Discontinuous Systems: Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions

Communications and Control Engineering

Series Editors E.D. Sontag • M. Thoma • A. Isidori • J.H. van Schuppen

Published titles include: Stability and Stabilization of Infinite Dimensional Systems with Applications Zheng-Hua Luo, Bao-Zhu Guo and Omer Morgul

Constrained Control and Estimation Graham C. Goodwin, María M. Seron and José A. De Doná

Nonsmooth Mechanics (Second edition) Bernard Brogliato

Randomized Algorithms for Analysis and Control of Uncertain Systems Roberto Tempo, Giuseppe Calafiore and Fabrizio Dabbene

Nonlinear Control Systems II Alberto Isidori L2-Gain and Passivity Techniques in Nonlinear Control Arjan van der Schaft Control of Linear Systems with Regulation and Input Constraints Ali Saberi, Anton A. Stoorvogel and Peddapullaiah Sannuti Robust and H∞ Control Ben M. Chen Computer Controlled Systems Efim N. Rosenwasser and Bernhard P. Lampe Control of Complex and Uncertain Systems Stanislav V. Emelyanov and Sergey K. Korovin Robust Control Design Using H∞ Methods Ian R. Petersen, Valery A. Ugrinovski and Andrey V. Savkin Model Reduction for Control System Design Goro Obinata and Brian D.O. Anderson Control Theory for Linear Systems Harry L. Trentelman, Anton Stoorvogel and Malo Hautus Functional Adaptive Control Simon G. Fabri and Visakan Kadirkamanathan

Switched Linear Systems Zhendong Sun and Shuzhi S. Ge Subspace Methods for System Identification Tohru Katayama Digital Control Systems Ioan D. Landau and Gianluca Zito Multivariable Computer-controlled Systems Efim N. Rosenwasser and Bernhard P. Lampe Dissipative Systems Analysis and Control (Second edition) Bernard Brogliato, Rogelio Lozano, Bernhard Maschke and Olav Egeland Algebraic Methods for Nonlinear Control Systems Giuseppe Conte, Claude H. Moog and Anna M. Perdon Polynomial and Rational Matrices Tadeusz Kaczorek Simulation-based Algorithms for Markov Decision Processes Hyeong Soo Chang, Michael C. Fu, Jiaqiao Hu and Steven I. Marcus Iterative Learning Control Hyo-Sung Ahn, Kevin L. Moore and YangQuan Chen

Positive 1D and 2D Systems Tadeusz Kaczorek

Distributed Consensus in Multi-vehicle Cooperative Control Wei Ren and Randal W. Beard

Identification and Control Using Volterra Models Francis J. Doyle III, Ronald K. Pearson and Babatunde A. Ogunnaike

Control of Singular Systems with Random Abrupt Changes El-Kébir Boukas

Non-linear Control for Underactuated Mechanical Systems Isabelle Fantoni and Rogelio Lozano

Nonlinear and Adaptive Control with Applications Alessandro Astolfi, Dimitrios Karagiannis and Romeo Ortega

Robust Control (Second edition) Jürgen Ackermann

Stabilization, Optimal and Robust Control Aziz Belmiloudi

Flow Control by Feedback Ole Morten Aamo and Miroslav Krstić

Control of Nonlinear Dynamical Systems Felix L. Chernous’ko, Igor M. Ananievski and Sergey A. Reshmin

Learning and Generalization (Second edition) Mathukumalli Vidyasagar

Periodic Systems Sergio Bittanti and Patrizio Colaneri

Yury V. Orlov

Discontinuous Systems Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions

123

Yury V. Orlov, PhD, DrSc Departamento de Electrónica y Telecomunicaciones Centro de Investigación Cientifica y Educación Superior de Ensenada (CICESE) Km.107 Carretera Tijuana-Ensenada 22860 Ensenada México

ISBN 978-1-84800-983-7

e-ISBN 978-1-84800-984-4

DOI 10.1007/978-1-84800-984-4 Communications and Control Engineering ISSN 0178-5354 A catalogue record for this book is available from the British Library Library of Congress Control Number: 2008939444 © 2009 Springer-Verlag London Limited MATLAB® and Simulink® are registered trademarks of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, USA. http://www.mathworks.com Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: le-tex publishing services oHG, Leipzig, Germany Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

Preface

A major problem in control engineering is a robust feedback design that stabilizes a nominal plant, while also attenuating the influence of parameter variations and external disturbances. In the last decade, this problem was heavily studied and considerable research efforts have resulted in the development of systematic design methodology for nonlinear feedback systems. This methodology was, however, basically confined to smooth feedback systems, whereas motivated by modern applications, significant interest has emerged in extending this methodology to complex, particularly, electromechanical systems with hard-to-model nonsmooth phenomena such as friction and backlash. Since ignoring these phenomena may severely limit the achievable performance, the practical utility of the existing smooth control algorithms becomes questionable for many electromechanical applications. Thus, nonsmooth feedback design is continuing to be developed. The primary concern of this book is stability analysis and robust control synthesis of uncertain discontinuous systems within the framework of methods of nonsmooth Lyapunov functions. Motivating examples for studying discontinuous systems are mechanical systems with complex nonlinear phenomena (collision, backlash, dry friction). Another motivation comes from the rapidly developed area of variable structure control. A discontinuous system is typically viewed as a simple model of hybrid systems, consisting of a finite family of continuous-time subsystems, equipped with a rule of switching between them. Whenever the system trajectory hits a switching surface, the continuous state makes a jump, specified by a restitution rule. A special case is when state jumps are absent. Just in case, the state trajectory is always continuous, but in general it is not differentiable when it hits a switching surface. After hitting a switching surface, the trajectory can either cross it or evolve along the surface on a finite time interval. In the latter case, sliding motions occur in the system. Although the Lyapunov methods have been widely used in practice and the need of nonsmooth Lyapunov functions has particularly been recognized for nonsmooth dynamic systems [94, 214], these methods do not admit a straightforward extension to discontinuous dynamic systems. In this regard, recall (see [146] for details) that Krasovskii–LaSalle’s invariance principle [118, 121], generally speaking, fails to v

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hold for dynamic systems, governed by differential inclusions and, in particular, by differential equations with discontinuous right-hand sides (see [6, 91, 97, 145, 165, 202, 223] for various extensions of the invariance principle). There has been work on the Lyapunov stability theory in discontinuous systems. Smooth Lyapunov functions have successfully been used by V. A. Yakubovich, G. A. Leonov, and A. Kh. Gelig [245] to analyze dynamic systems with discontinuous nonlinearities. For a class of discontinuous systems, admitting a finite frequency of switches only, the theory of multiple Lyapunov functions was initiated by M. Branicky [30] and then developed by D. Liberzon [132]. This theory does not capture discontinuous systems with sliding motions and it can be viewed as a complementary to analysis tools of sliding mode control theory pioneered by V. I. Utkin [227]. In order to analyze discontinuous systems with sliding modes, differentiable Lyapunov functions suffice in many cases. D. Shevitz and B. Paden [207] invoked the nonsmooth Lyapunov analysis for studying discontinuous systems where “kinks” form an essential part of dynamics. While being confined to a class of discontinuous systems, trajectories of which are unambiguously defined to the right, their analysis becomes restrictive in many practical situations. In order to avoid relating to this restrictive uniqueness condition, a novel technique, recently proposed in [165], is developed in the present book. In addition to a nonsmooth Lyapunov function with a non-positive time derivative along the system trajectories, the technique involves an auxiliary indefinite (rather than a definite) function that allows one to derive a certain integral inequality, which by Barbalat’s lemma ensures the asymptotical stability of the closed-loop system. In contrast to Krasovskii–LaSalle’s invariance principle [118, 121], the technique, which is referred to as the extended invariance principle, is applicable to general time-varying systems with discontinuous nonlinearities as well. The extended invariance principle, while being applied to a homogeneous discontinuous system, proves to be capable of revealing not simply asymptotic stability but also the finite time stability of such a system. In turn, the finite time stability of a homogenous discontinuous system is known [168, 169] to persist regardless of inhomogeneous perturbations. This fundamental property constitutes the quasihomogeneity principle, and along with the extended invariance principle, it forms the core of the stability analysis developed in the book. Based on these fundamental principles, a synthesis of globally stabilizing controllers of uncertain dynamic systems is subsequently developed. The present synthesis does not necessarily rely on the generation of sliding modes, while retaining robustness features, similar to those possessed by their sliding mode counterparts. The strategy of the discontinuous controllers constructed is to drive the system to the zero dynamics manifold in finite time and to maintain it there in spite of the parameter uncertainties and external disturbances, both with a priori known norm bounds. Desired robustness properties and an asymptotic stability of the the closedloop system are thus provided. Attractive features of the discontinuous controllers proposed are illustrated by applications to electromechanical systems. Allowing relatively strong Coulomb friction in these applications precludes the use of continuous regulators, because the

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closed-loop system, in that case, would have a nontrivial set of equilibrium points and it would therefore be driven to a wrong endpoint. As opposed to continuous controllers, the discontinuous controllers are demonstrated to be capable of providing the desired system performance in spite of significant uncertainties in the system description, as is typically the case in the control of electromechanical systems with complex backlash/friction phenomena. Besides the finite-dimensional treatment, robust discontinuous control algorithms, recently developed in [162, 171, 174] for infinite-dimensional systems such as time delay systems and distributed parameter systems, are also presented. The book is intended for graduate students and control specialists interested in the discontinuous systems theory and control applications. No background in discontinuous systems is required, as such systems are conceptually introduced at the appropriate level. Some related topics, however, are either not covered or only partially covered in this book. No specific study is proposed for impulsive systems, for switched systems with isolated switching events, and for sliding mode systems. These kinds of systems are treated within the general paradigm of discontinuous systems, whereas their rather comprehensive studies can be found in [13, 91, 123, 201, 204, 246], regarding impulsive systems, in [97, 132], regarding switched systems, and in [65, 99, 227, 228], regarding sliding modes (see also surveys [76, 77] and the special issue [208] for advanced results on higher-order sliding modes). General hybrid systems, to be involved into hybrid synthesis of underactuated systems in Sect. 10, are only briefly commented in Chap. 1. For a deeper insight on hybrid systems, see [34, 35, 82, 143, 202, 230]. Analysis, synthesis, and applications to electromechanical systems are developed under uncertainty conditions, including model inadequacies. While being less useful for modelers, this adopted framework is attractive for control engineers interested in coming up with schemes that allow one to successfully address stability/stabilization issues in all circumstances. Adequate models of complex phenomena in electromechanical systems can be found, e.g., in [10, 41, 56, 158] (friction modeling), [142, 156, 221] (backlash modeling), [33, 81, 125, 150] (post-impact restitution). The book consists of an introduction and four parts, and it is organized as follows. Chapter 1 previews issues that will arise in the sequel. Mathematical tools of discontinuous systems are reviewed in Part I of the book. Differential equations with piece-wise continuous right-hand sides are accepted in Chap. 2 as a basic mathematical model of such a system. While allowing Dirac functions in the coefficients, the equations admit instantaneous jumps of the state of the system. The instantaneous impulse response of the system is adequately defined according to Schwartz’ distribution theory in a nonlinear setting. Various solution concepts (Filippov solution [71], Utkin solution [227], and the vibroimpact solution [164]) are introduced for these equations. The existence and uniqueness of the solutions as well as their physical sense, and applications to modeling of nonlinear phenomena in electromechanical systems are discussed.

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In Chap. 3, the nonsmooth Lyapunov analysis of discontinuous systems and those of discrete-continuous dynamics are developed side by side. Several stability concepts such as stability and exponential/asymptotic stability are addressed locally and in large. Semiglobal Lyapunov functions are particularly introduced to subsequently address a relatively new kind of robust stability when the system is required to be asymptotically stable, equiuniformly in admissible non-vanishing external disturbances. In addition, L2 -gain analysis in a finite-dimensional setting and LMI-based analysis in an infinite-dimensional setting are presented. Chapter 4 deals with homogeneous systems. These systems are of a particular interest, because under appropriate conditions on the homogeneity degree, their equiuniform asymptotic stability ensures that the state of the system escapes to zero in finite time. The finite-time stability property persists, even if the system is affected by inhomogeneous external disturbances. This result constitutes the quasihomogeneity principle whose capabilities are illustrated by several examples. Part II of the book is devoted to robust discontinuous control synthesis. The quasihomogeneous design is constituted in Chap. 5. Sliding mode-based unit feedback controller design is developed in Chap. 6. Following the Lyapunov minmax approach, a discontinuous feedback controller, that counteracts non-vanishing disturbances and parameter variations, is synthesized to guarantee that the time derivative of a Lyapunov function, selected for a nominal system, is negative definite on the trajectories of the perturbed system. The approach gives rise to the so-called unit control feedback, the norm of which is equal to one everywhere but on the manifold where the feedback undergoes discontinuities. The resulting closed-loop system is shown to never pass through the switching manifold. The system stability is thus analyzed beyond the manifold. Once the trajectory is on the switching manifold, a smooth dynamic is restored and standard Lyapunov theory is in force. In addition, how undesired high frequency state oscillations, caused by fast switching in the unit controller, can be removed by smoothing the unit signal is discussed. In Chap. 7, H∞ -control synthesis is presented for nonsmooth time-varying systems via measurement feedback. Being in a certain sense an extension of the unit feedback synthesis, the H∞ -control design aims to guarantee both the internal asymptotic stability of the (disturbance-free) closed-loop system and the dissipative inequality (in that the size of an error signal is uniformly bounded with respect to the worst case size of an external disturbance signal). Similar to [107, 229], the approach here is to construct an energy or storage function, resulting in the dissipative inequality. Once the storage function is found, it can be used as a Lyapunov function, guaranteeing the internal stability requirement. Sufficient conditions for the storage function to exist are given in terms of the solvability of two nonsmooth Hamilton– Jacobi–Isaacs inequalities which arise in the state-feedback and, respectively, the output-injection design. The development follows the line of reasoning proposed in [4] where the corresponding Hamilton–Jacobi–Isaacs expressions are required to be negative definite rather than semidefinite. This feature allows one to develop an H∞ -design procedure with no a priori-imposed stabilizability-detectability conditions on the control system. Although the design procedure results in an infinite-

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dimensional problem, this difficulty is circumvented by solving the problem locally. The distribution-based formalism is then developed to straightforwardly derive a local solution of the sampled-data measurement feedback H∞ -control problem from that of the time continuous measurement feedback H∞ -control problem. Part III develops the unit feedback synthesis for infinite-dimensional systems driven in a Hilbert space. The presence of an unbounded operator in the state equation precludes from a simple extension of the finite-dimensional control algorithms. Theoretical results obtained in an abstract infinite-dimensional setting are then supported by applications to distributed parameter and time delay systems. In Chap. 8, the unit feedback synthesis is developed for a class of linear infinitedimensional systems with a finite-dimensional unstable part using finite-dimensional sensing and actuation. An output feedback controller is synthesized by coupling an infinite-dimensional Luenberger state observer and unit state feedback controller. In order to obtain the fully practical finite-dimensional framework for controller synthesis, a finite-dimensional approximation of the Luenberger observer as well as a continuous approximation of the unit feedback controller are carried out at the implementation stage. Implementation, performance, and robustness issues of the unit output feedback control design are illustrated in a simulation study of the linearization of the Kuramoto–Sivashinsky equation (KSE) around the spatially-uniform steady-state solution with periodic boundary conditions. While being unforced, the KSE describes incipient instabilities in a variety of physical and chemical systems and a control problem that occurs here is to avoid the appearance of the instabilities in the closed-loop system. In Chap. 9, the unit control approach is extended to Hilbert space-valued minimum phase semilinear systems. Control algorithms presented ensure global or local asymptotic stability, according as state feedback or output feedback is available. The desired robustness properties of the closed-loop system against external disturbances with a priori known norm bounds make the algorithms extremely suited for stabilization of the underlying system operating under uncertainty conditions. It is, in particular, shown that discontinuous feedback stabilization is constructively available in the case where complex nonlinear dynamics of the uncertain system do not admit factoring out a destabilizing nonlinear gain, and thus the destabilizing gain cannot be handled through nonlinear damping. The theory is applied to the stabilization of chemical processes around pre-specified steady-state temperature and concentration profiles corresponding to a desired coolant temperature. Performance issues of the unit feedback design are illustrated in a simulation study of the plug flow reactor. In Chap. 10, the unit feedback synthesis is generalized for a class of uncertain time-delay systems with nonlinear disturbances and unknown delay values whose unperturbed dynamics are linear. The controller constructed proves to be robust against sufficiently small delay variations and weak external (possibly, unmatched) disturbances. It is worth noticing that allowing unmatched disturbances is a step beyond a standard sliding mode control treatment. Specifically, the critical delay value when the closed-loop system, corresponding to this value, becomes asymptotically unstable, is explicitly calculated as a function of linear growth constants of the un-

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matched disturbances. Performance issues of the controller are illustrated by means of a numerical example. In Part IV of the book, performance issues of the developed controllers are experimentally tested in engineering applications to electromechanical systems with complex nonlinear phenomena. Chapter 11 develops the local nonsmooth H∞ -synthesis of Lagrangian systems which is capable of accounting for hard-to-model friction forces and backlash effects. The resulting control algorithms are illustrated in an experimental study of the position feedback regulation of a laboratory three-link robot manipulator with frictional joints and in that of the output feedback regulation of a servomechanism with backlash. Chapter 12 deals with the quasihomogeneous synthesis which is shown to be extremely suited for fully actuated systems with dry friction. Since external disturbances, affecting these systems, represent discontinuous functions in the state variables and meet the matching condition, their influence is not simply attenuated as it would be the case under H∞ -synthesis, but it is fully rejected under the quasihomogeneous synthesis. Moreover, global position regulation becomes possible provided that an upper bound on the magnitude of the external disturbances is known a priori. Attractive features of the quasihomogeneous synthesis are illustrated by means of orbital stabilization of a simple inverted pendulum, and by means of the global position regulation of a multi-link robot manipulator. In Chap. 13, the quasihomogeneous design is developed for 2-DOFs underactuated systems by including it into a unified hybrid synthesis framework. Being verified experimentally, the proposed hybrid synthesis appears to provide desired robustness properties against friction forces. Its capabilities are illustrated in experimental studies made for laboratory test beds such as a horizontal double pendulum, an inverted double pendulum (Pendubot), and a pendulum, located on a cart. The presentation in Chaps. 11-13 emphasizes the control algorithms, rather than their technical implementation. Such a technical consideration would somehow fall out of the general theoretical line of the book. A consideration of the specific applications, however, makes the book appropriately complete. Ensenada, Mexico, July 2008

Yury Orlov

Acknowledgements

This book summarizes the investigations I made in the area of discontinuous systems while working at the Institute of Control Sciences of the Russian Academy of Sciences (1980-1992) and at the Mexican research center CICESE (1993-2008). I wish to thank my scientific advisor Vadim Utkin and my colleagues from the laboratory of discontinuous control systems, affiliated with the former Institute, where I started my career, for the major influence on forming my scientific interests. This work benefited from a long-term stay with the CICESE Electronics and Telecommunication Department, headed by Joaquin Alvarez, and from supporting experiments, made by ex-PhD students Leonardo Acho, Luis Aguilar, and Raul Santiesteban. I also wish to thank Joseph Bentsman from the University of Illinois at UrbanaChampaign, USA, Emilia Fridman from Tel Aviv University, Israel, and Leonid Fridman from UNAM University, Mexico for stimulating discussions. The generous hospitality and fruitful exchanges of ideas with the nonlinear and delay systems team, headed by Jean-Pierre Richard, from Ecole Centrale de Lille, France, as well as with Denis Dochain from CESAME, Louven-la-Neuve, Belgium, Panagiotis Christofides from the University of California at Los Angeles, USA, and Miroslav Krstic from the University of California at San Diego, USA, are gratefully acknowledged. The work was partially supported by CONACYT under grant number 45900.

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Contents

Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Impulsive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Variable-structure Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Hybrid Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Sliding Modes in a Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 7 9

Part I Mathematical Tools 2

Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Nonlinear Differential Equations in Distributions . . . . . . . . . . . . . . . . 2.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Instantaneous Impulse Response in a Nonlinear Setting . . . . 2.1.3 Vibroimpact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Differential Equations with a Piece-wise Continuous Right-hand Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Filippov Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Equivalent Control Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Sliding Modes in a Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . 2.3 Modeling of Electromechanical Nonlinear Phenomena . . . . . . . . . . . 2.3.1 Friction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1.1 Static Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1.2 Dynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 The Multistable Backlash Model and Its Single-stability Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Limit Cycles and Nonlinear Asymptotic Harmonic Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Vibroimpact Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 13 14 21 23 24 26 28 33 34 34 35 36 39 41

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3

4

Contents

Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Stability and Nonsmooth Lyapunov Functions . . . . . . . . . . . . . . . . . . 3.3 Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Extension to a Class of Discontinuous Systems . . . . . . . . . . . 3.3.2 Illustrative Applications to a Mechanical Oscillator with Coulomb Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.1 Localization of the Equilibria Set . . . . . . . . . . . . . . . 3.3.2.2 Asymptotic Stabilization . . . . . . . . . . . . . . . . . . . . . . 3.3.2.3 Velocity Observer Design . . . . . . . . . . . . . . . . . . . . . 3.4 Extended Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Asymptotic Stability and Semiglobal Lyapunov Functions . . . . . . . . 3.6 L2 -Gain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Hamilton–Jacobi Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Time-varying Strict Bounded Real Lemma . . . . . . . . . . . . . . . 3.7 The Lyapunov Analysis of Discrete-continuous Dynamics . . . . . . . . 3.7.1 Global Asymptotic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Illustrative Example: Impulsive Stabilization of a Mechanical Oscillator with Coulomb Friction . . . . . . . . . . . . 3.7.2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2.2 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Linear Operator Inequalities in a Hilbert Space . . . . . . . . . . . . . . . . . . 3.8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1.1 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1.2 Background Material . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Linear Time-delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.3 Well-posedness Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.4 The Lyapunov–Krasovskii Method . . . . . . . . . . . . . . . . . . . . . 3.8.5 Linear Operator Inequalities in a Hilbert Space . . . . . . . . . . . 3.8.6 Illustrative Example: Delay Heat Equation . . . . . . . . . . . . . . .

45 45 47 48 49

66 67 68 72 72 72 72 74 75 77 79 81 82

Finite-time Stability of Uncertain Homogeneous and Quasihomogeneous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Uncertain Systems and Equiuniform Stability . . . . . . . . . . . . . . . . . . . 4.2 Homogeneity Degree and Homogeneity Dilation . . . . . . . . . . . . . . . . 4.3 Finite-time Stability of Locally Homogeneous Systems . . . . . . . . . . . 4.4 Quasihomogeneity Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Finite-time Stability of a First-order Quasihomogeneous System . . . 4.6 Finite-time Stability of a Second-order Quasihomogeneous System .

87 87 89 91 93 94 95

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Part II Synthesis 5

Quasihomogeneous Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.1 Quasihomogeneous Finite-time Stabilization of a Simple Oscillator 105 5.2 Quasihomogeneous Stabilization of Nonlinear Systems of Relative Degree (2, . . . , 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3 Local Quasihomogeneous Stabilization of Underactuated Mechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6

Unit Feedback Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.1 Unit Control and Disturbance Rejection . . . . . . . . . . . . . . . . . . . . . . . 114 6.2 Decomposition of Sliding-mode-based Synthesis Procedure . . . . . . 116 6.3 Global Asymptotic Stabilization of Uncertain Linear Systems . . . . . 117 6.3.1 State Feedback Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.3.2 Output Feedback Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.3.3 Practical Stabilization via Smoothed Unit Feedback . . . . . . . 129 6.4 Unit Feedback Control of Minimum Phase Nonlinear Systems . . . . . 131

7

Disturbance Attenuation via Nonsmooth H∞ -design . . . . . . . . . . . . . . . 133 7.1 Nonsmooth H∞ -control of Time-varying Systems . . . . . . . . . . . . . . . 134 7.1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 7.1.2 Global State-space Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.1.3 Local State-space Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.1.4 Autonomous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.2 Local H∞ -control Synthesis via Sampled-Data Measurements . . . . . 145 7.2.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.2.2 Conversion into the H∞ -control Synthesis via Continuous Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.2.3 Proof of the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.3 Unified SMF and CMF H∞ -control Synthesis . . . . . . . . . . . . . . . . . . . 152 7.3.1 Distribution Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.3.2 The H∞ -design Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.3.3 Illustrative Example: The H∞ -stabilization of an Inverted Pendulum via Nonlinear Sampled-Data Measurements . . . . . 154

Part III Unit Feedback Control of Infinite-dimensional Systems 8

Global Asymptotic Stabilization of Uncertain Linear Systems . . . . . . . 161 8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 8.2 State Feedback Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 8.3 Output Feedback Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.4 Applications to the Linearized Kuramoto–Sivashinsky Equation . . . 175 8.4.1 Numerical Results in the Disturbance-free Case . . . . . . . . . . . 178 8.4.2 Numerical Results Under Non-vanishing External Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

xvi

9

Contents

Asymptotic Stabilization of Minimum-phase Semilinear Systems . . . . 193 9.1 Stabilization in a Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.2 Application to Chemical Tubular Reactor . . . . . . . . . . . . . . . . . . . . . . 199 9.2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 9.2.2 Global Stabilization via State Feedback . . . . . . . . . . . . . . . . . 202 9.2.3 Concentration Observer Design . . . . . . . . . . . . . . . . . . . . . . . . 205 9.2.4 Regional Stabilization via Temperature Feedback . . . . . . . . . 209 9.2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

10 Global Asymptotic Stabilization of Uncertain Time-delay Systems . . . 215 10.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 10.2 Background Material on Discontinuous Time-delay Systems . . . . . . 217 10.3 Delay-/Disturbance-dependent Stability Criterion . . . . . . . . . . . . . . . . 218 10.4 Unit State Feedback Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 10.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Part IV Electromechanical Applications 11 Local Nonsmooth H∞ -synthesis Under Friction/Backlash Phenomena 229 11.1 Position Feedback Regulation of a Multi-link Robot Manipulator with Frictional Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 11.1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 11.1.2 Control Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 11.1.3 Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 11.1.3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 233 11.1.3.2 Dynamical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 11.1.3.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 236 11.2 Output Feedback Regulation of a Servomechanism with Backlash . . 240 11.2.1 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 11.2.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 11.2.3 Control Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 11.2.4 Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 11.2.4.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 244 11.2.4.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 246 12 Quasihomogeneous Stabilization of Fully Actuated Systems with Dry Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 12.1 Orbital Stabilization of an Inverted Pendulum . . . . . . . . . . . . . . . . . . . 249 12.1.1 Tracking of a Modified Van der Pol Oscillator . . . . . . . . . . . . 250 12.1.2 Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 12.2 Global Position Regulation of a Multi-link Robot Manipulator . . . . . 254 12.2.1 Position Control Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 12.2.2 Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

Contents

xvii

13 Hybrid Control of Underactuated Manipulators with Frictional Joints265 13.1 Stabilization of 2-DOF Systems Using Coulomb Friction Hierarchy 266 13.1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 13.1.2 Quasihomogeneity-based Control Synthesis . . . . . . . . . . . . . . 268 13.1.3 Application to a Horizontal Double Pendulum . . . . . . . . . . . . 272 13.1.3.1 State Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 13.1.3.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 273 13.2 Swing-up Control and Stabilization of Pendubot via Orbital Transfer Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 13.2.1 Orbital Stabilization of Pendubot . . . . . . . . . . . . . . . . . . . . . . . 275 13.2.1.1 Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 13.2.1.2 Control Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 13.2.2 Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 13.2.2.1 Pendubot Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . 281 13.2.2.2 Swinging Controller Design . . . . . . . . . . . . . . . . . . . 281 13.2.2.3 Locally Stabilizing Controller Design . . . . . . . . . . . 282 13.2.2.4 Hybrid Controller Design . . . . . . . . . . . . . . . . . . . . . 284 13.2.2.5 Experimental Verification . . . . . . . . . . . . . . . . . . . . . 284 13.3 Quasihomogeneous Swing-up Control and Stabilization of the Cart-pendulum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 13.3.1 State Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 13.3.2 Local Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 13.3.2.1 Transformation to a Regular Form . . . . . . . . . . . . . . 291 13.3.2.2 Locally Stabilizing Synthesis . . . . . . . . . . . . . . . . . . 292 13.3.2.3 Numerical and Experimental Verification . . . . . . . . 294 13.3.3 Swing-up Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 13.3.3.1 Partial Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . 297 13.3.3.2 Control Strategy and Synthesis . . . . . . . . . . . . . . . . . 298 13.3.4 Experimental Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 13.3.4.1 Swinging Controller Design . . . . . . . . . . . . . . . . . . . 302 13.3.4.2 Locally Stabilizing Controller Design . . . . . . . . . . . 303 13.3.4.3 Hybrid Controller Design . . . . . . . . . . . . . . . . . . . . . 304 13.3.4.4 Numerical Verification . . . . . . . . . . . . . . . . . . . . . . . . 304 13.3.4.5 Experimental Verification . . . . . . . . . . . . . . . . . . . . . 306 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

Abbreviations

CMF DOF DPS LMI LOI LTDS KdVB KSE ODE PDE TDS SOSM SMF VSS

Continuous measurement feedback Degree of freedom Distributed parameter system Linear matrix inequalities Linear operator inequalities Linear time-delay system Korteweg–de Vries–Burgers equation Kuramoto–Sivashinsky equation Ordinary differential equation Partial differential equation Time-delay system Second order sliding mode Sampled-data measurement feedback Variable structure system

xix

Chapter 1

Introduction

This book is intended to develop analysis tools and the control synthesis of discontinuous systems operating under uncertainty conditions. Throughout this book, discontinuous systems are regarded as the ones consisting of a family of continuoustime subsystems, equipped with the rule of switching between them. Conceptually, these systems are described as follows. Let the continuous state space, say an Euclidean space or a Hilbert space, be partitioned into an infinite number of operating domains by means of a family of switching surfaces. Inside an operating domain, such a system is governed by an ordinary differential equation with a differentiable right-hand side. Whenever the system trajectory hits a switching surface, the continuous state makes a jump, specified by a restitution rule. Such a rule is a map whose domain and range are the union of the switching surfaces and the entire state space, respectively. The state jumps are typically referred to as impulse effects. A special case is when impulse effects are absent. Just in case, the state trajectory is always continuous, but in general it is not differentiable when it hits a switching surface. After hitting a switching surface, the trajectory can either cross it or evolve along the surface on a finite time interval. In the latter case, sliding motions occur in the system. To summarize, a discontinuous system is specified by i) the family of switching surfaces determining the operating domains, ii) the family of continuous time subsystems, well-posed for each domain, iii) the restitution rule, determining potential jumps of the system, and iv) the sliding mode dynamics, governing potential state motions on the switching surfaces. We introduce here some basic models of discontinuous systems to preview issues that will arise in our study.

1.1 Impulsive Systems A rather general model of a dynamic system of class C1 with impulse effects can be introduced as follows. Let the continuous dynamics of the system be governed by

1

2

1 Introduction

x˙ = f (x,t)

(1.1)

where x ∈ Rn is the state vector, t is the time variable, the time-varying vector field f (x,t) is continuously differentiable in x and integrable in t. Consider a discrete set {τk } of time instants τ1 < τ2 < . . . < τk < τk+1 < . . . where the state of the system is admitted to make a jump, i.e., x(t0 −) = limt↑t0 x(t) = x(t0 +) = limt↓t0 x(t). Let the jump Δ x(τk ) = x(τk +) − x(τk −) (1.2) in the state vector at the time instant τk be determined by a restitution rule. Such a restitution rule U(τk , x(τk− )) is to be pre-specified, e.g., by using physical laws. Examples are impact dynamics (collisions, percussions, etc.) deduced for mechanical systems from Newton’s or Poisson’s restitution laws [33]. Then the impulsive system is described by  x˙ = f (x,t), t = τk (1.3) Δ x(τk ) = U(τk , x(τk −)), t = τk , k = 1, 2, . . . , and, for certainty, the solutions of the system are assumed to be left-continuous. Many aspects of such an impulsive differential system, including their well-posedness and stability, are now fully understood. There is already a large body of literature on this subject. In this regard, we refer to [13, 91, 123, 201, 246] to name a few references. In the present context of the synthesis of impulsive systems the main issue becomes as to how to construct accessible and manipulatable inputs resulting in the pre-scribed restitution rule in (1.3). The following example is borrowed from [220] to illustrate this issue. Example 1.1 (Impulsive stabilization of Chua’s circuit). A dimensionless version x˙1 = α0 [x2 − x1 − φ (x1 )] x˙2 = x1 − x2 + x3 x˙3 = −β0 x2 − γ0 x3

(1.4)

of Chua’s circuit is obtained from (1.1) by representing its right-hand side in the form f (x,t) = Ax + Φ (x) (1.5) where

⎤ ⎡ ⎤ −α0 φ (x1 ) −α0 α0 0 ⎦. 0 A = ⎣ 1 −1 1 ⎦ , Φ (x) = ⎣ 0 −β0 −γ0 0 ⎡

(1.6)

In the above representation, φ (x1 ) is the nonlinear characteristic of the Chua’s diode which is given by

1.1 Impulsive Systems

3

1 φ (x1 ) = b0 x1 + (a0 − b0)(|x1 + 1| − |x1 − 1|), 2

(1.7)

α0 , β0 , γ0 are positive constants, and a0 , b0 are negative constants such that a0 < b0 . Due to [220], the impulsive stabilization (1.3) of Chua’s circuit (1.5)–(1.7) can be achieved with a linear restitution rule U(τk , x(τk −)) = Bx(τk −).

(1.8)

Let d1 denote the largest eigenvalue of (I + BT )(I + B), and let B be a symmetric matrix with ρ (I + B) ≤ 1 where ρ (I + B) denotes the spectral radius of I + B. Let q be the largest eigenvalue of (A + AT ) and let the impulses be equidistant with time interval Δ > 0, i.e., τk+1 − τk = Δ for all k = 1, 2, . . .. If the following inequalities 0 ≤ q + 2α0|a0 | ≤ −

1 log(ξ d1 ) Δ

(1.9)

are satisfied with some ξ > 1, then by Corollary 1 of [220] the origin of the impulsively controlled Chua’s circuit (1.3)–(1.8) is globally asymptotically stable. The question then arises how the stabilizing restitution rule, given in the feedback form (1.8), can be implemented by using strong current sources to charge capacitors such that the capacitor voltages are changed according to (1.8) in a very short time. It is worth noting that a programming restitution rule U(τk ) is straightforwardly implemented by applying the external impulse (current) U(τk )δ (t − τk ) so that no implementation question (at least, theoretically) would arise in this case. The aforementioned implementation issue will be addressed later in the book within the framework of Schwartz’ distribution theory in a nonlinear setting [164]. Briefly, this framework is as follows. Let the system jump be caused by external inputs so that (1.3) can be represented in the form x˙ = f (x,t) + b(x,t)u

(1.10)

where the controlled vector input u ∈ Rm and the matrix function b(x,t) is of an appropriate dimension and continuous in all the arguments. In order to admit discontinuous dynamics of the system, the meaning of the differential equation (1.10) should be extended to impulse δ -wise inputs u(t). In a particular case where b(x,t) is a state-independent function, solutions of (1.10) subject to an impulse input are defined in the mild sense as those of the corresponding integral equation. In general, this way is hampered, however, by the irregularity of a product of the impulse function u(t) and the function b(x(t),t), being discontinuous in t if computed on potential solutions of (1.10). A generalized solution of (1.10) is then defined as a limiting result of standard solutions under a special approximation of the impulse input u(t) by integrable functions uk (t), k = 1, 2, . . .. While being unbounded, pre-limiting functions uk (t) can make trajectories xk (t) of (1.10) be arbitrarily close to discontinuous ones, which is why the generalized solution x(t) = limk→∞ xk (t) is expected to be discontinuous.

4

1 Introduction

It is the latter point of view on the modeling of impulsive systems that prevails in this book. Since different input approximations may result in different generalized solutions, the present modeling of impulsive systems accommodates the details of the realization of the impulse input, and admits the impulse response to depend upon the manner in which the impulse is implemented. The above model (1.10) of impulsive systems has attracted considerable research interest (see [32, 149, 161, 165, 247] and references quoted therein). The capability of (1.10) to produce a unique impulse response regardless of impulse approximations is referred to as vibrocorrectness, and it has been studied in [160, 165]. For vibrocorrect impulsive systems relations between the restitution rule U(τk , x(τk −)) in (1.3) and the input gain b(x,t) in (1.10) have been established in [164, 172]. These issues, among others, will be developed in the sequel.

1.2 Variable-structure Systems An autonomous switched system with no impulse effects serves as a prototypical example of a variable structure system (VSS). The operating domain of such a system, say Rn , is partitioned into a finite number of subdomains G j ⊂ Rn , j = 1, . . . , N with disjoint interiors and boundaries ∂ G j of measure zero. Within each of these subdomains, the system is governed by an ordinary differential equation x˙ = f j (x), x ∈ G j , j = 1, . . . , N

(1.11)

where the vector field f j (x) is continuous and it has a finite limit f j (x) as the argument x∗ ∈ G j approaches a boundary point x ∈ ∂ G j . A variable structure system yields a behavior that is significantly different from the behavior of each individual subsystem. For instance, it may be possible to stabilize a system by varying its structure, even if all individual subsystems are unstable. The following example, extracted from [228], illustrates this feature. Example 1.2 (VSS puzzle). Let the second-order system x¨ = u(x, x) ˙

(1.12)

consist of two linear unstable structures u = u1 and u = u2 , corresponding to u1 (x, x) ˙ = 6x˙ + 16x ˙ = 6x˙ − 16x. u2 (x, x)

(1.13) (1.14)

The phase portraits of these structures are shown in Fig. 1.1. The equation 2x + x˙ = 0

(1.15)

appears here for the stable eigenvector in the case of u = u1 where the equilibrium point is an unstable saddle.

1.2 Variable-structure Systems

5 x˙

x˙

x

(a)

x

(b)

Fig. 1.1 Phase portrait of a) the unstable structure (1.12), (1.13) (unstable saddle); b) the unstable structure (1.12), (1.14) (unstable focus)

A switching rule which forces the system structure to be asymptotically stable is given by  6x˙ + 16x i f xs(x, x) ˙ <0 u(x, x) ˙ = (1.16) 6x˙ − 16x i f xs(x, x) ˙ >0 where s(x, x) ˙ = x + x. ˙

(1.17)

The phase portrait of the resulting system (1.12), (1.16) is depicted in Fig. 1.2. It is concluded from this figure that the state trajectories cross the switching line x = 0 whereas they are pointing toward each other on the switching line s(x, x) ˙ = 0. Thus, the only possible behavior for the trajectories is to slide along this line. This motion is known as a sliding mode and for (1.12) and (1.16) it is unambiguously set by the switching line equation x˙ + x = 0. Since the latter equation is asymptotically stable, the overall system (1.12), (1.16) is asymptotically stable, too. Sliding modes are long recognized as a powerful control method to counteract non-vanishing disturbances and plant uncertainties. Due to these advantages they have become a principal operational mode in VSS’. A standard sliding mode control is synthesized to steer the system to a switching manifold in finite time; after that the system stays in this manifold forever. Typically [65, 227, 228], the switching manifold has a codimension equal to the number of independent controlled vector fields. Thus, another advantage of the order reduction is provided by keeping the system in the sliding mode along the switching manifold. Alternatively, by not forcing the state to evolve on the switching manifold, a higher-order reduction becomes possible for controlled systems whose vector relative degree has at least one component greater than one. To support this claim, we

6

1 Introduction x˙

x s=0

Fig. 1.2 Phase portrait of the closed-loop VSS (1.12), (1.16)

present the optimal synthesis from [248] for the two-dimensional Fuller problem [78]. Example 1.3 (The Fuller phenomenon). Minimize the integral  ∞ 0

x2 (t)dt

(1.18)

on the trajectories of (1.12) under the input constraint |u(t)| ≤ 1 f or all t ≥ 0.

(1.19)

By minimum principle, the switching curve for the optimal synthesis is governed by: s(x, x) ˙ = x + cx˙2 sign x˙ (1.20) for some constant c whereas the optimal synthesis is given by  1 i f s(x, x) ˙ <0 u(x, x) ˙ = . −1 i f s(x, x) ˙ >0

(1.21)

The qualitative behavior of the closed-loop system (1.12), (1.20), (1.21) is shown in Fig. 1.3. Rather than generate sliding modes on the switching curve, the optimal trajectories cross it at, countably, many points. The time intervals between successive switches form a decreasing geometric progression which is why the switching times have a finite accumulation point. Such a motion is known as Zeno behavior. Thus, (1.12) of relative degree 2 under the variable structure (1.20), (1.21) is steered to the origin in finite time; after that there appears a so-called sliding mode of the second order [76, 77]. As opposed to Example 1.2, the present sliding mode has codimension 2, i.e., it is confined to the origin only.

1.3 Hybrid Systems

7 x˙

0

x

0

Fig. 1.3 Fuller phenomenon (the dotted line is for the switching curve, the solid line is for an optimal trajectory)

In a more complicated VSS, the task of the deliberate introduction of Zeno trajectories and extending them beyond their accumulation point is far from trivial, and it is studied in the sequel.

1.3 Hybrid Systems Systems that inherently combine dynamic processes and discrete events are usually called hybrid. The interested reader can consult [34, 35, 82, 91, 143, 202, 230] for a comprehensive study of hybrid systems. Although these systems are of interest in themselves, specifics of the discrete dynamics are often ignored to analyze a more general description where the discrete behavior is involved as a particular model. Example 1.4 (Hysteresis switching). Let the operational domain of the VSS (1.11) consist of two subdomains G1 = {x ∈ Rn : s(x) < 0} and G2 = {x ∈ Rn : s(x) > 0} with the common boundary ∂ G1 = ∂ G2 = {x ∈ Rn : s(x) = 0} and s(x) : Rn → R1 being of class C1 . Assume that the switch between the structures f1 (x) and f2 (x) is performed according to  f1 (x) i f s(x) < −Δ f (x) = (1.22) f2 (x) i f s(x) > Δ and it has a hysteresis inside the stripe |s(x)| ≤ Δ of the width Δ > 0 where f (x) maintains the value it had when the trajectory hit the stripe boundary |s(x)| = Δ for the last time. Also assume that the vector fields f1 (x) and f2 (x) are directed toward each other whenever s(x) = 0 thereby ensuring that the ideal VSS, corresponding to Δ = 0, slides along the surface s(x) = 0. Then, given Δ > 0 is small enough, hysteresis switching results in oscillations of the state vector in the Δ -vicinity of this surface.

8

1 Introduction

f2Δt1

2Δ

f1Δt1

s(x)=0

Δx Fig. 1.4 Hysteresis switching

Since in the presence of the hysteresis, two consecutive switching events are always separated in time and the current system dynamics depends on the last preceding switch, (1.11) with hysteresis switching (1.22) becomes truly hybrid. Computing the average velocity f0 =

grad s · f1 grad s · f2 f2 − f1 grad s · ( f1 − f2 ) grad s · ( f1 − f2 )

(1.23)

in the region |s(x)| ≤ Δ one arrives at the approximate model x˙ = f0 (x)

(1.24)

of the hysteresis dynamics. The average velocity (1.23) has been derived by employing the relation Δx f0 = Δt where Δ t = Δ t1 + Δ t2 consists of the time intervals

Δ t1 =

2Δ 2Δ , Δ t2 = grad s · f1 grad s · f2

between consecutive switches of the structure of the system (see Fig. 1.4) and the state shift Δ x on the time interval Δ t is computed as follows:

Δ x = f1 Δ t1 + f2 Δ t2 . Since the same equations (1.23), (1.24) are invoked when Δ = 0 to describe Filippov’s solutions [71] of the corresponding VSS (1.11) on the switching surface s(x) = 0 , this VSS can be viewed as a higher-level abstraction of the hysteresis dynamics. In the present book, hybrid systems basically appear as such abstractions only. Although their general study is beyond the scope of the book some hybrid control algorithms are, however, developed for underactuated mechanical systems.

1.4 Sliding Modes in a Hilbert Space

9

1.4 Sliding Modes in a Hilbert Space Many important plants, such as time-delay systems, flexible manipulators, and structures, as well as heat transfer processes, combustion, and fluid mechanical systems are governed by functional and partial differential equations or, more generally, equations in a Hilbert space. As these systems are often described by models with a significant degree of uncertainty, it is of interest to develop consistent stabilization methods that are capable of utilizing time-delay and distributed parameter models and providing the desired system performance in spite of the model uncertainties. In the present book, this problem is treated within the discontinuous sliding mode-based control methods, recently developed for infinite-dimensional systems [162, 171, 174, 175]. A stabilizing discontinuous control law is obtained from the Lyapunov min-max approach, the origins of which may be found in [89, 90]. It is synthesized to guarantee the time derivative of a Lyapunov functional, selected for a nominal system, remains negative definite on the trajectories of the system with perturbations caused by uncertainties of a plant operator and environment conditions. The approach gives rise to a so-called unit feedback signal, whose norm is equal to one everywhere with the exception of the sliding surface where it undergoes discontinuities. Capability of an infinite-dimensional system to slide along its discontinuity manifold and attractive robustness features of the unit feedback synthesis are illustrated here with an extremely simple example. Example 1.5 (Unit feedback in a Hilbert space). Let an infinite-dimensional system x˙ = U(x), x(0) = x0 ∈ H

(1.25)

be driven in a real Hilbert space H by the unit control action U(x) = −

x , x

(1.26)

being discontinuous at x =0. Since the norm x = x, x in the Hilbert space is defined via the inner product ·, · , differentiating the function x(t)2 on the trajectories of the closed-loop system (1.25), (1.26) yields dx(t)2 = 2(x(t), x(t)) ˙ = −2(x(t). dt

(1.27)

Solving the differential equation (1.27) analytically, one derives that x(t) = x0  − t for t ≤ x0 . Thus, (1.25), (1.26) steer to the origin in finite time and due to (1.27) it never leaves the origin. Hence, starting from the time moment t = x0 , in the infinite-dimensional system (1.25), driven by (1.26), there appears a sliding mode on the trivial surface x = 0. Clearly, this sliding mode is unambiguously set by the surface equation x = 0 regardless of whichever additive dynamic disturbance h(x,t), uniformly bounded h(x,t)H ≤ ε in all arguments with a norm bound satisfying ε < 1, affects the

10

1 Introduction

system. Indeed, an influence of such a disturbance on the sliding mode is rejected by the unit feedback signal because the time derivative of the function V (t) = x(t)2 along the trajectories of the perturbed system

is estimated as follows:

x˙ = U(x) + h(x,t)

(1.28)

 V˙ (t) ≤ −2(1 − ε ) V (t).

(1.29)

By the comparison principle [110], an arbitrary solution V (t) of the differential inequality (1.29) is dominated V (t) ≤ V0 (t) f or all t ≥ 0 by a solution V0 (t) of the corresponding differential equation  V˙0 (t) = −2(1 − ε ) V0 (t)

(1.30)

(1.31)

subject to the same initial condition V0 (0) = V (0). By an analogy to the solution of (1.27), it follows that V0 (t) = 0 for all t ≥ (1 − ε )−1 x0  and by virtue of (1.30), V (t) vanishes after a finite time moment, too. Thus, the perturbed version (1.26), (1.28) of (1.25), (1.26) is still driven to the origin in finite time and it is maintained there forever. It is worth noticing, however, that in general neither the unit feedback to be constructed belongs to the state space nor the discontinuity manifold is pre-specified; that is why their synthesis is far from trivial and calls for further investigation.

Part I

Mathematical Tools

12

I Mathematical Tools

Differential equations with a piece-wise continuous right-hand side, which are accepted as a basic mathematical model of a discontinuous system, are under study. While allowing Dirac functions in the coefficients, the equations admit instantaneous jumps of the state of the system. The instantaneous impulse response of the system is adequately defined according to Schwartz’ distribution theory in a nonlinear setting. Various solution concepts, namely, Filippov solutions [71], Utkin solutions [227], and vibroimpact solutions [164] are introduced for these equations. The existence and uniqueness of the solutions, as well as their physical sense and applications to modeling of nonlinear phenomena in electromechanical systems, are discussed. The nonsmooth Lyapunov analysis of discontinuous systems and that of discretecontinuous dynamics are then developed side by side. Several stability concepts such as stability and exponential/asymptotic stability are addressed locally and in large. Particularly, semiglobal Lyapunov functions are introduced to subsequently address a relatively new kind of robust stability when the system is required to be asymptotically stable, equiuniformly in admissible non-vanishing external disturbances. In addition, L2 -gain analysis in a finite-dimensional setting and LMI-based analysis in infinite-dimensional setting are presented. A special attention is paid to homogeneous systems. These systems are of a particular interest because, under appropriate conditions on the homogeneity degree, their equiuniform asymptotic stability ensures that the state of the system escapes to zero in finite time. The finite-time stability property persists, even if the system is affected by inhomogeneous external disturbances. This result constitutes the quasihomogeneity principle whose capabilities are illustrated by several examples.

Chapter 2

Mathematical Models

2.1 Nonlinear Differential Equations in Distributions The present section develops a consistent modeling methodology which gives rise to mathematical models of nonlinear dynamical systems with nonlinear impulse responses. Modeling accommodates the details of the realization of the impulse input, and admits the impulse response to depend upon the manner in which the impulse is implemented. The set of all possible impulse responses is found from a certain auxiliary system with integrable inputs. Necessary and sufficient conditions are additionally obtained for the impulse response to be unique and independent of the impulse realization. The proposed modeling methodology is subsequently used to derive filtering equations over sampled-data measurements and to synthesize impulsive controllers.

2.1.1 Preliminaries A linear continuous functional, mapping the space D0 of continuous functions with compact support into the real line R1 , is referred to as a zero-order distribution. Thus, the distributions are defined indirectly, by specifying their effect on the test functions. Recall that the support of a function ϕ (t) specified point-wise is the closure of the set {t : ϕ (t) = 0}. As usual, D∗0 denote the dual space of zero-order distributions. The dual product ∗ < ∞u, ϕ > of a distribution u(t) ∗∈ D0 and a test function ϕ (t) ∈ D0 is denoted by u(t) φ (t)dt. Alternatively, D may be viewed as the space of all (Borel) measures −∞ 0 d μ (t) with locally bounded variations. ∞ The dual product is then explicitly defined by the Stieltjes integral < d μ , ϕ >= −∞ φ (t)d μ (t). A sequence of distributions uk (t) ∈ D∗0 , k = 1, 2, . . . converges to u(t) ∈ D∗0 in weak* topology if

13

14

2 Mathematical Models

lim

 ∞

k→∞ −∞

uk (t)ϕ (t)dt =

 ∞ −∞

u(t)ϕ (t)dt

for any test function ϕ (t) ∈ D0 .

2.1.2 Instantaneous Impulse Response in a Nonlinear Setting In what follows, we deal with affine systems, the dynamics of which are described by a nonlinear differential equation of the form x(t) ˙ = f (x,t) + b(x,t)u, x(0) = x0

(2.1)

where x(t) ∈ Rn is the state vector, x0 ∈ Rn is the initial state, u(t) ∈ Rm is the input, and t ≥ 0 is the time variable. Without loss of generality, we have assumed that the system in question is initialized at t = 0; otherwise, a time substitution s = t − t0 with an appropriate constant t0 should be applied. It is well known that the smoothness of f and b guarantees locally the existence of a unique Caratheodory trajectory, driven by an integrable input u(t). For the convenience of the reader the following definitions are reviewed. Definition 2.1. An absolutely continuous function x(t), defined on some interval [0, τ ), is said to be a Caratheodory solution of (2.1) iff it is initialized in accordance with (2.1), and it satisfies (2.1) for almost all t ∈ [0, τ ). Clearly, unbounded inputs can make trajectories of System 2.1 be arbitrary close to discontinuous ones. In order to admit discontinuous behavior of the system one should extend the description. If b(x,t) is a state-invariant function b(t), it appears that one can rigorously introduce discontinuous solutions into the equation by admitting the input to be a measure-type function (e.g., a δ -pulse). In general, this way is hampered, however, by the irregularity of a product of the impulsive input u(t) and the discontinuous (in t) function b(x(t),t). In order to avoid this we generalize the meaning of the differential equation as follows. Definition 2.2. A sequence {uk (t)} of integrable inputs, the L1 -norms of which are uniformly bounded, is a generalized system input, if the solutions xk (t), k = 1, 2, . . . of (2.1), corresponding to the inputs u(t) = uk (t), converge to a left-continuous function x(t) for all continuity points t ≥ 0 of x(t). The function x(t) is referred to as a generalized solution of (2.1). Relating the generalized control input {uk (t)} in the above definition to a peaking sequence of L1 inputs, which weakly* converges to a δ -pulse, one can define the impulse response of a nonlinear system as the corresponding generalized solution of (2.1). Generally speaking, different approximations of the δ -pulse result in different generalized solutions. Thus, the impulse response depends upon the implementation of the impulse. The symbol x{uk } (t) is used to denote the generalized solution of (2.1) corresponding to a generalized input {uk (t)}.

2.1 Nonlinear Differential Equations in Distributions

15

The result stated below describes the set X(γ ,t0 ) = {x{uk } (t0 +) : ∗ − lim uk (t) = γδ (t − t0 )} k→∞

of the instantaneous impulse responses x{uk } (t0 +) = limt↓t0 x{uk } (t) of (2.1) to all possible realizations {uk (t)} of the impulsive input γδ (t − t0 ), γ ∈ Rm , t0 ≥ 0. We shall prove that X (γ ,t0 ) can be specified by means of the reachability set R(y0 , γ ,t0 ) = {ηw (y0 , 1,t0 ) :

 1

w(t)dt = γ }

0

of the trajectories ηw (y0 ,t,t0 ) of the auxiliary dynamical system

η˙ = b(η ,t0 )w(t), η (0) = y0 with integrable inputs w(t) of fixed integral power lution of the unforced system

1 0

(2.2)

w(t)dt = γ . Denoting the so-

y˙ = f (y,t), y(0) = x0

(2.3)

as y(t), we arrive at the following [163]. Theorem 2.1. Let functions f (x,t), b(x,t) ∈ C1 satisfy the linear growth condition in x. Then X (γ ,t0 ) = R(y(t0 ), γ ,t0 ). Moreover, if y1 ∈ R(y(t0 ), γ ,t0 ) and the auxiliary system (2.2) is driven from the initial state y(t0 ) to the terminal state y1 for t = 1 by an admissible input w(t), i.e., ηw (y(t0 ), 1,t0 ) = y1 , then the instantaneous impulse response x{uk } (t0 +) = y1 of the affine system (2.1) is particularly forced by the generalized input {uk (t)} where  kw(k(t − t0 )) if t ∈ [t0 ,t0 + 1/k] uk (t) = 0 otherwise, k = 1, 2, . . . . (2.4) Proof. First, we shall demonstrate that functions (2.4) do converge to the impulsive input γδ (t − t0 ) in the weak* topology. Indeed,  ∞ −∞  1 0

ϕ (t)uk (t)dt =

 t0 +1/k t0

ϕ (t)kw(k(t − t0 ))dt =

t ϕ (t0 + )w(t)dt → ϕ (t0 ) k

 1 0

w(t)dt = ϕ (t0 )γ

as k → ∞ for all ϕ ∈ D0 . Now, in order to prove that {uk (t)} is a generalized input and x{uk } (t0 +) = ηw (y(t0 ), 1,t0 ), we introduce functions

16

2 Mathematical Models

⎧ if s ≤ 0 ⎨ xk (t0 + s) ηk (s) = xk (t0 + s/k) if s ∈ [0, 1] ⎩ xk (t0 + 1/k + s − 1) if s ≥ 1

(2.5)

where xk (t), k = 1, 2, . . . are the trajectories of (2.1) forced by the input actions (2.4). Then ηk (s), k = 1, 2, . . . satisfy

η˙ k (s) = f (ηk ,t0 + s), s ≤ 0 1 s η˙ k (s) = f (ηk ,t0 + ) + k k s b(ηk ,t0 + )w(s), 0 ≤ s ≤ 1 k η˙ k (s) = f (ηk ,t0 + 1/k + s − 1), s ≥ 1

(2.6)

and due to the continuous dependence of the solution on the small parameter 1/k, the functions ηk (s) converge point-wise to the solution of ⎧ if s ≤ 0 ⎨ f (η ,t0 + s) if s ∈ [0, 1] η˙ (s) = b(η ,t0 )w(s) (2.7) ⎩ f (η ,t0 + s − 1) if s ≥ 1, initialized with η (−t0 ) = x0 , as k → ∞. By virtue of (2.5) it follows that  η (t − t0) if t ≤ t0 x{uk } (t) = η (t − t0 + 1) if t > t0 and x{uk } (t+) = ηw (y(t0 ), 1,t0 ). Hence, if y1 ∈ R(y(t0 ), γ ,t0 ) and the auxiliary system (2.2) is driven from the initial state y(t0 ) to the terminal state y1 for t = 1 by an admissible input w(t), then the instantaneous impulse response x{uk } (t0 +) = y1 of (2.1) is particularly forced by the generalized input (2.4). Thus, the inclusion R(y(t0 ), γ ,t0 ) ⊂ X (γ ,t0 ) is shown. Proof of the converse inclusion follows the same method of time substitution. Let {uk (t)} be a generalized input such that the convergence ∗ − lim uk (t) = γδ (t − t0 ) k→∞

(2.8)

takes place in weak* topology and let x{uk } (t) be the generalized solution of (2.1) corresponding to the generalized input, i.e., x{uk } (t) = lim xk (t) f or all t ≥ t0 k→∞

(2.9)

where xk (t) is the trajectory of (2.1) driven by the input uk (t). By Helly’s theorem (see, e.g., [111]), the weak* convergence (2.8) means that   t 0 for all t < t0 lim uk (τ )d τ = (2.10) γ for all t > t0 k→∞ 0

2.1 Nonlinear Differential Equations in Distributions

17

and  t 0

uk (τ )d τ ≤ M(t)

(2.11)

where M(t) < ∞ for all t ≥ 0. Now define αk (t), βk (t), νk , μk (t), k = 1, 2, . . . by

αk (t) = t +

t 0

uk (s)ds, βk (t) = αk−1 (t)

νk = αk (t0 + 1/k) − αk (t0 − 1/k), μk (t) = αk (t0 − 1/k) + νkt and introduce functions

⎧ if s ≤ 0, ⎨ xk (t0 − 1/k + s) if s ∈ [0, 1] ζk (s) = xk (βk (μk (s))) ⎩ xk (t0 + 1/k + s − 1) if s ≥ 1.

(2.12)

Then ζk (s), k = 1, 2, . . . satisfy the differential equations

ζ˙k (s) = f (ζk ,t0 − 1/k + s) + b(ζk ,t0 − 1/k + s)uk (t0 − 1/k + s), s ≤ 0, ν f (ζ , β (μ (s))) + ζ˙k (s) = k k k k 1 + uk(βk (μk (s))) νk b(ζk , βk (μk (s)))uk (βk (μk (s))) + , 0 ≤ s ≤ 1, 1 + uk(βk (μk (s))) ζ˙k (s) = f (ζk ,t0 + 1/k + s − 1) + b(ζk ,t0 + 1/k + s − 1)uk(t0 + 1/k + s − 1), s ≥ 1.

(2.13)

(2.14)

(2.15)

Due to (2.8), (2.9), it follows that for all s ≤ 0 functions ζk (s) converge to the solution of the equation ζ˙ (s) = f (ζ ,t0 + s), ζ (−t0 ) = x0 as k → ∞, whereas for s ∈ [0, 1] the point-wise convergence of ζk (s) is guaranteed by their uniform boundedness and equicontinuity. Indeed, due to our assumptions regarding the functions f , b and by virtue of inequalities

νk = 2/k +

 t0 +1/k t0 −1/k

uk (τ )d τ ≤ 2 + M(t0 + 1) < ∞,

1 uk (βk (μk (s))) ≤ 1, ≤ 1, 1 + uk(βk (μk (s))) 1 + uk(βk (μk (s))) solutions of (2.14) are uniformly bounded, have uniformly bounded time derivatives, and converge at s = 0 as k → ∞. Then, Arzela’s theorem (see, e.g., [116]) ensures their point-wise convergence to a function that, according to Filippov’s lemma [70], satisfies the equation

18

2 Mathematical Models

ζ˙ (s) = f (ζ ,t0 )w0 (s) + b(ζ ,t0 )w(s). Since (2.10) implies that

νk = 0, k→∞ 1 + uk (βk (μk (s)))  1 νk uk (βk (μk (s))) lim ds = k→∞ 0 1 + uk (βk (μk (s))) lim

lim

 t0 +1/k

k→∞ t0 −1/k

uk (t)dt = γ ,



then w0 (s) ≡ 0 and 01 w(s)ds = γ . Therefore, in order to calculate the instantaneous impulse response x{uk } (t0 +) = ζ (1), we arrive at the same equation (2.2) subject to the same initial condition y0 = y(t0 ) and the same integral condition 01 w(s)ds = γ . Clearly, X(γ ,t0 ) ⊂ R(y(t0 ), γ ,t0 ) and Theorem 2.1 is completely proved. Thus, under the assumptions of Theorem 2.1, the set of all generalized solutions can be viewed as weak* closure of the set of the Caratheodory solutions corresponding to the integrable inputs. Due to the peaking phenomenon [219], the impulse response of the affine system (2.1) with no growth condition on the right-hand side can escape to infinity in infinitesimal time. The following example illustrates this property. Example 2.1. Let (2.1) be specified to x˙ = x2 u(t), x(0) = 1

(2.16)

where f = 0 and b = x2 , and let its generalized input u(t) = {uk (t)} be given by  kt if t ∈ [0, 0 + 1/k] uk (t) = 0 otherwise, k = 1, 2, . . . . (2.17) It is then straightforwardly verified that the weak* convergence ∗ − lim uk (t) = δ (t) k→∞

(2.18)

holds and the Caratheodory solution xk (t) of (2.16) with input (2.17), being substituted into (2.16) for u(t), is locally determined by xk (t) = (1 − kt)−1 for t ∈ [0, 1k ) so that x{uk } (t+) = lim xk (t+) = ∞. k→∞

Thus, the generalized solution x{uk } (t) of (2.16) with the δ -wise input (2.17) escapes to infinity in infinitesimal time. It is of interest to note that the Frobenius condition

2.1 Nonlinear Differential Equations in Distributions n Σk=1

19

∂ bli (x,t) n ∂ bl j (x,t) bk j (x,t) = Σk=1 bki (x,t), ∂ ξk ∂ ξk

(2.19)

imposed on the matrix function b(x,t) = {bki (x,t)} for all l = 1, . . . , n, i, j = 1, . . . , m, and x ∈ Rn , t ≥ 0, ensures the uniqueness of the impulse response [163]. Theorem 2.2. Let the assumptions of Theorem 2.1 be satisfied and let {uk (t)} be a generalized input of (2.1) such that ∗ − lim uk (t) = γδ (t − t0 ), γ ∈ Rm , t0 ≥ 0 k→∞

(2.20)

where γ ∈ Rm and t0 ≥ 0 are arbitrary. Then, a generalized solution x(t) = x{uk } (t) of (2.1) does not depend upon a choice of the approximating sequence {uk (t)} if and only if the Frobenius condition (2.19) holds for all l = 1, . . . , n, i, j = 1, . . . , m, and x ∈ Rn , t ≥ 0. Proof. Let the Frobenius condition (2.19) hold for all l = 1, . . . , n, i, j = 1, . . . , m, and x ∈ Rn , t ≥ 0. Then, the corresponding Pfaffian equation d ξ /dv = b(ξ ,t0 ), ξ ∈ Rn , v ∈ Rm , t0 ≥ 0

(2.21)

is well known [42] to possess a solution for arbitrary initial conditions ξ (0) = z ∈ Rn and t0 ≥ 0. Just in case, the Pfaffian equation (2.21) integrates to the function ξ (z, v,t0 ) regardless of a path v(s) between the initial point v(0) = y0 and the terminal point v(1) = γ . Hence, the reachability set R(y0 , γ ,t0 ) of (2.21), written in the parametric form (2.2) with w(t) = v(t), ˙ consists of the unique point ξ (y0 , γ ,t0 ) and, according to Theorem 2.1, the instantaneous impulse response is uniquely defined as x(t0 +) = ξ (y(t0 ), γ ,t0 )

(2.22)

where y(t) satisfies (2.3). To complete the proof it remains to demonstrate that the uniqueness of the impulse response guarantees the validity of the Frobenius condition. For this purpose, we specify an initial condition x0 ∈ Rn in (2.3) to ensure that the corresponding solution y(t) of this equation takes an a priori given value z ∈ Rn at t = t0 . By virtue of the assumptions, imposed on the function f (x,t), there always exists such an initial condition. Then the generalized solution x{uk } (t) of (2.1) takes the same value z at t = t0 before it makes the jump enforced by the generalized input (2.20), and by Theorem 2.1, x{uk } (t0 +) = ηγ (1) where ηγ (τ ) is a solution of

η˙ γ = b(ηγ ,t0 )γ , ηγ (0) = z.

(2.23)

Based on ηγ (τ ), thus specified, let us now introduce the function

ξ (z, v,t0 ) = ηv (1) = z +

 1 0

b(ηv (τ ),t0 )vd τ

(2.24)

20

2 Mathematical Models

that proves to meet the condition

∂ ξ (z, v,t0 ) |v=0 = b(z,t0 ), ∂v

(2.25)

computed at v = 0. Indeed, by taking into account the trivial relation

ξ (z, 0,t0 ) = z,

(2.26)

one derives that lim

v→0

ξ (z, v,t0 ) − ξ (z, 0,t0 ) − b(z,t0 )v v 

= lim

1

0 [b(ηv (τ ),t0 ) − b(z,t0 )]vd τ 

v

v→0

≤ lim  v→0

 1 0

[b(ηv (τ ),t0 ) − b(z,t0 )]d τ  = 0,

(2.27)

because limv→0 ηv (τ ) = z for all τ ∈ [0, 1]. Furthermore, the function ξ (z, v,t0 ), governed by (2.24), satisfies (2.21). To establish this, let us first note that

ξ (z, γ1 + γ ,t0 ) = ξ (ξ (z, γ1 ,t0 ), γ ,t0 ). In order to prove (2.28) it suffices to consider the generalized inputs  k(γ1 + γ ) if t ∈ [t0 ,t0 + 1/k] uk (t) = , 0 otherwise ⎧ ⎨ kγ1 if t ∈ [t0 ,t0 + 1/k] uk (t) = kγ if t ∈ [t0 + 1/k,t0 + 2/k] , ⎩ 0 otherwise k = 1, 2, . . . .

(2.28)

(2.29)

Similar to (2.4), both inputs in (2.29) are straightforwardly verified to converge to (γ1 + γ )δ (t − t0 ) in the weak* topology, so that they produce the same generalized solution x{uk } (t) = x{u } (t) of (2.1). It follows that k

ξ (z, γ1 + γ ,t0 ) = x{uk } (t0 +) = x{u } (t0 +) = ξ (ξ (z, γ1 ,t0 ), γ ,t0 ), k

thereby yielding (2.28). To this end, relations (2.25), (2.26), and (2.28), coupled together, lead to ξ (z, v + Δ v,t0) − ξ (z, v,t0 ) − b(z,t0)Δ v Δ v ξ (ξ (z, v,t0 ), Δ v,t0 ) − ξ (ξ (z, v,t0 ), 0,t0 ) − b(z,t0 )Δ v = lim = 0. (2.30) Δ v Δ v→0 lim

Δ v→0

2.1 Nonlinear Differential Equations in Distributions

21

Hence, the function ξ (z, v,t0 ), governed by (2.24), solves the Pfaffian equation (2.21) under arbitrary initial conditions ξ (0) = z ∈ Rn and t0 ≥ 0. It is well known [42] that this ensures the Frobenius condition (2.19) to hold for all l = 1, . . . , n, i, j = 1, . . . , m and x ∈ Rn , t ≥ 0. The proof is completed.

2.1.3 Vibroimpact Solutions As shown, modeling the nonlinear impulsive system (2.1) should, generally speaking, accommodate details of the impulse implementation, and the set of all possible instantaneous impulse responses may be found from the auxiliary system (2.2) with integrable inputs. In turn, imposing the Frobenius condition on the underlying system allows one to replace peak functions in modeling such a system by δ -pulses. The following definition is thus in order. Definition 2.3. System 2.1 is said to be vibrocorrect if given an initial condition x0 and a generalized input (2.20), the system possesses a unique generalized solution x(t), regardless of a choice of the δ -approximating sequence {uk (t)} in (2.20). The generalized solution x(t) is then referred to as a vibroimpact solution of (2.1) under the impulsive input u(t) = γδ (t − t0 ). By Theorem 2.2, the instantaneous impulse response (2.22) of the vibrocorrect system (2.1) is uniquely determined from the solution of the Pfaffian system (2.21). Particularly, in the case where b(x,t) = b(t) is a state-independent function, and hence the mild solution of (2.1) is well-defined, the solution ξ (z, v, s) of the Pfaffian system is given by ξ (z, v, s) = z + b(s)v and the instantaneous impulse response x(0+) = y(t0 ) + b(0)γ is the same as if the conventional mild solution would be under consideration. We are now in a position to address the problem of reconstructing a nonlinear vibrocorrect system (2.1), (2.20), based on its discrete-continuous representation x(t) ˙ = f (x,t), x(0) = x0 Δ x(t0 ) = θ (x(t0 −), γ ,t0 ),

(2.31) (2.32)

similar to that of (1.3), which we discussed in the introduction. Theorem 2.3. Let functions f (x,t), b(x,t) ∈ C1 satisfy the linear growth condition in x and let (2.1) be driven by the generalized input (2.20). Suppose that, given an arbitrary initial condition x0 ∈ Rn , and an arbitrary impulse magnitude γ ∈ Rm , (2.1), (2.20) and the discrete-continuous system (2.31), (2.32) possess the same solutions. Then (2.1) is vibrocorrect if and only if the following relations hold:

∂ θ (z, γ ,t0 ) |γ =0 = b(z,t0 ), ∂γ θ (z, γ1 + γ ,t0 ) = θ (z, γ1 ,t0 ) + θ (z + θ (z, γ1,t0 ), γ ,t0 ).

(2.33) (2.34)

22

2 Mathematical Models

Proof. Let us assume that (2.1) is vibrocorrect. Then relations (2.22) and (2.32), coupled together, yield ξ (z, v,t0 ) = z + θ (z, v,t0 ). (2.35) Taking (2.35) into account, relation (2.33) results from (2.25), whereas (2.34) is guaranteed by (2.28). Conversely, once relations (2.33), (2.34) are satisfied, the function ξ (z, v,t0 ), governed by (2.35), meets Conditions 2.25, 2.28. Then, following the line of reasoning, used in the proof of Theorem 2.2, this function is shown to solve the Pfaffian equation (2.21) under arbitrary initial conditions ξ (0) = z ∈ Rn and t0 ≥ 0. Since, due to [42], this ensures the validity of the Frobenius condition (2.19) for all l = 1, . . . , n, i, j = 1, . . . , m and x ∈ Rn , t ≥ 0 [42], the vibrocorrectness of (2.1) is straightforwardly verified by applying Theorem 2.2. The proof of Theorem 2.3 is completed. Clearly, Theorem 2.3 allows one to reconstruct an accessible input, resulting in the prescribed restitution rule (2.32). Indeed, provided that (2.34) is satisfied, applying the impulsive input b(x,t)γδ (t − t0 ) with the feedback gain b(x,t), determined by (2.33), ensures the desired state restitution (2.32) for (2.1), thus defined. A simple example, given below, illustrates the capabilities of the distributions theory in a nonlinear setting. Example 2.2. Let (2.1), (2.20) be specified to x˙ = xγδ (t − t0 ), x(0) = x0 ∈ R1

(2.36)

where f = 0 and b = x. Then, the Pfaffian equation (2.21) is simplified to the ordinary differential equation d ξ /dv = ξ , ξ , v ∈ R1 ,

(2.37)

which integrates to ξ (v) = ξ (0)ev . By taking into account Theorems 2.1 and 2.2, (2.36) is vibrocorrect and its instantaneous impulse response is uniquely defined by x(t0 +) = x0 eγ

(2.38)

whereas the discrete-continuous representation (2.31), (2.32) of the system is given by x(t) ˙ = 0, x(0) = x0 Δ x(t0 ) = x0 (eγ − 1).

(2.39) (2.40)

In turn, applying Theorem 2.3 to the discrete-continuous system (2.39), (2.40) γ yields ∂ {x(e∂ γ−1)} |γ =0 = x, thus converting the solutions of (2.39), (2.40) into the vibroimpact solutions of (2.36).

2.2 Differential Equations with a Piece-wise Continuous Right-hand Side

23

2.2 Differential Equations with a Piece-wise Continuous Right-hand Side Consider a non-autonomous differential equation x˙ = ϕ (x,t),

(2.41)

with the state vector x = (x1 , . . . , xn )T , with the time variable t ∈ R, and with a piece-wise continuous right-hand side ϕ = (ϕ1 , . . . , ϕn )T . Recall that the function ϕ : Rn+1 → Rn is piece-wise continuous iff Rn+1 is partitioned into a finite number of domains G j ⊂ Rn+1 , j = 1, . . . , N, with disjoint interiors and boundaries ∂ G j of measure zero such that each of the restrictions ϕ j = ϕ |G j , j = 1, . . . , N of ϕ to these domains is continuous within G j where it has a finite limit ϕ j (x,t) as the argument (x∗ ,t ∗ ) ∈ G j approaches a boundary point (x,t) ∈ ∂ G j . The solutions of (2.41) are defined in the conventional Caratheodory sense whenever they are within the domains G j j = 1, . . . , N. In order to describe possible generalized solutions of (2.41) on the boundaries ∂ G j , j = 1, . . . , N, which are further referred to as sliding modes, the following regularization technique is normally utilized. In a vicinity of the boundaries, the original equation is replaced by a related equation, the Caratheodory solutions of which are well-posed. The sliding modes are then defined by making the characteristics of the new equation approach those of the original one. The rigorous introduction of the solution concept is as follows. Definition 2.4. An absolutely continuous function xδ (t), defined on some interval I, is said to be an approximate δ -solution of (2.41) if it is a Caratheodory solution of x˙δ = φ δ (xδ ,t) with some φ δ (x,t) such that φ δ (x,t) − φ (x,t) ≤ δ for almost all (x,t) ∈ Rn × I, satisfying sup(ξ ,τ )∈∪N

j=1 ∂ G j

[x − ξ  + t − τ ] ≥ δ .

Definition 2.5. An absolutely continuous function x(t), defined on some interval I, is said to be a generalized solution of (2.41) if there exists a family of approximate δ -solutions xδ (t) of the equation such that lim xδ (t) − x(t) = 0 uni f ormly in t ∈ I.

δ →0

Being defined as a generalized solution, a sliding mode of (2.41) appears to depend on a particular approximation of the right-hand side of the equation, but it does not depend on the precise specification of the right-hand side on the discontinuity boundaries ∂ G j , j = 1, . . . , N.

24

2 Mathematical Models

2.2.1 Filippov Solutions A particular regularization of (2.41) with the discontinuous right-hand side occurs if the switch between the structures ϕ j = ϕ |G j , j = 1, . . . , N has a hysteresis. As mentioned in Sect. 1.3, such a regularization results in the meaning of the differential equation (2.41) to be defined in the sense of A. F. Filippov [71]. Definition 2.6. Given the differential equation (2.41) let us introduce for each point (x,t) ∈ Rn × R the smallest convex closed set Φ (x,t) which contains all the limit points of ϕ (x∗ ,t) as x∗ → x, t = const, and (x∗ ,t) ∈ Rn+1 \(∪Nj=1 ∂ G j ). An absolutely continuous function x(·), defined on an interval I, is said to be a Filippov solution of (2.41) if it satisfies the differential inclusion x˙ ∈ Φ (x,t)

(2.42)

almost everywhere on I. The set Φ (x,t) is further referred to as the Filippov set. Equation 2.41 with a piece-wise continuous right-hand side is well known [71] to possess a solution for arbitrary initial conditions x(t0 ) = x0 ∈ Rn , t0 ∈ R. This solution is locally defined on some time interval [t0 ,t1 ); however, generally speaking, it is nonunique. At any continuity point (x,t) ∈ ∪Ni=1 Gi of the function ϕ the Filippov set Φ (x,t) consists of the only point ϕ (x,t), and the Filippov solution satisfies (2.41) in the conventional sense. Given (x,t) ∈ ∩Lk=1 ∂ G jk from the intersection of the boundaries ∂ G jk of several domains G jk , k = 1, . . . , L, the Filippov set Φ (x,t) is either a segment, or a convex polygon, or a convex polyhedron with vertices

ϕ jk (x,t) =

lim

(ξ ,t)∈G jk , ξ →x

ϕ (ξ ,t), k = 1, . . . , L.

Apparently, the points ϕ jk (x,t), k = 1, . . . , L belong to the Filippov set Φ (x,t), but not all these points are the vertices, forming this set. Let us assume now that the function ϕ (x,t) undergoes discontinuities on a smooth surface S only, and let this surface be governed by the equation s(x) = 0. Then the discontinuity set S separates the x space into domains G− = {x ∈ Rn : s(x) < 0} and G+ = {x ∈ Rn : s(x) > 0}. Given t, the Filippov set Φ (x,t) would be a linear segment joining the endpoints of the vectors

ϕ − (x,t) =

lim

(ξ ,t)∈G− , ξ →x

ϕ (ξ ,t), ϕ + (x,t) =

lim

(ξ ,t)∈G+ , ξ →x

ϕ (ξ ,t).

Hereinafter, these vectors are assumed to have the same initial point x. If for t ∈ [t0 ,t1 ], the vectors ϕ − (x,t) and ϕ + (x,t) point toward the same region, the Filippov segment Φ (x,t) would be located on one side of the plane T , tangential to the discontinuity surface S so that the Filippov solutions of (2.41) would cross S for these t. Such a situation appears if either grad T s(x) ϕ − (x,t) < 0 or grad T s(x) ϕ + (x,t) > 0 where grad T s = ( ∂∂xs , . . . , ∂∂xsn ). 1

2.2 Differential Equations with a Piece-wise Continuous Right-hand Side

25

On the contrary, if grad T s(x) ϕ − (x,t) > 0, grad T s(x) ϕ + (x,t) < 0

(2.43)

for all t ∈ [t0 ,t1 ], the vectors ϕ − (x,t) and ϕ + (x,t) are directed to opposite directions, and the Filippov segment Φ (x,t) intersects the tangential plane T . Then a sliding mode occurs on the discontinuity surface S for these t. According to Definition 2.6, this mode is governed by x˙ = ϕ 0 (x,t) (2.44) where the intersection of the Filippov segment Φ (x,t) and the plane T , tangential to S, determines the endpoint of the vector ϕ 0 (x,t). Analytically, this vector is expressed in the form

ϕ 0 (x,t) = μ (x,t)ϕ + (x,t) + [1 − μ (x,t)]ϕ −(x,t), μ (x,t) ∈ [0, 1] where

μ (x,t) =

grad T s(x) ϕ − (x,t) grad T s(x) [ϕ − (x,t) − ϕ + (x,t)]

(2.45)

(2.46)

is found from the condition grad T s(x) { μ (x,t)ϕ + (x,t) + [1 − μ (x,t)]ϕ −(x,t)} = 0

(2.47)

that the velocity vector (2.45) is in the plane T , tangential to S. Recall that the same expression (2.45), specified with (2.46), has appeared in Example 1.4 of the introduction to describe the average velocity (1.23) of the VSS (1.11) subject to the hysteresis switching (1.22) between its two structures. It is clear that the velocity vector (2.45) is uniquely determined by (2.46) iff the segment with the ends ϕ − (x,t) and ϕ + (x,t) intersects the tangential plane T but it does not lie in it. If this segment however is entirely in the tangential plane T , i.e., grad T s(x) [ϕ − (x,t) − ϕ + (x,t)] = 0, the Filippov solution is not uniquely defined. The same line of finding the Filippov set Φ (x,t) applies to the (x,t) space with(n + 1)-dimensional vectors (ϕ − (x,t), 1) and (ϕ + (x,t), 1) in the case where the discontinuity surface S is governed by the time-dependent equation s(x,t) = 0. Setting st = ∂∂ st , Condition 2.43 of a sliding mode to exist is then modified to st (x,t) + grad T s(x,t) ϕ − (x,t) > 0, st (x,t) + grad T s(x,t) ϕ + (x,t) < 0,

(2.48)

whereas the velocity vector (2.45) is determined by

μ (x,t) =

st (x,t) + grad T s(x,t) ϕ − (x,t) , grad T s(x,t) [ϕ − (x,t) − ϕ + (x,t)]

being found from the condition

(2.49)

26

2 Mathematical Models

st (x,t) + grad T s(x,t) { μ (x,t)ϕ + (x,t) + [1 − μ (x,t)]ϕ −(x,t)} = 0, ds(x(t),t) dt

which ensures that continuity surface S.

(2.50)

= 0 for the sliding mode x(t), evolving along the dis-

2.2.2 Equivalent Control Method An alternative approach of defining the velocity vector on a discontinuity set is based on the equivalent control method, developed by V.I. Utkin [227] for controlled systems of the form x˙ = f (x, u(x,t),t) (2.51) with a continuous (in all arguments) right-hand side f = ( f1 , . . . , fn )T where the state vector x = (x1 , . . . , xn )T and the time variable t ∈ R are the same as before, whereas the controlled input u = (u1 , . . . , um )T is a piece-wise continuous function of the state and time variables. The components ui (x,t), i = 1, . . . , m of the control signal u(x,t) are assumed to undergo discontinuities on possibly intersecting smooth surfaces Si = {(x,t) ∈ Rn+1 : si (x,t) = 0}, and to vary within the state/time + dependent segments Ui (x,t) = [u− i (x,t), ui (x,t)] where u− i (x,t) =

lim

ui (ξ ,t), Si− = {(x,t) ∈ Rn+1 : si (x,t) < 0}

lim

ui (ξ ,t), Si+ = {(x,t) ∈ Rn+1 : si (x,t) > 0}.

(ξ ,t)∈S− i , ξ →x

and u+ i (x,t) =

(ξ ,t)∈S+ , ξ →x

Apparently, at the continuity points (x,t) of ui (x,t), the corresponding set Ui (x,t) consists of a unique point. According to the equivalent control method, the potential sliding modes of (2.51) at an intersection of some sets S jk , k = 1, . . . , r are governed by x˙ = f (x, ueq (x,t),t)

(2.52)

+ + ueq i (x,t) ∈ [ui (x,t), ui (x,t)], i = 1, . . . , m

(2.53)

where the components

of the equivalent control input ueq (x,t) are such that the velocity vector f in (2.52) is tangent to the sets S jk , k = 1, . . . , r, i.e., for (x,t) ∈ ∩rk=1 S jk , the equivalent control vector with components (2.53) is to satisfy

∂ s jk (x,t) + grad T s jk (x,t) f (x, ueq (x,t),t) = 0, k = 1, . . . , r. ∂t

(2.54)

2.2 Differential Equations with a Piece-wise Continuous Right-hand Side

27

Definition 2.7. An absolutely continuous function x(·), defined on an interval I, is said to be an Utkin solution of (2.51) if it satisfies (2.51) beyond the surfaces Si , i = 1, . . . , m and it satisfies equations of the form (2.52) on these surfaces and their intersections. The physical meaning behind Utkin solutions is as follows [228]. While approximating the discontinuous input signal u on the time interval I by its continuous counterpart uδ , providing system oscillations within a δ -vicinity of the discontinueq ity set S = ∪m i=1 Si , the equivalent control value u proves to be approached by the output of a lowpass filter τ z˙δ + zδ = uδ , (2.55) i.e., lim z(t) = ueq (t) uniformly in t ∈ I as τ ↓ 0 and δ /τ ↓ 0. In other words, the approximate δ -solution of (2.51), driven by the slow component u = zδ of the control signal uδ , passing through filter (2.55), approaches the sliding mode x(t), computed according to the equivalent control method: lim xδ (t) = x(t) uniformly in t ∈ I.

τ ↓0, δτ ↓0

(2.56)

Thus, Utkin solutions constitute a special class of generalized solutions under a particular regularization of (2.51) with the output z of filter (2.55), being substituted into (2.51) for the control input u. Generally speaking, Utkin solutions do not coincide with corresponding Filippov solutions. The following example drawn from [227] illustrates this feature. Example 2.3. Consider the discontinuous system x˙1 = 0.3x2 + x1u, x˙2 = −0.7x1 + 4x1u3 with the input function u, governed by  1 if (x1 + x2 )x1 < 0 u= . −1 if (x1 + x2 )x1 > 0

(2.57)

(2.58)

By inspection, the sliding mode existence condition (2.43), being specified for (2.57), (2.58), appears to hold on the discontinuity line x1 + x2 = 0, whereas it does not hold on the discontinuity line x1 = 0. Thus, sliding modes of the system in question occur along the line x1 + x2 = 0. The equivalent control value ueq = 0.5 is then obtained from (2.54), which is now specified to (−1 + ueq + 4(ueq)3 + ueq − 1 = 0. Substituting the equivalent control value into (2.57) for u yields the instable equation x˙1 = 0.2x1, governing Utkin solutions on the discontinuity line x1 + x2 = 0. In turn, Filippov solutions are governed by another equation x˙1 = −0.1x1, which is obtained by specifying the Filippov velocity (2.45), (2.46) for the present system and which is asymptotically stable.

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2 Mathematical Models

It is worth noticing that both Filippov sliding modes and Utkin sliding modes are described by reduced-order equations because they are confined to the onedimensional state subspace x1 + x2 = 0. The equivalent control method, applied to an affine system (2.51), which admits representation in the form x˙ = η (x,t) + b(x,t)u yields ueq (x) = −(

∂ s −1 ∂ s ∂ s b) ( + η ) ∂x ∂t ∂x

(2.59)

(2.60)

n+1 . provided that the matrix function ∂ s(x,t) ∂ x b(x,t) is nonsingular for all (x,t) ∈ R By substituting the equivalent control value (2.60) into the affine system (2.59) for the input function u, the equation

x˙ = η − b(

∂ s −1 ∂ s ∂ s b) ( + η ) ∂x ∂t ∂x

(2.61)

is derived to describe potential Utkin solutions of the affine system (2.59) on the discontinuity manifold s(x,t) = 0. As shown in [227, Sect. 3 of Chapt. II], the sliding mode equation (2.61) appears to hold for arbitrary Filippov solutions, and moreover, for arbitrary generalized solutions that can occur in the affine systems of the form (2.59). Our next goal is to extend this result to differential equations in a Hilbert space.

2.2.3 Sliding Modes in a Hilbert Space Following [162], consider a differential equation x˙ = Ax + f (x,t) + bu(x,t), t > 0, x(0) = x0 ∈ D(A)

(2.62)

where the state variable x(t) and the input signal u(x,t) are abstract functions with values in Hilbert spaces H and U, respectively, the infinitesimal operator A with domain D(A) generates a strongly continuous semigroup TA (t) on H, the operator function f (x,t) with values in H is of class C1 in all arguments, and b is a linear bounded operator, acting from U to H. For later use, recall the following (see, e.g., [55] for relevant background materials on Hilbert space-valued dynamical systems). A family {T (t)}t≥0 of linear bounded operators T (t), t ≥ 0 forms a strongly continuous semigroup on a Hilbert space H if the identity T (t + τ ) = T (t)T (τ ) is satisfied for all t, τ ≥ 0, and the functions T (t)x are continuous with respect to t ≥ 0 for all x ∈ H. The induced operator norm T (t) of the semigroup satisfies the inequality T (t) ≤ ω eβ t ,t ≥ 0 with some growth bound β and some ω > 0.

2.2 Differential Equations with a Piece-wise Continuous Right-hand Side

29

The domain of an operator A, generating a strongly continuous semigroup, forms the Hilbert space D(A) with the graph inner product ·, · D (A) defined by means of the inner product ·, · H of the underlying Hilbert space H: x, y D (A) = x, y H + Ax, Ay H , x, y ∈ D(A). If β is a growth bound of the semigroup, then given λ > β , there holds (A − λ I)−1 H = D(A) where I is the identity operator, and the norm of x ∈ D(A) given by (A − λ I)xH is equivalent to the graph norm xD (A) of D(A). In particular, xD (A) = AxH if A possesses a growth bound β < 0. It should be noted that D(A) → H, i.e., D(A) ⊂ H, D(A) is dense in H and the inequality xH ≤ ω0 xD (A) holds for all x ∈ D(A) and some constant ω0 > 0. If the input function u meets the same smoothness conditions as that imposed on the system nonlinearity f , the above equation locally has a unique strong solution x(t) which is defined as follows. Definition 2.8. A continuous function x(t), defined on [0, T ), is a strong solution of the initial value-problem (2.62) with a continuously differentiable input u(x,t) iff limt↓0 x(t) − x0 H = 0, and x(t) is continuously differentiable and satisfies the equation for t ∈ (0, T ). The precise meaning of the solutions of (2.62), for inputs which are only piecewise continuously differentiable, is defined as a limiting result obtained through the regularization procedure, similar to that proposed for finite-dimensional systems. Let the input u(x,t) be continuously differentiable beyond a linear manifold cx = 0

(2.63)

with c ∈ L (H, S) being a linear bounded operator from H to some Hilbert space S, and let u(x,t) undergo discontinuities on this manifold. Then the strong solutions of (2.62) are only considered whenever they are beyond the discontinuity manifold (2.63), whereas in a vicinity of this manifold, the original system is replaced by a related system, which takes into account all possible imperfections in the new input function uδ (x,t) (e.g., delay, hysteresis, saturation, etc.) and for which there exists a strong solution. A generalized solution of (2.62) is then obtained by making the characteristics of the new system approach those of the original one. As in the finite-dimensional case, a motion along the discontinuity manifold is referred to as a sliding mode. To rigorously introduce a sliding mode in the infinite-dimensional system, let us complement the subspace H1 = ker c = {x1 ∈ H : cx1 = 0} ⊆ H by the subspace H2 ⊆ H such that H = H1 ⊕ H2 .

30

2 Mathematical Models

Clearly, the discontinuity manifold (2.63), written through the new coordinates x1 (t) = P1 x(t) ∈ H1 and x2 (t) = P2 x(t) ∈ H2 , takes the form x2 = 0. Hereinafter, Pi is the projector on the subspace Hi , Ai = A|Hi is the operator restriction on Hi , i = 1, 2. Definition 2.9. An absolutely continuous function xδ (t), defined on some interval [0, τ ), is said to be an approximate δ -solution of (2.62) if it is a strong solution of x˙δ = Axδ + f (xδ ,t) + buδ (xδ ,t)

(2.64)

with some uδ (x,t) such that uδ (x,t) − u(x,t) ≤ δ

(2.65)

for all t ≥ 0 and for all x = (x1 , x2 ) ∈ H = H1 ⊕ H2 subject to x2 D (A2) ≥ δ . Definition 2.10. An absolutely continuous function x(t), defined on some interval [0, τ ), is said to be a generalized solution of (2.62) if there exists a family of approximate δ -solutions xδ (t) of the system such that lim xδ (t) − x(t)D (A) = 0 uni f ormly in t ∈ [0, τ ).

δ →0

Although beyond the discontinuity manifold, our investigation is confined to strong solutions of the initial-value problem, an extension to the case where such a solution is defined in a mild sense as a solution to a corresponding integral equation, is possible. By definition, the sliding motion, which is in general non-unique, does not depend on the precise specification of the discontinuous input on the discontinuity manifold. Moreover, an equivalent control value ueq (x,t), maintaining the system motion on this manifold, is imposed by the original system itself. To describe sliding modes in the infinite-dimensional system let us rewrite (2.62) in terms of variables x1 (t) ∈ H1 and x2 (t) ∈ H2 : x˙1 = A11 x1 + A12 x2 + f1 (x1 , x2 ,t) + b1u(x1 , x2 ,t), x1 (0) = x01 , x˙2 = A21 x1 + A22x2 + f2 (x1 , x2 ,t) + b2u(x1 , x2 ,t), x2 (0) = x02

(2.66) (2.67)

where Ai j = Pi A j , i, j = 1, 2 are the operators from H j to Hi , and fi = Pi f , bi = Pi b. If the operator b2 is non-singular and the inverse operator b−1 2 is bounded, there exists a unique solution of the algebraic equation A21 x1 + f2 (x1 , 0,t) + b2 u(x,t) = 0 with respect to u. This solution ueq (x1 ) = −b−1 2 [A21 x1 + f 2 (x1 , 0,t)]

(2.68)

is accepted as the equivalent control value because (2.68) is the only input ensuring that x˙2 = 0 on the discontinuity manifold x2 = 0, thereby maintaining (2.62) with appropriate initial conditions on the manifold x2 = 0. Setting A˜ = A11 − b1 b−1 2 A21 −1 and f0 (x1 , x2 ,t) = f1 (x1 , x2 ,t) − b1b2 f2 (x1 , x2 ,t), the sliding mode equation

2.2 Differential Equations with a Piece-wise Continuous Right-hand Side

˜ 1 + f0 (x1 , 0,t), x˙ = Ax

31

(2.69)

governing the system motion on the discontinuity manifold cx = 0, is then obtained by substituting the equivalent control value (2.68) into (2.67) for u. Under the assumptions 1. The linear operator b is bounded and its projection b2 on the subspace H2 is continuously invertible, i.e., the operator b−1 2 from H2 to U is bounded, too; 2. The infinitesimal operators A and A˜ = A11 − b1 b−1 2 A21 generate strongly continuous semigroups TA (t),t ≥ 0 and TA˜ (t),t ≥ 0 on the Hilbert spaces H and H1 , respectively; 3. The system nonlinearity f (x,t) is everywhere continuously differentiable in (x,t) and it satisfies the linear growth condition in x; 4. The input function u(x,t) is everywhere continuously differentiable but on the discontinuity manifold x2 = 0; ˜ 1 b−1 from H2 to H1 is governed by A12 in the sense that 5. The operator G0 = Ab 2 D(G0 ) ⊆ D(A12 ) and G0 y ≤ KA12 y for all y ∈ D(G0 ) and some K > 0; the result, given below, extends the equivalent control method to infinite-dimensional systems. Assumption 1, inherited from the finite-dimensional case, ensures the uniqueness of the sliding mode equation. Assumptions 2–4 are made for technical reasons. Coupled to Assumption 1, they ensure the existence and uniqueness of local strong solutions of (2.62) beyond the discontinuity manifold x2 = 0. As a matter of fact, the operator function f0 (x1 , x2 ,t) = f1 (x1 , x2 ,t) − b1 b−1 2 f 2 (x1 , x2 ,t) meets Assumption 4, too, so that the existence and uniqueness of local strong solutions of the sliding mode equation (2.69) are also guaranteed. Assumption 5 is intrinsic for infinitedimensional systems in the sense that if it would fail to hold other generalized solutions, not governed by (2.69), could appear. Theorem 2.4. Consider the dynamic system (2.62) under Assumptions 1–5. Let the system, being initialized in the discontinuity manifold x2 = 0 with x0 = (x01 , 0) ∈ H, start evolving in this manifold on some time interval [0, τ ). Then the initial value problem (2.62) possesses a unique generalized solution x(t), and on the time interval [0, τ ) this solution is governed by the sliding mode equation (2.69) under the initial condition x1 (0) = x01 . Proof. By definition 2.10, on the time interval [0, τ ) the generalized solution x(t) of the discontinuous system (2.62) is approximated by δ -solutions xδ (t), corresponding to inputs uδ (x,t) and evolving within the δ -vicinity x2 D (A2) ≤ δ of the discontinuity manifold x2 = 0. It follows that δ δ δ δ δ uδ (x,t) = −b−1 2 [A21 x1 + A22 x2 + f 2 (x1 , x2 ,t) − x˙2 ].

(2.70)

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2 Mathematical Models

By substituting the first component xδ1 (t) of the approximate δ -solution xδ (t) of the discontinuous system (2.62) is governed by ˜ δ1 + f0 (xδ1 , xδ2 ,t) + b1b−1 x˙δ2 + [A12 − b1 b−1 A22 ]xδ2 , xδ1 (0) = x01 . (2.71) x˙δ1 = Ax 2 2 Let on the time interval [0, τ ) the notation zδ (t) = xδ1 (t) − x1 (t) stands for the deviation of the strong solution x1 (t) of the sliding mode equation (2.69) from the approximate solution xδ1 (t). Then subtracting (2.69) from (2.71) yields the deviation equation ˜ δ + [ f0 (xδ1 , xδ2 ,t) − f0 (x1 , 0,t)] + b1b−1 x˙δ2 + [A12 − b1b−1 A22 ]xδ2 , z˙δ = Az 2 2 whose solutions are representable in the integral form zδ (t) = TA˜ (t)zδ (0) +

t 0

TA˜ (t − s){[ f0 (xδ1 , xδ2 , s) − f0 (x1 , 0, s)] +

δ −1 δ b1 b−1 2 x˙2 (s) + [A12 − b1 b2 A22 ]x2 (s)}ds.

(2.72)

Taking into account that zδ (0) = 0, employing integration by parts, and making use of the well-known property ˜ ˜ (t) = T ˜ (t)A˜ T˙A˜ (t) = AT A A ˜ one can rewrite (2.72) as follows: of the infinitesimal operator A, δ −1 δ zδ (t) = TA˜ (0)b1 b−1 2 x2 (t) − TA˜ (t)b1 b2 x2 (0) +

t 0

TA˜ (t − s){ f0 (xδ1 , xδ2 , s) −

˜ 1 b−1 − A12 + b1b−1 A22 ]xδ2 (s)}ds. f0 (x1 , 0, s) − [Ab 2 2 For the purpose of estimating the norm of the deviation zδ (t) we utilize Assump˜ tion 2 to note that the inequality TA˜ (t) ≤ ω eβ t holds for all t ≥ 0, for some growth bound β˜ , and for some constant ω > 0. Without loss of generality, we assume that β˜ > 0. Apart from this, we apply Assumptions 1–3, coupled together, to ensure that both the strong solution x1 (t) of (2.69) and the strong solution xδ1 (t) of (2.71) cannot escape to infinity on any finite time interval while the strong solution xδ2 (t) of (2.67) under u = uδ remains bounded in D(A2 ). With this in mind, we conclude that on the time interval [0, τ ) the deviation of the function f0 (xδ1 (t), xδ2 (t),t) from f0 (x1 (t), 0,t) can be estimated  f0 (xδ1 (t), xδ2 (t),t) − f0 (x1 (t), 0,t) ≤ L(xδ1 (t) − x1(t) + δ ) with some Lipschitz constant L > 0, the existence of which is guaranteed by Assumption 3. Summarizing, Assumptions 1–5 admit straightforward estimation of the state deviation zδ (t) on the time interval [0, τ ) when the approximate solution xδ = (xδ1 , xδ2 ) is assumed to evolve within the δ -vicinity (2.70) of the discontinuity manifold

2.3 Modeling of Electromechanical Nonlinear Phenomena

33

x2 = 0:  t 0

δ δ zδ (t) ≤ ω b1b−1 2 [x2 (0) + x2 (t)] +

˜ 1 b−1 xδ2 (s) + A12xδ2 (s) + TA˜ (t − s)[L(xδ1 (s) − x1 (s) + δ ) + Ab 2

δ −1 b1 b−1 2 A22 x2 (s)]ds ≤ ω (1 + b1b2 )δ + ω

 t β˜ (t−s) 0

e

[L(zδ1 (s) + δ ) +

δ −1 (K + 1)A12xδ2 (τ ) + b1b−1 2 A22 x2 (τ )]ds ≤ ω (1 + b1b2 )δ +

ω

 t ˜ δ eβ (t−s) [Lzδ1 (s) + Lδ + (K + 1 + b1b−1 2 )x2 (s)D (A2 ) ds ≤ 0  t −1 ω (1 + b1b−1 2 )δ + ω (L + K + 1 + b1b2 )δ

ωL

 t β˜ (t−s) 0

e

˜

0

zδ1 (s)ds ≤ L0 δ + L1 ˜

eβ (t−s) ds +

 t 0

zδ1 (s)ds ˜

−1 βτ βτ where L0 = ω [1 + b1b−1 2  + (L + K + 1 + b1 b2 )τ e ] and L1 = ω Le . By applying the Bellman–Gronwell lemma, it follows that the norm of the deviation zδ (t) tends to zero uniformly in t ∈ [0, τ ) as δ → 0. In other words, once a generalized solution x(t) = (x1 (t), x2 (t)) of the discontinuous system (2.62) starts evolving in the discontinuity manifold x2 = 0, its first component x1 (t) is governed by the sliding mode equation (2.69), strong solutions of which are uniquely defined. Theorem 2.4 is thus proved.

2.3 Modeling of Electromechanical Nonlinear Phenomena The dynamic model of an electromechanical system can be derived in a systematic way with a Lagrange formulation (see, e.g., [11]). The Lagrangian L = T −U

(2.73)

of such a system is defined in terms of the total kinetic energy T and potential energy U as a function of the vector q = (q1 , . . . , qn )T of generalized coordinates q1 , . . . , qn which effectively describe the state of the system. The Lagrange’s state equation is then expressed by d ∂L ∂L − = F(q, q) ˙ dt ∂ q˙ ∂q

(2.74)

where the component Fi , i = 1, . . . , n of the vector F = (F1 , . . . , Fn )T is the generalized force associated with the generalized coordinate qi . The contributions to the generalized forces are given by the nonconservative forces, such as the joint actuator torques and the joint friction torques. The Lagrange’s equation (2.74) establishes the relation between the generalized force F,

34

2 Mathematical Models

applied to the system, and the generalized position q, the generalized velocity v = q, ˙ and the generalized acceleration a = q. ¨

2.3.1 Friction Models Friction is a natural phenomenon, representing the tangential reaction force between two surfaces in contact. Since these reaction forces depend on many factors such as contact geometry and surface materials, and the displacement and relative velocities of contacting bodies, and the presence of lubrication, among others, it is hardly possible to deduce a general friction model from physical first principles. Instead, phenomenological models, capturing essential friction features, are normally brought into play. Some friction models of interest are reviewed below (see surveys [10, 158] for details and for other existing friction models). 2.3.1.1 Static Models The classical friction models are described by static maps between velocity and friction forces. The main idea of such a model is that friction opposes motion and its magnitude is independent of velocity v and contact area. The Coulomb friction model F(v) = FC sign v is an ideal relay model, multi-valued for zero velocity: ⎧ if v > 0 ⎨ 1, sign v = [−1, 1], if v = 0 . ⎩ −1, if v < 0

(2.75)

(2.76)

Since it does not specify the friction force F(v) for zero velocity, the static force F(0) is admitted to counteract external forces below the Coulomb friction level FC . Thus, stiction, describing the Coulomb friction force at rest, can take on any value in the segment [−FC , FC ], thereby yielding the meaning of the corresponding state equation (2.74) in the sense of Filippov. The viscous friction model F(v) = Fv v

(2.77)

with a viscous friction coefficient Fv > 0 is used for the friction force caused by the viscosity of lubricants. By combining with the Coulomb friction, it is often modified to F(v) = Fv v + FC sign v.

(2.78)

2.3 Modeling of Electromechanical Nonlinear Phenomena

35

In order to account for the observed destabilizing Stribeck phenomenon at very low velocities the latter model is augmented with the Stribeck friction model 2 σs e−(v/vs ) sign v where the constants σs > 0 and vs > 0 stand for the Stribeck level and for the Stribeck velocity, respectively. The resulting model is then given by v 2

F(v) = Fv v + [FC + σs e−( vs ) ]sign v

(2.79)

where the stiction force level FS = σs + FC is admitted to be higher than the Coulomb level FC . 2.3.1.2 Dynamic Models To better match experimental data, the dynamic modeling of friction is typically involved. Proposed by Dahl [56], the following dynamic model FD F˙D = σ1 v − σ1 |v| FC

(2.80)

where σ1 > 0, and FC > 0 are the stiffness and the Coulomb friction level, respectively, appears to describe the spring-like behavior of the friction force FD during stiction when the velocity v of the contacting body is infinitesimal. Formally setting σ1 = ∞, the above dynamic model (2.80) specializes to the static Coulomb model (2.75). Since the Dahl model (2.80) is nonsmooth rather than discontinuous, it can be viewed as a regularization of the discontinuous Coulomb model (2.75) as σ1 → ∞. While being essentially Coulomb friction with a lag in the change of the friction force when the motion direction is changed, the Dahl model (2.80) does not capture the Stribeck effect. In order to account for the Stribeck effect, the LuGre friction model from [41] FL = Fv v + σ1η + σ2

dη dt

(2.81)

can be utilized. In the LuGre model (2.81) the friction interface is thought of as a contact between bristles [41], Fv > 0 is a viscous friction coefficient, σ1 > 0 is the stiffness, σ2 > 0 is a damping coefficient, η is the average deflection of the bristles, whose dynamics are governed by dη σ1 |v| = v− v 2 η, dt FC + [FS − FC )e−( vs ) ]

(2.82)

where FC > 0 is the Coulomb friction level, FS > 0 is the level of the stiction force, vs > 0 is the Stribeck velocity, and v is the actual velocity of the contacting body. Thus, the complete LuGre model (2.81), (2.82) is characterized by six parameters Fv , σ1 , σ2 , FC , FS , vs . It reduces to the Dahl model (2.80) if Fv = 0, σ2 = 0, and FS = FC . In turn, for steady state motion when v is constant and η˙ = 0, the relation

36

2 Mathematical Models

between the velocity and the LuGre friction force (2.81) is given by the classical model (2.79).

2.3.2 The Multistable Backlash Model and Its Single-stability Approximation Backlash occurs in any mechanical system where a driving part (motor) is not directly connected with a driven part (load). An electrical actuator consists of a DC motor and a load coupled by a gear reduction part. The dynamics of the actuator are modeled as follows [144]: Jo N −1 q¨o + fo N −1 q˙o = T + wo Ji q¨i + fi q˙i + T = τm + wi .

(2.83)

Hereinafter, Jo , fo , q¨o , q˙o , and qo (t) are, respectively, the inertia of the load and the reducer, the viscous output friction, the load acceleration, the load velocity, and the angular position of the load. The inertia of the motor, the viscous motor friction, the motor acceleration, the motor velocity, and the angular of the motor are denoted by Ji , fi , q¨i , q˙i , and qi (t), respectively. The input torque τm serves as a control action, and T stands for the transmitted torque. The external disturbances wi (t), wo (t) have been introduced into the driver equation (2.83) to account for destabilizing model discrepancies due to hard-to-model nonlinear phenomena such as friction and backlash. The transmitted torque T through a backlash with an amplitude j is typically modeled by a dead-zone characteristic [156, p. 8]:  0 if |Δ q| ≤ j T (Δ q) = (2.84) K Δ q − K j sign(Δ q) otherwise where

Δ q = qi − Nqo ,

(2.85)

K is the stiffness, and N is the reducer ratio. Such a model is depicted in Fig. 2.1a. Provided the servomotor position qi (t) is the only available measurement on the system, the above model (2.83)–(2.85) appears to be non-minimum phase, because along with the origin, the unforced system possesses a multi-valued set of equilibria (qi , qo ) with qi = 0 and qo ∈ [− j, j]. To avoid dealing with a non-minimum phase system, the backlash model (2.84) is replaced with its monotonic approximation (see Fig. 2.1b) T = K Δ q + K η (Δ q) where

(2.86)

2.3 Modeling of Electromechanical Nonlinear Phenomena

37 T

T

−j

−j

j

Δq

j

(a)

Δq

(b)

Fig. 2.1 a) The dead-zone model of backlash; b) The monotonic approximation of the dead-zone model

η = −2 j

1 − e−

Δq j

(2.87) Δq . 1 + e− j The present backlash approximation is inspired from [144]. Coupled to the drive system (2.83), the motor position of which is only available for measurements, it is shown to constitute an internally minimum phase approximation of the underlying servomotor, operating under uncertainties wi (t), wo (t). As a matter of fact, these uncertainties involve discrepancies between the physical backlash model (2.84) and its approximation (2.86)–(2.87). For later use, let us introduce the state deviation vector x = [x1 , x2 , x3 , x4 ]T from a desired load position qo with x1 = q o − q d x2 = q˙o x3 = qi − Nqd x4 = q˙i where x1 is the load position error, x2 is the load velocity, x3 is the motor position deviation from its nominal value, and x4 is the motor velocity. The nominal motor position Nqd has been pre-specified in such a way to guarantee that Δ q = Δ x where

Δ x = x3 − Nx1 . Then (2.83)–(2.87), represented in terms of the deviation vector x, takes the form:

38

2 Mathematical Models

x˙1 = x2 x˙2 = Jo−1 [KNx3 − KN 2 x1 − fo x2 + KN η (Δ x) + wo ] x˙3 = x4 x˙4 = Ji−1 [τm + KNx1 − Kx3 − fi x4 − K η (Δ x) + wi ].

(2.88)

The internal zero dynamics x˙1 = x2 x˙2 = Jo−1 [−KN 2 x1 − fo x2 + NK η (−Nx1 )]

(2.89)

of the nominal system (2.88) with respect to the output y = x3

(2.90)

is formally obtained (see [104] for details) by setting wi = wo = 0 and specifying the control law that maintains the output identically zero. The following result, extracted from [5], guarantees that the error system (2.88), (2.90) is globally minimum phase. Theorem 2.5. Let the nonlinearity η be governed by (2.87). Then (2.89) is globally asymptotically stable. Proof. To begin with, let us consider a Lyapunov function of the form 1 1 V (x1 , x2 ) = x22 + KN 2 Jo−1 x21 + 2 2 2j Since 2 ln

e

2

e

KJo−1 [2 ln

Nx1 j

+1

2

≥

Nx1 j

2

+1

−

Nx1 ]. j

(2.91)

Nx1 f or all x1 ∈ R j

by inspection, the function V , governed by (2.91), appears to be positive definite. Next, let us compute the time derivative of the Lyapunov function on the trajectories of (2.89): V˙ =

×[

Nx1 j

e −Jo−1 (KN 2 x1 + fo x2 + 2 jKN Nx1

−1

)x2 e j +1 +KN 2 Jo−1 x1 x2 + 2 j2 KJo−1

4Ne 2 j(e

Nx1 j

Nx1 j

+ 1)

x2 −

Nx2 ] = −Jo−1 fo x22 ≤ 0. j

(2.92)

Let us now observe that the internal zero dynamics (2.89) of the system have no trivial solutions on the manifold x2 = 0 where the time derivative of the Lyapunov

2.3 Modeling of Electromechanical Nonlinear Phenomena

39

function equals to zero. Indeed, if x2 = 0 then due to (2.89) Nx1

1−e j Nx1 +2 Nx1 = 0, j 1+e j

(2.93)

thus concluding that x1 = 0. To reproduce the latter conclusion, it suffices to represent (2.93) in terms of z = Nxj 1 and note that the left-hand side of the inequality z+2

1 − ez =0 1 + ez

(2.94)

thus obtained is a strictly increasing function of z because its derivative is positive definite by inspection: 1−

(1 − ez)2 4ez = > 0 f or all z = 0. (1 + ez)2 (1 + ez)2

(2.95)

In order to complete the proof it remains to apply the LaSalle–Krasovskii invariance principle [110] to the system in question.

2.3.3 Limit Cycles and Nonlinear Asymptotic Harmonic Generators Oscillation is among the important phenomena that occur in electromechanical systems. While linear time-invariant systems can only generate oscillations of amplitude dependent on the initial state, nonlinear systems can go into an oscillation of a fixed amplitude irrespective of the initial conditions. Such an oscillation is referred to as a limit cycle [110, 213]. The Van der Pol equation, whose general representation is given by the secondorder scalar nonlinear differential equation ..

x +ε [(x − x0 )2 − ρ 2]x˙ + μ 2(x − x0 ) = 0

(2.96)

with positive parameters ε , ρ , μ , is a special case of the Lienard (circuit) equation (see, e.g., [110]) .. v +r(v)v˙ + g(v) = 0 (2.97) where the functions r(v) and g(v) are continuously differentiable. Equation 2.96, initially used by Van der Pol to study oscillations in vacuum tube circuits, presents a fundamental example in nonlinear oscillation theory. It possesses a stable limit cycle that attracts every other solutions except the unique equilibrium point (x, x) ˙ = (x0 , 0). The parameter ρ controls the amplitude of this limit cycle, the parameter

40

2 Mathematical Models

μ controls its frequency, the parameter ε controls the speed of the limit cycle transients, and the parameter x0 is for the offset of x (see [238] for details). Being proposed in [196], the Van der Pol modification ..

x +ε [(x2 +

x˙2 ) − ρ 2]x˙ + μ 2 x = 0 μ2

(2.98)

appears if (2.96) admits no offset of x, i.e., the parameter x0 = 0 is used, and if the additional term με2 x˙3 is involved. As opposed to the Van der Pol equation (2.96), the above modification has nothing to do with the Lienard equation (2.97). Meanwhile, it still possesses a stable limit cycle, being expressible in the explicit form x2 +

x˙2 = ρ2 μ2

(2.99)

(unlike that of the Van der Pol oscillator, exhibiting a nonsinusoidal periodic response in its limit cycle!). The following result is in order [181]. Theorem 2.6. Consider the modified Van der Pol equation (2.98) with positive parameters ε , μ , ρ . Then this equation has a stable limit cycle, given by (2.99), so that every other solution except the equilibrium point x = x˙ = 0 converges to the limit cycle (2.99) as t → ∞. Proof. By inspection, the origin x = x˙ = 0 is a unique equilibrium point of (2.98). Hence, Poincare–Bendixson criterion [110, p. 61] is applicable to the modified Van der Pol equation (2.98). By applying Poincare–Bendixson criterion, the existence of a periodic orbit is concluded for this equation. Its analytical representation (2.99) comes from the expression of the time derivative of the positive definite function 1 1 V (x, x) ˙ = x2 + 2 x˙2 , 2 2μ

(2.100)

computed along the trajectories of (2.98): x˙2 1 .. ε V˙ (x, x) ˙ = xx˙ + 2 x˙ x= 2 [ρ 2 − (x2 + 2 )]x˙2 . μ μ μ

(2.101)

⎧ 2 ⎪ > 0 if (x2 + μx˙ 2 ) < ρ 2 & x˙ = 0 ⎪ ⎨ 2 < 0 if (x2 + μx˙ 2 ) > ρ 2 & x˙ = 0 V˙ (x, x) ˙ ⎪ ⎪ ⎩ = 0 if [ρ 2 − (x2 + x˙2 )]x˙ = 0 μ2

(2.102)

It follows that

on the trajectories of (2.98). By applying the LaSalle–Krasovskii invariance principle [110] to (2.102), one concludes that a periodic solution of (2.98) has to oscillate within the set {(x, x) ˙ : V˙ (x, x) ˙ = 0} where

2.3 Modeling of Electromechanical Nonlinear Phenomena

[ρ 2 − (x2 +

x˙2 )]x˙ = 0. μ2

41

(2.103)

Since the origin is a unique equilibrium point of (2.98) all the trajectories of (2.98) cross the axis x˙ = 0 everywhere except the origin. Hence, the largest invariant manifold of set (2.103) coincides with ellipse (2.99) and it is straightforwardly verified that (2.99) is a limit cycle of the modified Van der Pol equation (2.98).  To complete the proof, it remains to note that due to (2.102), the norm x(t) = V (x(t), x(t)) ˙ of any trajectory of (2.98), initialized inside the limit cycle (2.99), must grow with time. Conversely, the norm of any trajectory of (2.98), initialized outside the limit cycle, must shrink with time. Thus, any trajectory of (2.98) except the equilibrium point x = x˙ = 0 is attracted by the limit cycle (2.99). The proof is completed. Now, it becomes clear that in (2.98), the parameter ρ stands for the amplitude of the limit cycle whereas μ is for its frequency. Furthermore, by substituting the orbit equation (2.99) into (2.98) we conclude that the limit cycle of the modified Van der Pol equation (2.98) is remarkably generated by a standard linear harmonic oscillator ..

x +μ 2 x = 0,

(2.104)

initialized on (2.99). Thus, we arrive at a nonlinear asymptotic harmonic generator (2.98) which naturally exhibits an ideal sinusoidal signal (2.104) in its limit cycle (2.99). The amplitude and frequency of this sinusoidal signal can be varied at will by tuning the parameters ρ and μ of the harmonic generator (2.98). The modified Van der Pol oscillator (2.98), the phase portrait of which is shown in Fig. 2.2 for the parameter values ε = 1000, ρ = 0.01, μ = 1, still belongs to a class of damped systems. In the region of negative damping, occurring within the limit cycle where the signals are small, the damping increases the energy level of the response (see the proof of Theorem 2.6). Conversely, outside the limit cycle, the damping becomes positive, thus decreasing the energy of the output signal. As a result, the motion approaches the limit cycle whose energy is determined by its amplitude ρ and frequency μ and therefore a desired level of the energy can be attained by assigning appropriate values of the oscillator parameters ρ and μ .

2.3.4 Vibroimpact Modeling Following [164], we demonstrate how impact dynamics (collisions, percussions, etc.) of mechanical systems can be treated within the framework of nonlinear differential equations in distributions. The vertical launch of a spacecraft, governed by

42

2 Mathematical Models

0.02

dx/dt

0.01

0

−0.01

−0.02 −0.02

−0.01

0 x

0.01

0.02

Fig. 2.2 Phase portrait of the modified Van der Pol oscillator

h˙ = V, h(0) = 0 p − mg − q(h,V) , V (0) = 0 V˙ = m p m˙ = − , m(0) = mo , c

(2.105)

serves here as an illustrative example. In the above equations, h(t),V (t), m(t) are, respectively, the altitude, the velocity, and the mass of the rocket at the time instant t, p is the thrust, q is the air resistance, g is the gravity acceleration, c is the specific impulse, and m0 is the initial mass. System 2.105 implies a continuous change of the mass m(t), whereas a discrete change m(0+) − m0 = Δ m0 < 0, corresponding to an instantaneous fuel combustion in the spacecraft jet, is also possible and it is caused by the impulsive thrust p(t) = −cΔ m0 δ (t).

(2.106)

From the physical point of view, the impulsive thrust (2.106) indicates that the cumulative power of the jet is limited while attaining a very high value during a very short period of time. With this in mind, the meaning of (2.105) subject to (2.106) has to be treated in the generalized sense because of the irregularity of the product cΔ m0 δ (t)m−1 (t) of the impulsive thrust (2.106) and the discontinuous function m−1 (t) that appears in the right-hand side of the second equation of (2.105). Since (2.105) is driven by the scalar impulsive action (2.106), the corresponding Frobenius condition is therefore satisfied, and by Theorem 2.2 the system has a unique vibroimpact solution. Apparently, the altitude h has no jump whereas the change

2.3 Modeling of Electromechanical Nonlinear Phenomena

43

Δ m0 of the mass m(t) at the initial time moment is straightforwardly computed by integrating the latter equation of (2.105) which has no irregular product. In turn, the instantaneous change Δ V0 = V (0+) of the velocity is found by applying Theorem 2.3 through the relation V (0+) = ξ2 (1), (2.107) where ξ (v) = (ξ1 (v), ξ2 (v), ξ3 (v))T solves the Cauchy problem d ξ1 = 0, ξ1 (0) = 0 dv c Δ m0 d ξ2 =− , ξ2 (0) = 0 dv ξ3 d ξ3 = Δ m0 , ξ1 (0) = m0 . dv

(2.108)

Since (2.108) integrates to ξ1 (v) = 0, ξ2 (v) = −c ln m0 +mΔ0m0 v , ξ3 (v) = m0 + Δ m0 v, relation (2.107) results in the formula V (0+) = −c ln(1 +

Δ m0 ), m0

(2.109)

for the instantaneous change of the velocity. Thus, the spacecraft dynamics enforced by the impulsive jet is modeled by vibroimpact solutions of the nonlinear differential equation (2.105) with the impulsive input (2.106). Such a nonlinear distribution formalism allows one to qualitatively analyze these dynamics. It is anticipated (but it is not the aim of the book to demonstrate) that the nonlinear distribution formalism relies on measure differential inclusions from [151] introduced for the adequate modeling of post-impact behavior of mechanical systems. Finding the post-impact velocity of such a system is, however, far from being trivial, as this represents a large combinatorial problem that can be formulated as a complementarity problem or as a normal cone inclusion (see [33, 81, 125, 150] for details).

Chapter 3

Stability Analysis

The analysis of discontinuous dynamic systems has attracted a considerable research interest in the last decades. Although the existing literature on this subject includes numerous books and papers such as [30, 52, 77, 108, 126, 132, 139, 212, 227], to name a few, these systems are far from being fully understood. In the present chapter, the analysis tools of such systems are developed within the framework of the second Lyapunov method with a focus on the use of nonsmooth Lyapunov functions, possessing non-positive time derivatives along the system trajectories.

3.1 Basic Definitions The following nonautonomous discontinuous system x˙ = ϕ (x,t)

(3.1)

is under study. Hereinafter, x = (x1 , . . . , xn )T is the state vector, t ∈ R is the time variable, ϕ (x,t) = (ϕ1 (x,t), . . . , ϕn (x,t))T is a piece-wise continuous function on Rn+1 , N undergoing discontinuities on the boundary set N = j=1 ∂ G j where the boundaries ∂ G j of the disjoint continuity domains G j ⊂ Rn+1 , j = 1, . . . , N of ϕ (x,t) are of zero measure. Throughout, the precise meaning of the differential equation (3.1) with a piecewise continuous right-hand side is defined in the sense of Filippov as that of the differential inclusion x˙ ∈ Φ (x,t) (3.2) with Φ (x,t) being the smallest convex closed set containing all the limit values of ϕ (x∗ ,t) for (x∗ ,t) ∈ Rn+1 \ N , x∗ → x, t = const. System 3.1 has a solution for arbitrary initial conditions x(t0 ) = x0 ∈ Rn . This solution is locally defined on some time interval [t0 ,t1 ), however, it is generally speaking non-unique (see Chap. 2 for details).

45

46

3 Stability Analysis

Apparently, the above model (3.1) of a discontinuous system admits sliding mo tions on the boundary set N = Nj=1 ∂ G j with an infinite number of switches on a finite time interval. Thus, the present model captures a larger class of discontinuous systems compared to that of [30, 132] where a finite frequency of switches is only taken into consideration. The stability of a discontinuous system (3.1) with possibly non-uniquely defined trajectories is introduced by means of that of the corresponding differential inclusion, whereas the stability of a differential inclusion (3.2) is defined as follows. Suppose x = 0 is an equilibrium point of the differential inclusion (3.2) and x(t,t0 , x0 ) denotes a solution x(t) of (3.2) under the initial conditions x(t0 ) = x0 . Definition 3.1. The equilibrium point x = 0 of the differential inclusion (3.2) is stable (uniformly stable) iff for each t0 ∈ R, ε > 0, there is δ = δ (ε ,t0 ) > 0, dependent on ε and possibly dependent on t0 (respectively, independent on t0 ) such that each solution x(t,t0 , x0 ) of (3.2) with the initial data x(t0 ) = x0 ∈ Bδ within the ball Bδ , centered at the origin with radius δ , exists for all t ≥ t0 and satisfies the inequality x(t,t0 , x0 ) < ε , t0 ≤ t < ∞. Definition 3.2. The equilibrium point x = 0 of the differential inclusion (3.2) is said to be (uniformly) asymptotically stable if it is (uniformly) stable and the convergence limt→∞ x(t,t0 , x0 ) = 0 (3.3) holds for all solutions of (3.2) initialized within some Bδ (uniformly in t0 and x0 ). Definition 3.3. If convergence (3.3) remains in force for all solutions of (3.2) regardless of the choice of the initial data (and, respectively, it is uniform in t0 and x0 ∈ Bδ for each δ > 0), the equilibrium point x = 0 of the differential inclusion (3.2) is said to be globally (uniformly) asymptotically stable. Definition 3.4. The equilibrium point x = 0 of the differential inclusion (3.2) is said to be exponentially stable if there exist positive constants δ , k and λ such that the inequality x(t,t0 , x0 ) ≤ kx0 e−λ (t−t0 ) (3.4) holds for all solutions of (3.2) initialized within Bδ and globally exponentially stable if (3.4) holds for any initial condition. The analysis to be developed is focused on the stability and global asymptotic stability. Other stability concepts, mentioned above, can readily be addressed within the framework developed in the chapter. For later use we also recall the following. A continuous scalar function v(x) is positive definite iff v(0) = 0 and v(x) > 0 for all x = 0. It is radially unbounded iff lim v(x) = ∞ as x → ∞. A continuous scalar function V (x,t) is positive definite iff V (x,t) ≥ v(x) for all (x,t) ∈ Rn+1 and some positive definite function v(x), radially unbounded iff v(x) is so, and decrescent iff V (x,t) ≤ v1 (x) for all (x,t) ∈ Rn+1 and some continuous function v1 (x). It is positive semidefinite iff V (x,t) ≥ 0 for all (x,t) ∈ Rn+1 and it is negative definite (semidefinite) iff −V (x,t) is positive definite (semidefinite).

3.2 Stability and Nonsmooth Lyapunov Functions

47

3.2 Stability and Nonsmooth Lyapunov Functions To begin with, we illustrate, with a help of a simple example, that for discontinuous systems the use of nonsmooth Lyapunov functions, including l1 -norms, is as natural as the use of quadratic Lyapunov functions for linear systems. Example 3.1. The time derivative of the nonsmooth function V = |x|+ |y|, computed along the trajectories (x(t), y(t)) of the system x˙ = sign x, y˙ = −2sign y beyond the manifold xy = 0, is as simple as V˙ = xsign ˙ x + ysign ˙ y = 1 − 2 = −1 so that the function V (x(t), y(t) is decreasing in t everywhere but the manifold xy = 0. This is, however, insufficient to ensure the system stability because yy˙ = −2 < 0 beyond the x-axis where y = 0 and the system generates unstable sliding modes on the xaxis. Indeed, according to the equivalent control method, these modes are governed by x˙ = sign x, y = 0, and the time derivative of V = |x| + |y|, computed along the sliding modes on the line y = 0, takes the positive value V˙ = xsign ˙ x = 1, thereby ensuring that the trajectories of the system escape to infinity along the x-axis. Thus, by using a nonsmooth Lyapunov function, the above system is established to be unstable. Motivated by this example, we intend to involve nonsmooth Lyapunov functions into our stability analysis of discontinuous systems. The analysis to be developed is based on the condition that there exists a positive definite continuous function V (x,t), nonincreasing along the trajectories of (3.1). Then, all the trajectories of (3.1) are bounded and, in accordance with [71], they are globally defined, possibly non-uniquely, in the direction of increasing t. By applying standard Lyapunov arguments, the system stability is also guaranteed. Moreover, if the function V (x,t) is Lipschitz continuous, i.e., it satisfies the Lipschitz condition in the neighborhood of each point of its domain then, for any solution of (3.1), the composite function V (x (t) ,t) is absolutely continuous and   d d V (x (t) ,t) = V (x (t) + hx(t) ˙ ,t + h) (3.5) dt dh h=0 almost everywhere. Indeed, d V (x (t + h),t + h) − V (x (t) ,t) V (x (t) ,t) = lim h→0 dt h V (x (t) + hx(t) ˙ ,t + h) − V (x (t) ,t) = lim h→0 h V (x (t + h),t + h) − V (x (t) + hx(t) ˙ ,t + h) − lim h→0 h   d V (x (t) + hx(t) ˙ ,t + h) = dh

(3.6)

h=0

provided that the derivatives x(t) ˙ and dV (x(t),t)/dt exist at a time instant t. In order to reproduce (3.6), one should take into account that the last limit in (3.6) equals

48

3 Stability Analysis

to zero because the function V (x,t) satisfies the Lipschitz condition and x(t + h) = o(h) x(t) + hx(t) ˙ + o(h) where o(h) is such that limh→0 h = 0. Summarizing, we arrive at the following. Theorem 3.1. Suppose that in a domain (x ∈ Bδ ,t ∈ R) there exists a Lipschitz continuous, positive definite, decrescent function V (x,t) such that its time derivative (3.5), computed along the trajectories x(t) of (3.1), which are initialized within Bδ , is negative semidefinite almost everywhere, i.e., d V (x (t) ,t) ≤ 0 dt

(3.7)

for almost all t. Then (3.1) is uniformly stable. Proof. The proof of Theorem 3.1 follows the same line of reasoning as that of the standard Lyapunov stability theorem. The main point is that the composite Lyapunov function V (x (t) ,t) is absolutely continuous and its time derivative is almost everywhere given by (3.5). Thus, the negative semidefiniteness of (3.5) ensures that V (x (t) ,t) does not increase along the trajectories of (3.1), thereby yielding the uniform stability of the system. To this end, we present two lemmas that allow one to simplify the verification of the conditions of Theorem 3.1. Lemma 3.1. Condition (3.7) of Theorem 3.1 holds if (3.5) is nonpositive at the points of the set NV where the function V (x,t) is not differentiable, and in the continuity domains of the function ϕ (x,t) where (3.5) is expressed in the standard form

∂ V (x,t) d V (x,t) = grad V (x,t) · ϕ (x,t) + , (x,t) ∈ Rn+1 \ (N ∪ NV ) . dt ∂t

(3.8)

Proof. At the discontinuity points (x,t) ∈ N of the function ϕ (x,t), the right-hand side Φ (x,t) of the corresponding differential inclusion (3.2) is obtained by closure of the graph of ϕ (x,t) and by passing over to a convex hull. As is shown in [71], these procedures do not increase the upper value of (3.8) and hence the negative semidefiniteness of (3.8) guarantees the validity of (3.7) for all (x,t) ∈ N . Lemma 3.2. Let no trajectory of (3.1) stay in NV \ {x = 0} within a finite time interval. Then (3.7) of Theorem 3.1 holds almost everywhere if (3.8) is nonpositive for all (x,t) ∈ Rn+1 \ (N ∪ NV ).   Proof. Since any trajectory of (3.1) stays in Rn+1 \ NV ∪ {x = 0} almost everywhere, (3.7) of Theorem 3.1 is also satisfied almost everywhere.

3.3 Invariance Principle Relating to autonomous continuous dynamic systems (3.1), Krasovskii–LaSalle’s invariance principle ensures the convergence of the state trajectories x (t) to the

3.3 Invariance Principle

49

largest invariant subset M of the manifold where the time derivative of the Lyapunov function takes no value. Recall that M ⊂ Rn is an invariant set of (3.1) if, for all x0 ∈ M, the trajectories initialized at x0 at some time t0 remain in M for all t > t0 . In general, the invariance principle does not hold for non-autonomous systems and it does not admit the extension to dynamic systems, governed by differential inclusions and, particularly, to discontinuous dynamic systems, possibly due to their ambiguous behavior (see e.g., [146] for details).

3.3.1 Extension to a Class of Discontinuous Systems Motivated by the goal to extend the invariance principle to discontinuous dynamic systems, we confine our present investigation to autonomous discontinuous systems for which right uniqueness of solutions holds. In other words, throughout this section we assume that (3.1) is autonomous, it is specified to x˙ = ϕ (x) ,

(3.9)

and any solution of (3.9) is uniquely continuable to the right. Some sufficient right uniqueness conditions for the solutions of (3.9) can be found in [71], where the continuous dependence of the solutions on their initial data is also proved. We are now in a position to reproduce the invariance principle from [6] for a discontinuous dynamic system governed by (3.9). Theorem 3.2. Consider an autonomous system (3.9), each solution of which is uniquely continuable to the right. Suppose that there exists a Lipschitz continuous, radially unbounded, positive definite function V (x) such that its time derivative along the trajectories of (3.9) is negative semidefinite. Let M ⊂ Rn be the largest invariant subset of the manifold where dtd V (x) = 0, and let V (x) → ∞ as dist (x, M) → ∞. Then all the trajectories x (t) of (3.9) converge to M, that is, lim dist (x (t) , M) = 0.

t→∞

Proof. First, let us note that the solutions of (3.9) generate a dynamical system in Rn through the relations S (t) x0 = x (t, x0 ) , t ≥ 0, where x (t, x0 ) stands for a solution of (3.9) subject to x (0) = x0 . Indeed, the operators S (t), t ≥ 0 thus defined possess the following properties: • S (t) x is continuous in x ∈ Rn for all t ≥ 0 (due to the continuous dependence of the solutions on the initial data). • S (t) x is continuous in t for all x ∈ Rn (due to the continuity of the solutions). • S (0) is the identity operator (by definition). • S (t + τ ) x = S (t) S (τ ) x for all x ∈ Rn and t, τ ≥ 0 (due to the semigroup property of the solutions).

50

3 Stability Analysis

Therefore, the operators S (t), t ≥ 0, form a dynamical system. Thus, the invariance principle, developed for abstract dynamical systems (see, e.g., [96, 146]), turns out to be applicable to (3.9) and, in order to complete the proof, it remains to note that the Dini derivative 1 Dt V (x (t)) = limh→0+ [V (x (t + h)) − V (x (t))] h

(3.10)

of the absolutely continuous function V (x (t)) along the trajectories of the system is represented in (3.5).

3.3.2 Illustrative Applications to a Mechanical Oscillator with Coulomb Friction The capabilities of the invariance principle are now illustrated for the vector field   x2 , x = (x1 , x2 )T ∈ R2 . ϕ (x) = (3.11) −x1 − sign x2 System 3.9, specified with (3.11), yields a dimensionless version     x˙1 x2 = , x˙2 −x1 − sign x2

(3.12)

of the mathematical model my¨ + P(y) ˙ + ky = 0,

(3.13)

describing the displacement y of a mechanical oscillator. In (3.13), m is the mass, k is the spring rigidity, and P (y) ˙ is the Coulomb friction given by  +P0 if y˙ > 0 P (y) ˙ = , −P0 if y˙ < 0 P0 > 0 is the Coulomb friction level. 3.3.2.1 Localization of the Equilibria Set   If we introduce the Lyapunov function V (x) = x21 + x22 /2, then its time derivative V˙ (x) = − |x2 | ,

(3.14)

calculated along the trajectories of (3.12), proves to be negative everywhere except the x1 -axis. By applying the invariance theory [146, Theorem 7.3.2], developed for

3.3 Invariance Principle

51

differential inclusions, one can conclude that any trajectory of (3.12) approaches the x1 -axis as t → ∞. x2

−1

x1

1

Fig. 3.1 Phase portrait of the dimensionless oscillator (3.12)

Let us now demonstrate how the invariance principle (Theorem 3.2), developed before, allows one to strengthen the latter conclusion. First,let us verify if Theorem  3.2 is applicable to (3.12). The discontinuity manifold S = x ∈ R2 : x2 = 0 of the vector field (3.11) partitions the phase plane R2 into two regions     G+ = x ∈ R2 : x2 > 0 and G− = x ∈ R2 : x2 < 0 . In the discontinuity manifold S the velocity vectors  ϕ + (xs ) = lim ϕ (x) = x→xs , x∈G+

ϕ − (xs ) =

 0 , −x1 − 1

 lim

x→xs , x∈G−

ϕ (x) =

0 −x1 + 1

 (3.15)

are directed to opposite directions inside the segment |x1 | ≤ 1 and they point toward the same region (G+ for x < −1 and G− for x > 1) outside the segment (see Fig. 3.1). Thus, the trajectories of (3.12) cross the x1 -axis for |x1 | > 1 and stay there for |x1 | ≤ 1. Moreover, within the segment I = {x : |x1 | ≤ 1, x2 = 0}, the velocity vectors ϕ + (xs ) and ϕ − (xs ) are normal to this segment, which is why I consists of the equilibrium points of (3.12) and forms the largest invariant subset of the x1 -axis. So, by Theorem 1 of [71, p. 106], (3.12) satisfies the right-uniqueness property and hence the invariance principle (Theorem 3.2) is applicable to the system.

52

3 Stability Analysis

Fig. 3.2 One-degree-offreedom mechanical oscillator

y k

m

u

According to the principle, any trajectory of the unforced system (3.12) converges to the segment I = {x : |x1 | ≤ 1, x2 = 0} rather than the whole x1 -axis. This result, however, is long recognized from the literature and it is presented here to illustrate the application of the invariance principle to discontinuous dynamic systems on one hand, and to motivate further investigation on asymptotic stabilization of the Coulomb friction manipulator, on the other hand. 3.3.2.2 Asymptotic Stabilization If oscillator (3.13) is enforced by a control input u (see Fig. 3.2) it can be asymptotically stabilized by the following control law u (x) = −sign x1 . Driven by (3.16), the closed-loop system is defined by the vector field   x2 , x = (x1 , x2 )T ∈ R2 , ϕ (x) = −x1 − sign x1 − sign x2

(3.16)

(3.17)

  which undergoes discontinuities on the manifolds S1 = x ∈ R2 : x1 = 0 and S2 =   x ∈ R2 : x2 = 0 . These manifolds partition the phase plane R2 into four regions G1 = {x : x1 > 0, x2 > 0} , G2 = {x : x1 > 0, x2 < 0} , G3 = {x : x1 < 0, x2 > 0} , G4 = {x : x1 < 0, x2 < 0} , and the velocity vectors in these regions are such that the trajectories of the closedloop system cross the discontinuity manifolds S1 and S2 everywhere except the origin x = 0, which is the only equilibrium point of the system. Thus, we conclude that the closed-loop system also satisfies the right uniqueness property. In order to prove the asymptotic stability of (3.12), (3.16), let us utilize the nonsmooth Lyapunov function V (x1 , x2 ) =

 1 2 x + x22 + |x1 | , 2 1

whose time-derivative along the trajectories of the closed-loop system is given by the same expression (3.14) everywhere beyond the discontinuity manifolds S1 and

3.3 Invariance Principle

53

S2 . Due to Lemmas 3.1 and 3.2, the invariance principle (Theorem 3.2) can be applied to the closed-loop system (3.12), (3.16) which, according to the principle, appears to be asymptotically stable since the largest invariant manifold is now reduced to the origin. 3.3.2.3 Velocity Observer Design Let us now analyze how to estimate the oscillator velocity over position measurement. A velocity observer of the enforced oscillator (3.9), (3.17) is constructed as follows     x˙ˆ1 xˆ2 + x1 − xˆ1 = . −xˆ1 − sign xˆ2 x˙ˆ2 Our aim is to prove the convergence of the observer output to that of the enforced oscillator (3.9), (3.17). We break up the proof into several simple steps. First, we rewrite the observer equations in terms of the estimation errors e1 = x1 − xˆ1 and e2 = x2 − xˆ2 ,     e2 − e 1 e˙1 (3.18) = e˙2 −e1 − sign x2 + sign (x2 − e2) and, following the same line of reasoning as before, we deduce the right uniqueness property for the overall system (3.9), (3.17), (3.18). We introduce now the Lyapunov function V (x) = |x1 | +

 1 2 x1 + x22 + e21 + e22 , 2

whose time-derivative is negative semidefinite along the trajectories of this system everywhere beyond the discontinuity manifolds, V˙ = −e21 − |x2 | − (x2 − xˆ2 ) (sign x2 − sign xˆ2 ) ≤ −e21 − |x2 | − |x2 | (1 − sign x2 sign xˆ2 ) − |xˆ2 | (1 − sign x2 sign xˆ2 ) ≤ −e21 − |x2 | . Next, we establish that the largest invariant subset of the manifold {(x1 , x2 , e1 , e2 ) ∈ R4 : x2 = e1 = 0} coincides with the origin x1 = x2 = e1 = e2 = 0. Indeed, the implication “x2 = 0 ⇒ x1 = 0” for the closed-loop system (3.9), (3.17) has been established before, whereas the implication “e1 = 0 ⇒ e2 = 0” for the error dynamics (3.18) results directly from the former equation of (3.18). Finally, the application of the invariance principle (Theorem 3.2) and Lemmas 3.1 and 3.2 to the overall system (3.9), (3.17), (3.18) ensures its asymptotic stability.

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3 Stability Analysis

3.4 Extended Invariance Principle As shown in Sect. 3.3, the invariance principle remains true if confined to a class of discontinuous dynamic systems whose trajectories are unambiguously defined. The uniqueness of the trajectories of discontinuous dynamic systems is, however, questionable in many practical situations. In order to avoid relating to this restrictive uniqueness condition, an alternative technique was developed in [165]. In addition to a nonsmooth Lyapunov function with non-positive time derivatives along the system trajectories, the technique involved an auxiliary indefinite (rather than definite) function that allowed one to derive a certain integral inequality, which by Barbalat’s lemma ensured the asymptotic stability of the system in question. It is worth noting that in contrast to the invariance principle, the technique, referred to as the extended invariance principle, appears to be applicable to general time-varying systems as well. Following [165], the global uniform asymptotic stability of the nonautonomous discontinuous system (3.1) is presently addressed in terms of a Lyapunov function with negative semidefinite time derivative along the system trajectories and a coupled indefinite auxiliary function. Conditions 1 and 2, given below, are involved for the stability analysis of (3.1). 1. There exists a Lipschitz continuous, radially unbounded, positive definite function V (x) such that the time derivative V˙ (x(t)) of the composite function V (x(t)), computed along the trajectories of (3.1), is negative semidefinite, i.e., V˙ (x(t)) ≤ −V1 (x(t)) f or almost all t ∈ R

(3.19)

where V1 (x) is a continuous, positive semidefinite function. 2. There exist a Lipschitz continuous (possibly, indefinite) function W (x) and a constant δ > 0 such that the time derivative W˙ (x(t)) of the composite function W (x(t)), computed along the trajectories of System 3.1, evolving within the set V1δ = {x ∈ Rn : V1 (x) ≤ δ }, satisfies the following inequality W˙ (x(t)) ≤ −W1 (x(t)) + c1V1 (x(t)) f or almost all t ∈ R

(3.20)

with some positive constant c1 and some continuous, positive semidefinite function W1 (x), whose combination V1 (x) + W1 (x) with V1 (x) is positive definite. Condition 1 is inherited from Krasovskii–LaSalle’s invariance principle, extended in Sect. 3.3 to autonomous discontinuous dynamic systems, whose trajectories are uniquely continuable on the right. Condition 2 is involved to avoid relating to the aforementioned uniqueness property, while being quite natural for smooth dynamic systems, but becoming restrictive for nonsmooth, particularly, discontinuous dynamic systems. Roughly speaking, Condition 1 can only guarantee that on the trajectories of (3.1) the composite function V1 (x(t)) → 0 as t → ∞ whereas Condition C2 additionally ensures that W1 (x(t)) → 0 as t → ∞. Thus, by virtue of the positive definiteness of the combination V1 (x) + W1 (x) a discontinuous system (3.1), satisfying Conditions

3.4 Extended Invariance Principle

55

1 and 2, turns out to be globally asymptotically stable even in the nonautonomous case where the piece-wise continuous function ϕ (x,t) is uniformly bounded in t, i.e., ϕ (x,t) ≤ M(D) for almost all (x,t) ∈ D × R (more precisely, for all (x,t) ∈ D × R except those in the discontinuity set N ), any bounded set D ∈ Rn , and some constant M(D) > 0, possibly dependent on D. Theorem 3.3 (Extended invariance principle). Let Conditions 1 and 2 be satisfied and let the right-hand side of (3.1) be uniformly bounded in t. Then the equilibrium point x = 0 of (3.1) is globally asymptotically stable. If, in addition, (3.1) is autonomous, then the equilibrium point is globally uniformly asymptotically stable. Proof. For the convenience of the reader, we recall Lemma 8.2 of [110], known as Barbalat’s lemma, to be used subsequently in proving Theorem 3.3. Lemma 3.3 (Barbalat’s lemma). Let φ (t) : R → R be a uniformly continuous func tion on [0, ∞). Suppose that limt→∞ 0t φ (τ )d τ exists and is finite. Then, φ (t) → 0 as t → ∞. To begin with, let us note that by Theorem 8 of [71], a solution of (3.1) is locally defined for arbitrary initial data x(t0 ) = x0 ∈ Rn . Due to Condition 1, all possible solutions of (3.1), which are initialized at t0 ∈ R within a bounded set {x ∈ Br : V (x) ≤ Vr } specified by positive constants r and Vr < minx=r V (x), are a priori estimated via sup V (x(t)) ≤ Vr . (3.21) t∈[t0 ,∞)

Thus, any solution, starting in {x ∈ Br : V (x) ≤ Vr }, remains bounded because it stays in the interior of the ball Br forever. Since V (x) is radially unbounded and the constants r and Vr can be chosen arbitrarily large, it follows that each solution of (3.1) is uniformly bounded in t, and it is therefore globally continuable on the right. Furthermore, x = 0 proves to be a stable equilibrium point of the system because r can be chosen as small as desired. Next, let us show that lim V1 (x(t)) = 0, (3.22) t→∞

regardless of the choice of the initial data of (3.1). In order to justify (3.22), it suffices to integrate (3.19) on solutions of (3.1), initialized at t0 ∈ R within a compact set D0 = {x ∈ Rn : V (x) ≤ V0 }

(3.23)

where V0 = V (x(t0 )). Due to (3.21), the integrand V1 (x(t)) of the resulting inequality  ∞ t0

V1 (x(t))dt ≤ V0

(3.24)

is uniformly bounded in t for any solution x(t) of (3.1). Moreover, the continuous function V1 (x(t)) is uniformly continuous in t because V1 (x) is uniformly continuous

56

3 Stability Analysis

on the compact set (3.23), and x(t) ˙ is uniformly bounded in t and x(t) is therefore uniformly continuous in t. Indeed, x(t) ˙ ∈ Φ (x,t) where, by Definition 1, the righthand side Φ (x,t) is obtained by closure of the graph φ (x,t) and by passing over to a convex set. Since φ (x,t) is uniformly bounded in t by a condition of the theorem, both Φ (x,t) and x(t) ˙ prove to be uniformly bounded in t with the same upper bound M(D0 ) as that of φ (x,t). Thus, the validity of (3.22) is straightforwardly obtained by applying Lemma 3.3 to the integral inequality (3.24). As a consequence of (3.22), each trajectory of (3.1) evolves in the set V1δ = {x ∈ n R : V1 (x) ≤ δ } starting from a finite time moment, and it remains to prove that any motion of the system within the set V1δ approaches the origin as t → ∞. For this purpose, let us preliminarily demonstrate that lim W1 (x(t)) = 0

(3.25)

t→∞

for each trajectory of (3.1), evolving within V1δ . Indeed, integrating (3.20) yields  t t0

W1 (x(t))dt ≤ [W (x(t0 )) − W (x(t))] + c1

 ∞ t0

V1 (x(t))dt ≤ c(V0 )

(3.26)

where the latter inequality with some positive constant c(V0 ), dependent on V0 = V (x(t0 )), is guaranteed for all t ≥ t0 by the continuity of the function W (x) and relations (3.21), (3.24). Hence,  ∞ t0

W1 (x(t))dt ≤ c(V0 )

where the integrand W1 (x(t)) is uniformly bounded and uniformly continuous in t for any solution x(t) of (3.1). The verification of the uniform boundedness and uniform continuity of the integrand W1 (x(t)) is similar to that made for V1 (x(t)). Thus, Lemma 3.3 is applicable to (3.26) as well, and by applying this lemma, the limiting relation (3.25) is obtained. Moreover, since the linear combination U(x) = V1 (x) +W1 (x) is positive definite by Condition 2, and due to (3.22), (3.25) lim U(x(t)) = 0,

t→∞

(3.27)

the desired state convergence (3.3) is thus guaranteed. This conclusion can be shown by contradiction, for if it were not true, there would exist a solution x(t) of (3.1), time instants tk , k = 0, 1, . . . , and ε > 0 such that tk → ∞ as k → ∞ and x(tk ) > ε for all k = 1, 2, . . . . In accordance with (3.21), it follows that the solution x(t) belongs to the compact set K = {x ∈ Rn : x ≥ ε and V (x) ≤ V (x(t0 ))} for all t ≥ t0 and the inequalities

3.5 Asymptotic Stability and Semiglobal Lyapunov Functions

inf U(x(tk )) ≥ inf U(x) > 0 k

K

57

(3.28)

hold due to the continuity and positive definiteness of U(x), whereas (3.28) contradicts the limiting relation (3.27). Finally, if (3.1) is autonomous, then it is, in addition, uniformly stable and by applying Lemma 1 of [71, p. 160], the state convergence (3.3) holds uniformly in the initial data t0 ∈ R, x0 ∈ Bδ for each δ > 0, thereby yielding the global uniform asymptotic stability of (3.1). Theorem 3.3 is thus completely proved. Example 3.2. Consider the system x˙ = y, y˙ = −α sign x − β sign y with the parameters

α > β > 0.

(3.29) (3.30)

Conditions 1 and 2 of Theorem 3.3 hold for (3.29) with the functions V (x, y) = α |x| + 12 y2 and W (x, y) = xy. Indeed, V˙ = α ysignx − y(α sign x + β sign y) = −β |y| = −V1 (y)

(3.31)

whereas W˙ = y2 − x(α sign x + β sign y) ≤ δ β −1 |y| − (α − β )|x| = δ β −2V1 (y) − W1 (x) within the set V1δ = {(x, y) : β |y| ≤ δ }. Since V1 (y) + W1 (x) = β |y| + (α − β )|x| is positive definite, Theorem 3.3 proves to be applicable to the autonomous system (3.29). Hence, the origin is a globally uniformly asymptotically stable equilibrium of (3.29). It is of interest to note that under the parameter subordination b ≥ a > 0,

(3.32)

opposite to (3.30), System 3.29 is not even asymptotically stable. Indeed, the velocity vectors in the case of (3.29) are normal to and directed towards each other while approaching the horizontal axis x where y = 0 from different sides. Due to this, the trajectories of (3.29) cannot leave the axis y = 0. Thus, the horizontal axis consists of equilibrium points of (3.29) which is why (3.29) is not asymptotically stable.

3.5 Asymptotic Stability and Semiglobal Lyapunov Functions In this section we deal with nonsmooth Lyapunov functions whose time derivative along the trajectories of the discontinuous system (3.1) are negative definite rather than simply negative semidefinite. Given such a Lyapunov function, Theorem 3.1 can be strengthened as follows.

58

3 Stability Analysis

Theorem 3.4. Suppose that 1. The right-hand side of System 3.1 is uniformly bounded in t and 2. In a domain (x ∈ Bδ ,t ∈ R) there exist a Lipschitz continuous, positive definite, decrescent function V (x,t) and continuous functions v0 (x), v1 (x), v(x) such that 0 < v0 (x) ≤ V (x,t) ≤ v1 (x) f or all (x,t) ∈ Bδ × R

(3.33)

and the time derivative (3.5) of the composite function V (x(t),t), computed along the trajectories x(t) of (3.1), which are initialized within Bδ , is negative definite almost everywhere, i.e., d V (x (t) ,t) ≤ −v(x(t)) < 0 dt

(3.34)

for almost all t. Then (3.1) is asymptotically stable. Proof. By Theorem 8 of [71, p. 85], a solution of (3.1) is locally defined for arbitrary initial data. Subject to r < δ and c < minx=r v0 (x), the set {x0 ∈ Bδ : v0 (x) ≤ c} appears in the interior of the ball Br . Let a time-dependent set be given by Ωc,t = {x ∈ Br : V (x,t) ≤ c}. Then, by virtue of (3.33), one concludes that {x ∈ Br : v1 (x) ≤ c} ⊂ Ωc,t ⊂ {x ∈ Br : v0 (x) ≤ c} ⊂ Br ⊂ Bδ

(3.35)

for all t ≥ t0 . Due to (3.34), each solution x(t) of (3.1) is a priori estimated via sup V (x(t),t) ≤ c

(3.36)

t∈[t0 ,∞)

if it is initialized at t0 ∈ R with x(t0 ) = x0 ∈ Ωc,t0 . It follows that x(t) ∈ Ωc,t for all t ≥ t0 . Coupled to (3.35), this ensures that any solution x(t) starting in {x ∈ Br : v1 (x) ≤ c} stays in {x ∈ Br : v0 (x) ≤ c} for all t ≥ t0 . Hence, x(t) is uniformly bounded in t and it is therefore globally continuable on the right. Moreover, by choosing r arbitrarily small, x = 0 is shown to be a stable equilibrium point of the system. It remains to demonstrate that lim x(t) = 0.

t→∞

(3.37)

For this purpose, let us integrate (3.34) on a solution x(t) of (3.1), initialized at t0 ∈ R with x(t0 ) = x0 within a compact set {x ∈ Br : v1 (x) ≤ c}. Due to (3.36), the integrand v(x(t)) of the resulting inequality  ∞ t0

v(x(t))dt ≤ c

(3.38)

3.5 Asymptotic Stability and Semiglobal Lyapunov Functions

59

is uniformly bounded in t for the solution x(t). Moreover, the continuous function v(x(t)) is uniformly continuous in t because v(x) is uniformly continuous on the compact set {x ∈ Br : v0 (x) ≤ c} and x(t) ˙ is uniformly bounded in t. Indeed, x(t) ˙ ∈ Φ (x,t) where the Filippov set Φ (x,t) is obtained by closure of the graph φ (x,t) and by passing over to a convex set. Since φ (x,t) is uniformly bounded in t by a condition of the theorem, both Φ (x,t) and x(t) ˙ prove to be uniformly bounded in t with the same upper bound as that of φ (x,t). Thus, the following convergence lim v(x(t)) = 0

t→∞

(3.39)

is straightforwardly obtained by applying Barbalat’s Lemma 3.3 to the integral inequality (3.38). By taking into account that v(x) is positive definite, convergence (3.37) is then guaranteed by (3.39). This conclusion is made by following the same line of reasoning used in the proof of Theorem 3.3. Theorem 3.4 is thus proved. For later use, we also analyze a parameterized family of semiglobal Lyapunov functions V δ (x,t), δ > 0 such that each V δ (x,t) is well-posed on the corresponding domain (x ∈ Bδ ,t ∈ R). In other words, the functions V δ (x, y), δ > 0 meet the condition 0 < vδ0 (x) ≤ V δ (x,t) ≤ vδ1 (x),

(3.40)

for all x ∈ Bδ , for almost all t ∈ R, and for some continuous functions vδ0 (x), vδ1 (x). In turn, their time derivatives V˙ δ (x(t),t), computed along the trajectories x(t) of (3.1), which are initialized within Bδ , are negative definite in the sense that the condition V˙ δ (x(t),t) ≤ −vδ (x(t)) < 0

(3.41)

holds almost everywhere for some vδ (x), positive definite on Bδ . Apart from this, the family {V δ (x,t)}δ >0 is assumed to be radially unbounded in the sense that lim min vδ0 (x) = ∞

δ →∞ x=δ

(3.42)

The following result is in order. Theorem 3.5. Suppose that the following conditions be satisfied: 1. There exist Lipschitz continuous functions V δ (x,t) , parameterized by δ > 0, such that (3.40), (3.41) hold for all x ∈ Bδ , for almost all t ∈ R, and for some continuous functions vδ0 (x), vδ1 (x), vδ (x); 2. The family {V δ (x,t)}δ >0 is radially unbounded in the sense of (3.42). Then (3.1) is globally asymptotically stable. Proof. The asymptotic stability of (3.1) is straightforwardly obtained by applying Theorem 3.4. As the family {V δ (x,t)}δ >0 is radially unbounded, the constant c,

60

3 Stability Analysis

used in the proof of Theorem 3.4, can be chosen arbitrarily large to include any initial state in the set {x ∈ R : vδ1 (x) ≤ c} for some δ > 0. Hence, convergence (3.37) holds for any initial conditions. The validity of Theorem 3.5 is thus established. Remark 3.1. It is worth noticing that the conditions of Theorem 3.5 admit that the family {V δ (x,t)}δ >0 is formed by a single globally defined function V (x,t), i.e., V δ (x,t) = V (x,t) for all δ > 0. In this particular case, Theorem 3.5 can be viewed as a generalization of Theorem 3.4 towards the global asymptotic stability of the system in question.

3.6 L2 -Gain Analysis The L2 -gain analysis, presented here, is based on the game-theoretic approach from [18] and extends the results from [38, 229], where investigations were confined to smooth autonomous systems only. An appropriate generalization of the Kalman– Popov–Yakubovich lemma to time-varying systems, made in [176], is additionally involved into analysis. The following perturbation x˙ = ϕ (x,t) + ψ (x,t)w(t)

(3.43)

of the discontinuous system (3.1) is presently under study. In the above equation, w ∈ Rr is the unknown disturbance, and ψ (·, ·) ∈ Rn×r is a continuous matrix function. The origin is assumed to be an equilibrium point of the nominal system (3.1) and the L2 -gain analysis is made with respect to the output z(t) = h(x(t),t)

(3.44)

where h(·, ·) ∈ Rp is a continuous vector function such that h(0,t) = 0 for all t ∈ R. Definition 3.5. Given a real number γ > 0, it is said that (3.43), (3.44) has L2 -gain less than γ if the response z, resulting from w for initial state x(t0 ) = 0 satisfies  t1 t0

z(t)2 dt < γ 2

 t1 t0

w(t)2 dt

(3.45)

for all t1 > t0 and all piecewise continuous functions w(t).

3.6.1 Hamilton–Jacobi Inequality The perturbed system (3.43), (3.44) is analyzed under the hypothesis that there exists a smooth, positive definite, decrescent, radially unbounded function V (x,t), such that the Hamilton–Jacobi inequality

3.6 L2 -Gain Analysis

61



∂V ∂V ∂V 1 ∂V + ϕ (x,t) + 2 ψ (x,t)ψ T (x,t) ∂t ∂x 4γ ∂ x ∂x

T

+ hT (x,t)h(x,t) ≤ −v(x)

(3.46) holds for all continuity points (x,t) ∈ Rn+1 \ N of the function ϕ (x,t), for some positive γ , and for some positive definite function v(x). Theorem 3.6. Let the above hypothesis be satisfied. Then the disturbance-free system (3.1) is globally asymptotically stable and its perturbed version (3.43), (3.44) has L2 -gain less than γ . Proof. To begin with, let us recall [71, p. 155] that the upper value of the time derivative d ∂ V (x(t),t) ∂ V (x(t),t) V (x,t) = + Φ (x(t),t) , (3.47) dt ∂t ∂x computed along the solutions of the differential inclusion (3.2), does not increase at the discontinuity points (x,t) ∈ N of the function ϕ (x,t) because the corresponding Filippov set Φ (x,t) is obtained by closure of the graph of ϕ (x,t) and by passing over to a convex hull. Thus, the Hamilton–Jacobi inequality (3.46) with Φ (x,t), being substituted for ϕ (x,t), remains in force for all (x,t) ∈ Rn+1 , thereby yielding the negative definiteness of the time derivative (3.47). It follows that the solution V (x,t) of the Hamilton–Jacobi inequality (3.46) forms a single-member family {V (x,t)} that meets the conditions of Theorem 3.5. By applying this theorem (cf. Remark 3.1), the global asymptotic stability of the disturbance-free system (3.1) is then concluded. It remains to show that the perturbed system (3.43), (3.44) has L2 -gain less than γ . For this purpose let us introduce the multi-valued function H(x, w,t) =

∂ V (x,t) ∂ V (x,t) + [Φ (x,t) + ψ (x,t)w] ∂t ∂x +hT (x,t)h(x,t) − γ 2wT w

(3.48)

which is quadratic in w. Then

∂ H (x, w,t) ∂ V (x,t) |w=α (x,t) = ψ (x,t) − 2γ 2 α T (x,t) = 0 (3.49) ∂w ∂x T  . Expanding the quadratic function H(x, w,t) in for α (x,t) = 21γ 2 ψ T (x,t) ∂ V∂(x,t) x Taylor series, we derive that H(x, w,t) = H(x, α (x,t),t) − γ 2 w − α (x,t)2

(3.50)

where H(x, α (x,t),t) ≤ −v(x) due to (3.46). Hence, H(x, w,t) ≤ −γ 2 w − α (x,t)2 − v(x) and employing (3.48) and (3.51) we arrive at

(3.51)

62

3 Stability Analysis

∂ V (x,t) ∂ V (x,t) + [Φ (x,t) + ψ (x,t)w] ∂t ∂x ≤ −γ 2 w − α (x,t)2 − v(x) − h(x,t)2 + γ 2 w2 .

(3.52)

Taking (3.52) into account, we estimate the time derivative of the solution V (x,t) of the Hamilton–Jacobi inequality (3.46) on the trajectories of the perturbed system (3.43), (3.44) as follows d V (x(t),t) ≤ −γ 2 w(t) − α (x(t),t)2 − v(x(t)) − z(t)2 + γ 2 w(t)2 .(3.53) dt Clearly, the latter inequality ensures that  t1 t0

(γ 2 w(t)2 − z(t)2)dt ≥ V (x(t1 ),t1 ) − V (x(t0 ),t0 ) +γ 2

 t1 t0

[w(t) − α (x(t),t)2 + v(x(t))]dt > 0

(3.54)

for any trajectory of (3.43), (3.44), initialized with x(t0 ) = 0. Thus, (3.45) is established and the proof of Theorem 3.6 is completed.

3.6.2 Time-varying Strict Bounded Real Lemma Being applied to a linear system x˙ = A(t)x + B(t)w, z(t) = C(t)x(t)

(3.55)

with piece-wise continuous matrix functions A(t), B(t),C(t) of appropriate dimensions, the Hamilton–Jacobi inequality (3.46), re-written in terms of the parameterized identical matrix v(x) = ε I, ε > 0 and the quadratic function V (x,t) = xT Pε (t)x, Pε (t) > 0, comes down to the differential Riccati inequality 1 P˙ε + Pε (t)A(t) + AT (t)Pε (t) + CT (t)C(t) + 2 Pε (t)BBT (t)Pε (t) + ε I ≤ 0. γ (3.56) For sufficiently small ε > 0 this inequality possesses a positive definite solution, uniformly bounded in t ∈ R, whenever the corresponding Riccati equation 1 P˙ + P(t)A(t) + AT (t)P(t) + CT (t)C(t) + 2 P(t)BBT (t)P(t) = 0 γ (3.57) has a uniformly bounded positive semidefinite solution P(t) such that system

3.6 L2 -Gain Analysis

63 .

x= [A + γ −2BBT P](t)x

(3.58)

is exponentially stable. The following result presents a natural extension of the strict bounded real lemma, also known as the Kalman–Popov–Yakubovich lemma, to time-varying systems. Lemma 3.4 (Time-varying strict bounded real lemma). Let the Riccati equation (3.57) have a uniformly bounded positive semidefinite solution P(t) such that (3.58) is exponentially stable. Then there exists ε0 > 0 such that the perturbed Riccati equation 1 P˙ε + Pε (t)A(t) + AT (t)Pε (t) + CT (t)C(t) + 2 Pε (t)BBT (t)Pε (t) + ε I = 0 γ (3.59) has a unique uniformly bounded, positive definite, symmetric solution Pε (t) for each ε ⊂ (0, ε0 ). Proof. The proof of Lemma 3.4 is preceded by an auxiliary result, extracted from [235, p. 181]. Lemma 3.5. Let the conditions listed below be satisfied: • A time-dependent n×n-matrix S (t) is piece-wise continuous, uniformly bounded, symmetric and positive definite; • A time-dependent n × n-matrix A (t) is piece-wise continuous and uniformly bounded; • The corresponding differential equation x˙ = A (t)x

(3.60)

is uniformly asymptotically stable. Then the matrix Q(t) =

 ∞ t

T ΦA (τ ,t)S (τ )ΦA (τ ,t)d τ ,

specified with the above matrix S (t) and the transition matrix ΦA (τ ,t) of (3.60), is uniformly bounded and positive definite. In order to prove that (3.59) has a unique bounded, positive definite solution Pε (t) for each ε > 0 small enough, let us represent this equation in the form

Γ (Pε ) + ε I = 0

(3.61)

where

Γ : P(t) → P˙ + P(t)A(t) + AT (t)P(t) + CT (t)C(t) +

1 P(t)B(t)BT (t)P(t), t ∈ R γ2 (3.62)

64

3 Stability Analysis

is a mapping from the space B of differentiable bounded symmetric matrix functions with bounded derivatives to the space B1 of piecewise continuous bounded symmetric matrix functions. Clearly, Equation 3.57 is equivalent to Γ (Pε =0 ) = 0. First, we demonstrate that the tangent map ˙ + [A(t) + 1 B(t)BT (t)P(t)]T Q(t) + DΓP : Q(t) → Q(t) γ2 1 Q(t)[A(t) + 2 B(t)BT (t)P(t)] γ

(3.63)

of Γ at Pε =0 is invertible. In other words, we demonstrate that for any S(t) ∈ B1 , the equation DΓP (Q) + S = 0 (3.64) has a unique solution Q(t) ∈ B. Indeed, it is straightforward to check that such a solution is given by Qs (t) =

 ∞ t

Φ T (τ ,t)S(τ )Φ (τ ,t)d τ

(3.65)

where Φ (τ ,t) is the transition matrix of (3.58), which under the lemma conditions satisfies the inequality Φ (τ ,t) ≤ μ e−ν (τ −t) (3.66) for all τ ≥ t and some positive μ and ν . Moreover, since the arbitrary solution of (3.64) admits the representation Q(t) = Φ T (0,t)Q(0)Φ (0,t) +

t 0

Φ T (τ ,t)S(τ )Φ (τ ,t)d τ ,

(3.67)

the difference Φ T (0,t)[Q1 (0) − Q2 (0)]Φ (0,t) between two uniformly bounded solutions Q1 (t) and Q2 (t) of (3.64) is uniformly bounded iff these solutions have the same initial conditions. Otherwise, there would exist a nonzero vector q ∈ Rn , satisfying the inequality Φ (0,t)q ≤ K with some K > 0 , which by virtue of (3.66) results in the false statement that 0 = q = Φ (t, 0)Φ (0,t)q ≤ Φ (t, 0) Φ (0,t)q ≤ K μ e−ν t → 0 as t → ∞. This contradiction proves the uniqueness of the uniformly bounded solution of (3.64). Due to Lemma 3.5, (3.65) is positive definite whenever S(t) is positive definite. By the implicit function theorem (see e.g., [110, p. 651]), it follows that (3.61) has a unique bounded, positive definite, symmetric solution Pε (t) for each ε > 0 small enough. Thus, Lemma 3.4 is proved.

3.7 The Lyapunov Analysis of Discrete-continuous Dynamics

65

3.7 The Lyapunov Analysis of Discrete-continuous Dynamics In this section, we admit that the underlying system x˙ = ϕ (x,t)

(3.68)

exhibits impulse effects on a countable set of time instants t1 < t2 < t3 . . . when the state of the system is instantaneously changed to the one re-computed according to x(ti +) = U(x(ti −),ti ), i = 1, 2, . . . .

(3.69)

In the above relations, the piece-wise continuous function ϕ (x,t) is the same as in (3.1), the time instants ti = ti (x), i = 1, 2, . . ., generally speaking, depend on the current state, and the restitution rule U(x,t) is a continuous function such that U(x,t) = 0 for all t ∈ R. It should be pointed out that the present investigation includes but is not confined to the impulse effects of Sect. 2.1, caused by an external impact g(x,t)δ (t − ti ).

3.7.1 Global Asymptotic Stability The second Lyapunov method is now developed for (3.68) with impulse effects (3.69). Note that the stability concepts, introduced in Sect.3.1, applies for the discrete-continuous system (3.68), (3.69) as well. Let discrete-continuous dynamics (3.68), (3.69) possess a globally defined, Lipschitz continuous, positive definite, decrescent function V (x,t) such that its time derivative, computed along the trajectories x(t) of (3.68) is negative semidefinite almost everywhere, i.e., d V (x (t) ,t) ≤ 0 for almost all t ∈ R. dt

(3.70)

Moreover, let at the time instants ti , i = 1, 2, . . ., this function, computed on the trajectories of (3.68) subject to the restitution rule (3.69), take the values V (x(ti +),ti ) that converge to zero as i → ∞, i.e., lim V (x(ti +),ti ) = 0.

i→∞

(3.71)

Then the following result is true. Theorem 3.7. Consider (3.68), (3.69) with the assumptions above. Suppose that there exists a Lipschitz continuous, positive definite, decrescent function V (x,t) such that any trajectory x(t) of (3.68), (3.69) meet (3.70), (3.71). Then (3.68) with impulse effects (3.69) is globally asymptotically stable.

66

3 Stability Analysis

Proof. By Theorem 3.1, (3.68) is stable and this property remains in force for (3.68) with impulse effects (3.69) because the function U(x,t) is continuous and U(0,t) = 0 for all t ∈ R. Since, due to (3.70), the composite function V (x(t),t) does not increase along the trajectories of (3.68), (3.69) between the successive time instants ti and ti+1 i = 1, 2, . . . whereas by virtue of (3.71), the values V (x(ti +),ti ) escape to zero as i → ∞, the convergence lim V (x(t),t) = 0

t→∞

(3.72)

follows, regardless whether the time instants ti , i = 1, 2, . . . possess a finite accumulation point or limi→∞ ti = ∞. Taking into account that V (x,t) is positive definite, (3.72) results in lim ν (x(t)) = 0 (3.73) t→∞

for some positive definite, time-independent function ν (x). As in the proof of Theorem 3.3, this ensures the global asymptotic stability of (3.68), (3.69). The proof of Theorem 3.7 is thus completed.

3.7.2 Illustrative Example: Impulsive Stabilization of a Mechanical Oscillator with Coulomb Friction The spring-mass system (Figure 3.2),affected by Coulomb friction, is governed by the differential equation my¨ = −ky − α signy+ ˙ w+u

(3.74)

where y is the displacement, y˙ is the velocity, m > 0 is the mass of the system, α > 0 is the Coulomb friction level, k > 0 is the stiffness coefficient, u denotes the control input, and w stands for external disturbances. In order to account for model discrepancies, an unknown external disturbance w(t) has been introduced into modeling. In what follows, the amplitude of the disturbance is assumed to be smaller than half the Coulomb level α , i.e., 1 |w(t)| ≤ α0 < α 2

(3.75)

for all t and some constant α0 > 0. Assumption 3.75 is made for a technical reason that becomes clear as far as the controller goes. Being represented in the state space form, the above system (3.74) is modified to: x˙1 = x2 x˙2 =

1 [−kx1 − α signx2 + w + u] m

(3.76)

3.7 The Lyapunov Analysis of Discrete-continuous Dynamics

67

where x1 = y and x2 = y. ˙ Since the unforced system (3.76) is dissipative and its dissipation is lower bounded by the positive constant α − α0 , (3.76) under u = 0 is getting stuck in finite time at the disturbance-dependent zone (see Sect. 3.3.2.1 for details): Sw ⊂ S = {(x1 , x2 ) : |x1 | ≤

α + α0 , x2 = 0} ⊂ R2 . k

(3.77)

In order to reduce this friction effect without resorting to high gain control loops, in Sect. 3.3.2.2 we presented the static discontinuous feedback (3.16), utilizing position measurements only. Now we develop a novel control approach counteracting friction effects. The approach is based on an impulsive actuation while the underlying system is getting stuck due to the presence of dry friction. The spring-mass system serves as a simple test bed. Asymptotic stabilization of this system is under study. Robustness against small parameter variations and external disturbances is additionally provided. 3.7.2.1 Problem Statement Assuming that the system is enforced by an impulsive actuator ∞

u = ∑ v(x1 )δ (t − ti )

(3.78)

i=1

utilizing a continuous position feedback v(x1 ) and being applied at some statedependent time instants ti (x1 , x2 ), i = 1, 2, . . ., the closed-loop system appears to exhibit discrete-continuous dynamics x˙1 = x2 , x˙2 =

1 (−kx − α signx2 + w) , t = ti , m

(3.79)

x1 (ti +) = x1 (ti −), x2 (ti +) = x2 (ti −) + v(x1(ti )), i = 1, 2, . . . .

(3.80)

It is clear that dealing with the position feedback v(x1 ) yields the above restitution rule (3.80) because the impulsive input (3.78) results in the corresponding instantaneous change of the velocity while the position dynamics and, hence, the position feedback v(x1 (t)) remain continuous in time. It is worth noticing that applying a state feedback v(x1 , x2 ) would impose a certain nonlinear restitution rule, caused by an ill-posed product of the Dirac function δ (t − τ ), localized at a time instant τ , and the function v(x1 (t), x2 (t)), discontinuous at t = τ (see Sect. 2.1.2 for details on the nonlinear impulse response). Our objective is to design an impulsive controller (3.78) such that the closedloop system (3.79), (3.80) is asymptotically stable around the origin, regardless of whichever external disturbance (3.75) affects the system. The current position x1 (t)

68

3 Stability Analysis

is assumed to be available for measurement, whereas the only available information on the velocity x2 (t) is the knowledge of whether it is nullified or it is not. 3.7.2.2 Controller Design Let the impulsive controller (3.78) be specified with  2α |x1 | − kx21 signx1 v(x1 ) = − m

(3.81)

and let it be applied to system (3.76) at the time instants ti , i = 1, 2, . . . such that |x1 (ti )| ≤

α + α0 , x2 (ti −) = 0. k

(3.82)

The dynamics of the closed-loop system (3.78)–(3.82) is then as follows. Once the underlying system (3.76) hits the stuck zone (3.77), it is enforced by the impulsive controller (3.78) that changes the velocity of the system instantaneously (see Fig. 3.3). The controller amplitude (3.81) has been pre-specified in such a manner as to bring the underlying system (3.76) from the stuck zone (3.77) to the phase trajectory that, while being disturbance-free, arrives at the origin without oscillations. It is shown that the asymptotic stabilization is thus achieved not only for the disturbance-free system (3.76) with w = 0 but also for its perturbed version, provided that external disturbances affecting the system meet the norm upper bound (3.75). The following result is in order. Theorem 3.8. Let the friction oscillator (3.76) be driven by the impulsive controller (3.78), specified with (3.81), (3.82). Then the closed-loop system (3.78)–(3.82) is globally asymptotically stable, provided that the norm upper bound (3.75) holds for the external disturbance affecting the system. Proof. Let us demonstrate that Theorem 3.7 is applicable to system (3.79), (3.80). To begin with, let us note that the smooth, positive definite, time-invariant function V (x1 , x2 ) =

kx21 + mx22 2

(3.83)

meets the condition (3.70) because its time derivative, computed along the trajectories of (3.76), is everywhere negative semidefinite: V˙ (x1 (t), x2 (t)) = −α |x2 (t)| + x2w(t) ≤ −(α − α0 )|x2 (t)|.

(3.84)

Next, let us verify that function (3.83) meets condition (3.71), too. Indeed, (3.76), as pointed out, possesses a globally finite time stable invariant manifold Sw , localized according to (3.77) within the estimated stuck zone S. Once the closed-loop

3.7 The Lyapunov Analysis of Discrete-continuous Dynamics

69

system (3.78)–(3.82) attains S, say at x1 (t1 ) = ξ1 such that |ξ1 | ≤ α +kα0 , the impulsive controller (3.78), (3.81), (3.82) is applied at the time instant t1 . The restitution rule (3.80) is then specified as  x2 (t1 +) = −

x1 (t1 ) = ξ1 , 2α |ξ1 | − k(ξ1 )2 signξ1 . m

(3.85)

If the closed-loop system is disturbance-free, the state trajectory, re-initialized with (3.85), would mono-directionally arrive at the origin in finite-time (see Fig. 3.3). However, if an admissible disturbance (3.75) affects the closed-loop system, the state trajectory would hit the stuck zone at x1 (t2 ) = ξ2 = 0 provided that x2 (t2 −) = 0. x2

α + α0 k

x1

− α +kα0

Fig. 3.3 Phase portrait of the impulsive closed-loop system (3.78)–(3.82): dotted lines are for the jumps of the velocity, solid lines are for the perturbed trajectories, and dashed lines are for the unperturbed trajectories

Our goal is to demonstrate that, even in the extreme disturbance case where w = α0 or w = −α0 , the following inequality holds: |ξ2 | ≤

2α0 |ξ1 |. α

(3.86)

Then by iteration on i, similar relations |ξi+1 | ≤

2α0 |ξi |, i = 1, 2, . . . . α

(3.87)

70

3 Stability Analysis

could be obtained for the state positions ξi = x1 (ti ) at the time instants ti , i = 1, 2, . . . of reaching the stuck zone (3.77). Since q = 2αα0 < 1 by assumption, it would follow that lim |x1 (ti )| = lim qi−1 |ξ1 | = 0, lim |x2 (ti +)| = 0

i→∞

i→∞

i→∞

(3.88)

 ξi )2 where the relations x2 (ti +) = − 2α |ξi |−k( signξi , i = 1, 2, . . . , similar to (3.85), m have been taken into account. Since (3.88) results in limi→∞ V (x1ti , x2ti ) = 0, condition (3.71) would thus be verified. For the purpose of validating (3.86), let us compute the value of ξ2 as a function of ξ1 , assuming, for certainty, that ξ1 < 0, and performing similar computation, otherwise. In the case where w = α0 the system dynamics between successive impacts are governed by x˙1 = x2 1 x˙2 = (−kx1 − α + α0 ), t ∈ (t1 ,t2 ). m

(3.89)

Initialized with (3.85), the solution to the perturbed system (3.89) is given by 1 α0 ξ1 x22 x2 (−k 1 − α x1 + α0 x1 ) = + . m 2 m 2

(3.90)

Being confined to the time instant t2 when x2 (t2 −) = 0, the above relation yields kx21 (t2 ) + 2(α − α0 )x1 (t2 ) + 2α0ξ1 = 0.

(3.91)

Setting ξ2 = x1 (t2 ) and taking into account that the case where ξ1 < 0 is under study, it follows that  α − α0 (α0 − α )2 2α0 |ξ1 | + . (3.92) ξ2 = − + k k2 k Substituting (3.92) into (3.86) for ξ2 , we arrive at the inequality  α − α0 (α0 − α )2 2α0 |ξ1 | 2α0 + ≤ − + |ξ1 | (3.93) k k2 k α to be verified. To validate (3.93) it suffices to represent it in the form   α − α0 2 (α0 − α )2 2α0 |ξ1 | 2α0 ≤ + | ξ | + , 1 k2 k α k

(3.94)

and to observe that (3.94) is equivalent to the inequality 2α0 |ξ1 | 4(α0 )2 4α0 (α − α0 ) ≤ |ξ1 | |ξ1 |2 + 2 k α αk

(3.95)

3.7 The Lyapunov Analysis of Discrete-continuous Dynamics

71

whose validation is reduced to the obvious inequality 1≤

2(α − α0 ) , α

(3.96)

resulted from (3.75). Thus, inequality (3.86) is verified in the case of w = α0 . It remains to verify inequality (3.86) in the case of w = −α0 with the system dynamics between successive impacts governed by x˙1 = x2 1 x˙2 = (−kx1 − α − α0 ), t ∈ (t1 ,t2 ). m

(3.97)

The solution of the above system, initialized with (3.85), is given by x2 1 α0 ξ1 x22 (−k 1 − α x1 − α0 x1 ) = − + . m 2 m 2

(3.98)

Specified at the next impact time instant t2 when x2 (t2 −) = 0, relation (3.98) yields kx21 (t2 ) + 2(α + α0 )x1 (t2 ) − 2α0ξ1 = 0.

(3.99)

Setting ξ2 = x1 (t2 ), it follows that

α0 + α + ξ2 = − k



(α0 + α )2 2α0 |ξ1 | − k2 k

(3.100)

provided that the case where ξ1 < 0 is under study. Substituting (3.100) into (3.86) for ξ2 , we arrive at the inequality  α0 + α (α0 + α )2 2α0 |ξ1 | 2α0 − ≤ − |ξ1 | k k2 k α

(3.101)

to be verified. To validate (3.101) it suffices to represent it in the form 

2 (α0 + α )2 2α0 |ξ1 | α + α0 2α0 |ξ1 | ≤ − − , k α k2 k

(3.102)

and to observe that (3.102) is equivalent to the inequality 2α0 |ξ1 | 4(α0 )2 4α0 (α + α0 ) + |ξ1 |. |ξ1 |2 ≤ 2 k α αk Since ξ1 is within the estimated stuck zone (3.77), i,e., |ξ1 | ≤ of (3.103) is reduced to the inequality 1+

2α0 (α + α0 ) 2(α + α0 ) ≤ , α2 α

α +α0 k ,

(3.103) the validation

(3.104)

72

3 Stability Analysis

straightforwardly resulted from (3.75). Thus, inequality (3.86) is verified in the case of w = −α0 , too. Hence, Theorem 3.7 is applicable to system (3.79), (3.80). By applying Theorem 3.7, the global asymptotic stability of the the closed-loop system (3.78)–(3.82) is established. This completes the proof of Theorem 3.8. 3.7.2.3 Numerical Results Performance issues and robustness properties of the proposed impulsive controller are additionally tested in numerical experiments. In the simulations, performed with MATLAB , the dimensionless spring-mass model (3.74) is studied with the parameters m = 1, k = 1, and α = 1. The initial position and the initial velocity are set to x1 (0) = 3.5 and x2 (0) = −4, respectively. The impulsive controller (3.78), (3.81), (3.82) is first applied to the disturbancefree system. In order to test the controller robustness, a harmonic external disturbance w = 0.7 sint is then applied to the closed-loop system. Good performance and desired robustness properties of the controller are concluded from Fig. 3.4.

3.8 Linear Operator Inequalities in a Hilbert Space Linear matrix inequalities (LMIs) represent a powerful method of analysis of uncertain dynamic systems. Here, a LMI-based approach is developed in infinitedimensional setting for linear time-delay systems, evolving in a Hilbert space.

3.8.1 Preliminaries 3.8.1.1 Historical Remarks Time-delay phenomenon is natural for many control systems, and it is frequently a source of instability [92]. In the case of distributed parameter systems, even arbitrarily small delays in the feedback may destabilize the system (see e.g., [57]). The stability issue of systems with delay is, therefore, of theoretical and practical value. By now, a considerable amount of attention has been paid to the stability analysis of dynamic systems governed by ordinary differential equations with uncertain, possibly, time-varying delays (see e.g., [29], [75], [88], [115], [155], [193]). The stability analysis of partial differential equations with delay is essentially more complicated. There are only a few works on a Lyapunov-based technique for such equations. In [236], the second Lyapunov method is extended to abstract nonlinear time-delay systems in Banach spaces. A delay-independent stability analysis of heat equations and wave equations, both with constant delays, is made in [102, 237].

73 4

3

3

2

2

1

1 1

4

0

x

x

1

3.8 Linear Operator Inequalities in a Hilbert Space

0

−1

−1

−2

−2 −3

−3 −4 0

2

4

6

8

−4 0

10

4

2

3

2

2

1

1

0

0 2

−1

x

2

x

3

−2

−3

−3

−5 0

2

4

6

8

−5 0

10

2

4

3

3

2

2

1

1

0

0

−1

−1

x2

2

x

6

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time

time

−2

−2

−3

−3

−4

−4

−3

−2

−1

1

0

2

3

−5 −4

4

−3

−2

−1

0

x1

1

2

3

4

x1

0.1

0.1

0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

u

u

10

−4

−4

−0.5

−0.4 −0.5

−0.6

−0.6

−0.7

−0.7

−0.8 −0.9 0

8

−1

−2

−5 −4

6

time

time

−0.8 2

4

6

time

8

10

−0.9 0

2

4

6

8

10

time

Fig. 3.4 Impulsive stabilization of the spring-mass system with Coulomb friction: the left column is for the no disturbance case, and the right column is for the disturbance case

74

3 Stability Analysis

The objective of the present section is to extend the Lyapunov method to a class of Hilbert space-valued time-delay systems where a bounded operator acts on the delayed state. The system delay is admitted to be unknown and time-varying with an a priori known upper bound on the delay derivative. Sufficient stability conditions are derived in the form of Linear operator inequalities (LOIs), where the decision variables are operators in the Hilbert space. Being applied to a parabolic heat equation, these conditions are represented in terms of standard finite-dimensional LMI. Although the development is confined to linear parabolic systems, the generalization to linear hyperbolic systems is indeed possible and can be found in [74]. 3.8.1.2 Background Material Let H be a Hilbert space equipped with the inner product ·, · and the corresponding norm | · |. L (H) stands for the space of linear bounded operators from H to H. The identity operator in H is denoted by I. Given a linear operator P : H → H with a dense domain D(P) ⊂ H, the notation P∗ stands for the adjoint operator. Such an operator P is strictly positive definite, i.e., P > 0, iff it is self-adjoint in the sense that P = P∗ and there exists a constant β > 0 such that x, Px ≥ β x, x for all x ∈ D(P),

(3.105)

whereas P ≥ 0 means that P is self-adjoint and nonnegative definite, i.e., x, Px ≥ 0 for all x ∈ D(P).

(3.106)

If an infinitesimal operator −A generates a strongly continuous semigroup T (t) on the Hilbert space H (see, e.g., [55] for details), the domain of the operator A forms another Hilbert space D(A) with the graph inner product (·, ·)D (A) defined as follows: (x, y)D (A) = x, y + Ax, Ay , x, y ∈ D(A). Moreover, the induced norm T (t) of the semigroup T (t) satisfies the inequality T (t) ≤ κ eδ t everywhere with some constant κ > 0 and growth bound δ , and the following relation A−1 H = D(A) holds for the operators A with negative growth bounds δ . Just in case, the norm of x ∈ D(A) given by |Ax| is equivalent to the graph norm xD (A) of D(A). The domain D(A) of such an operator A is thus continuously embedded into H, i.e., D(A) ⊂ H, D(A) is dense in H and the inequality |x| ≤ ω |Ax| holds for all x ∈ D(A) and some constant ω > 0. Apart from this, the square root √ A of the operator A is rigorously introduced on D(A) as a positive definite solution X of the algebraic operator equation X 2 √ = A. Being extended by continuity, this operator is well-posed on the domain D( A) and continuously √ embedded into H whereas D(A)√turns out to be continuously embedded into D( A). In other words, √ D(A) and D( A) are densely embedded into D( A) and H, respectively, and the following inequalities

3.8 Linear Operator Inequalities in a Hilbert Space

75

√ √ |x| ≤ ω | Ax| for all x ∈ D( A)

√ | Ax| ≤ ω |Ax|

for all x ∈ D(A)

(3.107) (3.108)

hold with a generic constant ω > 0. All relevant background material on fractional operator degrees can be found, e.g., in [117]. The space of the continuous H-valued functions x : [a, b] → H with the induced norm xC([a,b],H) = max |x(s)| s∈[a,b]

is denoted by C([a, b], H). The space of the continuously differentiable H-valued functions x : [a, b] → H with the induced norm xC1 ([a,b],H) = max(xC([a,b],H) , x ˙ C([a,b],H) ) is denoted by C1 ([a, b], H). L2 (a, b; H) is the Hilbert space of square integrable H-valued functions on (a, b) with the corresponding norm; L2 (a, b; R) := L2 (a, b). W l,2 ([a, b], H) is the Sobolev space of absolutely continuous square integrable H-valued functions on (a, b) with square integrable derivatives up to the order l ≥ 1 and with the corresponding norm; W l,2 ([a, b], R) := W l,2 (a, b). Given x(·) ∈ L2 ([a, b], H), we denote xt = x(t + θ ) ∈ L2 ([−h, 0], H) for t ∈ [a + h, b]; to reduce the notational burden the dependence xt on θ is subsequently suppressed. The following result from [237] is instrumental in the subsequent derivation. Lemma 3.6 (Wirtinger’s inequality). Let u ∈ W 1,2 (a, b) be a scalar function with u(a) = u(b) = 0. Then  b

(b − a)2 u (ξ )d ξ ≤ π2

 b

(u (ξ ))2 d ξ .

(3.109)

x(t) ˙ + Ax(t) = A0 x(t) + A1x(t − τ (t)),

(3.110)

a

2

a

3.8.2 Linear Time-delay Systems Consider a linear infinite-dimensional system

evolving in a Hilbert space H where x(t) ∈ H is the instantaneous state of the system, the system delay τ (t) > 0 is piece-wise continuous and it is such that inf τ (t) > 0, sup τ (t) ≤ h t

t

(3.111)

76

3 Stability Analysis

for all t and for some constant h > 0; A0 , A1 ∈ L (H) are linear bounded operators, −A is an infinitesimal operator, generating a strongly continuous semigroup T (t), and the domain D(A) of the operator A is dense in H. Since the operator A + λ I is strictly positive definite for the corresponding identity operator I and positive λ ∈ R large enough [117], without loss of generality we assume throughout that T (t) possesses a negative growth bound δ (otherwise, the bounded operator √ λ I is readily absorbed into A from A ). We also assume that the operators AA0 and 0 √ √ AA1 are subordinate to A in the sense that √ √ √ | AAi x| ≤ ωi | Ax| for any x ∈ D( A) (3.112) and some constants ωi > 0, i = 0, 1. The latter assumption is particularly satisfied if the operators A0 and A1 commute to A. Let the initial conditions xt0 = ϕ (θ ), θ ∈ [−h, 0], φ ∈ W

(3.113)

be given on a time interval [t0 − h,t0] in the space W = C([−h, 0], D(A)) ∩C1 ([−h, 0], H).

(3.114)

Throughout, solutions of such a system are defined in the classical sense. Definition 3.6. A function x(t) ∈ C1 ([t0 ,t0 + η ], H), η > 0 with range in D(A) for all t ∈ [t0 ,t0 + η ] is said to be a solution of the initial-value problem (3.110), (3.113) on [0, η ) if x(t) is initialized with (3.113) and it satisfies (3.110) for all t ∈ [t0 ,t0 + η ] where τ (t) is continuous. As a matter of fact, along with the above definition, the Caratheodory solutions, satisfying (3.110) for almost all t ∈ (t0 ,t0 + T ) only, and even generalized solutions, could be considered. This would particularly allow one to involve initial functions φ beyond the space W , thereby capturing a situation where the initial condition (3.113) is given for θ ∈ [−h, 0) while the instantaneous value x(t0 ) at time instant t0 does not necessarily coincide with φ (0) and it is introduced independently of (3.113). However, in order to facilitate the exposition, we prefer to confine the investigation to the classical solutions that captures all the essential features of the general treatment. By the same reason, a potential extension to multiple time-varying delays is also beyond the presentation. Our aim is to derive stability criteria for linear time-delay systems (3.110) thus defined. The stability concept to be studied is based on the initial data norm φ W = Aφ C([−τ ,0],H) + φ˙ C([−τ ,0],H)

(3.115)

in space (3.114). Suppose x(t,t0 , φ ) denotes a solution of (3.110), (3.113) at a time instant t ≥ t0 and x(t,t0 , φ )D (A) stands for its instantaneous norm. Definition 3.7. System 3.110 is said to be stable in D(A) iff for each ε > 0 and for each t0 ∈ R there exists δ > 0 such that for each solution x(t,t0 , φ ) with the initial

3.8 Linear Operator Inequalities in a Hilbert Space

77

condition (3.113), satisfying φ W < δ , the inequality x(t,t0 , φ )D (A) < ε

(3.116)

holds for all t ≥ t0 . Definition 3.8. System 3.110 is said to be globally asymptotically stable in D(A) iff it is stable in D(A), and for all initial data t0 ∈ R and φ (·) ∈ W solution x(t,t0 , φ ) of (3.110), (3.113) is such that x(t,t0 , φ )D (A) → 0 as t → ∞.

3.8.3 Well-posedness Issues To begin with, we demonstrate that the solutions of the initial-value problem (3.110), (3.113) are well-posed on the semi-infinite time interval [t0 , ∞) and they can be found as mild solutions, i.e., as those of the integral equation x(t) = T (t − t0 )x(t0 ) +

t

t0 T (t − s)[A0 x(s) + A1 x(s − τ (s))]ds,

t ≥ t0 .

(3.117)

Moreover, solutions x(t) of (3.110), (3.113), uniformly bounded in the state space H, remain uniformly bounded in D(A) and they possess uniformly bounded time derivatives x(t). ˙ Theorem 3.9. Let H be a Hilbert space and let the following assumptions be satisfied: 1. The operator −A generates an analytical semigroup T (t) with a negative growth bound δ ; 2. The linear operators A0 and √ √ A1 are bounded in H; √ 3. The operators AA0 and AA1 are subordinate to A in the sense of (3.112); 4. The function τ (t) is piecewise-continuous of class C1 on the closure of each continuity subinterval and it satisfies (3.111) for all t and some constant h > 0. Then there exists a unique solution of the initial-value problem (3.110), (3.113). This solution is also a unique solution of the integral initial-value problem (3.113), (3.117). Proof. To begin with, let us choose a positive η0 small enough to ensure that η0 < inft τ (t) and the first discontinuity point t01 > t0 of τ (t) is such that the difference t01 − t0 is multiple to η0 , i.e., t01 = t0 + k0 η0 for some integer k0 > 0. While being viewed over the time segment [t0 ,t0 + η0 ], the initial-value problem (3.110) is equivalent to x(t) ˙ + Ax(t) = A0 x(t) + A1 φ (t − t0 − τ (t)), x(t0 ) = φ (0)

(3.118)

where the inhomogeneous term A1 φ (t − t0 − τ (t)) is of class C1 on [0, η0 ]. By [55, Theorem 3.1.3], there exists a unique local solution of (3.118) and this solution satisfies the integral equation (3.117) on [t0 ,t0 + η0 ].

78

3 Stability Analysis

The same line of reasoning is step-by-step applied to the time segments [ti−1 ,ti−1 + η0 ], i = 1, · · · , k0 with ti = ti−1 + η0 and tk0 = t01 . Following this line, the initialvalue problem is demonstrated to possess a unique solution x(t,t0 , φ ) for t ∈ [t0 ,t01 ], which satisfies the integral equation (3.117) on [t0 ,t01 ]. The assertion of Theorem 3.9 is then concluded by iteration on the time segments [t0j ,t0j+1 ], j = 1, 2, . . . where t01 < t02 < . . . are the successive discontinuity points of the function τ (t). Theorem 3.10. Let the assumptions of Theorem 3.9 be satisfied and let a solution x(t,t0 , φ ) of the initial-value problem (3.110), (3.113) be uniformly bounded in the Hilbert space H for all t ≥ t0 . Then, this solution is uniformly bounded in D(A) for all t ≥ t0 , and it possesses a uniformly bounded time derivative x(t,t ˙ 0 , φ ) on the semi-infinite time interval [t0 , ∞). √ Proof. Let us first demonstrate that | Ax(t)| is uniformly bounded for all t ≥ t0 . Indeed, using the solution representation (3.117), taking into account Assumption 1 of Theorem 3.9, and applying the well-known estimate √ ce−δ s T (s) A ≤ √ , s > 0 s

(3.119)

of the induced operator norm with a positive constant c (cf. [96, Theorem 1.4.3]) yields  t √ √ √ | Ax(t)| ≤ T (t − t0 )| Aφ (0)| + |T (t − s) A[A0 x(s)

+A1 x(s − τ (s))]|ds ≤ N1 +

 t t0

t0

√ T (t − s) A[A0 |x(s)|

+A1|x(s − τ (s))|]ds ≤ N1 + N2 ≤ N1 + N2

 t t0

√ T (t − s) Ads

 t −δ (t−s) ce t0

√ ds < N t −s

(3.120)

for all t ≥ t0 and some positive constants N1 , N2 , N. Now, by employing the operator subordination (3.112) and by applying estimates (3.119), (3.120), the following inequalities |Ax(t)| ≤ T (t − t0 )|Aφ (0)| + ≤ M1 e−δ (t−t0 ) + ≤ M1 +

t t0

 t t0

t t0

|T (t − s)A[A0 x(s) + A1 x(s − τ (s))]|ds

√ √ T (t − s) A × | A[A0 x(s) + A1 x(s − τ (s))]|ds

√ √ √ T (t − s) A[ω0 | Ax(s)| + ω1 | Ax(s − τ (s))|]ds ≤ M1

+M2

 t t0

 t −δ (t−s) √ ce √ ds < M T (t − s) Ads ≤ M1 + M2 t −s t0 (3.121)

3.8 Linear Operator Inequalities in a Hilbert Space

79

are derived for some positive constants M1 , M2 , M and all t ≥ t0 . Hence, |Ax(t)| is uniformly bounded on [t0 , ∞). By taking into account Assumption 3 of Theorem 3.9 and by virtue of (3.110), it follows that the time derivative x(t,t ˙ 0 , φ ) is also uniformly bounded on [t0 , ∞). The proof is completed.

3.8.4 The Lyapunov–Krasovskii Method For later use, we extend the Lyapunov–Krasovskii method to Hilbert space-valued time-delay systems. Given a continuous functional V : R × W × C([−τ , 0], H) → R,

(3.122)

its upper right-hand derivative along solutions of the initial-value problem (3.110), (3.113) is defined as follows: 1 V˙ (t,φ ,φ˙ ) = lim sups→0+ [V (t+s, xt+s (t,φ ), x˙t+s (t, φ )) − V (t, φ , φ˙ )]. (3.123) s Theorem 3.11. Let the assumptions of Theorem 3.9 be in force. Then, (3.110) is globally asymptotically stable in D(A) if there exist positive numbers α , β , γ , and a continuous functional (3.122) such that 2 β |φ (0)|2 ≤ V (t, φ , φ˙ ) ≤ γ φ W , 2 ˙ ˙ V (t, φ , φ ) ≤ −α |φ (0)| ,

(3.124) (3.125)

and the function V¯(t) = V (t, xt , x˙t ) is absolutely continuous on solutions xt of (3.110). Proof. By Theorem 3.9, there exists a unique solution of the initial-value problem (3.110), (3.113), which is globally defined for all t ≥ t0 . Integrating (3.125), where φ = xs , with respect to s from t0 to t we have V (t, xt , x˙t ) − V (t0 , φ , φ˙ ) ≤ −α

 t t0

|x(s)|2 ds.

(3.126)

Relations (3.124) and (3.126), coupled together, result in 2 β |x(t)|2 ≤ V (t, xt , x˙t ) ≤ V (t0 , φ , φ˙ ) ≤ γ φ W

(3.127)

thereby yielding the system stability. Indeed, by applying (3.108) and (3.127), (3.120) is modified to

80

3 Stability Analysis

√ √ | Ax(t)| ≤ T (t − t0 )| Aφ (0)| +  +

 t t0

√ |T (t − s) A[A0 x(s)

+A1 x(s − τ (s))]|ds ≤ φ W {cω

γ [A0  + A1] β

 t −δ (t−s) ce

√ ds} ≤ N0 φ W , t ≥ t0 t −s

t0

(3.128)

where N0 is some positive constant. With this in mind, (3.121) can be strengthened to |Ax(t)| ≤ T (t − t0 )|Aφ (0)| +

t t0

√ √ T (t − s) A| A[A0 x(s)

+A1 x(s − τ (s))]|ds ≤ φ W {c + N0[ω0 + ω1 ]

 t −δ (t−s) ce

√ ds t −s ≤ M0 φ W , t ≥ t0 t0

(3.129)

where M0 is some positive constant, and subordination (3.112) has been taken into account. Hence, the solution norm x(t,t0 , φ )D (A) = |Ax(t)| is small for all t ≥ t0 if φ W is small. Thus, (3.110) is shown to be stable. It remains to be demonstrated that x(t,t0 , φ )D (A) → 0 as t → ∞. For this purpose, let us first demonstrate that |x(t)| → 0 as t → ∞. Similar to (3.126), the following inequality t

t

t0 +η

V (t, x , x˙ ) − V (t0 + η , x

t0 +η

, x˙

) ≤ −α

 t t0 +η

|x(s)|2 ds

(3.130)

is obtained for an arbitrary η > h. Since V (t0 + η , xt0 +η , x˙t0 +η ) ≤ V (t0 , φ , φ˙ ) < ∞ by virtue of (3.126), the implication |x(t)| ∈ L2 [t0 + η , ∞) is then guaranteed by (3.130). Apart from this, Theorem 3.10 ensures that x(t) ˙ is uniformly bounded on the semi-infinite interval [η , ∞). Thus, by applying Barbalat’s Lemma 3.3, the convergence |x(t)| → 0 as t → ∞ is established. It follows that for arbitrary ε1 > 0 there exists t1 > t0 such that |x(t)| < ε1 for all t ≥ t1 . Then taking into account the assumptions of Theorem 3.9, and employing (3.120)–(3.129) yield  √ √ | Ax(t)| ≤ T (t − t1 − h)| Ax(t1 + h)| +

t

t1 +h

√ T (t − s) A[|A0 x(s)|

+|A1 x(s − τ (s))|]ds ≤ Nce−δ (t−t1 −h) + ε1 [A0  + A1]

 ∞ −δ s ce 0

√ ds s (3.131)

for t ≥ t1 + h. By making ε1 small enough, we conclude √ from (3.131) that for arbitrarily small ε2 > 0 there exists t2 > t1 + h such that | Ax(t)| < ε2 for all t ≥ t2 .

3.8 Linear Operator Inequalities in a Hilbert Space

81

Once again, taking into account the assumptions of Theorem 3.9 and employing (3.120)–(3.129), we now derive that

+

 t t2 +h

|Ax(t)| ≤ T (t − t2 − h)|Ax(t2 + h)| √ √ √ T (t − s) A[| AA0 x(s)| + | AA1 x(s − τ (s))|]ds

≤ Mce−δ (t−t2 −h) + ε2 [ω0 A0  + ω1A1 ]

 ∞ −δ s ce 0

√ ds s

(3.132)

for t ≥ t2 + τ . By making ε2 small enough, we conclude from (3.132) that, for arbitrarily small ε > 0, there exists T > t2 + h such that |Ax(t)| < ε for all t ≥ T . With this in mind, the desired convergence x(t,t0 , φ )D (A) = |Ax(t)| → 0 as t → ∞ is established. Theorem 3.11 is thus proved.

3.8.5 Linear Operator Inequalities in a Hilbert Space In the sequel, delay-independent asymptotic stability conditions are derived for the case where the system delay τ (t) is additionally assumed to be differentiable with an a priori known derivative bound d < 1, i.e., τ˙ (t) ≤ d < 1 for all t. In the Hilbert space D(A), let us consider the following Lyapunov–Krasovskii functional t V (t, xt ) = x(t), Px(t) + t− (3.133) τ (t) x(s), Qx(s) ds where P > 0 and Q > 0 are subordinate to the positive definite operator A, i.e., x, Px ≤ γ0 x, Ax , x, Qx ≤ γ0 x, Ax

(3.134)

for all x ∈ D(A) and some constant γ0 > 0. In other words, P and Q are strictly positive definite and bounded on D(A). Since by virtue of (3.105) and (3.106) it follows that

β |x(t)|2 = β x(t), x(t) ≤ x(t), Px(t) +  t

t t−h

x(s), Qx(s) ds

√ x(s), Ax(s) ds ≤ γ0 | Ax(t)|2 t−h √ 2 +γ0 h max | Ax(t + s)|2 ≤ γ xt W

≤ γ0 x(t), Ax(t) + γ0

s∈[−h,0]

(3.135)

for all x ∈ D(A) and some positive β and γ , the Lyapunov–Krasovskii functional (3.133) appears to satisfy 3.124 of Theorem 3.11. Differentiating V in t along the trajectories of the time-delay system (3.110), we have

82

3 Stability Analysis

V˙ (t, xt ) = 2 x(t), PA0 x(t) + 2 x(t), PA1x(t − τ ) − 2 Ax(t), Px(t) + x(t), Qx(t) − (1 − τ˙ (t)) x(t − τ (t)), Qx(t − τ (t)) . (3.136) It follows that (3.125) is satisfied if the LOI   ∗ ∗ PA1 (A0 −A )P+P(A0 −A)+Q <0 Φ= A∗1 P −(1 − d)Q

(3.137)

holds in the Hilbert space D(A) × D(A) in the sense of (3.105). Thus, the following result is concluded. Theorem 3.12. Let the assumptions of Theorem 3.9 be in force. Then, (3.110) is globally asymptotically stable in D(A) for all differentiable delays τ (t) with τ˙ (t) ≤ d < 1 if there exist strictly positive definite operators Q and P subject to (3.134) such that the LOI (3.137) holds in the Hilbert space D(A) × D(A). Proof. First, let us note that by virtue of (3.135) (which results from (3.134), imposed on the Lyapunov–Krasovskii functional (3.133)), (3.124) of Theorem 3.11 is satisfied. Next, let us verify (3.125) of Theorem 3.11. Setting

ζ (t) = col{x(t), x(t − τ )} and employing (3.137) in accordance with (3.105) yields V˙ (t, xt ) ≤ ζ (t), Φζ (t) ≤ −α ζ (t), ζ (t) ≤ −α |x(t)|2

(3.138)

with some positive α where for the sake of reducing notational overload, the same symbol , is also used for the inner product in the Hilbert space H × H. Condition 3.125 is thus satisfied. To this end, we note that while being viewed on the solutions of (3.110), the Lyapunov–Krasovskii functional (3.133) is absolutely continuous as a function of t because the solutions are absolutely continuous in t. Thus, Theorem 3.11 proves to be applicable to (3.110). By applying this theorem, the assertion of Theorem 3.12 is established. General methods for solving LOI have not been developed yet. Some finite dimensional approximations were considered in [103]. In the rest of the section, we show that for delay heat equations the LOI (3.137) becomes feasible if a certain finite-dimensional LMI is feasible.

3.8.6 Illustrative Example: Delay Heat Equation Consider the heat equation ut (ξ ,t) = auξ ξ (ξ ,t) − a1u(ξ ,t − τ (t)), t ≥ 0, 0 ≤ ξ ≤ π

(3.139)

3.8 Linear Operator Inequalities in a Hilbert Space

83

with constant parameters a > 0 and a1 , with a time-varying delay τ (t) such that τ˙ (t) ≤ d < 1, and with the Dirichlet boundary condition u(0,t) = u(π ,t) = 0, t ≥ 0.

(3.140)

The boundary-value problem (3.139), (3.140) describes the propagation of heat in a homogeneous one-dimensional rod with a fixed temperature at the ends in the case of the delayed (possibly, due to actuation) heat exchange with the surroundings. Here a and a1 stand for the heat conduction coefficient and for the coefficient of the heat exchange with the surroundings, respectively, u(ξ ,t) is the value of the temperature field of the plant at time moment t and location ξ along the rod. In what follows, the state dependence on time t and spatial variable ξ is suppressed whenever possible. If, along with the trivial operator A0 = 0 and the bounded operator A1 = −a1 2 of the multiplication by the constant −a1 , we introduce the operator A = −a ∂∂ξ 2 of double differentiation with the dense domain D(

∂2 ) = {u ∈ W 2,2 (0, π ) : u(0) = u(π ) = 0}, ∂ξ2

(3.141)

then the boundary-value problem (3.139), (3.140) can be rewritten as the differential equation (3.110) in the Hilbert space H = L2 (0, π ) with the infinitesimal operator −A, generating an analytical semigroup with a negative growth bound (see, e.g., [55] for details). Let the Lyapunov–Krasovskii functional (3.133) be simplified to V (t, ut ) = p

 π 0

u2 (ξ ,t)d ξ + q

t

 π

t−τ (t) 0

u2 (ξ , s)d ξ ds

(3.142)

with some positive constants p and q. Then the operators P and Q in (3.137) take the form P = p, Q = q of the operators of the multiplication by positive constants p and q, respectively, and (3.134), imposed on these operators, is guaranteed by (3.107), being specified in the form of the Wirtinger’s inequality (3.109) according to Lemma 3.6. Integrating by parts and taking into account (3.140), we find that x, −(A∗ P + PA)x = 2a = −2a

 π 0

 π 0

pu2ξ d ξ ≤ −2a

puuξ ξ d ξ

 π 0

pu2 d ξ

(3.143)

for x ∈ D(A) where the last inequality has been derived by using the Wirtinger’s inequality (3.109). We thus obtain

84

3 Stability Analysis

 ×

x(t), Φ x(t) ≤ q − 2ap −a1 p −a1 p −(1 − d)q

 π

 0

[u(ξ ,t) u(ξ ,t − τ )]  u(ξ ,t) dξ < 0 u(ξ ,t − τ )

provided that the following LMI   q − 2ap −a1 p <0 −a1 p −(1 − d)q

(3.144)

(3.145)

is satisfied. Theorem 3.13. Let the LMI (3.145) holds for some positive scalars p and q. Then the Dirichlet boundary-value problem (3.139), (3.140) is globally asymptotically stable in the Sobolev space W 1,2 (0, π ) for all differentiable delays τ (t) with τ˙ (t) ≤ d < 1. Proof. As proposed before, let us represent the boundary-value problem (3.139), (3.140) in the form of the differential equation (3.110) in the Hilbert space H = L2 (0, 1). Since the operator −A generates an analytical semigroup with a negative growth bound, A0 = 0, and A1 is bounded and commutes to A, the assumptions of Theorem 3.9 are straightforwardly verified for the Hilbert space-valued equation, thus specified. Moreover, while specifying the Lyapunov–Krasovskii functional (3.133) as (3.142), (3.134) clearly results from the Wirtinger’s inequality (3.109). Now employing (3.143), the subordination   q − 2ap −a1 p Φ≤ −a1 p −(1 − d)q is established for Φ , defined according to (3.137). Thus, the LOI (3.137) is satisfied and Theorem 3.12 proves to be applicable to the present case. By applying Theorem 3.9, the assertion of Theorem 3.13 is concluded. By the Schur complements formula, LMI (3.145) is feasible iff the following inequality q2 − 2(a + a0 )pq + a21 p2 /(1 − d) < 0 holds for some p > 0 and q > 0. The left part of the latter inequality achieves its minimum at q = (a + a0 )p and, thus, the LMI holds iff a21 < (a + a0)2 (1 − d). It should be noted that the same LMI (3.145) appears to guarantee the scalar time-delay equation y(t) ˙ + (a + a0)y(t) + a1y(t − τ (t)) = 0,

(3.146)

to be exponentially stable for all delays with τ˙ ≤ d < 1. While being interpreted in terms of the modal representation of the Dirichlet boundary-value problem (3.139), (3.140), confined to the projection y˙ j (t) + (a j2 + a0)y j (t) + a1y j (t − τ (t)) = 0, j = 1, 2, . . . , n

(3.147)

3.8 Linear Operator Inequalities in a Hilbert Space

85

on the first n eigenfunctions, Theorem 3.13 essentially utilizes the structure of such a representation to reduce the corresponding LMI, derived for the diagonal n-thorder system (3.147), to the LMI (3.145), derived for the first-order system (3.146), that corresponds to the first modal dynamics (3.147) with j = 1. Actually, this interpretation becomes surprising since no similar results on LMI of a reduced order are known from the finite-dimensional theory. To this end, consider the heat equation (3.139) with a constant delay τ (that corresponds to d = 0 in the conditions above) and with the boundary condition (3.140). The characteristic equations of such a boundary-value problem are given by (see, e.g., [243]) λk + ak2 + a0 + a1e−λk τ = 0, k = 1, 2, . . . . (3.148) The exponential stability of (3.139), (3.140) is shown in [102] to be determined by (3.148) with k = 1, i.e., the system is exponentially stable if the roots of (3.148) with k = 1 have negative real parts. Since the aforementioned root condition is equivalent to the exponential stability of the first-order system (3.146), the constant-delay system proves to be delay-independently exponentially stable iff a1 ∈ (−a − a0 , a + a0 ] (see [155]).

Chapter 4

Finite-time Stability of Uncertain Homogeneous and Quasihomogeneous Systems

Until recently, the finite-time stability of asymptotically stable homogeneous systems has been well-recognized for only continuous vector fields [26, 100, 153]. Extending this result to switched systems has required proceeding differently [128, 169] because a smooth homogeneous Lyapunov function, whose existence was proven in [195] for continuous asymptotically stable homogeneous vector fields, can no longer be brought into play. The aforementioned work [169] forms a basis of the present chapter, where the finite-time stability property is established for homogeneous asymptotically stable discontinuous systems whose homogeneity degree is negative. Exemplified with a second-order system, the finite-time stability is then additionally demonstrated to remain in force regardless of inhomogeneous perturbations.

4.1 Uncertain Systems and Equiuniform Stability The following nonautonomous discontinuous system x˙ = ϕ (x,t)

(4.1)

is under study. Hereinafter, x = (x1 , . . . , xn )T is the state vector, t ∈ R is the time variable, the function ϕ (x,t) = (ϕ1 (x,t), . . . , ϕn (x,t))T is piece-wise continuous. Along with the differential equation (4.1), we deal with its perturbed version x˙ = ϕ (x,t) + ψ (x,t)

(4.2)

where ψ (x,t) is a piece-wise continuous function whose components ψi , . . . , ψn are locally uniformly bounded within a ball Bδ , centered at the origin with radius δ . In other words, we assume that an admissible perturbation ψ (x,t) is such that |ψi (x,t)| ≤ Mi , i = 1, . . . , n

(4.3)

87

88

4 Finite-time Stability of Uncertain Homogeneous and Quasihomogeneous Systems

for almost all (x,t) ∈ Bδ × R and some constants Mi ≥ 0, fixed a priori. The above equation is further viewed as a differential equation with rectangular uncertainties. A solution concept for such an uncertain differential equation is introduced as follows. Definition 4.1. An absolutely continuous function x(t), defined on an interval I, is said to be a solution of the uncertain differential equation (4.2) with the rectangular uncertainty constraints (4.3) iff it is a Filippov solution of (4.2) on the interval I for some piece-wise continuous function ψ (x,t), satisfying (4.3). As a matter of fact, the differential equation (4.1) with no uncertain term ψ is particularly represented in the form of (4.2), (4.3) with Mi = 0 for all i = 1, . . . , n. It should be pointed out that an uncertain system (4.2) with uncertainty constraints (4.3) can be represented as a differential inclusion of the form x˙ ∈ Φ (x,t) + Ψ

(4.4)

where Φ (x,t) is the Filippov set of the function ϕ (x,t), Ψ is the Cartesian product of the intervals Ψi = [−Mi , Mi ], i = 1, . . . , n, and the set

Φ (x) + Ψ = {φ + ψ : φ ∈ Φ (x), ψ ∈ Ψ }. If ϕ (x,t) = ϕ (x) is time-independent the uncertain system (4.2), (4.3) is governed by the autonomous differential inclusion (4.4), in spite of the presence of the uncertain time-varying term ψ (x,t). Stability of the nominal discontinuous system (4.1) has been introduced in Sect. 3. In application to the uncertain system (4.2), (4.3) the stability definitions are specified as follows. In order to emphasize that these definitions require such a system to be uniformly stable not only in the initial data but also in the uncertainty, the corresponding system will be referred to as equiuniformly stable. Suppose that x = 0 is an equilibrium point of the uncertain system (4.2), (4.3), i.e., x(t) = 0 is a solution of (4.2) for some function ψ0 (x,t), admissible in the sense of (4.3), and let xψ (t,t0 , x0 ) denote a solution x(t) of (4.2) for some admissible function ψ (x,t) under the initial conditions x(t0 ) = x0 . Definition 4.2. The equilibrium point x = 0 of the uncertain system (4.2), (4.3) is equiuniformly stable iff for each t0 ∈ R, ε > 0, there is δ = δ (ε ) > 0, dependent on ε and independent of t0 and ψ , such that each solution xψ (t,t0 , x0 ) of (4.2), (4.3) with the initial data x0 ∈ Bδ exists for all t ≥ t0 and satisfies the inequality xψ (t,t0 , x0 ) < ε , t0 ≤ t < ∞.

Definition 4.3. The equilibrium point x = 0 of the uncertain system (4.2), (4.3) is said to be equiuniformly asymptotically stable if it is equiuniformly stable and the convergence limt→∞ xψ (t,t0 , x0 ) = 0 (4.5)

4.2 Homogeneity Degree and Homogeneity Dilation

89

holds for all solutions of (4.2), (4.3) initialized within some Bδ , uniformly in the initial data t0 and x0 , and all the solutions xψ (·,t0 , x0 ). If this convergence remains in force for each δ > 0, the equilibrium point is said to be globally equiuniformly asymptotically stable. Definition 4.4. The equilibrium point x = 0 of the uncertain system (4.2), (4.3) is said to be globally equiuniformly finite-time stable if, in addition to the global equiuniform asymptotical stability, the limiting relation xψ (t,t0 , x0 ) = 0

(4.6)

holds for each solution xψ (·,t0 , x0 ) and all t ≥ t0 + T (t0 , x0 ) where the settling time function T (t0 , x0 ) =

sup xψ (·,t0 ,x0 )

inf{T ≥ 0 : xψ (t,t0 , x0 ) = 0 f or all t ≥ t0 + T }

(4.7)

is such that: T (Bδ ) = supt0 ∈R, x0 ∈Bδ T (t0 , x0 ) < ∞ f or each δ > 0. In the present chapter, we focus our analysis on the global equiuniform finitetime stability of discontinuous systems. The concept of homogeneity, earlier studied in [94, 254] and [195] for continuously differentiable and, respectively, continuous vector fields, plays a central role in our analysis.

4.2 Homogeneity Degree and Homogeneity Dilation The homogeneity concept for differential inclusions and, particularly, for nonautonomous discontinuous systems, is as follows. Definition 4.5. The differential inclusion x˙ ∈ Φ (x,t)

(4.8)

(the differential equation (4.1) or the uncertain system (4.2), (4.3)) is called locally homogeneous of degree q ∈ R with respect to dilation (r1 , . . . , rn ) where ri > 0, i = 1, . . . , n if there exist a constant c0 > 0, called a lower estimate of the homogeneity parameter, and a ball Bδ ⊂ Rn , called a homogeneity ball, such that any solution x(t) of (4.8) (respectively, that of (4.1) or (4.2), (4.3)), evolving within the ball Bδ , generates a parameterized set of solutions xc (t) with components xci (t) = cri xi (cqt) and parameter c ≥ c0 .

(4.9)

90

4 Finite-time Stability of Uncertain Homogeneous and Quasihomogeneous Systems

Definition 4.6. A piece-wise continuous function ϕ (x,t) is called locally homogeneous of degree q ∈ R with respect to dilation (r1 , . . . , rn ) where ri > 0, i = 1, . . . , n if there exist a constant c0 > 0 and a ball Bδ ⊂ Rn such that

ϕi (cr1 x1 , . . . , crn xn , c−qt) = cq+ri ϕi (x1 , . . . , xn ,t)

(4.10)

for all c ≥ c0 and almost all (x,t) ∈ Bδ × R. The global homogeneity concept for the differential inclusion (4.8) and that for the piece-wise continuous function ϕ (x,t) are formally introduced by setting δ = ∞ in the above definitions. It is worth noting that Definitions 4.5 and 4.6 are consistent in the sense that homogeneity of the function ϕ (x,t) ensures the homogeneity of the corresponding differential equation (4.1). Lemma 4.1. Let a piece-wise continuous function ϕ (x,t) be locally homogeneous of degree q ∈ R with respect to dilation (r1 , . . . , rn ). Then, the corresponding differential equation (4.1) is locally homogeneous of the same degree q ∈ R with respect to the same dilation (r1 , . . . , rn ). Proof. Let x(t) be a solution of (4.1), evolving within Bδ . Then it is straightforward to verify that due to (4.10), the function xc (t) with components (4.9) is also a solution of (4.1) for all c ≥ c0 . Thus, the differential equation (4.1) whose right-hand side is locally homogeneous in the sense of Definition 4.6, is also locally homogeneous in the sense of Definition 4.5. Lemma 4.1 is proved. To this end, we present conditions for the uncertain system (4.2), (4.3) to be locally homogeneous. Lemma 4.2. Let the following conditions be satisfied: 1. A piece-wise continuous function ϕ (x,t) is locally homogeneous of degree q ∈ R with respect to dilation (r1 , . . . , rn ); 2. Components ψi , i = 1, . . . , n of a piece-wise continuous function ψ are locally uniformly bounded by constants Mi ≥ 0; 3. Mi = 0 whenever q + ri > 0. Then, the uncertain differential equation (4.2) with the uncertainty constraints (4.3) is locally homogeneous of degree q ∈ R with respect to dilation (r1 , . . . , rn ). Proof. Let x(t) = (x1 (t), . . . , xn (t))T be a solution of (4.2) under some piece-wise continuous function ψ (x,t), satisfying (4.3), and let x(t) evolve within a ball Bδ where the homogeneity condition (4.10) holds almost everywhere for all c ≥ c0 . Then, it is straightforward to verify that for arbitrary c ≥ max(1, c0 ) the function xc (t) with components xci (t) = cri xi (cqt), i = 1, . . . , n is a solution of (4.2) with the piece-wise continuous function ψ (x,t) = ψ c (x,t) whose components are as follows

ψic (x,t) = cq+ri ψi (c−r1 x1 , . . . , c−rn xn , cqt).

4.3 Finite-time Stability of Locally Homogeneous Systems

91

Since by Condition 3 of the lemma one has cq+ri ≤ 1 whenever Mi > 0, the function ψ c (x,t) is also admissible in the sense of (4.3). Thus, any solution of the uncertain differential equation (4.2), evolving within a homogeneity ball Bδ , generates a parameterized set of solutions xc (t) with the parameter c large enough. Hence, (4.2), (4.3) is locally homogeneous of degree q ∈ R with respect to dilation (r1 , . . . , rn ). Lemma 4.2 is proved.

4.3 Finite-time Stability of Locally Homogeneous Systems When continuous, a globally homogeneous time-invariant vector field ϕ (x) of degree q < 0 with respect to dilation (r1 , . . . , rn ) is known to be globally finite-time stable whenever it is globally asymptotically stable. The proof of this fact, given in [100], is based on the result from [195] that an autonomous continuous homogeneous system, if asymptotically stable, possesses a homogeneous Lyapunov function. However, the existence of a homogeneous Lyapunov function is no longer guaranteed for systems governed by differential inclusions (even the converse of Lyapunov’s second theorem has not been successfully extended to this case). Therefore, the global finite-time stability of these systems is established by going through a different route, which is closely related to re-scaling of the time and state variables. Going through this route allows one to additionally obtain an upper estimate T (t0 , x0 ) ≤ τ (x0 , ER ) +

1 (δ R−1 )q s(δ ) 1 − 2q

(4.11)

of the settling-time function T (t0 , x0 ) = sup inf{T ≥ 0 : x(t,t0 , x0 ) = 0 f or all t ≥ t0 + T }

(4.12)

x(·,t0 ,x0 )

via the reaching-time functions

τ (x0 , ER ) = sup inf{T ≥ 0 : x(t,t0 , x0 ) ∈ ER f or all t0 ∈ R, t ≥ t0 + T } x(·,t0 ,x0 )

(4.13) and s(δ ) = sup τ (x0 , E 1 δ ) x0 ∈Eδ

where ER denotes an ellipsoid of the form  ER = {x ∈ Rn :

n Σi=1

2

 x 2 i ≤ 1}, R ri

(4.14)

(4.15)

92

4 Finite-time Stability of Uncertain Homogeneous and Quasihomogeneous Systems

ER is located within a homogeneity ball, δ ≥ c0 R, and c0 > 0 is a lower estimate of the homogeneity parameter. Theorem 4.1 (Homogeneity principle). Let the differential inclusion (4.8) be locally homogeneous of degree q < 0 with respect to dilation (r1 , . . . , rn ) and let the equilibrium point x = 0 of (4.8) be globally uniformly asymptotically stable. Then, the differential inclusion (4.8) is globally uniformly finite-time stable and an upper estimate of the settling-time function (4.12) is given by (4.11). Proof. Due to the global uniform asymptotic stability of (4.8), all the trajectories of the differential inclusion, initialized within a compact set, are uniformly steered towards an arbitrarily small ellipsoid (4.15). Then, the condition x(t) ∈ ER f or t ≥ t0 + τ (x0 , ER )

(4.16)

holds for an arbitrary solution x(t) of (4.8) initialized with x(t0 ) = x0 where the ellipsoid ER has been assumed to be small enough to be located within a homogeneity ball. Apart from this, given an a priori fixed δ ≥ c0 R where c0 > 0 is a lower estimate of the homogeneity parameter, there exists s(δ ) > 0 such that for each initial time moment t˜0 and all the solutions x(t) ˜ with x( ˜ t˜0 ) ∈ Eδ one has x(t) ˜ ∈ E 1 δ for t ≥ 2 c t˜0 + s(δ ). Since the function x (t), whose components (4.9) are specified with c = δ R−1 ≥ c0 , is a solution of (4.8) by homogeneity, and in addition, xc (t˜0 ) ∈ Eδ at t˜0 = c−q (t0 + τ (x0 , ER )), it follows that xc (t) ∈ E 1 δ f or t ≥ c−q (t0 + τ (x0 , ER )) + s(δ ). 2

(4.17)

The latter relation, rewritten in terms of x(t) by means of (4.9) subject to c = δ R−1 , is represented as follows: x(t) ∈ E 1 R f or t ≥ t1 = t0 + τ (x0 , ER ) + (δ R−1 )q s(δ ). 2

(4.18)

Now, by applying the same derivation to a solution of (4.8) with x(t1 ) ∈ E 1 R , one 2 obtains x(t) ∈ E 1 R f or t ≥ t2 = t1 + 2q(δ R−1 )q s(δ ). 4

(4.19)

In general, the following relations are derived x(t) ∈ E2−(i+1) R f or t ≥ ti+1 = ti + 2qi(δ R−1 )q s(δ ), i = 1, 2, . . .

(4.20)

by iterating on i. Since λ = 2q < 1 by virtue of q < 0, the convergence of the time instants tk , k = 1, 2, . . . to a finite limit takes place:

4.4 Quasihomogeneity Principle

93

k−1 i lim tk = t0 + τ (x0 , ER ) + lim Σi=0 λ (δ R−1 )q s(δ ) =

k→∞

k→∞

1−λk (δ R−1 )q s(δ ) = k→∞ 1 − λ 1 (δ R−1 )q s(δ ) < ∞. t0 + τ (x0 , ER ) + 1−λ

t0 + τ (x0 , ER ) + lim

(4.21)

Hence, relations (4.20) result in 0 x(t) ∈ ∩∞ i=1 E2−i R = {0} f or t ≥ t0 + τ (x , ER ) +

1 (δ R−1 )q s(δ ), (4.22) 1 − 2q

thereby establishing both the required finite-time convergence property for the locally homogeneous differential inclusion (4.8) and the upper estimate (4.11) of the settling-time function (4.12). Theorem 4.1 is thus proved. Remark 4.1. For a globally homogeneous differential inclusion (4.8) one can choose δ0 = c0 R0 and R0 sufficiently large to guarantee that x(t,t0 , x0 ) ∈ ER0 for all t ≥ t0 . Then τ (x0 , ER0 ) = 0 and the upper estimate (4.11) of the settling-time function (4.12) is simplified to T (t0 , x0 ) ≤

q

c0 s(δ0 ) 1 − 2q

(4.23)

with δ0 = δ0 (x0 ) dependent on x0 .

4.4 Quasihomogeneity Principle An important corollary of the homogeneity principle is obtained if the differential inclusion (4.8) is generated by an uncertain differential equation (4.2) with piecewise continuous functions ϕ (x,t) and ψ (x,t), locally homogeneous and locally uniformly bounded, respectively. By Lemma 4.2, the uncertain discontinuous system (4.2), (4.3), while being inhomogeneous for particular perturbations ψ (x,t), can be governed by a homogeneous differential inclusion. Playing with this quasihomogeneity property, the following specification of the homogeneity principle is obtained. Theorem 4.2 (Quasihomogeneity principle). Let the following conditions be satisfied: 1. The right-hand side of an uncertain differential equation (4.2) consists of a locally homogeneous piece-wise continuous function ϕ of degree q < 0 with respect to dilation (r1 , . . . , rn ) and a piece-wise continuous function ψ whose components ψi , i = 1, . . . , n are locally uniformly bounded by constants Mi ≥ 0 within a homogeneity ball; 2. Mi = 0 whenever q + ri > 0;

94

4 Finite-time Stability of Uncertain Homogeneous and Quasihomogeneous Systems

3. The uncertain system (4.2), (4.3) is globally equiuniformly asymptotically stable around the origin. Then the uncertain system (4.2), (4.3) is globally equiuniformly finite-time stable and its settling-time function (4.7) is estimated as (4.11). Proof. By Lemma 4.2, Conditions 1 and 2 of the theorem guarantee that the uncertain system (4.2), (4.3) is locally homogeneous of degree q < 0 with respect to dilation (r1 , . . . , rn ). Thus, coupled to Condition 3 of the theorem, these conditions ensure that Theorem 4.1 is applicable to the uncertain system (4.2), (4.3). By applying Theorem 4.1 to this system, the proof of Theorem 4.2 is completed. Apparently, Remark 4.1 remains valid for an uncertain system (4.2), (4.3), the right-hand side of which consists of a globally homogeneous piece-wise continuous function ϕ of degree q < 0 and a globally uniformly bounded piece-wise continuous function ψ .

4.5 Finite-time Stability of a First-order Quasihomogeneous System The first-order discontinuous system x˙ = −α sign x + w(x,t)

(4.24)

where α is a parameter and w(x,t) is a piece-wise continuous nonlinear perturbation, represents a simple example of a quasihomogeneous system. Indeed, while admitting particular inhomogeneous perturbations w(x,t), the uncertain discontinuous system (4.24) possesses solutions, which are governed by the corresponding homogeneous differential inclusion of degree q = −1 with respect to dilation r = 1. System 4.24 is long recognized to be globally equiuniformly finite time stable, provided that any admissible perturbation w(·) is uniformly bounded in amplitude with an upper bound, smaller than the parameter α , i.e., |w(x,t)| ≤ N

(4.25)

for all continuity points (x,t) of w(·) and for some N ∈ (0, α ). In order to make this sure it suffices to introduce the quadratic function V (x) = x2 and compute its time derivative along the solutions of (4.24): V˙ (x(t)) = −2|x(t)|[α − w(x(t),t)sign x(t)] ≤ −2(α − N)|x(t)|  = −2(α − N) V (x(t)).

(4.26)

The desired stability property is then established based on the following result wellknown from the literature (see, e.g., [26] and references quoted therein).

4.6 Finite-time Stability of a Second-order Quasihomogeneous System

95

Lemma 4.3. Let an everywhere non-negative function V (t) meet the differential inequality V˙ (t) ≤ −2γ V β (t)

(4.27)

for all t ≥ 0 and for some constants γ > 0 and β ∈ (0, 1). Then V (t) = 0 for all t ≥ [2γ (1 − β )]−1V 1−β (0). Proof. By the comparison principle [110, p.102], an arbitrary non-negative solution V (t) of (4.27) is dominated V (t) ≤ V0 (t) by the solution ⎧ ⎨ [V (1−β ) (0) − 2γ (1 − β )t] 1−1 β if t ∈ [0, V (1−β ) (0) ] 2γ (1−β ) (4.28) V0 (t) = (1−β ) (0) ⎩0 if t ≥ V 2γ (1−β )

of the differential equation

β V˙0 (t) = −2γ V0 (t),

specified with the same initial condition V0 (0) = V (0). Since V0 (t) = 0 for all t ≥ [2γ (1 − β )]−1V 1−β (0), the function V (t) vanishes after a finite time moment T ≤ [2γ (1 − β )]−1V 1−β (0), i.e., V (t) = 0 for all t ≥ [2γ (1 − β )]−1V 1−β (0). Lemma 4.3 is thus proved. By applying Lemma 4.3 to (4.26) with V (x) = x2 , computed along the solutions of (4.24), its global equiuniform finite time stability is concluded. Theorem 4.3. The discontinuous system (4.24), operating under uniformly bounded uncertainties (4.25), is globally equiuniformly finite-time stable around the origin provided that the subordination N < α holds. Clearly, the validity of Theorem 4.3 can also be verified by involving the quasihomogeneity principle, Theorem 4.2, whose applicability to the system in question follows from the aforementioned quasihomogeneity property the system exhibits and from its global equiuniform asymptotic stability, which is guaranteed by (4.26). A less trivial application of the quasihomogeneity principle comes with an uncertain system of the second order.

4.6 Finite-time Stability of a Second-order Quasihomogeneous System Capabilities of the quasihomogeneity principle are now illustrated by an application to a second-order discontinuous system of the form x˙ = y, y˙ = −asign x − bsign y.

(4.29)

To begin with, we recall that (4.29) is globally uniformly finite-time stable if the inequalities

96

4 Finite-time Stability of Uncertain Homogeneous and Quasihomogeneous Systems

a>b>0

(4.30)

hold for the parameters of the system (see Example 3.2). Along with (4.29), let us now study its inhomogeneous time-varying perturbation of the form x˙ = y, y˙ = −asign x − bsign y − hx − py + ω (x, y,t)

(4.31)

where h and p are parameters of the linear gain, and ω (x, y,t) is a piece-wise continuous nonlinear perturbation, uniformly bounded |ω (x, y,t)| ≤ M

(4.32)

for all continuity points (x, y,t) and some M > 0. If the bound M is small enough, namely, 0 < M < b < a − M,

(4.33)

and the linear gain is non-positive, i.e., h ≥ 0, p ≥ 0,

(4.34)

the uncertain system (4.31), (4.32) proves to be globally equiuniformly asymptotically stable. The global equiuniform finite-time stability of this system can then be demonstrated by invoking the quasihomogeneity principle. This principle turns out to be applicable to the uncertain system (4.31), (4.32) because its nominal model (4.29) has a globally homogeneous right-hand side of degree q = −1 with respect to dilation r = (2, 1), thus satisfying Condition 2 of Theorem 4.2. Summarizing, the following result is in force. Theorem 4.4. Let Conditions (4.33), (4.34) be satisfied. Then the uncertain discontinuous system (4.31), (4.32) is globally equiuniformly finite-time stable around the origin. The proof of Theorem 4.4 is presented in the next section. The qualitative behavior of (4.31) is depicted in Fig. 4.1. Due to the parameter subordination (4.33), the velocity vectors of (4.31) point toward the same region in the switching lines S1 = {(x, y) ∈ R2 : x > 0, y = 0}, S2 = {(x, y) ∈ R2 : x = 0, y < 0}, S3 = {(x, y) ∈ R2 : x < 0, y = 0}, S4 = {(x, y) ∈ R2 : x = 0, y > 0},

(4.35)

regardless of uncertainty (4.32) affecting the system. Hence, the uncertain system (4.31)–(4.34) and, particularly, its unperturbed version (4.29) rotate around the origin x = y = 0, while approaching the origin in a finite time. Thus, both the nominal

4.6 Finite-time Stability of a Second-order Quasihomogeneous System

97

system (4.29) and its uncertain counterpart (4.31)–(4.34) exhibit Zeno behavior with an infinite number of switches in a finite amount of time. These systems do not generate sliding motions everywhere except the origin. If a trajectory starts there at any given finite time, there appears the so-called sliding mode of the second order (see [16, 76, 77, 126, 127] for advanced results on second-order sliding modes).

Fig. 4.1 Phase trajectory of the second-order discontinuous system (4.31)

Proof of Theorem 4.4 We break up the proof into several simple steps. 1. Equiuniform stability. To demonstrate that the uncertain system (4.31)–(4.34) is equiuniformly stable, we introduce the Lyapunov function 1 V˜ (x, y) = a|x| + (y2 + hx2). 2

(4.36)

Since the trajectories (xω (t), yω (t)) of the uncertain system cross the vertical axis x = 0 everywhere but the equilibrium x = y = 0, the time derivative of V˜ (xω (t), yω (t))

98

4 Finite-time Stability of Uncertain Homogeneous and Quasihomogeneous Systems

is negative semidefinite for almost all t: V˙˜ (xω (t), yω (t)) = −b|yω (t)| − py2ω (t) + yω (t)ω (xω (t), yω (t),t) ≤ −(b − M)|yω (t)|.

(4.37)

It follows that the uncertain system (4.31)–(4.34) is equiuniformly stable. Indeed, the positive definite function (4.36) determines a norm in the Euclidean space R2 and due to (4.37), this norm does not increase along the solutions of (4.31)–(4.34). Thus, initialized in an arbitrarily small vicinity Dε = {(x, y) ∈ R2 : V˜ (x, y) ≤ ε }

(4.38)

of the origin, the uncertain system (4.31)–(4.34) can not leave this vicinity, regardless of whichever admissible uncertainty ω affects the system. 2. Global asymptotic stability. Now, let us prove that (4.31)–(4.34) is globally asymptotically stable, i.e., the limiting relations lim yω (t) = 0

(4.39)

lim xω (t) = 0

(4.40)

t→∞ t→∞

hold for each solution (xω (t), yω (t)) individually. The idea behind this proof is inspired from the Extended Invariance Principle, Theorem 3.3, and it is based on the use of an auxiliary indefinite function, coupled to a Lyapunov function with negative-semidefinite time derivatives along the system trajectories. To begin with, let us note that by virtue of (4.37) all possible solutions of (4.31)– (4.34), initialized at t0 ∈ R within a compact set DR = {(x, y) ∈ R2 : V˜ (x, y) ≤ R}

(4.41)

sup V˜ (xω (t), yω (t)) ≤ R.

(4.42)

are a priori estimated by t∈[t0 ,∞)

In order to justify (4.39) it remains to integrate (4.37) on solutions of (4.31)–(4.34), initialized at t0 ∈ R within the compact set (4.41) where R is arbitrarily large. Taking into account (4.32) and (4.42), one obtains  t t0

˜ ω (t), yω (t))] ≤ (b − M)−1R |yω (t)|dt ≤ (b − M)−1 [R − V(x

(4.43)

for all t ≥ t0 and for any solution (xω (t), yω (t)) of (4.31)–(4.34), initialized within D0 . Moreover, the integrand yω (t) is uniformly continuous on [t0 , ∞) for these solutions because both yω (t) and y˙ω (t) are uniformly bounded in t ≥ t0 . Thus, the validity of (4.39) is straightforwardly obtained by applying Barbalat’s lemma from [110] to the integral inequality (4.43).

4.6 Finite-time Stability of a Second-order Quasihomogeneous System

99

In turn, relation (4.40) is established by differentiating an auxiliary (indefinite!) function U(x, y) = xy (4.44) along the trajectories of the uncertain system (4.31)–(4.34): ˙ ω (t), yω (t)) = y2ω (t) − a|xω (t)| − bxω (t)sign yω (t) − hx2ω (t) U(x − pxω (t)yω (t) + xω (t)ω (xω (t), yω (t),t) ≤ y2ω (t) − |xω (t)|[a + bsign xω (t)sign yω (t) − ω (xω (t), yω (t),t)sign xω (t)] − pxω (t)yω (t) ≤ y2ω (t) − (a − b − M)|xω (t)| − pxω (t)yω (t).

(4.45)

Then integrating (4.45) on solutions of (4.31)–(4.34), initialized at t0 ∈ R within the compact set (4.41) with R arbitrarily large, and employing (4.42)–(4.43), one obtains (a − b − M) +

 t t0

t t0

|xω (τ )|d τ ≤ [U(xω (t0 ), yω (t0 )) − U(xω (t), yω (t))]

yω (τ )[yω (τ ) − pxω (τ )]d τ ≤ [xω (t0 )yω (t0 ) − xω (t)yω (t)] +{ max |yω (τ )| + p max |xω (τ )|} τ ∈[t0 ,t]

τ ∈[t0 ,t]

 t t0

|yω (τ )|d τ

1 2 [x (t0 ) + y2ω (t0 ) + x2ω (t) + y2ω (t)] 2 ω  t √ 1 1 +( 2R + R) |yω (τ )|d τ ≤ 2R + 2 R2 a a t0 √ 1 (4.46) +(b − M)−1R( 2R + R) a ≤

where, due to (4.42), the integrand xω (t) is uniformly continuous on [t0 , ∞) (indeed, both xω (t) and its time derivative x˙ω (t) = yω (t) are uniformly bounded in t ≥ t0 for the solutions of (4.31), initialized within (4.41)). Thus, Barbalat’s Lemma 3.3 is applicable to (4.46). By applying this lemma, the limiting relation (4.40) is obtained, and the proof of the global asymptotic stability of the uncertain system (4.31)–(4.34) is completed. 3. Semiglobal Lyapunov functions. Our next goal is to construct a parameterized family of local Lyapunov functions VR (x, y), R > 0 such that each VR (x, y) is wellposed on the corresponding compact set DR , governed by (4.41), i.e., VR (x, y) is positive definite on DR and its time derivative, computed along the trajectories of the uncertain system (4.31)–(4.34), initialized within DR , is equiuniformly negative definite in the sense that (4.47) V˙R (x, y) ≤ −WR (x, y) for all (x, y) ∈ DR and some WR (x, y), positive definite on DR .

100

4 Finite-time Stability of Uncertain Homogeneous and Quasihomogeneous Systems

Parameterized Lyapunov functions VR (x, y), R > 0 with the properties above, can be constructed by combining the aforementioned Lyapunov function (4.36), whose time derivative along the system motion is only semidefinite, with the indefinite function (4.44): 1 VR (x, y) = V˜ (x, y) + κRU(x, y) = a|x| + (y2 + hx2 ) + κRxy 2

(4.48)

where the weight parameter κR > 0 has been chosen small enough, namely,

κR < min{1,

2a2 a(b − M) , √ }. R a 2R + pR

(4.49)

Apparently, the function VR (x, y), thus constructed,√is positive definite on compacta (4.41) because (4.41) implies that |x| ≤ Ra , |y| ≤ 2R, and therefore 1 1 1 1 a|x| + (y2 + hx2 ) + κRxy ≥ a|x| + (y2 + hx2 ) − κR x2 − κR y2 2 2 2 2 R 1 1 2 2 ≥ (a − κR )|x| + (1 − κR)y + hx > 0 2a 2 2

(4.50)

for all (x, y) ∈ DR \ {(0, 0)} and κR > 0, satisfying (4.49). Moreover, the time derivative of (4.48), computed along the trajectories of the uncertain system (4.31)–(4.34), initialized within DR , is equiuniformly negative definite. Indeed, by employing (4.37) and (4.45) one has V˙R (xω (t), yω (t)) ≤ −(b − M)|yω (t)| +

κR y2ω (t) − κR(a − b − M)|xω (t)| − κR pxω (t)yω (t).

(4.51)

Since it has been shown at the first step that the trajectories of (4.31)–(4.34), starting in the region DR , cannot leave this region, by utilizing (4.42) it follows that √ pR )]|yω (t)| V˙R (xω (t), yω (t)) ≤ −[b − M − κR( 2R + a (4.52) −κR(a − b − M)|xω (t)| ≤ −cR [|yω (t)| + |xω (t)|] √ where cR = min{b − M − κR ( 2R + pR a ), κR (a − b − M)} > 0 by virtue of (6.53). To this end, (4.52) results in V˙R (xω (t), yω (t)) ≤ −KRVR (xω (t), yω (t)) where KR = and the upper estimate

2acR √ >0 max{2a2 + hR, a 2R + 2κRR}

(4.53)

4.6 Finite-time Stability of a Second-order Quasihomogeneous System

101

√ a 2R + 2κRR 2a2 + hR |x| + |y| VR (x, y) ≤ 2a 2a of the Lyapunov function (4.48) on compacta (4.41) has been used. Thus, the desired equiuniform negative definiteness (4.47) is obtained with WR (x, y) = KRVR (x, y). 4. Global equiuniform asymptotic stability. Since the differential inequality (4.53) holds on the solutions of the uncertain system (4.31)–(4.34), initialized within the compact set (4.41), the function VR (xω (t), yω (t)) exponentially decays VR (xω (t), yω (t)) ≤ VR (xω (t0 ), yω (t0 ))e−KR (t−t0 )

(4.54)

on these solutions with the decay rate KR , independent of the uncertainty ω . While viewed on compacta (4.41), the functions VR (x, y) and V˜ (x, y) are equivalent in the sense that LRV˜ (x, y) ≤ VR (x, y) ≤ MRV˜ (x, y)

(4.55)

for all (x, y) ∈ DR and positive constants LR , MR , satisfying LR < min{

2a2 − RκR 2a2 + RκR , 1 − κR}, MR > max{ , 1 + κR}. 2 2a 2a2

(4.56)

The above relations (4.54) and (4.55), coupled together, ensure that the function V˜ (x, y) exponentially decays −KR (t−t0 ) ˜ V˜ (xω (t), yω (t)) ≤ L−1 R MRV (xω (t0 ), yω (t0 ))e −KR (t−t0 ) ≤ L−1 R MR Re

(4.57)

on the solutions of (4.31)–(4.34), equiuniformly in the uncertainty ω and the initial data, located within an arbitrarily large set (4.41). Clearly, this proves that the uncertain system (4.31)–(4.34) is globally equiuniformly asymptotically stable. 5. Global equiuniform finite-time stability. Due to (4.32) the piece-wise continuous uncertainty ω (x, y,t) − hx − py is locally uniformly bounded whereas the righthand side of the nominal model (4.29) is piece-wise continuous and globally homogeneous of degree q = −1 with respect to dilation r = (2, 1). Hence, the condition q + r2 ≤ 0, required by Theorem 4.2, is satisfied, and Theorem 4.2 is applicable to the globally equiuniformly asymptotically stable uncertain system (4.31)–(4.34). By applying Theorem 4.2, the uncertain system (4.31)–(4.34) is thus globally equiuniformly finite-time stable. The proof of Theorem 4.4 is completed.

Part II

Synthesis

104

II Synthesis

In order to address the robust synthesis of uncertain systems, various approaches are developed. The quasihomogeneous design, resulting from the quasihomogeneity principle, is first constituted. This design does not rely on the generation of sliding motions whereas it provides robustness features similar to those possessed by their sliding mode counterparts. Next, a discontinuous feedback controller, following the Lyapunov minmax approach, is synthesized to guarantee that the time derivative of a Lyapunov function, selected for a nominal system, is negative definite on the trajectories of the perturbed system. The approach gives rise to the so-called unit control feedback, the norm of which is equal to one everywhere but on the manifold where the feedback undergoes discontinuities. The resulting closed-loop system is shown to never pass through the switching manifold. The system stability is thus analyzed beyond the manifold. Once the trajectory is on the switching manifold, a smooth dynamic is restored and standard Lyapunov theory is in force. In addition, it is discussed how undesired high frequency state oscillations, caused by fast switching in the unit controller, can be removed by smoothing the unit signal. For nonsmooth time-varying systems, H∞ -control synthesis is alternatively presented, using measurement feedback. Being in a certain sense an extension of the unit feedback synthesis, the H∞ -control design aims to guarantee both the internal asymptotic stability of the (disturbance-free) closed-loop system and the dissipative inequality (consisting in that the size of an error signal is uniformly bounded with respect to the worst case size of an external disturbance signal). The approach here is to construct an energy or storage function, which ensures the dissipative inequality. Once the storage function is found, it can be used as a Lyapunov function, guaranteing the internal stability requirement. Sufficient conditions for the storage function to exist are given in terms of solvability of two nonsmooth Hamilton–Jacobi–Isaacs inequalities which arise in the state-feedback and, respectively, output-injection design. Although the design procedure results in an infinite-dimensional problem this difficulty is circumvented by solving the problem locally. The distribution-based formalism is then applied to straightforwardly derive a local solution of the sampleddata measurement feedback H∞ -control problem from that of the time continuous measurement feedback H∞ -control problem.

Chapter 5

Quasihomogeneous Design

Quasihomogeneity-based control synthesis is presently developed to stabilize uncertain minimum phase systems of uniform m-vector relative degree (2, . . . , 2)T . The proposed synthesis does not rely on the generation of sliding motions while providing robustness features similar to those possessed by their sliding mode counterparts.

5.1 Quasihomogeneous Finite-time Stabilization of a Simple Oscillator The quasihomogeneous synthesis is first illustrated with a simple linear oscillator, operating under uncertainty conditions. The dynamics of the oscillator is governed by x¨ = ω (x, x,t) ˙ +u (5.1) where x is the oscillator position, x˙ is the velocity of the oscillator, u is the controlled input, ω (x, x,t) ˙ is a piece-wise continuous nonlinearity that captures all forces (viscous and Coulomb frictions, gravitation, etc.), affecting the oscillator. Operating under uncertainty conditions implies imperfect knowledge of the nonlinearity ω (x, x,t). ˙ This possibly destabilizing term

ω (x, x,t) ˙ = ω nom (x, x,t) ˙ + ω un(x, x,t) ˙

(5.2)

typically contains an a priori known nominal part ω nom (x, x,t) ˙ to be handled through nonlinear damping and an uncertainty ω un (x, x,t) ˙ to be rejected. It is assumed that ω un (x, x,t) ˙ is globally bounded, i.e., the upper estimate |ω un (x, x,t)| ˙ ≤N

(5.3)

105

106

5 Quasihomogeneous Design

holds for almost all t ≥ 0 and (x, x) ˙ ∈ IR2 and for some a priori known constant N > 0. Apart from this, both functions ω nom (x, x,t) ˙ and ω un (x, x,t) ˙ are assumed to be piece-wise continuous. The control law, given by u = −ω nom (x, x,t) ˙ − asign(x) − bsign(x) ˙ − hy − px, ˙

(5.4)

results in the quasihomogeneous closed-loop system (4.31). Provided that N < b < a − N,

h, p ≥ 0,

(5.5)

it appears to stabilize the uncertain system (5.1)–(5.3) in finite time. Indeed, being interpreted in terms of the uncertain oscillator (5.1), Theorem 4.4 yields the following. Theorem 5.1. Let the uncertain oscillator (5.1)–(5.3) be driven by the state feedback (5.4) such that (5.5) holds. Then the closed-loop system (5.1)–(5.5) is globally equiuniformly asymptotically stable. To this end, we note that the above controller (5.4), (5.5) consists of the linear gain −hx − px, ˙ the nonlinear damping −ω nom (x, x,t), ˙ and the homogeneous relay part ϕ (x, x) ˙ = −asign(x) −bsign(x) ˙ such that ϕ (cx, cx) ˙ = ϕ (x, x) ˙ for all c > 0. Since the closed-loop system, driven by the controller, meets the quasihomogeneity principle, such a controller is further referred to as quasihomogeneous. In a particular case, when the uncertainty ω (x, x,t) ˙ = ω un (x, x,t) ˙ has no nominal part and the control gains h, p are set to zero, the proposed control law (5.4) degenerates to the well-known homogeneous twisting algorithm [76, 77].

5.2 Quasihomogeneous Stabilization of Nonlinear Systems of Relative Degree (2, . . . , 2) The present section investigates the capability of a nonlinear system of the form z˙ = g(z, ξ , ξ˙ ,t), ξ¨ = f (z, ξ , ξ˙ ,t) + u

(5.6)

to be globally asymptotically stabilizable in spite of significant model uncertainties with a priori known norm bounds. Hereinafter, t ∈ R is the time variable, z ∈ Rn , ξ , ξ˙ ∈ Rm are the state components, u ∈ Rm is the input, ξ ∈ Rm is the output, and the nonlinear functions g and f have appropriate dimensions and involve the system uncertainties whose influence on the control process should be rejected. The above system often arises in practice, e.g., to describe controlled electromechanical plants. It can be obtained via a nonlinear change of state coordinates and a feedback transformation from a general affine control system

5.2 Quasihomogeneous Stabilization of Nonlinear Systems of Relative Degree (2, . . . , 2)

x˙ = a(x,t) + b(x,t)u, y = h(x,t),

107

x ∈ Rn+2m , u ∈ Rm ,

(5.7)

y ∈ Rm ,

of uniform m-vector relative degree (2, . . . , 2)T with the involutive distribution B = span{b1, . . . , bm }, the span of the columns of b(x) (see [38] for details). The following assumptions on (5.6) are made throughout. 1. The functions g(z, ξ , ξ˙ ,t) and f (z, ξ , ξ˙ ,t) are piece-wise continuous in all the arguments, and, in addition, the function g(z, ξ , ξ˙ ,t) is continuous in (ξ , ξ˙ ) locally around (ξ , ξ˙ ) = 0 for almost all z and t. 2. The function g(z, ξ , ξ˙ ,t) satisfies the linear growth condition g(z, ξ , ξ˙ ,t) ≤ k(ξ , ξ˙ ,t)(1 + z)

(5.8)

in z everywhere in its domain (i.e., (5.8) can only be violated on the set where g undergoes discontinuities) with some continuous function k(ξ , ξ˙ ,t). 3. The system z˙ = g(z, 0, 0,t) (5.9) has 0 as a globally uniformly asymptotically stable equilibrium. Under Assumption 1 on (5.6) the existence of a solution (possibly nonunique) of either equation with an arbitrary initial condition is guaranteed by Theorem 8 of [71, p. 85]. Other assumptions are made for technical reasons. Assumption 2 is introduced to avoid the destabilizing effect of the peaking phenomenon (a detailed treatment of the peaking phenomenon in the continuous setting can be found in [219]). Assumption 3 means that (5.6) is a globally uniformly minimum phase system. The role of this notion is well known from the theory of smooth fields [38] and it is now under study for discontinuous nonautonomous systems. System 5.6 is operating under uncertainty conditions that imply imperfect knowledge of the nonlinearities f and g. The nonlinear gain g cannot destabilize the closed-loop system because of the minimum phase hypothesis, which is why no more information is required for this gain. The destabilizing term f (z, ξ , ξ˙ ,t) = f nom (z, ξ , ξ˙ ,t) + f b (z, ξ , ξ˙ ,t)

(5.10)

typically consists of a nominal part f nom to be handled through nonlinear damping and an uncertain bounded gain f b to be rejected. It is assumed that the nominal part f nom is known a priori, whereas the components f jb , j = 1, . . . , m, of the bounded gain f b are upper estimated: | f jb (z, ξ , ξ˙ ,t)| ≤ Fj < ∞ for almost all (z, ξ , ξ˙ ,t) ∈ Rn+2m+1

(5.11)

by constants Fj , also known a priori. Apart from this, both functions f nom and f b are assumed to be piece-wise continuous.

108

5 Quasihomogeneous Design

The following control law u(z, ξ , ξ˙ ,t) = − f nom (z, ξ , ξ˙ ,t) − μ ξ˙ − νξ − β sign ξ − γ sign ξ˙ ,

(5.12)

with the parameter gains

μ = diag(μ j ),

ν = diag(ν j ),

β = diag(β j ),

γ = diag(γ j )

subject to

μ j ≥ 0,

ν j ≥ 0,

β j − Fj > γ j > Fj ,

j = 1, . . . , m,

(5.13)

is proposed to stabilize the uncertain system (5.6), (5.10), (5.11) whose state (z, ξ , ξ˙ ) is available for measurements. Hereafter, the notation diag is used to denote a diagonal matrix of an appropriate dimension; sign ξ with a vector ξ = (ξ1 , . . . , ξm )T stands for the column vector (sign ξ1 , . . . , sign ξm )T . In what follows, the control law (5.12), (5.13) is shown to constitute a quasihomogeneous controller, driving the uncertain system (5.6) to the zero dynamics manifold ξ = ξ˙ = 0 in finite time. Desired stability properties are thus imposed on the closed-loop system. Theorem 5.2. Let Assumptions 1–3 be satisfied and let the uncertain system (5.6), (5.10), (5.11) be driven by the state feedback (5.12) such that (5.13) holds. Then, the closed-loop system (5.6), (5.10)–(5.13) is globally equiuniformly asymptotically stable. Proof. The closed-loop system (5.6) driven by (5.12) is represented as follows: z˙ = g(z, ξ , ζ ,t), ˙ ξ j = ζ j, j = 1, . . . , m b ˙ ζ j = f j (z, ξ , ζ ,t) − ν j ξ j − μ j ζ j − β j sign ξ j − γ j sign ζ j .

(5.14) (5.15)

Due to Assumption 1, Theorem 8 of [71, p. 85] is applicable to (5.14), (5.15), and by applying this theorem, the system has a local solution for all initial data and uncertainties (5.11). Let us demonstrate that each solution of (5.14), (5.15) is globally continuable on the right. Similarly to (4.37), the time derivative of the function V j (ξ j , ζ j ) = β j |ξ j | + 1 2 2 ( ζ j + ν j ξ j ), j = 1, . . . , m, computed along the trajectories of the corresponding 2 subsystem (5.15), is negative semidefinite: V˙ j (ξ j (t), ζ j (t)) ≤ −(γ j − Fj )|ζ j (t)|.

(5.16)

As in the proof of Theorem 4.4, it follows that each solution of (5.15) subject to (5.11) is uniformly bounded in t. In turn, for given continuous, uniformly bounded functions ξ (t), ζ (t), all possible solutions of (5.14) remain bounded on any finite time interval due to the linear growth assumption (5.8) (Assumption 2) on the right-hand side of (5.14). Indeed,

5.3 Local Quasihomogeneous Stabilization of Underactuated Mechanical Systems

109

an arbitrary solution z(·) of (5.14) is a priori estimated as z(t) ≤ z(t0 ) +

t t0

k(ξ (t), ζ (t),t)(1 + z(t))dt,

(5.17)

and by applying the Bellman–Gronwall lemma to the integral inequality (5.17), z(·) is bounded on any finite time interval (t0 ,t1 ). Thus, all possible solutions of the overall uncertain system (5.11), (5.14), (5.15) remain bounded on any finite time interval, and by property B of Theorem 9 of [71, p. 86], these solutions are globally continuable on the right. Next let us observe that due to the uniform boundedness (5.11) of the uncertain terms f jb (z, ξ , ζ ,t), j = 1, . . . , m, and by virtue of the parameter subordination (5.13), each subsystem (5.15) satisfies all the conditions of Theorem 4.4, and by applying this theorem, (5.15) is globally equiuniformly finite-time stable. It follows that starting from a finite time moment, the overall uncertain system (5.11), (5.14), (5.15) evolves in the chattering mode on the zero dynamics manifold ξ = ζ = 0, where its behavior is governed by the zero dynamics equation (5.9). Now, to complete the proof, it remains to note that by Assumption 3, the zero dynamics (5.9) is globally uniformly asymptotically stable. Coupled to the global equiuniform finite-time stability of (5.15), this ensures that the closed-loop system (5.14), (5.15) is globally equiuniformly asymptotically stable, too. The proof of Theorem 5.2 is thus completed.

5.3 Local Quasihomogeneous Stabilization of Underactuated Mechanical Systems The quasihomogeneous synthesis is now developed for local stabilization of underactuated systems of the form q¨ = M −1 (q)[Bτ − C(q, q) ˙ q˙ − G(q) − F(q)]. ˙

(5.18)

In the above equation, that has been derived with Lagrange formulation [54], q ∈ IRn is the joint position vector, τ ∈ IRm , m < n is the input torque, q˙ and q¨ are the velocity and acceleration vectors, respectively, M(q) ∈ IRn×n is the inertia matrix, C(q, q) ˙ q˙ represents centrifugal and Coriolis terms, G(q) is the gravity vector, F(q) ˙ ∈ IRn is the friction force and B is the input matrix of rank m. Under certain conditions, (5.18) can locally be represented by means of a nonlinear change of state coordinates and a feedback transformation in the form

η¨ = g(η , η˙ , ξ , ξ˙ ) ξ¨ = f (η , η˙ , ξ , ξ˙ ) + u.

(5.19)

110

5 Quasihomogeneous Design

If, in addition, this system is locally minimum phase and sufficiently smooth, it can locally be stabilized by a quasihomogeneous controller similar to (5.12). The following assumptions, similar to those of the previous section, are made. 1. The functions g(η , η˙ , ξ , ξ˙ ) and f (η , η˙ , ξ , ξ˙ ) are piece-wise continuous in all the arguments and, in addition, g(η , η˙ , ξ , ξ˙ ) is continuous in (ξ , ξ˙ ) locally around (ξ , ξ˙ ) = 0 for almost all (η , η˙ ). 2. The system η¨ = g(η , η˙ , ξ (t), ξ˙ (t)) (5.20) is input-to-state stable. 3. The system

η¨ = g(η , η˙ , 0, 0)

(5.21)

has 0 as a locally asymptotically stable equilibrium. System 5.19 is operating under uncertainty conditions. The nonlinear vector field g cannot destabilize the closed-loop system because of the input to state stability and the minimum phase hypotheses, which is why no more information is required for this vector field. The possibly destabilizing term f (η , η˙ , ξ , ξ˙ ) = f nom (η , η˙ , ξ , ξ˙ ) + f b (η , η˙ , ξ , ξ˙ )

(5.22)

is partitioned into a nominal part f nom , known a priori, and an uncertain bounded gain f b whose components f jb , j = 1, . . . , m are globally upper estimated | f jb (η , η˙ , ξ , ξ˙ )| ≤ N j

(5.23)

by a priori known constants N j > 0. Apart from this, both functions f nom and f b are assumed to be piece-wise continuous. Being inspired from the quasihomogeneous controller (5.12), (5.13), the following control law u(η , η˙ , ξ , ξ˙ ) = − f nom (η , η˙ , ξ , ξ˙ ) − α sign ξ − β sign ξ˙ − H ξ − Pξ˙

(5.24)

H = diag{h j }, P = diag{p j }, α = diag{α j }, β = diag{β j },

(5.25)

N j < β j < α j − N j, h j , p j ≥ 0, j = 1, . . . , m

(5.26)

with

and

is proposed to locally stabilize the uncertain system (5.19), (5.22), (5.23) whose state (η , η˙ , ξ , ξ˙ ) is available for measurements. In what follows, the control law (5.24)–(5.26) is shown to drive the uncertain system (5.19) to the zero dynamics manifold ξ = ξ˙ = 0 in finite time thereby yielding desired stability properties of the closed-loop system.

5.3 Local Quasihomogeneous Stabilization of Underactuated Mechanical Systems

111

Theorem 5.3. Let Assumptions 1–3 of the present section be satisfied and let the uncertain system (5.19), (5.22), (5.23) be driven by the state feedback (5.24) such that (5.26) holds. Then the closed-loop system (5.19), (5.24)–(5.26) is locally asymptotically stable, uniformly in the admissible uncertainties (5.22), (5.23). Proof. The closed-loop system (5.19) driven by (5.24) is represented as follows:

η¨ = g(η , η˙ , ξ , ζ ) ξ˙ j = ζ j , j = 1, . . . , m, ζ˙ j = f jb (η , η˙ , ξ , ζ ) − α j sign ξ j − β j sign ζ j − h j ξ j − p j ζ j .

(5.27) (5.28)

Due to Assumption 1, Theorem 8 of [71, p. 85] is applicable to System 5.27, 5.28, and by applying this theorem, the system has a local solution for all initial data and globally bounded uncertainties (5.23). Let us demonstrate that each solution of (5.27), (5.28) is globally continuable on the right. First, let us note that given j ∈ (1, . . . , m), no motion appears on the axes ξ j = 0 and ζ j = 0 except their intersection ξ j = ζ j = 0. Indeed, if ξ j (t) = 0 on a trajectory of (5.27), (5.28) then it follows from (5.27) that ζ j (t) = 0 along the trajectory. In turn, if ζ j (t) = 0 on a trajectory of (5.27), (5.28) then due to the parameter subordination (5.26), the second equation of (5.28) fails to hold for ξ j = 0. Next, let us compute the time derivative of the function V j (ξ j , ζ j ) = α j |ξ j | + 1 2 2 (h j ξ j + ζ j ), j = 1, . . . , m along the trajectories of (5.15). Taking into account 2 (5.23), one derives that V˙ j (ξ j (t), ζ j (t)) = α j ζ j sign ξ j + h j ξ j ζ j + ζ j { f jb (η , η˙ , ξ , ζ ) − α j sign ξ j −β j sign ζ j − h j ξ j − p j ζ j } = −[β j − f jb (η , η˙ , ξ , ζ )sign ζ j ] × |ζ j | −p j ζ j2 ≤ −(β j − N j )|ζ j (t)| (5.29) everywhere but on the axis ξ j = 0 where the function V j (ξ j , ζ j ) is not differentiable. Since no sliding motion appears on the axis ξ j = 0 except the intersection ξ j = ζ j = 0 where V˙ j (ξ j (t), ζ j (t)) = 0, (5.29) remains in force for almost all t. By virtue of (5.26), it follows that each solution of (5.28) subject to (5.23) is uniformly bounded in t. Coupled to Assumption 2, this ensures that all possible solutions of the over-all uncertain system (5.23), (5.27), (5.28) remain bounded on any finite time interval and by the property B of Theorem 9 of [71, p. 86], these solutions are globally continuable on the right. Now let us observe that due to the global boundedness (5.23) of the uncertain terms f jb (η , η˙ , ξ , ζ ), j = 1, . . . , m and by virtue of the parameter subordination (5.26), each subsystem (5.27), (5.28) is similar to that considered in Sect. 5.1, and it is therefore globally finite-time stable, uniformly in the admissible uncertainties (5.23). It follows that starting from a finite time moment, the overall uncertain system (5.23), (5.27), (5.28) evolves on the zero dynamics manifold ξ = ζ = 0 where its behavior is governed by the zero dynamics equation (5.21). In order to complete the proof, it remains to note that by Assumption 3, the zero dynamics (5.21) is locally asymptotically stable. Coupled to the local uniform

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5 Quasihomogeneous Design

finite-time stability of (5.28), this ensures that the closed-loop system (5.27), (5.28) is locally uniformly asymptotically stable. The proof of Theorem 5.3 is thus completed. Summarizing, the following local quasihomogeneous synthesis procedure is proposed for underactuated systems. First, an output of the system is specified in such a way that the corresponding zero dynamics is locally asymptotically stable. Once such an output has been chosen, the underactuated system is locally transformed into the normal form (5.19), whose stabilization is achieved by applying the quasihomogeneous controller (5.24)–(5.26).

Chapter 6

Unit Feedback Design

The sliding mode control technique is long recognized as a powerful control method to counteract non-vanishing external disturbances and unmodeled dynamics. The standard sliding mode control is synthesized to steer the system to a submanifold in finite time, after that the system stays in this submanifold forever. Typically [227], there are several switching points as one coordinate after the other hits the discontinuity manifold, and in order to design such a controller, one needs to additionally follow the hierarchy of the control components that ensures the existence of the sliding motion. In order to simplify the stability analysis and avoid extra computations, required by the hierarchy procedure, the unit feedback controller, resulting from the Lyapunov min-max approach [89, 90], is developed in the present chapter. The controller is synthesized in such a manner that the time-derivative of a Lyapunov function, selected for a nominal, asymptotically stable system, remains negative definite in spite of parameter variations and external disturbances, affecting the system. The approach gives rise to the control action, referred to as a unit control, the norm of which is equal to one everywhere with the exception of the discontinuity manifold. The closed-loop system enforced by the unit control signal is shown to be asymptotically stable and robust against matched disturbances. The crucial point is that the motion of the closed-loop system never pass through the discontinuity manifold. The system stability is thus analyzed beyond the manifold. Once the trajectory is on the discontinuity manifold, a smooth dynamic is restored and standard Lyapunov theory is in force. Although the dynamics of the closed-loop system is enforced to be exponentially stable after the trajectory has entered the discontinuity manifold, the initial stage of the trajectory (before it hits the manifold) can peak to very large values, thus potentially being capable of destroying the exponential stability of the system. This peaking phenomenon is similar to its counterpart in smooth control systems that occurs when high-gain feedback is used [219]. A deeper insight is provided in the analysis of the peaking phenomenon in sliding mode control systems. Along with this, it is discussed how undesired high frequency state oscillations (the so-called

113

114

6 Unit Feedback Design

chattering phenomenon [15]), caused by fast switching in the unit controller, can be removed by smoothing the unit signal.

6.1 Unit Control and Disturbance Rejection In this section, we illustrate some attractive features of the unit control synthesis of affine systems of the form x˙ = f (x,t) + B(x,t)u + h(x,t)

(6.1)

with the state x ∈ Rn and the control input u ∈ Rm . Hereinafter, f (x,t), h(x,t), and B(x,t) are piecewise continuous functions of appropriate dimensions. The equation x˙ = f (x,t)

(6.2)

represents an open loop nominal system. For simplicity, the nominal system (6.2) is assumed to be asymptotically stable with some a priori known continuously differentiable Lyapunov function V (x) > 0 such that its time derivative, computed along the trajectories of (6.2), is negative definite, i.e., W0 (x,t) = grad T V (x) f (x,t) ≤ −W1 (x)

(6.3)

where W1 (x) is a continuous, positive definite function. The perturbation vector h(x,t) represents the system uncertainty and its influence on the control process should be rejected. It is assumed that h(x,t) satisfies the matching condition [63] h(x,t) ∈ span(B), i.e., there exists vector ω (x,t) ∈ Rm such that h(x,t) = B(x,t)ω (x,t).

(6.4)

Here ω (x,t) may be an unknown vector with an a priori known upper scalar estimate ω0 (x,t) such that  ω (x,t) < ω0 (x,t)

(6.5)

for almost all x ∈ Rn , t ∈ R. The time derivative of V (x) on the trajectories of the perturbed system (6.1), (6.4) is of the form W (x,t) =

dV = W0 + {grad(V)}T B(u + ω ). dt

Let (6.1) be driven by the control input

(6.6)

6.1 Unit Control and Disturbance Rejection

with a scalar function

115

u = −ρ (x,t)U(s(x,t))

(6.7)

ρ (x,t) > ω0 (x,t),

(6.8)

m-vector function s(x,t) = BT (x,t)grad V (x),

(6.9)

and unit signal U(s) =

s . s

(6.10)

Then due to (6.8), the time derivative of the Lyapunov function V (x), computed on the trajectories of the closed-loop system (6.1), (6.4), is negative definite: W (x,t) = W0 (x,t) − ρ (x,t)  BT grad V  +grad T V Bω (x,t) ≤ BT grad V −W1 (x)−  BT grad V  [ρ (x,t) − ω0 (x,t)] ≤ −W1 (x).  BT grad V  It follows that the perturbed system (6.1) with the control input (6.7) is asymptotically stable, too. Along with this, the proposed control algorithm possesses two important features that are subsequently utilized in the robust unit control design: 1. The control signal (6.7) is a discontinuous function of the system state and it undergoes discontinuities on the (n − m)-dimensional manifold BT (x,t)grad V (x) = 0.

(6.11)

2. The disturbance h(x,t) is rejected due to enforcing the sliding mode on the manifold (6.11). Note that the equivalent value ueq of the control input (6.7), that maintains the system on the discontinuity manifold (6.11), is equal to −ω (x,t), which is not, generally speaking, the case for the control law (6.7) beyond this manifold. Meanwhile, the norm s(x,t)    s(x,t)  of the control input (6.7) with the gain ρ (x,t) = 1 is equal to one whenever s(x,t) = 0. This explains the term unit control for the control signal (6.10). It is of interest to note, that in contrast to the conventional sliding mode control signals which undergo discontinuities whenever a component of the sliding manifold changes sign, the unit control action is a continuous state function until the manifold s(x,t) = 0 is reached. Due to this difference the unit control method is an appropriate tool of discontinuous synthesis not only in a finite-dimensional state space, but also in an infinite-dimensional state space where control inputs are not (or even cannot be) represented in a component-wise form.

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6 Unit Feedback Design

6.2 Decomposition of Sliding-mode-based Synthesis Procedure By analogy to the standard sliding mode control design [61, 197, 227], the unit control synthesis procedure consists of two steps. Firstly, prescribed dynamic properties are imposed on the sliding motion of the closed-loop system by a proper choice of the discontinuity manifold s = 0. And secondly, a unit feedback controller is constructed to guarantee the existence of the sliding motion along this manifold. Once a smooth manifold s(x) = 0 has been selected in compliance with some performance criterion, the control input is designed in the unit feedback form (6.7): u = −ρ (x,t)

GT s(x) ,  GT s(x) 

(6.12)

where G = { ∂∂ xs }B is assumed to be nonsingular. The equation of a motion projection of the system (6.1) on the subspace s is of the form s˙ = {

∂s }( f + h) + Gu. ∂x

(6.13)

The conditions for the trajectories to converge to the manifold s(x) = 0 and the sliding mode to exist on this manifold may be derived based on the Lyapunov function 1 V = sT s 2

(6.14)

with the time derivative

∂s V˙ = sT { }( f + h) − ρ (x,t)  GT s(x) ≤ ∂x ∂s T  G s(x)  ·[ G−1 { }( f + h)  −ρ (x,t)], ∂x

(6.15)

being computed in accordance with (6.12), (6.13). Given

ρ (x,t) > [G−1 {

∂s }( f + h)](x,t)  ∂x

the value of V˙ is negative definite, and therefore the state of the closed-loop system (6.1), (6.12) will reach the manifold s(x) = 0 in finite time, and then the sliding mode with the desired dynamics will occur. The boundedness of the interval preceding the sliding motion is established by applying Lemma 4.3 to the inequality √ V˙ ≤ −2γ0 V f or some constant γ0 > 0, (6.16) resulting from (6.14), (6.15). Thus, after a finite time interval, the closed-loop system (6.1), (6.12) evolves in the sliding mode on the manifold s(x) = 0.

6.3 Global Asymptotic Stabilization of Uncertain Linear Systems

117

In order to describe the sliding mode, one should substitute the equivalent control value ueq = −G−1 {

∂s }( f + h), ∂x

i.e., the continuous solution of the equation s˙ = 0 with respect to u, into (6.1) for u. Due to the equivalent control method, that has been presented in Chap. 2, the resulting equation x˙ = f + h − G−1{

∂s }( f + h), ∂x

(6.17)

referred to as a sliding mode equation, governs the system motion along the sliding manifold s = 0. Since the sliding mode equation is control- and matched disturbance-independent (indeed (6.17), subject to (6.4), takes the form x˙ = f − G−1 { ∂∂ xs } f ), this approach leads to a decomposition of the original design problem into two independent problems and permits the construction of a control system which is insensitive to matched disturbances.

6.3 Global Asymptotic Stabilization of Uncertain Linear Systems

The aim of this section is to demonstrate how the unit feedback approach can be used to globally asymptotically stabilize linear dynamic systems operating under uncertainty conditions. We consider a nominal linear system to be of the form x˙ = Ax + Bu

(6.18)

y = Cx,

(6.19)

with the state x ∈ Rn , the control input u ∈ Rm , and the measured output y ∈ R p . Throughout, (6.18), (6.19) are assumed to be stabilizable and detectable. This assumption is equivalent to the capability of the system to be asymptotically stabilized by means of a linear dynamic output feedback. In order to design such a stabilizing controller, one should partition the state of the system into stable modes x1 ∈ Rn1 and unstable ones x2 ∈ Rn2 , n1 + n2 = n, and represent the state equations (6.18), (6.19) in terms of x1 and x2 : x˙1 = A1 x1 + B1 u x˙2 = A2 x2 + B2 u

(6.20) (6.21)

y = C1 x1 + C2 x2

(6.22)

where the matrix A2 is Hurwitz by construction, and the matrix pairs {A1 , B1 } and {A1 ,C1 } are, respectively, controllable and observable due to the stabilizabil-

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6 Unit Feedback Design

ity/detectability assumption. Thus, there exist matrices D1 and L1 such that the matrices A1 + B1 D1 and A1 + L1C1 are Hurwitz and the synthesis of a stabilizing dynamic output feedback control law is obtained as follows: u(x˜1 ) = D1 x˜1 x˙˜1 = A1 x˜1 + B1 u(x˜1 ) − L1 (y − C1 x˜1 − C2 x˜2 ) x˙˜2 = A2 x˜2 + B2u(x˜1 ).

(6.23) (6.24) (6.25)

The above linear synthesis is well known to primarily serve for dominating vanishing perturbations only. Counteracting non-vanishing external disturbances is better handled by using discontinuous controllers. Sliding-mode-based unit controllers are subsequently utilized to additionally impose on the closed-loop system desired robustness features against non-vanishing matched disturbances. First, a unit state feedback is constructed and then the Luenberger observer (6.24), (6.25) of the unstable modes is involved into the discontinuous synthesis to derive a stabilizing output feedback law. Finally, continuous approximations of the unit controller, attractive from the implementation standpoint, are presented.

6.3.1 State Feedback Design The stabilizing unit state feedback is based on the deliberate introduction of sliding modes into the unstable subsystem (6.20). Taking into account that Subsystem (6.21) is stable and does not require to be corrected, the state feedback, stabilizing (6.20), simultaneously stabilizes the over-all system (6.20), (6.21). The stabilization problem for (6.20) is solved in the space of the new state variable ξ = Mx1 ∈ Rn1 (6.26) where M ∈ Rn1 ×n1 is a non-singular matrix such that MB1 = (0, B12 )T

(6.27)

with a non-singular matrix B12 ∈ Rm×m . To satisfy (6.27), the first n1 − m rows of M are composed of n1 − m linear independent vectors, orthogonal to the control subspace, whereas BT1 forms the remaining m rows of M so that the matrix B12 = BT1 B1 is nonsingular. System 6.20, rewritten in terms of the components ξ1 = P1 ξ ∈ Rn1 −m , ξ2 = P2 ξ ∈ m R of the new state variable ξ = (ξ1 , ξ2 )T , is given by

ξ˙1 = A11 ξ1 + A12ξ2 ξ˙2 = A21 ξ1 + A22ξ2 + B12u

(6.28) (6.29)

where P1 and P2 are the projectors on the subspaces spanned by the first n − m state components and by the remaining m components, respectively. By construction, the

6.3 Global Asymptotic Stabilization of Uncertain Linear Systems

119

matrix pair {A1 , B1 } is controllable, because the pair {A, B}, otherwise, would not be exponentially stabilizable. Due to the lemma of [227, p. 101], it follows that the matrix pair {A11 , A12 } is also controllable. Then, by an appropriate choice of K ∈ Rm×(n1 −m) , the matrix A11 + A12 K proves to be Hurwitz with an a priori fixed location of the eigenvalues. The unit-signal-based control law u(ξ ) = −[γ0 + γ1 ξ ]B−1 12 U(σ ξ )

(6.30)

γ1 > A21 − KA11 + A22 − KA12 , σ ξ = ξ2 − K ξ1 ,

(6.31)

with nonnegative γ0 ,

and U(σ ξ ) =

σξ σ ξ 

(6.32)

(6.33)

is synthesized to drive the system (6.28), (6.29) to the discontinuity hyperplane σ ξ = 0. In the above control law, the norm of the unit component U(σ ξ ) is equal to one everywhere, but on the hyperplane σ ξ = 0, where this component is permitted m to be multi-valued and take any value from the unit ball Bm 1 = {u ∈ R : u ≤ 1}. The linear boundedness of controller (6.30) prevents the closed-loop system (6.28), (6.29) to escape to infinity in finite time. Subject to the lower bound (6.31), this controller dominates the rest (A21 − KA11 )ξ1 + (A22 − KA12 )ξ2 of the right-hand side of the equation

σ ξ˙ = (A21 − KA11)ξ1 + (A22 − KA12)ξ2 + B12 u(ξ ),

(6.34)

thereby providing the flow of the closed-loop system to be directed toward the discontinuity hyperplane σ ξ = 0. Since the sliding mode equation σ ξ˙ = 0, or equivalently, ξ˙1 = (A11 + A12 K)ξ1 (6.35) is exponentially stable due to a special choice of K, (6.21), (6.28), (6.29), driven by the discontinuous control law (6.30)–(6.33), prove to be asymptotically or exponentially stable accordingly, as γ0 > 0 or γ0 = 0 is used. Summarizing, the following result is obtained. Theorem 6.1. Let (6.21), (6.28), (6.29) be driven by the unit control law (6.30)– (6.33). Then, given initial conditions x2 (0) = x02 ∈ Rn2 , ξ (0) = ξ 0 ∈ Rn1 , the closedloop system (6.21), (6.28)–(6.33) has a unique solution, globally defined for all t ≥ 0, and this system is globally asymptotically stable whenever γ0 > 0 and it is globally exponentially stable if γ0 = 0. Proof. We break up the proof into four simple steps. 1) By the theorem of [227, p. 16], a unique solution of (6.28)–(6.33) exists locally for all initial conditions. By the same theorem, this solution on the discontinuity

120

6 Unit Feedback Design

hyperplane σ ξ = 0, if any, is governed by the sliding mode equation (6.35). While being governed by the linear equation (6.35), such a motion is apparently uniformly bounded on an arbitrary finite time interval and globally continuable on the right. The same is also true for a motion, initialized beyond the discontinuity hyperplane. Indeed, beyond this hyperplane, the right-hand side of the state equation (6.28)– (6.33) is linearly bounded in the state variable so that ξ (t) ≤ ξ 0 +

t 0

[γ0 + Lξ (τ )]d τ

(6.36)

for some constant L > 0. By applying the Bellman–Gronwall lemma to Inequality (6.36), any solution of this equation is uniformly bounded ξ (t) ≤ [ξ 0  + γ0T ]eLT

(6.37)

on an arbitrary time interval [0, T ) before possibly getting on the discontinuity hyperplane where it has been shown to remain uniformly bounded on any finite time interval. Thus, given an arbitrary initial condition, there exists a unique solution of (6.28)–(6.33), which turns out to be globally continuable on the right. 2) Let us now assume that γ0 > 0 and let us demonstrate that the closed-loop system (6.28)–(6.33) in this case is globally asymptotically stable. For this purpose, we introduce the quadratic function V (σ ξ ) = (σ ξ )T σ ξ

(6.38)

and compute its time derivative along the trajectories of the closed-loop system (6.28)–(6.33): V˙ (t) = 2(σ ξ )T {(A21 − KA11)ξ1 + (A22 − KA12)ξ2 − [γ0 + γ1 ξ ]U(σ ξ )} σξ . (6.39) ≤ −2(σ ξ )T {γ0 + [γ1 − A21 − KA11 − A22 − KA12]ξ } σ ξ  Due to (6.31), it follows that  V˙ (t) ≤ −2γ0 σ ξ  = −2γ0 V (t),

(6.40)

thereby yielding V (t) = 0 after a finite time moment T ≤ T0 = γ0−1 σ ξ 0 . This conclusion is readily reproduced by employing Lemma 4.3. Thus, starting from the finite time moment T ≤ γ0−1 σ ξ 0 , the closed-loop system evolves in the sliding mode on the m-dimensional hyperplane σ ξ = 0. By the theorem of [227, p. 16] this motion is governed by the sliding mode equation (6.35) which is globally exponentially stable due to a special choice of the matrix K. While being governed on the time interval t ∈ [0, T ) by the ordinary differential equations (6.28), (6.29) with a smooth right-hand side, the initial stage of the trajectory, before it hits the discontinuity hyperplane σ ξ = 0, continuously depends on the initial conditions. Coupled to the exponential stability of the sliding motion after the time

6.3 Global Asymptotic Stabilization of Uncertain Linear Systems

121

moment T , this ensures the closed-loop system (6.28)–(6.33) with γ0 > 0 to be globally asymptotically stable on the semi-infinite time interval [0, ∞). 3) Setting γ0 = 0 in the control algorithm (6.30), the derivative estimate (6.39) for the quadratic function (6.38) takes the form  (6.41) V˙ (t) ≤ −2γ ξ (t) V (t) where γ = γ1 − A21 − KA11 − A22 − KA12 is positive by virtue of (6.31). In contrast to (6.40), equality (6.41) cannot guarantee that V (t) = 0 starting from a finite time moment, because (6.40) is now violated for ξ (t) → 0. However, if for some ζ ∈ (0, 1) the following state estimate ξ (t) > ζ ξ 0 

(6.42)

could be guaranteed on the time interval t ∈ (0, T ) with T=

σ ξ 0  , γζ ξ 0 

(6.43)

then by analogy to (6.40), the trajectory of the closed-loop system (6.28)–(6.33), corresponding to γ0 = 0, would also enter the hyperplane σ ξ = 0 in a finite time τ0 ≤ T , and starting from τ0 the system would evolve in the sliding mode on this hyperplane. If hypothesis (6.42) fails to hold on the entire time interval (0, τ0 ), then there exists a time moment t1 ∈ (0, τ0 ) such that ξ (t1 ) = ζ ξ 0 .

(6.44)

Taking into account the earlier obtained state estimate (6.37) subject to γ0 = 0, it follows that ξ (t) ≤ ξ 0 eLT f or t ∈ (0,t1 ). (6.45) Now, regarding t1 as the new initial time moment and following the same line of reasoning as before, one can obtain that either the state vector enters the hyperplane σ ξ = 0 at time instant τ1 < t1 + T and then it evolves in the sliding mode on this hyperplane, or ξ (t2 ) = ζ 2 ξ 0  for some t2 ∈ (t1 , τ1 ) and ξ (t) ≤ ζ ξ 0 eLT f or t ∈ (t1 ,t2 ).

(6.46)

Thus, by iterating on i, one concludes that either the state vector enters the hyperplane σ ξ = 0 in a finite time and after that it never leaves this hyperplane, or ξ (ti+1 ) = ζ i+1 ξ 0  for some ti+1 ∈ (ti , τi ), i = 0, 1, . . . and ξ (t) ≤ ζ i ξ 0 eLT f or t ∈ (ti ,ti+1 ) where t0 = 0.

(6.47)

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6 Unit Feedback Design

In the latter case, Inequalities 6.47, coupled together for i = 0, 1, . . ., ensure that for all t ≥ 0 there exists a decaying exponential state estimate ξ (t) ≤ L0 ξ 0 e−μ t

(6.48)

with μ = −T −1 ln ζ > 0 and L0 = eLT > 0. In turn, in the former case, (6.48) is also satisfied before the instant of hitting the hyperplane σ ξ = 0, whereas after this instant the sliding motion on σ ξ = 0 is governed by the globally exponentially stable equation (6.35). Thus, in any case the closed-loop system (6.28)–(6.33) with γ0 = 0 proves to be globally exponentially stable. 4) To complete the proof it remains to demonstrate that (6.21) is globally asymptotically stable for γ0 > 0 and it is globally exponentially stable for γ0 = 0. If γ0 > 0 then, as shown in step 2, after the finite time T ≤ γ0−1 σ ξ 0  subsystem (6.28)–(6.33) evolves in the sliding mode on the hyperplane σ ξ = 0. While being governed on the time interval [0, T ) by the differential equations (6.21), (6.28), (6.29), subject to the smooth control input (6.30), the initial stage of the trajectory x(t), before it hits the discontinuity hyperplane σ ξ = 0, continuously depends on the initial data x0 . In turn, for t ≥ T the closed-loop system (6.18), (6.30)–(6.33) is maintained on the hyperplane σ ξ = 0 and Subsystem (6.21) is governed by the sliding mode equation x˙2 = A2 x2 + B2 B−1 12 (KA11 + KA12 K − A21 − A22 K)ξ1 , t ≥ T,

(6.49)

which is obtained by substituting the equivalent control value ueq = B−1 12 (KA11 + KA12 K − A21 − A22 K)ξ1 ,

(6.50)

being a unique solution of the equation σ˙ (ξ (t)) = 0 with respect to u, into (6.21) for u. Due to a special choice of K, the solution of the sliding mode equation (6.35) on the variable ξ1 is exponentially decaying ξ1 (t) ≤ ω ξ1 (T )e−β (t−T ) , t ≥ T

(6.51)

with some ω > 0 and β > 0. Since the matrix A2 is Hurwitz by construction and hence eA2 (t − T ) ≤ ω0 e−α (t−T ) for all t ≥ T and some α , ω0 > 0, the solution  t T

x2 (t) = eA2 (t−T ) x2 (T ) + eA2 (t−T −τ ) B2 B−1 12 (KA11 + KA12 K − A21 − A22 K)ξ1 (τ )d τ ,

(6.52)

of (6.49) is estimated as follows x2 (t) ≤ ω0 x2 (T )e−α (t−T ) + κ ξ1(T )[e−β (t−T) + e−α (t−T ) ], t ≥ T (6.53)

6.3 Global Asymptotic Stabilization of Uncertain Linear Systems

where

κ=

123

ω0 ω e α T B2 B−1 12 (KA11 + KA12 K − A21 − A22 K) > 0 |α − β |

and (6.51) has been used. By taking into account (6.51) and employing that σ ξ = 0, or equivalently, ξ2 (t) = K ξ1 (t) for t ≥ T , it follows that x(t) ≤ ω1 x(T )e−ν1 (t−T ) , t ≥ T

(6.54)

for some ω1 > 0 and ν1 > 0. Coupled to the aforementioned property of the initial stage of the solution to continuously depend on the initial data, the exponential decay (6.54) after the finite time moment T ensures the global asymptotic stability of (6.21), corresponding to γ0 > 0. If γ0 = 0, then, apart from the exponential decay (6.54) on the discontinuity hyperplane σ ξ = 0, a similar state estimate takes place at the initial stage preceding a sliding motion on this hyperplane. Indeed, as shown at step 3, the component ξ (t) of the closed-loop system (6.18), (6.30)–(6.33) with γ0 = 0 is globally exponentially stable, whereas the solution x2 (t) = eA2 (t) x2 (0) + γ1

 t 0

ξ (τ )eA2(t−τ ) B2 B−1 12

σ ξ (τ ) d τ , 0 ≤ t < T (6.55) σ ξ (τ )

of Subsystem (6.21), before ξ (t) hits the hyperplane σ ξ = 0, is estimated as follows: x2 (t) ≤ ω0 x2 (0)e−α t + κ1 ξ 0 [e−μ t + e−α t ], 0 ≤ t < T where

κ1 =

(6.56)

L0 ω0 γ1 B2 B−1 12  > 0, |α − μ |

and (6.48) has been used. By taking into account (6.48), it follows that x(t) ≤ ω2 x0 e−ν2t , 0 ≤ t < T

(6.57)

for some ω2 > 0 and ν2 > 0. Since (6.57), coupled to (6.54), ensures that x(t) ≤ Ω x0 e−ν t , t ≥ 0

(6.58)

for Ω = max{ω1 , ω2 , ω1 ω2 } > 0 and ν = min{ν1 , ν2 } > 0, regardless of whether the closed-loop system enters the discontinuity hyperplane σ ξ = 0 in a finite time T < ∞ or it never hits this hyperplane, the global exponential stability of (6.21), corresponding to γ0 = 0, is guaranteed. Thus, Theorem 6.1 is completely proved. One can be surprised that according to Theorem 6.1, a stronger feedback (6.30) with γ0 > 0 can only guarantee the asymptotic stability of the closed-loop system whereas a weaker feedback (6.30) with γ0 = 0 exponentially stabilizes the system. To explain this, one should take into account that the peaking phenomenon [219] in the initial stage of the trajectory, before it hits the discontinuity hyperplane, is

124

6 Unit Feedback Design

capable of destroying the exponential stability of the system, but it can only occur when γ0 > 0 (cf. the general upper estimate (6.37) and the particular upper estimate (6.48), corresponding to γ0 = 0). Remark 6.1. It is worth noticing that the sign of the time derivative (6.39) of the quadratic function (6.38), computed along the trajectories of the closed-loop system (6.28)–(6.33) with γ0 > 0, remains negative even in the case when the system is affected by a piecewise continuous, sufficiently small (possibly, non-vanishing) matched disturbance ω (x,t) ∈ Rm , such that ω (ξ ,t) < γ0 for all ξ ∈ Rn1 and all t > 0. Thus, the line of reasoning, used in Sect. 6.1, applies to such a perturbed system as well, so that the non-vanishing matched disturbances are rejected by the unit controller (6.30)–(6.33) with γ0 exceeding a norm bound of these disturbances.

6.3.2 Output Feedback Design In this section, we complete the discontinuous output feedback treatment of (6.18), (6.19) by interconnecting the sliding mode state feedback law (6.30) to the Luenberger observer (6.24), (6.25). For this purpose we first represent (6.30) in terms of the original state component x1 : u(x1 ) = −[γ0 + γ1 Mx1 ]B−1 12

(P2 − KP1)Mx1 , (P2 − KP1)Mx1 

(6.59)

after that we substitute the observer output x˜1 into (6.59) for x1 and then we correct the nonlinear gain γ0 + γ1 M x˜1  by adding the error term γ2 y − Cx ˜ with

γ2 > (P2 − KP1)L1 .

(6.60)

The reason for the gain correction is to ensure the observer-based controller to be capable of steering the trajectories of the closed-loop system to the discontinuity hyperplane (P2 − KP1 )M x˜1 = 0 in spite of the observer error, thereby imposing on the system a desired stability property similar to that obtained in the state feedback design. By Theorem 6.2, stated below, the dynamic output feedback u(y, x) ˜ = −[γ0 + γ1 M x˜1  + γ2y − Cx]B ˜ −1 12

(P2 − KP1)M x˜1 (P2 − KP1)M x˜1 

(6.61)

thus constructed yields a proper solution of the stabilization problem in question. Theorem 6.2. Let (6.18) be driven by the observer-based dynamic output feedback (6.24), (6.25), (6.61) with gain parameters γ0 , γ1 , γ2 , satisfying (6.31) and (6.60), respectively. Then given initial conditions x(0) = x0 ∈ Rn , x(0) ˜ = x˜0 ∈ Rn , the closedloop system (6.18), (6.24), (6.25), (6.61) has a unique solution, globally defined for all t ≥ 0, and this system is globally asymptotically stable whenever γ0 > 0 and it is globally exponentially stable if γ0 = 0.

6.3 Global Asymptotic Stabilization of Uncertain Linear Systems

125

Proof. The proof parallels that of Theorem 6.1. 1) First, let us represent the over-all system (6.18), (6.24), (6.25), (6.61) in terms of the observer error e = (e1 , e2 )T , ei = xi − x˜i , i = 1, 2, and the observer state components ξ˜ = (ξ˜1 , ξ˜2 )T , ξ˜i = P1i M x˜1 and x˜2 : e˙1 = (A1 + L1C1 )e1 + L1C2 e2 e˙2 = A2 e2 ξ˙˜1 = A11 ξ˜1 + A12ξ˜2 − P1L1Ce ξ˙˜2 = A21 ξ˜1 + A22ξ˜2 + B12u(y, ξ˜ , x˜2 ) − P2L1Ce x˙˜2 = A2 x˜2 + B2 u(y, ξ˜ , x˜2 )

(6.62) (6.63) (6.64) (6.65)

where

σ ξ˜ u(y, ξ˜ , x˜2 ) = −[γ0 + γ1 ξ˜  + γ2y − C1M −1 ξ˜ − C2 x˜2 ]B−1 12 σ ξ˜ 

(6.66)

and σ ξ˜ = ξ˜2 − K ξ˜1 . Apparently, a solution of the observer error equation (6.62) under arbitrary initial conditions e1 (0) = e01 ∈ Rn1 , e2 (0) = e02 ∈ Rn2 is given by e1 (t) =

e(A1 +L1C1 )t e01 +

t 0

e(A1 +L1C1 )(t−τ ) L1C2 eA2 τ e02 d τ

e2 (t) = eA2t e02 , t ≥ 0.

(6.67) (6.68)

Since the matrices A1 + L1C1 and A2 are Hurwitz by construction, the following observer error estimate e(t) ≤ κ0 e(0)e−β0t , t ≥ 0, i = 1, 2,

(6.69)

is then straightforwardly obtained for some positive κ0 , β0 . It follows that (6.63), (6.64), (6.66) are enforced by the exponentially decaying inputs P1 L1Ce(t) and P2 L1Ce(t). By the theorem of [227, p. 16], whichever exponentially decaying inputs ϕ1 (t) = P1 L1Ce(t) and ϕ2 (t) = P2 L1Ce(t) affect this system, a unique solution of (6.63), (6.64), (6.66) exists locally for all initial conditions ξ˜1 (0) = ξ˜10 ∈ Rn1 −m , ξ˜2 (0) = ξ˜20 ∈ Rm . By the same theorem, a system motion on the discontinuity hyperplane σ ξ˜ = 0, if any, is governed by the sliding mode equation ξ˙1 = (A11 + KA12)ξ˜1 − P1L1Ce. (6.70) In this regard it should be noted that if (6.63), (6.64), (6.66) enter the discontinuity hyperplane then it never leaves it because the time derivative of the quadratic function V (σ ξ ) = (σ ξ˜ )T σ ξ˜ (6.71) remains negative on the trajectories of the system beyond this hyperplane:

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6 Unit Feedback Design

σ ξ˜ . V˙ (t) =≤ −2(σ ξ )T (γ0 + γ ξ˜  + γ˜e) σ ξ˜ 

(6.72)

where constants γ = γ1 − A21 − KA11  − A22 − KA12  and γ˜ = (γ2 − (P2 − KP1 )L1 )C are positive by virtue of (6.31), (6.60). While being governed by the linear equation (6.70) subject to (6.69), the sliding motion is apparently uniformly bounded on an arbitrary finite time interval and globally continuable on the right. The same is also true for a motion, initialized beyond the discontinuity hyperplane. Indeed, beyond this hyperplane the right-hand side of (6.63), (6.64), (6.66) is linearly bounded in the state variable so that ξ˜ (t) ≤ ξ˜ 0  +

t 0

[γ0 + N1 ξ˜ (τ ) + N2 e(τ )]d τ

(6.73)

for some constants N1 , N2 > 0. By applying the Bellman–Gronwall lemma to (6.73), any solution of this equation is uniformly bounded ξ (t) ≤ [ξ 0  +

κ0 N2 0 e  + γ0T ]eN1 T β0

(6.74)

on an arbitrary time interval [0, T ) before possible getting on the discontinuity hyperplane where it has been shown to remain uniformly bounded on any finite time interval. Thus, given arbitrary initial conditions e0 ∈ Rn , ξ˜ 0 ∈ Rn1 , there exists a unique solution of (6.62)–(6.64), (6.66), which turns out to be globally continuable on the right. While (6.63), (6.64), (6.66) remain beyond the discontinuity hyperplane σ ξ˜ = 0, the control function u(t), computed according to (6.66) on the trajectories of (6.63), (6.64), is continuously differentiable in t. Thus, before the solution of the subsystem hits the discontinuity hyperplane, (6.65), driven by the control signal (6.66) has a unique solution under arbitrary initial conditions x˜2 (0) = x˜02 ∈ Rn2 at least till a time instant when (6.63), (6.64), (6.66) enter the hyperplane σ ξ˜ = 0. In order to describe the behavior of (6.65), (6.66) on the hyperplane σ ξ˜ = 0, one should use the equivalent control method [227] that has been studied in Chap. 2. According to this method, the equivalent control value ˜ ueq = B−1 12 [(KA11 + KA12 K − A21 − A22 K)ξ1 + (P2 − KP1 )L1Ce],

(6.75)

a unique solution of the equation σ˙ (ξ˜ (t)) = 0 with respect to u, is substituted into (6.65) for u. Due to the equivalent control method, the equation ˜ x˙˜2 = A2 x˜2 + B2B−1 12 (KA11 + KA12 K − A21 − A22 K)ξ1 +B2 B−1 12 (P2 − KP1 )L1Ce,

(6.76)

thus obtained, describes the motion of (6.65), (6.66) after a time instant when (6.63), (6.64), (6.66) hit the discontinuity hyperplane σ ξ˜ = 0.

6.3 Global Asymptotic Stabilization of Uncertain Linear Systems

127

Apparently, the equivalent control function (6.75), computed along the sliding modes (6.70), is continuously differentiable in t, and hence, both the solutions of the sliding mode equation (6.76) and those of (6.65), (6.66) are unambiguously globally continuable on the right, even if the over-all system enters the discontinuity hyperplane σ ξ˜ = 0 in a finite time. Since (6.62)–(6.66) is nothing else than a representation of the original closedloop system (6.18), (6.24), (6.25), (6.61) in terms of the observer error and observer state, a solution of the original system is globally uniquely defined, too. 2) If γ0 > 0 then the time derivative estimate (6.40) of the quadratic function (6.71) is now guaranteed by (6.72). As shown in the proof of Theorem 6.1, this estimate ensures that V (t) = 0 for all t ≥ γ0−1 σ ξ˜ 0 , i.e., after a finite time moment T ≤ γ0−1 σ ξ˜ 0  subsystem (6.63), (6.64), (6.66) evolves in the sliding mode on the hyperplane σ ξ˜ = 0. While being governed on the time interval t ∈ [0, T ) by the differential equations (6.62)–(6.65) subject to the smooth control input (6.66), the initial stage of the trajectory of the system, before it hits the discontinuity hyperplane σ ξ˜ = 0, continuously depends on the initial conditions. In turn, for t ≥ T the closed-loop system (6.62)–(6.66) is maintained on the hyperplane σ ξ˜ = 0 and it is governed by the sliding mode equations (6.70), (6.76). Taking into account the observer error estimate (6.69) and employing that A11 + KA12 is a Hurwitz matrix due to a special choice of K, the solution of the former equation is proved to exponentially decay with some ω > 0 and β > 0: ξ1 (t) ≤ ω ξ1 (T )e−β (t−T ) , t ≥ T.

(6.77)

The matrix A2 is also Hurwitz, and therefore one has eA2 (t − T ) ≤ ω0 e−α (t−T ) for all t ≥ T and some α , ω0 > 0. Thus, the solution of the latter equation, given by  t  t T

T

x˜2 (t) = eA2 (t−T ) x˜2 (T ) + eA2 (t−T −τ ) B2 B−1 12 (P2 − KP1 )L1Ce(τ )d τ +

eA2 (t−T −τ ) B2 B−1 12 (KA11 + KA12 K − A21 − A22 K)ξ1 (τ )d τ ,

(6.78)

is estimated as follows: x˜2 (t) ≤ ω0 x˜2 (T )e−α (t−T ) + κ˜ e(T )[e−β0 (t−T ) + e−α (t−T ) ] +

κ ξ1 (T )[e−β (t−T ) + e−α (t−T ) ], t ≥ T where

κ˜ = κ=

ω0 κ 0 e α T B2 B−1 12 (P2 − KP1 )L1C > 0, |α − β0 |

ω0 ω e α T B2 B−1 12 (KA11 + KA12 K − A21 − A22 K) > 0, |α − β |

(6.79)

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6 Unit Feedback Design

and (6.69), (6.77) have been used. By taking into account (6.69) and (6.77), and employing that σ ξ˜ = 0, or equivalently, ξ˜2 (t) = K ξ˜1 (t) for t ≥ T , it follows that in the case when γ0 > 0 the closed-loop system (6.62)–(6.66) is exponentially decaying for t ≥ T . Coupled to the aforementioned property of the initial stage of the solution to continuously depend on the initial conditions, this ensures the global asymptotic stability of (6.62)–(6.66) as well as that of the original closed-loop system (6.18), (6.24), (6.25), (6.61) whenever γ0 > 0. 3) For later use, we present the following extension of Theorem 6.1. Lemma 6.1. Let (6.63), (6.64) be enforced by the unit controller (6.66) subject to γ0 = 0 and let the exponential estimate (6.69) hold for the input signal. Then the closed-loop system is globally exponentially stable. The proof of Lemma 6.1 is nearly the same as that of Theorem 6.1 and the details of the proof remain to the reader. If γ0 = 0 then apart from the exponential decaying of the closed-loop system (6.62)–(6.66) on the discontinuity hyperplane σ ξ˜ = 0, this is also true at the initial stage preceding a sliding motion on this hyperplane. Indeed, by Lemma 6.1, the component ξ˜ (t) of (6.62)–(6.66) with γ0 = 0 is globally exponentially stable: ξ˜ (t) ≤ L0 ξ˜ (0)e−μ t , t ≥ 0

(6.80)

for some μ > 0 and L0 > 0, whereas the solution x˜2 (t) = eA2 (t) x˜2 (0) +  t σ ξ˜ (τ ) [γ1 ξ˜ (τ ) + γ2Ce(τ )]eA2 (t−τ ) B2 B−1 dτ , 0 ≤ t < T 12 0 σ ξ˜ (τ )

(6.81)

of (6.25), before the time instant T when ξ˜ (t) hits the hyperplane σ ξ˜ = 0, is estimated as follows: x˜2 (t) ≤ ω0 x˜2 (0)e−α t + κ1 ξ˜ (0)[e−μ t + e−α t ] +

κ2 e(0)[e−β0t + e−α t ] 0 ≤ t < T where

κ1 =

(6.82)

κ0 ω0 γ2 L0 ω0 γ1 B2 B−1 B2 B−1 12  > 0, κ2 = 12 C > 0 |α − μ | |α − β0 |

and (6.69), (6.80) have been used. As in the proof of Theorem 6.1, relations (6.79) and (6.82), coupled together, ensure that (6.25) is also exponentially stable, regardless of whether the closed-loop system (6.62)–(6.66) enters the discontinuity hyperplane σ ξ˜ = 0 in a finite time T < ∞ or it never hits this hyperplane. Thus, the global exponential stability of (6.62)–(6.66) and consequently that of the original closed-loop system (6.18), (6.24), (6.25), (6.61) is guaranteed whenever γ0 = 0. This completes the proof of Theorem 6.2. Remark 6.2. In contrast to the state feedback design, the observer-based controller (6.61) is robust against vanishing disturbances only because the observer model

6.3 Global Asymptotic Stabilization of Uncertain Linear Systems

129

(6.24), (6.25) is capable of tracking the state of the system up to a certain estimation error, which decreases as the disturbance magnitude decreases. In order to enhance the controller performance and ensure the closed-loop system to be robust against non-vanishing matched disturbances, one can replace the linear Luenberger observer by its sliding mode counterpart. Due to duality, the construction of the sliding mode observer is similar to that of the sliding mode controller, and it is therefore omitted.

6.3.3 Practical Stabilization via Smoothed Unit Feedback The main drawback of the use of the unit state feedback controller (6.30)–(6.33) with positive γ0 is that fast switching in a practical implementation of such a controller may excite the unmodelled high frequency dynamics of the system that severely limit the achievable performance. A smoothed version of this controller appears when γ0 = 0 (in a particular case, where the number m of the control inputs equals to the dimension n of the system, this controller is a linear one). While being continuous at the origin, this controller reduces the amplitude of undesired oscillations around the equilibrium point; however, in contrast to its counterpart with positive γ0 , it cannot reject non-vanishing disturbances. In turn, the implementation of the unit state feedback (6.30)–(6.33) with positive γ0 can be performed through a continuous approximation U δ (σ ξ ) of the unit signal (6.33) such that U δ (σ ξ ) − U(σ ξ ) ≤ δ f or all σ ξ ∈ Rm sub ject to σ x ≥ δ

(6.83)

where δ > 0. In other words, the unit signal is smoothed in an arbitrary manner within the δ -vicinity of the discontinuity hyperplane σ x = 0, whereas beyond this vicinity it is approximated by a closely related signal. Thus, undesirable high frequency state oscillations that would be excited by fast switching in the discontinuous controller are avoided with the above approximation, while the resulting closed-loop system, corresponding to a sufficiently small δ , inherits desired dynamic properties with an arbitrary small accuracy ε . Such a practical stabilization is particularly obtained if the above approximation (6.83) is given by σξ U δ (σ ξ ) = . (6.84) σ ξ  + δ 3 . Theorem 6.3. Let a continuous approximation (6.84) of the discontinuous unit feedback signal (6.33) be substituted into the control law (6.30) for U(σ x). Then, given an initial condition x(0) = x0 ∈ Rn and sufficiently small δ > 0, the closed-loop system (6.18), (6.30)–(6.32), (6.84) has a unique solution xδ (t), globally defined for all t ≥ 0. Moreover, this system is stable, and for arbitrary ε > 0 there exist δ0 (ε )

130

6 Unit Feedback Design

and T0 (x0 , ε ) > 0 such that xδ (t) < ε f or all t ≥ T0 (x0 , ε ) and δ ∈ (0, δ0 ).

(6.85)

Proof. By Theorem 6.1, (6.18), driven by the unit feedback controller (6.30)–(6.33), has a unique solution x(t), globally defined for all t ≥ 0, and x(t) → 0 as t → ∞. The latter guarantees that for arbitrary ε > 0 there exists T0 (x0 , ε ) > 0 such that x(t) ≤

ε f or all t ≥ T0 (x0 , ε ). 2

(6.86)

In turn, the smoothed system (6.18), driven by the continuous, piece-wise smooth controller (6.30)–(6.32), (6.84), is well known to have a unique solution xδ (t), locally defined on some time interval (0, τ ). Then by the sliding mode definition, given in Chap. 2, one has x(t) − xδ (t) ≤

ε f or all t ∈ (0, τ ), δ ∈ (0, δ0 ), 2

(6.87)

and some δ0 (ε ) > 0. Thus, the solutions xδ (t) are uniformly bounded on (0, τ ) for all δ ∈ (0, δ0 ), and these solutions are therefore uniquely continuable on the right so that (6.87) remains in force for t ≥ τ . By virtue of (6.86), it follows stability of the closed-loop system (6.18), (6.30)–(6.32), (6.84) and validity of (6.85). Theorem 6.3 is proved. By analogy to the state feedback design, the implementation of the discontinuous output feedback (6.66) can be performed through its smoothed version u(y, ξ˜ , x˜2 ) = −[γ0 + γ1 ξ˜  + γ2y − C1M −1 ξ˜ − C2 x˜2 ]B−1 12

σ ξ˜ . (6.88) ˜ σ ξ  + δ 3

As opposed to the discontinuous feedback (6.66), its continuous approximation (6.88) does not excite undesirable high frequency state oscillations, whereas the resulting closed-loop system is still practically stabilized in the following sense. Theorem 6.4. Let (6.18) and observer (6.24), (6.25) be enforced by a continuous approximation (6.88) of the discontinuous dynamic output feedback (6.66) subject to (6.31), (6.60). Then given initial conditions x(0) = x0 , x˜1 (0) = x˜01 , x˜2 (0) = x˜02 and sufficiently small δ > 0, the closed-loop system (6.18), (6.24), (6.25), (6.66) has a unique solution xδ (t), x˜δ1 (t), x˜δ2 (t), globally defined for all t ≥ 0. Moreover, this system is stable, and for arbitrary ε > 0 there exist δ0 (ε ) > 0 and T0 (x0 , x˜01 , x˜02 , ε ) > 0 such that xδ (t), x˜δ1 (t), x˜δ2 (t) < ε f or all t ≥ T0 (x0 , x˜01 , x˜02 , ε ) and δ ∈ (0, δ0 ). (6.89) The proof of Theorem 6.4 is nearly the same as that of Theorem 6.3 and the details of the proof remain to the reader. As in the state feedback design, the output feedback controller (6.61) with γ0 = 0 appears to be continuous at the origin, thereby eliminating around the equilibrium

6.4 Unit Feedback Control of Minimum Phase Nonlinear Systems

131

point undesired high-frequency state oscillations caused by fast switching in the control signal.

6.4 Unit Feedback Control of Minimum Phase Nonlinear Systems The unit control approach is now extended to a class of cascaded nonlinear systems of the form x˙1 = Ax1 + Bu

(6.90)

x˙2 = f (x1 , x2 )

(6.91)

where x1 ∈ Rn1 and x2 ∈ Rn2 are the states, u ∈ Rm is the control input. The aforementioned form is a common point of departure for a large variety of stabilization methods in nonlinear control theory (see, e.g., [38, 219] and references therein). For technical reasons, we shall assume that 1. f (x1 , x2 ) is continuously differentiable and globally Lipschitz in (x1 , x2 ); 2. The system x˙2 = f (0, x2 )

(6.92)

has 0 as a globally exponentially stable equilibrium; 3. The pair {A, B} is controllable. Assumption 1 is introduced to guarantee the existence and uniqueness of solutions of the unforced system (6.90), (6.91) under u ≡ 0 and to avoid the destabilizing effect of the peaking phenomenon. Assumption 2 means that System 6.90, 6.91 is a globally exponentially minimum phase system. Due to [219, Theorem 3.1], Assumptions 2 and 3, coupled together, ensure that (6.90), (6.91) is globally stabilizable by means of a linear feedback. In what follows, the unit feedback design is used to additionally provide robustness of the closed-loop system against non-vanishing matched disturbances with norm bounds, known a priori. We demonstrate that the unit state feedback (6.30)– (6.33), given in terms of the new variables (6.26)–(6.29), is capable of stabilizing not only the linear subsystem (6.90) but also its cascaded connection to the nonlinear subsystem (6.91). Theorem 6.5. Consider the cascaded nonlinear system (6.90), (6.91) with the assumptions above. Let this system be driven by the unit control law (6.30)–(6.33), constructed in the space of the new state variables (6.26)–(6.29). Then, given initial conditions x1 (0) = x01 ∈ Rn1 , x2 (0) = x02 ∈ Rn2 , the closed-loop system (6.30)–(6.33), (6.90), (6.91), has a unique solution, globally defined for all t ≥ 0, and this system is globally asymptotically stable.

132

6 Unit Feedback Design

Proof. This consists of three simple steps. 1) By Theorem 6.1, given initial conditions, there exists a unique solution of the linear closed-loop subsystem (6.30)–(6.33), (6.90), which is globally defined and uniformly bounded on the semi-infinite time interval [0, ∞). Due to Assumption 1, it follows that the over-all system (6.30)–(6.33), (6.90), (6.91) has a unique solution, also globally defined for all t ≥ 0. 2) Let us assume that the linear subsystem (6.30)–(6.33), (6.90) never reaches the discontinuity hyperplane. Then, the over-all system (6.30)–(6.33), (6.90), (6.91) remains of class C1 beyond this hyperplane. Taking into account that by Theorem 6.1, the linear subsystem (6.30)–(6.33), (6.90) is globally asymptotically stable, the over-all system proves to be globally asymptotically stable due to the well known result from the smooth control theory [110, Section 4.9]. 3) If, as opposed to step 2, the linear subsystem (6.30)–(6.33), (6.90) reaches the discontinuity hyperplane in finite time, then, as shown in the proof of Theorem 6.1, after hitting the hyperplane the subsystem never leaves it, and the sliding motion along this hyperplane is governed by the globally exponentially stable equation (6.35). Thus, after the instant T of hitting the hyperplane, the smooth dynamics is restored and the same result from [110, Sect. 4.9] turns out to be applicable to the cascaded nonlinear system (6.30)–(6.33), (6.90), (6.91), viewed over the time interval [T, ∞). While being governed on the time interval [0, T ) by the differential equations (6.90), (6.91), subject to the smooth control input (6.30), the initial stage of the state trajectory, before it hits the discontinuity hyperplane, continuously depends on the initial data. Coupled to the aforementioned asymptotic stability on the time interval [T, ∞), this yields that the closed-loop system (6.30)–(6.33), (6.90), (6.91) remains globally asymptotically stable, even being viewed over the entire time interval [0, ∞). The proof is completed. Apparently, the desired robustness properties against non-vanishing matched disturbances, justified in Sect. 6.3 for the linear system (6.18), driven by the unit state feedback (6.30), are inherited by the cascaded connection of the linear closed-loop system (6.18), (6.30) to the minimum phase nonlinear system (6.90), (6.91). It is also clear that for the purpose of chattering avoidance in the nonlinear case, the implementation of the unit signal (6.33) can be performed through a continuous approximation, e.g., (6.84), like that in the linear case.

Chapter 7

Disturbance Attenuation via Nonsmooth H∞-design

H∞ -disturbance attenuation theory is fully understood when the underlying system is linear. In the state-space formulation, the problem of minimizing the H∞ -norm of a linear control system is viewed as a differential game of two antagonistic persons and a solution of the problem relates to certain solutions of the Riccati equations arising in linear quadratic differential game theory (see, e.g., [18, 62] for details). The formulation of the nonlinear H∞ -control problem requires the controller design which guarantees both the internal asymptotic stability of the closed-loop system and its dissipativity with respect to admissible external disturbances. The standard approach here is to construct a storage function for the closed-loop system that could also be used as a Lyapunov function for verifying the internal stability. This leads to a min-max criterion on the dissipation inequality in the nonlinear H∞ theory as opposed to a min-max criterion on the Lyapunov inequality in the unit feedback synthesis of Chap. 6 for which the assumed Lyapunov function for the nominal system serves as a Lyapunov function for the closed-loop system for all admissible uncertainties. The state-space approach has been developed for nonlinear systems at the same level of generality as that in the linear case, i.e., allowing, firstly, the existence of a desired controller to relate to the existence of suitable solutions of appropriate Hamilton–Jacobi–Isaacs equations, which replace the afore-mentioned Riccati equations and, secondly, to construct a desired controller which is associated with the solutions of the Hamilton–Jacobi–Isaacs equations (see, e.g., [95, 107, 229]). In the present section, nonlinear H∞ -synthesis is extended to nonsmooth time-varying systems. Sufficient conditions for the existence of a global solution of the problem are given in terms of the solvability of two Hamilton–Jacobi–Isaacs inequalities which arise in the state-feedback and, respectively, output-injection design. In our L2 -gain analysis we follow the line of reasoning proposed in Sect. 3.6 where the corresponding Hamilton–Jacobi–Isaacs expressions are required to be negative definite rather than semidefinite. This feature allows us to develop an H∞ -design procedure with no a priori-imposed stabilizability-detectability conditions on the control system. Although the design procedure results in an infinite-dimensional problem, this difficulty is circumvented by solving the problem locally. A local solution is 133

7 Disturbance Attenuation via Nonsmooth H∞ -design

134

derived by means of a certain perturbation of the differential Riccati equations, appearing in solving the H∞ -control problem for the linearized system, when these unperturbed equations have uniformly bounded positive semidefinite solutions. Stabilizability and detectability properties of the control system are thus ensured by the existence of the proper solutions of the unperturbed differential Riccati equations, and hence the proposed synthesis procedure obviates an extra work on verification of these properties that might present a formidable problem in the nonlinear case and is definitely so in the nonsmooth case. The nonsmooth H∞ -synthesis is first developed via continuous time measurement feedback and then via sampled-data measurement feedback. The vibroimpact modeling of Sect. 2.1 is brought into play to straightforwardly derive a solution of the sampled-data measurement feedback H∞ -control problem from that of the timecontinuous measurement feedback H∞ -control problem. Theoretical results are illustrated in a simulation study made for H∞ -control of an inverted pendulum via nonlinear sampled-data position measurements.

7.1 Nonsmooth H∞ -control of Time-varying Systems 7.1.1 Problem Statement Consider a nonautonomous nonlinear system of the form x(t) ˙ = f (x(t),t) + g1 (x(t),t)w(t) + g2 (x(t),t)u(t) z(t) = h1 (x(t),t) + k12 (x(t),t)u(t)

(7.1) (7.2)

y(t) = h2 (x(t),t) + k21 (x(t),t)w(t)

(7.3)

where x ∈ Rn is the state vector, t ∈ R is the time variable, u ∈ Rm is the control input, w ∈ Rr is the unknown disturbance, z ∈ Rl is the unknown output to be controlled, and y ∈ R p is the only available measurement on the system. The following assumptions are made throughout. A1. The functions f (x,t), g1 (x,t), g2 (x,t), h1 (x,t), h2 (x,t), k12 (x,t), k21 (x,t) are piecewise continuous in t for all x and locally Lipschitz continuous in x for almost all t. A2. f (0,t) = 0, h1 (0,t) = 0, and h2 (0,t) = 0 for almost all t. A3. T hT1 (x,t)k12 (x,t) = 0, k12 (x,t)k12 (x,t) = I T T (x,t) = I. k21 (x,t)g1 (x,t) = 0, k21 (x,t)k21

(7.4)

These assumptions are made for technical reasons. Assumption A1 guarantees the well-posedness of the above dynamic system, while being enforced by integrable exogenous inputs. Along with this, Assumption A1 admits nonsmooth nonlineari-

7.1 Nonsmooth H∞ -control of Time-varying Systems

135

ties. Assumption A2 ensures that the origin is an equilibrium point of the non-driven (u = 0) disturbance-free (w = 0) dynamic system (7.1)–(7.3). Assumption A3 is a simplifying assumption inherited from the standard H∞ -control problem. A causal dynamic feedback compensator u = K (y,t),

(7.5)

with internal state ξ ∈ Rs , is said to be a globally (locally) admissible controller if the closed-loop system (7.1), (7.5) is globally (uniformly) asymptotically stable when w = 0. Given a real number γ > 0, it is said that system (7.1)–(7.3), (7.5) has L2 -gain less than γ if the response z, resulting from w for initial state x(t0 ) = 0, ξ (t0 ) = 0, satisfies  t1  t1 z(t)2 dt < γ 2 w(t)2 dt (7.6) t0

t0

for all t1 > t0 and all piecewise continuous functions w(t). The nonsmooth H∞ -control problem is to find a globally admissible controller (7.5) such that L2 -gain of the closed-loop system (7.1), (7.5) is less than γ . In turn, a locally admissible controller (7.5) is said to be a local solution of the H∞ control problem if there exists a neighborhood U of the equilibrium such that (7.6) is satisfied for all t1 > t0 and all piecewise continuous functions w(t) for which the state trajectory of the closed-loop system starting from the initial point (x(t0 ), ξ (t0 )) = (0, 0) remains in U for all t ∈ [t0 ,t1 ].

7.1.2 Global State-space Solution Below, we list the hypotheses under which a global solution of the H∞ -control problem is derived. H1. There exist a positive definite function F(x) and a smooth, positive definite, decrescent, radially unbounded function V (x,t), such that the Hamilton–Jacobi– Isaacs inequality

∂V ∂V + f (x,t) + γ 2 α1T (x,t)α1 (x,t)− ∂t ∂x α2T (x,t)α2 (x,t) + hT1 (x,t)h1 (x,t) + F(x) ≤ 0

(7.7)

holds for almost all t ∈ R with

∂V T 1 T ) , g (x,t)( 2γ 2 1 ∂x

(7.8)

∂V T 1 ) . α2 (x,t) = − gT2 (x,t)( 2 ∂x

(7.9)

α1 (x,t) =

7 Disturbance Attenuation via Nonsmooth H∞ -design

136

H2. There exist a piecewise continuous function G(t), and a positive semidefinite function Q(x, ξ ) subject to Q(0, ξ ) is positive definite, and a smooth, positive semidefinite, decrescent function W (x, ξ ,t) subject to W (0, ξ ,t) is positive definite and radially unbounded, such that the Hamilton–Jacobi–Isaacs inequality

∂W + ( ∂∂Wx ∂t

∂W ∂ξ

) fe (x, ξ ,t) + hTe (x, ξ ,t)he (x, ξ ,t)

+γ 2 φ T (x, ξ ,t)φ (x, ξ ,t) + Q(x, ξ ) ≤ 0

(7.10)

holds for almost all t ∈ R with fe1

feT (x, ξ ,t) = (( fe1 )T , ( fe2 )T ) = f (x,t) + g1 (x,t)α1 (x,t) + g2(x,t)α2 (ξ ,t)

fe2 = f (ξ ,t) + g1(ξ ,t)α1 (ξ ,t) + g2(ξ ,t)α2 (ξ ,t) + G(t)(h2 (x,t) − h2 (ξ ,t)) he (x, ξ ,t) = α2 (ξ ,t) − α2(x,t)   ( ∂∂Wx )T 1 T φ (x, ξ ,t) = 2 ge (x,t) ( ∂∂Wξ )T 2γ   g1 (x,t) . ge (x,t) = G(t)k21 (x,t)

(7.11) (7.12) (7.13) (7.14)

Under Hypotheses H1 and H2 a solution of the H∞ -control problem stated above is as follows. Theorem 7.1. Let hypotheses H1 and H2 be satisfied. Then the output feedback .

ξ = f (ξ ,t) + g1(ξ ,t)α1 (ξ ,t) + g2(ξ ,t)α2 (ξ ,t) + G(t)[y(t) − h2(ξ ,t)], u = α2 (ξ ,t)

(7.15)

globally asymptotically stabilizes the disturbance-free system (7.1) and makes the L2 -gain of the closed-loop system less than γ . Proof. To begin with, let us introduce the function H(x, w, u,t) =

∂V ∂V + [ f (x,t) + g1 (x,t)w + g2(x,t)u] ∂t ∂x +hT1 (x,t)h1 (x,t) + uT u − γ 2wT w

which is quadratic in (w, u). Relations (7.8) and (7.9) result in   ∂H ∂V g1 (x,t) − 2γ 2 α1T = 0, = ∂ w (w,u)=(α1 ,α2 ) ∂ x

(7.16)

(7.17)

7.1 Nonsmooth H∞ -control of Time-varying Systems



∂H ∂u



(w,u)=(α1 ,α2 )

=

137

∂V g2 (x,t) + 2α2T = 0, ∂x

(7.18)

and expanding the quadratic function H(x, w, u,t) in Taylor series yields H(x, w, u,t) = H(x, α1 (x,t), α2 (x,t),t) − γ 2 w − α1 (x,t)2 + u − α2 (x,t)2 (7.19) where H(x, α1 (x,t), α2 (x,t),t) ≤ −F(x) in accordance with Hypothesis H1. Hence, H(x, w, u,t) ≤ u − α2(x,t)2 − γ 2 w − α1 (x,t)2 − F(x)

(7.20)

and utilizing (7.16), (7.20), we arrive at

∂V ∂V + [ f (x,t) + g1 (x,t)w + g2 (x,t)u] ≤ u − α2(x,t)2 ∂t ∂x −γ 2 w − α1 (x,t)2 − h1 (x,t)2 − u2 + γ 2 w2 − F(x).

(7.21)

Next, let us introduce the function He (x, ξ , r,t) =

∂W  ∂W + ∂x ∂t

∂W ∂ξ



[ fe (x, ξ ,t) + ge (x,t)r]

+hTe (x, ξ ,t)he (x, ξ ,t) − γ 2 rT r

(7.22)

which is quadratic in r. Then, we have     ∂ He = ∂∂Wx ∂∂Wξ ge (x,t) − 2γ 2φ T (x, ξ ,t) = 0 ∂ r r=φ (x,ξ ,t) and by applying Hypothesis H2, it follows that He (x, ξ , φ (x, ξ ,t),t) ≤ −Q(x, ξ ). Thereby, the quadratic function He (x, ξ , r,t) is expanded in the Taylor series as follows He (x, ξ , r,t) = He (x, ξ , φ (x, ξ ,t),t) − γ 2 r − φ (x, ξ ,t)2 ≤ −γ 2 r − φ (x, ξ ,t)2 − Q(x, ξ )

(7.23)

and, by virtue of (7.22), we establish that

∂W  ∂W + ∂x ∂t

∂W ∂ξ



[ fe (x, ξ ,t) + ge (x,t)(w − α1 (x,t)]

≤ −γ 2 w − α1 (x,t) − φ (x, ξ ,t)2 − α2 (ξ ,t) − α2(x,t)2 + γ 2 w − α1 (x,t)2 − Q(x, ξ ).

(7.24)

Let us now demonstrate that the function U(x, ξ ,t) = V (x,t) + W (x, ξ ,t),

(7.25)

7 Disturbance Attenuation via Nonsmooth H∞ -design

138

which is smooth, positive definite, radially unbounded, and decrescent by construction, has a negative time-derivative along the trajectories of the closed-loop system (7.1), (7.15) with w = 0. Indeed, (7.21) and (7.24), coupled together, yield dU ∂V ∂V ∂W = + [ f (x,t) + g1 (x,t)w + g2(x,t)u] + dt ∂t ∂x ∂t   + ∂∂Wx ∂∂Wξ [ fe (x, ξ ,t) + ge (x,t)(w − α1 (x,t)] ≤ u − α2 (x,t)2 − γ 2 α1 (x,t)2 − h1 (x,t)2 −γ 2 α1 (x,t) + φ (x, ξ ,t)2 − α2 (ξ ,t) − α2 (x,t)2 +γ 2 α1 (x,t)2 − u2 − F(x) − Q(x, ξ ) ≤ − h1 (x,t)2 − γ 2 α1 (x,t) + φ (x, ξ ,t)2 − α2 (ξ ,t)2 − F(x) − Q(x, ξ ) ≤ −F(x) − Q(x, ξ ).

(7.26)

Thus, Theorem 3.5 and Remark 3.1 appear to be applicable to the internal dynamics of the closed-loop system (7.1), (7.15), and (7.1), (7.15) is therefore internally globally asymptotically stable. Finally, to establish (7.6) we differentiate (7.25) along the trajectories of the perturbed system (7.1), (7.15) with w = 0 and using (7.21), (7.24) we then obtain that dU ≤ − z(t)2 + γ 2 w2 − F(x) − Q(x, ξ ) − γ 2 w − α1 (x,t) − φ (x, ξ ,t)2 . dt (7.27) Clearly, (7.27) ensures that  t1 t0

(γ 2 w2 − z(t)2 )dt ≥

 t1 t0

[F(x(t)) + Q(x(t), ξ (t))]dt+

U(x(t1 ), ξ (t1 ),t1 ) − U(x(t0), ξ (to ),t0 )+

γ2

 t1 t0

w(t) − α1 (x(t),t)− φ (x(t), ξ (t),t)2 dt > 0

(7.28)

for any trajectory of (7.1), (7.15) starting at x(t0 ) = 0, ξ (t0 ) = 0. Thus, the dynamic compensator (7.15) is shown to be a solution of the H∞ -control problem. This completes the proof of Theorem 7.1. It should be noted that Theorem 7.1 ensures the existence of a stabilizing measurement feedback controller under no stabilizability-detectability conditions, thereby removing an extra work on explicit verification of these conditions. Later on, Theorem 7.1 will be instrumental in the local analysis of the problem in question.

7.1 Nonsmooth H∞ -control of Time-varying Systems

139

7.1.3 Local State-space Solution In the subsequent local analysis, the function f (x,t) = f1 (x,t) + f2 (x,t) is decomposed into two components and for technical reasons, nonsmooth nonlinearities are assumed to only be absorbed into the term f2 (x,t) whereas the other terms, including f1 (x,t), are assumed to be smooth enough. The following assumptions are thus additionally made. A4. The function f (x,t) admits representation f (x,t) = f1 (x,t) + f2 (x,t) and for almost all t ∈ R the functions f1 (x,t), g1 (x,t), g2 (x,t), h1 (x,t), h2 (x,t), k12 (x,t), k21 (x,t) are twice continuously differentiable in x locally around the origin x = 0 whereas their first and second order state derivatives are piecewise continuous and uniformly bounded in t for all x in some neighborhood of the origin. A5. The vector ζ = 0 is a proximal time-uniform supergradient of the components f2i (x,t), i = 1, . . . , n of the function f2 (x,t) at x = 0. ˆ ∈ Rn is a proximal supergradient of Recall (see [51] for details) that a vector ζ (x,t) n+1 a scalar function f0 (x,t) at (x,t) ˆ ∈R if there exists some σ (x,t) ˆ > 0, such that f0 (x,t) ≤ f0 (x,t) ˆ + ζ T (x,t)(x ˆ − x) ˆ + σ (x,t)x ˆ − x ˆ 2

(7.29)

for all x in some neighborhood U(x,t) ˆ of x. ˆ The proximal supergradient ζ (x,t) ˆ is said to be time-uniform if (7.29) remains true for some time-independent σ (x) ˆ > 0 and all x in some time-invariant neighborhood U(x) ˆ of x. ˆ The set ∂ P f (x,t) ˆ of proximal supergradients of f at (x,t) ˆ is referred to as the proximal superdifferential of f at (x,t). ˆ So, the components f2i (x,t), i = 1, . . . , n of the function f2 (x,t) are such that

or equivalently,

0 ∈ ∂ P f2i (0,t) uni f ormly in t ∈ R

(7.30)

f2i (x,t) ≤ σ x2

(7.31)

for some σ > 0, almost all t ∈ R and all x ∈ U(0). Clearly, the latter implies that the nonsmooth function f2 (x) is concave in the origin. Assumptions A1–A5, coupled together, allow one to linearize the corresponding Hamilton–Jacobi–Isaacs inequalities, thereby yielding a local solution of the nonsmooth H∞ -control problem. It should be pointed out, however, that Estimate 7.31 is unilateral and it does not imply that the nonsmooth term f2 (x) becomes negligible in the subsequent local analysis. For instance, (7.31) holds with the scalar function f2 (x) = −|x| whose influence should be taken into account while locally analyzing the system behavior around its equilibrium point. Thus, the local analysis to be developed has to operate with unilateral estimates, and hence it does not represent a straightforward extension of the standard nonlinear H∞ -techniques. The subsequent local analysis involves the linear H∞ -control problem for the system . x= A(t)x + B1(t)w + B2 (t)u

7 Disturbance Attenuation via Nonsmooth H∞ -design

140

z = C1 (t)x + D12(t)u y = C2 (t)x + D21(t)w

(7.32)

where A(t) =

∂ f1 ∂ h1 (0,t), B1 (t) = g1 (0,t), B2 (t) = g2 (0,t), C1 (t) = (0,t), ∂x ∂x

∂ h2 (0,t), D12 (t) = k12 (0,t), D21 (t) = k21 (0,t). (7.33) ∂x Such a problem is now well understood if the linear system (7.32) is stabilizable and detectable from u and y, respectively. Under these assumptions, the following conditions are necessary and sufficient for a solution of the problem to exist (see, e.g., [190]): C2 (t) =

C1.

The equation .

− P= P(t)A(t) + AT (t)P(t) + C1T (t)C1 (t) +P(t)[

1 B1 BT1 − B2 BT2 ](t)P(t) γ2

(7.34)

possesses a uniformly bounded positive semidefinite symmetric solution P(t) such that the system .

x= [A − (B2 BT2 − γ −2 B1 BT1 )P](t)x(t)

(7.35)

is exponentially stable; C2. Being specified with A1 (t) = A(t) + γ12 B1 (t)BT1 (t)P(t), the equation .

Z = A1 (t)Z(t) + Z(t)AT1 (t) + B1 (t)BT1 (t)+ Z(t)[

1 PB2 BT2 P − C2T C2 ](t)Z(t), γ2

(7.36)

possesses a uniformly bounded positive semidefinite symmetric solution Z(t), such that the system .

x= [A1 − Z(C2T C2 − γ −2 PB2 BT2 P)](t)x(t)

(7.37)

is exponentially stable. According to Lemma 3.4, Conditions C1 and C2 ensure that there exists a positive constant ε0 such that the system of the perturbed Riccati equations −P˙ε = Pε (t)A(t) + AT (t)Pε (t) + C1T (t)C1 (t) +Pε (t)[

1 B1 BT1 − B2 BT2 ](t)Pε (t) + ε I, γ2

(7.38)

7.1 Nonsmooth H∞ -control of Time-varying Systems

141

.

Z ε = Aε (t)Zε (t) + Zε (t)ATε (t) + B1(t)BT1 (t) +Zε (t)[

1 Pε B2 BT2 Pε − C2T C2 ](t)Zε (t) + ε I γ2

(7.39)

has a unique uniformly bounded, positive definite symmetric solution (Pε (t), Zε (t)) for each ε ∈ (0, ε0 ) where Aε (t) = A(t) + γ12 B1 (t)BT1 (t)Pε (t). In what follows, (7.38) and (7.39) are utilized to derive a local solution of the nonlinear H∞ -control problem for (7.1). Theorem 7.2. Let Conditions C1 and C2 be satisfied and let (Pε (t), Zε (t)) be a uniformly bounded positive definite solution of (7.38), (7.39) under some ε > 0. Then Hypotheses H1 and H2 hold locally around the equilibrium (x, ξ ) = (0, 0) with V (x,t) = xT Pε (t)x, (7.40) W (x, ξ ,t) = γ 2 (x − ξ )T Zε−1 (t)(x − ξ ), ε F(x) = x2 , 2

(7.41)

G(t) = Zε (t)C2T (t), ε Q(x, ξ ) = γ 2 min Zε−1 (t)2 x − ξ 2, 2 t∈R and the output feedback

(7.43)

.

ξ = f (ξ ,t) + [

(7.42)

(7.44)

1 g1 (ξ ,t)gT1 (ξ ,t) − g2(ξ ,t)gT2 (ξ ,t)]Pε (t)ξ γ2 +Zε (t)C2T (t)[y(t) − h2(ξ ,t)],

(7.45)

u = −gT2 (ξ ,t)Pε (t)ξ

(7.46)

is a local solution of the H∞ -control problem. Proof. First, we demonstrate that in a neighborhood of the origin x = 0 and for almost all t ∈ R the function V (x,t) = xT Pε (t)x, being differentiable, positive definite, radially unbounded and decrescent by construction, satisfies the Hamilton–Jacobi– Isaacs inequality (7.7) subject to (7.42). Indeed,

∂V = xT P˙ε (t)x ∂t ∂V f1 (x,t) = xT [Pε A + AT Pε ](t)x + ot (x2 ) ∂x ∂V f2 (x,t) ≤ σ 0 x3 ∂x γ 2 α1T (x,t)α1 (x,t) =

1 T x Pε B1 (t)BT1 (t)Pε x + ot (x2 ) γ2

(7.47) (7.48) (7.49)

(7.50)

7 Disturbance Attenuation via Nonsmooth H∞ -design

142

−α2T (x,t)α2 (x,t) = −xT Pε B2 (t)BT2 (t)Pε x + ot (x2 )

(7.51)

hT1 (x,t)h1 (x,t) = xT C1T (t)C1 (t)x + ot (x2 )

(7.52)

where σ 0 = 2σ supt∈R Pε (t) and by virtue of Assumption A4, formly in t as

x2

ot

→ 0. Then, due to (7.38), we have

(x2 )

x2

→ 0 uni-

∂V ∂V + [ f1 (x,t) + f2 (x,t)] + γ 2α1T (x,t)α1 (x,t)− ∂t ∂x α2T (x,t)α2 (x,t) + hT1 (x,t)h1 (x,t) ≤ 1 B1 BT1 − B2BT2 ]Pε + C1T C1 }(t)x+ γ2 ε σ 0 x3 + ot (x2 ) ≤ −ε x2 + ot (x2 ) ≤ − x2 (7.53) 2 for almost all t ∈ R and x sufficiently small. Hence, Hypothesis H1 holds locally with V (x,t) and F(x) defined by (7.40) and (7.42). Next, let us observe that by construction, the function xT {P˙ε + Pε A + AT Pε + Pε [

W (x, ξ ,t) = γ 2 (x − ξ )T Zε−1 (t)(x − ξ ) is smooth, positive semidefinite, decrescent, and such that W (0, ξ ,t) is positive definite and radially unbounded. Moreover, in a neighborhood of the origin (x, ξ ) = (0, 0) and for almost all t ∈ R, W (x, ξ ,t) satisfies the Hamilton–Jacobi–Isaacs inequality (7.10) subject to (7.43) and (7.44). Indeed,

∂W = γ 2 (x − ξ )T Z˙ ε−1 (t)(x − ξ ) ∂t ( ∂∂Wx

∂W ∂ξ

(7.54)

) fe (x, ξ ,t) ≤ γ 2 (x − ξ )T {Zε−1 Aε + ATε Zε−1 − 2C2T C2 }(t)(x − ξ ) + ot (x − ξ 2)

hTe (x, ξ ,t)he (x, ξ ,t) = (x − ξ )T [Pε B2 BT2 Pε ](t)(x − ξ ) + ot (x − ξ 2)

γ 2 φ T (x, ξ ,t)φ (x, ξ ,t) = γ 2 (x − ξ )T {Zε−1 B1 BT1 Zε−1 + C2T C2 }(t)(x − ξ ) + ot (x − ξ 2)

(7.55) (7.56)

(7.57)

and taking into account the equation −Z˙ ε−1 = Zε−1 (t)Aε (t) + ATε (t)Zε−1 (t) + [

1 Pε B2 BT2 Pε − C2T C2 ](t) + γ2

Zε−1 (t)B1 (t)BT1 (t)Zε−1 (t) + ε Zε−2 (t), resulting from (7.39), we have

(7.58)

7.1 Nonsmooth H∞ -control of Time-varying Systems

∂W + ( ∂∂Wx ∂t

∂W ∂ξ

143

) fe (x, ξ ,t) + hTe (x, ξ ,t)he (x, ξ ,t) + γ 2 φ T (x, ξ ,t)φ (x, ξ ,t) ≤

γ 2 (x − ξ )T {Z˙ ε−1 + Zε−1 Aε + ATε Zε−1 [

1 Pε B2 BT2 Pε − C2T C2 ]+ γ2

Zε−1 B1 BT1 Zε−1 }(t)(x − ξ ) + ot (x − ξ 2) = ε −εγ 2 (x − ξ )T Zε−2 (t)(x − ξ ) + ot (x − ξ 2) ≤ − γ 2 min Zε−1 (t)2 x − ξ 2 2 t∈R for almost all t ∈ R and x, x − ξ  sufficiently small. Thus, Hypothesis H2 holds locally with W (x, ξ ,t), G(t), and Q(x, ξ ) defined by (7.41), (7.43), and (7.44), respectively. Finally, applying Theorem 7.1, we conclude that the output feedback (7.15) which, due to (7.40)–(7.44), is given by (7.45), (7.46), represents a local solution of the H∞ -control problem. Theorem 7.2 is proved. Remark 7.1. The existence of a local solution of the nonlinear H∞ -control problem is thus ensured by Conditions C1 and C2. It is worth noticing that in the smooth case these conditions are known from [176] to become, under an appropriate assumption, not simply sufficient but also necessary for such a solution to exist. Indeed, let f2 = 0 and let there exist a local solution of the nonlinear H∞ -control problem such that L2 -gain of the corresponding closed-loop system (7.1)–(7.3) is less than γ and the linear approximation of the disturbance-free version of this system is exponentially stable. Then it follows that (7.32) is stabilizable and detectable from u and y, respectively. Moreover, by [229, Proposition 6] L2 -gain of the corresponding linearized system (7.32) is less than γ , too1 , and hence (see [190, Theorem 3.1]) Conditions C1 and C2 are satisfied.

7.1.4 Autonomous Case In the autonomous case where all the functions in (7.1)–(7.3) and (7.33) are timeindependent, the interest is typically focused on the design of a time-invariant controller. In this case, the differential Riccati equations (7.34), (7.36) degenerate to the algebraic Riccati equations by setting P˙ = 0, Z˙ = 0 and Conditions C1 and C2 are simplified to the following: C1’.

The equation PA + AT P + C1T C1 + P[

1

1 B1 BT1 − B2BT2 ]P = 0 γ2

(7.59)

Although in [229] this result is shown for time-invariant systems only, its extension to timevarying systems is straightforward. Hopefully, the result also admits generalization to the nonsmooth case where f 2 = 0.

7 Disturbance Attenuation via Nonsmooth H∞ -design

144

possesses a positive semidefinite symmetric solution P such that the matrix [A − (B2 BT2 − γ −2B1 BT1 )P] has all eigenvalues with negative real part; C2’. Being specified with A1 = A + γ12 B1 BT1 P, the equation A1 Z + ZAT1 + B1 BT1 + Z[

1 PB2 BT2 P − C2T C2 ]Z = 0, γ2

(7.60)

possesses a positive semidefinite symmetric solution Z such that the matrix [A1 − Z(C2T C2 − γ −2 PB2 BT2 P)] has all eigenvalues with negative real part. Conditions C1’ and C2’ are known from [62] to be necessary and sufficient for a solution of the linear H∞ -control problem for the time-invariant version of (7.32) to exist. According to the strict bounded real lemma (see, e.g., [7]), Conditions C1’ and C2’ ensure that there exists a positive constant ε0 such that the system of the perturbed algebraic Riccati equations 1 B1 BT1 − B2BT2 ]Pε + ε I = 0, γ2

(7.61)

1 Pε B2 BT2 Pε − C2T C2 ]Zε + ε I = 0 γ2

(7.62)

Pε A + AT Pε + C1T C1 + Pε [ Aε Zε + Zε ATε + B1 BT1 + Zε [

has a unique positive definite symmetric solution (Pε , Zε ) for each ε ∈ (0, ε0 ) where Aε = A + γ12 B1 BT1 Pε . Based on this solution a time-invariant nonsmooth H∞ controller is constructed as follows. Theorem 7.3. Let Conditions C1’ and C2’ be satisfied for (7.1)–(7.3) which is assumed to be time-invariant and let (Pε , Zε ) be a positive definite solution of (7.61), (7.62) under some ε > 0. Then, the time-invariant output feedback   1 ξ˙ = f (ξ ) + 2 g1 (ξ )gT1 (ξ ) − g2(ξ )gT2 (ξ ) Pε ξ + Zε C2T [y − h2(ξ )], (7.63) γ u = −gT2 (ξ )Pε ξ

(7.64)

is a local solution of the H∞ -control problem in the autonomous case. The proof of Theorem 7.3 is similar to that of Theorem 7.2, and it is therefore omitted.

7.2 Local H∞ -control Synthesis via Sampled-Data Measurements

145

7.2 Local H∞ -control Synthesis via Sampled-Data Measurements Let us assume now that sampled-data measurements are available only at time instants τ0 < τ1 < . . . so that System 7.1–7.3 is specified as follows x(t) ˙ = f (x(t),t) + g1 (x(t),t)w(t) + g2 (x(t),t)u(t) z(t) = h1 (x(t),t) + k12 (x(t),t)u(t) y(τ j ) = h2 (x(τ j ), τ j ) + k21(x(τ j ), τ j )w(τ j ), j = 0, 1, . . . .

(7.65) (7.66) (7.67)

The goal of this section is to design a dynamic sampled-data measurement feedback (SMF) controller, with internal state ξ ∈ Rq , which internally uniformly asymptotically stabilizes (7.65)–(7.67) around the origin (x, ξ ) = (0, 0) and which is such that the closed-loop system has L2 /l2 -gain less than γ . Given a real number γ > 0, it is said that the closed-loop system (7.65)–(7.67) has L2 /l2 -gain less than γ if the response z, resulting from w for initial state x(t0 ) = 0, ξ (t0 ) = 0, satisfies  t1 t0

 t1

z(t)2 dt < γ 2 [

t0

w(t)2 dt +

∑

τ j ∈[t0 ,t1 ]

  w(τ j )2 ]

(7.68)

for all t1 > t0 and all continuous functions w(t). The right-hand side in (7.68) should be viewed as a mixed L2 /l2 -norm on the uncertain signals affecting the system and the sampled-data measurements. The nonsmooth H∞ -control problem over sampled-data measurements is to find a globally admissible controller (7.5) such that L2 /l2 -gain of the closed-loop system (7.65)–(7.67) is less than γ . Since the H∞ -norm translated to the continuous and discrete time domains is nothing else than the L2 - and l2 -induced norms, respectively, the above stated problem is a natural generalization of the standard H∞ control problem. In turn, a local solution to the above problem is defined as follows. A locally admissible controller (7.5) is said to be a local solution of the nonsmooth H∞ -control problem if there exists a neighborhood U of the origin such that (7.68) is satisfied for all t1 > t0 and all piece-wise continuous functions w(t) for which the state trajectory of the closed-loop system (7.5), (7.65)–(7.67), starting from the initial point (x(t0 ), ξ (t0 )) = (0, 0), remains in U for all t ⊂ [t0 ,t1 ].

7.2.1 Main Result Under Assumptions A1–A5, which are still assumed to be in force, we derive a local solution to the problem in question. The solution to be proposed invokes the differential equations

7 Disturbance Attenuation via Nonsmooth H∞ -design

146

Z˙ ε = A˜ ε (t)Zε (t) + Zε (t)A˜ Tε (t) + B1(t)BT1 (t)

.

+γ −2 Zε (t)Pε (t)B2 (t)BT2 (t)Pε (t)Zε (t) + ε I

(7.69)

. x= [A˜ + γ −2ZPB2 BT2 P](t)x(t)

(7.70)

1 g1 (ξ ,t)gT1 (ξ ,t) − g2(ξ ,t)gT2 (ξ ,t)]Pε (t)ξ γ2

(7.71)

ξ = f (ξ ,t) + [ with jumps

Zε (τ j +) = Zε (τ j −)[I + C2T (τ j )C2 (τ j )Zε (τ j −)]−1

(7.72)

x(τ j +) = x(τ j −) − Z(τ j −)[I + C2T (τ j )C2 (τ j )Z(τ j −)]−1C2T (τ j )C2 (τ j )x(τ j −) (7.73) ξ (τ j +) = ζ j (1), j = 0, 1, ... (7.74) where ζ j (t) satisfies

ζ˙ j (t) = Zε (τ j −)[I + C2T (τ j )C2 (τ j )Zε (τ j −)t]−1C2T (τ j )× [y j − h2(ζ j (t), τ j )],

ζ j (0) = ξ (τ j −).

(7.75)

These jumps describe instantaneous changes in the dynamics of the H∞ -controller at the sampling time moments τ j , j = 0, 1, .. .. In order to state the main result of this section, Condition C2 is modified to the following: C2”. Being specified with ε = 0 and a uniformly bounded, positive semidefinite, symmetric solution Pε =0 (t) = P(t) of (7.34), System 7.69, 7.72 possesses a uniformly bounded, positive semidefinite, symmetric solution Z(t) such that (7.70), (7.73) is exponentially stable. The result, given below, shows how the H∞ -control problem is locally solved via a sampled-data measurement feedback. Theorem 7.4. Let Conditions C1 and C2” be satisfied. Then there exists ε0 > 0 such that for each ε ∈ (0, εo ), System 7.38, 7.69, 7.72 has a unique continuous from the left, uniformly bounded, positive definite, symmetric solution (Pε (t), Zε (t)) and a solution of the nonsmooth H∞ -control problem for (7.65)–(7.67) is given by (7.46), (7.71), (7.74), (7.75). The proof of Theorem 7.4 appears in Sect. 7.2.3. To this end, we note that the controller output values ξ (τ j −) from the right at the sampling time moments τ j , j = 0, 1, ... are determined according to (7.74) through the evolution of the auxiliary dynamic system (7.75). Particularly, when dealing with a linear observation (7.67) where h2 (x,t) = C2 (t)x, System 7.75 is analytically solved by means of

ζ j (t) = Zε (τ j −)[I + C2T (τ j )C2 (τ j )Zε (τ j −)t]−1[Zε−1 (τ j −)ξ (τ j −) + C2T (τ j )y(τ j )], and (7.74) is represented in the explicit form

7.2 Local H∞ -control Synthesis via Sampled-Data Measurements

147

ξ (τ j +) = Zε (τ j −)[I + C2T (τ j )C2 (τ j )Zε (τ j −)]−1 [Zε−1 (τ j −)ξ (τ j −) + C2T (τ j )y(τ j )] = ξ (τ j −) + Zε (τ j −)[I + C2T (τ j )C2 (τ j )Zε (τ j −)]−1C2T (τ j )[y j − C2 (τ j )ξ (τ j −)], similar to that for the variable Zε (cf. (7.72) ).

7.2.2 Conversion into the H∞ -control Synthesis via Continuous Measurements We demonstrate now how to form an equivalent continuous measurement feedback (CMF) H∞ -control problem, and then, based on the equivalence idea, we prove Theorem 7.4. To begin with, let us introduce the functions  ∞ 0, t ≤ 0 χ (t) = , v(t) = ∑ χ (t − τk ), 1, t > 0 k=1

α (t) = t + v(t), β (s) = inf {t : α (t) > s} ,  s − t, if β (s) = t ∈ D ψ (s) = v(β (s)), otherwise

 ∞ where D = τ j j=0 . For the convenience of the reader, plots of these functions are depicted in Fig. 7.1. It is clear, that the functions β (s) and ψ (s) have the piecewise continuous derivatives   0, if β (s) ∈ D 1, if β (s) ∈ D ˙ , ψ˙ (s) = , (7.76) β (s) = 1, otherwise 0, otherwise and

β (α (t)) = t for all t ∈ R1 ,

(7.77)

1

α (β (s)) = s for all s ∈ R such that β (s) ∈ /D   α (β (s)−) = inf s : β (s ) = β (s) for all s ∈ R1 such that β (s) ∈ D, α (β (s)+) = α (β (s)−) + 1 for all s ∈ R1 such that β (s) ∈ D.

(7.78)

Along with the SMF H∞ -control problem for the nonlinear system (7.65)–(7.67), we shall consider the CMF H∞ -control problem for the auxiliary system ·

x (s) = f (x(s), s) + g1 (x(s), s)w(s) + g2(x(s), s)u(s) z(s) = h1 (x(s), s) + k12 (x(s), s)u(s) y(s) = h2 (x(s), s) + k21 (x(s), s)w(s)

(7.79)

148

7 Disturbance Attenuation via Nonsmooth H∞ -design

Fig. 7.1 Plots of the functions v(t), α (t), β (s), and ψ (s)

where x ∈ Rn is the state vector, s ∈ R1 is the time variable, u ∈ Rm is the control input, w ∈ Rr is the unknown disturbance, z ∈ Rl is the unknown output to be controlled, y ∈ R p is the only available continuous-time measurement on the system, f (x, s) = f (x, β (s))β˙ (s), g2 (x, s) = g2 (x, β (s))β˙ (s), h2 (x, s) = h2 (x, β (s))ψ˙ (s),

g1 (x, s) = g1 (x, β (s))β˙ (s), h1 (x, s) = h1 (x, β (s))β˙ (s), k12 (x, s) = k12 (x, β (s))β˙ (s),

k21 (x, s) = k21 (x, β (s))ψ˙ (s).

(7.80)

As stated below, the latter problem is equivalent to the former one. Such an equivalence will subsequently allow us to deduce a local solution of the SMF H∞ -control problem from that of the CMF H∞ -control problem. Theorem 7.5. There exists a (local and/or exponentially stabilizing) solution of the SMF H∞ -control problem for the nonlinear system (7.65)–(7.67) if and only if there exists a (local and/or exponentially stabilizing) solution of the CMF H∞ -control problem for the nonlinear system (7.79) specified with (7.80). Moreover, if u(t) is a (local and/or exponentially stabilizing) solution of the former problem then u(s) = u(β (s)) is a (local and/or exponentially stabilizing) solution of the latter problem. Conversely, if u(s) is a (local and/or exponentially stabilizing) solution of the latter problem then u(t) = u(α (t)) is a (local and/or exponentially stabilizing) solution of the former problem.

7.2 Local H∞ -control Synthesis via Sampled-Data Measurements

149

Proof. Let x(s) be a trajectory of (7.79) driven by an admissible dynamic controller u(s) and subjected to an external disturbance w(s). Then it is straightforward to check that in accordance with (7.77) and (7.80), x(α (t)) is a trajectory of (7.65) driven by the admissible dynamic controller u(t) = u(α (t)) and subjected to the external disturbance w(t) = w(α (t)). By the same reasoning, if x(t) is a trajectory of (7.65) enforced by u(t), w(t), then x(β (s)) is a solution of (7.79) enforced by u(s) = u(β (s)), w(s) = w(β (s)). It follows that a (local and/or exponentially stabilizing) solution u(t) of the SMF H∞ -control problem for (7.65)–(7.67) generates the (local and/or exponentially stabilizing) solution u(s) = u(β (s)) of the CMF H∞ control problem for (7.79), whereas a (local and/or exponentially stabilizing) solution u(s) of the latter problem generates the (local and/or exponentially stabilizing) solution u(t) = u(α (t)) of the former problem. Theorem 7.5 is thus proved. We conclude this section with providing sufficient conditions for a local solution of the H∞ -control problem for the auxiliary system (7.79) to exist. These conditions are as follows: ˜ C1. The equation .

− P (s) = P(s)A(β (s))β˙ (s) + AT (β (s))P(s)β˙ (s) + C1T (β (s))C1 (β (s))β˙ (s) +P(s)[

1 B1 (β (s))BT1 (β (s)) − B2 (β (s))BT2 (β (s))]P(s)β˙ (s) γ2

(7.81)

possesses a uniformly bounded, positive semidefinite, symmetric solution P(s) such that the system .

x (s) = {A(β (s)) − [B2(β (s))BT2 (β (s)) −γ −2 B1 (β (s))BT (β (s))]P(s)}β˙ (s)x(s) 1

(7.82)

is exponentially stable; ˜ C2. Being specified with A(s) = A(β (s)) + γ12 B1 (β (s))BT1 (β (s))P(s), the equation .

Z (s) = A(s)Z(s)β˙ (s) + Z(s)AT (t)β˙ (s) + B1 (β (s))BT1 (β (s))β˙ (s)+ Z(s)[

1 P(s)B2 (β (s))BT2 (β (s))P(s)β˙ (s) − C2T (β (s))C2 (β (s))ψ˙ (s)]Z(s) (7.83) γ2

possesses a uniformly bounded, positive semidefinite, symmetric solution Z(s) such that the system .

x (s) = {A(s)β˙ (s) − Z(s)[C2T (β (s))C2 (β (s))ψ˙ (s) −γ −2 P(s)B2 (β (s))BT2 (β (s))P(s)β˙ (s)]}x(s) is exponentially stable. The corresponding result is stated below in the form of lemma.

(7.84)

7 Disturbance Attenuation via Nonsmooth H∞ -design

150

˜ and C2 ˜ be satisfied. Then there exists εo > 0 such Lemma 7.1. Let Conditions C1 that system .

− Pε (s) = Pε (s)A(β (s))β˙ (s) + AT (β (s))Pε (s)β˙ (s) + C1T (β (s))C1 (β (s))β˙ (s) +Pε (s)[

1 B1 BT1 − B2 BT2 ](β (s))Pε (s)β˙ (s) + ε I, γ2

(7.85)

.

Zε (s) = A(s)Zε (s)β˙ (s) + Zε (s)AT (s)β˙ (s) + B1 (β (s))BT1 (β (s))β˙ (s)+ 1 Pε (s)B2 (β (s))BT2 (β (s))Pε (s)β˙ (s) − C2T (β (s))C2 (β (s))ψ˙ (s)]Zε (s) + ε I γ2 (7.86) has a unique, uniformly bounded, positive definite, symmetric solution (Pε (s), Zε (s)) for each ε ∈ (0, εo ) and a solution of the H∞ -control problem for the auxiliary system (7.79), (7.80) is given by Zε (s)[

.

ξ (s) = f (ξ , s) + [

1 g1 (ξ , s)gT1 (ξ , s) − g2(ξ , s)gT2 (ξ , s)]Pε (s)ξ γ2

+Zε (s)C2T (β (s))ψ˙ (s)[y(s) − h2 (ξ , s)],

(7.87)

u = −gT2 (ξ , s)Pε (s)ξ (s). The proof of Lemma 7.1 follows the same line of reasoning as that of Theorems 7.1 and 7.2, and it is therefore omitted. The above lemma is subsequently used in proving Theorem 7.4. Remark 7.2. Remark 7.1 applies to (7.79), (7.80) as well.

7.2.3 Proof of the Main Result In order to prove Theorem 7.4, we first demonstrate that Conditions C1 and C2” are ˜ and C2, ˜ respectively. equivalent to C1 Indeed, if P(s), Z(s) are solutions of (7.81), (7.83), then it is straightforward to check that in continuity intervals of α (t) the functions P(t) = P(α (t)), Z(t) = Z(α (t)) satisfy (7.38), (7.69) subject to ε = 0. Moreover, for s ∈ [α (τ j −), α (τ j +)] with τ j ∈ D and α (τ j +) = α (τ j −) + 1, we have that β˙ (s) = 0, ψ˙ (s) = 1, the following equations holds . P (s) = 0,

.

Z (s) = −Z(s)C2T (τ j )C2 (τ j )Z(s) and the functions

P(s) = P(α (τ j −)),

(7.88)

7.2 Local H∞ -control Synthesis via Sampled-Data Measurements

Z(s) = Z(τ j −)[I + C2T (τ j )C2 (τ j )Z(τ j −)s]−1

151

(7.89)

are solutions of (7.88). Thus, in spite of discontinuities in α (t), the function P(α (t)) is continuous for all t, thereby satisfying (7.38) for all t, whereas the jumps Z(α (τ j +)) − Z(α (τ j −)) = Z(τ j −)[I + C2T (τ j )C2 (τ j )Z(τ j −)]−1 −Z(τ j −),

j = 0, 1, ...

of the function Z(α (t)) are the same as those of (7.69), (7.72) and therefore Z(α (t)) is a solution of the differential equation (7.69) with jumps (7.72). Furthermore, if (P(t), Z(t)) is a solution of (7.38), (7.69), (7.72) subject to ε ≡ 0 then by inspecting, one proves that the pair P(s) = P(β (s)) Z(s) = Z(β (s)) satisfies ˜ C2, ˜ and C1, (7.81), (7.83). In order to conclude the equivalence of conditions C1, C2”, it remains to note that the same relations are also in force between solutions of (7.35), (7.70), (7.73) and (7.82), (7.84). By applying Theorem 7.5 and Lemma 7.1, it follows that conditions C1, C2” are sufficient for a local solution of the SMF H∞ -control problem for (7.65)–(7.67) to ˜ C2 ˜ are satisfied exist. Moreover, if these conditions are satisfied, the conditions C1, as well, and due to Lemma 7.1, there exists ε0 > 0 such that System 7.85, 7.86 has a unique uniformly bounded, positive definite, symmetric solution (Pε (s), Zε (s)) for each ε ∈ (0, εo ) . By Lemma 7.1, these functions generate the unique uniformly bounded, positive definite, symmetric solution (Pε (t) = Pε (α (t)), Zε (t) = Zε (α (t))) of (7.38), (7.69), (7.72) (the uniqueness is guaranteed by the invertibility of the time substitution in the above relations: Pε (s) = Pε (β (s)), Zε (s) = Zε (β (s))). Thus, by Theorem 7.5, the CMF solution (7.87), given by Lemma 7.1, generates the solution u(t) = u(α (t)) = −gT2 (! ξ (α (t)), α (t))Pε (α (t))ξ (α (t)) = −gT2 (ξ (α (t)),t)Pε (t)ξ (α (t)) of the SMF H∞ -control problem in question. To complete the proof, let us demonstrate that the function ξ (t) = ξ (α (t)) satisfies the differential equation (7.71) with jumps (7.74), (7.75). In continuity intervals of α (t) we have ψ˙ (s) = 0 and by inspection ξ (α (t)) is a solution of (7.71). If s ∈ [α (τ j −), α (τ j +)] where τ j ∈ D and α (τ j +) = α (τ j −) + 1, then the relations β˙ (s) = 0, ψ˙ (s) = 1 and (7.89) are in force and, consequently, .

ξ (s) = Zε (τ j −)[I + C2T (τ j )C2 (τ j )Zε (τ j −)s]−1C2T (τ j )× [y(τ j ) − h2(ξ (s), τ j )],

α (τ j −) ≤ s ≤ α (τ j −) + 1.

Thus, the jumps ξ (α (τ j +))− ξ (α (τ j −)), j = 0, 1, ... of the function ξ (t) = ξ (α (t)) are the same as those defined by (7.74), (7.75) and therefore ξ (α (t)) satisfies (7.71), (7.74), (7.75) for all t. Theorem 7.4 is proved.

7 Disturbance Attenuation via Nonsmooth H∞ -design

152

Remark 7.3. By Theorem 7.5, coupled to Remark 7.2, Remark 7.1 applies to (7.65)– (7.67) as well.

7.3 Unified SMF and CMF H∞ -control Synthesis As shown in Sect. 7.2, the SMF H∞ -control problem, rewritten in a new time variable obtained through a certain time substitution (the absolutely continuous, noninvertible substitution t = β (s)), becomes a CMF H∞ -control problem, and vice versa (the discontinuous time substitution s = α (t) is employed for the corresponding CMF H∞ -problem conversion). The distribution formalism in nonlinear setting of Sect. 2.1 is subsequently utilized to make such a connection between the SMF and CMF H∞ -control problems transparent.

7.3.1 Distribution Formalism Let us demonstrate that the SMF H∞ -controller equations (7.69)–(7.75) with jumps can be represented in the form .

"ε (t)Zε (t) + Zε (t)A "T (t) + B1(t)BT1 (t) + ε I Zε = A ε +Zε (t)[

1 Pε B2 BT2 Pε ](t)Zε (t) − Zε (t)C2T (t)C2 (t)Zε (t)ν (t), Zε (t0 ) = Zε0 γ2 .

ξ = f (ξ ,t) + [

(7.90)

1 g1 (ξ ,t)gT1 (ξ ,t) − g2(ξ ,t)gT2 (ξ ,t)]Pε (t)ξ γ2

+Zε (t)C2T (t)[y(t) − h2(ξ ,t)]ν (t), ξ (t0 ) = ξ 0

(7.91)

.

x= [A˜ ε = 0 + Z γ −2 PB2 BT2 P)](t)x(t) − Z(t)C2T (t)C2 (t)x(t)ν (t), x(t0 ) = x0 (7.92) of vibroimpact nonlinear differential equations in distributions. Due to Theorem 2.3, vibroimpact solutions of (7.90)–(7.92) are well-posed for arbitrary initial conditions, and satisfy the differential equations (7.69), (7.71), (7.70) for all t but the sampling time moments, while their values at t = τ j , j = 0, 1... are found through the relations Zε (τ j +) = μ j (1),

ξ (τ j +) = ζ j (1),

x(τ j +) = λ j (1)

(7.93)

μ j (0) = Zε (τ j −),

(7.94)

ζ j (0) = ξ (τ j −).

(7.95)

by solving the auxiliary differential equations .

μ j (t) = −μ j (t)C2T (τ j )C2 (τ j )μ j (t), ζ˙ j (t) = μ j (t)C2T (τ j )[y(τ j ) − h2(ζ j (t), τ j )],

7.3 Unified SMF and CMF H∞ -control Synthesis .

153

T

λ j (t) = −μ j (t)C2 (τ j )C2 (τ j )λ j (t),

λ j (0) = x(τ j −).

(7.96)

Since (7.94) and (7.96) integrate to

μ j (t) = Zε (τ j −)[I + C2T (τ j )C2 (τ j )Zε (τ j −)t]−1 , λ j (t) = {I − Z(τ j −)[I + C2T (τ j )C2 (τ j )Z(τ j −)t]−1C2T (τ j )C2 (τ j )t}x(τ j −), Relations 7.93–7.95 result in (7.72), (7.74), (7.75). So, the vibroimpact solutions of (7.90), (7.91) with the impulsive input

ν (t) =

∞

∑ δ (t − τ j )

(7.97)

j=0

coincide with the corresponding solutions of the differential equations (7.69), (7.71), specified with the same initial conditions, and with the jumps, given by (7.72), (7.74), (7.75). In other words, the output feedback (7.46) with the external state ξ (t), being a vibroimpact solution of (7.91), coupled to (7.90), (7.97), represents another form of the local solution of the SMF H∞ -control problem, whereas condition C2” is reformulated as follows: ˜ C2”. Being specified with ε = 0 and a uniformly bounded, positive semidefinite, symmetric solution Pε =0 (t) = P(t) of (7.34), System 7.90 possesses a uniformly bounded, positive semidefinite, symmetric vibroimpact solution Z(t) such that the nonlinear differential equation (7.92) in distributions is exponentially stable. Furthermore, substituting an arbitrary continuous strictly positive function σ (t) for ν (t) in the SMF H∞ -controller equations (7.46), (7.90)–(7.92), we arrive at the standard H∞ -controller equations for the nonlinear system (7.65), (7.66) with the continuous-time measurements y(t) = h2 (x(t),t) + k21 (x(t),t)w(t) subject to T k21 (x,t)k21 (x,t) =

1 I, σ (t)

(7.98)

(7.99)

whereas the last restriction in (7.4) is no longer in force. These H∞ -controller equations for the nonlinear system (7.65), (7.66) with the continuous-time measurements (7.98) subject to (7.99) can still be derived by applying Theorem 7.2 because (7.99) is readily simplified to the standard form of (7.4) by absorbing σ (t) into g2 (x,t) and k21 (x,t). Conversely, the SMF H∞ -controller equations in distributions are straightforwardly obtained from the aforementioned CMF H∞ -controller equations by substituting (7.97) for the function σ (t) in the latter equations. Hence, the SMF H∞ -controller thus constructed can be interpreted as a limiting result of the CMF H∞ -controllers under a weak* approximation of Distribution (7.97) by continuous strictly positive peaking functions σ (t). Indeed, Definition 2.3 of vibroimpact solutions admits such an interpretation of the SMF H∞ -controller

7 Disturbance Attenuation via Nonsmooth H∞ -design

154

equations in distributions. This interpretation, in turn, establishes the robustness of the H∞ -control synthesis to signal processing, regardless either sampled-data measurements or continuous measurements, made for a short time period, are accepted as an observation model.

7.3.2 The H∞ -design Procedure The following CMF H∞ -design procedure is proposed. According to this procedure, one should use the iteration on γ to compute H∞ -suboptimal solution as γ approaches the infimal achievable level γ ∗ in (2.11). By Theorem 7.2, γ ∗ is the infimum over all γ for which conditions C1 and C2 are in force. Thus, on the first step I. γ large enough is selected such that Conditions C1 and C2 are satisfied. II. Then, by iterating on γ (sub-) optimal value γ ∗ is determined, such that (7.6) is satisfied for all γ > γ ∗ . III. After that, a uniformly bounded, positive definite, symmetric solution of the perturbed system (7.38), (7.39) with γ = γ ∗ and ε > 0 small enough is constructed. IV. Finally, Controller 7.45, 7.46, which corresponds to γ and ε , thus selected, completes the locally (sub-) optimal H∞ -control synthesis of the nonlinear timevarying system (7.1)–(7.3). As a matter of fact, the distribution formalism of Sect. 7.3.1 allows one to extend the above CMF H∞ -control design procedure to both SMF and mixed CMF/SMF H∞ -controller designs.

7.3.3 Illustrative Example: The H∞ -stabilization of an Inverted Pendulum via Nonlinear Sampled-Data Measurements In order to illustrate the H∞ -controller performance a simulation study is conducted for an inverted pendulum, moving in the vertical plane (see Fig. 7.2). The pendulum is driven by an actuator u, using position measurements only. The measurements, which are available at the discrete time moments τ j = 0.5 j, j = 0, 1, ..., are made by a nonlinear potentiometer with the standard resistance function sin θ . Our aim is to design a SMF H∞ -suboptimal controller of the inverted pendulum, whose dimensionless model x˙1 = x2 .

x2 = sin(x1 ) + w1 + u z1 = u, z2 = x1 y j = sin x1 (τ j ) + w2 (τ j ),

j = 0, 1, ....

(7.100)

7.3 Unified SMF and CMF H∞ -control Synthesis

155

Fig. 7.2 Inverted pendulum

is given in terms of the angular position x1 = θ and the angular velocity x2 = θ˙ . According to Theorem 7.4, a local solution of the SMF H∞ -control problem for (7.100) is constructed as follows:    ˙       1 00 00 ξ1 ξ2 ξ1 }P − + { = , ε 01 sin(ξ1 ) ξ2 ξ˙2 γ2 0 1

ξi (τ j +) = ςi j (1),

i = 1, 2, j = 0, 1, ...   # $ ξ u = − 0 1 Pε 1 ξ2

where ςi j , Pε and Zε satisfy the equations       10 ς˙1 j 10 Z (τ −)t}−1 × + = Zε (τ j −){ 00 ε j 01 ς˙2 j       1 ς1 j (0) ξ (τ −) (y(τ j ) − sin(v1 )); = 1 j j = 0, 1, ... 0 ς2 j (0) ξ2 (τ j −) T     01 01 10 + Pε + Pε + 10 10 00       1 00 00 10 }Pε + ε − =0 Pε { 2 01 01 γ 01   . 00 Zε = A˜ ε Zε (t) + Zε (t)A˜ Tε + 01     1 00 10 Pε Zε (t) + ε + 2 Zε (t)Pε 01 01 γ

(7.101)

(7.102)



(7.103)

156

7 Disturbance Attenuation via Nonsmooth H∞ -design



Zε (τ j +) = Zε (τ j −){ with

   10 10 Z (τ −)}−1 , + 00 ε j 01

(7.104)

    1 00 01 ˜ Pε + 2 Aε = 10 γ 01

and γ such that (7.103), (7.104), corresponding to ε = 0, have uniformly bounded, positive semidefinite, symmetric solutions, such that the system       1 00 00 01 ˙ − 2 )P}X˙P −( XP = { 01 10 γ 01   1 00 P}XZ X˙Z = {A˜ + 2 ZP 01 γ       10 10 10 Z(τ j −)}−1 X (τ −) + XZ (τ j +) = XZ (τ j −) − Z(τ j −){ 00 00 Z j 01 is exponentially stable. In the simulation, performed with MATLAB , using the H∞ -design procedure yields γ ∗ ≈ 5.32. However, in the subsequent simulation γ = 10 is utilized to avoid an undesired high-gain controller design corresponding to a value of γ close to the optimum. For the Riccati equations (7.103)–(7.104), corresponding to this value of γ and ε = 0.1, we find the uniformly bounded, positive definite, symmetric solutions   3.261145 2.470038 Pε = 2.470038 2.256321 and Zε (t), the minors of which are presented in Fig. 7.3. These solutions result in the control law (7.101) which solves the problem in question. Two cases for the controller, with no disturbances and with permanent disturbances w1 = 0.08 and w2 = 0.07 are simulated. The initial conditions for the simulations are set to x1 (0) = 20◦ , x2 (0) = 0, ξ1 (0) = ξ2 (0) = 0. The resulting control inputs and trajectories are depicted in Figs. 7.4 and 7.5. From Fig. 7.4 we conclude that, in spite of the large position deviation x1 (0) = 20◦ from the desired one, the pendulum moves reasonably fast initially towards its endpoint; however, the final positioning requires more time. Fig. 7.5 shows that good system performance is still provided while the permanent disturbances enforce the system.

7.3 Unified SMF and CMF H∞ -control Synthesis

Fig. 7.3 Plot of minors of Zε (t): (a) the first order minor and (b) the second order minor

Fig. 7.4 The no-disturbance case: plots of (a) the angular position and (b) the control input

157

158

7 Disturbance Attenuation via Nonsmooth H∞ -design

Fig. 7.5 The permanent disturbance case: plots of (a) the angular position and (b) the control input

Part III

Unit Feedback Control of Infinite-dimensional Systems

160

III Unit Feedback Control of Infinite-dimensional Systems

Many important plants, such as flexible manipulators and structures, as well as heat transfer processes, combustion, and fluid mechanical systems, are governed by functional and partial differential equations or, more generally, differential equations in a Hilbert space. As these systems are often described by models with a significant degree of uncertainty, it is of interest to develop consistent control methods that are capable of utilizing distributed parameter and time-delay models, and providing the desired system performance in spite of significant uniformly bounded model uncertainties. The unit feedback control of finite-dimensional systems has been shown to guarantee a certain degree of robustness with respect to uniformly bounded unmodeled dynamics. First applications [31, 159, 173] of these algorithms to distributed parameter systems corroborated their utility for infinite-dimensional systems as well, and motivated further theoretical investigations [126, 127, 162, 171, 174, 175, 179, 180, 253]. The unit feedback synthesis is now extended to dynamic systems, evolving in a Hilbert space. The synthesis retains robustness features, similar to those possessed by its counterpart in the finite-dimensional case. Although the present development is reminiscent of the finite-dimensional design, it does not represent a simple extension of the finite-dimensional unit control algorithm to the infinite-dimensional case. Indeed, the discontinuous unit signal in the sliding mode control law is constructed to make the time derivative of the appropriate Lyapunov function positive definite. This is attained by setting the magnitude of the control signal to exceed the norm of the right-hand side of the plant equation without the control term. Such a construction, while being quite natural for the finite-dimensional case, becomes invalid in the infinite-dimensional case, due to the typical presence of unbounded operators in the plant equations. The effectiveness of the proposed synthesis in the infinite-dimensional setting is illustrated by applications to distributed parameter and time-delay systems.

Chapter 8

Global Asymptotic Stabilization of Uncertain Linear Systems

The unit feedback synthesis is developed for a class of linear infinite-dimensional systems with a finite-dimensional unstable part using finite-dimensional sensing and actuation. The present development is essentially from [180]. Initially, the class of infinite-dimensional systems is precisely formulated. Modal decomposition is then used to decompose the original infinite-dimensional system into an interconnection of a finite-dimensional (possibly unstable) system and an infinite-dimensional stable system. A stabilizing unit state feedback controller is constructed on the basis of the finite-dimensional system. Subsequently, an infinitedimensional Luenberger state observer, which utilizes a finite number of measurements, is constructed to provide estimates of the state of the infinite-dimensional system. Finally, an output feedback controller design is completed by coupling the infinite-dimensional Luenberger state observer and the unit state feedback controller. In order to obtain the fully practical finite-dimensional framework for controller synthesis, a finite-dimensional approximation of the Luenberger observer as well as a continuous approximation of the unit feedback controller are carried out at the implementation stage. The implementation, performance, and robustness issues of the unit output feedback control design are illustrated in a simulation study of a distributed parameter system governed by the linearization of the Kuramoto–Sivashinsky equation (KSE) around the spatially-uniform steady-state solution with periodic boundary conditions. While being unforced, the KSE describes incipient instabilities in a variety of physical and chemical systems and a control problem that occurs here is to avoid the appearance of the instabilities in the closed-loop system [49] (see also [48, 233] for other results on the control of distributed parameter systems). We also note several recent works on the control of the KSE including boundary control [136], as well as methods for robust [101] and adaptive [113] stabilization.

161

162

8 Global Asymptotic Stabilization of Uncertain Linear Systems

8.1 Preliminaries We consider infinite-dimensional systems of the form x˙ = Ax + Bu

(8.1)

y = Cx,

(8.2)

defined in a Hilbert space H, equipped with the inner product ·, · and the corresponding norm  · . Hereinafter, x ∈ H is the state, u ∈ Rm is the control input, y ∈ R p is the measured output, A is an infinitesimal operator with a dense domain D(A) which generates a C0 (strongly continuous) semigroup TA (t), B ∈ L (Rm , H) and C ∈ L (H, R p ), and the symbol L (U, H) stands for the set of linear bounded operators from a Hilbert space U to another Hilbert space H. Let the control input u(x) undergo discontinuities on the hyperplane

σ x = 0, σ ∈ L (H, Rm )

(8.3)

and let u(x) be continuously differentiable beyond this hyperplane. Then given an initial condition x(0) = x0 ∈ D(A), (8.4) the meaning of (8.1) is defined in the generalized sense of Definition 2.10. The trajectories of (8.1) are thus defined in the classical sense whenever they are beyond the discontinuity hyperplane (8.3) whereas by Theorem 2.4 sliding motions along the discontinuity hyperplane, if any, are governed by the so-called sliding mode equation x˙ = [A − B(σ B)−1σ A]x, (8.5) which is obtained by substituting the equivalent control value ueq (x) = −(σ B)−1 σ Ax

(8.6)

into (8.1) for u. Being inherited from the finite-dimensional case, several stability concepts for the resulting closed-loop system are given below. Suppose xu (t, x0 ) denotes a generalized solution of (8.1), initialized with (8.4) and driven by a discontinuous feedback u. Definition 8.1. A closed-loop system (8.1) is said to be stable iff for each ε > 0 there exists δ > 0 such that for each solution xu (t, x0 ) with the initial condition (8.4), satisfying x0  < δ , the inequality xu (t, x0 ) < ε

(8.7)

holds for all t ≥ 0. Definition 8.2. A closed-loop system (8.1) is said to be globally asymptotically (exponentially) stable iff it is stable and given an arbitrary initial condition (8.4), so-

8.1 Preliminaries

163

lution xu (t, x0 ) of (8.1), (8.4) is such that xu (t, x0 ) → 0 as t → ∞ (respectively, xu (t, x0 ) ≤ ω x0 e−α t for all t ≥ 0 and some positive ω , α ). For later use, we recall the definition of the generator of an exponentially stable semigroup as well as that of the exponential stabilizability/detectability (see, e.g., [55] for details). ˜ iff Definition 8.3. The operator A˜ generates an exponentially stable semigroup S(t) 0 ˜ ˜ the initial value problem x˙ = Ax, x(0) = x has a unique solution x(t) = S(t)x0 , and ˜ S(t) ≤ ω e−α t for all t ≥ 0 and some positive ω , α . Definition 8.4. System 8.1, 8.2, or equivalently, the pair {A, B} is said to be exponentially stabilizable if there exists D ∈ L (H, Rm ) such that the operator A + BD generates an exponentially stable semigroup. Definition 8.5. System 8.1, 8.2, or equivalently, the pair {A,C} is said to be exponentially detectable if there exists L ∈ L (R p , H) such that the operator A + LC generates an exponentially stable semigroup. Throughout this section, the following assumption is assumed. 1. System 8.1, 8.2 is exponentially stabilizable and detectable. Assumption 1 implies that System 8.1, 8.2 can be exponentially stabilized via linear dynamic output feedback control of the form: u(x) ˜ = Dx˜ x˙˜ = Ax˜ + Bu(x) ˜ − L(y − Cx) ˜

(8.8) (8.9)

where the internal state x˜ ∈ H, and the operators D and L satisfy the condition of the stabilizability/detectability definitions. Moreover, under the assumption of exponential stabilizability and detectability, H can be decomposed as H = H1 ⊕ H2 , and the state equation (8.1) can be rewritten in terms of x1 ∈ H1 and x2 ∈ H2 , as follows [55, Sect. 5.2]: x˙1 = A1 x1 + B1 u

(8.10)

x˙2 = A2 x2 + B2 u y = C1 x1 + C2 x2

(8.11) (8.12)

where H1 = Rn is a finite-dimensional subspace, A j = A|H j , and C j = C|H j are the operator restrictions on H j , j = 1, 2, B j = Pj B and Pj is the projector on H j . From Assumption 1, the matrix pairs {A1 , B1 } and {A1 ,C1 } are controllable and observable, respectively, and the operator A2 = A|H2 generates an exponentially stable semigroup SA2 (t) on H2 . The decomposition (8.10), (8.11) of the original system (8.1) into a finitedimensional, possibly unstable part (8.10) and an infinite-dimensional stable part

164

8 Global Asymptotic Stabilization of Uncertain Linear Systems

(8.11) has long been recognized as providing the framework for the design of stabilizing controllers for linear infinite-dimensional systems (see the notes and references quoted in [55, Chap. 5]). Using Decomposition 8.10, 8.11, the synthesis of dynamic output feedback control laws is obtained as follows: u(x˜1 ) = D1 x˜1 x˙˜1 = A1 x˜1 + B1 u(x˜1 ) − L1 (y − C1 x˜1 − C2 x˜2 ) x˜˙2 = A2 x˜2 + B2u(x˜1 )

(8.13) (8.14) (8.15)

where D1 = D|H1 , L1 = P1 L are such that the n × n- matrices A1 + B1 D1 and A1 + L1C1 are Hurwitz. Although the finite-dimensional state estimate x˜1 (t) is only required in the control law (8.13), the Luenberger observer design is carried out in the infinite-dimensional setting (8.14), (8.15) to avoid the appearance of the destabilizing spill-over effect in the closed-loop system (see also proof of Theorem 8.3). The implementation of the above observer can then be performed by obtaining a finite-dimensional approximation of the x˜2 -subsystem of a sufficiently high order. It is worth noting that there is a body of literature focusing on the construction of the associated Luenberger observers for stabilization of infinite-dimensional systems (see, e.g., [55, Chap. 5] and references therein). Such a treatment, based on a finite-dimensional approximation of the original infinite-dimensional system, becomes attractive in practice if instabilities due to spill-over effect can be prevented; it additionally ensures strong robustness properties against unmodeled dynamics and parameter variations. In order to enhance performance, the approximation-based treatment will subsequently be accompanied by the unit output feedback synthesis, potentially capable of providing this desired robustness feature. Remark 8.1. We note that under Assumption 1, the eigenspectrum of the operator A of (8.1) could have a hyperbolic-like or parabolic-like structure. As an example, in Fig. 8.1, we show the eigenspectrum of the linearized Korteweg–de Vries–Burgers (KdVB) equation (left plot), which is a third-order PDE with a hyperbolic-like spectrum and that of the linearized KSE (right plot), which is a fourth-order PDE with a parabolic-like spectrum. Specifically, the linearized KdVB equation takes the form 3

2

∂x ∂t

= − ∂∂ z3x + a1 ∂∂ z2x

∂ jx ∂zj

(0,t) =

∂ jx ∂zj

(8.16)

(+π ,t), j = 0, . . . , 2

and the linearized KSE takes the form 4

2

∂x ∂t

= −ν ∂∂ z4x − ∂∂ z2x

∂ jx ∂zj

(−π ,t) =

∂ jx ∂zj

(8.17)

(+π ,t), j = 0, . . . , 3.

Both the linearized KdVB and KS partial differential equations belong in the class of linear infinite-dimensional systems considered in this work and satisfy Assumption

8.2 State Feedback Design

165

1 (see [8] for results on finite-dimensional control design for the KdVB equation and the KSE). Im

Im

8000 6000 4000

Re

2000 −100

−80

−60

−40

−20

0 Re 0 −2000

−35.2

−7.2

0.8

−4000 −6000 −8000

Fig. 8.1 Eigenspectrum of the linearized KdVB equation with a1 = 0.05 (left plot) and that of the linearized KSE with ν = 0.2 (right plot)

8.2 State Feedback Design Our aim is to demonstrate how the approximation-based approach can be used to globally exponentially stabilize the infinite-dimensional system (8.1), (8.2) by means of discontinuous output feedback control. First, a discontinuous state feedback controller is constructed, and then the Luenberger observer (8.14), (8.15) of the unstable modes is involved to synthesize a stabilizing output feedback law. The stabilizing discontinuous state feedback controller to be developed in the present section is based on the deliberate introduction of sliding modes into the finite-dimensional subsystem (8.10). Taking into account that the infinite-dimensional subsystem (8.11) is stable and does not require to be stabilized, the discontinuous state feedback, stabilizing subsystem (8.10), is subsequently shown to simultaneously stabilize the infinite-dimensional closed-loop system (8.10), (8.11). Thus, the discontinuous stabilization of the infinite-dimensional system (8.10), (8.11) is reduced to the finite-dimensional treatment of Chap. 6. Recall that the stabilization problem for the finite-dimensional system (8.10) is solved in the space of the new state variable

ξ = Mx1 ∈ Rn

(8.18)

where M ∈ Rn×n is a non-singular matrix such that MB1 = (0, B12 )T

(8.19)

166

8 Global Asymptotic Stabilization of Uncertain Linear Systems

with a non-singular matrix B12 ∈ Rm×m . In order to satisfy (8.19), the first n − m rows of M are composed of n − m linearly independent vectors, orthogonal to the control subspace, whereas BT1 forms the remaining m rows of M so that the matrix B12 = BT1 B1 is nonsingular. System 8.10, rewritten in terms of the components ξ1 = P11 ξ ∈ Rn−m , ξ2 = P12 ξ ∈ Rm of the new state variable ξ = (ξ1 , ξ2 )T , is given by

ξ˙1 = A11 ξ1 + A12ξ2 ξ˙2 = A21 ξ1 + A22ξ2 + B12u

(8.20) (8.21)

where P11 and P12 are the projectors on the subspaces spanned by the first n − m state components and the remaining m components, respectively. By construction, the matrix pair {A1 , B1 } is controllable, because the pair {A, B}, otherwise, would not be exponentially stabilizable. Due to the lemma of [227, p.101], it follows that the matrix pair {A11 , A12 } is also controllable. Then, by an appropriate choice of K ∈ Rm×(n−m) , the matrix A11 + A12 K can be made Hurwitz with an a priori fixed location of the eigenvalues. In what follows, we demonstrate that the same unit control law u(ξ ) = −[γ0 + γ1 ξ ]B−1 12 U(σ ξ )

(8.22)

γ1 > A21 − KA11 + A22 − KA12 , σ ξ = ξ2 − K ξ1 ,

(8.23) (8.24)

with nonnegative γ0 ,

and U(σ ξ ) =

σξ σ ξ 

(8.25)

that stabilizes the finite-dimensional system (8.10) (cf. (6.30)–(6.33)), globally stabilizes (asymptotically when γ0 > 0 or exponentially when γ0 = 0) the infinitedimensional system (8.1), too. Theorem 8.1. Consider an infinite-dimensional system (8.1) for which Assumption 1 holds and let the unit control signal (8.22)–(8.25) be constructed via the finitedimensional approximation (8.10) of the system and its equivalent representation (8.18)–(8.21). Then, given an initial condition x(0) = x0 ∈ D(A), the closed-loop system (8.1), (8.22)–(8.25), has a unique solution, which is globally defined for all t ≥ 0, and this system is globally asymptotically stable when γ0 > 0 and it is globally exponentially stable when γ0 = 0. Proof. The proof is brought up into three simple steps. 1) By Theorem 6.1, a unique solution of the finite-dimensional system (8.20)– (8.25) exists for all initial conditions and this solution is globally defined for all t ≥ 0. While this system remains beyond the discontinuity hyperplane σ ξ = 0, the control function u(t), computed according to (8.22) on the trajectories of (8.20),

8.2 State Feedback Design

167

(8.21), is continuously differentiable in t. Thus, before the solution of the finitedimensional system (8.20)–(8.25) hits the discontinuity hyperplane, the infinitedimensional system (8.11), driven by this control signal, satisfies [188, p.109, Corollary 2.10]. By this corollary, System 8.11, 8.22 has a unique solution under arbitrary initial conditions (8.4) at least till a time instant when System 8.20–8.25 enters the hyperplane σ ξ = 0. As shown in the proof of Theorem 6.1, any solution of the finite-dimensional system (8.20)–(8.25), entering the discontinuity hyperplane in a finite time T > 0, never leaves it. In order to describe the behavior of the infinite-dimensional system (8.11), (8.22) on the hyperplane σ ξ = 0, one can use the infinite-dimensional extension of the equivalent control method. According to this extension, the equivalent control value ueq = B−1 (8.26) 12 (KA11 + KA12 K − A21 − A22 K)ξ1 , a unique solution of the equation σ˙ (ξ (t)) = 0 with respect to u, is substituted into (8.11) for u. By Theorem 2.4, the equation x˙2 = A2 x2 + B2 B−1 12 (KA11 + KA12 K − A21 − A22 K)ξ1 , t ≥ T,

(8.27)

thus obtained, describes the motion of the infinite-dimensional subsystem (8.11) while the finite-dimensional subsystem (8.10) evolves in the sliding mode on the discontinuity hyperplane σ ξ = 0. Apparently, the equivalent control function (8.26), computed along the sliding modes to be governed by (cf. (6.35))

ξ˙1 = (A11 + A12K)ξ1 ,

(8.28)

is continuously differentiable in t, and hence, the infinite-dimensional sliding mode equation (8.27) satisfies [188, p. 109, Corollary 2.10], too. By applying this corollary to (8.27), we conclude that the solutions of the infinite-dimensional part (8.11), (8.22) of the closed-loop system (8.1), (8.22)–(8.25) are globally continuable on the right, even if the finite-dimensional system enters the discontinuity hyperplane σ ξ = 0 in a finite time. 2) If γ0 > 0 then, as shown in the proof of Theorem 6.1, after the finite time T ≤ γ0−1 σ ξ 0  the finite-dimensional system (8.20)–(8.25) evolves in the sliding mode on the hyperplane σ ξ = 0. While being governed on the time interval [0, T ) by the differential equations (8.11), (8.20), (8.21), subject to the smooth control input (8.22), the initial stage of the trajectory x(t), before it hits the discontinuity hyperplane σ ξ = 0, continuously depends on the initial data x0 . In turn, for t ≥ T the closed-loop system (8.1), (8.22)–(8.25) is maintained on the hyperplane σ ξ = 0 and it is governed by the sliding mode equations (8.27), (8.28). Due to a special choice of K, the solution of the former equation is exponentially decaying ξ1 (t) ≤ ω ξ1 (T )e−β (t−T ) , t ≥ T (8.29) with some ω > 0 and β > 0. Since the semigroup SA2 (t) generated by the infinitesimal operator A2 is exponentially stable, i.e., SA2 (t − T ) ≤ ω0 e−α (t−T ) for all t ≥ T

168

8 Global Asymptotic Stabilization of Uncertain Linear Systems

and some α , ω0 > 0, the solution of the latter equation, given by  t T

x2 (t) = SA2 (t − T )x2 (T ) + SA2 (t − T − τ )B2 B−1 12 (KA11 + KA12 K − A21 − A22 K)ξ1 (τ )d τ ,

(8.30)

is estimated as follows x2 (t) ≤ ω0 x2 (T )e−α (t−T ) + κ ξ1(T )[e−β (t−T) + e−α (t−T ) ], t ≥ T (8.31) where

κ=

ω0 ω e α T B2 B−1 12 (KA11 + KA12 K − A21 − A22 K) > 0 |α − β |

and Inequality (8.29) has been used. By taking into account (8.29) and employing that σ ξ = 0, or equivalently, ξ2 (t) = K ξ1 (t) for t ≥ T , it follows that x(t) ≤ ω1 x(T )e−ν1 (t−T ) , t ≥ T

(8.32)

for some ω1 > 0 and ν1 > 0. Coupled to the afore-mentioned property of the initial stage of the solution to continuously depend on the initial data, the exponential decay (8.32) after the finite time moment T ensures the global asymptotic stability of the closed-loop system (8.1), (8.22)–(8.25) with γ0 > 0. 3) If γ0 = 0 then apart from the exponential decay (8.32) on the discontinuity hyperplane σ ξ = 0, a similar state estimate takes place at the initial stage preceding a sliding motion on this hyperplane. Indeed, by Theorem 6.1, the finite-dimensional component ξ (t) of the closed-loop system (8.1), (8.22)–(8.25) with γ0 = 0 is globally exponentially stable, i.e., ξ (t) ≤ L0 ξ 0 e−μ t

(8.33)

for some L0 , μ > 0, whereas the solution x2 (t) = SA2 (t)x2 (0) − γ1

 t 0

ξ (τ )SA2 (t − τ )B2 B−1 12

σ ξ (τ ) dτ , σ ξ (τ )

(8.34)

of the infinite-dimensional part (8.11), before ξ (t) hits the hyperplane σ ξ = 0, is estimated as follows: x2 (t) ≤ ω0 x2 (0)e−α t + κ1 ξ 0 [e−μ t + e−α t ], 0 ≤ t < T where

κ1 =

(8.35)

L0 ω0 γ1 B2 B−1 12  > 0, |α − μ |

and (8.33) has been used. By taking into account (8.33), it follows that x(t) ≤ ω2 x0 e−ν2t , 0 ≤ t < T

(8.36)

8.2 State Feedback Design

169

for some ω2 > 0 and ν2 > 0. Since (8.36), coupled to (8.32), ensures that x(t) ≤ Ω x0 e−ν t , t ≥ 0

(8.37)

for Ω = max{ω1 , ω2 , ω1 ω2 } > 0 and ν = min{ν1 , ν2 } > 0, regardless of whether the closed-loop system enters the discontinuity hyperplane σ ξ = 0 in a finite time T < ∞ or it never hits this hyperplane, the global exponential stability of (8.1), (8.22)– (8.25) with γ0 = 0 is guaranteed. Thus, the proof of Theorem 8.1 is completed. Remark 8.2. As in the finite-dimensional case (see Remark 6.1), the non-vanishing matched disturbances are rejected by the unit controller (8.22)–(8.25) with γ0 exceeding a norm bound of these disturbances. In Sect. 8.4, this conclusion will be supported by numerical simulations. The implementation of the discontinuous unit feedback (8.25) can be performed through its continuous approximation U δ (σ ξ ), e.g., U δ (σ ξ ) =

σξ , σ ξ  + δ 3

(8.38)

such that condition uδ (x) − u(x) ≤ δ f or all x ∈ H sub ject to σ x ≥ δ

(8.39)

holds with u(x) = σσ ξξ  and uδ (x) = U δ (σ ξ ). Undesirable high frequency state oscillations that would be excited by fast switching in the discontinuous controller are thus avoided, and the resulting closed-loop system, corresponding to a sufficiently small δ , can be shown to be practically stable (i.e., all the trajectories converge to an ε -vicinity of the equilibrium solution). Summarizing, we arrive at the following. Theorem 8.2. Let a continuous approximation (8.38) of the discontinuous unit feedback signal (8.25) be substituted into the control law (8.22) for U(σ x). Then, given an initial condition (8.4) and sufficiently small δ > 0, the closed-loop system (8.1), (8.22)–(8.24), (8.38) has a unique solution xδ (t), globally defined for all t ≥ 0. Moreover, this system is stable, and for arbitrary ε > 0 there exist δ0 (ε ) and T0 (x0 , ε ) > 0 such that xδ (t) < ε f or all t ≥ T0 (x0 , ε ) and δ ∈ (0, δ0 ).

(8.40)

Proof. By Theorem 8.1 (8.1), (8.4), driven by the discontinuous controller (8.22)– (8.25), has a unique solution x(t), globally defined for all t ≥ 0, and x(t) → 0 as t → ∞. The latter guarantees that for arbitrary ε > 0 there exists T0 (x0 , ε ) > 0 such that ε x(t) ≤ f or all t ≥ T0 (x0 , ε ). (8.41) 2 In turn, the smoothed system (8.1), (8.4), driven by the continuous, piece-wise smooth controller (8.22)–(8.24), (8.38), is well known [188] to have a unique solution xδ (t), locally defined on some time interval (0, τ ). Then Theorem 8.1 and

170

8 Global Asymptotic Stabilization of Uncertain Linear Systems

Definition 2.10, coupled together, ensure that x(t) − xδ (t) ≤

ε f or all t ∈ (0, τ ), δ ∈ (0, δ0 ), 2

(8.42)

and for some δ0 (ε ) > 0. Thus, the solutions xδ (t) are uniformly bounded on (0, τ ) for all δ ∈ (0, δ0 ), and these solutions are therefore uniquely continuable on the right so that (8.42) remains in force for t ≥ τ . By virtue of (8.41), it follows the stability of the closed-loop system (8.1), (8.22)–(8.24), (8.38) and validity of (8.40). Theorem 8.2 is thus proved.

8.3 Output Feedback Design In this section, we proceed with the design of the discontinuous output feedback controller by coupling the discontinuous state feedback law (8.22) with the Luenberger observer (8.14), (8.15). For this purpose, we first represent (8.22) in terms of the state component x1 : u(x1 ) = −[γ0 + γ1 Mx1 ]B−1 12

(P12 − KP11)Mx1 . (P12 − KP11)Mx1 

(8.43)

Then, we substitute x1 by the observer output x˜1 in (8.43) and modify the gain γ0 + γ1 M x˜1  by adding the error term γ2 y − Cx ˜ with

γ2 > (P12 − KP11)L1 .

(8.44)

The reason for the gain modification is to ensure that the observer-based controller is capable of steering the trajectories of the closed-loop system to the discontinuity hyperplane (P12 − KP11 )M x˜1 = 0 in spite of the observer error, thereby imposing on the system a desired stability property similar to that obtained under the state feedback controller. By Theorem 8.3, stated below, the resulting discontinuous output feedback control law, which yields a proper solution to the stabilization problem in question, takes the form u(y, x) ˜ = −[γ0 + γ1 M x˜1  + γ2y − Cx]B ˜ −1 12

(P12 − KP11)M x˜1 . (P12 − KP11)M x˜1 

Theorem 8.3. Let the infinite-dimensional system (8.1) for which Assumption 1 holds be driven by the observer-based dynamic output feedback controller (8.14), (8.15), (8.45) with gain parameters γ0 , γ1 , γ2 , satisfying (8.23) and (8.44), respectively. Then, given initial conditions x(0) = x0 ∈ D(A), x(0) ˜ = x˜0 ∈ D(A), the closed-loop system (8.1), (8.14), (8.15), (8.45) has a unique solution, globally defined for all t ≥ 0, and this system is globally asymptotically stable when γ0 > 0 and it is globally exponentially stable when γ0 = 0. Proof. The proof parallels that of Theorem 8.1.

8.3 Output Feedback Design

171

1) First, let us represent the over-all system (8.1), (8.14), (8.15), (8.45) in terms of the observer error e = (e1 , e2 )T , ei = xi − x˜i , i = 1, 2, and the observer state components ξ˜ = (ξ˜1 , ξ˜2 )T , ξ˜i = P1i M x˜1 and x˜2 : e˙1 = (A1 + L1C1 )e1 + L1C2 e2 e˙2 = A2 e2 ξ˙˜1 = A11 ξ˜1 + A12ξ˜2 − P11L1Ce ξ˙˜2 = A21 ξ˜1 + A22ξ˜2 + B12 u(y, ξ˜ , x˜2 ) − P12L1Ce x˙˜2 = A2 x˜2 + B2u(y, ξ˜ , x˜2 )

(8.45) (8.46) (8.47) (8.48)

where

σ ξ˜ u(y, ξ˜ , x˜2 ) = −[γ0 + γ1 ξ˜  + γ2y − C1M −1 ξ˜ − C2 x˜2 ]B−1 12 σ ξ˜ 

(8.49)

and σ ξ˜ = ξ˜2 − K ξ˜1 . Since the semigroup SA2 (t) generated by the infinitesimal operator A2 is exponentially stable and A1 + L1C1 is a Hurwitz matrix by construction, the observer error equation (8.45) has a unique globally defined solution under arbitrary initial conditions e1 (0) = e01 ∈ Rn , e2 (0) = e02 ∈ D(A) and this solution is given by e1 (t) =

e(A1 +L1C1 )t e01 +

t 0

e(A1 +L1C1 )(t−τ ) L1C2 SA2 (τ )e02 d τ

e2 (t) = SA2 (t)e02 , t ≥ 0.

(8.50) (8.51)

The following observer error estimate e(t) ≤ κ0 e(0)e−β0t , t ≥ 0, i = 1, 2,

(8.52)

is then straightforwardly obtained for some positive κ0 , β0 . It follows that (8.46)–(8.49) arre enforced by the exponentially decaying inputs P11 L1Ce(t) and P12 L1Ce(t). While Subsystem (8.46)–(8.49) remains beyond the discontinuity hyperplane σ ξ˜ = 0, the control function u(t), computed according to (8.49) on the trajectories of (8.46)–(8.48), is continuously differentiable in t. Thus, before the subsystem hits the discontinuity hyperplane, it satisfies [188, p.109, Corollary 2.10]. By this corollary, Subsystem (8.46)–(8.49) has a unique solution under arbitrary initial conditions at least till a time instant when the subsystem enters the hyperplane σ ξ˜ = 0. It should be noted that if (8.46)–(8.49) enter the discontinuity hyperplane, then it never leaves it, because the time derivative of the quadratic function V (σ ξ˜ ) = (σ ξ˜ )T σ ξ˜ remains negative on the trajectories of the system beyond this hyperplane:

(8.53)

172

8 Global Asymptotic Stabilization of Uncertain Linear Systems

σ ξ˜ V˙ (t) =≤ −2(σ ξ˜ )T (γ0 + γ ξ˜  + γ˜e) σ ξ˜ 

(8.54)

where constants γ = γ1 − A21 − KA11  − A22 − KA12  and γ˜ = (γ2 − (P12 − KP11 )L1 )C are positive by virtue of (8.23), (8.44). In order to describe the behavior of (8.46)–(8.49) on the hyperplane σ ξ˜ = 0, one can use the infinite-dimensional extension of the equivalent control method. According to this extension, the equivalent control value ˜ ueq = B−1 12 [(KA11 + KA12 K − A21 − A22 K)ξ1 + (P12 − KP11 )L1Ce],

(8.55)

a unique solution of the equation σ˙ (ξ˜ (t)) = 0 with respect to u, is substituted into (8.47), (8.48) for u. By Theorem 2.4, the equations

ξ˙˜1 = (A11 + KA12)ξ˜1 − P11L1Ce,

(8.56)

˜ ˜˙x2 = A2 x˜2 + B2B−1 12 (KA11 + KA12 K − A21 − A22 K)ξ1 +B2 B−1 12 (P12 − KP11 )L1Ce,

(8.57)

thus obtained, describe the motion of (8.46)–(8.49) after a time instant when the subsystem hits the discontinuity hyperplane σ ξ˜ = 0. While being governed by the linear equation (8.56) subject to (8.52), the finitedimensional component ξ˜ (t) of the sliding motion is apparently uniformly bounded on an arbitrary finite time interval and globally continuable on the right. The same is also true for a motion, initialized beyond the discontinuity hyperplane. Indeed, beyond this hyperplane, the right-hand side of (8.46), (8.47), (8.49) is linearly bounded in the state variable so that ξ˜ (t) ≤ ξ˜ 0  +

t 0

[γ0 + N1 ξ˜ (τ ) + N2 e(τ )]d τ

(8.58)

for some constants N1 , N2 > 0. Taking into account (8.52), it follows that

κ0 N2 0 ξ˜ (t) ≤ [ξ˜ 0  + e  + γ0T ] + N1 β0

 t 0

ξ˜ (τ )d τ

(8.59)

By applying the Bellman–Gronwall lemma to (8.59), any solution of this equation is uniformly bounded

κ0 N2 0 ξ˜ (t) ≤ [ξ˜ 0  + e  + γ0T ]eN1 T β0

(8.60)

on an arbitrary time interval [0, T ) before possible getting on the discontinuity hyperplane where it has been shown to remain uniformly bounded on any finite time interval. Thus, given arbitrary initial conditions e0 ∈ D(A), ξ˜ 0 ∈ Rn , there exists a

8.3 Output Feedback Design

173

unique solution of (8.45)–(8.47), (8.49), which turns out to be globally continuable on the right. Moreover, the equivalent control function (8.55), computed along the sliding modes (8.56), is continuously differentiable in t, and hence, the infinite-dimensional sliding mode equation (8.57) satisfies [188, p. 109, Corollary 2.10]. By applying this corollary to (8.57), one concludes that the solutions of the infinite-dimensional part (8.48), (8.49) of the closed-loop system (8.45)–(8.49) are unambiguously globally continuable on the right, even if the finite-dimensional system enters the discontinuity hyperplane σ ξ˜ = 0 in a finite time. Since (8.45)–(8.49) is nothing else than a representation of the original closedloop system (8.1), (8.14), (8.15), (8.45) in terms of the observer error and observer state, a solution of the original system is globally uniquely defined, too. 2) If γ0 > 0, then the time derivative estimate  (8.61) V˙ (t) ≤ −2γ0 σ ξ  = −2γ0 V (t), of the quadratic function (8.53) is guaranteed by (8.54). By Lemma 4.3, this estimate ensures that V (t) = 0 for all t ≥ γ0−1 σ ξ˜ 0, i.e., after a finite time moment T ≤ γ0−1 σ ξ˜ 0  the finite-dimensional system (8.46), (8.47), (8.49) evolves in the sliding mode on the hyperplane σ ξ˜ = 0. While being governed on the time interval t ∈ [0, T ) by the differential equations (8.45)–(8.48) subject to the smooth control input (8.49), the initial stage of the trajectory of the system, before it hits the discontinuity hyperplane σ ξ˜ = 0, continuously depends on the initial conditions. In turn, for t ≥ T the closed-loop system (8.45)–(8.49) is maintained on the hyperplane σ ξ˜ = 0 and it is governed by the sliding mode equations (8.56), (8.57). Taking into account the observer error estimate (8.52) and employing that A11 + KA12 is a Hurwitz matrix due to a special choice of K, the solution of the former equation is proved to exponentially decay with some ω > 0 and β > 0: ξ˜1 (t) ≤ ω ξ˜1 (T )e−β (t−T ) , t ≥ T.

(8.62)

Since the semigroup SA2 (t) generated by the infinitesimal operator A2 is exponentially stable, i.e., SA2 (t − T ) ≤ ω0 e−α (t−T ) for all t ≥ T and some α , ω0 > 0, the solution of the latter equation, given by  t  t T

T

x˜2 (t) = SA2 (t − T )x˜2 (T ) + SA2 (t − T − τ )B2 B−1 12 (P12 − KP11 )L1Ce(τ )d τ +

SA2 (t − T − τ )B2 B−1 12 (KA11 + KA12 K − A21 − A22 K)ξ1 (τ )d τ ,

(8.63)

is estimated as follows: x˜2 (t) ≤ ω0 x˜2 (T )e−α (t−T ) + κ˜ e(T )[e−β0 (t−T ) + e−α (t−T ) ] +

κ ξ1 (T )[e−β (t−T ) + e−α (t−T ) ], t ≥ T

(8.64)

174

8 Global Asymptotic Stabilization of Uncertain Linear Systems

where

κ˜ = κ=

ω0 κ 0 e α T B2 B−1 12 (P12 − KP11 )L1C > 0, |α − β0|

ω0 ω e α T B2 B−1 12 (KA11 + KA12 K − A21 − A22 K) > 0, |α − β |

and (8.52), (8.62) have been used. By taking into account (8.52) and (8.62), and employing that σ ξ˜ = 0, or equivalently, ξ˜2 (t) = K ξ˜1 (t) for t ≥ T , it follows that in the case when γ0 > 0 the closed-loop system (8.45)–(8.49) is exponentially decaying for t ≥ T . Coupled to the afore-mentioned property of the initial stage of the solution to continuously depend on the initial conditions, this ensures the global asymptotic stability of (8.45)–(8.49) as well as that of the original closed-loop system (8.1), (8.14), (8.15), (8.45) whenever γ0 > 0. 3) If γ0 = 0 then apart from the exponential decaying of the closed-loop system (8.45)–(8.49) on the discontinuity hyperplane σ ξ˜ = 0, this is also true at the initial stage preceding a sliding motion on this hyperplane. Indeed, by Lemma 6.1, the finite-dimensional component ξ˜ (t) of (8.45)–(8.49) with γ0 = 0 is globally exponentially stable, i.e., ξ˜ (t) ≤ L0 ξ˜ (0)e−μ t , t ≥ 0 (8.65) for some μ > 0 and L0 > 0, whereas the solution  t 0

x˜2 (t) = SA2 (t)x˜2 (0) − σ ξ˜ (τ ) [γ1 ξ˜ (τ ) + γ2 Ce(τ )]SA2 (t − τ )B2 B−1 dτ , 0 ≤ t < T 12 σ ξ˜ (τ )

(8.66)

of the infinite-dimensional part (8.15), before the time instant T when ξ˜ (t) hits the hyperplane σ ξ˜ = 0, is estimated as follows: x˜2 (t) ≤ ω0 x˜2 (0)e−α t + κ1 ξ˜ (0)[e−μ t + e−α t ] +

κ2 e(0)[e−β0t + e−α t ] 0 ≤ t < T where

κ1 =

(8.67)

L0 ω0 γ1 κ0 ω0 γ2 B2 B−1 B2 B−1 12  > 0, κ2 = 12 C > 0 |α − μ | |α − β0 |

and (8.52) and (8.65) have been used. As in the proof of Theorem 8.1, (8.64) and (8.67), coupled together, ensure that the infinite-dimensional part (8.15) is also exponentially stable, regardless of the closed-loop system (8.45)–(8.49) enters the discontinuity hyperplane σ ξ˜ = 0 in a finite time T < ∞ or it never hits this hyperplane. Thus, the global exponential stability of (8.45)–(8.49) and consequently that of the original closed-loop system (8.1), (8.14), (8.15), (8.45) is guaranteed whenever γ0 = 0. This completes the proof of Theorem 8.3. Remark 8.3. As in the finite-dimensional case (see Remark 6.2), the observer-based controller (8.45) is robust against vanishing disturbances only, because the observer

8.4 Applications to the Linearized Kuramoto–Sivashinsky Equation

175

model (8.14), (8.15) is capable of tracking the state of the system up to a certain estimation error, which decreases as the disturbance magnitude decreases. By analogy to the state feedback design, the implementation of the discontinuous output feedback (8.49) can be performed through its continuous version u(y, ξ˜ , x˜2 ) = −[γ0 + γ1 ξ˜  + γ2y − C1M −1 ξ˜ − C2 x˜2 ]B−1 12

σ ξ˜ . (8.68) σ ξ˜  + δ 3

As opposed to the discontinuous feedback (8.49), its continuous approximation (8.68) does not excite undesirable high frequency state oscillations, whereas the resulting closed-loop system is practically stabilized in the following sense. Theorem 8.4. Let (8.1) and (8.14), (8.15) be enforced by a continuous approximation (8.68) of the discontinuous dynamic output feedback (8.49) subject to (8.23), (8.44). Then, given initial conditions x(0) = x0 ∈ D(A), x˜1 (0) = x˜01 ∈ Rn , x˜2 (0) = x˜02 ∈ D(A) and sufficiently small δ > 0, the closed-loop system (8.1), (8.14), (8.15), (8.49) has a unique solution xδ (t), x˜δ1 (t), x˜δ2 (t), which is globally defined for all t ≥ 0. Moreover, this system is stable, and for arbitrary ε > 0 there exist δ0 (ε ) > 0 and T0 (x0 , x˜01 , x˜02 , ε ) > 0 such that xδ (t), x˜δ1 (t), x˜δ2 (t) < ε f or all t ≥ T0 (x0 , x˜01 , x˜02 , ε ) and δ ∈ (0, δ0 ). (8.69) Proof. The proof is similar to that of Theorem 8.2, and the details of the proof remain to the reader.

8.4 Applications to the Linearized Kuramoto–Sivashinsky Equation The proposed discontinuous output feedback controller design is additionally supported by applications to the linearization of the one-dimensional KSE around its spatially-uniform steady state. The KSE describes incipient instabilities in a variety of physical and chemical systems, and a control problem that occurs here is to avoid the appearance of the instabilities in the closed-loop system [8]. In order to smooth the response of the system the unit controller is approximated by its continuous δ −counterparts, (8.38) in the state feedback case and (8.68) in the output feedback case. The objectives of the numerical simulations are to illustrate that: (1) the proposed δ −unit output feedback controller is able to practically stabilize the linearized KSE at the spatially-uniform steady state solution with good robustness with respect to significant uncertainty in the instability parameter, ν , and (2) in the presence of non-vanishing external disturbance, the proposed δ −unit controller is superior to a linear feedback controller in terms of closed-loop performance. The two points are demonstrated in Sects. 8.4.1 and 8.4.2, respectively. The linearized KSE with external disturbance is of the form

176

8 Global Asymptotic Stabilization of Uncertain Linear Systems ∂x ∂t

4

2

= −ν ∂∂ z4x − ∂∂ z2x + ∑m i=1 bi ui (t) + d(z,t)

yκ =

π

−π sκ (z)xdz, κ = 1, . . . , p

(8.70)

subject to the periodic boundary conditions

∂ jx ∂ jx (−π ,t) = j (+π ,t) , j = 0, . . . , 3 j ∂z ∂z

(8.71)

and the initial condition x(z, 0) = x0 (z)

(8.72)

where x(z,t) is the state of the KSE, z ∈ (−π , π ) is the spatial coordinate, t is the time and 2π is the length of the spatial domain, ν is the instability parameter, x0 (z) is the initial condition. ui ∈ R denotes the i-th manipulated input, m is the total number of manipulated inputs, bi (z) ∈ L2 (−π , π ) is a square integrable function that represents the distribution function of the i−th control actuator, p is the total number of measurement sensors, yκ ∈ R denotes a measured output, sκ (z) ∈ L2 (−π , π ) is a square integrable function of z which is determined by the location and type of the measurement sensors, and d(z,t) is an unknown uniformly bounded external disturbance to be rejected. To simplify the presentation of our numerical results, we formulate (8.70)–(8.72) as an infinite-dimensional system in the Hilbert space of odd functions with spatial zero mean in the interval (−π , π ). This requires that the initial condition x0 (z), the actuator distribution functions bi (z), i = 1, . . . , m, and the external disturbance d(z,t) are odd functions in the spatial variable z (this assumption imposes certain practical constraints on the shape of the actuator distribution functions, which will be discussed below in Sect. 8.4.1). Under this assumption, we formulate (8.70)– (8.72) as an infinite-dimensional system in the Hilbert space Lodd 2 (−π , π ). To this end, we introduce the operator Ax = −ν

∂ 4x ∂ 2x − ∂ z4 ∂ z2

(8.73)

with the dense domain

 D(A) = x ∈ Lodd 2 (−π , π ) : ∂ jx ∂zj

(−π ) =

∂ jx ∂zj

∂ 4x ∂ z4

(z) ∈ Lodd 2 (−π , π ), %

(8.74)

(+π ), j = 0, . . . , 3

and the input and measured output operators Bu = ∑li=1 bi ui ,

Cx =

π

−π S (z)x(z)dz

(8.75)

where S (z) = (s1 (z), . . . , s p (z))T . Then, (8.70)–(8.72) can be written as follows:

8.4 Applications to the Linearized Kuramoto–Sivashinsky Equation

x˙ = Ax + Bu + d,

x(0) = x0

y = Cx.

177

(8.76)

m odd in the Hilbert space Lodd 2 (−π , π ) where the operators B : R → L2 (−π , π ) and odd p C : L2 (−π , π ) → R are bounded, and A is an unbounded operator, generating an analytical semigroup [188]. In order to decompose System 8.76 according to (8.10), (8.11) into stable and unstable parts, we formulate the following eigenvalue problem for A: 4

2

Aφn = −ν ∂∂ zφ4n − ∂∂ zφ2n = λn φn , n = 1, . . . , ∞

(8.77)

∂ j φn ∂ j φn (−π ) = (+π ), j = 0, . . . , 3 j ∂z ∂zj

(8.78)

subject to

where λn denotes an eigenvalue and φn denotes an odd eigenfunction. A direct computation of the solution of the above eigenvalue problem yields λn = −ν n4 + n2 with odd eigenfunctions φn (z) = √1π sin(nz), n = 1, . . . , ∞. The spectrum {λ1 , λ2 , . . . , } of A is defined as the set of all eigenvalues of A. We note that the fact that A has a real pure point spectrum is a result of the fact that the spatial differential operator of the linearization of KSE with periodic boundary conditions is self-adjoint and the problem is considered in a bounded domain. From the expression of the eigenvalues, it follows that, for a fixed value of ν > 0, the number of unstable eigenvalues of A is finite and the distance between two consecutive eigenvalues (i.e., λn and λn+1 ) increases as n increases. From these properties of the eigenspectrum of A, it follows that A generates an analytic semigroup [55, 224]. By expanding the solution of (8.70) in an infinite series in terms of the odd eigenfunctions of (8.77), we obtain x(z,t) =

∞

∑ αn (t)φn (z)

(8.79)

n=1

√ where αn (t) is a time-varying coefficient and φn (z) = (1/ π ) sin(nz). Substituting the above expansion for the solution x(z,t) into the system and taking the inner product in Lodd 2 (−π , π ) with the eigenfunction φn (z), the following infinite-dimensional system of ordinary differential equations is obtained: n n α˙n = (−ν n4 + n2 )αn + ∑m i=1 bi ui (t) + d (t), n = 1, . . . , ∞,

yκ = where and

π

∞ −π sκ (z)(∑n=1 αn (t)φn (z))dz, κ = 1, . . . , p

bni =

π

−π bi (z)φn (z)dz

(8.80)

(8.81)

178

8 Global Asymptotic Stabilization of Uncertain Linear Systems

d n (t) =

π

−π d(z,t)φn (z)dz.

(8.82)

All subsequent simulation runs are performed for ν = 0.2, using a thirtieth-order linear ordinary differential equation model obtained from the application of modal decomposition to (8.70) (the use of higher-order approximations led to identical numerical results, thereby implying that the following simulation runs are independent of the discretization).

8.4.1 Numerical Results in the Disturbance-free Case In this section, we demonstrate that the proposed unit feedback controller is capable of stabilizing the spatially-uniform steady state of the linearized KSE when there is no external disturbance (i.e., d(z,t) ≡ 0). We observe that (8.80) possesses two unstable eigenvalues, and thus for ν = 0.2, the spatially uniform steady-state x(z,t) = 0 is unstable. Therefore, we consider the first two modes of the linearization of KSE as the slow modes and use modal decomposition to construct a second-order ODE system. Two actuators are used to guarantee the exponential stabilizability of the system. At this point, it is important to note that our choice to consider the linearized KSE as an infinite-dimensional system in the Hilbert space of odd functions with spatial zero mean in order to simplify the presentation of the controller synthesis formulas imposes certain practical constraints on the shape of the actuator distribution functions; it requires, for example, that bi (z) are odd functions, e.g., bi (z) = φi (z), i = 1, 2. While it is possible to proceed with a such choice for bi (z), from a practical point of view as well as to study the effect of the control action on the fast modes (not accounted in the model used for controller design, i.e., the “spillover” effect), it is preferable to consider the use of point control actuators. Therefore, we assume that we have available for the control of the KSE two point control actuators, which are placed at the optimal locations za1 = 0.31π and za2 = 0.69π , found in [137], with the following actuator distribution functions:

bi (z) =

⎧ 1 ε ε ⎨ εb i f zai − 2b < z < zai + 2b ⎩

, i = 1, . . . , m.

(8.83)

0 i f otherwise

where εb is a constant representing the width of control actuation in the spatial interval (−π , π ) (note that the above selection of bi (z) ensures that bi (z) ∈ L2 (−π , π )). We use εb = 0.1 in all closed-loop simulations. The second-order system resulting from the application of modal decomposition to the linearized KSE is of the form       1 1   b b u1 α˙1 λ 0 α1 = 1 + 12 22 (8.84) 0 λ2 α˙2 α2 u2 b1 b2

8.4 Applications to the Linearized Kuramoto–Sivashinsky Equation

179

1.5 1

x

0.5 0 −0.5 −1 −1.5 0 1

3 2 2

1 3 4

Time

−2

−3

0

−1

z

5 u1 u 2

0

u

−5

−10

−15

−20

−25

0

0.5

1

1.5

2

2.5

3

3.5

4

Time

Fig. 8.2 Closed-loop spatio-temporal profile of x(z,t) (top figure) and manipulated input profiles (bottom figure) under the δ −unit state feedback controller

where bni (i, n = 1, 2) are computed by using (8.81). The first simulation run is performed to evaluate the ability of the unit state feedback controller to stabilize (8.70). The continuous approximation of the unit state feedback controller (8.22), (8.38) takes the form   u1 ξ = − (γ0 + γ1 ξ ) B−1 (8.85) 12 ξ +δc3 u2 T    1 1 T  1 1  α1 b1 b2 b11 b12 b1 b2 and B . The controller parame= 12 α2 b21 b22 b21 b22 b21 b22 ters used in the simulation are γ0 = 5.0, γ1 = 5.0, and δc3 = 0.005. 

where ξ =

180

8 Global Asymptotic Stabilization of Uncertain Linear Systems

The system is assumed to be at the initial condition x0 = 1.35 sin(z) ∈ Lodd 2 (−π , π ).

(8.86)

The closed-loop spatio-temporal profile of x(z,t) and the profiles of the two manipulated inputs are shown in Fig. 8.2. It is clear that the controller stabilizes the state of the system at x(z,t) = 0. The second simulation run is performed to evaluate the ability of the unit output feedback controller to stabilize (8.70). Two point measurement sensors are used in this simulation to guarantee the exponential detectability of the system. Following [137], the measurement sensors are optimally placed at zs1 = 0.35π and zs2 = 0.64π . Based on the two measurements, [ y1 y2 ]T , and the thirtieth-order approximation of the infinite-dimensional system (8.80), the state observer is of the form       1 1   b b u1 α˙˜ 1 λ1 0 α˜ 1 = + 12 22 0 λ2 α˜ 2 u2 α˜˙ 2 b1 b2 ⎛







α˜ ⎜ y −L1 ⎝ 1 − C1 1 y2 α˜ 2 ⎤ ⎡ α˙˜ 3 λ3 · · · ⎢ .. ⎥ ⎢ .. . . ⎣ . ⎦=⎣ . . 0 ··· α˙˜ 30 ⎡

where



⎡

⎤⎞ α˜ 3 ⎢ ⎥⎟ − C2 ⎣ ... ⎦⎠ α˜ 30

⎤⎡ ⎤ ⎡ 3 0 α˜ 3 b1 .. ⎥ ⎢ .. ⎥ + ⎢ .. . ⎦⎣ . ⎦ ⎣ . λ30 α˜ 30 b30 1

(8.87)

⎤ b32   .. ⎥ u1 . ⎦ u2 30 b2

 c11 c21 C1 = 1 2 , c2 c2   3 c · · · c30 1 , C2 = 13 c2 · · · c30 2 

(8.88) (8.89)

cnκ , (κ = 1, 2, n = 1, . . . , 30) is given by cnκ =

π

−π sκ (z)φn (z)dz,

the function sκ (z) is as follows: ⎧1 ε ε ⎨ εs i f zsκ − 2s < z < zsκ + 2s , sκ (z) = ⎩ 0 i f otherwise

(8.90)

(8.91)

zsκ is the location of the κ -th measurement sensor and εs is a constant representing the width of the distributed sensing in the spatial interval (−π , π ). We use εs = 0.1 in all closed-loop simulations, and L1 is set to be

8.4 Applications to the Linearized Kuramoto–Sivashinsky Equation

 L1 = −10

 2 −1

c11 c1 c12 c22

181

.

(8.92)

The observer outputs are used to compute the control action via the following unit feedback controller (based on (8.49) and (8.68)):        u1     ξ˜ = − γ0 + γ1 ξ˜  + γ2 ζ˜  B−1 (8.93) 12 ξ˜ +δ 3 u2 c where

ξ˜ =



b11 b21

T b12 b22







α˜ 1 , α˜ 2

and



α˜ y ζ˜ = 1 − C1 1 y2 α˜ 2 

B12 =



b11 b12 b21 b22

T 



⎡

⎤ α˜ 3 ⎢ ⎥ − C2 ⎣ ... ⎦ , α˜ 30

 b1α 1 b1α 2 . b2α 1 b2α 2

Controller 8.93 is specified with the parameters γ0 = 5.0, γ1 = 5.0, γ2 = 5.0, and δc3 = 0.005. The initial condition for the system is the same to that in (8.86) and the initial condition for the state observer is given by 0.1 x˜0 = 1.0 sin(z) + √ π

30

∑ sin(nz) ∈ Lodd 2 (−π , π ).

(8.94)

n=2

The closed-loop spatio-temporal profile of x(z,t) and the profiles of the two manipulated inputs are shown in Figure 8.3. It is clear that the proposed unit output feedback controller stabilizes the state of the system at x(z,t) = 0. The third simulation run is performed to test the robustness of the output feedback control law (8.87)–(8.93), for a 25% decrease in the value of the instability parameter (i.e., ν was taken to be equal to 0.15 in the high-order discretization of the linearization of KSE but it is used as 0.2 in the controller and state observer); this corresponds to a vanishing disturbance since x(z,t) = 0, for the uncertain system, continues to be an equilibrium solution. Figure 8.4 shows the closed-loop spatiotemporal profile of the state and profiles of the two manipulated inputs, using initial conditions for the system and for the state observer same to those in (8.86) and (8.94), respectively. Our simulation clearly shows that the proposed unit output feedback controller is capable of stabilizing the linearized KSE at the spatially-uniform steady state solution in the presence of significant uncertainty in ν .

182

8 Global Asymptotic Stabilization of Uncertain Linear Systems

1.5 1

x

0.5 0 −0.5 −1 −1.5 0 1

3 2 2

1 3 4

Time

−2

−3

0

−1

z u1 u2

2 0 −2 −4

u

−6 −8 −10 −12 −14 −16 −18 −20

0

0.5

1

1.5

2

Time

2.5

3

3.5

4

Fig. 8.3 Closed-loop spatio-temporal profile of x(z,t) (top figure) and manipulated input profiles (bottom figure) under the δ −unit output feedback controller

8.4.2 Numerical Results Under Non-vanishing External Disturbances In this section, we demonstrate that in the presence of external non-vanishing disturbances, the proposed sliding-mode feedback controller is superior to a linear feedback controller in terms of the closed-loop performance. All the simulation runs shown in this section are performed for ν = 0.2, using a thirtieth-order linear ordinary differential equation model obtained from the application of modal decomposition to (8.70) (the use of higher-order approximations led to identical numerical results, thereby implying that the following simulation runs are independent of the discretization). We consider the first five modes of the linearization of KSE as the

8.4 Applications to the Linearized Kuramoto–Sivashinsky Equation

183

1.5 1

x

0.5 0 −0.5 −1 −1.5 0 1

3 2 2

1 3 4

Time

−2

−3

0

−1

z

5 u1 u2

0

u

−5

−10

−15

−20

0

0.5

1

1.5

2

2.5

3

3.5

4

Time

Fig. 8.4 Closed-loop spatio-temporal profile of x(z,t) (top figure) and manipulated input profiles (bottom figure) under the δ −unit output feedback controller for 25% uncertainty in ν

slow modes and use modal decomposition to construct a fifth-order system. Five actuators are used to control the system. The resulting fifth-order system is of the form ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ 1 ⎤⎡ ⎤ ⎡ 1 ⎤ b1 · · · b15 d α˙1 λ1 . . . 0 α1 u1 ⎢ .. ⎥ ⎢ .. . . .. ⎥ ⎢ .. ⎥ ⎢ .. . . .. ⎥ ⎢ .. ⎥ ⎢ .. ⎥ (8.95) ⎣ . ⎦ = ⎣ . . . ⎦⎣ . ⎦ + ⎣ . . . ⎦⎣ . ⎦ + ⎣ . ⎦ α˙5 α5 u5 0 . . . λ5 b51 · · · b55 d5 where bni (i, n = 1, · · · , 5) are computed by using (8.81) with the actuator distribution functions bi (z) of the form as shown in (8.83) and d i (i = 1, · · · , 5) are computed by using (8.82) with d(z,t) specified by

184

8 Global Asymptotic Stabilization of Uncertain Linear Systems

1 d(z,t) = √ sin(z)[4 + sin(10t)]. π

(8.96)

The first simulation run is performed to compare the closed-loop performance, corresponding to the unit state feedback controller, and that corresponding to a linear state feedback controller while the non-vanishing external disturbance (8.96) affects the system. For System 8.95, the continuous approximation (8.38) of the unit state feedback controller (8.22) takes the form ⎡ ⎤ u1 ⎢ .. ⎥ ξd (8.97) ⎣ . ⎦ = − (γ0 + γ1 ξd ) B−1 d ξ +δ 3 d

u5

⎡

b11 . . . ⎢ .. . . where ξd = ⎣ . .

⎡ 1 ⎤T ⎡ ⎤ b1 · · · b15 α1 .. ⎥ ⎢ .. ⎥ and B = ⎢ .. . . ⎣ . . d . ⎦ ⎣ . ⎦

c

⎤T ⎡ 1 b15 b1 . . . .. ⎥ ⎢ .. . . . ⎦ ⎣ . .

⎤ b15 .. ⎥ . . ⎦

α5 b51 · · · b55 b51 . . . b55 b51 . . . b55 In order to reject the non-vanishing external disturbance, we assume that the upper bound of the external disturbance is known a priori and the parameter γ0 in (8.97) is larger than the upper bound of the external disturbance. The controller parameters used in this simulation are the same as those used in Sect. 8.4.1. The linear state feedback controller is designed as follows ⎡

⎤ ⎡ 1 b1 · · · u1 ⎢ .. ⎥ ⎢ .. . . ⎣ . ⎦ = −k ⎣ . . b51

u5

...

⎤−1 ⎡ ⎤ b15 α1 .. ⎥ ⎢ .. ⎥ . ⎦ ⎣ . ⎦ b55

(8.98)

α5

where the gain value k = 9.503 is selected for the simulation. To compare the closed-loop performance of the two controllers, we use a performance index of the form J(t f ) =

 tf  π 0

−π

x2 (z,t)dzdt

(8.99)

where x(z,t) is the state of the closed-loop system and t f is the total simulated time. Throughout, this time is set to be t f = 4. The initial condition of the system is the same to (8.86) and the locations of the five actuators are za1 = 0.15π , za2 = 0.31π , za3 = 0.5π , za4 = 0.69π and za5 = 0.85π . Figure 8.5 shows the spatio-temporal profile of x(z,t) and profiles of the five manipulated inputs under the unit state feedback controller (8.97). It is clear that this controller is capable of stabilizing the state of the system as x(z,t) = 0 in the presence of the non-vanishing external disturbance. The value of the performance index, corresponding to (8.97), is J(t f ) = 0.5731. Figure 8.6 shows the spatio-temporal profile of x(z,t) and profiles of the five manipulated inputs under the linear state feedback controller. We can see that the linear state feedback controller is not able to drive the state of the system to x(z,t) =

8.4 Applications to the Linearized Kuramoto–Sivashinsky Equation

185

1.5 1

x

0.5 0 −0.5 −1 −1.5 0 1

3 2 2

1 3 4

Time

−3

−2

0

−1

z u 1 u 2 u 3 u 4 u5

2

0

−2

u

−4

−6

−8

−10

−12

0

0.5

1

1.5

2

2.5

3

3.5

4

Time

Fig. 8.5 Closed-loop spatio-temporal profile of x(z,t) (top figure) and manipulated input profiles (bottom figure) under the δ −unit state feedback controller with the external disturbance (d(z,t) = √1 sin(z)[4 + sin(10t)]) π

0 in the presence of the non-vanishing external disturbance. Moreover, the value of the performance index is J(t f ) = 1.3263 which is much larger than that under the δ −unit state feedback controller (8.97). We note that the superior closed-loop performance achieved by the proposed δ −unit feedback controller is not due to the poor tuning of the linear feedback controller. The gain k of the linear state feedback controller (8.98) is tuned such that the performance under this linear controller is the same as that under the δ −unit state feedback controller (8.97) with respect to the performance index defined in (8.99), when there is no external disturbance. Figures 8.7 and 8.8 show the closed-loop

186

8 Global Asymptotic Stabilization of Uncertain Linear Systems

1.5 1

x

0.5 0 −0.5 −1 −1.5 0 1

3 2 2

1 3 4

Time

−3

0

−1

−2

z u1, u5 u ,u 2 4 u

4

3

2 0 −2

u

−4 −6 −8 −10 −12 −14 −16

0

0.5

1

1.5

2

2.5

3

3.5

4

Time

Fig. 8.6 Closed-loop spatio-temporal profile of x(z,t) (top figure) and manipulated input profiles (bottom figure) under the linear state feedback controller with the external disturbance (d(z,t) = √1 sin(z)[4 + sin(10t)]) π

spatio-temporal profile of x(z,t) and the profiles of the five manipulated inputs under the δ −unit state feedback controller and under the linear state feedback controller, respectively, while no external disturbances affect the system. The closed-loop per t formance J(t f ) = 0 f −ππ x2 (z,t)dzdt = 0.3567 is the same in both cases. Comparing Figs. 8.7 and 8.8 with Figs. 8.5 and 8.6, we can see that while both Controllers 8.97 and 8.98 can achieve the same performance without the presence of external disturbance, the closed-loop performance is significantly deteriorated by the non-vanishing disturbance under the linear state feedback control law (8.98) but it is not deteriorated under the δ −unit state feedback control law (8.97).

8.4 Applications to the Linearized Kuramoto–Sivashinsky Equation

187

1.5 1

x

0.5 0 −0.5 −1 −1.5 0 1

3 2 2

1 3 4

Time

0

−1

−2

−3

z u 1 u2 u3 u4 u

4

2

5

0

u

−2

−4

−6

−8

−10

−12

0

0.5

1

1.5

2

2.5

3

3.5

4

Time

Fig. 8.7 Closed-loop spatio-temporal profile of x(z,t) (top figure) and manipulated input profiles (bottom figure) under the δ −unit state feedback controller without external disturbances; J(t f ) = tf π 2 0 −π x (z,t)dzdt=0.3567

Finally, we compare the closed-loop performance under the unit output feedback controller and the linear output feedback controller in the presence of the non-vanishing external disturbance. Two measurement sensors are used in this simulation. The sensors are optimally placed according to [137] at zs1 = 0.35π and zs2 = 0.64π . Based on the two measurements, [ y1 y2 ]T , and the thirtieth-order approximation of the infinite-dimensional system (8.80), the state observer is of the form

188

8 Global Asymptotic Stabilization of Uncertain Linear Systems

1.5 1

x

0.5 0 −0.5 −1 −1.5 0 1

3 2 2

1 3 4

Time

−3

0

−1

−2

z u1, u5 u ,u 2 4 u

4

3

2 0 −2

u

−4 −6 −8 −10 −12 −14 −16

0

0.5

1

1.5

2

2.5

3

3.5

4

Time

Fig. 8.8 Closed-loop spatio-temporal profile of x(z,t) (top figure) and manipulated input profiles (bottom figure) under the linear state feedback controller without external disturbances; J(t f ) = tf π 2 0 −π x (z,t)dzdt=0.3567

⎡

⎤ ⎡ ⎤⎡ ⎤ ⎡ 1 bα 1 α˙˜ 1 λ1 . . . 0 α˜ 1 ⎢ .. ⎥ ⎢ .. . . .. ⎥ ⎢ .. ⎥ ⎢ .. = + ⎣ . ⎦ ⎣ . . . ⎦⎣ . ⎦ ⎣ . α˜ 5 0 . . . λ5 α˜˙ 5 b5α 1 ⎛



⎜ y −Ld1 ⎝ 1 y2 ⎤ ⎡ α˙˜ 6 λ6 ⎢ .. ⎥ ⎢ .. ⎣ . ⎦=⎣ . 0 α˙˜ 30 ⎡

··· .. . ···



⎤⎡ ⎤ · · · b1α 5 u1 ⎢ .. ⎥ . . .. ⎥ . . ⎦⎣ . ⎦ u5 · · · b5α 5

⎡

⎤ ⎡ ⎤⎞ α˜ 1 α˜ 6 ⎢ ⎥ ⎢ ⎥⎟ − Cd1 ⎣ ... ⎦ − Cd2 ⎣ ... ⎦⎠ , α˜ 5 α˜ 30

⎤⎡ ⎤ ⎡ 6 0 α˜ 6 bα 1 .. ⎥ ⎢ .. ⎥ + ⎢ .. . ⎦⎣ . ⎦ ⎣ . λ30 α˜ 30 b30 α1

··· .. . ···

⎤⎡ ⎤ u1 b6α 5 .. ⎥ ⎢ .. ⎥ . ⎦⎣ . ⎦ b30 α5

u5

(8.100)

8.4 Applications to the Linearized Kuramoto–Sivashinsky Equation

where

189

 c11 · · · c51 , c12 · · · c52   6 c1 · · · c30 1 , Cd2 = 6 c2 · · · c30 2 

Cd1 =

(8.101) (8.102)

and cnκ (κ = 1, 2. n = 1, . . . , 30) is given by (8.90). The constant Ld1 is set to be 

−61.503 −45.274 11.395 52.714 47.944 Ld1 = −64.149 44.218 17.044 −56.905 42.524

T .

(8.103)

The state estimates of (8.100) are used in the following δ −unit feedback controller: ⎡ ⎤ u1          ⎢ .. ⎥ ξ˜d (8.104) ⎣ . ⎦ = − γ0 + γ1 ξ˜d  + γ2 ζ˜d  B−1 d ξ˜d +δc3 u5 where ⎡

b11

b15

⎤T ⎡

⎤

··· α˜ 1 ⎢ .. . . .. ⎥ ⎢ .. ⎥ ˜ ξd = ⎣ . . . ⎦ ⎣ . ⎦ , α˜ 5 b51 · · · b55

⎤ α˜ 6 α˜ 1 ⎢ .. ⎥   ⎢ . ⎥ y1 ⎢ .. ⎥ ˜ ⎥ − Cd1 ⎣ . ⎦ − Cd2 ⎢ ζ= ⎢ . ⎥ y2 .. ⎦ ⎣ α˜ 5 α˜ 30 ⎡

⎤

⎡

⎡

⎤T ⎡ 1 ⎤ b11 . . . b15 b1 · · · b15 ⎢ ⎥ ⎢ ⎥ and Bd = ⎣ ... . . . ... ⎦ ⎣ ... . . . ... ⎦. The values of γ0 , γ1 , γ2 and δc3 are the same b51 · · · b55 b51 · · · b55 to those used in Sect. 8.4.1. In turn, the state estimates (8.100) are also used in the following linear controller ⎡

⎤ ⎡ 1 b1 · · · u1 ⎢ .. ⎥ ⎢ .. . . ⎣ . ⎦ = −k ⎣ . . u5

b51

...

⎤−1 ⎡ ⎤ b15 α˜ 1 .. ⎥ ⎢ .. ⎥ . ⎦ ⎣ . ⎦ b55

(8.105)

α˜ 5

where the gain value k = 9.503 is the same as before. Using as an initial condition for the system the same as that in (8.86) and an initial condition for the state observer the same as that in (8.94), the closed-loop spatio-temporal profile of x(z,t) and the profiles of the five manipulated inputs under unit output feedback controller (8.100)–(8.104) are shown in Fig. 8.9. We can clearly see that the δ −unit output feedback controller (8.100)–(8.104) is able to efficiently suppress the influence of the non-vanishing external disturbance to the state of the system and drive the state of the system very close to x(z,t) = 0 achieving, at the same time, very good performance (performance index J(t f ) = 0.7522). Figure 8.10 shows the closed-loop spatio-temporal profile of x(z,t) and the profiles

190

8 Global Asymptotic Stabilization of Uncertain Linear Systems

1.5 1

x

0.5 0 −0.5 −1 −1.5 0 1

3 2 2

1 3 4

Time

−3

0

−1

−2

z u 1 u 2 u 3 u4 u5

4

2

0

u

−2

−4

−6

−8

−10

−12

0

0.5

1

1.5

2

2.5

3

3.5

4

Time

Fig. 8.9 Closed-loop spatio-temporal profile of x(z,t) (top figure) and manipulated input profiles (bottom figure) under the δ −unit output feedback controller with the external disturbance (d(z,t) = √1 sin(z)[4 + sin(10t)]). π

of the five manipulated inputs under the linear output feedback controller (8.100)– (8.103), (8.105). The performance index under the linear output feedback control law is J(t f ) = 4.0886, which is much larger than that under the δ −unit output feedback controller. To this end, we also note that it is important to implement the designed unit state feedback controller (8.22) and the designed unit output feedback controller (8.45) by using their continuous approximations so that fast switching in the control actions can be avoided. To demonstrate the presence of fast switching in discontinuous unit controllers, we run a simulation to stabilize (8.70) using the discontinuous unit output feedback controller (8.49). The design of the output feedback controller and

8.4 Applications to the Linearized Kuramoto–Sivashinsky Equation

191

1.5 1

x

0.5 0 −0.5 −1 −1.5 0 1

3 2 2

1 3 4

Time

0

−1

−2

−3

z u1 u 2 u 3 u 4 u

2

0

5

−2

u

−4

−6

−8

−10

−12

0

0.5

1

1.5

2

2.5

3

3.5

4

Time

Fig. 8.10 Closed-loop spatio-temporal profile of x(z,t) (top figure) and manipulated input profiles (bottom figure) under the linear output feedback controller with the external disturbance (d(z,t) = √1 sin(z)[4 + sin(10t)]) π

that of the state observer follow the development in (8.87)–(8.94) except that the δc3 term is removed from the denominator of the controller. In Fig. 8.11, we show the closed-loop spatio-temporal profile of x(z,t) and the profiles of the two manipulated inputs. Comparing Fig. 8.3 and Fig. 8.11, we can see that the closed-loop profiles of x from both simulations are very close, but the manipulated inputs from the continuous δ -approximation of the unit output feedback controller are much smoother than those from the discontinuous unit output feedback controller. Therefore, the δ -unit controllers are preferable from a practical implementation point of view.

192

8 Global Asymptotic Stabilization of Uncertain Linear Systems

1.5 1 0.5

x

0

−0.5 −1 −1.5 0 3

1 2 1

2 3 4

Time

−3

0

−1

−2

z

15 u 1 u 2

10

5

u

0

−5

−10

−15

−20

0

0.5

1

1.5

2

2.5

3

3.5

4

Time

Fig. 8.11 Closed-loop spatio-temporal profile of x(z,t) (top figure) and manipulated input profiles (bottom figure) under the unit output feedback controller

Chapter 9

Asymptotic Stabilization of Minimum-phase Semilinear Systems

In the present section, the unit control approach is developed for minimum phase semilinear infinite-dimensional systems driven in a Hilbert space. Control algorithms presented ensure asymptotic stability, global or local according as state feedback or output feedback is available. The desired robustness properties of the closedloop system against external disturbances with a priori known norm bounds make the algorithms extremely suited for stabilization of the underlying system operating under uncertainty conditions. It is, in particular, shown that discontinuous feedback stabilization is constructively available in the case where complex nonlinear dynamics of the uncertain system does not admit factoring out a destabilizing nonlinear gain, and thus the destabilizing gain can not be handled through nonlinear damping. The theory is applied to the stabilization of chemical processes around pre-specified steady-state temperature and concentration profiles corresponding to a desired coolant temperature. Two specific cases, a plug flow reactor and an axial dispersion reactor, governed by hyperbolic and parabolic partial differential equations of the first order and of the second order, respectively, are under consideration. To achieve a regional temperature feedback stabilization around the desired profiles, with the region of attraction, containing a prescribed set of interest, a component concentration observer is constructed and included into the closed-loop system so that there is no need for measuring the process component concentration which is normally unavailable in practice. Performance issues of the unit feedback design are illustrated in a simulation study of the plug flow reactor.

9.1 Stabilization in a Hilbert Space Many recent studies in nonlinear stabilization have focused on finite-dimensional systems written, due to a nonlinear change of state coordinates and a feedback transformation, in a normal form (see, e.g., [38]). Following [171], these studies are now extended to an infinite-dimensional setting. We investigate the capability of a semilinear system 193

194

9 Asymptotic Stabilization of Minimum-phase Semilinear Systems

ξ˙1 = A1 ξ1 + f1 (ξ1 , ξ2 ) + ν , ξ1 (0) = ξ10 ∈ D(A1 ) ξ˙2 = A2 ξ2 + f2 (ξ1 , ξ2 ), ξ2 (0) = ξ20 ∈ D(A2 ),

(9.1) (9.2)

driven in a Hilbert space H = H1 ⊕ H2 , to be asymptotically stabilizable, globally or locally according as state feedback ν (ξ ) ∈ H1 or output feedback ν (ξ1 ) ∈ H1 is available. In the above equations, the Hilbert space H is equipped with the inner product ·, · , the state vector ξ = (ξ1 , ξ2 ) ∈ H is composed of the components ξ1 ∈ H1 , ξ2 ∈ H2 , the functions f1 (ξ ), f2 (ξ ) are continuously differentiable, the infinitesimal operators A1 and A2 generate analytical semigroups on the Hilbert spaces H1 and H2 , respectively, and the symbols D(A1 ) and D(A2 ) stand for the domains of these operators. Throughout, actuation, distributed through the state subspace H1 of, possibly, an infinite dimension, and distributed sensing are assumed to be available. Due to technological developments of micro-electro mechanical systems (MEMS), manufacturing large arrays of micro-sensors and actuators with integrated control circuitry has become feasible (for control applications of such devices see [14] and references quoted therein), and therefore, the latter assumption is justifiable for a large class of physical processes with sensor and actuator arrays. For technical reasons, we also assume the following: 1. The infinitesimal operators A1 and A2 generate analytical semigroups T1 and T2 on the Hilbert spaces H1 and H2 , respectively. 2. A1 is nonpositive definite on its domain D(A1 ), i.e., A1 ξ1 , ξ1 ≤ 0 for all ξ1 ∈ D(A1 ). 3. System

ξ˙2 = A2 ξ2 + f2 (0, ξ2 )

(9.3)

has 0 as a globally asymptotically stable equilibrium. 4. f1 (0, 0) = 0, f2 (0, 0) = 0. 5. f2 (ξ1 , ξ2 ) satisfies the linear growth condition in ξ2 for all ξ1 ∈ H1 :  f2 (ξ1 , ξ2 ) ≤ k(ξ1 )(1 + ξ2)

(9.4)  where the induced norm in the Hilbert space is given by ξ  = ξ , ξ and k(·) is a continuous function. 6. The functions f1 (ξ1 , ξ2 ) and f2 (ξ1 , ξ2 ) are continuously differentiable in all arguments. Assumptions 1 and 6, coupled together, guarantee the existence, uniqueness, and well-posedness of strong solutions ξ1 , ξ2 of the unforced plant equations (9.1), (9.2) with ν = 0. Moreover, under these assumptions the strong solutions may be found as mild (generalized) solutions, i.e., as those of the integral equations

ξi (t) = Ti (t)ξi0 +

 t 0

Ti (t − τ ) fi (ξ (τ ))d τ , i = 1, 2.

(9.5)

9.1 Stabilization in a Hilbert Space

195

Remark 9.1. An important relaxation that is a consequence of Assumption 1 is that the operators A1 and A2 generate just C0 (strongly continuous rather than analytical) semigroups on the Hilbert spaces H1 and H2 . Although in that case the existence of the strong solutions is no longer guaranteed, however, the results stated below, Theorems 9.1 and 9.2, and their proofs remain true for the mild solutions whose existence is still in force. Assumption 2 imposes a useful stability property on the linear system ξ˙1 = A1 ξ1 evolving in the Hilbert space H1 . This can straightforwardly be established via utilizing the inner product ξ1 , ξ1 as a Lyapunov functional. Assumption 2 simplifies the stabilizing control law to be worked out, but if it is relaxed, the controller can be modified so that the closed-loop system is still asymptotically stable. Considering y = ξ1 as the system output, Assumption 3 means that (9.1), (9.2) is a globally minimum phase system. The role of this notion is well known from the finite-dimensional theory [38] and its extension to the infinite-dimensional setting is given in [162]. Assumption 4 ensures that the origin is an equilibrium of the unforced system (9.1), (9.2) with ν = 0. Assumption 5 is introduced to avoid the destabilizing effect of the peaking phenomenon (a detailed treatment of the peaking phenomenon in the finite-dimensional setting can be found in [219]). The functions f1 and f2 represent the system nonlinearities and, along with the linear unbounded operators A1 and A2 , they include the system uncertainties. The influence of these functions on the control process should be rejected. The only available information on the system uncertainties is an upper scalar estimate F(ξ ) of the function f1 (ξ ) such that F(ξ ) is continuous, F(0) = 0, and  f1 (ξ ) ≤ F(ξ ) < ∞ f or all ξ ∈ H.

(9.6)

One might be surprised by the absence of a matching condition on the system uncertainties, as is typically the case in the sliding mode control theory [227]. However, since in our case the zero dynamics equation (9.3) contains the unmatched uncertainties f2 , they can not destabilize the closed-loop system by the minimum phase hypothesis. The following control law

ν (ξ ,t) = −M(ξ ,t)U(ξ1 )

(9.7)

with a constant gain M(ξ ,t) = M > 0 or a nonlinear gain M(ξ ,t) subject to M(ξ ,t) ≥ F(ξ ) + cξ1α f or some c > 0 and α ∈ [0, 1)

(9.8)

is proposed to stabilize the uncertain system (9.1), (9.2). Hereinafter, U(ξ1 ) = ξξ1  1 is a unit control signal, the norm of which is equal to one everywhere but on the hyperplane ξ1 = 0, where it undergoes discontinuities; the function M(ξ ,t) is continuously differentiable everywhere, possibly except the hyperplane ξ1 = 0. Certainly, the above control law can also be applied to (9.1), (9.2) whose nonlinearities are known a priori. In this case, however, there appear alternative stabilizing controllers

196

9 Asymptotic Stabilization of Minimum-phase Semilinear Systems

that may have advantages (e.g., the absence of switching) over the discontinuous feedback. Meanwhile, it should be pointed out that the state feedback (9.7) is continuous if M(ξ ,t) tends to zero as ξ1 → 0; otherwise (9.7) specifies a discontinuous control law. As before, the precise meaning of the Hilbert space-valued differential equations (9.1), (9.2) with the discontinuous input function (9.7) is defined in the generalized sense of Definition 2.10. The design methodology proposed is based on deliberate use of, generally speaking, a discontinuous feedback ν (ξ ) which drives (9.1) towards the origin ξ1 = 0 and retains useful robustness features similar to those possessed by its counterpart in the finite-dimensional case. The discontinuous feedback mainly serves to counteract non-vanishing external disturbances. If the system is affected by vanishing perturbations only, the methodology can be applied for constructing a continuous feedback which dominates these perturbations and drives Subsystem (9.1) to its origin while eliminating undesirable effects of switching. Stability of the over-all system (9.1), (9.2) is achieved provided that the zeroinput system (9.3) has desired stability properties and the input ξ1 (t) does not destabilize (9.2). The destabilizing effect of the input ξ1 (t) on (9.2) is avoided by assuming the nonlinearity f2 (ξ ) to have a sufficiently slow growth. In what follows, the control law (9.7) with a nonlinear state-dependent gain M(ξ ,t) subject to (9.8) is proved to globally stabilize the system in question. Theorem 9.1. Let Assumptions 1–6 be satisfied and let the infinite-dimensional system (9.1), (9.2) be driven by the state feedback (9.7) with a nonlinear gain M(ξ ,t), continuosly differentiable for all (ξ ,t) ∈ H × R+ , possibly except those with ξ1 = 0, and such that (9.8) holds. Then, (9.1), (9.2) subject to (9.7) has a unique strong solution, globally defined for all t ≥ 0, and the closed-loop system is globally asymptotically stable. Proof. We break up the proof into five simple steps. 1. The operators Aˆ 1 = λ I − A1 and Aˆ 2 = λ I − A2 with λ > 0 large enough are of a strongly positive type [117, p.304]. Thus, the state equations (9.1), (9.2) can be rewritten as the semilinear equations

ξ˙1 + Aˆ 1ξ1 = fˆ1 (ξ1 , ξ2 ) + ν , ξ1 (0) = ξ10 ξ˙2 + A2ξ2 = fˆ2 (ξ1 , ξ2 ), ξ2 (0) = ξ20

(9.9) (9.10)

with the strongly positive type operators Aˆ 1 and Aˆ 2 , and the nonlinear functions fˆ1 (ξ ) = λ ξ1 + f1 (ξ ), fˆ2 (ξ ) = λ ξ2 + f2 (ξ ). Because the nonlinearities fˆ1 (ξ ) and fˆ2 (ξ ), and the control input ν (ξ ,t) are continuously differentiable as soon as ξ1 = 0, (9.9), (9.10) subject to (9.7) satisfies [117, Theorem 23.3 and Theorem 23.4] beyond the discontinuity hyperplane ξ1 = 0 with δ = 1, ρ = 1 and by these theorems a unique strong solution of (9.9), (9.10), (9.7) and, consequently, that of (9.1), (9.2), (9.7) exists locally for all initial conditions such that ξ10 = 0. 2. Now, using the functional

9.1 Stabilization in a Hilbert Space

197

V (ξ1 ) = ξ1 2 , we can show that, given a continuous (possibly, unbounded as t → ∞) function ξ2 (t), t ≥ 0, a unique solution of (9.1), (9.7) is globally defined for all t ≥ 0. Indeed, the computation of the derivative of this functional along the trajectories of (9.1) subject to (9.6)–(9.8) yields V˙ (t) = 2 ξ˙1 , ξ1 = 2 A1 ξ1 , ξ1 + 2 f1 + ν , ξ1 ≤

2 f1 ξ1  − 2M(ξ ,t)ξ1 ≤ −2cξ1α +1 = −2cV α0 (t) where α0 =

α +1 2

(9.11)

∈ (0, 1). Clearly, (9.11) implies the a priori estimate V (t) ≤ |ξ10 2 , t ≥ 0

(9.12)

of a possible solution ξ1 (t) of (9.1). Moreover, by applying Lemma 4.3 to (9.11), ξ 0 2(1−α0 )

1 one concludes that V (t) vanishes after a finite time moment t0 ∈ [0, 2c(1− α0 ) ]. Thus, on the time interval [0,t0 ), while (9.1), (9.7) remain beyond the discontinuity hyperplane ξ1 = 0, it satisfies [117, Theorem 23.5] and by this theorem the strong solution of (9.1) subject to (9.7) exists on the entire interval [0,t0 ]. Starting from t0 System 9.1, 9.7 is maintained within the discontinuity hyperplane so that the state of the system is uniquely defined by the hyperplane equation, i.e., ξ1 (t) = 0 for all t ≥ t0 . 3. Next, we demonstrate that given a continuous, uniformly bounded function ξ1 (t), t ≥ 0, a unique strong solution of (9.2) is globally defined for all t ≥ 0. To prove this, we use Representation (9.10) of (9.2) where operator Aˆ 2 is of a strongly positive type and consequently it generates an analytical bounded semigroup Tˆ2 [117, p. 304]. On a finite time interval [0,t1 ) with arbitrary positive t1 a possible generalized solution of (9.10) can a priori be estimated. Indeed, due to Assumption 6, we have

ξ2 (t) ≤ Tˆ2 (t)ξ20  + c1 ξ20  + c1

 t 0

t 0

Tˆ2 (t − τ ) fˆ2 (ξ1 (τ ), ξ2 (τ ))d τ ≤

k(ξ1 (τ ))[1 + (1 + λ )|ξ2(τ )]d τ ≤

c1 (ξ20  + k1t1 ) + c1 k1 (1 + λ )

t 0

|ξ2 (τ )d τ

(9.13)

where c1 is an upper bound of the norm of the bounded semigroup Tˆ2 and k1 is an upper bound of the continuous function k(ξ1 ) over all ξ1  ≤ R with R being an upper bound of the norm of the uniformly bounded function ξ1 (t). Now, applying the Bellman–Gronwall inequality to (9.13), the following estimate ξ2 (t) ≤ c1 (ξ20  + k1t1 )exp{c1 k1 (1 + λ )t1} is derived.

(9.14)

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9 Asymptotic Stabilization of Minimum-phase Semilinear Systems

Thus, (9.10) satisfies [117, Theorem 23.5] and, by this theorem, there exists a unique generalized solution of (9.10), defined on the entire interval [0,t1 ). Due to [117, Theorem 23.4], whose applicability to (9.10) has been shown at the first step, this generalized solution is also a unique strong solution of (9.10). Since t1 is arbitrary the unique strong solution of (9.10) is globally defined for all t ≥ 0. 4. By analogy to steps 2 and 3, we obtain that a possible solution of the overall system (9.1), (9.2) driven by (9.7) admits the a priori estimate (9.12), (9.14) and therefore there exists a unique strong solution of (9.1), (9.2) subject to (9.7), globally defined for all t ≥ 0. 5. To complete the proof of Theorem 9.1 it remains to establish that the closedloop system (9.1), (9.2), (9.7) is globally asymptotically stable. Following the line of reasoning used at step 2, we conclude that trajectories of the system, starting within a ball BR ⊂ H with radius R > 0 and center at the origin, are steered towards R2(1−α0 ) the discontinuity hyperplane ξ1 = 0 for at most t = 2c(1− α0 ) and then maintained on the hyperplane. According to Theorem 2.4, the sliding motion of (9.1), (9.2) on the hyperplane ξ1 = 0 is governed by the zero dynamics (9.3), which is globally asymptotically stable by Assumption 3. Taking into account that trajectories of (9.1), (9.2), (9.7) uniformly bounded at the initial time moment, converge to the discontinuity hyperplane ξ1 = 0 within a finite time interval, and on this interval the trajectories depend on initial conditions continuously, it follows that the closed-loop system (9.1), (9.2), (9.7) is globally asymptotically stable. Theorem 9.1 is thus proved. To this end, we note that replacing the state-dependent gain M(ξ ,t) in the above control law (9.7) with a constant gain M(ξ ,t) = M > 0 yields an output feedback which stabilizes (9.1), (9.2) locally. Certainly, the output feedback thus constructed is always discontinuous. Theorem 9.2. Let Assumptions 1-6 be satisfied and let the infinite-dimensional system (9.1), (9.2) be driven by the discontinuous output feedback (9.7) with a stateindependent gain M(ξ ,t) = M > 0. Then, there exists a neighborhood of the origin such that given initial conditions in the neighborhood, (9.1), (9.2) subject to (9.7) has a unique strong solution, globally defined for all t ≥ 0. Moreover, the closedloop system is locally asymptotically stable. Proof. Since the function F(ξ ) is continuous and F(0) = 0 there exists R > 0 such that F(ξ ) + cξ α ≤ M for c > 0 and α ∈ [0, 1), given a priori, and all ξ ∈ BR . For R thus selected, let us fix R0 ∈ (0, R) such that trajectories of the globally asymptotically stable system (9.3), initialized within BR0 , do not leave the ball BR . As shown in proving Theorem 9.1, the state of the closed-loop system (9.1), (9.2), (9.7) with the nonlinear gain M(ξ ,t) is driven to the discontinuity hyperplane ξ 0 2(1−α0 )

1 ξ1 = 0 for a finite time moment t1 ≤ 2c(1− α0 ) and for t ≤ t1 the state estimates (9.12), (9.14) are satisfied. These estimates ensure that for initial conditions, small enough, trajectories of (9.1), (9.2), (9.7) remains within the ball BR0 ⊂ BR so that for these initial conditions the required disturbance rejection (9.8) is still provided when the nonlinear gain M(ξ ,t) in the control law (9.7) is replaced with the state-

9.2 Application to Chemical Tubular Reactor

199

independent gain M. Thus, the line of reasoning used in the proof of Theorem 9.1 applies here as well, and the validity of Theorem 9.2 is thus established.

9.2 Application to Chemical Tubular Reactor The controller earlier developed for the infinite-dimensional system (9.1), (9.2) specializes here for a tubular chemical reactor. The mathematical model of the reactor is given by the following mass and energy balance equations (e.g., [232]): E 4h ∂T λea ∂ 2 T ∂T ΔH = − −u k0Ce− RT − (T − Tw ), 2 ∂t ρ Cp ∂ z ∂ z ρ Cp d ρ Cp T (z, 0) = T0 (z) E ∂C ∂ 2C ∂C = Dma 2 − u − k0Ce− RT , ∂t ∂z ∂z C(z, 0) = C0 (z)

(9.15)

(9.16)

with the following boundary conditions:

λea ∂ T (z = 0) = u(T (z = 0) − Tin) ρ Cp ∂ z ∂C (z = 0) = u(C(z = 0) − Cin) Dma ∂z λea ∂ T (z = L) = 0 ρ Cp ∂ z ∂C (z = L) = 0. Dma ∂z

(9.17) (9.18) (9.19) (9.20)

The boundary-value problem (9.15)–(9.20) describes a non-isothermal reaction, taken within a tubular reactor, with first-order kinetics with respect to the reactant concentration and Arrhenius-type dependence with respect to the temperature. In the above equations t is the time (s), z is the spatial variable (0 ≤ z ≤ L), T > 0 is the temperature (K), C > 0 is the process component concentration (kg.m−3 ), the initial distributions T0 (z) > 0 and C0 (z) > 0 are twice continuously differentiable functions, satisfying the boundary conditions (9.17)–(9.20), L (m) is the length of the reactor, u is the fluid superficial velocity (m/s), λea is the axial energy dispersion coefficient (kJ.m−1 .s−1 .K −1 ), Dma is the axial mass dispersion coefficient (m2 /s), Δ H is the reaction heat (kJ.kg−1 )(Δ H < 0 for exothermic reactions), ρ is the fluid density (kg/m3 ), C p is the specific heat (kJ.kg−1 .K −1 ), k0 is the kinetic constant (s−1 ), E is the activation energy (kJ.kg1 ), R is the gas constant (kJ.kg−1 .K −1 ), h is the wall heat transfer coefficient (kJ.m−2 .K −1 .s−1 ), d is the reactor diameter (m), Tw > 0 is the coolant temperature (K), Tin > 0 is the inlet temperature (K), and Cin > 0 is the inlet reactant concentration (kg.m3 ).

200

9 Asymptotic Stabilization of Minimum-phase Semilinear Systems

The present process example is particularly interesting, since it is well known that the existence of Arrhenius type nonlinearities can generate multiple steady states, either stable or unstable, and that in practical applications, the unstable steady states may correspond to the operating points of interest. The study of the steady state multiplicity and stability has been the object of intensive research activity in the 1960s and 1970s, and most of the results are gathered in the book chapter written by Varma and Aris [232]. Multiple steady states have been observed experimentally, e.g., in adiabatic tubular reactors [79, 234]. If the steady state multiplicity and stability of stirred tank reactors is now well understood, the tubular reactor case is still the object of research works. As a matter of example, Varma and Aris [232] emphasize sufficient conditions for the uniqueness of steady states for adiabatic reactors (i.e., when h = 0 in (9.15) above) in the very particular case when the energy diffusion coefficient ρλCeap is equal the mass diffusion coefficient Dma . They also show that in presence of multiple steady states for the above tubular reactor, these are alternatively stable and unstable. The case of nonequal diffusion coefficients has been considered by Deimling [59] who emphasized steady-state multiplicity, yet in unrealistic conditions, i.e., when the reactor temperature is lower than the inlet and cooling temperature in an exothermic reactor. More recently, the steady-state multiplicity in non-isothermal reactors has been shown in [120] by using similar arguments as those considered by Deimling as well as compactness and nonlinear operator arguments. However, although there is a strong convergence of results in the literature in the direction of an alternance of stable and unstable steady states in the presence of multiple equilibrium points, the stability of the multiple steady states in the general non adiabatic reactor with different diffusion coefficients remains an open question. Two specific cases, the hyperbolic model (9.15)–(9.20) with λea = 0 and Dma = 0, corresponding to the plug flow reactor, and the parabolic model (9.15)–(9.20) with λea > 0 and Dma > 0, corresponding to the axial dispersion reactor, are under consideration. To insure that the physical variables T (z,t) and C(z,t), the process temperature and the component concentration, are always positive initial distributions T (z, 0) and C(z, 0) are required to be positive for all z ∈ [0, L]. The temperature T along the reactor is enforced by acting on the coolant temperature Tw with the objective to track the reference steady-state temperature and concentration profiles corresponding to a desired (optimal) coolant temperature. Both the discontinuous state feedback design and the discontinuous output (temperature) feedback design are provided. The state feedback solves the tracking problem globally, whereas the output feedback yields a local solution of the problem. To achieve a regional temperature feedback stabilization around the desired profiles, with the region of attraction containing a prescribed set of interest, a component concentration observer, driven by a discontinuous input, is constructed and included in the closed-loop system. This observer-based controller also appears to smooth the control signal, thereby eliminating undesirable high-frequency temperature oscillations (the so-called chattering phenomenon [15, 16], caused by fast switching in the discontinuous control signal).

9.2 Application to Chemical Tubular Reactor

201

9.2.1 Problem Statement In order to have a homogeneous formulation of the plant equations (and, in particular, of the boundary conditions), we shall consider here a dimensionless version of the above model (9.15)–(9.20):

∂ x1 ∂t x1 (z, 0) ∂ x2 ∂t x2 (z, 0)

∂ 2 x1 ∂ x1 + k0 δ (1 − x2)r(x1 ) − β (x1 − xw ), −u 2 ∂z ∂z = x01 (z) ∂ 2 x2 ∂ x2 = D2 2 − u + k0 (1 − x2)r(x1 ), ∂z ∂z = x02 (z) = D1

∂ xi (z = 0) = uxi (z = 0), i = 1, 2 ∂z ∂ xi (z = L) = 0, i = 1, 2 Di ∂z Di

(9.21)

(9.22)

(9.23) (9.24)

where x = (x1 , x2 )T and xw are dimensionless variables defined as follows: x1 =

T − Tin Cin − C Tw − Tin , x2 = , xw = . Tin Cin Tin

(9.25)

The parameters D1 , D2 , α , β , δ and γ are related to the parameters in the original model as follows:

λea , D2 = Dma ρ Cp Δ H Cin 4h E β= ,δ =− ,γ= d ρ Cp ρ C p Tin RTin D1 =

(9.26)

whereas the initial distributions x01 (z) and x02 (z), and the function r(x1 ) are given by x01 =

T0 − Tin 0 Cin − C0 − γ , x2 = , r(x1 ) = e 1+x1 . Tin Cin

The properties of the above tubular reactor model have been studied in particular in [119] which discusses the existence and uniqueness of the state trajectories. As already mentioned in the introduction, the main question remains the existence of multiplicity of steady states and their stability in the general case of the non-adiabatic reactor model with different diffusion coefficients D1 and D2 . If the multiplicity is discussed in [120], their stability remains an open question. It is worth noting that the steady state multiplicity implies that the open loop process model is not globally asymptotically stable.

202

9 Asymptotic Stabilization of Minimum-phase Semilinear Systems

In the present example, we aim to control the temperature T and the concentration C (or equivalently, x1 and x2 ) along the reactor by acting on the coolant temperature Tw (or equivalently, xw ). For this, we have admitted that Tw is also a function of the space variable z. So, it is required to synthesize a control signal xw (z,t) such that the closed-loop system (9.21)–(9.24) with the state vectors x1 , x2 ∈ L2 (0, L) is asymptotically stable around the desired profiles y1 (z) and y2 (z). These profiles are governed by the steady state equations D1

∂ 2 y1 ∂ y1 + k0 δ (1 − y2)r(y1 ) − β (y1 − v∗ ) = 0 −u ∂ z2 ∂z ∂ 2 y2 ∂ y2 + k0 (1 − y2)r(y1 ) = 0 D2 2 − u ∂z ∂z ∂ yi (z = 0) = uyi (z = 0), i = 1, 2 ∂z ∂ yi (z = L) = 0, i = 1, 2 Di ∂z Di

(9.27) (9.28)

(9.29) (9.30)

corresponding to (9.21)–(9.24) under the prescribed (certainly, integrable) distribution v∗ (z) of the coolant temperature xw that ensures the (optimal) profiles y1 (z) and y2 (z) in the steady state (an appropriate optimal control problem is solved in advance and, as a matter of fact, the function v∗ (z) is assumed to be bounded).

9.2.2 Global Stabilization via State Feedback Controller 9.7, developed in the abstract setting, specializes here for the dimensionless reactor model (9.21)–(9.24): xw (z,t) = v∗ (z) − M(x1 , x2 ) =

M(x1 , x2 )Δ x1 (z,t) , Δ x1 (·,t)L2

k0 δ ψ (x1 , x2 ) + cΔ x1(·,t)αL2 , c > 0, α ∈ (0, 1) β

(9.31) (9.32)

where Δ x1 (z,t) = x1 (z,t) − y1 (z) and Δ x2 (z,t) = x2 (z,t) − y2 (z) are the state deviations from the desired profiles, the norm  x1 (·,t) − y1(·)L2 =

 L 0

[x1 (z,t) − y1 (z)]2 dz

is in the Hilbert space L2 (0, L), and

ψ (x1 , x2 ) = (1 − x2)r(x1 ) − (1 − y2)r(y1 )L2 .

9.2 Application to Chemical Tubular Reactor

203

Theorem 9.3. The chemical process (9.21)–(9.24) enforced by the discontinuous state feedback (9.31), (9.32) is globally asymptotically stable around the desired steady-state profiles (9.27)–(9.30) whereas the discontinuous output feedback (9.31) with the permanent state-independent gain M(x) = M > 0 stabilizes this process locally. Proof. Subtracting (9.27)–(9.30) from (9.21)–(9.24), let us rewrite the plant equations (9.21)–(9.24) in terms of the state deviations Δ x1 (z,t) = x1 (z,t) − y1 (z) and Δ x2 (z,t) = x2 (z,t) − y2 (z) from the desired profiles y1 (z), y2 (z), corresponding to the prescribed value v∗ (z) of the coolant temperature:

∂ Δ x1 ∂ 2 Δ x1 ∂ Δ x1 −u = D1 − β Δ x1 + ∂t ∂ z2 ∂z k0 δ (1 − Δ x2 − y2 )r(Δ x1 + y1 ) − k0δ (1 − y2 )r(y1 ) + β (xw − v∗ ), ∂ Δ x2 ∂ 2 Δ x2 ∂ Δ x2 = D2 + −u ∂t ∂ z2 ∂z k0 (1 − Δ x2 − y2)r(Δ x1 + y1 ) − k0 (1 − y2)r(y1 ), ∂ Δ xi ∂ Δ xi (z = 0) = uΔ xi (z = 0), Di (z = L) = 0, i = 1, 2. Di ∂z ∂z

(9.33)

After that, let us represent the resulting distributed parameter system (9.33) as a dynamical system (9.1), (9.2) in the Hilbert space H = L2 (0, L) ⊕ L2 (0, L) driven by the control signal ν = β (xw − v∗ ) ∈ L2 (0, L). The operators Ai , i = 1, 2 are now defined on the domains d 2 ξi (z) ∈ L2 (0, L), dz2 d ξi (z) d ξi (z) |z=0 = uξi (0), Di |z=L = 0} Di dz dz

D(Ai ) = {ξi (z) ∈ L2 (0, L) :

and specified as follows: A 1 = D1

d d d2 d2 − − u β , A = D −u 2 2 dz2 dz dz2 dz

(9.34)

whereas the nonlinearities f1 , f2 are given by f1 (ξ ) = k0 δ (1 − ξ2 − y2)r(ξ1 + y1 ) − k0 δ (1 − y2)r(y1 ), f2 (ξ ) = k0 (1 − ξ2 − y2 )r(ξ1 + y1 ) − k0(1 − y2)r(y1 ). Next, let us verify that (9.21)–(9.24) thus specified satisfies Theorems 9.1 and 9.2. Verification of Assumption 1. The infinitesimal differential operators A1 and A2 given by (9.34) are known (see, e.g., [241]) to generate C0 semigroups on L2 (0, L). Furthermore, if Di = 0, i = 1, 2, then the operators −A1 and −A2 are strongly ellip-

204

9 Asymptotic Stabilization of Minimum-phase Semilinear Systems

tic [188, p. 209], and by [188, Theorem 2.7, p.211] A1 and A2 generate analytical semigroups on L2 (0, L). Verification of Assumption 2. The following relations justify that the operator A1 is nonpositive definite on its domain:  L

d 2 Δ x1 (z) d Δ x1(z) − β Δ x1 (z)]Δ x1 (z)dz = −u dz2 dz 0  L d Δ x1 (z) d Δ x1 (z) 2 1 ] + D1 D1 [ Δ x1 (z)|L0 − u[Δ x1 ]2 (z)|L0 − − dz dz 2 0  L 1 1 2 2 2 β [Δ x1 ] (z)dz ≤ −u[Δ x1 ] (0) − u[Δ x1 ] (L) + u[Δ x1 ]2 (0) = 2 2 0 1 1 − u[Δ x1 ]2 (L) − u[Δ x1 ]2 (0) ≤ 0 f or all Δ x1 ∈ D(A1 ). 2 2 [D1

Verification of Assumption 3. The zero dynamics (9.3), rewritten in terms of Δ x2 , is given by

∂ Δ x2 ∂ 2 Δ x2 ∂ Δ x2 = D2 − k0Δ x2 r(y1 ), −u 2 ∂t ∂z ∂z

∂ Δ x2 (z = 0) = uΔ x2 (z = 0) ∂z ∂ Δ x2 (z = L) = 0. D2 ∂z D2

(9.35)

Differentiating the functional V (t) =

 L 0

[Δ x2 (z,t)]2 dz

along the trajectories of (9.35) yields  L

d 2 Δ x2 (z,t) d Δ x2 (z,t) − k0 Δ x2 (z,t)r(y1 (z))] × −u 2 dz dz 0  L d Δ x2 (z,t) d Δ x2 (z,t) 2 ] dz + 2D2 Δ x2 (z,t)dz = −2 D2 [ Δ x2 (z,t)|L0 − dz dz 0 V˙ (t) = 2

[D2

u[Δ x2 ]2 (z,t)|L0 − 2k0 −2k0r0

 L 0

 L 0

r(y1 (z))[Δ x1 ]2 (z,t)dz ≤

[Δ x1 ]2 (z,t)dz = −2k0r0V (t)

where r0 = min r(y1 (z)). 0≤z≤L

Taking into account that from a physical point of view the desired temperature distribution T1 (z), corresponding to the steady state y1 (z) = (T1 (z) − Tin )/Tin , is uniformly greater than the absolute zero (the solution y1 (z) of (9.27) is a continuous

9.2 Application to Chemical Tubular Reactor

205

function!), i.e., minz T1 (z) > 0, and hence r0 > 0, it follows the exponential stability of the zero dynamics (9.35). The verification of Assumptions 4 and 6 is straightforward. E − RT in Verification of Assumption 5. Since the function r(x1 ) = r( T −T is Tin ) = e bounded in the physical domain T > 0, the verification of Assumption 5 is also straightforward. Finally, we note that the discontinuous state feedback (9.31) is nothing else than the unit controller (9.7) applied to the chemical reactor (9.33) and definitely, the disturbance rejection condition (9.8) holds with the nonlinear gain (9.32). So, the state deviation equations (9.33) satisfy Theorems 9.1 and 9.2 and by these theorems the chemical process (9.21)–(9.24) enforced by the discontinuous state feedback (9.31), (9.32) is globally asymptotically stable around the desired steadystate profiles (9.27)–(9.30), whereas the discontinuous output feedback (9.31) with the permanent state-independent gain M(x) = M > 0 stabilizes this process locally. Theorem 9.3 is proved. Remark 9.2. To retain the proof of Theorem 9.3 in the case of the plug flow reactor (9.21)–(9.24) when Di = 0, i = 1, 2, one should confine the investigation to mild solutions of (9.21)–(9.24) and invoke Remark 9.1, while applying Theorems 9.1 and 9.2. Thus, the state feedback controller (9.31) with the nonlinear gain (9.32) is shown to globally asymptotically stabilize system (9.21)–(9.24). In turn, the local asymptotic stability of the closed-loop system (9.21)–(9.24), (9.31) is concluded for the output (temperature) feedback controller (9.31) with a permanent gain M(x,t) = M > 0, so that there is no need for measuring the process component concentration, which is usually unavailable in practice. The larger M the greater the attraction domain of the steady-state solution, and hence, the realistic temperature output feedback design becomes feasible. However, while designing such a controller, one should prevent undesired high-frequency temperature oscillations, caused by fast switching in the discontinuous control input. In the sequel, we demonstrate how these oscillations, with amplitudes proportional to M, can be eliminated by running parallel a component concentration observer driven by a discontinuous signal.

9.2.3 Concentration Observer Design A simple reduced-order component concentration observer for the dimensionless model (9.21)–(9.24) when the temperature x1 (z,t) is the only available measurement on the system is constructed as follows:

∂ x˜2 ∂ 2 x˜2 ∂ x˜2 = D2 2 − u + k0 (1 − x˜2)r(x1 ), ∂t ∂z ∂z x˜2 (z, 0) = x˜02 (z)

(9.36)

206

9 Asymptotic Stabilization of Minimum-phase Semilinear Systems

∂ x˜2 (z = 0) = ux˜2 (z = 0) ∂z ∂ x˜2 (z = L) = 0. D2 ∂z D2

(9.37) (9.38)

The reconstruction error e2 (z,t) = x2 (z,t) − x˜2 (z,t) is shown to converge to zero lim e2 (·,t)L2 = 0

t→∞

(9.39)

if a twice continuously differentiable initial distribution x˜02 (z), satisfying the boundary conditions (9.37), (9.38), is imposed on the observer state and the temperature of the controlled process remains uniformly isolated from the absolute zero, i.e., T (z,t) > T0 for all z ∈ [0, L], t ≥ 0 and a sufficiently small T0 > 0. Moreover, in that case, Convergence 9.39 is exponential with the decay rate μ = k0 rmin where in )=e rmin = r( T0T−T in

E − RT

0

.

Theorem 9.4. Consider the reduced-order observer (9.36)–(9.38) for the controlled system (9.21)–(9.24). Let for all t ≥ 0 (9.21)–(9.24) evolve in the region x1 > (T0 − Tin )/Tin . Then the reconstruction error e2 (z,t) has the property that e2 (·,t)L2 ≤ le−μ t for all twice continuously differentiable initial errors e2 (z, 0), satisfying the boundary conditions (9.37), (9.38) and such that the condition e2 (·, 0)L2 ≤ l holds with a positive constant l. Proof. Representing the observer dynamics (9.36)–(9.38) in terms of the reconstruction error e2 (z,t) and, as in proving Theorem 9.3, differentiating the functional Ve (t) =

 L 0

e22 (z,t)dz

along the trajectories of the resulting error system, we obtain V˙e (t) ≤ −2k0

 L 0

r(x1 (z,t))e22 (z,t)dz ≤ −2k0rminVe (t) = −2 μ Ve(t).

Apparently, the latter inequality ensures the exponential stability Ve (t) ≤ Ve (0)e−2μ t of the error system, thereby yielding that e2 (·,t)L2 ≤ le−μ t for all initial errors e2 (z, 0), satisfying the condition e2 (·, 0)L2 ≤ l. The proof of Theorem 9.4 is thus completed. One of the main drawbacks of the above reduced-order observer of the chemical process is that the convergence rate μ cannot be made as large as desired because it is completely determined by the internal dynamics of the process. To have an a priori given convergence rate μ ∗ > k0 rmin , larger than that of the reduced-order observer, one can utilize the following full-order observer:

9.2 Application to Chemical Tubular Reactor

∂ x¯1 ∂t x¯1 (z, 0) ∂ x¯2 ∂t x¯2 (z, 0)

207

∂ 2 x¯1 ∂ x¯1 + k0 δ (1 − x¯2)r(x1 ) − β (x¯1 − xw ) + v, −u ∂ z2 ∂z = x¯01 (z) ∂ 2 x¯2 ∂ x¯2 = D2 2 − u + k0 (1 − x¯2)r(x1 ) − Nv, ∂z ∂z = x¯02 (z) = D1

∂ x¯i (z = 0) = ux¯i (z = 0), i = 1, 2 ∂z ∂ x¯i (z = L) = 0, i = 1, 2 Di ∂z Di

where N =

μ ∗ −k0 rmin k0 rmin δ ,

(9.40)

(9.41)

(9.42) (9.43)

the observer input is given by v=

M0 r(x1 (·,t)L2 e¯1 (z,t) , e¯1 (·,t)L2

(9.44)

the initial distributions x¯01 (z) and x¯02 (z) are twice continuously differentiable and satisfy the boundary conditions (9.42), (9.43), and M0 is a sufficiently large constant. In what follows, the reconstruction errors between the plant outputs and the observer outputs are denoted by e¯1 (z,t) = x1 (z,t) − x¯1 (z,t) and e¯2 (z,t) = x2 (z,t) − x¯2(z,t). Theorem 9.5. Consider the full-order observer (9.40)–(9.44) for the controlled system (9.21)–(9.24). Let the following conditions be satisfied: in with some T0 > 0; 1. For all t ≥ 0, (9.21)–(9.24) evolves in the region x1 > T0T−T in 2. The initial distribution of the model variable x¯1 (z,t) is the same as that of the measured variable x1 (z,t), i.e., x¯01 (z) = x01 (z); 3. M0 > k0 δ l for some constant l > 0.

Then for all t ≥ 0, (9.40)–(9.44) is driven along the hyperplane e¯1 = 0 and the reconstruction error e¯2 (z,t) has the property that e¯2 (·,t)L2 ≤ le−μ

∗t

(9.45)

for all initial errors e¯2 (·, 0) satisfying the condition e¯2 (·, 0)L2 ≤ l. Proof. Since a detailed proof is similar to that of Theorem 9.3, here we provide a sketch only. To begin with, let us represent the full-order observer dynamics (9.40)– (9.49) in terms of the reconstruction errors e¯1 and e¯2 :

208

9 Asymptotic Stabilization of Minimum-phase Semilinear Systems

∂ e¯1 = ∂t e¯1 (z, 0) = ∂ e¯2 = ∂t e¯2 (z, 0) =

D1

∂ 2 e¯1 ∂ e¯1 − k0 δ e¯2 r(x1 ) − β e¯1 − v, −u ∂ z2 ∂z

0

(9.46)

∂ 2 e¯2 D2 2 ∂z

∂ e¯2 − k0 e¯2 r(x1 ) + Nv, ∂z x02 (z) − x¯02 (z) −u

∂ e¯i (z = 0) = ue¯i (z = 0), i = 1, 2 ∂z ∂ e¯i (z = L) = 0, i = 1, 2 Di ∂z Di

(9.47)

(9.48) (9.49)

Differentiating the functional V¯1 (t) =

 L 0

e¯21 (z,t)dz

along the trajectories of the error system (9.46)–(9.49) subject to e¯2 L2 ≤ l, utilizing the observer algorithm (9.44), and applying the conditions of the theorem, yields  L

M0 r(x1 (·,t)L2 e¯1 (z,t) + k0δ e¯2 (z,t) × e¯1 (·,t)L2  r(x1 (z,t))]dz ≤ −(M0 − k0 δ l)r(x1 (·,t)L2 V¯1 (t) < 0 i f V¯1 (t) = 0, (9.50) V˙¯1 (t) ≤ −

0

e¯1 (z,t)[

thereby proving that the reconstruction error e¯1 (t) is maintained within the discontinuity hyperplane e¯1 = 0 (9.51) whenever e¯2 (t) remains in the ball e¯2 L2 ≤ l. By Theorem 2.4, the system motion along the discontinuity hyperplane (9.51) is governed by equation

∂ e¯2 ∂ 2 e¯2 ∂ e¯2 = D2 2 − u − k0(1 + N δ )r(x1 ), ∂t ∂z ∂z ∂ e¯i (z = 0) = ue¯i (z = 0), i = 1, 2 ∂z ∂ e¯i (z = L) = 0, i = 1, 2 Di ∂z Di

(9.52)

(9.53) (9.54)

which is derived by substituting the equivalent continuous observer input value veq = k0 δ e¯2 r(x1 ),

9.2 Application to Chemical Tubular Reactor

209

resulted from (9.46), (9.51), into (9.47). By applying Theorem 9.4 to (9.52)–(9.54) the sliding motion e¯2 (z,t) has Property (9.45) with μ ∗ = k0 (1 + N δ ) for all initial errors e¯2 (·, 0) satisfying the condition e¯2 (·, 0)L2 ≤ l. To complete the proof, it remains to note that the above property (9.45) ensures that the sliding motion e¯2 (t) does not leave the ball e¯2 L2 ≤ l. Thus, the reconstruction error e¯1 (t) is actually maintained along the discontinuity hyperplane (9.51) for all trajectories of the error system (9.47)–(9.49) initialized within the ball e¯2 (·, 0)L2 ≤ l. Theorem 9.5 is proved. Thus, Theorem 9.5 yields the conditions which ensure that the output of the fullorder observer converges to that of the chemical process as fast as desired. The validity of Condition 1 can be insured by selecting T0 small enough. In that case, Condition 1 is nothing else than a well-known unreachability property of the Kelvin absolute zero. Condition 2 is involved for a technical reason: it allows us to guarantee the exponential convergence (9.45) of the reconstruction error with an explicitly given multiplier l. According to Condition 2, a certain initial distribution should be imposed on the observer. If this condition is relaxed, the exponential convergence of the reconstruction error holds with the same decay rate; however, an uncertain multiplier should be substituted in (9.45). Condition 3 specifies a lower bound for the gain of the discontinuous observer input.

9.2.4 Regional Stabilization via Temperature Feedback In this section, the full-order concentration observer is involved into the closed-loop system to achieve a local temperature feedback stabilization of (9.21)–(9.24), with the region of attraction containing a prescribed set of the form  Bl (y1 , y2 ) = {x1 (z), x2 (z) : x1 − y1 2L2 + x2 − y2 2L2 ≤ l} with l > 0 such that Bl (y1 , y2 ) is within the physical domain where the process temperature T (z) and the component concentration C(z) are everywhere positive, i.e., Bl (y1 , y2 ) ⊂ Ω = {x1 (z), x2 (z) : x1 > −1, x2 < 1}. In the sequel, such a stabilization is referred to as a regional stabilization that has become standard in the literature [110]. The following control law is shown to regionally stabilize (9.21)–(9.24) around the desired steady state: M1 (x1 , x¯2 ,t)[Δ x1 (t, z)] , Δ x1 (t, ·)L2

(9.55)

∗ k0 δ l r(x1 (·,t))L2 e−μ t , c > 0, α ∈ (0, 1) β

(9.56)

xw (t, z) = v∗ (z) − M1 (x1 , x¯2 ,t) =

k0 δ ψ (x1 , x¯2 ) + cΔ x1(t, ·)αL2 + β

210

9 Asymptotic Stabilization of Minimum-phase Semilinear Systems

where x¯2 (t, z) is an output of the full-order observer (9.40)–(9.44). Theorem 9.6. Let the conditions of Theorem 9.5 be satisfied. Then the chemical process (9.21)–(9.24) is regionally asymptotically stabilized around the desired steadystate profiles (9.27)–(9.30) by the dynamic temperature feedback controller (9.40)– (9.44), (9.55), (9.56). Proof: Analyzing the proof of Theorem 9.3, one can note that the assertion of this theorem remains true if the nonlinear gain M(x1 , x2 ) in the control law (9.31), (9.32) is replaced with a larger one. Since, due to Theorem 9.5, the reconstruction error e¯2 (z,t) satisfies (9.45), and hence M1 (x1 (·,t), x¯2 (·,t),t) = M(x1 (·,t), x2 (·,t)) +

∗ k0 δ l r(x1 (·,t))L2 e−μ t + β

k0 δ k0 δ ψ (x1 (·,t), x¯2 (·,t)) − ψ (x1 (·,t), x2 (·,t)) = M(x1 (·,t), x2 (·,t)) + β β ∗ k0 δ [lr(x1 (·,t))L2 e−μ t + (1 − x¯2)r(x1 ) − (1 − y2)r(y1 )L2 − β (1 − x2)r(x1 ) − (1 − y2)r(y1 )L2 ] ≥ M(x1 (·,t), x2 (·,t)) + k0 δ ∗ [lr(x1 (·,t))L2 e−μ t − e¯2 (·,t)r(x1 (·,t))L2 ] ≥ M(x1 (·,t), x2 (·,t)) β for all t ≥ 0, the validity of Theorem 9.6 is thus established. The detailed proof of Theorem 9.6 is similar to that of Theorem 9.3 and it is therefore omitted. To conclude this section, we note that, due to Theorem 9.5, the discontinuous controller (9.40)–(9.44), (9.55) is such that the nonlinear gain (9.56) approaches zero as x1 → y1 , x2 → y2 ,t → ∞. Thus, one can expect that in comparison to the discontinuous temperature feedback (9.31) with a permanent gain M(x) = M > 0 the above controller eliminates the aforementioned chattering phenomenon around the desired steady state profile. This conclusion is further supported by numerical simulations.

9.2.5 Simulation Results As already mentioned, exothermic reactors represent an interesting class of systems that may exhibit multiple steady states, either stable or unstable. As a matter of illustration, Fig. 9.1 shows the three steady state temperature profiles of an adiabatic (i.e., for which β = 0 in (9.21)) reactor for the following set of parameter values: D1 = D2 = 0.167, δ = 0.5, k0 = 2.426 107, γ = 20

(9.57)

By using the arguments in [232], we can deduce that the two extreme (upper and lower) steady state profiles are stable while the intermediate one is unstable.

9.2 Application to Chemical Tubular Reactor

211

450

T (K)

400

350

300

0

0.1

0.2

0.3

0.4

0.5 z

0.6

0.7

0.8

0.9

1

Fig. 9.1 Multiple equilibrium profiles in a non-isothermal adiabatic tubular reactor

Figure 9.1, which corresponds to a particular situation, is indeed easy to obtain, because the two nonlinear ordinary differential equations that represent the steady state profile can be decoupled by using the state transformation ζ = x1 - δ x2 (indeed related to the notion of reaction invariants, e.g., [80]). However, either in the nonadiabatic reactor or when the two diffusion coefficients D1 and D2 are not equal to each other, this decoupling is not possible anymore. If there exist now conditions for exhibiting multiple steady states [120], their numerical computation has not yet been solved. In order to be as close as possible to a tubular reactor model with possibly unstable equilibrium profiles, we have selected the above model in the non-adiabatic conditions (with β = 0.05) to test the performance of the observers and the feedback control algorithm. Figures 9.2–9.5 illustrate the behavior of Observers 9.40–9.44 and 9.36–9.38, and that of the feedback controller (9.55), (9.56) in numerical simulation for the same set of parameters as above and a heat exchange coefficient β = 0.05 s−1 . The process model has been initialized with two constant profiles x1 (z, 0) = 1, x2 (z, 0) = 0, i.e., there is no reactant initially in the reactor and the initial temperature is the temperature in the feed. The observers have been initialized with the correct value for x2 and 10% error on the estimation of x1 (x1 = 0.9). The reference coolant temperature in a dimensionless format, v∗ (z) corresponds to the reactor steady-state with Tw = 300K, and is therefore set to (Tw∗ − Tin )/Tin = -40/340. The equations have been integrated by considering a backward finite difference approximation for the first-order space derivative ∂ /∂ z and a central difference for the second-order space derivative:

212

9 Asymptotic Stabilization of Minimum-phase Semilinear Systems

Fig. 9.2 Control simulation results of the two variables x1 and x2 as a function of time and space

∂ x ∼ x(t, zi ) − x(t, zi−1 ) ∂ 2 x ∼ x(t, z + 1) − 2x(t, zi) + x(t, zi−1 ) , 2 = = ∂z Δz ∂z (Δ z)2

(9.58)

where Δ z is the spatial step (here, equal to 0.01). The controller parameters have been set to the following values in line of the conditions emphasized in the preceding sections:

μs = α eγ , c = 0.25, l = 1

(9.59)

Figure 9.2 shows the time evolution of the two profiles of x1 and x2 with the feedback controller connected to the full-order observer. Figure 9.3 shows the profiles of x1 at three time instants: t = 0, 100 s, 500s. Figures 9.4 and 9.5 compare the convergence properties of the observers, the fullorder observer (9.40)–(9.44) and the reduced-order observer (9.36)–(9.38). Figure 9.4 shows the 3D-plot of the observation errors e2 for both observers (the full-order one at the top, the reduced-order one at the bottom) while Fig. 9.5 compares the space distribution of the observation errors e2 at the final simulation time (t = 500s). The design parameters of the full-order observer have been set by following Assumptions 3 in Theorem 9.5: M0 = 10 k0 δ l, and N = 10. Note that, as expected, the full-order observer converges faster than the reduced-order one.

9.2 Application to Chemical Tubular Reactor

213

−3

1

x 10

t =0

0

t = 100 s

x1

−1

−2

−3

t = 500 s −4

−5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

z (m)

Fig. 9.3 Control simulation results of the variable x1 at different time instants

Fig. 9.4 Comparison of the two observers in 3D

0.9

1

214

9 Asymptotic Stabilization of Minimum-phase Semilinear Systems 0.1

0.09

0.08

0.07

Reduced Order Observer

e2

0.06

0.05

Full Order Observer

0.04

0.03

0.02

0.01

0

0

0.1

0.2

0.3

0.4

0.5

0.6

z (m)

Fig. 9.5 Comparison of the two observers at time t = 500 s

0.7

0.8

0.9

1

Chapter 10

Global Asymptotic Stabilization of Uncertain Time-delay Systems

To this end, the unit feedback synthesis is developed for a class of uncertain timedelay systems with nonlinear disturbances and unknown delay values whose unperturbed dynamics are linear. Being inspired from [179], the present synthesis is based on the delay-dependent stability criterion, which is derived within the framework developed in Sect. 3.8. The controller constructed proves to be robust against sufficiently small delay variations and weak external (possibly, unmatched) disturbances. It is worth noticing that allowing unmatched disturbances is a step beyond a standard sliding mode control treatment. Specifically, the critical delay value when the closed-loop system, corresponding to this value, becomes asymptotically unstable, is explicitly calculated as a function of linear growth constants of the unmatched disturbances. Performance issues of the controller are illustrated in a simulation study.

10.1 Problem Statement The present investigation is focused on time-delay systems that, by means of a nonlinear change of state coordinates and a feedback transformation, are representable in the form ⎧ 2 ⎪ ⎪ dz1 (t) ⎪ ∑ (A1i zi (t) + B1izi (t − τ )) ⎪ = ⎪ i=1 ⎪ ⎪ dt ⎪ +p1 (t, z1 (t)) + p2(t, z1 (t − τ )), ⎨ 2 (10.1) dz2 (t) ⎪ ∑ (A2i zi (t) + B2i zi (t − τ )) ⎪ = ⎪ i=1 ⎪ ⎪ dt ⎪ +u + f (t, zt ), ⎪ ⎪ ⎩ z(t) = φ (t) for t ∈ [−τ , 0] . Hereinafter, z(t) = (z1 , z2 )T is the state vector with components z1 ∈ IR n z2 ∈ IR r , Ai j , Bi j i, j = 1, 2 are constant matrices of corresponding dimensions, u ∈ IR r is the input vector; p1 , p2 ∈ IR n and f ∈ IR r are external disturbances, φ is the initial

215

216

10 Global Asymptotic Stabilization of Uncertain Time-delay Systems

piecewise continuous function defined on [−τ , 0], zt (θ ) is the function associated with z and defined on [−τ , 0] by zt (θ ) = z(t + θ ). Since the delay value τ is typically unknown a priori, in the sequel we shall also use the notation zt,τmax (θ ) for the function zt,τmax (θ ) = z(t + θ ) defined on some larger interval [−τmax , 0] with τmax > τ . As usual, the symbol I will stand for the identity matrix of an appropriate dimension. Setting     A11 A12 B11 B12 A0 = , B0 = , A21 A22 B21 B22       0 0 pi , (10.2) ,F= , i = 1, 2, C = Pi = f I 0 (10.1) is simplified to z˙(t) = A0 z(t) + B0z(t − τ ) + Cu + P1(t, z(t)) + P2 (t, z(t − τ )) + F(t, zt ).

(10.3)

Although the investigation is confined to time-delay systems with delayed states, however, the extension to the case of delayed inputs is straightforward. Indeed, let a system with an input delay be governed by ⎧ ˙ = Ax(t) + Bx(t − τ ) + Du(t − τ ), t > 0 ⎨ x(t) x(t) = ψ1 (t), t ∈ [−τ , 0] , (10.4) ⎩ u(t) = ψ2 (t), t ∈ [−τ , 0] where x ∈ IR n ; A, B are constant n × n matrices, D is a n × r matrix, u ∈ IR r is the input vector, and ψ1 and ψ2 are piecewise continuous functions. Then, an additional input integrator transforms the system into the above form ⎧ ⎪ ⎨ dz1 (t) = Az1 (t) + Bz1 (t − τ ) + Dz2 (t − τ ), dt (10.5) ⎪ ⎩ dz2 (t) = v dt with the control input v ∈ IR r . Throughout, the following assumptions are made for technical reasons. 1. p1 (t, z(t)) and p2 (t, z(t − τ )) are Lipschitz continuous and satisfy the linear growth conditions p1 (t, z(t)) ≤ α1 z(t), p2 (t, z(t − τ )) ≤ α2 z(t − τ ) with some positive constants α1 , α2 . 2. f is Lipschitz continuous and it is bounded  f (t, zt ) ≤ Ψ (t, zt,τmax ) by a continuous functional Ψ (t, zt,τmax ), known a priori. 3. (A11 + B11, A12 + B12) is controllable. Assumptions 1 and 2, coupled together, guarantee the unforced functional differential equation (10.1) with u = 0 to have a unique solution for all t ≥ 0 [115]. Apart from this, Assumption 2 allows one to reject the uncertain disturbance f by a bounded-gain sliding mode feedback. Assumption 3 implies the controllability of

10.2 Background Material on Discontinuous Time-delay Systems

217

the delay-/disturbance-free system (10.1) subject p1 = p2 = 0, τ = 0, f = 0 (see [227, p. 101] for details). Thus, the above assumptions ensure that the delay-free system (10.1) with τ = 0 turns out to be stabilizable under sufficiently small external disturbances p1 , p2 , f . Due to conservatism, the time-delay system (10.1) remain stabilizable for any positive τ ≤ τmax not exceeding some positive, sufficiently small τmax . The goal of the present investigation is to construct a stabilizing unit feedback controller of (10.1) that makes the value of τmax as large as possible.

10.2 Background Material on Discontinuous Time-delay Systems Since the closed-loop system, driven by a unit feedback controller, is governed by a functional differential equation with a discontinuous right-hand side, the precise meaning of such an equation should first be defined. For this purpose, let us represent the time-delay system (10.3) as a dynamic system d ζt = A ζt + C u + P1(t, μ (t)) + P2(t, ν (−τ )) + F (t, νt ) dt

(10.6)

evolving in the Hilbert space M2n+r = IR n+r × Ln+r 2 [−τ , 0]. The instantaneous state ζt = (μ (t), νt (·))T of (10.6) at a time moment t consists of the components μ (t) = z(t) ∈ IR n+r and νt (θ ) = zt (θ ) ∈ Ln+r 2 [−τ , 0]. The linear operators A : D(A ) ⊂ M2n+r → M2n+r , C : IR r → M2n+r and nonlinear operators Pi : IR 1+n+r → M2n+r , i = 1, 2, F : M21 → M2n+r , involved into the state equation (10.6), are given by     A0 μ + B0 ν (−τ ) μ = , A dν ν (θ ) dθ       F Pi Cu . , Pi = ,F= C= 0 0 0

(10.7)

Apparently, C is a bounded operator on IR r whereas the nonlinear operators Pi , i = 1, 2 and F are Lipschitz continuous on their domains due to Assumptions 1 and 2. In turn, it is well known [55], that the infinitesimal operator A generates a (strongly continuous) C0 -semigroup on M2n+r and its domain D(A ) = {ζ = (μ , ν )T ∈ M2n+r : is dense in M2n+r .

dν ∈ Ln+r 2 [−τ , 0] and ν (0) = μ } dθ

218

10 Global Asymptotic Stabilization of Uncertain Time-delay Systems

Thus, solutions of the Hilbert space-valued dynamic system (10.6) with a discontinuous right-hand side can rigorously be defined in the generalized sense of Definition 2.10. It is worth noting that given an initial condition ζ0 = (φ (0), φ (·)) ∈ M2n+r , the Cauchy problem, thus stated for (10.6) with the assumptions above, proves to have a unique solution whenever the operator G C is continuously invertible. Just in case, the sliding motion of the system on the discontinuity hyperplane is governed by the sliding mode equation derived through the equivalent control method. The validity of the equivalent control method in the present case is straightforwardly established by applying Theorem 2.4. According to this method, the sliding mode equation is obtained by substituting the solution ueq = −[G C ]−1 G [A ζt + P1 (t, μ (t)) + P2(t, ν (−τ )) + F (t, νt )] of the equation G ζ˙ = 0 into (10.6) for u. The resulting sliding mode equation plays an important role in the subsequent stability analysis of the discontinuous time delay system in question. Since this equation contains no discontinuities in the right-hand side, its stability is established via standard techniques.

10.3 Delay-/Disturbance-dependent Stability Criterion For later use, we derive delay-/disturbance-dependent stability conditions for the time-delay system dx(t) = Ax(t) + Bx(t − τ ) + p1(t, x(t)) + p2(t, x(t − τ )) dt x ∈ IR n , x(t) = ζ (t) for t ∈ [−τ , 0]

(10.8)

with a piece-wise continuous initial function ζ (t), some constant matrices A, B, and the same nonlinearities as before. Our objective is to find an upper bound τmax of admissible delay values τ such that the above system, while being asymptotically stable with τ = 0, is so for all delays 0 ≤ τ < τmax . This upper bound τmax = τmax (α1 , α2 ) depends on the linear growth constants α1 , α2 of the disturbances p1 , p2 and relates to a positive solution of the following optimization problem: τmax (α1 , α2 ) = sup τ subject to the constraints ⎞ ⎛ H(τ ) P P τ PB τ PB τ PBA τ PB2 ⎜ P −γ1 I 0 0 0 0 0 ⎟ ⎟ ⎜ ⎜ P 0 −γ2 I 0 0 0 0 ⎟ (10.9) ⎟ ⎜ ⎟<0 ⎜ τ BT P 0 0 − γ τ I 0 0 0 3 ⎟ ⎜ ⎜ τ BT P 0 0 0 −γ4 τ I 0 0 ⎟ ⎟ ⎜ T T ⎝ τA B P 0 0 0 0 − τ R1 0 ⎠ τ B2T P 0 0 0 0 0 − τ R2

10.3 Delay-/Disturbance-dependent Stability Criterion

219

over all positive constants γi , i = 1, 2, 3, 4 and symmetric positive definite matrices P, R1 , R2 ∈ IR n×n , and H(τ ) = (E T P + PE) + (γ1 α12 + γ2 α22 )I + τ (γ3 α12 + γ4 α22 )I + τ (R1 + R2) with E = A + B. Theorem 10.1. Let the matrix E = A + B be Hurwitz. For α1 , α2 > 0 sufficiently small, the above optimization problem has a positive solution τmax (α1 , α2 ) and (10.8) is globally asymptotically stable for each delay value τ ∈ [0, τmax (α1 , α2 )). Proof. Consider the following Lyapunov–Krasovskii functional V (zt ) = V1 (zt ) + V2(zt ) + V3(zt ) where

(10.10)

V1 (zt ) = zT (t)Pz(t), V2 (zt ) =

 t  t t−τ s  t−τ

+

t−2τ

zT (w)(R1 + γ3 α12 In )z(w)dwds  t s

zT (w)(R2 + γ4 α22 In )z(w)dwds,

V3 (zt ) = γ2 α22

 t t−τ

zT (w)z(w)dwds,

P, R1 , R2 are symmetric positive definite matrices, and γi , i = 1, 2, 3, 4 are positive constants. The functional V (zt ) is positive definite and radially unbounded. Using t the representation z(t − τ ) = − t− τ z˙(θ )d θ + z(t), we can rewrite the system in the form dz(t) = (A + B)z(t) dt −B

 t

t−τ

[Az(w) + Bz(w − τ ) + p1(w, z(w)) + p2 (w, z(w − τ ))]dw

+ p1 (t, z(t)) + p2 (t, z(t − τ )).

With this in mind, let us calculate the derivative of V (zt ) along the trajectories of (10.8):

220

10 Global Asymptotic Stabilization of Uncertain Time-delay Systems

V˙1 = zT (E T P + PE)z + 2zT Pp1 + 2zT Pp2 − 2zT PBA − 2zT PB2 − 2zT PB − 2zT PB

 t

t−τ  t t−τ

 t

t−τ  t t−τ

z(w)dw z(w − τ )dw

p1 (w, z(w))dw p2 (w, z(w − τ ))dw,

V˙2 = τ zT (R1 + γ3 α12 I)z + τ zT (R2 + γ4 α22 I)z − −

t

t−τ t t−τ

zT (w)(R1 + γ3 α12 I)z(w)dw zT (w − τ )(R2 + γ4 α22 I)z(w − τ )dw,

V˙3 = γ2 α22 zT (t)z(t) − γ2 α22 zT (t − τ )z(t − τ ).

(10.11)

Due to the well known inequality 2uT v ≤ α uT Ru + α −1 vT R−1 v, valid for any vectors u, v ∈ IR n , any symmetric positive definite matrix R ∈ IR n×n and a positive T T constant α , the following six inequalities are in force with Z1 = PBAR−1 1 A B P, 2T T Z2 = PB2 R−1 2 B P, Z3 = PBB P, and non-negative γi , i = 1, 2, 3, 4: 2zT Pp1 ≤ zT (γ1−1 P2 + γ1 α12 I)z, 2zT Pp2 ≤ γ2−1 zT P2 z + γ2 α22 z(t − τ ) T z(t − τ ), t t T T −2zT PBA t− τ z(w)dw ≤ τ z Z1 z + t−τ z (w)R1 z(w)dw, t t T 2 T −2z PB t−τ z(w − τ )dw ≤ τ z Z2 z + t−τ zT (w − τ )R2 z(w − τ )dw, t −1 T 2 t T −2zT PB t− τ p1 (w, z(w))dw ≤ τγ3 z Z3 z + γ3 α1 t−τ z (w)z(w)dw, t t −1 T T 2 −2z PB t−τ p2 (w, z(w − h))dw ≤ τγ4 z Z3 z + γ4α2 t−τ zT (w − τ )z(w − τ )dw. Now it follows from (10.11) that the inequality V˙ (zt ) ≤ zT M(τ )z holds for M(τ ) = (E T P + PE) + (γ1−1 + γ2−1 )P2 + (γ1 α12 + γ2 α22 + τγ3 α12 + τγ4 α22 )I + τ (R1 + R2) −1 −1 T T 2 −1 2T T T +τ PBAR−1 1 A B P + τ PB R2 B P + τγ3 PBB P + τγ4 PBB P. The matrix E is Hurwitz and hence M(0) = (E T P + PE) + (γ1−1 + γ2−1 )P2 + (γ1 α12 + γ2 α22 )I is negative definite if P is a symmetric positive definite solution of the Lyapunov equation E T P + PE = −I and positive constants γ1 , γ2 , are sufficiently small. By continuity, M(τ ) remains negative definite for sufficiently small positive τ . Since by Schur’s lemma [29], M(τ ) < 0 if and only if (10.9) holds, it follows that the aforementioned optimization problem has a positive solution τmax (α1 , α2 ).

10.4 Unit State Feedback Controller

221

Thus, the matrix M(τ ) is negative definite for all τ ∈ [0, τmax (α1 , α2 )), and (10.8) is, therefore, globally asymptotically stable for all delays 0 ≤ τ < τmax (α1 , α2 ). The proof of Theorem 10.1 is completed. It is worth noticing that (10.9) represents a linear matrix inequality in P, R1 , R2 ,γ1 ,γ2 ,γ3 ,γ4 , and it can therefore efficiently be solved by convex optimization algorithms (see [29]). Example 10.1 (Comparison with other criteria). Let (10.8) be specified by     −1 0 −2 0 (10.12) ,B = A= −1 −1 0 −1 and let p1 and p2 be such that p1  ≤ 0.2 z1 and p2  ≤ 0.2 z1 (t − τ ). Using Theorem 10.1, we find τmax (0.2, 0.2) = 0.623, whereas the largest upper bound of time-delay available in the literature (see [131, 217, 218]) is τmax = 0.4428. It should also be noted that neither the disturbance-free criteria from [84, 85] nor the delay-independent criterion from [244] is capable of providing any conclusion. Thus exemplified, the proposed criterion is capable of enhancing the existing results.

10.4 Unit State Feedback Controller The following state feedback control law s(z(t)) , s(z(t)) g > 0, ε ∈ [0, 1)

u = −[Ω0 (t, zt,τmax , α1 , α2 ) + Ψ (t, zt,τmax ) + gsε ]

(10.13)

is proposed to drive the uncertain time-delay system (10.1) to a linear hyperplane s(z) = 0 with s(z) = z2 + Kz1 (10.14) in finite time, thereby globally asymptotically stabilizing the system for each τ ∈ [0, τmax ). Hereinafter, Ψ (t, zt,τmax ) is the same upper bound of the norm of the external disturbance f (t, zt ) as it is in Assumption 2, zt,τmax (θ ) stands for the function z(t − θ ) with θ ∈ [−τmax , 0] and τmax > τ , the linear functional 2

Ω (t, zt ) = ∑ {A2i zi (t) + B2izi (t − τ ) + i=1

K[A1i zi (t) + B1izi (t − τ ) + p1(t, z1 (t)) + p2(t, z1 (t − τ ))]} is viewed on the trajectories of the controlled system (10.1), and

Ω0 (t, zt,τmax , α1 , α2 ) =

sup

(Ω (t, zt (θ ))

θ ∈[−τmax ,0]

222

10 Global Asymptotic Stabilization of Uncertain Time-delay Systems

is its upper bound1 on a time interval θ ∈ [−τmax , 0]. In the proposed synthesis, the upper bound τmax = τmax (α1 , α2 ) and the matrix K ∈ IR n×n are subject to the optimization

⎛

τmax (α1 , α2 ) = sup τ under the constraints

J(τ ) S S τS τS γ5 L(τ ) γ6 L(τ ) ⎞ 0 0 0 0 0 S −γ1−1 α1−2 In ⎜ S 0 0 0 0 ⎟ 0 −γ2−1 α2−2 In ⎜ τS ⎟ 0 0 −τγ3−1 α1−2 In 0 0 0 ⎟ ⎜ ⎝ τS 0 0 0 −τγ4−1 α2−2 In 0 0 ⎠ γ5 L(τ )T 0 0 0 0 −τ Q1 0 γ6 L(τ )T 0 0 0 0 0 −τ Q2

<0

(10.15)

(A11 − A12 K)S < γ5 S (B11 − B12 K)S < γ6 S S > 0, Q1 > 0, Q2 > 0, γi > 0, i = 1, . . . , 6

with symmetric positive definite matrices S, Q1 , Q2 ∈ IR (n−m)×(n−m) , J(τ ) = [(A11 + B11 ) − (A12 + B12 )K]S + S[(A11 + B11) − (A12 + B12)K]T +[γ1−1 + γ2−1 + τ (γ3−1 + γ4−1)γ62 ]I + τ (Q1 + Q2 ), and L(τ ) = τ (B11 − B12K)S. It is of interest to note that if (10.1) is only affected by external disturbances f (t, zt ), vanishing in the origin, then an upper bound Ψ (t, zt,τmax ), such that Ψ (t, 0) = 0 for all t, becomes available so that the corresponding controller (10.13) with a positive parameter ε is continuous in the origin with no undesirable effects of switching in the steady state. It should also be pointed out that the control law (10.13) appears to be applicable even if the system delay is unknown a priori. We are now in a position to state the main result of this section. Theorem 10.2. Let Assumptions 1–3 be satisfied. Then for α1 , α2 sufficiently small the optimization problem (10.15) has a positive solution τmax (α1 , α2 ) and the closed-loop system (10.1), (10.13) is globally asymptotically stable for each delay value τ ∈ [0, τmax (α1 , α2 )). Proof. We break up the proof into three simple steps. Firstly, we prove that the discontinuity hyperplane s(z) = 0 is reached in finite time; secondly, we derive the state equations along the hyperplane s(z) = 0, and thirdly, we prove the asymptotic stability of the system motion on this hyperplane. Since beyond the discontinuity hyperplane s(z) = 0 there exists a local solution of (10.1), and due to the linear growth condition, imposed on the system, this solution cannot escape to infinity before it hits the discontinuity hyperplane, this will complete the proof of Theorem 10.2. 1

Due to Assumption 1, such an upper bound can be chosen to only depend upon the linear growth constants α1 , α2 of the external disturbances p1 and p2 regardless of a certain realization of these disturbances.

10.4 Unit State Feedback Controller

223

1) Attractiveness of the discontinuity hyperplane. As in the proof of Theorem 9.1, (10.6), which represents the time-delay system (10.1) and which is initialized beyond the discontinuity hyperplane s(z) = 0, possesses a local solution. While evolving beyond this hyperplane, such a solution cannot escape to infinity in finite time due to the linear growth condition that the underlying system meets. Taking into account that beyond the discontinuity hyperplane the time derivative of s(z(t)), computed along the solutions of (10.1), is given by s(z(t)) ˙ = Ω (t, zt ) + u + f (t, zt ), differentiating the Lyapunov–Krasovskii functional V (t) = sT (z(t))s(z(t)).

(10.16)

on the trajectories of the closed-loop system under Assumptions 1–3 yields ˙ = 2sT (z(t)){Ω (t, zt ) + f (t, zt ) V˙ (t) = 2sT (z(t))s(z(t)) s } − [Ω0(t, zt,τmax , α1 , α2 ) + Ψ (t, zt,τmax ) + g sε ] s ≤ −2g s(z(t))ε +1 = −2gV (t)

ε +1 2

.

(10.17)

Since by Lemma 4.3, an arbitrary solution of the latter inequality vanishes after a finite time moment, the finite-time convergence of the trajectories of the closed-loop system to the discontinuity hyperplane s(z) = 0 is concluded. 2) Sliding mode equation. By Theorem 2.4, the system motion on the discontinuity hyperplane s(z) = 0 is derived according to the equivalent control method, applied to Representation 10.6 of the functional differential equation (10.1) in the Hilbert space M2n+r = IR n+r × Ln+r 2 [−τ , 0]. Due to this method, the continuous solution ueq = −[Ω (t, zt ) + f (t, zt )] of the equation s˙ = 0 is substituted into (10.1) for u. Thus, on the discontinuity hyperplane, the system proves to be governed by the resulting differential equation: dz1 (t) = (A11 − A12K)z1 (t) + (B11 − B12 K)z1 (t − τ ) dt +p1 (t, z1 (t)) + p2 (t, z1 (t − τ )).

(10.18)

3) Asymptotic stability of the reduced system. Due to Assumption 3, the matrix E = (A11 − A12 K) + (B11 − B12 K) is Hurwitz by a proper choice of the matrix K. Applying Theorem 10.1 to the sliding mode equation (10.18), we obtain that the optimization problem (10.15) has a positive solution τmax (α1 , α2 ). Moreover, as in proving Theorem 10.1, (10.18) is asymptotically stable if (SE T + ES) + (γ1−1 + γ2−1 )In−r + (γ1 α12 + γ2 α22 + τγ3 α12 + τγ4 α22 )S2 T T 2 −1 2T +τ (SR1 S + SR2S) + τ BAR−1 1 A B + τ B R2 B −1 −1 T +(τγ3 + τγ4 )BB < 0

(10.19)

224

10 Global Asymptotic Stabilization of Uncertain Time-delay Systems

where A = A11 − A12K and B = B11 − B12 K. In turn, under constraints Q1 = SR1 S, Q2 = SR2S, A11 S − A12KS < γ5 S, B11 S − B12KS < γ6 S

(10.20)

with some positive γ5 , γ6 , (10.19) results from (SE T + ES) + (γ1−1 + γ2−1)In−r + τ (γ3−1 + γ4−1 )γ62 In−r + τ (Q1 + Q2 ) +(γ1 α12 + γ2 α22 + τγ3 α12 + τγ4 α22 )S2 T +γ52 τ (B11 S − B12W )Q−1 1 (B11 S − B12W ) −1 2 +γ6 τ (B11 S − B12W )Q2 (B11 S − B12W )T < 0.

(10.21)

Since (10.21) is equivalent to (10.15) by Schur’s lemma [29], the global asymptotic stability of the sliding mode equation (10.18) is then concluded for each delay value τ ∈ [0, τmax (α1 , α2 )). Theorem 10.2 is thus completely proved. Apparently, with γi > 0, i = 1, . . . , 6 being fixed, (10.15) represents a generalized eigenvalue problem which can efficiently be solved by convex optimization. A suboptimal upper bound τmax (α1 , α2 ) can thus be found by relaxation type algorithms.

10.5 Numerical Example In order to illustrate the synthesis procedure, consider (10.3), specified with ⎞ ⎞ ⎛ ⎞ ⎛ ⎛ 0 2 0 1 −1 0 0 A0 = ⎝ 1.4 0.25 0.8 ⎠ , B0 = ⎝ −0.1 0.25 0.2 ⎠ , C = ⎝ 0 ⎠ , 1 −1 0 1 −0.2 4 5 ⎛ ⎞ 0 ⎠ , P1 = P2 = 0 0 F =⎝ (10.22) sin(z1 (t − τ )) and the initial condition

⎤ −1 z(t) = ⎣ −0.5 ⎦ for t ∈ [−τ , 0]. 3 ⎡

(10.23)

Let the discontinuity surface (10.14) be specified with K = (1.317, 3.979). Then, by using semidefinite programming, (10.3), (10.22), driven by the unit feedback controller (10.13) subject to ε = 0, Ψ (t, zt,τmax ) = maxθ ∈[t,t−τmax ] z(t), g = 1.1, are globally asymptotically stable for each delay value τ ∈ [0, τmax ]) where τmax = 0.63. The simulation results depicted in Fig. 10.1 are obtained with a fifth-order integration scheme of step 10−3 .

10.5 Numerical Example

225

Fig. 10.1 Response of the closed-loop system (10.3), (10.13), (10.22) with τ = 0.63 and ε = 0

By setting ε = 12 , the control input becomes continuous in the origin and the chattering phenomenon is thus drastically reduced. As shown in Fig. 10.2, the amplitudes of the input oscillations, coming in this simulation run, are reduced from 2 to 0.02 without apparent changes in the convergence rate. Remark 10.1 (Comparison with available results). The sliding mode controllers proposed in [47, 210] cannot stabilize System 10.22 since the pair (A0 ,C) is not controllable in this case. With our approach, (A0 + B0 ,C)-controllability relaxes this requirement. The standard linear state feedback, developed in [131, 244], can only attenuate vanishing disturbances. Controllers from [53], which are based on linear models over rings and reject non-vanishing disturbances, require the delay value to be known a priori. At last, controllers from [84, 85] are not applicable to the perturbed system (10.3), (10.22) when P1 = 0, P2 = 0 thereby being extremely sensitive to the external disturbances P1 and P2 .

226

10 Global Asymptotic Stabilization of Uncertain Time-delay Systems

Fig. 10.2 Response of the closed-loop system (10.3), (10.13), (10.22) with τ = 0.63 and ε =

1 2

Part IV

Electromechanical Applications

228

IV Electromechanical Applications

Performance issues of the developed controllers are experimentally tested in engineering applications to electromechanical systems with complex nonlinear phenomena. Capabilities of the local nonsmooth H∞ -synthesis of accounting for hard-tomodel friction forces and backlash effects are investigated in experimental studies made for the position feedback regulation of a three-link robot manipulator with frictional joints and for the output feedback regulation of a servomechanism with backlash. Attractive features of the quasihomogeneous synthesis to reject uniformly bounded external disturbances, affecting fully actuated systems (rather than to simply attenuate them as it would be the case under the nonsmooth H∞ -synthesis) are then illustrated by means of orbital stabilization of a simple inverted pendulum and by means of global position regulation of a multi-link robot manipulator. After that, the quasihomogeneous design is developed for 2-DOFs underactuated systems by including it into a unified hybrid synthesis framework. Being verified experimentally for laboratory test beds, such as a horizontal double pendulum, an inverted double pendulum (Pendubot), and a pendulum, located on a cart, the proposed hybrid synthesis yields the desired robustness properties against friction forces.

Chapter 11

Local Nonsmooth H∞-synthesis Under Friction/Backlash Phenomena

The robust control of mechanical systems has attracted considerable research interest (see, e.g., related surveys compiled in [10] and [199]). The existing controllers from [19, 20, 44, 68, 225], which were derived via the nonlinear H∞ -control approach coupled to the feedback linearization techniques, as well as passivity-based controllers from [187], are widely used in practice due to their robustness and simplicity of implementation. However, the frictional influence and backlash effects, as well as the fact of the incompleteness of the state measurements, were ignored in these studies, that severely limited the achievable performance and their practical utility. The nonsmooth H∞ -synthesis of Chap. 7 is, in principle, capable of accounting for hard-to-model friction forces and backlash effects. It is the latter approach that is applied in the present chapter to regulation problems for a frictional mechanical manipulator and for a servomechanism with backlash, both with incomplete measurements. Due to the nature of the approach, the resulting regulators yield the desired robustness properties against the discrepancy between the real friction/backlash phenomenon and that described in the model.

11.1 Position Feedback Regulation of a Multi-link Robot Manipulator with Frictional Joints The local nonsmooth H∞ -synthesis of a multi-link manipulator with frictional joints is under study. The manipulator is required to move from the origin to a desired position. To facilitate exposition, the friction model chosen for treatment is confined to the Dahl model augmented with viscous friction. This simplest dynamic model captures all the essential features of the general treatment, thereby making the extension to other friction dynamic models straightforward. Since in robotic applications, velocity sensors are often omitted to considerably save in cost, volume, and weight [112], the position is assumed to be the only available measurement on the system and the resulting nonsmooth H∞ -regulator necessarily includes velocity and friction compensators. 229

230

11 Local Nonsmooth H∞ -synthesis Under Friction/Backlash Phenomena

Performance issues of the regulator developed are illustrated in an experimental study made for a 3-DOFs robot manipulator with frictional joints. In this study, better performance of the nonsmooth H∞ -regulator, compared to its smooth version derived via the standard nonlinear H∞ approach, is obtained. It should be pointed out that implementation of the local regulator developed is of the same level of simplicity as that of the aforementioned nonlinear H∞ -regulators derived via the feedback linearization techniques. Moreover, in comparison to the latter H∞ -regulators, the proposed regulator seems to be of a higher degree of robustness in the sense that it has the larger definition domain of external disturbances attenuated by the regulator.

11.1.1 Problem Statement A mathematical model for a frictional mechanical manipulator, whose links are joined together with revolute joints, is given by M (q) q¨ + C (q, q) ˙ q˙ + G (q) + F (q) ˙ = τ + w1

(11.1)

where q ∈ Rn is a position, τ ∈ Rn is a control input, w1 ∈ Rn is an external disturbance, F (q) ˙ , G(q) M(q), and C(q, q) ˙ are matrix functions of appropriate dimensions. From the physical point of view, q is the vector of generalized coordinates, τ is the vector of external torques, M(q) is the inertia matrix, symmetric and positive definite for all q ∈ Rn , C(q, q) ˙ q˙ is the vector of Coriolis and centrifugal torques, G(q) is the vector of gravitational torques, the components Fi (q˙i ) , i = 1, . . . , n of F(q) ˙ are the friction forces acting independently in each joint. The friction forces are represented as a combination Fi = σ0i q˙i + Fdi , i = 1, . . . , n

(11.2)

of viscous friction σ0i q˙i and the Dahl friction Fdi governed by the following dynamic model: Fdi F˙di = σ1i q˙i − σ1i |q˙i | + w2i Fci

(11.3)

where σ0i > 0, σ1i > 0, and Fci > 0 are the viscous friction coefficient, the stiffness, and the Coulomb friction level, respectively, corresponding to the i-th manipulator joint; w2i is an external disturbance which is involved to account for inadequacies of the friction modeling. Clearly, the above component-wise relations can be rewritten in the vector form F = σ0 q˙ + Fd (11.4) F˙d = σ1 q˙ − σ1 diag{|q˙i|}Fc−1 Fd + w2

(11.5)

11.1 Position Feedback Regulation of a Multi-link Robot Manipulator with Frictional Joints 231

where F = col{Fi }, Fd = col{Fdi}, q = col{qi } σ0 = diag{σ0i }, σ1 = diag{σ1i}, Fc = diag{Fci }, w2 = col{w2i }, the notations diag and col are used to denote a diagonal matrix and a column vector, respectively. Let qd = col{qdi } be the desired position. Then, if there were no initial and external disturbances, the desired position would be an equilibrium point of the closed loop system driven by τ = G(qd ). Our objective is to design a regulator of the form:

τ = G(qd ) + u

(11.6)

that imposes on the disturbance-free manipulator motion desired stability properties around qd while also locally attenuating the effect of the disturbances. Thus, the regulator to be constructed consists of a gravity compensator and a disturbance attenuator u, internally stabilizing the closed-loop system around the desired position. For certainty, we confine our investigation to the position regulation problem where 1. The output to be controlled is given by     0 1 z=ρ u + q − qd 0

(11.7)

with a positive weight coefficient ρ , and 2. The position measurements y = q + w0,

(11.8)

corrupted by the error vector w0 (t) ∈ Rn , are only available. The extension to a general case is straightforward. The H∞ position regulation problem for robot manipulators with friction can formally be stated as follows. Given a mechanical system (11.1)–(11.8), a desired position qd , and a real number γ > 0, it is required to find (if any) a causal dynamic feedback regulator (7.5) with internal state ξ ∈ Rs such that the undisturbed closedloop system is uniformly asymptotically stable around the desired position and its L2 -gain is locally less than γ , i.e., (2.11) is satisfied for all T > 0 and all piecewise continuous functions w = (w0 , w1 , w2 )T for which the state trajectory of the closedloop system starting from the initial point (q(0), q(0), ˙ Fd (0), ξ (0)) = (qd , 0, 0, 0) remains in some neighborhood of this point.

11.1.2 Control Synthesis To begin with, let us introduce the state deviation vector x = (x1 , x2 , x3 )T where x1 = qd − q is the position deviation from the desired position qd , x2 = q˙ is the velocity, and x3 = Fd is the Dahl friction. After that let us rewrite the state equations (11.1)–(11.8) in terms of the state vector x:

11 Local Nonsmooth H∞ -synthesis Under Friction/Backlash Phenomena

232

x˙1 = x2 x˙2 = −M −1 (qd − x1)[C(qd − x1 , x2 )x2 +G(qd − x1 ) − G(qd ) + σ0x2 + x3 − u − w1] x˙3 = σ1 x2 − σ1 diag{|x2i|}Fc−1 x3 − w2 .

(11.9)

Setting f (x) = f1 (x) + f2 (x), ⎡ ⎤ x2 f1 (x) = ⎣ −M −1 (qd − x1)[C(qd − x1 , x2 )x2 − G(qd − x1 )] ⎦ σ1 x2 ⎤ ⎡ 0 + ⎣ −M −1 (qd − x1 )[G(qd ) + σ0 x2 + x3 ] ⎦ , 0 ⎡ ⎤ 0 ⎦, 0 f2 (x) = ⎣ −σ1 diag{|x2i|}Fc−1 x3 ⎡ ⎤ 0 0 0 g1 (x) = ⎣ 0 −M −1 (qd − x1 ) 0 ⎦ , 0 0 −1 ⎡ ⎤ 0 g2 (x) = ⎣ −M −1 (qd − x1) ⎦ , 0   0 , h2 (x) = x1 , h1 (x) = ρ x1   1 , k21 (x) = [ 1 0 0 ] k12 (x) = 0

(11.10)

(11.11)

where by virtue of the well known inequality 2|a|b ≤ a2 + b2 , a, b ∈ R the function f2 (x) satisfies (7.31) for σ = 0.5 maxi σ1i Fci−1 and all x ∈ Rn , a local solution of the H∞ -regulation problem can be derived by applying Theorem 7.3 to (7.1)–(7.3), specified with (11.10), (11.11). Theorem 11.1. Let Conditions C1’ and C2’ of Sect. 7.1.4 hold for the matrix functions A, B1 , B2 ,C1 , C2 , governed by (7.33), (11.10), (11.11), and let (Pε , Zε ) be the corresponding positive definite solution of (7.61), (7.62) under some ε > 0. Then the output feedback .

ξ = f (ξ ) + [

1 g1 (ξ )gT1 (ξ ) − g2(ξ )gT2 (ξ )]Pε ξ + Zε C2T [y(t) − h2(ξ )], (11.12) γ2

11.1 Position Feedback Regulation of a Multi-link Robot Manipulator with Frictional Joints 233

u = −gT2 (ξ )Pε ξ ,

(11.13)

specified with (11.10), (11.11), is a local solution of the H∞ -position regulation problem. Proof. Since, by inspection, Assumptions A1–A5 of Chap. 7 hold for (11.9), which represents the manipulator equations (11.1)–(11.8) in terms of the state deviation with respect to the desired position qd , the validity of Theorem 11.1 is concluded by applying Theorem 7.3.

11.1.3 Experimental Study 11.1.3.1 Experimental Setup The experimental setup, designed in the research laboratory of CITEDI-IPN and shown in Fig. 11.1, involves a 3-DOFs industrial robot manipulator manufactured by Amatrol. Figure 11.2 describes the entire hardware setup configuration of the system. The base of the mechanical robot has a horizontal revolute joint (q1 ), whereas two links have vertical revolute joints q2 and q3 . Parameters of mechanical manipulator are summarized in Table 11.1. The worm gear set, the helicon gear set, and the roller chain are used for torque transmission to joints q1 , q2 and q3 , respectively; there is a DC gear motor for each joint with a reduction ratio of 19.7:1 for q1 and q2 , and 127.8:1 for q3 . These gears are the main source of friction. The ISA Bus servo I/0 card from the company Servo To Go is employed for the real time control system and it consists of eight channels of 16-bits D/A output, 32 bits of digital I/O, and an interval timer capable of interrupting the PC. The controller is implemented using C++ programming language running on a 486 PC. Position measurements of each articulation of the robot are obtained using the channels of quadrature encoders available on each DC gearmotors which are connected to the I/0 card, programmed to provide the encoder signal processing each one millisecond; resolution of each encoder is 52 × 10−3 rad, 62 × 10−3 rad and 34 × 10−3 rad for q1 , q2 and q3 , respectively. Along with this, a digital oscilloscope is utilized to store control signals. Linear power amplifiers are installed in each servomotor, which applies a variable torque to each joint. These amplifiers accept control inputs from D/A converter in the range of ±10 volts. 11.1.3.2 Dynamical Model The motion of the experimental manipulator, governed by (11.1), was specified by applying the Euler-Lagrange formulation, where:

234

11 Local Nonsmooth H∞ -synthesis Under Friction/Backlash Phenomena

Fig. 11.1 3-DOF robot manipulator

Fig. 11.2 Experimental setup

⎡

⎤ M11 0 0 M(q) = ⎣ 0 M22 M23 ⎦ 0 M23 M33 M11 (q) = m2 l12 cos2 (q2 ) + 2m2l1 l2 cos(q2 ) cos(q2 + q3) +m2 l22 cos2 (q2 + q3) + m1 l12 cos2 (q2 ) + I1 , M22 (q) = m1 l12 + m2 l12 + 2m2l1 l2 cos(q3 ) + m2 l22

+I2 + I3 , M23 (q) = m2 l1 l2 cos(q3 ) + m2l22 + I3 , M33 (q) = m2 l22 + I3 ;

11.1 Position Feedback Regulation of a Multi-link Robot Manipulator with Frictional Joints 235

⎡

⎤ C11 C12 C13 C(q, q) ˙ = ⎣ C21 C22 C23 ⎦ C31 C32 0 C11 = −m2 l12 S2 C2 q˙2 − m2 l1 l2 S2 C23 q˙2 − m1l12 S2 C2 q˙2 −m2 l1 l2 C2 S23 (q˙2 + q˙3) − m2 l22 S23 C23 (q˙2 + q˙3)

C12 = −m2 l12 S2 C2 q˙1 − m2 l1 l2 C2 S23 q˙1 − m2l1 l2 S2 C23 q˙1 −m2 l22 S23 C23 q˙1 − m1 l12 S2 C2 q˙1 ,

C13 = −m2 l1 l2 C2 S23 q˙1 − m2 l22 S23 C23 q˙1 , C21 = m2 l12 S2 C2 q˙1 + m2 l1 l2 C2 S23 q˙1 + m2 l1 l2 S2 C23 q˙1 C22

+m2 l22 S23 C23 q˙1 + m1 l12 S2 C2 q˙1 = −m2 l1 l2 S3 q˙3 ,

C23 = −m2 l1 l2 S3 (q˙2 + q˙3), C31 = m2 l1 l2 C2 S23 q˙1 + m2 l22 S23 C23 q˙1 , C32 = m2 l1 l2 S3 q˙2 ; with Si = sin qi , Ci = cosqi , Si j = sin(qi + q j ), Ci j = cos(qi + q j ); G(q) = ⎡

⎤ 0 g ⎣ m1 l1 cos q2 + m2 l1 cos q2 + m2 l2 cos(q2 + q3) ⎦ . m2 l2 cos(q2 + q3 ) The physical constant parameters, mi , li , i = 1, 2 and I j , j = 1, 2, 3 are given in Table 11.1. Table 11.1 Parameters of the mechanical manipulator Description Notation Value Length of link 1 l1 0.297 Length of link 2 l2 0.297 Mass of link 1 m1 0.38 Mass of link 2 m2 0.34 Inertia 1 I1 .243 × 10−3 Inertia 2 I2 .068 × 10−3 Inertia 3 I3 .015 × 10−3 Gravity acceleration g 9.8

Units m m Kg Kg Kg m2 Kg m2 Kg m2 m/seg2

236

11 Local Nonsmooth H∞ -synthesis Under Friction/Backlash Phenomena

11.1.3.3 Experimental Results The regulator performance was studied experimentally. The robot manipulator was required to move in the space from the origin q1 = q2 = q3 = 0 (the references for each joint are shown in Fig. 11.1) to the desired position qd1 = qd2 = qd3 = 30 degrees. The viscous friction coefficient σ0 , stiffness σ1 , and Coulomb friction level Fc were obtained by applying the procedure from [109]:

σ0 = diag{9.84, 13.02, 9.87}Nms/rad, σ1 = diag{0.054, 0.053, 0.039}Nm, Fc = diag{2.1, 1.02, 0.78}Nm. The initial velocity q(0) ˙ and compensator state ξ (0) were set to zero for all the experiments. The control goal was achieved by implementing the nonsmooth H∞ -regulator with the weight parameter ρ = 1 on the 3-DOF manipulator. By iterating on γ , we found the infimal achievable level γ ∗  14. However, in the subsequent experiment γ = 20 was selected to avoid an undesirable high-gain controller design that would appear for a value of γ close to the optimum. With γ = 20 we obtained that for ε = 0.15 the corresponding Riccati equations (7.38), (7.39) have positive definite solutions. By Theorem 11.1, these solutions result in the control law (11.12), (11.13) solving the regulation problem. For the sake of comparison, the smooth version of the H∞ -regulator, synthesized for the friction-free model of the robot, has also been tested on the laboratory manipulator. The resulting trajectories are depicted in Fig. 11.3. This figure demonstrates that the nonsmooth H∞ -regulator does asymptotically stabilize the system motion around the desired position, as opposed to the standard smooth H∞ -regulator. Thus, a better performance of the nonsmooth H∞ -regulator is concluded. In addition, the nonsmooth H∞ -regulator was successively applied to the robot manipulator with 5-Nm overload in each joint (thereby modeling permanent external disturbances), and to the manipulator whose nominal parameter values (length, mass, Coulomb level, stiffness, and viscous friction) had been varied in the ±10% range. The resulting motion of the closed-loop system with the overload and that with the modified parameters are shown in Figs. 11.4 and 11.5, respectively. These figures carry out favorable robustness properties of the closed-loop system against external disturbances and parameter variations that underscore the attraction of the regulator developed.

11.1 Position Feedback Regulation of a Multi-link Robot Manipulator with Frictional Joints 237 q1 [deg] 40

20

0

−20

40

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

1

2

q2 [deg]

20

0

−20

40

q3 [deg]

20

0

−20

40

q˙1 [deg/s]

20

0

−20

50

q˙2 [deg/s]

0

−50

40

q˙3 [deg/s]

20

0

−20

10

τ1 [Nm]

0

−10

−20

10

τ2 [Nm]

0

−10

−20

10

τ3 [Nm]

0

−10

−20

0

Time[sec]

Fig. 11.3 Experimental results for the nonsmooth H∞ regulator (solid lines) and its smooth version (dotted lines)

11 Local Nonsmooth H∞ -synthesis Under Friction/Backlash Phenomena

238 q1 [deg] 40

20

0

−20

40

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

1

2

q2 [deg]

20

0

−20

40

q3 [deg]

20

0

−20

50

q˙1 [deg/s]

0

−50

50

q˙2 [deg/s]

0

−50

50

q˙3 [deg/s]

0

−50

10

τ1 [Nm]

0

−10

−20

10

τ2 [Nm]

0

−10

−20

10

τ3 [Nm]

0

−10

−20

0

Time[sec]

Fig. 11.4 Disturbance attenuation provided by the nonsmooth H∞ regulator

11.1 Position Feedback Regulation of a Multi-link Robot Manipulator with Frictional Joints 239 q1 [deg] 10

0

−10

−20

10

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

0

1

2

1

2

q2 [deg]

0

−10

−20

10

q3 [deg]

0

−10

−20

40

q˙1 [deg/s]

20

0

−20

50

q˙2 [deg/s]

0

−50

40

q˙3 [deg/s]

20

0

−20

40

τ1 [Nm]

20

0

−20

40

τ2 [Nm]

20

0

−20

40

τ3 [Nm]

20

0

−20

0

Time[sec]

Fig. 11.5 Robustness of the nonsmooth H∞ regulator against parametric uncertainties: solid lines are for nominal parameter values, dotted lines for ±10% parameter variations

240

11 Local Nonsmooth H∞ -synthesis Under Friction/Backlash Phenomena

11.2 Output Feedback Regulation of a Servomechanism with Backlash In the present section, the output regulation problem is studied for an electrical actuator consisting of a motor part driven by a DC motor and a reducer part (load) operating under uncertainty conditions in the presence of nonlinear backlash effects, resulting in the multi-stability of the system. The objective is to drive the load to a desired position while providing the boundedness of the system motion and attenuating external disturbances. Due to practical requirements [122], the motor angular position is assumed to be the only information available for feedback. The above problem is locally resolved within the nonsmooth H∞ -control approach of Chap. 7. It is worth noticing that this approach does not admit a straightforward application to the above problem because a partial state stabilization (i.e., the asymptotic stabilization of the output of the system) is required only, provided that the complementary variables remain bounded. The fact that under motor position measurements the servomotor with backlash presents a non-minimum phase system is another challenge to be overcome. The output feedback synthesis to be developed is based on a related singlestability backlash approximation of Sect. 2.3.2 replacing the underlying system by a minimum phase one. Being designed for such a minimum phase approximation of the servomotor, a corresponding nonlinear H∞ -regulator is experimentally shown to properly attenuate backlash model discrepancies, thereby yielding an appropriate solution to the regulation problem in question.

11.2.1 Dynamic Model The following dynamic model Jo N −1 q¨o + fo N −1 q˙o = T + wo Ji q¨i + fi q˙i + T = τm + wi

(11.14)

presented in Sect. 2.3.2 for the angular position qi (t) of the DC motor and for the load qo (t), is under study. Hereinafter, Jo , fo , q¨o , and q˙o are, respectively, the inertia of the load and the reducer, the viscous output friction, the output acceleration and the output velocity. The inertia of the motor, the viscous motor friction, the motor acceleration, and the motor velocity are denoted by Ji , fi , q¨i and q˙i respectively. The input torque τm serves as a control action, and T stands for the transmitted torque. The external disturbances wi (t), wo (t) have been introduced into the driver equation (11.14) to account for destabilizing model discrepancies due to hard-tomodel nonlinear phenomena such as friction and backlash. The transmitted torque T through a backlash with an amplitude j is modeled by a dead-zone characteristic

11.2 Output Feedback Regulation of a Servomechanism with Backlash

 T (Δ q) =

0 if |Δ q| ≤ j K Δ q − K j sign(Δ q) otherwise

241

(11.15)

where

Δ q = qi − Nqo ,

(11.16)

K is the stiffness, and N is the reducer ratio. Provided that the servomotor position qi (t) is the only available measurement on the system, the above model (11.14)– (11.16) appears to be non-minimum phase because, along with the origin, the unforced system possesses a multi-valued set of equilibria (qi , qo ) with qi = 0 and qo ∈ [− j, j]. To avoid dealing with a non-minimum phase system, the backlash model (11.15) is replaced with its monotonic approximation T = K Δ q + K η (Δ q)

(11.17)

where

η = −2 j

1 − e−

Δq j

1 + e−

Δq j

.

(11.18)

As shown in Sect. 2.3.2, this approximation, coupled to the drive system (11.14), constitutes a minimum phase approximation of the underlying servomotor under motor position measurements. Discrepancies between the physical backlash model (11.15) and its approximation (11.17)–(11.18) are thus involved into the uncertain terms wi (t), wo (t). While controlling the servomotor, the influence of the uncertainties wi (t), wo (t) on the performance of the closed-loop system is to be attenuated.

11.2.2 Problem Statement The objective of the H∞ -output regulation of the nonlinear drive system (11.14) with backlash effects (11.17)–(11.18) is to design a nonlinear H∞ -controller so as to obtain the closed-loop system in which all these trajectories are bounded and the output qo (t) asymptotically decays to a desired position qd as t → ∞ while also attenuating the influence of the external disturbances wi (t), wo (t). To formally state the problem, let us introduce the state deviation vector x = [x1 , x2 , x3 , x4 ]T with x1 = qo − qd , x2 = q˙o , x3 = qi − Nqd , x4 = q˙i where x1 is the load position error, x2 is the load velocity, x3 is the motor position deviation from its nominal value, x4 is the motor velocity. The nominal motor position Nqd has been prespecified in such a way to guarantee that Δ q = Δ x where

11 Local Nonsmooth H∞ -synthesis Under Friction/Backlash Phenomena

242

Δ x = x3 − Nx1 . Then (11.14)–(11.18), represented in terms of the deviation vector x, takes the form: x˙1 = x2 x˙2 = Jo−1 [KNx3 − KN 2 x1 − fo x2 + KN η (Δ x) + wo ] x˙3 = x4 x˙4 = Ji−1 [τm + KNx1 − Kx3 − fi x4 − K η (Δ x) + wi ].

(11.19)

The zero dynamics x˙1 = x2 x˙2 = Jo−1 [−KN 2 x1 − fo x2 + NK η (−Nx1 )]

(11.20)

of the undisturbed version of (11.19) with respect to the output y = x3

(11.21)

are formally obtained by specifying the control law that maintains the output identically zero. By Theorem 2.5, the error system (11.19), (11.21) appears to be globally minimum phase. Due to this, the H∞ -output regulation problem for the drive system with backlash can formally be stated as a standard H∞ -control problem for the deviation system (11.19). In the sequel, we confine our investigation to the H∞ -position regulation problem where 1. The output to be controlled is given by     0 1 z=ρ + τ x1 0 m

(11.22)

with a positive weight coefficient ρ . 2. The motor position qi is the only available measurement and this measurement is corrupted by the error vector wy (t) ∈ R, i.e., y = x3 + Nqd + wy .

(11.23)

The H∞ -control problem in question is thus stated as follows. Given the system representation (11.19)–(11.23) and a real number γ > 0, it is required to find (if any) a causal dynamic feedback controller u = K (ξ )

(11.24)

with internal state ξ ∈ R4 such that the undisturbed closed-loop system is uniformly asymptotically stable around the origin and its L2 -gain is locally less than γ , i.e., the inequality

11.2 Output Feedback Regulation of a Servomechanism with Backlash

 T 0

z(t)2 dt < γ 2

 T 0

w(t)2 dt

243

(11.25)

holds for all T > 0 and all piecewise continuous functions wT (t) = [wo , wi , wy ](t) for which the corresponding state trajectory of the closed-loop system, initialized at the origin, remains in some neighborhood of this point.

11.2.3 Control Synthesis Setting

⎡

⎤ x2 ⎢ Jo−1 (KNx3 − KN 2 x1 − fo x2 + KN η ) ⎥ ⎥, f (x) = ⎢ ⎣ ⎦ x4 −1 Ji (KNx1 − Kx3 − fi x4 − K η ) ⎤ 0 0 ⎢ Jo−1 0 0 ⎥ ⎥ g1 (x) = ⎢ ⎣ 0 0 0⎦, 0 J −1 0  i 0 h1 (x) = ρ , x1   1 , k12 (x) = 0 ⎡

0

(11.26)

 03×1 g2 (x) = , Ji−1 

h2 (x) = x3 + Nqd , # $ k21 (x) = 0 0 1 .

(11.27)

a local H∞ -position regulator of the servomechanism with backlash is synthesized by applying Theorem 7.3 to (7.1)–(7.3) specified with (11.26), (11.27). Theorem 11.2. Let Conditions C1’ and C2’ of Sect. 7.1.4 hold for the matrix functions A, B1 , B2 , C1 , C2 governed by (7.33), (11.26), (11.27) and let (Pε , Zε ) be the corresponding positive definite solution of (7.61), (7.62) under some ε > 0. Then the output feedback 1 ξ˙ = f (ξ ) + [ 2 g1 (ξ )gT1 (ξ ) − g2(ξ )gT2 (ξ )]Pε ξ γ +Zε C2T [y − h2(ξ )], u = −gT2 (ξ )Pε (t)ξ

(11.28) (11.29)

specified with (11.26), (11.27) presents a local solution of the H∞ -position regulation problem (11.19), (11.22)–(11.25). Proof. By applying Theorem 7.3 to the state deviation equations (11.19), (11.22), (11.23) the validity of the present theorem is straightforwardly verified.

244

11 Local Nonsmooth H∞ -synthesis Under Friction/Backlash Phenomena

u qi

qo

Fig. 11.6 Experimental test bench

11.2.4 Experimental Study 11.2.4.1 Experimental Setup The experimental setup, installed in Robotics & Control Laboratory of CITEDIIPN, involves a DC motor linked to a mechanical load through an imperfect contact gear train. Figure 11.6 shows the location of the sensors, the actuator and the load. The input-output motion graph of Fig. 11.7 reveals the gear backlash effect. A schematic diagram of the closed-loop system, presenting the monotonic approximation of the dead zone model and the servomotor position, being the only measured output available for feedback, is depicted in Fig. 11.8. The maximum allowable torque is 1.24 N-m. The ISA Bus servo I/O card from the company Servo To Go allows one to control the servomotor in real time. It consists of eight channels of 13-bits D/A converter, 32 bits of digital I/O, and an interval timer capable of interrupting the PC. The controller runs in a 486 PC using C++ pro-

11.2 Output Feedback Regulation of a Servomechanism with Backlash

245

2

Load position [rad]

1

j

0

−j

−1

−2

−3 −2

−1

0 1 Motor position [rad]

2

3

Fig. 11.7 Backlash hysteresis before compensation

-

Controller

Servo mechanism

Load

Fig. 11.8 Schematic diagram of the closed-loop system

gramming language. A shaft mounted encoder gets the motor angular position and provide the encoder signal to the I/O card each millisecond (sampling time). The resolution of the encoder is 2000 ppr. A high resolution potentiometer has been placed in the load side to support the results. A linear power amplifier is installed in the servomotor accepting control signals from D/A converter in the range of ±10 volts. For experimental purposes, we add a coupler to increase the backlash level to j = 0.2 rad. The gear reduction ratio is of 65.5:1 and it is the main source of friction. The stiffness coefficient K is computed experimentally by differentiating the transmitted torque measurements in the state variable at the zone of the linear growth. As follows from Fig. 11.9, with the stiffness value K = 5 N-m/rad, thus computed, the monotonic backlash model (11.17)–(11.18) yields an appropriate approximation of the transmitted torque. Table 11.2 presents the parameters of the motor, taken from the manufacturer data specifications, and the nominal load parameters, identified in accordance with the experimental procedures proposed in [109].

246

11 Local Nonsmooth H∞ -synthesis Under Friction/Backlash Phenomena 2.5 2 1.5

Torque [N−m]

1 0.5 −j 0 j −0.5 −1 −1.5 −2 −2.5 −1

−0.5

0 Δ q [rad]

0.5

1

Fig. 11.9 Transmitted torque, computed from the experiment (solid line), and the modeled one (dotted line) Table 11.2 Nominal parameters Description Notation Value Units Motor inertia Ji 2.8 × 10−6 Kg-m2 Load inertia Jo 1.07 Kg-m2 −7 Motor viscous friction fi 7.6 × 10 N-m-s/rad Load viscous friction fo 1.73 N-m-s/rad

11.2.4.2 Experimental Results The experiments were carried out for the closed loop system (11.26)–(11.29) with a position sensor located at the motor side, thus considering the angular motor position as the only information available for feedback. In the experiments, the load was required to move from the initial static position qo (0) = 0 to the desired position qd = π /2 rad. In order to illustrate the size of the attraction domain the initial load position was chosen reasonably far from the desired position whereas the initial velocity q˙o (0) of the load and that q˙i (0) of the motor as well as the initial compensator state ξ (0) were set to zero for all experiments. Despite disturbances and uncertainties always presented in the implemented closed-loop system (such as a sensor noise and modeling error), we additionally apply an unknown but bounded torque perturbation governed by wi (t) = 0.25e−0.1t cos(10t).

(11.30)

For the selected γ = 50, ρ = 1, ε = 12, the corresponding Riccati equations (7.61), (7.62) have the positive definite solutions:

11.2 Output Feedback Regulation of a Servomechanism with Backlash

247

⎤ ⎡ 203.8105 95.7354 40.5855 0.5357 ⎢ 95.7354 55.6470 32.6991 4.6956⎥ ⎥ Pε = ⎢ ⎣ 40.5855 32.6991 42.0288 8.4923⎦ , 0.5357 4.6956 8.4923 3.7658 ⎤ 1.8709 0.6252 0.0498 −0.1190 ⎢ 0.6252 0.2669 0.0222 −0.0240⎥ ⎥ Zε = 103 ⎢ ⎣ 0.0498 0.0222 0.0119 −0.0008⎦ ., −0.1190 −0.0240 −0.0008 0.0291 ⎡

that have been numerically found using MATLAB . The resulting trajectories and applied torque are depicted in Figs. 11.10 and 11.11, respectively. These figures demonstrate that the regulator stabilizes the disturbance-free load motion around the desired position and attenuates external load disturbances. As predicted, Fig. 11.10 shows that the load position goes asymptotically to the desired position while the motor position remains bounded. In Fig. 11.11, the load position shows a 5% error around its reference position when torque is perturbed by (11.30). It should be noted that the potentiometer induces noise in the load position measured signal as reflected in Figs. 11.10 and 11.11.

Position [rad]

3 2 1 motor position load position desired position

0 −1 0

1

2

3

4

5

6

4

5

6

Time [s]

Input torque [N−m]

1 0 −1 −2 −3 0

1

2

3 Time [s]

Fig. 11.10 Experimental results for the H∞ -regulator considering only motor position information available for feedback

248

11 Local Nonsmooth H∞ -synthesis Under Friction/Backlash Phenomena

Position [rad]

3 2 1 0

motor position load position desired position

−1 −2 0

2

4 Time [s]

6

8

Input torque [N−m]

1 0 −1 −2 −3 −4 0

1

2

3

4 Time [s]

5

6

7

8

Fig. 11.11 Experimental results for the H∞ - regulator considering only motor position information available for feedback: the perturbed case

Chapter 12

Quasihomogeneous Stabilization of Fully Actuated Systems with Dry Friction

Quasihomogeneity-based synthesis appears to be extremely suited for fully actuated systems with dry friction. This claim is supported by several arguments. Firstly, mechanical systems with relatively strong Coulomb friction require discontinuous controllers for adequate regulation. Secondly, external disturbances, affecting these systems, meet the matching condition so that their influence on the underlying system is not simply attenuated as it would be the case under H∞ -synthesis, but it is also rejected under the quasihomogeneous synthesis. Thirdly, the global position regulation becomes possible provided that an upper bound on the magnitude of the external disturbances is known a priori. These features are subsequently illustrated by means of the orbitally stabilizing synthesis of a simple inverted pendulum and by means of the global position regulation of a multi-link robot manipulator.

12.1 Orbital Stabilization of an Inverted Pendulum Research interest in the orbital stabilization of mechanical systems is motivated by applications where the natural operation mode is periodic [40, 209]. For these systems the orbital stabilization paradigm differs from typical formulations of output tracking where the reference trajectory to follow is known a priori. The control objective for the orbital stabilization (e.g., periodic balancing a walking rabbit [46] or trajectory planning for industrial robot manipulators [66]), is to result in the closedloop system that generates its own limit cycle similar to that produced by a nonlinear oscillator. Apart from this, the closed-loop system should be capable of moving from one orbit to another by modifying the orbit parameters such as the frequency and/or the amplitude. In the present section, the quasihomogeneous control synthesis of Chap. 5 is developed towards the orbital stabilization and tested on an inverted pendulum with both Coulomb and viscous frictions. The development is essentially from [185] and it is confined to the full state feedback case where the one-link friction pendulum is to be driven to a model orbit in finite time. The resulting closed-loop system is 249

250

12 Quasihomogeneous Stabilization of Fully Actuated Systems with Dry Friction

shown to track the model orbit in a sliding mode of the second order, even in the presence of external disturbances with an a priori known magnitude bound. The proposed synthesis is thus provided with desired robustness properties against the discrepancy between the real friction and that described in the model. In particular, the advantage of the controller constructed may be outstanding if Coulomb friction, typically ignored in the existing controllers design, is relatively strong for the actuator’s power. The modified Van der Pol oscillator ..

x +ε [(x2 +

x˙2 ) − ρ 2]x˙ + μ 2 x = 0, μ2

(12.1)

that has been studied in Sect. 2.3.3, is introduced into the synthesis as a reference model. The modification has been made to shape the oscillator limit cycle to a harmonic one. The parameters ρ , μ , ε > 0 of the modified Van der Pol oscillator specify the amplitude and frequency of the limit cycle production, and damping of nonlinear oscillations, respectively. Amplitude, frequency, and damping can thus be modified dynamically by simply changing these parameters. Due to this, the proposed oscillator is well suited for addressing the periodic stabilization problem in question. Along this line, the proposed Van der Pol modification serves as an asymptotic harmonic generator of the to-be-enforced motion of the pendulum. It is introduced in a conceptually different way to the integral form proposed in [209] for the reference model that possesses a center, whose neighborhood is filled with multi-cycles, and the target orbit is a priori to be specified among the cycles. Another example of an asymptotic harmonic generator (nearly the only one available in the literature) is the variable structure Van der Pol oscillator from [211]. However, it is hardly possible to use that oscillator for generating a reference signal because the system response would be contaminated by high frequency oscillations (a so-called chattering effect) caused by rapidly switching the structure of the Van der Pol oscillator. The control law, enforcing the system to slide along a periodic orbit of the phase space, and an asymptotic harmonic generator of this orbit, being coupled together, yield a unified framework for the orbital stabilization of a friction pendulum. The resulting closed-loop system possesses its own limit cycle, producing a prescribed harmonic, whose frequency and amplitude can be modified at the designer’s will dynamically. Capabilities of the proposed orbitally stabilizing synthesis are illustrated in an experimental study of a laboratory pendulum. Good controller performance is concluded from this study in spite of the presence of friction forces and actuator dynamics.

12.1.1 Tracking of a Modified Van der Pol Oscillator The state equation of the controlled one-link pendulum, depicted in Fig. 12.1, is given as

12.1 Orbital Stabilization of an Inverted Pendulum

(ml 2 + J)q¨ = mglsin(q) − F(q) ˙ +τ +w

251

(12.2)

where q is the angle made by the pendulum with the vertical, m is the mass of the pendulum, l is the distance to the center of mass, J is the moment of inertia of the pendulum about the center of mass, g is the gravity acceleration, F is the friction force, τ is the control torque, and w is the external disturbance. In order to describe the friction force F the classical model is utilized: F(q) ˙ = αv q˙ + αc sign(q). ˙

(12.3)

The above model comes with the viscous friction coefficient αv > 0, the Coulomb friction level αc > 0, and the standard notation sign(q) ˙ for the signum function of the angular velocity, multi-valued for zero velocity. Because the phenomenon of friction is hard to model, we have introduced the uncertain term w(t) into the dynamic equation (12.2) to account for destabilizing model discrepancies such as the Stribeck effect and backlash. An upper bound M > 0 for the magnitude of this term is normally known a priori: |w(t)| ≤ M f or all t.

q

(12.4)

m l

τ Fig. 12.1 Schematic diagram of pendulum

The objective is to design a controller that causes the friction pendulum (12.2), (12.3) to track the modified Van der Pol oscillator (12.1) with the parameters ρ , μ , ε , which are admitted to depend on the state of the pendulum, while also attenuating the effect of an external disturbance (12.4). Let the tracking error be given by y(t) = q(t) − x(t). Due to (2.98), (12.2), (12.3), the error dynamics are then governed by

(12.5)

252

12 Quasihomogeneous Stabilization of Fully Actuated Systems with Dry Friction ..

(ml 2 + J) y = mglsin(q) − αv q˙ − αc sign(q) ˙ +τ +w +(ml 2 + J){ε [(x2 +

x˙2 ) − ρ 2 ]x˙ + μ 2 x}. μ2

(12.6)

Under the following control law x˙2 ) − ρ 2]x˙ + μ 2x} − mglsin(q) μ2 ˙ − hy − py˙ −αv x˙ − α sign(y) − β sign(y)

τ = −(ml 2 + J){ε [(x2 +

(12.7)

with parameters such that h, p ≥ 0, β > M + αc , α > β + M + αc ,

(12.8)

the error dynamics (12.6) are feedback transformed to ..

(ml 2 + J) y= w − αc sign(q) ˙ − α sign(y) − β sign(y) ˙ − hy − (p + αv)y. ˙ (12.9) Relating the quasihomogeneous synthesis of Chap. 5, the above controller has been composed of the nonlinear compensator uc = −(ml 2 + J){ε [(x2 +

x˙2 ) − ρ 2]x˙ + μ 2 x} − mglsin(q), μ2

(12.10)

the homogeneous relay part uh = −α sign(y) − β sign(y), ˙ and the linear remainder

ul = −hy − py˙ − αv x. ˙

A block diagram of the complete control system, including a potential feedback f from the pendulum to the asymptotic harmonic generator, is depicted in Fig. 12.2. By virtue of Theorem 4.4, the quasihomogeneous closed-loop system (12.7), (12.9) with the parameter subordination (12.8), proves to be globally finite time stable regardless of which external uniformly bounded disturbance (12.4) affects the system. The control objective is thus achieved, even in finite time. Theorem 12.1. Let the modified Van der Pol equation (12.1) with positive parameters ε , μ , ρ , possibly dependent on the state of the pendulum, be a reference model of the pendulum dynamics (12.2), enforced by the quasihomogeneous controller (12.7), (12.8), and let the tracking error be given by (12.5). Then the error system (12.9) is globally finite time stable, uniformly in admissible disturbances (12.4). Proof. The validity of the theorem is straightforwardly verified by applying Theorem 4.4 to the closed-loop system (12.7)–(12.9).

12.1 Orbital Stabilization of an Inverted Pendulum

253

Fig. 12.2 Block diagram for the quasihomogeneous controller (12.7)

So, starting from a finite time moment T > 0, the pendulum, driven by the quasihomogeneous controller (12.7), (12.8), stays in the intersection of the switching lines y = 0, y˙ = 0 forever, regardless of which external disturbance (12.4) affects the closed-loop system. This means that the enforced pendulum copies the motion, generated by the modified Van der Pol oscillator, i.e., q(t) = x(t) for all t ≥ T .

12.1.2 Experimental Study Performance issues of the quasihomogeneous synthesis were tested on the laboratory pendulum (see Fig. 12.3) of the mass m = 0.5234 kg, centered at l = 0.108 m, and the inertia J = 0.006 kg · m2 about the center of mass. The friction in the motor brushes and bearings was identified with the parameters αv = 0.00053 N · m · s/rad and αc = 0.05492 N · m. The initial position of the pendulum, and that of the modified Van der Pol oscillator, selected for the experiments, were q(0) = 3.14 rad and x(0) = −3.14 rad, whereas all the velocity initial conditions were set to zero. In the experimental study, the controller gains in (12.7) were set to h = 0, p = 0, α = 20 N · m, β = 5 N · m, whereas the reference parameters were tuned to ε = 8.7 [rad]−2 s−1 , ρ = 0.15 rad, μ = 10 s−1 . A hyperbolic tangent approximation of the relay input was implemented to reduce high frequency oscillations of the pendulum, caused by fast switching in the control signal.

254

12 Quasihomogeneous Stabilization of Fully Actuated Systems with Dry Friction

Fig. 12.3 Laboratory pendulum

In order to test the robustness of the orbitally stabilizing controller (12.7) external disturbances were randomly added by lightly hitting the pendulum at time instants t1 ≈ 10s and t2 ≈ 16s. For demonstrating the capability of the controller to move the pendulum from one orbit to another the parameter ε was reset to the higher value 100 [rad]−2 s−1 to speed up the limit cycle transient. Then, the amplitude ρ of the model limit cycle was changed from its initial value ρ = 0.15 rad to the new one ρ = 0.5 rad at a random time instant t0 (it was t0 ≈ 12s in the experiment). Experimental results for the resulting pendulum motion, enforced by the orbitally stabilizing controller, are depicted in Fig. 12.4 on pp. 256–257. This figure demonstrates that despite modeling errors, ever present in mechanical systems with hardto-model nonlinear phenomena such as friction and backlash, the closed-loop system still generates an asymptotic periodic motion and exhibits a fast recovery of this motion when quick disturbances are applied to the pendulum. In addition, the desired orbital transfer is shown to be achieved by simply changing the reference parameters, as was predicted by theory. Thus, good performance of the orbitally stabilizing controller is concluded from Fig. 12.4.

12.2 Global Position Regulation of a Multi-link Robot Manipulator The position feedback regulation of a fully actuated mechanical manipulator, each link of which is driven by its “own” actuator, is under study. In contrast to Sect. 11.1, the Coulomb frictions among the manipulator joints are assumed to be relatively

12.2 Global Position Regulation of a Multi-link Robot Manipulator

255

strong for the actuators’ power, which is why the n-link manipulator dynamics M(q)q¨ + C(q, q) ˙ q˙ + G(q) + F(q) ˙ = τ,

(12.11)

are augmented with the classical friction model F(q) ˙ = Kb q˙ + K f sign(q), ˙ sign(q) ˙ = (sign (q˙1 ) , sign(q˙2 ), . . . , sign(q˙n ))T

(12.12)

where Kb = diag{kbi } and K f = diag{k fi } are diagonal positive definite matrices, and parameters kbi and k fi , i = 1, 2, . . . , n represent coefficients of viscous and Coulomb frictions, respectively. As before, the position q ∈ Rn is the vector of generalized coordinates, the control input τ ∈ Rn is the vector of external torques, and the inertia matrix M(q), the vector C(q, q) ˙ q˙ of Coriolis and centripetal torques, the vector G(q) of gravitational torques, and the vector F(q) ˙ of friction torques possess appropriate dimensions.

12.2.1 Position Control Synthesis Based on the extended invariance principle (Theorem 3.3), the following control law

τ = G(q) − Kd x˙ − K p e − Kα sign(e)

(12.13)

x˙ = −Lx + Kd e

(12.14)

is derived to globally asymptotically stabilize the friction system (12.11) around the desired position qd ∈ Rn . In the above control law, L ∈ Rn×n is a symmetric positive definite matrix, Kd ∈ Rn×n is a symmetric positive semi-definite matrix, K p = diag(k pi ) ∈ Rn×n is a diagonal positive definite matrix, Kα = diag(kαi ) ∈ Rn×n is a diagonal matrix such that Kα > K f , and e = q − qd represents the position error with respect to the desired position qd . One can see that Regulator 12.13, 12.14 is quasihomogeneous as it consists of a gravitational compensation part, a linear dynamic output feedback part, and a homogeneous switching part, which imposes on the closed-loop system to have a unique equilibrium point. Particularly, by letting Kd = 0 the dynamic part in (12.13), (12.14) is omitted, thereby yielding the static position regulation

τ = G(q) − K pe − Kα sign(e),

(12.15)

which becomes feasible, in contrast to the standard sliding mode approach where no static position regulation is available. The idea behind the derivation is as follows. First, the position feedback (12.13), (12.14) is constructed to guarantee the quadratic/magnitude function

256

12 Quasihomogeneous Stabilization of Fully Actuated Systems with Dry Friction

6

6 q x

q x

4

4

2 [rad]

[rad]

2

0

−2

−2

−4

−4

−6 0

5

10

15

−6 0

20

40

10

15

20

10

10 [rad/s]

20

0

0

−10

−10

−20

−20

−30

−30

5

10

15

dq/dt dx/dt

30

20

−40 0

5

40 dq/dt dx/dt

30

[rad/s]

0

20

−40 0

5

10

15

20

Fig. 12.4 Orbital stabilization of the pendulum: the left column for the orbital recovery under randomly added quick disturbances, and the right column for the transfer from one orbit to another

V (e, q, ˙ x) =

1 T 1 1 q˙ M(e + qd )q˙ + eT K p e + (Kd e − Lx)T (Kd e − Lx) 2 2 2 (12.16) +kα1 |e1 | + kα2 |e2 | + ... + kαn |en | ,

to have a non-positive definite time derivative along the trajectories of the closedloop system (12.11), (12.13), (12.14). After that, the auxiliary cross-term function

8

8

6

6

4

4

2

2 y [rad]

y [rad]

12.2 Global Position Regulation of a Multi-link Robot Manipulator

0

0

−2

−2

−4

−4

−6

−6

−8 0

5

10

15

−8 0

20

257

5

10

15

20

5

10 time [s]

15

20

30

30

20

10

10 Torque τ [Nm]

Torque τ [Nm]

Disturbances 20

0

0

−10

−10

−20

−20

−30

0

5

10

15

20

−30 0

time [s]

Fig. 12.4 (continued)

W (e, q) ˙ = eT M(e + qd )q˙

(12.17)

is involved to validate the desired stability property by applying the extended invariance principle to the closed-loop system. Thus, the following result is obtained.

258

12 Quasihomogeneous Stabilization of Fully Actuated Systems with Dry Friction

Theorem 12.2. Let the friction manipulator (12.11), (12.12) be driven by the position feedback regulator (12.13), (12.14) with the assumptions above (thereby admitting the static position feedback regulator (12.15), in particular). Then the closed-loop system (12.11), (12.12), (12.13), (12.14) (or (12.11), (12.12), (12.15)) is globally asymptotically stable at the equilibrium point (q, ˙ e, x) = 0 (respectively, at (q, ˙ e) = 0). Proof. Let us demonstrate that the functions V (e, q, ˙ x) and W (e, q), ˙ given by (12.16) and (12.17), satisfy Conditions 1 and 2 of Theorem 3.3. For this purpose, let us first note that the function V (e, q, ˙ x) is a Lipschitz continuous, radially unbounded, and positive definite. Then, let us compute the time derivative of the composite function V (e(t), q(t), ˙ x(t)) along the trajectories of the closed-loop system (12.11), (12.12), (12.13), (12.14). Employing the well known ˙ property q˙T [ 12 M(q) − C(q, q)] ˙ q˙ = 0, (see, e.g., [54]), we derive that 1 V˙ = q˙T M (q) q¨ + q˙T M˙ (q) q˙ + eT K p e˙ + q˙T Kα sign (e) 2 1 1 + (Kd e − Lx)T (Kd e˙ − Lx) ˙ + (Kd e˙ − Lx) ˙ T (Kd e − Lx) = −q˙T Kb q˙ 2 2 − q˙T K f sign (q) ˙ − e˙T Kd x˙ + eT Kd Kd e˙ + xT LLx˙ − xT LKd e˙ − eT Kd Lx˙ = −q˙T Kb q˙ − q˙T K f sign (q) ˙ − (Kd e − Lx)T L (Kd e − Lx) n = −q˙T Kb q˙ − Σi=1 k fi |q˙i | − x˙T Lx˙ ≤ −α q ˙ − L × Kd e − Lx2

(12.18)

for some constant α > 0 everywhere but on the hyperplanes ei = 0, i = 1, . . . , n where the function V (e, q, ˙ x) is not differentiable. Apparently, sliding motions, possibly occurring on the discontinuity hyperplanes ei = 0, i = 1, . . . , n, including their intersections, are governed by a lower order system of the same form (12.11) because these motions represent behavior of the manipulator, some links of which are “rigidly” connected to others in the sense that corresponding generalized positions qi remain constant in the sliding motion. Hence, a derivation, similar to (12.18), applies here as well and (12.18) remains true for almost all t. Thus, Condition 1 of Theorem 3.3 is shown to hold with the function V (e, q, ˙ x), specified in (12.16), and V1 (e, q, ˙ x) = eα q ˙ − L × Kd e − Lx2.

(12.19)

In turn, the time derivative of the indefinite function (12.17) along the trajectories of the closed-loop system is computed as follows: W˙ = eT M˙ q˙ + e˙T M q˙ + eT M q¨ = eT M˙ q˙ + e˙T M q˙ ˙ + eT [−Kd x˙ − K p e − Kα sign (e)] + eT [−Cq˙ − Kb q˙ − K f sign (q)] n ei kαi sign (ei ) = eT [M˙ − C − Kb ]q˙ + q˙T M q˙ − eT K p e − Σi=1 n ei k fi sign (q˙i ) − eT Kd x˙ ≤ eT [M˙ − C − Kb ]q˙ − Σi=1

n (kαi − k fi )|ei | − eT Kd x. ˙ + q˙T M q˙ − eT K p e − Σi=1

Employing the well-known property

(12.20)

12.2 Global Position Regulation of a Multi-link Robot Manipulator

259

1 2gT h ≤ ε gT g + hT h, g, h ∈ Rn , ε > 0 ε of the Euclidean inner product it follows that ˙ ≤ −eT K p e + ε eT e + 1 q˙T [M˙ − C − Kb ]2 q˙ + q˙T M q˙ W 2ε 1 + (Kd e − Lx)T Kd2 (Kd e − Lx) 2ε

(12.21)

for all ε > 0. Since sup e:V (e,q,x)≤V ˙ 0

M(e + qd ) = M0 < ∞,

sup e:V (e,q,x)≤V ˙ 0

˙ + qd ) = M1 < ∞ (12.22) M(e

due to the smoothness of the inertia matrix, and sup (e,q):V ˙ (e,q,x)≤V ˙ 0

C(e + qd , q) ˙ = C0 < ∞

(12.23)

due to the smoothness of the Coriolis and centripetal torques, the following estimates ˙ M(q(t)) ≤ M0 , M(q)(t) ≤ M1 , C(q(t), q(t)) ˙ ≤ C0

(12.24)

hold for all t ≥ t0 because by virtue of (12.18) one derives that V (e(t), q(t), ˙ x(t)) ≤ V (e(t0 ), q(t ˙ 0 ), x(t0 )) = V0 , for all t ≥ t0 . Apart from this, (12.25) ensures that  q(t) ˙ ≤

2V0 for all t ≥ t0 . M0

(12.25)

(12.26)

Now by applying (12.24)–(12.26), inequality (12.21) results in 2

Kd  ˙ + Kd e − Lx2 W˙ ≤ −eT (K p − ε I)e + α0q 2ε ≤ −W1 ((e(t)) + α1V1 (e(t), q(t), ˙ x(t)) (12.27)  2 −1 1 2 −1 0 where α0 = 21ε 2V M0 (M1 + C0 + Kb ) , α1 = max{α0 α , 2ε Kd  L }, W1 (e(t)) = eT (K p − ε I)e,

(12.28)

I is the identity matrix of an appropriate dimension, V1 (e, q, ˙ x) is given by (12.19). If ε > 0 is chosen small enough to ensure that K p − ε I > 0, the linear combination V1 (e, q, ˙ x) + W1 (e) of Functions (12.19) and (12.28) is positive definite, and Condition 2 of Theorem 3.3 is satisfied with these functions, arbitrary δ > 0, and c1 = α1 .

260

12 Quasihomogeneous Stabilization of Fully Actuated Systems with Dry Friction

Thus, Theorem 3.3 is applicable to the friction n-link manipulator (12.11), (12.12), driven by regulator (12.13), (12.14), and by applying this theorem, the closed-loop system is globally uniformly asymptotically stable at the equilibrium point (e, x, ˙ η ) = 0. Theorem 12.2 is proved. It is worth noticing that letting Kα = 0 and thus omitting the switching part in Regulator 12.13, 12.14 one arrives at the well-known controller

τ = G(q) − Kd x˙ − K p e

(12.29)

x˙ = −Lx + Kd e

(12.30)

developed in [22] to asymptotically stabilize the frictionless manipulator (12.11) subject to F = 0. However, Regulator 12.29, 12.30 becomes invalid to asymptotically stabilize a manipulator with a non-zero Coulomb friction level. Indeed, the friction manipulator (12.11), (12.12) driven by (12.29), (12.30) with no switching part has a nontrivial set of equilibrium points around the desired position, thereby yielding an inappropriate solution to the regulation problem. This theoretical conclusion is additionally supported by experimental results presented in the next section. Thus, high gain regulators are only capable among those with no switching part to provide an appropriate regulation of the manipulator.

12.2.2 Experimental Study The controller performance was studied experimentally. Several experiments were performed with the same 3-DOF robot manipulator as that presented in Sect. 11.1.3. In the experiments, the 3-DOF robot manipulator was required to move in the space from the origin q1 = q2 = q3 = 0 (the references for each articulation are shown in Fig. 11.1) to the desired position qd1 = qd2 = qd3 = π /2 rad. In order to select the constant matrix Kα the knowledge of the Coulomb friction level Fc is required. The procedure of [109] was utilized to obtain the Coulomb friction coefficients Fc = diag{2.1, 1.02, 0.78}Nm. The controller and compensator gains were selected as follows ⎤ ⎤ ⎡ ⎡ 10 0 0 20 0 0 Kd = ⎣ 0 5 0 ⎦ , K p = ⎣ 0 20 0 ⎦ , 0 05 0 0 20 ⎤ ⎤ ⎡ ⎡ 700 500 L = ⎣ 0 5 0 ⎦ , Kα = ⎣ 0 3 0 ⎦ . 005 003 The initial velocities q˙1 (0), q˙2 (0), q˙3 (0) were set to zero in all experiments.

12.2 Global Position Regulation of a Multi-link Robot Manipulator

261

For the sake of comparison, both the dynamic regulator (12.13), (12.14) and its static version (12.15), as well as Regulator 12.29, 12.30 with no switching part, that was proposed in [22], were implemented to drive the manipulator to the desired position. Experimental results for these regulators are depicted in Figs. 12.5–12.7. Figure 12.5 shows that, as predicted by theory, Regulator 12.29, 12.30 with no switching part drives the friction manipulator to a wrong position. t"

t#

@#

t$

@#

t>"

t> #

"

#

@#

t> $

$

Fig. 12.5 Experimental results for the dynamic regulator with no switching part

Figures 12.6 and 12.7 correspond to the quasihomogeneous static and dynamic regulators, respectively. These regulators appear to move the manipulator to the desired position in finite time while having an infinite number of the switches. Figure 12.8 compares the performance of these static and dynamic controllers. From Fig. 12.8 better performance (minimal cumulative actuator energy) is concluded for the dynamic controller.

262

12 Quasihomogeneous Stabilization of Fully Actuated Systems with Dry Friction t" @#

t# @#

t$ @#

t>"

t>#

t>$

"

#

$

Fig. 12.6 Experimental results for the static switched regulator t"

t#

t$

@#

@#

@#

t>"

t>#

t>$

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Fig. 12.7 Experimental results for the dynamic switched regulator

12.2 Global Position Regulation of a Multi-link Robot Manipulator

263

0

vwdwlf uhjxodwru

g|qdplf uhjxodwru

M

Fig. 12.8 Plots of E =

t

2 2 2 0 [τ1 (s) + τ2 (s) + τ3 (s)]ds

for the static and dynamic switched regulators

Chapter 13

Hybrid Control of Underactuated Manipulators with Frictional Joints

The stabilization of underactuated systems, forced by fewer actuators than DOFs, is more complex than that of fully actuated systems [67, 186]. As well known (see, e.g., [23, 249]), these systems possess nonholonomic properties, caused by nonintegrable differential constraints, and therefore, they cannot be stabilized by means of smooth feedback. Due to this, the present investigation is based on hybrid synthesis, which is recognized as an effective tool of controlling underactuated mechanical systems [250, 251]. Representative papers analyzing some control problems for underactuated systems include the study of accessibility [191], the stabilization of equilibria through passivity techniques [186] and energy shaping [27], stabilization and tracking via backstepping control [206], the use of virtual constraints to produce stable oscillations [209], path planning [37], and the control of mechanical systems with an unactuated cyclic variable [86], among others. Most papers on underactuated mechanical systems have typically neglected a fundamental issue in modeling and control, as it is the nonlinear friction effect [69], [250]. Friction forces have been repeatedly left unaddressed or confined to linear damping, thereby severely limiting the achievable performance of the closedloop system. The classical approach regards solving the control problem for an undamped, open-loop model. In [242], the effect of linear damping (viscous friction) on the stability of equilibria which has been stabilized previously in the absence of damping, with the method of controlled Lagrangians, is described. The interconnection and damping assignment passivity-based control technique for underactuated mechanical systems was extended in [83] to incorporate open-loop damping. There, the conditions for recovering stability via damping injection in the presence of highly uncertain, smooth approximations of Coulomb friction effects, were additionally presented. The hybrid synthesis to be developed is centered around the quasihomogeneous design of Chap. 5. In Chap. 12, this design was demonstrated to provide the desired system performance of fully actuated friction mechanical systems in spite of significant uncertainties in the system description and friction modeling. In the present chapter, the quasihomogeneous design is developed for 2-DOFs underactuated systems by including it into a unified hybrid synthesis framework. Capabili265

266

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

ties of the proposed synthesis are illustrated in numerical and experimental studies. These studies are made for laboratory test beds such as a horizontal double pendulum, an inverted double pendulum (Pendubot), and a pendulum, located on a cart. These test beds were initially used to illustrate ideas in linear control such as the stabilization of non-minimum phase unstable systems. Due to the nonlinear nature, they have also maintained their utility of highlighting many ideas in the nonlinear control field. A recent application to smart two-wheel vehicles has demonstrated an increasing practical interest towards underactuated inverted pendulums.

13.1 Stabilization of 2-DOF Systems Using Coulomb Friction Hierarchy In the present section, hybrid control synthesis is proposed for a class of 2-DOF underactuated mechanical systems with dry friction in both joints, governed by the Coulomb model often used to describe these friction forces. The control objective is to regulate both actuated and unactuated joints to desired positions. As opposed to the most existing approaches, which either ignore friction effects or compensate for them, the present synthesis is based on a pure technical solution of imposing a relatively stronger dry friction on the unactuated joint compared to that of the actuated joint. This design feature, proposed in [141], allows one to decouple the hybrid synthesis procedure into two steps. At the first step, a quasihomogeneous controller, driving the unactuated link to the desired position in finite time, is utilized. Once this task is achieved, the control input is switched to another quasihomogeneous algorithm which drives the actuated link to its desired position, also in finite time. As the amplitude of the controller used at the second step is smaller than the Coulomb friction level in the unactuated link, but is greater than that in the actuated link, the unactuated link remains in the rest, whereas the actuated link is regulated to the endpoint of interest. Along with the finite time regulation, the developed hybrid controller exhibits the desired robustness features against parameter variations in friction modeling. Performance issues of the resulting closed-loop system are illustrated in an experimental study made for a laboratory horizontal 2-DOF pendulum.

13.1.1 Problem Statement Consider a 2-DOF underactuated mechanical system whose state-space representation, derived with Lagrange formulation, is given by

∂ V1 (q1 ) + C1 (q˙1 ) α (q˙1 ) = 0 ∂ q1 I3 q¨1 + I4q¨2 + C2 (q˙2 ) α (q˙2 ) = u.

I1 q¨1 + I2q¨2 +

(13.1)

13.1 Stabilization of 2-DOF Systems Using Coulomb Friction Hierarchy

267

Hereinafter, q1 ∈ R, q2 ∈ R are the generalized positions of the unactuated and actuated links, respectively, I1 , I2 , I3 , and I4 are positive entries of the constant inertia matrix subject to the positive definiteness condition I1 > I2I4I3 ; V1 (q1 ) is a potential energy of the unactuated link, ∂ V∂1q(q1 ) is a gravitational torque, and it is a smooth 1  ∂ V1 (q1 )  = 0 and V1 (0) = 0, function satisfying ∂ q  1 q1 =0

α (z) = and u is a control input. The static friction model with

⎧ ⎨

1 z>0 [−1, 1] z = 0 , ⎩ −1 z < 0



q˙2 Ci (q˙i ) = Cc,i + (Cs,i − Cc,i ) exp − 2i vs,i

(13.2)

 , i = 1, 2

(13.3)

Cc,i and Cs,i being the Coulomb friction level and the stiction level such that 0 < Cc,i ≤ Cs,i , and vs,i > 0 being the Stribeck velocity, has been accepted for describing dry friction forces of the unactuated and actuated joints, respectively. Apparently, the friction modulus (13.3) meets the condition Cc,i ≤ Ci (q˙i ) ≤ Cs,i for all q˙i ∈ R.

(13.4)

For technical reasons, the following assumptions are made throughout. 1. A relatively strong friction level is assumed in the unactuated joint link, whereas friction forces, affecting the actuated link, are assumed to be small enough to meet the condition I4 2Cs,2 < Cc,1 . (13.5) I2 2. The gravitational torque

∂ V1 (q1 ) ∂ q1

|

is uniformly bounded, i.e.,

∂ V1 (q1 ) | ≤ V10 for all q1 ∈ R ∂ q1

(13.6)

and some constant V10 > 0. Another assumption on the absence of the gravitational torque of the actuated link, is implicitly made to simplify the controller to be worked out. The above assumptions are particularly satisfied for an underactuated horizontal 2-DOF pendulum experimentally studied in the sequel. The inertial wheel from [186] is another example where the above assumptions hold. The control objective is to steer the generalized coordinates q1 , q˙1 , q2 , and q˙2 to zero. Since the system in question is underactuated, the hybrid control synthesis is invoked. Due to the presence of the friction, our hybrid controller is composed of

268

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

discontinuous control laws which are capable of rejecting friction influence on the system. Setting x1 = q1 , x2 = x˙1 , x3 = x1 + II43 q2 , x4 = x˙3 and .



q˙2 ψ = (Cs,1 − Cc,1 ) exp − 21 vs,1



/ − (Cs,1 − Cc,1 ) α (q˙1 ) ,

(13.7)

(13.1) is represented in the form x˙1 = x2 ,

 −1 ∂ V1 (x1 ) + Cs,1 α (x2 ) x˙2 = I1 − I2I3 /I4 ∂ x1  I2 I2 +ψ − C2 (x4 ) α (x4 ) + u , I4 I4 x˙3 = x4 , 1 1 x˙4 = − C2 (x4 ) α (x4 ) + u. I3 I3

(13.8)

(13.9)

Note that the function ψ (x2 ) is continuous, ψ (0) = 0, and |ψ (x2 )| ≤ Cs,1 − Cc,1 < Cs,1 for all x2 ∈ R.

(13.10)

In the new variables, the control aim is that of steering the state x = (x1 , x2 , x3 , x4 )T to zero with a hybrid control law u = u(x1 , x2 , x3 , x4 ). The idea behind the synthesis to be developed is as follows. Let us consider a hybrid control law u of the form  u1 (x1 ) if x21 + x22 = 0 u= (13.11) u2 (x3 , x4 ) if x1 = x2 = 0 where a position controller u1 (x1 ) is applied to guarantee the finite time stability of (13.8). Once (13.8) is in the origin, the controller is switched to another algorithm u2 (x3 , x4 ) which steers the variables x3 and x4 to zero. As the amplitude of the controller u2 is chosen to be smaller than the Coulomb friction level in the unactuated link, but is greater than that in the actuated link, Subsystem (13.8) remains in the rest whereas (13.9) is regulated to the endpoint of interest.

13.1.2 Quasihomogeneity-based Control Synthesis The discontinuous components u1 (x1 , x2 ) = k1 sign(x1 ) u2 (x3 , x4 ) = −k2 sign(x3 ) − k3sign(x4 )

(13.12) (13.13)

13.1 Stabilization of 2-DOF Systems Using Coulomb Friction Hierarchy

269

of the hybrid controller (13.11) with constant parameters k1 > 0, k2 > 0, and k3 ≥ 0 such that   I4 I2 k1 > Cs,1 + Cs,2 + V10 , (13.14) I4 I2 k2 + k3 + Cs,2 < Cc,1

I4 , I2

k3 + Cs,2 < k2 ,

(13.15) (13.16)

are inherited from the quasihomogeneous synthesis of Sect. 5.1 to stabilize the underactuated system (13.8), (13.9) in finite time. It should be pointed out that the selection of the parameters k2 and k3 , satisfying (13.15) and (13.16), becomes possible due to the subordination condition (13.5), imposed on the friction forces. Theorem 13.1. Let (13.8), (13.9) with the above assumptions be driven by the hybrid controller (13.11) composed of (13.12), (13.13) and let the controller parameters meet (13.14)–(13.16). Then the the closed-loop system is globally finite time stable. Proof. First let us consider (13.8) driven by u = u1 , where u1 is given by (13.12). Note that x1 = x2 = 0 is the only equilibrium point of the closed-loop subsystem subject to (13.14). Indeed, if x2 = 0 on a trajectory of this subsystem, then due to (13.14), the second equation of (13.8) with (13.12) fails to hold for any x1 = 0. This particularly means that the subsystem trajectories cross the axes x1 = 0 and x2 = 0 everywhere but the origin. Thus, it is sufficient to analyze the behavior of the trajectories beyond the discontinuity lines 0 1 S1 = (x1 , x2 )T ∈ R2 : x1 = 0 and

0 1 S2 = (x1 , x2 )T ∈ R2 : x2 = 0 .

With this in mind, let us consider the Lipschitz continuous, radially unbounded, positive definite function   I2 1 I2 I3 2 I1 − V2 (x1 , x2 ) = V1 (x1 ) + k1 |x1 | + x2 (13.17) I4 2 I4 on the trajectories of the closed-loop system (13.8), (13.12) with the input x4 (t), being viewed as an external disturbance. The time-derivative of V2 along (13.8), (13.12) under the lower bound (13.14) outside the discontinuity surfaces S1 and S2 is computed as follows:   I2 (13.18) V˙2 (x1 , x2 ) = −x2 Cs,1 α (x2 ) + ψ − C2 (x4 ) α (x4 ) , I4 I2 V˙2 (x1 , x2 ) ≤ −Cs,1 |x2 | + |ψ | |x2 | + Cs,2 |x2 | , I4

(13.19)

270

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

and from (13.5) one has I2 V˙2 (x1 , x2 ) ≤ −Cc,1 |x2 | + Cs,2 |x2 | ≤ 0. I4

(13.20)

By the extended invariance principle, the non-autonomous closed-loop system (13.8), (13.12), (13.14) proves to be globally asymptotically stable, regardless of which input x4 (t) affects the system. The validity of the extended invariance principle for the system in question follows the line of reasoning used in the proof of Theorem 3.3 and its verification is given below. Clearly, (13.20) ensures that the closed-loop system (13.8), (13.12), (13.14) is stable and the state variables x1 (t) and x2 (t), initialized within a bounded set {(x1 , x2 ) ∈ R2 : V2 (x1 , x2 ) ≤ V 0 } with an arbitrarily large positive constant V 0 , are uniformly bounded: sup V2 (x1 (t), x2 (t)) ≤ V0 . (13.21) t∈[0,∞)

Apart from this, it follows that lim x2 (t) = 0,

t→∞

(13.22)

regardless of the choice of initial conditions of the closed-loop system. To justify (13.22) it suffices to integrate (13.20) and then represent the resulting inequality in the form  ∞ γ |x2 (t)|dt ≤ V0 (13.23) 0

I2 I4 Cs,2

with γ = Cc,1 − > 0. Since due to (13.8) the time-derivative of the trajectories of the closed-loop system are also uniformly bounded, the integrand in (13.23) is equicontinuous, and the validity of (13.22) is straightforwardly obtained by applying Barbalat’s lemma 3.3. To justify that lim x1 (t) = 0 (13.24) t→∞

another auxiliary (indefinite!) function W (x1 , x2 ) = x1 x2 is differentiated along the trajectories of the system:  ∂ V1 (x1 ) −x1 2 ˙ W = x˙1 x2 + x˙2 x1 = x2 + I1 − I2 I3 /I4 ∂ x1  I2 I2 +C1 α (x2 ) − C2 (x4 ) α (x4 ) + k1 sign(x1 ) I4 I4   | I |x I 1 2 2 2 0 k1 − (Cs,1 + Cs,2 + V1 ) . (13.25) ≤ x2 − I1 − I2I3 /I4 I4 I4 Integrating (13.25) and employing (13.14), (13.21), (13.23) yields  ∞ 0

|x1 (t)|dt ≤ Vˆ0

(13.26)

13.1 Stabilization of 2-DOF Systems Using Coulomb Friction Hierarchy

271

for some positive constant Vˆ0 . Similar to (13.22), the position convergence (13.24) follows by applying Barbalat’s lemma 3.3 to the integral inequality (13.26). The global asymptotic stability of the non-autonomous closed-loop system (13.8), (13.12), (13.14) is thus concluded. Hence, any trajectory of (13.8), (13.12), (13.14) is driven to the vicinity 1 0 Dε = (x1 , x2 )T ∈ R2 : V2 (x1 , x2 ) ≤ ε , (13.27) with an arbitrarily small ε > 0 in a finite time t1 which depends on the initial conditions xi (0) = x0i , i = 1, 2, 3, 4, and once the trajectory arrives in the vicinity it remains there forever. Let us now demonstrate that the quasihomogeneity principle, Theorem 4.2 is applicable to the closed-loop system (13.8), (13.12), (13.14), evolving within such a vicinity with arbitrary external inputs x4 (t) and a sufficiently small ε > 0. For this purpose, let us note that, due to (13.5), (13.14),    ∂ V1 (x1 (t))  I2  + Cs,1 + |ψ (x2 (t))| + I2 Cs,2 k1 >  (13.28)  I4 ∂ x1 I4 and

   ∂ V1 (x1 (t))   + |ψ (x2 (t))| + I2 Cs,2  Cs,1 ≥ Cc,1 >   ∂ x1 I4

(13.29)

for all t ≥ t1 because the gravitational torque ∂ V∂1x(x1 ) so that ψ (x2 ) are smooth func1 tions, vanishing in zero, and starting from the time instant t1 , the state variables x1 (t) and x2 (t) are evolving within (13.27) with ε > 0 small enough. Setting y1 = x1 , y2 = x2 , a =

I2 I4 k1 I1 −I2 I3 /I4

w(t) =

,b=

Cs,1 I1 −I2 I3 /I4 ,

h = p = 0, and

  I2 −1 ∂ V1 (x1 (t)) + ψ − C2 (x4 )α (x4 ) , t ≥ t1 , I1 − I2 I3 /I4 ∂ x1 I4

the conditions of Theorem 4.2 are straightforwardly verified, and by applying this theorem, the closed-loop system (13.8), (13.12), (13.14) is established to be globally finite time stable, thereby ensuring that x1 (t) = x2 (t) = 0 at the settling time instant t = T (x0 ), which depends on the initial condition x0 = (x01 , x02 , x03 , x04 )T . To this end, let us consider (13.9) with u = u2 , where u2 is given by (13.13). In order to avoid escaping (13.8) from the origin after the settling time instant T (x0 ), the parameters of (13.11) have been pre-specified to meet (13.15), thereby ensuring that the controller magnitude is smaller than the Coulomb friction level. Apparently, the friction joint, enforced by a sufficiently small power actuation, remains in the rest, and this conclusion can formally be reproduced by following the same line of reasoning as before while computing the time derivative of the Lipschitz continuous, radially unbounded, positive definite function (13.17) after the settling time instant T (x0 ) and making sure that

272

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

I2 V˙2 (x1 , x2 ) ≤ −[Cc,1 − (k2 + k3 + Cs,2 )] |x2 | ≤ 0 I4

(13.30)

on the solutions of (13.8), driven by (13.13), (13.15), (13.16). Moreover, the verification of the applicability of Theorem 4.2 to (13.9), (13.13), (13.15), (13.16) is similar to that made for (13.8), (13.12), (13.14), and by applying this theorem, the closed-loop subsystem (13.9), (13.13), (13.16), (13.15) is established to be globally finite time stable while (13.8), (13.12), (13.14) remains in the rest. Thus, the hybrid controller (13.11), composed of (13.12), (13.13) under (13.14)–(13.16), globally stabilizes (13.8), (13.9) in finite time. It is worth noticing that due to Theorem 4.2, the proposed controller appears to be robust against friction discrepancies. In the remainder, our development is supported by an application to a physical system.

13.1.3 Application to a Horizontal Double Pendulum 13.1.3.1 State Equations Let us consider the system shown in Fig. 13.1, whose motion is constrained to the horizontal plane. The first joint pivots around a point O, and a driver is applied to the second joint. This system is modeled by the equations Jo q¨1 + m2 lo q¨2 + 2m2q2 q˙2 q˙1 + C1 (q˙1 ) α (q˙1 ) = 0, m2 lo q¨1 + m2 q¨2 − m2 q2 q˙21 = u,

(13.31)

where q1 is the pendulum angle, q2 is the position of mass 2, m1 and m2 are the masses, Jo = J1 + J2 + m2 lo2 + q22 + m1 lc2 is the system moment of inertia about O, J1 and J2 are the moments of inertia, α (·) and C1 (·) are given by (13.2) and (13.3), lc and lo are positives constants, and u is the control input. The objective is to design a hybrid control law u which steers the generalized positions q1 , q˙1 , q2 and q˙2 to zero. In order to apply the previous result let us simplify this model by assuming that q1 , q˙1 , q2 and q˙2 are small. Under this condition, (13.31) can be represented in the form Joe q¨1 + m2 lo q¨2 + C1 (q˙1 ) α (q˙1 ) = 0, m2 lo q¨1 + m2 q¨2 = u,

(13.32)

with Joe = J1 + J2 + m2 lo2 + m1 lc2 . For this system one has I1 = Joe , I2 = I3 = m2 lo , I4 = m2 , I1 > I2I4I3 , x1 = q1 , x2 = q˙1 , x3 = x1 + l1o q2 , x4 = q˙1 + l1o q˙2 , and, according to (13.11) with (13.12), (13.13), the following algorithm is proposed:

13.1 Stabilization of 2-DOF Systems Using Coulomb Friction Hierarchy

u

273

q2

m2 J2

m1

J1

q1 O

lo

lc

Fig. 13.1 Horizontal pendulum

u1 = k1 sign (q1 ) ,     1 1 u2 = −k2 sign q1 + q2 − k3sign q˙1 + q˙2 , lo lo (13.33) where, according to (13.14), (13.15) and (13.16), k1 > (2Cs,1 − Cc,1 ) l1o , Cc,1 > lo (k2 + k3) , and k2 > k3 > 0. 13.1.3.2 Experimental Results Figure 13.2 depicts the experimental results conducted on the system manufactured by ECP, model 505 with the parameters lo = 0.330 m, m1 = 0.785 kg, m2 = 0.213 kg, Joe = 0.0477957 kg · m2 , Cs,1 ≈ Cc,1 ≈ 0.0004 N · m, k1 = 0.0024, k2 = 0.0005, k3 = 0.0004, and with the initial conditions q1 (t0 ) = −0.045 rad, q˙1 (t0 ) = 0 rad/s, q2 (t0 ) = 0 m, and q˙2 (t0 ) = 0 m/s. The controller is turned on in t = 2.5 sec. The proposed controller was designed assuming that q1 , q˙1 , q2 and q˙2 are small. However, one can take advantage of the finite time convergence if steering to an arbitrary position of the unactuated link is desired. Experimental results, shown in Fig. 13.3, correspond to the zero initial conditions q1 (t0 ) = 0 rad, q˙1 (t0 ) = 0 rad/s, q2 (t0 ) = 0 m, q˙2 (t0 ) = 0 m/s, and to a second order step filter used as a reference of the unactuated link q1 . The step of the magnitude 0.31416 rad is applied in t = 1.5 sec. Here, q1 is capable of tracking a step signal of the magnitude not exceeding 0.31416 rad. The time between each step is required to be greater than the time convergence at each step. In Fig. 13.3, the control input u represents a sequence of the control inputs similar to that of Fig. 13.2. Thus, the capability of the closed-loop system to steer to an arbitrary position in a large time is experimentally established.

274

q1 (rad)

13 Hybrid Control of Underactuated Manipulators with Frictional Joints 0.06 0.03 0 −0.03 −0.06

q2 (m)

0

2

4

6

8

10

0 −3 x 10

2

4

6

8

10

0

2

4

6

8

10

0.12 0.06 0 −0.06 −0.12

u (N)

6 3 0 −3 −6

t (sec) Fig. 13.2 Local regulation of the horizontal pendulum

q1 (rad)

0.4 0.2 0 −0.2

q2 (m)

0

5

10

15

20

25

30

0 −3 x 10

5

10

15

20

25

30

0

5

10

15

20

25

0.04 0.02 0 −0.02

u (N)

4 2 0 −2

30

t (sec) Fig. 13.3 Non-local regulation of the horizontal pendulum

13.2 Swing-up Control and Stabilization of Pendubot via Orbital Transfer Strategy

275

13.2 Swing-up Control and Stabilization of Pendubot via Orbital Transfer Strategy In the present section, a periodic balancing problem is under study for an underactuated two-link pendulum robot that appears as a special test bed for this purpose. The pendulum robot, typically abbreviated as Pendubot, is a simple underactuated mechanical manipulator, whose first link (shoulder) is actuated, whereas the second one (elbow) is not actuated. Throughout, the positions of both links and their angular velocities are assumed to be available for measurements. We demonstrate that the quasihomogeneous synthesis of Chap. 5 is applicable to the underactuated Pendubot to design a variable structure controller that drives the shoulder of the Pendubot to a periodic reference orbit in finite time, in spite of the presence of external disturbances with an a priori known magnitude bound. As in orbital stabilization, developed in Sect. 12.1 for the periodic tracking of a simple one-link inverted pendulum, a modified Van der Pol oscillator is involved in the present synthesis as a reference model. The effectiveness of the orbitally stabilizing synthesis is illustrated in an experimental study of the swing up/balancing control problem, where the Pendubot is required to move from its stable downward position to the unstable upright position and to be stabilized about the vertical. The orbital transfer strategy has been utilized in [183] to swing up the Pendubot to its upright position. Following this strategy, a swinging controller is composed by an inner loop controller, partially linearizing the Pendubot, and an orbitally stabilizing outer loop controller, that completes the generation of the swing up motion. Switching from the swinging controller to a locally stabilizing one, when the Pendubot enters the attraction basin of the latter, yields a unified framework for the orbital transfer of the Pendubot from the downward position and its stabilization around the unstable equilibrium. The locally stabilizing controller is obtained by applying the quasihomogeneous robust synthesis from [182]. Being verified experimentally, the proposed framework presents an interesting alternative to the energybased approach from [12, 67] to the stabilization of underactuated systems. While the energy-based approach is limited to models with negligible friction forces, the present approach does not suffer from this drawback.

13.2.1 Orbital Stabilization of Pendubot The state equation of the Pendubot, depicted in Fig. 13.4, is derived according to [228, p. 55]: M(q)q¨ + N(q, q) ˙ = τ +w (13.34) where

276

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

 M(q) =

m11 m12 m12 m22 

τ=

τ1 0



 , N(q, q) ˙ =



 , w=

w1 w2

N1 N2

 ,

 (13.35)

and m11 = θ1 m12 = θ3 cos(q1 − q2) m22 = θ2

(13.36)

N1 = θ3 sin(q1 − q2)q˙22 − gθ4 sin(q1 ),

N2 = −θ3 sin(q1 − q2)q˙21 − gθ5 sin(q2 ),

(13.37)

θ1 = m1 l12 + m2L21 + J1 + Jm , θ2 = m2 l22 + J2 , θ3 = m2 L1 l2 , θ4 = m1 l1 + m2L1 , θ5 = m2 l2 .

(13.38)

q2

m2 L2 l2

m1 g

q1

?

l1 L1

τ1

Fig. 13.4 Pendubot

In the above equations, m1 is the mass of link 1, m2 the mass of link 2, L1 and L2 are respectively the lengths of link 1 and link 2; l1 and l2 are the distances to the center of mass of link 1 and link 2; J1 and J2 are the moments of inertia of link 1

13.2 Swing-up Control and Stabilization of Pendubot via Orbital Transfer Strategy

277

and link 2 about their centroids; Jm is the motor inertia, τ1 is the control torque, w is the external disturbance, and g is the gravity acceleration. We assume throughout that the external disturbance is of class L∞ (0, ∞) with an a priori known norm bound K > 0 such that ess sup w(t) ≤ K

(13.39)

t∈[0,∞)

where  ·  stands for the standard Euclidean norm. Our objective is to design a controller that causes the actuated link of the Pendubot to track a trajectory x(t), generated by the modified Van der Pol oscillator ..

x +ε [(x2 +

x˙2 ) − ρ 2]x˙ + μ 2 x = 0 μ2

(13.40)

with positive parameters ε , μ , ρ , while attenuating the effect of an admissible external disturbance (13.39), i.e., the limiting relation lim [q1 (t) − x(t)] = 0

t→∞

(13.41)

is to hold. 13.2.1.1 Control Strategy In order to present a control strategy that allows us to achieve the above objective, let us partially linearize the Pendubot dynamics in accordance with [216]. For this purpose, let us rewrite (13.34) in the form ..

..

m11 q1 +m12 q2 +N1 = τ1 + w1 ..

..

m12 q1 +m22 q2 +N2 = w2 .

(13.42) (13.43)

Then using (13.43), the following equation is derived: ..

..

q2 = −m−1 q 22 [m12 1 +N2 − w2 ].

(13.44)

Now substituting (13.44) into (13.42) yields ..

−1 q (m11 − m12m−1 22 m12 ) 1 −m12 m22 (N2 − w2 ) + N1 = τ1 + w1 .

(13.45)

Finally, setting |M| = m11 − m12m−1 22 m12 and

τ1 = |M|u − m12m−1 22 N2 + N1 where u is the new control input, the desired linearization is obtained:

(13.46)

278

13 Hybrid Control of Underactuated Manipulators with Frictional Joints ..

q1 = u + |M|−1[w1 − m12m−1 22 w2 ] ..

(13.47)

−1 −1 −m−1 22 {m12 [u + |M| (w1 − m12 m22 w2 )]

q2 =

+N2 − w2 }.

(13.48)

In (13.47), (13.48), the positive definiteness of the inertia matrix M(q) has been used to ensure that |M| = 0. Since the above system describes the linearized actuated joint model, it is referred to as collocated linearization [215]. The control strategy is now formalized as follows. Let the control input (13.46) be composed of an inner loop controller, partially linearizing the Pendubot, and an outer loop controller u to be constructed. Given the system output y(t) = q1 (t) − x(t),

(13.49)

that combines the actuated state q1 (t) of the system and the reference variable x(t) governed by the modified Van der Pol equation (13.40), the outer loop controller u is to drive the system output (12.5) to the surface y = 0 in finite time and maintain it there in spite of a uniformly bounded external disturbance w, affecting the system. 13.2.1.2 Control Synthesis Due to (13.40), (13.47), (13.49), the output dynamics is given by ..

2 y = u + |M|−1 w1 − m12 m−1 22 w2 + ε [(x +

x˙2 ) − ρ 2]x˙ + μ 2x. μ2

(13.50)

The following control law u = −ε [(x2 +

x˙2 ) − ρ 2]x˙ − μ 2 x − α sign(y) − β sign(y) ˙ − hy − py˙ (13.51) μ2

with the parameters such that h, p ≥ 0, α − β > (|M|−1 + |m12m−1 22 |)K

(13.52)

is proposed. The closed-loop system (13.50), (13.51) is then feedback transformed to the one ..

y = |M|−1 w1 − m12 m−1 22 w2 −α sign(y) − β sign(y) ˙ − hy − py˙

(13.53)

with a piece-wise continuous right-hand side. As in the quasihomogeneous orbital synthesis of a simple one-link pendulum, developed in Sect. 12.1, the above controller is composed of the nonlinear compensator

13.2 Swing-up Control and Stabilization of Pendubot via Orbital Transfer Strategy

uc = −ε [(x2 +

x˙2 ) − ρ 2]x˙ − μ 2x, μ2

279

(13.54)

the homogeneous relay part ˙ uh = −α sign(y) − β sign(y), and the linear remainder ul = −hy − py˙ that vanishes in the origin y = y˙ = 0. By Theorem 4.4, the quasihomogeneous system (13.53) with the parameter subordination (13.52) is finite time stable regardless of which external uniformly bounded disturbance subject to (13.39) affects the system. The control objective is thus achieved. So, starting from a finite time moment the Pendubot evolves in the second order sliding mode on the zero dynamics surface y = 0. While being restricted to this surface, the system dynamics is given by m22 q¨2 = m12 {ε [(x2 +

x˙2 ) − ρ 2]x˙ + μ 2x} − N2 (−x, q2 ) + w2 μ2

(13.55)

where x(t) is a reference trajectory governed by the modified Van der Pol equation (13.40). For the orbits x(t) of the Van der Pol modification (13.40), initialized on the limit cycle x˙2 (x2 + 2 ) = ρ 2 , (13.56) μ the zero dynamics (13.55) is simplified to m22 q¨2 = m12 μ 2 x − N2 (−x, q2 ) + w2 . (13.57) To formally derive (13.55) one should utilize the equivalent control method and substitute a unique solution ueq of the algebraic equation 2 u + |M|−1w1 − m12 m−1 22 w2 + ε [(x +

x˙2 ) − ρ 2]x˙ + μ 2 x = 0 μ2

(13.58)

with respect to u (i.e., the equivalent control input ueq that ensures equality y¨ = 0) into (13.48). The following result is thus obtained. Theorem 13.2. Let the modified Van der Pol equation (13.40) with positive parameters ε , μ , ρ be a reference model of the Pendubot dynamics (13.34) and let the system output y be given by (13.49). Then, the quasihomogeneous controller (13.46), (13.51), (13.52) drives the Pendubot to the zero dynamics surface y = 0 in finite time, uniformly in admissible disturbances (13.39). After that, the actuated part

280

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

q1 (t) follows the output x(t) of the modified Van der Pol equation (13.40) whereas the non-actuated part q2 (t) is governed by the zero dynamics equation (13.55). Proof. By Theorem 8 of [71, p. 85] the closed-loop system (13.47), (13.48), (13.51) has a local solution for all initial data and admissible disturbances (13.39). Let us demonstrate that each solution of this system is globally continuable on the right. Due to (13.39), the magnitude |W (t)| of the uncertainty W (t) = |M(t)|−1 w1 (t) − m12 (t)m−1 22 w2 (t) that appears in the right-hand side of (13.53) is upper estimated as follows: |W (t)| ≤ (|M|−1 (t) + |m12(t)m−1 (13.59) 22 |)K. Since this estimate, coupled to (13.52), ensures that

α − β > |W (t)|,

(13.60)

Theorem 4.4 turns out to be applicable to the quasihomogeneous system (13.52), (13.53). By applying this theorem, (13.52), (13.53) is proved to be finite time stable, uniformly in admissible disturbances (13.39). Now employing (12.5) and taking into account Theorem 2.6, it follows that along with a solution y(t) of (13.53), an arbitrary solution q1 (t) = y(t) + x(t) of (13.48) is globally continuable on the right and uniformly bounded in t. Moreover, due to the uniform boundedness of y(t), the control signal (13.51) is uniformly bounded, too. Thus, an arbitrary solution q2 (t) of (13.47) is also globally continuable on the right. To complete the proof, it remains to derive the sliding mode system dynamics that has been shown to appear on the surface y = 0 in finite time. For this purpose, let us apply the equivalent control method of Sect. 2.2 and substitute the only solution ueq = −ε [(x2 +

x˙2 ) − ρ 2 ]x˙ − μ 2 x − |M|−1 w1 + m12 m−1 22 w2 μ2

of the algebraic equation (13.58), resulting from the equality y¨ = 0, into (13.48). The sliding mode equation (13.55) on the surface y = 0 is thus validated for the non-actuated variable q2 . Theorem 13.2 is proved. Analyzing the proof of Theorem 13.2, one can conclude that the actuated variable q1 (t) remains bounded regardless of which admissible disturbance w(t) affects the closed-loop system. For practical reasons, the non-actuated variable q2 (t) is also required to remain bounded under an arbitrary disturbance w2 of a sufficiently small magnitude. Due to this, (13.55), governing the zero dynamics, is locally (in w2 ) required to be a bounded input–bounded state system; therefore, its simplified version (13.57) appears to be a bounded input–bounded state system, too. This item is not studied here in details, but only experimental evidences, demonstrating that this is indeed the case, are presented in the next section (for an analysis of bounded input–bounded state systems see, e.g., [105, 133, 219]). In the remainder, capabilities of the synthesis procedure, constituted by Theorem 13.2, are tested in an experimental study of the swing up/balancing control problem.

13.2 Swing-up Control and Stabilization of Pendubot via Orbital Transfer Strategy

281

13.2.2 Experimental Study The orbitally stabilizing synthesis is now utilized to swing up the Pendubot from its downward position to the upright position and it is then switched to a quasihomogeneous controller from [182], locally stabilizing the Pendubot about the vertical. The hybrid control strategy, to be tested in an experimental study, is to select the amplitude ρ and the frequency μ of the model limit cycle (13.56) reasonably small and the parameter ε , controlling the speed of the limit cycle transient in the modified Van der Pol equation (13.40), reasonably large to ensure that the Pendubot enters the attraction basin of the quasihomogeneous locally stabilizing controller. Proper switching from the swinging quasihomogeneous controller to the stabilizing one yields the generation of a swing-up motion, asymptotically stable about the vertical. 13.2.2.1 Pendubot Prototype Performance issues of the quasihomogeneous synthesis are tested on the laboratory Pendubot, manufactured by Mechatronics Systems Inc. and installed in the CICESE Research Center. The values of the Pendubot parameters (13.38), supplied by the manufacturer [1], are listed in Table 13.1. Table 13.1 Parameters of the Pendubot Notation θ1 θ2 θ3 θ4 θ5

Value 0.0308 0.0106 0.0095 0.2087 0.0630

Units kg m2 kg m2 kg m2 kg m kg m

13.2.2.2 Swinging Controller Design In order to apply the orbitally stabilizing synthesis (13.40), (13.46), (13.51) to swinging the Pendubot up from the downward position to the upright position the cumulative energy, pumped into the Pendubot, should be at an appropriate level. According to the energy-based approach [67], such a level is to ensure that the total energy 1 E(q, q) ˙ = q˙T M(q)q˙ + θ4 g cos q1 + θ5 g cos q2 2

(13.61)

of the closed-loop system near the upright position approaches the nominal energy value

282

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

E0 = (θ4 + θ5 )g

(13.62)

that corresponds to the inverted equilibrium of the Pendubot. Being crucial to a successful swing-up, this is achieved by tuning both the controller parameters α , β , h, p and the reference parameters ε , ρ , μ of the Van der Pol modification (13.40). Appropriate values of the parameters to be tuned are carried out by trial and error in successive experiments where the total energy (13.61) of the Pendubot, driven by the developed controller (13.51), is required to enter an interval BR (E0 ) of a sufficiently small radios R, centered around the nominal energy value (13.62). In our experimental study, the controller gains were set to α = 140, β = 40, h = 0, p = 0 whereas the reference parameters were tuned to ε = 8.7, ρ = 0.013, μ = 10. With these parameters, the total energy (13.61) of the Pendubot, driven by the developed controller (13.51), enters the ball BR (E0 ) of the radios R = 0.127, centered at E0 = 2.665. Detailed experimental results, supporting the orbitally stabilizing synthesis, are presented in Sect. 13.2.2.5. 13.2.2.3 Locally Stabilizing Controller Design The quasihomogeneous synthesis, used above for the orbital stabilization, is now applied to the Pendubot block-canonical form, well known from [228], to derive a robust controller, locally stabilizing the Pendubot around its upright position. As reported in [228], the disturbance-free Pendubot system (13.34) is locally minimum phase if its output is given by z = sin q2 + k1 ψ + k2ψ˙

(13.63)

where the output parameters k1 and k2 are positive, and

ψ (q1 , q2 ) = q2 − ϕ (q1 − q2), 2γ ϕ (ν ) = −ν +  tan−1 γ2 − 1



ν 

γ −1 tan γ +1 2

(13.64)

 .

(13.65)

For later use, let us denote A=

m2 gl2 1 , ν = q1 − q2 , η = , m2 L1 l2 γ + cos ν (13.66)

2 3 δ = η sin ν (ψ˙ + γη ν˙ )2 − γη ν˙ 2 + Aη sin(q2 ), (13.67) where

ν˙ = q˙1 − q˙2, η˙ = η 2 sin(ν )(q˙1 − q˙2)

(13.68)

13.2 Swing-up Control and Stabilization of Pendubot via Orbital Transfer Strategy

283

and by virtue of (13.64), (13.65)

ψ˙ = q˙2 − η cos(ν )(q˙1 − q˙2).

(13.69)

Then differentiating (13.63) along the solutions of (13.34) yields z˙ = cos(q2 )q˙2 + k1 ψ˙ + k2 δ ,

(13.70)

z¨ = F(q1 , q2 , q˙1 , q˙2 ) + u

(13.71)

u = Φ (q1 , q2 , q˙1 , q˙2 )τ ,

(13.72)

where

F = cosq2 f2 − sin(q2 )q˙22 + k1δ + k2 [η˙ sin(ν ) + η cos(ν )ν˙ ] (ψ˙ + γη ν˙ )2 +2k2 η sin(ν )(ψ˙ + γη ν˙ )(δ + γ η˙ ν˙ ) − 2γ k2η sin(ν )η˙ ν˙ 2 − γ k2 η 2 cos(ν )ν˙ 3 +Ak2 η˙ sin(q2 ) + Ak2η cos(q2 )q˙2 2γ k2 η 2 sin ν [ψ˙ + ν˙ (γη − 1)]( f1 − f2 ), (13.73) 1 m12 cos q2 , Δ

(13.74)

˙ = f1 (q, q)

m12 (C21 q˙1 + G2 ) − m22 (C12 q˙2 + G1 ) , Δ

(13.75)

˙ = f2 (q, q)

m12 (C12 q˙2 + G1 ) − m11 (C21 q˙1 + G2 ) , Δ

(13.76)

Φ = 2γ k2 η 2 sin ν [ψ˙ + ν˙ (γη − 1)] −

and

Δ = m11 m22 − m212 > 0

(13.77)

because the inertia matrix M is positive definite. Taking into account (13.68), (13.69), one concludes that

Φ (q1 , q2 , q˙1 , q˙2 )|(0,0,0,0) = 0

(13.78)

which is why the locally minimum phase system (13.34), (13.63) has relative degree 2 at the origin. Thus, by applying the quasihomogeneous synthesis to the local stabilization of the Pendubot around its upright position, the following control law is derived:

τ (q, q) ˙ = Φ −1 (q, q)u(q, ˙ q) ˙ where

(13.79)

284

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

u(q, q) ˙ = −F(q, q) ˙ − α1 sign z − β1sign z˙ − h1 z − p1z˙,

(13.80)

the controller gains are such that p1 , h1 ≥ 0, α1 > β1 > 0,

(13.81)

F(q, q) ˙ is governed by (13.73), and z, z˙ are viewed as functions of (q, q), ˙ which are defined by relations (13.63)–(13.70). By analogy to the orbitally stabilizing controller (13.46), the above controller is composed by a partially stabilizing inner loop controller and an outer loop controller that nullifies the system output (13.63) in finite time. Due to Theorem 5.3, the disturbance-free Pendubot dynamics (13.34), enforced by the quasihomogeneous state feedback (13.79)–(13.81), is locally asymptotically stable about the upright position, uniformly in matched disturbances w1 (t), whose magnitude is less than α1 − β1 . Apart from this, the closed-loop system proves to have a certain degree of robustness against vanishing mismatched disturbances w2 (t). Robustness features of the proposed locally stabilizing controller were tested in an experimental study. Quite impressive experimental results were obtained for the local stabilization of the Pendubot about the upright position with the controller parameters α1 = 40, β1 = 20, h1 = 0, p1 = 0 and the output parameters k1 = 0.4, k2 = 0.2. The results can be observed in Sect. 13.2.2.5 where the proposed controller is involved into a hybrid synthesis of the Pendubot to swing up and be balanced about the vertical. The size of the attraction domain of the locally stabilizing controller was additionally evaluated by adding quick disturbances using a metal stick. 13.2.2.4 Hybrid Controller Design In order to accompany swinging up the Pendubot by the subsequent stabilization around the upright position, the model orbit-based swinging controller, presented in Sect. 13.2.2.2, is switched to the locally stabilizing controller from Sect. 13.2.2.3 whenever the Pendubot enters the basin of attraction, experimentally found for the latter controller. The problem of choosing a proper switching time moment is thus logically resolved. While not studied in detail, the capability of the closed-loop system of entering the attraction basin of the locally stabilizing controller is supported by experiments. 13.2.2.5 Experimental Verification The initial conditions of the Pendubot position and those of the modified Van der Pol oscillator, selected for all experiments, were q1 (0) = 3.14 rad, q2 (0) = 3.14 rad, and x(0) = 3.14 rad, whereas all the velocity initial conditions were set to zero.

13.2 Swing-up Control and Stabilization of Pendubot via Orbital Transfer Strategy

285

To begin with, we separately implemented the orbitally stabilizing controller from Sect. 13.2.2.2. In order to test the robustness of this controller, an external disturbance, similar to that of [249], was randomly added by lightly hitting the links of the Pendubot. For demonstrating the capability of the controller to move the Pendubot from one orbit to another by modifying the orbit parameters, we then introduced a random time instant t0 (it was t0 ≈ 10s in the experiment), when the amplitude ρ of the model limit cycle was changed from its initial value ρ = 0.013 to the new one ρ = 0.5. Finally, the hybrid controller from Sect. 13.2.2.4 was implemented to swing the Pendubot up and stabilize it about the vertical. To better demonstrate robustness features of the proposed hybrid synthesis, some external disturbances, again similar to those of [249], were randomly added by lightly hitting the links of the Pendubot. For a robustness comparison, the same hybrid controller was additionally applied to the Pendubot while a mass of 0.0542kg was detached from its shoulder. Since the manufacturer’s user manual [1] identified the Pendubot parameters (13.38) only, while the physical parameters of the Pendubot such as l1 , l2 , J1 , J2 remained unknown, parameters (13.38), corresponding to the modification of the Pendubot, appeared unavailable for tuning the controller gains. Experimental results for the resulting Pendubot motion, enforced by the orbitally stabilizing controller and by the hybrid controller, are depicted in Figs. 13.5 and 13.6, respectively. Figure 13.5 demonstrates that while being driven by the orbitally stabilizing controller, the closed-loop system generates a bounded, quasi-periodic motion and exhibits fast recovery of this motion when the quick disturbance is successively applied to each link of the Pendubot. As predicted by theory, the desired orbital transfer is achieved by simply changing the amplitude of the model2 limit cycle. Thus, good performance of the orbitally stabilizing controller is concluded from Fig. 13.5. In turn, Fig. 13.6 demonstrates that the hybrid controller swings the Pendubot up and stabilizes it about the upright position while also attenuating the parameter variations and external disturbances. Switching from the model orbit-based swinging controller to the locally stabilizing one occurred at the time instant ts ≈ 2.1s. As opposed to the hybrid controller from [249], whose application to a modified model of the Pendubot required the knowledge of the modified parameters to tune the controller parameters, the proposed controller is successfully applied not only to the nominal Pendubot model but also to its modification.

286

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

Fig. 13.5 Orbital stabilization of the Pendubot: left column for the orbital recovery under the randomly added quick disturbance, right column for the transfer from one orbit to another

13.2 Swing-up Control and Stabilization of Pendubot via Orbital Transfer Strategy

287

Fig. 13.6 Swing-up and stabilization of the Pendubot: left column for the nominal model and right column for the modified model, operating under randomly added quick disturbances

288

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

13.3 Quasihomogeneous Swing-up Control and Stabilization of the Cart-pendulum System To better utilize the mechanical structure of underactuated systems, the hybrid synthesis of Sect. 13.2 is now modified by coupling the orbital transfer strategy to the energy-based approach which is adapted for dry friction systems. Such a modification, made in [203], appears to reduce a required cumulative actuator energy at the swinging-up phase. Attractive features of the modified synthesis procedure are illustrated in numerical and experimental studies made for an unactuated pendulum, located on an actuated cart. A control problem under study is to swing the pendulum up and to stabilize it around an unstable equilibrium position while driving the cart to a desired position. Many papers, e.g., [43, 138, 215, 239, 251] among others, have been devoted to this problem. The traditional approach is to develop two controllers. The first controller is based on the nonlinear model and it is dedicated to the swing-up phase, whereas the second controller is derived from the linearized model to locally stabilize the pendulum around the unstable equilibrium position. A hybrid strategy is then determined to switch the swinging controller to the balancing controller when the system enters the attraction domain of the latter controller. It should be noted that while being a linear static feedback (similar to that of the aforementioned works), it remains of limited practical utility because of the size of the attraction domain. Apart from this, ignoring the frictional influence severely limits the achievable performance of the existing hybrid controllers, combining swing up and balancing controllers. The hybrid controller to be developed is based on the orbital transfer strategy, similar to that used in Sect. 13.2 to swing up a Pendubot to its upright position. The strategy is, in addition, combined with the energy approach to enhance the performance of the closed-loop system. First, a swinging controller is composed by an inner loop controller, partially linearizing the cart-pendulum system, and by an orbitally stabilizing outer loop controller, that pumps into the system as much energy as required to approach a homoclinic orbit with the same energy level as that corresponding to the desired equilibrium point. Once the cart-pendulum system reaches this homoclinic orbit, the orbitally stabilizing controller is turned off, and the system is therefore obliged to evolve along the homoclinic orbit. Finally, turning on a locally stabilizing controller, when the homoclinic motion enters the attraction basin of the latter controller, completes a unified framework for the swing-up/balancing control of the pendulum, located on the cart. The locally stabilizing controller from [192], which is also based on quasihomogeneous considerations, is involved in the present hybrid controller design. The locally stabilizing synthesis provides a large domain of attraction, efficient for the hybrid strategy developed subsequently. The resulting hybrid controller is shown to possess the desired robustness properties against friction forces. Being verified experimentally, the proposed framework presents an interesting addition to the energy-based approach from [12, 67] to the stabilization of mechanical systems with underactuation degree one.

13.3 Quasihomogeneous Swing-up Control and Stabilization of the Cart-pendulum System 289

13.3.1 State Equations By applying Newton’s second law of motion, the following dynamical equations (M + m)x¨ + ml sin θ θ˙ 2 − ml cos θ θ¨ = τ + w1 (t) − ψ (x), ˙ 4 2¨ ml θ − ml cos θ x¨ − mgl sin θ = w2 (t) − ϕ (θ˙ ) 3

(13.82) (13.83)

of a cart-pendulum system are derived. In the above relations, x and θ are the generalized coordinates (see Fig. 13.7), M and m are the cart and pendulum mass, respectively, l is the distance from the pendulum center of mass to its pivot, g is the gravitational acceleration, τ is the input force, w1 (t) and w2 (t) are unknown bounded perturbations which are involved to account for model uncertainties and external disturbance; ψ (x) ˙ and ϕ (θ˙ ) are cart and pendulum friction forces, respectively. Coulomb and viscous frictions are modeled as follows:

Fig. 13.7 The cart-pendulum in generalized coordinates x and θ and the experimental test bed.

290

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

ψ (x) ˙ = ψDahl (x) ˙ + ψv x, ˙

ϕ (θ˙ ) = ϕDahl (θ˙ ) + ϕv θ˙ ,

(13.84)

˙ ϕDahl (θ˙ ) stand for where ψv , ϕv denote viscous friction coefficients, and ψDahl (x), dry friction forces. As seen hereafter, the control design requires that the unknown perturbations and the friction forces are differentiable. Thus, it is assumed here that the time evolution of the Coulomb friction is described by the Dahl model     ψDahl (x) ˙ ϕDahl (θ˙ ) ˙ ˙ ˙ ψ˙ Dahl (x) ˙ = σx x˙ − |x| ˙ and ϕ˙ Dahl (θ ) = σθ θ − |θ | (13.85) ψc ϕc where σx , σθ are stiffness coefficients, and ψc and ϕc represent cart Coulomb friction force (linear motor) and rod Coulomb friction torque, respectively. Then the system equations take the form q¨ = M −1 (q)[Bτ − C(q, q) ˙ q˙ − G(q) − F(q)] ˙

(13.86)

with the generalized position q = (x, θ )T and     M + m −ml cos θ 0 ml sin θ θ˙ , M(q) = , C(q, q) ˙ = 4 2 0 0 −ml cos θ 3 ml  G(q) =

     ψ (x) ˙ − w1 (t) 0 1 , F(q) ˙ = . , B= −mgl sin θ 0 ϕ (θ˙ ) − w2 (t)

After some mathematical manipulations, the following state space representation ⎤ ⎡ ⎡ ⎤ ⎡ ⎤ x˙ 0 x˙ 2 2 (3mgl cos θ −4ml θ˙ ) sin θ ⎥ ⎢ 4l ⎥ ⎢ x¨ ⎥ ⎢ ⎥ ⎢ D ⎥ D ⎢ ⎥=⎢ ˙ + w1 (t)] [τ − ψ (x) ⎥+⎣ ⎢ ⎣ θ˙ ⎦ ⎣ 0 ⎦ θ˙ ⎦ 2 3 cos θ 3((M+m)g−ml cos θ θ˙ ) sin θ θ¨ D D ⎡ ⎤ 0 ⎢ 3 cos θ ⎥ D ⎥ ˙ −⎢ (13.87) ⎣ 0 ⎦ [ϕ (θ ) − w2 (t)] 3(M+m) mlD

is obtained with D = l(4M + m + 3m sin2 θ ) > 0. By examining (13.87), it can be seen that ϕ (θ˙ ) and w2 (t) are unmatched friction and perturbation terms. We will see that the developed controller provides attenuation of their perturbing effect.

13.3.2 Local Stabilization The key idea to the local stabilization of a cart-pendulum system is to find a diffeomorphic state space transformation which brings the system into a regular form of two integrator chains.

13.3 Quasihomogeneous Swing-up Control and Stabilization of the Cart-pendulum System 291

13.3.2.1 Transformation to a Regular Form For the purpose of representing the cart-pendulum system in the regular form, the following local change 4 η = x − l ρ (θ ) (13.88) 3 of the state variable with   π 1 + sin θ (13.89) , |θ | < ρ (θ ) = ln cos θ 2 is introduced. System 13.87 is thus transformed into two chains of integrators ⎤ ⎡ η˙   η˙ 4 θ˙ 2 ⎢ η¨ ⎥ ⎢ − g + l 3 cos θ tan θ ⎢ ⎥=⎢ ⎣ θ˙ ⎦ ⎢ ⎣ θ˙ 3((M+m)g−ml cos θ θ˙ 2 ) sin θ θ¨ ⎡

D

⎤

⎡ ⎥ ⎢ ⎥ ⎢ ⎥+⎣ ⎦

0 0 0

3 cos θ D

⎤

⎡

⎥ ⎢ ⎥ (τ − ψ + w1 ) − ⎢ ⎦ ⎣

0

1 ml cos θ

0

3(M+m) mlD

⎤ ⎥ ⎥ (ϕ − w2 ). (13.90) ⎦

with the control input τ acting on the second integrator only. One can see that the unmatched terms ϕ and w2 affect the (η , η˙ ) subsystem of (13.90). Although the influence of these terms on the closed-loop system cannot be rejected, their effects are, however, attenuated under the quasihomogeneous synthesis. This claim is subsequently supported by numerical and experimental simulations. Now, a fictitious output ξ is to be chosen to ensure that the (η , η˙ ) subsystem with ϕ = w2 = 0 is minimum phase with respect to this output. Let us demonstrate that the required output ξ can be locally chosen as

ξ = tan θ − λ1η − λ2 η˙ ,

(13.91)

with some parameters λ1 , λ2 > 0. Taking into account (13.88), (13.89), and (13.91), both θ and x prove to converge to zero whenever η , η˙ and ξ tend to zero. Therefore, enforcing a sliding motion on ξ = 0 solves the problem of the asymptotic stabilization of x and θ provided that the (η , η˙ ) subsystem is minimum phase. The problem is thus reduced to designing a quasihomogeneous controller which guarantees the finite time convergence of ξ to zero. Differentiating ξ twice, one gets

ξ¨ = μ + ζ + u, with

(13.92)

292

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

×

    tan θ ˙ 2 λ2 ϕv 1 8l λ2 θ˙ tan θ ˙ μ θ,θ = 2 2 θ + + − cos θ cos2 θ 3 cos θ ml cos θ / . M+m 2 ˙ −3 ϕ (θ˙ ) 3[(M + m)g − ml cos θ θ˙ ] sin θ − 3 cos θ ψ (x) ml

D     θ˙ 4 l θ˙ 2 4 l θ˙ 2 (1 + sin2 θ ) + g+ λ1 tan θ + g + λ2 2 3 cos θ 3 cos θ cos θ   ˙ λ1 + λ2θ tan θ λ2 − ϕ (θ˙ ) − ϕ˙ Dahl (θ˙ ), ml cos θ ml cos θ  u=

 3ml + 8ml 2λ2 θ˙ sin θ − 3λ2 cos θ ϕv τ, mlD cos θ

(13.93)

and the uncertain term    3 cos θ 1 ml 8ml 2 λ2 θ˙ tan θ 3(M + m) − λ2ϕv w1 + w2 ζ= { + ml cos θ cos θ 3 D mlD $ # ˙ + λ1 + λ2θ tan θ w2 + λ2 w˙ 2 }. Due to physical constraints, λ2 can be chosen small enough to ensure that 3ml + 8ml 2 λ2 θ˙ sin θ − 3λ2 cos θ ϕv = 0 in (13.93). Coupled to (13.92), the disturbancefree system   4 θ˙ 2 1 ¨ (ξ + λ1η + λ2η˙ ) + η = − g+ l (ϕ − w2 ). (13.94) 3 cos θ ml cos θ with ϕ = w2 = 0 is in a regular form1 . 13.3.2.2 Locally Stabilizing Synthesis Following the quasihomogeneous synthesis of Sect. 5.3, the control input u is specified as follows:   u = −μ θ , θ˙ − α1 sign(ξ ) − β1sign(ξ˙ ) − hξ − pξ˙ , thereby yielding

1

In order to put the cart-pendulum system into its regular form given by (5.6), one has to represent the state equations in terms of η and ξ . The full regular form is not explicitly given here, because it is rather lengthy from a computational point of view, and because it is not essential for the design of the controller. As a matter of fact, (13.92) and (13.94) constitute the regular form since the variables θ and θ˙ are implicit functions of the new state variables η , η˙ , ξ and ξ˙ .

13.3 Quasihomogeneous Swing-up Control and Stabilization of the Cart-pendulum System 293

mlD cos θ $ τ=# 2 3ml + 8ml λ2 θ˙ sin θ − 3λ2 cos θ ϕv     × −μ θ , θ˙ − α1 sign(ξ ) − β1sign(ξ˙ ) − hξ − pξ˙ .

(13.95)

Provided that w1 , w2 , w˙2 and θ˙ are bounded, the matched disturbance ζ appears to be uniformly bounded |ζ | < Δ by some positive Δ for almost all t, θ and θ˙ . Thus, letting Δ < β1 < α1 − Δ h, p ≥ 0, (13.96) the quasihomogeneous controller (13.95), (13.96) meets the conditions of Theorem 5.3, and by applying Theorem 5.3, it ensures the uniform finite time stability of the (ξ , ξ˙ ) system. By applying the equivalent control method, the dynamics on the manifold ξ = 0 are given by        0 0 1 η˙ η = + (ϕ − w2 ). (13.97) 1 −λ1 γ −λ2 γ η¨ η˙ ml cos θ   $ π π# θ˙ 2 where γ = g + 43 l cos θ > 0, for all θ ∈ − 2 , 2 . Finally, it remains to investigate the asymptotic stability of (13.97), when ϕ = w2 = 0. Note that there exists a state transformation     Ω1 η =P , Ω= Ω2 η˙ with

 10 , κ > 0, P= κ1 

such that (13.97) is specialized to the form   −κ 1 Ω˙ = Ω, κσ δ

(13.98)

where

−κ 2 + λ2 γκ − λ1γ and δ = κ − λ2 γ . κ   |Ω 1 | as a vector Lyapunov function [28, 87], the Viewing V (Ω ) = |Ω | = |Ω2 | right-handside Dini derivative of V (Ω ) satisfies D+V (Ω ) ≤ Γ V (Ω ) with Γ =  −κ |1| . If |κσ | δ 4λ1 λ22 > , (13.99) γ

σ=

then there exists κ > 0 such that κ 2 − λ2 γκ + λ1γ < 0 and   −κ 1 . Γ= κσ δ

(13.100)

294

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

Note that the non-constant terms in (13.100) appear in the last row only, and Γ is a Metzler matrix (−M matrix). This allows one to apply the linear stability theory (see [28], [87]). Therefore, under condition (13.99), Ω = 0 is an asymptotically stable equilibrium for the nonlinear system (13.98). Now, by taking into account pendulum friction ϕ (θ˙ ) and the non-matched perturbation w2 (t), it is obvious that we loose the asymptotic stability property, but it is proven by the linear stability theory that the perturbed linear system (13.97) converges into a ball around the origin, thus obtaining practical stability. The radius of this ball can be decreased by the tuning of λ1 and λ2 . This will be supported by experimental results. 13.3.2.3 Numerical and Experimental Verification In order to illustrate the effectiveness of the above controller, numerical simulations and experiments were conducted on a cart-pendulum system, installed in the LAGIS Laboratory of Ecole Centrale de Lille (see Fig. 13.7).The following numerical values M = 3.4Kg m = 0.147Kg; l = (0.351/2)m; λ1 = 0.2m−1 ; λ2 = 0.25 sec/m.rad K1 = 40rad/sec2; K2 = 20rad/sec; h = 30rad/sec2; p = 0rad/sec

ψv = 8.5Nsec/m; ϕv = 0.0015Nmsec/rad; σx = 10000N/m; σθ = 50Nm/rad ψc = 6.5N; ϕc = 0.00115Nm were used. The simulation runs were conducted without any unmatched frictions and unknown perturbations, and with the following initial conditions: (x, x, ˙ θ , θ˙ )T (0) = (0.2m, 1m/sec, 0.8rd, 2rd/sec). The initial condition used for θ in the simulation was chosen as θ (0) = 0.8rd = 45.8◦ in order to illustrate the reasonably large size of the attraction domain of the controller. The results are depicted in Fig. 13.8. As it is theoretically expected, both the cart position and the pendulum position are asymptotically stabilized. In addition, an experimental test was conducted on the laboratory cart-pendulum system with both unmatched pendulum joint friction and matched cart friction. To clearly carry out the controller robustness, a significant disturbance was also manually applied to the pendulum during the time interval from about 6 sec to 10 sec. Under the parameters tuned to λ1 = 0.2 and λ2 = 0.25, practical stability and good performance were obtained (see Fig. 13.9).

13.3 Quasihomogeneous Swing-up Control and Stabilization of the Cart-pendulum System 295

Fig. 13.8 Numerical simulation without unmatched friction: top figure for the cart position and for the rod angle, bottom figure for the control input

296

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

Fig. 13.9 Experimental results: top figure for the cart position and for the rod angle, bottom figure for the control input

13.3 Quasihomogeneous Swing-up Control and Stabilization of the Cart-pendulum System 297

13.3.3 Swing-up Control The control objective for the periodic balancing results in the closed-loop system that generates its own periodic orbit similar to that produced by a nonlinear oscillator, e.g., by a modified Van der Pol oscillator. The quasihomogeneous controller of Sect. 13.2, enforcing a Pendubot to periodically balance, is now modified for the periodic balancing the cart-pendulum system (13.82), (13.83). Since, in contrast to locally stabilizing synthesis, the periodically balancing design does not require the differentiability of the friction forces ψ (x), ˙ ϕ (θ˙ ), the friction Dahl model (13.84), (13.85) is subsequently simplified to the classical model

ψ (x) ˙ = ψv x˙ + ψc sign(x), ˙ ϕ (θ˙ ) = ϕv θ˙ + ϕc sign(θ˙ )

(13.101)

by formally substituting σx = σθ = ∞ into (13.85). The uncertain terms w1 (t), w2 (t), introduced into the dynamic equations (13.82), (13.83), still account for destabilizing model discrepancies such as the Stribeck effect and backlash. Due to the dissipative properties of mechanical systems, upper bounds Ni > 0, i = 1, 2 for the magnitudes of these terms can normally be estimated a priori: |wi (t)| ≤ Ni (13.102) for all t. Our nearest goal is to design a controller that causes the actuated part of the cart-pendulum system to track lim [x(t) − z(t)] = 0,

t→∞

(13.103)

a trajectory z(t) generated by the modified Van der Pol equation ..

z + ε [(z2 +

z˙2 ) − ρ 2]˙z + ν 2 z = 0, ν2

(13.104)

while also attenuating the effect of the friction forces (13.101) and external disturbances (13.102). 13.3.3.1 Partial Linearization In order to present a control strategy that allows one to achieve the control goal let us partially linearize the cart-pendulum dynamics. For this purpose, let us rewrite the state equation (13.83) in the form ..

θ=

3 .. [ml cos θ x + mgl sin θ + w2 (t) − ϕ (θ˙ )]. 4ml 2

Now substituting n (13.105) into (13.82) yields

(13.105)

298

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

3 .. [(m + M) − m cos2 θ ]x = τ + w1 (t) − ψ (x) ˙ − ml sin θ θ˙ 2 4 3 3 + mg cos θ sin θ + [w2 (t) − ϕ (θ˙ )] cos θ . 4 4l

(13.106)

Finally, setting J = (m + M) − 34 m cos2 θ = 14 m + M + 34 m sin2 θ and 3 τ = Ju + ml sin θ θ˙ 2 − mg cos θ sin θ 4

(13.107)

where u is the new control input, and taking into account that the relation J = 0 holds for all values of θ , the desired linearization is obtained: 3 cos θ 1 [w2 (t) − ϕ (θ˙ )] + [w1 (t) − ψ (x)] ˙ 4lJ J .. 3 3m cos2 θ + 4J [w2 (t) − ϕ (θ˙ )] θ = {u cos θ + 4l 4mlJ cos θ [w1 (t) − ψ (x)] + ˙ + g sin θ }. J ..

x = u+

(13.108)

(13.109)

Since (13.108), (13.109) describes the linearized actuated joint model it is referred to as collocated linearization [215]. 13.3.3.2 Control Strategy and Synthesis The control strategy is then formalized as follows. The control input (13.107) is composed by an inner loop controller, partially linearizing the cart-pendulum, and an outer loop controller u to be constructed. Given the system output y(t) = x(t) − z(t),

(13.110)

that combines the actuated state x(t) of the system and the reference variable z(t) governed by the modified Van der Pol equation (13.104), the outer loop controller u is to drive the system output (13.110) to the surface y = 0 in finite time, and maintain it there in spite of the friction forces ψ (x), ˙ ϕ (θ˙ ) and external disturbances w1 (t), w2 (t), affecting the system. The following control law u=

3ϕv cos θ ˙ ψv z˙2 θ + x˙ − ε [(z2 + 2 ) − ρ 2 ]˙z − ν 2 z 4lJ J ν ˙ − hy − py˙ −α sign(y) − β sign(y) (13.111)

with the parameters such that

13.3 Quasihomogeneous Swing-up Control and Stabilization of the Cart-pendulum System 299

h, p ≥ 0, α − β >

3(ϕc + N2 ) ψc + N1 + 4lJ J

(13.112)

is proposed. Since by virtue of (13.104), (13.108), (13.110), the output dynamics are given by ..

3 cos θ 1 [w2 (t) − ϕ (θ˙ )] + [w1 (t) − ψ (x)] ˙ 4lJ J z˙2 +ε [(z2 + 2 ) − ρ 2]˙z + ν 2 z, ν

y = u+

(13.113)

the closed-loop system (13.101), (13.111), (13.113) is feedback transformed to ..

y=

3 cos θ 1 [w2 (t) − ϕc sign(θ˙ )] + [w1 (t) − ψc sign(x)] ˙ 4lJ J ˙ − hy − py. ˙ −α sign(y) − β sign(y)

(13.114)

Relating the quasihomogeneous synthesis of Chap. 5, the above controller has been composed of the linear viscous friction compensator uf =

3ϕv cos θ ˙ ψv ˙ θ + x, 4lJ J

(13.115)

the nonlinear trajectory compensator uc = −ε [(z2 +

z˙2 ) − ρ 2]˙z − ν 2 z, ν2

(13.116)

the homogeneous switching part uh = −α sign(y) − β sign(y), ˙

(13.117)

ul = −hy − py˙

(13.118)

and the linear remainder that vanishes in the origin y = y˙ = 0. By Theorem 5.3, the quasihomogeneous system (13.114) with the parameter subordination (13.112) is finite time stable, regardless of which friction forces (13.101) and which uniformly bounded external disturbances subject to (13.102) affect the system. The control goal is thus achieved and the following result is obtained. Theorem 13.3. Let the modified Van der Pol equation (13.104) with positive parameters ε , ν , ρ be a reference model of the cart-pendulum dynamics (13.82), (13.83), (13.101) and let the system output be given by (13.110). Then the quasihomogeneous controller (13.46), (13.111), (13.112) drives the cart-pendulum system to the zero dynamics surface y = 0 in finite time, uniformly in friction forces (13.101) and admissible disturbances (13.102). After that, the actuated part x(t) follows the output z(t) of the modified Van der Pol equation (13.104) whereas the non-actuated part θ (t) is governed by the zero dynamics equation

300

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

4 .. 3 cos θ 1 l θ = cos θ { [ϕ (θ˙ ) − w2 (t)] + [ψ (x) ˙ − w1 (t)] 3 4lJ J x˙2 3m cos2 θ + 4J [w2 (t) − ϕ (θ˙ )] + ε [(x2 + 2 ) − ρ 2]x˙ + ν 2 x} + ν 4mlJ x˙2 cos θ [w1 (t) − ψ (x)] ˙ + g sin θ = cos θ {ε [(x2 + 2 ) − ρ 2 ]x˙ + ν 2 x} + J ν 1 [w2 (t) − ϕ (θ˙ )] + g sin θ . + (13.119) ml Proof. By Theorem 8 of [71, p. 85] the closed-loop system (13.108), (13.109), (13.111) has a local solution for all initial data and admissible disturbances (13.102). Let us demonstrate that each solution of this system is globally continuable on the right. Due to (13.102), the magnitude |W (t)| of the uncertainty W (t) =

3 cos θ 1 [w2 (t) − ϕc sign(θ˙ )] + [w1 (t) − ψv sign(x)] ˙ 4lJ J

that appears in the right-hand side of (13.114) is upper estimated as follows |W (t)| ≤

3(ϕc + N2 ) ψc + N1 + . 4lJ J

(13.120)

Since this estimate, coupled to (13.112), ensures that

α − β > |W (t)|,

(13.121)

Theorem 5.3 turns out to be applicable to the quasihomogeneous system (13.112), (13.114). By applying this theorem, (13.114) subject to (13.112) is proved to be finite time stable, uniformly in admissible disturbances (13.102). Now, employing (13.110), it follows that along with a solution y(t) of (13.114), an arbitrary solution x(t) = y(t) + z(t) of (13.108) is globally continuable on the right and uniformly bounded in t. Moreover, due to the uniform boundedness of y(t), the control signal (13.111) is uniformly bounded, too. Thus, an arbitrary solution θ (t) of (13.109) is also globally continuable on the right. So, starting from a finite time moment, the cart-pendulum system evolves in the second order sliding mode on the zero dynamics surface y = 0, and to complete the proof, it remains to derive the sliding mode system dynamics. For this purpose, let us apply the equivalent control method and substitute the only solution ueq of the algebraic equation u+

3 cos θ 1 x˙2 [w2 (t) − ϕ (θ˙ )] + [w1 (t) − ψ (x)] ˙ − ε [(x2 + 2 ) − ρ 2]x˙ − ν 2 x = 0 4lJ J ν

with respect to u (i.e., the equivalent control input ueq that ensures equality y¨ = 0) into (13.109). It follows that, while being restricted to this surface, the system

13.3 Quasihomogeneous Swing-up Control and Stabilization of the Cart-pendulum System 301

dynamics are given by (13.119). The sliding mode equation (13.119) on the surface y = 0 is thus validated for the non-actuated variable θ (t), restricted to any finite time interval. The proof of Theorem 13.3 is completed. Analyzing the proof of Theorem 13.3, one can conclude that the actuated variable x(t) remains bounded regardless of which admissible disturbances affect the closedloop system. For practical reasons, the non-actuated variable θ (t) is also required to remain bounded under arbitrary disturbances of sufficiently small magnitudes. Due to this, the zero dynamics (13.119) are required to locally (in (w1 , w2 )) form a bounded input - bounded state (BIBS) system. This item is not studied here in details, but only numerical and experimental evidences, demonstrating that this is indeed the case, are presented in the next section. For analysis of BIBS systems see, e.g., [105, 133, 219]). For the orbits z(t) of the Van der Pol modification (13.104), initialized on the limit cycle x˙2 x2 + 2 = ρ 2 , (13.122) ν the zero dynamics (13.119) are simplified to the equation 4 .. 1 l θ = cos θ ν 2 x + g sin θ + [w2 (t) − ϕ (θ˙ )], 3 ml

(13.123)

whose BIBS analysis is much easier. In the remainder, capabilities of the synthesis procedure, constituted by Theorem 13.3, are tested in numerical and experimental studies of the laboratory cartpendulum system.

13.3.4 Experimental Study In the test swing up/balancing control problem, the pendulum, located on the cart, is required to move from its stable downward position to the unstable upright position and to be stabilized about the vertical while the cart is stabilized about a desired endpoint. An orbitally stabilizing controller is designed to swing up the pendulum to make it reach a homoclinic orbit of the energy level, corresponding to the cart-pendulum system at the desired equilibrium point. Once the homoclinic orbit is reached at a state where the cart velocity is infinitesimal, the orbitally stabilizing controller is turned off, thereby letting the system evolve along the homoclinic orbit. A locally stabilizing controller of Sect. 13.3.2.3 is then turned on when the homoclinic motion enters the attraction basin of the latter controller. The hybrid control strategy, to be tested in an experimental study, is to select the amplitude ρ and the frequency ν of the model limit cycle (13.122) reasonably small and the parameter ε , controlling the speed of the limit cycle transient in the modified Van der Pol equation (13.104), reasonably large to ensure that the cartpendulum system reaches its homoclinic orbit of the corresponding energy level. Turning off the orbitally stabilizing controller, once the system is synchronized with

302

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

the homoclinic orbit, and proper switching to the stabilizing controller when the subsequent homoclinic motion enters the attraction basin of the latter controller yields the generation of a swing-up motion, asymptotically stable about the vertical. In order to observe the performance of the proposed synthesis we made simulations on MATLAB using Simulink . We considered the real parameters of the laboratory cart-pendulum system, presented in Section 13.3.2.3. The parameters of the cart-pendulum system are listed in Table 13.2. Table 13.2 Parameters of the Cart-pendulum System. Notation M m l ψv ϕv ψc ϕc

Value 3.4 0.147 0.175 8.5 0.0015 6.5 0.00115

Units kg kg m N · s/m N · m · s/rad N N ·m

13.3.4.1 Swinging Controller Design By a suitable choice of the reference parameters ε , ρ , ν , the orbitally stabilizing synthesis (13.104), (13.107), (13.111) can be applied to the cart-pendulum system to swing the pendulum up from the downward position to the upright position. To successfully apply this synthesis the cumulative energy, pumped by the resulting controller into the cart-pendulum system, is to be of an appropriate level. According to the energy-based approach [67], such a level in the friction-free case must ensure that the total energy 1 2 E(q, q) ˙ = (M + m)x˙2 − ml x˙θ˙ cos θ + ml 2 θ˙ 2 + mgl(cos θ − 1) (13.124) 2 3 of the closed-loop system increases from the negative value E0 = −2mgl,

(13.125)

at the initial time moment to the zero-value energy E0 = 0,

(13.126)

at a time instant when the cart velocity becomes infinitesimal and ready to change the rotation direction. Thus synchronized, the unforced friction-free system generates a homoclinic orbit converging to the desired equilibrium that has the same energy level (13.126).

13.3 Quasihomogeneous Swing-up Control and Stabilization of the Cart-pendulum System 303

In order to take into account friction forces, resulting in an energy loss of the laboratory cart-pendulum system, the required energy level (13.126) has to deliberately be increased to a positive value E0 = δ > 0.

(13.127)

Under a certain δ , found experimentally, the unforced friction system also generates an orbit, converging to the desired equilibrium. This orbit is further referred to as a quasihomoclinic orbit. So, tuning the energy parameter δ and the reference parameters ε , ρ , ν is crucial to a successful swing up. Appropriate values of these parameters are carried out in successive simulations. The parameter ρ , responsible for the amplitude of the cart oscillations, is simply set slightly smaller than the admissible road length of the laboratorial equipment. The parameters ε and ν are responsible for making a reasonably high convergence rate of the modified Van der Pol oscillator (13.104) and, respectively, for providing the desired energy level (13.127) of the system at a position of the cart where it possesses an infinitesimal velocity (the higher ε the faster convergence; the larger ν the greater the energy of the limit cycle and hence the greater the energy of the cart, tracking the limit cycle). These parameters are iteratively tuned to approach the quasihomoclinic orbit. Turning off the controller at the time instant, when the system attains the quasihomoclinic orbit, makes the system follow this orbit, thereby approaching the desired endpoint. In our simulation study, the controller gains were set to α = 3 N · m, β = 1 N · m, h = 0, p = 0. With these parameters, (13.52) holds and the output tracking of the modified Van der Pol oscillator (13.104) is therefore guaranteed. The reference parameters were tuned to ε = 40 [rad]−2s−1 , ρ = 0.5 rad, ν = 1 s−1 . Being found numerically for the above parameters, the energy parameter took the value δ = 0.8 N · m. Once the system energy attained this value under x = ρ , x˙ = 0, the orbitally stabilizing controller was turned off and the unforced pendulum was swing up along the quasihomoclinic orbit. Detailed experimental results, supporting the proposed swing up synthesis, are presented in Sect. 13.3.4.5. 13.3.4.2 Locally Stabilizing Controller Design Being developed in Sect. 13.3.2.3, the locally stabilizing controller 2 3 4lJ cos θ $ −μ (θ , θ˙ ) − α1 sign(ξ ) − β1sign(ξ˙ ) − h1ξ − p1 ξ˙ 3 + 8l λ2θ˙ sin θ (13.128) with α1 , β1 , h1 , p1 > 0 and 32 3 2 (M+m)g−ml cos θ θ˙ 2 3+8l λ2 θ˙ sin θ tan θ ˙ 2 tan θ μ (θ , θ˙ ) = 2 cos 2θ θ + cos θ D     (13.129) 2 2 ˙ 2 ˙ ˙ θ 4 θ (1+sin θ ) θ +λ1 g + 43 l cos , θ tan θ + λ2 g + 3 l cos θ cos2 θ

τ=#

304

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

ξ = tan θ − λ1η − λ2 η˙ , λ1 , λ2 > 0 η = x−

  π 1 + sin θ 4l ln , |θ | < , 3 cos θ 2

(13.130) (13.131)

was tested in a simulation study. Quite impressive numerical results were obtained for the local stabilization of the pendulum about the upright position with the controller parameters α1 = 55 m/s2 , β1 = 35 m/s2 , h1 = 0, p1 = 0, λ1 = 0.2 1/m, λ2 = 0.26 s/m. The results can be observed in Sect. 13.3.4.4 where the proposed controller is introduced into our hybrid synthesis of swinging the pendulum up and balancing it about the vertical. 13.3.4.3 Hybrid Controller Design In order to accompany swinging the pendulum up by the subsequent stabilization around the upright position, the swinging controller, presented in Sect. 13.3.4.1, is turned off, once the system reaches the corresponding homoclinic orbit, and then the locally stabilizing controller of Sect. 13.3.2.3 is turned on once the pendulum, evolving along the quasihomoclinic orbit, enters the basin of attraction, numerically found for the latter controller. The resulting hybrid controller moves the inverted pendulum, located on the cart, from its downward position to the upright position and stabilizes it about the vertical, whereas the cart is stabilized at the desired endpoint. The capability of the closed-loop system of reaching the homoclinic orbit and entering the attraction basin of the locally stabilizing controller is additionally supported by experimental results. 13.3.4.4 Numerical Verification The initial conditions of the position of the cart-pendulum system and that of the modified Van der Pol oscillator, selected for all experiments, were x(0) = 0, θ (0) = 3.14 rad, and z(0) = 0.05 rad, whereas all the velocity initial conditions were set to zero. To begin with, we separately studied the orbitally stabilizing controller of Sect. 13.3.4.1. For demonstrating the capability of the controller to move the pendulum from one orbit to another by modifying the orbit parameters, we then introduced a random time instant t0 (it was t0 ≈ 10s in the experiment), when the amplitude ρ and frequency ν of the model limit cycle were changed from their initial values ρ = 0.1 rad, ν = 10 s−1 to the new one ρ = 0.5 rad, ν = 1 s−1 . Finally, the hybrid controller from Sect. 13.3.4.3 was implemented to swing the pendulum up and to stabilize it about the vertical, while the cart is stabilized around the initial position. Numerical results for the resulting motion, enforced by the orbitally stabilizing controller, are depicted in Fig. 13.10. This figure demonstrates that while being

13.3 Quasihomogeneous Swing-up Control and Stabilization of the Cart-pendulum System 305

driven by the orbitally stabilizing controller, the closed-loop system, perturbed by the permanent external disturbances w1 (t) ≡ 0.5 N, w2 (t) ≡ 0.5 N · m, generates a bounded, quasi-periodic motion. As predicted by theory, the desired orbital transfer is achieved by simply changing the amplitude of the orbit limit cycle. Thus, good performance of the orbitally stabilizing controller is concluded from Fig. 13.10.

Fig. 13.10 Orbital stabilization of the cart-pendulum system: left column for the orbital recovery under the permanent disturbances, right column for the transfer from one orbit to another

306

13 Hybrid Control of Underactuated Manipulators with Frictional Joints

13.3.4.5 Experimental Verification The initial conditions of the position of the cart-pendulum system and those of the modified Van der Pol oscillator, selected for the experiment, were x(0) = −0.3 m, θ (0) = π rad (downward position), and z(0) = 0.3 m, whereas all the velocity initial conditions were set to zero. The hybrid controller from Sect. 13.3.4.3 was implemented to swing the pendulum up and stabilize it about the vertical while the cart is stabilized around the desired endpoint. Experimental results for the closed-loop system, enforced by the hybrid controller, are depicted in Fig. 13.11. It is concluded from the figure that while being driven by the orbitally stabilizing controller, the pendulum approaches the homoclinic orbit in 2 seconds and then the controller is turned off and the pendulum travels along the orbit within the interval (2, 2.5) towards the unstable equilibrium. Once the pendulum is close enough to the equilibrium, the local controller is turned on. The local controller possesses a large domain of attraction (see Sect. 13.3.2.3), and it is turned on at 10◦ from the upright position. In order to appreciate the performance of the developed controller, the experimental results were compared to those obtained in [67], where switching from the swinging controller to a local state feedback was done after approximately 30 seconds. The proposed approach admits switching to the local quasihomogeneous controller in approximately 2.5 seconds, and the stabilization is accomplished in a shorter time compared to that of [67]. It should also be noted that the energy-based approach of [67] is limited to models with negligible friction forces, whereas the present approach does not suffer from this drawback.

13.3 Quasihomogeneous Swing-up Control and Stabilization of the Cart-pendulum System 307 cart position and rod angular position 7

6

5

4

3

2

1

0

−1 0

0.5

1

1.5

2

2.5 time

3

3.5

4

4.5

5

3

3.5

4

4.5

5

control input 0.2

0.15

0.1

0.05

0

−0.05

−0.1

−0.15

−0.2

−0.25 0

0.5

1

1.5

2

2.5 time

Fig. 13.11 Experimental results: top figure for the cart position and for the rod angle, bottom figure for the control input

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Index

H∞ -design procedure, 154 L2 -gain analysis, 60 approximate δ -solution, 30 asymptotic harmonic generator, 41

finite-dimensional approximation, 164 friction, 34 Frobenius condition, 18 Fuller phenomenon, 6 fully actuated system, stabilization of, 249

backlash, 36, 240 Barbalat’s lemma, 55

generalized input, 14 generalized solution, 14, 30

Caratheodory solution, 14 cart-pendulum system, swing up control and stabilization, 288 characteristic equation, 85 Chua’s circuit, 2 concentration observer, 205 Coulomb model, 34

Hamilton–Jacobi inequality, 60 Hamilton–Jacobi–Isaacs inequality, 133, 135, 139, 141 Hilbert space-valued system, 28, 75, 162, 193 homogeneity degree (dilation), 89 homogeneity principle, 92 horizontal double pendulum, regulation of, 272 hybrid system, 7 hysteresis switching, 7

Dahl model, 35 delay-/disturbance-dependent stability criterion, 218 discontinuous system, 1 discrete-continuous system, 21, 65, 145 disturbance attenuation, 133 disturbance rejection, 115 equilibria set, localization of, 51 equiuniform (asymptotic) stability, definition of, 88 equiuniform finite-time stability, definition of, 89 equivalent control method, 26 exponential stabilizability/detectability in Hilbert space, definition of, 163 extended invariance principle, 55 Filippov set, 24 Filippov solution, 24

impulsive actuator, 67 impulsive stabilization, 68 impulsive system, 1 infinite-dimensional Luenberger observer, 161, 164 instantaneous impulse response, 15 invariance principle, 49 inverted pendulum, orbital stabilization of, 249 Kalman–Popov–Yakubovich lemma, 63 Korteweg–de Vries–Burgers equation, 164 Kuramoto–Sivashinsky equation, 164, 175 limit cycle, 39 linear matrix inequalities, 72 linear operator inequalities, 82 LuGre model, 35 Lyapunov–Krasovskii method, 79, 219 319

320 mass-energy balance equation, 199 mild solution, 30 minimum phase nonlinear system, stabilization of, 107 minimum phase semilinear infinitedimensional system, stabilization of, 193 modified Van der Pol oscillator, 40, 250, 277, 297 multi-link robot manipulator, regulation of, 229, 254 Pendubot, swing up control and stabilization of, 275 quasihomogeneity principle, 93 quasihomogeneous synthesis procedure, 112 regional stabilization, 209 restitution rule, 1–3, 65 Riccati equation, 62, 140, 144 sampled-data measurement, 145 servomechanism with backlash, regulation of, 240 sliding mode, 9, 29 sliding mode synthesis procedure, 116 Sobolev space, 75

Index stability (global asymptotic) in Hilbert space, definition of, 76, 162 stability (global, uniform, exponential, asymptotic), definition of, 46 Stribeck model, 35 strong solution, 29 time-delay system, 75, 215 uncertain equation, solution of, 88 uncertain system, (practical) stabilization of, 117, 129, 161, 175, 221 underactuated system, hybrid control of, 265 unit feedback, 9, 115, 166, 175, 221 Utkin solution, 27 Van der Pol oscillator, 39 variable structure system, 4 velocity observer, 53 vibroimpact solution, 21, 43 vibroimpact system, 21 viscous friction, 34 Wirtinger’s inequality, 75 Zeno mode, 6 zero-order distribution, 13