1 L∗2 |ua − ub | which by the definitions of L∗1 and L∗2 (see lemma 4) and condition (2) shows that g is a contraction mapping on the complete metric space U2 = R+ . Then a contraction mapping argument shows the small-gain condition is indeed satisfied, which concludes the proof of this theorem.
3 Coexistence for 2 species In this section we provide a coexistence result for system (1) with n = 2. A coexistence result in case of n species is deferred to an extended version of this paper. Definition 3. System (1) with n = 2 is coexistent if there exists some ² > 0 such that for i = 1, 2 holds: lim inf xi (t) > ² whenever x1 (0) > 0 and x2 (0) > 0 t→∞
where (x1 (t), x2 (t), S(t))T denotes the solution of system (1) with initial condition (x1 (0), x2 (0), S(0))T ∈ R3+ . In fact we will prove the much stronger result that coexistence takes the form of a globally attracting interior equilibrium point. This contrasts the competitive exclusion principle which holds for the classical chemostat model. Since crowding effects are the only difference between the classical chemostat and the model presented here, this suggests they may be responsible for the observed coexistence of several species competing for a single nutrient. We make the following additional -but fairly natural; see [12]- assumptions: • H1 fi (S1 ) < fi (S2 ) if S1 < S2 , where S1 , S2 ∈ R+ and i = 1, 2. • H2 For i = 1, 2 there exist numbers λi ∈ (0, 1) satisfying fi (λi ) − Di = 0. Notice that if H1 holds, then the numbers λi , i = 1, 2 are unique. It is noteworthy that the numbers λi are independent of the ai , i = 1, 2. For i = 1, 2, we define the functions Fi : R+ → R as follows: Fi (S) = 1 − S −
fi (S) − Di fi (S) for i = 1, 2 ai
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Obviously, both functions Fi (S) are continuously differentiable. Claim: If H1 and H2 are satisfied, then there exist unique roots λ∗i ∈ R+ such that Fi (λ∗i ) = 0 for i = 1, 2. In addition, λ∗i ∈ (λi , 1) for i = 1, 2. (The proof of this claim is deferred to an extended version of this paper) A final additional and non-trivial assumption is expressed in terms of these roots λ∗i : • H3 max(λ1 , λ2 ) < min(λ∗1 , λ∗2 ). Later we will impose H1, H2 and H3 on system (1) with n = 2, so it is important to know whether these assumptions can be satisfied simultaneously. The next lemma shows that this can always be arranged by choosing the crowding effect parameters a1 and a2 large enough. The proof is left out and will be included in an extended version of this paper. Lemma 5. Assume that two uptake functions f1 , f2 and two numbers D1 and D2 are given such that H1 and H2 hold. Interpret both a1 and a2 as variables in int(R+ ). Then for i = 1, 2, the λ∗i are differentiable functions of ai taking values in (λi , 1): λ∗i : int(R+ ) → (λi , 1) and lim λ∗i (ai ) = 1 ai →∞
In particular, this implies that for a∗i large enough, H3 is satisfied. Under the 3 additional assumptions it turns out that system (1) with n = 2, possesses 4 equilibria in R3+ . Exactly one of these equilibria lies in int(R3+ ) and is locally asymptotically stable as we show next. Again, the proof is deferred to an extended version of this paper. Lemma 6. If H1, H2 and H3 are satisfied, then system (1) with n = 2 possesses the following equilibria in R3+ : E0 = (0, 0, 1)T , E1 = (x∗1 , 0, λ∗1 )T , E2 = (0, x∗2 , λ∗2 )T and Ee = (xe1 , xe2 , λe )T where x∗1 , x∗2 , xe1 , xe2 and λe are positive numbers. The equilibrium point Ee is locally asymptotically stable. The previous lemma and our main result suggest a mechanism to achieve coexistence for system (1) with n = 2: Suppose that it is possible to satisfy both the small-gain condition (2) and the three conditions expressed by H1, H2 and H3. Then lemma 6 guarantees the existence of a locally asymptotically stable equilibrium point E e ∈ int(R3+ ), while Theorem 1 ensures the existence of an equilibrium point for system (1) with n = 2 which attracts almost every solution initiated in R3+ . Obviously this equilibrium point must be E e . It can be shown that the set of non-converging initial conditions (note that although they are not converging to E e , they might be converging to other equilibria) is: B = {(x1 , x2 , S)T ∈ R3+ | x1 = 0 or x2 = 0}
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In particular, this implies that all solutions initiated in P := {(x1 , x2 , S) ∈ R3+ | x1 > 0, x2 > 0} do converge to E e and consequently that system (1) with n = 2 is coexistent. The main problem is thus whether the small-gain condition (2) and H1, H2 and H3 can be satisfied simultaneously for system (1) with n = 2. But from lemma 5 and (2) is follows that this is possible if crowding effects are large enough. Combining theorem 1 and lemma 6 we conclude: Theorem 3. Assume that two uptake functions f1 , f2 , two numbers D1 and D2 are given such that H1 and H2 hold. Consider system (1) with n = 2 and interpret the ai , i = 1, 2 as positive parameters. If the ai are chosen large enough then H3 and (2) are satisfied. Then system (1) with n = 2 possesses an equilibrium point E e ∈ int(R3+ ) which is almost globally asymptotically stable with respect to initial conditions in R3+ . Moreover, every solution initiated in P converges to E e implying that system (1) with n = 2 is coexistent.
References 1. D. Angeli and E.D. Sontag, Monotone control systems, arXiv.org math.OC/0206133 and submitted (Prelim. vers. in Proc. 41st CDC, 2002) 2. D. Angeli, P. De Leenheer and E.D. Sontag, A small-gain theorem for almost global convergence of monotone systems, in preparation. 3. G.J. Butler, S.B. Hsu, and P. Waltman, A mathematical model of the chemostat with periodic washout rate, SIAM J. Appl. Math. 45 (1985) 435-49. 4. P. De Leenheer and H.L. Smith, Feedback control for chemostat models, J. Math. Biol. 46, 48-70 (2003). 5. P. De Leenheer, D. Angeli and E.D. Sontag, On predator-prey systems and small-gain theorems, submitted. 6. P. De Leenheer and H.L. Smith, Virus dynamics: a global analysis, to appear in SIAM J. Appl. Math. 7. J.K. Hale and A.S. Somolinas, Competition for fluctuating nutrient, J. Math. Biol. 18 (1983) 255-80. 8. M. Hirsch, Systems of differential equations which are competitive or cooperative II: convergence almost everywhere. SIAM J. Math. Anal. 16, 423-439 (1985) 9. S.B. Hsu, A competition model for a seasonally fluctuating nutrient, J. Math. Biol. 9 (1980) 115-32. 10. M.Y. Li and J.S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci. 125, 155-164, 1995. 11. H.L. Smith, Monotone Dynamical Systems, AMS, Providence, 1995. 12. H.L. Smith and P. Waltman, The theory of the chemostat, Cambridge University Press, Cambridge, 1995. 13. E.D. Sontag, Asymptotic amplitudes and Cauchy gains: A small-gain principle and an application to inhibitory biological feedback, Systems Control Lett. 47, 167-179 (2002). 14. G. Stephanopoulos, A.G. Fredrickson, R. Aris, The growth of competing microbial populations in CSTR with periodically varying inputs, Amer. Instit. of Chem. Eng. J. 25 (1979) 863-72.
Positive Control for a Class of Nonlinear Positive Systems Ludovic Mailleret Comore, Inria, bp 93, 06 902, Sophia Antipolis, France, [email protected] Abstract. In this contribution, we focus on a class of nonlinear positive systems, arising especially in biological processes. These are SISO input-affine systems with a nonlinear measurable drift. We develop a positive output feedback control law for these systems and prove the global asymptotic stability of an equilibrium point for the closed loop systems, that can be set on a model-dependent surface. Finally, we consider a special case of such systems, a biological waste water treatment plant. Real life experiments illustrate our approach and show its interest for industrial bio-processes.
1 Introduction Our goal in this contribution is to globally stabilize, with a positive control, some nonlinear positive systems in Rn towards a positive equilibrium, that can be tuned in some way. First, we consider “almost” linear positive systems of the form: x˙ = u(Kx + L) + M φ(x) In a very same way, we will then consider “almost” cooperative systems (that includes our first case) of the form: x˙ = u(f (x) + L) + M φ(x) Further details and hypotheses will be given in the following. Since we will only assume very loose hypotheses on the function φ(.), the systems can have very complex dynamics. Our aim is to stabilize the systems, using techniques related to positivity [2, 8]. Finally, we propose a real-life example of such systems: a biological anaerobic waste water treatment plant (WWTP). Real-life experiments show the interest and efficiency of the proposed regulation procedure.
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Notations In the sequel we will deal with positive systems, and positive cooperative systems. Then, as other authors [5, 8, 3, 2], we will use the following notations for Rn vectors: x ≤ y ⇔ ∀i, xi ≤ yi x ¿ y ⇔ ∀i, xi < yi 1l is the (n × 1) vector: (1, 1, ..., 1)T
2 “Almost” linear positive systems The properties of linear positive systems are central to this section; see [5, 2] for a survey of results. 2.1 The system We will consider a class of SISO (u, input, y, output) nonlinear dynamical systems in Rn of the form: x˙ = u(Kx + L) + M φ(x) x(t0 ) À 0 (1) y = φ(x) Matrix K is a (n × n) matrix, L and M are (n × 1) vectors, u and y = φ(x) are scalars. Moreover, we assume the following on system (1). Hypotheses (H1): a: K is a stable Metzler matrix (off-diagonal non-negative and stable) b: φ(.) is C 1 such that: ∀i ∈ [1..n], Mi φ(x|xi =0 ) ≥ 0 and ∀x À 0, φ(x) > 0. c: L ≥ 0 d: ∃βm > 0, ∀β ≥ βm , βL + M À 0 e: the input u is non-negative First, we check that, under hypotheses (H1), system (1) is a positive system (its state variables remain non-negative). To guarantee this property, according to [2, 5], we only have to check that ∀i ∈ [1..n], x˙i (xi = 0) ≥ 0: X Ki,j xj ) + Mi φ(x|xi =0 ) (2) x˙i |xi =0 = u(Li + j6=i
This quantity is non-negative since K is a Metzler matrix (H1a), L and u are non-negative (H1c,e) and Mi φ(x|xi =0 ) is non-negative (H1b). ¤
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2.2 Positive output feedback control Theorem 1. The positive output feedback control law: u = γy = γφ(x) with γ ≥ βm > 0
(3)
ensures that the closed loop system (1) will have a single strongly positive equilibrium point x? . Moreover, x? is globally asymptotically stable on the positive orthant and such that: x? = −K −1 (L +
1 M) γ
Proof. First note that the control u = γφ(x) fulfills hypothesis (H1e). Now, we consider the closed loop system. The control law (3) leads to the following dynamical system: ½ x˙ = φ(x)[γKx + (γL + M )] (4) x(t0 ) À 0 First, we show that the state variables remain positive for system (4). Consider a small ² > 0; we focus on the dynamics of xi (∀i ∈ [1..n]) near 0 (i.e. at xi = ²), as the state is in the cone x ≥ ².1l. Then according to system (4), we have: x ≥ ².1l ³ ´ X x˙ = φ(x) γK ² + γ Ki,j xj + γLi + Mi i i,i | {z } |{z} | {z } (5) j6=i >0 >0 ≥0 or ≤0 | {z } P ≥γ
j6=i
Ki,j ²≥0
We summarized the implications of hypotheses (H1) on system (5). Since the state variables are positive, φ(x) is positive (H1b). Now consider the terms inside the brackets. Since matrix K is a Metzler matrix (H1a), Ki,i is not of fixed sign, while the off-diagonal terms are non-negative. Last term positivity holds since γ ≥ βm (H1d). Then it is straightforward that, for a small enough positive ², the bracketted expression will be positive and therefore that we have: for ² small enough x˙i |xi =² > 0 Then xi is lower bounded by a positive ². Since our reasoning holds for all i ∈ [1..n], using (H1b) we prove that the function φ(x) is lower bounded by a positive constant. Since φ(x) does not cancel we are able to make the time change (see e.g. [4]): Z t φ(x(τ ))dτ t0 = 0
0
We express system (4) with t as the new time unit, we have:
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(
dx = γKx + (γL + M ) dt0 0 x(t0 ) À 0
(6)
We know that system (4) and system (6) have the same orbits and therefore have the same asymptotic behavior. The dynamical system (6) is linear, the vector (γL + M ) is strongly positive (since (H1d) holds and γ ≥ βm ) and the matrix γK is a stable Metzler matrix (H1a). Then according to [5], it is straightforward that the matrix γK is invertible, that (γK)−1 is a positive matrix (∀i, j, (γK)−1 i,j ≥ 0) and that the point: x? = −K −1 (L +
1 M) À 0 γ
is GAS for system (6). The control law (3) globally stabilizes system (4) towards the strongly positive set point: x? = −K −1 (L +
1 M ) with γ ≥ βm γ
¤
Remark 1 It is important to notice that it is possible to change the gain γ in order to tune the equilibrium point x? , provided its desired value belongs to the set Γ = {x = −K −1 (L + γ1 M ), γ ≥ βm }
3 “Almost” cooperative positive systems The properties of cooperative systems are central to this section; see [8] for a survey of results. 3.1 The system In this section we will consider a class of SISO (u, input, y, output) nonlinear dynamical systems in Rn of the form: x˙ = u(f (x) + L) + M φ(x) x(t0 ) À 0 (7) y = φ(x) Function f (x) ∈ Rn , L and M are (n × 1) vectors, u and y = φ(x) are scalars. Moreover, we assume the following on system (7). Hypotheses (H2): a: L ≥ 0 b: ∃βm > 0, ∀β ≥ βm , βL + M ¡À 0 ¢ ∂fi is off-diagonal non-negative c: f (.) is C 1 such that Df (x) = ∂x j ∀i,j d: if 0 ≤ y ≤ x, then ∀i, j Df (y)i,j ≥ Df (x)i,j (concavity of f (.))
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e: f (0) ≥ 0 and ∃λm > 0, ∀β ≥ βm , (f (λm .1l) + (L + β1 M )) ¿ 0 f: φ(.) is C 1 such that: ∀i ∈ [1..n], Mi φ(x|xi =0 ) ≥ 0 and ∀x À 0, φ(x) > 0 g: u is non-negative We check that, with hypotheses (H2), system (7) is a positive system: x˙i|xi =0 = u(fi (x|xi =0 ) + Li ) + Mi φ(x|xi =0 ) Let Dfi (.) be the ith line of the matrix Df (.), we have: hZ 1 i fi (x|xi =0 ) = fi (x = 0) + Dfi (σ.x|xi =0 )dσ .x|xi =0 ≥ 0 0
(8)
(9)
Since f (.) is cooperative (H2c), the only possible negative term of Dfi (.) is the ith one which is canceled by xi = 0. Moreover fi (x = 0) ≥ 0 (H2e). Then, since u and L are non-negative (H2g,a) and since Mi φ(x|xi =0 ) is non-negative (H2f), we have x˙i |x =0 ≥ 0. ¤ i
Notice that from (H2c,d,e), it can be shown that: ∀λ ≥ λm , ∀β ≥ βm , (f (λ.1l) + (L + β1 M )) ¿ 0 3.2 Positive output feedback control Theorem 2. The positive output feedback control law: u = γy = γφ(x) with γ > βm > 0
(10)
ensures that the closed loop system (7) will have a single strongly positive equilibrium point x? . Moreover, x? is globally asymptotically stable on the positive orthant and is the single solution of: f (x? ) = −(L +
1 M) γ
Proof. First note that the control u = γφ(x) fulfills hypothesis (H2g). Now, we consider the closed loop system. The control law (10) leads to: ½ x˙ = γφ(x)[f (x) + (L + γ1 M )] (11) x(t0 ) À 0 As in the first section, we show that the state variables remain positive for system (11). Consider a small ² > 0; we focus on the dynamics of xi (∀i ∈ [1..n]) near 0 (i.e. xi = ²), as the state is in the cone x ≥ ².1l. Then according to system (11), we have: x ≥ ².1l ´ ³ 1 x˙i |xi =² = γφ(x) fi (x|xi =² ) + (Li + Mi ) (12) | {z } γ | {z } >0 >0
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Remember that fi (x|xi =0 ) ≥ 0; then since f (.) is a continuous function of the state (H2c), we have: ∃²M > 0, ∀² ∈]0, ²M [, (fi (x|xi =² ) + (Li +
1 Mi )) > 0 βm
Then it is straightforward that for a small enough positive ², x˙i |xi =² > 0. Thus, the function φ(x) is lower bounded by a positive constant (H2f). Now Rt we make the time change t0 = 0 γφ(x(τ ))dτ , we have: ( dx = f (x) + (L + 1 M ) γ dt0 0 (13) x(t0 ) À 0 Note that system (13) is a cooperative system (H2c). To finish the proof we have to show that system (13) has a single, GAS strongly positive equilibrium. This proof is very similar to the one given by Smith in [7]. First, we consider the state box Bλ = {x, λ1l ≥ x ≥ 0} with λ ≥ λm . From hypotheses (H2b) and (H2e), and since (13) is cooperative, it is straightforward that Bλ is a positively invariant set. The Brouwer Fixed-Point Theorem [10] ensures that there exists at least an equilibrium in Bλ since it is positively invariant. Since all trajectories of (13) remain (strongly) positive, equilibria of (13) must be strongly positive. Let x? be an equilibrium of (13), from (H2e,b) we have: 0 À f (x? ) − f (0) =
³Z 0
1
´ Df (sx? )ds x? = Ax?
(14)
Since A is off-diagonal non-negative, we use Theorem 1.2 from [7], which ensures, from equation (14), that matrix A has only eigenvalues with negative real parts. Then, from (H2d), Df (x? )i,j ≤ Ai,j and Theorem 1.1 from [7] ensures that Df (x? ) is a stable matrix, then x? is asymptotically stable. We still have to show the uniqueness of x? . Consider two steady states x?1 and x?2 such that x?1 ≥ x?2 . We have: 0 = f (x?1 ) − f (x?2 ) =
³Z 0
1
´ Df (sx?2 + (1 − s)x?1 )ds (x?2 − x?1 ) = A0 (x?2 − x?1 )
As above, from (H2d), matrix A0 is such that A0i,j ≤ Df (x?1 ); since Df (x?1 ) is stable, A0 is stable too. Thus A0 is invertible and: x?1 = x?2 . Now consider two positive steady states, x?1 and x?2 , but not related by ≤. Consider x3 , such that x3,i = min(x?1,i , x?2,i ). The box Bx3 = {x, 0 ≤ x ≤ x3 } is positively invariant since Bx3 = Bx?1 ∩ Bx?2 that are both positively invariant. Then, there exists at least an equilibrium x?3 in Bx3 . x?3 is related by ≤ to both x?1 and x?2 , then: x?1 = x?3 = x?2 . Then there exists x? , a single (strongly) positive equilibrium in Bλ , asymptotically stable for system (13). It remains true ∀λ ≥ λm , therefore system (13) has a single equilibrium x? on the positive
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orthant, moreover x? is asymptotically stable. To finish the proof, consider the trajectories (of (13)) initiated at x = 0 and at x = λ.1l (with λ ≥ λm ), we have: x(x ˙ = 0) À 0 and x(x ˙ = λ.1l) ¿ 0 Consider now the new variable z = x. ˙ We have: z˙ = Df (x)z. The matrix Df (x) being off-diagonal non-negative (H2c), if z(t0 ) ≥ 0 (resp. ≤ 0), then ∀t > t0 , z(t) ≥ 0 (resp. ≤ 0). Then the trajectory initiated at time t0 at x = 0 (resp. at x = λ.1l) is non-decreasing (resp. non-increasing) for all time t > t0 . Moreover, since this trajectory is upper bounded by λ.1l (resp. lower bounded by 0) it will converge to the (single) equilibrium x? belonging to Bλ . Since (13) is cooperative, every trajectory initiated in Bλ will converge to x? too. Then, we conclude that x? is GAS for system (11). ¤ Remark 2 It is important to notice that it is possible to change the gain γ to tune the equilibrium point x? , provided its desired value is solution of: f (x) = −(L +
1 M ) with γ ≥ βm γ
4 Example: anaerobic WWTP We consider an anaerobic wastewater treatment process: it consumes organic pollution to create insoluble biogas (CH4 ). We propose a simple model, describing the key features of anaerobic digestion, derived from [1]; this model is unstable since two locally stable equilibrium points exists. x and sT denotes respectively the bacterial and the pollution concentration in the reactor, r(.) the bacterial growth speed (positive, increasing in x, non-monotone in sT ), D the flow per volume passing through the reactor (the input), sT in the inflow pollution concentration, (1−α) the proportion of bacteria fixed in the reactor, k and k 0 are yield coefficients, the methane outflow QCH4 is measured online. We have the following model: µ ¶ ·µ ¶µ ¶ µ ¶¸ µ ¶ x˙ −α 0 x 0 1 =D + + r(sT , x) s˙T 0 −1 sT sT,in −k y = QCH4 = k 0 r(sT , x) Note that this system is a particular case of systems (1) and that we have not assumed any analytical form of the bio-reaction rate r(sT , x) (model’s biological uncertainty is located in this term). Moreover hypotheses (H1) hold. Experimental validation of the control law (3) has been performed on the anaerobic digester (see [9]), located in Narbonne (France), at the LBE-INRA. We have applied, with different values of γ, the feedback: D(.) = γy = γQCH4
with γ >
k k 0 ST,in
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Ludovic Mailleret
We show two transient behaviors with two different values of γ for the input variable D(.) and for the variable sT on figure 1. Both experiments agree with our predictions: a single GAS positive equilibrium, that can be tuned via the choice of the gain γ; see [6] for further details. 45
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References 1. J. E. Bailey and D. F. Ollis. Biochemical engineering fundamentals, second edition. McGraw-Hill chemical engineering series, 1986. 2. L. Farina and S. Rinaldi. Positive linear systems, theory and applications. John Wiley and Sons, 2000. 3. M. W. Hirsch. The dynamical systems approach to differential equations. Bulletin of the American mathematical society, 11:1–64, 1984. 4. J. Hofbauer and K. Sigmund. The Theory of Evolution and Dynamical Systems. Cambridge University Press, 1988. 5. D. G. Luenberger. Introduction to Dynamic Systems. Theory, Models and Applications. John Wiley and Sons, New York, 1979. 6. L. Mailleret, O. Bernard, and J.P. Steyer. Robust regulation of anaerobic digestion processes. to appear in Water Science and Technology, to appear. 7. H. L. Smith. On the asymptotic behavior of a class of deterministic models of cooperating species. SIAM Journal on Applied Mathematics, 46:368–375, 1986. 8. H. L. Smith. Monotone dynamical systems, an introduction to the theory of competitive and cooperative systems. Mathematical Surveys and Monographs. American mathematical society, 1995. 9. J. P. Steyer, J. C. Bouvier, T. Conte, P. Gras, and P. Sousbie. Evaluation of a four year experience with a fully instrumented anaerobic digestion process. Water Science and Technology, 45:495–502, 2002. 10. E. Zeidler. Nonlinear Functional Analysis and its Applications. I: Fixed-Point Theorems. Springer-Verlag, 1985.
Competitive and Cooperative Systems: A Mini-review Morris W. Hirsch1 and Hal L. Smith2 1 2
Department of Mathematics, University of California, Berkeley, CA 94720, USA, [email protected] Department of Mathematics, Arizona State University, Tempe, AZ 85287, USA, [email protected]
Abstract. The theory of competitive and cooperative dynamical systems has had some remarkable applications to the biological sciences. The interested reader may consult the monograph [18] and lecture notes [19] of Smith, and to a forthcoming review by the authors [7] for a more in-depth treatment.
1 Strong monotonicity for ODEs In this brief review we give some of the main results in the theory of competitive and cooperative systems. But first, we give some new strong monotonicity results for odes. Let J be a nontrivial open interval, D ⊂ IRn be an open set, f : J × D → IRn be a locally Lipschitz function, and consider the ordinary differential equation x0 = f (t, x) (1) Denote by x(t, t0 , x0 ) the non-continuable solution of the initial value problem x(t0 ) = x0 for t0 ∈ J. A cone K in IRn is a non-empty, closed subset of IRn satisfying K +K ⊂ K, IR · K ⊂ K and K ∩ (−K) = {0}. We hereafter assume K nonempty interior in IRn . The order relations ≤, <, ¿ are induced by K as follows: x ≤ y if and only if y −x ∈ K; x < y if x ≤ y and x 6= y, and x ¿ y whenever y −x ∈ IntK. A cone is a polyhedral cone if it is the intersection of a finite family of half spaces. The standard cone IRn+ = ∩ni=1 {x : hei , xi ≥ 0} is polyhedral (ei is the unit vector in the xi -direction) while the ice cream cone K = {x ∈ IRn : x21 + x22 + · · · + x2n−1 ≤ x2n , xn ≥ 0} is not. The dual cone K ∗ is the set positive linear functionals, i.e., linear functionals λ ∈ (IRn )∗ , the dual space of IRn , such that λ(K) ≥ 0. If we adopt the standard inner product h, i on IRn then we can identify (IRn )∗ with IRn since for each λ ∈ K ∗ we can find a ∈ IRn such that λ(x) = ha, xi for all x. We use the following easy result; see e.g. Walcher [24]. L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 183-190, 2003. Springer-Verlag Berlin Heidelberg 2003
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Lemma 1. Let x ∈ K. Then x ∈ IntK if and only if λ(x) > 0 for all λ ∈ K ∗ \ {0}. We say that (1) is monotone, or order-preserving, if whenever x0 , x1 ∈ D satisfy x0 ≤ x1 and the solutions x(t, t0 , x0 ) and x(t, t0 , x1 ) are defined on [t0 , t1 ], t1 > t0 , then x(t, t0 , x0 ) ≤ x(t, t0 , x1 ) holds for t ∈ [t0 , t1 ]. The vector field f : J × D → IRn is said to satisfy the quasimonotone condition in D if for every (t, x), (t, y) ∈ J × D we have (Q)x ≤ y and φ(x) = φ(y) for some φ ∈ K ∗ implies φ(f (t, x)) ≤ φ(f (t, y)). The quasimonotone condition was introduced by Schneider and Vidyasagar [16] for finite dimensional, autonomous linear systems and used later by Volkmann [23] for nonlinear infinite dimensional systems. The following result is certainly inspired by a result of Volkmann [23] and work of W. Walter [25]. See also Uhl [22] and Walcher [24]. The proof appears in [7]. Theorem 1.1 Let f satisfies (Q) in D, t0 ∈ J, and x0 , x1 ∈ D. Let t0 are such that both x(t, t0 , xi ), i = 0, 1 are defined, then x(t, t0 , x0 )
for all x ∈ ∂K and λ ∈ K ∗ such that λ(x) = 0, λ(A(t)x) ≥ 0.
Therefore, we have the following Corollary of Theorem 1.1. Corollary 1.1 The matrix solution X(t) satisfies X(t)K ⊂ K for t ≥ t0 for each t0 ∈ J if and only if for all t ∈ J, (M) holds for the function x → A(t)x. In fact, (M) implies that X(t)(K \ {0}) ⊂ (K \ {0}) and X(t)IntK ⊂ IntK for t > t0 . A matrix A is K-nonnegative if A(K) ⊂ K. Corollary 1.1 says that X(t) is K-nonnegative for t ≥ t0 if (M) holds. The domain D is p-convex if for every x, y ∈ D satisfying x ≤ y the line segment joining them also belongs to D. Let ∂f ∂x (t, x) be continuous on J × D. We say that f (or (1)) is K-cooperative if for all t ∈ J, y ∈ D, (M) holds for the function x → ∂f ∂x (t, y)x. By Corollary 1.1 applied to the variational equation ∂f (t, x(t, t0 , x0 ))X, X(t0 ) = I X 0 (t) = ∂x ∂x we conclude that if f is K-cooperative then X(t) = ∂x (t, t0 , x0 ) is K-positive. 0 Straightforward arguments lead to the following result.
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Theorem 1.2 Let ∂f ∂x (t, x) be continuous on J × D. Then (Q) implies that f is K-cooperative. Conversely, if D is p-convex and f is K-cooperative, then (Q) holds. If K = IRn+ , then it is easy to see by using the standard inner product that we may identify K ∗ with K. The quasimonotone hypothesis reduces to the Kamke condition [13, 10]: x ≤ y and xi = yi implies that fi (t, x) ≤ fi (t, y). This holds by taking φ(x) = hei , xi and noting that every φ ∈ K ∗ can be represented as a positive linear combination of these functionals. If f is differentiable, the Kamke condition implies ∂fi (t, x) ≥ 0, i 6= j. ∂xj
(3)
Conversely, if ∂f ∂x (t, x) is continuous on J × D and satisfies (3) and if D is p-convex, then the Kamke condition holds. Stern and Wolkowicz [21] give necessary and sufficient conditions for (M) to hold for matrix A relative to the ice cream cone K = {x ∈ IRn : x21 + x22 + · · · + x2n−1 ≤ x2n , xn ≥ 0}. Let Q denote the n × n diagonal matrix with first n − 1 entries 1 and last entry −1. Then (M) holds for A if and only if QA + AT Q + αQ is negative semidefinite for some α ∈ IR. Their characterization extends to other ellipsoidal cones. Additional hypotheses are required for establishing the strong order preserving property and here we provide full details following [7]. Recall that the matrix A is strongly positive if A(K \ {0}) ⊂ IntK. The following hypothesis for the matrix A follows Schneider and Vidyasagar [16]. (T) for all x 6= 0, x ∈ ∂K, there exists ν ∈ K ∗ such that ν(x) = 0 and ν(Ax) > 0. Our next result was proved by Elsner [3] for the case of constant matrices, answering a question in [16]. Our proof follows that of Theorem 4.3.26 of Berman et al [1]. Proposition 1.1 Let the linear system (2) satisfy (M). Then the fundamental matrix X(t1 ) is strongly positive for t1 > t0 if there exists s satisfying t0 < s ≤ t1 such that (T) holds for A(s). Proof: If not, there exists x > 0 such that the solution of (2) given by y(t) = X(t)x satisfies y(t1 ) ∈ ∂K \ {0}. By Corollary 1.1, y(t) > 0 for t ≥ t0 and y(t) ∈ ∂K for t0 ≤ t ≤ t1 . Let s ∈ (t0 , t1 ] be such that (T) holds for A(s). Then there exists ν ∈ K ∗ such that ν(y(s)) = 0 and ν(A(s)y(s)) > 0. As ν ∈ K ∗ and y(t) ∈ K, h(t) := ν(y(t)) ≥ 0 for t0 ≤ t ≤ t1 . But h(s) = 0 and d dt |t=s h(t) = ν(A(s)y(s)) > 0 which, taken together, imply that h(s − δ) < 0 for small positive δ, giving the desired contradiction. Proposition 1.1 leads immediately to a result on strong monotonicity for the nonlinear system (1).
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Theorem 1.3 Let D be p-convex, ∂f ∂x (t, x) be continuous on J × D, and f be K-cooperative. Let B = {(t, x) ∈ J × D : (T) does not hold for ∂f ∂x (t, x)}. Suppose that for all (t0 , x0 ) ∈ J × D, the set {t > t0 : (t, x(t, t0 , x0 )) ∈ B} is nowhere dense. Then x(t, t0 , x0 ) ¿ x(t, t0 , x1 ) for t > t0 for which both solutions are defined provided x0 , x1 ∈ D satisfy x0 < x1 . In particular, this holds if B is empty. Proof: We apply the formula Z x(t, t0 , x1 ) − x(t, t0 , x0 ) =
0
1
∂x (t, t0 , sx1 + (1 − s)x0 )(x1 − x0 )ds ∂x0
∂x where X(t) = ∂x (t, t0 , y0 ) is the fundamental matrix for (2) corresponding 0 to the matrix A(t) = ∂f ∂x (t, x(t, t0 , y0 )). The left hand side belongs to K \ {0} if x0 < x1 by Theorem 1.2 and Theorem 1.1 but we must show it belongs to IntK. For this to be true, it suffices that for each t > t0 there exists s ∈ [0, 1] such that the matrix derivative in the integrand is strongly positive. In fact, it is K-positive by Corollary 1.1 for all values of the arguments with t ≥ t0 so application of any nontrivial φ ∈ K ∗ to the integral gives a nonnegative numerical result. If the condition mentioned above holds then the application of φ to the integrand gives a positive numerical result for all s0 near s by continuity and Lemma 1 and hence the integral belongs to IntK by Lemma 1. ∂x (t, t0 , y0 ) is strongly positive for t > t0 if (T) holds for By Proposition 1.1, ∂x 0 ∂f A(r) = ∂x (r, x(r, t0 , y0 )) for some r ∈ (t0 , t]. But this is guaranteed by our hypotheses.
The somewhat stronger condition of irreducibility may be more useful in applications because there is a large body of theory related to it [2, 1]. A closed subset F of K that is itself a cone is called a face of K if x ∈ F and 0 ≤ y ≤ x (inequalities induced by K) implies that y ∈ F . For example, the faces of K = IRn+ are of the form {x ∈ IRn+ : xi = 0, i ∈ I} where I ⊂ {1, 2, · · · n}. For the ice-cream cone K = {x ∈ IRn : x21 + x22 + · · · + x2n−1 ≤ x2n , xn ≥ 0}, the faces are the rays issuing from the origin and passing through its boundary vectors. A K-positive matrix A is K-irreducible if the only faces F of K for which A(F ) ⊂ F are {0} and K. The famous Perron-Frobenius Theory is developed for K-positive and K-irreducible matrices in Berman and Plemmons [2]. In particular, the spectral radius of A is a simple eigenvalue of A with corresponding eigenvector in IntK. The next result is adapted from Theorem 4.3.17 of Berman et al. [1]. Proposition 1.2 Let A be a matrix such that B := A + αI is K-positive for some α ∈ IR. Then B is K-irreducible if and only if (T) holds for A. Motivated by Proposition 1.2, we introduce the following hypothesis for matrix A.
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(I)
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there exists α ∈ IR such that A + αI is K-positive and K-irreducible.
In the special case that K = IRn+ , n ≥ 2, matrix A satisfies (I) if and only if aij ≥ 0 for i 6= j and for every non-empty, proper subset I of N := {1, 2, · · · , n}, there is an i ∈ I and j ∈ N \ I such that aij 6= 0. This is equivalent to the assertion that the incidence graph of A is strongly connected. See Berman and Plemmons [2]. The following is a direct corollary of Theorem 1.3. Corollary 1.2 Let D be p-convex, ∂f ∂x (t, x) be continuous on J × D and f ˜ = {(t, x) ∈ J × D : (I) does not hold for ∂f (t, x)}. be K-cooperative. Let B ∂x ˜ Suppose that for all (t0 , x0 ) ∈ J × D, the set {t > t0 : (t, x(t, t0 , x0 )) ∈ B} is nowhere dense. Then x(t, t0 , x0 ) ¿ x(t, t0 , x0 ) for t > t0 for which both solutions are defined provided x0 , x1 ∈ D satisfy x0 < x1 . In particular, this ˜ is empty. holds if B Corollary 1.2 is an improvement of the restriction of Theorem 10 of Kunze and Siegel [11] to the case that K has nonempty interior. Their results also treat the case that K has empty interior in IRn but nonempty interior in some subspace of IRn . Theorem 4.3.40 of Berman et al. [1] implies that for polyhedral cones a matrix A satisfies (M) and (T) if and only if there exists α ∈ IR such that A + αI is K-positive and K-irreducible. Therefore, for polyhedral cones like IRn+ , Corollary 1.2 and Theorem 1.3 are equivalent.
2 Competitive and cooperative systems We now focus on the autonomous system of ordinary differential equations x0 = f (x)
(4)
where f is continuously differentiable on an open subset D ⊂ IRn . Let φt (x) denote the solution of (4) that starts at the point x at t = 0. φt will be referred to as the flow corresponding to (4). We sometimes refer to f as the vector field generating the flow φt . We introduce following mild dissipativity condition for our next result. (A) for each x ∈ D, φt (x) is defined for all t ≥ 0 and φt (x) ∈ D. Moreover, for each bounded subset A of D, there exists a compact subset B = B(A) of D such that for each x ∈ A, φt (x) ∈ B for all large t. The following result should be viewed as prototypical of the generic convergence result that may be proved using general results in [20, 18]. Theorem 2.1 Let the hypotheses of Theorem 1.3 hold for (4), assume D = IRn or D = IRn+ , and assume that (A) holds. Then the set C of convergent points contains an open and dense subset of D.
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We say that (4) is K-competitive in D if the time-reversed system x0 = −f (x) is K-cooperative. Observe that if (4) is a K-competitive system with flow φt then the time reversed system above is a K-cooperative system with flow ψt where ψt (x) = φ−t (x), and conversely. Therefore, by time reversal, a competitive system becomes a cooperative system and vice-versa. We will sometimes drop the K from K-cooperative (competitive) when no confusion may result. Let A be an invariant set for (4) with flow φt (i.e. φt (A) = A for all t) and let B be an invariant set for the system y 0 = F (y) with flow ψt . We say that the flow φt on A is topologically equivalent to the flow ψt on B if there is a homeomorphism Q : A → B such that Q(φt (x)) = ψt (Q(x)) for all x ∈ A and all t ∈ IR. The relationship of topological equivalence says, roughly, that the qualitative dynamics of the two flows are the same. With these definitions, we can state a result of Hirsch [4]. Theorem 2.2 The flow on a compact limit set of a competitive or cooperative system in IRn is topologically equivalent to a flow on a compact invariant set of a Lipschitz system of differential equations in IRn−1 . The Poincar´e-Bendixson Theorem for three dimensional cooperative and competitive systems is the most notable consequence of Theorem 2.2. It was proved by Hirsch [6] who improved earlier partial results [4, 17]. Theorem 2.3 (Poincar´e-Bendixson Theorem for 3-Dimensional Competitive and Cooperative Systems) A compact limit set of a competitive or cooperative system in IR3 that contains no equilibrium points is a periodic orbit. The following result of Smith [18] is useful for verifying that an omega limit set is a periodic orbit. Theorem 2.4 Suppose that D ⊂ IR3 contains a unique equilibrium p for the competitive system (4) and it is hyperbolic. Suppose further that its stable manifold W s (p) is one-dimensional and tangent at p to a vector v À 0. If the orbit of q ∈ D \ W s (p) has compact closure in D, then ω(q) is a nontrivial periodic orbit. The existence of v À 0 usually follows from the Perron-Frobenius Theorem. Zhu and Smith [26] establish the existence of an orbitally asymptotically stable periodic orbit if (4) is dissipative and f is analytic. Ortega and S´anchez [14] observed that the above results hold for general cones. Competitive systems arise naturally from models in the biological sciences, not just in population biology. The following, taken from de Leenheer and Smith [8], illustrates this point. Consider an individual infected with a virus V which attacks target cells T producing infected cells T ∗ which in turn each produce on average N virus particles during their lifetimes. Following Perelson et al. [15], who focus on HIV, we obtain the following system for the dynamics of the vector of blood-concentrations (T, T ∗ , V ) ∈ IR3+ .
Competitive and Cooperative Systems: A Mini-review
T˙ = f (T ) − kV T T˙ ∗ = −βT ∗ + kV T V˙ = −γV + N βT ∗ − kV T.
189
(5)
T ) with δ, α, p, Tmax Perelson et al. [15] take f (T ) ≡ δ − αT + pT (1 − Tmax positive and denote by T¯ the positive root of f (T ) = 0. The basic reproductive number for the model, R0 = k T¯(N − 1)/γ, gives the number of infected T cells produced by a single infected T cell in a healthy individual. Among other results, de Leenheer and Smith [8] prove the following.
Theorem 2.5 If R0 > 1, in addition to the unstable virus-free state E0 ≡ (T¯, 0, 0), there is a “chronic disease” steady state Ee ≡ (Te , Te∗ , Ve ) given by Te = T¯/R0 ,
Te∗ = γVe /(N − 1)β,
Ve = f (Te )/kTe .
which is locally attracting if f 0 (Te ) ≤ 0. The omega limit set of every solution with initial conditions satisfying T ∗ (0) + V (0) > 0 either contains Ee or is a nontrivial periodic orbit. There exist parameter values for which Ee is unstable with a two dimensional unstable manifold. In this case, there exists an orbitally asymptotically stable periodic orbit; every solution except those with initial data on the one-dimensional stable manifold of Ee or on the T axis converges to a non-trivial periodic orbit. System (5) is competitive with respect to the cone K := {T, V ≥ 0, T ∗ ≤ 0}. The change of variables T ∗ → −T ∗ results in a system the Jacobian for which has non-positive off-diagonal terms on the relevant domain and hence is competitive in the IR3+ -sense. In [8], it is shown that Ee is unstable with a twodimensional unstable manifold when kTmax > β +γ + N2γ −1 and p is sufficiently large. The final assertion of Theorem 2.5 follows from Theorem 2.4; domain D is chosen to exclude E0 . The existence of an orbitally asymptotically stable periodic orbit uses the analyticity of the system and results of [26].
References 1. A. Berman, M. Neumann and R. Stern, Nonnegative matrices in dynamic systems, John Wiley& Sons, New York (1989). 2. A. Berman and R. Plemmons, Nonnegative matrices in the mathematical sciences, Academic Press, New York (1979) 3. L. Elsner, Quasimonotonie und Ungleichungen in halbgeordneten Raumen, Linear Alg. Appl. 8(1974), 249-261. 4. M.W. Hirsch, Systems of differential equations which are competitive or cooperative 1: limit sets. SIAM J. Appl. Math. 13(1982) , 167-179. 5. M.W. Hirsch, Systems of differential equations which are competitive or cooperative II: convergence almost everywhere. SIAM J. Math. Anal. 16(1985), 423-439.
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6. M.W. Hirsch, Systems of differential equations that are competitive or cooperative. IV: Structural stability in three dimensional systems. SIAM J.Math. Anal.21(1990), 1225-1234. 7. M.W. Hirsch and H.L. Smith, Monotone dynamical systems, in preparation. 8. P. de Leenheer and H.L. Smith, Virus Dynamics: a global analysis, to appear, SIAM J. Appl. Math (2003). 9. R. Loewy and H. Schneider, Positive operators on the n-dimensional ice cream cone, J. Math. Anal. & Appl. 49(1975), 375-392. 10. E. Kamke, Zur Theorie der Systeme Gewoknlicher Differentialgliechungen, II, Acta Math. 58(1932), 57-85. 11. H. Kunze and D. Siegel, Monotonicity with respect to closed convex cones II, Applicable Analysis 77(2001), 233-248. 12. R. Lemmert and P. Volkmann, On the positivity of semigroups of operators, Comment. Math. Univ. Carolinae 39(1998), 483-489. 13. M. Muller, Uber das fundamenthaltheorem in der theorie der gewohnlichen differentialgleichungen, Math. Zeit. 26(1926), 619-645. 14. R. Ortega and L. S´ anchez, Abstract competitive systems and orbital stability in IR3 , Proc. Amer. Math. Soc. 128(2000), 2911-2919. 15. A.S. Perelson and P.W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev. 41(1999), 3-44. 16. H. Schneider and M. Vidyasagar, Cross-positive matrices, SIAM J. Numer. Anal. 7(1970), 508-519. 17. H.L. Smith, Periodic orbits of competitive and cooperative systems, J.Diff.Eqns. 65(1986), 361-373. 18. H.L. Smith, Monotone Dynamical Systems, an introduction to the theory of competitive and cooperative systems, Math. Surveys and Monographs, 41, American Mathematical Society, Providence, Rhode Island(1995). 19. H.L. Smith, Dynamics of Competition, in Mathematics Inspired by Biology, Springer Lecture Notes in Math. 1714, 1999, 191-240. 20. H.L. Smith and H.R. Thieme, Convergence for strongly ordered preserving semiflows, SIAM J. Math. Anal. 22(1991), 1081-1101. 21. R. Stern and H. Wolkowicz, Exponential nonnegativity on the ice cream cone, SIAM J. Matrix Anal. Appl. 12(1991), 160-165. 22. R. Uhl, Ordinary differential inequalities and quasimonotonicity in ordered topological vector spaces, Proc. Amer. Math. Soc. 126(1998), 1999-2003. 23. P. Volkmann, Gewohnliche Differentialungleichungen mit quasimonoton wachsenden Funktionen in topologischen Vektorraumen, Math. Z. 127(1972), 157164. 24. S. Walcher, On cooperative systems with respect to arbitrary orderings, Journal of Mathematical Analysis and Appl. 263(2001), 543-554. 25. W. Walter, Differential and Integral Inequalities, Springer-Verlag, Berlin(1970). 26. H.-R. Zhu and H.L. Smith , Stable periodic orbits for a class of three dimensional competitive systems, J.Diff.Eqn. 110(1994), 143-156.
Small-gain Theorems for Predator-prey Systems Patrick De Leenheer1 , David Angeli2 , and Eduardo D. Sontag3 1 2 3
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA, [email protected] Dip. di Sistemi e Informatica Universit´ a di Firenze, Via di S. Marta 3, 50139 Firenze, Italy, [email protected] Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA, [email protected]
Abstract. We present a global stability result for Lotka-Volterra systems of the predator-prey type. It turns out that these systems can be interpreted as feedback interconnections of two monotone control systems possessing particular input-output properties. The proof is based on a small-gain theorem, adapted to a setting of systems with multiple equilibrium points. Our main result provides a sufficient condition to rule out oscillatory behavior which often occurs in predator-prey systems.
1 Introduction Predator-prey systems have been -and still are- attracting a lot of attention [6, 11, 8] since the early work of Lotka and Volterra. It is well-known that these systems may exhibit oscillatory behavior, the best known example being the classical Lotka-Volterra predator-prey system, see e.g. [6, 7], defined by µ ¶ µ ¶µ ¶ µ ¶ x˙ 0 +a12 x −r1 = diag(x, z)( + ) z˙ −a21 0 z r2 where x and z denote the predator, respectively the prey concentrations and a12 , a21 , r1 and r2 are positive constants. The phase portrait consists of an infinite number of periodic solutions centered around an equilibrium point. It is also well-known that this system is not structurally stable and perturbations in the coefficients destroy this qualitative picture. However, structurally stable predator-prey systems with isolated periodic solutions can be found as well. One example (which is still low-dimensional but not of the Lotka-Volterra type) is Gause’s model [7] which admits isolated periodic solutions under suitable conditions [8]. Oscillatory behavior is possible for systems in the class of Lotka-Volterra predator-prey systems, but then the number of predator and prey species is necessarily greater than two. L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 191-198, 2003. Springer-Verlag Berlin Heidelberg 2003
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To illustrate this we provide an example with 2 predator species and 1 prey species. We also assume that the predator species are mutualistic [7], which is the case for instance if the predator population is stage-structured. (e.g. consisting of young and adults) Example Consider the parameterized (parameter k > 0) Lotka-Volterra predatorprey system with 2 predator species x1 and x2 and 1 prey species z : x˙ 1 −1 1 1 x1 −1 x˙ 2 = diag(x1 , x2 , z)( 1 −2 0 x2 + 1 ) (1) z˙ 0 −k −3 z k+3 One could interpret x1 as the immature and x2 as the mature predators. For every k > 0 there is a nontrivial equilibrium point at (1, 1, 1) and the Routh-Hurwitz criterion reveals that it is locally asymptotically stable if k ∈ (0, kc ) where kc := 57. For k > kc however, the linearization at (1, 1, 1) possesses 1 stable (and hence real) eigenvalue and 2 unstable eigenvalues. It can be shown that for k − kc > 0 but small, the unstable eigenvalues must be complex conjugate with nontrivial imaginary part. In [4] we have shown that a supercritical Hopf bifurcation occurs a at the critical value kc . The example suggests that oscillatory behavior is to be expected in the following Lotka-Volterra predator-prey system: .. x˙ . B A r1 x . . . = diag(x, z)( . . . . . . . . . . . . + . . .) (2) .. z˙ z r2 −C . D where x is k-dimensional and z is (n − k)-dimensional. Throughout this paper we make the following assumption: H: For system (2), A and D are Metzler and stable and B, C ≥ 0 where the inequalities on the matrices B and C should be interpreted entrywise. (A matrix is a Metzler matrix if its off-diagonal entries are non-negative and stable if it only has eigenvalues with negative real part.) The given example satisfies this assumption. Here we consider whether oscillations or more complicated behavior of system (2) can be ruled out. To system (2) one can associate two Input/Output (I/O) systems:
and
z˙ = diag(z)(Dz + r2 + Cu(t)), w = z
(3)
x˙ = diag(x)(Ax + r1 + Bv(t)), y = x
(4)
where u(t) is a (component-wise) non-positive and v(t) a (component-wise) non-negative input signal and w and y are output signals. These I/O-systems
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are monotone in the sense of [1] (a precise definition of such systems is given later). To each I/O system we associate an I/O quasi-characteristics kw , respectively ky (see Definition 2). This is a mapping between the input and output space capturing the ability of an I/O system to convert a constant input in a converging output with a limit which is (almost) independent of initial conditions. The I/O quasi-characteristic assigns to every input its corresponding output limit. Notice that system (2) is a negative feedback interconnection of system (3) and system (4) by setting: v = w, u = −y.
(5)
This allows the use of results from theories on interconnected control systems -in particular small-gain theorems- to prove global stability. Our main result can informally be stated as follows: Theorem 1. The feedback system (3), (4) and (5) possesses an (almost) globally attractive equilibrium point provided the discrete-time system uk+1 = −(ky ◦ kw )(uk ) possesses a globally attractive fixed point. Our results illustrate a recently developed theory for monotone control systems [1, 2]. Important note: Due to space constraints we leave out all proofs. They can be found in an extended version of this paper; see [4].
2 Preliminaries First we will review a small-gain theorem which applies to a particular class of I/O systems. Consider the following I/O system: x˙ = f (x, u), y = h(x)
(6)
where x ∈ Rn is the state, u ∈ U ⊂ Rm the input and y ∈ Y ⊂ Rp the output. It is assumed that f and g are smooth (say continuously differentiable) and that the input signals u(t) : R → U are Lebesgue measurable functions and locally essentially bounded. Solutions are then defined and unique and we denote the solution with initial state x0 ∈ Rn and input signal u(.) by x(t, x0 , u(.)), t ∈ I where I is the maximal interval of existence. We will also assume that a forward invariant set X ⊂ Rn is given, meaning that for all inputs u(.) and for every x0 ∈ X it holds that x(t, x0 , u(.)) ∈ X, for all t ∈ I ∩ R+ . Initial conditions shall be restricted to the X in the sequel. The usual partial order on Rn , denoted by ¹, is to be understood component-wise, i.e. x ¹ y means that xi ≤ yi for i = 1, ..., n. As a subset
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of Rn (Rm , Rp ), the state space X (input space U , output space Y ) inherits its partial order. Similarly, the set of input signals then also has a (obvious) partial order: u(.) ¹ v(.) if u(t) ¹ v(t) for almost all t ≥ 0. Next we define the concept of a monotone I/O system. Definition 1. The I/O system (6) is monotone if the following conditions hold: x1 ¹ x2 , u(.) ¹ v(.) ⇒ x(t, x1 , u(.)) ¹ x(t, x2 , v(.)), ∀ t ∈ (I1 ∩ I2 ) ∩ R+ . (7) and
x1 ¹ x2 ⇒ h(x1 ) ¹ h(x2 ).
(8)
A key role in our main result is played by the following concept. Definition 2. Assume that X has positive measure. System (6) has an Input/State (I/S) quasi-characteristic kx : U → X if for every constant input u ∈ U (and using the same notation for the corresponding u(.)), there is a zero-measure set Bu such that: lim x(t, x0 , u) = kx (u), ∀ x0 ∈ X \ Bu .
t→+∞
(9)
If system (6) possesses an I/S quasi-characteristic kx then it also possesses an Input/Output (I/O) quasi-characteristic ky : U → Y defined as ky := h ◦ kx . The following result can be found in [2]. A system possesses an almost globally attractive equilibrium point if it has an equilibrium point that attracts all solutions not initiated in a set of measure zero. If in addition, this equilibrium point is stable, we call it almost globally asymptotically stable. Theorem 2. Consider two I/O systems: x˙ 1 = f1 (x1 , u1 ), x˙ 2 = f2 (x2 , u2 ),
y1 = h1 (x1 ) y2 = h2 (x2 )
(10) (11)
where xi ∈ Xi ⊂ Rni , ui ∈ Ui ⊂ Rmi and yi ∈ Yi ⊂ Rpi for i = 1, 2. Assume that Y1 = U2 and Y2 = −U1 and that these systems are connected via a negative feedback loop: u2 = y1 , u1 = −y2 .
(12)
Suppose that: 1. Systems (10) and (11) are monotone I/O systems. 2. Systems (10) and (11) have continuous I/S quasi-characteristics kx1 and kx2 respectively (and also I/O quasi-characteristics ky1 and ky2 ). 3. The forward solutions of the full system (10) − (12) are bounded.
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If the following discrete-time system, defined on U1 : uk+1 = −(ky2 ◦ ky1 )(uk )
(13)
possesses a globally attractive fixed point u ¯ ∈ U1 , then the full system has an almost globally attractive equilibrium point (¯ x1 , x ¯2 ) ∈ X1 × X2 and is such that (¯ x1 , x ¯2 ) = (kx1 (¯ u), (kx2 ◦ ky1 )(¯ u)). This result is called a small-gain theorem and the last condition will be referred to as a small-gain condition. Next we specialize to (autonomous) Lotka-Volterra systems and provide a boundedness and a stability result. Consider the classical Lotka-Volterra system: x˙ = diag(x)(Ax + r) n
n
(14)
Rn+
where x ∈ R and r ∈ R . It is well-known that is a forward invariant set for (14) and thus we always assume that initial conditions are restricted to Rn+ . Recall that a Lotka-Volterra system is uniformly bounded [7] if there exists a compact, absorbing set K ⊂ Rn+ , i.e. for all x0 ∈ Rn+ , there is a T (x0 ) ≥ 0 such that x(t, x0 ) ∈ K for all t ≥ T (x0 ). Below we use the notation int(Rn+ ) for the interior points of Rn+ . Lemma 1. (Exercise 15.2.7, p.188 in [7]) System (14) is uniformly bounded if and only if ∃c ∈ int(Rn+ ) : −Ac ∈ int(Rn+ ). (15) and every principal sub-matrix of A has the same property. We will soon specialize to Lotka-Volterra systems with a Metzler interaction matrix A. First we recall some facts about the stability of these matrices [7] which are based on the Perron-Frobenius Theorem [9, 7]. Lemma 2. (Theorem 15.1.1, p.181 in [7]) A Metzler matrix is stable if and only if it is diagonally dominant, i.e. ∃d ∈ int(Rn+ ) : −Ad ∈ int(Rn+ ).
(16)
If A is a stable Metzler matrix then (16) holds for every principal sub-matrix of A as well, implying that every principal sub-matrix of A is stable and thus that system (14) is uniformly bounded. The following result is an immediate application of results in [10, 7]. The support set of x ∈ Rn+ is defined as supp(x) := {y ∈ Rn+ | yi > 0 if xi > 0}. Lemma 3. (Theorem 15.3.1, p.191 in [7]) If A is a stable Metzler matrix, then system (14) possesses a unique equilibrium point x ¯ which is globally asymptotically stable with respect to initial conditions in its support set supp(¯ x). Suppose that xe is an equilibrium point of (14). Then xe is globally asymptotically stable with respect to initial conditions in supp(xe ) (and hence xe = x ¯) if and only if the following condition is satisfied: Axe + r ≤ 0
(17)
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The previous results allow us to state a boundedness result for system (2). Lemma 4. The solutions of system (2) are uniformly bounded provided H holds. Now we consider Lotka-Volterra systems with inputs: x˙ = diag(x)(Ax + r + Bu)
(18)
m where x ∈ Rn , u ∈ U is the input. We assume that U = Rm + or U = −R+ . The input signals u(.) : R → U are Lebesgue measurable and locally essentially bounded functions. It can be shown that Rn+ is forward invariant, see [4] and therefore we restrict initial conditions to Rn+ .
Lemma 5. If A is a stable Metzler matrix, then system (18) possesses a continuous I/S quasi-characteristic kx : U → Rn+ . Finally, we consider a scalar discrete-time system: xk+1 = g(xk )
(19)
where g : R+ → R+ is some given, possibly non-smooth map. Lemma 6. Suppose that x ¯ is a fixed point of system (19) in R+ . If there exists an α ∈ [0, 1) such that for all x ∈ R+ with x 6= x ¯: |g(x) − x ¯| ≤ α |x − x ¯|
(20)
then x ¯ is globally asymptotically stable.
3 Main results We return to the study of system (2) or equivalently, (3) − (5) and summarize some of its properties assuming H holds. 1. Following [1], the I/O systems (3) and (4) are monotone. 2. The systems (3), (4) have continuous I/S quasi-characteristics kz , respectively kx (and I/O quasi-characteristics kw ≡ kz , respectively ky ≡ kx ) by lemma 5. 3. By lemma 4 the solutions of system (2) are uniformly bounded. Next we state and prove the main result of this paper. Theorem 3. If H holds, then system (2) possesses an almost globally attractive equilibrium point (¯ z, x ¯) ∈ Rn+ , provided that the discrete-time system uk+1 = −(ky ◦ kw )(uk )
(21)
which is defined on −Rk+ , possesses a globally attractive fixed point u ¯. In that case (¯ z, x ¯) = (kz (¯ u), (kx ◦ kw )(¯ u)).
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In general it is hard to determine whether the discrete-time system (21) has a globally attractive fixed point, but easier under the following condition: R: Rank (B) = Rank (C) = 1. The biological interpretation is that to a prey species it is irrelevant by which predator its individuals are eaten and, there is no prey-selection by the predator species. (n−k) If H and R hold, one can find nonzero vectors b, γ ∈ Rk+ and c, β ∈ R+ with B = bβ T and C = cγ T and such that system (2) simplifies to: z˙ = diag(z)(Dz + r2 + cu), w = β T z x˙ = diag(x)(Ax + r1 + bv), y = γ T x
(22) (23)
v = w, u = −y
(24)
where u ∈ −R+ and v ∈ R+ . Then another application of theorem 2 yields: Corollary 1. If H and R hold, then system (22) − (24), possesses an almost globally attractive equilibrium point (¯ z, x ¯) ∈ Rn+ , if the scalar discrete-time system uk+1 = −(ky ◦ kw )(uk ) (25) which is defined on −R+ , has a globally attractive fixed point u ¯. In this event, (¯ z, x ¯) = (kz (¯ u), (kx ◦ β T kz )(¯ u)). Example (continued) Defining b = (1 0)T , β = 1, c = k and γ = (0 1)T , system (1) can be re-written in the form (22)-(24). The characterization (17) in lemma 3 allows to compute the I/O quasi-characteristics kw and ky . Then the transformation u ˜k = −uk , transforms system (25) to: ( 3 uk + (1 + k3 ) for u ˜k ∈ [0, 1 + 2k ] (− k )˜ (26) u ˜k+1 = 1 3 3 for u ˜ > k 2 2k ¯ It is easy to verify that system (26) has a fixed point u ˜ in the interval (0, 1 + 3 ). If we choose α > 0 as follows:: 2k α=
k <1 3
(27)
the conditions of lemma 6 are satisfied. Note that condition (27) is close to ¯ a necessary condition for global asymptotic stability of u ˜. (indeed, if k3 > 1 ¯ then u ˜ is unstable) By corollary 1, we get that system (1) possesses an almost globally attractive equilibrium point at (1, 1, 1)T under condition (27). The small-gain condition (27) also yields that the equilibrium point is locally stable by recalling that (1, 1, 1)T is locally asymptotically stable if 0 < k < kc = 57. It can be shown that the domain of attraction of (1, 1, 1)T is the interior of R3+ , see [4]. Simulations performed in [4], suggest that the equilibrium point remains almost globally asymptotically stable for intermediate k-values (i.e. k ∈ (3, 57)).
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References 1. D. Angeli and E.D. Sontag, Monotone control systems, arXiv.org math.OC/0206133 and submitted (prelim. version in cdrom proc. 41st CDC, 2002). 2. D. Angeli, P. De Leenheer and E.D. Sontag, A small-gain theorem for almost global convergence of monotone systems, in preparation. 3. A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. 4. P. De Leenheer, D. Angeli and E.D. Sontag, On predator-prey systems and smallgain theorems, submitted. 5. M.W. Hirsch, Systems of differential equations which are competitive or cooperative II: convergence almost everywhere. SIAM J. Math. Anal. 16, 423-439 (1985). 6. M.W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1974. 7. J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998. 8. S.B. Hsu, T.B. Hwang and Y. Kuang, Global analysis of the Michaelis-Mententype ratio-dependent predator-prey system, J. Math. Biol. 42, 489-506 (2001). 9. H.L. Smith, Monotone Dynamical Systems, AMS, Providence, 1995. 10. Y. Takeuchi and N. Adachi, The existence of globally stable equilibria of ecosystems of the generalized Volterra type, J. Math. Biol. 10, 401-415 (1980). 11. Y. Takeuchi and N. Adachi, Oscillations in prey-predator Volterra models. In: H.I. Freedman and C. Strobeck (eds), Population Biology. Lect. Notes in Biomath. 52, Springer-Verlag, Heidelberg, 320-326 (1983).
Positive Particle Interaction Ulrich Krause FB Mathematik und Informatik, Universit¨ at Bremen, 28334 Bremen, Germany, [email protected]
Abstract. This paper treats interaction between finitely many particles where the future state of each particle is obtained from the present states of all other particles by a positive linear combination with time variant coefficients. The main result provides conditions for a common globally asymptotically stable equilibrium to exist. These conditions are, in particular, satisfied if the particles show slowly decaying interation. Since “particles” can be many things, there are many applications, for example, heat diffusion in an inhomogeneous medium, a many body problem under pseudo–gravity and consensus formation under bounded confidence.
1 Introduction: the model “It is for positive systems, therefore, that dynamic systems theory assumes one of its most potent forms.” (D.G. Luenberger, Introduction to Dynamic Systems, page 188) Consider finitely many particles in some space with positive interaction between them in the sense that the future state of each particle is a positive linear combination of the present states of all other particles. The dynamics of such a system is quite well–understood if the coefficients of the combination do not depend on time but there are many open problems concerning the dynamics for time variant interaction. The present paper contains a stability theorem for positive linear and time variant interaction. This theorem provides conditions for a common globally asymptotically stable equilibrium that is, for any given initial states, all particles approach asymptotically the same state. Examples and applications abound because “particles” can mean many things from moving bodies in real space to inhomogeneous plates cooling by heat diffusion to human beings exchanging opinions. “Approaching the same state” then means collision of bodies or equalization of temperature or reaching a consensus, respectively. L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 199-206, 2003. Springer-Verlag Berlin Heidelberg 2003
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Ulrich Krause
Consider n interacting particles in a closed convex region D ⊂ IRd and let xi (t) ∈ D the state of particle i ∈ {1, 2, . . . , n} at time t ∈ IN = {0, 1, 2, . . .}. The dynamics of interaction for particle i is modeled as X aij (t)(xj (t) − xi (t)) (1) xi (t + 1) − xi (t) = j6=1
where the coefficients of interaction aij (t) are nonnegative P numbers. We shall assume that forces are bounded in the sense that aij (t) ≤ 1 for j6=i
all iP and P we admit the possibility of selfinteraction by defining aii (t) = 1− aij (t). Thus, we can (1) rewrite as j6=i
xi (t + 1) =
n X
aij (t)xj (t) for 1 ≤ i ≤ n
(2)
j=1
or, in matrix form, as 1
x(t + 1) = A(t)x(t) n
0
where x(t) = (x (t), . . . , x (t)) ∈ D
(3) n×n
n
and A(t) = (aij (t)) ∈ IR is the n P row–stochastic matrix of coefficients, i.e., 0 ≤ aij (t) and aij (t) = 1 j=1
for 1 ≤ i, j ≤ n and t ∈ IN. In other words, the model as given by (2) exhibits not only positive inn P teraction but positive and convex interaction because of aij (t) = 1. This j=1
happens often to be the case, as for instance in the applications mentioned above where also selfinteraction makes sense (see Section 3). For various kinds of positive systems see [4, 6, 9, 13, 14, 15], for the particular positive systems of consensus formation in one dimension see [1, 2, 3] and for opinion dynamics under bounded confidence, also in one dimension, see [5, 7, 11, 12]. In Section 2 we shall present the main result of the paper, a stability theorem for time variant interaction. In Section 3 we specialize this result to the case of slowly decaying interaction and illustrate this by some examples.
2 A stability theorem for time variant particle interaction Let k · k be an arbitrary vector space norm on IRd which is fixed in what follows. For a subset M ⊆ IRd the diameter of M is defined by ∆(M ) = sup{k m − m0 k | m, m0 ∈ M }. The convex hull of M , denoted by convM , n n P P is the set of all convex combinations αk xk , where 0 ≤ αk , αk = 1 k=1
k=1
and xk ∈ M . The following lemma extends a useful inequality known for one dimension (cf. [15, Theorem 3.1]) into higher dimensions.
Positive Particle Interaction
Lemma 1. Let x1 , . . . , xn ∈ IRd and let y i =
n P k=1
201
aik xk for 1 ≤ i ≤ n and
A = (aij ) a row–stochastic matrix. Then the following inequality holds ∆(conv{y 1 , . . . , y n }) ≤ (1 − min
1≤i,j≤n
n X
min{aik , ajk })∆(conv{x1 , . . . xn })
k=1
(4) P
Proof. First, we show that for any two convex combinations αk uk and P d k k k βk v of points u and v , respectively, from IR for k ∈ I finite, one has that X X k αk uk − βk v k k≤ max{k ui − v j k | i, j ∈ I}. (5) P For, βk v k then P if wk= P k αk u − w k = kP αk (uk − w) k≤ max{k ui − w k | i ∈ I} and k w − ui k = k βk (v k − ui ) k≤ max{k v j − ui k | j ∈ I}. From (5) we have that ∆(conv{y 1 , . . . , y n }) ≤ max{k y i − y j k | i, j ∈ I} with I = {1, . . . , , n}. (6) P Let λhk = ahk − min{aik , ajk } for h = i, j. Obviously, λhk ≥ 0 and λik = k∈I P P λjk = rij with rij = 1 − min{aik , ajk }. For rij > 0 let αhk = λrhk and, ij k∈I k∈I P P hence, αik = αjk = 1. k∈I
Now
k∈I
P P P (aik − ajk )xk k=k λik xk − λjk xk k P P = rij k αik xk − αjk xk k for rij > 0
k yi − yj k = k
and, by (5),
k y i − y j k≤ rij max{k xk − xl k | k, l ∈ I}.
(7)
If rij = 0 then λik = λjk for all k ∈ I and, hence, aik = ajk for all k. In this case, y i = y j and (7) holds trivially. Equations (6) and (7) together prove inequality (4). For the model of positive particle interaction x(t + 1) = A(t)x(t) as introduced in the previous section we obtain the following stability result. Theorem 1. Denote for s, t ∈ IN with s < t the matrix product A(t − 1)A(t − 2) · · · A(s) by B(t, s) with entries bij (t, s). Suppose there exist a sequence 0 = t0 < t1 < t2 < . . . in IN and a sequence δ1 , δ2 , . . .in [0, 1] ∞ P δm = ∞ such that for all 1 ≤ i, j ≤ n and all m ≥ 1 the following with m=1
inequality holds
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Ulrich Krause n X
min{bik (tm , tm−1 ), bjk (tm , tm−1 )} ≥ δm
(8)
k=1
Then for arbitrary starting points x1 (0), . . . , xn (0) in D there exists an equilibrium x∗ ∈ conv{x1 (0), . . . , xn (0)} ⊆ D such that for all 1 ≤ i ≤ n lim xi (t) = x∗ .
(9)
t→∞
Furthermore, for any other starting points y 1 (0), . . . , y n (0) in D with equilibrium y ∗ it holds that k x∗ − y ∗ k≤ max{k xi (0) − y j (0) k | 1 ≤ i, j ≤ n}.
(10)
Proof. From x(t + 1) = A(t)x(t) for t ∈ IN we have for s, t ∈ IN with s < t x(t) = A(t − 1)A(t − 2) · · · A(s)x(s) = B(t, s)x(s) and, hence, x(tm ) = B(tm , tm−1 )x(tm−1 ) for m ≥ 1. n P For y i = xi (tm ) = bik (tm , tm−1 )xk (tm−1 ) and M (t) = conv{x1 (t), . . . , xn (t)} k=1
from Lemma 1 we obtain, taking assumption (8) into account, that ∆M (tm ) ≤ (1 − δm )∆M (tm−1 ). By iteration ∆M (tm ) ≤ (1 − δm )(1 − δm−1 ) · · · (1 − δ1 )∆M (0). By the mean value theorem 1 − r ≤ exp(−r) for r ≥ 0 and, hence, Ã m ! X δi ∆M (0) for all m. ∆M (tm ) ≤ exp − i=1
By assumption
∞ P m=1
δm = ∞ and, hence, lim ∆M (tm ) = 0. From x(t + 1) = m→∞
A(t)x(t) it follows that M (t + 1) ⊆ M (t) and, hence, ∆M (t + 1) ≤ ∆M (t). Since ∆M (tm ) converges to 0 this shows that ∆M (t), too, converges to 0. ∞ T Furthermore, since M (t), t ∈ IN, is compact we have that M (t) is non– ∗
empty. For x ∈
∞ T t=0
t=0
M (t) it follows for every 1 ≤ i ≤ n that
k x∗ − xi (t) k≤ ∆M (t) for all t and, hence, lim xi (t) = x∗ for all i. t→∞
Obviously, x∗ ∈ M (0) = conv{x1 (0), . . . , xn (0)} and, similarly, y ∗ ∈ conv{y 1 (0), . . . , y n (0)}. Therefore, inequality (5) implies inequality (10). The above theorem and its crucial condition (8) are inspired by the treatment of consensus formation in one dimension as in [1] and [2]. (Other extensions can be found in [11] and [12].) In the next section we present more easy–to–use criteria for condition (8) to hold. Roughly speaking, these criteria require that interaction between the particles does not decay too fast.
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203
3 Stability for slowly decaying interaction One cannot expect a common globally asymptotically stable equilibrium if there is almost no interaction between the particles. Similarly, for time variant interaction, one cannot expect conclusion (9) of Theorem 1 to hold if interaction is vanishing too fast. More precisely, we say that for the model given as before by x(t + 1) = A(t)x(t), t ∈ IN, there holds slowly decaying interaction of degree p if there exists a base matrix A ∈ IRn×n and a + decreasing function f : IR+ → IR+ such that A(t) ≥ f (t)A for all t ≥ s for some fixed s ∈ IN and
Z∞
(11)
f (t)p dt = ∞ where p ≥ 1.
(12)
1
Thus, by this definition the interaction between the particles may become weaker in the course of time but a lower ceiling f (t) of interaction should exist which on the average is big enough. Theorem 2. Suppose that interaction is slowly decaying of degree p and that (p) for any two particles i and j there exists a third one k such that aik > 0 (p) and ajk > 0 for the entries of the p–th power of the base matrix A. Then the conclusions (9) and (10) of Theorem 1 hold. Proof. We show that the assumptions made imply condition (8) of Theorem 1. For m ∈ IN let tm = pm and suppose that m ≥ p1 + 1. From (11) we get B(tm , tm−1 ) = A(tm − 1) · · · A(tm−1 ) ≥ f (tm )p Ap taking into account that f is decreasing. For the entries bij (tm , tm−1 ) of B(tm , tm−1 ) this implies n X
min{bik (tm , tm−1 ), bjk (tm , tm−1 )} ≥ f (tm )p
n X k=1
k=1
(p)
(p)
min{aik , ajk }.
By assumption on Ap ( δ 0 = min
n X k=1
) (p) (p) min{aik , ajk }
| 1 ≤ i, j ≥ n
> 0.
Thus, inequality (8) is satisfied for δm = δ 0 f (tm )p and we have to show that ∞ P δm = ∞. m=1
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According to a general relationship between the convergence of series and ∞ P f (tm )p = ∞ if and only integrals (cf. [8, Theorem 3, p. 64]) we have that if
R∞ 1
m=1
p
f (pt) dt = ∞. The latter follows from (12) and, hence, we arrive at ∞ X m=1
δm ≥ δ
0
∞ X
f (tm )p = ∞.
s m≥ p
To conclude we mention a few examples which will be discussed in detail elsewhere. Examples 3 a) Obviously, constant interaction, i.e., A(t) = A for all t, provides an example for f identically equal to 1 and A as in Theorem 2. In particular, for d = 1, D = IR+ one obtains the Basic Limit Theorem for Markov Chains (cf. [13, page 230])which states that lim At = B for each t→∞ regular stochastic matrix, where the rows of B are all equal to a vector n P v 0 ≥ 0 with vi = 1 and v 0 A = v 0 . “Regular” means that all entries of i=1
some power Aq of A are all (strictly) positive. Obviously, a regular matrix A satisfies the assumption made on A in Theorem 2 whereas ¸ converse · the 1 0 is not true as can be seen from the simple example A = 1 1 . 2 2
b) Consider Jacobi–interaction, where the particles can be labelled in such a way that each particle interacts (strictly) positively with its direct neighbours, that is, A(t) has for every t the structure ++ + + + 0 +++ A(t) = where + indicates +++ .. 0 . ++
a (strictly) positive entry. Suppose that the smallest positive entry of A(t) 1√ for t big enough. Choosing the latter as f (t) and p = n−1 is at least n−1 t the conclusions of Theorem 2 hold. It is easy to give examples where the entries of A(t) decay too fast for the conclusions of Theorem 2 to hold. In the field of consensus formation this phenomenon is known as a fast “hardening of positions” (cf. [3]). c) To treat heat diffusion in an inhomogeneous medium consider an agglomeration of n pieces of different materials which has been heated from the outside and for which we will study the movement of heat through all the pieces. Denote by xi (t) the temperature (in Kelvin) of piece of material i at time t ∈ IN. Obviously, d = 1, and let D denote the range of relevant termperatures. By Newton’s law of cooling we have that
Positive Particle Interaction
xi (t + 1) − xi (t) =
X
aij (t)(xj (t) − xi (t))
205
(13)
j6=i
for the change in temperature of piece i, where coefficient aij (t) measures heat transfer from piece j to i and may depend on various circumstances as the materials of pieces i and j, their boundaries, the states xj (t) and time t directly via, e.g., changing room temperature. d) Consider finitely many bodies in Euclidean space which attract each other by some pseudo–gravitational force which goes inversely with a certain power of the bodies distance. If the continuous dependence on distance is replaced by an appropriate step function one obtains the nonlinear system in discrete time X xi (t + 1) = αij (x(t)xj (t) (14) j∈I(i,x(t))
where I(i, t) is the set of bodies “near” to body i at state x(t) = (x1 (t), . . . , xn (t))0 and the αij make a row–stochastic matrix by adding zeroes. Actually, equation (13) has been originally obtained within a model of consensus formation under bounded confidence ([5, 7, 11]). There the ”particles” are experts who assess a certain issue and interact by exchanging opinions.
References 1. S. Chatterjee. Reaching a consensus: Some limit theorems. Proc. Int. Statist. Inst., pages 159–164, 1975. 2. S. Chatterjee and E. Seneta. Toward consensus: some convergence theorems on repeated averaging. J. Applied Probability, 14:89–97, 1977. 3. J.E. Cohen, J. Hajnal, and C.M. Newman. Approaching consensus can be delicate when positions harden. Stochastic Proc. and Appl., 22:315–322, 1986. 4. J. Conlisk. Stability and monotonicity for interactive Markov chains. J. Math. Sociology, 17:127–143, 1992. 5. J.C. Dittmer. Consensus formation under bounded confidence. Nonlinear Analysis, 47:4615–4621, 2001. 6. L. Farina and S. Rinaldi Positive Linear Systems: Theory and Applications. Wiley & Sons, New York, 2000. 7. R. Hegselmann and U. Krause. Opinion dynamics and bounded confidence: Models, analysis and simulation. J. Artificial Societies and Social Simulation, 5 (3), 2002. http://jasss.soc.surrey.ac.uk/5/3/2.html. 8. K. Knopp. Infinite Sequences and Series. Dover Publ., New York, 1956. 9. U. Krause. Positive nonlinear systems: Some results and applications. In V. Lakshmikantham, editor, World Congress of Nonlinear Analysts, pages 1529– 1539. De Gruyter, Berlin, 1996. 10. U. Krause and T. Nesemann. Differenzengleichungen und diskrete dynamische Systeme. Teubner, Stuttgart, 1999.
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11. U. Krause. A discrete nonlinear and non–autonomous model of consensus formation. In S. Elaydi, G. Ladas, J. Popenda, and J. Rakowski, editors, Communications in Difference Equations, 227–236. Gordon and Breach Science Publ., Amsterdam, 2000. 12. N. Kruse. Semizyklen und Kontraktivit¨ at nichtlinearer positiver Differenzengleichungen mit Anwendungen in der Populationsdynamik. Ph. thesis, dissertation.de, Verlag im Internet. Bremen, 1999. 13. D.G. Luenberger. Introduction to Dynamic Systems. Theory, Models, and Applications. Wiley & Sons, New York, 1979. 14. T. Nesemann. Stability Behavior of Positive Nonlinear Systems with Applications to Economics. PhD thesis, Wissenschaftlicher Verlag, Berlin, 1999. 15. E. Seneta. Non–negative Matrices and Markov Chains, 2nd. edition. Springer, New York, 1980.
Stability of Linear Systems and Positive Semigroups of Symmetric Matrices Tobias Damm Institute of Applied Mathematics, TU Braunschweig, Germany, [email protected]
Abstract. The role of Lyapunov operators in stability theory is well-known. In this paper we present an interesting characterization of Lyapunov operators. We show that an operator generates a positive group on the real space of real or complex Hermitian matrices, if and only if it is a Lyapunov operator.
1 Introduction There is an interesting relation between stability of linear dynamical systems in Rn and positive semigroups on the space S n of symmetric matrices in Rn×n . A famous result due to Lyapunov states that the linear system x˙ = Ax is asymptotically stable if and only if there exists a positive definite matrix V ∈ S n , such that AT V + V A ∈ S n is negative definite. In honour of Lyapunov, the matrix operator LA : X 7→ AT X +XA is called a Lyapunov operator. It is easy to see that by the negativity of√AT V + V A the positive definite matrix V defines a Euclidean norm kxkV = xT V x (an energy norm) on Rn with respect to which all non-zero solutions of x˙ = Ax are strictly decreasing. This is the usual interpretation of Lyapunov’s result. But there is another useful point of view. Obviously, a solution x(t) converges to zero, if and only if the family of rank-1 positive semidefinite matrices X(t) = x(t)x(t)T does. This family is subject to the differential equation X˙ = L∗A (X), where L∗A = LAT is the operator adjoint to LA . Since X(t) ≥ 0 for all t ∈ R+ , it follows that for all A ∈ Rn×n the Lyapunov operator L∗A generates a positive semigroup on S n . We note the simple fact that the given system is stable, if and only if the positive semigroup generated by LAT is stable. It turns out (cf. Remark 1) that, in general, positive semigroups on S n can be used to characterize stability and robust stability properties of various types of linear dynamical systems in Rn . This unifying way of characterization constitutes a useful connection between stability problems and spectral properties of generators of positive semigroups, which we discuss in Theorem 2. To classify different stability problems it would be useful to have a general representation result L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 207-214, 2003. Springer-Verlag Berlin Heidelberg 2003
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for generators of positive semigroups on S n . To the author’s best knowledge, however, this is an open problem. As a step in this direction, in Section 3 we characterize the set of generators of positive groups on S n . More precisely, we prove that this set coincides with the set of Lyapunov operators.
2 Exponentially positive operators on S n and stability of linear systems in Rn n Let S n denote the set of n × n symmetric matrices and S+ ⊂ S n the cone of nonnegative definite matrices X ≥ 0. It is well-known that S n together with n the inner product hX, Y i = trace XY is a Hilbert space and S+ is a self-dual closed normal solid convex cone (e.g. [14]). We write AT for the conjugate transpose of a matrix A ∈ Rn×n .
Definition 1. A linear operator T : S n → S n is said to be n n (i) positive, if T (S+ ) ⊂ S+ , (ii) a generator of a positive group, if etT : S n → S n is positive for all t ∈ R, (iii) a Lyapunov operator, if there exists an A ∈ Rn×n , such that
∀X ∈ S n :
T (X) = AX + XAT .
In this case we write T = LA . If etT : S n → S n is positive only for all t ∈ R+ , then T generates a positive semigroup. We recall a number of equivalent properties from [18, 7, 1]. Definition 2. A linear operator T : S n → S n is said to be (i) exponentially positive, if etT : S n → S n is positive for all t ≥ 0 (i.e. T generates a positive semigroup). (ii) resolvent positive, if (αI − T )−1 is positive for sufficiently large α > 0, n n (iii) quasi-monotonic, if for all X ∈ S+ there exists an Y ∈ S+ , such that hX, Y i = 0 and hT (X), Y i ≥ 0, n implies hT (X), Y i ≥ 0, (iv) cross-positive, if hX, Y i = 0 for X, Y ∈ S+ (v) essentially positive, if T ∈ cl{S − αI | S : S n → S n is positive, α ∈ R}. The equivalence of these properties was established in [7] (cf. also [1]) for general finite-dimensional vector spaces ordered by a closed normal solid cone. Theorem 1. For a linear operator T : S n → S n the properties (i)–(v) from Definition 2 are equivalent. It follows immediately from property (v), that every positive operator is exponentially positive, and so are arbitrary linear combinations of exponentially positive operators with nonnegative coefficients. In other words, the set of exponentially positive operators forms a closed solid wedge (cf. [14]) in the space of endomorphisms on S n .
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One easily verifies that all Lyapunov operators generate positive groups. In Section 3 we will show that, in fact, Lyapunov operators are the only generators of positive groups on S n . Lemma 1. Every Lyapunov operator generates a positive group on S n . Proof. Let T = LA for some A ∈ Rn×n . By Lyapunov’s Theorem (e.g. [11]), L−1 A is positive if all eigenvalues of LA have positive real part. Thus (αI − LA )−1 = (L−A+ α2 I )−1 is positive for sufficiently large α ∈ R. Hence T is resolvent positive and thus exponentially positive. Since −T = L−A is a Lyapunov operator, too, it is exponentially positive as well. Thus, e±T t is positive for all t ≥ 0. Corollary 1. Let A, B1 , . . . , BN ∈ Rn×n . Then the operator X 7→ T (X) = AT X + XA +
N X
BjT XBj
(1)
j=1
is exponentially positive. Proof. The operator is the sum of a Lyapunov operator and a positive operator and hence exponentially positive. Remark 1. Operators of the form (1) occur in the stability analysis of different types of linear systems on Rn : 1. The continuous-time deterministic system x˙ = Ax is asymptotically stable, if and only if (cf. [9, 19]) ∃X > 0 : T1 (X) = AT X + XA < 0 . 2. The discrete-time deterministic system xk+1 = Axk is asymptotically stable, if and only if (cf. [19]) ∃X > 0 : T2 (X) = AT XA − X < 0 . 3. The PN continuous stochastic differential equation of Itˆo-type dx = Ax dt + j=1 Bj x dwj with independent normed Wiener processes wj is asymptotically mean-square stable, if and only if (cf. [13]) ∃X > 0 : T3 (X) = AT X + XA +
N X
BjT XBj < 0 .
j=1
4. The discrete stochastic system x(k + 1) = Ax(k) + mean-square stable, if and only if (cf. [15]) ∃X > 0 : T4 (X) = AT XA − X +
N X j=1
PN
j=1
Bj x(k) wj (k) is
BjT XBj < 0 .
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5. The deterministic delay system x(t) ˙ = Ax(t)+Bx(t−h) is asymptotically stable for all delays h > 0, if (cf. [12]) ∃X > 0 : T5 (X) = AT X + XA + X + B T XB < 0 . PN 6. The uncertain linear system x˙ = (A + j=1 δj (t, x)Bj )x is asymptotically stable for arbitrary measurable functions δ : R × Rn → [−1, 1], if (cf. [3]) ∃X > 0 : T6 (X) = AT X + XA + nX +
N X
BjT XBj < 0 .
j=1
Based on these criteria, the following theorem (compiled from [17, 6, 16, 4, 8, 5]) constitutes a useful tool in the analysis of stability and stabilizability. PN Theorem 2. Let A, B1 , . . . , BN ∈ Rn×n and set ΠB : X 7→ j=1 BjT XBj . By σ and ρ we denote the spectrum and the spectral radius of a linear operator. The following are equivalent: ¯ (a) σ (LA + ΠB ) ⊂ C− = {z ∈ C ¯ < z < 0}. (b) max σ(LA + ΠB ) ∩ R < 0. (c) σ (LA ) ⊂ C− and ∀τ¡ ∈ [0, 1] ¢: det (LA + τ ΠB ) 6= 0. (d) σ (LA ) ⊂ C− and ρ L−1 A ΠB < 1. (e) ∀Y < 0 : ∃X > 0 : LA (X) + ΠB (X) = Y . (f ) ∀Y ≤ 0 with (A, Y ) observable: ∃X > 0 : LA (X) + ΠB (X) = Y . (g) ∃X > 0 : LA (X) + ΠB (X) < 0. (h) ∃X ≥ 0 : LA (X) + ΠB (X) < 0. (i) ∃Y ≤ 0 with (A, Y ) observable: ∃X ≥ 0 : LA (X) + ΠB (X) ≤ Y . n (j) ∃Y ≤ 0 s.t. LA (X) + ΠB (X) ≤ Y for some X ∈ S+ , and XY 6= 0 for n every pair (X, λ) ∈ (S+ \ {0}) × C+ satisfying (LA + ΠB )∗ (X) = λX. In a sense, this theorem parallels [2, Theorem 2.3], which provides even fifty equivalent conditions for a Z-matrix to be an M -matrix. (Here, exponentially positive operators correspond to Z-matrices, while stable exponentially positive operators correspond to M -matrices.)
3 Generators of positive groups on S n All exponentially positive operators considered in the previous section are of the form LA +Π, where Π : S n → S n is some positive operator. It is a natural question whether every exponentially positive operator on S n can be written in this form. As a partial result in this direction and the main result of this paper, we prove that every generator of a positive group on S n is a Lyapunov operator.
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211
Remark 2. For general solid regular cones in a finite-dimensional real vector space, the question, whether every exponentially positive operator is the sum of a positive operator and the generator of a positive group was posed in [20]; an affirmative answer was given for important classes of cones. On the other hand, it was shown in [10] that such a representation is impossible for almost all cones in a certain categorial sense. Nevertheless, the question seems to be n still open for the cone S+ in S n (see also Section 4). Theorem 3. A linear operator T : S n → S n is a generator of a positive group, if and only if T is a Lyapunov operator. Before proceeding with the proof of Theorem 3, we verify that the situation is different for discrete-time Lyapunov operators TA : X 7→ AT XA − X (cf. Rem. 1). While TA is exponentially positive, −TA , in general, is not: · ¸ · ¸ 00 01 and Xt = for t > 0. Example 1. Let A = 10 1 ·t ¸ t α−1 For all α > 0 we have αXt + TA (Xt ) = , which is positive for α − 1 αt large t, though Xt is always indefinite. Hence (αI + TA )−1 is not positive for any α, and hence −TA is not resolvent positive. By definition, T generates a positive group on S n , if and only if both T and −T are exponentially positive. Hence we have the following criterion. Lemma 2. A linear operator T : S n → S n generates a positive group, if and n only if hX, Y i = 0 for X, Y ∈ S+ implies hT (X), Y i = 0. Proof. By Theorem 1 the operator T : S n → S n generates a positive group, if and only if both T and −T are cross-positive. The latter holds if and only if n hX, Y i = 0 for X, Y ∈ S+ implies both hT (X), Y i ≥ 0 and −hT (X), Y i ≥ 0. We use this criterion in the proof Theorem 3. Proof of Theorem 3. Let T generate a positive group. By Lemma 2 this is equivalent to ³ ´ X, Y ≥ 0 and hX, Y i = 0 ⇒ hT X, Y i = 0 . (2) If ej denotes the j-th canonical unit vector in Rn , then the set n B := {ej eTk + ek eTj | j, k = 1, . . . , n} ⊂ S+
(3)
forms a basis of S n ⊂ Rn×n . It suffices to find an A ∈ Rn×n , such that T (X) = LAT (X) for all X ∈ B. Let X = ej eTj (i.e. 2X ∈ B). To apply criterion (2) we characterize all matrices n such that hX, Y i = 0 . Y ∈ S+
(4)
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Let Y ≥ 0 and hX, Y i = yjj = 0. Then necessarily the j-th row and column in Y vanish. Hence, (4) is true if and only if in Y ≥ 0 the j-th row and column vanish. Criterion (2) in turn implies that in T (X) everything vanishes except for the j-th row and column. Otherwise we could choose some Y satisfying (4) and hT (X), Y i 6= 0. For j = 1, . . . , n we thus have T (ej eTj ) = aj eTj +ej aTj with vectors a1 , . . . , an ∈ Rn . If we build the matrix A = (a1 , . . . , an ), then T (X) = AX + XAT for all X = ej eTj . In other words, we have found a unique candidate for the Lyapunov operator. It remains to show, that also for Xjk = ej eTk + ek eTj with j < k we have T (Xjk ) = AXjk + Xjk AT = aj eTk + ak eTj + ej aTj + ek aTk
a1k .. .
···
(5)
a1j .. .
a1k · · · 2ajk · · · ajj + akk · · · ank .. .. = . . . a1j · · · ajj + akk · · · 2akj · · · anj .. .. . . ank ··· anj Let j and k be fixed. A matrix Y satisfies condition (4) with X = Xjk + Xjj + Xkk ≥ 0 if in Y the j-th and k-th row and column vanish. As above we conclude from criterion (2), that in T (X) and hence also in T (Xjk ) everything vanishes except for the j-th and k-th row and column. Thus T (Xjk ) is of the general form T (Xjk ) = bj eTj + ej bTj + bk eTk + ek bTk ,
b1j .. .
···
with bj , bk ∈ Rn
b1k .. .
(6)
b1j · · · 2bjj · · · bjk + bkj · · · bnj .. .. = . . . b1k · · · bjk + bkj · · · 2bkk · · · bnk .. .. . . bnj · · · bnk Now we consider matrices of the form X = xxT with x = xj ej + xk ek where xj , xk ∈ R are arbitrary real numbers. Writing X = xj xk Xjk + x2j ej eTj + x2k ek eTk , and exploiting the linearity of T we have the decomposition
Stability of Linear Systems and Positive Semigroups
T (X) = xj xk T (Xjk ) + x2j T (ej eTj ) + x2k T (ek eTk ) =
xj xk (bj eTj + x2j (aj eTj
Let y⊥x, for instance y1 .. y = . ∈ Rn yn
+ ej bTj + bk eTk + ek bTk ) + ej aTj ) + x2k (ak eTk + ek aTk )
213
(7) .
yj = xk , yk = −xj , with y` arbitrary for ` 6∈ {j, k} .
(8)
Then Y = yy T satisfies condition (4), and by (2) we have hT (X), Y i = 0. If we write T (X) like in equation (7) we obtain: 1 1 1 hT (X), Y i = trace(T (X)Y ) = y T T (X)y 2 2 2 Ã ! n n n n X X X X b`j y` + yk a`j y` + x2k yk = xj xk yj b`k y` + x2j yj a`k y`
0=
`=1
=
`=1
`=1
`=1
xj x3k (bjj − ajk ) + x3j xk (bkk − akj ) + x2j x2k (−bkj − bjk + ajj + akk ) X X + xj x2k y` (b`j − a`k ) + x2j xk y` (−b`k + a`j ) . `6∈{j,k}
`6∈{j,k}
The right hand side is a homogeneous polynomial in the real unknowns xj , xk , and y` for ` 6∈ {j, k}. Since these unknowns can be chosen arbitrarily, all the coefficients of the polynomial necessarily vanish, i.e. bjj = ajk , bkk = akj , bkj + bjk = ajj + akk , b`j = a`k , b`k = a`j . Inserting these data into (6), we see that (5) holds.
4 Open questions We conclude this paper with two open questions: 1. Can every exponentially positive operator T on S n be written in the form T = LA +Π with some Lyapunov operator LA and some positive operator Π? 2. A positive operator Π : S n → S n is called completely positive, PNif for some N ∈ N there exist matrices B1 , . . . , BN , such that T (X) = j=1 BjT XBj for all X ∈ S n . Let us call T completely resolvent positive if αI − T is completely positive for α À 1. Then we ask: Can every completely resolvent positive operator T on S n be written in the form T = LA + Π with some Lyapunov operator LA and some completely positive operator Π?
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References 1. A. Berman, M. Neumann, and R. J. Stern. Nonnegative Matrices in Dynamic Systems. John Wiley & Sons, New York, 1989. 2. A. Berman and R. J. Plemmons. Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics. SIAM, 1994. 3. D. S. Bernstein. Robust static and dynamic output-feedback stabilization: Deterministic and stochastic perspectives. IEEE Trans. Autom. Control, AC32(12):1076–1084, 1987. 4. T. Damm and D. Hinrichsen. Matrix (in)equalities for linear stochastic systems. In Proceedings of MTNS-98, Padova, Italy, 1998. Il Poligrafio. 5. T. Damm and D. Hinrichsen. Newton’s method for a rational matrix equation occuring in stochastic control. Linear Algebra Appl., 332–334:81–109, 2001. 6. L. Elsner. Monotonie und Randspektrum bei vollstetigen Operatoren. Arch. Ration. Mech. Anal., 36:356–365, 1970. 7. L. Elsner. Quasimonotonie und Ungleichungen in halbgeordneten R¨ aumen. Linear Algebra Appl., 8:249–261, 1974. 8. M. D. Fragoso, O. L. V. Costa, and C. E. de Souza. A new approach to linearly perturbed Riccati equations in stochastic control. Applied Mathematics and Optimization, 37:99–126, 1998. 9. F. R. Gantmacher, The Theory of Matrices (Vol. II). Chelsea, New York, 1959 10. P. Gritzmann, V. Klee, and B.-S. Tam. Cross-positive matrices revisited. Linear Algebra Appl., 223/224:285–305, 1995. 11. R. A. Horn and C. R. Johnson. Topics in Matrix Analysis. Cambridge University Press, Cambridge, 1991. 12. V. L. Kharitonov. Robust stability analysis of time delay systems: A survey. In Commande et Structure des Syst`emes, pages 1–12, Nantes, 1998. Conference IFAC. 13. R. Z. Khasminskij. Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Alphen aan den Rijn, NL, 1980. 14. M. A. Krasnosel’skij, J. A. Lifshits, and A. V. Sobolev. Positive Linear Systems - The Method of Positive Operators, volume 5 of Sigma Series in Applied Mathematics. Heldermann Verlag, Berlin, 1989. 15. T. Morozan. Stabilization of some stochastic discrete-time control systems. Stochastic Analysis and Application, 1:89–116, 1983. 16. T. Sasagawa and J. L. Willems. Parametrization method for calculating exact stability bounds of stochastic linear systems with multiplicative noise. Automatica, 32(12):1741–1747, 1996. 17. H. Schneider. Positive operators and an inertia theorem. Numerische Mathematik, 7:11–17, 1965. 18. H. Schneider and M. Vidyasagar. Cross-positive matrices. SIAM J. Numer. Anal., 7(4):508–519, 1970. 19. E. D. Sontag. Mathematical Control Theory, Deterministic Finite Dimensional Systems. Springer-Verlag, New York, 2nd edition, 1998. 20. R. J. Stern and H. Wolkowitz. Exponential nonnegativity on the ice-cream cone. SIAM J. Matrix. Anal. Appl., 12:755–778, 1994.
Digraph-based Conditioning for Markov Chains Stephen J. Kirkland Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, S4S 0A2, Canada, [email protected]
Abstract Let T be an irreducible stochastic matrix, so that we can consider T to be the transition matrix for a Markov chain; one of the central quantities of interest for that chain is the stationary vector for T , i.e. the left Perron vector π t for T , normalized so that its entries sum to 1. It is well-known that in the case that T is primitive then the iterates of the chain converge to π t . It is natural to consider the stability of π t under perturbation of T , and so we focus on the following quantity associated with T , known as a condition number for the chain. Given an n × n stochastic matrix T that has 1 as an algebraically simple eigenvalue (this holds if T is irreducible, for instance), let Q = I − T , and denote the group generalized inverse of Q by Q# , i.e. the unique matrix X such that XQX = X, QXQ = Q and QX = XQ. Define the condition number c(T ) by c(T ) =
1 max max (Q# − Q# i,j ). 2 1≤j≤n 1≤i≤n j,j
For matrices T and T˜ as above, we have max1≤i≤n |πi − π ˜i | ≤ c(T )||T − T˜||∞ , where for matrices, || • ||∞ is the maximum absolute row sum norm (see [2]). Further, for any irreducible stochastic T, there is a family of perturbation matrices E of arbitrarily small norm so that for each such E, the matrix T˜ = T − E is irreducible and stochastic, and in addition, max1≤i≤n |πi − π ˜i | > c(T )||E||∞ /2 (see [1]). Thus the quantity c(T ) provides a reasonable measure of the stability of the stationary distribution when T is perturbed. There is no upper bound on c(T ) as T ranges over the entire set of n × n irreducible stochastic matrices, so it is natural to look for subclasses of transition matrices for which c(T ) is bounded as T ranges over such a class. The present piece of work proceeds in this direction by considering the directed graph associated with T, which is denoted by ∆(T ), and addressing the following two problems: 1. Given a strongly connected directed graph D, let L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 215-216, 2003. Springer-Verlag Berlin Heidelberg 2003
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SD = {T |T is n × n, stochastic and ∆(T ) is a subgraph of D}; characterize the digraphs D such that c(T ) bounded from above as T ranges over the matrices in SD that have 1 as an algebraically simple eigenvalue. 2. For those digraphs D such that c(T ) is bounded on SD , find sup{c(T )|T ∈ SD }. The following result, which is established using algebraic and combinatorial techniques, deals with problem 1. Theorem 1. Let D be a strongly connected directed graph on n vertices. If D has at least two vertex-disjoint directed cycles, then for each m ∈ R, there is an irreducible matrix T ∈ SD such that c(T ) > m. Conversely, if any two cycles in D intersect in at least one vertex, then c(T ) is bounded on SD . It turns out that if D is a strongly connected digraph with the property that any two cycles intersect in at least one vertex, then c(T ) attains its maximum value on SD at a (0, 1) matrix in SD . Further, for any (0, 1) matrix T ∈ SD , ∆(T ) contains a unique cycle. These considerations, along with a formula for the group inverse associated with such a (0, 1) matrix, yield the following. Theorem 2. Let D be a strongly connected digraph on n vertices such that ˜D ˜ is a subgraph of any two cycles intersect in at least one vertex. Let G = {D| ˜ ˜ ˜ denote the D and each vertex of D has outdegree 1}. For each D ∈ G, let k(D) ˜ ˜ length of the single cycle in D, and let p(D) denote the length of the longest ˜ p(D) ˜ Then maxT ∈S c(T ) = 1 max ˜ path in D. D ˜ . In particular, letting g be D∈G k(D) 2 the length of the shortest cycle in D and p be the length of the longest path in p D, we have that for each T ∈ SD , c(T ) ≤ 2g ≤ n−1 2g . Corollary 2.1. Suppose that D is a strongly connected directed graph on n vertices and let g be the length of its shortest cycle. If g ≥ n+1 2 , then for each (n−1) n−1 T ∈ SD , we have c(T ) ≤ 2g ≤ n+1 .
References 1. S. Kirkland, Conditioning properties of the stationary distribution for a Markov chain, Electronic Journal of Linear Algebra 10:1-15 (2003). 2. S. Kirkland, M. Neumann and B. Shader, Applications of Paz’s inequality to perturbation bounds for Markov chains, Linear Algebra and its Applications 268: 183-196 (1998).
Paths and Cycles in the Totally Positive Completion Problem Cristina Jord´an and Juan R. Torregrosa Dpto. de Matem´ atica Aplicada, Universidad Polit´ecnica de Valencia, Valencia, Spain, {cjordan,jrtorre}@mat.upv.es Abstract. An n × n real matrix is said to be totally positive if every minor is nonnegative. In this paper, we are interested in totally positive completion problems, that is, when a partial totally positive matrix has a totally positive matrix completion. This problem has, in general, a negative answer when the graph of the specified entries of the partial matrix is a path or a cycle. For these cases, we obtain necessary and sufficient conditions in order to obtain the desired completion.
1 Introduction A partial matrix over R is an array in which some entries are specified, while the remaining entries are free to be chosen from R. We make the assumption throughout that all diagonal entries are prescribed. A completion of a partial matrix is the conventional matrix resulting from a particular choice of values for the unspecified entries. A matrix completion problem asks which partial matrices have completions with some desired property. An n×n partial matrix is said to be combinatorially symmetric if the (i, j) entry is specified if and only if the (j, i) entry is. The specified positions in an n × n combinatorially symmetric partial matrix A = (aij ) can be represented by an undirected graph GA = (V, E), where the set of vertices V is {1, ..., n} and {i, j}, i 6= j, is an edge if and only if the (i, j) entry is specified. Since all diagonal entries are specified, we omit loops. A path is a sequence of edges {i1 , i2 }, {i2 , i3 }, ..., {ik−1 , ik } in which all vertices are distinct. A cycle is a closed path, that is a path in which the first and the last vertices coincide. A graph is chordal if it has no minimal induced cycles of length 4 or more (see [2]). A graph is connected if there is a path from any vertex to any other vertex. A graph is complete if it includes all possible edges between its vertices. A clique is an induced subgraph that is complete. An n × n real matrix is called a totally positive matrix (T P -matrix) if every minor is nonnegative, that is, det A[α|β] ≥ 0, for all α, β ⊆ {1, ..., n}
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such that |α| = |β|, where A[α|β] denotes the submatrix of A lying in rows α and columns β. The principal submatrix A[α|α] is abbreviated to A[α]. These matrices are getting an increasing importance in approximation theory, combinatorics, statistic, economics, computer aided geometric design and wavelets. To mark that in interactive design the variation diminishing properties of totally positive matrices allows that the curve imitates the shape of its control polygon; thus we can predict or manipulate the shape of the curve by choosing or changing the control polygon suitably, [3]. See [1] for a comprehensive survey from an algebraic point of view and historical references, and [6] for many classical applications of T P -matrices. The following simple facts are very useful in the study of T P -matrices. Proposition 1. Let A = (aij ) be an n × n T P - matrix. Then, 1. If D is a positive diagonal matrix, then DA, AD are T P -matrices. 2. If D is a nonsingular positive diagonal matrix, then DAD−1 is a T P matrix. 3. If P is the permutation matrix P = [n, n − 1, . . . , 2, 1], then P AP T is a T P -matrix. · ¸ A 0 4. If B is a T P -matrix, of size m × m, then the matrix , is totally 0 B positive. 5. Any submatrix of A is a T P -matrix. The last property of Proposition 1 allows us to give the following definition. Definition 1. A partial matrix is said to be a partial totally positive matrix if every completely specified submatrix is a totally positive matrix. Our interest here is the totally positive matrix completion problem, that is, to know when a partial T P -matrix has a T P -matrix completion. The first natural question is: for which graphs G does every partial T P -matrix, the graph of whose specified entries is G, have a totally positive completion? When the diagonal entries of the partial matrix are nonzero, property (1) of Proposition 1 allows us to assume that them are equal to 1. In addition, from property (4) of the same proposition, we can work, without loss of generality, with connected graphs. On the other hand, because total positivity is not preserved by permutation similarity we must consider labeled graphs, that is, graphs in which the numbering of the vertices is fixed. The graph G is said to be 1-chordal graph [2] if G is a chordal graph in which every pair of maximal cliques Ci , Cj , Ci 6= Cj intersect in at most one vertex. A monotonically labeled 1-chordal graph is a labeled 1-chordal graph in which the maximal cliques are labeled in natural order, that is, for every pair of maximal cliques Ci , Cj in which i < j and Ci ∩ Cj = {u} the labeling within the two cliques is such that every element of {v : v ∈ Ci − u} is labeled less than u and every element of {w : w ∈ Cj − u} is labeled greater than u.
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In [4] the totally positive completion problem is resolved for connected monotonically labeled 1-chordal graphs. The proof of sufficiency, however, relies on the invertibility of the specified principal submatrices. We can relax, as the authors suggest, this invertibility assumption and we obtain the following result. Theorem 1. Let G be a labeled graph on n vertices. Every partial T P -matrix, the labeled graph of whose specified entries is G, has a totally positive completion if and only if G is a monotonically labeled 1-chordal graph. In section 2 we analyze the totally positive completion problem for a partial matrix whose associated graph is a path. We obtain necessary and sufficient conditions for the existence of a totally positive completion when the path is not monotonically labeled. In section 3 we show that the ”cycle condition” is a necessary and sufficient condition in order to obtain a totally positive completion of a partial T P -matrix whose graph is a cycle. In order to simplify the reasoning we consider partial T P -matrix with all specified entries positive. If some specified entries were zero our matrix becomes one of the several degenerate cases [7].
2 Paths Let A be a partial T P -matrix, the graph of whose specified entries is a path GA . If GA is monotonically labeled, from Theorem 1 we can assure that there exists a totally positive completion Ac of A. If GA is not monotonically labeled the completion problem has, in general, a negative answer, as we can see in the following example. 1x2 Example 1. The partial T P -matrix A = y 1 1 , whose graph is the non112 monotonically labeled path {1, 3}, {3, 2}, does not have a totally positive completion because det A[{1, 2}|{2, 3}] = x − 2 ≥ 0 and det A[{2, 3}|{1, 2}] = y − 1 ≥ 0, then xy ≥ 2. However, det A[{1, 2}] = 1 − xy ≥ 0, then xy ≤ 1. The partial T P -matrix, whose associated graph is the path of the above example, has the form 1 x12 a13 A = x21 1 a23 . a31 a32 1 In order to obtain a totally positive completion in a similar way to 1chordal case of Theorem 1, we complete x12 = a13 /a23 and x21 = a31 /a32 . It is easy to see that Ac is a T P -matrix if a13 a31 ≤ a23 a32 . Therefore, we introduce the following definition.
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Definition 2. Let G be a non-monotonically labeled path G with tree vertices, i, j, j, k. We say that the matrix A associated to G satisfies the ”edge condition” if: (a) j < i < k then ajk akj ≤ aij aji . (b) i < k < j then aij aji ≤ akj ajk . That is, an edge which lies ”inside” another edge has an edge product greater than or equal to the edge product of the ”outside” edge. In general, we say that a matrix A, of size n × n, whose associated graph is a path or a cycle, satisfies the ”edge condition” if every principal submatrix of size 3×3, whose associated graph is a non-monotonically labeled path, satisfies the above inequalities. Using this definition we can establish the following result. Proposition 2. Let A be a partial T P -matrix, of size 3 × 3, whose associated graph GA is a non-monotonically labeled path. There exists a totally positive completion if and only if A satisfies the edge condition. Proof. Taking into account property (3) of Proposition 1, we can reduce all the cases of GA to two: path {1, 3}, {3, 2} and path {2, 1}, {1, 3}. We are going to study the first case. The other is completely analogous. We can assume that matrix A has the form 1 x12 a13 A = x21 1 a23 . a31 a32 1 We use the edge condition to complete A by taking x12 = a13 /a23 , x31 = a31 /a32 . It is easy to verify that the 2 × 2 submatrices are totally positive and a31 ) det A[{2, 3}], which is nonnegative by the edge condition. det Ac = (1− aa13 23 a32 Now, let’s see the necessity of the edge condition. Suppose that there exists a totally positive completion Ac of A, x12 = c12 and x21 = c21 . Then det Ac [{1, 2}|{2, 3}] = c12 a23 − a13 ≥ 0 =⇒ c12 a23 ≥ a13 , det Ac [{1, 3}|{1, 2}] = a32 − a31 c12 ≥ 0 =⇒ a32 ≥ a31 c12 . We multiply the first inequality by a31 and use the second inequality to generate the edge condition a13 a31 ≤ a23 a32 . This completes the proof. We extend, in a natural way, Definition 2 for a non-monotonically labeled path with n vertices, n > 3, naming it ”path condition”. By using an induction process we obtain the following result. Proposition 3. Let A be a partial T P -matrix, of size n × n, n > 3, whose associated graph GA is a non-monotonically labeled path. If A satisfies the path condition then there exists a totally positive completion Ac of A.
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Example 2. Consider the following partial T P -matrix whose graph is a nonmonotonically labeled path u 3
u 1
u 4
u 2
1 x21 A= a31 a41
x12 1 x32 a42
a13 x23 1 x43
a14 a24 . x34 1
In this case, the path condition gives the following inequalities: (a) a14 a41 ≤ a13 a31 , (b) a14 a41 ≤ a24 a42 , (c) a14 a41 ≥ a13 a31 a24 a42 . First, we complete the principal submatrix A[{1, 2, 4}] by putting x12 = a14 /a24 and x21 = a41 /a42 . Then we obtain, by means of a similar process to used in Theorem 1, the totally positive completion: 1 a14 /a24 a13 a14 a41 /a42 1 a13 a24 /a14 a24 . Ac = a31 a31 a42 /a41 1 a14 /a13 a41 a42 a41 /a31 1
3 Cycles In this section we solve the totally positive completion problem for partial matrices whose associated graphs are cycles. In general, the mentioned problem has a negative answer for monotonous and non-monotonically labeled cycles, as we can see en the following examples. Example 3. The following partial T P -matrix, whose associated graph is a monotonically labeled cycle 1 u
u 2
4 u
u 3
1 0.8 A= x31 0.1
1 1 0.2 x42
x13 1 1 0.2
0.8 x24 , 0.7 1
has no T P -matrix completion, since det A[{1, 2}|{2, 4}] = x24 − 0.8 ≥ 0, then x24 ≥ 0.8. On the other hand, det A[{2, 3}|{3, 4}] = 0.7 − x24 ≥ 0, then x24 ≤ 0.7
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Example 4. The following partial T P -matrix, whose associated graph is a nonmonotonically labeled cycle 1 u
u 2
3 u
u 4
1 1 0.01 1 1 x23 A= 0.01 x32 1 x41 0.02 1
x14 0.02 , 1 1
has no totally positive completion, since from det A[{1, 2}|{3, 4}] ≥ 0 and det A[{2, 3}|{3, 4}] ≥ 0 we obtain x14 ≤ 0.01. On the other hand, from det A[{3, 4}|{1, 4}] ≥ 0 we have x41 ≤ 0.01. These values give det A[{1, 2, 4}] < 0. We can observe in both examples that the corresponding matrix satisfies the edge condition. So this condition is not sufficient. Moreover, we can extend Example 3 for n×n matrices. The matrix A = (aij ) such that aij = 1, ∀i, j, |i− j| ≤ 1, except an−1n = 0.7; a1n = 0.8, an1 = 0.1, and the remaining entries unspecified, is a partial T P -matrix whose associated graph is a monotonically labeled cycle and it satisfies the edge condition. Using a similar reasoning to Example 3 we prove that this matrix has no a totally positive completion. The matrix of size 3 × 3 whose associated graph is a cycle is completely specified. Therefore, we are going to work with partial T P -matrix of size n×n, n ≥ 4, the graph of whose specified entries is a monotonically labeled cycle. We can assume, without loss of generality, that this type of matrices have the form: 1 a12 x13 · · · x1n−1 a1n a21 1 a23 · · · x2n−1 x2n x31 a32 1 · · · x3n−1 x3n A= . .. .. .. .. . .. . . . . xn−11 xn−12 xn−13 · · · 1 an−1n an1 xn2 xn3 · · · ann−1 1 Definition 3. Let A = (aij ) a partial T P -matrix, of size n × n, whose associated graph is a monotonically labeled cycle. We say that A satisfies the ”cycle condition” if a12 a23 · · · an−1n ≥ a1n and ann−1 an−1n−2 · · · a21 ≥ an1 . Lemma 1. Let A be a partial T P -matrix, of size 4 × 4, whose graph is a monotonically labeled cycle. There exists a T P -matrix completion Ac of A if and only if A satisfies the cycle condition.
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Proof. Suppose that there exists a T P -matrix completion Ac of A, 1 a12 c13 a14 a21 1 a23 c24 Ac = c31 a32 1 a34 . a41 c42 a43 1 From the requirement det A[{1, 2}|{2, 3}] ≥ 0, we have the condition a12 a23 ≥ c13 , and from det A[{1, 3}|{3, 4}] ≥ 0 we have c13 a34 ≥ a14 . Combining these inequalities we obtain a12 a23 a34 ≥ a14 . An analogous reasoning for det A[{2, 3}|{1, 2}] and det A[{3, 4}|{1, 3}] gives the condition a43 a32 a21 ≥ a41 . So the cycle condition is necessary for the existence of a totally positive completion. For the sufficiency we take the completion x13 = a12 a23 , x24 = a14 /a12 , x31 = a41 /a43 and x42 = a43 a32 . (Note that if a12 = 0 or a43 = 0 we can give a easier completion). We must verify whether this completion is totally positive. The 2 × 2 and 3 × 3 submatrices are easily verified to be totally positive, with the cycle condition sometimes used to show the nonnegativity of its determinants. Finally, it is easy to see that det Ac = (1 − a43 a34 ) det Ac [{1, 2, 3}] ≥ 0. So Ac is a T P -matrix. Theorem 2. Let A be a partial T P -matrix, of size n × n, n ≥ 4, whose graph is a monotonically labeled cycle. There exists a totally positive completion Ac of A if and only if A satisfies the cycle condition. Proof. The proof is by induction on n. If n = 4 we apply Lemma 1. Let A be an n×n, n > 4, partial T P -matrix, whose associated graph is a monotonically labeled cycle. We generate the necessary condition a12 a23 · · · an−1n ≥ a1n combining the conditions obtained from det A[{1, 2}|{2, 3}] ≥ 0, det A[{1, 3}|{3, 4}] ≥ 0, . . ., det A[{1, n − 1}|{n − 1, n}] ≥ 0. Analogously, from det A[{2, 3}|{1, 2}] ≥ 0, det A[{3, 4}|{1, 3}] ≥ 0, . . ., det A[{n − 1, n}|{1, n − 1}] ≥ 0 we obtain the necessary condition ann−1 an−1n−2 · · · a21 ≥ an1 . Conversely, suppose that the cycle condition is satisfied. We take x1n−1 = a1n an1 ¯ an−1n and xn−11 = ann−1 , and the new matrix is denoted by A. The principal ¯ submatrix A[{1, 2, . . . , n − 1}] is a partial T P -matrix, of size (n − 1) × (n − 1), whose associate graph is a monotonically labeled cycle. By induction hypothesis, there exists a totally positive completion C of that submatrix. Let A˜ be the partial T P -matrix obtained from A¯ by completing the prin¯ cipal submatrix A[{1, 2, . . . , n−1}] to C. The associated graph of A˜ is a monotonically labeled 1-chordal with two maximal cliques. By applying Theorem 1 we obtain the desired completion. If we analyze the existence of a totally positive completion of a partial T P -matrix whose graph is a non-monotonically labeled cycle, we can observe that the ”edge condition” and the SP P -condition, introduced by Johnson and Smith in [5], are necessary conditions. But, what are sufficient conditions?
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Let A be a partial T P -matrix and GA its associated graph. In the next table we show the current status of the totally positive completion problem, denoting with ”yes” or ”no” the existence of the desired completion. It is an open problem the search of sufficient conditions for the existence of totally positive completions when the totally positive completion problem has a negative answer.
GA cycle, n ≥ 4
mon. labeled
non-mon. labeled
Yes, with cond.
No, in general
No, in general
No, in general
Yes
Yes, with cond.
GA non chordal GA non cycle
GA path GA chordal
GA 1-chordal GA non path Yes
No, in general
References 1. T. Ando, Totally positive matrices, Linear Algebra and Applications, 90, 165219, (1987). 2. J.R.S. Blair, B. Peyton, An introduction to chordal graphs an clique trees, The IMA volumes in Mathematics and its Applications, vol. 56, Springer, New York, 1-31, 1993. 3. J.M. Carnicer, J.M. Pe˜ na, Total positivity and optimal bases, in: Gasca, M. and Micchelli, C.A.,eds., Total positivity and its applications, Kluwer Academic, Dordrecht, 133-155, 1996. 4. C.R. Johnson, B.K. Kroschel, M. Lundquist, The totally nonnegative completion problem, Fields Institute Communications, American Mathematical Society, Providence, RI, 18:97-108, (1998). 5. C.R. Johnson, R. Smith, Path product matrices, Linear and multilinear Algebra, 46:177-191 (1999) 6. S. Karlin, Totally positive matrices, Stanford University Press, Stanford, 1968. 7. C.E. Radke, Classes of matrices with distinct real characteristic values, SIAM Journal of Applied Mathematics, 16: 1192-1207, (1968).
Completion Problems for Positive Matrices with Minimal Rank ? Rafael Cant´ o and Ana M. Urbano Dept. Matem` atica Aplicada, Univ. Polit`ecnica de Val`encia, 46071 Val`encia, Spain, {rcanto, amurbano}@mat.upv.es Abstract. Matrix completion problems studies the partial matrices, that is, a rectangular matrix some of whose entries are specified, and the remainder entries are free variables of some indicated set. By a completion of a partial matrix we consider a specification of the free variables obtaining a conventional matrix. The basic type of these problems try to obtain conditions for the existence of a completion for a given partial matrix in a class of interest. In this work, we study the minimal rank completion problem when the partial matrix P has the specified entries equal to zero, and the remaining entries are positive real numbers. By a graph theoretic approach we introduce some approximations to the question. Furthermore, we obtain completions for some classes of positive pattern matrices with minimal rank.
1 Introduction We point out that the historical motivation for the study of matrix completion problems appears in subjects as mathematical economics, biology, chemistry, social sciences, etc. in which models some matrix elements give qualitative instead of quantitative information [1]. Concretely, Johnson [8] in an appreciate survey, collects the relation between Matrix Theory and Combinatorics and introduce a separate research topic: Combinatorial Matrix Analysis, which includes some subjects that fit within it and help to define it. These are qualitative matrix analysis, matrix completion problems, combinatorial issues in matrix inequalities via optimization, the role of the longest simple circuit in spectral analysis and an attenuation of matrix products. Some motivation for the study of these subjects appear in economics, biology, ecology and chemistry, in which models whose parameters (matrix entries) are at best qualitative known occur. We consider a partial matrix P with the specified entries equal to zero and the remaining entries are positive real free variables, i.e., P is a zero-nonzero ?
Supported by the Spanish DGI grant number BFM2001-0081-C03-02.
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pattern matrix, and we ask for those completions of P with the lowest possible rank. This question has applications to minimal representations for linear input/output discrete time systems [10], for the partial realization problem [5], and for the problem of completing a matrix and its inverse [3, 14]. In fact, Rumchev [12] pointed out that the reachability and the controllability properties of positive discrete time linear control systems depends on some zero-nonzero patterns of the pair (A, B) and do not depend on the values of their entries. It is a structural generic property of positive linear systems.
2 Definitions This section contains most of the notation and definitions. We recall some concepts given in [4, 6, 7, 11] for general fields. We only consider the real field for the historical motivation in the study of matrix completion problems as we are commented in the previous section. For a positive integer n we denote by hni the set {1, 2 . . . , n}. Let A be an m × n matrix and let α ⊆ hmi and β ⊆ hni. We denote by A [α | β] the submatrix of A whose rows are indexed by α and whose columns are indexed by β in their natural order. Two matrices A and B are said to be permutationally equivalent if there exist permutation matrices P1 and P2 such that A = P1 BP2 . Definition 1. Any m × n real matrix A = (aij ) has associated a bipartite graph GA = {V (GA ), E(GA )}. The vertex-set V (GA ) has m + n vertices denoted by V (GA ) = {v1 , v2 . . . , vm , w1 , w2 , . . . , wn } and it is divided into two disjoint subsets V (GA ) = VR (GA ) ∪ VC (GA ) where VR (GA ) = {vi , i ∈ hmi} is associated with the rows of A, and VC (GA ) = {wj , j ∈ hni} is associated with the columns of A. The edge-set of GA is the set E(GA ) = {(vi , wj ) | aij 6= 0}. Definition 2. Let G = {V (G), E(G)} be a bipartite graph where V (G) = VR (G) ∪ VC (G), VR (G) = {vi , i ∈ hmi}, and VC (GA ) = {wj , j ∈ hni}. A sequence of vertices [vi1 , wj1 , vi2 , wj2 , . . . , vir , wjr , vi1 ] with vis 6= vit and wjs 6= wjt for s 6= t, is called a simple cycle (or simple closed path) of length 2r if one of the following conditions holds: (vi1 , wjr ) ∈ E(G), and (vis , wjt ) ∈ E(G) for s − t ≤ 1, s, t ∈ hri, or (vir , wj1 ) ∈ E(G), and (vis , wjt ) ∈ E(G) for t − s ≤ 1, s, t ∈ hri. G is a chordal bipartite graph if every (simple) cycle of length strictly greater than 4 has a chord, that is, an edge joining two nonconsecutive vertices of the cycle. Definition 3. A matrix (vector) P is said to be a (zero-nonzero) pattern matrix (pattern vector) if each nonzero element is a real free variable. We denote the unspecified entries by stars. The pattern of a matrix A, pattern(A), is
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the pattern matrix obtained by replacing every nonzero entry of A by an unspecified element. If pattern(A) = P , then A is a completion matrix of P . Similarly we define the pattern of a vector y and denote it by pattern(y). By Definition 1, any m × n pattern matrix P = (pij ) has associated a bipartite graph GP , where (vi , wj ) ∈ E(GP ) if and only if pij 6= 0. Furthermore, we use the same notation P [α | β] for a subpattern of P as we introduce above for a standard submatrix. Definition 4. Let x be an m × 1 pattern vector. We define the support of x as the set s(x) = {i ∈ hmi | xi 6= 0}. A set of r pattern vectors {x1 , x2 , . . . , xr } is said to be combinatorially dependent if either xi = 0, for some i ∈ hri, or there exist nonempty disjoint subsets α, β of hri such that ∪i∈α s(xi ) = ∪i∈β s(xi ). Otherwise, we call the set of pattern vectors combinatorially independent. Note that {x1 , x2 , . . . , xr } is combinatorially dependent if and only if there exist linearly dependent nonnegative vectors {y1 , . . . , yr } with pattern(yi ) = xi , i ∈ hri [11, Lemma 5.2]. Definition 5. We define ccr(P ) by the combinatorial column rank of P , that is, the maximal number of combinatorially independent columns of P . The combinatorial row rank of P , crr(P ), is similarly defined. Example 1. If P is an 10 × 5 pattern matrix with two zero entries and three nonzero entries in each row, and the rows are pairwise distinct, then we have the inequality ccr(P ) = 5 > crr(P ) = 4 (see [11, pp. 224]). Definition 6. The minimal rank of a pattern matrix P , mr(P ), is defined by the number mr(P ) = min{rank(A) | pattern(A) = P }. Similarly, MR(P ) = max{rank(A) | pattern(A) = P } denotes the maximal rank of P . In this paper, we study the positive minimal rank completion problem for a given pattern matrix P , so we introduce the next definition. Definition 7. The positive minimal rank of a pattern matrix P , mr+ (P ), is defined by the number mr+ (P ) = min{rank(A) | pattern(A) = P, A ≥ O}. Obviously, for a pattern matrix P 6= O we have 1 ≤ mr(P ) ≤ mr+ (P ) ≤ MR(P ) ≤ min{m, n} and max{ccr(P ), crr(P )} ≤ mr+ (P ). µ By Example 1, one can construct a 15 × 15 pattern matrix Q = with mr+ (Q) = 10 but ccr(Q) = crr(Q) = 9 (see [7, Example 5.6]).
P O O PT
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3 Previous results In this section we recall previous results proved in [2, 4]. For instance, next Theorem is given by Cant´ o [4, Theorem 4]. Theorem 1. Let P be an m × n pattern matrix, then MT(P ) ≤ mr(P ) ≤ bi(P ) ≤ MR(P ). We denote by MT(P ) the maximum triangle size of P , i.e., the maximum nonnegative integer r such that P has an r × r subpattern permutationally equivalent to a triangular pattern matrix with nonzero diagonal entries. By bi(P ) we denote the minimum covering number of P , that is, the number of complete subpatterns of a minimum collection which cover the nonzero elements of P . A complete subpattern of P is any subpattern with all nonzero entries. Observe that if P is a pattern matrix with MT(P ) = bi(P ), then by Theorem 1 mr(P ) is characterized. Nevertheless, there exist different examples where the inequalities in Theorem 1 can be stricted.
4 Main results: positive minimal rank Similar to Theorem 1 we can obtain the following result, Theorem 2. Let P be an m × n pattern matrix, then mr+ (P ) ≤ bi(P ) ≤ MR(P ). By Definitions 4 and 5, and the definition of the maximum triangle size given in Section 3, we obtain Proposition 1. Let P be an m × n pattern matrix, then MT(P ) ≤ min{ccr(P ), crr(P )}. By Theorem 2, Proposition 1 and Definition 7 we conclude the following theorem, Theorem 3. Let P be an m × n pattern matrix, then MT(P ) ≤ max{ccr(P ), crr(P )} ≤ mr+ (P ) ≤ bi(P ) ≤ MR(P ). Remark 1. Let P be pattern matrix. We are looking for the equality MT(P ) = bi(P ) in Theorem 3, because in this case mr+ (P ) is characterized. For the trivial cases MT(P ) = 1 or MT(P ) = min{m, n} the equality holds. In fact, MT(P ) = 1 when P is a complete matrix or when P has a row (or column) with, at least, one nonzero entry and the other rows (columns) have
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the same pattern, so P [nonzero rows | nonzero columns] is a complete subpattern that covers all the nonzero entries of P , then bi(P ) = 1. Otherwise, the case MT(P ) = min{m, n} = m occurs when P is an m × m diagonal (or triangular) pattern matrix with nonzero diagonal entries. Nevertheless, the inequality MT(P ) ≤ bi(P ) can be stricted. For instance, if we recall Example 1, then MT(P ) = 3, crr(P ) = 4, and ccr(P ) = bi(P ) = 5. Therefore, we do not know the value for the minimal rank of P because by Theorem 1 we have 3 ≤ mr(P ) ≤ 5, but the positive minimal rank of P is completely determined because by Theorem 3, max{crr(P ), ccr(P )} = bi(P ) = 5 implies mr+ (P ) = 5. All examples that we can found give the following inequality for a nonzero pattern matrix P , 1 ≤ MT(P ) ≤ mr(P ) ≤ min{ccr(P ), crr(P )} ≤ max{ccr(P ), crr(P )} ≤ ≤ mr+ (P ) ≤ bi(P ) ≤ MR(P ) ≤ min{m, n} Note that we only need to prove mr(P ) ≤ min{ccr(P ), crr(P )}
(1)
Case 1. If MT(P ) = mr(P ), then by Proposition 1 the inequality (1) holds. Case 2. If min{ccr(P ), crr(P )} = mr+ (P ), then by Definition 7 the equation (1) holds. 2.1. Note that if ccr(P ) = n − 1 ≤ crr(P ), or crr(P ) = m − 1 ≤ ccr(P ), then min{ccr(P ), crr(P )} = mr+ (P ) and Case 2 can be applied. For proving this observation suppose, for instance, that ccr(P ) = n − 1 ≤ crr(P ), by Definitions 4 and 5 the n column vectors of P are combinatorially dependent and there exist n linearly dependent nonnegative vectors with the same support, therefore mr+ (P ) < n and the equality ccr(P ) = mr+ (P ) holds.
5 Applications From now on, let P = (pij ) ≥ O be an n × n pattern matrix with nonzero diagonal entries, that is, pii 6= 0 for all i ∈ hni. We study when the equality max{ccr(P ), crr(P )} = mr+ (P ) = bi(P )
(2)
in Theorem 3 holds, therefore the positive minimal rank of P is characterized. We consider some classes of positive pattern matrices. Definition 8. A pattern matrix P = (pij ) is said to be combinatorially symmetric when pij 6= 0 if and only if pji 6= 0. If P is combinatorially symmetric, we consider its bipartite graph GP , and introduce a nondirected graph HP = {V (HP ), E(HP )}, where the vertex-set is V (HP ) = hni, and the edgeset is E(HP ) = {(i, j) | pij 6= 0, i 6= j}. In other case, P is said to be
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noncombinatorially symmetric pattern matrix, and we consider the associated directed graph (or digraph) FP = {V (FP ), E(FP )}, where the vertex-set is V (FP ) = hni, and the arc-set is E(FP ) = {(i, j) | pij 6= 0, i 6= j}. Note that HP and FP have no loops nor multiple arcs. We recall that a nondirected graph HP is said to be a chordal graph if every cycle of length strictly greater than 3 has a chord. Observe that if GP is a chordal bipartite graph, then HP is a chordal graph [6, pp. 265]. 5.1 Noncombinatorially symmetric pattern matrices We study the equality (2) when P is a noncombinatorially symmetric pattern matrix. Consider two cases, depending on the digraph FP associated with P . FP is an acyclic directed graph In this case, FP has not any cycle. If P is a triangular pattern matrix, then the equality ccr(P ) = crr(P ) = mr+ (P ) = bi(P ) = n (3) holds. If P is not a triangular pattern matrix, since FP is an acyclic graph, then the topological order algorithm (see [13]) gives a total order relation and permit to transform P , up to a permutation, in a triangular pattern matrix [9, Theorem 3.4]. Therefore, the equality (3) holds. FP is a nonacyclic directed graph Consider that FP is a cycle and suppose, up to a permutation, that the nonzero entries of P are the main diagonal, the superdiagonal and the entry in the position (n, 1). Note that P has the following structure ∗ ∗ 0 0 ··· 0 0 0 ∗ ∗ 0 ··· 0 0 0 0 ∗ ∗ ··· 0 0 P =. . .. .. .. .. . . 0 0 0 0 ··· ∗ ∗ ∗ 0 0 0 ··· 0 ∗ and FP is the graph
• ··· ··· • @ I ¡ @
¡ ¡ n •ª @ @ @ R• 1
- •¡ 2
@ •3 ¡ µ ¡
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Note that bi(P ) = n because a smallest covering by complete subpatterns of P is given by P [i | i, i + 1], i ∈ hn − 1i, and P [n | 1, n]. Now, we consider two cases depending if the order n is even or odd. 1. n ≥ 4, even. In this case, ccr(P ) = crr(P ) = n − 1 because ∪i∈α s(pi ) = ∪i∈β s(pi ) where pi denotes the i-th row (or column) of P , α = {1, 3, . . . , n−1} and β = {2, 4, . . . , n}. Then, the equality (2) does not hold. Observe that the (0, 1)-completion matrix of P has rank equal to n − 1, i.e., mr+ (P ) = n − 1. 2. n ≥ 3, odd. One can obtain that ccr(P ) = crr(P ) = n, and the equality (3) holds. 5.2 Combinatorially symmetric pattern matrices We study the equality (2) for this class of pattern matrices depending on the nondirected graph HP associated with P . HP is a nonchordal graph Let HP be a nonchordal graph, i.e., HP has a cycle of length greater than 3. Now we consider two cases in order to prove that the equality (2) does not hold. 1. n = 4. Let P be the 4 × 4 combinatorially symmetric pattern matrix ∗∗0∗ ∗ ∗ ∗ 0 P = 0 ∗ ∗ ∗ ∗0∗∗ then HP is the nonchordal graph 1 •
2 •
• 4
• 3
One can obtain MT(P ) = mr(P ) = 2, ccr(P ) = crr(P ) = mr+ (P ) = 3, but bi(P ) = 4. 2. n ≥ 5. It can be proved that MT(P ) = mr(P ) = ccr(P ) = crr(P ) = mr+ (P ) = n − 2, but bi(P ) = n.
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HP is a chordal graph If HP is a chordal graph, then we can construct some examples where the equality (2) does not hold. Nevertheless, there exists a class of chordal graphs where the equality (2) is satisfied. These are the interval graphs (see [6, Chapters 1 and 8]). Definition 9. HP is an interval graph if its vertices can be put into one-toone correspondence with a set of intervals S of the real line such that two vertices are connected by an edge in HP if and only if their corresponding intervals have nonempty intersection. Theorem 4. If HP is an interval graph associated with the n × n combinatorially symmetric pattern matrix P , then MT(P ) = mr(P ) = ccr(P ) = crr(P ) = mr+ (P ) = bi(P ).
References 1. L. Basset, J. Maybee, J. Quirk. Qualitative economics and the scope of the correspondence principle, Econometrica 26 (1968), 544-563. 2. A. Borobia, R. Cant´ o. Ranks of Patterns of Nonnegative Matrices, Proceedings of the XVI Cedya, VI CMA Applied Math. Conference, Las Palmas de Gran Canaria (1999), 955-962. 3. A.A. Bostian, H.J. Woerdeman. Unicity of minimal rank completions for tridiagonal partial block matrices, Linear Algebra and its Applications 325 (2001), 23-55. 4. R. Cant´ o. On the minimal rank completion problem for Generic Matrices, submitted. 5. I. Gohberg, M.A. Kaashoek, L. Lerer. On minimality in the partial realization problem, Systems Control Lett. 9 (1987), 97-104. 6. M.C. Golumbic. Algorithmic Graph Theory and Perfect Graphs, Academic Press, (USA), 1980. 7. D. Hershkowitz and H. Schneider. Ranks of zero patterns and sign patterns, Linear and Multilinear Algebra 34 (1993), 3-19. 8. C.R. Johnson. Combinatorial matrix analysis: An overview, Linear Algebra and its Applications 107 (1988), 3-15. 9. C. Jord´ an, J.R. Torregrosa, A.M. Urbano. Graphs and controllability completion problems, Linear Algebra and its Applications 332-334 (2001), 355-370. 10. M.A. Kaashoek, H.J. Woerdeman. Unique minimal rank extensions of triangular operators, J. Math. Anal. Appl. 131 (1988), 501-516. 11. D.J. Richman and H. Schneider. On the singular graph and the Weyr characteristic of an M –matrix, Aequationes Mathematicae 17 (1978), 208–234. 12. V.G. Rumchev. On controllability of discrete-time positive systems, Proceedings of the VI Int. Conf. on Control, Automation, Robotics and Vision, Singapore (2000), cd-rom. 13. S. Sahni. Concepts in Discrete Mathematics, Camelot, second ed., 1985. 14. H.J. Woerdeman. Minimal rank completions of partial banded matrices, Linear and Multilinear Algebra 36 (1993), 59-68.
Some Problems about Structural Properties of Positive Descriptor Systems ? Rafael Bru, Carmen Coll, Sergio Romero-Vivo, and Elena S´anchez Departament de Matem` atica Aplicada, Universitat Polit`ecnica de Val`encia, 46071, Valencia, Spain, {rbru, mccoll, sromero, esanchezj}@mat.upv.es Abstract. The reachability and controllability of positive descriptor systems are analyzed. Without nonnegative restrictions on the system, the reachability property is invariant under state feedbacks, but this is not true when a positive system is considered. In this work, some conditions on feedbacks are studied to obtain closedloop positive systems preserving a structural property. In this way, it is worthwhile to obtain the corresponding reachable canonical form.
1 Introduction A descriptor discrete-time linear control system is given by a system of difference equations in the following way: Ex(k + 1) = Ax(k) + Bu(k)
(1)
where E, A ∈ Rn×n , and B ∈ Rn×m , k ∈ Z+ , x(k) ∈ Rn is the state vector and u(k) ∈ Rm is the control vector. This system is denoted by (E, A, B). If E = I, the system (1) is called standard. It is well-known (see [4]) that if the pair (E, A) is regular, that is, there exists a λ ∈ C such that det(λE − A) 6= 0, then the system has a solution, which can be obtained in terms of Drazin inverses of suitable matrices. When the trajectory of the system is positive from any positive initial state and any positive control sequence, it is said that (E, A, B) is positive. Recently, Bru et al. [1] established that the positive structural properties of these kind of systems are characterized by the monomial vectors contained in the reachability matrix R(E, A, B). In the standard case (E = I), a reachable canonical form for the positive reachability property was constructed in [3] (the null-controllable canonical form is also given in [3]). This canonical form is obtained under monomial matrices. However, in general, it is not easy to find a reachable canonical form for positive descriptor systems (E, A, B) since ?
Partially supported by Spanish DGI grant BFM2001-2783
L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 233-240, 2003. Springer-Verlag Berlin Heidelberg 2003
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the reachability matrix involves Drazin inverses. This fact complicates the search of monomial vectors. In this work, we focus our attention on a positive forward-backward system. Given the system (E, A, B), there exist two nonsingular matrices Q and the system into an equivalent forward-backward system ´ ³ P transforming e e e E, A, B (see [4]) in the following way: x1 (k + 1) = A1 x1 (k) + B1 u (k) forward subsystem N x2 (k + 1) = x2 (k) + B2 u (k)
backward subsystem,
e = QEP =diag[In , N ], A e = QAP =diag[A1 , In ] and B e = QB = with ¸E 1 2 · B1 , where n1 + n2 = n, A1 ∈ Rn1 ×n1 and N ∈ Rn2 ×n2 is nilpotent. The B2 solution of the initial system (E, ³ ´ A, B) can be obtained from the solution of e A, e B e . the equivalent system E, Moreover, it is worth to study under what kind of transformations the positive structural properties are invariant. If there are not nonnegative restrictions it is well-known that the reachability property is also invariant under state transformations and state feedbacks ([11]). However, this is not true when the positiveness condition is considered. In [2] it was proven that the positive reachability property is invariant under state transformations given by monomial matrices. The construction of feedbacks is an interesting problem for solving the pole-assignment problem. Rumchev and James (see [9] and [12]) characterized the set of complex numbers being the spectrum of a state matrix and they derived some results for single-input single-output systems (with and without nonnegative restrictions). Recently, in [8], a new study of the pole-assignment problem was given for a class of positive linear systems. In this paper, we raise two problems: firstly, we construct a reachable canonical form for positive forward-backward systems; secondly, we study when the structural properties are invariant under nonnegative state feedbacks. Definition 1. (see [1]) Let (E, A, B) be the system (1) and X0 the set of all admissible initial states. This system is positive if, for every x(0) ∈ X0 ∩ Rn+ , and for every nonnegative control sequence u (·) ≥ 0, the state trajectory belongs to Rn+ , that is, x (k) = x (k, x0 , u (·)) ∈ Rn+ , ∀k ≥ 0. A positive descriptor system ³ is denoted ´ by (E, A, B) ≥ 0. The positiveness of a e e e forward-backward system E, A, B equivalent to (E, A, B), is characterized as follows. Theorem 1. (see [1]) A system n1 ×n1 R+ ,
Rn+1 ×n1 ,
i
B1 ∈ −N B2 ∈ is the nilpotence index of N .
³
e A, e B e E,
Rn+2 ×n2 ,
´
is positive if and only if A1 ∈
for all i = 0, 1, . . . , q − 1, where q
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Now, structural properties for a positive descriptor system (E, A, B) ≥ 0 are introduced. Definition 2. Let (E, A, B) be a positive system. It is said that a) The state w ∈ Rn+ is positively reachable if there exist k ∈ Z+ and a control sequence u(j) ≥ 0, j = 0, 1, ..., k + q − 1, transferring the state of the system from the origin at time 0, x(0) = 0, to w at time k. b) The system is positively controllable if for any pair of nonnegative states x0 ∈ X0 and xf there exist k ∈ Z+ and a control sequence u(j) ≥ 0, j = 0, 1, ..., k+q−1 transferring x0 at time 0 to xf at time k. In particular, when xf = 0, the system is called positively null-controllable. In general, the positiveness of the solution and structural properties is lost if the forward-backward system is used instead of the initial descriptor system. This property holds if the transformation matrices P and Q are monomial matrices. When the system is forward-backward then the positively reachable property is equivalent to the positively reachable property of the forward and backward subsystems, respectively. Hence, the positively reachable property of a positive forward-backward system is characterized as follows (see [1]). Theorem 2. A positive forward-backward descriptor system is positively reachable if and only if the matrices £ ¤ R(A1 , B1 ) = B£1 | A1 B1 | · · · | An1 1 −1 B1 ¤ R(N2 , B2 ) = − N n2 −1 B2 | · · · | N B2 | B2 have a monomial submatrix of order n1 and n2 , respectively.
2 Reachable canonical form for positive forward-backward systems Throughout this work, we denote by (E, A, B) a positive forward-backward system to simplify the notation. Since one can prove that the positive reachability property is invariant under monomial matrices, in this section a positive reachable canonical form is constructed under this kind of transformation. Consider a positively reachable forward-backward system (E, A, B), x1 (k + 1) = A1 x1 (k) + B1 u (k) forward subsystem N x2 (k + 1) = x2 (k) + B2 u (k) backward subsystem. From this system we construct the new system · ¸ · ¸ · ¸· ¸ In1 O A1 O B1 O u(k) x(k + 1) = x(k) + , k ∈ Z, O N O In2 O B2 u(k) ˆ where which is denoted by (E, A, B)
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·
E=
¸ · ¸ · ¸ A1 O In1 O ˆ = B1 O . , A= , B O N O In2 O B2
When E ≥ O, analyzing the directed graph of the matrices A1 and N , a reachable canonical form similar to this positively reachable system can be constructed as follows. Consider the positively reachable subsystems (A1 , B1 ) and (N, −B2 ) (the columns without influence in the characterization of the structural property are not considered). It is known (see [3]) that there exist a monomial matrix M1 and a permutation matrix Q1 such that the forward subsystem (A1 , B1 ) is similar to (A1c , B1c ) = (M1−1 A1 M1 , M1−1 B1 Q1 ), where
[M1−1 A1 M1 || M1−1 B1 Q1 ] = C O O O O
O B O O O
O O A O O
O Σ ∆B AB O
∆R1 ∆R2 ∆R3 ∆R4 AR
O O O O B1 R
O O O B1 B O
O B1 C O O B1 A B1 AC O O O O
(2)
being – C and B are block diagonal matrices with blocks in the following way: 0 + 0 ... 0 . 0 0 + . . . .. .. .. (3) . . 0 ... 0 , . . . . .. .. .. . . + + 0 0 ... 0
where + denotes a positive entry. – Σ = diag [Φ, . . . , Φ] and ∆B , ∆Rj = [Ψ, . . . , Ψ ], j = 1, 2, 3, 4, where 0 0 ... 0 ∗ 0 ... 0 .. .. .. . . .. , . Φ = . . , and Ψ = .. .. . 0 0 ... 0 ∗ 0 ... 0 + 0 ... 0 where ∗ denotes a nonnegative entry. – A is a block upper triangular matrix in the following way A1 ∆ · · · ∆ ∆ O A2 · · · ∆ ∆ .. .. . . . .. , . A= . . . . . O O · · · An−2 ∆ O O · · · O An−1
(4)
(5)
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where each diagonal block is at the same time a block diagonal matrix with blocks as follows: 0 + 0 ... 0 . 0 0 + . . . .. .. .. (6) . . 0 ... 0 . . . . .. .. .. . . + 0 0 0 ... 0 and ∆ = [Ψ, . . . , Ψ ] where Ψ is such as given in (4). – AB is a block diagonal matrix with blocks such as given in (6). – AR is a block matrix, with all their off-diagonal blocks Ψ and their diagonal blocks in the following way: ∗ + 0 0 ... 0 ∗ 0 + 0 ... 0 .. .. . . . 0 . . . . . . .. (7) . ∗ 0 0 ... ... 0 . . . .. .. .. 0 . . . + ∗ 0 0 ... 0 0 – Finally, the block B1 AC has all its entries greater than or equal to zero, where there is at least a positive entry in each column, and the blocks GC , B1 A , B1 B and B1R are antidiagonal block matrices, that is, O O ··· M .. .. .. . . . (8) O M ··· O M O ··· O where M is formed by a unique column of the type col [0 0 · · · 0 +], that is, there is a monomial vector associated with each diagonal block of the submatrices C, A, AB and AR . In the same way, if the pair (N, −B2 ) is considered there exist a monomial matrix M2 and a permutation matrix Q2 such that the backward subsystem (N, −B2 ) is similar to (Nc , B2c ) = (M2−1 N M2 , M2−1 (−B2 )Q2 ), where Nc and −B2c have the structure as in (5) and (8), respectively. Now, applying M = diag [M1 , M2 ] and Q = diag [Q1 , Q2 ] to the forwardˆ one obtains the similar positively reachable backward system (E, A, B), ˆc ) = (M EM −1 , M AM −1 , M BQ) ˆ forward-backward system (Ec , Ac , B where · ¸ · ¸ · ¸ I O A1c O ˆc = B1c O . Ec = n1 , Ac = , B O Nc O In2 O B2c
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Hence, the following system is called reachable canonical form for positively reachable forward-backward systems: x1 (k + 1) = A1c x1 (k) + B1c u (k) forward subsystem Nc x2 (k + 1) = x2 (k) + B2c u (k)
backward subsystem.
3 Positive reachability and feedbacks It is well-known that the reachability property is invariant under state feedbacks when systems without nonnegative restrictions are considered. However, this is not true when the positive reachability property for positive systems is considered. The next example shows this fact, that is, given a positively reachable forward-backward system we construct a positive state feedback such that the closed-loop forward-backward system is positive but it is not positively reachable. Example 1. Let (E, A, C) be a positive forward-backward system, £ ¤T E = diag [In1 , N ] , A = diag [A1 , In2 ] and B = B1T B2T , where
001 101 020 0 −1 0 A1 = 1 0 1 , B1 = 0 0 1 , N = 0 0 3 and B2 = 0 −1 0 . 010 000 000 −1 0 0
Constructing the reachability matrices ¯ ¯ 1 0 1 ¯¯ 0 0 0 ¯¯ 0 0 1 £ ¤ B1 | A1 B1 | A21 B1 = 0 0 1 ¯¯ 1 0 1 ¯¯ 0 0 1 , 0 0 0¯ 0 0 1¯ 1 0 1 ¯ ¯ ¯ 0 0 0¯ 6 0 0 0 1 0 ¯ ¯ £ ¤ B2 | N B2 | N 2 B2 = − 0 1 0 ¯¯ 3 0 0 ¯¯ 0 0 0 , 1 0 0¯ 0 0 0¯ 0 0 0 we can check that both reachability matrices contain a suitable monomial submatrix. Thus, the system (E, A, B) is positively reachable. Now, we apply the following positive state feedback to this system, u(k) = [F1 O] x(k) + [O K2 ] x(k + 1) + v(k), where v(k) ∈ R2+ is a new nonnegative control vector, and F1 ∈ R2×2 + , K2 ∈ R3×3 are given by + 000 000 F1 = 0 0 0 and K2 = 0 0 1 . 000 111
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Hence, the closed-loop system (E − BK, A + BF, B) is positive but it is not positively reachable since the reachability matrices of both subsystems, (A1 + B1 F1 , B1 ) and (N − B2 K2 , B2 ), do not contain all the monomial vectors generating R3+ . Note that such matrices are ¯ ¯ 1 0 1 ¯¯ 1 0 2 ¯¯ 3 0 7 £ ¤ B1 | (A1 + B1 F1 )B1 | (A1 + B1 F1 )2 B1 = 0 0 1 ¯¯ 2 0 3 ¯¯ 4 0 9 0 0 0¯ 0 0 1¯ 2 0 3 ¯ ¯ 0 1 0 ¯¯ 1 2 0 ¯¯ 8 0 0 £ ¤ B2 | (N − B2 K2 )B2 | (N − B2 K2 )2 B2 = − 0 1 0 ¯¯ 4 0 0 ¯¯ 0 0 0 . 1 0 0¯ 0 0 0¯ 0 0 0 The goal of this section is to find conditions on state feedbacks in order to get a positive closed-loop system preserving the positive controllability, positive null-controllability, or positive reachability property. Firstly, note that given a positive forward-backward system (E, A, B) with E ≥ O, and a positive state feedback u(k) = F x(k) + Kx(k + 1) + v(k), F, K ≥ O, v(k) ∈ Rn+ , k ∈ Z, if we have K = [K1 , K2 ] ≥ 0 and F = [F1 , F2 ] ≥ 0 such that # " # " 0 0 B1 F1 0 , BK = , with N − B2 K2 nilpotent, BF = 0 B 2 K2 0 0
(9)
then we can assure that the closed-loop system (E − BK, A + BF, B) is also a positive forward-backward system. Without loss of generality we can consider that the system is given in the canonical form. In the following theorem the positive controllability property is considered and a characterization for positive state feedbacks preserving this property is given. Theorem 3. Consider a positively controllable forward-backward canonical system (E, A, B) with E ≥ O. The positive forward-backward closed-loop system (E − BK, A + BF, B) with F, K satisfying (9) is positively controllable if and only if the blocks F1 and K2 have the structure O O ··· O O O O ··· O Ψ .. .. . . .. .. (10) . . . . . , O O ··· Ψ Ψ O Ψ ··· Ψ Ψ where
£ ¤ Ψ = ∗ 0 ··· 0
with suitable size of the blocks.
(11)
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Theorem 4. Consider a positively null-controllable forward-backward system (E, A, B) with E ≥ O. If F1 and K2 are given by (10), then the positive forward-backward closed-loop system (E − BK, A + BF, B) is also positively null-controllable. Finally, a positive state feedback is constructed, which preserves the positive reachability property. Theorem 5. Consider a positive reachability forward-backward system (E, A, B) with E ≥ O. If K2 is given by (10) and O O O O ΩR O O O O ∆B F1 = O O ΩA ∆A1 ∆A2 , O O O O ∆C where ΩA has the structure as (10), ΩR is a block matrix with all blocks of type Ψ given in (11), and ∆A1 , ∆A2 , ∆B and ∆C are diagonal matrices with diagonal blocks structured as in (6), then the positive forward-backward closed-loop system (E − BK, A + BF, B) is also positively reachable.
References 1. R. Bru, C. Coll and E. S´ anchez, Structural properties of positive linear systems time-invariant difference-algebraic equations. Linear Algebra and its Applications, 349: 1-10 (2002). 2. R. Bru, C. Coll, S. Romero and E. S´ anchez, Reachability indices of positive linear systems submitted, (2002). anchez, Canonical forms for positive discrete-time 3. R. Bru, S. Romero and E. S´ linear systems. Linear Algebra and its Applications, 310: 49-71 (2000). 4. L. Dai, Singular Control Systems, Springer-Verlag, New York, 1989. 5. L. Caccetta and V. G. Rumchev, A survey of reachability and controllability for positive linear systems. Annals of Operations Research, 98: 101-122 (2000). 6. P. G. Coxson and H. Shapiro, Positive reachability and controllability of positive systems. Linear Algebra and its Applications, 94: 35-53 (1987). 7. L. Farina and S. Rinaldi, Positive Linear Systems, John Wiley & Sons, New York, 2000. 8. D.J.G. James, S.P. Kostova and V.G. Rumchev, Pole–assignment for a class of positive linear systems, International Journal of Systems Science, 32(12): 13771388 (2001). 9. D. J. G. James y V. G. Rumchev. Controllability of single-input single-output positive linear discrete-time systems and the pole-assignment problem, Systems Science, Vol. 21, No. 4, pp. 17-26, 1995. 10. T. Kaczorek, Positive 1D and 2D Systems, Springer, London, 2002. 11. V.M. Popov, Invariant description of linear time–invariant controllable systems. SIAM J. Control, 15(2): 252–264 (1972). 12. V. G. Rumchev y D. J. G. James. Spectral characterization and pole assignment for positive linear discrete-time systems, Int. J. Systems Sci., Vol. 26, No. 2, pp. 295-312, 1995.
Positive Linear Systems Reachability Criterion in Digraph Form Ventsi G. Rumchev Department of Mathematics and Statistics, Curtin University of Technology, GPO Box U 1987, Perth, WA 6845, Australia, [email protected] Abstract. Reachability criterion in digraph form for discrete-time positive linear systems in terms of the non-negative pair (A, B) is obtained. The criterion (and the proof) is based on a simple (vertex) decomposition of the system matrix digraph into monomial subdigraphs: simple monomial paths, blossoms, cycles, monomial trees and bunches. The reachability criterion gives not only a deeper insight into the structure of reachable positive systems but its graph theoretic nature makes it easy to solve large-scale problems.
1 Introduction Consider the positive linear discrete-time system (PLDS) [4, 6] x(t + 1) = Ax(t) + Bu(t), A∈
, Rn×n +
B∈
, Rn×m +
t = 0, 1, 2, . . .
(1)
Rm +,
(2)
u(t) ∈
where x(t) is the state of the system at time t, u(t) is the control (decision) denotes vector and for each pair of positive integers (n,s) the symbol Rn×s + the set of all n × s non-negative real matrices (vectors) with non-negative entries. Positive linear systems are defined on cones and not on linear spaces and that is why the well-known reachability and controllability criteria for linear systems [7] fail to give the correct answer. The system (1)–(2) (and the pair (A, B) ≥ 0) is said to be reachable [8] if for any state x ∈ Rn+ , x 6= 0, and some finite t there exists a non-negative control sequence {u(s), s = 0, 1, 2, . . . , t − 1} that transfers the system from the origin into the state x(t) = x. It is well known [3, 8] that the positive linear system (1)–(2) is reachable if its reachability matrix, which is non-negative, contains n linearly independent £ ¤
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pair (A, B) ≥ 0 and does not depend on the values of their entries. Indeed, reachability is a generic structural property of PLDS, which exhibit highly combinatorial behaviour with respect to this property. At the same time efficient graph-theoretic (combinatorial) criteria for recognizing the reachability property in terms of the nonnegative pair (A, B) have not been paid the deserved attention yet. The aim of this paper is to develop such a criterion.
2 Monomial subdigraphs and the digraph D(Ao ) Let D(A) be the digraph of an n × n non-negative matrix A constructed as follows. The set of vertices of D(A) is denoted as N = {1, 2, . . . , n}. There is an arc (i, j) in D(A) if and only if aji > 0. The set of arcs will be denoted by U so D(A) = (N, U ). A walk in D(A) is an alternating sequence of vertices and arcs of D: {i1 , a1 , i2 , . . . , ak , ik }, in which each arc as is (is−1 , is ) [5]. The walk is called closed if io = ik and spanning if {i1 , i2 , . . . , ik } = N . A walk is termed a trail if all of its arcs are distinct, a path if all of its vertices are distinct, and a cycle if it is a closed path. The path length is defined to be equal to the number of arcs it contains. The number of arcs away from a vertex i is called outdegree of i and is written od(i), whilst the number of arcs directed toward a vertex i is called indegree of i and is written id(i). Notice that zero columns in A correspond to vertices j with od(j) = 0 in D(A); respectively, zero rows correspond to vertices with id(i) = 0. A digraph D0 = (N 0 , U 0 ) is termed a subdigraph of a digraph D = (N, U ) if N 0 ⊆ N and U 0 ⊆ U . The positive entries in the columns of B ≥ 0 will be associated with the corresponding vertices in D(A). Monomial subdigraphs of a digraph are defined as subdigraphs with outdegrees of each of their vertices equal to at most one. The following monomial subdigraphs are distinguished in this paper: simple monomial paths, blossoms, monomial trees and bunches. A path {(i1 , i2 ), . . . , (ik , ik+1 )} is called an i1 -monomial path if each vertex of the path (except possibly the last one) has unitary outdegree. Thus, for any i1 -monomial path, od(is ) = 1 for s = 1, 2, . . . , k, id(is ) ≥ 1 for s = 2, . . . , k, k + 1 but od(ik+1 ) and id(i1 ) are not specified (they can be zero, one or greater than one). An i1 -monomial path is termed simple if od(ik+1 ) = 0, id(i1 ) = 0 and id(is ) = 1 for s = 2, . . . , k, k + 1, see Fig. 1. A simple monomial path (s.m.p.) starting from i1 will be denoted as Ms . Moreover, in order to point out that i1 (s) is the initial vertex of the s.m.p. Ms the notation i1 is used instead of i1 . The (s) initial vertex i1 is called origin, and the final vertex ik+1 (with od(ik+1 ) = 0) — end of the s.m.p. An isolated vertex j is considered as a particular kind of an s.m.p. (of zero length) with id(j) = od(j) = 0. A trail {(i1 , i2 ), . . . , (ik , ik+1 ), (ik+1 , ik+2 )} is termed a blossom if all of its vertices is , s = 1, 2, . . . , k, k + 1 are different and ik+2 = is for some s, see Fig. 2. Each vertex of a blossom has unitary in- and outdegrees except
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i
t -t -t i1
i2
i3
-t -t ik
ik+1
k+1 tY ¾ - ÿti = i
t -t i1
s
i2
k+2
-t ½
is+1
Fig. 1. Simple monomial path
Fig. 2. Blossom
the initial vertex i1 and the vertex is , for which id(i1 ) = 0 and id(is ) = id(ik+2 ) = 2. Any blossom contains a cycle, but only becomes one if ik+2 = i1 and, consequently, id(is ) = od(is ) = 1 for s = 1, 2, . . . , k, k + 1. Note also that every blossom becomes a s.m.p. of length k if the arc (ik+1 , ik+2 ) is removed. A blossom (which is not a cycle) starting from some vertex il1 will be denoted as Bl . Moreover, in order to underline that i1 is the initial vertex of the blossom (l) Bl , the notation i1 will be used instead of i1 . Cycles will be denoted as Ck . A monomial tree is a union of monomial paths (with possible common parts, see Fig. 3(a)) originating at different vertices called origins and terminating at a single common vertex called top of the tree. Any monomial tree can be decomposed into a union of disjoint simple monomial paths. A “breadth first search” based process leads to (vertex) decomposition in which:
t6 S w S t7 S w St - t t -t -t -t 2 7 5 8 9 1 4 t 3
(a)
6
7
3
t -t
t -t -t -t -t -t 1
2
4
5
8
9
(b)
Fig. 3. Monomial tree and its decomposition into simple monomial paths
1. each vertex of the monomial tree belongs to one and only one s.m.p., and 2. the number of simple monomial paths is equal to the number of origins of the monomial tree, is proposed in [2]. The monomial tree in Fig. 3(a) is decomposed on Fig. 3(b). A bunch is a union of a blossom (possibly a cycle) and monomial trees (possibly s.m.p.) joined with the tops (ends) only to the vertices of the cycle of the blossom. Bunches are denoted as Γj . Clearly, od(s) = 1 for s ∈ Γj but id(s) = 0 if s is an origin and id(s) ≥ 1 if s is any other vertex of the bunch. Since monomial trees can be decomposed into a union of disjoint s.m.p., any bunch can be decomposed into a blossom and disjoint simple monomial paths. A procedure that leads to (vertex) decomposition, in which
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6 t10 ¿ ti t t/ t t t - t - tÿ 2 3 5 7 8 9 1 ÁÀ qt 4
6 t10 ¿ ti t t-t-t t - t - tÿ 2 3 5 9 8 7 1 ÁÀ qt
(a)
4
(b)
Fig. 4. Bunch and its decomposition
1. each vertex of the bunch belongs to one and only one s.m.p. or to the blossom, and 2. the number of s.m.p. and the blossom is equal to the number of origins in the bunch, is developed in [2]. The bunch given in Fig. 4(a) is decomposed on Fig. 4(b). An i-monomial (column) vector is a scalar multiple of the basis unit vector ei . It is not difficult to see that if b ≥ 0 is an i-monomial vector and D(A) contains an i-monomial path of length k then the vectors b, Ab, A2 b, . . . , Ak b
(4)
are ν = k + 1 linearly independent monomials. Note that ν is equal to the number of vertices in this monomial path. Consider now an i1 -monomial path {(i1 , i2 ), . . . , (ik , ik+1 )} and let b ≥ 0 be an i1 -monomial vector. Then all of the vectors in the sequence (4) are linearly independent monomial vectors such that As b = αs+1 eis+1 ≥ 0 for αs+1 > 0 and s = 0, 1, 2 . . . , k. Moreover, Ak+1 b will be an ik+1 -monomial vector if and only if od(ik+1 ) = 1. If od(ik+1 ) = 0 then Ak+1 b = 0. If od(ik+1 ) > 1 then Ak+1 b is a nonmonomial vector with a number of positive entries equal to od(ik+1 ). The number of linearly independent monomial columns in (4) generated by an i1 -monomial path of length k is clearly ν = k+1 and this is the maximal number a monomial (or non-monomial) vector b can produce in the sequence (4). If a blossom with origin i1 contains l arcs then for k ≥ l the number of monomials in the sequence (4) generated by an i1 -monomial b ≥ 0 is (k + 1), only ν = l +1 being linearly independent. The cyclic behaviour of the columns of reachability matrix (3) is due to the blossoms (cycles) in D(A). Actually, all monomial columns (except those in B) present in the reachability matrix
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Consider now non-monomial columns. Let J = {j1 , j2 , . . . , jr } be the set of indices of all positive entries of b ≥ 0. For r ≥ 2 (r ≤ n) denote the corresponding non-monomial column as bJ and for any jm ∈ J, m = 1, . . . , r, the corresponding monomial column that is contained in bJ as bjm . So, bJ = bj1 + bj2 + · · · + bjr . The set J is identified with the same subset of vertices in the digraph D(A). The proof of the following results is given in [2]. Lemma 1. If As bJ is an i-monomial vector for some s = 1, 2, . . . , k then As bjm , m = 1, 2, . . . , r is an i-monomial vector too. The converse, in general, does not hold true. Corollary 1. Let (A, b) ≥ 0 and As bJ be an i-monomial for some s. Then As+1 bJ is a j-monomial vector if and only if (i, j) ∈ D(A) and od(i) = 1. Moreover, when od(i) 6= 1 the number of positive entries of As+1 bJ is exactly equal to the od(i). Remark 1. Lemma 1 and Corollary 1 simply tell us that a monomial vector b can generate in the sequence (4) at least as many monomial columns as a nonmonomial or, in other words, a non-monomial column can generate in (4) at most as many monomial columns as any monomial column that is contained in it. So, for reachability it makes sense to consider monomial columns only, particularly, if od(i) ≥ 1 for every vertex i ∈ D(A). The result below can be found in [9]. Lemma 2. If the pair (A, B) ≥ 0 is reachable all the monomial columns that . are in the reachability matrix < are in the matrix [A..B]. n
The idea of decomposing the digraph D(A) into monomial subdigraphs is quite simple. If all of the outward arcs from vertices with od(i) ≥ 2 in D(A) = (N, U ) are removed from D(A) then the reduced digraph D(o) (Au ) = (N, U (o) ) becomes a union of disjoint canonical monomial subdigraphs (s.m.p., blossoms or cycles, monomial trees and bunches) since od(i) < 2 for any vertex i ∈ D(o) (Au ). Then, by decomposing every monomial tree into a union of disjoint s.m.p. and every bunch into a union of disjoint blossom and s.m.p. (as shown above) the digraph D(o) (Ao ) can be reduced to a union D(Au ) of simplest disjoined canonical monomial subdigraphs: simple monomial paths Ms , blossoms Bl , and, possibly, cycles Ck : [ [ \ \ D(Ao ) = M B C with M B C 6= ∅, where
M=
X (s)
Ms , B =
X (l)
Bl and C =
X
Ck .
(k)
The matrix Ao of the reduced digraph D(Ao ) is contained in A. It represents the canonical monomial components of A only. Note that all of the monomial subdigraphs that are in D(Ao ) are in D(A) and, since od(i) ≥ 1 for every i ∈ D(A), all of the monomial subdigraphs that are in D(A) are in D(Ao ).
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3 Reachability Theorem 1. (Reachability criterion in digraph form) Let A ≥ 0 and its di(µ) (1) (2) graph D(A) have no vertices with od(i) = 0. Let also I1 = {i1 , i1 , . . . , i1 } (σ) (1) (2) and J1 = {j1 , j1 , . . . , j1 } be, respectively, the set of all origins (of simple (k) monomial paths and blossoms) and any set of vertices such that j1 ∈ Ck for k = 1, 2, . . . , σ, where Ck are cycles in the reduced digraph D(Ao ). Then the pair (A, B) ≥ 0 (and the positive linear system (1)–(2)) is reachable if and only if the matrix B contains an n × (µ + σ) monomial submatrix Bo = DE
(5)
D = diag{α1 , . . . , αµ+σ }, αj > 0, j = 1, 2, . . . , µ + σ,
(6)
· ¸ . . . . . E = ei(1) .. · · · ..ei(ω) ..ej(µ) .. · · · ..ej(σ) .
(7)
such that
and
1
1
1
1
Proof. It suffices to consider the reduced digraph D(Ao ) only since the digraphs D(A) contains the same monomial subdigraphs as D(Ao ). If part. Let B contain a monomial submatrix Bo = DE with D and E given respectively by (6) and (7). The digraph D(Ao ) is a union of disjoined canonical monomial S subdigraphs: simple monomial paths, blossoms and cycles. Let S = M B be the set of all simple monomial paths and blossoms (µ) (1) (2) (but not cycles) and let I1 = {i1 , i1 , . . . , i1 } be the set of their origins. (s) Consider a s.m.p. Ms (respectively, a blossom Bs ) originating at i1 . An (s) i1 -monomial column generates exactly ν(Ms ) (respectively, ν(Bs )) linearly independent monomial columns in the following sequence, as well as that in b, Ao b, Ao 2 b, . . . , Ao k b,
(8)
(4), where ν(Ms ) (respectively, ν(Bs )) as in the previous section is the number of vertices in the s.m.p. Ms (the blossom Bs respectively) and no other column (monomial or not) can produce the same number of monomials (it could, actually, produce only less) along this s.m.p. (blossom). Thus, the (maximal) number of linearly independent monomials that can be generated along Ms (or Bs respectively) is ν(Ms ) (or ν(Bs )). Since S is a disjoined set, each vertex in S belongs to one and only one s.m.p. (or blossom). Then the number of linearly independent monomials generated by the columns ei(s) , s = 1, 2, . . . , µ, in the 1 sequences (4) and (8) is ν(S) = ν(M ) + ν(B), and this is the maximal number of linearly independent columns that can be generated on S. (k) (k) If the blossom is a cycle Ck then any j1 -monomial such that j1 ∈ Ck generates along Ck exactly ν(Ck ) linearly independent monomials in (8) (and
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in (4)), where ν(Ck ) is the number of vertices in the cycle Ck (notice that the cycle length is exactly ν(Ck ) − 1). This is the maximal number of linearly independent monomials in the sequences (4) and (8) that can be generated (k) on Ck . Since all cycles Ck are disjoined and each vertex j1 , k = 1, 2, . . . , σ, belongs to one and only one cycle Ck , the number Pσ of linearly independent monomials produced by ej(k) is exactly ν(C) = k=1 ν(Ck ). This is the max1 S imal number that can be generated on C. Because C S = ∅ and every vertex of D(Ao ) (that is a vertex of D(A) as well) belongs to one and only one canonical monomial structure (s.m.p., blossom or cycle), the number of linearly independent monomials in the reachability matrix (3), generated by (s) any B ≥ 0 that contains µ columns which are i1 -monomials, s = 1, 2, . . . , µ, (k) and σ columns which are j1 -monomials, k = 1, 2, . . . , σ, is exactly equal to ν(S) + ν(C) = n. Thus, the pair (A, B) ≥ 0 is reachable. Only If part. Assume that (A, B) ≥ 0 is reachable. Then the reachability matrix
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Corollary 3. If the digraph D(A) associated with the matrix A ≥ 0 contains vertices with od(i) = 0, then the conditions (5)–(7) are sufficient but not necessary for reachability of the pair (A, B) ≥ 0. Example 1. Consider the example given in [9]. The system matrix A and its digraph D(A) are presented in Fig. 5. Note that od(i) ≥ 1 for every vertex i ∈ D(A), and also that od(4) = od(6) = 2. Removal of the outward arcs from vertices 4 and 6 leads to the reduced digraph shown in Fig. 6. The monomial subdigraphs of D(Ao ) are the s.m.p. 5 → 4 and the isolated vertex.
1 A=
- 4t '
1 1 1
- 4t $' st5 3 t
3 t
1
t ? t ? t8 2 t 6A 6 6 6 A A UA t %& t t &
t5 t? 8
2 t
1 1 1
1
1
1
7
D(A)
1
Fig. 5. The digraph D(A)
D(Ao )
t % 7
Fig. 6. Reduced digraph D(Ao )
According to Theorem 1, the pair (A, B) ≥ 0 is reachable if and only if . the matrix B contains the submatrix B = [e ..e ]. The minimum number of o
5
6
controls required for reachability of the system is p = 2.
References 1. Bru, R., Romero S., Sanchez, E.: Canonical forms for positive linear discretetime linear systems. Linear Algebra and its Applications, 310, 49–71 (2000) 2. Caccetta, L., Rumchev, V.G.: Reachable discrete-time positive systems with minimal dimension control sets. Dynamics of Continuous, Discrete and Impulsive Systems, 4, 539–552 (1998) 3. Coxson, P.G., Shapiro, H.: Positive input reachability and controllability of positive systems. Linear Algebra and its Applications, 94, 35–53 (1987) 4. Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications. Wiley, New York (2000) 5. Foulds, L.R.: Graph Theory Applications. Springer-Verlag, New York (1992) 6. Kaczorek, T.: Positive 1D and 2D Systems. Springer, London (2002) 7. Lin, C.T.: Structural controllability. IEEE Transactions on Automatic Control, AC 19–3, 201–208 (1974) 8. Rumchev, V.G., James, D.J.G.: Controllability of positive linear discrete-time systems. Intern. J. Control, 50, 845–857 (1989) 9. Valcher, E.M.: Controllability and reachability criteria for discrete-time positive systems. Intern. J. Control, 65, 511–536 (1996)
A Characterization of Reachable Positive Periodic Descriptor Systems ? Bego˜ na Cant´ o, Carmen Coll, and Elena S´anchez Departament de Matem` atica Aplicada, Universitat Polit`ecnica de Val`encia, 46071, Valencia, Spain, {becanco1, mccoll, esanchezj}@mat.upv.es Abstract. Periodic descriptor system with nonnegative restrictions and its associated invariant cyclically augmented system (ICAS) are analyzed. The weakly cyclic structure of the state matrix in ICAS plays an important role in developing the theory of structural properties in the periodic case. In this paper, the influence of the structure of state matrices and monodromy matrices on the positively reachable periodic descriptor system has been analyzed. Finally, a characterization of the positive reachability property for positive periodic descriptor systems is derived when the matrix properties are used.
1 Introduction and notation We consider an N −periodic descriptor system denoted by (E(·), A(·), B(·))N and given by E(k)x (k + 1) = A(k)x (k) + B(k)u (k) , (1) where E (k) , A (k) ∈ Rn×n , and B (k) ∈ Rn×m , k ∈ Z are N -periodic matrices, x (k) is an n-dimensional vector of state variables and u (k) is an ndimensional vector of control variables. In the case of positive invariant systems (that is, when N = 1 and the solution takes only nonnegative values) the reachability and controllability properties have been studied for the standard case in [5] and for the descriptor case in [1]. Moreover, some characterizations of the positiveness of the solution and the structural properties for N −periodic standard systems (that is, where E (k) = I, k ∈ Z) have been discussed in [2] and [3]. In this paper we extend the results obtained in [1] to the case of N −periodic descriptor systems with nonnegative restrictions. That is, we study structural properties for positive N -periodic descriptor systems denoted by (E(·), A(·), B(·))N ≥ 0. ?
Partially supported by Spanish grants BFM2001-2783 and BFM2001-0081-C0302
L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 249-255, 2003. Springer-Verlag Berlin Heidelberg 2003
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An N −periodic descriptor system with nonnegative restrictions appears in various applications such as biological or economic processes where the variables must always be nonnegative. In [4, 5, 6, 7] different problems and applications of positive standard systems have been studied. The paper is organized as follows. In this section we review the invariant cyclically augmented descriptor system (ICAS) and we give some definitions used in the paper. In the following section we study structural properties for the positive periodic case considering the ICAS and a special structure of the Drazin inverse of the matrix Ae . For obtaining the solution and some properties of the system (1) the Invariant Cyclically Augmented System (ICAS) is usually used. This system is associated with the system (1), denoted by (Ee , Ae , Be ) and defined as Ee z (k + 1) = Ae z(k) + Be ue (k) where
−1 N −1 Ee = diag [E (i)]N i=0 , Be = diag [B (i)]i=0 ,
and Ae is a weakly cyclic matrix of index N (see [8]), given by · ¸ O A (0) Ae = , −1 diag [A (i)]N O i=1
(2)
(3)
with
k z (k) = Mnk−1 x b (k) and ue (k) = Mm u b (k) , (4) where x b (k) and u b (k) are in the stacked-form of the vectors x (k) and u (k) respectively, that is
x b (k) = col[x(k) · · · x(k + N − 1)], u b (k) = col[u(k) · · · u(k + N − 1)], and Mj is a weakly cyclic matrix of index N given by ¸ · O I(N −1)j , j ∈ Z+ . Mj = Ij O If the pair (Ee , Ae ) is regular (that is, det (λEe − Ae ) 6= 0, for some λ), then the ICAS offers a solution in terms of Drazin inverses of suitable matrices (see [1]). From the relationship between the ICAS state and the N -periodic system state given in (4), we can make the following remark. Remark 1. The N -periodic system is positive if and only if the ICAS is positive. For studying the positiveness of the solution and the positive structural properties of the system we also need the following definition. Definition 1. Consider {M (k) , k ∈ Z+ } and {N (k) , k ∈ Z+ } two collections of matrices. M (·) and N (·) satisfy the σ−property if (σM ) (k) (σN ) (k) = (σN ) (k) M (k) , k ∈ Z, where the shift operation σ is defined by (σx) (k) = x (k + 1) , k ∈ Z.
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2 Structural properties Now, we define the positive reachability property and positive controllability properties of a positive N −periodic descriptor system. Denoting by X0 (s) the set of admissible initial conditions of the N -periodic system at time s, for every s = 0, 1, . . . , N − 1. Definition 2. Consider the system (E (·) , A (·) , B (·))N ≥ 0 given in (1) and the set of admissible initial conditions X0 (s), s = 0, 1, . . . N − 1. (i) For each s = 0, 1, . . . , N − 1, the system is positively reachable at time s, if for any state x ∈ Rn+ there exist k ∈ Z+ and a control sequence u(j) ≥ 0, j = s, ..., k + s + q − 1 transferring the state of the system from the origin at time s, x(s) = 0, to x at time k. (ii) For each s = 0, 1, . . . , N − 1, the system is positively controllable at time s, if for any pair of nonnegative states x(s) ∈ X0 (s) and xf there exist k ∈ Z+ and a control sequence u(j) ≥ 0, j = s, ..., k + s + q − 1 transferring x(s) at time s, to xf at time k. (iii) In this last case, when xf = 0, the system is called positively nullcontrollable. To study the positive reachability property, it is worthwhile introducing reachability cones for a positive N -periodic descriptor system. Definition 3. For each s, s = 0, 1, . . . , N −1, the cone Rk (E(·), A(·), B(·), s)N is the set of positively reachable states at time k from the origin x(s) = 0. Remark 2. The system (E(·), A(·), B(·))N ≥ 0 is positively reachable at time s if and only if the set of all reachable states in finite time is Rn+ , that is R∞ (E(·), A(·), B(·), s)N =
∞ [
Rk (E(·), A(·), B(·), s)N = Rn+ .
k=1
Because of the way the ICAS is constructed the positively reachable states of the invariant system (Ee , Ae , Be ) and the positively reachable states of the N periodic system (E(·), A(·), B(·))N are related. If we denote by Rk (Ee , Ae , Be ) and R∞ (Ee , Ae , Be ) the set of positively reachable states at time k and the set of all reachable states in finite time, respectively, we have the following result. be where x = col[x0 · · · xN −1 ], with xs ∈ Rn+ , Proposition 1. Let x ∈ RnN + s = 0, 1, . . . , N − 1. Then, (i) x ∈ Rk (Ee , Ae , Be ) if and only if xs ∈ Rk (E(·), A(·), B(·), s)N , s = 0, 1, . . . , N − 1. n (ii) R∞ (Ee , Ae , Be ) = RnN + if and only if R∞ (E(·), A(·), B(·), s)N = R+ , s = 0, 1, . . . , N − 1.
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Proof. Taking into account the relationship between states of the N periodic system and the state of the ICAS given in (4) we have x = n n col[x0 · · · xN −1 ] ∈ RN + is reachable if and only if the entries of x, xs ∈ R+ are reachable at time s, for every s = 0, 1, . . . , N − 1. ¤ To obtain a characterization of the positive reachability property, we study the product of matrices involved in these cones. Using [1] and by the construction of the matrix Ae given in (3) we can see that in cones Rk (Ee , Ae , Be ), k ∈ Z appear products of matrices A(0), A(1), . . . , A(N −1). Thus, we use the state transition matrix ΦA (k, j) = A(k − 1)A(k − 2) · · · A(j) if k 6= j ΦA (k, k) = I, and monodromy matrices As = ΦA (N + s, s), s = 0, 1, . . . , N − 1. In the following proposition we give some properties about the relation between the Drazin inverses of the monodromy matrices As , s = 0, 1, . . . , N − 1. Proposition 2. Given an N −periodic collection of matrices {A (k) , k ∈ Z+ } , then the Drazin inverses AD s for every s = 0, 1, . . . , N − 1, satisfy D D (i) AD s+1 = A(s)As As ΦA (N + s, s + 1). D (ii) As As = ΦA (N + s, s + 1)AD s+1 A(s). D (iii) ΦA (N + s, s + 1)AD s+1 = As ΦA (N + s, s + 1).
Proof. Using the Drazin inverse properties: P D P P D = P D , P P D = P D P , P k+1 P D = P k , where ¡k =¢ ind (P ),¡ that is, ¢it is the smallest nonnegative integer such that rank P k =rank P ( k + 1) , and (P Q)D = P [(QP )2 ]D Q, we prove the results. (i)
AD s+1 = [ΦA (N + s + 1, s + 1)]
D
D
= [A(s)ΦA (N + s, s + 1)] .
Then, D
AD s+1 = [A(s)ΦA (N + s, s + 1)] = h iD 2 = A(s) [ΦA (N + s, s + 1)A(s)] ΦA (N + s, s + 1) D = A(s)AD s As ΦA (N + s, s + 1).
(ii) By the above result ΦA (s + N, s + 1)AD s+1 A (s) = D = ΦA (s + N, s + 1)A(s)AD s As ΦA (N + s, s + 1)A (s) D D = As AD s As As = As As
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253
(iii) Moreover, ΦA (N + s, s + 1)AD s+1 = D = ΦA (N + s, s + 1)A(s)AD s As ΦA (N + s, s + 1)
D D = As AD s As ΦA (N + s, s + 1) = As ΦA (N + s, s + 1).
¤ In the next result, some properties of the weakly cyclically matrix Ae and its Drazin inverse are given. Proposition 3. Consider the matrix Ae given in (3). If N = 2r , r ∈ Z+ , then · ¸ N −1 O diag [ΦA (N + i, i + 1)AD i+1 ]i=1 AD = e ΦA (N, 1)AD O 1 The following characterization of the positive reachability property of a positive N -periodic descriptor system can be proven using the above propositions. Theorem 1. Consider the system (E(·), A(·), B(·))N ≥ 0 given in (1) with N = 2r , r ∈ Z+ . If E(·) and A(·) satisfy the σ-property and E(s)E D (s) ≥ 0, s = 0, 1, . . . , N − 1, then R∞ (E(·), A(·), B(·), s)N = Rn+ if and only if the matrix £ ¤n−1 row E D (s)k+1 ΦA (N + s + 1, N + s − k + 1)B(N + s − k) k=0
¡ ¢ has an n × ns1 monomial submatrix, where rank E(s)E D (s) = ns1 , and the matrix h iq−1 k row −(I − E D (s)E(s)) [E(s)] AD s+1 ΦA (N + s + 1, s + k + 2)B(s + k + 1)
k=0
has an n × (n − ns1 ) monomial submatrix. Proof. From proposition 1 we have R∞ (Ee , Ae , Be ) = RnN + if and only if R∞ (E(·), A(·), B(·), s)N = Rn+ , s = 0, 1, . . . , N − 1. Moreover, taking into account the solution of ICAS from the zero initial state given by
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Bego˜ na Cant´ o, Carmen Coll, and Elena S´ anchez
z(k) = EeD Ee z(k) + (I − EeD Ee )z(k) = +
k−1 X
(5)
¡ ¢k−i−1 EeD EeD Ae Be ue (i)
i=0 q−1 ¡ ¢X ¡ ¢i D − I − EeD Ee Ee AD Ae Be ue (k + i) , e i=0
the ICAS is reachable (see [1]) if and only if conditions (i)
£ D ¤ Ee Be EeD EeD Ae Be · · · EeD (EeD Ae )n−1 Be ¡ ¢ has an n × n1 monomial submatrix, where rank Ee )EeD = n1 , and
(ii)
£ ¤ D D q−1 D −(I − EeD Ee )AD Ae Be e Be · · · − (I − Ee Ee )(Ee Ae ) has an n × (n − n1 ) monomial submatrix,
are hold. By the structure of matrices Ee and Ae given in (2) and (3), we have −1 EeD = diag [E D (s)]N s=0
and N −1
Ae k = diag [ΦA (N + i + 1, N + i − k + 1)]i=0 Mn−k N −1
= Mn−k diag [ΦA (N + i + 1 + k, N + i + 1)]i=0 . Hence,
¡ ¢k EeD EeD Ae Be =
£ ¤N −1 N −1 = diag E D (s)k+1 φA (N + s + 1, N + s − k + 1) s=0 Mn−k diag [B(s)]s=0 £ ¤N −1 k = diag E D (s)k+1 φA (N + s + 1, N + s − k + 1)B(N + s − k) s=0 Mm . By the structure of the matrix AD e given in proposition 3 we have ¤N −1 £ D ¡ D ¢k = diag Ai+1 ΦA (N + i + 1, i + k + 1) i=0 Mnk . Ae and then
¡ ¢k D Ee AD Ae Be = e
£ ¤N −1 k+1 N −1 = diag E(s)k AD diag [B(s)]s=0 s+1 φA (N + s + 1, s + k + 2) s=0 Mn £ ¤N −1 −k−1 = diag E(s)k AD . s+1 φA (N + s + 1, s + k + 2)B(s + k + 1) s=0 Mn Using above expressions, computing the solution (5) and taking into account −1 that n1 = n01 + · · · + nN , then the conditions (i) and (ii) are equivalent, 1 respectively, to
Reachable Positive Periodic Descriptor Systems
(i’)
(ii’)
255
£ ¤n−1 row E D (s)k+1 ΦA (N + s + 1, N + s − k + 1)B(N + s − k) k=0 ¡ ¢ has an n × ns1 monomial submatrix, where rank E(s)E D (s) = ns1 , and h iq−1 k row −(I − E D (s)E(s)) [E(s)] AD Φ (N + s + 1, s + k + 2)B(s + k + 1) s+1 A
k=0
has an n × (n − ns1 ) monomial submatrix.
¤
3 Conclusions Structural properties of positive N −periodic descriptor systems have been studied in this work. The invariant cyclically augmented system associated with the periodic system is used. A characterization of the positive reachability property is made using the structure of the Drazin inverse of a weakly cyclic matrix.
References 1. Bru, R., Coll, C. and S´ anchez, E., Structural properties of positive linear timeinvariant difference-algebraic equations, Linear Algebra and its Applications, 349, 1-10, (2002). 2. Bru, R., Coll, C., Hern´ andez V. and S´ anchez, E., Geometrical conditions for the reachability and realizability of positive periodic discrete systems, Linear Algebra and its Applications, 256, 109-124, (1997). 3. R. Bru and V. Hern´ andez. Structural properties of discrete-time linear positive periodic systems, Linear Algebra and its Applications, 121, 171-183, (1989). 4. Caccetta, L. and Rumchev, V. G., A survey of reachability and controllability for positive linear systems . Annals of Operations Research, 98, 101-122, (2000). 5. Coxson, P.G. and Shapiro, H, Positive reachability and controllability of positive systems. Linear Algebra and its Applications, 94, 35-53, (1987). 6. Farina, L. and Rinaldi, S. Positive linear systems, John Wiley & Sons, (2000). 7. Kaczorek,T., Positive 1D and 2D systems. Springer, (2002). 8. Varga, R.S., Matrix iterative analysis. Prentice-Hall, INC, (1962).
A PLDS Model of Pollution in Connected Water Reservoirs Snezhana P. Kostova Institute of Control and System Research, Bulgarian Academy of Sciences, P.O.B. 79, Sofia 1113, Bulgaria, [email protected] Abstract. In this paper a PLDS (positive linear discrete time system) model of pollution in connected water reservoirs is described. At the beginning, the description of the model is given: state variables, control, parameters, initial conditions, constraints and some assumptions. On the basis of these notations the dynamics of the system is written in matrix form and its positive and compartmental nature is assessed. The real example: five Great Lakes is given. State (pollution) constraints describe the admissible set in the state space. The hit and hold problem is solved by using linear programming approach.
1 Introduction Recently the theory and applications of positive linear systems [5, 6, 7] and in particular compartmental systems [1, 2, 8] have attracted the interest of several authors. One reason for this is the possibility for application in many different fields. Many real-life environmental problems, for example water and soil contamination, mobile and stationary source air pollution, are inherently positive dynamic systems problems. It is very important to control human activities in order to prevent those systems from potential upset. In this paper our interest is focused on the modelling and control of pollution in connected water reservoirs by using positive linear discrete-time systems (PLDS). Our goal is to describe and analyze the system of water reservoirs by means of positive systems in order to control the amount of pollution in each reservoir. At the beginning, the description of the model is given: state variables, control, parameters, initial conditions, constraints and some assumptions. State variables express the amount of pollutant (industrial, agricultural, residential and other) in the reservoirs and control expresses the concentration of pollution in the inflow of incoming water. The parameters of the system are the flows into, between and out of reservoirs and the volume of water in each reservoir. On the basis of these notations the dynamics of the system is written in matrix form and its positive and compartmental nature is assessed. The real L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 257-263, 2003. Springer-Verlag Berlin Heidelberg 2003
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Snezhana P. Kostova
example is given-five Great Lakes. State (pollution) constraints describe the admissible set in the state space. The hit and hold problem is solved by using linear programming approach. In the paper, for each pair of positive integers (n,m) the symbol Rn×m + denotes the set of all n × m non-negative real matrices (matrices with nonnegative entries).
2 Model description Let us consider the system of n connected water reservoirs and with xi (t), i = 1, 2, . . . , n denoting the quantity (in tone, kg or other in dependence of the size of the reservoirs) of pollutant in the i-th reservoir in period t. Let also xi (0), i = 1, 2, . . . , n be the quantity of pollutant at the beginning of the period under consideration. The values of xi (t), i = 1, 2, . . . , n must satisfy the following constraints: xi (t) ≤ x ¯i , i = 1, 2, . . . , n where x ¯i are maximal admissible values of pollutant. With Vi , i = 1, 2, . . . , n we denote the quantity of water (measured in km3 , m3 and other) in each reservoir. Let assume that in m reservoirs (m ≤ n) there are inflows from sources out of the system (incoming flows). We denote the amount of this water with fi , i = 1, . . . , m . The flow from j − th to i − th reservoir is denoted with fi,j , i, j = 1, 2, . . . , n and the flow leaving the system from i − th reservoir with f0i . We assume that fi , fij and f0j , are constant for the considered period. There are two sources of pollution for each water reservoir: Incoming flows fi , i = 1, 2, . . . , m. If we denote the concentration of pollutants in these flows with pi (t), i = 1, 2, . . . , m; t = 0, 1, 2, . . . (measured in t/km3 , kg/m3 ,and other), then the quantity of pollutant coming from sources out of the system in the i − th water reservoir at time t is equal to pi (t)fi . Water flows between reservoirs fij , i, j = 1, 2, . . . , n. f At moment t the quantity of pollutant in these flows is equal to Vijj xj (t) = f
aij xj (t),where aij = Vijj , i, j = 1, 2, . . . , n. When there is no flow from j − th to i − th reservoir fij = 0 and therefore aij = 0. In vector form the system variables and system parameters are: State vector x(t) = (x1 (t), x2 (t), . . . , xn (t))T , x ∈ R+ n denotes the quantity of pollutant in water reservoirs at moment t; Vector initial state x(0) = (x1 (0), x2 (0), . . . , xn (0))T , denotes the quantity of pollutant in water reservoirs at the initial moment ; Control vector u(t) = (ui (t), u2 (t), . . . , um (t))T = (p1 (t), p2 (t), . . . , pm (t))T , u ∈ Rm + denotes the concentration of pollutant in the incoming flows; Parameters Vi , fij , i, j = 1, 2, . . . , n; fi , i = 1, 2, . . . m; Constraint vector x ¯ = (¯ x1 , x ¯2 , . . . , x ¯n ) ∈ Rn+ n Admissible set ∆ = (x ∈ R+ |x ≤ x ¯, ) The model is based on the assumption that any pollutant introduced into a reservoir is immediately and perfectly mixed within the water of the reservoir.
A PLDS Model of Pollution in Connected Water Reservoirs
259
By using the above notations and assumption we can describe the dynamic of the system.
3 System dynamics For each water reservoir we can describe the following balance equation: xi (t + 1) = xi (t) +
n X fij j=1
and by using aij =
fij Vj ,
xi (t + 1) = xi (t) +
Vj
xj (t) −
n X fji j=0
Vi
xi (t) + fi ui (t), i = 1, 2, . . . , n
we have:
n X
aij xj (t) −
n X
aji xi (t) + fi ui (t), i = 1, 2, . . . , n
(1)
j=0
j=1
The system dynamics can be described by a matrix equation x(t + 1) = Ax(t) + Bu(t),
t = 0, 1, 2, . . .
Rn×n +
Rm +.
(2)
and the matrix B ∈ where the matrix A ∈ Taking into account the introduced parameters and variables, each of the water reservoirs of the system can be represented as follows: ' fi ui (t)
-
xi (t) Vi , fi , fij
&
$ aji xi (t)
aij xj (t)
% a0i xi (t) ?
Fig.1. Scheme of water flows in the i-th reservoir
The system under consideration is governed by a low of mass conservation and both the state variables and the control input are physically constrained to remain non-negative along the system trajectories. Consequently, the system is a compartmental PLDS and the system matrices have several structural properties : f 1. The matrix A is nonnegative, 0 ≤ aij = Vijj < 1 for all i, j = 1, 2, . . . , n
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Snezhana P. Kostova
Pn 2. All column sums of A are less than or equal to one: i=1 aij ≤ 1 for all j = 1, 2, . . . , n. 3. The matrix B is nonnegative and consists of monomial columns (columns with exactly one nonzero element) with m nonzero elements equal to f1 , f2 , . . . , fm . A nonnegative matrix with property 2 is said to be a compartmental matrix. From property (2) and [4] it follows that σ(A) = (λ ∈ C; |λ| ≤ 1) where σ is the spectrum of A. If f0i 6= 0 for all i = 1, 2, . . . , n, i.e. for each reservoir there Pn is a flow leaving the system, all column sums of A are less than one, i=1 aij < 1 for all j = 1, 2, . . . , n. This means [4] that σ(A) = (λ ∈ C; |λ| < 1). This is important for the stability analysis of the system. A matrix with property 3 is said to be a monomial matrix. The zero rows of B correspond to the water reservoirs without incoming flow and hence the matrix B have n − m zero rows. In the case where n = m, i.e. each reservoir has incoming flow, the matrix B is a diagonal nonnegative matrix B = diag(f1 , f2 , . . . , fm ) ∈ Rn×n . + Example 1. Using the above notations we will describe the system of five Great Lakes as a positive linear discrete time system. The parameters of the system are given in the following table: Number 1 2 3 4 5
Name
Volume Inflow mi3 mi3 /year Superior 2900 15 Michigan 1180 38 Huror 850 15 Erie 116 17 Ontario 393 14
The flows between lakes measured in mi3 /year are: f31 = 15, f32 = 38, f43 = 68, f54 = 85, f05 = 99. The system matrices are: 1 − a31 0 0 0 0 15 0 0 0 0 0 0 38 0 0 0 1 − a32 0 0 0 a32 1 − a43 0 0 ,B = A = a31 0 0 15 0 0 0 0 a43 1 − a54 0 0 0 0 17 0 0 0 0 a54 1 − a05 0 0 0 0 14 We calculate the coefficients aij =
fij Vj
:
a11 = 0.995, a22 = 0.968, a33 = 0.92, a44 = 0.267, a55 = 1 − 0.252 = 0.748 a31 = 0.005, a32 = 0.032, a43 = 0.08, a54 = 0.733
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261
Then, the PLDS model of the system is: 0.0995 0 0 0 0 15 0 0 0 0 0 0.968 0 0 38 0 0 0 0 0 0 x(t) + x(t + 1) = 0.005 0.032 0.92 0 0 0 15 0 0 u(t) 0 0 0.08 0.267 0 0 0 0 17 0 0 0 0 0.733 0.748 0 0 0 0 14 If as a consequence of one time disaster, for example a terroristic attack, train or ship wreck the levels of pollutants exceed the admissible levels it is important to know how quickly reduction of pollution should be expected. In this case it is necessary to control the concentration of pollutant in the incoming flows in order to achieve admissible set (Fig.2.). The control is achieved by means of turning on operation the refining installation, technology change, reduction or stopping the dirt production and other activity. In the next section of the paper we will solve the hit and hold problem by using linear programming approach. x3 6 xr0 A A x¯3 ¾ x¯2 ´ x 2 ´ ´ ´ ´ ´ x¯1 ´ + ´ x1 Fig.2. Admissible set ∆ and initial state out of ∆
4 Hit and hold problem Consider PLDS system described by the deference equation x(t + 1) = Ax(t) + Bu(t) A∈
,B Rn×n +
∈
,x Rn×m +
∈
Rn+ , u
(3) ∈
Rm +
where the state vectors is subject to the constraint 0 ≤ x(t) ≤ x ¯. Hit and hold problem: For the system (3) with initial condition x(0) = x0 ∈ / ∆ to find control u(t), t = 1, 2, . . .such that the system trajectory hit and hold the admissible set ∆. We will use linear programming approach to solve above problem. Let define the following linear programming problem:
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Snezhana P. Kostova
LP Problem:
Pm max i=1 fi ui s.t Ax0 + Bu ≤ x ¯ ui ≥ 0, i = 1, 2, . . . , m
Note that linear objective function represent the maximal admissible total quantity of pollution incoming into the system. We maximize the upper bound of the control vector u ensuring that x = Ax0 + Bu ∈ ∆. Let the optimal solution of the above problem is umax . If obtained optimal solution is nonnegative, umax ≥ 0, the control objective may be achieved in one step by using control u(t) = λT umax , where λ is arbitrary design vector, such that 0 < λi < 1, i = 1, 2, . . . , m. There are different ways to choose the elements of design vector λ in dependance of physical possibilities to reduce the level of pollution in each of incoming flows. If umax = 0 the system will hit admissible set for minimal number of steps(time intervals) when the control is zero (zero concentration of the incoming water flows). Then the system stay unforced x(t + 1) = Ax(t) since for some k, x(k) = Ak x0 ∈ ∆. Consequence, the minimal number of steps to hit the admissible set is k. After that we will hold the system in admissible set ∆ by solving above LP problem for each time period. The initial state at period (t + 1) is equal to final state at period t. Then every control vector satisfying u(t + 1) = λT umax (t), 0 < λi < 1, i = 1, 2, . . . m will hold the system in the admissible set. Another way to hold the system in ∆ is by using linear state feedback control low for a given system in such a way that the admissible set in the state space to be positively invariant for the closed loop system while the control action does not violate nonnegativity constraints. Consider PLDS system described by the deference equation (3). Feedback holdability problem: For the system (3) with admissible set ∆ = (x ∈ Rn+ |x ≤ x ¯), and with the initial condition x(0) = x0 ∈ ∆, find a linear nonnegative state feedback control low u(t) = Kx(t) ≥ 0, K ∈ Rm×n
(4)
such that ∆ is positively invariant with regard to the motion of the closed loop system x(t + 1) = (A + BK)x(t) = Ac x(t). (5) Following [3] we will prove sufficient condition for a set ∆ to be positively invariant. The set ∆ = (x ∈ Rn+ |x ≤ x ¯) can be expressed as ∆ = (x ∈ Rn+ |x = Dα, α ∈ Rn+ , αi ≤ 1, i = 1, 2, . . . , n)
(6)
where the matrix D = (d1 , d2 , . . . , dn ) = diag(¯ x1 , x ¯2 , . . . , x ¯n ), d1 , d2 , . . . , dn are columns of D and generators of the set ∆.
A PLDS Model of Pollution in Connected Water Reservoirs
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Theorem 1. The set ∆ is positively invariant with regards to the motion of n×n the closed loop system (5) if there exists Pan nonnegative matrix L ∈ R+ with all row sums less or equal to one, i.e. j=1 lij ≤ 1, i = 1, 2, . . . n , such that Ac D = DL
(7)
Proof. Let there exist a nonnegative matrix L with elements lij , such that P n j=1 lij ≤ 1, i = 1, 2, . . . n and Ac D = DL. We have to prove that ∆ is positively invariant set for the system (5), or equivalently that from x ∈ ∆ ⇒ Ac x ∈ ∆. From the expression (6) of the set ∆ it follows that x = Dα, Pn 0 ≤ α ≤ 1. The transformed vector is Ac x = Ac Dα = DLα. But from j=1 lij ≤ 1, i = 1, 2, . . . n and 0 ≤ αi ≤ 1 it follows that all elements of β = Lα are less than or equal to one. Consequently, Ac x = Dβ ∈ ∆, i.e. the transformed vector belongs to ∆ and the set ∆ is positively invariant. Taking into account the above theorem, the feedback holdability problem for the considered system can be solved as a linear programming problem.
References 1. G.Bastin and A. Provost, Feedback stabilization with positive control of dissipative compartmental systems, Int. Symp. MTNS 2002, Notre Dame, 2002. 2. L. Benvenuti and L. Farina Positive and compartmental systems, IEEE Trans. Autom. Contr. Vol. 47, No. 2, 2002. 3. L. Benvenuti and L. Farina, Linear programming approach to constrained feedback control, Int. J. of Systems Science, vol.33, No. 1,pp.45-53, 2002. 4. A. Berman and R.J. Plemmons, Nonnegative matrices in the Mathematical Sciences, SIAM Press, Philadelphia, 1994. 5. L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications, Wiley, New York, 2000. 6. T. Kaczorek, Positive 1D and 2D systems, Springer-Verlag 2002. 7. D.G. Luenberger, Introduction to Dynamic Systems, Wiley,New York, 1979. 8. V. G. Rumchev, D.J.G. James, Reachability and Controllability of Compartmental Systems, Systems Science, 26, No.1, pp.5-13, 2000.
Positivity for Matrix Systems: A Case Study from Quantum Mechanics Claudio Altafini SISSA-ISAS, International School for Advanced Studies, via Beirut 2-4, 34014 Trieste, Italy, [email protected] Abstract. We discuss an example from quantum physics of “positive system” in which the state (a density operator) is a square matrix constrained to be positive semidefinite (plus Hermitian and of unit trace). The positivity constraint is captured by the notion of complete positivity of the corresponding flow. The infinitesimal generators of all possible admissible ODEs can be characterized explicitly in terms of cones of matrices. Correspondingly, it is possible to determine all linear timevarying systems and bilinear control systems that preserve positivity of the state space.
1 Introduction For a matrix system, i.e., a system of ODEs having as state space a set of square matrices, the idea of state that can assume only positive values naturally generalizes to operator positivity i.e., to positive (semi)definiteness of the square matrix representing the state variables. Such a concept is not new in applied mathematics and for example it has long been used in modeling quantum dissipative systems. Assume that the system obeys to a timeinvariant linear (matrix) ODE. To preserve positivity of the state it is not sufficient to have a flow which is itself positive; it is necessary to impose the stronger concept of complete positivity of the flow. Infinitesimally, for positive (semi)definite state matrices that are hermitian and have trace equal to 1 i.e., densities in quantum mechanical language, the set of generators that satisfy complete positivity is well-known under the name of Lindbladian and the type of equations resulting is normally referred to as quantum Markovian master equation [8, 6]. In this note we first expose the general form of the Lindbladian for the particular case of density operators and then we reformulate the matrix system as a vector system in Rn , for suitable n, using a variant of the vec(·) operation. At the same time, we rewrite the constraints that the complete positivity imposes on such a system on Rn . Finally we show how the control inputs can enter into such a system (thus making it a bilinear control L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 265-272, 2003. Springer-Verlag Berlin Heidelberg 2003
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system) while preserving positivity. In fact, the description we obtain is in terms of invariant cones of matrices and therefore all time-varying generators contained in the admissible cone for all times are compatible with the problem. Likewise, control inputs (some constrained, some unconstrained) can be added that also preserve positivity of the state matrix making the system into a bilinear control system. Remarkably, all the Lindbladians admissible by the complete positivity assumption form the set of all possible affine dynamics that leave the unit ball invariant. In the particular case that the dynamics are linear, rather than affine, we obtain the set of semistable matrices belonging to the cone generated by choosing the identity as quadratic form in the Lyapunov equation i.e., {A ∈ Rn×n s.t. A + AT ≤ 0}. In other words, all possible linear dynamics obtained in this way are contracting in the natural inner product of Rn .
2 Completely positive linear maps Assume that the state matrix ρ ∈ CN ×N of our system is positive semidefinite (and therefore Hermitian): ρ ≥ 0. We want to compute what kind of ODE ρ˙ = L(ρ) or control system ρ˙ = L(ρ, u) are compatible with the assumption ρ(t) ≥ 0 ∀ t ≥ 0. A theorem by Choi [5] completely characterizes all linear transformations Λt : CN ×N → CN ×N such that ρ(0) ≥ 0 implies Λt (ρ(0)) ≥ 0. Imposing Λt ≥ 0 is not enough, as we have the following well-known fact (see Appendix A of [6] for a practical example): ¡ (1) ¢ 6 Λ ⊗ Λ(2) (ρ) ≥ 0 Remark 1. ρ = ρ(1) ⊗ ρ(2) , Λ(1) (·) ≥ 0, Λ(2) (·) ≥ 0 =⇒ The stronger requirement that must be imposed on Λt is known under the name of complete positivity. The symbol “⊗” indicates the tensor product (for matrices it coincides with the so-called Kroneker product, see [4]). Definition 1. The map Λt : M (N ) → M (N ) is completely positive if the tensor product map Λ˜t = Λt ⊗ Im is positive ∀ m ∈ N The general form of a completely positive map is also known as the StinePN 2 spring representation [9] and is given by Λ(ρ) = k=1 Wk ρWk† , where the N 2 T operators Wk are N × N matrices and W † = (W ∗ ) (conjugate transposed). To verify complete positivity it is enough to test positive semidefiniteness on the elementary matrices Ejk (having 1 in the (j, k) slot and 0 elsewhere): Theorem 1. ([5], Theorem 2) The map Λt is completely positive if and only if (Λt (Ejk ))1≤j,k≤N ≥ 0. PN 2 If, in addition, we impose Λt (I) = k=1 Wk† Wk = I, then Λt is also trace preserving. This is the case discussed here thereafter.
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3 Quantum dynamical semigroups The state of a quantum mechanical system in an N -dimensional complex space CN ×N can be described in terms of a positive semidefinite Hermitian operator ¡ ¢ ρ, called the density matrix, having trace tr (ρ) = 1 and tr ρ2 ≤ 1. Assume that the system of state ρ is governed by a time-invariant matrix ODE with flow at time t Λt . Λt is a one-parameter semigroup ρ(t) = Λt (ρ(0)), t ∈ R+ , such that: • Λt (Λs (·)) = Λt+s (·) (semigroup property of the flows); • for ρ ≥ 0 Λt (ρ) ≥ 0 (positive semidefiniteness of the state); • tr (Λt (ρ)) = tr (ρ) = 1 (preservation of the trace). Problem 1. Characterize Λt infinitesimally i.e., provide an explicit expression for the infinitesimal generators¡ L(·) ¢ such that ρ˙ = L(ρ) has ∀ t ≥ 0, ρ(t) = ρ† (t) ≥ 0, tr (ρ(t)) = 1 and tr ρ2 ≤ 1. If n = N 2 − 1, the infinitesimal form corresponding to Λt is known under the name of Lindblad (or Gorini-Kossakowski-Sudarshan) form [6] LD (ρ) and is given by n ³ ´ 1 X ajk [λj , ρλ†k ] + [λj ρ, λ†k ] LD (ρ) = 2 j,k=1 (1) n 1 X ajk (2λj ρλk − {λk λj , ρ}) = 2 j,k=1
†
where A = A = (ajk ) ≥ 0 i.e., n × n Hermitian positive semidefinite matrix, [·, ·] is the commutator of matrices, [A, B] = AB − BA, {·, ·} is the anticommutator, {A, B} = AB + BA, and λk the N -dimensional Pauli matrices λj , see for example [7] for their explicit expression (or Appendix A of Part II of [1] for N = 2, 3, 4). Example 1. For N = 2 · ¸ 1 01 λ1 = √ , 2 10
· ¸ 1 0 −i λ2 = √ , 2 i 0
· ¸ 1 1 0 λ3 = √ 2 0 −1
√
The anticommutator for the λk matrices is given by {λj , λk } = 2N 2 δjk λ0 + Pn − 12 I. l=1 djkl λl , with djkl fully symmetric two-tensor and λ0 = N To LD (which is normally called the relaxation or dissipation part of the dynamics) one must add the Hamiltonian part given by LH = −i[H, ρ]
(2)
where −iH is a skew-Hermitian matrix in the special unitaryPLie algebra n su(N ) that can be expressed in terms of basis elements as H = m=1 hm λm (see [2] for details). The set of admissible infinitesimal generators for our problem is then L(ρ) = LH (ρ) + LD (ρ) and is known under the name of d (Λt )t=0 . quantum Markovian master equation. In fact L = dt
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4 Density operators and vectors of coherences From (1) and (2) the ODE ρ˙ = L(ρ) does not look linear (neither affine) at first sight. However, its linearity can be emphasized by transforming the matrix system into a vector system with larger state update matrices. Operations like vec(·) (which stacks the columns of ρ thus obtaining a vector of dim N 2 ) are well-known in control theory, see e.g. [4], and can be used for this scope. For density operators, a physically motivated variant of vec(·) is the so-called vector of coherences formulation [1]. By dimension counting, the state matrix ρ depends on n = N 2 −1 real parameters, since ρ = ρ† and tr(ρ) = 1. Up to the imaginary unit, N × N traceless Hermitian matrices like our λk form the Lie algebra su(N ) of dimension exactly n. If to it we add the (properly normalized) unit vector λ0 , then we obtain a complete basis for the density operator of an N -dimensional quantum mechanical system. In fact, the N -dimensional Pauli matrices λj and the identity matrix λ0 , form a complete orthonormal set of basis operators for ρ (orthonormal in the sense that tr(λj λk ) = δjk ). Pn Pn 1 In particular, then, ρ = j=0 tr(ρλj )λj = j=0 ρj λj , with ρ0 = N − 2 fixed constant and the n real parameters ρj giving the parameterization of ρ. Example 2. In the case of N = 2 ¸ · ρ ρ ρ = 00 01 = ρ0 λ0 + ρx λx + ρy λy + ρz λz ρ10 ρ11 where ρ0 =
√1 , 2
ρ1 =
√ √ 2 Re[ρ01 ], ρ2 = − 2 Im[ρ01 ] and ρ3 =
√1 2
(ρ00 − ρ11 ).
Call ρ = [ρ1 . . . ρn ]T such vector of coherences of ρ. Due to the constant component along λ0 , ρ is living on an affine space characterized by the ex1 tra fixed coordinate ρ0 = N − 2 . Such n dimensional space of affine vectors T ρ¯ = [ρ0 ρ1 . . . ρnp ]T = [ρ0 ρT ]p has Euclidean inner product given by the trace ¯ ρi ¯ = tr (ρ2 ). The condition tr(ρ2 ) ≤ 1 then transla¯ = hρ, metric: kρk ¯ tes in ρ-space as ρ¯ belonging to the solid affine ball of radius 1 centered at ¯ n , for all positive times. [ρ0 0 . . . 0]T , call it B
5 Admissible infinitesimal generators in the vector of coherences form ¯ In ρ-space, the terms LH (·) and LD (·) are linear n × n operators, also called superoperators in the Physics literature. Their expression is as follows:
Positivity for Matrix Systems
· ¸ ¸ · n X 0 0 0 0 ¯ hl ρ¯ = −i ρ¯ LH (ρ) = LH ρ¯ = −i 0 adH 0 adλl l=1 · ¸ n n X X 0 0 ¯= hl =− hl C¯l ρ¯ k ρ 0 clj l=1 l=1 · ¸ X 1 1 X 0 0 ¯ ¯ jk ρ¯ LD (ρ) = LD ρ¯ = ajk ajk L ρ¯ = vjk Ljk 2 2 1≤j,k≤n
269
(3) (4)
1≤j,k≤n
where cklj are the (real and fully skew-symmetric) structure constants of the Lie algebra su(N ) (adλl = (adλl )kj = icklj ), the n × n matrices Ljk are given by: n ¢ 1 X¡ m l m m l Ljk = (Ljk )lr = (cjr − idm jr )ckm + (ckr + idkr )cjm 4 m=1 and the n × 1 real vectors vjk are i vjk = √ [c1jk . . . cnjk ]T N These expressions were obtained in [1], Part II, eq. (II.4.18)-(II.4.19). The partition lines in the matrices above emphasize the use of homogeneous coordinates for the treatment of affine vector fields. The sum (4) contains only real linear combinations of real matrices, as it is shown in [3] Sec. IV.B by properly rearranging A and Ljk . Example 3. In the case N = 2, from (3) one obtains the three 4 × 4 real skew symmetric matrices 0 0 00 0 0 0 0 0 00 0 ¯ 2 = 0 0 0 1 , M ¯ 3 = 0 0 −1 0 ¯ 1 = 0 0 0 0 , M M 0 0 0 0 0 1 0 0 0 0 0 −1 0 01 0 0 −1 0 0 0 0 0 0 while from (4) one gets the nine 4×4 real symmetric or “purely” affine matrices (see again [3] for the details) 0 000 0 000 ¯ 5 = 0 0 0 0 ¯ 4 = 0 0 1 0 , M M 0 0 0 0 0 1 0 0 0 000 −2 0 0 0 0 000 0 000 ¯ 6 = 0 0 0 1 , M ¯ 7 = 0 0 0 0 M 0 0 0 0 2 0 0 0 0 100 0 000
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¯ 10 M
0 0 = 0 0
0 0 0 0
0 000 0 −2 0 0 0 0 ¯ ¯ M8 = 0 0 0 1 , M9 = 0 0 010 0 0 0 0 0 0 0 0 −1 0 0 0 0 ¯ 11 = , M 0 0 0 0 , −1 0 0 −1 0 0 0 −1
000 0 0 0 0 0 0 000 ¯ 12 M
0 0 0 0 0 −1 0 0 = 0 0 −1 0 0 0 0 0
¯ jk and the As it is clear from the example of N = 2, the n2 matrices L n matrices C¯l form a basis of the semidirect product of real Lie algebras M (n, R)sRn (of dimension n2 + n). ¡ ¢ From the Physics of the problem, the condition tr ρ2 ≤ 1 can be reformulated as follows: Proposition 1. The dynamical system ¡ ¢ ¯˙ = L¯H + L¯D ρ¯ ρ
(5)
¯ n. is defined and invariant on B
6 The case of linear generators Due to the presence of affine generators, the system (5) may have an equilibrium point which is different from the origin, which substantially complicates the intuition of the problem. For sake of simplicity, we only treat here the case of L¯D linear, rather than affine: · ¸ 0 0 L¯D = 0 LD Consider the inner product h·, ·i in Rn+1 . Notice first that L¯H is linear ¯ L¯H ρi ¯ = 0. If also L¯D is linear, then and skew-symmetric and therefore hρ, ¯ L¯D ρi ¯ = ρT LD ρ. Furthermore, if LD = LDs + LDk with LDs symmetric hρ, ¯ n to be ¯ L¯D ρi ¯ = ρT LDs ρ. In order for B and LDk skew-symmetric, then hρ, invariant, it has to be ρT LDs ρ ≤ 0. To verify this, it is enough to consider any ρ¯◦ such that kρ¯◦ k = 1: the only (linear) infinitesimal generators admissible ¯ in this case in (5) ¡ ¢are those determined by hρ¯◦ , LD ρ¯◦ i ≤ 0, which, again, correspond to tr ρ2 ≤ 1 in the original problem formulation. Since this must ¯ = 1, then it must be LD ≤ 0. hold for the entire sphere kρk
7 Linear time-varying systems and bilinear control systems preserving positivity The condition A ≥ 0 on the L¯D part of the dynamics identifies a cone of matrices that can serve as Lindbladian for our matrix system. Each choice
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of the n2 parameters ajk of A satisfying the constraint A ≥ 0 preserve the positivity of the state space matrix ρ. Concerning the L¯H part, any linear ¯ k is admissible. combination of the L ¯ jk linear already For sake of simplicity, here we discuss only the case of L treated in Section 6. Proposition 2. Consider the system ¯˙ = L¯H ρ¯ + L¯D ρ¯ = ρ
n X l=1
1 hl C¯l ρ¯ + 2
X
¯ jk ρ¯ ajk L
(6)
1≤j,k≤n
· ¸ · ¸ 0 0 0 0 ¯ ¯ with Cl = , Ljk = and A = (ajk ) ≥ 0. 0 Cl 0 Ljk 1. Any linear time-varying system hl = hl (t) and ajk = ajk (t) is admissible for the problem provided that A(t) = (ajk )(t) ≥ 0, ∀t ≥ 0; 2. Any of the time-varying parameters hl (t) and ajk (t) such that (ajk )(t) ≥ 0, ∀t ≥ 0 can be intended as a control input. Proof. From the discussion at the end of Section 6 ¯ ρ ¯˙ i = ρT LH ρ + ρT LD ρ = ρT LDs ρ ≤ 0 hρ, Thus the dynamics can at most contract toward ρ = 0, regardless of the values of the hl (t) and ajk (t) provided that A(t) = (ajk )(t) ≥ 0, ∀t ≥ 0. u t Corollary 1. All linear dynamics admissible by the problem are semistable and must belong to the cone generated by the identity as quadratic form in the ¢T ¢ ¡ ¡ Lyapunov equation i.e., L¯H + L¯D + L¯H + L¯D ≤ 0.
8 Conclusion The main goal of this paper is to present the general class of linear dynamics that are preserving the positivity (and the trace) of a square Hermitian matrix. Such a model is well-studied in quantum mechanics and is of fundamental importance to avoid formal inconsistencies in the dynamical evolution of a density operator. It is arguable that such a characterization (and its intuitive interpretation given by the coherence vector) may be of interest also in other contexts where positive semidefiniteness of operators under dynamics (also controlled dynamics) is an issue.
References 1. R. Alicki and K. Lendi. Quantum dynamical semigroups and applications, volume 286 of Lecture notes in physics. Springer-Verlag, 1987.
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2. C. Altafini. Controllability of quantum mechanical systems by root space decomposition of su(N). Journal of Mathematical Physics, 43(5):2051–2062, 2002. 3. C. Altafini. Controllability properties for finite dimensional quantum Markovian master equations. To appear in J. of Mathematical Physics. Also preprint arXiv:quant-ph/0211194, 2002. 4. J. W. Brewer. Kroneker products and matrix calculus in system theory. IEEE Transaction on Circuits and Systems, CAS-25(9):772–781, 1978. 5. M. D. Choi. Completely positive linear maps on complex matrices. Linear algebra and its applications, 10:285–290, 1975. 6. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan. Completely positive dynamical semigroups of n-level systems. Journal of Mathematical Physics, 17:821–825, 1976. 7. D. B. Lichtenberg. Unitary symmetry and elementary particles. Academic Press, 1978. 8. G. Lindblad. On the generators of quantum dynamical semigroups. Comm. in Mathematical Physics, 48:119–130, 1976. 9. W. F. Stinespring. Positive functions on C ∗ algebras. Proc. American Mathematical Society, 6:211–216, 1955.
A Simple Food Chain Model with Delay Mario Cavani and Sael Romero Departamento de Matem´ aticas, N´ ucleo de Sucre, Universidad de Oriente, Cuman´ a 6101, Venezuela, [email protected]
Abstract. In this paper we consider a chemostat-like model for a simple food chain of microorganisms. This consists of a well stirred nutrient substance that is the food for a prey population of microorganism. At the bottom of the food chain there is a predator population of miroorganisms that grows up on the prey. The nutrientuptake of each population is of Holling type I (or Lotka-Volterra) form. We show the existence of the global attractor for the solutions of the model and also we show that the positive globally asymptotically stable equilibrium point of the system undergoes a Hopf bifurcation when we suppose that the dynamic of the microorganisms at the bottom of the chain depend on the past history of the prey population by mean a distributed delay that take an average of the microorganisms in the middle of the chain.
1 Introduction In this paper we consider the following simple food chain model b S 0 (t) = (S 0 − S (t))D − S (t) X (t) , γ ¶ µ d 0 X (t) = X(t) bS (t) − D − Y (t) , η 0 Y (t) = Y (t) (dX(t) − D) ,
(1)
S(0) = S0 ≥ 0, X(0) = X0 ≥ 0, Y (0) = Y0 ≥ 0. These equations are in the form of the chemostat model according with the theory given in [4]. The meaning of the variables and parameters is as follows: S(t) denotes the substrate concentration, X(t) and Y (t) denote the concentration of a prey population that grows eating the substrate and the concentration of a predator population that eats the prey respectively. The functional responses of the species X(t) and Y (t) are of the so called Holling type I (or Lotka-Volterra) form (see [3]) where the parameters b and d denote the per capita growth rate of the prey and the predator respectively; γ and η represent the growth yield L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 273-280, 2003. Springer-Verlag Berlin Heidelberg 2003
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constants of the microorganisms X(t) and Y (t) respectively. The parameter S 0 denotes the concentration in the feed bottle and D denotes the input rate from the feed bottle and the washout rate from the growth chamber. We describe the global dynamic of this model in order to point out the differences with the dynamic of a simple food chain model with a distributed delay that we introduce here. The recognition of time lag in the growth response of a population to changes in the environment has led to extensive theoretical and experimental studies but there has been little emphasis in distributed delays in chemostat models, the paper of [5] and its extensive bibliography is important for this subject. Thus, we are assuming in a more realistic fashion that the growth of the predator is influenced by the amount of prey in the past. More precisely, we suppose as, for example, in [1] or [5] that the predator grows up depending on the weight average over the past by mean of the following integro-differential equation Z t ³ ´ Y 0 (t) = dX (τ ) Y (τ ) e−D(t−τ ) αe−α(t−τ ) dτ − DY (t). (2) −∞
In this case, X(0) = X0 (t) = φ(t) ≥ 0, Y (0) = Y0 (t) = ϕ(t) ≥ 0 (t ≤ 0). Clearly this assumption implies that the influence of the past is fading away exponentially and the number α1 might be interpreted as the measure of the influence of the past. So, to smaller α > 0, longer is the interval in the past in which the values of X are taken into account, see [2], [5]. The initial functions φ, ϕ ∈ BC+ , belong to the Banach space of the bounded and continuous functions from (−∞, 0] to IR+ . In order to obtain a dimensionless system (1) and equation (2) we need the following substitutions: t = tD, S =
S X Y bS 0 γdS 0 α , , , X = Y = b = , d = ,α = . S0 γS 0 ηγS 0 D D D
Omitting the bars we have: S 0 (t) = 1 − S (t) − bS (t) X (t) , X 0 (t) = X(t) (bS (t) − 1 − dY (t)) , Y 0 (t) = Y (t) (dX(t) − 1) , and 0
Y (t) =
Z
t
−∞
dX (τ ) Y (τ ) αe−(α+1)(t−τ ) dτ − Y (t).
(3)
(4)
In this paper we show the existence of the global attractor for the solutions of both systems (3) and the corresponding system with delay taking (4) as the third equation in (3). Also we show that the positive globally asymptotically stable equilibrium point of the system (3) losts stability when we model the predator equation of the food chain by mean of (4). In this case the equilibrium of positive coordinates undergoes a Hopf bifurcation and more realistic periodic solutions occur.
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275
2 The simple food chain without delay With respect to system (3) we have the following properties. Lemma 1. The positive cone, IR3+ , is positively invariant with respect to the system (3). Proof. As we can see, if S(t∗ ) = 0 for some t∗ ≥ 0 then S(t) ≥ 0 for all t ≥ t∗ . By other hand Rtaking into account their respective equationsR we can see that t t X(t) = X0 exp 0 (bS (s) − 1 − dY (s))ds and Y (t) = Y0 exp 0 (dY (s) − 1)ds and the positiveness of these functions hold. Also, if we adds the three equations of system (3) and defines W (t) = 1 − S(t) − X(t) − Y (t), then we obtain the single equation W 0 (t) = −W (t) with W (0) > 0. It follows that limt→∞ W (t) = 0 and the convergence is exponential. This implies that the system (3) has the property of pointwise dissipativity in the sense that there exists a bounded set B to which the solutions eventually enter and remain. Thus we have the following lemma. Lemma 2 (dissipativity). The system (3) is pointwise dissipative. Moreover the attractors of the solutions of the system are concentrated on the manifold ª © (5) Σ = (S, X, Y ) ∈ IR3+ : S + X + Y = 1 . The pointwise dissipative property implies the existence of a unique global attractor of the system (3) which must lie in the manifold Σ. Lemma 3. If d ≤ 1 then the predator population Y (t) dies out. Proof. Taking into account the equation for Y (t) in the system (3) and the fact that X(t) ≤ 1 then applying comparative arguments the result is strightforward. By virtue of the previous lemma we suppose in the sequel that d>1
(6)
Also, the dissipativity lemma implies that the system can be reduced in one equation. So, taking S(t) = 1−X(t)−Y (t) we obtain the following system of two ordinary differential equations: X 0 (t) = (b − 1)X(t) − bX 2 (t) − (b + d)X(t)Y (t) , Y 0 (t) = Y (t) (dX(t) − 1) .
(7) (8)
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Lemma 4. If b ≤ 1 then the prey and predator populations X(t) and Y (t) die out. Proof. Taking into account the equation for X(t) and applying comparative arguments the result is strightforward. As a consequence of the previous lemma we suppose in the sequel that b > 1. The system (7-8) has three equilibrium point given by µ ¶ µ ¶ 1 d(b − 1) − b b−1 , 0 , E2 = , . E0 = (0, 0), E1 = b d d(b + d) The stability properties of these equilibrium points are summarized in the following theorem. Theorem 1. (i) If b < 1, then E0 is the unique equilibrium point of system (7)-(8) in the positive cone and is globally asymtotically stable. (ii) If b = 1, the point E0 undergoes a node-saddle bifurcation, and when b > 1 the equilibrium point E1 appears and is globally asymptotically stable d . for 1 < b < d−1 d , the point E1 undergoes a node-saddle bifurcation, and (iii) If b = d−1 when d b> . (9) d−1 the equilibrium point E2 appears and is globally asymptotically stable for this parameters constellation. Proof. Parts (i ) and (ii ) follow inmediately from a linear analysis of the equilibria solutions E0 and E1 . Note that in part (ii ) there are no equilibria with positive coordinates, in this case the Poincar´e-Bendixson theorem implies the result. To show the part (iii ) of the theorem we need to apply the Dulac’s Criterion (see [2]). We call f1 (X, Y ) and f2 (X, Y ) the corresponding functions in the right hand side of the system (7)-(8) for X 0 (t) and Y 0 (t) respectively. In our case we look for a function of the form h(X, Y ) = X α Y δ such that the ∂hf2 1 expression ∂hf ∂X + ∂Y is not zero and does not change its sign while X > 0 and Y > 0. In doing this, we can see that ∂(hf1 )(X, Y ) ∂(hf2 )(X, Y ) + = [(α + 1)(b − 1) − (δ + 1)] X α Y δ ∂X ∂Y + [(δ + 1)d − b(α + 2)] X α+1 Y δ −(b + d)(α + 1)X α Y δ+1 . Therefore, while X > 0 and Y > 0, the previous expression will be always of negative sign if we can find out values of α and δ such that
A Simple Food Chain Model with Delay
(α + 1) −
d δ+1 ≤ 0, (α + 2) − (δ + 1) ≥ 0 b−1 b
277
(10)
hold simultaneously. It is easy to check out that always is possible guarantee the existence of values α∗ and δ ∗ such that the previous inequalities hold, ∗ ∗ and therefore the function h(X, Y ) = X α Y δ satisfies the conditions that we was looking for. So, by applying the Dulac’s Criterion with this function we can see that the system (7)-(8) has no periodic orbits and therefore the Poincar´e-Bendixson theory implies that the equilibrium point E2 is globally asymptotically stable.
3 The simple food chain with delay Now we take in account the model with delay given by taking the equation (4) as the third equation in (3). If we take into accont the new variable Z t ´ ³ (11) dX (τ ) Y (τ ) e−D(t−τ ) αe−α(t−τ ) dτ, α > 0, Z(t) := −∞
then we get the following system. S 0 (t) = 1 − S (t) − bS (t) X (t) , X 0 (t) = X(t) (bS (t) − 1 − dY (t)) , Y 0 (t) = Z(t) − Y (t),
(12)
Z 0 (t) = αdX(t)Y (t) − (α + 1)Z(t), S(0) = S0 ≥ 0, X(0) = φ(0) ≥ 0, Y (0) = ϕ(0) ≥ 0, Z(0) = ϕ(0) ≥ 0. The relations between the solutions of this system and those of the intrgro-differential system with the equation (4) are as in the corresponding description given in [1]. The properties of positiveness and pointwise dissipativeness hold as in the case of the model without delay. So, we have the following lemmata whose proofs follow as in the same way that before. Lemma 5. The positive cone, IR4+ , is positively invariant with respect to the system (12). Now let U (t) = 1 − S(t) − X(t) − Y (t) − Z(t) α , then we obtain the single equation U 0 (t) = −U (t), where U (0) > 0. It follows that limt→∞ U (t) = 0 and the convergence is exponential. This implies, as before, that the system (12) has the property of pointwise dissipativity. Lemma 6 (dissipativity). The system (12) is pointwise dissipative. Moreover the attactors of the system are concentrated on the manifold ¾ ½ Z 4 (13) Λ = (S, X, Y, Z) ∈ IR+ : S + X + Y + = 1 . α
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The pointwise dissipative property implies the existence of a unique global attractor of the system (12) which must lie in the manifold Λ. As before the system can be simplified. In this case we take S(t) = 1 − X(t)−Y (t)− Z(t) to obtain the following system of three ordinary differential α equations: X 0 (t) = (b − 1)X(t) − bX 2 (t) − (b + d)X(t)Y (t) − Y 0 (t) = Z(t) − Y (t),
b X(t)Z(t), α
(14)
Z 0 (t) = αdX(t)Y (t) − (α + 1)Z(t). The system (14) has three equilibrium point given by ¶ µ b−1 , 0, 0 , P2 = (X ∗ , Y ∗ , Y ∗ ) , P0 = (0, 0, 0), P1 = b α+1 ∗ αd(b − 1) − b(α + 1) X∗ = ,Y = . αd d(b(α + 1) + αd) The equilibrium point P2 has sense if and only if αd(b − 1) − b(α + 1) > 0, and this inequality is equivalent to X ∗ < b−1 b , and for b > 1, the right hand side of the last inequality implies that X ∗ < 1. The stability properties of these equilibrium points are summarized in the following theorem. Theorem 2. (i) If b < 1, then P0 is the unique equilibrium point of system (14) in the positive cone and is globally asymtotically stable. (ii) If b = 1, the point P0 undergoes a node-saddle bifurcation, and for b > 1 the equilibrium point P1 appears and is globally asymptotically stable for (α+1)b . b(α + 1) − αd(b − 1) ≥ 0, or equivalently 1 < d ≤ α(b−1) (iii) If d = for
(α+1)b α(b−1) ,
the point E1 undergoes a node-saddle bifurcation, and d>
(α + 1)b α(b − 1)
(15)
the equilibrium point with positive coordinates P2 appears. Proof. Follow inmediately from a linear analysis of the equilibria solutions. The stability properties of the equilibrium point E2 are given in the following theorem. Theorem 3. There exists a value of d∗ such that satisfies the inequality (15) and for d < d∗ the equilibrium point E2 is locally asymptotically stable. If d = d∗ the equilibrium undergoes a Hopf bifurcation and there exist periodic solutions for d > d∗ .
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Proof. The Jacobian matrix of the system (14) evaluated in the equilibrium point E2 is given by −bX ∗ −(b + d)X ∗ − αb X ∗ A = 0 −1 1 αdY ∗ α + 1 −(α + 1) and the charateristic polinomial of this matrix is given by p(λ) = λ3 + a2 λ2 + a1 λ + a0 ,
(16)
where a2 = (α + 2 + bX ∗ ), a1 = bX ∗ (α + 2 + dY ∗ ), a0 = (α + 1)(b − 1 − bX ∗ ). We are going to apply the Routh-Hurwitz Criterion. Note that a0 , a1 , a2 are positive quantities. So we have to check the sign of Φ = a2 a1 − a0 , or equivalently the sign of the expression Ψ (d) =
α2 d2 (1 + bX ∗ ) (a2 a1 − a0 ), X∗
that after some manipulations is possible to write in the form Ψ (d) = c3 d3 + c2 d2 + c1 d + d0 , where c3 = −α3 (b − 1), c2 = α2 b((α + 2)b + (α + 1)(α + 3)), c1 = α(α + 1)b2 (b + (α + 1)(α + 3)), c0 = (α + 1)3 b3 . The polynomial Ψ (d) has the following properties: limd→−∞ Ψ (d) = +∞, Ψ (0) > 0, and limd→+∞ Ψ (d) = −∞. And the derivative Ψ 0 (d) satisfies that limd→−∞ Ψ 0 (d) = −∞, Ψ 0 (0) > 0, and limd→+∞ Ψ 0 (d) = −∞, therefore there exists a unique value d1 such that Ψ 0 (d) > 0 for 0 < d < d1 , Ψ 0 (d1 ) = 0 and Ψ 0 (d) < 0 for d > d1 . This implies that Ψ (d) increases for 0 ≤ d < d1 and decreases for d > d1 . So, accordingly with the afore mentioned properties of Ψ (d), we deduce the existence of a unique value d∗ > d1 , such that Ψ (d) > 0 for 0 < d < d∗ , Ψ (d∗ ) = 0, and Ψ (d) < 0 for d > d∗ . Also, taking into account the explicit value of d1 and that c2 > 2α2 (α + 1) we get d∗ > d1 > (α+1)b α(b−1) > 1. Thus, the Routh-Hurwitz Criterion implies that E2 is locally asymptotically stable for d < d∗ and unstable for d > d∗ . For d = d∗ the equilibrium undergoes a Hopf bifurcation.In fact, λ3 + a2 (d∗ )λ2 + a1 (d∗ )λ + a0 (d∗ ) = (λ2 + a1 (d∗ ))(λ + a2 (d∗ )). Let us denote by λ1 (d∗ ) the root of (16) that assume the value iω, ω 2 = a1 (d∗ ), at d∗ and by
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F (λ, d) = λ3 + a2 (d)λ2 + a1 (d)λ + a0 (d) the characteristic polynomial (16) as a function of the parameter d. Thus the derivative of the implicit function λ1 at d∗ is given by λ01 (d∗ ) = − =−
Fd0 (iω, d∗ ) Fλ0 (iω, d∗ )
a02 (d∗ )(iω)2 + a01 (d∗ )(iω) + a00 (d∗ ) , 3(iω)2 + 2a2 (d∗ )(iω) + a01 (d∗ )
and (Re(λ1 (d∗ ))0d = Re((λ1 )0d (d∗ )) = −
(a1 (d∗ )a2 (d∗ ) − a0 (d∗ ))0d > 0, a1 (d∗ )(1 + a22 (d∗ ))
therefore (Re(λ1 (d∗ ))0d , and so the transversality condition required for the Hopf bifurcation hold and therefore periodic solutions come in sight.
References 1. M. Cavani, M. Lizana, H. L. Smith, Stable Periodic Orbits for a Predator-Prey Model with Delay, J. of Math. Anal. and Appl., 249 (2000), 324-339. 2. M. Farkas, Periodic Motion, Springer-Verlag, New York, 1994. 3. C. S. Holling, The components of predation as revealed by a study of smallmammal predation of the European pine sawfly, Canadian Entomologist, 91 (1959), 293-320. 4. H. L. Smith, P. Waltman, The Theory of the Chemostat, Cambridge University Press, Cambridge, UK, 1995. 5. G. Wolkowicz, H. Xia, S. Ruan, Competition in the Chemostat: A distributed delay model and its global asymptotic behavior, SIAM J. Appl. Math. 57(1997), 1281-1310.
Linear Positive Systems and Phase-type Representations Christian Commault Laboratoire d’Automatique de Grenoble, ENSIEG-BP 46, 38402 Saint Martin d’H`eres Cedex, France, [email protected] Abstract. In this paper we try to put a bridge between the theory of phase-type distributions and the general theory of positive linear systems. The phase-type distributions correspond to the random hitting time of an absorbing Markov chain. The representation problem which consists of finding a Markov chain associated with some phase-type distribution is a positive realization problem. It turns out that this realization problem may be seen as concerning a large class of positive problems. Therefore a lot of results obtained in the field of phase-type distributions may be extended to a large class of positive systems.
1 Introduction This paper is an attempt to make connections between two fields of engineering science which had important but parallel developments in the last twenty years, namely the theory of linear positive systems and the theory of phasetype distributions. The theory of linear positive systems was developed in the community of control with strong connections with the applications areas which are in biology, ecology, production systems, etc [13, 12]. It appeared rapidly that the constraint of positivity makes very difficult problems which are quite easy for usual linear systems. This is the case for the question of existence of a positive realization for a system which has an externally positive behavior [11] and for the problem of finding the minimal dimension of such a realization [2]. A phase-type distribution is the distribution of the time to absorption in a finite state Markov chain, this Markov chain is called a representation of the distribution [15]. It allows the modelling of positive random variables, and in particular random times. The interest in using representations with several stages has been recognized for a long time [10]. The same questions as for linear positive systems appeared for phase-type distributions, namely the questions of existence of a realization [16] and the minimal dimension of such realizations [18]. L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 281-288, 2003. Springer-Verlag Berlin Heidelberg 2003
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Starting from a common root which is the Perron-Frobenius theory [4], these two fields were developed by different communities which seem to have very few communication. In this paper we try to initiate such a communication. We prove first that phase-type representations, which seem to be very special positive realizations, correspond, up to a constant, to the set of exponentially stable positive realizations. Thus all the results which were obtained in the framework of phase-type distributions may be translated in terms of general stable linear positive systems. As an example, we prove that a particular canonical form called the Cox form may be associated with any exponentially stable positive system whose transfer function has only real poles. We also derive a characterization of the relative degree of a positive transfer function in terms of a shortest input-output path in the graph associated with any positive realization of this transfer function. The paper is organized as follows. In section 2 we give the definition of phasetype distributions and representations. Section 3 presents the general problem of positive realization and introduces their graph. In section 4 we show that the phase-representation problem and the positive realization problem are essentially the same problem. In section 5 we introduce the Cox canonical form and give some applications. In section 6 we give a characterization of the relative degree of a positive transfer function in terms of paths in the graph associated with any realization. Some concluding remarks and perspectives end the paper.
2 Introduction to phase-type distributions and representations In this section we present the main definitions concerning phase-type distributions and phase-type representations. Definition 1 [15] A phase-type distribution is the distribution of the absorption time in a finite state absorbing Markov chain. Consider now a Markov chain with (n + 1) states, the first n states being transient and the last one absorbing. This Markov chain is governed by a state equation as follows ˙ x(t) = x(t)Q (1) x(0) = α. where the ith component of the row vector x(t) is the probability of being in state i at time t. The vector α represents the initial probability distribution and is a stochastic vector. The (n + 1) × (n + 1) matrix Q represents the infinitesimal generator of the Markov process. From now on we will assume that α = (α, 0) where α is n dimensional. This means that we assume that the probability of being in the absorbing state at time 0 is 0. We will also denote
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x(t) = (x(t), xn+1 (t)). It can be proved that the probability density function of the absorption time, f (t), can be read from the state equation, x(t) ˙ = x(t)T f (t) = x˙ n+1 (t) = x(t)v.
(2)
Tij ≥ 0 for i 6= j, i = 1, . . . , n; j = 1, . . . , n, is the transition rate from state i to state j. vi for i = 1, . . . , n, represents the transition rate from state i to the absorbing state. Due to the special structure of a Markov generator we must have v = −T 1 where 1 is the n dimensional column vector whose entries are all equal to one.The solution is f (t) = αetT v. We also frequently make use of the Laplace transform of f (t) that we will denote by f˜(s). f˜(s) = α(sIn − T )−1 v, where In denotes the order n identity matrix.This function has the familiar aspect of the transfer function of a single input-single output (SISO) linear system. The order of a representation is the number of transient states of the Markov chain which is the dimension of the matrix T .
3 Positive realizations In this section we will introduce the positive realization problem and recall some results on this problem, see [12] for a fairly complete introduction to this field. We will then prove that the phase-type representation problem is in fact a particular positive realization problem. 3.1 Positive systems A state space representation is said to be positive if it has the form x(t) ˙ = Ax(t) + Bu(t) y(t) = Cx(t).
(3)
where A, B, C are real matrices of respective dimensions n × n, n × m, p × n with aij ≥ 0 for i 6= j; i = 1, . . . , n; j = 1, . . . , n (4) bij ≥ 0 for i = 1, . . . , n; j = 1, . . . , m
(5)
cij ≥ 0 for i = 1, . . . , p; i = 1, . . . , n.
(6)
This state space representation is such that for any initial state x(0) ≥ 0 and any input function u(t) ≥ 0 for t ≥ 0, the state and output functions remain non-negative. Positive representations appear naturally in modelling of a number of physical, economical and ecological systems [13]. A linear system defined by its rational transfer function g˜(s) is said to be (externally) positive if its impulse response g(t), the inverse Laplace transform of g˜(s), is such that g(t) ≥ 0 for t ≥ 0. We assume that the transfer function is strictly proper, that is g(0) = 0.
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3.2 Graph of a positive realization To a positive state space representation one can associate a graph. The vertices are associated with input, state and output variables. Therefore we have m input vertices u1 , . . . , um , p output vertices y1 , . . . , yp and n state vertices x1 , . . . , xn . There is an edge (xi , xj ) (resp. (ui , xj ), (xi , yj )) if aji > 0 (resp. bji > 0, cji > 0). This graph is called the influence graph in [12]. A circuit is a closed path and a graph is said to be acyclic when it has no circuit. We say that the representation is irreducible if any state vertex belongs to an input-output path in the associated graph. Irreducibility is equivalent to both excitability and transparency in the terminology of [12]. It is clear that when a representation is not irreducible it can be simplified by discarding some states, therefore we may restrict our attention, without loss of generality, to irreducible representations.
4 Some properties of stable positive systems The main results of this section was presented first in [7], it is given here without proof. Theorem 1 [7] Consider a linear system with transfer function g˜(s) and impulse response g(t) which admits a positive irreducible realization of order n, (F, G, H): 1. if g˜(s) is asymptotically stable there exists an order n positive irreducible realization of g˜(s), (A, B, C), which satisfies: n X
aij + bi = 0 for i = 1, . . . , n,
(7)
j=1
and the graph of (A, B, C) is the same as the graph of (F, G, H), 2. if moreover g˜(0) = 1 then n X ci = 1.
(8)
i=1
Remark 1 It is clear that we can get as well a realization which is the dual of the previous one. Therefore, Theorem 1 is a generalization of a result in [1] in which it is proved that any asymptotically stable positive system which has a positive realization may be seen as a compartmental system. Remark 2 A very important consequence of Theorem 1 is that most of the results in the realization theory of positive systems may be used for phase-type representations and vice versa. With this last remark in mind we will continue the paper by the study of results obtained in the framework of phase-type representations and which may be transferred to classes of positive systems.
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5 The Cox canonical form for linear positive systems In the field of phase-type distributions and representations an important effort has been devoted to the study of particular canonical forms [8, 10]. In the theory of general linear systems we are also used to some classical canonical forms as the reachable and observable forms or the diagonal (or Jordan) form. It happens that these usual canonical forms may not be adapted for some important classes of positive systems. • The reachable or observable canonical forms show up on the last row of the A matrix the coefficients of its characteristic polynomial. Since for stable positive systems these coefficients are negative, the corresponding representation is not positive. • For a positive systems, the residues need not be positive for all the poles, then the diagonal realization need not be positive. The aim of this section is to prove that the Cox canonical form presents an interesting alternative to these usual canonical forms in the case of positive stable SISO systems. Definition 2 We call Cox canonical form a SISO state space representation (A, B, C) such that β1 −λ1 α1 0 ... 0 β2 0 −λ2 α2 . . . 0 ¡ ¢ .. .. . . . . . . . . , B = A= . , C = δ1 0 0 . . . 0 . . . . . . βn−1 0 . . . 0 −λn−1 αn−1 0 ... ... 0 −λn λn (9) where the parameters satisfy the following conditions αi , βi ≥ 0 for i = 1, . . . , n.; λ1 ≥ λ2 ≥ . . . ≥ λn > 0; δ1 > 0 and moreover
αi + βi = λi for i = 1, . . . , n − 1.
(10) (11)
Some properties and limitations immediately follow from the definition of the Cox canonical form. • It is a positive representation, • The number of independent parameters is 2n as in the reachable canonical form, • From the triangular form of the A matrix, the eigenvalues must be real, therefore this canonical form can only represent transfer functions with real poles, • There is no hope to extend this canonical form to unstable systems since condition (11) would induce a non positive representation.
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We are now in order to use some results of the phase-type representation theory for our purpose. Theorem 2 Consider an asymptotically stable transfer function f˜(s) with real poles. Then this transfer function has a Cox realization. Proof. Denote f˜(s) = f˜(0)f¯(s). f¯(s) can be considered as the Laplace transform of a phase-type distribution. From [17] this distribution has a Cox representation which has the form of definition (2) with δ1 = 1. Hence f˜(s) has a realization of the same form with δ1 = f˜(0). Remark 3 Some canonical forms similar to the Cox form have appeared in the literature on positive systems. A canonical form less constrained that the Cox form was given in [3] for order 3 systems. A canonical form with the same structure but different constraints was given in [14] for compartmental systems. Remark 4 Theorem 2 means that one can find a Cox realization. There is no guarantee that the order of this realization is the minimal order for a positive realization. A positive state space representation is said to be triangular when its graph is acyclic. The term triangular comes from the fact that, in an acyclic graph, there is a natural order among the vertices and renumbering the states according to this order induces a representation where the A matrix is upper triangular. Theorem 3 Consider an asymptotically stable transfer function f˜(s) which has a triangular positive realization (A, B, C). Then f˜(s) has a realization of at most the same order which is in Cox canonical form Proof. Again, denote f˜(s) = f˜(0)f¯(s) and (A, B, C 0 ) a realization of f¯(s) with 1 C 0 = f˜(0) C. From Theorem 1, (A, B, C 0 ) which is a triangular realization can be transformed in another triangular realization which is a phase-type representation. In [9] it is proved that this triangular representation can be transformed in a Cox form of at most the same order (F, G, H), see also [7]. f˜(s) has then a Cox canonical realization (F, G, H 0 ) where H 0 = f˜(0)H. Theorem 3 says that, given a positive realization which graph is acyclic, one can find a positive realization with a simpler graph and therefore defined by a lesser number of parameters.
6 A structural invariant We give in this section a general result concerning positive systems which again was found first in the framework of phase-type distributions [5]
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Theorem 4 Consider a linear positive system with transfer function f˜(s) = p(s) q(s) , where p(s) and q(s) are coprime polynomials. Then the difference deg(q(s))− deg(p(s)) is equal to the minimal number of state vertices on an input-output path in the graph of any realization of f˜(s). Proof. The proof is given in [5] for phase-type distributions. It is based on the graph interpretation in terms of input-output paths of CAk B. In this proof the fact that the diagonal terms in A are negative (which is a characteristic of stability) for phase-type representations is not relevant, therefore the result stands for arbitrary positive realizations. This number, deg(q(s)) − deg(p(s)), is known as the relative degree or the infinite zero order for a usual transfer function. However, the equality holds only generically [6] without the positivity assumption.
7 Conclusion In this paper we have tried to put a bridge between the phase-type representation problem and the positive realization theory. These two fields had their own development starting with a common root which was the PerronFrobenius theory. Our contribution, in particular through Theorem 1, is to make clear that the problems are essentially the same and that most of the results obtained in one of these fields could be translated in the other one. As examples we have shown that the Cox canonical form may be of interest for general linear positive systems and that a simple structural result on the relative degree proved for phase-type representations is also true in general. It is clear that a lot remains to be done to exploit and interpret in each field the results which are coming from the other one.
References 1. L. Benvenuti and L. Farina. Positive and compartmental systems. IEEE Trans. Autom. Cont., 47(2):370–373, 2002. 2. L. Benvenuti and L. Farina. Minimal positive realizations: a survey of recent results and open problems. Kybernetika, To appear in 2003. 3. L. Benvenuti, L. Farina, B.D.O. Anderson, and F. de Bruyne. Minimal positive realizations of transfer functions with positive real poles. IEEE Trans. on Circ. and Syst.,I, 47:1370–1377, 2000. 4. A. Berman and R. J. Plemmons. Nonnegative matrices in the mathematical sciences, volume 9 of Classics in Applied Mathematics. SIAM, Philadelphia, 1994. 5. C. Commault and J-P. Chemla. An invariant of representations of phase-type distributions and some applications. J. of Appl. Prob., 33(2):368–381, 1996.
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6. C. Commault, J.M. Dion, and A. Perez. Disturbance rejection for structured systems. IEEE Trans. on Aut. Cont., 36:884–887, 1991. 7. C. Commault and S. Mocanu. Phase-type distributions and representations: some results and open problems for system theory. Int. J. of Control, To appear, 2003. 8. D.R. Cox. A use of complex probabilities in the theory of stochastic processes. Proc. Camb. Phil. Soc., 51:313–319, 1955. 9. A. Cumani. On the canonical representation of homogenous Markov processes modelling failure-time distributions. Microelectronics and reliability, 22(3):583– 602, 1982. 10. A.K. Erlang. Solution of some problems in the theory of probabilities of significance in automatic telephone exchanges. P.O. Elect. Eng. J., 10:189–197, 1917. 11. L. Farina. On the existence of a positive realization. Syst. and Cont. Letters, 28:219–226, 1996. 12. Farina L. and Rinaldi S. Positive Linear Systems. Wiley, New York, 2000. 13. D. Luenberger. Introduction to dynamical systems: theory, models and applications. Wiley, New York, 1979. 14. H. Maeda, S. Kodama, and F. Kajiya. Compartmental system analysis: realization of a class of linear systems with physical constraints. IEEE Trans. on Circ. and Syst.,I, 28:39–47, 1981. 15. M. Neuts. Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach. The John Hopkins University Press, Baltimore and London, 1981. 16. C. A. O’Cinneide. Characterization of phase-type distributions. Commun. Stat., Stochastic Models, 6(1):1–57, 1990. 17. C. A. O’Cinneide. Phase-type distributions and invariant polytopes. Adv. Appl. Probab., 23(3):515–535, 1991. 18. C. A. O’Cinneide. Phase-type distributions: open problems and a few properties. Commun. Stat., Stochastic Models, 15(4), 1999.
Blending Positive Matrix Pencils with Economic Models Teresa P. de Lima Faculdade de Economia, Universidade de Coimbra, Av Dias da Silva, 165, 3004-512 Coimbra, Portugal, [email protected] Abstract. We consider E(x)x(k + 1) = F (k)x(k), k ∈ Z0+ , where E(k) and F(k) are real square matrices of order n, not necessarily invertible. Assuming the regularity of the matrix pencils λE(x) − F (k)x(k), k ∈ Z0+ and the existence of ˆ a nonzero common eigenvector of the family of n × n real matrices {E(k) = [αk E(k) − F (k)]−1 E(k), αk ∈ R, k ∈ Z0+ }, we will obtain a solution to the above descriptor system. We also analyse a particular case related with a positive equation — the closed dynamic Leontief model.
1 Introduction The plan of this paper is twofold. First we apply some results from spectral theory for matrix pencils to the study of the solutions of discrete linear timevariant descriptor systems described by E(x)x(k + 1) = F (k)x(k), k ∈ Z0+ n×n
(1)
where the matricial coefficients E(k), F (k) ∈ R can be singular. We are particulary interested in the study of the situation of (1) possessing nonnegative solutions. Second, because the general problem just described is difficult, we shall also consider a particular case related with a positive equation — the closed dynamic Leontief model. The paper is organized as follows: section 2 is devoted to some definitions and results required to our study. In section 3 we exhibit certain conditions under which a particular kind of matrix pencils has a common eigenvector, and we present a technique to obtain a special type of solutions to a specific homogeneous system (with time-variant coefficients) related with an economic model. Finally we conclude with an open problem.
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2 Preliminaries In this section we briefly introduce the notation and summarize some concepts and properties of matrix pencils. Let Z0+ be the set of nonnegative integers. The set of m × n real (complex) matrices will be denoted by Rm×n (C n×n ) and Rm = Rm×1 (C m = C m×1 ). A matrix A is nonnegative (positive) if each entry of the matrix is nonnegative (positive), in which case we write A ≥ 0(A > 0). Definition 1. [6] A family of real (complex) matrices n × n is an arbitrary (finite or infinite) set of matrices, and a commuting family is one in which each pair in the set commutes under multiplication. Lemma 1. [6] If Mc is a commuting family of n × n complex matrices, then there is a nonzero vector v ∈ C n that is an eigenvector of every A ∈ Mc . Definition 2. [12] A nonzero vector v ∈ C n is a common eigenvector of the n-square complex matrices E and F if there exists two complex numbers, λ and µ, such that Ev = λv and F v = µv. By Lemma 1, whenever two matrices E and F commute they possess at least one common eigenvector. Definition 3. If E and F are real n × n matrices, the family of matrices {λE − F, λ ∈ R} is called a real matrix pencil of order n. If E ≥ 0 and F ≥ 0, λE − F will be called a positive matrix pencil of order n. Definition 4. A real matrix pencil of order n, λE − F , is regular if there is a real number λ0 such that det [λ0 E − F ] 6= 0. Definition 5. Let λE − F be a regular real matrix pencil of order n. If there exists two real numbers α and β such that αE + βF = I then λE − F is a standard real matrix pencil. Remark 1. Furthermore, if the real matrix pencil λE − F is regular, there exists a scalar λ0 such that det [λ0 E − F ] 6= 0 and it can be easily transformed −1 into a standard pencil multiplying, on the left, λE − F by (λ0 E − F ) . The −1 ˆ − Fˆ where E ˆ = (λ0 E − F ) E and standard pencil obtained will be λE −1 ˆ F = (λ0 E − F ) F and will be called a standard pencil associated to the regular real matrix pencil. Definition 6. The regular real matrix pencil of order n, λE − F , is almostpositive (and denoted a-positive) if there exists a positive standard pencil associated to the regular real matrix pencil λE − F . Proposition 1. Let λE − F be a regular real matrix pencil of order n and ˆ = (λ0 E − F )−1 E and Fˆ = λ0 ∈ R such that det [λ0 E − F ] 6= 0. Define E −1 ˆ Fˆ = Fˆ E. ˆ (λ0 E − F ) F . Then λ0 E − F = I and E
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Definition 7. Let λE − F be a real regular matrix pencil of order n. Then there are r distinct (real or complex) numbers,λ1 , λ2 , . . . λr , r ≤ n, for which det [λj E − F ] = 0, j = 1, 2, . . . , r.These numbers are called the (finite) eigenvalues — or latent roots — of the real pencil. Any nontrivial ¡ solutions of the¢ homogeneous equation [λj E − F ] q = 0, j = 1, 2, . . . , r, rT [λj E − F ] = 0 are known as right (left) eigenvectors — or latent vectors — of the real matrix pencil λE − F corresponding to λj . If matrix E is singular then it is said that λE − F has an eigenvalue at infinity. The spectrum of λE − F , written σ(E, F ), is the set of all eigenvalues of λE −F (including that at infinity when it exists). Definition 8. A nonzero vector v ∈ C n is a common eigenvector of the set of matrix pencils {λE(k) − F (k), k ∈ Z0+ }, if there exists a sequence {ρk ∈ C, k ∈ Z0+ } such that [ρk E(k) − F (k)] v = 0, k ∈ Z0+ . Remark 2. The above results suggest the next Problem 1: Can every real matrix pencil be transformed in a real standard one?
3 Solutions to discrete linear time-variant descriptor systems 3.1 Homogeneous case Let
E(k)x(k + 1) = F (k)x(k), k ∈ Z0+
(2)
be a discrete linear time-variant descriptor system where x(k) is the semistate vector and where the matricial coefficients E(k), F (k) ∈ Rn×n can be singular. We presume the existence of: (i) a real number αk such that det [αk E(k) − F (k)] 6= 0, k ∈ Z0+ ;and (ii) a ˆ common eigenvector, v, of the family of n × n real matrices {E(k), k ∈ Z0+ }. Under the above conditions we can obtain the following solution x(k) = ρk−1 ρk−2... ρ1 ρ0 v, 1 ˆ b = F(k)v and , µk = ak −ρ , j = 0, 1, . . . , k − 1, k ∈ Z0+ , with ρj 6= 0,ρk E(k)v k to the time-variant descriptor system (2). In the subsequent lemma it will be seen that, for k ∈ Z0+ , the eigenproblems
ˆ ˆ = Fˆ (k)v, E(k)w = µk w and ρk E(k)v share their eigenvectors and that the corresponding non-null finite eigenvalues are related. Lemma 2. Let λE(k) − F (k), for k ∈ Z0+ , be a real regular matrix pencil of order n and αk a real number such that det [αk E(k) − F (k)] 6= 0, k ∈ Z0+ . −1 −1 ˆ Define E(k) = (αk E(k) − F (k)) E and Fˆ (k) = (αk E(k) − F (k)) F (k).
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ˆ ˆ Then the real matrix E(k) and the standard pencil λE(k) − Fˆ (k) share the eigenvectors; and if µk , k = 1, 2, . . . , r1 are the real non-null eigenvalues ˆ ˆ of E(k) and ρk , k = 1, 2, . . . , r2 those of λE(k) − Fˆ (k) which are real and 1 different from αk , then µk = ak −ρk . ˆ Proof. Suppose that ρ is a real, different from αk , eigenvalue of λE(k) − Fˆ (k). ˆ ˆ Then ρE(k)v = F (k)v, where v is an eigenvector corresponding to ρ. So we can write ´ ³ ˆ ˆ ˆ ˆ ˆ − I v ⇐⇒ ρE(k)v = αk E(k)v− ρE(k)v = Fˆ (k)v ⇐⇒ ρE(k)v = αk E(k) ˆ ˆ ˆ ˆ v ⇐⇒ αk E(k)v − ρE(k)v = v ⇐⇒ (αk − ρ) E(k)v = v ⇐⇒ E(k)v = 1 v. αk −ρ
Consequently, if ρ 6= αk , we can conclude that µ = αk1−ρ is a real non-null ˆ ˆ eigenvalue of E(k) and v is an eigenvector of E(k) corresponding to µ. Now, using Lemma 2, we state and prove our main result which is the following
Proposition 2. Let λE(k) − F (k), for k ∈ Z0+ , be a real regular matrix pencil of order n and αk a real number such that det [αk E(k) − F (k)] 6= 0, k ∈ −1 −1 ˆ Z0+ . Define E(k) = (αk E(k) − F ) E and Fˆ = (αk E − F ) F . If Mc is a ˆ commuting family of n × n real matrices and E(k) ∈ Mc , then: ˆ (i) there exists a nonzero v ∈ C n such that E(k)v = µk v, µk ∈ C, k ∈ Z0+ ; (ii) if v ∈ Rn , µk ∈ R and µk 6= 0, we are able to affirm that the set ˆ of matrix pencils {λE(k) − Fˆ (k), k ∈ Z0+ } has a common real eigenvector v associated with the real, non-null and different i from αk , eigenvalue, ρk , such h 1 ˆ ˆ that µk = αk −ρk ∈ R , i.e., ρk E(k) − F (k) v = 0, k ∈ Z0+ ; (iii) if ρk ∈ / {0, αk }, we can set up the following real solution, x(k) = ρk−1 ρk−2... ρ1 ρ0 v ∈ Rn ,
(3)
ˆ to E(k)x(k + 1) = Fˆ (k)x(k) and, therefore, to the descriptor time-variant system (2). Proof. (i) Clear from Lemma 1. (ii) This follows from Lemma 2. ˆ (iii) Suppose that v is a real eigenvector of the matrix pencil λE(k) − Fˆ (k) associated with the non-null eigenvalue ρk . Therefore, assuming that x(0) = v, we can write x(k) = ρk−1 ρk−2... ρ1 ρ0 v ∈ Rn , because E(k)x(k+1) = F (k)x(k) i ρk−2... ρ1 ρ0 v = F (k)ρk−1 ρk−2... ρ1 ρ0 v ⇔ h ⇔ E(k)ρk ρk−1 ˆ ˆ [ρk E(k) − F (k)] v = 0 ⇔ ρk E(k) − F (k) v = 0. Remark 3. a) Proposition 2 generalizes the results we have obtained for the periodic case in [10]; b) If (4) E(k)x(k + 1) = F (k)x(k), k ∈ Z0+ ,
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is a positive descriptor system — i.e., E(k) ≥ 0 and F (k) ≥ 0, k ∈ Z0+ [7] — and if it is possible to choose a real sequence {ak , k ∈ Z0+ } such that −1 [αk E(k) − F (k)] ≥ 0, we can change (4) in another positive descriptor system ˆ E(k)x(k + 1) = Fˆ (k)x(k), k ∈ Z + , (5) 0
−1 −1 ˆ where E(k) = (λ0 E(k) − F (k)) E(k) and Fˆ (k) = (λ0 E(k) − F (k)) F (k) ˆ ˆ ˆ ˆ and E(k)F (k) = F (k)E(k). The study of this new positive system (5) will be simpler than the case when the matricial coefficients don’t commute. So we pose the related Problem 2: Under what conditions can we transform a positive matrix pencil into a positive standard one? c) Finally, in this subsection, some other questions arise: (c.i) When does a (commuting, or not) family of n × n real matrices have a common real eigenvector? (c.ii) Under what conditions can we guarantee the existence of a common real eigenvector to the set of real matrix pencils {λE(k) − F (k), k ∈ Z0+ }?
3.2 A particular case - the closed dynamic Leontief model The discrete-time standard version of the dynamic input-output model is of the form x(k) = A(k)x(k) + B(k)x(k + 1) − x(k) + y(k), k ∈ Z0+
(6)
where x(k) ∈ Rn is the vector of output (production) for period k, y(k) ∈ Rn is the vector of final demand (excluding investment) at time period k, A(k) = [aij (k)] ∈ Rn×n is the matrix of input-output coefficients at time period k, and B(k) = [bij (k)] ∈ Rn×n is the capital coefficient matrix for period k. In the case A(k) ≡ A and B(k) ≡ B, for k ∈ Z0+ , i.e., when no change in the technology is assumed over time, the model (6) becomes Bx(k + 1) = (I − A + B)x(k) − y(k), k ∈ Z0+
(7)
and sufficient conditions for the existence of nonnegative solutions to the above nonhomogeneous model are presented in [13]. If the final demand is completely endogenous we obtain a homogeneous system that can be interpreted as a closed Leontief dynamic input-output model. [14] and [16] provide some answers (about the existence of a balanced growth solution) to the (homogeneous) invariant case. In previous work [2],[3] we have also supposed that A and B are timeinvariant, since in the original version of the static Leontief model no change in the technology is assumed over time.
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However in the study of the dynamic version of the referred model the above assumption has become less realistic than before — [11], [14], [1]. So, in [9] and [10], we have analysed a generalization of the invariant case — the periodic case — hoping that our approach will contribute to create interactions with scientists from economics (and/or other areas) and develop interdisciplinary research, although we didn’t know any economic arguments for the periodicity of the matricial coefficients. Unfortunately most economists don’t agree with the periodicity of the technical and capital coefficient matrices. As a consequence we will treat the time-variant case, i.e., we will presume, in this paper, that matrices A(k) and B(k) are time-variant. Before we start, we must remember that (in the economic context): (i) the matrices A(k) and B(k), as well as the vector x(k), are nonnegative; (ii) the spectral radius of A(k) is less then one (and thus the inverse −1 [I − A(k)] exists and is nonnegative); (iii) the capital coefficient matrices B(k) can be singular. So, the above economic model (6) can be described by the homogeneous descriptor discrete time-variant system B(k)x(k + 1) = H(k)x(k), k ∈ Z0+
(8)
where H(k) = I − A(k) + B(k) ∈ Rn×n , B(k) is not necessarily invertible, −1 and [I − A(k)] ≥ 0, k ∈ Z0+ . We verify that the real matrix pencil λB(k) − H(k) is not a positive one. However if λ = 1 then det [λB(k) − H(k)] 6= 0. So λB(k) − H(k) is regular and [4] the descriptor system (7) is solvable, which means that if we start with an initial condition x(0) we obtain a unique solution. Moreover it is possible to construct the following positive standard matrix −1 pencil λU (k) − [I + U (k)], where U (k) = [I − A(k)] B(k) ≥ 0, k ∈ Z0+ , associated to the real matrix pencil λB(k) − H(k). So we can conclude that although λB(k) − H(k) is not positive it is an a-positive matrix pencil. The application of Proposition 2 to the particular economic case can be summarized in Proposition 3. If Mc is a commuting family of n × n real matrices and A(k), B(k) ∈ Mc , for k ∈ Z0+ , then: −1 (i) U (k) ∈ Mc , where U (k) = [I − A(k)] B(k) ≥ 0, k ∈ Z0+ ; (ii) we can n assure the existence of a nonzero v ∈ C such that U (k)v = µk v, k ∈ Z0+ . That means, v is a common eigenvector of the matrices U (k), k ∈ Z0+ ; (iii) if v ∈ Rn , µk ∈ R and µk 6= 0, we are able to affirm that the set of matrix pencils {λB(k) − H(k), k ∈ Z0+ } has a common real eigenvector v associated with the real, non-null and different from one, eigenvalue, ρk , such that µk 1 ∈ R , i.e., [ρk B(k) − H(k)] v = 0, k ∈ Z0+ ; = 1−ρ k
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(iv) if ρk ∈ / {0, 1}, we can set up the following real solution, x(k) = ρk−1 ρk−2... ρ1 ρ0 v ∈ Rn .
(9)
for the anterior homogeneous descriptor discrete-time economic system (8). Proof. It is known [14] that the eigenvectors of the matrix pencil λB(k) − H(k)(corresponding to the eigenvalue ρk = 1) coincide with eigenvectors of −1 1 ), the matrix U (k) = [I − A(k)] B(k) (corresponding to the eigenvalues 1−ρ k + k ∈ Z0 . Therefore, if A(k), B(k) ∈ Mc , we observe that there exists a nonzero vector v ∈ C n such that [ρk B(k) − H(k)] v = 0, ρk ∈ C, k ∈ Z0+ . So, if A(k), B(k) ∈ Mc then v is a common eigenvector of the matrix pencils λB(k) − H(k), k ∈ Z0+ . Remark 4. It may be objected that we have impose strong and unrealistic restrictions on the matrices A(k) and B(k) — such as commutativity — as well as (9) is a solution with no economic meaning. Thus an important circle of questions concerns the study of the existence of nonnegative solutions to the homogeneous descriptor discrete time-variant economic system (8).
4 Conclusion In order to investigate what sorts of conditions lead to the existence of a nonnegative solution to the positive descriptor system E(k)x(k + 1) = F (k)x(k), k ∈ Z0+ ,
(10)
by means of an eigenvector and the spectrum of the matrix pencil λE(k) − F (k), k ∈ Z0+ , we must be able to give some answers to the Problem 3: Let λE(k) − F (k), k ∈ Z0+ , be a regular real matrix pencil of ˆ order n, αk a real number such that det [αk E(k) − F (k)] 6= 0, k ∈ Z0+ , E(k) = −1 −1 ˆ (λ0 E(k) − F (k)) E(k) ≥ 0 and F (k) = (λ0 E(k) − F (k)) F (k) ≥ 0. Is it possible to choose a vector v ∈ Rn satisfying the condition i h ˆ − Fˆ (k) v = 0, k ∈ Z0+ , ρk E(k) i h ˆ − Fˆ (k) = 0, such that where ρk ∈ R\{0}, det ρk E(k) ρk−1 ρk−2... ρ1 ρ0 v ≥ 0 ? Thus a major step in this direction will be taken when it will be possible to provide some responses to the problems/questions referred in Remarks 2-4.
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References 1. Amaral, J.F. (1991). Curso Avan¸cado de An´ alise Econ´ omica Multi-Sectorial, Escher, Lisbon, Portugal.(in Portuguese) 2. Borges, A. and T. P. de Lima (1994). O Estudo de Sistemas Lineares Discretos na Economia - Modelo Dinˆ amico de Leontief, vol.II, pp.25-29. Proc. 1st Portuguese Conf. Aut. Control, Lisboa, Portugal.(in Portuguese) 3. Borges, A. and T. P. de Lima (1996). Nonsingular Formulation and Reachability for Leontief Dynamic Systems, pp.593-598. Proc. 2nd Portuguese Conf. Aut. Control, Porto, Portugal. 4. Campbell, S.L. (1980). Singular Systems of Differential Equations, Pitman, vol I. 5. Dai, L. (1989). Singular Control Systems, Lecture Notes in Information Sciences, Springer-Verlag, Berlin. 6. Horn, R. A. and C. A. Johnson (1985). Matrix Analysis, Cambridge University Press. 7. Kaczorek, T. (1997). Positive Singular Discrete Linear Systems, Bulletin of the Polish Academy of Sciences: Tech., Vol.45, No 4, pp. 619-631. 8. Lancaster, P. and L. Rodman D.A. (1995). Algebraic Riccati Equations, Oxford Science Publications. 9. Lima, T.P. (2001). Studying a Basic Price Equation as a Periodic System, Proceedings of the IFAC Workshop on Periodic Control Systems, 27-28th August, 2001, Cernobio-Como, Italy. 10. Lima, T.P. and A. Borges (2002). Matrix Pencils with Joint Eigenvectors versus Homogeneous Periodic Systems, Proceedings of the 5th Portuguese Conference on Automatic Control (Controlo 2002), 5-7 September, 2002, Aveiro, Portugal. 11. Livesey, D.A. (1973). The Singularity Problem in the Dynamic Input-Ouput Model, Int.Journal Systems Sci., Vol.4, No 3, pp. 437-440. 12. Shemesh, D. (1984). Common Eigenvectors for Two Matrices, Linear Algebra Appl. 62, pp 11-18. 13. Silva, M.S. and T.P. de Lima (2002). Looking for Nonnegative Solutions of a Dynamic Leontief Model, Linear Algebra Appl. to appear. 14. Szyld, D.B. (1985). Conditions for the Existence of a Balanced Growth Solution for the Leontief Dynamic Input-Output, Econometrica, vol.53, No 6, pp.14111419. 15. Takayama, A. (1996). Mathematical Economics, 2nd ed, Cambridge University Press. 16. Zeng, L. (2001). Some Applications of Spectral Theory of Nonnegative matrices to Input-output Models, Linear Algebra Appl. 336, pp. 205-218.
On the Positive Reachability of 2D Positive Systems Ettore Fornasini and Maria Elena Valcher Dipartimento di Ingegneria dell’Informazione, Universit` a di Padova, via Gradenigo 6B, 35131 Padova, Italy, {fornasini,meme}@dei.unipd.it Abstract. Local reachability of two-dimensional (2D) positive systems, by means of positive scalar inputs, is addressed by means of a graph theoretic approach. Some results concerned with equivalent conditions for local reachability as well as upper and lower bounds on the reachability indices are provided.
1 Introduction Recent years have seen a growing interest in two-dimensional (2D) systems that are subject to a positivity constraint on their dynamical variables [2, 3, 4, 5]. There are actually several different motivations for this interest, coming from various domains of science and technology. Positive 2D systems arise, for instance, when discretizing pollution and self-purification processes along a river stream, or when providing a discrete model for the traffic flow in a motorway. More generally, the positivity assumption is a natural one when describing distributed processes whose variables represent quantities that are intrinsically nonnegative, like pressures, concentrations, population levels, etc. In this paper we address the positive local reachability property for 2D positive systems with scalar inputs. To this end, we assume a combinatorial point of view. 2D influence graphs (namely direct graphs which exhibit two types of arcs and two types of input flows [3, 4]) are the appropriate tools for formalizing and solving the problem. The results presented here are preliminary and the general solution of the problem seems nontrivial. 2D positive systems considered in this paper are described by the following state-updating equation [1]: x(h+1, k+1) = A1 x(h, k+1)+A2 x(h+1, k)+B1 u(h, k+1)+B2 u(h+1, k), (1) where the local states x(·, ·) and the scalar input u(·, ·) take nonnegative values, A1 and A2 are nonnegative n × n matrices, B1 and B2 are nonnegative n-dimensional column vectors, and the initial conditions are assigned by L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 297-304, 2003. Springer-Verlag Berlin Heidelberg 2003
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specifying the nonnegative values of the state vectors on the separation set C0 := {(h, k) : h, k ∈ Z, h + k = 0}, namely by assigning all local states of the initial global state X0 := {x(h, k) : (h, k) ∈ C0 }. Hurwitz products of two n × n matrices A1 and A2 are inductively defined as A1 i
j
i
0
A1 A1 i
A2 = 0,
when either i or j is negative,
A2 = Ai1 , for i ≥ 0, A1 0 j A2 = A1 (A1 i−1 j A2 ) + A2 (A1 i
A2 = Aj2 , for j ≥ 0, j−1 A2 ), for i, j > 0.
j
A 2D influence graph D(2) is a sextuple (s, V, A1 , A2 , B1 , B2 ), where s is the source, V = {v1 , v2 , . . . , vn } is the set of vertices, A1 and A2 are subsets of V ×V whose elements are called A1 -arcs and A2 -arcs, respectively, meanwhile B1 and B2 are subsets of s × V whose elements are called B1 -arcs and B2 -arcs, respectively. To every 2D positive system (1), of size n, with scalar inputs we associate a 2D influence graph D(2) (A1 , A2 , B1 , B2 ) of source s, with n vertices, v1 , v2 , . . . , vn . There is an A1 -arc (an A2 -arc) from vj to vi iff the (i, j)th entry of A1 (of A2 ) is nonzero. There is a B1 -arc (a B2 -arc) from s to vi iff the ith entry of B1 (of B2 ) is nonzero. A path p in D(2) (A1 , A2 , B1 , B2 ) is a sequence of adjacent arcs and, in particular, an s-path is a path which originates from the source s. A path (in particular, an s-path) p is specified by assigning its vertices and the type of arcs they are connected by. If we denote by |p|1 the number of A1 -arcs and B1 -arcs, and by |p|2 the number of A2 -arcs and B2 -arcs occurring in p, then [|p|1 |p|2 ] is the composition of p and |p| = |p|1 + |p|2 its length. A path whose extreme vertices coincide is a cycle. In particular, if each vertex appears exactly once as the first vertex of an arc, the cycle is a circuit. A 2D influence graph is strongly connected is for any two vertices vi and vj there is a path (of arbitrary composition) connecting vi to vj . D(2) (A1 , A2 , B1 , B2 ) is strongly connected iff A1 + A2 is an irreducible matrix. Two matrices M and N , of the same size, are said to have the same nonzero pattern if mij 6= 0 implies nij 6= 0 and viceversa. A vector v is said to be an ith monomial vector if it can be expressed as αi ei , where ei denotes the ith canonical vector and αi is some positive real coefficient. A monomial matrix is a nonsingular (square) matrix whose columns are monomial vectors.
2 Reachability and positive reachability definitions Definition 1. A 2D state-space model (1) is positively locally reachable n [1] if, upon assuming X0 = 0, for every x∗ ∈ R+ there exists (h, k) ∈ Z × Z with h+k > 0 and a nonnegative input sequence u(·, ·) such that x(h, k) = x∗ . When so, we will say that x∗ is reached in h + k steps. As for standard (i.e., not necessarily positive) 2D systems, positive local reachability analysis can be reduced to the analysis of the reachability matrix in k steps [1]
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Rk = [ B1 B2 A1 B1 A1 B2 + A2 B1 A2 B2 A21 B1 . . . Ak−1 B2 ] 2 ¤ £ i j−1 i−1 j A2 )B1 + (A1 A2 )B2 i,j≥0, 0
vm 1 @ I @ ? vm 2
sm
- vm 4
vm 7 6
@ @
@ - vm 3
? vm 5
- vm 6
Fig. 2.1 D(2) (A1 , A2 , B1 , B2 ) corresponding to Example 1 In this case, the reachability index proves to be 13 while the system dimension is n = 7. The above structure can be generalized. If the 2D influence graph of a 2D positive system consists of two loops, including n1 vertices and n1 + 1 vertices, respectively, connected by two arcs of type 2, while all the remaining are of type 2, just like in Fig. 2.1, IR turns out to be of the same order as n1 ·(n1 +1), namely of the same order as n2 /4, since n = n1 +(n1 +1). Example 1 has proved that for a locally reachable 2D positive system the reachability index may reach the value n2 /4. It seems reasonable to conjecture that n2 /4 represents an upper bound for the reachability index of every
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2D positive system, however, up to now a formal proof of this result is not available. A necessary condition for positive reachability is the following one. Proposition 1. If the positive system (1) is positively locally reachable then the matrix [ A1 A2 B1 B2 ] includes an n × n monomial submatrix. Proof. If the system is locally reachable, there exist n pairs (hi , ki ) ∈ Z+ × ki A2 )B1 + (A1 hi ki −1 A2 )B2 is an ith monomial vector. If hi + ki = 1, the ith monomial vector is a column of B1 or of B2 , otherwise, if hi + ki > 1, it is a column of A1 or of A2 (possibly both).
Z+ , i = 1, 2, . . . , n, such that (A1 hi −1
As for 1D positive systems, local reachability property admits an interesting and useful characterization in terms of the 2D influence graph associated with the system. Indeed, saying that (A1 hi −1 ki A2 )B1 + (A1 hi ki −1 A2 )B2 is an ith monomial vector just means that every s-path p of composition [|p|1 |p|2 ] = [hi ki ] necessarily reaches the vertex vi alone. If so, we will say that the vertex vi is deterministically reached by all s-paths of composition [hi ki ]. As a consequence, the 2D system (1) is positively locally reachable iff for every i ∈ {1, 2, . . . , n} the vertex vi is deterministically reached by all s-paths of a given composition [hi ki ]. Moreover, IR coincides with max min {hi + ki : all s-paths of composition [hi ki ] deterministically reach vi }. i
hi ,ki
In the sequel, we will confine our attention to 2D positive systems (1) having one of the two input-to-state matrices which is zero, and assume w.l.o.g. B2 = 0 and, consequently, denote B1 as B, for the sake of simplicity. These systems are described by the following equation: x(h + 1, k + 1) = A1 x(h, k + 1) + A2 x(h + 1, k) + Bu(h, k + 1), n×n
where A1 , A2 are in R+
(2)
n
and B is in R+ .
3 2D influence graphs devoid of cycles In this section we consider 2D positive systems (2) whose 2D influence graph is devoid of cycles. This amounts to saying that the system (1) is finite memory or, equivalently [2], by the positivity assumption, that A1 + A2 is nilpotent. Proposition 2. Given a 2D positive system (1), its 2D influence graph D(2) (A1 , A2 , B1 , B2 ) is devoid of cycles iff the system is finite memory. Proof. Observe, first, that since the source exhibits no incoming arcs, D(2) (A1 , A2 , B1 , B2 ) is devoid of cycles iff D(2) (A1 , A2 , 0, 0) is. On the other hand, if γ is a cycle in D(2) (A1 , A2 , 0, 0) and the vertex vi belongs to γ, then [(A1 +
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A2 )m·|γ| ]ii > 0 for every positive integer m. So, if (1) is finite memory, namely A1 + A2 is nilpotent, then (A1 + A2 )k = 0, ∀ k ≥ n. Therefore, no cycle γ can exist in D(2) (A1 , A2 , 0, 0). Conversely, if there is a cycle γ in D(2) (A1 , A2 , 0, 0) then condition (A1 + A2 )k = 0 for every k ≥ n cannot be satisfied. Proposition 3. If a 2D positive system (2), with 2D influence graph D(2) (A1 , A2 , B, 0) devoid of cycles, is positively locally reachable then i) B is a canonical vector, and
o n Pk ii) the reachability index IR satisfies min k ∈ N : i=1 i ≥ n ≤ IR ≤ n. Proof. i) Since A1 + A2 is (positive and) nilpotent, it entails no loss of generality [2] assuming that A1 + A2 (and hence A1 and A2 , separately) is in upper triangular form with zero diagonal. So, if A1 and A2 have the structure 0 + + .. . + 0
and the system is positively locally reachable, then, by Proposition 1, in [ A1 A2 B 0 ] there must appear also the nth canonical vector en . This necessarily implies B = en . ii) Since (A1 +A2 )n = 0, all Hurwitz products A1 i j A2 are zero whenever + i+j ≥ n. So, Xn+1 = Xn+ , and, in general, Xk+ = Xn+ , ∀ k ≥ n. If B = en , it is easily seen that after one step the only outgoing arc from the source reaches vertex vn . On the other hand, due to the fact that only two types of arcs are available, paths of length 2 with a common initial arc (from the source to vertex vn ) and distinct compositions may reach deterministically at most two vertices. Again, paths of length 3 with a common initial arc and distinct compositions may deterministically reach at most three vertices, and so on. This means that the minimum number of steps required to deterministically reach each vertex is the smallest positive integer k such that 1+2+. . .+k ≥ n.
4 2D influence graphs consisting of either one or two disjoint circuits In this section we consider, first, systems (2) with 2D influence graphs consisting of a single circuit, by this meaning that all vertices v1 , v2 , . . . , vn belong to a circuit (and each pair of adjacent vertices is connected by one single arc). This assumption amounts to saying that A1 + A2 is a permutation matrix, while A1 ∗ A2 = 0, where ∗ denotes the Hadamard product. So, by resorting to a suitable permutation of the state components we can always obtain
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0
0 A1 + A2 = +
+ 0
0
0 + .. .
.. ..
0 0 . .
+ 0
,
(3)
where + represents a strictly positive entry and each nonzero entry + appears only in one of the two matrices A1 and A2 . Notice that vertex vi+1 accesses vertex vi , for i = 1, 2, . . . , n − 1, while vertex v1 accesses vn . Differently from the 1D case, the positive local reachability of such a system (A1 , A2 , B, 0) does not require B to be a monomial vector. When D(2) (A1 , A2 , B, 0) consists of a single circuit, every monomial vector B makes (A1 , A2 , B, 0) positively locally reachable with reachability index IR = n. When B exhibits k nonzero entries and the system is locally reachable, the reachability index may take quite smaller values. Proposition 4. Consider a 2D positive system (2) such that D(2) (A1 , A2 , B, 0) consists of a single circuit and assume w.l.o.g. that A1 + A2 is expressed as in (3) with A1 ∗ A2 = 0. If the system is positively locally reachable and B has k > 1 nonzero entries, say 1 ≤ i1 < i2 < . . . < ik ≤ n, then IR ≥ max{i2 − i1 , i3 − i2 , . . . , ik − ik−1 , n − ik + i1 } + 1. Proof. Suppose, for the sake of simplicity, that max{i2 − i1 , i3 − i2 , . . . , ik − ik−1 , n − ik + i1 } = i2 − i1 . By the ordering assumptions introduced on the system vertices and on the labels i1 , i2 , . . . , ik , it is clear that the minimum h1 + k1 such that all s-paths of composition [h1 k1 ] deterministically reach vi1 (keeping in mind that at the first step we get B and hence not a monomial vector) coincides with the length of the s-path that, starting from the source, reaches vertex vi2 at the first step and later enters vertex vi1 without passing through the other vertices vi` for ` 6= 1, 2. Such an s-path has length i2 − i1 + 1. Condition IR = maxi minhi ,ki {hi + ki : all s-paths of composition [hi ki ] deterministically reach vi ,} ≥ i2 − i1 + 1 completes the proof. In particular, when k = 2 the minimum value of max{i2 −i1 , i3 −i2 , . . . , ik − ik−1 , n − ik + i1 } = max{(i2 − i1 ), n − (i2 − i1 )} is just n/2 and therefore the minimum value of the reachability index is n2 + 1. For a 2D influence graph consisting of two disjoint circuits we have the following result. Proposition 5. Let (A1 , A2 , B, 0) be a 2D positive system such that D(2) (A1 , A2 , 0, 0) consists of two disjoint circuits γ and γ 0 of length n and n0 , respectively. If B has only two nonzero entries, one for each cycle, then IR ≤ l.c.m{n, n0 } + max{n, n0 }.
On the Positive Reachability of 2D Positive Systems
303
Proof. Assume that the vertices in γ are (ordinately) v1 , v2 , . . . , vn , the vertices in γ 0 are (ordinately) v10 , v20 , . . . , vn0 0 , and that the two nonzero entries in B correspond to the vertices v1 and v10 , as depicted in Figure 4.2. vm 2
vm 1
sm
0 - vm 1
vn0m 0
6 ? vm 3
... - vnm
6 .. . ?
0 vm 2
0 - vm 3
Fig. 4.2 D(2) (A1 , A2 , B, 0) in Proposition 5 Due to the previous assumptions, any vertex vj ∈ γ (vj0 ∈ γ 0 ) is periodically visited after j, j + n, j + 2n, . . . steps (j, j + n0 , j + 2n0 , . . . steps, respectively). Moreover, for every k ∈ N there exist exactly two s-paths of length k in D(2) (A1 , A2 , B, 0), and they reach vertices vk mod n in γ and vk0 mod n0 in γ 0 , respectively. Such vertices are reached deterministically iff the two s-paths have distinct compositions. Set N := l.c.m.{n, n0 } and suppose that none of the paths of length j, j + n, . . . , j + N deterministically reaches vj . Since after j + N steps we reach, at the same time and with the same composition, vj and vj0 just like after j steps, the subsequent evolution will periodically repeat the same nonzero pattern, thus preventing the possibility of deterministically reaching vj . As this reasoning applies to all vertices of γ and γ 0 (in particular to vn and vn0 0 ), the given bound immediately follows.
5 Strongly connected 2D influence graphs including only two circuits In this section we aim at addressing 2D positive systems with a strongly connected 2D influence graph that includes only two circuits, γ1 and γ2 . Even though these assumptions are undoubtedly restrictive, an extension to the general case of 2D positive systems with a strongly connected 2D influence graph seems reasonable. On the other hand, when D(2) (A1 , A2 , B, 0) is not strongly connected, which amounts to saying that the matrix A1 +A2 is not irreducible, possible bounds on the reachability index of the system (A1 , A2 , B, 0), based on the reachability indices of the irreducible subsystems of D(2) (A1 , A2 , B, 0), can be obtained. We first derive a lemma, whose proof is omitted for the sake of brevity. Lemma 1. If D(2) (A1 , A2 , B1 , B2 ) is a strongly connected 2D influence graph with n vertices and that it includes only two circuits, say γ1 and γ2 , then
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i) every vertex belongs either to γ1 or to γ2 ; ii) there exists at least one vertex which belongs both to γ1 and to γ2 ; iii) each path p, with |p| ≥ |γ1 |, includes at least one vertex v2 ∈ γ2 and, conversely, each path p, with |p| ≥ |γ2 |, includes at least one vertex v1 ∈ γ1 ; iv) N := |γ1 | + |γ2 | ≥ n + 1. Proposition 6. Let (A1 , A2 , B, 0) be a 2D positive system such that D(2) (A1 , A2 , B, 0) is strongly connected and includes only two circuits γ1 and γ2 . If (A1 , A2 , B, 0) is positively locally reachable, then IR ≤ N := |γ1 | + |γ2 |. Proof. Suppose, by contradiction, that there exists some vertex r which is deterministically reached by all s-paths of composition [h+1 k], with h+1+k ≥ N + 1, and cannot be reached deterministically in a smaller number of steps. This amounts to saying that (A1 h k A2 )B is an r-monomial vector and for every i, j, with i + j < h + k, (A1 i j A2 )B is not an r-monomial vector. All paths of composition [h k] from the vertices corresponding to the nonzero entries of B to the vertex r have length greater than or equal to N ≥ n+1. This implies that each of these paths contains at least one circuit. We can assume w.l.o.g. that there exists one such path p the vertex r of which includes the circuit γ1 . It is easily seen that there exists a path p0 , from at least one vertex corresponding to the nonzero entries of B to the vertex r of composition [h − α1 (γ1 ) k − α2 (γ1 )]. Since (A1 h−α1 (γ1 ) k−α2 (γ1 ) A2 )B is not an r-monomial vector, it means that there exists also a path q 0 , of composition [h − α1 (γ1 ) k − α2 (γ1 )], from at least one vertex corresponding to the nonzero entries of B to some other vertex s. Since |q 0 | = (h+k)−|γ1 | ≥ N −|γ1 | = |γ2 |, this implies that at least one vertex of q 0 belongs to γ1 . But then, by suitably adding a circuit γ1 , we can obtain from q 0 a new path q, from some vertex corresponding to the nonzero entries of B to the vertex s, of composition [h k]. This implies that in (A1 h k A2 )B both the rth and the sth entries are nonzero, thus contradicting the original assumption. Therefore h + k < N .
References 1. Fornasini E., Marchesini G. : Doubly indexed dynamical systems. Math. Sys. Theory, 12, 1978, pp. 59-72. 2. Fornasini E., Valcher M.E. : On the spectral and combinatorial structure of 2D positive systems. Lin. Alg. & Appl., 245, 1996, pp. 223–258. 3. Fornasini E., Valcher M.E. : Directed graphs, 2D state models and characteristic polynomial of irreducible matrix pairs. Lin. Alg. & Appl., 263, 1997, pp.275–310. 4. Fornasini E., Valcher M.E. : Primitivity of positive matrix pairs: algebraic characterization, graph-theoretic description, 2D systems interpretation. SIAM J. Matrix Analysis & Appl., 19, no.1, 1998, pp.71–88. 5. Kaczorek T. : Reachability and controllability of 2D positive linear systems with state feedback. Control and Cybernetics, 29, no.1, 2000, pp. 141–151.
On Nonnegative Realizations
?
Karl-Heinz F¨orster1 and B´ela Nagy2 1 2
Department of Mathematics, Technical University Berlin, Sekr. MA 6-4, Strasse des 17. Juni 136, D-10623 Berlin, Germany, [email protected] Department of Analysis, Institute of Mathematics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary, [email protected]
Abstract. Problems of nonnegative finite dimensional realizations of a nonnegative impulse response (or, equivalently, Markov sequence) or of a given rational transfer function are studied. We first apply the nonnegative factorization and nonnegative rank approach introduced by Cohen and Rothblum [4], and employed to nonnegative realizations by [6]. Theorem 3 proves that the finiteness of the so called c + Rank of a matrix formed from the Markov coefficients is a necessary and sufficient condition for the nonnegative realizability of the nonnegative Markov sequence (when the coefficients are c × b nonnegative matrices). We show that if a matrix-valued transfer function has only nonnegative and simple poles with nonnegative coefficient matrices (residues), then (nonnegative realizations exist, and) the order p of any nonnegative-minimal (M P ) realization lies between the sum of the ranks and the sum of the nonnegative ranks of the residues. We present an example showing that both inequalities simultaneously may be strict.
1 Introduction The problems of nonnegative finite dimensional realizations of an (entrywise) nonnegative impulse response (≡ Markov sequence) or of a given rational transfer function have been very interesting, useful and highly nontrivial (see, e.g., the paper [1] by B.D.O. Anderson). The first results concerning the first problem are contained in the thesis [6] and the paper [7] by van den Hof. The present paper uses the nonnegative factorization and nonnegative rank approach introduced by Cohen and Rothblum [4], and developed further and employed to nonnegative realizations by [6] and [7]. Our first main result (Theorem 3) proves that the finiteness of the so called c + Rank of a matrix formed (in the natural way) from the Markov coefficients is a necessary and sufficient condition for the nonnegative realizability of the nonnegative Markov sequence (in the matrix case, when the coefficients are c × b entrywise ?
The work of the second author was supported by the Hungarian National Scientific Grant OTKA No. T-030042.
L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 305-312, 2003. Springer-Verlag Berlin Heidelberg 2003
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nonnegative matrices). The reader may compare this characterization with pertaining results by [6], Th. 5.4.10, p. 117 and by Benvenuti and Farina in [2], Theorem 1. It is well-known that the minimal dimension of nonnegative (equivalently, we sometimes say positive) realizations of a Markov sequence or of a transfer function (assuming their existence) may be strictly larger than the McMillan degree (i.e. the minimal dimension of any realization). Any realization of the former class will be called a PM realization, whereas a minimal (or canonical) realization, if it happens to be positive, can be called an M ∩ P realization. In the end of the paper we show that if a matrix-valued transfer function has only nonnegative and simple poles with nonnegative coefficient matrices (residues), then nonnegative realizations exist, and the order p of any PM realization lies between the sum of the ranks and the sum of the nonnegative ranks (in the sense of [4]) of the residues. An example, showing that both inequalities simultaneously may be strict, gives also some ideas how the nonnegative ranks can be calculated: admittedly, a nontrivial task in many cases.
2 Nonnegative rank and factorization The nonnegative (row and column) rank of a finite nonnegative matrix A (over an ordered field G) was defined by Cohen and Rothblum in [4], and they obtained the basic results concerning these concepts. We shall need variants of these results for possibly infinite matrices, so we shall reproduce them indicating a proof only when it seems to be essentially different from its counterpart in [4]. We emphasize that the novelty of Theorems 1 and 2 lies in the possibility of the matrices to be infinite, which will play an important role in the new characterization result (Theorem 3). For the applications the cases G := R (the field of reals) or G := Q (the field of rationals) are very important, so we shall consider only these possibilities, and G+ will denote the positive part of (one of) these fields. . Define the nonnegative Let N∗ := N ∪ ∞, m, n ∈ N∗ , and let A ∈ Gm×n + column rank of A, denoted by colrank+ (A), as the smallest nonnegative integer q for which there are nonnegative column vectors v 1 , . . . , v q ∈ Gm such that each column of A is a linear combination with nonnegative coefficients of v 1 , . . . , v q . (Note that, by the usual convention, the empty sum or product is [the] zero [matrix].) Any such system {v 1 , . . . , v q } will be called a nonnegative basis for the column space of A. It is clear that colrank+ (A) ≤ n, the cardinal of the columns of A. For any infinite column vector x = (x1 , x2 , . . .)T ∈ G∞ define the shift operator s : G∞ → G∞ by sx = s(x1 , x2 , . . .)T := (x2 , x3 , . . .)T , where T denotes transpose. Also let (for m := ∞ ) sA := (sA(1) , sA(2) , . . . , sA(n) ) ∈ G∞×n ,
On Nonnegative Realizations
307
with the clearly needed modification if n = ∞. (Here A(k) denotes the kth column vector of A.) Let c be a nonnegative integer, and let sc denote (as usual) the cth power of the operator s. Let k be a positive integer. We shall say that the matrix V ∈ G∞×k + is c-shift-nonnegative, if there is a nonnegative matrix Q ∈ Gk×k satisfying sc V = V Q. The c-shift nonnegative column rank of a row-infinite nonnegative matrix A, denoted by c + Rank(A), is defined as the smallest nonnegative integer p for which there are nonnegative matrices V ∈ G∞×p , U ∈ Gp×n such that V is c-shift nonnegative, and + + A = V U . Note that it can happen that there is no such integer p: in that case we set c + Rank(A) := ∞. Clearly, if it is finite, then c + Rank(A) = c − pos − rank(A), i.e. coincides with the c-positive rank of the matrix A as defined in [6], p.115. The following basic theorem can be proved exactly as Lemma 2.1 and Corollary 2.2 for finite matrices in [4]. Theorem 1. Let A be a nonnegative matrix in Gm×n , and let q be a nonnegative integer. The following are equivalent: (1) colrank+ (A) = q, (2) q is the smallest nonnegative integer for which there exist nonnegative matrices V ∈ Gm×q and U ∈ Gq×n satisfying A = V U , (3) q is the smallest nonnegative integer for which there exist nonnegative column vectors v 1 , . . .P , v q in Gm and nonnegative column vectors u1 , . . . , uq in q n G such that A = j=1 v j (uj )T . The next result is the c-shift-nonnegative version of the above. Theorem 2. Let A, q, c be as above. The following are equivalent: (1) there exists a nonnegative basis {v 1 , . . . , v q } for the column space of A such that each sc v j (j = 1, . . . , q) is a nonnegative linear combination of the vectors of the above basis, (2) c + Rank(A) = q, (3) q is the smallest nonnegative integer for which there exist nonnegative column vectors v 1 , . . . P , v q ∈ Gm and nonnegative column vectors u1 , . . . , uq ∈ q n G such that A = j=1 v j (uj )T , and the matrix V := ( v 1 . . . v q ) is c-shift-nonnegative. The proof of Theorem 2 is a natural completion of the proof of Theorem 1. We note only that each vector sc v j in (1) is a nonnegative linear combination of the vectors of the basis if and only if the matrix V in(3) is c-shift-nonnegative.
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Now we are in a position to state and prove the following basic result on nonnegative (rational) realizability of (entrywise nonnegative) Markov sequences of matrices (i.e. impulse responses in the MIMO case). Note that we shall compare it below to a related result of [6]. Theorem 3. Let b,c denote positive integers, and for every nonnegative k let Mk ∈ Gc×b + . Let M0 M1 M := M2 .. . denote the row-infinite block matrix formed of the Markov sequence. The Markov sequence {Mk } has a nonnegative realization if and only if r := c + Rank(M ) < ∞,
(1)
and this number r is the order (=dimension of the state space) of any nonnegative-minimal (i.e. PM) realization of the Markov sequence. Proof. (Sufficiency.) Assume that the Markov sequence M has a nonnegative realization (C, A, B), and the dimension of the space of A is d. Then clearly C CA M = CA2 B, .. . and the c-shift of the left-hand side factor equals C CA CA2 A. .. . Hence c + Rank(M ) is not greater than the dimension d of the space of A, cf. [6], 5.4.9. (Necessity.) In the converse direction let us assume that for the rowinfinite block matrix M of the Markov sequence (1) holds. We shall show that there is a nonnegative realization, and the number r equals the order of any nonnegative-minimal realization of the given Markov sequence. By (1), there are nonnegative matrices P, A, B such that P ∈ G∞×r , M = P B,
A ∈ Gr×r ,
B ∈ Gr×b ,
sc P = P A.
On Nonnegative Realizations
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Define the nonnegative matrix C ∈ Gc×r to be the uppermost block matrix of P . Then µ ¶ C P = , sc P therefore
M0 µ ¶ µ ¶ C CB M1 = M = P B = B= . PA P AB .. .
Hence M0 = CB. We proceed by induction. Assume now that Mj = CAj B for j = 0, 1, . . . , k. Then sc (P B) = (sc P )B = P AB implies that M k+1
Mk+2 c(k+1) M = sc(k+1) P B = Mk+3 = s .. .
= sck P AB = · · · = sc P Ak B = P Ak+1 B =
µ
C PA
¶
Ak+1 B =
µ
CAk+1 B P Ak+2 B
¶ .
By induction, for every nonnegative integer j we obtain Mj = CAj B. Moreover, the dimension d of the space of A is equal to r = c + Rank(M ). Finally, if there existed a nonnegative realization (C0 , A0 , B0 ) of the given Markov sequence such that the dimension of the space of A0 was d0 < r, then the sufficiency part of this proof (or [6], 5.4.9) would yield that c+Rank(M ) ≤ d0 < r, a contradiction. Hence the realization (C, A, B) is a PM realization, and the proof is complete. Corollary 1. Under the conditions of Theorem 3 for every positive integer q consider the row-infinite block matrix H(q) := ( M Then
sc M
s2c M
···
c + Rank[H(q)] = r
. s(q−1)c M ) ∈ G∞×qb + (q = 1, 2, . . .).
Proof. Assume that H(q +1) can be factorized as V U with nonnegative factor matrices such that for some nonnegative square matrix Q the equality sc V = V Q holds. U can be partitioned as (Uq U1 ), where Uq has as many columns as H(q), and U1 has b columns. Therefore H(q) = V Uq . Hence c + Rank[H(q)] ≤ c + Rank[H(q + 1)]
(q = 1, 2, . . .).
On the other hand, the existence of a nonnegative-minimal realization of order d implies similarly as the sufficiency part of the proof of Theorem 3 (cf. also [6], 5.4.9) that for every q = 1, 2, . . . we have c+Rank[H(q)] ≤ d. By Theorem 3, for q = 1 we have here equality. By the preceding paragraph, we obtain then equality for every q as stated.
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In what follows we shall work with finite nonnegative matrices, for which nonnegative column and row ranks are defined and shown to be equal to their joint value rank+ (A) = colrank+ (A) (see [4]). We shall need the following Lemma 1. If A is a (finite) nonnegative idempotent matrix, then its nonnegative rank is the same as its usual rank: rank+ (A) = rank(A). The proof makes use of [3], Theorem (3.1), p.65, but for reasons of space we must omit it. Theorem 4. Assume that all the poles zk (k = 1, . . . , n) of the transfer function W are simple and nonnegative, and the corresponding spectral projections (residues) Wk are also all nonnegative, i.e. W (z) =
n X Wk z − zk
(Wk ≥ 0, zk ≥ 0).
k=1
Then the transfer function has a nonnegative realization, and the order p of any PM realization satisfies n X k=1
rank(Wk ) ≤ p ≤
n X
rank+ (Wk ),
k=1
where rank and rank+ denote the rank and the nonnegative rank of a matrix, respectively. Further, there are PM realizations for which the inequalities on both sides are simultaneously strict. Proof. Gilbert [5], Theorem 7 proved that if W has the above form (even without zk and Wk being nonnegative), the left-hand side inequality holds for the order p of any realization of W , and equality stands there for any minimal realization. Under the assumptions of our theorem let the matrices Wk have the sizes r × s, and let r+ (k) := rank+ (Wk ). Then there are nonnegative matrices Ck and Bk of sizes r × r+ (k) and r+ (k) × s, respectively, such that Wk = Ck Bk (k = 1, . . . n). Define the block matrices B1 B2 C := ( C1 C2 . . . Cn ) , A := z1 I1 ⊕z2 I2 ⊕. . .⊕zn In , B := ... , Bn
where Ik denotes the unit (identity) matrix of dimension r+ (k). It is straightforward to see that (C, A, B) is a nonnegative realization of W , clearly of order Pn r (k). This proves the right-hand side inequality. In order to see that + k=1 there is a PM realization with the stated strict inequalities (there are then, trivially, many), consider the following
On Nonnegative Realizations
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Example. Let 1 Q1 := 2
µ
1 1 1 1
¶ ,
1 Q2 := I − Q1 = 2
P1 := Q1 ⊕ Q1 ⊕ Q1 ⊕ 1,
µ
1 −1
−1 1
¶ ,
P2 := I − P1 = Q2 ⊕ Q2 ⊕ Q2 ⊕ 0.
Here the the matrices I are identity matrices of the appropriate orders, and the numbers 1 and 0 in the direct sums denote the corresponding 1 × 1 matrices. Then the matrices Q1 , P1 ≥ 0 and Q2 , P2 are projections, and the 7 × 7 matrix A := P1 is an entrywise nonnegative symmetric matrix with eigenvalues (which are simple poles of the resolvent) 1 and 0. Note that the spectral projection of A belonging to 1 is nonnegative, whereas that to 0 is not. The resolvent of A is P1 P2 + . (z − A)−1 = z−1 z Consider now the matrices 2 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 0 2 1 0 0 1 1 0 1 C := B := 0 0 0 0 , , 0 0 1 0 0 0 0 0 0 2 2 0 0 0 0 1 0 0 0 0 0 0 2 0 2 0 and the transfer function defined by W (z) := C(z − A)−1 B. Then the coefficients in the partial fraction decomposition of W are W (z) =
W2 W1 + , z−1 z
Wj = CPj B (j = 1, 2).
Thus we obtain
1 3 W1 = 0 0
1 2 1 0
0 3 0 1
0 2 , 1 1
1 1 W2 = 0 0
1 0 1 0
0 1 0 1
0 0 . 1 1
It is clear that rank(W1 ) = rank(W2 ) = 3, and the argument in [4], pp.152153 shows that rank+ (W2 ) = 4. By [5], Theorem 7, the order of any minimal realization (the McMillan degree) is rank(W1 ) + rank(W2 ) = 6. It can be shown that rank+ (W1 ) + rank+ (W2 ) = 8. Then the order q = 7 of the nonnegative realization (C, A, B) satisfies rank(W1 ) + rank(W2 ) < q < rank+ (W1 ) + rank+ (W2 ).
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Finally, we show that the realization (C, A, B) is a PM realization of the transfer function W . The order of any minimal realization (C0 , A0 , B0 ) is, as noted already, rank(W1 )+rank(W2 ) = 6. The order of (C, A, B) is 7. If it were not a PM realization, then we should have a PM realization (C+ , A+ , B+ ) of order 6. The well-known connection between the realizing matrices and the Markov coefficients would then yield C+ A+ B+ = C0 A0 B0 = CAB = CP1 B = W1 , C+ B+ = C0 B0 = CB = C(P1 + P2 )B = W1 + W2 for any minimal realization (C0 , A0 , B0 ). Consider now (and denote by the same triple (C0 , A0 , B0 )) the special minimal realization constructed in the proof of Gilbert [5], Theorem 7. Then, in particular, A0 is the direct sum of the 3×3 identity matrix I3 and of the 3×3 zero matrix 03 , hence is an idempotent matrix of rank 3. The minimal (and also PM) realization (C+ , A+ , B+ ) is known to be system similar to (C0 , A0 , B0 ). Hence there is an invertible 6×6 matrix H such that, in particular, A+ = HA0 H −1 . Therefore A+ is also an idempotent matrix of rank 3, and also nonnegative. By the Lemma, we have rank+ (A+ ) = rank(A+ ) = 3. Applying [4], Lemma 2.6, then rank+ (W1 ) = rank+ (C+ A+ B+ ) ≤ rank+ (A+ ) = 3, a contradiction. Hence we see that the order of any PM realization of W must be ≥ 7. Therefore (C, A, B) is a PM realization with order p = 7. We have seen that it satisfies the inequalities in the Theorem in the form 6 < p < 8. The example and the proof are complete.
References 1. B.D.O. Anderson, New developments in the theory of positive systems, in: Systems and Control in the Twenty-First Century, C. Byrnes, B. Datta, M. Gilliam, C. Martin eds., Birkh¨ auser, Boston, 1997. 2. L. Benvenuti and L. Farina, A note on minimality of positive realizations, IEEE Trans. Circuits and Systems, I: Fundamental Theory, 45/6 (1998), 676-677. 3. A. Berman and R.J. Plemmons, Nonnegative matrices in the mathematical sciences, Academic Press, New York, 1979. 4. J.E. Cohen and U.G. Rothblum, Nonnegative ranks, decompositions, and factorizations of nonnegative matrices, Linear Algebra Appl., 190 (1993), 149-168. 5. E.G.Gilbert, Controllability and observability in multivariable control systems, SIAM J. Control 1 (1963), 128-151. 6. J. van den Hof, System theory and system identification of compartmental systems, Thesis, Groningen, 1996. 7. J. van den Hof, Realization of positive linear systems, Linear Algebra Appl., 256 (1997), 287-308.
Estimation and Strong Approximation of Hidden Markov Models L´ aszl´ o Gerencs´er and G´abor Moln´ar-S´ aska MTA SZTAKI, Computer and Automation Institute Hungarian Academy of Sciences 13-17 Kende u., Budapest 1111, Hungary, {gerencser,molnarsg}@sztaki.hu Abstract. We give novel conditions for the existence of the limit of the normalized log-likelihood function for a finite-state continuous read-out Hidden Markov Model. Our results complements those of [8]. The result is then applied to derive a strong approximation result for the parameter estimates of the Hidden Markov Model.
1 Introduction Hidden Markov Models have become a basic tool for modeling stochastic systems with a wide range of applicability. For a general introduction see [11]. The estimation of the dynamics of a Hidden Markov Model is a basic problem in applications. A key element in the statistical analysis of HMM-s is a strong law of large numbers for the log-likelihood function, see [8], [7], [9], [3]. An alternative tool that has been widely used in linear system identification is theory of L-mixing processes. The relevance of this theory is established in [6] using a random-transformation representation for Markov-processes. The advantage of this approach is that, under suitable conditions a more precise characterization of the estimation error-process can be obtained. The purpose of this paper is to give conditions which can be used in practice, and compare our conditions with that of [8].
2 Hidden Markov Models We consider Hidden Markov Models with a finite state space X and a possibly continuous observation or read-out space Y. The latter is assumed to be a measurable set of Rd as in [8] or [9]. The pair (Xn , Yn ) is a Hidden Markov process if (Xn ) is a homogenous Markov chain, with state space X and the observations Yn are conditionally independent given (Xn ), i.e. for any integer n ≥ 0, and for any i0 , . . . in ∈ X
L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 313-320, 2003. Springer-Verlag Berlin Heidelberg 2003
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L´ aszl´ o Gerencs´er and G´ abor Moln´ ar-S´ aska
P (Yn ∈ dyn , . . . , Y0 ∈ dy0 |Xn = in , . . . , X0 = i0 ) =
n Y
P (Yk ∈ dyk |Xk = ik ),
k=0
and the conditional distributions, considered as functions of y and x, are independent of k. Let Q∗ be the transition matrix of the unobserved Markov process (Xn ), i.e. Q∗ij = P (Xn+1 = j|Xn = i), and let the read-outs have a density with respect to a fixed nonnegative, σ-finite measure λ on Rd P (Yn ∈ dy|Xn = i) = b∗i (y)λ(dy), and denote
B ∗ (y) = diag(b∗i (y)).
Let the predictive filter be defined by p∗j n+1 = P (Xn+1 = j|Yn , . . . Y0 ). ∗N T Writing p∗n+1 = (p∗1 n+1 , . . . , pn+1 ) , the filter process satisfies the Baumequation p∗n+1 = π(Q∗T B ∗ (Yn )p∗n ), (1)
where π is the normalizing operator, to ensure that p∗n+1 is a probability P vector: for x ≥ 0, x 6= 0 set π(x)i = xi / j xj , see [1]. Here p∗j 0 = P (X0 = j). In practice, the transition probability matrix Q∗ and the initial probability distribution p∗0 of the unobserved Markov chain (Xn ) and the conditional densities b∗i (y) of the observation sequence (Yn ) are possibly unknown. For this reason we consider the Baum-equation with running parameters Q, bi (y) pn+1 = π(QT B(Yn )pn ),
(2)
with initial condition p0 = q, where Q is a stochastic matrix, pn is a probability vector on X , and B(y) = diag(bi (y)) is a collection of density functions. In the sequel we write (2) shortly as p∗n+1 = f (Yn , p∗n ).
(3)
Taking an arbitrary probability vector q as initial condition the solution of the Baum equation will be denoted by pn (q). Let the parameter of the model be denoted by θ∗ . Usually θ∗ is assumed to be composed of the elements of the unknown transition probability matrix and the parameters of the densities of the read-outs.
3 Exponential forgetting A key property of the Baum-equation is its exponential stability with respect to the initial condition. This has been established in [8], see Theorem 2.2. Recall that a stochastic matrix Q is primitive if there exists an r ≥ 1 such that Qr is positive.
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Theorem 1. If the stochastic matrices Q, Q∗ are primitives and the read-out densities are positive, i.e. bi (y), b∗i (y) > 0 then for any two initializations q, q 0 kpm (q) − pm (q 0 )kT V ≤ C(ω)(1 − δ)m kq − q 0 kT V almost surely under the true probability measure with 0 < δ < 1, where C(ω) is a finite random variable and k kT V denotes the total variation norm. Theorem 2. If the matrix Q is positive in Theorem 1, then with some constant C and for all ω kpm (q) − pm (q 0 )kT V ≤ C(1 − δ)m kq − q 0 kT V . In this case we say that the prediction filter generated by the Baum-equation is uniformly exponentially stable, i.e. any initial condition for the prediction filter is forgotten exponentially fast.
4 The log-likelihood function For the consistency of the maximum-likelihood estimation we need to show the existence of the limit log p(yn−1 , . . . y0 , θ)/n as n goes to infinity. We can write log p(yn−1 , . . . y0 , θ) as n−1 X
log p(yk |yk−1 , . . . y0 , θ) + log p(y0 , θ).
k=1
The k-th term is log
P i
bi (yk )P (i|yk−1 , . . . , y0 , θ) = log g(y, p) = log
X
P i
bi (yk )pik (θ) . Let
bi (y)pi .
(4)
i
Thus we get
n
1 1X g(yk , pk (θ)). log p(yn , . . . , y0 , θ) = n n
(5)
k=1
Definition 1. (see [10]) We say that the Markov process (Xn ) is geometrically ergodic if there exists a stationary distribution π and there exists an x-dependent constant Mx and an x-independent constant 0 < ρ < 1 for all initialization x ∈ X such that kP n (x, ·) − πkT V ≤ Mx ρn . If the transition probability matrix Q∗ is primitive then the process (Xn , Yn ) is geometrically ergodic (see [8], pp. 70) under the true probability measure, with a unique invariant probability distribution π on X × Y, and for any i ∈ X
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π i (dy) = ν i b∗i (y)λ(dy), where ν denotes the stationary distribution on X . In addition the geometrically ergodicity of (Xn , Yn , pn ) is proved (see Theorem 4 below). To describe the results of [6] we are going to introduce the notion of Doeblin-condition, see [2]: Definition 2. Given a Markov chain (Xn ) with state space X . We say that the Doeblin-condition is satisfied if there exists an integer m ≥ 1 such that P m (x, A) ≥ δν(A) is valid for all x ∈ X and A ⊂ B(X ), where B(X ) denotes the Borel measurable sets, with δ > 0 and some probability measure ν. Lemma 1. (see [6]) Assume that the Doeblin-condition holds for the Markov process (Xn ), and the mapping f defined in (3) is uniformly exponentially stable. Then the process (Xn , Yn , pn ) has a stationary distribution. Theorem 3. Consider a Hidden Markov Process (Xn , Yn ), where the state space X is finite and the observation space Y is continuous, a measurable subset of Rd . Let the running value of the transition probability matrix Q and the running value of the conditional read-out densities be positive, i.e. Q > 0 and bi (y) > 0, respectively. Assume that the process (Xn , Yn ) satisfies the Doeblin-condition. Let the initialization of the process (Xn , Yn ) be random, where the Radon-Nikodym derivate of the initialized distribution π0 w.r.t the stationary distribution π is bounded, i.e. dπ0 ≤ K. dπ Assume that for all i, j ∈ X Z | log bj (y)|q b∗i (y)λ(dy) < ∞.
(6)
(7)
Then the process g(Yn , pn ) is L-mixing. For the definition of L-mixing see the Appendix. The law of large numbers is valid for L-mixing processes, see [4], which implies the existence of the limit in (5). The corresponding result in [8] is the following, given as Theorem 3.5 of [8], which also implies the existence of the limit in (5). Theorem 4. Consider a Hidden Markov Process (Xn , Yn ), where the state space X is finite and the observation space Y is continuous, a measurable subset of Rd . Let the true and the running value of the conditional read-out densities be positive, i.e. bi (y), b∗i (y) > 0, respectively. Let the true and the
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running transition probability matrices Q∗ and Q be primitive, and for all i, j ∈ X Z max bj (y) j∈X b∗i (y)λ(dy) < ∞, (8) min bj (y) Rd j∈X
and
Z
| log bj (y)|b∗i (y)λ(dy) < ∞
(9)
Rd
be satisfied. Then the process g(Yn , pn ) is geometrically ergodic. We compare the results of Theorem 3 and Theorem 4. In Theorem 4 only the weaker primitivity condition is assumed for the transition probability matrix and condition (7) is relaxed to condition (9) where the exponent q becomes 1. On the other hand in Theorem 3 condition (8), which is a type of Lipschitzcondition is not needed. To see that (8) is a type of Lipscitz-condition consider the following inequality for an arbitrary fix y ∈ Y: k
1 ∂g(y, p) (b1 (y), . . . bN (y))T k ≤ k = kP j ∂p b (y)pj √
(10)
j
N max bi (y) √ max bi (y) P ji ≤ N i i . min b (y) b (y)pj
(11)
i
j
Since the positivity of Q implies that the stationary distribution of (Xn ) is strictly positive in every state and the densities of the read-outs are strictly positive Condition (6) is not a strong condition. The main advantage of Theorem 3 introduced in [6] is that, it is potentially useful for deriving strong approximation results, see [5], which are in turn applicable to analyze adaptive predictors. Consider a Hidden Markov Model which satisfies the conditions of Theorem 3. Let the parameters of the model be the elements of the transition probability matrix and the parameters of the read-out densities, say bi (y) = f (y, θi ), θi ∈ Θ. We will use the following identifiability condition, see [9]: Identifiability Condition: The family of mixtures of f (y, θi ) is identifiable, i.e. N N X X 0 α0 f (y, θ j ) λ-a.e. αj f (y, θj ) = j=1
j=1
implies
N X j=1
αj δ
θj
=
N X j=1
αj0 δθ0 j ,
where δθ denotes the distribution function of a point mass at θ.
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Theorem 5. Consider a Hidden Markov Model under the conditions of Theorem 3. Let the parameters of the model be the elements of the transition probability matrix and the parameters of the read-out densities. Assume that the identifiability condition above is satisfied. Let θˆN be the ML estimate of θ∗ . Then N 1 X ∂ log p(Yn |Yn−1 , . . . , Y0 , θ∗ ) + OM (N −1 ), θˆN − θ∗ = −(R∗ )−1 N n=1 ∂θ
where R∗ is the Fisher-information matrix and rN = OM (N −1 ) means that rN N is M -bounded, see the Appendix. A key point here is that the error term is OM (N −1 ). This ensures that all basic limit theorems, that are known for the dominant term, which is a martingale, are also valid for θˆN − θ∗ .
5 Gaussian read-outs Consider the most commonly used example, the conditionally Gaussian observations. Let the read-out space be R. Assume that the process (Xn ) satisfies the Doeblin-condition with m = 1 and let the true and the running value of the transition probability matrix be positive, i.e. Q, Q∗ > 0. Let the read-outs be continuous with normal density functions, i.e. ¾ ½ (y − mi )2 1 , exp − bi (y) = √ 2σi 2πσi where (mi , σi )-s are the parameters. Assume that σ1 ≤ . . . ≤ σN . Let denote the true parameter by (m∗i , σi∗ ). Since log bi (y) is quadratic in y, (7) is satisfied as the momentums of the normal distribution exist. Hence Theorem 3 is applicable, and the limit of the log-likelihood function (5) exists and Theorem 5 is applicable. On the other hand Condition (8) of Theorem 4 may not be satisfied if σ1 < σN . For large y-s the integrand of (8) is µ ¶ (y − m1 )2 (y − m∗i )2 (y − mN )2 + − C exp − , 2 2σN 2σ12 2(σi∗ )2 where C is a constant, which is integrable only if −
1 1 1 2 + σ 2 − (σ ∗ )2 < 0 σN 1 i
for all i, i.e. if σi∗ >
(σ1 σN )2 , (σN )2 − (σ1 )2
(12)
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6 Appendix In this section we summarize some basic definitions of [4]. Definition 3. A stochastic process (Xn ) (n ≥ 0) taking its values in an Euclidean space is M -bounded if for all q ≥ 1 Mq = sup E 1/q kXn kq < ∞. n≥0
Let (Fn ) and (Fn+ ) be two sequences of monoton increasing and monoton decreasing σ-algebras, respectively such that Fn and Fn+ are independent for all n. Definition 4. A stochastic process (Xn ) taking its values in a finite-dimensional Euclidean space is L-mixing, if it is M -bounded and with + γq (τ ) = sup E 1/q kXn − E(Xn |Fn−τ )kq n≥τ
Γ (q) =
∞ X
γq (τ ) < ∞.
τ =0
holds. A simple application of L-mixing processes in the context of Markovian processes is given by the following result, see [6] Proposition 1. Let (Xn ) be a Markov chain with state space X , where X is a Polish space, and assume that the Doeblin condition is valid. Furthermore let g : X −→ R be a bounded, measurable function. Then g(Xn ) is an L-mixing process.
7 Acknowledgement The authors acknowledge the support of the National Research Foundation of Hungary (OTKA) under Grant no. T 032932.
References 1. L.E. Baum and T. Petrie. Statistical inference for probabilistic functions of finite state Markov chains. Ann. Math. Stat., 37:1559–1563, 1966. 2. R. Bhattacharya and E. C. Waymire. An approach to the existence of unique invariant probabilities for markov processes. 1999. 3. R. Douc and C. Matias. Asymptotics of the maximum likelihood estimator for general hidden markov models. Bernoulli, 7:381–420, 2001. 4. L. Gerencs´er. On a class of mixing processes. Stochastics, 26:165–191, 1989.
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5. L. Gerencs´er. On the martingale approximation of the estimation error of ARMA parameters. Systems & Control Letters, 15:417–423, 1990. 6. L. Gerencs´er, G. Moln´ ar-S´ aska, Gy. Michaletzky, and G. Tusn´ ady. New methods for the statistical analysis of Hidden Markov Models. In Proceedings of the 41th IEEE Conference on Decision & Control, Las Vegas, pages WeP09–6 2272– 2277., 2002. 7. F. LeGland and L. Mevel. Basic Properties of the Projective Product with Application to Products of Column-Allowable Nonnegative Matrices. Mathematics of Control, Signals and Systems, 13:41–62, 2000. 8. F. LeGland and L. Mevel. Exponential forgetting and geometric ergodicity in hidden Markov models. Mathematics of Control, Signals and Systems, 13:63–93, 2000. 9. B.G. Leroux. Maximum-likelihood estimation for hidden Markov-models. Stochastic Processes and their Applications, 40:127–143, 1992. 10. S.P. Meyn and R.L. Tweedie. Markov Chains and Stochastic Stability. SpringerVerlag, London, 1993. 11. J. H. van Schuppen. Lecture notes on stochastic systems. Technical report. Manuscript.
A Paradigm for Derivatives of Positive Systems Bernd Heidergott Vrije Universiteit Amsterdam, Department of Econometrics and Operations Research, De Boelelaan 1105, 1081 HV Amsterdam, the Netherlands, [email protected]
Abstract. We develop a framework for differentiation of positive operators, such as Markov kernels, through interpreting derivatives of positive operators as differences between positive operators. This new paradigm allows to deal with differentiability issues while retaining the framework of positive systems.
1 Introduction In this paper we show how the dichotomy between positivity and differentiation can be overcome through a concept of ”weak” differentiation. The basic idea will be to write the derivative of a positive operator as re-scaled difference of two positive operators. Our main object of study will be Markov chains. The first part of the paper illustrates our concept of differentiation with finite state Markov chains: ”weak differentiability” for finite Markov chains is introduced and it is shown that differentiability of the transition matrix of a finite state Markov chain implies differentiability of its stationary distribution. The proof elaborates on the product rule of differentiation (for real–valued mappings) and is different from the proofs put forward in [8] and [4], respectively, where this result has been shown by using the fact that the stationary distribution of a Markov chain is an invariant distribution of the Markov kernel. In the second part of the paper, a review of the theory of ”weak differentiation” for general Markov chains will be given. Eventually, we discuss the situation for general positive operators and identify topics of further research.
2 Finite state Markov chains Let Θ = (a, b) ⊂ R, with a < b. Let Xθ (n) ∈ {1, . . . , N } be a discrete-time Markov chain depending on a control parameter θ with deterministic initial value Xθ (0) = x0 ∈ {1, . . . , N }. Let Pθ denote the transition probability matrix of Xθ (n), i.e.: L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 321-328, 2003. Springer-Verlag Berlin Heidelberg 2003
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(Pθ )ij = P ( Xθ (n + 1) = j | Xθ (n) = i ) ,
i, j ∈ {1, . . . , N } ,
for n ≥ 0. For example, Xθ (n) may model the queue length in a M/M/1 queue where θ represents the service rate. 2.1 Differentiability Assume that the elements of Pθ are differentiable and denote the derivative of Pθ by Pθ0 , i.e., (Pθ0 )ij =
d (Pθ )ij , dθ
1 ≤ i, j ≤ N .
(1)
Notice that while Pθ acts as a mapping on the set of probability vectors in Rn , In other words, the image space of Pθ0 contains vectors with negative elements. P for any probability distribution µ = (µ1 , · · · , µN ) (i.e., µk = 1 and µk ≥ 0), µPθ is again a probability distribution whereas µPθ0 fails to be one. However, as we will show in the following, µPθ0 can be written as difference between positive vectors. The key observation is that a matrix CPθ and Markov kernels Pθ+ and Pθ− exists such that Pθ0 = CPθ (Pθ+ − Pθ− ). Typically, CPθ turns out to be a diagonal matrix with identical elements on the diagonal, which yields µPθ0 = CPθ (ν + − ν − ) with ν + = µPθ+ and ν − = µPθ− probability P vectors. ExaminingPthe situation in (1) more closely, one notices that j (Pθ )ij = 1 implies that j (Pθ0 )ij = 0, for 1 ≤ i ≤ N . In words, because Pθ has row sums equal to one (and thus independent of θ), Pθ0 has row sum zero, or, equivalently: X X max(−(Pθ0 )ij , 0) , max((Pθ0 )ij , 0) = j
j
for any row i. For 1 ≤ i ≤ N , let cPθ (i) = matrices Pθ+ and Pθ− defined through (Pθ+ )ij
( max((P 0 ) =
and (Pθ− )ij =
θ ij ,0)
cPθ (i)
(Pθ )ij
( max(−(P 0 )
θ ij ,0)
cPθ (i)
(Pθ )ij
P j
max((Pθ0 )ij , 0), then the
for cPθ (i) > 0 for cPθ (i) = 0 for cPθ (i) > 0 for cPθ (i) = 0
are transition matrices, i.e., their row sum equals one. Moreover, the derivative of Pθ has the following representation ³ ´ Pθ0 = CPθ Pθ+ − Pθ− , (2) where
A Paradigm for Derivatives of Positive Systems
(Cθ )ij
cPθ (i) = 1 0
323
for cPθ (i) > 0 and j = i for cPθ (i) = 0 and j = i otherwise.
The representation of Pθ0 in (2) allows to interpret the derivative of the transition matrix as re-scaled difference of two transition matrices. Remark 1. Let P, Q denote transition matrices on a common state space. In the theory of singularly perturbed Markov chains, the situation is studied when Pθ = θ(Q − P ) + P , for θ ∈ [0, 1], see Chapter 4 in [2] for details on singularly perturbed Markov chains. Hence, Q − P is the derivative of Pθ with respect to θ and formulae for singularly perturbed Markov chains can be interpreted as particular derivative expressions. For an interpretation of the above model in terms of infinitesimal perturbation analysis we refer to [3]. Example 1. Let Xθ (n) be the discrete-time queue length process of an M/M/1/N queue with arrival rate λ and service rate θ, with θ > λ > 0. The transition matrix is then given in matrix form by 0 1 0 λ θ λ+θ 0 λ+θ 0 θ λ 0 0 0 · · · Pθ = λ+θ λ+θ . .. 1 0 The matrix Pθ is differentiable with −λ (λ+θ)2 d λ (Pθ )i j = (λ+θ) 2 dθ 0
respect to θ with derivatives for 2 ≤ i ≤ N − 1 , j = i + 1 for 2 ≤ i ≤ N − 1 , j = i − 1 otherwise.
1 Let Cλ,θ be a matrix with diagonal elements (λ+θ) 2 and zero elements elsewhere, then ´ ³ d (3) Pθ = Cλ,θ P + − P − , dθ with 0 1 0 1 1 0 0 1 + − . . .. .. P = P = , 1 0 0 1 1 0 1 0
where the definitions of the elements of the first and the last rows of Cλ,θ , P + and P − have been chosen in order to obtain a simple representation. Notice that P + and P − are transition matrices. The triple (Cλ,θ , P + , P − ) may serve as matrix–valued representation of Pθ0 .
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For i, j ∈ {1, . . . , N }, the probabilities P ( Xθ (n + 1) = j | Xθ (1) = i ) are given through the elements of nth power of Pθ , denoted by Pθn := (Pθ )n , where Pθ0 is the identity matrix, i.e., P ( Xθ (n + 1) = j | Xθ (1) = i ) = (Pθ )nij . We have assumed that Pθ is differentiable. Therefore, Pθn is differentiable as well. Specifically, n−1 X j d n Pθ Pθ 0 Pθn−j−1 . Pθ = (4) dθ j=0 Example 2. We revisit the M/M/1/N queue as introduced in Example 1. Inserting (3) in (4) and noticing that Cλ,θ is a matrix that only has elements on its diagonal and these elements are identical, yields n−1 n−1 X X d n Pθj P + Pθn−j−1 − P = Cλ,θ Pθj P − Pθn−j−1 . dθ θ j=0 j=0 In words, the derivative of the nth power of a differentiable transition matrix admits a representation like (2) as well. 2.2 Differentiating a stationary distribution In this section we show that, under some mild additional conditions, differentiability of Pθ implies differentiability of the unique invariant distribution of Pθ (existence is assumed here), denoted by πθ , and that the derivative of πθ can be obtained as difference between appropriate Markov chains. We denote by N 1 X n Pθ Πθ = lim N →∞ N + 1 n=0 the ergodic projector associated to Pθ . Specifically, Πθ is a matrix with rows equal to πθ and it holds that πθ = µΠθ , for any initial distribution µ. Assume that µ is independent of θ. Hence, d d π θ = µ Πθ dθ dθ In the following we calculate à ! N 1 X n d lim P . dθ N →∞ N + 1 n=0 θ The key conditions for our analysis is the following. The space of transition probabilities on {1, . . . , N } can be equipped with a norm, denoted by || · ||, such that for an open neighborhood Θ0 ⊂ Θ of θ it holds that:
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(C1) ||πθ || is finite on Θ0 (local stability), (C2) finite constants cθ0 and ρθ0 , with supθ0 ∈Θ0 cθ0 < ∞ and supθ0 ∈Θ0 ρθ0 , ρ < 1, exist such that ∀θ0 ∈ Θ0 :
||Pθm0 − Πθ0 || ≤ cθ0 ρm θ0 ,
(local geometric ergodicity at uniform rate) (C3) Pθ0 is Lipschitz continuous at θ, i.e., ¡ ¢0 ∀θ0 ∈ Θ0 : ||Pθ 0 − Pθ0 || < |θ − θ0 | K 0 , for some finite number K, and ||Pθ0 || is finite on Θ0 . A typical choice for || · || is the supremum norm on Rn , which implies that ≤ 1, for any n, and ||Πθ || ≤ 1 provided that πθ exists. By (4), Ã ! N N n−1 1 X X j 0 n−j−1 d 1 X n P = lim Pθ Pθ Pθ lim N →∞ N + 1 N →∞ dθ N + 1 n=0 θ n=1 j=0
||Pθn ||
and the fact that Pθ 0 has row sum zero implies N n−1 N n−1 1 X X j 0 n−j−1 1 X X j 0 n−j−1 = − Πθ ) . Pθ Pθ Pθ P Pθ (Pθ N + 1 n=1 j=0 N + 1 n=1 j=0 θ
(5)
By conditions (C1) – (C3), for any N , the supremum norm of the expression 1 on the right–hand side of the above equation is bounded by c ||Pθ0 || 1−ρ , which is finite. Hence, the limit exists and we compute N n−1 1 X X j 0 n−j−1 Pθ Pθ (Pθ − Πθ ) N →∞ N + 1 n=1 j=0
lim
N ∞ 1 X nX 0 j Pθ Pθ (Pθ − Πθ ) N →∞ N + 1 n=0 j=0
= lim
= Πθ P θ 0
∞ X j=0
with
(Pθj − Πθ ) = Πθ Pθ 0 Dθ ,
∞ X (Pθj − Πθ ) , Dθ = j=0
where Dθ is known as deviation matrix in the literature, see for example [7]. We have thus shown that à ! N X 1 X n d Pθ Pθ 0 Dθ = Πθ lim N →∞ dθ N + 1 n=0 j=0
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and elaborating on (C1) - (C3) it follows that à ! N d 1 X n 0 P . Πθ = lim N →∞ dθ N + 1 n=0 θ
(6)
For a proof use the fact that we may choose Θ0 small enough such that supθ0 ∈Θ ||Pθ0 || =: K < ∞. Hence, the expression in equation (5) is uniformly bounded on Θ0 in N . By Lipschitz continuity of Pθ0 and Pθ (which follows from K < ∞), we obtain that the expression in equation (5) is uniformly continuous as well. The theorem of Arzela-Ascoli applies and the right–hand side of (6) converges uniformly, which implies that interchanging the order of differentiation and limit is justified. Hence, d πθ = µΠθ Pθ 0 Dθ = πθ Πθ 0 Dθ , dθ
(7)
or, equivalently, d πθ = πθ CPθ Pθ+ Dθ − πθ CPθ Pθ− Dθ . dθ In words, the derivative of the stationary distribution (the fix-point of the positive operator Pθ ) can be represented as the difference of two well–defined positive systems. Specifically, the above result recovers the result in [9] for the case of finite state space. The above formula can be translated in various ways into unbiased gradient estimators for the stationary performance, see [6, 4] for details. Example 3. We revisit the M/M/1 example. If the system is stable on Θ0 with Θ0 an open neighborhood of θ, then πθ0 = πθ Cλ,θ P + Dθ − πθ Cλ,θ P − Dθ .
3 General state–space Markov chains In this section we review the theory of differentiation for Markov chains on a general state–space S. Let (S, T ) denote a measurable space, i.e., T is a σ–field over S, and consider a family of Markov kernels (Pθ : θ ∈ Θ) on (S, T ), with Θ = (a, b) ⊂ R, for a < b. Let LR1 (Pθ ; Θ) ⊂ RS denote the set of measurable mappings g : S → R such that S Pθ (s; du) |g(u)| is finite for all θ ∈ Θ and s ∈ S. A first complication arises when one tries to define what ”differentiability” of Pθ should mean. The following definition has been fruitful in applications. Let D ⊂ L1 (Pθ ; Θ). We call Pθ D–differentiable if a transition kernel Pθ0 exists such that for any s ∈ S and any g ∈ D Z Z d (8) Pθ (s; du) g(u) = Pθ0 (s; du) g(u) . dθ S S
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For example, the Markov kernel Pθ in Example 1 is RN –differentiable. Let Cb (S) denote the set of continuous bounded mappings from S to R. Then, Cb (S) ⊂ D implies that Pθ0 in (8) is uniquely defined. Notice that uniqueness of Pθ0 comes for free in the finite state–space case. Any triple (cPθ (·), Pθ+ , Pθ− ), with Pθ± Markov kernels and cPθ a measurable mapping from S to R, that satisfies ¶ µZ Z Z + − 0 Pθ (s; du) g(u) = cPθ (s) Pθ (s; du) g(u) − Pθ (s; du) g(u) , S
S
S
for any g ∈ D, is called a D–derivative of Pθ . Notice that D–derivatives are not unique. For example, the Markov kernel in Example 1 has RN –derivative (cPθ , P + , P − ) with P + and P − as defined in Example 1 and cPθ (s) = λ/(λ + θ)2 . Does D–differentiability of Pθ already imply the existence of a D–derivative of Pθ ? For the finite state space case, the answer is ”yes” as we have shown in Section 2.1. For a general state–space, however, the situation is more complicated. Provided that (S, T ) is such that T is countable, it holds that if D contains for any A ∈ T its indicator function, then D–differentiability of Pθ implies the existence of a D–derivative. For general (S, T ), we have the following result. Denote the total variation norm of a transition kernel Q on (S, T ) by Z kQktv , sup sup
s∈S f ∈Cb (S) |f |≤1
f (z) Q(s; dz) .
(9)
If Cb (S) ⊂ D and if kPθ0 ktv < ∞, then D–differentiability of Pθ implies the existence of a D–derivative, see [5]. The key ingredients for our proof of differentiability of the stationary distribution in Theorem 1 was that (i) a product rule of differentiation holds, and that (ii) there exists a norm, say || · ||, such that ||Pθ0 || is finite, Pθ0 is Lipschitz and Pθ is geometrically ergodic with coefficient ρθ , such that supθ0 ∈Θ0 ρθ0 , ρ < 1 for some neighborhood Θ0 of θ, i.e., conditions (C1) (C3) hold. It can be shown that, provided the Markov kernel satisfies a weak Lipschitz condition, the product of D–differentiable Markov kernels is again D– differentiable, see [6]. To find good candidates for the norm, we have to resort to stability theory for Markov kernels. A first choice is the total variation norm, see equation (9) for a definition. This is the choice in [8] where it is shown that, under suitable conditions, the stationary distribution is Cb (S)–differentiable. Of course, Cb (S)–differentiability of π is not satisfactory in applications where one is also interested in unbounded performance indicators. Fortunately, the concept of normed ergodicity allows to overcome this restriction. The key idea is to find a Lyapunov function g for Pθ and to consider v = eλg , for some positive λ. The norm is then the weighted supremum norm with respect to v, in symbols:
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¯R ¯ ¯ g(z) Q(s; dz)¯ , ||Q||v , sup sup v(s) s∈S g |g|≤v
with Q a transition kernel on (S, T ), see [4] for details. In [4] sufficient conditions are established such that Pθ is ergodic with coefficient ρθ , such that supθ0 ∈Θ0 ρθ0 , ρ < 1, and ||Πθ0 || < ∞, for θ0 ∈ Θ0 , where Θ0 is neighborhood of θ (i.e., condition (C1) and (C2) hold for || · ||v ). Under these conditions, it holds true that if Pθ is Dv –differentiable with Pθ0 Lipschitz continuous at θ and ||Pθ0 ||v finite, then the stationary distribution is Dv –differentiable as well and its derivative is given by equation (7), we refer to [4] for details. Facilitating this formula for gradient estimation is discussed in [4].
4 General positive operators Let λθ denote an eigenvalue and xθ an eigenvector (associated to λθ ) of the positive operator Tθ , i.e., λθ xθ = Tθ xθ , θ ∈ Θ, for some suitable set Θ. For example, Tθ may represent the transition operator in a (max,+)– or (min,+)– linear system and λθ the unique eigenvalue (existence is assumed here) and xθ an eigenvalue, see [1] for details. For the analysis in the previous sections, we relied on the fact that, for Markov chains, the maximal positive eigenvalue of Tθ is independent of θ (in fact, λθ = 1 for θ ∈ Θ). For general operators eigenvector(s) as well as eigenvalue(s) will depend on θ. The development of an approach for general positive operators is topic of future research. An application of these results might, for example, lead to a sensitivity analysis of the spectral gap of a Markov chain.
References 1. F. Baccelli, G. Cohen, G. Olsder, and J.–P. Quadrat. Synchronization and Linearity. John Wiley and Sons, New–York, 1992. 2. E. Feinberg and A. Schwartz (eds.) Handbook of Markov decision processes. Kluwer, Boston, 2002. 3. B. Heidergott and X. Cao. A note on the relation between weak derivatives and perturbation realization. IEEE Trans. Aut. Control, 47:1112-1115, 2002. 4. B. Heidergott, A. Hordijk and H. Weisshaupt. Measure–Valued Differentiation for Stationary Markov Chains. EURANDOM report 2002-027, 2002. 5. B. Heidergott, A. Hordijk and H. Weisshaupt. A Jordan type decompostion for weak derivatives of Markov kernels. EURANDOM report 2003-001, 2003. 6. B. Heidergott and F. V´ azquez-Abad. Measure–valued differentiation for stochastic processes: the finite horizon case. EURANDOM report 2000-033, 2000. 7. S. Meyn and R. Tweedie Markov Chains and Stochastic Stability. Springer, London, 1993. 8. G. Pflug. Optimization of Stochastic Models. Kluwer Academic, Boston, 1996. 9. F. V´ azquez–Abad and H. Kushner. Estimation of the derivative of a stationary measure with respect to a control parameter. J. Appl. Prob., 29:343–352, 1992.
Nonlinear Positive 2D Systems and Optimal Control Dariusz Idczak and Marek Majewski Faculty of Mathematics, University of L #o ´d´z, Banacha 22, 90-238 L #o ´d´z, Poland, {idczak,marmaj}@math.uni.lodz.pl Abstract. In the paper, a nonlinear positive 2-D control system given by the Goursat-Darboux problem is considered. First, we give sufficient conditions for the system to be positive. Next, we derive an existence result for an optimal control problem of Lagrange type under a convexity assumption.
1 Introduction In many applications of the optimal control theory of one-dimensional (1D) systems a very important role is played by the following time-invariant systems i) discrete linear system xi+1 = Axi + Bui , i ∈ Z+ , restricted to the nonnegative values of the state variable xi ∈ Rn+ (by Rn+ we mean the set of n-dimensional vectors with real nonnegative entries) and control variable ui ∈ Rm +, ii) continuous linear system x0 (t) = Ax(t) + Bu(t), t ∈ [0, ∞), x(0) = x0 restricted to the nonnegative values of the state variable x(t) ∈ Rn+ and control variable u(t) ∈ Rm +. Such systems are called positive. More precisely, the discrete system is called positive if for every x0 ∈ Rn+ and sequence of inputs (ui )i∈Z+ ⊂ Rm +, xi ∈ Rn+ for i ∈ N. Similarly, the continuous system is called positive if for n every x0 ∈ Rn+ and control u : [0, ∞) → Rm + , x(t) ∈ R+ for t ∈ [0, ∞). A rich survey of the results and methods concerning the controllability, reachability, L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 329-336, 2003. Springer-Verlag Berlin Heidelberg 2003
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observability, minimum energy control and realization of the above positive systems can be found in the first part of monograph [9]. The second part of [9] is devoted to the theory of positive two-dimensional discrete linear time-invariant systems of Roesser and Fornasini-Marchesini type (continuous-discrete systems are also considered). A discrete system of Fornasini-Marchesini type, introduced in [4], has the form z(i + 1, j + 1) = Az(i, j) + Bz(i + 1, j) + Cz(i, j + 1) + Du(i, j), i, j ∈ Z+ , where z ∈ Rn , u ∈ Rm and A, B, C, D are matrices of appropriate dimensions. Our aim is to investigate a continuous nonlinear analog of the above system. Namely, we consider system zxy (x, y) = f (x, y, z(x, y), zx (x, y), zy (x, y), u(x, y)), (x, y) ∈ P a.e., z(x, 0) = z(0, y) = 0, x, y ∈ [0, 1], 2
n
m
(1) (2)
n 3
m
where P = [0, 1] × [0, 1] ⊂ R , z ∈ R , u ∈ R , f : P × (R ) × R → Rn . Boundary value problem of the above type is referred as Goursat-Darboux problem and, in the automatic control theory, as continuous Fornasini-Marchesini problem. Such systems have many applications, for example, in a model of a process of the gas absorbtion (cf. [14] and also [7]). In optimal control theory the following two questions play the fundamental role: (a) existence of an optimal control and (b) its calculation. In our paper, we shall study question (a) for a positive system of type (1) (see (3) below). First of all, we shall give sufficient conditions under which system (1) is positive, i.e., roughly speaking, the response (output) z : P → Rn has the nonnegative values whereas the control (input) u : P → Rm has. Next, we shall give sufficient conditions for the existence of an optimal solution to the Lagrange optimal control problem connected with a particular case of the positive system (1), namely zxy (x, y) = f (x, y, z(x, y), u(x, y)) + A1 (x, y)zx (x, y) + A2 (x, y)zy (x, y), (3) with boundary conditions (2), cost functional Z J(u) = f 0 (x, y, z(x, y), u(x, y))dxdy → min
(4)
P
and the control constrain n
m
u(x, y) ∈ Ω, n
n×n
(5) n×n
0
n
where f : P ×R ×R → R , A1 : P → R , A2 : P → R , f : P ×R × Rm → R, and Ω ⊂ Rm is some fixed set. More precisely, we give conditions under which there exists a pair (z∗ , u∗ ) of functions with nonnegative values and belonging to some functional spaces (given in section 1.2), such that it satisfies (3), (2), (5) and for any pair (z, u) of functions with nonnegative
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values, belonging to the functional spaces mentioned above, and satisfying (3), (2), (5), we have J(z∗ , u∗ ) ≤ J(z, u). These conditions are based on a convexity assumption. A result of such type has been obtained, in the case of 1-D continuous systems, by Fillipov and Roxin (cf. [3], [13], [11]).
2 Existence of positive solutions By AC(P, R) we shall denote the space ofR allR functions z : P → R that x y have the integral representation z(x, y) = 0 0 g(s, t)dsdt for (x, y) ∈ P , where g ∈ L1 (P, R) (the space of all real Lebesgue integrable functions on P ). A function z : P → R belonging to the set AC(P, R) is called absolutely continuous. The notion of an absolutely continuous function (of two variables) z : P → R, based on the notion of a function of an interval, associated with z, has been introduced in [16]. The theorem on the integral representation is also derived there. One can show (cf. [16]) that absolutely continuous function z : P → R possesses a.e. on P the partial derivatives zx , zy , zxy and Z y Z x zx (x, y) = g(x, t)dt, zy (x, y) = g(s, y)ds, zxy (x, y) = g(x, y) 0
0
for (x, y) ∈ P a.e. Of course, z satisfies boundary conditions (2). By AC(P, Rn ) we shall denote the set of all functions z = (z 1 , ..., z n ) : P → Rn that have absolutely continuous coordinate functions z i , i = 1, ..., n. By L we shall denote the following set ½ Z xZ y Z y g ∈ L1 (P, Rn ) : g(s, t)dsdt ≥ 0, g(x, t)dt ≥ 0, 0
Z
x 0
0
0
¾ g(s, y)ds ≥ 0 f or (x, y) ∈ P a.e. .
Now, let us consider in the space L1 (P, Rn ) the Bielecki metric (cf. [1]) Z ρk (g, h) = e−k(x+y) |g(x, y) − h(x, y)| dxdy P
for g, h ∈ L1 (P, Rn ), where k > 0 is any fixed number. It is easy to Rsee that the above metric is equivalent to the classical one (ρ(g, h) = |g(x, y) − h(x, y)| dxdy). Consequently, the metric space (L, ρk |L ) is comP plete (it suffices to observe that the set L is closed in the metric space (L1 (P, Rn ), ρ) and use the equivalence of ρ and ρk ). In the next, we shall assume that the function f : P × (Rn )3 × Rm → Rn (appearing in (1)) satisfies the following conditions
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(2.1) f is measurable in (x, y) ∈ P , continuous in u ∈ Rm and lipschitzian in (z, z1 , z2 ) ∈ (Rn )3 , i.e. there exists a constant K ≥ 0 such that |f (x, y, z, z1 , z2 , u) − f (x, y, w, w1 , w2 , u)| ≤ K(|z − w| + |z1 − w1 | + |z2 − w2 |) for (x, y) ∈ P a.e., z, z1 , z2 , w, w1 , w2 ∈ Rn , u ∈ Ω, (2.2) for any control u ∈ L1 (P, Ω) (the set of integrable functions taking their values in Ω) the function P 3 (x, y) 7→ f (x, y, 0, 0, 0, u(x, y)) ∈ Rn is integrable, (2.3) f (·, ·, 0, 0, 0, 0) ∈ L, (2.4) f (x, y, z, z1 , z2 , u) ≥ f (x, y, 0, 0, 0, 0) for (x, y) ∈ P a.e., z, z1 , z2 ∈ Rn+ , u∈Ω . From [5, Theorem 1] it follows that if f satisfies (2.1) and (2.2), then, for any fixed control u ∈ L1 (P, Ω ), the operator Fu : L1 (P, Rn ) → L1 (P, Rn ) given by Z xZ y Z y Z x (Fu g)(x, y) = f (x, y, g(s, t)dsdt, g(x, t)dt, g(s, y)ds, u(x, y)) 0
0
0
0
is well-defined. Moreover, one can choose a number k0 > 0 such that Fu is contracting in the metric space (L1 (P, Rn ), ρk0 ). In an elementary way one obtains Lemma 1. If f satisfies (2.1)-(2.4) and u ∈ L1 (P, Ω ), then Fu g ∈ L for any g ∈ L. So, if u ∈ L1 (P, Ω ) and f satisfies (2.1)-(2.4), then Fu |L : L → L. Of course, Fu |L is contracting in the metric space (L, ρk0 |L ). From the completeness of (L, ρk0 |L ) and Banach contraction principle it follows that there exists exactly one function gu ∈ L such that Fu (gu ) = gu . Adopting Z xZ y zu (x, y) = gu (s, t)dsdt 0
0
for (x, y) ∈ P , we obtain a unique solution of the problem (1)-(2) in the space AC(P, Rn ). In fact, zu ∈ AC+ (P, Rn ) where AC+ (P, Rn ) = {z ∈ AC(P, Rn ) : z, zx , zy ≥ 0 a.e. on P } . We say in this case that the problem (1)-(2) is positive. More precisely, problem (1)-(2) is called positive if for any control u ∈ L1 (P, Ω ) there exists a solution zu ∈ AC+ (P, Rn ). Thus, we have proved Theorem 1. If f satisfies (2.1)-(2.4) and u ∈ L1 (P, Ω ), then the GoursatDarboux problem (1)-(2) has a unique solution zu in the set AC+ (P, Rn ) and, in consequence, the problem is positive.
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Remark 1. If system (1) is linear, time-invariant, i.e. zxy (x, y) = Az(x, y) + A1 zx (x, y) + A2 zy (x, y) + Bu(x, y) m and, for example, Ω = Rm + or Ω = [0, 1] × ... × [0, 1] ⊂ R , then from (2.4) it follows that A, A1 , A2 , B are matrices with nonnegative entries.
3 Existence of an optimal solution In this part we shall assume that Ω ⊂ Rm is a fixed compact set. Let us consider optimal control problem (2)-(5). On the functions A1 , A2 : P → Rn×n we assume that they have nonnegative and essentially bounded (on P ) entries. A function f : P × Rn × Rm → Rn (appearing in (3)) is assumed to be measurable in (x, y) ∈ P , continuous in u ∈ Rm and lipschitzian in z ∈ Rn . Moreover, we assume that for any control u ∈ L1 (P, Ω ) the function P 3 (x, y) 7→ f (x, y, 0, u(x, y)) ∈ Rn is integrable. We also assume that f (·, ·, 0, 0) ∈ L and f (x, y, z, u) ≥ f (x, y, 0, 0) for (x, y) ∈ P a.e., z ∈ Rn+ , u∈Ω . The above assumptions guarantee that the right-hand side of the system (3) satisfies the conditions (2.1)-(2.4). Consequently, problem (3)-(2) possesses a unique solution zu ∈ AC+ (P, Rn ) for any u ∈ L1 (P, Ω ). On the integrand f 0 : P × Rn × Rm → R we assume that it is measurable in (x, y) ∈ P , continuous in (z, u) ∈ Rn × Rm and for any r ≥ 0 there exists a function cr ∈ L1 (P, R) such that ¯ 0 ¯ ¯f (x, y, z, u)¯ ≤ cr (x, y) (6) for (x, y) ∈ P a.e., |z| ≤ r, u ∈ Ω . In the next, we shall assume that the responses of (3)-(2) satisfy a priori bound : (3.2) there exists a constant r ≥ 0 such that |zu (x, y)| , |(zu )x (x, y)| , |(zu )y (x, y)| , |(zu )xy (x, y)| ≤ r for (x, y) ∈ P a.e., u ∈ L1 (P, Ω ). Remark 2. One can show (cf. [12]) that if there exists a constant b ≥ 0 such that |f (x, y, 0, u)| ≤ b for (x, y) ∈ P a.e., u ∈ Ω , then the condition (3.2) is satisfied. In the case of nonlinear system (1), some conditions guarantying a priori bound are given in [6].
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Before we prove the main result of this part of the paper we shall introduce some notions. Let us consider the so-called extended problem zbxy (x, y) = fb(x, y, zb(x, y), zbx (x, y), zby (x, y), u(x, y)), (x, y) ∈ P a.e., zb(x, 0) = zb(0, y) = 0 x, y ∈ [0, 1], where fb(x, y, zb, zb1 , zb2 , u) = (f 0 (x, y, z, u), f (x, y, z, u)+A1 (x, y)z1 +A2 (x, y)z2 ), for (x, y) ∈ P a.e., zb = (z 0 , z), zb1 = (z10 , z1 ), zb2 = (z20 , z2 ) ∈ R × Rn , u ∈ Rm . It is easy to see that if zu is a solution of problem (3)-(2), corresponding to a control u, then zbu (x, y) = (zu0 (x, y), zu (x, y)) R R x y where zu0 (x, y) = 0 0 f 0 (s, t, zu (s, t), u(s, t))dsdt, is a solution of the extended problem, corresponding to u. Of course, zu0 (1, 1) = J(u). We have the following Theorem 2. If all of the assumptions given in this part are satisfied and the set © W (x, y, z, z1 z2 ) = (ξ 0 , ξ) ∈ R × Rn : there exists v ∈ Ω such that ª ξ 0 = f 0 (x, y, z, v) and ξ = f (x, y, z, v) + A1 (x, y)z1 + A2 (x, y)z2 is convex for (x, y) ∈ P a.e., z, z1 , z2 ∈ Rn+ , then optimal control problem (2)-(5) has a solution. Proof. From the assumptions (6), (3.2) it follows that inf{J(u); u ∈ L1 (P, Ω)} ∈ R. Let (ul )l∈N be a minimizing sequence for J (i.e. liml→∞ J(ul ) = inf{J(u); u ∈ L1 (P, Ω )} and (zl )l∈N - the sequence of corresponding solutions to problem (3)-(4), (b zl )l∈N - the sequence of corresponding solutions to extended problem. We shall show that there exists a control u0 ∈ L1 (P, Ω ) such that ¡ ¢ zbu0 (1, 1) = zu0 0 (1, 1), zu0 (1, 1) = (J(u0 ), zu0 (1, 1)) ¡ © ª ¢ = inf J(u) : u ∈ L1 (P, Ω) , zu0 (1, 1) . Assumptions (6), (3.2) and [5, Theorems 3.3, 3.4] imply that from the sequence (b zl )l∈N one can choose a subsequence, say still (b zl )l∈N , uniformly converging to some absolutely continuous function zb0 : P → R × Rn . It is easy to see, using Dunford-Pettis theorem (cf. [2, Theorem 10.3.i]), that one can choose a subsequence, say still ((b zl )xy )l∈N , such that (b zl )xy * (b z0 )xy , (b zl )y * (b z0 )y , (b zl )x * (b z0 )x
(7)
weakly in L1 (P, R × Rn ) when l → ∞. Moreover, one can check that z0 ∈ AC+ (P, Rn ). From the separation theorem (cf. [15, IV.5, Theorem 1]) it follows that (b z0 )xy (x, y) ∈ fb(x, y, zb0 (x, y), (b z0 )x (x, y), (b z0 )y (x, y), Ω)}, (x, y) ∈ P a.e.
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Applying the implicit function theorem for multivalued mapping (cf. [10, II, Theorem 3.12]) we assert that there exists a measurable function u0 : P → Ω such that (b z0 )xy (x, y) = fb(x, y, zb0 (x, y), (b z0 )x (x, y), (b z0 )y (x, y), u0 (x, y)) for (x, y) ∈ P a.e. In particular, (z0 )xy (x, y) = f (x, y, z0 (x, y), u0 (x, y))+A1 (x, y)(z0 )x (x, y)+A2 (x, y)(z0 )y (x, y) for (x, y) ∈ P a.e. and, as we have showed, z0 ∈ AC+ (P, Rn ). Moreover (we recall that zbl → zb0 uniformly on P when l → ∞ ), inf{J(u) : u ∈ L1 (P, Ω)} = lim J(ul ) = lim zl0 (1, 1) = z00 (1, 1) Z =
P
(z00 )xy (x, y)dxdy =
Z P
l→∞
l→∞
f 0 (x, y, z0 (x, y), u0 (x, y))dxdy = J(u0 ).
This completes the proof.
u t
Remark 3. Let us point that the set W (x, y, z, z1 , z2 ) is assumed to be convex only for z, z1 , z2 ∈ Rn+ (not for all z, z1 , z2 ∈ Rn ). Example 1. Let us consider the following optimal control problem zxy = d(x, y) + zu2 cos u1 + exy zx +
1 zy 1 + x + y2
for (x, y) ∈ P a.e., z(x, 0) = z(0, y) = 0, x, y ∈ [0, 1], Z J(u) = f 0 (x, y, z(x, y), u(x, y))dxdy → min, P
u(x, y) ∈ M = [0, where
π ] × [0, 1] × [1, 2] ⊂ R3 , 2
1 1 d : P 3 (x, y) 7−→ (x − )2 + (y − )2 − c ∈ R 2 2 with c > 0 such that d ∈ L (it is easy to see that such constant c > 0 exists), ½ zu3 ; z≥0 0 f : P × R × M 3 (x, y, z, u) 7→ p ∈ R. |z| u2 cos u1 ; z < 0 In an elementary way one can check that all of the assumptions of Theorem 2 (including the assumption contained in the first part of the Remark 2) are satisfied. Consequently, there exists a solution u ∈ L1 (P, M ) to the above optimal control problem, such that zu ∈ AC+ (P, Rn ).
Acknowledgement. The preparation of this paper was supported by the State Committee for Scientific Research under Grant 7T11A00421.
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References 1. A. Bielecki, Une remarque sur l’application de la methode de Banach-CocciopoliTichonov dans la theorie de l’equation s = f (x, y, z, p, q), Bull. Acad. Pol. Sci. 4 (1956), 265-268. 2. L. Cesari, Optimization - Theory and Applications, Springer - Verlag, New York, 1983. 3. A. F. Fillipov, On certain questions in the theory of optimal control, SIAM J. Control, 1 (1959), 76-84. 4. E. Fornasini, G. Marchesini, State space realization of two dimensional filters, IEEE Trans. Autom. Control, AC-21 (1976), 484-491. 5. D. Idczak, K. Kibalczyc, S. Walczak, On an optimization problem with cost of rapid variation of control, J. Austral. Math. Soc. Ser. B, vol. 36 (1994), 117-131. 6. D. Idczak, M. Majewski, S. Walczak, Stability of solutions to an optimal control problem for a continuous Fornasini-Marchesini system, Proceedings of the Second Internation Workshop on Multidimensional (nD) Systems, Czocha Castle, Lower Silesia, Poland, 2000. 7. D. Idczak, S. Walczak, On Helly’s theorem for functions of several variables and its applications to variational problems, Optimization, 30 (1994), 331-343. 8. A. D. Ioffe, V. M. Tikhomirov, Theory of Extremal Problems, Moskov, 1974 (in Russian). 9. T. Kaczorek, Positive one- and two-dimensional systems, Technical University of Warsaw, Warsaw, 2000. 10. M. Kisielewicz, Differential Inclusions and Optimal Control, PWN with Kluwer Academic Publishers, Warsaw-Dordrecht, 1991. 11. J. Macki, A. Strauss, Introduction to Optimal Control Theory, Springer-Verlag, New York, 1982. 12. M. Majewski, Stability of solutions to differential systems and control systems, Doctoral thesis, University of L # ´ od´z, L #o ´d´z, 2003 (in Polish). 13. E. Roxin, The existence of optimal controls, Michigan Math. J., 9 (1962), 109119. 14. A. N. Tikhonov, A. A. Samarski, Equations of Mathematical Physics, Moscov, 1978 (in Russian). 15. F. P. Vasiliev, Numerical Methods of Solving of Extremal Problems, Moscov, 1988 (in Russian). 16. S. Walczak, Absolutely continuous functions of several variables and their applications to differential equations, Bull. Polish Acad. Sci. Math., vol. 35, no 11-12 (1987), 733-744.
State Feedback Set Stabilization for a Class of Nonlinear Systems Lars Imsland and Bjarne A. Foss Department of Engineering Cybernetics, Norwegian Univ. of Science and Technology, 7491 Trondheim, Norway, {Lars.Imsland,Bjarne.A.Foss}@itk.ntnu.no Abstract. A controller for a class of multiple input systems is proposed, with a subset of nonlinear positive systems as a special case. The controller is shown to obtain set stability of a certain subset of the state space. A simple example illustrates the theory.
1 Introduction The class of nonlinear systems is very diverse, and encompasses many types of applications and behaviors. It seems apparent that for control purposes, one should focus on specific nonlinear system classes to exploit the structure of these, rather than make an all-encompassing theory for nonlinear systems. One class of nonlinear systems that has strong structural constraints that can be exploited, is the class of positive systems. The typical class of nonlinear positive systems seems to be systems based on material balances, for example compartmental systems (Jacquez and Simon, 1993) and systems based on mass balances (Bastin, 1999). These system classes enjoy in addition to positivity structural constraints related to the flow of “mass” between the states. Similar structural constraints are assumed on the system class in Imsland and Foss (2002), Imsland (2002). In this paper, the situation compared to Imsland and Foss (2002) is generalized. As an outset, we consider a (non-positive) system which have a set of positive inventories (which can be interpreted as mass, energy, entropy, ...). This allows more general processes than mass balance systems. Furthermore, the structural constraints are related to a general nonlinear function (rather than the sum) of the states, and the controller aims at a constant value of this nonlinear function. Hence, the “total mass”-approach used in Imsland and Foss (2002) becomes a special case of the approach used herein. Similarly to Imsland and Foss (2002) we allow multiple-input systems by dividing the states into what we call phases.
L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 337-344, 2003. Springer-Verlag Berlin Heidelberg 2003
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The considered system class is presented in Section 2, while we in Section 3 first recapitulates the necessary Lyapunov theorems before we give the main result (Theorem 3). After a brief discussion in Section 4, we present a simple example to illustrate the results. The notation is fairly standard. |z|A denotes the distance from z to the set A, defined by |z|A := inf z˜∈A kz − z˜k where k · k is the Euclidean norm. Furthermore, R+ = [0, ∞).
2 System class Consider the input affine state-space system x˙ = f (x) + g(x)u,
(1)
where x ∈ Rn and u ∈ Rm + . The inputs are positive and governs in some sense the inflow of the process system. For this system, there exists of a positive vector function of “inventories” (Farsham, Viswanath and Ydstie, 1998) ν(x) = (ν1 (x), . . . , νp (x)) ⊂ Rp+ , which behave according to ν˙ =
∂ν ∂ν f (x) + g(x)u := F (x) + G(x)u. ∂x ∂x
(2)
At this point, note that the positive system ν˙ = F (ν) + G(ν)u is a special case. We will assume that the inventories are divided into m different groups (one for each input), which for convenience will be denoted phases. Phase Pm j will consist of rj inventories (with j=1 rj = p), and have the control uj associated with it (controlling the inflow¡ to that phase). The inventories in ¢ phase j will be denoted ν j , such that ν = ν 1 , . . . , ν m . Note that m ≤ p, such that ν j of phase j can consist of several inventories νi . Corresponding to this structure, we write the vector function F (x) and the matrix function G(x) as ´ ³ ¡ ¢ F (x) = F 1 (x), . . . , F m (x) , F j (x) = φj1 (x), . . . , φjrj (x) , ³ ´ ¡ ¢ G(x) = blockdiag G1 (x), . . . , Gm (x) , Gj (x) = γ1j (x), . . . , γrjj (x) . For each phase, we will choose a positive scalar function Hj (ν j ) such that the control objective is that the inventories of phase j should converge to the “level set” Ωj , given by Ωj := {z ∈ Rrj | Hj (z) = Cj }. The inventories can typically be interpreted as mass, or energy, and the functions Hj (ν j ) as total mass or energy (of a “subprocess”). In some sense, the functions Hj (ν j ) are also inventories, but we distinguish these since the control objective is related to them. 2 Define the scalar function Vj (z) := 21 (Hj (z) − Cj ) , on which we impose
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Assumption 1 The sets Ωj are compact, and there exist constants cj < cj such that for all z ∈ Rrj , cj |z|Ωj ≤ Vj (z) ≤ cj |z|Ωj .
(3)
Remark 1. The above assumption holds (see Imsland (2002)) for example if we for a positive system (with states ν) choose the “total mass” Hj (ν j ) = Prj j i=1 νi , which is used e.g. in Bastin and Praly (1999) and Imsland and Foss r (2002). The positivity of the system (ν j ∈ R+j ) ensures in this case that Ωj is compact, which is essential. The system assumptions are made with respect to the set D ⊆ Rn : Assumption 2 The function F (x) : D → Rn is locally Lipschitz and Prj ∂Hj (ν j ) j φi (x) ≤ 0 and the set for x ∈ {x ∈ D | Hj (ν j (x)) > Cj }, i=1 ∂νij Prj ∂Hj (ν j ) j {x ∈ D | φi (x) = 0 and Hj (ν j ) > Cj } does not contain an ini=1 ∂νij variant set. This assumption means that F (x) consists of interconnection terms (terms that have zero net contribution to Hj (ν j ), that is, “flow” between different inventories in a phase), and the rest of the terms are dominantly outflow (at least for large Hj (ν j )). Assumption 3 The block diagonal matrix function G(x) : D → Rn×m is Prj ∂Hj j γ (x) > 0 for all x ∈ D. locally Lipschitz and i=1 ∂ν j i i
Note that further assumptions are needed to ensure that the inventories remain positive, but these are implicitly required to hold.
3 State feedback set stabilization A Lyapunov condition for asymptotic stability of a compact set A is: Theorem 1. The compact set A is asymptotically stable (in the sense of Lyapunov) for x˙ = f (x) if there on an open set O containing A exists a C 1 function V and positive constants κ1 and κ2 such that κ1 |x|A ≤ V (x) ≤ κ2 |x|A ∂V f (x) < 0 for |x|A 6= 0 ∂x
(4) (5)
on the set O. See e.g. Rouche, Habets and Laloy (1977) for a treatment of set stability. The above condition is not the most general, but sufficient for our purpose. In some cases, it is hard to find a Lyapunov function where the derivative is negative definite. If it is semidefinite, a straightforward adoption of LaSalle’s invariance principle (similar to the Barabashin-Krasovskii Theorem) can be used:
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Theorem 2. If the conditions of Theorem 1 hold on the compact set D (containing A) except that (5) is replaced with ∂V f (x) ≤ 0 ∂x
(6)
on D, and the largest invariant set contained in the set where ∂V ∂x f (x) = 0 is contained in A, then the set A is asymptotically stable for x˙ = f (x). Recalling that asymptotic stability is equivalent to stability and convergence (Rouche et al., 1977), this result is obtained by noting that the semidefiniteness of the derivative of the Lyapunov function implies (set) stability, while convergence follows from LaSalle’s invariance principle (LaSalle, 1960). The controller we suggest for the given system class, is #−1 Ã rj ! " rj X ∂Hj j X ∂Hj j j γ (x) − φ (x)−λj (Hj (ν )−Cj ) } (7) uj (x) = max{0, j i j i i=1 ∂νi i=1 ∂νi Sm where the λj s are positive constants. Defining Ω := j=1 Ωj , we make the following assumption: Assumption 4 There exists a (not necessarily compact) set D that is i) invariant for the dynamics (1) in closed loop with control (7), ii) has a nonempty intersection with Ω, and iii) for x ∈ Ω ∩ D, uj (x) > 0. The stability properties can be summarized as follows: Theorem 3. Under Assumptions 1-4, the set Ω ∩ D is asymptotically stable for the closed loop system controlled with (7). Moreover, convergence to Ω ∩ D and boundedness of trajectories holds for initial conditions x(0) ∈ D. Proof. The set D is by Assumption 4 invariant, hence Assumptions 2 and 3 hold along closed loop trajectories. By Assumption 3, the control (7) is well defined on D. Pm j Define the positive semidefinite function V (x) := j=1 Vj (ν (x)), with time derivative along system trajectories V˙ (x) =
m X £ ¤ Hj (ν j (x)) − Cj H˙ j (ν j (x)) j=1
à rj ! rj m X X £ ¤ X ∂Hj i ∂Hj i j = Hj (ν (x)) − Cj φ (x) + γ (x)uj (x) . j j j j j=1 i=1 ∂νi i=1 ∂νi For Hj (ν j ) 6= Cj , we have one of the following cases: 1. If uj (x) > 0, summand j is ! à rj rj X £ ¤2 £ ¤ X ∂Hj i ∂Hj i j φ (x)+ γ (x)uj (x) = −λj Hj (ν j )−Cj < 0. Hj (ν )−Cj j j j j i=1 ∂νi i=1 ∂νi
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£ ¤ Prj ∂Hj i 2. If uj (x) = 0, then summand j is Hj (ν j ) − Cj i=1 ∂ν j φj (x). Thus, if i P r ∂H j j Hj (ν j ) > Cj , then by Assumption 2 i=1 ∂ν j φij (x) ≤ 0, and i
r
j £ ¤X ∂Hj i ψ (x) ≤ 0. Hj (ν j ) − Cj j j i=1 ∂νi
Prj ∂Hj If Hj (ν j ) < Cj , then since uj (x) = 0 and i=1 > 0 by Assumption 3, ∂ν j £ ¤ i Prj ∂Hj i j we see that i=1 ∂ν j φj (x) ≥ −λj Hj (ν )−Cj , which gives i
£
r
Hj (ν j ) − Cj
j ¤X ∂Hj
j i=1 ∂νi
£ ¤2 φij (x) ≤ −λj Hj (ν j ) − Cj < 0.
If Hj (ν j ) = Cj , ν j ∈ Ωj and V˙ j ≡ 0 due to Assumption 4 iii). This holds for all summands (phases), and we conclude that V˙ (x) ≤ 0. This implies that V (x(t)) ≤ V (x(t0 )) along system trajectories. From the construction of V and Assumption 1, we see that kxk → ∞ if and only if V (x) → ∞, hence bounded V (x(t)) implies that kx(t)k is bounded (stays in the compact set given by {x | V (x) ≤ V (x(t0 ))}). We conclude by Theorem 2 that Ω ∩ D is asymptotically stable, since by Assumption 1, V fulfills (4) with respect to Ω, and as shown above, V˙ (x) ≤ 0 for x ∈ D. Moreover, from the above and Assumption 2, the only invariant set where V˙ (x) = 0 is (within) Ω ∩ D. u t Remark 2. For positive systems, the set D can in many cases be taken as the positive orthant, D = Rn+ . Another option can be “Lyapunov level sets” of the type {x ∈ Rn | Cj − β j ≤ Hj (ν j (x)) ≤ Cj + β j , j = 1, . . . , m}. Similarly to De Leenheer and Aeyels (2002), we can state the following result on asymptotic stability of equilibria. It can be proved the same way as in De Leenheer and Aeyels (2002), or seen as a consequence of the theory of semidefinite Lyapunov functions (Chabour and Kalitine, 2002), hence we state it without proof: Theorem 4. Let the conditions of Theorem 3 hold. If the closed loop (1) with control (7) has a single equilibrium in the interior of Ω ∩ D that is asymptotically stable with respect to initial conditions in Ω ∩ D and attractive for all initial conditions in Ω ∩ D, the equilibrium is asymptotically stable for the closed loop with a region attraction (of at least) D.
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4 Discussion of controller 4.1 Connection with similar control schemes The trained eye will see that the proposed controller is similar to feedback linearizing controllers (as, for instance, in Isidori (1995)). However, while feedback linearization linearizes the whole state space, (7) linearizes (in the unconstrained case) only the “phase dynamics” (the dynamics of the functions Hj (ν j (x))), while the total dynamics remain nonlinear. Feedback linearization requires in general solving a set of PDEs to find the right “full relative degree” output, while for the approach herein, the functions Hj are given by assumption. Furthermore, the present approach can preserve stability under some input constraints, due to the system properties. The controller (7) is the same as proposed in Bastin and Praly (1999) (see also Bastin and Provost (2002)). Herein, a larger class of systems is studied, for instance the concept of phases allow to handle systems with multiple inputs. This idea is also explored in Imsland and Foss (2002), Imsland (2002) in the setting of positive systems, but herein the concept of “inventories” allows us to move beyond positive systems. Moreover, the approach in Imsland and Foss (2002) is generalized herein in allowing more general functions Hj . The inventory concept is borrowed from Farsham et al. (1998), where interesting connections are tied between the first and second law of thermodynamics and system theoretic properties of “process systems”. The control problem therein is to steer the inventories to their setpoints, and this is done by assuming it is possible to manipulate the flows corresponding to each inventory. Only convergence of the inventories are considered, the underlying system behavior is not examined. One major difference to the controller developed herein, is that we do not assume a one-to-one relationship between inventories and manipulated flows. Constraints are considered in Farsham et al. (1998) by augmenting the controller to a PID type controller with antireset windup. The “original” controller is in this setting a P-controller, in the same way as the one proposed herein (interpreting the λj s as gains). 4.2 Outflow controlled systems and upper constraints In view of Assumption 3 (and the sign of Hj ), it can be said that the system class treated herein is inflow controlled (with respect to Hj ). However, systems that in a corresponding sense are outflow controlled (or having a mixture of inflow or outflow controlled phases), can be treated in a similar way.PThis is rj νij . done in Imsland and Foss (2002) for positive systems with Hj (ν j ) = i=1 In addition to more rigid conditions on the “uncontrolled” flows, one must then in general have an upper saturation on the control to ensure that the inventories remain positive.
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5 Example For mass balance systems where Hj (ν j ) is the sum of the phase states (masPrj j ses), Hj (ν j ) = i=1 νi , several examples can be found in Imsland (2002), for instance the stabilization of a gas lifted oil well (Imsland and Foss, 2002). Here we will consider an example which is a (slightly modified) system taken from De Leenheer and Aeyels (2002), which illustrate another type of Hj for a positive system (states are inventories). The system equations are x˙ 1 = −x1 x2 + x22 − x1 + u x21
x˙ 2 = −x1 x2 +
(8)
+ u.
(9)
∂H 2 We choose H(x) = 12 (x22 + x22 ), which gives ∂H ∂x F (x) = −x2 and ∂x G(x) = 2 x1 + x2 . The assumptions 2-3 hold on D = R+ , hence by Theorem 3 the control law given by (7) globally (on the positive orthant) stabilizes the set 1 2 2 2 (x1 + x2 ) = C, for any positive C. Furthermore, it is rather easy to show that on the set H(x) = C the √ p 1+16C−1 ? system has a single equilibrium at x1 = , x?2 = 2C − (x?1 )2 which 4 is asymptotically stable with respect to the set H(x) = C. By Theorem 4, this equilibrium is stable for all initial conditions in R2+ . A simulation confirming these results is shown in Figure 1. Note the initial input saturation.
5
0.35 0.3
4.5
0.25 u(x)
4 3.5
0.2 0.15 0.1 0.05
3
0
0
2
4
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10
12
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16
0
2
4
6
8 Time
10
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x
2
2.5 10
2
8
H(x)
1.5 1 H(x) = C 0.5 0
6 4 2
0
0.5
1
1.5 x1
2
2.5
3
0
Fig. 1. To the left, a phase-plot, to the right u(x) and H(x). The simulation had initial condition (2, 4), and the parameters C = 1 and λ = 1.
6 Concluding remarks The concept of set stability is used to demonstrate the stability properties of a certain positive controller, for a special system class. One important restriction of the system class is the assumptions that ensure that the “Lyapunov
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function” used in the proof of the main result is decreasing when the input saturates. According to Bastin and Praly (1999), this condition can be seen as a dissipativity property of the system.
Acknowledgment The NTNU Natural Gas Research Center is acknowledged for financial support.
References 1. Bastin, G. (1999). Issues in modelling and control of mass balance systems, in D. Aeyels, F. Lamnabi-Lagarrigue and A. van der Schaft (eds), Stability and stabilization of Nonlinear Systems, Springer. 2. Bastin, G. and Praly, L. (1999). Feedback stabilisation with positive control of a class of dissipative mass-balance systems, Proceedings of the 14th IFAC World Congress, Beijing, P. R. China. 3. Bastin, G. and Provost, A. (2002). Feedback stabilisation with positive control of dissipative compartmental systems, Proceedings of MTNS. 4. Chabour, R. and Kalitine, B. (2002). Semi-definite Lyapunov functions - stability and stabilizability, IEEE Trans. Aut. Control. To appear. 5. De Leenheer, P. and Aeyels, D. (2002). Stabilization of positive systems with first integrals, Automatica 38(9): 1583–1589. 6. Farsham, C. A., Viswanath, K. P. and Ydstie, B. E. (1998). Process systems and inventory control, AIChE Journal 44(8): 1841–1857. 7. Imsland, L. and Foss, B. A. (2002). A state feedback controller for a class of positive systems: Application to gas lift stabilization, Submitted to European Control Conference, 2003, Cambridge, England. 8. Imsland, L. S. (2002). Topics in Nonlinear Control - Output Feedback Stabilization and Control of Positive Systems, PhD thesis, Norwegian University of Science and Technology, Department of Engineering Cybernetics. 9. Isidori, A. (1995). Nonlinear control systems, third edn, Springer-Verlag, Berlin. 10. Jacquez, J. A. and Simon, C. P. (1993). Qualitative theory of compartmental systems, SIAM Rev. 35(1): 43–79. 11. LaSalle, J. P. (1960). Some extensions of Liapunov’s second method, IRE Trans. CT-7: 520–527. 12. Rouche, N., Habets, P. and Laloy, M. (1977). Stability theory by Liapunov’s direct method, Springer-Verlag, New York. Applied Mathematical Sciences, Vol. 22.
Some Recent Developments in Positive 2D Systems Tadeusz Kaczorek Institute of Control and Industrial Electronics, Warsaw Technical University, Faculty of Electrical Engineering 00-662 Warszawa, Koszykowa 75, Poland, [email protected] Abstract. Necessary and sufficient conditions for the reachability and controllability of positive 2D Roesser model are established. It is shown that the reachability and observability of the positive 2D Roesser model are not invariant under the state-feedbacks. New canonical forms of matrices of singular 2D Roesser model are introduced. Conditions for the existence of a pair of non-singular diagonal matrices transforming the matrices of singular 2D Roesser model to their canonical forms are established and a procedure for computation of the matrices is given.
1 Introduction The most popular models of two-dimensional (2D) linear systems are models introduced by Roesser [21], Fornasini-Marchesini [3, 4] and Kurek [20]. The models have been extended for singular 2D models [9, 6] and singular 2D continuous-discrete models [10]. The positive (non-negative) 2D Roesser type model has been introduced in [7]. The internal stability and asymptotic behavior of 2D positive systems have been considered by Valcher in [22]. An overview of 2-D positive systems theory is given in the monograph [5]. The controllability of 2D linear systems has been investigated in many papers and books [16, 19, 7, 11, 8, 6]. In this paper some recent developments in positive 2D systems theory will be presented, in particular: 1. Reachability and controllability of 2D linear systems described by Roesser model, without feedbacks and with state-feedbacks. 2. Transformation of singular 2D Roesser model to its canonical form.
2 Externally and internally positive Roesser model Let Rn×m be the set of n × m real matrices with non-negative entries and + Rn+ := Rn×1 + . The set of non-negative integers will be denoted by Z+ . L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 345-352, 2003. Springer-Verlag Berlin Heidelberg 2003
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Consider the 2D Roesser model [21,8] ¸ · ¸· h ¸ · ¸ · h xij xi+1,j A11 A12 B1 = + u A21 A22 B2 ij xvij xvi,j+1 ¸ · £ ¤ xhij + Duij , i, j ∈ Z+ := {0, 1, ...} yij = C1 C2 xvij
(1) (2)
where xhij ∈ Rn1 and xvij ∈ Rn2 are the horizontal and vertical state vectors at the point (i, j), respectively, uij ∈ Rm is the input vector, yij ∈ Rp is the output vector and Akl ∈ Rnk ×nl , Bk ∈ Rnk ×m , Ck ∈ Rp×nk , k, l = 1, 2, D ∈ Rp×m . Definition 1 The Roesser model (1), (2) is called externally positive if for zero boundary conditions xh0j = 0, j ∈ Z+ , xvi0 = 0, i ∈ Z+ and all inputs p uij ∈ Rm + , i, j ∈ Z+ we have yij ∈ R+ for i, j ∈ Z+ . Theorem 1 [5] The Roesser model (1), (2) is externally positive if and only if the matrix coefficients of its impulse response gij ∈ Rp×m for i, j ∈ Z+ . + Definition 2 The Roesser model (1), (2) is called internally positive (shortly positive) if for all boundary conditions xh0j ∈ Rn+1 , j ∈ Z+ and xvi0 ∈ Rn+2 , i ∈ Z+ · and all uij ∈
Rm +,
i, j ∈ Z+ we have
i, j ∈ Z+ .
(3)
¸ xhij , n = n1 + n2 and yij ∈ Rp+ for all xvij
Theorem 2 [5] The Roesser model (1), (2) is positive if and only if ¸ · ¸ · B1 A11 A12 p×m , C = [C1 C2 ] ∈ Rp×n ∈ Rn×m , B = ∈ Rn×n A= + + , D ∈ R+ + B2 A21 A22 The transition matrix Tij for (1), (2) is defined as follows In for i = j = 0 Tij = T10 Ti−1,j + T01 Ti,j−1 for i, j ≥ 0 (i + j 6= 0) Tij = 0 for i < 0 or/and j < 0 where
(4)
¸ · ¸ 0 0 A11 A12 , T01 := := 0 0 A21 A22 ·
T10
From (4) it follows that the transition matrix Tij of the positive model (1), (2) is a positive matrix, Tij ∈ Rn×n for all i, j ∈ Z+ . +
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3 Reachability and controllability Definition 3 The positive Roesser model (1), (2) is called reachable for zero boundary conditions (3) (ZBC) at the point (h, k), (h, k ∈ Z+ , h, k > 0), if for every xf ∈ Rn+ there exists a sequence of inputs uij ∈ Rm + for (i, j) ∈ Dhk such that xhk = xf , where ¾ ½ (i, j) ∈ Z+ × Z+ : 0 ≤ i ≤ h, 0 ≤ j ≤ k Dhk := i + j 6= h + k Definition 4 The positive Roesser model (1), (2) is called controllable to zero (shortly controllable) at the point (h, k), (h, k ∈ Z+ , h, k > 0) if for any nonzero boundary conditions xh0j ∈ Rn+1 , 0 ≤ j ≤ k and xvi0 ∈ Rn+2 , 0 ≤ i ≤ h there exists a sequence of inputs uij ∈ Rm + for (i, j) ∈ Dhk such that xhk = 0. Theorem 3 [11, 5] The positive Roesser model (1), (2) is reachable for ZBC at the point (h, k) if and only if there exists a monomial matrix Rn consisting of n linearly independent columns of the reachability matrix £ ¤ Rhk := Mhk Mh−1,k Mh,k−1 · · · M01 M10 where
· Mij := Ti−1,j
· ¸ ¸ 0 B1 + Ti,j−1 B2 0
and Tij is defined by (4). Theorem 4 [11, 5] The positive Roesser model (1), (2) is controllable if and only if the matrix A is nilpotent matrix, i.e. ¸ · In1 z1 − A11 −A12 = z1n1 z2n2 det −A21 In2 z2 − A22 To simplify the notation we assume that m = 1 (the single-input systems) and the matrices A and B of the positive model (1), (2) have the canonical form 0 1 0 ··· 0 0 0 0 · · · 0 0 0 1 ··· 0 .. 0 0 ··· 0 . A11 = . . . . . . . . . . . . . . . , A12 = . . . . . . . . , B1 = 0 0 0 0 ··· 1 1 0 ··· 0 1 a1 a2 a3 · · · an1 (5) b1 1 0 · · · 0 b1 a11 · · · a1n1 b2 0 1 · · · 0 b2 a21 · · · a2n1 , A22 = . . . . . . . . . . . . . . , B2 = A21 = .. .............. . bn2 −1 0 0 · · · 1 an2 1 · · · an2 n1 bn2 bn 0 0 · · · 0 2
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where al ≥ 0, akl ≥ 0, bk ≥ 0 for k = 1, ..., n2 , l = 1, ..., n1 . Consider the Roesser model (1) with the state-feedback · h¸ x uij = vij + K ij , i, j ∈ Z+ (6) xvij ¤ £ where K = K1 K2 , K1 ∈ R1×n1 , K2 ∈ R1×n2 and vij ∈ Rm is a new input vector. Substitution of (6) into (1) yields · h ¸ · h¸ xi+1,j x = Ac ij + Bvij (7) xvi,j+1 xvij where
·
A11 + B1 K1 A12 + B1 K2 Ac = A + BK = A21 + B2 K1 A22 + B2 K2
¸ (8)
The standard closed-loop system (7) is reachable (controllable) if and only if the standard open-loop system (2D Roesser model (1), (2)) is reachable (controllable). It is easy to show that if at least one of al 6= 0, l = 1, ..., n1 or bk 6= 0, k = 1, ..., n2 then the condition of theorem 3 is not satisfied and the positive model (1), (2) is not reachable at the point (n1 , n2 ). Let the positive system (1), (2) with (5) be unreachable at the point (n1 , n2 ). In [12] it was shown that there exists the state-feedback gain matrix £ ¤ K = −a1 −a2 · · · −an −1 0 · · · 0 (9) such the closed-loop system (7) is reachable at the point (n1 , n2 ). Theorem 5 [12] Let the positive system (1), (2) with (5) be unreachable at the point (n1 , n2 ). Then the closed-loop system (7) is reachable at the point (n1 , n2 ) if the state-feedback gain matrix K has the form (9). The reachability of positive Roesser model (1), (2) with (5) is not invariant under the state-feedback (6). According the theorem 4 the positive system is controllable (to zero) if and only if the matrix A is nilpotent. It is said that the state-feedback (6) violetes the nilpotency of A if and only if the closed-loop matrix (8) is not nilpotent. From theorem 4 the following theorem follows. Theorem 6 The closed-loop system (7) is uncontrollable at the point (n1 , n2 ) if the state-feedback (6) violetes the nilpotency of A. The controllability of positive Roesser model (1), (2) is not invariant under the state-feedback (6).
4 Transformation to canonical form Let Rn×m (z1 , z2 ) be the set of n × m rational (proper or improper) matrices in variables z1 and z2 .
Some Recent Developments in Positive 2D Systems
Consider the single-input single-output 2D Roesser model ¸ · h¸ · h xij x + Buij = A E i+1,j xvij xvi,j+1 · h¸ x yij = C ij , i, j ∈ Z+ xvij
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(10) (11)
where xhij ∈ Rn1 and xvij ∈ Rn2 are horizontal and vertical state vectors at the point (i, j), respectively, uij ∈ Rm is the input vector, yij ∈ Rm is the output vector and · ¸ ¸ · ¸ · £ ¤ E11 E12 B1 A11 A12 E= , A= , C = C1 C2 (12) , B= E21 E22 B2 A21 A22 Ekl , Akl ∈ Rnk ×nl , Bk ∈ Rnk , Ck ∈ R1×nk for k, l = 1, 2. It is assumed that detE = 0 and · ¸ E11 z1 − A11 E12 z2 − A12 det 6= 0 for some z1 , z2 ∈ C × C (13) E21 z1 − A21 E22 z2 − A22 where C denotes the field of complex numbers. The transfer matrix of the system (10), (11) is given by ¸−1 · E11 z1 − A11 E12 z2 − A12 B= T (z1 , z2 ) = C E21 z1 − A21 E22 z2 − A22 Pm1 Pm2 m1 −i m2 −j z2 j=0 bij z1 i=0 (14) = Pn1 Pn2 n1 −i n2 −j z2 j=0 −aij z1 i=0 Definition 5 It is said that the matrices (12) have the canonical form if · ¸ ¯12 = 0, E ¯21 = 0, E ¯11 = Im1 0 ∈ R(m1 +1)×(m1 +1) , E ¯22 = I2m E 2 0 0 0 0 ··· 0 ¸ · ........ 0 Im1 (m1 +1)×2m2 ∈ R(m1 +1)×(m1 +1) , A¯12 = A¯11 = 0 0 ··· 0 ∈ R 0 0 1 0 ··· 0 0 0 ··· 0 0 ............................. 0 0 ··· 0 0 an1 0 an1 −1,0 · · · a00 0 2m2 ×(m1 +1) A¯21 = ............................. ∈ R an1 n2 an1 −1,n2 · · · a0n2 0 bm1 1 bm1 −1,1 · · · b11 b01 ............................. bm1 m2 bm1 −1,m2 · · · b1m2 b0m2
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0 0 Im2 −1 0 0 .. 0 0 0 0 ¯1 = ∈ R2m2 ×2m2 , B A¯22 = . ∈ Rm1 +1 0 0 0 Im2 −1 0 0 0 0 0 1 0 ¤ £ .. ¯ B2 = . ∈ R2m2 , C¯1 = bm1 0 bm1 −1,0 · · · b00 ∈ R1×(m1 +1)
(15)
0
C¯2 = [ 0 · · · 0 1 0 · · · 0 ] ∈ R1×2m2 | {z } m2 +1
The problem can be stated as follows. For given matrices (12) establish the conditions under which they can be transformed to their canonical forms (15) and find nonsingular matrices P = diag [P1 , P1 ] , Q = diag [Q1 , Q2 ] , Pk , Qk ∈ Rnn ×nk for k = 1, 2
(16)
such that the matrices ¸ ¸ · ¸ · · ¸ · ¯1 ¯11 0 P1 B1 B E P1 E11 Q1 P1 E12 Q2 ¯ ¯ E= ¯2 = P2 B2 ¯22 = P2 E21 Q1 P2 E22 Q2 , B = B 0 E ¸ · ¸ · £ ¤ £ ¤ A¯11 0 P1 A11 Q1 P1 A12 Q2 ¯ , C¯ = C¯1 C¯2 = C1 Q1 C2 Q2 A= = P2 A21 Q1 P2 A22 Q2 0 A¯22 (17) have the canonical forms (15). Theorem 7 [5] The matrices (12) can be transformed by the nonsingular matrices (16) to their canonical forms (15) only if E12 = 0, E21 = 0, rank E11 = m, rank E22 = 2m2 rank A11 = m1 , rank A12 = 1, rank A22 = 2(m2 − 1), B2 = 0
(18)
If (13) holds then ·
E11 z1 − A11 E12 z2 − A12 E21 z1 − A21 E22 z2 − A22
¸−1 =
∞ X
∞ X
−(i+1) −(j+1) z2
Tij z1
(19)
i=−µ1 j=−µ2
where the pair (µ1 , µ2 ) is the nilpotence index and Tij are the transition matrices defined by ½ ¤ £ ¤ £ In for i = j = 0 E1 0 Ti,j−1 + 0 E2 Ti−1,j − ATi−1,j−1 = (20) 0 i 6= 0 or/and j 6= 0 Let
Some Recent Developments in Positive 2D Systems
·
¯11 z1 − A¯11 E −A¯12 ¯ ¯ −A21 E22 z2 − A¯22
¸−1 =
∞ X
∞ X
−(i+1) −(j+1) z2 T¯ij z1
351
(21)
i=−µ1 j=−µ2
Then from (17), (19) and (21) we have Tij = QT¯ij P for i, j ∈ Z+ Theorem 8 [5] Let the matrices (12) satisfy the assumption (18). Then there exist nonsingular matrices (16) such that the matrices (17) have the canonical forms (5) if rank Rn1 n2 = n and rank On1 n2 = n where £ ¤ Rn1 n2 := Tn1 −1,n2 −1 B, · · · , T00 B, T−1,0 B, T0,−1 B, · · · , T−µ1 ,−µ2 B £ ¤T On1 n2 := CTn1 −1,n2 −1 , · · · , CT00 , CT−1,0 , CT0,−1 , · · · , CT−µ1 ,−µ2 C and T denotes the transpose. The desired matrices are given by £
¯ n−1 , P = O ¯ n−1 On Q = Rn R
(22) ¤
¯ T¯−1,0 B, ¯ · · · , T¯00 B, ¯ · · · , T¯−µ ,−µ B ¯ T¯0,−1 B, ¯ ¯ n n := T¯n1 −1,n2 −1 B, R 1 2 1 2 ¤ £ ¯ n n := C¯ T¯n1 −1,n2 −1 , · · · , C¯ T¯00 , C¯ T¯−1,0 , C¯ T¯0,−1 , · · · , C¯ T¯−µ1 ,−µ2 C¯ T O 1 2 ¯ n (O ¯ n ) are square matrices consisting of n linearly inwhere Rn (On ) and R ¯n n dependent corresponding columns of the matrices Rn1 n2 (On1 n2 ) and R 1 2 ¯ n n ), respectively. Matrices P and Q can be found by the use of the follo(O 1 2 wing procedure. Procedure 1
1. Knowing E, A, B, C find the transfer matrix (14). 2. Using the procedure presented in [8], find the realization in canonical form (15) of the transfer matrix. 3. Using (19) and (21) find the fundamental matrices Tij and T¯ij for i = −µ1 , ..., n1 + 1, j = −µ2 , ..., n2 + 1. ¯ n , On and O ¯n . 4. Find Rn , R 5. Using (22) find the desired matrices P and Q.
5 Concluding remarks It is well-known [8] that the first Fornasini-Marchesini model can be recasted in the 2D Roesser model. Therefore, the considerations can be immediately extended for the positive first Fornasini-Marchesini model. Extensions of the considerations for the positive second Fornasini-Marchesini model [5] and general 2D model [6, 5] are also possible. In the paper [15] the stabilization problem is formulated and solved for singular 2D linear systems described the Fornasini-Marchesini model. In [14] a new concept of the perfect singular observer for singular 2D linear systems has been proposed. Sufficient conditions for the existence of minimal order deadbeat functional observers for singular 2D linear systems described by the general singular 2D model have been established in [13] and a procedure for computation of matrices of the observers has been given.
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References 1. Ga"lkowski K., Elementary operation approach to state-space realizations of 2-D systems, IEEE Trans. Circ. and Syst. Vol. 44, No 2, 1997. 2. Ga"lkowski K., State-space realizations of Linear 2-D Systems with Extensions to the General nD Case, Springer London, 2001. 3. Fornasini E. and Marchesini G., State-space realization theory of two-dimensional filters, IEEE Trans. Autom. Contr. Vol. AC-21, 1976, pp. 484-491. 4. Fornasini E. and Marchesini G., Doubly-indexed dynamical systems: State-space models and structural properties, Math. Syst. Theory, vol. 12, 1978, pp. 59-72. 5. Kaczorek T., Positive 1D and 2D Systems, Springer-Verlag London 2002. 6. Kaczorek T., Linear Control Systems, vol. 2, New York, Wiley, 1992. 7. Kaczorek T., Reachability and controllability of non-negative 2-D Roesser type models, Bull. Acad. Pol. Sci. Ser. Sci. Techn., vol. 44, No 4, 1996, pp. 405-410. 8. Kaczorek T., Two-Dimensional Linear Systems, Springer-Verlag, Berlin 1985. 9. Kaczorek T., Singular general model of 2-D systems and its solution, IEEE Trans. on Autom. Contr., vol. AC-33, No 11, 1988, pp. 1060-1061. 10. Kaczorek T., Singular 2-D continuous-discrete linear systems, Dynamics of continuous, discrete and impulse systems, Advances in Systems Science and Applications, 1995, pp. 103-108. 11. Kaczorek T., Reachability and controllability of positve 2D Roesser type models, 3rd International Conference on Automation of Hybrid Systems, ADPM 98 , Reims - France ,19-20 March 1998, pp. 164-168. 12. Kaczorek T., Reachability and Controllability of 2D Positive Linear Systems with State Feedbacks, Control and Cybernetics, vol. 29. No 1, 2000, pp. 141-151. 13. Kaczorek T., Minimal order deadbeat functional observers for singular 2D linear systems, Control Cybernetics, 2003 (in press) 14. Kaczorek T., Perfect observers for singular 2D linear systems, Bull. Pol. Acad. Techn. Sci., vol. 49, No 1, 2001, pp. 141-147. 15. Kaczorek T. and Nguyen Bang Giang, Stabilisation problem of singular 2-D model by state feedback, 6th International Conference on Methods and Models in Automation and Robotics, 28-31.08.2000, Mi¸edzyzdroje, Poland, pp. 153-158. 16. Klamka J., Complete controllability of singular 2-D system, Proc. 13 IMACS World Congress, Dublin, 1991, pp. 1839-1840. 17. Klamka J., Constrained controllability of 2-D linear systems, Proc. 12 World IMACS Congress, Paris, 1988, vol. 2, pp. 166-169. 18. Klamka J., Controllability of 2-D Linear Systems, Advances in Control Highlights of ECC 99, Springer, 1999, pp. 319-326. 19. Klamka J., Controllability of dynamical systems, Kluwer Academic Publ., Dordrecht, 1991. 20. Kurek J., The general state-space model for a two-dimensional linear digital system, IEEE Trans. Autom. Contr. AC-30, June 1985, pp. 600-602. 21. Roesser R.B., A discrete state space model for linear image processing, IEEE Trans. Autom. Contr. AC-20, 1975, pp. 1-10. 22. Valcher M.E., On the internal stability and asymptotic behavior of 2-D positive systems, IEEE Trans. on Circuits and Systems I, vol. 44, No 7, 1997, pp. 602-613
Nonnegative Infinite Hankel Matrices having a Finite Rank Andrea Morettin Dipartimento di Ingegneria dell’Informazione, Universit` a degli Studi di Padova, via Gradenigo 6/B, 35131 Padova, Italy, [email protected] Abstract. A nonnegative infinite Hankel matrix is an infinite Hankel matrix which has nonnegative entries. The paper studies the relationship between rational functions and such matrices.
1 Preliminaries A nonnegative inputs–outputs system 1 (IO–system) [1, 5, 6] is defined as a linear time invariant system which has nonnegative outputs in consequence of nonnegative inputs. For a discrete system, where inputs and outputs are trajectories defined in the time axis of nonnegative integers (Z+ ) IO–non– negativity is equivalent to require that the impulse responses of the system is nonnegative. This means that if G(z) ∈ Rp×m (z) is the the rational transfer function of a discrete IO–nonnegative system, writing G(z) = P−n z n + P−n+1 z n−1 + · · · + P0 +
P2 P1 + −2 + · · · , z −1 z
we have that Pi ∈ Rp×m , for all i, are matrices such that no element of Pi is negative: Pi are nonnegative matrices. In this paper we restrict our attention on the proper rational transfer functions of discrete IO–nonnegative scalar systems. In order to describe the proper rational transfer function of a scalar system which has a nonnegative impulse response we introduce the infinite Hankel matrices. Let p1 , p2 , p3 , . . . be a sequence of complex numbers. From this sequence we construct the (infinite) Hankel matrix p1 p2 p3 . . . p2 p3 p4 . . . H = p3 p4 p5 . . . . .. .. .. . . . 1
They are also called external positive systems.
L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 353-360, 2003. Springer-Verlag Berlin Heidelberg 2003
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Andrea Morettin
We shall write briefly H = [pt ]+∞ t=1 . By definition, an Hankel matrix has a finite rank equal to r [7, I–§1, and XV–§10] if the largest among the orders of the non–zero minors is equal to r. Next fact holds [7, XV–§10 Theorem 7]. 1.1 Proposition The infinite Hankel matrix H = [pt ]+∞ t=1 is of finite rank r if and only if α0 , α1 , . . . , αr−1 exist such that ph+r =
r−1 X
αk pk+h+1 = 0 ,
h≥0,
(1)
k=0
and r is the least number having this property. We shall now show the connection between the Hankel matrices and the strictly proper rational functions. Let g(z) =
n(z) , d(z)
where d(z) = dr z r + dr−1 z r−1 + · · · + d0 , and n(z) = nr−1 z r−1 + · · · + n0 . Consider the expansion of g(z) in power series g(z) = pz1 + pz22 + pz33 + · · · , then we obtain that dr p1 = nr−1 dr p2 + dr−1 p1 = nr−2 .. (2) . dr pr + dr−1 pr−1 + · · · + d1 p1 = n0 dr pk + dr−1 pk−1 + · · · + d0 pk−r = 0 ,
k = r + 1, r + 2, · · · .
−1 The Hankel matrix H = [pt ]+∞ t=1 has a finite rank. Putting αi = −dr di , (1) is satisfied. Conversely, given a rank r Hankel matrix H = [pt ]+∞ t=1 , i = 0, . . . , r − 1, (1) is satisfied for a suitable r–tuple (α0 , . . . , αr−1 ), then put dr = 1, di = −αi , and from (2) we can obtain ni , i = 0, . . . , r −1. If d(z) = dr z r +dr−1 z r−1 +· · ·+d0 , and n(z) = nr−1 z r−1 + · · · + n0 , being r minimal, n(z)/d(z) is irreducible. Then next proposition is proved.
1.2 Proposition Consider a finite rank Hankel matrix H = [pt ]+∞ t=0 . A rational function g(z) = n(z)d−1 (z) corresponds to H such that if (n(z), d(z)) is a coprime pair, then the rank of H is equal to deg d(z). We define the shift operator as follows. Given an Hankel matrix H = [ps ]+∞ s=1 , we denote with σ t H the Hankel matrix that corresponds to pt+1 , pt+2 , pt+3 , . . . . If the (irreducible) fraction n(z)d−1 (z) corresponds to H = [ps ]+∞ s=1 , then to σ t H corresponds r(z)d−1 (z), where z t n(z) = q(z)d(z) + r(z). Before proceeding we need some definitions and facts. A subset C of a real vector space is a cone if v1 , v2 ∈ C implies that v1 + v2 ∈ C, and w ∈ C implies that αw ∈ C for all nonnegative α. Now suppose C ⊆ Rn .
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C is pointed if v, −v ∈ C implies that v = 0n . C is full if span C = Rn . C is closed if it is a closed set. A pointed, full, and closed cone in Rn is called a proper cone. A cone C is said finitely generated or polyhedral if there exist p1 , . . . , pn+ ∈ C such P that, for any vector v ∈ C, there are αi ≥ 0, i = 1, . . . , n+ so that v = αi pi . If pi , i = 1, . . . , n+ generate C we shall write C = span+ {pi i = 1, . . . , n+ }. Let F ∈ Rn×n , λ1 , . . . , λs , be its eigenvalues, and deg λi , i = 1, . . . , s, be the multiplicity of λi as a root of the minimal polynomial of F . ρ(F ) = maxi=1,...,s |λi | is said the spectral radius of F . F satisfies the Perron Schaefer condition (PS condition) if PS1 ρ(F ) is an eigenvalue of F , and PS2 if |λi | = ρ(F ) then deg λi ≤ deg ρ(F ). It is well known that if F ∈ Rr×r maps a proper cone into itself F satisfies the PS condition, and conversely if F satisfies the PS condition a F –invariant proper cone can be constructed [4, (3.2)Theorem, and (3.5)Theorem].
2 Nonnegative finite rank Hankel matrices A nonnegative infinite scalar Hankel matrix of finite rank H = [pt ]+∞ t=0 is a finite rank Hankel matrix with infinite nonnegative entries. From now on we shall briefly write Hankel matrix, meaning infinite scalar Hankel matrix +∞ with a finite rank. Let H1 = [p1 t ]+∞ t=1 and H2 = [p2 t ]t=1 two nonnegative Hankel matrices, and let g1 (z) and g1 (z) be the corresponding P irreducible fractions. Let’s define the convolution product H1 ∗ H2 as [ τ p1 (t−τ ) p2 t ], where pt = 0 iff t < 0. We have that g1 (z)g2 (z) corresponds to H1 ∗ H2 . The set of nonnegative Hankel matrices is a cone in the vector space of the Hankel matrices. It is also a semigroup with the convolution product. To these algebraic structures there correspond a cone and a semigroup of rational functions where the product is the usual one. Let H = [pt ]+∞ t=1 , a nonnegative Hankel matrix. From Prop. 1.1 there are d0 , . . . , dr−1 , dr = 1, such that pk + dr−1 pk−1 + · · · + d0 pk−r = 0, k = r + 1, r + 2, . . . , where r is equal to the rank of the matrix H. Define the cones n o Ck = span+ col(ph , · · · , ph+r−1 ) h = k, . . . , k + r ⊂ Rr . Ck are polyhedral cones which are also full [7, Corollary XV–S10]. Let 1 .. . F = ∈ Rr×r 1 −d0 −d1 · · · −dr−1
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the companion matrix corresponding to the polynomial d(z) = z rS +dr−1 z r−1 + +∞ · · · + d0 . From (2) we obtain that F Ck ⊂ Ck+1 . Putting C = cl k=1 Ck , we obtain that C is a proper cone by construction. Next fact holds. 2.3 Proposition Let g(z) = n(z)/d(z) be the irreducible fraction corresponding to a finite rank nonnegative Hankel matrix H = [pt ]+∞ t=1 . The distinct roots λi of d(z) satisfy the next conditions: i) ρ = maxi |λi | is a root of d(z). ii) If |λj | = ρ then the multiplicity of λj as a root of d(z) is less or equal the multiplicity of ρ. Proof Without loss of generality we suppose that d(z) is a monic polynomial. Let F ∈ Rr×r the companion square matrix corresponding to d(z). We can construct a proper cone C such that F C ⊆ C. Then, thanks to P F 1 and P F 2, putting ρ(F ) = ρ, the result is proved. 2 Notation We shall say that a polynomial p(z) ∈ C[z] satisfies the PS condition if i) and ii) as in the previous proposition are satisfied. Next example shows as, starting from a polynomial d(z) which satisfies a particular PS condition, it is possible to construct n(z) such that the Hankel matrix corresponding to n(z)/d(z) is nonnegative. 2.4 Example Let d(z) a real polynomial which satisfies the following particular PS condition. i) Let λi , i = 1, . . . s, be the roots of d(z), ρ = maxi |λi | is a root of d(z). ii) Each peripheral root of d(z) is simple, and it is one of the complex k numbers ρej r 2π , k = 1, 2, . . . , r. j r1 2π iii) % , 0 < % < ρ, is not a root of d(z). It is known [18, Theorem 5] that if d(z) satisfies i), ii), iii), it divides a monic polynomial z r+ − βr+ −1 z r+ −1 − · · · − β0 , where rS+ ≥ r, and βi ≥ 0 i = 0. . . . , n+ − 1. Constructed as above the cone +∞ C = cl k=1 Ck , and being pr+ +1 = βr+ −1 pr+ + · · · + β0 p1 , it follows that C is a polyhedral cone (it has r+ generators). In order to obtain that, for a suitable n(z), the Hankel matrix corresponding to n(z)/d(z) is nonnegative, it is enough to choose p1 , p2 , . . . , pr+ −1 such that they are nonnegative real numbers satisfying (2), and construct n(z) by (2). Consider the irreducible fraction g(z) = n(z)/d(z), and suppose that d(z) satisfies the PS condition: d(z) = (z − ρ)m (z − λ2 )m (· · · )(z − λh )m (z − λh+1 )mh+1 (· · · )(z − λs )ms , ρ = |λ2 | = · · · = |λh | ≥ |λh+1 | ≥ · · · , mi = m for 1 ≤ i ≤ h, and mi < m if |λi | = ρ and i > h. We can decompose d(z) in partial fractions:
Nonnegative Infinite Hankel Matrices having a Finite Rank
g(z) =
357
β1 m−1 β2 m βh m β1 m + + ··· + + ··· + + ···+ m m−1 m (z − ρ) (z − ρ) (z − λ2 ) (z − λh )m βi m i βi mi −1 + + + ··· . (z − λi )mi (z − λi )mi −1
The Hankel matrix corresponding to n(z)/d(z) is X Hij , H=
(3)
1≤i≤s 1≤j≤mi −j where Hij = [pij t ]+∞ t=1 is the Hankel matrix that corresponds to βij (z − λi ) . Being X µ t − j ¶ λt−j 1 = , j − 1 zt (z − λi )j t=j
we have that pij t ∼ βij
(t − j)j−1 t−j λ . (j − 1)!
Then [4, theorem 3.2], for large values of t, the dominant summands in pt will be those that correspond to µ ¶ t − mi t−mi , |λi | = ρ and mi = m . λ mi − 1 i Given a finite rank scalar nonnegative Hankel matrix H = [pt ]+∞ t=1 , next proposition offers a condition on the coefficients β1j of a partial fraction decomposition of any irreducible fraction which corresponds to H. To prove it we need a lemma. 2.5 Lemma Let ξ1 , ξ2 , . . . , ξk be real numbers, and take any positive ε. An integer q exists such that qξi , i = 1, . . . , k, differs from an integer by less than ε. Moreover, fixed ε, there is an infinite number of such q. Proof See [10, chapter XI], in particular [11.3, theorem 200 and 201].
2
2.5 Proposition Suppose that the irreducible fraction g(z) = n(z)/d(z) corresponds to the nonnegative Hankel matrix H = [pt ]+∞ t=1 . Decompose g(z) in partial fraction X βi j , g(z) = (z − ρ)mi −j 1≤i≤s 0≤j≤mi −1
and order (see prop. 2.3) the roots λi of d(z) with the respective multiplicity mi in such a way: ρ = λ1 = |λ2 | = · · · = |λh | ≥ . . . |λh+1 | ≥ · · · |λs |, mi = m = m1 , for 1 ≤ i ≤ h, and mi < m if |λi | = ρ and i > h. The following inequality holds β1 m + β2 m + · · · + βh m ≥ 0 (4)
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Andrea Morettin
Proof We can assume that βi m 6= 0, otherwise βi j = 0. Being m the multiplicity of ρ as root of d(z), the sequence 1 1 1 pm , m−1 2 pm+1 , . . . , pk , . . . . ρ 2 ρ (k − m)m−1 ρk−m
(∗)
turns out bounded and not convergent to zero (β1 m 6= 0). From the previous lemma (see also [21, theorem 7.1]) there exists a sequence {tk : tk ∈ Z+ } such that (ejtk θ1 , . . . , ejtk θh ) converges to (1, . . . , 1) as k increases. h
X (m − 1)! βi m ≥ 0 . ptk = m−1 t −m k→+∞ (tk − m) ρk i=1
2
lim
2.6 Remark Suppose that ρ > 0 is the unique peripheral root of d(z), and that to n(z)/d(z) corresponds a nonnegative Hankel matrix. From the previous proposition we obtain that β1m > 0. Next proposition gives an answer to the question: take an irreducible fraction n(z)/d(z) such that d(z) satisfies the PS condition, does a polynomial n(z) exist such that H corresponding to n(z)/d(z) is a nonnegative Hankel matrix? 2.7 Proposition Let (ρ, m) = (λ1 , m1 ), (λ2 , m2 ), . . . , (λs , ms ), (λi , mi ) ∈ C × N, be the pairs of the distinct roots of a real polynomial with mi = deg λi . Suppose that next conditions hold. i) ρ > 0. ii) |λi | ≤ ρ, i = 2, . . . , s, and if ρ = |λi | then mi ≤ m. A real polynomial n(z) can be found such that the Hankel matrix corresponding to the fraction n(z)/d(z) is nonnegative. Proof Put g(z) =
X 1≤i≤n 1≤j≤mi
p1 p2 p3 βi j = + 2 + 3 + ... (z − λi )j z z z
,
take βi h such that n(z) := g(z)d(z) is a real polynomial, and β1 m , . . . , βh m such that h X |βi m | . (5) β1 m > i=2
Consider the sequence (∗). Following the proof of the previous proposition, in view of the fact that for large¡ values of t the dominant summands in pt will ¢ t−m t−mi i , |λi | = ρ and mi = m, there is a λ be those that correspond to m i i −1 suitable large T such that (∗) becomes nonnegative (pk ≥ 0 for all k ≥ T ). In other words, if H is the Hankel matrix corresponding to g(z), a suitable large T exists such that σ T H is nonnegative. 2
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3 Conclusions In this paper we have studied a class of scalar transfer functions under the framework of infinite Hankel matrices with finite rank. A Condition has been found on the location of the poles of the transfer function to guarantee that, for some suitable zeros, it has a nonnegative impulse response.
Acknowledgment The author would like to thank the reviewers for their helpful comments as well as Dr. Gianluigi Pillonetto, University of Padova, for his useful suggestions.
References 1. B.D.O. Anderson, M. Deistler, L. Farina, and L. Benvenuti. Nonnegative Realization of a Linear System with Nonnegative Impulse Response. IEEE Transactions on Circuits and Systems, 43(2). 2. A. Morettin. Modelli dinamici per coni di Traiettorie Laurea thesis. DEI University of Padova, 1999. 3. A. Morettin. Cones of trajectories as subsets of discrete linear systems: the autonomous case. MTNS 2002. 4. A. Bermann and R.G. Plemmons. Nonnegative matrices in the mathematical sciences. Academic Press, 1979. 5. L. Farina. On the existence of a positive realization. System & Control Letters 28:219–226, 1996. 6. L. Farina and L. Benvenuti. Polyhedral reachable set with positive control. Math. Control Signals System 10, 1997. 7. F.R. Gantamacher. The Theory of Matrices. Chelsea Publishing Company NY, 1959. 8. R.A. Horn and C.R. Johnson. Matrix Analysis. Cambridge University Press, 1990. 9. R.A. Horn and C.R. Johnson. Topics in Matrix Analysis. Cambridge University Press, 1999. 10. G.H. Hardy and E.M. Wright. An Introduction to the theory of numbers. Oxford, 1960. 11. D.G. Luenberger. Introduction to Dynamic Systems. Wiley N.Y., 1979. 12. P. Lancaster and M. Tismenetsky. The theory of matrices. Academic Press, 1984. 13. H. Minc. Nonnegative matrices. John Wiley & Sons, 1998. 14. H. Maeda and S. Kodama. Positive Realization of Differential Equations. IEEE Transaction on Circuits and Systems, vol cas. 28(1):39–47, 1981. 15. J.W. Nieuwenhuis. Modern Linear Systems Theory. Linear Algebra and its Applications, 122/123/124:655-680, 1989. 16. Y. Ohta, H. Maeda and S. Kodama. Reachability, Observability, and realizability of continuous–time positive systems. SIAM J. Control and Optimization, 22(2), 1984.
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17. N.J. Pullman. A Geometric Approach to the Theory of Nonnegative Matrices. Linear Algebra and its Applications 4:297–312, 1971. 18. M. Roitman and Z. Rubinstein On linear recursion with nonnegative coefficients. Linear Algebra and its Applications, 167:151-155, 1992. 19. H.H. Schaefer. Topological Vector Spaces. The Macmillan Company, New York, 1966. 20. J. St¨ oer and R. Witzgal. Convexity and Optimization in finite dimensional. Springer Verlag, 1970. 21. B. Tam and H. Schneider. On the core of a cone preserving map. Transaction of the American Mathematical Society, 343(2), 1994. 22. G.M. Ziegler. Lectures on Polytopes. Springer Verlag, 1994.
The Character of an Idempotent-analytic Nonlinear Small Gain Theorem Henry G. Potrykus1 , Frank Allg¨ower2 , and S. Joe Qin1 1 2
Department of Chemical Engineering, The University of Texas at Austin, Austin, TX 78712, USA, [email protected], [email protected] Institut f¨ ur Systemtheorie Technischer Prozesse, Universit¨ at Stuttgart, 70550 Stuttgart, Germany, [email protected]
Abstract. In this paper a general nonlinear input-to-state stability small gain theory is described using idempotent analytic techniques. The theorem is proved within the context of the idempotent semiring K ⊂ End⊕ 0 (R≥0 ), and may be regarded as an application of theoretical computer science techniques to systems and control theory. We show that particular to power law input-to-state gain functions the deduction of the resulting sufficient condition for input-to-state stability may be performed efficiently, using any suitable dynamic programming algorithm. We indicate, through an example, how an analysis of the (weighted, directed) graph of the system complex gives a computable means to delimit (in an easily understood form) robust input-to-state stability bounds.
1 Introduction By many, the input-to-state stability (ISS) paradigm is considered an elegant, comprehensive formalism in which one may describe robust stability concepts. Once the ISS gain and decay functions are known, the comprehension of stability properties of the system is intuitive and quite clear. To our knowledge, however, all computations of ISS gain functions for concrete systems have exploited either low-dimensionality or considerable symmetry in order to deduce the gain function.3 It is of considerable practical interest, so that the ISS concept may be applied to industrial/practical process models, to detail a workable methodology allowing for the ISS characterization of higher state-space dimension systems which may a priori lack a high degree of symmetry. The purpose of the present paper is to describe one way such a characterization may be undertaken. Here we prove an extended small gain theorem applicable to systems without symmetry–systems represented as an interconnected complex of more basic subsystems. 3
Cf. [5] for examples on how easily the ISS paradigm applies to systems with one state dimension.
L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 361-368, 2003. Springer-Verlag Berlin Heidelberg 2003
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The ISS paradigm was introduced by Sontag in [10]. See also [11] for an overview of the concept. In the present work, based on the development in [12], we exploit a formal, or algebraic, similarity between traditional transportation problems, e.g. the Bellman dynamic programming problem, formulated algebraically, and the standard formulation of ISS, as applied to the nonlinear small gain theorem in particular [4]. This similarity allows us to formulate a computationally efficient means of determining sufficient conditions for input/output stability of an interconnected set of i/o systems once the ISS gain functions relating each basic system to the others are given. Since this methodology allows for an efficient way to determine the ISS of interconnected systems algebraically, it may be used, among other applications, to determine parametrically-demarked regions of robust stability. Otherwise stated, we present a systematic theory that allows us to deduce, under a small gain hypothesis, the stability properties of a complicated system given the gain and interconnection data of its constituents. In the standard formulation of ISS [11], a K and KL function are joined (in the mathematical sense, i.e. γ(||d||) ∨ β(||x0 ||, t)) in order to describe the (asymptotic) stability properties of the input/output system: the K function describes the radius of the attractor of practical stability (= γ −1 (||d||)) as a function of the magnitude of the input, d, the KL function describes the rate of attraction (β(||x0 ||, t) → 0 as t → ∞) as a function of the initial condition, x0 , of the state. The formal, algebraic similarity that is exploited here is precisely that the K functions comprise the (continuous) endomorphisms of the (R≥0 , max, 0) monoid; dynamic programming may be adapted to the enn n domorphism ring End⊕ 0 (R≥0 ) over (R≥0 , max, 0) in order to verify the ‘small gain property’ of the system complex. That the Bellman problem, as well as other problems in positive, discrete event systems theory such as Petri net modelling, can be given an effective, clear presentation within the algebraic formalism of idempotent semirings, and more specifically that of the max,+ semiring, is by now well documented; see [1], [6], [2] and references therein. Of special note, Kleene’s Lemma, on the acceptability of regular expressions by finite automata, is elegantly proved in [3] using system-theoretic ideas by means of idempotent analysis. In this paper we work in the other direction: we apply theoretical computer science techniques (idempotent algebra, the discrete Bellman algorithm) to prove and characterize a novel theorem of use in the analysis of complex, networked systems. Thus, this work may be partially viewed as work along the direction of theoretically “. . . integrating control, computer science, communications, and networking,” as called for in [7]. We now proceed with a short explanation of the algebra employed, our basic set-up for the analysis of system complexes, and a development of the theory for “power-law” systems. The major reference for this work is [9], which is a much more complete exposition on the ideas in this paper.
Idempotent-analytic Small Gain Theorem
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2 Idempotent semirings A grammar for nonlinear systems theory An {idempotent} [abelian] monoid is a set A, an operation ⊕ : A×A → A, and a distinguished element 0 ∈ A such that 0⊕a = a = a⊕0, a⊕(b⊕c) = (a⊕b)⊕c, {a ⊕ a = a} [and a ⊕ b = b ⊕ a] for all a, b, c ∈ A. For us, · ⊕ · = · ∨ · = max{·, ·} (on some directed partially ordered set), and, as indicated, we will use ⊕ and ∨ as infix operations. An [idempotent] semiring R is an [idempotent] abelian monoid with another operation, multiplication, · : R×R → R (juxtaposition often replacing the dot: rs = r ·s), such that r(st) = (rs)t, r(s⊕t) = rs⊕rt, and (s⊕t)r = sr ⊕tr for all r, s, t ∈ R. Our idempotent semiring will be End⊕ 0 (A), the endomorphisms of (A, 0, ⊕); maps from A to itself preserving 0, and commuting with ⊕ (= preserving addition): γ ∈ End⊕ 0 (A) ⇔ γ(0) = 0 and γ(a ⊕ b) = γ(a) ⊕ γ(b) End⊕ 0 (A) is a semiring under the additive operation ‘pointwise addition’ and the multiplicative operations ‘functional composition:’ (γ⊕δ)(a) := γ(a)⊕δ(a) and γ · µ := γ ◦ µ (the zero is the zero map which sends any element of A to 0). Of course, (R≥0 , max, 0) is an idempotent monoid, and the order relation on R≥0 induces a partial order on Rn≥0 : x, y ∈ Rn≥0 , x ≤ y ⇔ xi ≤ yi ∀i. x < y means simply x ≤ y, x 6= y. Rn≥0 is also an idempotent monoid once endowed with component-wise operations. Rn≥0 = {x ∈ Rn : x ≥ 0}. We will write x ¿ y ⇔ xi < yi ∀i. Similary for functions: γ ≤ δ ⇔ ∀ a ∈ R≥0 γ(a) ≤ δ(a), γ, δ ∈ K. And, most importantly, E ¿ Γ ⇔ ∀x ∈ Rn≥0 \{0} Ex ¿ Γ x where E, Γ ∈ Kn×n .4 ⊕ We have K ⊂ End⊕ 0 (R≥0 ), the subset of continuous functions in End0 (R≥0 ). ⊕ Analogously, K/0 ⊂ End (R≥0 ). We will refer to either of these two classes of functions as ISS (disturbance) gain functions; when all gain functions are of class K we say that the system is ISS stable, if we only know that they fall in the larger class K/0 we say that the system is merely ISS bounded. K∞ = Aut⊕ 0 (R≥0 ), the set of automorphisms of R≥0 (ρ ∈ K∞ ⇔ ρ(0) = 0, ρ 1-1, onto), will be employed in future work. We note for completeness that a function β : R≥0 × R≥0 → R≥0 is a KL function if it is separately continuous in each variable, a ‘K function in the first variable’ (holding the second argument fixed), and for each a ∈ R≥0 , β(a, t) → 0 as t → ∞. These are the canonical ISS decay functions.
3 Stability of system complexes Figure 1 is the schematic presentation of the basic ISS relation x1 ≤ β(x01 , t) ∨ δ11 (d1 ). In words this relation states that the state signal’s norm, x1 , is boun4
For endomorphism, when employing ‘¿’ we always ignore the trivial fixed point 0.
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Henry G. Potrykus, Frank Allg¨ ower, and S. Joe Qin x01
Σ (1)
d1
δ11
x1
x1
d1
Fig. 1. Representation of an elementary system
ded by the maximum of the inputs signal’s norm d1 , with nonlinear gain coefficient δ11 ∈ K/0 and the ‘β−decay rate’ β(x01 , t) → 0, as t → ∞, β ∈ KL. Our system complexes will be represented by weighted digraphs, each vertex i ∈ V representing an elementary system component, each weighted (by an element of K or K/0 ) arc (ij) 7→ γji representing that xj ≤ γji (xi ) ∨ · · · . Here the xi is an appropriate numerical sup-seminorm of the output of system i. It is important to note that we work with the semi-norms of system output signals and not with the signals themselves. References to appropriate seminorms may be found in [12]. So, we dispense with, and leave ambiguous, the norming of a L∞ (R≥0 ; Rn ), say, signal. That is, x represents both x : R≥0 7→ Rn and ||x||. It is here that we are dealing with positive systems; we consider how seminorms of signals are “transferred” through a system complex via our formalism. As the actual signals are not used in any of the mathematics of the methodology, the convention is sensible. If a vertex i has no immediate predecessor, we give it the trivial arc (ii) 7→ Id, i.e. employ the inequality xi ≤ xi (this is also the way in which all input disturbances, di , are to be represented), we then have that the entire dependence of the output signal norm on the input norms, as well as the output norms themselves, is given by ⊕i xi ≤
µM i
∞ ^ |v|=1
v
¶
(Γ ) x ⊕ (⊕i Gd) ⊕ B(x0 , t) ¡
¢ =: ⊕i Ex ⊕ Gd ⊕ B(x0 , t).
(1)
In Inequality (1) we have used the notation Γ v = Γ v(1) Γ v(2) · · · Γ v(N ) where v is a finite sequence of subsets of vertices, and the meet (=“minimize”) runs over all such finite sequences. Here Γ {i,...,k} is the matrix with rows i, . . . , k being the vector of non-linear gain functions which bound xi , . . . , xk , identities on the diagonal excepting these rows, and zeros elsewhere. The matrix operations are induced from the semiring K. Γ {i,...,k} is to be regarded as the elementary ‘simultaneous step back from vertices i, . . . , k’ matrix— it corresponds to that particular ISS stability dependence query. Note that Γ {1,...,n} = Γ V is the (weighted) adjacency matrix of the system complex graph. G are the “residual gain functions;” the collection of all dependence on
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d gained by “stepping back along the graph” to form E. G is not, a priori, defined independent of the infinitary ‘meet’ operation performed above. It is found, e.g., only once a concrete cut-off, |v| ≤ N has been chosen (this is not the only way to cut-off the meet).5 B(x0 , t) is, similarly, the compound ISS decay function for the system complex. Inequality (1) reads, in words, “to bound ⊕i xi we may take the minimum of costs of stepping back along the arcs, so long as we always include all contributing inputs input to a given elementary system = vertex.” The right action of Kn×n on itself effects the “stepping back” procedure. If E is a contraction (i.e. ⊕i E ¿ ⊕i Id, or to be completely explicit: ¡ ⊕i E(x) < ⊕i x ∀x ∈ Rn≥0 ) then because ∀², a, b ∈ R≥0 , ² < 1 : a ≤ ²a ∨ b =⇒ ¢ a ≤ b , we may conclude that ⊕i xi ≤ ⊕i Gd. We state this formally as our extended nonlinear small gain theorem: Theorem 1. If E is a contraction then ⊕i xi ≤ ⊕i Gd ⊕ B(x0 , t). Note that we have not concerned ourself with B or G. This is due to the algebra involved in our formalism: multiplication distributes over addition, so the contraction property may be checked (first) for E; if it holds B and G may then be computed as another (separate) task.
4 Power-law gain graph-theoretic characterization We say a function γ : R≥0 → R≥0 is a power law function if it is of the form γ(x) = r · xa where r, a ∈ R≥0 , r > 0, a ≥ 0. We term r the coefficient and a the power or exponent of the power law function. Proposition 1. Given a complex system whose system link-up graph is strongly connected and weighted by power law ISS gain functions, and letting Λ be the set of all elementary cycles of the link-up graph, consider the two following tuples: • (λµ )µ∈Λ , the list of evaluated products of the coefficients of the gain functions on each cycle µ, • (lµ )µ∈Λ , the list of evaluated products of the powers of the gain functions on each cycle µ, then we have the following cases: Case 1: ∃ ν ∈ Λ s.t. lν > 1, in which case ISS stability may not, by any means corresponding to this theory, be concluded. Case 2: ∀ µ ∈ Λ lµ < 1, but ∃ ν ∈ Λ s.t. λν > 1, in which case the system complex is ISS bounded. 5
G is well defined in the case “where the theory works,” i.e. when we may apply the theorem as detailed in [9].
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Case 3: ∀ µ ∈ Λ lµ < 1 & λµ ≤ 1, in which case the system complex is ISS bounded. Remark 1. An analysis of the ‘boundary’ cases is straightforward. We do not list these cases to keep the proposition uncluttered: Remark 2. Calculating the resulting B, G requires some work. See [9] for further discussion on this point. Power-Law Gain Example To demonstrate the above result, we submit: S1:
x˙ 1 = −x31 + x23 ∨ d1
S2:
x˙ 2 = −x2 − 3x32 + (1 + x22 )x1 + x22 x3
S3:
x˙ 3 = −x3 + x2
1/4
whose digraph is given in Figure 2. x3
d1
x1
x2
Fig. 2. The 4 vertex graph.
Following Theorem 5.2 and the subsequent examples given in [5], we take a quadratic ISS-Liapunov function for all three elementary systems Si : Vi (xi ) = x2i . Then V˙ 1 = −x41 + (x23 ∨ d1 )x1 = −(1 − θ)x41 − θx41 + (x23 ∨ d1 )x1 ≤ −(1 − θ)x41 , ∀ |x1 | ≥ (|x23 ∨ d1 |/θ)1/3 1/4 V˙ 2 = −x22 − 3x42 + x2 (1 + x22 )x1 + x32 x3 1/4
≤ −x42 , ∀ |x2 | ≥ x1 + x3
⇐ (|x2 | ≥ 2|x1 | ∨ 2|x3 |1/4 ) V˙ 3 = −x23 + x2 x3 = −(1 − τ )x23 − τ x23 + x2 x3 ≤ −(1 − τ )x23 , ∀ |x3 | ≥ |x2 |/τ.
Idempotent-analytic Small Gain Theorem
Thus δ11 (d) = (|d|/θ)1/3 γ21 (x1 ) = 2|x1 | γ32 (x2 ) = |x2 |/τ.
367
√ γ13 (x3 ) = (|x3 |)/ θ)2/3 γ23 (x3 ) = 2|x3 |1/4
Collecting exponents and coefficients we find, for the outer cycle, λout = 2 · (1/τ ) · (1/θ2/3 ) > 1, lout = 1 · 1 · 2/3 < 1. Similarly, for the inner cycle, λin = 2/τ > 1, lin = 1/4·1 < 1. We may immediately apply Proposition 1 (case 2) to conclude ISS boundedness with the disturbance gain’s exponent being 1/3. Thus, starting with the system complex depicted in Figure 2, we have come to the point where Proposition 1 (case 2) may be applied to conclude ISS boundedness. It is not difficult to work out through dynamic programming ¡ ¢1/3 that the ISS bound for the compound system is x1 ⊕ x2 ⊕ x3 ≤ θτ8 3 ⊕ 8|d| . θ Even for so elementary an example, it is not clear to us how we would otherwise deduce such input-output bounds.
5 Conclusions We have developed an extended nonlinear ISS small gain theorem for the determination of ISS gains of an interconnected complex of systems, they themselves having a priori an ISS characterization. The route to this theorem was via an idempotent analytic presentation of the ISS paradigm. This theorem allows one, under hypotheses of a contractive condition, to characterize complex, many state dimension systems according to ISS. Since the analysis is (idempotent) algebraic, and even formally linear, one can see simply the dependence of the resulting ISS gain functions on all parameters of the model. Further, the algebraic nature of the theory interacts harmoniously with the graph-theoretic representation of a system complex; for favorable classes of gain functions a complete, graph-theoretic characterization of the theory is thus possible. The theory lends itself naturally to algorithmization. The main limitation of the theory is its hinging on the small gain criteron. Such small gain criteria have been known to be conservative and we simply cannot completely alleviate this condition. Even given this constraint, we expect our theory to be profitably applicable to a wide variety of interesting dynamical systems. To wit, the theory has been applied successfully to complex process models of chemical engineering systems; The theory yields results that have otherwise not been derived for these systems. We refer to [8] for details on the application of this theory to such systems. Acknowledgement This research leading to this paper was performed while the first author was supported through a DAAD fellowship. He would like to thank C. Ebenbauer of the IST (Stuttgart) for helpful discussions.
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References 1. F. L. Baccelli, G. Cohen, G. J. Olsder, and J.-P. Quadrat. Synchronization and Linearity. Wiley, New York, NY, 1992. 2. J. Cochet-Terrasson and S. Gaubert. Policy iteration algorithm for shortest path problems. Submitted, 2000. http://amadeus.inria.fr/gaubert/papers.html. 3. J. Gunawardena. Idempotency, chapter An Introduction to Idempotency. Cambridge University Press, Cambridge, U.K., 1998. 4. Z.-P. Jiang, A. R. Teel, and L. Praly. Small-gain theorem for ISS systems and applications. Mathematics of Control, Signals, and Systems, 7:95–120, 1994. 5. H. K. Khalil. Nonlinear Systems. Prentice-Hall, Inc., Upper Saddle River, NJ, 2nd edition, 1996. 6. V. N. Kolokoltosov and V. P. Maslov. Idempotent Analysis and Its Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. 7. R. M. Murray, editor. Control in an Information Rich World. Panel on Future Directions in Control, Dynamics, and Systems, 2002. 8. H. G. Potrykus, F. Allg¨ ower, and S. J. Qin. Applications of a small gain theorem to 3 chemical engineering systems. Chem. Eng. Sci., 2002. Submitted. 9. H. G. Potrykus, F. Allg¨ ower, and S. J. Qin. The characterization of an idempotent-analytic nonlinear small gain theorem. IEEE Trans. Auto. Cont., 2002. Submitted. 10. E. D. Sontag. Smooth stabilization implies coprime factorization. IEEE Trans. Auto. Cont., 34(4):435–443, 1989. 11. E. D. Sontag. The ISS philosophy as a unifying framework for stability-like behavior. In A. Isidori et al., editor, Nonlinear Control in the Year 2000, volume 2, pages 443–468. Springer, Heidelberg, 2000. 12. A. R. Teel. On graphs, conic relations, and input-output stability of nonlinear feedback systems. IEEE Trans. Auto. Cont., 41(5):702–709, 1996.
Positive Systems with Nondecreasing Controls. Existence and Well-posedness Stanis#law Walczak and Dariusz Idczak Department of Mathematics, University of L #o ´d´z, Banacha 22, 90-238 L #o ´d´z, Poland, {stawal,idczak}@math.uni.lodz.pl Abstract. In the paper we consider a linear system with nondecreasing controls and a convex cost functional. Basing it on classical methods and the theory of Γ -convergence we prove an existence result and formulate a sufficient condition for well-posedness and necessary conditions for optimality. We also explain the physical meaning of the problem under study.
1 Introduction Consider the control system x˙ (t) = Ax (t) + Bu (t) , x (0) = a0 ∈ Rn+ .
(1)
System (1) is called positive if, and only if, for each positive control u (·) ∈ U, n n u (t) ∈ Rm + , the trajectory of the system is positive; i.e., x (t) ∈ R+ where R+ is the positive cone of Rn and U is the set of admissible controls. Positive systems, discrete and continuous, serve a major role in science, engineering, medicine, economics, and many other fields. Many physical quantities such as pressure, humidity, and molecular mass are positive; similarly, income and dosage are expressed as positive numbers. The theory of positive systems has a short history. Its development has been undertaken mainly by Italian and Polish mathematicians, who have been working on it over the past decade (see [3], [4], [5], [7], [13], [14], [15], [17], and [18]; see also [8] and [9]). In the first part of the paper we examine one-dimensional continuous systems with positive nondecreasing controls. We state necessary and sufficient conditions for the existence of optimal controls and sufficient conditions for the continuous dependence of optimal processes on variable parameters. That is to say, we address ourselves to the question of well-posedness. In the second part we study two-dimensional continuous systems described by hyperbolic equations. These systems are the continuous counterparts of the discrete 2D Fornasini-Marchesini systems (see [6]). L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 369-376, 2003. Springer-Verlag Berlin Heidelberg 2003
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The proof of the stability theorem involves the theory of Γ -convergence. We give the definition and basic properties of Γ -convergent functionals. Definition 1. Let X be a Banach space and Fi (·) : X → R, i = 0, 1, 2, . . . , a sequence of functionals. We write F0 = Γ − lim Fi iff i→+∞
F0 (x) = sup lim inf inf Fi (y) = sup lim sup inf Fi (y) U ∈N (x) i→+∞ y∈U
U ∈N (x) i→+∞ y∈U
where N (x) denotes the set of all neighborhoods of x in the space X (see also [2]). In the space X the following proposition holds true ([2], Prop. 8.1). +∞
Proposition 1. A sequence {Fi (·)}i=1 Γ -converges to F0 (·) iff the following conditions are satisfied: +∞ (a) for every x ∈ X and for every sequence {xi }i=1 converging to x in X F0 (x) ≤ lim inf Fi (xi ) ; i→+∞
+∞
(b) for every x ∈ X there exists a sequence {xi }i=1 converging to x in X such that F0 (x) = lim Fi (xi ) . i→+∞
Further we shall use +∞
Proposition 2. Let X be a Banach space. Assume that {Fi (·)}i=1 Γ - converges to a function F0 (·) in X and that xi is a minimizer of Fi (·) in X, +∞ i = 1, 2, . . . . If x is a cluster point of {xi }i=1 then x is a minimizer of F0 in X and F0 (x) = lim sup Fi (xi ) . i→+∞
2 Formulation of one-dimensional problem and its interpretation Let U be a convex compact nonempty subset of the positive cone ©¡ 1 ¢ ª Rm u , . . . , um ∈ Rm : ui ≥ 0 f or i = 1, . . . , m ; + := the set U is supposed to consist of more than one point. Let U denote the set of all vector functions u : [0, T ] → Rm such that u (t) ∈ U for t ∈ [0, T ] and that each coordinate ui (·) , i = 1, . . . , m, is a nondecreasing function on the interval [0, T ] , T > 0. Let f0 be a real function defined on the set [0, T ] × Rn+ × U. Consider the control system
Positive Systems with Nondecreasing Controls
x˙ (t) = A0 x (t) + B0 u (t) , x (0) = a0 ∈ Rn+
371
(2)
with the cost functional ZT J0 (x, u) :=
f0 (t, x (t) , u (t)) dt + l0 (x (T ))
(3)
0 1, 1
where x (·) ∈ W ([0, T ] , Rn ) ; W 1, 1 ([0, T ] , Rn ) denotes the space of all absolutely continuous vector functions on [0, T ]; u (·) ∈ U ; A0 ∈ Rn×n is a Metzler matrix; B0 ∈ Rn×m ; and l0 : Rn → R is a continuous function. Remark 1. Recall that a matrix A := [aij ]1≤i, j≤n is a Metzler matrix iff its off-diagonal elements are nonnegative. It can be proven that the solution x (·) of system (2) is positive (i.e., x (t) ∈ Rn+ ) provided A0 is a Metzler matrix, n B0 ∈ Rn×m , u (t) ∈ Rm + , and the initial value a0 is in R+ (see [14]). + The optimization problem formulated above bears a simple physical meaning. Consider an axisymmetric object moving in a plane. The object has four engines, which are denoted by e1 , e2 , e3 , and e4 . The first engine can produce a force directed to the right of the symmetry axis of the object; the second one, to the left; the third engine can increase the speed of the object along the symmetry axis; and the fourth one reduces it. Let ui (t) denote the amount of fuel used by the engine ei , i = 1, 2, 3, 4, within the time interval [0, t]. Suppose that the engines are supplied with fuel from a common tank of capacity q and that each engine uses no more fuel than 2q . In this case o n q U := u ∈ R4+ : 0 ≤ ui ≤ for i = 1, 2, 3, 4 and u1 + u2 + u3 + u4 ≤ q . 2 Assume that the state x ∈ R2 of the object at a moment t ∈ [0, T ] is described by the differential system x˙ (t) = Ax (t) + Bu (t) , x (0) = a ∈ R2 where x (t) ∈ R2 , A := [aij ] ∈ R2×2 , B := [bij ] ∈ R2×4 , and u (t) ∈ U. Using admissible controls we want to bring the object from the point a at the moment t = 0 to the point xT at the moment t = T and to minimize the 2 functional in (3) with l0 (x (T )) := k |x (T ) − xT | where k > 0 is a constant. 1 If a12 ≥ 0, a21 ≥ 0, and bij ≥ 0 then x (t) ≥ 0 and x2 (t) ≥ 0 for all t ∈ [0, T ] (see Remark 1). Further we shall prove that, in the set of admissible controls, there exists an optimal control and that the set of optimal processes depends continuously on variable parameters.
3 Existence of optimal controls Consider the optimal control system given by (2) and (3). We prove
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Theorem 1. Suppose that 1◦ A0 ∈ Rn×n is a Metzler matrix, and B0 ∈ Rn×m ; + 2◦ the integrand f0 is measurable in t ∈ [0, T ] for each (x, u) ∈ Rn+ ×U and continuous in (x, u) for a.e. t ∈ [0, T ] ; the function l0 : Rn+ → R is continuous; 3◦ for each ball B (0, ρ) ⊂ Rn there exists a function h (·) ∈ L1 ([0, T ] , R+ ) such that |f0 (t, x, u)| ≤ h(t) for all x ∈ B (0, ρ) and u ∈ U ; 4◦ the input set U ⊂ Rm + is compact. Then there exists at least one positive control u∗ (·) ∈ U and a positive trajectory x∗ (·) such that the process (x∗ , u∗ ) is optimal for the system of (2) and (3). Proof. Let {(xn , un )} be a minimizing sequence for the system of (2) and (3); i.e., un (·) ∈ U, xn is the trajectory of system (2) corresponding to the control un , and lim J0 (xn , un ) = inf J0 (x, u) . Using Helly’s theorem (see [11] n→+∞
u(·)∈U
and [16]), the dominated convergence theorem, and the following Cauchy’s formula xn (t) = e
A0 t
Zt a0 +
eA0 (t−τ ) B0 un (τ ) dτ,
n = 1, 2, . . . ,
(4)
0
it is easy to show that the minimizing sequence is compact and the cost functional (3) is continuous with respect to the topology of pointwise convergence, which implies that there exists an optimal control u∗ (·).
4 Stability of positive systems In this section we prove sufficient conditions for stability. By the stability of an optimal control system we understand the (semi)continuous dependence of optimal processes on the variable parameters of the system. In addition to the system of (2) and (3) consider the system x˙ (t) = A (ε) x (t) + B (ε) u (t) , x (0) = a (ε) ∈ Rn+
(5)
with the cost functional ZT Jε (x, u) :=
fε (t, x (t) , u (t)) dt + lε (x (T ))
(6)
0
where x (·) ∈ W 1, 1 ([0, T ] , Rn ) ; u (·) ∈ U; A (ε) ∈ Rn×n is a Metzler matrix; ; fε : [0, T ] × Rn × Rm → R; lε : Rn → R is a continuous B (ε) ∈ Rn×m + function; and ε ∈ (−ε0 , ε0 ) with ε0 > 0.
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We term the system of (5) and (6) the perturbed system (P Sε ); the system of (2) and (3), the unperturbed system (U S). We put P S0 := U S, which means that the case ε = 0 corresponds to the unperturbed system by assumption. Let Xε denote the set of trajectories of system (5); that is, © ¡ ¢ ª Xε := xu ∈ W 1, 2 [0, T ] , Rn+ ; xu satisfies (5) with some u ∈ U . X0 denotes the set of trajectories of system (2). Let Xε∗ ⊆ Xε and Uε∗ ⊆ U denote the set of optimal trajectories and the set of optimal controls for the system of (5) and (6) whenever ε 6= 0 or for the system of (2) and (3) whenever ε = 0. Put mε := Jε (x∗ , u∗ ) if x∗ ∈ Xε∗ and u∗ ∈ Uε∗ . The value mε will be referred to as the optimal value for the control problem P Sε , ε ∈ (−ε0 , ε0 ) . In this chapter we set forth conditions under which the set of optimal trajectories X0∗ and the set of optimal controls are stable. Assume that (A) the matrix functions A (ε) , B (ε) , and aε are continuous on the interval (−ε0 , ε0 ). Cauchy’s formula (4) and the definition of the set U imply that there exists a ball B (0, ρ) ⊂ Rn such that xu (t) ∈ B (0, ρ) for all u (·) ∈ U, all ε ∈ (−ε0 , ε0 ), and a.e. t ∈ [0, T ] where xu (·) is the solution of system (5) corresponding to u (·) ∈ U . The integrands fε and the functions lε must fulfill the following conditions: (B) fε (t, x, u) → f0 (t, x, u) on [0, T ] × B (0, ρ) × U and lε (x) → l0 (x) on B (0, ρ) uniformly. (C) for each ε ∈ (−ε0 , ε0 ) the function fε is measurable in t ∈ [0, T ] and continuous in (x, u) ∈ B (0, ρ) × U and there exists a function h (·) ∈ L1 ([0, T ] , R+ ) such that |fε (t, x, u)| ≤ h (t) for all ε ∈ (−ε0 , ε0 ) and all (x, u) ∈ B (0, ρ) × U. Let {εk } be a sequence converging to zero. Put Xε∗k := Xk∗ and Uε∗k := Uk∗ . The set of all cluster points of the sequences {xk (·)} where xk (·) ∈ Xk∗ for k = 1, 2, . . . (taken with respect to the topology of uniform convergence) is called the upper limit of the sequence {Xk∗ } and denoted by lim sup Xk∗ or k→+∞
lim sup Xε∗ . ε→0
Similarly, the set of all cluster points of the sequences {uk } where uk (·) ∈ Uk∗ for k = 1, 2, . . . (taken with respect to the topology of pointwise convergence) is termed the upper limit of the sequence {Uk∗ } and denoted by lim sup Uk∗ or lim sup Uε∗ (see [1]). k→+∞
ε→0
We prove the following sufficient condition for stability of optimal processes.
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Theorem 2. If the perturbed system (5), (6) and the unperturbed system (2), (3) satisfy assumptions (A)-(C) then (a) lim sup Xε∗ is nonempty and lim sup Xε∗ ⊆ X0∗ , ε→0
ε→0
(b) lim sup Uε∗ is nonempty and lim sup Uε∗ ⊆ U0∗ , ε→0
(c) lim sup mε = m0 .
ε→0
ε→0
Remark 2. If for each ε ∈ (−ε0 , ε0 ) there exists exactly one optimal control for the system of (5) and (6), that is, Xε∗ = {x∗ε } and Uε∗ = {u∗ε }, then conditions (a) and (b) of Theorem 2 imply that x∗ε → x∗0 uniformly on [0, T ] and u∗ε → u∗0 pointwise on [0, T ] as ε → 0. Proof of Theorem 2. See Step 1 and Step 2, below. Step 1. Let {εk } ⊂ (−ε0 , ε0 ) be a sequence converging to zero. Put fεk =: fk , Jεk =: Jk , lεk =: lk , and aεk =: ak , k = 1, 2, . . . . Consider a sequence of controls {uk } ⊂ U converging pointwise to u0 ∈ U and the sequence of trajectories {xk } of system (5) corresponding to controls {uk } and initial values {ak }. From Cauchy’s formula (4) it follows that xk (t) → x (t) uniformly on [0, T ] where x (·) is the trajectory of system (5) corresponding to control u0 (·) and initial value a0 , which implies that Jk (xk , uk ) → J0 (x0 , u0 ). Step 2. ¡ ¢ Let V = V ([0, T ] , Rm ) denote the space of functions v = Pv 1 , v 2 , . . . , v m m of bounded variation on [0, T ] with the norm kvk = |v (0)|+ i=1 var v i where i i var v denotes the variation of the function v on the interval [0, T ]. The space V is a Banach space. Put ½ Jk (x (v) , v) if v ∈ U, J¯k (v) := +∞ if v ∈ V \ U where x (v) denotes the solution of system (5) with u = v, ε = εk and aεk = ak . It is easy to see that the optimal control problem (5), (6) can be reduced to the minimization of the functional J¯k (·) over the space V . The sequence of functionals J¯k (·) Γ -converges to J¯0 (·) on V . Indeed, let v0 be a point of V and {vk } ⊂ V a sequence converging to v0 . If v0 does not belong to U then, for a sufficiently large k, vk ∈ / U because the set U is compact. Thus lim inf J¯k (vk ) = J¯0 (v0 ) = +∞. Suppose that v0 ∈ U and {vk } ⊂ V is a sequence converging to v0 . Two cases can occur: A. There exists a finite number of vk such that vk ∈ U, which means that lim inf J¯k (vk ) = +∞ > J¯0 (v0 ). B. There exists a subsequence {vki } such that vki ∈ U for i = 1, 2, . . . . Then, by Step 1, lim inf J¯k (vk ) = J¯0 (v0 ). The set U is convex and has more than one point. Thus, for each v0 ∈ U there exists a sequence {vk } ⊂ U, vk 6= v0 , vk → ªv0 such that lim J¯k (vk ) = J¯0 (v0 ), which means that the sequence © J¯k (·) Γ -converges to J¯0 (·) in the space V . Applying Proposition 2 ends the proof.
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5 Two-dimensional continuous systems © ª Let g : P 2 → R where P 2 := (x, y) ∈ R2 ; 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 . Let Fg denote a function of intervals defined by the formula ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ Fg (Q) = g x2 , y 2 − g x1 , y 2 + g x1 , y 1 − g x2 , y 1 £ ¤ £ ¤ where Q = x1 , x2 × y 1 , y 2 ⊆ P 2 . The function Fg is called the function of intervals associated with the function g (see [16]). We say that the function Fg is nonnegative iff Fg (Q) ≥ 0 for each Q ⊆ P 2 . The function g : P 2 → R is called a nondecreasing function on P 2 iff the associated function Fg is nonnegative and g (·, y), g (x, ·) do not decrease on the interval [0, 1] (see [10] and [11]). If the function Fg is an absolutely continuous function of intervals (see [16]) and g (x, ·), g (·, y) are absolutely continuous functions of one ¡variable¢ then g is termed an absolutely continuous function on P 2 . Let AC P 2 , R denote the space of all absolutely continuous functions on P 2 . Consider the optimal control problem 0 zxy (x, y) = A0 (ε) z (x, y) + A1 (ε) zx0 (x, y) + A2 (ε) zy0 (x, y) + B (ε) u (x, y) , (7) with the boundary conditions
z (x, 0) = ϕε (x) ,
z (0, y) = ψε (y)
and with the cost functional Z Jε (z, u) = fε (x, y, z (x, y) , u (x, y)) dxdy
(8)
(9)
P2 2 where u ∈ U, U is a set of all nonnegative ¡ 2 ¢ nondecreasing functions on P such m that u (x, y) ∈ U ⊂R , z ∈ AC P , R , ϕε and ψε are absolutely continuous on [0, 1] , ϕε (0) = ψε (0), Ai (ε) ∈ Rn×n , B (ε) ∈ Rn×n , fε : P 2 ×Rn ×Rm → R, and ε ∈ (−ε0 , ε0 ). For the two-dimensional control problem with perturbations described by (7)-(9), we can obtain results analogous to those stated in Theorem 1 and Theorem 2.
Acknowledgement. The preparation of this paper was supported by the State Committee for Scientific Research under Grant 7T11A00421 and by the University of L #o ´d´z under Grant 505/714.
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References 1. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkh¨ auser, Boston (1990). 2. Dal Maso, G.: An Introduction to Γ -Convergence. Birkh¨ auser, Boston (1993). 3. Fanti, M.P., Maione, B., Turchiano, B.: Controllability of linear single-input positive discrete-time systems. Internat. J. Control, 50, 2523–2542 (1989). 4. Farina, L.: On the existence of a positive realization. Systems Control Lett., 28, 219–226 (1996). 5. Farina, L., Benvenuti, L.: Positive realizations of linear systems. Systems Control Lett., 26, 1–9 (1995). 6. Fornasini, E., Marchesini, G.: Doubly-indexed dynamical systems: state-space models and structural properties. Math. Systems Theory, 12, 59–72 (1978/79). 7. Fornasini, E., Valcher, M.E.: On the spectral and combinatorial structure of 2D positive systems. Linear Algebra Appl., 245, 223–258 (1996). 8. Hof, J.M. van den: Realization of positive linear systems. Linear Algebra Appl., 256, 287–308 (1997). 9. Hof, J.M. van den, Schuppen, J.H. van: Realization of positive linear systems using polyhedral cones. In: Proceedings of the 33rd IEEE Conference on Decision and Control, Lake Buena Vista (1994). 10. Idczak, D.: Distributional derivatives of functions of two variables of finite variation and their application to an impulsive hyperbolic equation. Czechoslovak Math. J., 48 (123), 145–171 (1998). 11. Idczak, D., Walczak, S.: On Helly’s theorem for functions of several variables and its applications to variational problems. Optimization, 30, 331–343 (1994). 12. Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North-Holland, Amsterdam New York (1979). 13. Kaczorek, T.: Positive realization in canonical form of the 2D Roesser type model. In: Proceedings of the 36th IEEE Conference on Decision and Control, San Diego (1997). 14. Kaczorek, T.: Realisation problem for 2-D positive systems. In: ICSE’97, Coventry (1997). 15. Kaczorek, T.: Realisation problem for positive 2-D Roesser type model. Bull. Polish Acad. Sci. Tech. Sci., 45, 607–619 (1997). 16. L # ojasiewicz, S.: An Introduction to the Theory of Real Functions. Wiley, Chichester (1988). 17. Valcher, M.E.: On the internal stability and asymptotic behavior of 2-D positive systems. IEEE Trans. Circuits Systems I Fund. Theory Appl., 44, 602–613 (1997). 18. Valcher, M.E., Fornasini, E.: State models and asymptotic behaviour of 2D positive systems. IMA J. Math. Control Inform., 12, 17–36 (1995). 19. Walczak, S.: Absolutely continuous functions of several variables and their application to differential equations. Bull. Polish Acad. Sci. Math., 35, 733– 744 (1987).
Reachability and Controllability of Positive Linear Discrete-time Systems with Time-delays Guangming Xie and Long Wang Center for Systems and Control, Dept. of Mechanics and Eng. Sci., Peking Univ., Beijing, 100871, China, {xiegming,longwang}@mech.pku.edu.cn?? Abstract. Many practical systems in engineering, management science, economics, social sciences, compartmental analysis in biology and medicine, can be modelled as positive systems in which the state trajectory is always positive whenever the initial state is positive. This paper first introduces the time-delay phenomenon into positive linear discrete-time systems and studies reachability and controllability of such systems. Necessary and sufficient criteria for reachability and controllability are established, respectively. It is also shown that for time-delay positive systems, reachability and null controllability together imply complete controllability.
1 Introduction Positive linear systems(PLSs) are such a class of systems whose states and inputs are nonnegative. Many practical systems in engineering, management science, economics, social sciences, compartmental analysis in biology and medicine, can be modelled as positive systems [1],[2],[3]. Positive systems are defined on cones, not on linear spaces. Consequently, many well known properties of linear systems can not be applied to positive systems. There are a lot of works on structural properties of positive linear systems(see [4], [5], [6], [7] and the references there in). On reachability of positive linear discrete-time systems, [8] gave a conjecture that the system x(t + 1) = Ax(t) + Bu(t) is reachable if and only if the n-step reachability matrix [B, AB, . . . , An−1 B] contains an n × n monomial submatrix, and gave an algebraic proof with the assumption that the system matrix A is a diagonal matrix. Next, [9] proved this conjecture by a graphtheoretic approach. Then, [10] gave an algebraic proof. [11] presented another ??
Supported by National Natural Science Foundation of China (No. 69925307 and No. 60274001), National Key Basic Research Special Fund (2002CB312200), National Hi-Tech R&D 863 Project (No. 2002AA755002) and China Postdoctoral Program Foundation.
L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 377-384, 2003. Springer-Verlag Berlin Heidelberg 2003
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condition which is equivalent to the above one. On reachability of positive linear continuous-time systems, only sufficient condition is available [12]. Time-delay phenomena are very common in practical systems, for instance, in economic, biological and physiological systems. Studying the time-delay phenomena in control systems has become an important topic in control theory, there are many work on controllability and reachability of general linear discrete-time systems with time-delays [13] [14] [15]. Surprisingly, there has been relatively little work on the study of PLSs with time-delays. Motivated by this fact, in this paper, we first introduce time-delay phenomena into the positive system model and investigate the reachability and controllability of positive linear discrete-time systems with time-delays. Before proceeding further, we first introduce some preliminaries. Denote Z+ the non-negative integer set. Denote
2 Delay in input In this section, we first consider the input time-delay case, the system model is described as follows: x(t + 1) = Ax(t) + B0 u(t) + B1 u(t − 1), where x(t) ∈
t ∈ Z+
is the input, A ∈
(1)
Definition 1 (Reachability). System (1) is said to be reachable (from zero initial condition, i.e. zero initial state and zero initial input), if for any nonzero terminal state xf ∈
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Denote if t = 1; [B0 ], if t = 2; Rt = [B1 + AB0 , B0 ], t−2 [A (B1 + AB0 ), At−3 (B1 + AB0 ), . . . , B1 + AB0 , B0 ], if t > 2. (2) Rt is called the reachability matrix at time t. It is easy to see that the general solution of system (1) is given by x(t) = At x(0) + At−1 B1 u(−1) + Rt [u(0), . . . , u(t − 1)]0
(3)
Theorem 1. The system (1) is (i) reachable if and only if for some t ≥ n, the reachability matrix Rt contains an n × n monomial submatrix, i.e., n linearly independent monomial columns; (ii) null controllable if and only if A is a nilpotent matrix; (iii) controllable if and only if it is both reachable and null controllable. Proof. We only prove (i), since (ii) and (iii) are obvious. (Sufficiency of (i)) It is obvious. (Necessity of (i)) If the system is reachable, it means there exists t such that the reachable set Rt = {x|x = Rt [u(0), . . . , u(t − 1)]0 , u(i) ≥ 0, i = 0, . . . , t − 1} =
K X
Bk u(t − k),
t ∈ Z+
(4)
k=0
are , B0 , . . . , BK ∈
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[B0 ], if t = 1; [B1 + AB0 , B0 ], if t = 2; t−1 [Bt−1 + ABt−2 + . . . + A B0 , . . . , B1 + AB0 , B0 ], if 2 < t ≤ K + 1; K Rt = [A P Aj B if t = K + 2; K−j , RK+1 ], j=0 K K P P [At−K−1 Aj BK−j , . . . , A Aj BK−j , RK+1 ], if t > K + 2. j=0
j=0
(6)
It is easy to see that Theorem 1 is still filled for system (4).
3 Delay in state In this section, we investigate the reachability and controllability of PLS with delay in state x(t + 1) = A0 x(t) + A1 x(t − 1) + Bu(t),
t ∈ Z+
(7)
where x(t) ∈
as ⊆ 2.
(8)
Lemma 1. The general solution of the system (7) is given by x(t) = G0t x(0) + G1t x(−1) + Rt [u(0), . . . , u(t − 1)]0
(9)
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where the reachability matrix is given by Rt = [G0t−1 B, . . . , G01 B, G00 B]
(10)
Proof. See Appendix A. Theorem 2. The system (7) is (i) reachable if and only if for some t ≥ n, the reachability matrix Rt defined by (10) contains an n × n monomial submatrix; (ii) null controllable if and only if the matrix ¸ · A0 A1 (11) Π= I 0 is nilpotent; (iii) controllable if and only if it is both reachable and null controllable. Proof. Since the proof of (i) is similar to that of Theorem 1 and (iii) is obvious, we only prove (ii). In fact, it is easy to see that · 0 · 0 ¸ ¸ Gt−1 G1t−1 Gt G1t = Π , t = 1, 2, . . . . G0t−1 G1t−1 G0t−2 G1t−2 Then we have · 0 ¸ ¸ · Gt G1t t−1 A0 A1 = Π t, =Π G0t−1 G1t−1 I 0
t = 1, 2, . . . .
Consider the system (7) with zero inputs, we have that x(t) = [G0t , G1t ][x(0), x(−1)]0 (Sufficiency of (ii)) Since Π is nilpotent, there exists t such that Π t = 0, this means that G0t = G1t = 0. Thus, the system is null controllable. (Necessity of (ii)) Since the system is null controllable, there must exist t ¸ · 0 0 such that G0t = G1t = 0. It follows that Π t = . Then, on one hand, we ∗∗ ¸ · ∗∗ . On the other hand, we have have Π t+1 = 00 · ¸ · ¸ · ¸ · ¸ A0 A1 00 A0 A1 00 Π t+1 = Π t × = × = I 0 ∗∗ I 0 ∗∗ · ¸ 00 Thus, we get Π t+1 = . This implies that Π is nilpotent. 00 Remark 3. If A1 = 0, then the system (7) is reduced to the general PLS without time-delay. It is easy to see that, in this case, Theorem 2 is reduced to the existent results(Theorem 1 in [7]).
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Remark 4. Similar to the input delay case, Theorem 2 can be extended to the case of PLS with multiple state time-delays described as follows: x(t + 1) =
K X
Ak x(t − k) + Bu(t),
t ∈ Z+
(12)
k=0
are where x(t), u(t) are defined as before, A0 , . . . , AK ∈ K +1. Ai GK k−1−i ] i=0
i=0
(13)
Lemma 2. The general solution of the system (12) is given by x(t) =
K X
Gkt x(−k) + Rt [u(0), . . . , u(t − 1)]0
(14)
k=0
where the reachability matrix is given by Rt = [G0t−1 B, . . . , G01 B, G00 B]
(15)
Proof. Omitted since it is analogous to the proof of Lemma 1. Theorem 3. The system (12) is (i) reachable if and only if for some t ≥ n, the reachability matrix Rt defined by (15) contains an n × n monomial submatrix; (ii) null controllable if and only if the matrix A0 A1 . . . AK−1 AK I 0 ... 0 0 0 I ... 0 0 Π= (16) .. .. . . .. .. . . . . . 0 0 ... I 0 is nilpotent; (iii) controllable if and only if it is both reachable and null controllable. Proof. Omitted since it is analogous to the proof of Theorem 2. Remark 5. Necessary and Sufficient Conditions for reachability and controllability of positive discrete-time linear systems with delays in both states and controls can be established by simply combining Theorem 1-3.
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4 Conclusion In this paper, we have investigated reachability and controllability of positive discrete-time linear systems with time-delays. Necessary and sufficient conditions have been established for positive discrete-time linear systems with state delay or control delay, respectively.
Acknowledgements The authors are very grateful to the anonymous reviewers for their helpful and valuable comments and suggestions for improving this paper.
Appendix A Proof (Proof of Lemma 1). Proceeding by mathematical induction. First, for s = 1, we have that x(1) = A0 x(0) + A1 x(−1) + Bu(0). It is obvious that (9) and (10) hold. Secondly, suppose that (9) and (10) hold for s ≤ t−1, i.e., x(s) = G0s x(0)+ 1 Gs x(−1) + Rs [u(0), . . . , u(s − 1)]0 and Rs = [G0s−1 B, . . . , G01 B, G00 B]. We prove they hold for s = t. In fact, x(t) = A0 x(t − 1) + A1 x(t − 2) + Bu(t − 1) = A0 (G0t−1 x(0) + G1t−1 x(−1) + Rt−1 [u(0), . . . , u(t − 2)]0 ) +A1 (G0t−2 x(0) + G1t−2 x(−1) + Rt−2 [u(0), . . . , u(t − 3)]0 ) + Bu(t − 1) = (A0 G0t−1 + A1 G0t−2 )x(0) + (A0 G1t−1 + A1 G1t−2 )x(−1) +A0 Rt−1 [u(0), . . . , u(t − 2)]0 + A1 Rt−2 [u(0), . . . , u(t − 3)]0 + Bu(t − 1) = G0t x(0) + G1t x(−1) +[A0 G0t−2 B + A1 G0t−3 B, . . . , A0 B, B][u(0), . . . , u(t − 1)]0 = G0t x(0) + G1t x(−1) +[G0t−1 B, . . . , G01 B, G00 B][u(0), . . . , u(t − 1)]0 Thus, (9) and (10) hold for s = t. Hence, (9) and (10) hold for any t.
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References 1. R. Bru, J. M. Carrasco and L. Costa. Unsteady state fugacity model by a dynamic control systems, Applied Mathematical Modelling. Vol.22, 1998, p665-670. 2. N. G. Kalaitzandonakes and J. S. Shonkwiler. A state-space approach to perennial crop supply analysis, Amer. J. Agr. Econ. 2. Vol. 74, 1989, p343-352. 3. L. Caccettaand V. G. Rumchev. A Survey of Reachability and Controllability for Positive Linear Systems, Annals of Operations REsearch. Vol. 98, 2000, p101122. 4. L. Farina and S. Rinaldi. Positive Linear Systems: Theory and Applications. John Wiley and Sons Inc., New York, 2000. 5. D. G. Luenberger. Introduction to Dynamical Systems: Theory, Models and Applications. Jonh Wiley and Sons Inc., New York, 1979. 6. A. Berman, M. Neumann and R. J. Stern. Nonnegative Matrices in Dynamic Systems. Wiely Interscience, 1989. 7. R. Bru. L. Cassetta, et. al. Recent Developments in Reachability and Controllability of Positive Linear Systems. Proceedings of the 15th Triennial World Congress. Barcelona, Spain, 2002. 8. P. G. Coxson and H. Shapiro. Positive Input Reachability and Controllability of Positive Systems. Linear Algebra Appls. Vol.94, 1987, p35-53. 9. P. G. Coxson and L. C. Larson and H. Schneider Shapiro. Monomial patterns in the sequence Ak b. Linear Algebra Appls. Vol.94, 1987, p89-101. 10. V. G. Rumchev. On Reachability of Positive Linear Discrete-Time Systems with Scalar Controls. Proceedings of the 39th IEEE Conference on Decision and Control. Sydney, Australia, 2000, p3159-3162. 11. M. Fanti, B. Maione and B. Turchiano. Controllability of linear single-input positive discrete-time systems, Internation Journal of Control. Vol.50, 1989, p2523-2542. 12. T. Kaczorek. Reachability and Minimum Energy Control of Positive Continuoustime Linear Systems. Proceedings of 6th International Conference on Methods and Models in Automation and Robotics. Miedzyzdroje, Poland, 2000, p177-181. 13. J. Klamka. Relative and absolute controllability of discrete systems with delay in control, Int. J. Control, v 26, n 1, 1977, p65-74. 14. K. Watanabe, Further study of spectral controllability of systems with multiple commensurate delays in state variables, Int. J. Control, V 39, n 3, 1984, p497505. 15. V. N. Phat, Controllability of discrete-time systems with multiple delays on controls and states”, Int. J. Control, v 49, n 5, 1989, p1645-1654.
Countercurrent Double-pipe Heat Exchangers are a Special Type of Positive Systems Arturo Zavala-R´ıo1 , Ricardo Femat1 , and Ricardo Romero-M´endez2 1
2
Instituto Potosino de Investigaci´ on Cient´ıfica y Tecnol´ ogica, Av. V. Carranza 2425-A, Lomas 78210, San Luis Potos´ı, S.L.P., Mexico, {azavala,rfemat}@ipicyt.edu.mx Universidad Aut´ onoma de San Luis Potos´ı, Av. Dr. Manuel Nava 8, 78290 San Luis Potos´ı, S.L.P., Mexico, [email protected]
Abstract. Countercurrent liquid-liquid double-pipe heat exchangers prove to be a special type of positive systems: namely, they are compartmental and cooperative irreducible. Each of these characterizations define particular dynamical properties of the system solutions. This work brings to the fore such characterizations and dynamical properties.
1 Introduction Heat exchangers are frequently used in industrial processes. Nevertheless, their dynamical models have not yet been completely characterized under a formal analytical framework. Some works have used empirical correlations to describe the heat transfer phenomenon [7]; others have presented analyses in the frequency domain [3]. In [19], a dynamical model of countercurrent double-pipe liquid-liquid heat exchangers has been treated under a dynamical systems setting. Some analytical and stability properties of such systems have been formally demonstrated. In this work, we emphasize on the fact that such type of heat exchangers are an irreducible cooperative and compartmental type of positive systems. Under such a framework, some results gotten in [19] are corroborated. The mathematical models of heat exchangers are constructed from an energy conservation principle [7]. That taken in the present study was obtained by considering the total volume of the heat exchanger (see Fig. 1). Such a model is well accepted in literature and, even though its dynamical properties have not yet been proved, it has been used to design temperature controllers [12]. The following assumptions are considered: A.1 The heat exchange takes place between two liquids which are flowing in contrary sense (countercurrent flow). A.2 There is no heat exchange between the system and its surroundings. L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 385-392, 2003. Springer-Verlag Berlin Heidelberg 2003
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Fig. 1. Sketch of the heat exchanger under countercurrent flow configuration.
A.3 The system parameters (physical properties) are time-invariant. A.4 The flow rates and masses are constant, finite, and positive. Such assumptions are not restrictive as thoroughly supported in [19]. Moreover, most of them are frequently made in control system frameworks [4] and modeling/estimation settings [5]. The text is organized as follows: in section 2, positive systems are characterized. Section 3 presents the system state-space model, proves some of its analytical properties, and outlines its positive, compartmental, and cooperative irreducible characters. Concluding remarks are given in section 4.
2 Positive systems Positive dynamical systems are those whose state variables remain nonnegative [16]. Control systems are included in such a classification whenever nonnegativity of their states holds for any nonnegative input sequence [2]. This type of systems are very frequently found in a variety of applied areas such as biology, chemistry, ecology, economics, and sociology among many others (see [1, 16] and references therein). n-dimensional dynamical systems Tn leaving
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© ª Tl © ª system state space as W = x ∈ 0 = i=1 x ∈ 0 , for some continuously differentiable h : 0 i and ∂h ∂x f (x) > 0 for all (i, x) ∈ [1 . . . l] × ∂W such that hi (x) = 0. i Imposing ∂h ∂x f (x) ≤ 0 for all (i, x) ∈ [1 . . . l] × ∂W such that hi (x) = 0 states forward invariance of W with respect to (1): such a condition implies that, at every point on ∂W , f points towards a nondecreasing direction of every vanished hi ; in other words, the vector field f aims inwards W at any point on its boundary. Consequently, ∂W is unreachable by any orbit of (1) with initial conditions in W , concluding that x0 ∈ W ⇒ x(t; x0 ) ∈ W , ∀t ∈ Ix0 , where x(t; x0 ) denotes the forward solution of (1) with initial condition x0 at t = 0 and Ix0 stands for the maximal forward interval of existence of x(t; x0 ). Since x0 can only be taken in W ⊂
3 The system dynamics Since, according to Assumption A.2, the heat transfer is carried out only between fluids (no heat dissipation takes place), the heat exchanger dynamical model can be obtained from the following energy balance equation [7] applied to each of both fluids time heat leaving entering heat flow heat flow transfer rate ± (2) between = of heat from − from exchange fluids outlet fluid inlet fluid where the positive sign corresponds to the energy balance in the cold liquid and means absorbed heat whereas the negative sign concerns the hot fluid and stands for the heat removed. Thus, considering the total volume inside the tubes (see Fig. 1), a simplified but realistic state-space dynamical model of a liquid-liquid double-pipe heat exchanger is given by [12]
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2 [Fc (Tci − x1 ) + γc ∆T (x)] Mc 2 x˙ 2 = [Fp (Tpi − x2 ) − γp ∆T (x)] Mp x˙ 1 =
(3)
where the state variables, x1 and x2 , stand for the output temperature of the coolant and hot fluids, respectively; Tci and Tpi are, respectively, the coolant and hot fluid input temperatures, considered to be constant values satisfying: 0 < Tci < Tpi ; x := (x1 , x2 )T ; Mc , Mp , Fc , and Fp are, respectively, the cold and hot masses and flow rates (see Assumption A.4); ∆T (x) stands for the mean temperature difference which is the nonlinear term of the system (its modeling and analytical properties are treated in subsection 3.1); and γc and γp are positive constant parameters (see Assumption A.3). These are actually Ae and defined in terms of the system physical properties as follows: γc := UCep,c
Ai γp := UCip,p , where Ue and Ui denote, respectively, the overall heat transfer coefficient (OHTC) for the external and internal tubes; Cp,c and Cp,p are, respectively, the heat capacitance of the cold and hot fluids; and Ae and Ai denote the external and internal heat transfer surface area (HTSA), respectively. As stated in Assumption A.3, all these parameters are considered to be constant, and are by nature positive [7]. More detailed information about the nature (and calculation) of each of these physical properties (as well as other complementary information on heat exchangers) can be found, for instance, in [7, ©14]. Furthermore, a physically reasonable state-space domain for (3) is ª D := x ∈ <2 | Tci < xj < Tpi , j = 1, 2 (this is supported in subsection 3.2).
3.1 The mean temperature difference model From a general point of view, (3) is suitable for either countercurrent or parallel flow heat exchangers. It is the mean temperature difference, ∆T (x), that defines the configuration type that (3) refers to [7]. This term is approached from the temperature differences in the inlet and outlet position of the heat exchanger (see Fig. 1). For countercurrent heat exchangers (recall Assumption A.1), the mean temperature difference is commonly modeled as an arithmetical relation of x: ∆T (x) =
(x2 − Tci ) + (Tpi − x1 ) := ∆Ta (x) 2
(4)
(x2 − Tci ) − (Tpi − x1 ) ³ ´ := ∆Tl (x) 2 −Tci ln Txpi −x1
(5)
or in its logarithmic form: ∆T (x) =
[7]. Model (4) is simplistic, while (5) is not only the appropriate average of the temperature difference over the tube length [7] but also what permits to approach (closely enough to reality) the output temperature dynamics of
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heat exchangers through a set of ordinary differential equations [18] as in (3).3 However, existence, continuity, and differentiability of ∆Tl (x) on S := {x ∈ D | x1 + x2 = Tpi + Tci } are not well-defined. In [19], it is shown that lim ∆Tl (x) = ∆Ta (x? ); this fact leads to the complementation of (5) ? x→x ∈S
with ∆Ta (x) on S, resulting in a well-defined continuously differentiable mean temperature difference logarithmic model on D: Lemma 1. The mean temperature difference complemented logarithmic model ( ∆Tl (x) ∀x ∈ D \ S ∆T (x) = (6) ∆Ta (x) ∀x ∈ S (see (5) and (4)) is continuously differentiable and positive on D. The proof is given in [19].4 Remark 1. In [19], it is proved © ª that the extension of ∆T (x) in (6) to Ω := x ∈ <2 | x1 ≤ Tpi , x2 ≥ Tci ⊃ D keeps its continuous differentiability and positivity properties in the interior of Ω and vanishes on ∂Ω. Remark 2. Let us denote the system dynamics in (3) as x˙ = f (x), with ! Ã ! Ã 2 f1 (x) Mc [Fc (Tci − x1 ) + γc ∆T (x)] f (x) = = 2 f2 (x) Mp [Fp (Tpi − x2 ) − γp ∆T (x)]
(7)
Observe that, as a direct consequence of Lemma 1, f (x) is continuously differentiable on D. An additional analytical feature of ∆T in (6), that will prove to be useful, is outlined next. Lemma 2. The complemented logarithmic mean temperature difference, (6), is strictly decreasing in x1 for any fixed value of x2 , and strictly increasing in ∂∆T x2 for any fixed value of x1 , i.e. ∂∆T ∂x1 < 0 and ∂x2 > 0, ∀x ∈ D. The proof is given in [19]; see footnote 4. Remark 3. Let us consider the Jacobian matrix of f (x) in (7), ! Ã ∂f1 ∂f1 ! Ã 2Fc 2γc ∂∆T c ∂∆T − Mc + 2γ ∂f Mc ∂x1 Mc ∂x2 ∂x1 ∂x2 = = ∂f ∂f 2F 2γ c ∂∆T 2 2 ∂x − 2γ − M p − Mp ∂∆T M ∂x ∂x ∂x1
∂x2
c
Notice that, from Lemma 2, we have: (a) ∂fi ∂xi
1
∂fi ∂xj
p
p
(8)
2
> 0, i 6= j, ∀x ∈ D, and (b)
< 0, i = 1, 2, ∀x ∈ D. This characterizes the system dynamical behavior as exposed in subsection 3.4. 3
4
More detailed and complex modeling of heat exchangers considers the different temperatures at every point within the tubes, giving rise to infinite dimensional dynamical models through a set of partial differential equations; see for instance [4, 14, 18]. The proof is not reproduced here in order to cope with space limitations. Interested readers may contact the authors.
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3.2 Positive dynamics The nature of the heat transfer process renders physically reasonable the definition of D as the system state space: Due to the heat absorbed by the coolant in its way along the external tube, its output temperature, x1 , cannot be lower than or even equal to Tci . It cannot either be greater than or equal to Tpi because the coolant cannot absorb the whole (or a greater) amount of heat energy conveyed by the process liquid along the internal tube (this would imply an eventual inversion of the heat transfer process, i.e. the coolant becoming the hot liquid while the process liquid the cold one). Following an analog reasoning concerning the heat removed from the process liquid, one sees that x2 cannot take values out of (Tci , Tpi ) either. x1 − Tci Tpi − x1 © ª Now, notice that D = x ∈ <2+ | h(x) > 0 with h(x) = x2 − Tci . Tpi − x2 ∂hi Furthermore, from (7) and Remark 1, one can easily verify that ∂x f (x) > 0 for all (i, x) ∈ [1 . . . 4] × ∂D such that hi (x) = 0. Consequently, D is forward invariant with respect to (3) (see section 2). Furthermore, since D ⊂
In fact, (2) represents the special case for heat exchangers (under Assumption A.2) of the general (compartmental) material balance equation (1) in [8].
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are composed by a finite number of subsystems, called compartments, interacting by exchanging material [2]. Since their state variables actually represent the material contained in each compartment, these cannot ever take negative values. Therefore, compartmental processes are a special class of positive systems. In the particular case of (3), each flow constitutes a compartment and heat (expressed as temperature in (3)) is the exchanged material. Let us now comment on the system dynamical features characterized by ∂f the sign properties of ∂f ∂x in (8) exposed in Remark 3. Property (a) defines ∂x as a Metzler matrix [11] and characterizes it as irreducible [10], both (such matrix properties) on D. Dynamical systems, x˙ = F (x), such that ∂F ∂x is Metzler on the system state space, W , are called cooperative systems [13]. If ∂F ∂x is additionally irreducible on W , they are called cooperative irreducible systems (or, equivalently, irreducible cooperative systems). Cooperative dynamical models have been used to describe networks of reservoirs, chemical processes, human relationships, demographic systems, neural networks, digital filters, and socialeconomic systems among many others (see [15] and references therein). They give rise to solutions with particular dynamical properties. For instance, cooperative (irreducible) systems on p-convex state-space domains have (strong) monotone flows [13]. In the particular case of (3), given the forward invariance and boundedness of D, for any x0 ∈ D [17, Thrm. 3.2.2]: (i) there exists T ≥ 0 such that for each i = 1, 2, xi (t; x0 ) is monotone on t ≥ T , and (ii) x(t; x0 ) converges to a single equilibrium. Notice that (i) implies non-oscillating solutions and (ii) entails the absence of closed orbits in D. Furthermore, it is shown in [19] that system (3) possesses a unique equilibrium point on D. Consequently, such unique equilibrium point is globally attractive on D (which can be as well corroborated through [10, Thrm. 1]).
4 Conclusions In this work, the dynamics of countercurrent liquid-liquid double-pipe heat exchangers have been characterized. Some analytical properties of their statespace model and its solutions have been outlined. Since the heat (expressed as temperature in the model) at the outlet point of the external tube and that of the internal pipe are taken as state variables, x1 and x2 respectively, these cannot take negative values; therefore heat exchangers are positive systems. Moreover, x1 and x2 are proved to remain within a bounded set D fully contained in <2+ ; in other words, the state-space domain D ⊂ <2+ is forward invariant with respect to the system dynamics, while state values on <2 \ D cannot actually take place. A characterization of positive systems that takes into account such state space restriction was proposed (see Proposition 1). Since the dynamical model is gotten from a heat balance equation (applied to each fluid), the system is compartmental. Each flow constitutes a compartment and heat is the exchanged material. Furthermore, the analysis of the system dynamics has shown the positive influence that the state
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variables have on each other (see Remark 3), which brings to the fore the irreducible cooperative character of the model. Finally, such characterization has permitted to highlight some dynamical properties of the system solutions, like monotonicity (after a finite time) and convergence to a single (actually unique) equilibrium point.
References 1. Aeyels, D., and P. de Leenheer. Extension of the Perron-Frobenius theorem to homogeneous systems. SIAM J. on Control and Optim., 41(2):563–582, 2002. 2. Benvenuti, L. and L. Farina. Positive and compartmental systems. IEEE Transactions on Automatic Control, 47(2):370–373, 2002. 3. Cohen, W.C., and Johnson, E.F. Dynamic characteristic of double-pipe heat exchangers. Ind. and Eng. Chem., 48:1031–1034, 1956. 4. Fazlur Rahman, M.H.R., and R. Devanathan. Feedback linearization of a heat exchanger. Systems and Control Letters, 26:203–209, 1995. 5. Fossard, A.J., and D. Normand-Cyrot (Eds.). Nonlinear Systems. Vol. 1: Modeling and Estimation. Chapman & Hall, London, 1995. 6. Hirsch, M.W., and S. Smale. Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, New York, 1974. 7. Incropera, F.P., and D.P. DeWitt. Fundamentals of Heat and Mass Transfer. Wiley, New York, 3rd edition, 1990. 8. Jacquez, J.A., and C.P. Simon. Qualitative theory of compartmental systems. SIAM Review, 35(1):43–79, 1993. 9. Khalil, H.K. Nonlinear Systems. Prentice-Hall, Upper Saddle River, NJ, 2nd edition, 1996. 10. Leenheer, P. de, and D. Aeyels. Stability properties of equilibria of classes of cooperative systems. IEEE Trans. on Automatic Control, 46(12):1996–2001, 2001. 11. Luenberger, D.G. Introduction to Dynamic Systems: Theory, Models & Applications. John Wiley & Sons, USA, 1979. 12. Malleswararao, Y.S.N., and M. Chidambaram. Nonlinear controllers for a heat exchanger. J. of Proc. Control, 2:17–21, 1992. 13. Mierczy´ nsky, J. Cooperative irreducible systems of ordinary differential equations with first integral. In Proc. of the 2nd Marrakesh Int. Conf. on Diff. Eqs., 1995. Available at: http://www.im.pwr.wroc.pl/ mierczyn/publications.html 14. Munz-Brienza, B., J.B. Gandy, and L. Lackenbach (Eds.). Heat Exchanger Design Handbook. Vols. 1–5. Hemisphere Publishing Corporation, USA, 1983. 15. Piccardi, C., and S. Rinaldi. Remarks in excitability, stability, and sign of equilibria in cooperative systems. Systems and Control Letters, 46:153–163, 2002. ´ 16. Silva-Navarro, G., and J. Alvarez-Gallegos. Sign and stability of equilibria in quasi-monotone positive nonlinear systems. IEEE Trans. on Autom. Cont., 42(3):403–407, 1997. 17. Smith, H.L. Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems. Amer. Math. Soc., Providence, RI, 1995. 18. Walas, S.M. Modelling with Differential Equations in Chemical Engineeing. Butterworth-Heinemann, Stoneham, MA, 1991. 19. Zavala-R´ıo, A., R. Femat, and G. Sol´ıs-Perales. Analysis, stability, and control of countercurrent liquid-liquid double-pipe heat exchangers. Submitted to Automatica, 2001.
Note on Structural Properties and Sizes of Eigenspaces of Min-max Functions Qianchuan Zhao and Da-Zhong Zheng Department of Automation, Tsinghua University, Beijing 100084, China, {zhaoqc,dauzdz}@mail.tsinghua.edu.cn Abstract. In this paper, we study relationships among structural properties of min-max functions and complexity on deciding the sizes of the eigenspaces of minmax functions. We show that strong connectivity implies inseparability; Olsder’s separated systems are separable. A relaxed condition under which inseparability remains equivalent to the balance condition is also presented. The decision problem of whether the size of the eigenspace of a min-max function is greater than one is then report to be NP-hard based on the connection between the separability of a min-max function and the size of the eigenspace of its skeleton.
1 Introduction Recently, several authors (e.g., [9],[2] and [6]) studied the min-max systems (also known as min-max-plus systems) in which the systems are described by functions containing operations min, max and plus. Such systems are natural extensions of timed discrete event systems described by max-plus (or minplus) algebra (see e.g. [2]). The dynamics of min-max systems are closely related to the functions (known as min-max functions) defining the systems. For a given min-max function F , a real λ and a real vector X are called an (additive, non-linear) eigenvalue and an eigenvector of F respectively, if F (X) = λ + X. The eigenvectors are also called (generalized) fixed points [6]. The eigenspace for a min-max function F is defined as the set of all its fixed points. The study of eigen-structure of min-max functions has become a very important research topic. It has been proved that min-max functions permit at most one eigenvalue. For a given min-max function, several structural properties (e.g., irreducibility , strong connectivity, the balance condition, inseparability, see next section for details) are known to imply the existence of such an eigenvalue. It is interesting to know the relationships among these structural properties and the size of the eigenspace of a min-max function. By size we mean the number of different eigenvectors up to an additive constant. L. Benvenuti, A. De Santis, and L. Farina (Eds.): Positive Systems, LNCIS 294, pp. 393-400, 2003. Springer-Verlag Berlin Heidelberg 2003
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In this paper, by employing the conjunctive and disjunctive normal forms of min-max expressions, we will derive the relationships among several structural properties and formulate a new condition under which inseparability remains equivalent to the balance condition. The decision problem of whether the size of the eigenspace of a given min-max function is greater than one will be shown to be NP-hard.
2 Basic definitions We follow the notations used in [6] and [3]. The operations a ∨ b and a ∧ b are used to stand for maximum and minimum respectively: a ∨ b = max(a, b) and a ∧ b = min(a, b). We also write R for the set of real numbers and B for the Boolean space {0, 1}. We use a prime (0 ) to denote the transpose of a vector, i.e., (x1 , . . . , xn )0 is the column vector formed by transposing (x1 , . . . , xn ). The upper case English letters X, Y and Z will be used in the rest of this paper to stand for vectors (x1 , . . . , xn )0 ,(y1 , . . . , yn )0 and (z1 , . . . , zn )0 . The notations 1 and 0 represent the vectors whose components are uniformly 1 and 0 respectively. Definition 1. A min-max function of type (n, 1) is any function f : Rn → R, which can be written as a term in the grammar: f := x1 , x2 , . . . , xn , |f + a|f ∧ f |f ∨ f,
(1)
where a ∈ R will be understood as parameters. Notice, expressions like 1 ∧ x, 1 ∨ x should not be regarded as min-max functions according to this definition. Monotone Boolean functions are min-max functions when AND and OR are understood as ‘∧’ and ‘∨’ respectively. By D(f ) we refer to the largest number of plus nodes contained in paths from the root to the leaves in the derivation tree (also known as parse tree)[8] of f . It is not hard to see that every min-max function can be converted into such a form in which the operands of ‘plus’ include only parameters and one of the unknowns x1 , . . . , xn . Definition 2. A min-max function of type (n, m) is any function F : Rn → Rm , such that each component Fi is a min-max function of type (n, 1). The set of min-max functions of type (n, m) will be denoted by MM(n, m). Max-plus functions are special types of min-max functions. Now we define eigen-structure of min-max functions and the decision problem that will be studied in this paper. Definition 3. For a given min-max function F ∈ MM(n, n), X ∈ Rn is said to be an eigenvector associated with an eigenvalue λ ∈ R if F (X) = λ + X. The vector X is also called a (generalized) fixed point of F . The set of fixed points of F ,i.e. {X|F (X) = λ + X}, will be denoted by E(F ). E(F ) is also called the eigenspace of F .
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We define an equivalence relation ∼ on an eigenspace E(F ), by X ∼ Y if and only if there exists an h ∈ R such that X = h + Y . It is easy to check that ∼ is indeed an equivalence relation on E(F ). If for X, Y ∈ E(F ) we have X ∼ Y , we will say that X and Y are equivalent fixed points; otherwise, X and Y are inequivalent fixed points. Definition 4. The size of the eigenspace of F is defined as the number of distinct equivalence classes of fixed points of F and will be denoted by size(E(F )). Note size(E(F )) = 1 corresponding to the case that F has a unique eigenvector in the context of [1]. The decision problem of interests is as follows. Definition 5 (Size of Eigenspace). For a given min-max function F ∈ MM(n, n), whether size(E(F )) > 1? Next, let us introduce structural properties that will be used in this paper. It is clear that min-max functions depend on parameters. We use the notation F (X; P ) to emphasize the role of parameters. P is the vector of parameters which appear in Fi (X) as coefficients. Note that the dimensions of P and X may be different. It is interesting to note that for every min-max function F (X; P ), one can always obtain a so called monotone Boolean function F (X; 0) by simply setting all parameters to 0. This Boolean function plays an important role in revealing structural properties of min-max functions. In fact, we have the following definitions. Definition 6. [11] A min-max function is inseparable, if the only solutions of the Boolean equation F (X; 0) = X are the trivial solutions 0 and 1. A min-max function is separable if it is not inseparable. We shall call F (X; 0) the skeleton of F (X; P ). An example of min-max function and its skeleton is ¶ µ ¶ µ (p1 + x1 ) ∨ (p2 + x2 ) x1 ∨ x2 , (2) F (X; P ) = , F (X; 0) = x1 ∧ x2 (p3 + x1 ) ∧ (p4 + x2 ) where X = (x1 , x2 ) and P = (p1 , p2 , p3 , p4 ). Next we introduce more structural properties of min-max functions. Define a graph G(F ) = (V, E) such that V = {1, . . . , n} and (i, j) ∈ E if and only if limv→∞ F (vej )i = ∞. Here vX is the vector whose components are vxi for v ∈ R and X ∈ Rn . The vector ej is the j-th vector in the canonical basis of Rn . The adjacency matrix for the graph G(F ) = (V, E) is the n−dimensional square matrix M (G(F )) whose elements are mij = 1 if (i, j) ∈ E and mij = 0, otherwise. Definition 7. [5] A min-max F is strongly connected, if the graph G(F ) is a strongly connected graph.
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Note that in [5], strong connectivity was defined for a broader class of functions known as topical functions [7]. Also notice that the graph of a max-plus matrix is strongly connected if and only if it is irreducible [2]. Definition 8. [6] A min-max function is balanced, if F (X; P ) has a generalized fixed point for every P ∈Rm . In other words, a min-max function is balanced if and only if it has an eigenvalue for every assignment of parameters. Note the balance condition is descriptive. The definition itself does not provide an algorithm to test the balance condition.
3 Main results We begin with expansions of min-max functions. The min-max functions F (X; P )i have equivalent finite expressions called conjunctive normal form (CNF) expansions in which the minimum operations are outermost and connect the maximum items whose operands are summations of parameters and a single argument as the following. F (X; P )i =
mi l(k,i) _ ^
m n (fC (P )i,k t + xst ), i ∈ I, ∀P ∈ R , X ∈ R ,
(3)
k=1 t=1
are summations of where st ∈ {1, . . . , n}, l(k, i) < ∞, mi < ∞ and fC (P )i,k t some components of the parameter vector P . The summations fC (P )i,k t over parameters will be called the coefficients of xst . Notice the expansion does not depend on P or X. That means for any other parameter vector Q ∈ Rm Vmi Wl(k,i) i,k and vector Y ∈ Rn , we have F (Y ; Q)i = k=1 t=1 (fC (Q)t + yst ). Dually, the expressions F (X; P )j can also be expressed in the so-called disjunctive normal form (DNF) F (X; P )j =
mj l(k,j) ^ _
(fD (P )j,k t + xst ).
(4)
k=1 t=1
In the CNF or DNF of a min-max function, the subscripts C or D in coefficients fC (P )t., k or fD (P ).,t k are used to indicate the type of the expansion. When only one form is used for an expression, we will simply use f (P ).,t k . Observe that f (0).,t k = 0. Proposition 1. Strong connectivity implies inseparability for min-max functions. The reverse of Proposition 1 is not always true. One example is the not strongly connected example given in [5].
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Proof. Suppose the graph of F is strongly connected. Give a Z ∈ Bn \{0, 1}. Let I = {i|zi = 1}. I 6= {1, . . . , n}. There must be is a j ∈ {1, . . . , n}\I such that zj = 0. Since graph G(F ) is strongly connected, there must be a node i(j) in I so that the edge (j, i(j)) ∈ E. This implies that zi(j) = 1 and limv→∞ F (vei(j) ; P )j = ∞. In terms of CNF of Fj , we have Vmj Wl(k,j) j,k limv→∞ k=1 + (vei(j),st )) = ∞. This means for every k, t=1 (fC (P )t there exists a t(k) such that ei(j),st(k) = 1. Hence st(k) = i(j). F (Z; 0)j = Vmj Vmj Wl(k,j) Vmj t=1 zst ≥ k=1 k=1 zst(k) = k=1 zi(j) = 1. If F (Z; 0) = Z were true, we would have that zj = 1. But this is impossible since it contradicts the assumption that j ∈ / I. Remark 1. The separated min-max systems studied in [9] are separable. In fact, Let Z = (1A , 0B ), where 1A is the unit vector of dimension dim(A) and 1B is the zero vector of dimension dim(B). Note F (Z; 0) = ( A(0)1A ∨ C(0)0B , B(0)0B ∧ D(0)1A )0 , since A is an irreducible max-plus matrix and B is an irreducible min-plus matrix, A(0)1A = 1A and B(0)0B = 0B . Since 1A is the maximal dim(A)-dimensional Boolean vector and 0B is the minimal dim(B)-dimensional Boolean vector, it also follows F (Z) = (1A , 0B )0 = Z. Next, we will give a condition under which inseparability is equivalent to the balance condition. Introduce a partial order on R+n . Here R+ is the set of non-negative real numbers. Let X, Y ∈ R+n , we define X º Y if xj = 0, for j s.t. yj = 0; xi ≥ yi , for i s.t. yi > 0, i, j ∈ {1, . . . , n}. For F ∈ MM(n, n), D(F ) is defined as maxi D(Fi ). Condition C. For every Z ∈ Bn \{0, 1}, there is a P ∈ {0, 1/D(F )}m so that F (Z; P ) º (1 + 1/D(F ))F (Z; 0).
(5)
It should be pointed out that D(F ) in Condition C can be calculated directly from the original min-max function instead of its CNF or DNF. Theorem 1. Under Condition C, inseparability is equivalent to the balance condition for min-max functions. This theorem improves Theorem 2 in [11]. Proof. If F is inseparable, then it follows from Theorem 1 in [11] that F is balanced. Suppose F is separable. This implies there exists a Z ∈ Bn \{0, 1} so that F (Z; 0) = Z. We shall prove that F is not balanced. Let I = {i|zi = 1, i = 1, . . . , n} and J = {j|zj = 0, j = 1, . . . , n}. Under Condition C, for this Z, we have a parameter vector P ∈ {0, 1/D(F )}m , F (Z; P ) º (1 + 1/D(F ))F (Z; 0). Define another parameter vector P 0 from P as follows: p0s = 1 if ps > 0 and p0s = 0 if ps = 0. We are about to show that F r (Z; P 0 ) º (r + 1)Z,
(6)
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for r = 1, 2, . . ., which implies that there is no eigenvalue for F (X; P 0 ), i.e., F is not balanced. For the CNF expansions of components of F in I, it holds that F (Z; P )i = Vmi Wl(k,i) i,k k=1 t=1 (f (P )t + zst ) > 1 + 1/D(F ), i ∈ I. So, for each i, k, at least one tk,i ∈ l(k, i) is such that f (P )i,k tk,i + zstk,i ≥ 1 + 1/D(F ). By noting that
i,k zstk,i ∈ B and f (P )i,k tk,i ∈ [0, 1] we know that zstk,i = 1 and f (P )tk,i ≥ 1/D(F ).
0
F (Z; P )i =
mi l(k,i) _ ^ k=1 t=1
(f (P 0 )i,k t +zst ) ≥
mi ^ k=1
(f (P 0 )i,k tk,i +1) ≥ 1+
mi ^
f (P )i,k tk,i , i ∈ I,
k=1
i,k 0 i,k since P 0 ≥ P and furthermore f (P 0 )i,k tk,i ≥ f (P )tk,i . By noting that f (P )t
0 i,k can only take integer values and that f (P 0 )i,k tk,i ≥ 1 whenever f (P )tk,i > 0 (in
i,k 0 fact f (P 0 )i,k tk,i ≥ f (P )tk,i ≥ 1/D(F ) > 0), we can further obtain F (Z; P )i ≥ 0 2, i ∈ I. Since F (Z; P )j = 0, j ∈ J, it follows that F (Z; P )j = 0, j ∈ J. This establishes (6) for r = 1. Make the induction hypothesis that (6) is true for every r ≤ u where u > 0 Vmi Wl(k,i) 0 i,k is any given number. We have for i ∈ I, F u+1 (Z; P 0 )i = k=1 t=1 (f (P )tk,i + V V mi mi u 0 0 i,k F u (Z; P 0 )stk,i ) ≥ k=1 (f (P 0 ))i,k tk,i + F (Z; P )stk,i ) ≥ k=1 (f (P )tk,i + V m i F u (Z, P 0 )stk,i ≥ 1 + (u + 1), i ∈ I. Here we F u (Z; P 0 )stk,i ) ≥ 1 + k=1
use the fact that f (P 0 )i,k tk,i ≥ 1. We have for the DNFs of F components Wmj V l(k,j) j,k j ∈ J, F (Z; P )j = k=1 t=1 (f (P )t + zst ) = 0. Hence for every k, j there j,k exists a tk,j ∈ l(k, j) such that f (P )j,k tk,j + zstk,j = 0. Since f (P )tk,j ≥ 0
and zstk,j ≥ 0, it must hold that f (P )jtk,j = 0 and zstk,j = 0. Furthermore, since f (P )j,k tk,j are summations of some components of P ≥ 0, from
0 j,k 0 f (P )j,k tk,j = 0 it is clear that f (P )tk,j = 0 according to the definition of P . V W m l(k,j) j,k j For case of r = u + 1, F u+1 (Z, P 0 )j = k=1 t=1 (f (P 0 )t + F u (Z; P 0 )st ) Wmj u+1 u 0 (Z; P 0 )j (f (P 0 )j,k ≤ k=1 tk,j + F (Z; P )stk,j ) = 0. Since all components of F are non-negative, we have F u+1 (Z; P 0 )j = 0, j ∈ J. Put together, we have shown that F u+1 (Z; P 0 ) º (1 + (u + 1))Z. By the induction hypothesis we can say that (6) is true for all r ≥ 1. The proof is completed.
Next we establish the relationship between the separability of a min-max F and the size of the eigenspace of its skeleton F (X; 0) and present the complexity result. Lemma 1. Let F be a monotone Boolean function. size(E(F )) > 1 if and only if there exists a real vector X ∈ E(F ))\{0 + h|h ∈ R}. Proof. The min-max function F has a unique eigenvalue 0. Hence size(E(F )) ≥ 1. Since 0 ∈ E(F ), the conclusion follows from the definition of the sizes of eigenspaces of min-max functions.
Note on Min-max functions
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Theorem 2. Let F be a min-max function. F is separable if and only if size(E(F (X; 0))) > 1. Proof. We begin with the ‘only if’ direction. If F is separable, there is a nontrivial solution Z ∈ Bn to F (Z; 0) = Z. Obviously, there is no h ∈ R so that Z = 0 + h. It follows from Lemma 1 that size(E(F (X; 0))) > 1. We proceed with the ‘if’ direction. Suppose size(E(F )) > 1. Then, according to Lemma 1 there exists a fixed point Z of F satisfying Z ∈ Rn \{0 + h|h ∈ R}. It is convenient to introduce the notation b(X) to refer to the value of the minimal elements of X. Based on Z, we shall find a non-trivial solution to the Boolean equation F (X; 0) = X. To do so, we introduce the following map ρ(X) = (ˆ x1 , . . . x ˆn ), where x ˆi = 1, if xi > b(X); x ˆi = 0, if xi = b(X), for i = 1, . . . , n. Obviously, ρ(Z) is a Boolean vector not equivalent to 0. Let us prove it is a solution to the Boolean equation F (X; 0) = X. For i such that zˆi = 0, consider a CNF of Fi : Vmi Wl(k,i) (zst ). F (Z; 0)i = zi = b(Z) implies there exists a F (Z; 0)i = k=1 Wl(k,i) t=1 Wl(k,i) k such that t=1 (zst ) = b(Z). Thus, 0 ≤ F (ρ(Z); 0)i ≤ t=1 (ˆ zst ) = 0. Wmi Vl(k,i) For i such that zˆi = 1, consider a DNF of Fi : F (Z; 0)i = k=1 t=1 (zst ). Vl(k,i) F (Z; 0)i = zi > b(Z) implies there exists a k such that t=1 (zst ) > b(Z). Vl(k,i) Thus, 1 ≥ F (ρ(Z); 0)i ≥ t=1 (ˆ zst ) = 1. In summary, we have shown that F (ρ(Z); 0) = ρ(Z). It follows that F is separable. Corollary 1. Size of Eigenspace is NP-hard. Proof. Let f be an instance of SAT(the Boolean Satisfiability Problem)[4], namely a Boolean formula in CNF. Let f be over n Boolean variables. We view f as a monotone Boolean function over 2n variables (X, Y ) by adding the restriction that xi = ¬yi , ∀i. For example, we view f = x1 ∨ ¬x2 as f (X, Y ) = x1 ∨ y2 with restriction that x1 = ¬y1 , x2 = ¬y2 . It is clear that X is a satisfiable assignment of f if and only if (X, Y ) is such that f (X, Y ) = 1 and xi = ¬yi , ∀i. For f , Boolean variables A, D, x ∈ B and Boolean vectors X, Y ∈ Bn , define new monotone Boolean functions Φ(A, D, x) = (A ∨ x) ∧ Wn Vn D, Ωa (X, Y ) = i=1 (xi ∧ yi ), Ωb (X, Y ) = i=1 (xi ∨ yi ), and Ωdf (X, Y ) = f (X, Y ) ∧ Ωb (X, Y ). It has been proved in [10] that f is satisfiable if and only if the monotone Boolean function Ff defined below is separable (when viewed as a min-max function). Ff (X, Y ) = (Ff1 (X, Y ), Ff2 (X, Y ))0
(7)
where Ff1 (X, Y )i = Φ(Ωa (X, Y ), Ωdf (X, Y ), xi ), i = 1, 2, . . . , n. Ff2 (X, Y )i = Φ(Ωa (X, Y ), Ωdf (X, Y ), yi ), i = 1, 2, . . . , n. Notice that Ff only needs a linear increase in input data, namely, 2n variables. The conclusion then follows from Theorem 2 and the NP-hardness of SAT[4].
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Qianchuan Zhao and Da-Zhong Zheng
4 Conclusions In this paper, by using the conjunctive and disjunctive normal forms of minmax functions, we have derived the relationships among several structural properties and formulated a relaxed condition under which inseparability remains equivalent to the balance condition. The decision problem of whether the size of the eigenspace of a given min-max function is greater than one is shown to be NP-hard.
Acknowledgement The authors would like to thank the reviewers for helpful suggestions. This work was supported in part by NSFC(Grant No.60074012,60274011), National Key Project of China, Fundamental Research Funds from Tsinghua University and Chinese Scholarship Council, Ministry of Eduction of China.
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