Advanced Strategies in Control Systems with Input and Output Constraints (Lecture Notes in Control and Information Sciences)

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Lecture Notes in Control and Information Sciences 346 Editors: M. Thoma, M. Morari

Sophie Tarbouriech, Germain Garcia, Adolf H. Glattfelder (Eds.)

Advanced Strategies in Control Systems with Input and Output Constraints

ABC

Series Advisory Board F. Allgöwer, P. Fleming, P. Kokotovic, A.B. Kurzhanski, H. Kwakernaak, A. Rantzer, J.N. Tsitsiklis

Editors Dr. Sophie Tarbouriech

Adolf Hermann Glattfelder ETH Zürich Dept.Informationstechnol./Elekt Inst. Automatik Physikstr. 3 8092 Zürich Switzerland

LAAS-CNRS 7 Avenue du Colonel Roche 31077 Toulouse cedex 4 France

Professor Germain Garcia LAAS-CNRS 7 Avenue du Colonel Roche 31077 Toulouse cedex 4 France

Library of Congress Control Number: 2006930671 ISSN print edition: 0170-8643 ISSN electronic edition: 1610-7411 ISBN-10 3-540-37009-9 Springer Berlin Heidelberg New York ISBN-13 978-3-540-37009-3 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, 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 protective laws and regulations and therefore free for general use. Typesetting: by the authors and techbooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

SPIN: 11616344

89/techbooks

543210

Preface

Physical, safety or technological constraints induce that the control actuators can neither provide unlimited amplitude signals nor unlimited speed of reaction. The control problems of combat aircraft prototypes and satellite launchers offer interesting examples of the difﬁculties due to these major constraints. Neglecting actuator saturations on both amplitude and dynamics can be source of undesirable or even catastrophic behavior for the closed-loop system (such as loosing closed-loop stability) [3]. Such actuator saturations have also been blamed as one of several unfortunate mishaps leading to the 1986 Chernobyl nuclear power plant disaster [12], [10]. For these reasons, the study of the control problem (its structure, performance and stability analysis) for systems subject to both amplitude and rate actuator saturations as typical input constraints has received the attention of many researchers in the last years (see, for example, [13], [8], [7], [6]). Anti-windup is an empirical approach to cope with nonlinear effects due to input constraints, and override is a related technique for handling output constraints, [6]. The anti-windup approach consists of taking into account the effect of saturations in a second step after a previous design performed disregarding the saturation terms. The idea is then to introduce control modiﬁcations in order to recover, as much as possible, the performance induced by a previous design carried out on the basis of the unsaturated system. In particular, anti-windup schemes have been successfully applied to avoid or minimize the windup of the integral action in PID controllers. This technique is largely applied in industry. In this case, most of the related literature focuses on the performance improvement in the sense of avoiding large and oscillatory transient responses (see, among others, [1], [5]).

VI

Preface

More recently, special attention has been paid to the inﬂuence of the anti-windup schemes on the stability and the performance of the closed-loop system (see, for example, [2], [9], [11]). Several results on the anti-windup problem are concerned with achieving global stability properties. Since global results cannot be achieved for open-loop exponentially unstable linear systems in the presence of actuator saturation, local results have to be developed. In this context, a key issue concerns the determination of stability domains for the closed-loop system. If the resulting basin of attraction is not sufﬁciently large, the system can present a divergent behavior depending on its initialization and the action of disturbances. It is worth to notice that the basin of attraction is modiﬁed (and therefore can be enlarged) by the anti-windup loop. In [4], or in the ACC03 Workshop “T-1: Modern Anti-windup Synthesis”, some constructive conditions are proposed both to determine suitable anti-windup gains and to quantify the closed-loop region of stability in the case of amplitude saturation actuator. The override technique uses the same basic approach of a two-step design. A linear control loop is designed for the main output ﬁrst without regard of the output constraints. It normally performs control for small enough deviations form its design operating point. Then one or more additional feedback control loops are designed for the system trajectory to run along or close to those output constraints. The transfer between the loops is automatic (for example by Min-Max-Selectors) and bumpless (by using antiwindup) and constitutes the dominant nonlinear element in the control system, Much less research results have been published on this topic so far, [6]. The book is organized as follows. • Part 1 is devoted to anti-windup strategies and consists of chapters 1 through 6. • Part 2 is devoted to model predictive control (MPC) and consists of chapters 7 through 10. • Part 3 is devoted to stability and stabilization methods for constrained systems and consists of chapters 11 through 15. Note that this partition is somewhat arbitrary as most of the chapters are interconnected, and mainly reﬂects the editors’ biases and interests. We hope that this volume will help in claiming many of the problems for controls researchers, and to alert graduate students to the many interesting ideas

Preface

VII

Proposal of Benchmarks (Common Application Examples) We decided to provide two benchmark problems, hoping that this will make reading the book more attractive. One of them or both are considered in several chapters. However the given benchmark problems may not be ideally suited to each design method. Then in some chapters, an additional case has been supplemented. Both examples are abstracted from their speciﬁc industrial background, but conserve the main features which are relevant in this speciﬁc context. The main focus is on plants which are exponentially unstable systems. They are known to be sensitive to constraints. We elected this feature as it seems currently to be the most interesting and also the most challenging one. Hurwitz systems have been covered in many publications in recent years, and that area seems to have matured. In operation, both plants have a strong persistent disturbance z, which moves the steady state value u ¯ of the control variable u close to the respective saturation. As one control engineer put it: ‘operate as close as possible to the top of the hill (to maximize revenue), but do not fall off the cliff on the other side’. The plants are described by models in continuous form. Model variables are scaled and given in ‘per unit’ form. All state variables xi are measured and are available as outputs yi . No provision has been made for fast non-modelled dynamics. However it is recommended to check for possible bandwidth limitations. The same holds for the inevitable measurement noise. The controllers may be implemented in continuous or time discrete form. Integral action or a suitable substitute shall be provided, in order to drive the control error to zero for steady state loads z¯ = 0. A typical electro-hydraulic actuator subsystem is considered. The mechanical end stops of the servomotor piston are at ulo , uhi (stroke constraints), and vlo , vhi represent the ﬂow constraints from the pilot servovalve (slew constraints), and ks is the ﬁnite, moderate gain of the P-controller. This model is embedded in the typical cascade structure, where the main controller outputs the position reference uc = rs . In the contribution we expect the contributor to

VIII

Preface 2

1

v_up

u_hi

u_c k_s

1/T_s

u

1 s

v_dn

u_lo k_a_s

-

show that the speciﬁcations are met,

-

document the system responses to a given test sequence, and indicate the maximum stabilizable inputs,

-

provide a stability analysis,

-

and optionally to comment on the implementation requirements.

The Inverted Pendulum Consider the inverted pendulum around its upright position for small inclination angles. The pendulum mass m1 shall be concentrated at its center of gravity (cg), and the connecting member of length L to the slider at its bottom is mass-free and rigid. The slider mass m3 may move horizontally without friction. A horizontal force is applied by the actuator subsystem to the slider, in order to control the speed error to zero. Alternatively you may investigate the case of driving the position error to zero. Denote as state variables the horizontal speed of the pendulum cg as x1 , the horizontal displacement between the cg’s of pendulum and slider as x2 and the horizontal speed of the slider as x3 . The load z1 represents a horizontal force on the pendulum cg, which is persistent and with unknown but bounded magnitude. And u represents the force by the actuator subsystem, which is constrained ﬁrst in magnitude (stroke) only and then in rate (slew) as well. -

The linearized model in appropriately scaled variables is given by

Preface

IX

v1

x1

m1

Fu

Fz

m3 x3

v3

d x1 = −x2 + z1 ; y1 = x1 dt d τ2 x2 = −x1 + x3 ; y2 = x2 dt d τ3 x3 = +x2 + u; y3 = x3 dt τ1

-

with parameter values τ3 = τ2 = τ1 = 1.0 s.

-

The transfer functions have one pole at the origin, one on the negative real axis, and one symmetrical on the positive real axis.

-

The closed loop bandwidth in the linear range is speciﬁed at ≥ 2.0 rad/s.

-

The actuator magnitude (stroke) saturations are at ulo = −1.25 and uhi = +1.25 and on the actuator rate (slew) are: (a) |du/dt| ≤ 10.0/s

-

and then (b) |du/dt| ≤ 2.0/s

The test sequence is deﬁned as follows -

always start in closed loop operation at initial conditions x(0) = 0, reference r1 = 0 and load z1 = 0.

-

apply a large reference ‘step’ of size r1 up to max. stabilizable or to r1 = 2.0, whichever is smaller, with slew rate dr1 /dt = +0.5/s, after stabilization apply a small additional reference Δr1 = 0.10 and then back to Δr1 = 0.0. and set back r1 = 0., with slew rate dr1 /dt = −0.5/s

X

Preface

-

apply a large load ‘step’ of size z1 up to z1 = 1.0 or to max. stabilizable, whichever is smaller with slew rate dz1 /dt = +0.25/s after stabilization apply a small additional load ‘step’ Δz1 = 0.10 and then back to Δz1 = 0.0. and set back z1 = 0., with slew rate dz1 /dt = −0.25/s 2

Continuous Stirred Tank Reactor (CSTR) with a Strong Exothermal Reaction Consider a batch reactor with a continuous feed ﬂow of reactant and control of the contents temperature by means of a heating/cooling jacket. The main variable to be controlled is the ﬂuid temperature x2 , to r2 = 1.0, with zero steady state error e2 = r2 − x2 → 0. The main disturbance to the temperature control loop is the thermal power production. It should be as high as possible without the temperature x2 running away, in order to maximize the production rate. The limit to this is set by the maximum cooling heat ﬂow though the wall to the jacket, and thus by the minimum temperature of the heating/cooling ﬂuid (u2 ) entering the jacket. Both magnitude and rate constraints on u2 are speciﬁed from the subsystem delivering the heating/cooling ﬂuid. The reactant concentration x1 is to be controlled to its setpoint r1 through the reactant feed ﬂow u1 , with no perceptible magnitude or rate constraints here. Both the temperature controller R2 and the concentration controller R1 are to be designed. The model of the plant shall be given by the nonlinear equations (in ‘per-units’): d x1 = −a1 f τ1 dt

+ u1 ;

d τ2 dt x2 = −a2 f

+ a3 (x3 − x2 ); y2 = x2

y1 = x1

d τ3 dt x3 = −a3 (x3 − x2 ) + a4 (u2 − x3 ); y3 = x3

where f = a0 · x1 · max[0, (x2 − x2s )]

Preface

XI

with x1 for the reactant mass or concentration in the vessel content ﬂuid where the reaction takes place, x2 for the content temperature, x3 for the jacket temperature (lumping both metal walls and heating/cooling ﬂuid temperatures), u2 for the entry temperature of the heating/cooling ﬂuid from the supply subsystem to the jacket, and u1 for the reactant feed ﬂow. Below the ignition temperature x2s , f is zero. Above, f is proportional to the reactant concentration x1 and the ﬂuid contents temperature x2 − x2s . The mass ﬂow disappearing from the mass balance of reactant x1 is set proportional to f (with coefﬁcient a1 ). The thermal power production is also set proportional to f , with coefﬁcient a2 . The exothermal reaction relates to a2 < 0. During production, the contents temperature x2 must always stay above the ignition temperature x2s , which calls for high closed loop performance. The inventory of reactant must also be closely controlled, for safety reasons. The following numerical values are speciﬁed: 1 = 10.; r2 − x2s a1 = 1.0; a2 = −1.0; a3 = 1.0; a4 = 1.0;

x2s = 0.90; a0 =

τ1 = 0.20; τ2 = 1.00; τ3 = 0.20; where a small τ1 /τ2 signiﬁes a relatively small inventory of reactant in the reactor, and where a large τ3 /τ2 signiﬁes a relatively large heat capacity of the jacket. The values for τ1 , τ2 , τ3 imply a time scaling of typically 1000 → 1. For the nominal steady state operating conditions at x ¯1 = 1.0; x ¯2 = 1.0; x ¯3 = 0.0; that is u ¯2 = −1.0; u ¯1 = 1.0; the open loop poles are at s1 = +3.4640; s2 = +0.7049; s3 = −10.1734 The closed loop bandwidth for the temperature control in its linear range is speciﬁed at ≥ 15.0 rad/s. The actuator stroke is saturated at u2lo = −1.10 and u2hi = +1.10

XII

Preface

and the maximum actuator slew rate is (a) |du2 /dt| = 200.0/s or (b) |du2 /dt| = 5.0/s The proportional gain of the concentration controller R1 is to be: kp1 ≥ 10.0 No slew saturations are speciﬁed for both r1 (t) and r2 (t). The test sequence is deﬁned as follows: -

start in closed loop operation with all initial conditions and inputs r1 , r2 at zero,

-

apply a reference step of size r2 = 1.0; and wait for equilibration

-

then add a load step of size r1 up to max. stabilizable or to r1 = 1.00, whichever is smaller.

The control targets are -

to attain the full equilibrium production at r1 = 1.0; r2 = 1.0 as fast as possible,

-

and then control with the closed loop bandwidth speciﬁed above, with - an additional reactant inﬂow step

Δr1 = +0.020

- and after equilibration a step change in a2 of

Δa2 = +0.020

- and after equilibration an additional temperature reference step Δr2 = −0.020 -

and fast shut-down of production by resetting ﬁrst r1 to zero, and then r2 to zero. 2

Acknowledgements The idea of this edited book was formed through a series of e-mail exchanges and face-to-face discussions when the three of us met at conferences or during visiting stay in each of the lab in each side. First and foremost, we would like to thank all the contributors of the book. Without their encouragement, enthusiasm, and patience, this book would have not been pos-

Preface

XIII

sible. A list of contributors is provided at the end of the book. We also thank also Isabelle Queinnec (LAAS-CNRS) for her help regarding latex problems. We also wish to thank Springer for agreeing to publish this book. We wish to express our gratitude to Dr. Thomas Ditzinger (Engineering Editor), and Ms. Heather King (International Engineering Editorial) for their careful consideration and helpful suggestions regarding the format and organization of the book.

Toulouse, France, June 2006,

Sophie Tarbouriech

Toulouse, France, June 2006,

Germain Garcia

Z¨urich, Switzerland, June 2006 ,

Adolf H. Glattfelder

References ˚ om and L. Rundqwist. Integrator windup and how to avoid it. In Proc. of the 1. K. J. Astr¨ American Control Conference, Pittsburgh, USA, pp.1693-1698, 1989. 2. C. Barbu, R. Reginatto, A. R. Teel and L. Zaccarian. Anti-windup for exponentially unstable linear systems with inputs limited in magnitude and rate. In Proc. of the American Control Conference, Chicago, USA, 2000. 3. J.M. Berg, K.D. Hammett, C.A. Schwartz and S.S. Banda. An analysis of the destabilizing effect of daisy chained rate-limited actuators. IEEE Trans. Control Syst. Tech., vol.4, no.2, pp.375-380, 1996. 4. Y-Y. Cao, Z. Lin and D.G. Ward. An antiwindup approach to enlarging domain of attraction for linear systems subject to actuator saturation. IEEE Transactions on Automatic Control, vol.47, no.1, pp.140-145, 2002. 5. H. A. Fertik and C. W. Ross. Direct digital control algorithm with anti-windup feature. ISA Transactions, vol.6, pp.317-328, 1967. 6. A.H. Glattfelder and W. Schaufelberger. Control Systems with Input and Output Constraints. Springer-Verlag, London, 2003. 7. J.M. Gomes da Silva Jr., S. Tarbouriech and G. Garcia. Local stabilization of linear systems under amplitude and rate saturating actuators. IEEE Transactions on Automatic Control, vol.48, no.5, pp.842-847, 2003. 8. V. Kapila and K. Grigoriadis. Actuator saturation control. Marcel Dekker, Inc., 2002. 9. N. Kapoor, A. R. Teel and P. Daoutidis. An anti-windup design for linear systems with input saturation. Automatica, vol.34, no.5, pp.559-574,1998.

XIV

Preface

10. M. Kothare. Control of systems subject to constraints. PhD thesis, Division of Engineering and Applied Science, California Institute of Technology, USA, 1997. 11. M. V. Kothare and M. Morari. Multiplier Theory for stability analisys of Anti-windup control systems. Automatica, vol.35, pp.917-928, 1999. 12. G. Stein. Bode lecture: Respect to unstable. In Proc. of the 28th IEEE Conference on Decision and Control, Tampa, USA, 1989. 13. F. Tyan and D. Bernstein. Dynamic output feedback compensation for linear systems with independent amplitude and rate saturation. International Journal of Control, vol.67, no.1, pp.89-116, 1997.

Contents

Anti-windup Augmentation for Plasma Vertical Stabilization in the DIII-D Tokamak Eugenio Schuster, Michael Walker, David Humphreys, Miroslav Krsti´c . . . . . . .

1

Stable and Unstable Systems with Amplitude and Rate Saturation Peter Hippe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 An Anti-windup Design for Linear Systems with Imprecise Knowledge of the Actuator Input Output Characteristics Haijun Fang, Zongli Lin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Design and Analysis of Override Control for Exponentially Unstable Systems with Input Saturations Adolf Hermann Glattfelder, Walter Schaufelberger . . . . . . . . . . . . . . . . . . . . . . . 91 Anti-windup Compensation using a Decoupling Architecture Matthew C. Turner, Guido Herrmann, Ian Postlethwaite . . . . . . . . . . . . . . . . . . . 121 Anti-Windup Strategy for Systems Subject to Actuator and Sensor Saturations Sophie Tarbouriech, Isabelle Queinnec, Germain Garcia . . . . . . . . . . . . . . . . . . 173

XVI

Contents

Sampled-Data Nonlinear Model Predictive Control for Constrained Continuous Time Systems Rolf Findeisen, Tobias Raff, and Frank Allg¨ower . . . . . . . . . . . . . . . . . . . . . . . . . 207 Explicit Model Predictive Control Urban Maeder, Raphael Cagienard, Manfred Morari . . . . . . . . . . . . . . . . . . . . . 237 Constrained Control Using Model Predictive Control J.M. Maciejowski, P.J. Goulart, E.C. Kerrigan . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Risk Adjusted Receding Horizon Control of Constrained Linear Parameter Varying Systems M. Sznaier, C. M. Lagoa, X. Li, A. A. Stoorvogel . . . . . . . . . . . . . . . . . . . . . . . . . 293 Case Studies on the Control of Input-Constrained Linear Plants Via Output Feedback Containing an Internal Deadzone Loop Dan Dai, Tingshu Hu, Andrew R. Teel, Luca Zaccarian . . . . . . . . . . . . . . . . . . . 313 Set Based Control Synthesis for State and Velocity Constrained Systems Franco Blanchini, Stefano Miani, Carlo Savorgnan . . . . . . . . . . . . . . . . . . . . . . 341 Output Feedback for Discrete-Time Systems with Amplitude and Rate Constrained Actuators J.M. Gomes da Silva Jr., D. Limon, T. Alamo, E.F. Camacho . . . . . . . . . . . . . . . 369 Decentralized Stabilization of Linear Time Invariant Systems Subject to Actuator Saturation Anton A. Stoorvogel, , Ali Saberi, Ciprian Deliu, , Peddapullaiah Sannuti . . . . . 397 On the Stabilization of Linear Discrete-Time Delay Systems Subject to Input Saturation Karim Yakoubi, Yacine Chitour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Contents

XVII

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

Anti-windup Augmentation for Plasma Vertical Stabilization in the DIII-D Tokamak Eugenio Schuster1 , Michael Walker2 , David Humphreys3 , and Miroslav Krsti´c4 1

Lehigh University [email protected]

2

General Atomics [email protected]

3

General Atomics [email protected]

4

University of California San Diego [email protected]

Summary. In the Advanced Tokamak (AT) operating mode of the DIII-D tokamak, an integrated multivariable controller takes into account highly coupled inﬂuences of plasma equilibrium shape, proﬁle, and stability control. Time-scale separation in the system allows a multiloop design: the inner loop closed by the nominal vertical controller designed to control a linear exponentially unstable plant and the outer loop closed by the nominal shape controller designed to control a nonlinear stabilized plant. Due to actuator constraints, the nominal vertical controller fails to stabilize the vertical position of the plasma inside the tokamak when large or fast disturbances are present or when the references coming from the shape controller change suddenly. Anti-windup synthesis is proposed to ﬁnd a nonlinear modiﬁcation of the nominal vertical controller that prevents vertical instability and undesirable oscillations but leaves the inner loop unmodiﬁed when there is no input saturation.

1 Introduction Demands for more varied shapes of the plasma and requirements for high performance regulation of the plasma boundary and internal proﬁles are the common denominator of the Advanced Tokamak (AT) operating mode in DIII-D [1]. This operating mode requires multivariable control techniques [2] to take into account the highly coupled inﬂuences of equilibrium shape, proﬁle, and stability control. Current efforts are focused on providing improved control for ongoing experimental

2

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krsti´c

operations, preparing for future operational control needs, and making advances toward integrated control for AT scenarios. Vertical and Shape Control: The initial step toward integrating multiple individual controls is implementation of a multivariable shape and vertical controller for routine operational use which can be extended to integrate other controls. The long term goal is to integrate the shape and vertical control with control of plasma proﬁles such as pressure, radial E-ﬁeld, and current proﬁles using feedback commands to actuators such as gas injectors, pumps, neutral beams (NB), electron cyclotron heating (ECH), and electron cyclotron current drive (ECCD). The problem of vertical and shape control in tokamaks was and is still extensively studied in the fusion community. A recent summary of the existing work in the ﬁeld can be found in [3]. Several solutions [4, 5, 6, 7, 8, 9, 10] for the design of the nominal controller were proposed for different tokamaks using varied control techniques based on linearized models. Although the saturation of coil currents and voltages (actuators) is a common problem in tokamaks and there were efforts to minimize the control demand for shape and vertical control and to avoid saturation [11, 12], the saturation of the actuators was rarely taken into account in the design of the controllers until recently [13, 14]. Although similar in concept, our work uses a different approach to the problem: anti-windup compensator. The input constraints are not taken into account at the moment of designing the nominal controller. The goal is not the design of the nominal controller but the design of an anti-windup compensator that blends any given nominal controller, which is designed to fulﬁl some local (saturation is not considered) performance criterion, with a nonlinear feedback designed to guarantee stability in the presence of input saturation but not necessarily tuned for local performance. This nonlinear modiﬁcation of the nominal controller also aims at keeping the nominal controller well-behaved and avoiding undesirable oscillations. The anti-windup augmentation must in addition leave the nominal closed loop unmodiﬁed when no saturation is present. Shape Control Methodology: The shape control methodology at DIII-D is based on “isoﬂux control”. The isoﬂux control method, now in routine use on DIII-D, exploits the capability of the real time

Anti-Windup Augmentation in the DIII-D Tokamak

3

Fig. 1. DIII-D controlled plasma parameters in isoﬂux control (X-point R and Z positions, and ﬂux at control points on control segments).

EFIT plasma shape reconstruction algorithm to calculate magnetic ﬂux at speciﬁed locations within the tokamak vacuum vessel. Figure 1 illustrates a lower single null plasma which was controlled using isoﬂux control. Real time EFIT can calculate very accurately the value of ﬂux in the vicinity of the plasma boundary. Thus, the controlled parameters are the values of ﬂux at prespeciﬁed control points along with the X-point R and Z positions, deﬁned in Figure 1. By requiring that the ﬂux at each control point be equal to the same constant value, the control forces the same ﬂux contour to pass through all of these control points. By choosing this constant value equal to the ﬂux at the X-point, this ﬂux contour must be the last closed ﬂux surface or separatrix. The desired separatrix location is speciﬁed by selecting one of a large number of control points along each of several control segments. An X-point control grid is used to assist in calculating the X-point location by providing detailed ﬂux and ﬁeld information at a number of closely spaced points in the vicinity of the X-point. Several problems make practical implementation of shape and vertical position controllers on DIII-D challenging: (P1) Computational speed is insufﬁcient to do both vertical stabilization and shape control with the same controller. (P2) Vertical stability control and shape control share the same actuators, i.e., shaping coils (F-coils)

4

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krsti´c

2A, 2B, 6A, 6B, 7A, and 7B (Figure 1). This is a particular problem for the outer coils because they become the only coils which perform shape control for the outer plasma boundary. (P3) The shape control power supply system (choppers) is extremely nonlinear. (P4) Limitations on actuator voltage imply that commands to shaping power supplies (choppers) often saturate, particularly with large or fast disturbances. (P5) Multiple current and voltage constraints imply that the range of accessible plasma equilibria is constrained. Programmed attempts to reach equilibria outside these constraints can lead to problems such as exceeding coil current limits or saturation of actuators leading to loss of control. (P6) Changes in linearized response of the nonlinear plasma during the course of a discharge can degrade control by linear controllers. This chapter is organized as follows. Section 2 introduces the challenges and the strategy used for plasma shape and vertical position control. Section 3 introduces the basics of the anti-windup method. The characteristics of our plant and its controllable region are presented respectively in Section 4 and Section 5. The design of the antiwindup compensator is detailed in Section 6. Section 7 discusses some simulation results and the conclusions are presented in Section 8.

2 Control Strategy Computational Speed Limitation: Time-scale separation of vertical and shape control appears to be critical for DIIID, since multivariable shape controllers can require signiﬁcant computation [15]. Figure 2 shows the closed loop system comprised of the DIII-D plant and stabilizing controller. This system is stable and the 6 coil currents F2A, F2B, F6A, F6B, F7A, and F7B are approximately controlled to a set of input reference values. As a result, this system can act as an inner control loop for shape control. Figure 3 shows that the inner loop’s inputs are the 6 vertical coil current reference signals rI , the centroid (center of mass of plasma current) vertical position reference signal rZ , and up to 12 shape coil command voltages v.

Anti-Windup Augmentation in the DIII-D Tokamak

5

Fig. 2. Inner loop: Vertical Stabilization.

Actuator Sharing: A method implemented for sharing actuators involves constructing a linear controller which simultaneously performs vertical stabilization and shape control (by controlling the currents of the 6 vertical control coils F2A, F2B, F6A, F6B, F7A, and F7B). The former is done on a fast time scale, as seen by comparing Figures 2 and 3. By integrating current control of the vertical control coils into a stabilizing controller, conﬂicts between shape and vertical control use of these coils is eliminated. “Frequency sharing” is accomplished explicitly with an H∞ loop shaping design by weighting low frequencies to regulate the coil currents and high frequencies to stabilize the plasma. The design technique ensures that the overall system remains robustly stable. Power Supply Nonlinearities: The problem of highly nonlinear outer power supplies (choppers) was addressed previously by constructing closed loop controllers [16] using a nonlinear output inversion. However, this solution, for the outer loop, is not fast enough to be implemented in the inner loop. A possible approach to deal with the inner choppers is shown in Figure 4. To take into account the nonlinear nature of the choppers we incorporate them into an augmented saturation block. The nominal linear vertical controller is synthesized without using a model of the choppers and its output yc is equal to the coil voltages in the absence of saturation. A chopper inverse function, which is part

6

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krsti´c

Fig. 3. Outer loop: Shape Control.

of the vertical controller, computes the necessary command voltages Vc within the saturation levels to make u equal to yc . When |yc | is large, the saturation block will obviously result in |u| being smaller than |yc |. Although the saturation levels of the command voltages Vc are still ﬁxed values (±10 V), the saturation levels of the augmented saturation block are functions of time, i.e., of the coil load currents IL and DC charging supply voltage Vps . Constrained Control: In order to make this approach successful the inner controller (vertical controller, Figure 2) must guarantee stability of the plant for all commands coming from the outer controller (shape controller, Figure 3). However, the constraints on the input of the inner plant due to the saturation of the actuators may prevent this goal from being achieved. The saturation of the coil voltages can not only degrade the performance of the inner closed loop system but also impede the vertical stabilization when the synthesis of the nominal inner controller does not account for plant input saturation. Although the saturation of coil currents and voltages is a common problem in tokamaks and there are efforts to minimize the control demand for shape and vertical control and to avoid saturation [12, 13], the saturation of the actuators are generally not taken into account in the design of the nominal controllers in present works.

Anti-Windup Augmentation in the DIII-D Tokamak

7

Fig. 4. Plant Architecture.

The inner loop design must take care then of the windup of that loop and ensure vertical stability for any command coming from the outer controller. We understand as windup the phenomenon characterized by degradation of nominal performance and even loss of stability due to magnitude and/or rate limits in the control actuaction devices. The anti-windup synthesis problem is to ﬁnd a nonlinear modiﬁcation of the predesigned nominal linear controller that prevents vertical instability and undesirable oscillations but leaves the nominal closed loop unmodiﬁed when there is no input saturation. This problem is different from the problem of synthesizing a controller that accounts for input saturation without requiring it to match a given predesigned arbitrary controller locally. Several survey papers [17, 18, 19] describe early ad-hoc anti-windup methods. Recently several other approaches have been proposed [20, 21, 22, 23, 24, 25, 26]. Due to the characteristics of our problem we follow the ideas introduced in the companion papers [27, 28] and also discussed in [29, 30] for exponentially unstable systems. The method is modiﬁed to fulﬁll the performance requirements of our system.

8

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krsti´c

3 Anti-Windup Compensator Fundamentals We consider exponentially unstable linear plants with control input u ∈ m and measurements y ∈ p . We write the model of our system in state-space form, (1)

x˙ = Ax + Bu,

separating the stable modes (xs ∈ ns ) from the exponentially unstable modes (xu ∈ nu ),

x˙ s x˙ u

=

As Asu

0 Au

xs xu

+

Bs

Bu

u (2)

y = Cx + Du,

where the dimension of the state vector x is n = ns +nu . The eigenvalues of As have non-positive real part, and the eigenvalues of Au have positive real part. In addition, we consider that a nominal linear controller with state xc ∈ nc , input uc ∈ p , output yc ∈ m and reference r ∈ p , x˙ c = Ac xc + Bc uc + Gr

(3)

yc = Cc xc + Dc uc + Hr

(4)

has been already designed so that the closed loop system with interconnection conditions u = yc ,

(5)

uc = y

is well posed and internally stable. When the controller output is subject to saturation, i.e., the interconnection conditions (5) are changed to u = sat(yc ),

(6)

uc = y,

the synthesis of an anti-windup scheme is necessary. In this case the interconnection conditions are modiﬁed to u = sat(yc + v1 ),

(7)

u c = y + v2 ,

where the signals v1 and v2 are the outputs of the anti-windup compensator [29] x˙ aw = Axaw + B[sat(yc + v1 ) − yc ] v1 = (β(xu ) − 1)yc + α (xu − β(xu )(xu − xawu ), β(xu )κ(xaw )) v2 = −Cxaw − D[sat(yc + v1 ) − yc ],

(8) (9) (10)

Anti-Windup Augmentation in the DIII-D Tokamak

9

Fig. 5. Anti-windup scheme.

and where the state xaw is also divided into stable (xaws ) and unstable (xawu ) modes. The anti-windup scheme is illustrated in Figure 5. In addition to modifying the nominal controller when input saturation is encountered, the anti-windup compensator modiﬁes the closed loop if the exponentially unstable modes get close to the boundary of some reasonably large subset of the region where these unstable modes are controllable with the given bound on the control (controllable region). The “distance” from this boundary is measured by the function β, deﬁned as [29] 1, xu ∈ χlower β(xu ) = 0, xu ∈ χupper

(11)

and interpolated in between, where χlower ⊂ χupper are subsets of χ, the domain of attraction of the disturbance-free system subject to the saturation of the output controller or what we call controllable region. The freedom to deﬁne χlower and χupper is a tool the designer has to deal with the disturbances that although not modeled are present in the system. The smaller χlower and χupper , the bigger the disturbances tolerated by the system without escaping the controllable region χ (at the same time, unnecessarily small choices of χlower and χupper result in the reduction of the region in which desired linear performance is achieved). When the unstable modes get close to the boundary of the controllable region χ, the closed loop is modiﬁed by the function α, which takes over control of the plant (β = 0 ⇒ v1 = −yc +α ⇒ u = sat(α)). One choice of the function α : nu × m → m is given by [29]

10

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krsti´c

α(ζ, ω) = Ku ζ + ω

(12)

where Ku is such that Au + Bu Ku is Hurwitz. The function κ(xaw ) is designed to improve the performance of the antiwindup scheme when the controller output is not saturating. It is important to note at this point that this scheme requires the measurement or estimation of the exponentially unstable modes xu . Several comments are in order here: • When xu ∈ χupper the unstable mode is close to escape from the controllable region χ. In this case the antiwindup must modify in a nonlinear way the output of the controller in order to guarantee the stability of the system. Noting that xu ∈ χupper implies that β(xu ) = 0, and in turns that v1 = −yc + Ku xu , we can write the dynamics of the exponentially unstable mode as x˙ u = Au xu + Bu sat(yc + v1 ) = Au xu + Bu sat(Ku xu ). The goal is to design the feedback gain Ku to stabilize the unstable modes before they reach the boundary of the controllable region χ. The anti-windup, through the signal v1 , is ensuring that the unstable modes do not escape the controllable region and guaranteeing in this way that the plant is well behaved. • Let us denote P (s) = C(sI − A)−1 B + D. Then, we can write y = P (s)sat(yc + v1 ) v2 = −P (s)[sat(yc + v1 ) − yc ], and consequently note that uc = y + v2 = P (s)yc , which, in view of (5), is the output of the nonsaturated model. Thus, the antiwindup, through the signal v2 , is hiding the saturation from the nominal controller and guaranteeing in this way that the controller is well behaved. • When xu ∈ χlower and there is no saturation of the controller output, i.e., sat(yc + v1 ) = yc + v1 , we do not want the antiwindup either through v1 or through v2 to affect the nominal closed loop system. Noting that xu ∈ χlower implies that β(xu ) = 1, and in turns that v1 = Ku xawu + κ(xaw ), we can write the dynamics of the anti-windup in this case as

Anti-Windup Augmentation in the DIII-D Tokamak

x˙ aws x˙ awu

= =

As Asu 0 Au

xaws xawu

As Asu + Bs Ku 0 Au + Bu Ku

+

Bs Bu

xaws xawu

11

v1

+

Bs Bu

κ(xaw )

v1 = Ku xawu + κ(xaw ) v2 = −Cxaw − Dv1 . If κ = 0, since Au + Bu Ku is Hurwitz, we have the freedom through a proper design of Ku , to make xawu , and consequently v1 , converge to zero as fast as desired. However this does not guarantee a fast convergence of v2 because this signal is also a function of xaws which is evolving in open loop. A problem arises when some of the stable eigenvalues are very slow because the system will be affected by the anti-windup for an unnecessarily long time. In this case, a proper design of κ can make v1 and v2 converge to zero as fast as possible.

4 Plant Model Figure 4 illustrates the architecture of our plant. The dynamics of our plant can be written as As Asu xs Bs Es x˙ s = x˙ = Ax + Bu + Ev = + u+ v x˙ u 0 Au xu Bu Eu

(13)

y = Cx + Du + Gv, where there are n ≈ 50 states, the vector u of dimension m = 6 are the voltage commands for power supplies on the vertical coils F 2A, F 2B, F 6A, F 6B, F 7A and F 7B, the vector v of dimension q ≤ 12 are the voltage demands for the shape coils, the vector y of dimension p = 7 consists of the six vertical coil currents IL and the plasma centroid position. Due to the composition of the output vector y it is convenient to write the reference for the nominal controller as r = [rIT

rZ ]T ,

where rI are the current references for the six vertical coils and rZ is the centroid position reference. The main characteristics of our plant can be summarized as: • There is only one unstable eigenvalue, i.e., nu = 1. The ns = n − 1 stable eigenvalues all have strictly negative real parts. However, some of them are very close to zero (slow modes).

12

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krsti´c

• Deﬁning the saturation function max

sataamin (b) =

⎧ ⎪ ⎪ amax if amax < b ⎨

b if amin ≤ b ≤ amax ⎪ ⎪ ⎩ amin if b < amin

(14)

the saturation of the channel i of the controller, for i = 1, . . . , m, will be denoted as M max (t)

sat(yci ) = satMimin (t) (yci ) i

where the saturation levels

Mimin (t)

and Mimax (t) are functions of time (i.e., of

coil load current IL (t) and DC supply voltage Vps (t)). • There is no direct measurement of the unstable mode xu . • The control input u is not the only input of the inner plant. In addition to the voltage commands u for the vertical coil from the vertical controller, there are voltage demands v for the shape coils coming from the shape controller.

5 Controllable Region Given the dynamics of the unstable mode (from 13) by the scalar equation x˙ u = Au xu + Bu u + Eu v,

(15)

we can compute the minimum and maximum values of the unstable mode that can be reached without losing control authority to stabilize the system, min

− Eu v − (Bu u) = Au max − Eu v − (Bu u) = , Au

xmax u xmin u

and deﬁne the controllable region as

. ≤ xu ≤ xmax χ = xu : xmin u u

(16) (17)

(18)

The maximal and minimal control are given by min

(Bu u)

max

(Bu u)

= =

m i=1 m i=1

Bui gi (−Bui )

(19)

Bui gi (Bui )

(20)

Anti-Windup Augmentation in the DIII-D Tokamak

13

6β(xu ) 1

@

@ @

@ @

xmin u

xmin,u u

xmin,l u

xmax,l u

xmax,u u

xmax u

xu

Fig. 6. Beta function.

where Bui is the i-th component of Bu and the function gi is deﬁned as Mimax if a > 0 gi (a) = Mimin if a < 0.

(21)

6 Anti-Windup Design Design of Function β: Once χ is determined, we can deﬁne

< xu < xmax,l χlower = xu : xmin,l u u

< xu < xmax,u χupper = xu : xmin,u u u

(22) (23)

= f l xmin , xmax,l = f l xmax , xmin,u = f u xmin , xmax,u = f u xmax , where xmin,l u u u u u u u u and 0 < f l < f u < 1. Once χ, χlower and χupper are deﬁned, the function β in (11) adopts the shape shown in Figure 6. Design of Gain Ku : The feedback gain Ku in (12) is designed such that Au + Bu Ku < 0 sgn(Bui ) = −sgn(Kui )

(24) (25)

| > max(|Mimin |, |Mimax |) |Kui xmax,u u

(26)

| > max(|Mimin |, |Mimax |), |Kui xmin,u u

(27)

14

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krsti´c

for i = 1, . . . , m, where Bui and Kui are the i-th components of Bu and Ku respectively. With (26),(27), we guarantee that for xmax,u ≤ xu < xmax (where u u β(xu ) = 0) we have Bu u = Bu sat(Ku xu ) = (Bu u)min

(28)

and consequently that sgn(x˙ u ) = sgn(Au xu + Bu sat(Ku xu ) + Eu v) = sgn(Au xu + (Bu u)min + Eu v) (29)

<0

≤ xu < xmin,u (where by deﬁnition (16). In similar way we can show that for xmin u u β(xu ) = 0) we have sgn(x˙ u ) > 0.

(30)

Conditions (29) and (30) ensure stabilization of the unstable mode when β(xu ) = 0 through the signal v1 = −yc + Ku xu . Design of the Function κ: We want to make the states xaw of the anti-windup compensator converge to zero as fast as possible when the unstable mode is in the “safe” region, deﬁned by the condition β(xu ) = 1, and no channel of the controller output is saturating. The function κ(xaw ) in (9) can be designed toward this goal. However, at this point we depart from the original method and follow an alternative procedure for simplicity and effectiveness. We make κ(xaw ) = 0 and modify the structure of the anti-windup compensator (8) as follows x˙ aw = Axaw + B[sat(yc + v1 ) − yc ] − [1 − γ(yc + v1 )]δxaw ,

δ > 0 (31)

where the function γ is deﬁned as γ=

m

γi

(32)

i=1

and the function γi , for each channel i = 1, . . . m,, is deﬁned in Figure 7. With min,γ

0 ≤ fγ ≤ 1, we deﬁne the threshold values Mi

max,γ

and Mi

as

Anti-Windup Augmentation in the DIII-D Tokamak

-

-

1

6γi (a)

?

6

?

15

-

6 -

Mimin,γ

Mimin

Mimax,γ

Mimax

a

Fig. 7. Gamma function for each channel i = 1, . . . , m.

min,γ Mi max,γ

Mi

max

Mi − Mimin Mimin + Mimax − fγ = 2 2 max

min max Mi + Mi − Mimin Mi + fγ = . 2 2

Note that γi becomes one when the signal coming from the controller and antiwindup goes above Mimax or below Mimin . However, the function γi recovers its zero value only when the signal coming from the controller and anti-windup bemax,γ

comes lower than Mi

min,γ

< Mimax or higher than Mi

> Mimin . These his-

teresis loops are introduced to avoid chattering in the scheme. The function γ is an indication of saturation, being zero if none of the input channel is saturating and one otherwise. In this way, when the unstable mode is in the “safe” region and there is no saturation the dynamics of the anti-windup can be written as x˙ aw = Axaw + Bv1 − δxaw As − δIs xaws Asu + Bs Ku = 0 Au + Bu Ku − δIu xawu

(33)

v1 = Ku xawu

(34)

v2 = −Cxaw − Dv1 ,

(35)

where Is and Iu are identity matrices of appropriate dimension. The rate of convergence of xaw to zero can be regulated now by the gain δ. A scheme of the anti-windup design is shown in Figure 8. The Saturation Limits block computes the saturation levels Mimin and Mimax , for i = 1, . . . , m, which are functions of time (i.e., of coil load currents IL (t) and DC supply voltage Vps (t)). These saturation levels are used by the Saturation Indication and Anti-Windup blocks to compute the function γ and the controllable region χ respectively. The Observer block estimates the unstable mode which cannot be measured. This estimate is used

16

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krsti´c

Fig. 8. Inner loop anti-windup scheme.

by the Anti-Windup block to compute the function β. The Saturation Indication computes γ to speed up the convergence of slow modes (and v2 ) to zero. Once the controllable region χ and the function β are computed, and the function γ provided, the Anti-Windup block is able to achieve stability through the signal v1 and keep the nominal controller well-behaved through the signal v2 . Rate Limiter: It is necessary at this point to recall that we are controlling the current in the vertical coils by modulating their imposed voltages. It is clear now that there will be a minimum integration time constant which will depend on the inductance of the coils. Given a maximum coil voltage value dictated by the saturation level we will have then a maximum rate of variation for the coil current. Any reference rI (imposed by the shape controller) that exceeds this physical rate limit will only cause performance deterioration due to the windup of the controller.

Anti-Windup Augmentation in the DIII-D Tokamak

17

The rate limit on the vertical coil current references aims at preventing the shape controller from asking the system for a response rate that cannot be physically fulﬁlled. The dynamics of the rate limiter for the vertical coil current references can be written as r˙I = Rsgn(˜ rI − rI )

(36)

where M max

r˜I = satMImin (rI∗ )

(37)

I

is the saturated version of the current references rI∗ imposed by the shape controller. The saturation levels MImin and MImax are the minimum and maximum currents allowed in the coils. As long as rI = r˜I , the output of the rate limiter rI follows the target reference r˜I with a rate limit R = diag(R1 , R2 , . . . , Rm ) according to (36). The components Ri , for i = 1, . . . , m, can be ﬁxed or varied adaptively as it will be shown below. The centroid position reference is also limited by physical constraints due to the ﬁnite size of the tokamak represented by MZmin and MZmax , and is written as M max

∗ rZ = satMZmin (rZ ).

(38)

Z

Figure 8 shows that the vertical coil current references and the centroid position reference r∗ = [rI∗

T

∗ T rZ ] imposed by the shape controller go through a double

saturation stage. The ﬁrst saturation is in magnitude and is due to constraints imposed on the coil currents and centroid position due to physical considerations. The second saturation is in rate and is due to the rate limits imposed on the coil current references as part of the inner anti-windup design (rate limiter). To take advantage of all the actuation force available and increase the response speed of our system, when rI = r˜I we vary Ri , for i = 1, . . . , m, adaptively between maximum and minimum values (Rimax and Rimin ) according to the available control, ⎧ min(Mimax − yci , yci − Mimin ) if Mimin ≤ yci ≤ Mimax ⎪ ⎪ ⎨0 if Ri > Rimax and Mimin ≤ yci ≤ Mimax ˙ if Ri < Rimin and (yci < Mimin or yci > Mimax ) R i = Ki 0 ⎪ max ⎪ yci ) if yci > Mimax and Ri > Rimin ⎩ (Mi −min min min (yci − Mi

)

if yci < Mi

and Ri > Ri

(39) where Ki is a constant gain. The maximum and minimum values Rimin and Rimax deﬁne the zone where Ri is allowed to move. These limits are imposed to avoid the windup of Ri . Basically, Ri is increased when there is available control, is decreased when the channel is saturating, and is frozen when one of its limits is reached.

18

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krsti´c

Watch Dog: As it is shown in Figure 4 and stated in (13), our plant is also governed by the voltage demands for the shape coils v coming from the shape controller. This represents a potential risk of instability for our plant because the unstable mode can be potentially pushed outside the controllable region by shape control voltages. The goal of the watch-dog is to limit the voltage demands to the shape coils to avoid the loss of controllability of the unstable mode due to the sudden shrinkage of the controllable region. We call this block ”Watch Dog” because it is permanently monitoring and regulating the value of the shape coil voltage demands v to mantain the size of the controllable region of the system above certain level. We present in this section our ﬁrst approach to the problem. There is room for further development aimed at reducing the conservatism of the algorithm. Figure 8 shows that the shape coil voltage demands v ∗ coming from the shape controller go through a double saturation stage. The ﬁrst magnitude saturation is due to the watch-dog that is part of the inner anti-windup augmentation. The second magnitude saturation comes from the hardware limitations of the power supply stage. We can write the shape coil voltage demands as M max (t)

v = satMvmin (t) (˜ v) , v

(40)

where N max (t)

v˜ = satNvmin (t) (v ∗ ) v

(41)

is the output of the watch-dog block. The saturation levels Mvmin and Mvmax are determined by the hardware limitations of the power supply stage and are functions of time because they depend on the coil load current and DC supply voltage. The saturation levels Nvmin and Nvmax , imposed by the watch-dog as it will be explained below, are also functions of time because they depend on the value of the unstable mode xu and the controllable region χ. The limiting values of the controllable region given by (16) and (17) are plotted as functions of Eu v in Figure 9. We cannot allow the controllable region to shrink to a null set and we impose a minimum size to it deﬁned as

max . χmin = xu : (xmin ) ≤ x ≤ (x ) max u min u u

(42)

This minimum controllable region deﬁnes hard limits on Eu v allowing this signal to lim be between (Eu v)lim min and (Eu v)max . In addition we need to impose variable limits

Anti-Windup Augmentation in the DIII-D Tokamak

19

xmax (Eu v) u

aa xmax , xmin u u 6 aa aa aa aa aa aa max (xu )min aa aa lim aa (Eu v)min ? aaa aa 6 a min lim aa Eu v aa (xu )max (Eu v)max a aa aa aa aa aa aa aa xmin (Eu v) a u Fig. 9. Controllable region as a function of Eu v.

on Eu v depending on the position of the unstable mode xu . Toward this goal we deﬁne (xmax )allowed = max(fu xu , (xmax )min ) u u max

= − (Bu u) (Eu v)allowed max

− Au (xmax )allowed u

)allowed = min(fu xu , (xmin )max ) (xmin u u min

= − (Bu u) (Eu v)allowed min

− Au (xmin )allowed u

where fu > 1 is a design constant and for each channel i we compute ⎧ (E v)allowed max ⎪ if sgn(Eui ) > 0 ⎨ Nucoils |Euj | max j=1 (Nv )i = (E v)allowed min ⎪ if sgn(Eui ) < 0 ⎩ − Nucoils |Euj | j=1 ⎧ (E v)allowed max ⎪ if sgn(Eui ) < 0 ⎨ − Nucoils |Euj | min j=1 (Nv )i = (E v)allowed min ⎪ if sgn(Eui ) > 0. ⎩ Nucoils j=1

(43) (44) (45) (46)

(47)

(48)

|Euj |

Observer: The stability of the anti-windup scheme is guaranteed when the unstable mode is directly measured [29]. When there is no such direct measurement, a bad estimation of the unstable mode can prevent the anti-windup compensator to stabilize the system. A high gain observer is required in this case to ensure that the estimation is fast enough to prevent any excursion of the unstable mode outside the controllable

20

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krsti´c

region. The estimate x ˆu is substituted for xu in the anti-windup dynamics, which, sumarizing (9), (10), (12), (31), is given by x ˆ˙ s =x ˆ˙ = Aˆ x + Bu + Ev + L(y − yˆ) ˙x ˆu

(49)

yˆ = C x ˆ + Du + Gv. The estimate x ˆu is substituted for xu in the anti-windup dynamics, which, sumarizing (9), (10), (12), (31), is given by x˙ aw = Axaw + B[sat(yc + v1 ) − yc ] − [1 − γ(yc + v1 )]δxaw v1 = (β(ˆ xu ) − 1)yc + Ku [ˆ xu − β(ˆ xu )(ˆ xu − xawu )]

(50)

v2 = −Cxaw − D[sat(yc + v1 ) − yc ]. For our simulation studies, a convential Luenberger observer was implemented. However, during the implementation stage other types of observers more suitable for noisy environments will be considered. From the observer we do not need much accuracy, we only need to know if the unstable mode is inside χlower or outside χupper . Certain level of noise in the estimation can be tolerated because it is always possible to compensate the inaccuracy of the observer with a convenient selection (reduction) of the design parameters f l and f u , paying the price on the other hand of reducing conservatively the region χupper where we allow the states to move.

7 Simulation Results Simulations were run to show the effect of the anti-windup. Figure 10 shows the response of the augmented closed loop when rIi , for i = 1, . . . , m, are step functions of magnitude approximately equal to 840 Amps. Compared to the results without anti-windup, the excursion of the controller output yc (Figure 10-a) is reduced and the stability is recovered but the step response for the coil currents (y1 , . . . , y6 ) (Figure 10-c) is far from the desired one. In addition, the centroid vertical position (y7 ) (Figure 10-d) exceeds physical limits (±20cm approximately; 1V = 10cm). For all the simulations in this paper we have made rZ = 0 which means that the centroid vertical position must be regulated around its equilibrium value. However, the nominal vertical controller is not designed for a perfect regulation of this variable because we do not care about the centroid vertical position as long as it stays within

Anti-Windup Augmentation in the DIII-D Tokamak Controller Output (total): y

c

5000

21

Plant input (Chopper output) (total): u

1000

500

0

0

−500

−5000 0.05

0.1

0.15

0.2

0.25

0.3

−1000 0.05

0.35

0.1

0.15

0.2

0.25

Time (sec)

Time (sec)

(a)

(b)

1

1000

6

0.35

Centroid vertical position (total): y7

Vertical F−coil currents (rel) (Amps): y ,...,y 2000

0.3

0

−2

0 −4

−1000

−6

−2000 −3000 0.05

0.1

0.15

0.2 0.25 Time (sec)

0.3

0.35

−8 0.05

0.1

0.15

0.2

(c)

0.3

0.35

(d)

β(xu)

1.2

0.25

Time (sec)

Unstable Mode x −− Controllable Region u

10000 1

0.8

x u min xu max x

5000

u

0.6

0 0.4 0.2 0 0.05

−5000

0.1

0.15

0.2

0.25

Time (sec)

(e)

0.3

0.35

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (sec)

(f)

Fig. 10. System response with anti-windup to step changes of approximately 840 Amps in rIi , for i = 1, . . . , m, at t = 0.1 sec (rZ = 0 and v = 0). Low gain Ku , δ = 0.

22

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krsti´c v1

6000

v2

400

4000

300

2000 200 0 100 −2000 0

−4000 −6000 0.05

0.1

0.15

0.2

0.25

0.3

0.35

−100 0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (sec)

Time (sec)

(a)

(b)

Vertical F−coil currents (rel) (Amps): y1,...,y6

1500

Centroid vertical position (total): y7

0

1000

−0.5 500

−1

0 −500

−1.5 −1000 −1500 0.05

0.1

0.15

0.2 0.25 Time (sec)

0.3

0.35

−2 0.05

0.1

0.15

(c)

0.3

0.35

(d)

Vc

10

0.2 0.25 Time (sec)

10000

5

Unstable Mode xu −− Controllable Region x u xmin u xmax u

5000

0 0

−5 −5000

−10 0.05

0.1

0.15

0.2

0.25

Time (sec)

(e)

0.3

0.35

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (sec)

(f)

Fig. 11. System response with anti-windup to step changes of approximately 840 Amps in rIi , for i = 1, . . . , m, at t = 0.1 sec (rZ = 0 and v = 0). High gain Ku , δ = 0.

Anti-Windup Augmentation in the DIII-D Tokamak v

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Fig. 12. System response with anti-windup to step changes of approximately 840 Amps in rIi , for i = 1, . . . , m, at t = 0.1 sec (rZ = 0 and v = 0). High gain Ku , δ > 0.

the physical limits. The β(xu ) = 1 value (Figure 10-e) shows the intervention of the anti-windup compensator in order to keep the unstable mode inside χlower . It is possible to note that in this case the unstable mode (Figure 10-f) gets really close to the boundary of the controllable region and it requires a great effort from the anti-windup compensator, manifested in the shape of input signal u (Figure 10-b), to return it to the safe region. We can avoid this effect either by reducing the size of χlower , χupper in (22), (23) (by reducing the values of f l and f u ), or by increasing the anti-windup gain Ku in (12).

24

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krsti´c *

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Fig. 13. System response with anti-windup and rate limiter to step changes of approximately 840 Amps in rIi , for i = 1, . . . , m, at t = 0.1 sec (rZ = 0 and v = 0). Fixed rate limit R.

Figure 11 shows the response of the augmented closed loop when rIi , for i = 1, . . . , m, are step functions of magnitude approximately equal to 840 Amps but this time with a higher gain Ku . Comparing Figures 10-f and 11-f, it is possible to note that the excursion of the unstable mode xu is much smaller (β(xu ) = 1 for all time in this case). In addition, the centroid vertical position (y7 ) (Figure 11-d) is kept within it physical limits, and the step response for the coil currents (y1 , . . . , y6 ) (Figure 11-c) is improved. The intervention of the anti-windup compensator through β(xu ) = 1 is inversely proportional to the gain Ku . A high value of Ku helps to keep the unstable mode inside χlower and makes less frequent a value of β(xu )

Anti-Windup Augmentation in the DIII-D Tokamak R

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Fig. 14. System response with anti-windup and rate limiter to step changes of approximately 840 Amps in rIi , for i = 1, . . . , m, at t = 0.1 sec (rZ = 0 and v = 0). Variable rate limit R.

distinct from one. Although better than in Figure 10-c, the step response for the coil currents (y1 , . . . , y6 ) in Figure 11-c is still far from the desired one. It is possible to note that even after the system leaves the saturation zone (the chopper commands Vc (Figure 11-e) come back within their limits (±10V )) the nominal controller exhibits a sluggish behavior. This is explained by the fact that the signal v2 (Figure 11-b) still modiﬁes the nominal closed loop when there is no saturation. From (33), when δ = 0 the stable modes of the anti-windup are evolving in open loop and the time it takes for them and for v2 to converge to zero depends on the magnitude of their eigenvalues. As it was discussed in previous sections, it is possible to note that although the

26

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krsti´c Vertical F−coil currents (rel) (Amps): y1,...,y6

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Fig. 15. System response with anti-windup to step changes of approximately 280 Amps in rIi , for i = 1, . . . , m, at t = 0.1 sec (rZ = 0 and v = 0). (a) no anti-windup, (b) anti-windup δ = 0, (c) anti-windup δ > 0, (d) anti-windup δ > 0 + rate limiter with variable rate limit R.

signal v1 (Figure 11-a) converges to zero arbitrarily fast once the saturation is absent, the signal v2 converges to zero very slowly due to the presence of very slow modes in our system. Figure 12 shows the importance of the function γ modifying the structure of the antiwindup system. When δ = 0 it is possible to make the signal v2 converge to zero very fast (once the chopper commands Vc (Figure 12-a) come back within their limits (±10V )) as it is shown in Figure 12-b. In this way we allow the system to evolve according to the nominal design. In our system the difference between the δ = 0 and δ = 0 cases is huge because the plant has stable eigenvalues with real part

Anti-Windup Augmentation in the DIII-D Tokamak

27

very close to zero. The step response for the coil currents (y1 , . . . , y6 ) (Figure 12-c) is improved greatly and their steady state values are reached much sooner, while the centroid vertical position (y7 ) (Figure 12-d) is kept within it physical limits. Figure 13 shows how smooth the response of the system is when a ﬁxed rate limit is imposed on each channel of the reference rI . The reference r∗ (Figure 13-a) imposed by the shape controller, which in this case turns to be equal to r˜ because of the absence of saturation, is modiﬁed by the rate limiter to ensure that the reference signal r (Figure 13-b) reaching the vertical controller has a ﬁxed rate limit. The improvement in the response is manifested by the shape of the coil currents signals (y1 , . . . , y6 ) (Figure 13-c). However the conservatism of this approach can be appreciated when we examine the chopper commands Vc (Figure 13-d). These signals show that there is appreciable available actuation force that is not used. Figure 14 shows how the system becomes much faster thanks to the adaptation of the reference rate limit but keeps the smoothness of the response. A quick comparison between Figure 12-c (or Figure 13-c) and Figure 14-c shows the effectiveness of the approach. It is possible to note how the adaptation of the rate limit R between Rmin and Rmax (Figure 14-a) for each coil according to the available actuation force (Figure 14-d) allows the system to reach the steady state for the coil currents much sooner. The current references in r (Figure 14-b) depend now on the available actuation force in each one of the vertical coils. The rate limit is reduced for those channels more affected by saturation. Although it is not possible to compare the performance of the system without and with anti-windup compensators when the current references are steps of magnitude equal to 840 Amps (because the system without anti-windup compensator is unstable in this case), it is possible to make such comparison when the current references are steps of magnitude equal to 280 Amps as it is shown in Figure 15. When δ = 0 the step response (b) exhibits a deterioration of performance with respect to the step response without anti-windup (a). However, when δ = 0 the step response (c) is smoother and shows a settling time that is approximately the same as shown in the step response without anti-windup (a). When the rate limiter is activated (d), the step response is not only smoother but also the settling time is reduced considerably for some of the coils.

28

Eugenio Schuster, Michael Walker, David Humphreys, and Miroslav Krsti´c

8 Conclusions The proposed scheme has been shown in nonlinear simulations to be very effective in guaranteeing stability of the inner loop in the presence of voltage saturation of the vertical coils. The scheme is being implemented and will be tested in experimental conditions. After succeeding in the vertical stabilization of the plasma in experimental conditions, efforts will be concentrated on the design of the outer shape control loop. The necessity of a similar anti-windup scheme for the outer loop is anticipated; not only due to the inherent limitations of its actuators but also due to the fact that the inner loop will modify, through the watch-dog and rate limiter, the control signals of the outer loop in order to preserve stability of the inner plant and improve performance. In this case we will deal with a stable (stabilized by the inner loop design) but nonlinear plant.

References 1. J. L. Luxon, “A design retrospective of the DIII-D tokamak,” Nuclear Fusion, vol. 42, pp. 614-33, 2002. 2. D. A. Humphreys, M. L. Walker, J. A. Leuer and J. R. Ferron, “Initial implementation of a multivariable plasma shape and position controller on the DIII-D tokamak,” Proceedings of the 2000 IEEE International Conference on Control Applications, Anchorage, Alaska, USA, pp. 385-94, September 2000. 3. R. Albanese and G. Ambrosino, “Current, position and shape control of tokamak plasmas: A literature review,” Proceedings of the 2000 IEEE International Conference on Control Applications, Anchorage, Alaska, USA, pp. 412-18, September 2000. 4. Y. Mitrishkin and H. Kimura, “Plasma vertical speed robust control in fusion energy advanced tokamak,” Proceedings of the 40th Conference on Decision and Control, Orlando, Florida, USA, pp. 1292-97, December 2001. 5. M. Ariola, A. Pironti and A. Portone, “Vertical stabilization and plasma shape control in the ITER-FEAT tokamak,” Proceedings of the 2000 IEEE International Conference on Control Applications, Anchorage, Alaska, USA, pp. 401-405, September 2000. 6. M. Ariola, A. Pironti and A. Portone, “A framework for the design of a plasma current and shape controller in next-generation tokamaks,” Fusion Technology, Vol. 36, pp. 263-277, Nov. 1999.

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7. M. Ariola, G. Ambrosino, J. B. Lister, A. Pironti, F. Villone and P. Vyas, “A modern plasma controller tested on the TCV tokamak,” Fusion Technology, Vol. 36, pp. 126-138, Sep. 1999. 8. G. Ambrosino, M. Ariola, Y. Mitrishkin, A. Pironti and A. Portone, “Plasma current and shape control in tokamaks using H∞ and μ-synthesis,” Proceedings of the 36th Conference on Decision and Control, San Diego, California, USA, pp. 3697-3702, December 1997. 9. A. Portone, R. Albanese, Y. Gribov, M. Hughet, D. Humphreys, C. E. Kessel, P. L. Mondino, L. D. Pearlstein and J. C. Wesley, “Dynamic control of plasma position and shape in ITER,” Fusion Technology, Vol. 32, pp. 374-88, Nov. 1997. 10. A. Portone, Y. Gribov, M. Hughet, P. L. Mondino, R. Albanese, D. Ciscato, D. Humphreys, C. E. Kessel, L. D. Pearlstein and D. J. Ward, “Plasma position and shape control for ITER,” Proceedings of the 16th IEEE/NPSS Symposium on Fusion Engineering, Champaign, IL, USA, pp. 345-348, Sep. 1995. 11. G. Ambrosino, M. Ariola, A. Pironti, A. Portone and M. Walker, “A control scheme to deal with coil current saturation in a tokamak,” IEEE Transactions on Control Systems Technology, Vol. 9, no. 6, p. 831, November 2001. 12. R. Albanese, G. Ambrosino, A. Pironti, R. Fressa and A. Portone, “Optimization of the power supply demand for plasma shape control in a tokamak,” IEEE Transactions on Magnetics, Vol. 34, no. 5, p. 3580, September 1998. 13. L. Scibile and B. Kouvaritakis, “A discrete adaptive near-time optimum control for the plasma vertical position in a tokamak,” IEEE Transactions on Control Systems Technology, Vol. 9, no. 1, p. 148, January 2001. 14. J-Y. Favez, Ph. Mullhaupt, B. Srinivasan, J.B. Lister and D. Bonvin, “Improving the region of attraction of ITER in the presence of actuator saturation,” IEEE Conference on Decision and Control, 4616-21, 2003. 15. M.L. Walker, D.A. Humphreys and E. Schuster, “Some nonlinear controls for nonlinear processes in the DIII-D tokamak,” Proceedings of the 42th IEEE Conference on Decision and Control, Maui, Hawaii, USA, pp. 6565-6571, Dec. 2003. 16. M.L. Walker, J.R. Ferron, B. Penaﬂor, D.A. Humphreys, J.A. Leuer, A.W. Hyatt, C.B. Forest, J.T. Scoville, B.W. Rice, E.A. Lazarus, T.W. Petrie, S.L. Allen, G.L. Jackson, R. Maingi, “Status of DIII-D plasma control,” Proceedings of the 16th IEEE/NPSS SOFE, p. 885, 1995. ˚ om and L. Rundqwist, “Integrator windup and how to avoid it,” Proceedings of 17. K. J. Astr¨ the American Control Conference, Pittsburgh (PA), USA, vol. 2, pp. 1693-98, June 1989. 18. R. Hanus, “Antiwindup and bumpless transfer: a survey,” Proceedings of the 12th IMACS World Congress, Paris, France, vol. 2, pp. 59-65, 1988.

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19. M. Morari, “Some control problems in the process industries,” H. L. Trentelman and J. C. Willems, editors, Essays on Control: Perspectives in the Theory and its Applications, pp. 55-77. Birkhauser, Boston (MA), USA, 1993. 20. E. G. Gilbert and I. V. Kolmanovsky, “Set-point control of nonlinear systems with state and control constraints: A Lyapunov-function, Reference-Governor approach,” Proceedings of the 38th Conference on Decision and Control, Phoenix (AZ), USA, pp. 2507-12, Dec. 1999. 21. A. Bemporad and M. Morari, “Control of systems integrating logic, dynamics, and constraints,” Automatica, vol. 35, pp. 407-427, 1999. 22. L. Scilibe and B. Kouvaritakis, “Stability region for a class of open-loop unstable linear systems: theory and application,” Automatica, vol. 36, pp. 37-44, 2000. 23. J. S. Shamma, “Anti-windup via constrained regulation with observers,” Systems and Control Letters, vol. 40, pp. 1869-83, 2000. 24. A. Zheng, M. V. Kothare and M. Morari, “Anti-windup design for internal model control,” International Journal of Control, vol. 60, no. 5, pp. 1015-1024, 1994. 25. A. Miyamoto and G. Vinnicombe, “Robust control of plants with saturation nonlinearity based on coprime factor representation,” Proceedings of the 36th Conference on Decision and Control, Kobe, Japan, pp. 2838-40, 1996. 26. E. F. Mulder, M. V. Kothare and M. Morari, “Multivariable anti-windup controller synthesis using linear matrix inequalities,” Automatica, vol. 37, no. 9, pp. 1407-16, 2001. 27. A. R. Teel and N. Kapoor, “The L2 anti-windup problem: Its deﬁnition and solution,” Proceedings of the 4th ECC, Brussels, Belgium, July 1997. 28. A. R. Teel and N. Kapoor, “Uniting local and global controllers,” Proceedings of the 4th ECC, Brussels, Belgium, July 1997. 29. A. R. Teel, “Anti-Windup for exponentially unstable linear systems,” International Journal of Robust and Nonlinear Control, vol. 9, pp. 701-716, 1999. 30. C. Barbu, R. Reginatto, A.R. Teel and L. Zaccarian, “Anti-windup for exponentially unstable linear systems with rate and magnitude limits,” V. Kapila and K. Grigoriadis (editors), Actuator Saturation Control, Marcel Dekker, 2002.

Stable and Unstable Systems with Amplitude and Rate Saturation Peter Hippe Lehrstuhl f¨ur Regelungstechnik, Universit¨at Erlangen-N¨urenberg [email protected]

1 Introduction In nearly all control applications, the output signal of the compensator cannot be transferred to the system with unlimited amplitudes and arbitrarily fast. Such limitations in magnitude and rate can lead to performance degradations and they may even cause an unstable behaviour of the closed loop. These undesired effects of actuator saturation are called windup. The problem most intensively investigated is that of amplitude saturation. The prevention of windup effects due to input magnitude restrictions is the subject of a great variety of papers. The references in [3], [8] and [26] give a good overview of the existing literature. Controllers with badly damped or unstable modes tend to “wind up” during input saturation, giving rise to big overshoots and slowly settling transients. Since these effects are related to the dynamics of the controller, they are called controller windup [15]. Most of the early windup literature was devoted to the prevention of this kind of windup. A uniﬁed framework incorporating the various existing schemes is presented in [19]. If the observer technique [15], [13] is applied, controller windup is prevented and the dynamics of the controller no longer inﬂuence the effects of input saturation. Then all undesired effects of input saturation can be attributed to plant windup [15], caused by the dynamics of the controlled plant which are assigned by constant state feedback. When the characteristic polynomial of the state observer appears in the numerator

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of the reference channel, the conditioning technique yields the same results as the observer technique. In the framework of the conditioning technique, the remaining effects of plant windup are called the “short sightedness of the conditioning technique” [10]. Whereas controller windup can be removed by mere structural methods (appropriately add a model of input saturation to the controller and stabilize it when its output signal becomes constraint), the prevention of plant windup usually calls for additional dynamic elements like the ﬁltered setpoint [25] or the additional dynamic network [15]. Consequently, the prevention of windup is possible in a two-step approach. First remove the inﬂuence of the controller dynamics, and if this does not remove all undesired effects of input saturation (i.e., if the danger of plant windup exists), introduce additional dynamic elements. A one-step approach to the joint prevention of controller and plant windup was presented in [31]. This anti-windup scheme contains a model of the plant (additional dynamics), and its major advantage is the automatic elimination of controller windup, regardless of whether the compensator is a classical P, PI or PID controller or an observer-based controller possibly containing signal models for disturbance rejection. The only design parameter of the scheme is the feedback of the model states, assuring the prevention of plant windup. There is a slight drawback insofar, as the additional dynamics are also present if the undesired effects of input saturation can solely be attributed to controller windup. Without additional measures, the above-discussed methods for the prevention of plant windup are only applicable to open loop stable systems. Magnitude constraints always limit the maximum sustainable output amplitudes. In stable systems no control input can drive the outputs beyond these amplitudes, which are ﬁnite in the absence of eigenvalues at s = 0. For unstable systems, these amplitudes have to be avoided under all circumstances. If the output of an unstable system were to reach this limit, it could increase exponentially for any controller action. Consequently, stability can only be guaranteed as long as a sufﬁcient safety margin is observed. For “slow” dynamics of the stabilizing control, step-like reference inputs could be applied close to the maximum sustainable amplitudes, i.e., the safety margins could be kept small. For faster dynamics of the closed loop, overshoots can appear, requiring considerably larger margins. This is the effect of plant windup. Virtually all known measures for the prevention of plant windup only become active after a saturation limit has been hit for the ﬁrst time. Disturbances acting during

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33

reference transients can obviously destabilize such schemes in the presence of exponentially unstable systems. There are papers on the control of constrained unstable systems. In [17] a reference “governor” is presented which, via a time varying rate operator, shapes the reference input so that the control input does not saturate. The computations for constructing the time varying rate are, however, quite complicated. In [34] it is proposed to approximate the linear constraints on the input (and states) by conservative quadratic constraints. In [27] the set point tracking problem was investigated using an internal model control conﬁguration, and rules of thumb were presented giving bounds on amplitude and frequency for sinusoidal references ensuring stable saturated operation of the nonlinear reference ﬁlter. The tracking of constant reference signals and the computation of regions of initial conditions for which stability is guaranteed is considered in [32],[33]. In [32] a nonlinear term is added to the control law whereas in [33], arbitrary LQ control laws are allowed. Disturbance rejection for unstable systems with input saturation has also been investigated by various authors. The rejection of input additive disturbances is discussed for state feedback control in [26] and [21] whereas [20] solves the H∞ -almost disturbance decoupling problem. The windup problem for exponentially unstable systems in the presence of reference and disturbance inputs is considered in [30]. The disturbances, however, are restricted not to act on the exponentially unstable modes. The joint problem of tracking a certain reference level in the presence of (constant) disturbances was investigated in [29]. The objective of this approach is to null the steady state error due to disturbances by using a saturated controller containing nonlinear integrating action. Beyond the above discussed windup effects, there is an additional problem caused by saturating actors in MIMO systems, namely that of directionality. “For MIMO controllers, the saturation may cause a change in the direction of the plant input resulting in disastrous consequences” [7]. There are also various attempts to solve the directionality problem. An anti-windup scheme with stability augmentation and appropriate modiﬁcations of the saturation limits to prevent directionality problems was presented in [28]. In [36] and [24] the conditioning technique is enhanced for windup prevention by an artiﬁcial nonlinearity, optimized by nonlinear programming such that the realizable reference signal stays to the actual one as closely as possible. A similar approach has also been used in

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Peter Hippe

[23]. An artiﬁcial multivariable state dependent saturation element preserving the directionality of the input vector is used in [18]. An optimization of the internal model control setup to minimize the effects of input saturation on the directionality problem is presented in [37]. All these approaches constitute approximate solutions. The only scheme seemingly preserving the decoupled behaviour in spite of input saturation is the directionality preserving control of [4]. Unfortunately it maintains directionality only in very speciﬁc applications. Its basic feature, loop gain reduction to prevent input signal saturation, makes the scheme time variant so that stability may be at stake especially in the case of unstable plants. The methods discussed so far have been developed for amplitude saturated actuators. If one has to cope with an additional rate saturation, the prevention of its destabilizing inﬂuence calls for novel measures. Also the case of joint amplitude and rate constraints has been investigated intensively (see e.g., the references in [9], [2] or [1], and the discussions of the existing approaches therein). The solutions presented so far use an actuator model incorporating two nonlinear elements (for amplitude and rate saturation). Consequently the problem becomes much more involved as compared to the well-known techniques for amplitude-restricted actuators. In this contribution, a new scheme for the prevention of windup is presented which is applicable both to stable and unstable SISO and MIMO systems. It uses a nonlinear model-based reference shaping ﬁlter for feedforward tracking such that feedback signals are not caused by reference inputs. The feedback controller is used for system stabilization and disturbance rejection. Therefore, one can reserve part of the constrained input for disturbance rejection while the remaining amplitude region is available for tracking. This allows a proof of stability for constrained unstable systems subject to reference and disturbance inputs. Also a perfect directionality preservation becomes possible by the presented approach. Also presented is a solution to the problem of joint amplitude and rate saturation, which allows to use the well-known results for amplitude saturated actuators. This is made possible by the introduction of a simple ersatz model for such actuators, consisting of an input saturation and a ﬁrst-order dynamic element, whose time constant is chosen so that the rate constraint is automatically met. By adding the dynamic element to the plant, the windup prevention for amplitude- and rate-constrained actuators boils down to the well-known problem of amplitude saturation for the augmented plant.

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Two examples demonstrate the application of the proposed scheme for the prevention of windup.

2 The Problem Considered Given a strictly proper, linear, time invariant MIMO system having state x(t) ∈ n , control input us (t) ∈ p , controlled variables y(t) ∈ p and measurements ym (t) ∈ m , and a completely controllable and observable state space representation x(t) ˙ = Ax(t) + Bus (t) + Bd d(t) y(t) = Cx(t) + Cd d(t)

(1)

ym (t) = Cm x(t) + Cmd d(t) where d(t) ∈ q is a disturbance input. In the sequel, the tracking of constant reference signals is investigated. This is only feasible when the system has no transmission zeros at the origin. Therefore, it is assumed that the (square) transfer function matrix G(s) = C(sI − A)−1 B = N (s)D−1 (s)

(2)

is such that det N (s) does not have zeros at s = 0. There is a saturating nonlinearity us (t) = satu0 (u(t)) at the input of the system and its components are described by ⎧ ⎪ ⎪ u if ui > u0i ⎪ ⎨ 0i satu0i (ui ) = ui if − u0i ≤ ui ≤ u0i ; u0i > 0, i = 1, 2, ..., p ⎪ ⎪ ⎪ ⎩−u if ui < −u0i 0i

(3)

Due to input saturation the maximum sustainable output amplitudes are conﬁned to the regions yimin ≤ yi (∞) ≤ yimax , i = 1, 2, ..., p

(4)

and it is assumed that these limits are ﬁnite, which is the case in systems without eigenvalues at s = 0. The case of systems with eigenvalues in the origin of the s-plane will be addressed in Remark 5.

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In order to get a meaningful control problem, the controlled variables must stay within the maximum usable amplitude ranges yil ≤ yi (∞) ≤ yiu , i = 1, 2, ..., p

(5)

characterized by the requirement that arbitrary combinations of the output signals yi (t) can be sustained on the boundaries (5). In single input systems the equalities |yimin | = yil and |yimax | = |yiu | always hold. In MIMO systems, however, one usually has |yimin | > yil and |yimax | > |yiu | and the usable regions not only depend on the saturation limits u0i , but also on each other. Example 1. As a demonstrating example consider the system with transfer matrix ⎡ ⎤ 1 1 ⎢ ⎥ ⎢ s + 4 s + 1⎥ ⎢ ⎥ G(s) = ⎢ ⎥ ⎣ 1 1 ⎦ s+2

s+5

and saturation limits u01 = u02 = 10. The maximum sustainable output amplitudes are y1min = −12.5, y1max = 12.5, y2min = −7 and y2max = 7, because the input us1 (∞) 10 y1 (∞) 12.5 vector gives rise to an output vector . = = 10 7 us2 (∞) y2 (∞) y1l , y1u −7.5, +7.5 One maximum usable output range is, for example, = . In −3, +3 y2l , y2u order to ﬁnd this range, one has to check for the worst case situation that can still be handled with the given input limits. If both outputs y1∞ = y1 (∞) and y2∞ = y2 (∞) were at the upper or the lower limits, the necessary components of the input vector

t would be |us1∞ | = 10/3 and |us2∞ | = 20/3. A vector y∞ = 7.5, −3 , however, requires an input vector uts∞ = −10, 10 . An output range with symmetric lim −3.75, 3.75 y1l , y1u as the vector its of equal magnitude is characterized by l u = −3.75, 3.75 y 2 , y2 t = 3.75, −3.75 requires an input vector uts∞ = −10, 6.25 . If, as another y∞

example, only positive outputs were desired, the existing in with equal magnitudes l u y 1 , y1 0, 4.5 t put saturation limits allow l u = , because the vector y∞ = 0, 4.5 0, 4.5 y 2 , y2 t requires an input vector us∞ = 10, −2.5 .

Stable and Unstable Systems with Amplitude and Rate Saturation

37

For the sake of simplicity, the limits of the maximum usable amplitudes are assumed to be symmetric, i.e., yiu = −yil = y0i , i = 1, 2, ..., p. In the case of stable plants, the limits y0i also deﬁne the maximum used ranges. In the case of unstable plants, these limits must be avoided because a non-recoverable unstable behaviour could occur if an output yi (t) reached the limit y0i . In view of safety consideration and of unknown disturbance inputs, the outputs must be conﬁned to the maximum used ranges −r0i < yi (t) < r0i , i = 1, 2, ..., p

(6)

where the r0i > 0 satisfy r0i < y0i . Tis also speciﬁes the regions −r0i ≤ ri (t) ≤ r0i > 0, i = 1, 2, ..., p

(7)

of the maximum applied reference signal changes. The r0i depend on the dynamics assigned to the loop. For slow dynamics, the margins Mi = y0i − r0i may be small, whereas for faster dynamics, they have to be increased. This is the effect of plant windup due to input saturation. Now consider the case where the signals usi (t) not only differ from the ui (t), i = 1, 2, ..., p whenever |ui (t)| ≥ u0i > 0 (amplitude saturation), but also whenever |u˙ i (t)| ≥ uvi > 0 (rate saturation), i.e., there is an actuator with combined amplitude and rate saturation. In [1] a model according to u˙ si (t) = uvi sgn {satu0i [ui (t)] − usi (t)} , i = 1, 2, ..., p

(8)

has been proposed, where satu0i (·) is the above-deﬁned saturation function (3) and sgn(·) is the standard sign function. In numerical simulations, this actuator model causes problems whenever the input to the sign function vanishes and whenever |u˙ i (t)| < uvi . A model better suited for simulations of an amplitude- and raterestricted actuator component usi (t) = satuuvi (ui (t)) is shown in Fig. 1. 0i The gains Ri , i = 1, 2, ..., p should be chosen such that the resulting time constants 1/Ri do not have a signiﬁcant inﬂuence on the behaviour of the closed loop. The nonlinear elements in Fig. 1 are again of the saturation type, namely u ¯si (t) = satu0i (ui (t)) = sign(ui (t)) min {u0i , |ui (t)|} , u0i > 0 and wsi (t) = satuvi (wi (t)) = sign(wi (t)) min {uvi , |wi (t)|} , uvi > 0, i = 1, 2, ..., p.

38

Peter Hippe

satu0i ui

satuvi

usi

wsi

wi

u ¯si

1

Ri

s

Fig. 1. Simulation model of an amplitude- and rate-restricted actuator component

In the case of mere amplitude saturation, one inserts a model of the restricted actuator (see (3)) at the output of the compensator. Thus, the actual actuator never saturates, so that only the model has to be taken into account. If the actuator contains an additional rate saturation, the ersatz models shown in Fig. 2 can be inserted at each output ui (t), i = 1, 2, ..., p of the compensator. Due to the saturation element u ¯si (t) = satu0i (ui (t)) the amplitude of the output signal usi (t) never exceeds the limit u0i . When choosing the time constant Tvi of the ﬁrst-order system as Tvi =

2u0i , i = 1, 2, ..., p uvi

(9)

one also assures |u˙ si (t)| ≤ uvi , because the maximum speed |u˙ si (t)| resulting in the worst case situation (namely a sudden change of u ¯si (t) from one limit to the opposite) is uvi . Consequently, when inserting these models at the compensator output, the plant input signal neither exceeds the rate nor the amplitude constraints, so that the actual actuator nonlinearities do not have to be taken into account. When adding the ﬁrst-order elements of the p actuator models x˙ v (t) = Av xv (t) + Bv u ¯s (t)

(10)

us (t) = xv (t)

satu0i ui

usi

u ¯si 1 Tvi s + 1

Fig. 2. Ersatz model of an actuator component with joint amplitude and rate constraints

Stable and Unstable Systems with Amplitude and Rate Saturation

39

t with the ¯ts (t) = [¯ us1 (t), · · · , u ¯sp (t)], vectors xv (t) = [xn+1 (t), · · · , xn+p (t)] , u 1 Tv1 , · · ·

ttv =

, T1vp , and Av = diag(−tv ), Bv = diag(tv ) to the system (1), one

obtains the state equations ¯x(t) + B ¯u ¯d d(t) x ¯˙ (t) = A¯ ¯s (t) + B y(t) = C¯ x ¯(t) + Cd d(t)

(11)

¯(t) + C¯md d(t) y¯m (t) = C¯m x of the augmented system with x ¯(t) =

x(t)

xv (t)

, y¯m (t) =

ym (t)

us (t)

(since the vector

us (t) = xv (t) is measurable) and the parameters

A A¯ = 0

C¯ = C

B

Av 0 ,

,

¯= B

C¯m =

0

Bv Cm 0

¯d = B

,

0 Ip

Bd 0

,

C¯md =

Cmd

0

0

0

This augmented plant has an input nonlinearity u ¯s (t) = satu0 (u(t)), so that the problem of joint amplitude and rate constraints can be reformulated as a problem with mere input saturation for the system (11). Remark 1. The following discussions are valid for systems with amplitude saturation only and for systems with joint amplitude and rate saturation. In the case of mere amplitude restraints, the state equations (1), and in the case of additional rate constraints, the state equations (11) constitute the description of the plant. In order to keep the developments simple, the case of mere amplitude saturation is considered in the sequel, as it is now obvious, how the case of additional rate constraints can be handled. Remark 2. The above-suggested procedure is somehow outside the usual design paradigm of windup prevention, namely: “Design the linear control regardless of the input saturations and then add appropriate measures to remove the problems caused by these restrictions.” However, the introduction of the ersatz model for an amplitude- and rate-restrained actuator considerably facilitates the design of antiwindup measures and it has the advantage of being applicable in the presence of stable and unstable systems.

40

Peter Hippe

When using a slightly modiﬁed ersatz model, the usual design paradigm is also applicable in the case of additionally rate-resticted actuators (see [14], Section 7.4). This modiﬁed ersatz model, however, should only be applied in the case of stable systems.

3 The Underlying Control Scheme Given the system description (1), it is assumed that there is a nominal state feedback (12)

uC (t) = Kx(t)

This nominal linear control, characterized by the feedback interconnection us (t) = −uC (t) is supposed to stabilize the possibly unstable system, and to give a desired disturbance attenuation such that even worst case disturbances do not give rise to control signals ui (t) with amplitudes beyond the saturation limits u0i . Reference tracking is assured by feedforward control as shown in Fig. 3. The corresponding input signal uCr (t) drives a model of the system in the reference shaping ﬁlter (see Section 5) and it is assumed, that this model is correct. Therefore, after the initial transients have decayed, reference inputs lead to coinciding states xM (t) = x(t) in the model and the system. Consequently, the feedback signal uCd (t) vanishes unless xM (t) = x(t) is caused by external disturbances.

d

satu0 uCr

u

us

Linear System

uCd r

x

Nonlinear reference

K

shaping ﬁlter xM

Fig. 3. Proposed control scheme

y

Stable and Unstable Systems with Amplitude and Rate Saturation

41

The reference shaping ﬁlter is designed such that only the safely applicable reference signal amplitudes are effective and all control signals uCri (t) stay in predeﬁned amplitude ranges. Assume that the worst case disturbances cause feedback signals |uCdi (t)| ≤ βi u0i , i = 1, 2, ..., p, 0 < βi < 1

(13)

In exponentially unstable systems (13) is a necessary assumption. If disturbances generate feedback signals |uCdi (t)| > u0i , the stability of the loop cannot be guaranteed. By conﬁning the signals uCri (t) to the regions |uCri (t)| ≤ (1 − βi )u0i = uβi , i = 1, 2, ..., p

(14)

the saturation us (t) = satu0 (u(t)) at the input of the system never becomes active in spite of jointly acting reference and disturbance signals. Consequently, even in the presence of exponentially unstable systems the stability of the scheme in Fig. 3 is guaranteed. Remark 3. A design of the scheme in Fig. 3 such that the input saturation us (t) = satu0 (u(t)) never becomes active gives rather conservative results. If one knew the exact forms of the reference and the disturbance signals, one could allow for some over-saturation in the sense of [27]. For arbitrary signal forms, however, a stable behaviour of a loop containing an unstable system can only be guaranteed if input saturation never becomes active (consider, e.g., the case of constant disturbances driving one or more input signals ui (t) beyond the saturation limits). Remark 4. Disturbances acting during reference transients could be allowed to cause saturating input signals ui (t) in stable systems. For a nominal state feedback (12) such that K(sI − A)−1 B violates the circle criterion, this would, however, entail the danger of plant windup. For stable systems, plant windup can be prevented by an additional dynamic element (ADE). In the scheme of Fig. 3 this ADE is equivalent to substituting the input u(t) = uCr (t) − uCd (t) to the saturating element by u(t) = uCr (t) − uCd (t) − η(t) with

(15)

42

Peter Hippe

˙ = (A − BKS )ξ(t) + B [u(t) − us (t)] ξ(t) η(t) = (K − KS )ξ(t)

(16)

where KS is a “safe state feedback” such that KS (sI −A)−1 B meets the circle criterion [13]. This ADE is the time domain version of the “additional dynamic network” in [15]. The maximum amplitude of the output signal of a SISO system due to arbitrary input signals of know maximum amplitudes can be obtained by the L1 -norm of the corresponding transfer function (see also [12]). Given a linear, asymptotically stable, time invariant SISO system with transfer function G(s) and a corresponding impulse response h(t). Then, when an arbitrary input signal u(t) with known maximum amplitude |u(t)| ≤ ulim > 0 is applied to the input, the maximum amplitude ylim of the output of this system is bounded by ∞ ylim ≤ ulim |h(τ)| dτ = ulim G(s) 1

(17)

0

where G(s) 1 is the L1 -norm of G(s) (see, e.g., [5]). For a MIMO system with a p × m-transfer function matrix G(s), the notion G(s) 1 will be used in the sense ⎡

⎤ G11 (s) 1 · · · G1m (s) 1 ⎢ ⎥ .. .. ⎥ G(s) 1 = ⎢ . . ⎣ ⎦

(18)

Gp1 (s) 1 · · · Gpm (s) 1 Denote the transfer function matrix of the closed loop between the disturbances d and the controller output uCd (see Fig. 3) by uCd (s) = Gd (s)d(s)

(19)

Then, given the worst case amplitudes dimax , i = 1, 2, ..., q of the disturbances di (t), the signals uCdi (t) caused by these disturbance inputs are limited by uCdimax =

p j=1

Gij (s) 1 djmax i = 1, 2, ..., p

(20)

Stable and Unstable Systems with Amplitude and Rate Saturation

43

4 Modiﬁcations in the Presence of State Observers and Signal Models for Disturbance Rejection The scheme shown in Fig. 3 uses constant state feedback control. Usually such controllers also incorporate state observers and signal models for disturbance rejection. In view of reduced-order observers, the measurement vector ym (t) of the system (1) is subdivided according to

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 1 1 1 C C ym (t) ⎦ = ⎣ m ⎦ x(t) + ⎣ md ⎦ d(t) ym (t) = ⎣ 2 2 2 (t) Cm Cmd ym

(21)

2 where the vector ym (t) contains the κ measurements directly used to obtain the state 1 estimate x ˆ(t) and the vector ym (t) the remaining (m − k) measurements.

The (reduced-order) state observer of order nO = n−κ with 0 ≤ κ ≤ m is described by z(t) ˙ = F z(t) + Dym (t) + T Bus (t)

(22)

where the observer technique for the prevention of controller windup has been applied, i.e., the observer is driven by the constrained input signal. For vanishing disturbances d(t) ≡ 0 and if the initial transients have decayed, the state estimate ⎡ ⎤⎡ ⎡ ⎤ ⎤ 2 2 2 ym Cm ym (t) (t) ⎦ = Ψ2 Θ ⎣ ⎦ = Ψym (t) + Θz(t) x ˆ(t) = ⎣ ⎦ ⎣ (23) T z(t) z(t) coincides with the state x(t) of the system if the relations T A − F T = DCm

(24)

ΨCm + ΘT = In

(25)

and

are satisﬁed, where the abbreviation Ψ = [0n,m−κ Ψ2 ] has been used. Substituting x(t) in Fig. 3 by x ˆ(t), only the disturbance rejection of the loop is modiﬁed. The reference behaviour remains unchanged since the constrained input us (t) does not cause observation errors. Consequently, for reference inputs r(t) one has x ˆ(t) = x(t) = xM (t), so that uCd (t) vanishes. This allows the above-discussed subdivision of the limited input signal into one part for disturbance rejection and the remaining part for tracking.

44

Peter Hippe

If constantly acting disturbances, modeled in a signal process with characteristic polynomial det(sI − Sd ) = sq + ψq−1 sq−1 + · · · + ψ0

(26)

have to be compensated asymptotically, one can use a slightly modiﬁed version of Davison’s approach [6]. In its original version, Davison’s approach assures reference tracking and disturbance rejection for the same type of signals. On the one hand, this is more than required here and on the other hand, it can have an unfavourable inﬂuence on the tracking behaviour [11]. Therefore, Davison’s approach is modiﬁed such that the signal model is only used for disturbance rejection. To achieve this, the signal model is no longer driven by Bε [y(t) − r(t)], but by Bε y(t) and Bσ [ˆ uC (t) + us (t)], where u ˆC (t) is the output of the observer-based controller and us (t) is the saturated input signal to the plant. By an appropriate choice of Bσ the dynamics of the modiﬁed signal model do not inﬂuence the tracking behaviour. The modiﬁed signal model has the form v(t) ˙ = Sv(t) + Bε y(t) − Bσ [ˆ uC (t) + us (t)]

(27)

with S = diag [Sd , · · · , Sd ] , Bε = diag [bε , · · · , bε ] and ⎤ 1 0 ··· 0 ⎢ . .. ⎥ .. ⎥ ⎢ . ⎢ . . . ⎥ Sd = ⎢ ⎥, ⎥ ⎢ 0 0 · · · 1 ⎦ ⎣ −ψ0 −ψ1 · · · −ψq−1 ⎡

⎡ ⎤ 0 ⎢.⎥ ⎢.⎥ ⎢.⎥ bε = ⎢ ⎥ ⎢0⎥ ⎣ ⎦ 1

Thus the vector v(t) has pq components. The quantity u ˆC (t) is deﬁned by x ˆ(t) u ˆC (t) = Kx Kv v(t)

(28)

(29)

The matrices Kx and Kv will be chosen to obtain a controlled signal model with desired eigenvalues and a reference behaviour that only depends on the nominal state feedback Kx(t). Given the solution X of the Sylvester equation X(A − BK) − SX = Bε C and the matrix

(30)

Stable and Unstable Systems with Amplitude and Rate Saturation

45

Bσ = −XB

(31)

With this Bσ choose the feedback matrix Kv such that the eigenvalues of (S − Bσ Kv ) have desired locations. When ﬁnally computing the feedback matrix Kx according to K x = K − Kv X

(32)

the eigenvalues of the loop are the zeros of the characteristic polynomial det(sI − A + BK)det(sI − F )det(sI − S + Bσ Kv ), and neither the state observer nor the signal models inﬂuence the reference behaviour (for a proof of this see, e.g., [13]). In the absence of disturbances and after the initial transients have decayed, reference inputs cause u ˆC (t) = K x ˆ(t) = Kx(t). Therefore, the scheme in Fig. 3 needs to be slightly modiﬁed as shown in Fig. 4. d satu0 uCr

y x˙ = Ax+Bus +Bd d

us

u

y = Cx + Cd d

ym

ym = Cm x + Cmd d

r

Nonlinear uCd

reference

x ˆ

Kx

z˙ = F z +Dym +T Bus x ˆ = Θz + Ψym

shaping ﬁlter σ xM v Kv

K

v˙ = Sv+Bε y−Bσ σ

u ˆC

Fig. 4. The control structure of Fig. 3 in the case of observer-based compensators containing signal models for disturbance rejection

5 The Model-based Reference Shaping Filter Given is a model x˙ M (t) = AM xM (t) + BM uas (t) yM (t) = CM xM (t)

(33)

46

Peter Hippe

of the system (1) with an external input uas (t), and we assume that this model is correct, i.e., (AM , BM , CM ) = (A, B, C). At the input of the model is a nonlinearity uas (t) = satuβ (ua (t)), where the saturation limits uβi are deﬁned in (14). Further assume for the moment that AM has no eigenvalues at s = 0. The case of systems having eigenvalues at s = 0 will be discussed in Remark 5. Based on this model of the system, the nonlinear reference shaping ﬁlter shown in Fig. 5 is considered. First we design a state feedback ua (t) = −Ka xM (t) + La ubs (t)

(34)

for the model (33) so that in the linear case (i.e., uas (t) = ua (t)) the possibly unstable model is stabilized, and the steady state errors yM i (∞) − ubsi (∞), i = 1, 2, ..., p vanish for constant input signals ubsi (t). This is assured by

−1 La = CM (−AM + BM Ka )−1 BM

(35)

Given the limits uβi for the maximum amplitudes of the input signals uasi (t) = ˆ uCri (t) to the model of the system, the zeros of det(sI − AM + BM Ka ) are chosen such that the ranges

r Reference shaping ﬁlter L−1 a Lb

uCr satuβ

satr0 ub

ubs

ua La

L−1 a Kb

yM

uas x˙ M = AM xM +BM uas

CM

Ka

xM

Fig. 5. Reference shaping ﬁlter

Stable and Unstable Systems with Amplitude and Rate Saturation

−r0i ≤ ubsi (t) ≤ r0i , i = 1, 2, ..., p

47

(36)

can be made as large as possible without causing signals |uai (t)| > uβi (see (14)). This tuning of (34) has two important implications. On the one hand, actuator saturation does not become active when using uCr (t) = uas (t) as an input to the system (see Figs. 3 and 4), and on the other hand, also the input saturation uas (t) = satuβ (ua (t)) in front of the model in the scheme of Fig. 5 is never active. Thus, the possibly unstable dynamics of the model are always stabilized by the linear control (34). For arbitrary input signals ri (t), the restriction (36) can be assured by a saturation element ubs (t) = satr0 (ub (t)) as shown in Fig. 5. The transfer behaviour between ubs and uas = ua is characterized by ua (s) = Gua (s)ubs (s) with Gua (s) = −Ka (sI − AM + BM Ka )−1 BM La + La

(37)

Given arbitrary input signals within the limits (36), one obtains the upper limits of the signals uai (t) with the aid of the L1 -norm Gua (s) 1 (see also (17), (18) and (20)) as uaimax

p u G a (s) r0j , i = 1, 2, ..., p = ij 1

(38)

j=1

and these have to meet the restrictions uaimax ≤ uβi . Due to the above construction, the inner loop is a linear stable system with input saturation ubs (t) = satr0 (ub (t)). Consequently, one can add an outer loop

with

−1 ub (t) = −L−1 a Kb xM (t) + La Lb r(t)

(39)

−1 −1 Lb = CM [−AM + BM (Ka + Kb )] BM

(40)

assuring vanishing tracking errors yM i (∞) − ri (∞), i = 1, 2, ..., p for constant reference signals. Now consider the transfer function matrix −1 GL (s) = L−1 BM La a Kb (sI − AM + BM Ka )

(41)

characterizing the transfer behaviour ub (s) = −GL (s)ubs (s) in Fig. 5 (recall that saturation uas (t) = satuβ (ua (t)) never becomes active). If GL (s) satisﬁes the circle

48

Peter Hippe

criterion [35], the cascaded nonlinear loop in Fig. 5 is globally asymptotically stable for arbitrary reference inputs. This, of course, is only true for vanishing initial conditions xM (0). As r(t) is the only input to the reference shaping ﬁlter, a start from xM (0) = 0 can be assured. Remark 5. The above-presented design also holds for systems with eigenvalues at the origin of the s-plane, and it then allows for the chosen maximum reference signal amplitudes r0i . If, however, one wants to exploit the potential of such systems to reach unlimited output amplitudes in spite of input saturation, eigenvalues in the inner loop (i.e., roots of det(sI − AM + BM Ka )) should be placed at s = 0. Only in the second cascade (b) these eigenvalues are also shifted to the left. Such reference shaping ﬁlters allow one to obtain unlimited amplitudes yM (∞) in a stable manner. Remark 6. In the case of stable systems, the inner stabilizing loop in the reference shaping ﬁlter is no necessity. Then the trajectory generator in Fig. 5 could be realized with one loop only (i.e., Ka = 0, La = Ip , and the nonlinearity ubs (t) = satr0 (ub (t)) removed), and the outer cascade such that Kb (sI−AM )−1 BM satisﬁes the circle criterion. Then, however, directionality problems (see Section 6) could occur, which is not the case, when the above-discussed ﬁlter design is also carried out for stable systems.

6 The Problem of Directionality An important design goal in MIMO systems is to assure a decoupled reference behaviour. The change of one set point should not have a disturbing inﬂuence on the remaining ones. The decoupling controller generates speciﬁc plant input signals ui (t), deﬁning the direction of the input vector u(t). When one or more input signals saturate, the direction of this input vector is changed, giving rise to possibly strong and undesired coupling effects, the so-called directionality problem. Another important issue related to the decoupling control of constrained systems is the fact, that there are plants that cannot be decoupled without violating all the existing stability criteria for systems with input saturation. Consequently, if saturation becomes active in such a loop, an unstable behaviour can occur (see Example 2).

Stable and Unstable Systems with Amplitude and Rate Saturation

49

The above-presented design procedure allows an exact preservation of directionality. This becomes possible by designing the inner loop of the reference shaping ﬁlter to obtain a decoupled transfer behaviour between ubs and yM such that arbitrary signals ubsi (t) within the ranges −r0i ≤ ubsi (t) ≤ r0i do not cause saturated signals uai (t). Seen from the input ub (t) this leads to p linear decoupled subsystems with input saturations ubsi (t) = satr0i (ubi (t)). When adding p SISO loops ubi (t) = −kbi yM i (t) + li ri (t), a saturation ubsi (t) = satr0i (ubi (t)) does not cause directionality problems. Given the feedback factors kbi , the rows of L−1 a Kb in (39) have the forms diag[kb1 , . . . , kbp ]Cm and L−1 a Lb = diag [l1 , · · · , lp ]. The shaping ﬁlter exhibits a stable behaviour for arbitrary reference inputs if the transfer matrix (41) satisﬁes, i.e., the circle criterion. If the system cannot be decoupled by static state feedback, the necessary preﬁlter (see, e.g., [22]) could be incorporated in the reference shaping ﬁlter and when the preﬁltered input signal to the model also drives the system, the scheme of Figure 3 would again be applicable.

7 Two Examples Example 2. Considered is a stable system with ⎡

0

1

0

0

0

0

⎡ 0 ⎢ ⎢1 ⎢ ⎢0 ⎢ B=⎢ ⎢0 ⎢ ⎢ ⎣0 0

⎤

⎥ ⎢ ⎢−3 −2 2 0 −1 −1⎥ ⎥ ⎢ ⎢ 0 0 0 1 0 0⎥ ⎥ ⎢ A=⎢ ⎥, ⎢−2 2 −1 −3 −2 −2⎥ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 1 ⎦ ⎣ −1 −3 2 2 0 −1

0 0 0 1 0 0

C = Cm

⎤

⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ ⎥ 0⎦ 1

⎤

⎡ 3 ⎢ =⎢ ⎣−2 1

0

1

1

0

10.5

0

1

1 −18

⎥ 0⎥ ⎦

1

0

0

1

18.5

0

and input saturation limits u01 = 6, u02 = 12 and u03 = 9. If one wants to use symmetric output regions with equal bounds y0i = y0 , the maximum sustainable amplitudes are y0 = 1.248, since the stationary output vector y(∞) = t

[−1.248, 1.248, 1.248] requires an input vector u(∞) where the ﬁrst component

50

Peter Hippe

is at its boundary u01 = 6. In what follows it is assumed that the maximum applied reference signals are limited by r0i = 1, i = 1, 2, 3. The problems caused by input saturation in this system are ﬁrst discussed in a standard state feedback setting with u(t) = −Kx(t) + Lr(t). The system can be decoupled by static state feedback, but each decoupling K violates the circle criterion. Assigning, e.g., a decoupling control u(t) = −Kx(t) + Lr(t) with ⎡ ⎤ ⎡ 7 0 18 8 9 1 72.5 9.5 ⎢ ⎥ ⎢ ⎢ ⎥ K=⎢ ⎣−16 0 6 5 −128 −20 ⎦ , L = ⎣ 0 7 −15 −5 −5

1

56

−7

14

0

⎤ 0 ⎥ 0⎥ ⎦ 7

the (linear) closed loop ⎞ has a decoupled reference behaviour of the form y(s) = ⎛ diag⎝

7

7 ⎠r(s). Using a step-like reference input sequence ri (t) = , s+7 s+7 s+7 ,

7

rsi 1(t) − 2rsi 1(t − 5), i = 1, 2, 3, (here 1(t) is the unit step function), the transients of the loop already exhibit an unstable behaviour for amplitudes rs1 = 0.5, rs2 = −0.5 and rs3 = −0.5. Also the directionality preserving approach [4] does not stabilize the transients in this case. A stable behaviour of the nonlinear loop can be obtained when the state feedback satisﬁes the Kalman-Yakubovich lemma [35] At P + P A = −Qt Q B t P + W t Q = KKY W t W = 2Ip With

⎤

⎡ 0.8

0.4

⎢ Q=⎢ ⎣−0.8 −0.8 0.4

0.4

the solution for KKY is ⎡ 1.290 2.978 ⎢ KKY = ⎢ ⎣−2.176 −2.465 2.900 −1.718

0

0

0

0.8

0.4

0

1.6

0

2.8

0

⎥ 0⎥ ⎦ 0.4

0.8359 −1.333 −2.787 −2.284

⎤

3.195

2.480

3.354

⎥ 1.799⎥ ⎦

1.659

1.799

8.665

4.037

and the corresponding reference injection matrix is

Stable and Unstable Systems with Amplitude and Rate Saturation

⎡ LKY =

29.51705 −66.76825 −84.80765

1 ⎢ ⎢ 35.5475 98.6925 32 ⎣ 28.503 −39.415

51

⎤

⎥ 85.1105 ⎥ ⎦

−39.539

Figure 6 shows the resulting reactions of the closed loop to the above reference sequences (i.e., for amplitudes rs1 = 0.5, rs2 = −0.5 and rs3 = −0.5). It is deteriorated by large and long lasting overshoots due to unfavorable zeros and severe cross-couplings. A perfect preservation of directionality for all reference signals within the amplitude limits r0i , i = 1, 2, 3 can be obtained by the approach presented in Section 5. In a ﬁrst step the inner loop is parametrized by ⎤ ⎡ −1.35 1.55 2.55 1 4.775 9.5 ⎥ ⎢ ⎥, Ka = ⎢ −3.4 0 −0.3 −1.3 −14.6 −20 ⎦ ⎣ −1.95 −4.85 1.45 1 7.175 7.7

⎤

⎡ 0.55

⎢ La = ⎢ ⎣ 0 −0.55

0 0.7 0

⎛

assuring a decoupled behaviour yM (s) = diag⎝

0

⎥ 0⎥ ⎦ 0.7

⎞ 0.55

0.7

0.7

⎠ubs (s). , , s + 0.55 s + 0.7 s + 0.7

This tuning was done by trial and error, which is facilitated by the decoupling design having only three free parameters, namely the pole locations in the three channels. The above pair (Ka , La ) leads to a transfer matrix Gua (s) (see (37)), whose L1 norm is

Outputs y1 (t), y2 (t) and y3 (t)

4

y2 (t) 2

y1 (t) 0

−2

−4

y3 (t) 0

2

4

6

8

10

Time

Fig. 6. Reference step responses with a control that satisﬁes the circle criterion

52

Peter Hippe

⎤ ⎡ 1.44 2.26 2.30 ⎥ ⎢ ⎥ Gua 1 = ⎢ 2.72 2.67 3.83 ⎦ ⎣ 3.49 1.81 3.69 Inserting the r0i = 1, i = 1, 2, 3 in (38), one obtains the upper limits of the resulting input signals uai (t) as ua1max = 6, ua2max = 9.22 and ua3max = 8.99, so that for arbitrary signals ubi (t) the input signals uai (t) never exceed the saturation limits uβi = u0i (i.e., no amplitude margin for disturbance rejection has been reserved). Now adding an outer (decoupling) cascade with ⎤ ⎡ 58.35 19.45 19.45 0 204.225 0 ⎥ ⎢ ⎥, Kb = ⎢ −38.6 0 19.3 19.3 −347.4 0 ⎦ ⎣ −39.05 −0.15 −19.45 0 152.825 19.3

⎤

⎡ 20

⎢ Lb = ⎢ ⎣ 0 −20

0

0

20

⎥ 0⎥ ⎦

0

20

! [2723, 2123, 2123] Cm and diag[l1 , l2 , l3 ] = ! 1 L−1 L = diag [2800, 2200, 2200] . The transfer matrix (41) satisﬁes the cirb a 77

one obtains L−1 a Kb = diag

1 77

cle criterion, guaranteeing a stable behaviour of the nonlinear reference shaping ﬁlter for all input signals r(t). For small reference inputs, the linear behaviour of the ⎛ reference shaping ﬁlter ⎞ is characterized by the transfer matrix yM (s) = diag⎝

20

,

20

,

20

s + 20 s + 20 s + 20

⎠r(s).

Figure 7 shows the reference behaviour of this ﬁlter, which is identical with that of the controlled system in a scheme of Fig. 3, if, e.g., the feedback matrix K = KKY is used in a “nominal” control (12). There are no directionality problems for any reference signal changes within the regions −1 ≤ ri (t) ≤ 1, i = 1, 2, 3 and the transient behaviour is satisfactory.

Example 3. Considered is the benchmark system of Section A1. Case a) Amplitude constraint u0 = 1 and rate constraint uv = 10. To handle the joint amplitude and rate constraints, the augmented system must be used. With the given values of the constraints, the time constant (9) is Tv = 0.2. Consequently, the state equations of the augmented system (11) are (to comply with the benchmark the notation d(t) = ˆ z1 (t) is used)

Stable and Unstable Systems with Amplitude and Rate Saturation

53

Outputs y1 (t), y2 (t) and y3 (t)

1

y1 (t)

0.5

0

y2 (t)

−0.5 y3 (t)

−1

0

2

4

6

8

10

Time

Fig. 7. Reference behaviour of the directionality preserving ﬁlter

⎡

⎢ ⎢−1 x ¯˙ (t) = ⎢ ⎢ 0 ⎣

⎡ ⎤ ⎡ ⎤ 0 1 ⎢ ⎥ ⎥ ⎢ ⎥ ⎢0⎥ ⎢0⎥ 0 1 0⎥ ⎢ ⎥u ⎥x ⎢ ⎥z1 (t) ¯ ¯ (t) + (t) + s ⎥ ⎢ ⎥ ⎢0⎥ 1 0 1⎦ ⎣0⎦ ⎣ ⎦ 5 0 0 −5 0 ¯(t) 0 0 x

0 −1

0

y(t) = 1 0

0

0

⎤

y¯m (t) = x ¯(t) To allow input disturbances z1 (t) = z1s 1(t) with relatively large amplitudes z1s , a nominal state feedback K x ¯(t) = [−31.66, 35.72, 25.66, 3.26] x ¯(t) is chosen, placing the eigenvalues of the closed loop at s1 = −0.3, s2 = −1 and s3/4 = −10. The integral action in the nominal controller is represented by the signal model v(t) ˙ = y(t) − Bσ [uC (t) + u ¯s (t)] (see (27)), i.e., S = 0 and Bε = 1, and the characteristic polynomial of the controlled signal model is chosen as det(sI − S + Bσ Kv ) = s + 0.05. The Sylvester equation (30) is solved by X =

1 3

[−15.73, 12.83, 2.13, 0.1], and

with this solution (31) gives Bσ = −0.5/3 and consequently Kv = −0.3. Using (32), the feedback Kx = [−33.233, 37.003, 25.873, 3.27] results. Since the states of the system are all measurable, the design of the feedback control is complete.

54

Peter Hippe 1

Output y(t)

0.8

0.6

0.4

0.2

0

0

5

10

15 Time

20

25

30

Fig. 8. Reference step responses

When choosing the parameters in the inner loop of the reference shaping ﬁlter as Ka = [−1.24992, 1.6896, 1.168, −0.36] and La = −0.08192 (giving det(sI − ¯ a ) = (s + 0.8)4 ), the L1 -norm of (37) is 1.01892. Restricting the input A¯ + BK signal ubs (t) to r0 = 1/1.01892, the inner saturation uas (t) = satu0 (ua (t)) (i.e., we have assumed β = 0 in (14)) never becomes active for arbitrary reference inputs r(t). Thus, the inner loop of the reference shaping ﬁlter is a stable, linear system with input nonlinearity ubs (t) = satr0 (ub (t)). In the second cascade, the feedback is chosen to obtain the characteristic polyno¯ b ) = (s + 1.8)4 , which corresponds to Kb = ¯ a + BK mial det(sI − A¯ + BK [−5.1376, 5.856, 3.12, 0.8] and leads to Lb = −2.09952. With this Kb the transfer function (41) of the linear part of the loop satisﬁes the circle criterion so that the reference shaping ﬁlter is asymptotically stable for all reference inputs. With this parameterization of the trajectory generator, the loop of Fig. 4 exhibits the reference behaviour shown in Fig. 8. The input signals uCr (t) to the plant produced by the reference shaping ﬁlter never pass the saturation limit u0 = 1 and the rate of duCr (t) always stays well below the upper limit uv = 10. change dt

Stable and Unstable Systems with Amplitude and Rate Saturation

55

The broken lines in Fig. 9 show the disturbance step responses up to the maximum allowable amplitude z1s = 0.744. For a step amplitude z1s = 0.745 the loop exhibits an unstable behaviour.

Output y(t)

3.5 3

tuning for z1s = 0.744

2.5

tuning for z1s = 0.5

2 1.5 1 0.5 0 −0.5

0

10

20

30

40

50

Time

Fig. 9. Disturbance step responses with control settings allowing maximum amplitudes z1s = 0.744 and z1s = 0.5

A much better disturbance rejection in view of the resulting amplitudes of y(t) is obtained by the nominal state feedback K = [−25, 31.4, 19, 5.6] placing the eigenvalues of the controlled augmented system at s1 = −30 and s2/3/4 = −1, and when det(sI − S + Bσ Kv ) = (s + 1) is chosen. This leads to Bσ = −0.5/3, Kv = −6 and Kx = [−49.8, 50.4, 25.6, 5.8]. This parameterization, however, reduces the amplitudes of the step disturbances which do not destabilize the loop. The solid lines in Figure 9 show the disturbance step responses with this modiﬁed control up to the maximum allowable step amplitude z1s = 0.5.

Case b) Amplitude constraint u0 = 1 and rate constraint uv = 2. With the given constraint values, the time constant (9) is now Tv = 1. Consequently, the description (11) of the augmented system has the form

56

Peter Hippe

⎡

⎢ ⎢−1 x ¯˙ (t) = ⎢ ⎢ 0 ⎣

⎡ ⎤ ⎡ ⎤ 0 1 ⎢ ⎥ ⎥ ⎢ ⎥ ⎢0⎥ ⎢0⎥ 0 1 0⎥ ⎢ ⎥u ⎥x ⎢ ⎥z1 (t) ¯ ¯ (t) + (t) + s ⎥ ⎢ ⎥ ⎢0⎥ 1 0 1⎦ ⎣0⎦ ⎣ ⎦ 1 0 0 −1 0 ¯(t) 0 0 x

0 −1

0

y(t) = 1 0

0

0

⎤

y¯m (t) = x ¯(t) To obtain a stable behaviour for step disturbances up to an amplitude z1s = 0.38, the nominal state feedback K x ¯(t) = [−45, 57, 35, 12] x ¯(t) (giving det(sI − A¯ + ¯ BK) = (s + 1)3 (s + 10)) and det(sI − S + Bσ Kv ) = s + 0.05 were chosen. The Sylvester equation (30) is now solved by the row vector X = [−4.4, 3.5, 1.3, 0.1], and with this solution (31) gives Bσ = −0.1 and, consequently, Kv = −0.5 and (32) leads to Kx = [−47.2, 58.75, 35.65, 12.05]. The model-based reference shaping ﬁlter is parameterized by the vector Ka = [−4.71350625, 6.2985, 4.535, 1.6] which yields the characteristic polynomial det(sI− ¯ a ) = (s + 0.65)4 and La = −0.17850625. With this feedback the L1 A¯ + BK norm of (37) is 1.0785 so that r0 = 1/1.0785 assures that the inner saturation uas (t) = satu0 (ua (t)) is never active. When ﬁnally choosing Kb = [−14.322, 17.496, 10.08, 3.2] (which leads to det(sI− ¯ BK ¯ a + BK ¯ b ) = (s+1.45)4 and Lb = −4.42050625), the transfer function (41) A+ satisﬁes the circle criterion. The loop of Fig. 4 now exhibits the reference behaviour shown in Fig. 10. The input signals uCr (t) producing these transients never hit the saturation limit u0 = 1 and the rate of change |u˙ Cr (t)| always stays well below the upper limit uv = 2. Figure 11 shows the disturbance step responses up to the maximum allowable step amplitude z1s = 0.38. Again, the disturbance rejection could be improved in view of the resulting amplitudes of the output y(t), but this would again reduce the maximum allowable amplitude z1s of the disturbance signal. According to the benchmark speciﬁcations, the potential of the new scheme to handle the joint action of reference and disturbance signals in a stable manner has not been exploited. If disturbance rejection would require α% of the restricted input signals, this could be handled by reducing the value of uβ (see Fig. 4) by α%.

Stable and Unstable Systems with Amplitude and Rate Saturation

57

1

Output y(t)

0.8

0.6

0.4

0.2

0

0

5

10

15

20

25

30

Time

Fig. 10. Reference step responses of the loop

2

Output y(t)

1.5

1

0.5

0

0

10

20

30

40

50

Time

Fig. 11. Disturbance step responses up the maximum amplitude z1s = 0.38

8 Conclusions A windup prevention scheme was presented which is equally applicable to stable and unstable constrained SISO and MIMO systems. Beyond windup prevention, the new scheme also allows a perfect directionality preservation for stable and unstable MIMO systems. The scheme uses feedback control for system stabilization and disturbance rejection, and a nonlinear model-based reference shaping ﬁlter for feedfor-

58

Peter Hippe

ward tracking. For arbitrary reference inputs this ﬁlter generates plant input signals such, that only the safely applicable reference signal amplitudes are transferred to the system, and that the plant input signals stay within user-deﬁned limits. The structure of the closed loop assures vanishing feedback signals for reference inputs, so that the input signals to the system result from a superposition of the signals necessary for disturbance rejection and the signals used for reference tracking. For exponentially unstable systems, a stable behaviour of the loop in spite of jointly acting reference and disturbance inputs can be obtained when splitting the constrained input range into a part used for disturbance rejection while only the remaining part is utilized for reference tracking. For stable systems, disturbances could be allowed to drive the input signals into saturation. If the danger of plant windup exists it can easily be removed by an additional dynamic element. Since the plant input signals generated in the reference shaping ﬁlter do not cause saturation, a systematic prevention of the possibly existing directionality problems becomes also feasible in the new scheme. Using a special model for an actuator with joint amplitude and rate saturation, the results presented are readily applicable also to systems with such actuators. A MIMO example and an unstable system were used to demonstrate the design procedures.

References 1. Barbu C, Reginatto R, Teel AR, Zaccarian L (2002) Anti-windup for exponentially unstable linear systems with rate and magnitude constraints. In: Kapila V, Grigoriadis KM (eds), Actuator saturation control. Marcel Dekker, N.Y. 2. Bateman A, Lin Z (2002) An Analysis and Design Method for Linear Systems Under Nested Saturations. In Proc. 2002 ACC, Anchorage, AK:1198–1203 3. Bernstein DS, Michel AN (1995) Intern. J. Robust and Nonlinear Control 5:375–380 4. Campo PJ, Morari M (1990) Computers Chem. Engrng. 14:343–358 5. Dahleh MA, Pearson JB (1987) IEEE Trans. on Automatic Control 32:889–895 6. Davison EJ (1976) IEEE Trans. on Automatic Control 21:25–34 7. Doyle JC, Smith RS, Enns DF (1987) Control of Plants with Input Saturation NonLinearities. In Proc. 1987 ACC, Minneapolis:1034–1039 8. G¨okc¸ek C, Kabamba PT, Meerkov SM (2001) IEEE Trans. on Automatic Control 46:1529–1542

Stable and Unstable Systems with Amplitude and Rate Saturation

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9. Gomes da Silva Jr. JM, Tarbouriech S, Garcia G (2003) IEEE Trans. on Automatic Control 48:842–847 10. Hanus R, Kinnaert M, Henrotte JL (1987) Automatica 23:729–739 11. Hippe P (1992) Automatica 28:1003–1009 12. Hippe P (2003) Automatica 39:1967–1973 13. Hippe P (2005) Windup prevention when using Davison’s approach to disturbance rejection. In Proc. 2005 IFAC World Congress Prague. 14. Hippe P (2006) Windup in control - Its effects and their prevention. Springer Berlin Heidelberg New York London 15. Hippe P, Wurmthaler C (1999) Automatica 35:689–695 16. Johnson CD (1971) IEEE Trans. Automatic Control 16:635–644 17. Kapasouris P, Stein G (1990) Design of Feedback Control Systems for Unstable Plants with Saturating Actuators. In Proc. IFAC Symposium on Nonlinear Control System Design. 18. Kendi TA, Doyle FJ (1997) J. Proc. Control 7:329–343 19. Kothare MV, Campo PJ, Morari M, Nett CN (1994) Automatica 30:1869–1883 20. Lin Z (1997) IEEE Trans. on Automatic Control 42:992–995 21. Lin Z, Saberi A, Teel AR (1996) Automatica 32:619–624 22. Moness M, Amin MH (1988) Int. J. Control 47:1925–1936 23. Mhatre S, Brosilow C (1996) Multivariable Model State Feedback. In Proc. IFAC World Congress, San Francisco:139 24. Peng Y, Vrancic D, Hanus R, Weller SSR (1998) Automatica 34:1559–1565 25. R¨onnb¨ack S, Walgama KS, Sternby J (1991) An Extension to the Generalized AntiWindup Compensator. In Proc. 13th IMACS World Congress on Scientiﬁc Computation, Dublin, Ireland. 26. Saberi A, Lin Z, Teel AR (1996) IEEE Trans. on Automatic Control 41:368–378 27. Seron MM, Goodwin GC, Graebe SF (1995) IEE Proc. Control Theory and Applications 142:335–344 28. Shim JH, Goodwin GC, Graebe SF (1996) MIMO Design with Saturating Actuators. In Proc. CESA’96 Multiconference, Lille, France:1026–1031 29. Tabouriech S, Pittet C, Burgat C (2000) Int. J. Robust and Nonlinear Control 10:489–512 30. Teel AR (1999) Int. J. Robust and Nonlinear Control 9:701–716 31. Teel AR, Kapoor N (1997) The L2 Anti-Windup Problem: Its Deﬁnition and Solution. In Proc. 4th ECC, Brussels, Belgium. 32. Turner MC, Postlethwaite I, Walker DJ (2000) Int. J. Control 73:1160–1172 33. Turner MC, Postlethwaite I (2001) Int. J. Control 74:1425–1434 34. Verriest EI, Pajunen GA (1992) Analysis of Constrained Tracking Problem. In Proc. 1992 American Control Conference:676–680

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35. Vidyasagar M (1993) Nonlinear Systems Analysis. Prentice Hall, Englewood Cliffs, N.J. 36. Walgama KS, Sternby J (1993) IEE Proc. Control Theory and Applications 140:231–241 37. Zheng A, Kothare MV, Morari M (1994) Int. J. Control 60:1015–1024

An Anti-windup Design for Linear Systems with Imprecise Knowledge of the Actuator Input Output Characteristics Haijun Fang1 and Zongli Lin2 1

Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, P. O. Box 400743, Charlottesville, VA 22904-4743, U.S.A. [email protected]

2

Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, P. O. Box 400743, Charlottesville, VA 22904-4743, U.S.A. [email protected]

Summary. This chapter proposes an anti-windup design for linear systems in the absence of precise knowledge of the actuator nonlinearity. The actuator input output characteristic is assumed to reside in a nonlinear sector bounded by two convex/concave piecewise linear curves. For the closed-loop system with a given anti-windup compensation gain matrix, stability properties, both in the absence and in the presence of external disturbances are characterized in terms of matrix inequalities. The estimation of the domain of attraction with a contractive invariant ellipsoid as well as the assessment of the disturbance tolerance and disturbance rejection properties of the closed-loop system are then formulated and solved as LMI optimization problems. The design of the anti-windup compensation gain matrix for a large domain of attraction and/or a large disturbance tolerance/rejection capability is then carried out by viewing the anti-windup compensation gain as an additional free parameter in these optimization problems. Numerical examples as well as an experimental setup are used to illustrate the proposed analysis and design approach and its effectiveness.

This work was supported in part by NSF grant CMS-0324329.

62

Haijun Fang and Zongli Lin

1 Introduction Saturation nonlinearity is ubiquitous in engineering systems. In a control system, physical limitations on its actuator cause the control input to saturate. It has been long recognized that actuator saturation, when occurs, degrades system performance and even causes the loss of stability. Even though there has been a surge of research on control design that directly takes actuator nonlinearity into account (see, e.g., [2, 7, 10, 12] and the references therein), the anti-windup design remains an important approach to dealing with actuator saturation and continues to attract attention from control research community (see [1, 3, 6, 5, 9, 13, 11, 17, 15, 16] for a small sample of the literature). While the earlier work on anti-windup design was more ad hoc in nature, recent literature has witnessed the emergence of systematic approaches to the design of anti-windup compensators. The both performances and stability are quantiﬁed in a rigorous way and the synthesis of the anti-windup compensators are formulated as optimization problems, usually with linear or bilinear matrix inequality constraints (see, e.g., [3, 9, 5, 14, 16]). In the recent literature on control systems with actuator saturation, there is a separate line of work started in [8], which aims to accommodate the uncertainties that commonly exist in the actuator input output characteristics. This is motivated by the fact that the behavior of a physical actuator usually cannot be represented by a standard saturation function and its actual input output characteristics are often not precisely known and may even vary as time. More speciﬁcally, in [8], the concept of a generalized sector bounded by two piecewise linear convex/concave curves (see Fig. 1) is introduced and the actuator input output characteristics are assumed to reside in such a generalized sector. By exploring the relationship between the piecewise linear functions and a saturation function, necessary and sufﬁcient conditions under which an ellipsoid is contractively invariant for any nonlinearity that resides in the given generalized sector are derived in terms of linear matrix inequalities. Based on these conditions, a constrained optimization problem is formulated and solved, in which the largest contractively invariant ellipsoid is sought for use as an estimate of the domain of attraction. By allowing the feedback gain as an additional free parameter, this optimization problem is also readily adapted for the synthesis of feedback gain. By its analogy

Robust Anti-windup Design

63

5

v 4

v=ψ (u) 1

3

v=ψ(u)

2

1

v=ψ2(u)

u

0

o −1

−2

−3

−4

−5 −8

−6

−4

−2

0

2

4

6

8

Fig. 1. A generalized sector as bounded by a convex and a concave curve.

with the classical absolute stability theory, such an approach to the stability analysis and design is referred to as a generalized absolute stability approach. Since the generalized sector can describe actuator nonlinearity more ﬂexibly than the classical sector and circle criterion, and the set invariant conditions established in [8] are necessary conditions, less conservative results are expected from the generalized absolute stability approach. This generalized absolute stability approach of [8] has also be adopted recently in the analysis and design for disturbance tolerance and disturbance rejection [4]. The objective of this chapter is to develop a generalized absolute stability approach to the design of an anti-windup gain for a given closed-loop system with imprecise knowledge of the actuator input output characteristics. We will address the problems of stability analysis and disturbance tolerance/rejection assessment. For the closedloop system under a given anti-windup compensation gain and in the absence of disturbance, we will establish the conditions under which an ellipsoid is contractively invariant. The determination of the largest such ellipsoid for use as an estimate of the domain of attraction is then formulated and solved as an LMI optimization problem. The same optimization problem is then adapted for the design of the anti-windup by

64

Haijun Fang and Zongli Lin

viewing the anti-windup gain as an additional design parameter. As the optimization problem with this additional free parameter is no longer an LMI problem, an iterative LMI algorithm is derived for its solution. In the presence of disturbances, the goal is to design an anti-windup compensation gain to improve the system disturbance tolerance and rejection capabilities, while ensuring a certain size of the stability region. As with the design for stability, we will start with the assessment of the disturbance tolerance and rejection capabilities of the system under a given anti-windup gain by formulation and solution of LMI optimization problems. The design of the anti-windup gain is again carried out by allowing the anti-windup gain matrix in these optimization problems as an additional free parameter and solving them via iterative LMI algorithms. The remainder of this chapter is organized as follows. Section 2 gives the problem statements and some necessary preliminaries. The analysis of the stability and disturbance tolerance/rejection capabilities is carried out in Section 3. Iterative LMI algorithms for solving the optimization problems involved in the design of the antiwindup gain are developed in Section 4. Numerical examples are used to demonstrate the effectiveness of these algorithms. In Section 5, the proposed design approach is applied to an experimental setup. Finally, conclusions are drawn in Section 6.

2 Problem Statement and Preliminaries We consider the following system, ⎧ ⎪ ⎪ x˙ = Ax + Bψ(u, t) + Kd, ⎨ y = C1 x, ⎪ ⎪ ⎩ z = C x,

(1)

2

where x ∈ Rn is the system state, u ∈ R is the control input, y ∈ Rm is the measurement output, z ∈ Rp is the controlled output, d ∈ Rq is the external disturbance, with ||d||∞ ≤ α, for some α ≥ 0. The function ψ(u, t) is a saturation type function that represents the actuator input output characteristics and is not precisely known. The function ψ(u, t) is however assumed to reside in a generalized sector deﬁned by two piecewise linear concave functions ψ1 (u) and ψ2 (u),

Robust Anti-windup Design

⎧ ⎪ ki0 u, if u ∈ [0, bi1 ], ⎪ ⎪ ⎪ ⎪ ⎨ ψi1 (u) = ki1 u + ci1 , if u ∈ (bi1 , bi2 ), ψi (u) = .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎩ ψ (u) = k u + c , if u ∈ (b , ∞), iNi iNi iNi iNi

65

(2)

where for each i ∈ [1, 2], ki0 > ki1 > ki2 > · · · > kiNi , ki(Ni −1) > 0 (see Fig. 2).

v v = k i0 u

ψ i1(u )

ψ i2(u ) ψ i3(u )

ψi(u )

ki0v3 k i0v2 k i0v1

o

bi1 = v1 v 2

v3

bi2

b i3

u

Fig. 2. Concave piecewise linear functions, ψi (u), i ∈ [1, 2].

We will assume that a linear dynamic compensator of the form x˙ c = Ac xc + Bc y, xc (0) = 0 ∈ Rnc , u = Cc xc + Dc y

(3)

has been designed that stabilizes the system (1) in the absence of actuator nonlinearity, i.e., φ(u, t) ≡ u, and meets the performance speciﬁcation. The anti-windup compensation we consider in this chapter is a correction term Ec (ψ(u, t) − u) in the above pre-designed dynamic output feedback law. The output feedback compensator with the anti-windup compensation thus takes the following form, x˙ c = Ac xc + Bc y + Ec (ψ(u, t) − u), xc (0) = 0, u = Cc xc + Dc y.

(4)

The closed-loop system (1) under the output feedback compensator (4) can be written as,

66

Haijun Fang and Zongli Lin

⎧ ˆx + B(ψ(u, ˆ ˆ ⎪ ˆ˙ = Aˆ t) − u) + Kd, ⎪ ⎨x u = Fˆ x ˆ, ⎪ ⎪ ⎩z = C 0 x ˆ,

(5)

2

where x ˆ= and

Aˆ =

A + BDc C1 BCc Bc C1

Ac

ˆ= ,B

x

xc

B

Ec

,

ˆ = , Fˆ = Dc C1 Cc , K

K 0

.

With the control input substituted, the system (5) can be readily rewritten as ⎧ ⎨x ˆ Fˆ )ˆ ˆ Fˆ x, t) + Kd, ˆ ˆ˙ = (Aˆ − B x + Bψ( ⎩ z = C2 0 x ˆ.

(6)

We will propose design of the anti-windup gain Ec , in the presence and in the absence of the external disturbance, respectively. In the absence of the disturbance, we will design an anti-windup compensation gain Ec such that the closed-loop system has a large domain of attraction. In the presence of the external disturbance, we will design an anti-windup compensation gain to improve the tolerance and rejection capabilities of the closed-loop system. We will measure the disturbance tolerance capability by the bound α on disturbances, under which any trajectory of the closed-loop system starting from a given set of initial conditions will remain bounded. An alternative measurement of the disturbance tolerance capability is the size of the maximum ellipsoid that remains invariant under the inﬂuence of disturbances bounded in magnitude by a given number α. We will measure the disturbance rejection capability by the bound on the l∞ norm of the controlled output z. To achieve the above design objectives by the approach summarized in Section 1, we need to recall some preliminaries from our earlier work [4, 8]. Given a positive deﬁnite matrix P ∈ R(n+nc )×(n+nc ) and a scalar ρ > 0, denote

ˆT P x ˆ≤ρ , ε(P, ρ) := x ˆ ∈ Rn+nc , x which is an ellipsoid in the state space Rn+nc . For a vector H ∈ R1×(n+nc ) , denote L(H) := x ˆ ∈ Rn+nc ,

|H x ˆ| ≤ 1 ,

Robust Anti-windup Design

67

which is a subset of the state space where the signal H x ˆ is bounded by 1. The convex hull of a given set of vectors x1 , x2 , · · · , xl , is denoted as co{x1 , x2 , · · · , xl }. Given a set XR ∈ R(n+nc )×(n+nc ) and a β > 0, the set βXR is deﬁned as x, βXR := {βˆ

∀ˆ x ∈ XR } .

The following theorem, adapted from [8], forms the basis for the design of antiwindup compensation gain in the absence of external disturbances. Theorem 1. Consider the system (6) in the absence of the disturbance d ≡ 0, and let the positive deﬁnite matrix P ∈ R(n+nc )×(n+nc ) be given. If there exist matrices ˆ ij ∈ R1×(n+nc ) , j ∈ [1, Ni ], i ∈ [1, 2], such that, for j ∈ [1, Ni ], i ∈ [1, 2], H ˆ Fˆ )T P + P (Aˆ − B ˆ Fˆ ) < 0, ˆ Fˆ + ki0 B ˆ Fˆ + ki0 B (Aˆ − B (7) ˆ Fˆ + B ˆH ˆ ij )T P + P (Aˆ − B ˆ Fˆ + B ˆH ˆ ij ) < 0, (Aˆ − B # "ˆ Hij −kij Fˆ , then the ellipsoid ε(P, 1) is contractively invariant, and ε(P, 1) ⊂ L cij and thus is an estimate of the domain of attraction. The next two theorems are adapted from [4] and are needed in the design of antiwindup compensation gain in the presence of external disturbances. Theorem 2. Consider the system (6), and let the positive deﬁnite matrix P ∈ ˆ ij ∈ R1×(n+nc ) , j ∈ [1, Ni ], i ∈ R(n+nc )×(n+nc ) be given. If there exist matrices H [1, 2] and a positive number η, such that, for j ∈ [1, Ni ], i ∈ [1, 2], ˆ Fˆ )T P +P (Aˆ − B ˆ Fˆ ) + 1 P K ˆ Fˆ + ki0 B ˆK ˆ T P + ηαP < 0, ˆ Fˆ + ki0 B (Aˆ − B η (8) ˆ Fˆ + B ˆH ˆ ij ) + 1 P K ˆK ˆ T P + ηαP < 0, ˆ Fˆ + B ˆH ˆ ij )T P +P (Aˆ − B (Aˆ − B η # "ˆ Hij −kij Fˆ , then the ellipsoid ε(P, α) is an invariant set. and ε(P, α) ⊂ L cij Theorem 3. Consider the system (6), and let P ∈ R(n+nc )×(n+nc ) be a given positive deﬁnite matrix and ζ > 0 be a given positive scalar. If there exist matrices ˆ ij ∈ R1×(n+nc ) , j ∈ [1, Ni ], i ∈ [1, 2] and a positive number η, such that, for H j ∈ [1, Ni ], i ∈ [1, 2], ⎧ ˆ Fˆ + ki0 B ˆ Fˆ )T P + P (Aˆ − B ˆ Fˆ ) + 1 P K ˆ Fˆ + ki0 B ˆK ˆ T P + ηαP < 0, ⎪ (Aˆ − B ⎪ η ⎪ ⎪ ⎨ (Aˆ − B ˆ Fˆ + B ˆH ˆ ij ) + 1 P K ˆ ˆ T P + ηαP < 0, ˆ Fˆ + B ˆH ˆ ij )T P + P (Aˆ − B η K ζ2 ⎪ C2T ⎪ ⎪ ⎪ C2 0 ≤ P , ⎩ α 0 (9)

68

Haijun Fang and Zongli Lin

and ε(P, α) ⊂ L

"

ˆ ij −kij F H cij

# . Then the maximal l∞ norm of the system output z is

less than or equivalent to ζ.

3 System Analysis under a Given Anti-windup Gain In this section, we assume that the anti-windup compensation gain Ec is given. The problems of estimating the domain of attraction in the absence of disturbances and disturbance tolerance/rejection capabilities are formulated and solved as optimization problems with LMI constraints. The estimation of the domain of attraction: Based on Theorem 1, the estimation of the domain of attraction of the system (6) in the absence of the disturbance can be formulated into the following optimization problem, max

ˆ ij ,j∈[1,Ni ],i∈[1,2] P >0,H

β,

(10)

s.t. a) βXR ⊂ ε(P, 1), b) Inequalities (7), i c) ε(P, 1) ⊂ ∩N j=1 L

$

ˆ ij − kij Fˆ H cij

% ,

where XR is a shape reference which controls the shape of the resulting ellipsoid. The shape reference set XR could be either an ellipsoid ε(R, 1), R ∈ R(n+nc )×(n+nc ) , or a polyhedron XR = co {ˆ x1 , x ˆ2 , · · · , x ˆl } ,

(11)

where x ˆi ∈ R(n+nc )×(n+nc ) , i ∈ [1, l]. ˆ ij Q, j ∈ [1, Ni ], i ∈ To solve the optimization problem (10), let Q = P −1 , Yˆij = H [1, 2] and γ = β12 . Then, the condition (7) is equivalent to ⎧ ˆ ˆˆ ˆˆ T ˆˆ ˆ ˆˆ ⎪ ⎪ ⎨ Q(A − B F + ki0 B F ) + (A − B F + ki0 B F )Q < 0, ˆ Fˆ )T + (Aˆ − B ˆ Fˆ )Q + (B ˆ Yˆij )T + B ˆ Yˆij < 0, Q(Aˆ − B ⎪ ⎪ ⎩ j ∈ [1, N ], i ∈ [1, 2]. i

And the condition c) is equivalent to

(12)

Robust Anti-windup Design

⎡ ⎣

1 (Yˆij −kij Fˆ Q)T cij

Yˆij −kij Fˆ Q cij

Q

69

⎤ ⎦ > 0, j ∈ [1, Ni ], i ∈ [1, 2].

If XR = ε(R, 1), the condition a) is equivalent to γR I > 0. I Q

(13)

(14)

Thus, the optimization problem (10) can be transformed into the following LMI optimization problem, min

Q>0,Yˆij ,j∈[1,Ni ],i∈[1,2]

γ,

(15)

s.t. (14), (12), (13). On the other hand, if XR = co {ˆ x1 , x ˆ2 , · · · , x ˆl }, then the condition a) is equivalent to

γ x ˆTi x ˆi Q

> 0, i ∈ [1, l],

(16)

and the optimization problem (10) is equivalent to the following LMI optimization problem, min

Q>0,Yˆij ,j∈[1,Ni ],i∈[1,2]

γ,

(17)

s.t. (16), (12), (13). Disturbance tolerance: Let the set of initial conditions be speciﬁed by an ellipsoid ε(S, 1), S ∈ R(n+nc )×(n+nc ) . The disturbance tolerance capability of the closed-loop system can be measured by the maximum level of the disturbances, α, under which all trajectories of the closed-loop system starting from inside ε(S, 1) remain bounded. In view of Theorem 2, the determination of such a maximum value of α can be formulated into the following constrained optimization problem, max

ˆ ij ,j∈[1,Ni ],i∈[1,2],η>0 P >0,H

α,

(18)

s.t. a) ε(S, 1) ⊂ ε(P, α), b) Inequalities (8), i c) ε(P, α) ⊂ ∩N j=1 L

$

ˆ ij − kij Fˆ H cij

% .

70

Haijun Fang and Zongli Lin

ˆ ij Q, j ∈ [1, Ni ], i ∈ To solve this optimization problem, let Q = αP −1 , Yˆij = H [1, 2] and ηˆ = ηα. Then the constraint a) is equivalent to αS αI > 0, αI Q

(19)

and the condition (8) is equivalent to ⎧ ˆ−B ˆ Fˆ + ki0 B ˆ Fˆ )T + (Aˆ − B ˆ Fˆ )Q + ηˆQ αK ˆ ˆ Fˆ + ki0 B ⎪ Q( A ⎪ ⎪ ≤ 0, ⎪ ⎪ ˆT ⎪ α K −ˆ η ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ T T ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Q(A − B F ) + (A − B F )Q + B Yij + (B Yij ) + ηˆQ αK ⎪ ≤ 0, ⎪ ⎪ ˆT ⎪ αK −ˆ η ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ j ∈ [1, N ], i ∈ [1, 2]. i

(20)

Thus, the optimization problem (18) is equivalent to max

Q>0,Yˆij ,j∈[1,Ni ],i∈[1,2],ˆ η >0

α,

(21)

s.t. (19), (20), (13). For a ﬁxed value of ηˆ, all constraints of the problem (21) are LMIs. By sweeping ηˆ over the interval (0, ∞), we can obtain the global maximum of α. An alternative measurement of the disturbance tolerance capability is the size of the largest ellipsoid that remains invariant under the inﬂuence of disturbances whose magnitudes are bounded by a given positive number α. In view of Theorem 2, the size of the largest such ellipsoid can be determined by solving the following constrained optimization problem, max

ˆ ij ,j∈[1,Ni ],i∈[1,2],η>0 P >0,H

β,

(22)

s.t. a) βXR ⊂ ε(P, α), b) Inequalities (8), i c) ε(P, α) ⊂ ∩N j=1 L

$

ˆ ij − kij Fˆ H cij

% .

ˆ ij Q, the condition b) in (22) is Under the change of variable, Q = P −1 , Yˆij = H equivalent to

Robust Anti-windup Design

71

⎧ 1 ˆ ˆT ˆ ˆˆ ˆˆ T ˆˆ ˆ ˆˆ ⎪ ⎪ ⎨ Q(A − B F + ki0 B F ) + (A − B F + ki0 B F )Q + η K K + ηαQ < 0, ˆ Yˆ T +ki0 B ˆ Yˆij + 1 K ˆK ˆ T +ηαQ < 0, (23) ˆ Fˆ )+(Aˆ − B ˆ Fˆ )T Q+ki0 B Q(Aˆ − B ij η ⎪ ⎪ ⎩ j ∈ [1, N ], i ∈ [1, 2], i

and the constraint c) is equivalent to ⎡ ⎣

Yˆij −kij Fˆ Q 1 α cij (Yˆij −kij Fˆ Q)T Q cij

⎤ ⎦ > 0, j ∈ [1, Ni ], i ∈ [1, 2].

(24)

Let XR be an ellipsoid ε(R, 1), R ∈ R(n+nc )×(n+nc ) . By the additional change of variable, γ =

1 , β2

the optimization problem (22) can be transformed into, min

Q>0,Yˆij ,j∈[1,Ni ],i∈[1,2],η>0

s.t.

γαR I I

Q

γ,

(25)

> 0,

Inequalities (23), (24). Similarly, if XR is a polyhedron deﬁned in (11), then the problem (22) is equivalent to min

Q>0,Yˆij ,j∈[1,Ni ],i∈[1,2],η>0

s.t.

γα x ˆi x ˆTi Q

γ,

(26)

> 0,

Inequalities (23), (24). In both optimization problems (25) and (26), all constrains are LMIs for a ﬁxed value of η. Therefore, the minimum of γ, hence the maximum of β, can be achieved by sweeping the value of η over the interval (0, ∞). Disturbance rejection: We measure the disturbance rejection capability of a closed-loop system by the maximal l∞ norm of its controlled output within a given set of initial conditions, say an ellipsoid ε(S, 1). In view of Theorem 3, this problem can be formulated into the following optimization problem,

72

Haijun Fang and Zongli Lin

max

ˆ ij ,j∈[1,Ni ],i∈[1,2],η>0 P >0,H

(27)

ξ,

s.t. a) ε(S, 1) ⊂ ε(P, α), b) Inequalities (9),

$

i c) ε(P, α) ⊂ ∩N j=1 L

ˆ ij − kij Fˆ H cij

% .

ˆ ij Q, ∀j ∈ [1, Ni ], i ∈ [1, 2], the Under the change of variable, Q = P −1 , Yˆij = H optimization problem (27) is equivalent to min

Q>0,Yˆij ,j∈[1,Ni ],i∈[1,2],η>0

s.t. a)

b)

(28)

S I I αQ

> 0,

ξ,

C2 0 Q

C2 0

≤

ζ2 I, α

c) Inequalities (23), (24). Again, all the constraints in (28) are LMIs for a ﬁxed value of η. The global minimum of ζ can be obtained by sweeping η over the interval (0, ∞).

4 Anti-windup Compensation Gain Design In this section, we will propose algorithms for designing the anti-windup compensation gain to improve system stability and disturbance tolerance/rejection capabilities. On the basis of optimization problems proposed in Section 3, the design of antiwindup compensation gain can be formulated into optimization problems by viewing Ec as an additional variable. However, with Ec as an additional variable, constraints in these optimization problems can no longer be transformed into LMIs when P , Ec ˆ ij , j ∈ [1, Ni ], i ∈ [1, 2], are all variables. We will develop iterative algorithms and H to solve these optimization problems. Design for a large domain of attraction: We will ﬁrst consider the design of Ec to achieve a large domain of attraction for the closed-loop system. This involves the optimization problem (10), where constraint (7), with Ec as an additional variable, cannot be transformed into an LMI. Following the idea of [3], let

Robust Anti-windup Design

P=

P1 P12

˜= ,B

T

P12 P22

(n+nc )×(n+nc )

ˆ B

0

, P˜2 =

(n+nc )×1

P12 P22

73

. (29) (n+nc )×nc

Then, (7) can be rewritten as # " ⎧ ˆ + P AˆT + (ki0 − 1) P B ˜ T P + P˜2 Ec Fˆ + Fˆ E T P˜ T < 0, ˜ Fˆ + Fˆ T B ⎪ AP c 2 ⎪ ⎪ ⎪ ⎨ ˆ ˜ TP ˜ Fˆ − H ˆ ij ) − (Fˆ − H ˆ ij )T B AP + P AˆT − P B( (30) ⎪ ˜2 Ec (Fˆ − H ˆ ij ) − (Fˆ − H ˆ ij )T EcT P˜ T < 0, ⎪ − P 2 ⎪ ⎪ ⎩ j ∈ [1, Ni ], i ∈ [1, 2], which are LMIs in P1 and Ec . Thus, for a given P˜2 , the optimization problem (10) can be adapted to the following one for the design of Ec : max β,

(31)

P1 >0,Ec

s.t. a) P > 0, b) βXR ⊂ ε(P, 1), c) Inequalities (30), %T $ ˆ ij − kij Fˆ ˆ ij − kij Fˆ H H , d) P ≥ cij cij Let γ =

1 . β2

j ∈ [1, Ni ], i ∈ [1, 2].

If XR is an ellipsoid ε(R, 1), constraint b) in (31) is equivalent to P ≤ γR.

(32)

On the other hand, if XR is a polyhedron as deﬁned in (11), then b) in (31) is equivalent to γ − xTi P xi ≥ 0,

i ∈ [1, l].

(33)

Consequently, the optimization problem (31) is equivalent to min γ,

(34)

P1 >0,Ec

s.t. a) P > 0, b) (32) or (33), c) Inequalities (30), %T $ ˆ ij − kij Fˆ ˆ ij − kij Fˆ H H , d) P ≥ cij cij which, for a given P˜2 , is an LMI problem.

j ∈ [1, Ni ], i ∈ [1, 2],

74

Haijun Fang and Zongli Lin

Motivated by the above development, we propose the following iterative algorithm for the design of an Ec that results in a large domain of attraction. Algorithm 1 (Design of Ec for a large domain of attraction): Step 1. Let XR be the shape reference set and set Ec = 0. If XR is an ellipsoid ε(R, 1), solve the optimization problem (15). If XR is a polyhedron as deﬁned in (11), solve the optimization problem (17). Denote the solution as γ0 , Q and Yˆij , j ∈ [1, Ni ], i ∈ [1, 2]. Set XR = √1 XR . γ0

Step 2. Set the initial value for Ec , and let k = 1 and γopt = 1. Step 3. Using the value of Ec obtained in the previous step. If XR is an ellipsoid ε(R, 1), solve the problem (15) for γk , Q, Yˆij , j ∈ [1, Ni ], i ∈ [1, 2]. If XR is a polyhedron as deﬁned in (11), solve the optimization problem (17) for γ , Q, Yˆij , j ∈ [1, Ni ], i ∈ [1, 2]. If γ > 1, go to Step 2. Else, let γ = γ γ , k

XR =

k

√1 XR , P γk

−1

=Q

opt

k opt

ˆ ij = Yˆij P, j ∈ [1, Ni ], i ∈ [1, 2]. ,H

Step 4. If 1 − γk < δ, a pre-determined tolerance, go to Step 6. ˆ ij , j ∈ [1, Ni ], i ∈ [1, 2], P12 , P22 obtained in Step 3, Step 5. Using the value of H solve the LMI problem (34). Let k = k + 1, go to Step 3. Step 6. Let βopt = √γ1

opt

. The solution Ec is a feasible solution. End of the algo-

rithm. Remark 1. To prevent the anti-windup compensation gain Ec to reach an undesirably large value, an additional constraint can be incorporated into theoptimization prob

lems involved in the algorithm above. More speciﬁcally, Ec = e1 , e2 , · · · , enc the following constraint can be imposed: −φmin ≤ ei ≤ φmax ,

T

,

i ∈ [1, nc ],

where φmin and φmax are the bounds imposed on the anti-windup compensation gain. Example. Consider the system (1), where 0.1 −0.1 5 A= , B= , C1 = 1 0 , 0.2 −3 0

C2 = 1 1 ,

K=

1 1

,

Robust Anti-windup Design

75

and ψ(u, t) is a locally Lipschitz function that reside in a generalized sector bounded by ψ1 (u) = u and ψ2 (u) of the form (2) with N = 5 and (k20 , k21 , k22 , k23 , k24 , k25 ) = (0.7845, 0.3218, 0.1419, 0.0622, 0.0243, 0), (c21 , c22 , c23 , c24 , c25 ) = (0.4636, 0.8234, 1.0625, 1.2517, 1.4464). Such a function ψ(u) = tan−1 (u) is depicted in Fig. 3) in a dashed curve. The two solid curves in the ﬁgure are the bounding function ψ1 (u) and ψ2 (u). 2

v

v=u v=tan−1(u)

1.5

v=ψ2(u)

1

0.5

u 0

o −0.5

−1

−1.5

−2 −10

−8

−6

−4

−2

0

2

4

6

8

10

Fig. 3. The function ψ(u) = tan−1 (u) in the generalized sector deﬁned by ψ1 (u) and ψ2 (u) in the example.

Also, assume that a dynamic controller of the form (3) has been designed with −171.2 27.06 −598.2 Ac = , Bc = , Cc = 0.146 0.088 , Dc = 0, −68 −626.8 −4.567 We would like to design an Ec that results in a large domain of attraction. Each element of this Ec is restricted to bebetween −100 and 100. Using Algorithm 1, we 1 choose the initial value of Ec to be , and obtain 1

76

Haijun Fang and Zongli Lin

Ec =

P1 = 10−3

,

0.8102

with

−0.6946

0.1681 −0.0054 −0.0054

0.0002

.

Shown in Figs. 4 and 5 are the cross section of the ellipsoid ε(P, 1) at xc = 0 and V (ˆ x(t)) = x ˆT P x ˆ along several trajectories that start from the boundary of the ellipsoid ε(P, 1). In the simulation, ψ(u) = tan−1 (u). The smaller ellipsoid also shown in Fig. 4 is the cross section of the invariant ellipsoid at xc = 0, which is obtained for the closed-loop system in the absence of the anti-windup compensation, as computed in Step 1 of Algorithm 1.

5000

4000

3000

2000

x2

1000

0

−1000

−2000

−3000

−4000

−5000 −200

−150

−100

−50

0

50

100

150

200

x1

Fig. 4. The cross section of the ellipsoid ε(P, 1) at xc = 0.

Design for large disturbance tolerance capability: We now consider the design of an anti-windup compensation gain Ec to achieve large disturbance tolerance capability. We will ﬁrst consider the case when the disturbance tolerance capability is measured by the largest bound on the disturbance

Robust Anti-windup Design

77

1

0.9

0.8

0.7

V

0.6

0.5

0.4

0.3

0.2

0.1

0

0

5

10

15

20

25

30

t

Fig. 5. V (ˆ x(t)) along several trajectories that start from the boundary of the ellipsoid ε(P, 1).

under which the closed-loop system trajectories starting from a given set of initial conditions ε(S, 1), S ∈ Rn+nc )×(n+nc ) , remain bounded. The design of such an Ec can be carried out by solving the optimization problem (18) with Ec viewed as an additional variable. To develop an algorithm for the solution of this optimization problem, we ﬁrst have that constraint (8) is equivalent to # " ⎤ ⎧⎡ ˆ +P AˆT +(ki0 −1) P B ˜ T P + P˜2 Ec +E T P˜ T + ηˆP P K ˆ ˜ Fˆ + Fˆ T B ⎪ AP c 2 ⎪ ⎪ ⎣ ⎦ < 0, " #T ⎪ ⎪ ⎪ ˆ ⎪ P K −ˆ η γ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎡ ⎤ ˆ TP ˆ +P AˆT −P B( ˜ Fˆ − H ˆ ij )−(Fˆ − H ˆ ij )T B (35) AP ˆ⎥ ⎪ P K ⎪ ⎢ ⎪ ˆ ij )−(Fˆ − H ˆ ij )T EcT P˜ T + ηˆP ⎪ ⎥ < 0, ⎢ − P˜2 Ec (Fˆ − H ⎪ 2 ⎪ ⎦ ⎣ " # ⎪ T ⎪ ⎪ ˆ ⎪ −ˆ ηγ PK ⎪ ⎪ ⎪ ⎩ j ∈ [1, Ni ], i ∈ [1, 2], ˜ and P˜2 are as deﬁned in (29). Also, the constraint where γ = α1 , ηˆ ="ηα, and P1#, B i ε(P, α) ⊂ ∩N j=1 L

ˆ ij −kij Fˆ H cij

is equivalent to

78

Haijun Fang and Zongli Lin

⎡ ⎣

γ ˆ ij −kij Fˆ )T (H cij

ˆ ij −kij Fˆ H cij

⎤ ⎦ > 0,

P

j ∈ [1, Ni ], i ∈ [1, 2].

(36)

We observe that the inequalities in (35) and (36) are all linear in the variables P1 and Ec . This motivates us to develop the following algorithm for the design of an Ec that leads to a large disturbance tolerance capability, as measured by the level of allowable disturbances without causing unbounded trajectories. Algorithm 2 (Design of Ec for a high level of allowable disturbance): Step 1. Set Ec = 0, and solve the optimization problem (21). Denote the solution as α0 . Step 2. Set the initial value for Ec . Let k = 1, αopt = α0 . Step 3. Using the value of Ec obtained in the previous step, solve the optimization problem (21). Denote the solution as αk , Q and Yˆij , j ∈ [1, Ni ], i ∈ [1, 2]. Then, ˆ ij = Yˆij P, j ∈ [1, Ni ], i ∈ [1, 2]. If αk < αopt , go to Step 2). P = αk Q−1 , H Else if αk /αopt < δ, a pre-determined tolerance, let αopt = αk and go to Step 5. Else, let αopt = αk and k = k + 1. ˆ ij , j ∈ [1, Ni ], i ∈ [1, 2], P12 , P22 obtained in Step 3, Step 4. Using the value of H solve the following optimization problem min

P1 >0,Ec ,ˆ η

γ,

(37)

s.t. a) P > 0, b) P < αopt S, c) Inequalities (35) and (36). Denote the solution as αk =

1 γ,

P1 and Ec . If αk /αopt < δ, a pre-determined

tolerance, let αopt = αk and go to Step 5. Else, let αopt = αk and k = k + 1. Go to Step 3. Step 5. The estimate of the largest disturbance bound is αopt , with a corresponding Ec . End of algorithm. Remark 2. In Steps 2-4 of Algorithm 2, ∀k > 1, it is easy to observe that

αk αopt

> 1.

Thus, the pre-determined tolerance δ should be a number which is slightly larger than 1. The algorithm ends when the difference between αk and αopt is small enough.

Robust Anti-windup Design

79

Remark 3. In Algorithm 2, the optimization problems (21) and (37) are LMI problems for each ﬁxed value of ηˆ. Thus, both optimization problems can be solved by sweeping ηˆ over the interval (0, ∞). Example (continued). Let T S = I. Following Algorithm 2, we choose the initial value of Ec as −5 −20 , and obtain Ec =

−96.0117 −82.9739

,

with the corresponding αmax = 6.135 and 0.0626 −0.0021 . P1 = −0.0021 0.0001 Shown in Figs. 6 and 7 are the cross section of the ellipsoid ε(P, αmax ) at xc = 0 and V (ˆ x(t)) along several trajectories under the inﬂuence of a disturbance bounded by α = 6.0 sin(t). All trajectories start from the boundary of the ellipsoid ε(P, αmax ). 500

400

300

200

x2

100

0

−100

−200

−300

−400

−500 −20

−15

−10

−5

0

5

10

15

x1

Fig. 6. The cross section of the ellipsoid ε(P, αmax ) at xc = 0.

20

80

Haijun Fang and Zongli Lin 1

0.9

0.8

0.7

V

0.6

0.5

0.4

0.3

0.2

0.1

0

0

2

4

6

8

10

12

14

16

18

20

t

Fig. 7. V (ˆ x(t)) along several trajectories that start from the boundary of the ellipsoid ε(P, α).

Next, we consider the situation when the disturbance tolerance capability is measured by the size of the largest ellipsoid that remain invariant under the inﬂuence of disturbances whose magnitudes are bounded by a given number α. This problem can be formulated into the optimization (22) with Ec being an additional variable. Unlike (22) with a given Ec , the optimization problem with Ec as an additional variable is no longer transformable into an LMI problem and an effective algorithm is needed for its solution. To this end, we note that, by Schur complement, (8) is equivalent to ⎧⎡ # " ⎤ T T ˜T T ˜T ˆ ˆ ˆ ˜ ˆ ˆ ˜ ⎪ AP +P A +(ki0 −1) P B F + F B P + P2 Ec +Ec P2 +ηαP P K ⎪ ⎪ ⎪ ⎣ ⎦ < 0, " #T ⎪ ⎪ ⎪ ˆ ⎪ PK −η ⎪ ⎪ ⎪ ⎨ (38) ⎡ ⎤ ⎪ ˆ +P AˆT −P B( ˆ ij )−(Fˆ −kij H ˆ ij )T B ˜ TP ˜ Fˆ −kij H ⎪ AP ⎪ ˆ⎥ ⎪ PK ⎢ ⎪ ⎪ T T ˜T ˆ ˆ ˜ ˆ ˆ ⎢ ⎥ < 0, ⎪ − P2 Ec (F −kij Hij )−(F −kij Hij ) Ec P2 +ηαP ⎪ ⎦ ⎪⎣ " #T ⎪ ⎪ ⎩ ˆ −η PK ˜ where " P1 , P˜2 and # B are as deﬁned in (29). Also, the constraint ε(P, α) ⊂ ˆ ˆ Hij −kij F i is equivalent to ∩N j=1 L cij

Robust Anti-windup Design

⎡ ⎣

ˆ ij −kij Fˆ H 1 α cij ˆ ij −kij Fˆ )T (H P cij

81

⎤ ⎦ > 0,

j ∈ [1, Ni ], i ∈ [1, 2].

(39)

It is easy to see that, for a ﬁxed value of η, inequalities (37) and (38) are LMIs in P1 and Ec . The above development motivates us to develop the following algorithm for the design of Ec that leads to a large invariance set in the presence of bounded disturbances. Algorithm 3 (Design of Ec for a large invariant set in the presence of bounded disturbances): Step 1. Let XR be the shape reference set and Ec = 0. If XR is an ellipsoid ε(R, 1), solve the optimization problem (25). If XR is a polyhedron as deﬁned in (11), solve the optimization problem (26). Denote the solution as γ0 . Set XR =

1 √ γ XR .

Step 2. Set the initial value for Ec . Set k = 1 and γopt = 1. Step 3. If XR is an ellipsoid ε(R, 1), solve the optimization problem (25) for γ , Q, Yˆij , j ∈ [1, Ni ], i ∈ [1, 2]. If XR is a polyhedron as deﬁned in (11), solve k

the optimization problem (26) for γk , Q, Yˆij , j ∈ [1, Ni ], i ∈ [1, 2]. If γk > 1, ˆ ij = Yˆij P, j ∈ go to Step 2. Else let γopt = γk γopt , XR = √1γ XR , P = Q−1 , H k

[1, Ni ], i ∈ [1, 2].

Step 4. If 1 − γopt < δ, a pre-determined tolerance, go to Step 6. Else, let k = k + 1. ˆ ij , j ∈ [1, Ni ], i ∈ [1, 2], P12 , and P22 obtained in Step Step 5. Using the value of H 3, solve the following LMI problem, min

P1 >0,Ec ,η

γ,

s.t. a) P > 0, b) (32) or (33), c) Inequalities (38) and (39). Denote the solution as γk , Ec and P1 . Let γopt = γk and XR =

√1 XR . γk

Go to

Step 3. Step 6. Let βopt =

1 . γ2opt

The solution Ec is a feasible solution. End of the algorithm.

82

Haijun Fang and Zongli Lin

T Example (continued). Let α = 4, and the initial value of Ec be −1 −10 . Following Algorithm 3, we arrive at −99.7956 0.2078 −0.0078 Ec = , P1 = . −99.0753 −0.0078 0.0005 Shown in Figs. 8 and 9 are the cross section of the maximized ellipsoid ε(P, α) at x(t)) along several trajectories starting from the boundary of the xc = 0, and V (ˆ ellipsoid ε(P, α) under the inﬂuence of a disturbance d(t) = 4 sin(t). Also shown in Fig. 8 for comparison is the cross section of the corresponding maximized ellipsoid when Ec = 0, which is much smaller than the ellipsoid ε(P, α) under the obtained anti-windup gain. Design for disturbance rejection: Finally, we consider the design of Ec to increase the disturbance rejection ability of the closed-loop system. We will do this by minimizing the maximum l∞ norm of the system output z within a given set of initial conditions, say, ε(S, 1), and under the inﬂuence of disturbances whose magnitudes are bounded by an α > 0. The 150

100

x2

50

0

−50

−100

−150 −8

−6

−4

−2

0

2

4

6

8

x1

Fig. 8. The cross section of the maximized invariant ellipsoid ε(P, α) at xc = 0.

Robust Anti-windup Design

83

1

0.9

0.8

0.7

V

0.6

0.5

0.4

0.3

0.2

0.1

0

0

2

4

6

8

10

12

14

16

18

20

t

Fig. 9. V (ˆ x(t)) along several trajectories starting from the boundary of ε(P, α).

underlining optimization problem will be (28), with Ec as an additional variable. The following is an algorithm for the solution of this optimization problem. Algorithm 4 (Design of Ec for minimizing the l∞ norm of the output): Step 1. Set Ec = 0, and solve the optimization problem (28). Denote the solution as ζ0 . Step 2. Set the initial value for Ec . Let k = 1 and ζopt = ζ0 . Step 3. Using the value of Ec obtained in the previous step, solve the optimization problem (28). Denote the solution as ζk , Q and Yˆij , j ∈ [1, Ni ], i ∈ [1, 2]. Let ˆ ij = Yˆij P, j ∈ [1, Ni ], i ∈ [1, 2]. If ζk > ζopt , go to Step 2. Else, P = Q−1 , H if

ζk ζopt

> δ, a predetermined tolerance, let ζopt = ζk , and go to Step 5, else, let

ζopt = ζk and k = k + 1. ˆ ij , j ∈ [1, Ni ], i ∈ [1, 2], P12 and P22 obtained in Step Step 4. Using the value of H 3, solve the following optimization problem,

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Haijun Fang and Zongli Lin

max

P1 >0,Ec ,η

γ,

s.t. a) P > 0, b) P < αS, c) Inequalities (38) and (39), C2T 1 d) γ C2 0 ≤ P. α 0 Denote the solution as ζk =

1 √ γ,

P1 and Ec . If

ζk ζopt

> δ, a predetermined

tolerance, let ζopt = ζk , and go to Step 5). Else let ζopt = ζk and k = k + 1, go to Step 3. Step 5. The maximal l∞ norm of the output z is less than ζopt . The solution Ec is a feasible solution. End of the algorithm. Remark 4. It is clear from the algorithm that ζk < ζopt . Hence the tolerance δ should be set as a positive number slightly smaller that 1. Example (continued). Let S = I and Tα = 4. Following Algorithm 4, we choose the initial value of Ec = and obtain the estimated bound of the −50 0 output z as 3.8368, with T the following anti-windup compensation gain Ec = −99.1173 −96.0495 . Shown in Fig. 10 is the response of the system output, under zero initial condition, to a disturbance d = 4sign(sin(t)). The two straight lines in the ﬁgure represent the estimated bound on the output z.

5 Application to an Inverted Pendulum on a Cart: In this section, we will consider the problem of stabilizing an inverted pendulum on a cart. This system was constructed at Carnegie Mellon University3 and is shown in Fig. 11. The parameters of the system are given in the following table. A linearized model for this system is given by, 3

http://www.library.cmu.edu/ctms/ctms/examples/pend/invpen.htm

Robust Anti-windup Design 4

3

2

z

1

0

−1

−2

−3

−4

0

2

4

6

8

10

12

14

16

t

Fig. 10. A trajectory of the output z.

m,I

θ

M F

x

Fig. 11. An inverted pendulum on a cart.

18

20

85

86

Haijun Fang and Zongli Lin Table 1. The parameters of the inverted pendulum and the cart. M mass of the cart

0.5kg

m mass of the pendulum

0.2kg

b friction of the cart

0.1 N/m/sec

l

length to pendulum center of mass 0.3 m 0.006 kgm2

I inertia of the pendulum F force applied to the cart x cart position coordinate θ pendulum angle from vertical

⎡ ⎤ ⎡ 0 1 x˙ 2 ⎢ ⎥ ⎢ )b ⎥ ⎢ 0 I(M−(I+ml ⎢x ¨ +m)+M ml2 ⎢ ⎥=⎢ ⎢ θ˙ ⎥ ⎢ 0 ⎣ ⎦ ⎣0 −mlb θ¨ 0 I(M +m)+M ml2 ⎡ 0 ⎢ I+ml2 ⎢ I(M +m)+M ml2 +⎢ ⎢ 0 ⎣

0

0

0

m2 gl2 I(M +m)+M ml2

0

1

−mgl(M +m) I(M +m)+M ml2

0

⎤

⎤⎡ ⎤ x ⎥⎢ ⎥ ⎥ ⎢ x˙ ⎥ ⎥⎢ ⎥ ⎥⎢θ⎥ ⎦⎣ ⎦ θ˙

⎥ ⎥ ⎥ψ(u), ⎥ ⎦

ml I(M +m)+M ml2

⎡ ⎤ x ⎥ ⎢ ⎢ x˙ ⎥ ⎥ y= 1100 ⎢ ⎢θ⎥, ⎣ ⎦ θ˙

(40)

where ψ(·) is deﬁned in the example in Section 4. The unit of x, ˙ x, θ˙ and θ are m/s, m, rad, and rad/s, respectively. Using the data in Table 1, (40) becomes ⎤ ⎤⎡ ⎤ ⎡ ⎡ ⎤ ⎡ 0 0 1.0000 0 0 x x˙ ⎥ ⎥⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ x˙ ⎥ ⎢ 1.8182 ⎥ ⎥ ⎢ 0 −0.1818 2.6727 ⎢x 0 ¨ ⎥ψ(u), ⎥ ⎢ ⎥+⎢ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎢ θ˙ ⎥ ⎢ 0 0 0 1.0000 ⎦ ⎣ θ ⎦ ⎣ 0 ⎥ ⎦ ⎣ ⎦ ⎣ θ¨

0 −0.4545 31.1818 ⎡ ⎤ x ⎥ ⎢ ⎢ x˙ ⎥ ⎥ y= 1100 ⎢ ⎢θ ⎥. ⎣ ⎦ θ˙

0

θ˙

4.5455

(41)

Robust Anti-windup Design

87

By neglecting the actuator nonlinearity, the following controller is designed such that the closed-loop system (41) is stabilized at the origin, ⎤ ⎤ ⎡ ⎡ −5.6316 −4.6316 0 0 5.6316 ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ 2.5073 ⎢ 12.1866 ⎥ 1.7726 −63.2776 −12.7837 ⎥ xc + ⎢ ⎥ x˙ c = ⎢ ⎢ −51.2478 −51.2478 ⎢ 51.2478 ⎥ y 0 1.0000 ⎥ ⎦ ⎦ ⎣ ⎣ −249.5222 −251.3589 −133.6939 −31.9592 286.2569 u = 8.0816 7.7776 −36.2727 −7.0310 xc . However, as shown in Fig. 12, in the presence of the actuator nonlinearity, the controller fails to stabilize the system at T x(0) = 0 0 0.01 0 .

200

0

−200

states of the system

−400

−600

−800

−1000

−1200

−1400

−1600

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t

Fig. 12. State trajectories of the closed-loop system (41) without anti-windup compensation, x(0) = [0 0 0.01 0]T .

By applying Algorithm 1, we obtain the following anti-windup compensation gain, Ec = 0.0031 1.8963 0.1473 6.0010 ,

88

Haijun Fang and Zongli Lin 0.8

0.6

states of the system

0.4

0.2

0

−0.2

−0.4

−0.6

−0.8

0

1

2

3

4

5

6

7

8

9

10

t

Fig. 13. State trajectories of the closed-loop system (41) with anti-windup compensation, x(0) = [0 0 0.06 0]T . 3

2

1

u and ψ(u)

0

−1

−2

−3

−4

−5

−6

0

1

2

3

4

5

6

7

8

9

10

t

Fig. 14. ψ(u)(solid line) and u(dotted line) of the closed-loop system (41) with anti-windup compensation, x(0) = [0 0 0.06 0]T .

Robust Anti-windup Design

89

under which the domain of attraction is signiﬁcantly enlarged. Shown in Figs. 13 and 14 are the closed-loop trajectories with T x(0) = 0 0 0.06 0 , and the corresponding control input, respectively.

6 Conclusion In this chapter, we have addressed the problem of anti-windup compensator design for linear systems with imprecise knowledge of the actuator input output characteristics. Several problems in the analysis and design of a control system with anti-windup compensation are considered. These problems are formulated into constrained optimization problems, for which numerical algorithms are proposed. Throughout the paper, numerical examples are used to illustrate the effectiveness of the proposed analysis and design procedure. Finally, the proposed anti-windup design was applied to an inverted pendulum on a cart experimental setup to demonstrate that a carefully designed anti-windup compensation gain leads to signiﬁcant improvement on the closed-loop stability.

References 1. Astrom KJ, Rundqwist L (1989) Integrator windup and how to avoid it. Proceedings of American Control Conference 1693-1698 2. Bernstein DS, Michel AN (1995) A chronological bibliography on saturating actuators. International Journal of Robust and Nonlinear Control 5:375-380 3. Cao Y, Lin Z, Ward DG (2002) An antiwindup approach to enlarging domain of attraction for linear systems subject to actuator saturation. IEEE Transactions on Automatic Control 47:140-145 4. Fang H, Lin Z, Shamash Y (2005) Disturbance tolerance and rejection for linear systems with saturation: disturbance attenuation using output feedback. Proceedings of the 2005 Joint IEEE Conference on Decision and Control and European Control Conference 82948299 5. Gomes JM, Tarbouriech S (2005) Antiwindup design with guaranteed regions of stability: an LMI-based approach. IEEE Transactions on Automatic Control 50:106-111

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Haijun Fang and Zongli Lin

6. Grimm G, Hatﬁeld J, Postlethwaite I, Teel AR, Turner MC, Zaccarian L (2003) Antiwindup for stable linear systems with input saturation: an LMI-based synthesis. IEEE Transactions on Automatic Control 30:1509-1525 7. Hu T, Lin Z (2001) Control Systems with Actuator Saturation: Analysis and Design. Birkh¨auser, Boston 8. Hu T, Huang B, Lin Z (2004) Absolute stability with a generalized sector condition. IEEE Transactions on Automatic Control 49:535-548 9. Kothare MV, Campo PJ, Morari M, Nett CN (1994) A uniﬁed framework for study of anti-windup designs. Automatica 30:1869-1883 10. Lin Z (1998) Low Gain Feedback. Lecture Notes in Control and Information Sciences 240, Springer-Verlag, London 11. Morabito F, Teel AR, Zaccarian L (2004) Nonlinear anti-windup design applied to eulerlagrange systems. IEEE Transactions on Robotics and Automation 20:526-537 12. Tarbouriech S, Garcia G (1997) Control of Uncertain Systems with Bounded Controls. Springer, London 13. Takaba K, Tomida Y (2000) Analysis of anti-windup control system based on LPV system representation. Proceedings of American Control Conference 1760-1765 14. Teel AR, Kapoor N (1997) The L2 anti-windup problem: Its deﬁnition and solution. Proceedings of European Control Conference 15. Turner MC, Postlethwaite I (2004) A new prospective on static and low order anti-windup synthesis. International Journal of Control 77:27-44 16. Wu F, Grigoriadis KM, Packard A (2000) Anti-windup controller design using linear parameter-varying control methods. International Journal of Control 73:1104-1114 17. Zaccarian L, Teel AR (2004) Nonlinear scheduled anti-windup design for linear systems. IEEE Transactions on Automatic Control 49:2055-2061

Design and Analysis of Override Control for Exponentially Unstable Systems with Input Saturations Adolf Hermann Glattfelder and Walter Schaufelberger Automatic Control Lab, ETH Z¨urich, Switzerland, [email protected], [email protected] Summary. The antiwindup and override design techniques have been developed in industrial design practice for controlling plants with input and output constraints. At present, various analysis and synthesis techniques are well established. These techniques have shown to be both effective and efﬁcient for plants of dominant low order, which are either asymptotically stable or at least marginally stable. However, for exponentially unstable plants with input saturations, the analysis for the standard antiwindup design reveals a strongly limited radius of attraction, which is often smaller than what is required from the speciﬁcations. In this contribution it is shown how the standard override technique can be used in a systematic way to extend the radius of attraction signiﬁcantly, such as to meet the speciﬁcations. The design is based on nonlinear stability analysis. The generic design approach developed here is applied to both benchmark cases.

1 Introduction In the last two decades the control of asymptotically stable LTI-plants or open integrator chains with input constraints in the form of actuator stroke s has been extensively investigated. Several analytical and computational approaches have been developed (see the introduction to this book).

92

Adolf Hermann Glattfelder and Walter Schaufelberger

Their current state as extended to the control of exponentially unstable plants is given in the contributions in this book. The main characteristic for this class of plants is a severely restricted basin of attraction (stability radius) of the closed loop system, which often does not cover the speciﬁcation pertaining to the particular application. One such approach is feedback for input constraints and feedback for output constraints. It has been developed in industrial control systems design and is routinely applied in practice. Basically the approach is an intuitive one. It uses simple but frequent cases as a starting point, and then expands the techniques to more complex situations. – This approach has been been extensively applied in [3]. There, suitable generic structures for both the and cases are described, and corresponding implementation variants often used in practice. It also prepares a framework for the analysis of their nonlinear stability properties, which is based on the . This leads to a simple, straightforward and transparent stability-based design. However the nonlinear loop performance is not addressed in a formal way there, but rather using established linear design methods in conjunction with simulations. However the many cases in [3] demonstrate that this produces results, which are quite acceptable from a practical point of view1 . Finally, the focus there clearly is on exponentially stable plants and on chained-openintegrator plants. The case of exponentially unstable plants has been mentioned (see sect. 6.4.2, ‘stability charts’), but not investigated in further detail. This shall be done here. The aim is to explore how far these basic methods carry in this particular area, and more speciﬁcally if they are able to extend the basin of attraction sufﬁciently in order to cover the typical speciﬁcations from practical cases. The basic approach is the same as for the non-exponentially-unstable plants. It is a two step design, mixing both analytic and intuitive elements. It starts from closed loop equilibrium at nominal operating conditions. First, small deviations are applied, such that all elements in the loop evolve within their linear range. Then a linear controller is designed which stabilizes the loop and yields the closed loop performance as speciﬁed. 1

Note that the relative strength and weakness regarding stability and performance properties are in contrast to the other available methods, and may be seen to complement them

Antiwindup and Override Control Solutions

93

Next, the deviations are increased until the actuator constraints are met, and the performance will deteriorate. The usual countermove is to augment the linear controller structure by a standard algebraic antiwindup feedback (awf), while the linear controller part is not changed (in order to conserve the small signal performance). This works quite well for plants of dominant ﬁrst order. But for plants of dominant higher order further modiﬁcations are needed to attain an acceptable performance. As the intuitive element of the design procedure relies on details observed on the speciﬁc case, the two benchmark cases shall be addressed separately. A more general view may evolve from this later. For each benchmark, a linear controller is designed ﬁrst to meet the small signal speciﬁcations. Then a standard awf is added. If the dynamic response does not meet the large signal speciﬁcations, then further structural modiﬁcations are designed and tested in simulations. In parallel, the nonlinear stability properties are investigated at each step.

2 Case A: The Inverted Pendulum The control problem has been fully stated in the ‘speciﬁcations’ (see the corresponding chapter of this book), and shall not be repeated here. The following is an extended version of [4].

2.1 The Standard AWF-Approach The Control Structure The details are shown in Fig. 1.

94

Adolf Hermann Glattfelder and Walter Schaufelberger

ka 4

r_1(t)

k0/T0

1 s

r_2 r_4

k1 k2

x00

k3

Setpoint sequencer

k4

1 s

1/T4

2

x40 u_lo to u_hi

v_lo to v_hi

x_1

1

u x_2 z

z(t)

x_3 − x_1

Load sequencer

Plant 3

Fig. 1. The control system, consisting of the plant, the nonlinear actuator and the linear augmented state feedback controller with standard algebraic awf

Within the linear range the transfer function of the plant is 1 1 x1 = 2 u sT1 s T2 T3 − (1 + (T3 /T1 )) & & 1+(T3 /T1 ) 3 /T1 ) with poles at s1 = − ; s2 = 0; s3 = + 1+(T T2 T3 T2 T3 Gu =

(1)

Consider now the steady state conditions for the plant. From Fig. 1 for all r → u = 0; for all z → u = −z;

(2)

Thus, z must be restricted to within the working range of u, while r has no such restrictions. In other words the z(t)-sequence can be expected to be more difﬁcult to stabilize than the r(t)-sequence. The next step is to design the linear feedback controller. The method to be used is to Ω1 , as this relates in a straightforward way to the small signal closed loop bandwidth from the speciﬁcations.

Antiwindup and Override Control Solutions

95

More speciﬁcally (s + Ω1 )4 · (s + Ωa ) := 0 with Ωa := α1 · Ω1

(3)

where Ωa is the pole associated to the actuator loop in Fig. 1. Here α1 := 10. And by way of the closed loop characteristic equation for the feedback gains k4s = (4 + α1 ) · (Ω1 T4 ) k3s = (6 + 4α1 ) · (Ω1 T3 )(Ω1 T4 ) + (T4 /T2 ) · (1 + (T3 /T1 )) k2s = (4 + 6α1 ) · (Ω1 T2 )(Ω1 T3 )(Ω1 T4 ) + k4s · (1 + (T3 /T1 )) k1s = (1 + 4α1 ) · (Ω1 T1 )(Ω1 T2 )(Ω1 T3 )(Ω1 T4 ) k0s = (α1 ) · (Ω1 T0 )(Ω1 T1 )(Ω1 T2 )(Ω1 T3 )(Ω1 T4 ) ; with T0 := T2

(4)

Then the augmented state feedback is transformed to the cascade structure Fig. 1, with - actuator loop r4 → x4 , - inclination feedback r2 → x2 , using x3 , - speed control r1 → x1 , with x0 that is k4 = k4s; k3 = k3s/k4s; k2 = k2s/k3s; k1 = k1s/k2s; k0 = k0s/k1s;

(5)

For the resulting dominant fourth order closed loop transfer function r1 → x1 having four poles at −Ω1 , the −3dB-crossover is at ≈ 0.425Ω1 . Thus to comply with the speciﬁcation: Ω−3dB > 1.0, set Ω1 := 2.5

The Nonlinear Stability Properties They are investigated here with the , [5]. Its basic form allowing a graphical test is for a single input single output linear subsystem and one nonlinear element with one input and one output. However the actuator in the control system Fig. 1 has two nonlinear elements, that is the stroke s uhi , ulo and the slew s vhi , vlo . This may be circumvented by using an approximate model of the actuator, Fig. 2.

96

Adolf Hermann Glattfelder and Walter Schaufelberger 2

1 a_s

w_lin

u_c

w_hi

w

k_s

u 1/T_s

1 s

w_lo a_s

1

0.8

0.6

0.4

0.2

0

0

0.5

1

1.5

Fig. 2. The actuator subsystem, and its step responses r4 → x4

This model now has a single nonlinear element. It is similar to the one used in [1], but avoids the very high gain. – Let as = 1.0 Then whi = uhi ; wlo = ulo d 1 d 1 uup = whi , udn = wlo and dt τs dt τs

(6)

This determines τs and assumes symmetric values. In numerical values whi = 1.25; → τs = 0.125 s for slew rate 10.0 /s; and τs = 0.625 s for slew rate 2.0 /s. The approximate model is conservative with respect to the rate constraint, especially near to the stroke constraints. But it reproduces the position constraint correctly, see Fig. 2. The assumes the nonlinear subsystem to be asymptotically stable. However here, both the plant and the controller in Fig. 1 are not asymptotically stable. Thus the element must be replaced by a parallel arrangement of a unity gain and a dead-span element with unity gain slopes and breakpoints at wlo , whi . Then for the stability test (cf. [3], p.284ff): Δmin =

whi wlinmax − whi

(7)

Antiwindup and Override Control Solutions

97

and for the transfer function of the linear subsystem 1 + ka 1s dG F +1= (s + ka ) = 1 D1 1 + R sG

(8)

where dG , D1 denote the characteristic polynomials of the plant and of the linear closed loop. F + 1 is now asymptotically stable due to D1 , that is to the small signal closed loop design. For the benchmark case: F +1=

(s +

1 T4 )

(s2 −

1+(T3 /T1 ) ) T3 T2

s2 s +

(s + Ω1 )4 (s + Ωa )

kas T0

s

(9)

If a compensating awf gain kas is used, i.e. kas := Ωa T0 = α1 Ω1 T2 ; then F +1=

s(s +

1 T4 )

(s2 −

(s + Ω1

1+(T3 /T1 ) ) T3 T2 )4

(10)

and ﬁnally for the cascade structure (see Fig. 1): ka1 = kas/(k1k2k3k4s). The unstable root factor produces a strong phase shift of the Nyquist contour F + 1 into the LHP, which indicates a substantial reduction of the radius of attraction, Fig. 4. And the reduction is stronger for the slow actuator case. 1.5 Ts = 0.125 T = 0.625 s

1

0.5

0

−0.5 −0.5

0

0.5

1

1.5

Fig. 3. For the circle test: Nyquist contours of F + 1 for the standard awf approach

98

Adolf Hermann Glattfelder and Walter Schaufelberger 2.5 x 1 x 2 x 3 x

2

4

z r

1

1.5

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 4. Response of the system Fig. 1 to the speciﬁed r1 (t)-sequence, slow actuator

Transient Responses to the Input Sequences The case with the slow actuator will be documented here, as it is the more critical one. First the setpoint sequence r1 (t) from the speciﬁcations is applied ﬁrst to the control structure Fig. 1. The size of the ﬁrst step of r1 may be increased until its maximum value from the speciﬁcations without diverging. From Fig. 4 the transient is linear everywhere, and thus the full range for r can be applied. If for instance just the slew rate of r(t) would be increased from ±0.5/s to ±1.0/s then the size of r1 must be reduced considerably, Fig. 5, to 0.78 for the fast actuator and to 0.26 for the slow one. Next the load sequence z(t) is applied. Again the size of the ﬁrst step of z is increased until close to the border of stability, Fig. 6, to 0.75 for the fast actuator and to 0.71 for the slow one. Figure 7 documents the response for the slow original actuator to the z(t)-sequence up to 0.975, but with its slew rate reduced from the speciﬁed ±0.25/s to ±0.125/s. By inspection of Figs. 6 and 7, the response to the ﬁrst large z ramp is the critical one. During the ramp the controller has x2 converge to z, while x1 stabilizes at a nonzero value, which is proportional to the ramp speed selected (this follows directly from

Antiwindup and Override Control Solutions

99

2.5 x 1 x 2 x 3 x

2

4

z r

1

1.5

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 5. Response to r1 (t)-sequence with faster ramp

1.5 x 1 x 2 x 3 x

1

4

z r

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 6. Response of the system Fig. 1 to the speciﬁed z(t)-sequence

the plant equations). After the ramp on z ends, the controller moves to reduce x1 to its setpoint at zero again. – For the nominal ramp speed and the associated large offset on all xi , i = 1...4, the margin on the resulting u to the value ulo is no longer sufﬁcient for stabilization. In other words the controller output moves far beyond the value. Then the control loop is cut open for too long, and cannot recover in time

100

Adolf Hermann Glattfelder and Walter Schaufelberger 1.5 x 1 x 2 x 3 x

1

4

z r

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 7. Response to z(t)-sequence with ramps slowed down to half of speciﬁed

to stabilize the unstable inclination dynamics. But this is (barely) sufﬁcient for the slow ramp with its smaller offset on x1 , where the controller output saturates for a relatively shorter time span. – Note that this behavior corresponds to the stability properties following from Fig. 3.

2.2 The Override Approach The Basic Idea However the analysis of the responses in Figs. 6 and 7 suggests a possible solution: If the inclination feedback would stay closed ‘for most of’ the transient (‘for all of it’ would be safest) by simply avoiding the actuator s, then the asymptotically stable closed loop dynamics would be conserved. And this can be implemented by placing a on the inclination setpoint r2 (t), which is the output of the lateral speed controller, see Fig. 1. This idea has been investigated in [6]. It generalizes the cascade-limiter design from DC-servodrives where e.g. the speed constraint is implemented by a corresponding on the speed setpoint, which is the position controller output. Note however that the saturation values must correspond as closely as possible to the actual operating span

Antiwindup and Override Control Solutions

101

of the actuator but may not exceed it. In other words this approach requires exact a priori knowledge of additional persistent disturbances, which will lead to u = 0, and thereby shift the actuator operating span. Thus the saturation values must be shifted accordingly. This drawback can be avoided with the technique: the input to the actuator loop is taken as the secondary output yc which is to be constrained by override feedback controllers with appropriate setpoints. In essence this replaces the feedforward concept described above by a feedback one, and as such is less sensitive to persistent disturbances. Note that the different approaches are most popular (see the other chapters of this book), while explicit use of overrides is not common. Note also that both approaches are related, [7].

The Override Control Structure Details are shown in Fig. 8. Note that the feedback acts not on r 2 1 but further downstream on r 4 1. This provides a separate set of feedback gains for its independent tuning.

y_c

r_c_hi

r_c_hi x_2

u_c_hi

x_3 awf

R_c_hi

7

r_1(t)

Setpoint sequencer

k0/T0

ka_1

min

1 s

x_1

1

x_2

2

x_3

3

k1 k2

x3

k3

r_4_1

6

ka_C

r_2_c k0_C/T0 r_c_lo

4

r_4

max

r_2_1

x_4

1 s

k3_C k2_C

z(t)

Load sequencer

z

x_3 − x_1

5

x00_lo

Fig. 8. The structure of the override control system

Plant with Actuator Loop

102

Adolf Hermann Glattfelder and Walter Schaufelberger

The linear constraint feedback is designed as follows (see also [4]). -

To shorten the design process, the actuator dynamics are neglected. From Fig. 8, the same actuator feedback gain k4 will be inserted here, as it has been determined for the main control loop in the previous section.

-

A stabilizing state feedback for the inclination dynamics Go =

s2 T

1 2 T3 − (1 + (T3 /T1 ))

(11)

is designed. This yields for the closed loop response from r 2 c to yc = r4 in Fig. 8, with feedback gains k2s c on x2 and k3s c on x3 = sT2 · x2 : 1 1 + Go · (sT2 k3s c + k2s c) s2 T2 T3 − (1 + (T3 /T1 )) = 2 s T2 T3 + sT2 k3sc + [k2sc − (1 + (T3 /T1 ))]

Gc =

-

(12)

For the overall constraint loop this is the ‘plant’. It has a negative sign on the constant term, therefore the sign on the controller has to be inverted as well for negative feedback in the overall loop. It also has one zero in the right half plane, ' at + [1 + (T3 /T1 )]/[T1 T2 ].

-

A well established design rule states that a pure integral controller is best suited for such inverse-unstable plants. Then the overall closed loop characteristic equation is 0 = 1+

k0sc · Gc sT0

( ) T 3 = s3 T0 T2 T3 + s2 T0 T2 k3sc − k0sc + ... T0 +sT0 [k2sc − (1 + (T3 /T1 ))] + k0sc (1 + (T3 /T1 )) -

(13)

Another such design rule is that the zero in the right half plane should be matched by one closed loop pole no further left than at its mirror position in the left half ' √ plane, that is at − [1 + (T3 /T1 )]/T1 T2 = − 2. Here this pole is set at Ωz = −1.0.

-

The other two closed loop poles may now be assigned, such as by (s + Ωc )2 := 0.

Antiwindup and Override Control Solutions

-

103

With βc := Ωz /Ωc k0sc = (βc ) (Ωc T0 )(Ωc T2 )(Ωc T3 ) ; with T0 := T2 T3 k3sc = (2 + βc )(Ωc T0 ) + k0sc T0 k2sc = (1 + 2βc ) (Ωc T2 )(Ωc T3 ) + [1 + (T3 /T1 )]

(14)

Then the augmented state feedback is transformed into the cascade structure Fig. 7, yielding k0C = k0sc /k2sc ; k2C = k2sc /k3sc ; k3C = k3sc ;

-

The awf gain is set as ‘compensating’, that is kaC = α1 Ωc T4 /k2sc

-

Then the setpoints are selected. Set rchi = uhi − Δrc ; rclo = ulo + Δrc ;

(15)

(16)

where Δrc is an additional design parameter. It should be selected such that the s uhi , ulo are not met for the usual disturbances while the constraint loop is in operation. Nor should it be taken too large, as then the available range on u1 is no longer usable. A numerical example shall illustrate this. Let Δrc = −0.06 · ulo = 0.075. This leads to rclo = −1.175. Now compare this to the maximum z = 1.10, taken from the speciﬁcation, which leads to u = −1.10. Thus the margins for maneuvering in both directions (down to ulo and up to u from z) are the same. -

Finally the controller outputs must be implemented into the main speed loop. As the plant transfer function sign is negative, the direction of u is reversed, and therefore the output of the rchi -controller must be connected to the Max-selector and the rclo -controller to the Min-selector, see [3], p. 376 ff.

The nonlinearity in Fig. 8 has three inputs. But the canonical structure required for the stability test only allows one input. However the Min-Max-selector block is under some weak assumptions equivalent to a one-input dead span nonlinearity, [3], p. 114ff. Then for the linear subsystem

104

Adolf Hermann Glattfelder and Walter Schaufelberger

F +1 =

≈

(1 + F )1 (1 + F )c (1+(T3 /T1 )) )·s·s T2 T3 (s + Ωa )(s + Ω1 )4 (s + Ωa )(s + Ωc )2 (s + Ωz ) · 3 /T1 )) (s + T14 )(s2 − (1+(T ) · s T2 T3

(s + ·

1 2 T4 )(s

−

·

s+

kas1 T0

s

·

s s+

kasc T0

(17)

In this equation, the ≈ is due to the approximative2 separation of the closed loop polynomial for

-

the loop into (s + Ωa )(s + Ωc )2 . The open loop poles at −(1/T4 ) from the actuator, and the poles from the in' clination dynamics (s2 − [1 + (T3 /T1 )]/[T2 T3 ] belong to the same physical

-

subsystems, and thus the corresponding root factors in (F + 1)1 and (F + 1)c cancel without generating any controllability or observability problems. -

The same holds for the poles at zero from the s in the main controller and the constraint controllers with those in their respective awf loops.

-

And the awf gain kas1 /T0 has been set to the value Ωa in the corresponding closed loop factor, such that this also cancels in the (1 + F )1 part

-

The same holds for the corresponding root factors in the (1 + F )c part.

Thus ((s + Ωc )2 (s + Ωz ) s · (s + Ω1 )4 1 2 (s + Ωc ) (s + Ωz ) s = (s + Ω1 ) (s + Ω1 )2 (s + Ω1 )

F +1 ≈

Setting now Ωc := Ω1

(18)

3

this reduces further to a very simple relation F +1 ≈ 2 3

(s + Ωz ) s (s + Ω1 ) (s + Ω1 )

(19)

Ωa /Ωc is large but not inﬁnite which is reasonable as the as the dynamic properties of both systems 1 and c are not widely different

Antiwindup and Override Control Solutions

105

The ﬁrst part generates a half circle from the origin to +1; j0 as ω runs from 0 to ∞. In other words it evolves in the right half plane. This indicates an ‘unbounded’ radius of attraction. The second part in the equation above modiﬁes this shape by a lead-lag element (note that Ωz < Ω1 ) in the direction of the left half plane. This gets more pronounced for Ω1 Ωz and ultimately will restrict the radius of attraction. – Note that the approximation from above will show up at high frequencies, ω Ω1 , see Fig. 9. 1.5 T = 0.10 4 T4 = 0.50

1

0.5

0

−0.5 −0.5

0

0.5

1

1.5

Fig. 9. Nyquist contours for the override system,if no saturation is met

The shape of the Nyquist contour in Fig. 9 indicates a very large radius of attraction. This allows either large reference steps r1 or, for step loads, resulting steady state offsets u, which are close to the constraint setpoints rchi ; rclo . Transient Responses to the Speciﬁed Input Sequence Only the responses with the slow actuator subsystem are shown here. The setpoint response to the nominal sequence r1 (t) is not shown, as it does not differ signiﬁcantly from Fig. 4. However the response to the faster setpoint ramp from Fig. 5 with the system is shown in Fig. 10. And Fig. 11 shows the response for the nominal load sequence z(t) from the speciﬁcations.

106

Adolf Hermann Glattfelder and Walter Schaufelberger 2.5 x 1 x 2 x 3 x

2

4

z r

1

1.5

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 10. Responses of the override control system Fig. 8 to the r(t)-sequence from the speciﬁcations, but with increased slew rate 1.5 x 1 x 2 x 3 x

1

4

z r

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 11. Responses of the override control system Fig. 8 to the full z(t)-sequence from the speciﬁcations

In other words, the solution now covers the full speciﬁcations. From Fig. 11, the critical phase is no longer the ﬁrst large loading, but rather the last large unloading in the z(t)-sequence. The actuator position x4 is moved to its

Antiwindup and Override Control Solutions

107

value as soon as the z-ramp starts for a long time interval. After the ramp has come to its end x4 is driven to the opposite , and therefore barely manages to stabilize the response. This is less critical if the ramp slew rate would be −0.20/s instead of −0.25/s, see Fig. 12 (top). If the slew rate is ﬁxed, then the only way out seems to be to increase the actuator working range, from −1.25 to −1.35, Fig. 12 (bottom). 1.5 x 1 x 2 x 3 x

1

4

z r

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

1.5 x 1 x 2 x 3 x

1

4

z r

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 12. Responses of the override control system Fig. 8 to the z(t)-sequence, (top) with slower unload ramp −0.25 /s → −0.20 /s, (bottom) with extended working range −1.25 → −1.35

108

Adolf Hermann Glattfelder and Walter Schaufelberger

3 Case B: The Continuous Stirred Tank Reactor (CSTR) The statement of the control problem has been given in the speciﬁcations (see the corresponding chapter of this book), and shall not be repeated here. The bandwidth speciﬁcation is changed to > 6 rad/s to comply with the usual −3dB notion, as for case A.

3.1 Designing The Linear Control System The plant given in the speciﬁcations is a multivariable one, with two inputs (reactant inﬂow u1 and heat ﬂow u2 into the jacket subsystem) and two outputs (measured reactant concentration y1 and measured temperature y2 of the content). Thus the control structure should be multivariable as well. Here the separation method often used in practice is applied. It consists of designing two independent single input single output loops, thereby neglecting the interaction within the plant, and separating the closed loop bandwidths of both loops as far as feasible. In this case the ﬁrst loop is given by safety considerations: the measured concentration y1 is fed back to the reactant inﬂow u1 in order to keep the inventory of reactant under rather tight control: The controller gain of the concentration loop is speciﬁed as kp1 > 10. If the contents temperature x2 were set constant (which would imply an inﬁnite bandwidth on the temperature control loop), then for the x1 -closed loop If x1 = 1.0 ∀ t, then 0 = sT1 + (kcc + 1)

(20)

and setting kcc := 15 yields as the closed loop bandwidth Ω1 = 80 rad/s, which is approx. ﬁve times faster than the bandwidth speciﬁcation for the temperature control loop Ω2 := 15 rad/s, and thus complies with the separation rule. Next the temperature control loop is designed. The basic structure is a cascade arrangement with an inner loop (with gain k4) for the servomotor position x4 . Its pole shall be placed at −10 · Ω2 . The plant response is considerably different during runup mode (where x1 = 0) and during production mode (where x1 ≈ 1.0). This is taken into account by designing two separate controllers, and selecting by the current

Antiwindup and Override Control Solutions

k_cc

u_1

109

y_1 1 s

1/T1

1

x10

Tstep_r1

a1 x_2_S

max

a0 a2

0 1 s

2

x20

y_1

1/T2

a3

1 s

1/T3

x30

a4

Tstep_u2

k4

x40 1/T4

1

1 s

u_2

Fig. 13. Model of the plant with the concentration control loop (gain kcc ) and the heat input servo loop (gain k4), with no saturations on slew and stroke

concentration setpoint: if r1 < 0.05, then the runup mode controller is selected, else the production mode controller. In production mode the plant has one additional state variable (x1 (t)), which is however already controlled by way of kcc . A temperature controller with full state feedback and would have to interfere with this loop. Therefore an approximate design procedure is pursued. It considers the concentration to be constant x1 = 1.0 ∀ t (which implies an inﬁnite bandwidth of the y1 -loop, in other words kcc → ∞). This will eliminate both the state variable x1 and the control variable u1 , and the design now becomes the standard one for a scalar control variable (u2 ) 4 . Thus three further closed loop poles have to be assigned for state variables x2 , x3 , and the x0 of the temperature master controller. Here the three poles are assigned at −Ω2 with Ω2 = 15 rad/s. The design procedure is the same as for case A, and its details are omitted here. 4

Note that this again implements the separation design idea

110

Adolf Hermann Glattfelder and Walter Schaufelberger

Then for the original system with ﬁnite kcc , the closed loop poles will move away from the assigned locations, speciﬁcally for this case with kcc := 15 to −79.86 −8.23 −18.48 + j8.33 −18.48 − j8.33 From transient responses (not shown here)for small setpoint steps on r2 starting from steady state x2 = 1.0 at x1 = 0.0 and at x1 = 1.0, this is considered to be still acceptable. It also complies with the speciﬁcations (> 6 rad/s at −3dB).

3.2 The Standard Anti Windup Feedback Approach The Control Structure Details are shown in Fig. 14.

7 concentration setpoint ramper

k_cc Plant with actuator

r_1 (t)

1

x_2

2

x_3

3

x_4

4

u_1

6

kaa

1 s

r_2 (t)

k0a/T0

x00 x20 k2a

u_2

x30

u_lo to u_hi

k4a k3a

5

x_4

Temp. Controller for run_up x_3

x_2

r_2 (t)

u_2 Lin

Switch

u_2 (awf)

temp setpoint ramper

x_1

Fig. 14. The control system with standard awf

Antiwindup and Override Control Solutions

111

The Nonlinear Stability Properties The general result of [3], p. 70ff, is applied. For the Nyquist contour F +1=

1 + ka2 (1/sT0 ) d2 · sT0 sT0 + ka2 = · 1 + R2 G2 D2 sT0

(21)

where d2 is the polynomial with the open loop roots, and D2 is the closed loop polynomial including the . The second fraction is from the awf loop. First, the terms sT0 in the numerator and denominator cancel, as they are due to the same integral action. Then consider d2 . Its roots are functions of kcc . For kcc = 15: +8.7037 −10.2695 −79.4341 −5.0 The fastest mode is associated with the concentration loop, the mode with pole at ≈ −10 with the jacket dynamics, and the unstable mode with the content temperature dynamics and the exothermal reaction. The root at −5.0 is from the slow actuator open loop. Further for the closed loop polynomial D2 : D2 :≈ T0 T2 T3 T4 (s + Ω2 )3 · (s + αΩ2 )

(22)

The ≈ is due to the fact that the to three times −Ω2 is performed for kcc → ∞ instead of its ﬁnite design value. Further, α := 10 is associated with the actuator positioning feedback, following the usual design rule in practice. Finally design the awf loop as with ‘compensating awf gain’ ka∗2 : ka∗2 := (s + αΩ2 ) → ka∗2 = αΩ2 T0 s+ T0

(23)

Figure 15 shows the Nyquist contour. For the runup phase of the temperature loop it starts in the right hand half plane. And the yields an unbounded radius of attraction. However for the production phase it starts for ω = 0 well in the left hand plane, which indicates a strongly delimited region of attraction, see [3], p.290 ff. This is even more so as the loop operates at its production equilibrium u = −1.0 close to the saturation ulo = −1.10.

112

Adolf Hermann Glattfelder and Walter Schaufelberger 1.5

0.15

0.1

1

0.05

0.5

ω=∞ 0

0 ω=0

−0.05 −0.1

−0.08 −0.06 −0.04 −0.02

0

0.02

ω=0

0.04

0.06

0.08

0.1

−0.5 −0.5

0

0.5

1

1.5

Fig. 15. Nyquist contour for the graphic stability test (circle criterion) of the control system with standard awf with compensating gain ka∗2 and slow actuator (left) for the runup phase (x1 = 0): (right) for the production phase (x1 = 1.0)

Transient Responses to the Benchmark Input Sequence A ﬁrst intuitive solution to avoid run away would be to limit the concentration runup rate. Here this would have to be set to ≈ +1/6 s−1 . But such a clamp on runup rate of r1 is only needed in the ﬁnal approach phase. Therefore it may be set larger for concentrations below say 0.6 of nominal. This simple feedforward strategy has been implemented in Fig. 16. It reduces the runup time from 6 s to 3 s. However the simulation also shows that the subsequent small disturbances to the nominal production given in the speciﬁcations lead to divergent responses. To avoid this they have to be reduced as follows: for the concentration (Δr1 )

from 0.02 to 0.01

and for the reaction rate (Δa21 ) from 0.02 to 0.01 Thus a better solution is needed.

Antiwindup and Override Control Solutions

113

1 0.8 0.6 0.4 0.2 0 x1 x2 x3 x4 u2 r2 r1

−0.2 −0.4 −0.6 −0.8 −1

0

5

10

15

20

1.1

1.05

1

x1 x2 x3 x4 u2 r2 r1

0.95

0.9

0.85

0

5

10

15

20 x 4 u

−0.95

2

−1

−1.05

−1.1 0

5

10

15

20

Fig. 16. Transient response of the control system with standard awf, slow actuator (top) overall view, (center) zoom-in around +1.0, (bottom) zoom-in at −1.0

114

Adolf Hermann Glattfelder and Walter Schaufelberger

3.3 The Override Approach The Basic Idea The basic mechanism leading to instability is that if the concentration increases too fast then the temperature control is pushed to the actuator , and then the temperature can no longer be controlled and will diverge. To avoid this the concentration must be manipulated such that the temperature controller will no longer push the actuator beyond its saturation, but keep it within its operating range. In other words this means constraining the actuator position x4 to a value sufﬁciently close to the saturation value. And this can be done -

by standard feedback techniques,

-

designating as constrained output yc the reference to the u2 -actuator loop,

-

setting the constraint controller setpoints to rclo = ulo + Δ; rchi = uhi − Δ;

-

with Δ := 0.01, such that the loss in operating performance is small,

-

and modifying by selection the concentration reference input r1 .

Note that here (in contrast to case A) the ‘disturbance’ is accessible for control. Designing the Override Loop The design procedure is demonstrated on a reduced order system, where the dynamics of the concentration loop and the temperature actuator loop are neglected and the jacket dynamics are disregarded, that is T1 = T4 = T3 := 0, and thus 1 G2r (s) = sT2 − a21

r c _ R + r

c

y c

2

+ _

+ R

2

G +

2

y 2

Fig. 17. The reduced order override loop

(24)

Antiwindup and Override Control Solutions

115

Read from Fig. 17 for the closed loop characteristic equation ( ) R2 0 = 1 − Rc − 1 + R2 G2r k1 sTc + k0c k12 sT0 + k02 · 2 = 1+ c sTc s T0 T2 + sT0 [k12 − a21 ] + k02 = s3 Tc T0 T2 + s2 Tc T0 [k12 − a21 + k1c k12 ] ( ) T0 + sTc k02 + k1c k02 + k12 k0c + k0c k02 Tc

(25)

Using now 1 k12 − a21 k1c k12 and 1 k02 k1c k02 +

T0 k1 k0 Tc 2 c

(26)

and setting Tc = T0 = T2 := 1.0 yields in the low frequency range, which is of interest here: 0 ≈ s2 Tc T0 [k1c k12 ] + sTc [k1c k02 + k12 k0c ] + k0c k02

(

) k1c k12 k1c k12 2 = s Tc T0 · + sTc + +1 k0c k02 k0c k02

(27)

Also note k1c 2 = k0c Ωc T k1 2 := 2Ω2 T ; that is 2 = k02 Ω2 T

for Rc : k0c := Ω2c T 2 ; and k1c := 2Ωc T ; that is and for R2 : k02 := Ω22 T 2 ; and k12 and select Ωc := Ω2 ﬁnally 0 ≈ 4

s Ω2

2 +4

s Ω2

( +1 =

s (Ω2 /2)

)2 (28) +1

for the characteristic equation of the closed loop while the is active. This simple but approximate design rule shall be applied from here on, but it will have to be validated by the transient responses.

The Override Control Structure Details are shown in Fig. 18. As for case A, a sign inversion is required in the controller to conserve negative feedback, and the constraining action for the low setpoint must be applied to the Min-selection.

116

Adolf Hermann Glattfelder and Walter Schaufelberger 7

Override controller

ka_c

1 s

k0_c/Tc

− r_c_lo

k_cc

x_c0 k1_c

min

x_1

−1

1

u_1 r_1 (t)

Concentration setpoint sequencer

x_2

2

x_3

3

x_4

4

5

r_2(t)

y_c Temperature setpoint generator

u_2

kaa

6

1 s

k0a/T0

Switch runup to production at r_1 = +0.05

x00 x20 k2a

Plant with actuator

k4a

x30

8

k3a

awf in

x_4

x_2

r2(t)

Temperature controller for runup

x_3

u2_runup

u_lo to u_hi

Fig. 18. The control system with override feedback control from the temperature controller output (i.e. actuator position setpoint) to the concentration setpoint r1 (t)

Nonlinear Stability Properties Applying the general result for systems (without active s) from [3], p.142, F +1=

1 + (kam /sT0 ) 1 + Rc Gc 1 = (F + 1)m · · 1 + Rm Gm 1 + (kac /sTc ) (F + 1)c

(29)

with index m for the main loop and index c for the loop. Here the part with index m is a special case, as there is no feedback loop: Rm = 0; kam = 0; → (F + 1)m = 1.0; Dc sTc Dc and thus F + 1 = · = dc sTc sTc + kac dc (sTc + kac )

(30)

Antiwindup and Override Control Solutions

117

where for Dc from the previous section: Dc ≈ s3 Tc T0 T2 + · · · + k0c k02

k02 k0c 3 = Tc T0 T2 s + · · · + 2 Tc Tc T0 T2 ! 3 2 = Tc T0 T2 s + · · · + (Ωc ) Tc (Ω2 )2

(31)

and for dc · (sTc + kac ): dc ≈ s2 T0 T2 + · · · + k02

! ka = Tc T0 T2 s2 + · · · + (Ω2 )2 · (s + c ) Tc

(32)

From this intuitively set ka∗c := (Ωc Tc )2 to obtain a low frequency gain of (F + 1) of about 1. The Nyquist contour for F + 1 is plotted in Fig. 19. It starts on the positive real axis (at +2 + j0), and evolves well into the right half plane, except for frequencies above the closed loop bandwidth (ω > 10Ω2 ), where F (jω + 1) crosses over into the left half plane. So stability can no longer shown by the on-axis . However if the less conservative Popov test is used (see right hand plot in Fig. 19), then an inclined straight line may be plotted through the origin, and approx. parallel to the Popov contour there, which does not intersect the contour anywhere. – And this indicates an unbounded radius of attraction. 15 2.5

2

ω to ∞

10 1.5

1

5 ω=0

ω=0

0.5

0

0

ω to ∞

−0.5

−5 −1

−1.5

−10 −2

−15

0

5

10

15

20

25

−2.5 −0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Fig. 19. Graphic stability test for the override control system override, slow actuator kac := (Ωc Tc )2 ; (left) Nyquist contour, (right) Popov contour zoom-in to the origin

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Adolf Hermann Glattfelder and Walter Schaufelberger

1 0.8 0.6 0.4 0.2 0 x1 x2 x3 x 4 yc r2 r1

−0.2 −0.4 −0.6 −0.8 −1

0

5

10

15

20 x1 x2 x3 x4 yc r2 r1

1.1

1.05

1

0.95

0.9

0.85

0

5

10

15

20 x 4 y c u

−0.95

2

−1

−1.05

−1.1 0

5

10

15

20

Fig. 20. Transient response, override control system, slow actuator, kac := (Ωc Tc )2 ; (top) overall view, (center) zoom-in around +1.0, (bottom) zoom-in at −1.0

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119

Transient Response to the Speciﬁed Input Sequence Figure 20 shows that the rise time from hot standby to production has been reduced from 3 s to approx. 1.8 s, that is to about 60%. It also documents convergence for the full speciﬁed variations to the nominal operating conditions. The smooth switch-in and transient response of the loop is documented in Fig. 20 (bottom), at 7.5 s, 12.5 s and at 14 to 15 s.

4 Summary Both cases A and B have been successfully solved and the speciﬁcations given for the benchmarks have been fulﬁlled. This has been achieved in both cases by applying the technique in a suitably modiﬁed way, whereas the standard awf design well known in practice has been shown to fail. – Note that the design process used here is not a purely algorithmic or automatic one, but essential elements arise from careful inspection of the speciﬁc system’s behavior. However, insight alone is not sufﬁcient, as the number of design parameters and choices is such that analytic parts become mandatory in order to produce rationally founded choices and solid parameter values. For this, nonlinear stability theory has again proven to be a powerful tool. But again it cannot produce all necessary design elements by itself. It must be supplemented by carefully designed experiments for transient responses. Thus solutions by applying the and techniques so far always arise through strong interaction of insight, intuition, analysis and experiments.

References 1. Barbu C., R. Reginatto, A.R. Teel, and L. Zaccharian (2002), Anti-Windup for Exponentially Unstable Linear Systems with Rate and Magnitude Input Limits, in Actuator Saturation Control, Marcel Dekker Inc., ISBN 0-8247-0751-6, Editors V. Kapila and K.M. Grigoriadis, p.1 – 31 2. Tarbouriech S., and G. Garcia (2002), Output Feedback Compensators for Linear Systems with Position and Rate Bounded Actuators, in Actuator Saturation Control, Marcel Dekker Inc., ISBN 0-8247-0751-6, Editors V. Kapila and K.M. Grigoriadis, p. 247 – 272

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3. Glattfelder A.H., and W. Schaufelberger (2003), Control Systems with Input and Output Constraints, Springer Verlag Limited, ISBN 1-85233-387-1, London 4. Glattfelder A.H., and W. Schaufelberger (2005), Antiwindup and Override Control for Exponentially Unstable Systems with Actuator Constraints, Proceedings 16th IFAC World Congress, Prague, Tu-A20-TO, 2005, paper 02398 5. Khalil H.K. (2002), Nonlinear Systems, 3rd ed., Prentice Hall 6. Hippe P. (2003), Windup Prevention for Unstable Systems, Automatica 39, 1967 - 1973 7. Glattfelder A.H., and W. Schaufelberger (2004), A Path from Antiwindup to Override Control, Proc. 6th IFAC Symposium - Nolcos, Stuttgart, 1379 - 1384

Anti-windup Compensation using a Decoupling Architecture Matthew C. Turner, Guido Herrmann, Ian Postlethwaite Control and Instrumentation Research Group, Department of Engineering, University of Leicester, Leicester, LE1 7RH, UK. Email: {mct6,ixp,gh17}@le.ac.uk Summary. This chapter describes the theory and application of a ‘decoupled’ approach to anti-windup compensation. One of the salient features of this approach is that the anti-windup problem can be posed in a manner which decouples the nominal linear closed-loop from the nonlinear stability problem associated with actuator saturation and the anti-windup compensator. This process also reveals a logical and intuitive performance criterion which one can optimise by means of linear matrix inequalities. Details of how the anti-windup problem can be solved using various forms of compensator are given, together with some robustness considerations. Initially, the problem is posed for globally asymptotically stable plants, although this condition is later relaxed to allow the consideration of plants with exponentially unstable modes. Finally, the results are demonstrated on a multivariable aircraft model and the benchmark inverted pendulum.

1 Introduction 1.1 “Anti-Windup” and Its Philosophy The problem of “windup” was probably ﬁrst discovered by practical control engineers when designing integral and PID controllers for tracking problems with strict

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limits on the control signal amplitude. In these types of situation, if saturation occurs it can cause the integrator state to attain a large value (or to “windup”) during saturation and then this energy has to be dissipated once saturation has ceased. Such saturation problems were generally observed as excessive overshoot and/or a slow settling time at the plant output. This early interpretation of the word “windup” was easily justiﬁed as it corresponded to the accumulation and dissipitation of charge on the capacitor of an analogue integrator. Thus, at ﬁrst, windup tended to be associated with saturation problems with integrator type controllers, particularly the popular PID controller. To overcome these saturation problems, a number of largely ad hoc methodologies - so called “anti”-windup techniques - were developed and most have had a fair amount of success attributed to them. Among these ad hoc techniques are high gain anti-windup, reset anti-windup and Hanus anti-windup. In high gain anti-windup, the controller is prevented from producing too large a control signal during saturation by means of a large feedback gain ‘steered’ by a deadzone nonlinearity. In reset antiwindup, the integral state is reset to zero following saturation to prevent the windup problem. Finally, the Hanus technique works by partially inverting the controller, in an attempt to force the controller to produce an output no greater than the saturation level. A good summary of these can be found in [4]. The basic anti-windup architecture is shown in Figure 1. Up until, roughly, the time of reference [7], all anti-windup techniques were ad hoc, had no stability or performance guarantees but were simple to implement and hence were useful to the practising engineer. A particularly useful property of these so-called anti-windup techniques is that they were designed to work in conjunction with the existing controllers and only became active once saturation occurred. Research in optimal control in the 60s and 70s and then again in the mid 90s pointed to a different source of saturation problems. In papers such as [5], [30] and [22], it was observed that the fundamental limitation in (linear) control systems subject to input saturation was the plant not the controller (This observation was also made in less formal manner in the anti-windup literature - see [18]). Importantly, it was established that global stability is only possible for systems with poles only in the closed-left-half complex plane. Furthermore, global asymptotic stability with ﬁnitegain stability is not possible using linear control laws if the system contains repeated poles on the imaginary axis (see [27] for a good summary of this). Also, if a system

Anti-windup Compensation using a Decoupling Architecture

123

disturbance

reference Controller

anti−windup compensator

Plant

+ _ saturation detection (deadzone)

Fig. 1. Basic anti-windup architecture

contains poles in the open-right-half complex plain, the most which can be achieved is local stability, with local gain properties ([30]). In contrast to the anti-windup techniques though, the solutions offered by these optimal control techniques tended to be complex, unintuitive and, typically, one-step solutions, i.e. they would not work with an existing controller, they would replace it. Hence, although more theoretically sound, these techniques found less favour with the practising engineer than the aforementioned anti-windup techniques. Some of the weaknesses of the “ﬁrst generation” of anti-windup compensators were lack of stability guarantees, restriction to single-input-single-output systems (typically) and imprecise tuning rules. In addition, this generation of anti-windup compensators often made the inaccurate connection that saturation problems were always associated with integrators in the controller. The “second generation” of anti-windup compensators attempted to address at least the ﬁrst two of these problems by appealing to some of the control theory developed for the one-step solutions to the antiwindup problem. That is, the basic architecture of the anti-windup problem would be retained, but ideas from optimal and nonlinear control (see [7] for an early reference) would be used to guarantee stability of the resulting (nonlinear) closed-loops and, as these tools were not necessarily insistent on SISO systems, the second restriction was also removed. Some of the ﬁrst examples of “second generation” anti-windup techniques can be found in [4], [23], [24] and [31], for example. Today, we are probably still in the era of “second generation” anti-windup technology. Many methods have been proposed based on the underlying idea of augmenting

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a linear controller during saturation to obtain improved performance [37, 25, 2]. Indeed recently, constructive techniques have become available which allow the tuning and optimisation of various aspects of closed-loop system behaviour e.g. L2 gain, performance recovery, basin of attraction [9, 35, 8]. Probably the main weakness in this generation of AW compensators is that most have been demonstrated in simulation or on toy laboratory examples and few have been tried on real engineering systems (One exception is [14] where the results in [35] were tested on a multivariable hard disk servo system). A possible reason for this is that some of this generation of AW compensators have discarded the simplicity and intuition found in many earlier schemes and have instead embraced mathematical rigour. Our intention in this chapter is to show that the two are not mutually exclusive and to describe and demonstrate techniques which are both practically appealing and endowed with stability and performance guarantees.

1.2 The Goal The aim of this chapter is to motivate and introduce a particular form of anti-windup compensator which we believe is ﬂexible, intuitive and suitable for implementation in real engineering systems. The basic architecture which we use is taken from [38] in which a special parameterisation of anti-windup compensators is proposed (this is related to the work of [21] and [2], and [31] as well). The structure of the chapter is as follows. After the notation is introduced, we describe the architecture of the problem we consider, and show how this has an appealing structure. The next section is devoted to global anti-windup, particularly for open-loop stable plants and a number of different types of anti-windup compensator are proposed (full-order, reduced-order and static). Robustness is also considered. The following section introduces local anti-windup compensation, which must be used if the plant in question is unstable. Based on reduced sector bounds (used in [17] and [26]) similar results are developed for local stability/performance guarantees. Two example sections are then given, where the theoretical results are demonstrated in detail. The chapter concludes with some ﬁnal remarks.

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125

1.3 Notation The notation is standard throughout and mainly complies with that listed at the beginning of this book. However, we speciﬁcally deﬁne: • The L2 norm of a vector-valued function x(t) as x 2 :=

&*

∞ 0

x(t) 2 dt. Fur-

thermore, whenever x 2 < ∞, we say that x ∈ L2 . • The induced L2 norm or the L2 gain of a (nonlinear) operator Y is deﬁned as Y i,2 := sup

x∈L2

Y x 2 x 2

(1)

If this quantity exists, we say that Y is ﬁnite-gain L2 stable, or simply L2 stable.

• Given the sets Dx := x(t) ∈ Rnx : supt≥0 x(t) ≤ βx and

Dy := y(t) ∈ Rny : supt≥0 y(t) ≤ βy , and if Y : Dx → Dy is a nonlinear operator and W ⊂ Dx , then we deﬁne the local induced L2 norm, or small signal L2 gain as Y i,2,W :=

sup x∈L2 ,x∈W

Y x 2 x 2

(2)

If this quantity exists, we say that Y is small-signal ﬁnite-gain L2 stable, or simply L2 stable. Note that sup x∈L2 ,x∈W

Y x 2 Y x 2 ≤ sup x 2 x∈L2 x 2

(3)

implies that Y i,2,W ≤ Y i,2 and therefore we can expect the small-signal L2 gain to be smaller than the actual L2 gain (which may not exist for certain systems). • We denote the space of real rational p × q transfer functions by R p×q ; the subset of these which permit analytic continuity in the closed-right-half complex plane is denoted RH p×q ∞ ; the space of all such real rational transfer functions is RH ∞ . Note that the H∞ norm of a real rational transfer function, G(s) ∈ RH ∞ , can be deﬁned either in the frequency domain as G ∞ := supω σ ¯ (G(jω)) or in the time domain as G ∞ := G i,2 where G is the time-domain linear operator associated with G(s) and σ ¯ (.) denotes maximum singular value.

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2 Architecture Consider the system with input saturation and anti-windup compensation shown in Figure 2. This is a reasonably generic form of anti-windup compensation and allows the anti-windup compensator Θ(s) to inject signals, which modify the controller’s behaviour, at the controller output and also one of its inputs. Here G(s) = [G1 (s)

G2 (s)] is the ﬁnite-dimensional linear time invariant (FDLTI) plant which

we have assigned the following state-space realisation G(s) ∼

x˙ p = Ap xp + Bp um + Bpd d

(4)

y = Cp xp + Dp um + Dpd d

where xp ∈ Rnp is the plant state, um ∈ Rm is the control input to the plant, y ∈ Rq is the plant (measurement) output and d ∈ Rnd is the disturbance. The FDLTI controller K(s) is assumed to be implemented as K(s) = [K1 (s)

K2 (s)]

and is assigned the following state-space realisation x˙ c = Ac xc + Bc ylin + Bcr r K(s) ∼ ulin = Cc xc + Dc ylin + Dcr r

(5)

where xc ∈ Rnc is the controller state, ulin ∈ Rm is the linear controller output r ∈ Rnr is the disturbance on the controller, normally the reference. The overall controller output is given by u = ulin + θ1 ∈ Rm where θ1 is a signal produced by

d

r K(s)

+ -

u

um G(s)

θ1 + + +

θ2

Θ (s)

Fig. 2. Basic anti-windup architecture

y

Anti-windup Compensation using a Decoupling Architecture

127

the anti-windup compensator; if the anti-windup compensator is inactive, u = ulin . ylin ∈ Rq is the linear input to the controller and is given by y + θ2 ∈ Rq ; if the anti-windup compensator is inactive ylin = y. The plant input, um and the controller output are related through the nonlinear function sat : Rm → Rm which is deﬁned as

⎡

⎤ sat1 (u1 ) ⎢ ⎥ .. ⎥ sat(u) = ⎢ . ⎣ ⎦ satm (um )

where sati (ui ) = sign(ui ) min(|ui |, u ¯i )

(6)

∀i and Dzi (ui ) = sign(ui ) max(0, |ui | −

u ¯i ) ∀i. Also u ¯i > 0 ∀i ∈ {1, . . . , m}. The anti-windup compensator is driven by the function Dz : Rm → Rm

⎡

⎤ Dz1 (u1 ) ⎢ ⎥ .. ⎥ Dz(u) = ⎢ . ⎣ ⎦ Dzm (um )

(7)

sat(u) = u − Dz(u).

(8)

and

Both the deadzone and saturation functions satisfy ‘sector bounds’. A nonlinear function N (.) is said to belong to the Sector[0, E] for some diagonal matrix E if the following inequality holds for some diagonal matrix W (see [20] for a full description): N (u) W [Eu − N (u)] ≥ 0

∀u ∈ Rm

(9)

Both the saturation and deadzone functions lie in the Sector[0, I], as they are decentralised functions each of which has a graph lying below the unity gradient line. It is often convenient to use the set U := [−¯ u1 , u ¯1 ] × [−¯ u2 , u ¯2 ] × . . . × [−¯ um , u ¯m ] This set has the interpretation that sat(u) = u,

∀u ∈ U, and Dz(u) = 0,

(10) ∀u ∈

U. As it stands, the scheme depicted in Figure 2 does not have any illuminating features. However, consider a particular case of the generic scheme, as shown in Figure 3 (introduced in [38]), where the anti-windup compensator Θ(s) has been parameterised in terms of a transfer function M (s) ∈ RH ∞ and a copy of the plant G2 (s). Using the identity (8), Figure 3 can be re-drawn as the decoupled scheme in Figure 4. Notice that this system exhibits an attractive decoupled structure:

128

Matthew C. Turner, Guido Herrmann, Ian Postlethwaite d r K

u lin + − ud M−I

y lin + +

y

G

+ −

yd

G2 M

~ u

Fig. 3. Conditioning with M (s) M−I

ud

Disturbance Filter

Nonlinear Loop

− ~ u

+

r

d K

G

u lin

yd

G 2M

+ ylin

− y

Nominal Linear Transfer Function

Fig. 4. Equivalent representation of Figure 3

1. Nominal linear system. This represents the linear system which would have resulted if no saturation was present. 2. Nonlinear loop. This contains the nonlinear stability problem and assuming that the linear loop is stable and the linear operator G2 M is also stable, the entire nonlinear stability problem is contained within this part of the system 3. Disturbance ﬁlter. This part of the system determines how the system recovers after a saturation event has stopped. This part is responsible for both the speed and manner of recovery after saturation has ceased. In [39], it was shown that most anti-windup schemes can be interpreted as certain choices of M (s) and therefore schemes such as the Hanus conditioning scheme ([12]) and the high gain approach ([4, 19]) can be analysed in terms of Figure 4. The advantages of viewing anti-windup in terms of Figure 4 is that the nominal lin-

Anti-windup Compensation using a Decoupling Architecture

129

ear performance is separated from the nonlinear part of the scheme. Moreover, the stability of the scheme is dependent on the stability of the nonlinear loop, assuming stability of the nominal linear closed loop and stability of the plant. This leads to our ﬁrst assumption. Assumption 1 •

The poles of

I

−K2 (s)

−G2 (s)

I

−1 are in the open left-half

complex plane. • The limit lims→∞ (I − K2 (s)G2 (s))−1 exists.

The ﬁrst point in this assumption ensures that all nominal closed loop transfer functions are asymptotically stable (in RH ∞ ) and if r = 0 and d = 0, then limt→∞ [xp (t) xc (t)] = 0. The second assumption ensures that the nominal linear system is well-posed, i.e. unique solutions exist to the feedback equations. These assumptions will be satisﬁed in most circumstances. From Figure 4, it can be seen that the performance of the overall closed loop system, including the anti-windup compensator, is closely related to the mapping Tp : ulin → yd . This mapping represents the deviation from nominal linear behaviour in response to a saturation event and can be used as a measure of the anti-windup compensator’s performance. If some appropriate norm of this mapping is small, then the anti-windup compensator is successful at keeping performance close to linear (which we assume is the desired performance). In fact, as the AW compensator is parameterised by M (s), the choice of M (s) dictates the system’s stability properties under saturation. This chapter ultimately proposes different approaches to choosing M (s) such that performance during saturation is improved. In [35] (see also [32, 13]), the L2 gain of Tp was minimised using a system of linear matrix inequalities and, furthermore, M (s) was chosen such that it corresponded to static or low order anti-windup compensators. The result of [35] demonstrated, using suitable examples, that direct minimisation of Tp was central to good anti-windup performance, and compensators designed according to the ideas in [35] seemed to perform at least as well, and often better, than most other anti-windup compensators. This idea was extended to more general cases in [15].

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3 Global Anti-Windup Results In this section, we treat the case where we are interested in global stability of the closed-loop system and in the global (possibly sub-optimal) minimisation of the L2 gain of the operator Tp . From Figure 4, note that in order to have the widest possible choices of M (s), it would seem sensible to require that G2 (s) ∈ RH ∞ , thus M (s) could be chosen as any M (s) ∈ RH ∞ such that the nonlinear loop is asymptotically stable and well-posed. In fact, this intuition is in agreement with results cited earlier where it was proven that in order to obtain global results we must ensure that G2 (s) ∈ RH ∞ 1 . Our second assumption is thus. Assumption 2 Re(λmax (Ap )) < 0 We say that an anti-windup compensator has solved the anti-windup compensation problem if it is stable, not active until saturation occurs, and if, after saturation has occured, it allows the system to recover, asymptotically, linear behaviour. Formally, this is deﬁned as follows: Deﬁnition 1 The anti-windup compensator Θ(s) is said to solve the anti-windup problem if the closed loop system in Figure 4 is internally stable, well-posed and the following hold: 1. If dist(ulin , U) = 0,

∀t ≥ 0, then yd = 0,

∀t ≥ 0 (assuming zero initial

conditions for Θ(s)). 2. If dist(ulin , U) ∈ L2 , then yd ∈ L2 . The anti-windup compensator Θ(s) is said to solve strongly the anti-windup problem if, in addition, the following condition is satisﬁed: 3. The operator Tp : ulin → yd is well-deﬁned and ﬁnite gain L2 stable. Remark 1: In previous papers ([35], [32]), we have also considered more general Lp norms, p ∈ [0, ∞). The results given in this chapter are easily extendable to these cases, although the optimisation is simple with the L2 norm. 1

Recall that we require a ﬁnite L2 gain to hold; this is only possible globally if G2 (s) ∈ RH ∞ , although local L2 gains hold for a wider class of systems

Anti-windup Compensation using a Decoupling Architecture

131

Note that in our parameterisation of anti-windup compensators, stability of the system in Figure 4 is equivalent to stability of the system in Figures 2 and 3.

3.1 Full Order Compensators An appealing choice of M is as a part of a right comprime factorisation of G2 (s) = N (s)M (s)−1 where M, N ∈ RH ∞ . This means that the disturbance ﬁlter is given as yd (s) = N (s)˜ u(s) and is hence a stable transfer function. The central reason for the appeal of this choice of M (s) is that this means that the anti-windup compensator M (s) − I Θ(s) = (11) N (s) has order equal to that of the plant, providing the coprime factors share the same state-space. A state-space realisation of the operator Tp is then easily calculated (using [42], for example) as 2

M (s) − I N (s)

⎡

⎤⎡

⎤

Ap + Bp F Bp x˙ ⎢ 2⎥⎢ ⎥ x2 ⎥⎢ ⎥ ∼⎢ u F 0 d ⎣ ⎦⎣ ⎦ u ˜ yd Cp + Dp F Dp

(12)

where we have used the state x2 ∈ Rnp to represent the anti-windup compensator’s state and to distinguish it from the plant state. Another endearing property of this choice of AW compensator is that there is no direct-feedthrough from u ˜ to ud which prevents algebraic loops. The following is the main result of the section: Theorem 1. Under Assumptions 1 and 2, there exists a full anti-windup compensator Θ(s), described by equations (11) and (12), which solves strongly the anti-windup problem if there exist matrices Q > 0, U = diag(μ1 , . . . , μm ) > 0,L ∈ R(m×n and a postive real scalar γ such that the following LMI is satisﬁed 2

We could in fact use a more general coprime factorisation as discussed in [36] and [15], although it is not clear that this gives us much advantage over the simple factorisation described above

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Matthew C. Turner, Guido Herrmann, Ian Postlethwaite

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

QAp + Ap Q + L Bp + Bp L Bp U − L 0 QCp + L Dp

−2U

I

U Dp

−γI

0

−γI

⎤ ⎥ ⎥ ⎥<0 ⎥ ⎦

(13)

Furthermore, if this inequality is satisﬁed, a suitable F achieving Tp i,2 < γ is given by F = LQ−1 .

Proof: The idea behind the proof is also germane to many results in this chapter. Well-posedness. Note from equation (12), all equations which comprise the operator Tp are explicit; hence the sytem is, trivially, well-posed. Stability and L2 gain. If there exists a positive deﬁnite symmetric matrix P > 0 which satisiﬁes J=

d x2 (t) P x2 (t) + yd 2 − γ2 ulin 2 < 0 ∀[x2 dt

u ˜

ulin ] = 0

(14)

it follows that 1. If ulin = 0, then there exists a P > 0 such that v(x2 ) = x2 P x2 is a Lyapunov function for the system; hence the system is globally stable. 2. Integrating J from 0 to T and taking the limit as T → ∞ y 22 < γ2 ulin 22 + (x2 (0) P x2 (0) − x2 (∞) P x2 (∞))

(15)

which implies L2 gain less than γ from ulin to yd (and thus Item 3). The remainder of this part of the proof involves simplifying the expression for J. First, note that as the deadzone function is such that Dz ∈ Sector[0, I] we have that ˜2i , u ˜ i ui ≥ u

∀i ∈ {1, . . . , m}

(16)

From this, it follows that for some matrix W = diag(w1 , . . . , wm ) > 0 ˜) ≥ 0 u ˜ W (u − u Thus it follows that a sufﬁcient condition for J < 0 is if

(17)

Anti-windup Compensation using a Decoupling Architecture

d J˜ = x2 (t) P x2 (t)+ yd 2 −γ2 ulin 2 +2˜ u W (u−˜ u) < 0 ∀[x2 dt

u ˜

133

ulin ] = 0 (18)

where u = ulin − ud . Hence using the state-space realisation (12), this becomes ⎡ ⎢ ⎢ ⎣

(Ap + Bp F ) P + P (Ap + Bp F ) + (Cp + Dp F ) (Cp + Dp F )

P BP + (Cp + Dp F ) Dp − F W −2W +

Dp Dp

⎤ 0

⎥ W ⎥ ⎦<0 −γ2 I

(19)

The LMI in the theorem then follows using the standard Schur complement and congruency transformation arguments, as reported, for example, in [25] (see also [35, 32], although the details are not the same). Item 1 of Deﬁnition 1. This follows from the stability of the nonlinear loop above, assuming that x2 (0) = 0. Item 2 of Deﬁnition 1. This is implied directly from Item 3. . Remark 2: Notice that a full-order anti-windup compensator always exists for any stable plant G2 (s). Simply choosing F = 0 recovers the internal model control (IMC) solution ([41]), although this may not yield good performance if the plant has slow or lightly damped modes.

3.2 Static Compensators The main problem with full-order anti-windup compensators is that they require more on-line computation to implement due to them having np states which require updating every cycle, assuming that the AW compensator will be implemented in discrete-time. Thus, it is more appealing to implement static anti-windup compensators - compensators which have no dynamics. In order to do this, we can no longer choose M (s) as part of a coprime factorisation of the plant, but must choose M (s) so that it essentially ensures that Θ is purely static. Comparing Figure 2 with Figure 3 we can see that if we choose Θ as

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Matthew C. Turner, Guido Herrmann, Ian Postlethwaite

θ1 θ2

= Θ˜ u=

Θ1 Θ2

u ˜,

θ1 θ2

∈ Rm+q ,

u ˜ ∈ Rm

(20)

then u = K1 r + K2 y + (K2 Θ2 − Θ1 )˜ u

(21)

˜ u = K1 r + K2 y + (−[(I − K2 G2 )M − I]) u

(22)

Obviously, for the two schemes to be equivalent, we must have M = (I − K2 G2 )−1 (−K2 Θ2 + Θ3 )

(23)

where Θ3 := Θ1 + I. Note that M (s) is well deﬁned by virtue of Assumption 1. This implies that M − I = (I − K2 G2 )−1 (−K2 Θ2 + Θ3 ) − I

(24)

G2 M = G2 (I − K2 G2 )−1 (−K2 Θ2 + Θ3 )

(25)

A state-space realisation for the transfer functions M (s) − I and G2 (s)M (s) can then be given as

M (s) − I G2 (s)M (s)

⎧ ¯x + (B0 + BΘ)˜ ¯ u ⎪ ¯˙ = A¯ ⎪ ⎨x ¯ 1 Θ)˜ ∼ ud = C¯1 x ¯ + (D01 + D u ⎪ ⎪ ⎩ y = C¯ x ¯ u d 2 ¯ + (D02 + D2 Θ)˜

(26)

where Θ is a static matrix by virtue of Θ1 , Θ2 being static and the precise form of the state-space matrices is described in Appendix A. Using this, we can then derive conditions which ensure the existence of a static compensator. Theorem 2. Under Assumptions 1 and 2, there exists a static anti-windup compensator Θ = [Θ1 Θ2 ] ∈ R(m+q)×m which solves strongly the anti-windup problem if there exist matrices Q > 0, U = diag(μ1 , . . . , μm ) > 0,L ∈ R(m+q)×m and a postive real scalar γ such that the following LMI is satisﬁed ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

¯ + AQ ¯ QA

⎤ ¯ − QC ¯1 B0 U + BL 0 QC2 ¯ 1 − D ¯ 2 ⎥ ¯ 1 L I U D02 ⎥ −2U − U D01 − D01 U − L D + L D ⎥<0 ⎥ −γI 0 ⎦ −γI

(27)

Furthermore, if this inequality is satisﬁed, a suitable Θ achieving Tp i,2 < γ is given by Θ = LU −1 .

Anti-windup Compensation using a Decoupling Architecture

135

Proof: Much of the proof follows in a similar manner to the full-order results. Stability and L2 gain. Again stability and L2 gain of the operator Tp follow if we can ensure that d x ¯(t) P x ¯(t) + yd 2 − γ2 ulin 2 < 0 ∀[¯ x u ˜ ulin ] = 0 (28) dt Appending the sector condition (17) to this, we have that J < 0 if J˜ < 0, where J=

d J˜ = x ¯(t) P x ¯(t) + yd 2 − γ2 ulin 2 + 2˜ u W (u − u ˜) < 0 ∀[¯ x u ˜ dt

ulin ] = 0 (29)

and u = ulin − ud . Hence using the state-space realisation (26) this becomes ⎡ ⎤ ⎡ ⎤ ⎤⎡ x ¯ A¯ P + P A¯ + C¯2 C¯2 V12 0 x ¯ ⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎢ u ⎥ ⎢ ⎥<0 ⎥ ⎢ ˜ u ˜ V W 22 ⎣ ⎦ ⎣ ⎦ ⎦⎣ ulin

−γ2 I

ulin

where ¯ − W C¯1 + C¯2 (D02 + D ¯ 2 Θ) V12 := P B0 + P BΘ ¯ 2 Θ) (D02 + D ¯ 2 Θ) V22 := −2W + (D02 + D ¯ 1 Θ) − (D01 + D ¯ 1 Θ) W −W (D01 + D The LMI in the theorem then follows using standard Schur complement and congruency transformation arguments, as reported, for example, in [25] or [35]. Well-posedness. It transpires that well-posedness is guaranteed by the (2,2) element of the LMI being negative deﬁnite. Hence satisfaction of this LMI ensures wellposedness of Tp . See [35] for this part of the proof (another way of proving this is given in [9]). Item 1 of Deﬁnition 1. This follows from the AW compensator being static and linear, and from internal stability of the nonlinear loop (guaranteed in the ﬁrst part of the proof). Item 2 of Deﬁnition 2. This follows directly from Item 3, which holds by virtue of the L2 gain property proved earlier. . Remark 3: Note that, in contrast to full-order anti-windup compensators, there is no guarantee that a static compensator will exist for an arbitrary linear controller or

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Matthew C. Turner, Guido Herrmann, Ian Postlethwaite

anti-windup compensator. Furthermore, the L2 gain of such a compensator may not

be as low as the full-order compensator.

4 Low-Order Anti-Windup Compensator Synthesis Globally stabilising full-order compensators always exist for linear plants satisfying Assumption 2. (i.e. G(s) ∈ RH ∞ ). It was proved in [9] and [15] that such compensators yield “optimal” anti-windup performance in terms of the achievable L2 gain. However, a number of shortcomings can be present in such compensators. Firstly, the synthesis of such compensators is purely dependent on the plant, and no frequency domain information is used explicitly. This may give rise to anti-windup compensators with very fast poles and ones which are unrealistically responsive to saturation. Furthermore, as such compensators have order equal to that of the plant, their implementation may not be practicable. On the other hand, it is well documented that static compensators can be unreliable or infeasible ([3, 11]) in some circumstances. Thus, there is some middle ground to be explored, in particular the design of “reduced” order compensators - dynamic - AW compensators with state dimension less than that of the plant. The basis for our work is that of [35], although other work has been contributed by [6] and elsewhere. Our proposal retains much of the simplicity of the static approach, except we now let our compensator be described by the dynamic equations

Θ(s) =

Θ1 (s) Θ2 (s)

∈ R (m+q)×m

(30)

where ˜1 Θ1 (s) := F1 (s)Θ

(31)

˜2 Θ2 (s) := F2 (s)Θ

(32)

˜ 1 ∈ Rm×m F1 (s) ∈ R m×m and F2 (s) ∈ R q×q are transfer function matrices and Θ ˜ 2 ∈ Rq×m are constant matrices. A possible way of synthesising Θ1 (s) and and Θ Θ2 (s) would be to ﬁnd them directly using some form of optimisation (although this in general is non-convex - see [9]). However, this is fairly complicated and could lead to unsuitable compensators suffering from large poles and a sensitive numerical structure.

Anti-windup Compensation using a Decoupling Architecture

137

Instead we propose that F1 (s), F2 (s) be chosen using intuition and experience, and ˜ 1, Θ ˜ 2 are synthesised in an optimal way, similar to the pure static synthesis dethen Θ scribed earlier. Obviously, the resulting compensator will be sub-optimal in terms of its L2 gain but simulation results have shown that using relatively simple choices for F1 (s), F2 (s), such as ﬁrst order low pass ﬁlters (which suppress high frequency signals in yd ), very good responses can be obtained. Furthermore, bearing in mind that in the full order case, the choice for Θ2 (s) would be G2 (s)M (s), a sensible choice ˆ 2 (s), a reduced order version of G2 (s). This approach was for Θ2 (s) would be G used in [14] where discrete-time versions of the formulae here were implemented on a hard-disk drive system.

4.1 A Sub-Optimal Synthesis Routine As before we obtain that M = (I − K2 G2 )−1 (−K2 Θ2 + Θ3 )

(33)

˜ 1 and Θ2 (s) = F2 (s)Θ ˜ 2 are dynamic. Similar except that this time Θ1 (s) = F1 (s)Θ to the static case, a minimal state-space realisation for M (s) − I and G2 (s)M (s) can be derived as

M (s) − I G2 (s)M (s)

⎧ ¯x + (B0 + B ¯ Θ)˜ ˜ u ⎪ ¯˙ = A¯ ⎪ ⎨x ˜ u ¯ 1 Θ)˜ ∼ ud = C¯1 x ¯ + (D01 + D ⎪ ⎪ ⎩ y = C¯ x ¯ ˜ u d 2 ¯ + (D02 + D2 Θ)˜

˜ is given as Θ ˜ := [Θ ˜ where the matrices3 are described in Appendix B and Θ 1

(34)

˜ ] . Θ 2

Similar to the static case we have the following theorem: Theorem 3. Let Assumptions 1 and 2 be satisﬁed. Given F1 (s), F2 (s), where ˜ ] ∈ R(m+q)×m , then ˜ = [Θ ˜ Θ deg(F1 (s)) = n1 and deg(F2 (s)) = n2 , and Θ 1

2

there exists an (n1 + n2 )’th order anti-windup compensator of the form ˜1 F1 (s)Θ Θ(s) = ˜2 F2 (s)Θ 3

(35)

The matrices A¯ etc. are different from those in the previous section because they also incorporate the dynamics of Θ(s).

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which solves strongly the anti-windup problem if there exist matrices Q > 0, U = diag(μ1 , . . . , μm ) > 0, L ∈ R(m+q)×m and a positive real scalar γ such that the LMI (27) is satisﬁed (with the state-space realisation of equation (34)). Further˜ achieving Tp i,2 < γ is given by more, if inequality (27) is statisﬁed, a suitable Θ ˜ = LU −1 . Θ Proof: The proof is similar, mutatis mutandis, to that of Theorem 2.

As with static AW compensators, there is no guarantee that a low-order compensator will exist for a given plant-controller combination, although there is probably more likelihood of one existing. There is no exact formula for computing a low-order compensator although we have found that one of the following procedures can work well ˆ 2 (s) such that deg(G ˆ 2 (s)) < 1. Compute a reduced order approximation of the plant, G ˆ 2 (s). deg(G2 (s)). Set F2 (s) as G 2. Choose F1 (s) as a low pass ﬁlter and set its bandwidth to ensure high frequency anti-windup action is prevented. ˜2 . ˜ 1 and Θ 3. Use Theorem 3 to choose Θ 4. Iterate, using different ﬁlter bandwidths and higher order plant approximations. or ˆ 2 (s) such that 1. Compute a reduced order approximation of the plant G ˆ 2 (s)) < deg(G2 (s)). deg(G ˆ −1 (s)N ˆ (s). ˆ 2 (s) = M 2. Form a right comprime factorisation G ˆ (s) − I; set F2 (s) = N ˆ (s). 3. Set F1 (s) = M ˜ 2. ˜ 1 and Θ 4. Use Theorem 3 to choose Θ 5. Iterate using higher order plant approximations and different coprime factorisations Note that the second procedure could also involve the synthesis of F in the same way as with the full-order case.

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139

5 Robustness The techniques described in the foregoing sections should allow the synthesis of antiwindup compensators which perform well during saturation; the results, however, do not guarantee good performance in the presence of saturation and plant uncertainty, which is inevitable in practice. Preserving our decoupled and coprime factor based ˜ is the true thinking, we must now consider the situation depicted in Figure 5, where G ˜ plant given by G(s) = [G1 (s) G2 (s) + ΔG (s)], where G(s) = [G1 (s) G2 (s)] is the model of the plant with which we work and ΔG is additive uncertainty to the feedback part4 which is assumed to be stable and linear. Other types of uncertainty ˜ 2 (s) = (I + Δo (s))G2 (s), and such as output-multiplicative uncertainty, where G ˜ 2 (s) = G2 (s)(I + Δi (s)) could be used input multiplicative uncertainty, where G instead. However, it is easy to see that both these uncertainties can be captured by additive uncertainty (ΔG = Δo G2 or ΔG = G2 Δi ), although the converse is not always true (unless G2 is invertible), so we prefer to work with additive uncertainty. We assume that little is known about the arbitrary, linear time invariant uncertainty ΔG (s) other than the fact that it belongs to a certain ball of uncertainty Δ deﬁned as follows

+ , 1 Δ = Δ ∈ RH ∞ : Δ(s) ∞ < γr

(36)

When uncertainty is present in the system, the appealing decoupled structure of the original scheme is lost. Figure 6 shows an equivalent representation of Figure 5 . Note that the mapping from u ˜ to yΔ (i.e. ΔG (s)M (s)) destroys the decoupling and therefore even if the uncertainty linear system is stable, and the nonlinear loop is stable, this does not guarantee stability of the whole closed loop system. It can be observed, that in order to tackle the problem of anti-windup for uncertain systems we have to work under different assumptions to the standard anti-windup problem and solve the problem in a slightly different manner. Thus, Assumption 3 is Assumption 3 The system in Figure 6 is well-posed and 1. G(s) ∈ RH ∞ 4

It is likely that there will also be a perturbation of the disturbance feedforward portion of the plant, G1 , although this will have no bearing on stability, so for simplicity we do not consider it

140

Matthew C. Turner, Guido Herrmann, Ian Postlethwaite d r

~ G

u lin + − ud

K

M−I

y lin + +

y

+ −

yd

G2 M

~ u

Fig. 5. Anti-windup with uncertainty

M−I

Disturbance Filter

ud Nonlinear Loop

− +

~ u

M − +

r

d G K

u lin

G 2M

ΔG ++

+

yΔ ylin

− y d y

Fig. 6. Equivalent representation of Figure 5

2. Δ(s) ∈ Δ 3. The poles of all Δ ∈ Δ.

I −K2 (s) ˜ 2 (s) −G I

−1 are in the open left-half complex plane for

The ﬁrst point of the assumption is identical to Assumption 2; the second is necessary for us to obtain the small-gain type results we consider here. The third ensures that the linear system itself (i.e. with no saturation) is stable for all relevant Δ(s) essentially this assumes that a K(s) is a suitable robust linear controller. If Assumption 3 is satisﬁed, four features are evident from Figure 6: 1. If ΔG is small in some sense, then the robustness of the anti-windup scheme is similar to that of the nominal, unconstrained linear system (via a Small Gain argument).

Anti-windup Compensation using a Decoupling Architecture

141

2. If the mapping from ulin → M u ˜ is small, again, the robustness of the antiwindup system is similar to that of the nominal linear system (again using a Small gain argument). So in other words the map ulin → M u ˜ contains important robustness information. 3. The robustness of the system with anti-windup compensation can never be better than the robustness of the linear system. Denoting the ‘modiﬁed’ uncertainty by ˜ G : ulin → yΔ , this follows by noting that ΔG ∞ = ΔG i,2 ≤ Δ ˜ G i,2 Δ (by using a contradiction argument). So, in a sense, the retention of the linear system’s robustness can be considered as an optimal property (discussed in more detail later). 4. In order to obtain ‘maximal’ robustness for a given Δ, it is desirable that the map from ulin to the input of the uncertainty is as small as possible in some appropriate norm.

Stability Robustness Optimisation As noted above, in order to achieve maximal robustness we would like the map from ulin to the input of the uncertainty to have as small gain as possible. This can be depicted graphically in Figure 7. Thus, we would like to minimise the map Tr i,2 := [I − M F (ulin )] i,2 where F (.) : Rm → Rm denotes the nonlinear map from ulin to u ˜. As mentioned in [35], this optimisation is typically a difﬁcult problem to solve, so instead we seek to ensure a certain L2 gain bound holds for the map Tr .

M−I ud u lin

+

− ~ u

M

uΔ − +

z

Fig. 7. Robustness optimisation for general anti-windup schemes: graphical representation of Tr

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Matthew C. Turner, Guido Herrmann, Ian Postlethwaite

To make our discussion brief, we only consider full-order anti-windup compensation explicitly; similar (although not as strong) results can be obtained for low-order and static anti-windup compensation. Thus we choose M (s) to be part of a right-coprime factorisation of G2 (s) = N (s)M −1 (s) and we let the anti-windup compensator have the state-space realisation given in equation (12). In this case, it is obvious that the map F has state-space realisation: ⎤ ⎤ ⎡ ⎡ Ap + Bp F Bp x˙ 2 ⎥ x2 ⎥ ⎢ ⎢ M (s) − I ⎥ ⎥=⎢ ∼⎢ u F 0 d ⎣ ⎦ u ⎦ ⎣ M (s) ˜ F uΔ I

(37)

To ensure robustness, that is to ensure that Tr i,2 < γr , it sufﬁces for the following inequality to hold for sufﬁciently small γr : J=

d x P x2 + z 2 − γ2r ulin 2 < 0, dt 2

∀ [x2 u ˜ ulin ] = 0

(38)

where x2 is the state vector associated with the realisation of [M − I, M ]. As shown in, for example [10, 35] this ensures that the L2 gain from ulin to z is less than γr and that the system in Figure 7 is asymptotically stable. As before, we use the sector bound (17) to get d J˜ = x2 P x2 + z 2 − γ2r ulin 2 + 2˜ u W (ulin − ud − u ˜) dt

(39)

If J˜ < 0, this implies that J < 0. Evaluating J˜ in a similar manner to before (see [33]) yields the following LMI ⎤ ⎡ QAp + Ap Q + L Bp + Bp L Bp U − L 0 L ⎥ ⎢ ⎥ ⎢ −2U I U ⎥<0 ⎢ ⎥ ⎢ −μI −I ⎦ ⎣ −I

(40)

in Q > 0, U = W −1 = diag(ν1 , . . . , νm ) > 0, L, μ > 0. Satisfaction of this LMI means that inequality (39) is satisﬁed and hence that the √ L2 gain from ulin to z is less than γr = μ and a suitable choice of F is given by −μI −I F = LQ−1 . From the term of this LMI we can see that, as anticipated −I earlier, the L2 gain can be no less than unity, which is the robustness of the nominal linear loop without uncertainty.

Anti-windup Compensation using a Decoupling Architecture

143

Remark 4: An important example of a robust anti-windup compensator is the much maligned IMC scheme. In [33] (see also [1]) it was proved that the IMC scheme corresponded to an ‘optimally robust’ anti-windup compensator. In the framework considered here, the IMC scheme results from choosing M (s) = I. This implies that Tr (ulin ) = sat(ulin ) (see Figure 7) and thus this choice corresponds to the

recovery of ‘linear’ robustness.

Optimisation for Robustness and Performance The primary goal of anti-windup compensation is to provide performance improvement during saturation, but optimising the LMI (40) alone does not guarantee this. Indeed, there is little point in optimising (40) when an optimal solution can be found by inspection as the IMC anti-windup solution. The real use of (40) and the arguments of the previous subsection is to use them in conjunction with performance optimisation, the goal being to optimise performance and robustness together, although there will often be a trade-off. Earlier, we argued that a logical choice of operator for which we would like to minimise the L2 gain was given by Tp , that is the map from ulin to yd . For good performance and robustness, it therefore follows that we would like to optimise according to min Tp i,2

s.t. Tr i,2 < γr

This is generally difﬁcult so what we shall be content to ensure that W T p p <γ Wr Tr

(41)

(42)

i,2

where Wp and Wr are static weighting matrices which can be chosen to capture the required trade-off in performance and robustness. Note that satisfaction of the above ensures that Wp Tp i,2 < γ and Wr Tr i,2 < γ

(43)

To guarantee robustness to all uncertainty within the set Δ we need to ensure that γσ(Wr−1 ) ≤ γr . The minimsation procedure of (42) can be approximated by the following LMI optimisation

144

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Matthew C. Turner, Guido Herrmann, Ian Postlethwaite

QAp + Ap Q + L Bp + Bp L Bp U − L 0 QCp + L Dp

L

−2U

I

U Dp

U

−γI

0

−I

−γWp−1

0

−γWr−1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎦ (44)

in the variables Q > 0, U = diag(ν1 , . . . , νm ) > 0, L, γ > 0. Wp and Wr are positive deﬁnite weighting matrices which reﬂect the relative importance of performance and robustness respectively and are chosen by the designer. The derivation of this LMI is carried out in a similar way to that of the previous section in the spririt of that done in [35]. Remark 5: Throughout this section, we have mainly discussed full-order antiwindup compensation for two reasons: (i) A full-order anti-windup compensator always exists, and (ii) the expressions and derivations of formulae for static and low-order anti-windup compensators are more complex, although the same ideas are certainly applicable to these types of compensator.

Remark 6: Another advantage of using LMI (44) to synthesise full-order compensators is that it tends to prevent fast poles appearing in the compensator dynamics. If a robustness weight (Wr ) was not included in the optimisation - or if Wr was only chosen small - the poles of the anti-windup compensator would tend to be rather fast, lying far to the left of the imaginary axis. Obviously this would require a very high sampling frequency for implementation, which is not always possible in practice. However, when simultaneously optimising performance and robustness using (44), the poles are placed in regions more comparable to that of the controller. This feature is reminiscent of solving ‘singular’ H∞ problems with LMI’s, where poles tend to get placed far from the imaginary axis.

Similar results can be obtained for robustness of static and low order compensators although the results are not as strong. As low order and static AW compensators which guarantee L2 gains are not guaranteed to exist, it follows that robust versions of these compensators will not exist either. However, AW compensators which guarantee locally robust performance may well exist, see [33] for more remarks on this.

Anti-windup Compensation using a Decoupling Architecture

145

6 Local Anti-windup The previous sections considered the design of global anti-windup compensators, that is AW compensators which ensure global (quadratic) stability of the complete closed-loop system together with certain L2 gain properties. Unfortunately, for many systems, it is impossible to obtain global results. This class of systems includes all anti-windup problems in which the plant is exponentially unstable, as well as static anti-windup problems which are (globally) infeasible. Also, it is often possible to obtain improved local performance by relaxing the restriction of global feasibility. In practice, this is a fairly mild assumption as states and exogenous inputs are normally restricted to some subset of the space in which they are deﬁned. Thus, antiwindup compensators which only guarantee local stability and performance warrant consideration.

6.1 Sharper Sector Bounds Essentially, we use ideas introduced in [17] and [26], where it was observed that, locally, the deadzone nonlinearity inhabits a narrower sector than Sector[0, I]. Consider Figure 8. Dz i(u i)

_β_ i ui

__ ui

Sector(0,α i)

Sector(0,α i) _ ui

_ βi ui

ui

Fig. 8. The deadzone and Sector bounds

For any u, the deadzone nonlinearity remains in the Sector[0, I], that is, the graph of each component remains below the unity gradient line. However, let us suppose that ui never becomes greater than βi u ¯i ,

∀i, where βi > 1,

∀i, then locally we

can see that the graph of Dzi (ui ) remains below the gradient αi :=

βi −1 βi

< 1.

Thus for all u u ¯ it follows that the deadzone actually inherits the ‘sharper’ sector

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Matthew C. Turner, Guido Herrmann, Ian Postlethwaite

Sector[0, A] where A := diag(α1 , . . . , αm ) and αi :=

βi −1 βi

∈ (0, 1). This can be

seen in Figure 8. Note that A < I. Thus, it follows that if we can ensure that our control signal is such that u β¯ u

∀t ≥ 0 (where β = [β1 β2 . . . βm ] ), then our deadzone can be considered to

be in Sector[0, A] and we can obtain tighter bounds on the small-signal L2 gain and, potentially, prove local stability for systems where global stability is not achievable. Let us deﬁne the set V (β) := {u ∈ Rm : u β¯ u} .

(45)

Thus, for small signal L2 gain we need to consider sets W of the following form W = {ulin ∈ Rm : u(t) ∈ V (β) ∀t ≥ 0} .

(46)

Remark 7: W cannot simply be considered as the set W× := ulin ∈ Rm :

ulin β¯ u because u = ulin − ud and hence W and W× will not have the same geometry and may not satisfy any simple subset properties.

Similarly in order to prove local stability we need to consider positively invariant sets, X such that when ulin = 0 for all xnl (0) ∈ X , u β¯ u, where xnl ∈ Rnnl is the state vector of the nonlinear loop. Hence we deﬁne X := {xnl (0) ∈ Rnnl : u(t) ∈ V (β) ∀t ≥ 0}

(47)

where xnl (t) is the state of the dynamic system M (s) in the nonlinear loop. This would be x2 in the case of full-order AW, or x ¯ in the case of static or low order AW. It is now appropriate to deﬁne the local anti-windup problem Deﬁnition 2 The anti-windup compensator Θ(s) is said to solve the A-local anti-windup problem if the closed loop system in Figure 4 satisﬁes the following properties: 1. It is well-posed. 2. There exists a non-empty compact set Y such that the closed-loop system is internally stable for all xnl (0) ∈ Y ⊂ X 3. If dist(ulin , U) = 0, conditions for Θ(s)).

∀t ≥ 0, then yd = 0,

∀t ≥ 0 (assuming zero initial

Anti-windup Compensation using a Decoupling Architecture

147

4. There exists a non-empty set W such that if dist(ulin , U) ∈ L2 ∩ W , then yd ∈ L2 . The anti-windup compensator Θ(s) is said to solve strongly the A-local anti-windup problem if, in addition, the following condition is satisﬁed: 5. The operator Tp : ulin → yd is well-deﬁned and small signal ﬁnite gain L2 stable, that is Tp i,2,W ≤ γ for some γ > 0. The next section demonstrates how these concepts can be used in the synthesis of full-order anti-windup compensators. Remark 8: Trivially, the set Y¯ = {xnl (0) ∈ Rnnl : u(t) ∈ U, ∀ t ≥ 0} ⊂ X is a true subset of {xnl (0) ∈ Rnnl : u(t) ∈ V (β) ∀t ≥ 0} (actually V (1) = U) and the trivial solution for Y is Y = Y¯ in Deﬁnition 2. However, it is required that Y is a true superset of Y¯ , i.e. Y ⊂ Y¯ for Deﬁnition 2 to be useful.

6.2 Full Order Anti-Windup Synthesis As before, a logical choice for M (s) is to be part of a coprime factorisation of G(s) and thus the state-space realisation of the nonlinear-loop and disturbance ﬁlter remains the same as before. However, in contrast to the last section we shall assume that we require local results, that is for systems where the control signal u remains in the set V (β). The main result is the follwing theorem. Theorem 4. Under Assumption 1, there exists a full anti-windup compensator described by equations (11) and (12) which solves strongly the A-local anti-windup problem if there exist matrices Q > 0, U = diag(μ1 , . . . , μm ) > 0, L ∈ R(m+q)×m and a positive real scalar γ such that the following LMI is satisﬁed ⎤ ⎡ QAp + Ap Q + L Bp + Bp L Bp U − L A 0 QCp + L Dp ⎥ ⎢ ⎥ ⎢ −2U A U D p ⎥<0 ⎢ ⎢ ⎥ −γI 0 ⎣ ⎦ −γI

(48)

Furthermore, if this inequality is satisﬁed, a suitable Θ achieving Tp i,2,W < γ is given by F = LQ−1 .

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Matthew C. Turner, Guido Herrmann, Ian Postlethwaite

Proof: The theorem follows in much the same way as the proof for the global case presented earlier. The main difference is that the LMI which guarantees local stability and small-signal ﬁnite L2 gain has been modiﬁed. Consider

J=

d x2 (t) P x2 (t) + yd 2 − γ2 ulin 2 < 0 dt

∀[x2 u ˜ ulin ] = 0

(49)

where the state of the nonlinear loop xnl = x2 as we consider full order anti-windup. Let us assume that x2 and ulin are sufﬁciently small such that u ∈ V (β). From this it follows that Dz(u) ∈ Sector[0, A] and hence that for some matrix W = diag(w1 , . . . , wm ) > 0 we have ˜) ≥ 0 u ˜ W (Au − u

(50)

This can then be adjoined to equation (49) to get d J˜ = x2 (t) P x2 (t) + yd 2 − γ2 ulin 2 + u ˜ W (Au − u ˜) < 0 ∀[x2 u ˜ ulin ] = 0 dt (51) As in the foregoing proofs, it can be proved that J˜ < 0 if the LMI (48) is satisﬁed. Noting that J˜ ≥ J, and picking v(x2 ) = x2 P x2 > 0 as a Lyapunov function, local stability and small-signal L2 gain follow if J˜ < 0. This will be discussed next. First, local exponential stability will be shown. With ulin = 0, it follows from the

LMI (48) that there exists a level set L = x2 ∈ Rnp : x2 P x2 ≤ l2 , l > 0, for which

dv dt

≤ −δv for small δ > 0. Hence, local exponential stability follows. This

implies that there is a nontrivial, compact, positively invariant set Y , L ⊆ Y ⊂ X , which ensures local internal stability. Local exponential stability of the anti-windup compensated system is implies, from Theorem 6.5 of [20], that the operator ulin → yd is small-signal ﬁnite L2 gain stable. Thus for small enough ulin , it follows that u = ulin − F x2 ∈ V (β) and hence that there exists a non-empty set W such that a small-signal ﬁnite L2 gain holds.

Remark 9: Note that if Ap is exponentially unstable, there will never exist a globally stabilising anti-windup compensator; if Ap is critically stable, a compensator achieving a small-signal L2 gain only is guaranteed, although a globally stabilising compensator will exist. Thus, for such cases, it is convenient (and necessary) to consider a locally stabilising AW compensator. Such a compensator is guaranteed to exist for small enough βi > 1.

Anti-windup Compensation using a Decoupling Architecture

149

Remark 10: Many authors have, quite logically, taken as a measure of anti-windup compensator performance, the size of the domain of attraction of the system. However, it can be difﬁcult to relate this back to the behaviour of the saturating control signal. Furthermore, such a measure is complicated in the case of dynamic antiwindup compensators and, for state vectors with large dimensions, the analysis of such regions of attraction becomes fraught with further difﬁculty. Having said this, in principle such an approach is relatively easy to accommodate within our framework. To estimate the region of attraction for our system we follow the approach of [8] and assume that we seek an ellipsoid approximation of the form E = {x2 ∈ Rnp : x2 P x2 ≤ 1}

(52)

Then, in order that our control signal resides in the reduced sector we must have, when ulin = 0, that ¯i |ud,i | = |Fi x2 | ≤ βi u

∀i

(53)

¯2i ⇔ x2 Fi Fi x2 ≤ β2i u

∀i

(54)

Now if x2 ∈ E, the above holds if ¯2i x2 Fi Fi x2 ≤ x2 P x2 β2i u P Fi ⇔ ≥ 0 ∀i β2i u ¯2i I

∀i

Using the congruence transformation diag(P −1 , I), we have the LMI’s: Q Li ≥ 0 ∀i β2i u ¯2i I

(55) (56)

(57)

These can easily be incorporated into the LMI optimisation schemes advocated earlier, perhaps with the aim of minimising the following objective Γ = ηγ + (1 − η)det(Q)

η ∈ [0, 1]

(58)

where η is used to trade of the importance between local performance and region of attraction and, as the volume of an ellipsoid is inversely proportional to det(Q), minimising this quantity helps to maximise the region of attraction.

Remark 11: Even if a globally stabilising AW compensator exists (i.e. for all exponentially stable plants), it may not always be desirable to use this in practice. This is

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Matthew C. Turner, Guido Herrmann, Ian Postlethwaite

because it may be accompanied by unacceptably large L2 gains which may be indicative of poor performance. In practise though, if we impose a limit on the control signal magnitude and hence consider local results, it is possible to improve signiﬁcantly the local performance at the expense of abandoning global stability guarantees.

It is possible to guarantee local robustness i.e. to guarantee robust stability within the set W in a similar way to before, if we replace the LMI in Theorem 4 with an LMI similar to that in inequality (44).

6.3 Static and Low-Order Anti-Windup Synthesis The motivation behind static and low-order anti-windup synthesis is similar to that behind full-order AW synthesis: locally stabilising AW compensators may exist when globally stabilising ones do not; and improved local performance can often be obtained at the expense of discarding global stability guarantees. The main results of the section are the following two Theorems. Theorem 5. There exists a static anti-windup compensator Θ = [Θ1 Θ2 ] ∈ R(m+q)×m which solves strongly the A-local anti-windup problem if there exist matrices Q > 0, U = diag(μ1 , . . . , μm ) > 0,L ∈ R(m+q)×m and a postive real scalar γ such that the following LMI is satisﬁed ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

¯ + AQ ¯ QA

⎤ ¯ − QC ¯1 A B0 U + BL 0 QC2 ¯ 1 A − A D ¯ 2 ⎥ ¯ 1 L A U D02 ⎥ −2U − U D01 A − AD01 U − L D + L D ⎥<0 ⎥ −γI 0 ⎦ −γI (59)

Furthermore, if this inequality is satisﬁed, a suitable Θ achieving Tp i,2,W < γ is given by Θ = LU −1 .

Theorem 6. Given F1 (s), F2 (s), where deg(F1 (s)) = n1 and deg(F2 (s)) = n2 , ˜ ] ∈ R(m+q)×m , then there exists an (n1 + n2 )’th order anti˜ = [Θ ˜ Θ and Θ 1

2

windup compensator of the form Θ(s) =

˜1 F1 (s)Θ ˜2 F2 (s)Θ

(60)

which solves strongly the A-local anti-windup problem if there exist matrices Q > 0, U = diag(μ1 , . . . , μm ) > 0, L ∈ R(m+q)×m and a positive real scalar γ such that

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151

the LMI (59) is satisﬁed (with the state-space realisation of equation (34)). Further˜ achieving Tp i,2,W < γ is given more, if inequality (59) is statisﬁed, a suitable Θ ˜ = LU −1 . by Θ Proofs: These theorems are proved in a similar way to Theorems 2 and 3.

Remark 12: It is important to point out that, for an arbitrary linear closed loop system, there is no guarantee that a static or low order AW compensator yielding global performance exists (see [9]) - even if the plant is asymptotically stable. However, it can easily be shown (see [34]) that for a small enough sector bound, that is for small enough β, there will always exist a locally stabilising AW compensator, with local performance

Remark 13: As before, these performance based AW compensators can be combined with the ideas on robustness from the foregoing section to get compensators which are locally robustly stable.

6.4 Limiting the Control Signal Magnitude The last three theorems essentially provide a compensator which guarantees that the closed loop system will be stable provided the magnitude of the control signal u does not exceed a certain bound and, therefore, stray outside the local sector, Sector[0, A]. As the control signal is a function of the anti-windup compensator output, u = ulin − ud in the full order case, it appears logical to attempt to minimise, in a weighted sense, this signal in the optimisation process. In other words, we would like to minimise the L2 gain of the operator Wp y d ulin → . Wu (ulin − ud ) where Wp ∈ Rp×p is a static weighting matrix used to penalise the performance map and Wu ∈ Rm×m is a static weighting matrix used to penalise the control signal. Thus for the full order case, the following LMI can be used in Theorem 4

152

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Matthew C. Turner, Guido Herrmann, Ian Postlethwaite

QAp + Ap Q + L Bp + Bp L Bp U − L A 0 QCp + L Dp

L

−2U

A

U Dp

0

−γI

0

−I

−γWp−1

0

−γWu−1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎦ (61)

Similar LMI’s can be derived for the static and low-order compensator cases.

7 Examples 7.1 Global Anti-Windup Compensation: High Angle-of-Attack Aircraft Example To demonstrate and to make a comparative analysis of our results, we study a simple model of an aircraft, linearised at a high angle-of-attack, α (equations A2-A4 in [40]). This is an interesting model to study, partly due to the poor performance it exhibits under saturation constraints on the control magnitude and partly due to the highly coupled system characteristics which makes this plant an ideal test-bed for multivariable anti-windup techniques. The aircraft is described by a sixth-order state-space model consisting of the nonlinear rigid body dynamics, which have been linearised at a high angle-of-attack. At this ﬂight condition, the aircraft is exponentially stable. The controlled variables (which ˙ α, β can be obtained from the rigid body equations using simple trigonometry) are μ, which represent the stability-axis roll rate (μ˙ = p cos α+r sin α), the angle-of-attack and the side-slip angle, respectively. The plant inputs are δh , δa , δr which represent the elevator deﬂection, the aileron deﬂection and the rudder deﬂection, respectively. These actuators are subject to the following constraints: |δh | ≤ 25◦ , |δa | ≤ 20◦ and |δr | ≤ 30◦ . Due to interaxis coupling, it is suggested in [40] that H∞ control methods be used to control this aircraft. We therefore used the ﬁrst design approach advocated in [40] for our nominal controller design, which guarantees robustness against input and output perturbations of the plant. Hence, a twelfth order H∞ controller was designed from which an eighth order controller was obtained using balanced residualization methods.

Anti-windup Compensation using a Decoupling Architecture

153

100 μ α β

80

200

δh δa δ r

60

150

40 100 20 50

0 −20

0 −40 −50 −60 −80 −100 0

−100

5

10

15 time [sec]

20

25

30

0

5

Output Response

10

15 time [sec]

20

25

30

Control Responses

Fig. 9. Nominal linear response due to pulse demand in μ˙ μ α β

15

150 100

10 50

5

0

0

−50 −100

−5

−150

−10 −200

−15 0

δ h δa δ

−250

5

10

15 time [sec]

20

25

30

Output Response

0

r

5

10

15 time [sec]

20

25

30

Control Responses

Fig. 10. Nominal linear response due to pulse demand in β

Nominal Controller Performance Figures 9 and 10 show the response of the high-α aircraft to pulse demands in the μ˙ and β channels without saturation constraints (read μ˙ for μ in the graphs). Note that good nominal performance is obtained, but that there is high actuator activity in most of the control channels, which indicates a signiﬁcant amount of coupling. In particular, note that two of the three actuators transiently exceed the actuator constraints. Had these constraints been included in the simulation one might expect the results to deteriorate substantially.

Performance Subject to Saturation Figures 11 and 12 show the response of the high-α aircraft when the saturation constraints are imposed on the control signal for the same pulse demands as those used

154

Matthew C. Turner, Guido Herrmann, Ian Postlethwaite δ h δ a δ

30

100

r

20 50

10 0

0

−10

−50

−20 −100

0

μ α β 5

−30 10

15 time [sec]

20

25

30

0

5

Output Response

10

15 time [sec]

20

25

30

Control Responses

Fig. 11. Saturated responses due to pulse demands in μ˙ μ α β

30

δ h δ a δ

30

r

20

20

10 10 0 0 −10 −10

−20

−20 0

−30 5

10

15 time [sec]

20

Output Response

25

30

0

5

10

15 time [sec]

20

25

30

Control Responses

Fig. 12. Saturated responses due to pulse demands in β

for the unconstrained responses. The performance has deteriorated in both channels. The demand in μ˙ (Figure 11) causes the controlled aircraft to lose its precise tracking ability in the μ˙ channel; the demand in β (Figure 11) gives rise to a highly coupled and oscillatory type of behaviour. This undesirable behaviour is also conﬁrmed by the long extended periods of saturation of the control signals in Figures 11 and 12. The controller effectively loses authority which affects controller performance. Stability of the control system is not lost in this case, although this is not guaranteed. One might expect somewhat poorer results had the aircraft been unstable. This response can be easily improved by introducing suitable anti-windup compensation.

Anti-windup Compensation using a Decoupling Architecture

155

Full-Order Anti-Windup Compensation A full-order anti-windup compensator was synthesised using the LMI from (44) for Wr = 0.2 and Wp = 0.9798 (Wr2 + Wp2 = 1). This yielded the matrix F as ⎤ ⎡ −0.0039 0.0001 0.0023 −0.0004 0.5218 0.0108 ⎥ ⎢ ⎥ F = 1000 ⎢ ⎣ 0.0005 −0.0060 0.0000 1.0807 −0.0006 1.1299 ⎦ 0.0003 −0.0046 0.0000 −0.2222

(62)

0.0036 3.9672

with an accompanying optimal performance value of γ ≈ 5.67. Figures 13 and 14 show the aircraft responses for the same demands applied previously but with full order anti-windup compensation. Observe how the tracking performance in μ˙ has been tightened and that, apart from an initial transient, the coupling in the β-pulse response has been eliminated. On the negative side, the anti-windup compensator does slow down the tracking performance in this channel (Figures 14 versus 12). More importantly, the control system with full-order anti-windup recovers swiftly from saturation and allows the linear controller to take over quickly. This contrasts starkly with the response without anti-windup compensation (Figures 13 and 14 versus 11 and 12). Similar results are also obtained for the static anti-windup compensator, as discussed below.

Static Anti-Windup Compensation The main problem with full-order anti-windup is that it requires a signiﬁcant amount of extra on-line computation to implement. For this problem, the plant has six states, 100 μ α β

80

δ h δ a δ

30

r

60

20

40

10 20

0

0 −20

−10 −40

−20

−60 −80 −100 0

−30 5

10

15 time [sec]

20

Output Response

25

30

0

5

10

15 time [sec]

20

Control Responses

Fig. 13. Responses with full-order AW due to pulse demands in μ˙

25

30

156

Matthew C. Turner, Guido Herrmann, Ian Postlethwaite μ α β

15

r

10

20

5

10

0

0

−5

−10

−10

−20

−15 0

δ h δ a δ

30

−30 5

10

15 time [sec]

20

25

30

0

5

Output Response

10

15 time [sec]

20

25

30

25

30

Control Responses

Fig. 14. Responses with full-order AW due to pulse demands in β 100 μ α β

80

30 20

60

10 40

0 20

−10 0

−20 −20

−30 −40

−40

−60

−50

−80 −100 0

−60 5

10

15 time [sec]

μ˙ Response

20

25

30

0

μ α β 5

10

15 time [sec]

20

β Response

Fig. 15. Responses with static AW due to pulse demands in μ˙ and β

and therefore the anti-windup compensator has six states in the full-order case. Static-anti-windup, although not guaranteed to be feasible can be an alternative to this as it requires no additional states. For this plant and controller combination static anti-windup was indeed feasible and, using the design parameters Wr = 0.2 and Wp = 0.9798, a static anti-windup compensator was computed as ⎤ ⎡ −1.0000 −0.2165 −0.4713 ⎥ ⎢ ⎢ −0.0863 19.2532 17.4708 ⎥ ⎥ ⎢ ⎢ −0.2429 −3.8156 19.6683 ⎥ ⎥ ⎢ Θ=⎢ ⎥ ⎢ −0.1228 −38.2983 −35.3663 ⎥ ⎥ ⎢ ⎥ ⎢ ⎣ −0.0014 −0.1082 −0.1326 ⎦ 0.0444 1.9752 7.5235

(63)

which yielded an optimal performance value of γ ≈ 8910. Despite this surprisingly high value, compared to the full-order case, the anti-windup compensator still

Anti-windup Compensation using a Decoupling Architecture

157

performed well 5 . The responses of the aircraft due to the same reference demands as before is depicted in Figure 15. The improvement in the response due to the demand in μ˙ is very similar to that of the full-order anti-windup compensator, although the rise time is slightly slower for the static case. For the demand in β, we can again see that the coupling has been improved, except for sharp initial peaks following every pulse change. In fact, the tracking performance of the compensator seems somewhat better than for the full order case, with a sharper response in β. Discussion of Global Anti-Windup Compensation Results For comparison, we designed an IMC compensator (equivalent to choosing F = 0 in equation (12)). Figure 16 shows the response using the IMC compensator. While it is clear that the system is certainly stable with such a compensator, the response is somewhat poorer than with either the optimal full or static compensation. Furthermore, it is interesting to note that the design results of the anti-windup compensation resemble those of the discrete and sampled-data cases as reported in [16, 32, 13]. For a sampling frequency of 10 ms, the uncompensated discrete/sampled-data controller loses its tracking capabilities for μ˙ demands while interaction increases for demand changes in β. As for the continuous-time case, the discrete full-order antiwindup controller improves on the tracking behaviour for μ˙ but creates a slightly slower response in β with signiﬁcantly decreased interaction in the controller channels. Similar parallels are observed for the static anti-windup scheme.

7.2 Local Anti-Windup Compensation for Unstable Plants: Inverted Pendulum Benchmark Example The control of a linearised third order model for the inverted pendulum benchmark is considered:

⎡ ⎤ ⎡ ⎤ 1 0 ⎥⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ x˙ 2 ⎥ = ⎢ −1 0 1 ⎥ ⎢ x2 ⎥ + ⎢ 0 ⎥ z1 + ⎢ 0 ⎥ u z ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎣ ⎦ 1 ∼ G1 G2 0 1 0 0 1 x˙ 3 x3 u y = x1 x2 x3 ⎤

⎡

x˙ 1

5

⎡

0 −1 0

⎤⎡

⎤

x1

(64)

There currently is no intuitive interpretation for the absolute value of the L2 gain, although minimising it appears logical and helpful

158

Matthew C. Turner, Guido Herrmann, Ian Postlethwaite

100 μ α β

80

30 20

60

10 40

0 20

−10 0

−20 −20

−30 −40

−40

−60

−50

−80 −100 0

μ α β

−60 5

10

15 time [sec]

20

25

30

0

5

μ˙ Response

10

15 time [sec]

20

25

30

β Response

Fig. 16. Responses with IMC AW due to pulse demands in μ˙ and β

The control signal is to be limited by three different sets of constraints: The control signal rate du/dt is subjected to (a) no limit i.e. |du/dt| ≤ ∞, (b) a limit of |du/dt| ≤ 10/sec or (c) a limit of |du/dt| ≤ 2/sec. In all cases, it is expected that the control signal amplitude remains bounded by |u| ≤ 1.25. The target is for the horizontal speed x1 of the pendulum to track a demand signal r1 of prescribed proﬁle, while rejecting a speciﬁc disturbance (z1 ) proﬁle (see Figure 17). The demand changes at 1 sec to a value of 2 with a rate of 0.5/sec and at 70 sec the demand is set back to 0 with a rate of -0.5/sec. A small rectangular pulse is added to the demand in the interval [18 sec, 40 sec]. The disturbance is activated at a later point, rising at 110 sec from 0 to 1 with a rate of 0.25/sec. The disturbance is set back to 0 at 170 sec with a rate of -0.25 sec. A small rectangular pulse of amplitude 0.1 is added to the slowly changing disturbance in the interval [125 sec, 160 sec] (see Figure 17).

Nominal Controller Design and Control Signal Limits As complete information about the character of the disturbance is given and the controller requirements are well deﬁned in terms of frequency response requirements, the design of the nominal controller using an H∞ -approach appears to be attrac-

Anti-windup Compensation using a Decoupling Architecture

159

1.2 2 1 1.5

r1

z1

0.8

0.6

1 0.4 0.5

0.2

0

0 0

20

40

60

80

100 120 time [sec]

140

160

180

200

0

20

Demand

40

60

80

100 120 time [sec]

140

160

180

200

Disturbance

Fig. 17. Nominal demand and disturbance proﬁle

Se1 Sz1 r1 -

+

Su z1

K

u

x1

[G1G2] x

x2 3

Fig. 18. H∞ -control problem for pendulum control

tive. Thus, a mixed sensitivity H∞ -control problem is posed as in Figure 18. The controller error dynamics are shaped by the weight S e1 =

0.5s + 0.7 s + 7 · 10−5

This ensures good low frequency error rejection so that any constant demand r1 for the horizontal speed x1 results in virtually asymptotic tracking of the demand. A ﬁrst order weight

0.7 s+1 is used to characterise the reasonably slow behaviour of the disturbance z1 , while Sz1 =

0.5s + 3 · 10−5 Su = 0.5s + 0.6 is used to limit the amplitude of the control signal in particular in the frequency region above 1 rad/sec. A linear H∞ -controller of sixth order was designed which was reduced to third order using balanced residualization. The resulting controller

160

Matthew C. Turner, Guido Herrmann, Ian Postlethwaite

has a closed loop sensitivity response r1 → (x1 − r1 ) with bandwidth6 of 0.365 rad/sec and a bandwidth for the complementary sensitivity r1 → x1 of 1 rad/sec (see Figure 19). Note that the co-sensitivity bandwidth can be a poor measure of the tracking capability of the system as the phase lag may be quite large at this frequency (see [28]). This is the case in this example as can be seen in Figure 19 where the phase of the co-sensitivity is approximately −150◦ around the bandwidth. A second 0

10 Log Magnitude

Log Magnitude

10

−2

10

10

0

−2

−4

10

−5

10

−4

10

−3

10

−2

−1

10 10 Frequency (radians/sec)

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10 10 Frequency (radians/sec)

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80 60 40 20

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10

−4

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0

10

1

10

10

−5

10

−4

Sensitivity

−3

10

10 10 Frequency (radians/sec)

0

10

10

1

Co-Sensitivity

Fig. 19. Frequency responses of ﬁrst nominal controller

H∞ -controller has also been tested using a slightly different set of weights allowing faster dynamics in terms of error and disturbance rejection. For this controller, the sensitivity bandwidth in this case is 1 rad/sec while the co-sensitivity has a bandwidth of 3.1 rad/sec (see Figure 20). The following weights, chosen along similar lines for the slower controller, were used 0.5s + 5.6 s + 8 · 10−4 8 Sz1 = s+8 0.65s + 4.457 · 10−4 Su = 0.5s + 4.8 S e1 =

(65) (66) (67)

The nominal controller behaviour without any control signal limitations for the ﬁrst investigated controller is presented in Figure 21 using the prescribed demand and disturbance proﬁle. The controller is designed to have small controller amplitudes and small control signal rates. However, note the large control signal amplitude of 6

The cut-off level is -3 dB to determine the bandwidth for both the sensitivity and the cosensitivity.

Anti-windup Compensation using a Decoupling Architecture 0

10 Log Magnitude

Log Magnitude

10

−2

10

10

10 −5

−4

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60 40 20 0

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Sensitivity

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−3

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10

0

10

1

Co-Sensitivity

Fig. 20. Frequency responses of second nominal controller 1.5 2.5 1

2 1.5

0.5

1

0

0.5 −0.5 0 −1 −0.5 −1.5 −1 −2

−1.5 −2 0

20

40

60

80

100 120 time [sec]

Output x1

140

160

180

200

−2.5 0

20

40

60

80

100 120 time [sec]

140

160

180

200

Control

Fig. 21. Nominal controller response of ﬁrst controller

2.3 at about 115 sec. This signiﬁcant amplitude is caused by the disturbance rate of 0.25/sec. For the control signal limited control problem, it will be necessary to consider this disturbance rate. Indeed, it is the control signal peak due to the disturbance which causes the control system to destabilise, once any of the control signal amplitudes are imposed. However, note that the actuator limits for tracking of the demand proﬁle do not destabilise the control system. Thus, any of the control signal limit options a), b) or c) will not affect stability of the demand tracking task and tracking performance is almost identical to the nominal control between 0 sec and 110 sec. Hence, anti-windup control appears necessary only for the disturbance rejection case. The second, fast-settling controller (Figure 22) shows larger control signal amplitudes and is also faster acting. Hence the control signal limited problem is signiﬁ-

162

Matthew C. Turner, Guido Herrmann, Ian Postlethwaite 2.5

2.5 2 1.5

2

1 1.5

0.5 0

1 −0.5 0.5

−1 −1.5

0 −2 −0.5 0

20

40

60

80

100 120 time [sec]

140

160

180

−2.5 0

200

20

40

60

Output x1

80

100 120 time [sec]

140

160

180

200

Control

Fig. 22. Nominal controller response of second controller

cantly more difﬁcult than for the ﬁrst considered controller. In fact, the control system is destabilised for all discussed control limit cases (a), (b) and (c).

Anti-Windup for Controller 1 For anti-windup compensator design, the full-order compensation was preferred in this case. As for global anti-windup compensation schemes for asymptotically stable plants, a feasible, well-performing full order anti-windup scheme is easier to ﬁnd than a static or low order scheme. This was vital in this case. The general conﬁguration for the full-order anti-windup control scheme is given in Figure 23 and the compensator is designed using the LMI (61). As shown in [29], this also ensures local stability for an amplitude and rate limited actuator.

r +

K(s)

+ ulin

amplitude rate limit limit

u -

ud M(s)-I yd(t) + G2(s)M(s) +

z1(t)

[G1G2]

y(t)

+ ~ u

Fig. 23. Full-order anti-windup compensation with rate limits

Anti-windup Compensation using a Decoupling Architecture

163

As stated in Section 6.4, in the design of this anti-windup controller, the L2 -gain of the operator

ulin →

Wp y d Wu (ulin − ud )

.

is minimized. The rationale for this is to retain the conditioned signal (ulin − ud ) small so that the sector constraint of equation (9) deﬁned by the value of E = A < 1 is not violated for the considered class of control signals and local stability is ensured. Considering this set-up, an anti-windup controller with parameters A = 0.001, Wp = 1 and Wu = 1 has been designed achieving F = 8.6 −10.9 −6.8

(68)

and γ = 1.00. For the unstable inverted pendulum, it is not possible to achieve stability for an arbitrarily large region of attraction. Thus, a trade-off in terms of the magnitude and the rate of change of the disturbance has to be made. The amplitude of the large disturbance has to be decreased from 1 to 0.57 and the rate of increase from 0.25/sec to 0.56 × 0.25/sec, while the rate of decrease and the small rectangular disturbance of amplitude 0.1 retain their original characteristics. The nominal controller subjected to the control signal limits of case (c) will remain unstable after this modiﬁcation, while for cases (a) and (b) stability of the control scheme is achieved without anti-windup compensation. For case (c), the anti-windup compensator is needed to retain stability. Good performance is assured via the design approach (Figure 24). For case (b) as illustrated in 1.5 2.5 2

1

1.5 0.5

1 0.5

0 0 −0.5

−0.5 −1

−1 −1.5 −2 0

20

40

60

80

100 120 time [sec]

Output x1

140

160

180

200

−1.5 0

20

40

60

80

100 120 time [sec]

140

160

Control

Fig. 24. Full-order anti-windup for amplitude and rate limitation case (c)

180

200

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Matthew C. Turner, Guido Herrmann, Ian Postlethwaite

Figure 25, the anti-windup compensator is partially better able to minimise the effect of the disturbance on the output measurement, in particular in the interval [110 sec, 140 sec]. Similar results are observed in case (a). 2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

−0.5

−0.5

−1

−1

−1.5

−1.5

−2 0

20

40

60

80

100 120 time [sec]

140

160

180

Output x1 (Full order anti-windup)

200

−2 0

20

40

60

80

100 120 time [sec]

140

160

180

200

Output x1 (No anti-windup)

Fig. 25. Control results for amplitude and rate limitation case (b)

Anti-Windup for Controller 2 In the case of the second, faster nominal controller, without an anti-windup compensator, the control system will become unstable for both the demand tracking and the disturbance rejection task for any of the control signal limits (a,b,c). Hence, a full-order compensator is to be designed. A set of design parameters A = 0.99999, Wp = 10000 · I and Wu = 0.00000001 gives F = 108 1.9586 −2.2800 −1.2617 and γ = 10.02. For control signal limit case (a), i.e. amplitude limits only, the anti-windup compensator retains good tracking and disturbance rejection for the prescribed demand and disturbance proﬁle (see Figure 26). In comparison to the nominal control behaviour (Figure 22), a slight increase in overshoot and a small increase of the controller error due to the disturbance is observed but in general the controller characteristics are almost identical. When the rate limit of 10/sec is introduced for the control signal, the demand proﬁle and also the disturbance proﬁle need to be adjusted so that the locally stable

Anti-windup Compensation using a Decoupling Architecture

165

1.5 2.5 1 2 0.5 1.5

0 1

−0.5

0.5

−1

0

−0.5 0

20

40

60

80

100 120 time [sec]

140

160

180

200

−1.5 0

20

40

60

Output x1

80

100 120 time [sec]

140

160

180

200

Control

Fig. 26. Control results for amplitude limitation case (a)

anti-windup compensator is able to guarantee stability and good performance. The rate of the demand, when decreasing from 2 to 0, is changed from -0.5/sec to 0.33/sec, while the large disturbance of amplitude 1 is decreased in rate and amplitude to 0.73×0.25/sec and 0.73. The small rectangular impulse disturbance remains unchanged. Using this compromise for the anti-windup compensated controller, overshoots are slightly increased in comparison to the former case and the disturbance rejection dynamics are comparable to the control signal amplitude limited case. However, note the fast controller action (in Figure 27) which is caused by the large anti-windup compensator gain F in combination with the rate limit of 10/sec for the control signal. The large magnitude of F results in fast poles, [−1.26 · 108 , −0.39, −1.41], for the ﬁlter M (see equation (12)), which imply fast control action of the anti-windup scheme. 1.5 2.5 1 2 0.5 1.5

0 1

−0.5

0.5

−1

0

−0.5 0

20

40

60

80

100 120 time [sec]

Output x1

140

160

180

200

−1.5 0

20

40

60

80

100 120 time [sec]

Control

Fig. 27. Control results for amplitude and rate limitation case (b)

140

160

180

200

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The even more severe control signal limit case (c) has also been investigated. However, in this case it was not possible to obtain a good controller result without compromising too strongly in terms of the disturbance and demand proﬁle.

Discussion of Local Anti-Windup Compensation Results The target for the local anti-windup control scheme is to achieve, in a reasonable region of attraction, output measurement performance close to the nominal controller despite rate and amplitude limits on the control signal. For those cases, where stability was ensured, this goal has been achieved. However, this target does not necessarily relate to the actual characteristics of the control signal, that is the target does not clearly determine the speed and magnitude of the control signal. In the case of the unstable pendulum example, it has been found that the values of A and Wu inﬂuence control action. The parameter Wu has been explicitly introduced to minimize the control signal (ulin − ud ), which enters the amplitude and rate limiting ‘saturation’ elements. Hence, Wu is chosen large enough to obtain small values for (ulin − ud ) so that the sector non-linearity of (12) remains satisﬁed and local stability is guaranteed. However, too large a value for Wu may create a slow and unsatisfactory controller behaviour so that a compromise between Wu and Wp is necessary. The parameter A also has an inﬂuence on the control signal. A small value for the scalar A appears to imply slow poles of small magnitude for the ﬁlter M (12), while a large value permits very fast dynamics in M . This has been observed for the two investigated nominal controllers in Section 7.2. For the ﬁrst controller, a small A delivers poles of [−5.09, −0.25, −1.42] which are are slightly faster but of the same order as the closed loop poles of the nominal controller. This is conﬁrmed with respect to controller action in Figure 24. In contrast, for the second controller, a large value of A creates a high gain F and one fast pole [−1.26 · 108 , −0.39, −1.41]. This creates anti-windup compensator action for which, in the control rate limited case (b), controller speed appears to be limited by the rate limit (Figure 27) only. It has been also observed that the amplitude of (ulin − ud ) increases with A. For this, it must be noted that a large value of A increases the sector to a region approaching

Anti-windup Compensation using a Decoupling Architecture

167

Sector[0, I], although this seems to create larger control signals which may violate this sector bound. The region of attraction for the anti-windup scheme shows a rather complex relationship in connection to A and a large or small value for A may not necessarily imply instability for a certain demand or disturbance proﬁle. In addition, the incorporation of the LMI’s (57) into the optimisation complicates things further because the ellipsoid approximation of the region of attraction is not necessarily close to the actual region of attraction and the scalar η represents another degree of freedom which exerts some inﬂuence over the design. Thus, for now, the choice of A is left to the designer and the special requirements raised for the control task. More advanced strategies for the choice of A, and a more detailed investigation into its relationship with the system’s region of attraction remain to be completed. A summary of the main features of the results of the local anti-windup compensation for this example is shown in Table 1. It is worth mentioning that the initial controller design is of paramount importance. Although for the full order designs, the design of the AW compensator is not dependent on the controller, the responses of the system are highly dependent on this initial controller design. In particular, for the limits in conﬁguration (a), Controller 2 drives the system unstable without AW, whereas Controller 1 maintains stability in this case without AW. Table 1. Summary of anti-windup results for the fast and slow controllers: - magnitude and rate of disturbance reduced;

- rate of reference and magnitude and rate of disturbance

reduced. Controller Bandwidth S

T

1

0.37

1

2

1

3.1

Control limit conﬁguration (a) stable even without

(b) AW

stable only with AW

stable even without

(c) AW

stable only with AW

stable only with AW unstable

8 Conclusion This chapter has considered the design of anti-windup compensators based on the decoupling of the saturated system into three distinct parts. Several different types of anti-windup compensator were proposed, all accompanied with stability and L2 performance guarantees. When the open-loop system under consideration

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is stable, the compensators advocated in this chapter appear to work well and require only rough tuning. For plants which are exponentially unstable, the compensators proposed appear to work well up to a point, but as there is no precise way in which to choose the sector bound, A, require more tuning to achieve maximal performance. This feature of the design still requires some reﬁnement and as many important practical systems (such as ﬁghter aircraft) are unstable, is a crucial topic for future research.

References 1. J.A.G. Akkermans, A.G. Wills, and W.P. Heath. Robust cross-directional control of paper making machines with saturating actuators. Proc. Control Systems 2004, 2004. 2. S. Crawshaw and G. Vinnicombe. Anti-windup synthesis for guaranteed L2 performance. Proc. IEEE Conference on Decision and Control, 2000. 3. J.C. Doyle, D.F. Enns, and R.S. Smith. Control of plants with input saturation nonlinearities. Proc. American Control Conference, pages 2147–2152, 1987. 4. C. Edwards and I. Postlethwaite. Anti-windup and bumpless transfer schemes. Automatica, 34(2):199–210, 1998. 5. A.T. Fuller. In-the-large stability of relay and saturating control systems with linear controllers. International Journal of Control, 10(4):457–480, 1969. 6. S. Galeani, M. Massimetti, A.R. Teel, and L. Zaccarian. Reduced order linear anti-windup augmentation for stable linear systems. International Journal of Systems Science, to appear, 2005. 7. A. H. Glattfelder and W. Schaufelberger. Stability of discrete override and cascade-limiter single-loop control systems. IEEE Transactions on Automatic Control, 33(6):532–540, 1988. 8. J. M. Gomes da Silv Jr. and S. Tarbouriech. Anti-windup design with guaranteed regions of stability: an LMI based approach. IEEE Transactions on Automatic Control, 50(1):106–111, 2005. 9. G. Grimm, J. Hatﬁeld, I. Postlethwaite, A.R. Teel, M.C. Turner, and L. Zaccarian. Antiwindup for stable linear systems with input saturation: an LMI based synthesis. IEEE Transactions on Automatic Control, 48(9):1509–1525, 2003. 10. G. Grimm, I. Postlethwaite, A.R. Teel, M.C. Turner, and L. Zaccarian. Linear matrix inequalities for full and reduced order anti-windup synthesis. Proc. American Control Conference, 2001.

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11. G. Grimm, I. Postlethwaite, A.R. Teel, M.C. Turner, and L. Zaccarian. Case studies using linear matrix inequalities for optimal anti-windup compensator synthesis. European Journal of Control, 9:463–373, 2003. 12. R. Hanus, M. Kinnaert, and J.L. Henrotte. Conditioning technique, a general anti-windup and bumpless transfer method. Automatica, 23(6):729–39, 1987. 13. G. Herrmann, M.C. Turner, and I. Postlethwaite. Discrete time anti-windup - part 2: extension to sampled data case. Proceedings of the European Control Conference, 2003. 14. G. Herrmann, M.C. Turner, and I. Postlethwaite. Practical implementation of a novel antiwindup scheme in a HDD-dual-stage servo-system. IEEE Transactions on Mechatronics, 9(3):580–592, 2004. 15. G. Herrmann, M.C. Turner, and I. Postlethwaite. Some new results on anti-windup conditioning using the Weston-Postlethwaite approach. Proc. Conference on Decision and Control, 2004. 16. G. Herrmann, M.C. Turner, and I. Postlethwaite. Discrete-time and sampled-data antiwindup synthesis: stability and performance. International Journal of Systems Science, to appear. 17. H. Hindi and S. Boyd. Analysis of linear systems with saturation using convex optimisation. Proc. IEEE Conference on Decision and Control, pages 903–907, 1998. 18. P. Hippe and C. Wurmthaler. Systematic closed-loop design in the presence of input saturation. Automatica, 35(4):689–695, 1999. 19. R.A. Hyde. The Application of Robust Control to VSTOL Aircraft. PhD thesis, Department of Engineering, University of Cambridge, 1991. 20. H.K. Khalil. Nonlinear Systems. Prentice Hall, New Jersey, 1996. 21. M.V. Kothare, P.J. Campo, M. Morari, and C.N. Nett. A uniﬁed framework for the study of anti-windup designs. Automatica, 30(12):1869–1883, 1994. 22. Z. Lin and A. Saberi. A semi-global low and high gain design technique for linear systems with input saturation - Stabilisation and Disturbance rejection. International Journal of Robust and Nonlinear Control, 5:381–398, 1995. 23. V.R. Marcopoli and S.M. Phillips. Analysis and synthesis tools for a class of actuatorlimited multivariable control systems: a linear matrix inequality approach. International Journal of Robust and Nonlinear Control, 6(9-10):1045–1063, 1996. 24. S. Miyamoto and G. Vinnicombe. Robust control of plants with saturation nonlinearity based on comprime factor representations. Proc. IEEE Conference on Decision and Control, pages 2838–2840, 1996. 25. E.F. Mulder, M.V. Kothare, and M. Morari. Multivariable anti-windup controller synthesis using linear matrix inequalities. Automatica, 37:1407–1416, 2001. 26. C. Pittet, S. Tarbouriech, and C. Burgat. Stability regions for linear systems with saturating control via Circle and Popov criteria. Proc. IEEE Conference on Decision and Control, 5:4518–4523, 1997.

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27. G. Shi, A. Saberi, and A.A. Stoorvogel. On the Lp (lp ) stablisation of open-loop neutrally stable linear plants with input subject to amplitude saturation. International Journal of Robust and Nonlinear Control, 13:735–754, 2003. 28. S. Skogestad and I. Postlethwaite. Multivariable Feedback Control - Analysis and Design. Wiley, Chichester, 2005. 29. J. Sofrony, M.C. Turner, and I. Postlethwaite. Anti-windup for systems with rate-limited actuators. in preparation, 2005. 30. H.J. Sussmann, E.D. Sontag, and Y. Yang. A general result on the stabilisation of linear systems using bounded controls. IEEE Transactions on Automatic Control, 39(12):2411– 2425, 1994. 31. A.R. Teel and N. Kapoor. The L2 anti-windup problem: Its deﬁnition and solution. Proceedings of the European Control Conference, 1997. 32. M.C. Turner, G. Herrmann, and I. Postlethwaite. Discrete time anti-windup - part 1: stability and performance. Proceedings of the European Control Conference, 2003. 33. M.C. Turner, G. Herrmann, and I. Postlethwaite. Accounting for uncertainty in antiwindup synthesis. Proc. American Control Conference, 2004. 34. M.C. Turner, G. Herrmann, and I. Postlethwaite. Improving local anti-windup performance: preliminary results on a two-stage approach. IFAC World Congress, to appear, 2005. 35. M.C. Turner and I. Postlethwaite. A new perspective on static and low order anti-windup synthesis. International Journal of Control, 77(1):27–44, 2004. 36. M.C. Turner, I. Postlethwaite, and G. Herrmann. Further results on full-order anti-windup synthesis: exploiting the stability multiplier. Proc. IFAC Nonlinear Controller Symposium, 2004. 37. N. Wada and M. Saeki. Design of a static anti-windup compensator which guarantees robust stability. Transactions of the Institute of Systems, Control and Instrumentation Engineers, 12(11):664–670, 1999. 38. P.F. Weston and I Postlethwaite. Analysis and design of linear conditioning schemes for systems containing saturating actuators. IFAC Nonlinear Control System Design Symposium, 1998. 39. P.F. Weston and I Postlethwaite. Analysis and design of linear conditioning schemes for systems with nonlinear actuators. Internal Report, Dept. of Engineering, Leicester Univ., 98-6, 1998. 40. J.-S. Yee, J.L. Wang, and Sundararajan. Robust sampled data H∞ ﬂight controller design for high α stability-axis roll manoeuver. Control Enigneering Practice, 8:735–747, 2001. 41. A. Zheng and M. Morari. Anti-windup using internal model control. Interntional Journal of Control, 1994. 42. K. Zhou, J.C. Doyle, and K. Glover. Robust and Optimal Control. Prentice Hall, 1996.

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A State-space Realisations Static Compensator: A minimal realisation (assuming the realisations of K2 and G2 to be minimal) is given by

x ¯ :=

xp

xc

A¯ :=

,

˜ c Cp Ap + Bp ΔD

Bc ΔCp Ac + Bc ΔDp Cc ˜ ˜ B Θ p Δ −Bp ΔDc 1 ¯ := B , Θ := , Bc ΔDp −Bc Δ Θ2 ¯1 := [ΔD ˜ c Cp C ¯2 := [ΔCp C

˜ c ], ΔC

ΔDp Cc ],

˜ c Dp , D01 := ΔD D02 := ΔDp ,

˜ c Bp ΔC

¯ 1 := [Δ ˜ D

¯ 2 := [ΔDp D

,

B0 :=

˜ Bp Δ Bc ΔDp

,(69)

(70) ˜ c ], − ΔD

(71)

− ΔDp Dc ]

(72)

Low order compensator: If the ﬁlters F1 and F2 are given the following realisations ˜1 ˜2 A1 A2 B2 Θ B1 Θ (73) F1 ∼ F2 ∼ ˜2 ˜ 1 + I) C1 D1 (Θ C2 D2 Θ where A1 ∈ Rn1 ×n1 and A2 ∈ Rn2 ×n2 , then a state-space realisation for the low order compensator can be given as ⎡ ⎤ ˜ c Cp ˜ c ˜ 1 −Bp ΔD ˜ c C2 Ap + Bp ΔD Bp ΔC Bp ΔC ⎢ ⎥ ⎢ ⎥ B ΔC A + B ΔD C B ΔD C −B ΔC c p c c p c c p 1 c 2 ⎥ (74) A¯ = ⎢ ⎢ ⎥ 0 0 A1 0 ⎣ ⎦ 0 0 0 A2 ⎡ ⎤ ⎡ ⎤ ˜ ˜ 1 −Bp ΔD ˜ c D2 Bp ΔD Bp Δ ⎢ ⎥ ⎢ ⎥ ˜1 ⎢ Bc ΔDp D1 −Bc ΔD2 ⎥ ⎢ Bc ΔDp ⎥ Θ ¯=⎢ ˜ = ⎥ B ⎥ Θ B0 = ⎢ (75) ⎢ ⎥ ⎢ ⎥ ˜2 Θ 0 B 0 ⎣ ⎦ ⎣ ⎦ 1

0

0

B2

˜ c Cp ΔC ˜ c ΔC ˜ 1 −ΔD ˜ c C2 C¯1 = ΔD ¯ C2 = ΔCp ΔDp Cc ΔDp C1 −ΔDp Dc C2 ˜ ¯ ˜ ˜ D01 = [ΔDc Dp ] D1 = ΔD1 −ΔDc D2 ¯ 2 = ΔDp D1 −ΔDp Dc D2 D02 = [ΔDp ] D

(76) (77) (78) (79)

¯ ∈ Rnx ×(m+q) , C¯1 ∈ Rm×nx , D01 ∈ Rm×m , where A¯ ∈ Rnx ×nx , B0 ∈ Rnx ×m , B ¯ 1 ∈ Rm×(m+q) , C¯2 ∈ Rq×nx , D02 ∈ Rq×m , D ¯ 2 ∈ Rq×(m+q) and nx := np + D nc + n1 + n2 .

Anti-Windup Strategy for Systems Subject to Actuator and Sensor Saturations Sophie Tarbouriech, Isabelle Queinnec, and Germain Garcia LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse cedex 4, FRANCE tarbour,queinnec,[email protected] Summary. This chapter addresses the problems of design of anti-windup schemes for linear systems subject to amplitude limitations in actuator and sensor. The design of anti-windup loops, using both measured and estimated variables, is considered in order to enlarge the region of stability of the closed-loop system, whereas some performance in terms of bounded controlled output is guaranteed. Based on the modelling of the closed-loop saturated system including the observer loop as a linear system with dead-zone nested nonlinearities, constructive stability conditions are formulated in terms of linear matrix inequalities by using Finsler’s Lemma. Keywords. Anti-windup scheme, actuator and sensor saturations, observer, stability regions, LMI

1 Introduction During the past years, the problem of stabilization for linear systems subject to amplitude and dynamics restricted actuators and/or sensors has attracted the attention of the control community, due to its practical interest and its inherent difﬁculties (see, for example, [3]). The studies concerning these systems have been done in terms of modelling, properties analysis (as stability and performance analysis) or still control design problem [11]. Most of these theoretical studies are motivated by the fact that physical, safety or technological constraints generally induce that the actuators and sensors cannot provide unlimited amplitude signal neither unlimited speed of reaction. The control problems of combat aircraft prototypes and

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Sophie Tarbouriech, Isabelle Queinnec, and Germain Garcia

launchers offer interesting examples of the difﬁculties due to these major constraints [20], [19]. Neglecting actuator and sensor limitations can be source of undesirable even catastrophic behaviors for the closed-loop system (as the lost of the stability). The case of actuator saturation has been addressed in much details ([14] and references therein). Few results however concern the case of sensor saturation. Among these results one can cite the studies of the effects of sensor saturation on plant observability [16]. In [18], the global stabilization of a linear SISO system is carried out via the use of dead beat controller. In [5], the authors use the circle criterion to design an output H∞ feedback controller for linear systems with sensor nonlinearity. If few results are available on systems with sensor actuator still less results concern the case of systems with both actuator and sensor limitations. In [7], adaptive integral control design for linear systems with actuator and sensor nonlinearities is addressed by considering the asymptotic stability of the open-loop. See also [9], in which different methods of analysis and synthesis are presented (like the PID controllers design). The anti-windup techniques consist in taking into account the effects of saturation in a second stage after previous design performed disregarding the actuator limitations. The idea is to introduce some control modiﬁcation in order to recover, as much as possible the properties induced by the previous design carried out for the unsaturated system. In particular, anti-windup schemes have been successfully applied in order to minimize the windup due the integral action in PID controllers. In this case, most of the related literature focuses on the performance improvement in the sense of avoiding large and oscillatory transient responses (see, among others, [1]). The inﬂuence of the anti-windup schemes in the stability and the performances of the closed-loop system has been also studied (see, for example, [2], [17]). Several results on the antiwindup problem are concerned with achieving global stability properties [13], [27]. Since global results cannot be achieved for open-loop unstable linear systems in the presence of actuator saturation, local results have to be developed. In this context, a key issue concerns the determination of domains of stability for the closed-loop system by noting that the basin of attraction is modiﬁed by the anti-windup loop. More recently, in [4], [10], [25], in the ACC03 Workshop “T-1: Modern Anti-windup Synthesis”, in the ACC04 (Session FrP04 “Anti-windup”) or in [26], some constructive conditions are proposed both to determine suitable anti-windup loops and to quantify the closed-loop region of stability in the case of amplitude saturation actuator. Differently from the references cited above, in this chapter we focus our attention

Anti-Windup Strategy for Systems Subject to Actuator and Sensor Saturations

175

on linear systems with amplitude saturation on actuator and sensor, and bounded controlled outputs. Our aim is the design of the suitable anti-windup gains in order to ensure the closed-loop stability for regions of admissible initial states as large as possible. Based on the modelling of the closed-loop system resulting from the controller plus the anti-windup loop as a linear system with dead-zone nested nonlinearities, original constructive stability conditions are directly formulated as LMI conditions. Moreover, in order to use in the anti-windup loops some unmeasured signals, a solution based on the addition of an observer step is considered. Thus, in order to address inherent structural constraints on some decision variables the Finsler’s Lemma is used. The chapter is organized as follows. In section 2, the system under consideration with its assumptions is described. The main anti-windup problem to be addressed is then summarized. Some preliminaries about sector nonlinearities and Finsler’s Lemma are quickly exposed in section 3. The main theoretical anti-windup conditions are formulated in section 4. Some important remarks are given in section 5, whereas the optimization issues are discussed in section 6. Examples are provided in section 7 to illustrate the potentialities and the drawbacks of the approach proposed. Finally, some concluding remarks end the chapter. Notation. For any vector x ∈ n , x 0 means that all the components of x, denoted x(i) , are nonnegative. For two vectors x, y of n , the notation x y means that x(i) − y(i) ≥ 0, ∀i = 1, . . . , n. 1 and 0 denote respectively the identity matrix and the null matrix of appropriate dimensions. A(i) , i = 1, ..., m, denotes the ith row of matrix A ∈ m×n . For two symmetric matrices, A and B, A > B means that A − B is positive deﬁnite. A denotes the transpose of A. For any vector u of m one deﬁnes each component of satu0 (u) by satu0 (u(i) ) = sign(u(i) )min(u0(i) , |u(i) |), with u0(i) > 0, i = 1, ..., m.

2 Problem Statement The saturated closed-loop system under consideration is described in Fig. 1. The open-loop model is given by: x˙ = Ax + Bu y = saty0 (Cx)

(1)

176

Sophie Tarbouriech, Isabelle Queinnec, and Germain Garcia uc

controller (η)

yc

u

nonlinear actuator

sensor ±y0

system (x)

y

Fig. 1. Saturated closed-loop system

where x ∈ n , u ∈ m and y ∈ p are the state, the input and the measured output vectors, respectively. A, B and C are real constant matrices of appropriate dimensions. Let us consider the controller dynamics in nc : η˙ = Ac η + Bc uc

(2)

yc = Cc η + Dc uc

where η ∈ nc is the state of the controller, yc ∈ m is the output of the controller and uc ∈ p is the input of the controller. Ac , Bc , Cc , Dc are matrices of appropriate dimensions. Such a controller has generally been designed such that the linear closed-loop system resulting from the linear interconnection conditions (3)

u = yc ; uc = Cx is asymptotically stable.

The real interconnections are however obtained through the actuator and sensor elements. The nonlinear actuator scheme is given in Fig. 2. The actuator model is described as follows: v˙ = Tv satv0 (Tyc (yc − satu0 (v))) + Tu (satu0 (v) − v) yc

+

Tyc −

±v0

Tv

+

v˙

u

v ±u0

+

− Tu

Fig. 2. Nonlinear actuator

(4)

+

Anti-Windup Strategy for Systems Subject to Actuator and Sensor Saturations

177

and the real interconnection, for which some windup problems may arise when saturations occur, is: u = satu0 (v) ; uc = y = saty0 (Cx)

(5)

The vectors u0 , v0 and y0 are the positive saturation levels. Hence, from the above description, the complete closed-loop system reads: x˙ = Ax + Bsatu0 (v) v˙ = Tv satv0 (Tyc (yc − satu0 (v))) + Tu (satu0 (v) − v) η˙ = Ac η + Bc saty0 (Cx)

(6)

yc = Cc η + Dc saty0 (Cx) The set of measured or accessible variables of this system is: yc ; u ; y ; v

(7)

i.e. the output of the controller, actuator and sensor, and the internal state of the actuator. By extension, one can consider that the input and output signals at the bounds of the saturation block v0 are known. On the other hand, one cannot consider that the input of the sensor saturation block is accessible. However, a way to use this information is to build a state observer described as: x ˆ˙ = Aˆ x + Bu + L(saty0 (C x ˆ) − saty0 (Cx))

(8)

=x ˆ−x

(9)

with the error

Then, in order to avoid the undesirable effects of the windup, or at least to mitigate them, we want to build static anti-windup loops, which act on the dynamics and possibly on the output of the controller. Thus, consider the addition of terms related to the nonlinearities: φv0 = satv0 (Tyc (yc − satu0 (v))) − Tyc (yc − satu0 (v)) φu0 = satu0 (v) − v φy0 = saty0 (C x ˆ) − C x ˆ = saty0 (Cx + C) − (Cx + C) and denote also: φy0 = saty0 (Cx) − Cx

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Sophie Tarbouriech, Isabelle Queinnec, and Germain Garcia

Tu yc + − Tyc

±v0 −

Tv

+ + v˙

+ − ±u0

v −

+

u

system

+

±y0

observer

AW block

controller

±y0 −

+

Fig. 3. Saturated closed-loop system with anti-windup strategy

Then, the saturated closed-loop system with anti-windup strategy can be described in Fig. 3, and reads: x˙ = Ax + Bv + Bφu0 v˙ = Tv Tyc (yc − v − φu0 ) + Tv φv0 + Tu φu0 η˙ = Ac η + Bc Cx + Bc φy0 + Ec φv0 + Fc φu0 + Gc φy0

(10)

˙ = A + LC + Lφy0 − Lφy0 with yc = Cc η + Dc Cx + Dc φy0 + Hc φv0 + Jc φu0 + Kc φy0

(11)

Remark 1. It could be possible to consider a simpler observer as follows: x ˆ˙ = Aˆ x + Bu + L(C x ˆ − saty0 (Cx))

(12)

In such a case, from (9) and (12), the part due to in (10) would read: ˙ = (A + LC) − Lφy0

(13)

Remark 2. This system involves nested saturations (φv0 = f (φu0 , φy0 )), but also, if the gain Hc is not set equal to 0, an implicit function (φv0 = f (φv0 )), which increases the complexity of the conditions. We will see in the sequel how to deal with such a complexity.

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179

Thus, by deﬁning the extended vector ⎡ ⎤ x ⎢ ⎥ ⎢v⎥ 2n+m+nc ⎥ ξ=⎢ ⎢η⎥ ∈ ⎣ ⎦ and the following matrices ⎤ ⎡ A B 0 0 ⎥ ⎢ ⎢ Tv Tyc Dc C −Tv Tyc Tv Tyc Cc 0 ⎥ ⎥ ∈ (2n+m+nc )×(2n+m+nc ) A=⎢ ⎥ ⎢ B C 0 A 0 ⎦ ⎣ c c 0 0 0 A ⎤ ⎡ ⎤ ⎡ 0 B ⎥ ⎢ ⎥ ⎢ ⎢ Tv ⎥ ⎢ T u − T v T yc ⎥ (2n+m+n )×(m) c ⎥ ⎥ ∈ (2n+m+nc )×(m) Bv0 = ⎢ ; Bu0 = ⎢ ⎥ ⎢ 0 ⎥∈ ⎢ 0 ⎦ ⎣ ⎦ ⎣ 0 0 ⎤ ⎡ ⎡ ⎤ 0 0 (14) ⎥ ⎢ ⎢ ⎥ ⎢ Tv Tyc Dc ⎥ ⎢ ⎥ ⎥ ∈ (2n+m+nc )×(p) ; RL = ⎢ 0 ⎥ ∈ (2n+m+nc )×n By0 = ⎢ ⎥ ⎢ B ⎢0⎥ ⎦ ⎣ ⎣ ⎦ c 1 0 ⎤ ⎡ ⎡ ⎤ 0 0 ⎥ ⎢ ⎢ ⎥ ⎢ T v T yc ⎥ ⎢ ⎥ ⎥ ∈ (2n+m+nc )×n ; R2 = ⎢ 0 ⎥ ∈ (2n+m+nc )×nc R1 = ⎢ ⎢ 0 ⎥ ⎢1⎥ ⎦ ⎣ ⎣ ⎦ 0 0 CL = 0 0 0 C ∈ p×(2n+m+nc ) the closed-loop system reads: ξ˙ = (A + RL LCL )ξ + (Bv0 + R1 Hc + R2 Ec )φv0 + (Bu0 + R1 Jc + R2 Fc )φu0

(15)

+ (By0 − RL L)φy0 + (RL L + R1 Kc + R2 Gc )φy0 The basin of attraction of system (15), denoted Ba , is deﬁned as the set of all ξ ∈ 2n+m+nc such that for ξ(0) = ξ the corresponding trajectory converges asymptotically to the origin. In particular, when the global stability of the system is ensured the basin of attraction corresponds to the whole state space. However, in the general case, the exact characterization of the basin of attraction is not possible.

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In this case, it is important to obtain estimates of the basin of attraction. For this, consider ﬁrst the following deﬁnition. Deﬁnition 1. A set S0 , containing the origin in its interior, is said to be a region of asymptotic stability for the system (15) if for all ξ(0) ∈ S0 the corresponding trajectory converges asymptotically to the origin. From Deﬁnition 1, if S0 is a region of stability, then one gets S0 ⊆ Ba . Hence, regions of asymptotic stability can be used to estimate the basin of attraction [15]. Moreover, in some practical applications one can be interested in ensuring the stability for a given set of admissible initial conditions. This set can be viewed as a practical operation region for the system or a region where the states of the system can be steered by the action of temporary disturbances. The problem we want to solve can be summarized as follows. Problem 1. Determine the anti-windup matrices Ec , Fc , Gc , Hc , Jc , Kc and the observer gain L in order to enlarge the basin of attraction of the closed-loop system. Since in general (even for very simple systems), the analytical characterization of the basin of attraction is not possible, the exact solution of Problem 1 is impossible to obtain. The problem can therefore be indirectly addressed by searching for antiwindup gains Ec , Fc , Gc , Hc , Jc , Kc and the observer gain L that lead to a region of asymptotic stability as large as possible. Problem 2. Given a set of admissible initial states X0 , determine the anti-windup gains Ec ∈ nc ×m , Fc ∈ nc ×m , Gc ∈ nc ×p , Hc ∈ m×m , Jc ∈ m×m , Kc ∈ m×p , the observer gain L ∈ n×p and an associated region of asymptotic stability S0 , such that X0 ⊂ S0 . Remark 3. Note that in Problem 2, the convergence of the observer is implicitly guaranteed. Moreover, when the state x of the system is assumed to be accessible, the observation block may be removed and the problem to be solved can then be stated as follows;

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181

Problem 3. Given a set of admissible initial states X0 , determine the anti-windup gains Ec ∈ nc ×m , Fc ∈ nc ×m , Gc ∈ nc ×p , Hc ∈ m×m , Jc ∈ m×m , Kc ∈ m×p and an associated region of asymptotic stability S0 , such that X0 ⊂ S0 . By exploiting quadratic Lyapunov functions and ellipsoidal regions of stability, the maximization of the region of stability can be accomplished by using well-known size optimization criteria. On the other hand, when the open-loop system is asymptotically stable, it can be possible to search for anti-windup matrices Ec , Fc , Gc , Hc , Jc , Kc (and observer gain L) in order to guarantee the global asymptotic stability of the closed loop system. In this last case the basin of attraction will be the whole state space.

3 Preliminaries Let us consider the generic nonlinearity ϕ(α) = satα0 (α) − α, ϕ(α) ∈ m and the following set: S(α0 ) = {α ∈ m , ω ∈ m ; −α0 α − ω α0 }

(16)

Lemma 1. [24] If α and ω are elements of S(α0 ) then the nonlinearity ϕ(α) satisﬁes the following inequality: ϕ(α) S −1 (ϕ(α) + ω) ≤ 0

(17)

for any diagonal positive deﬁnite matrix S ∈ m×m . Moreover, in order to treat in a potentially less-conservative framework the possibility of considering structural conditions, the technique developed in the sequel is based upon the use of the Finsler’s Lemma, recalled below [6]. Lemma 2. Consider a vector ζ ∈ n , a symmetric matrix P ∈ n×n and a matrix B ∈ m×n , such that rank(B) < n. The following statements are equivalent: (i) ζ P ζ < 0, ∀ζ such that Bζ = 0, ζ = 0. (ii)(B ⊥ ) PB ⊥ < 0. (iii)∃μ ∈ : P − μB B < 0. (iv)∃F ∈ n×m : P + FB + B F < 0.

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4 Theoretical Anti-Windup Conditions 4.1 Quadratic Anti-Windup Conditions First denote the following: Ku = 0 1 0 0 ; Ky = C 0 0 0 K = C 0 0 C ; Kv = Tyc Dc C −Tyc Tyc Cc 0 The following proposition is stated based upon the use of Lemma 1. Proposition 1. If there exist a symmetric positive deﬁnite matrix W , matrices Yu , Yv , Yy , Y , ZE , ZF , ZG , ZH , ZJ , ZK , Xu , Xv , Xy , X , L, and four diagonal matrices Su , Sv , Sy , S , all of appropriate dimensions, satisfying1 : ⎤ ⎡ W (A + RL LCL ) + (A + RL LCL )W ⎥ ⎢ ⎥ ⎢ −2Su ⎥ ZF R2 + ZJ R1 + Su Bu0 − Yu ⎢ ⎢ ⎥ ⎢ ⎥ −2Sv ⎢ ⎥ ⎢ −Xu ⎥<0 ZE R2 + ZH R1 + Sv Bv0 − Yv ⎥ ⎢ −Xv − Xv ⎥ ⎢ ⎥ ⎢ ⎢ 0 −Xy −2Sy ⎥ Sy By0 − Sy L RL − Yy ⎦ ⎣ 0 −X 0 −2S ZG R2 + ZK R1 + S L RL − Y (18) ⎡ ⎤ W ⎣ ⎦ ≥ 0, i = 1, ..., m (19) 2 Ku(i) W − Yu(i) u0(i) ⎤ ⎡ W ⎦ ≥ 0, i = 1, ..., p ⎣ (20) 2 Ky(i) W − Yy(i) y0(i) ⎤ ⎡ W ⎦ ≥ 0, i = 1, ..., p ⎣ (21) 2 K(i) W − Y(i) y0(i) 1

The symbol stands for symmetric blocks.

Anti-Windup Strategy for Systems Subject to Actuator and Sensor Saturations

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

W Yu

Yv

2Su

Xu

Yy Y 0

2Sv

0

ZJ Ty c (i)

−

Su Ty c (i)

−

⎤

ZH Ty c (i) − Xv(i)

2Sy 0

Sy Dc Ty c (i) − Xy(i)

2S

⎥

⎥ Xu(i) ⎥

Xy X

+Xv + Xv

W Kv(i) − Yv(i)

183

ZK Ty c (i) − X(i)

⎥ ⎥ ⎥ ⎥ ⎥ ≥ 0, i = 1, ..., m ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

2 v0(i)

(22) then the anti-windup gains Ec = ZE Sv−1 , Fc = ZF Su−1 , Gc = ZG S−1 , Hc = ZH Sv−1 , Jc = ZJ Su−1 , Kc = ZK S−1 , the observer gain L and the set S0 = {ξ ∈ 2n+m+nc ; ξ W −1 ξ ≤ 1} are solutions to Problem 2. Proof. According to the nonlinearities φu0 , φy0 , φy0 and φv0 , Lemma 1 applies as follows. • In the case of φu0 , one can deﬁne S(u0 ) according to (16) with α = v = Ku ξ ; ω = Gu ξ ; α0 = u0 Then, by denoting Yu = Gu W , the satisfaction of relation (19) guarantees that the set S0 is included in S(u0 ), and relation (17) applies with S = Su . • In the case of φy0 , one can deﬁne S(y0 ) according to (16) with α = Cx = Ky ξ ; ω = Gy ξ ; α0 = y0 Then, by denoting Yy = Gy W , the satisfaction of relation (20) guarantees that the set S0 is included in S(y0 ), and relation (17) applies with S = Sy . • In the case of φy0 , one can deﬁne S (y0 ) according to (16) with α = Cx + C = K ξ ; ω = G ξ ; α0 = y0 Then, by denoting Y = G W , the satisfaction of relation (21) guarantees that the set S0 is included in S (y0 ), and relation (17) applies with S = S . • In the case of φv0 , one can deﬁne S(v0 ) according to (16) with α = Kv ξ + (Tyc Jc − Tyc )φu0 + Tyc Hc φv0 + Tyc Dc φy0 + Tyc Kc φy0 ω = Gv ξ + Gvu φu0 + Gvv φv0 + Gvy φy0 + Gv φy0 α0 = v0

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Sophie Tarbouriech, Isabelle Queinnec, and Germain Garcia

Denote Yv = Gv W , Xu = Gvu Su , Xy = Gvy Sy , X = Gv S and Xv = Gvv Sv , and apply relation (17) with S = Sv . We have to prove that S0 is included in S(v0 ), for all ξ, φu0 , φv0 , φy0 and φy0 satisfying their respective sector condition (17). Using the S-procedure and the Schur’s complement [24], it follows that the satisfaction of relation (22) guarantees that the set S0 is included in S(v0 ). Consider now the quadratic Lyapunov function V (ξ) = ξ W −1 ξ. The time-derivative of V (ξ) along the trajectories of system (15) reads: V˙ (ξ) = ξ [(A + RL LCL ) W −1 + W −1 (A + RL LCL )]ξ + 2ξ W −1 (Bv0 + R1 Hc + R2 Ec )φv0 + 2ξ W −1 (Bu0 + R1 Jc + R2 Fc )φu0 + 2ξ W −1 (By0 − RL L)φy0 + 2ξ W −1 (RL L + R1 Kc + R2 Gc )φy0 Moreover, since (19)-(22) are satisﬁed, sector conditions hold and it follows, for all ξ ∈ S0 , that: V˙ (ξ) ≤ V˙ (ξ) − 2φu0 Su−1 (φu0 + Gu ξ) −2φv0 Sv−1 (φv0 + Gv ξ + Gvu φu0 + Gvv φv0 + Gvy φy0 + Gv φy0 ) −1 −2φy0 Sy−1 (φy0 + Gy ξ) − 2φ y0 S (φy0 + G ξ)

Then, the inequality above may be equivalently rewritten as: ⎡ V˙ (ξ) ≤ ξ W −1 φu0 Su−1 φv0 Sv−1 φy0 Sy−1

−1

⎤

W ξ ⎥ ⎢ −1 ⎢ S u φu ⎥ 0 ⎥ ⎢ ⎢ ⎥ φ S−1 M ⎢ Sv−1 φv0 ⎥ ⎢ ⎥ ⎢ S −1 φ ⎥ ⎣ y y0 ⎦ S−1 φ

where M is the left-hand term of inequality (18). Hence, it follows that, if relation (18) is satisﬁed, one has V˙ (ξ) < 0, ∀ ξ ∈ S0 , ξ = 0, and one can conclude that the set S0 is a set of asymptotic stability for the closed-loop system. Therefore, one can conclude that the conditions of Proposition 1 allows to exhibit solutions to Problem 2.

2

Condition (18) in Proposition 1 involves some nonlinear terms due to the products in the decision variables (L, W ), (Sy , L) and (S , L). A relaxation procedure may then

Anti-Windup Strategy for Systems Subject to Actuator and Sensor Saturations

185

be used to alternatively search for L or (W , Sy , S ). The main difﬁculty in using such a strategy is to initialize the relaxation procedure. In the present case, we can consider as an initial solution any observer gain L a priori determined independently of the anti-windup strategy, but with no guarantee that the procedure will converge to a ”good solution”. Another possibility would be to consider a coordinate-descent algorithm [21], which does not prevent from using a relaxation scheme, but at least allows to search for (W , Sy , S ) at each iteration step. Finally, as a more elegant solution, one could prefer modify Proposition 1 via the use of Finsler’s lemma to recover LMIs conditions.

4.2 Anti-Windup Conditions Revisited Via Finsler’s Lemma Let us denote the following: ⎤ ⎡ W A + AW ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ZF R2 + ZJ R1 + Su Bu0 − Yu −2Su ⎢ ⎥ ⎢ ⎥ −2Sv ⎢ ⎥ M1 = ⎢ ZE R2 + ZH R1 + Sv Bv0 − Yv −Xu ⎥ ⎢ ⎥ −Xv − Xv ⎢ ⎥ ⎢ ⎥ ⎢ 0 −Xy −2Sy ⎥ Sy By0 − Yy ⎣ ⎦ ZG R2 + ZK R1 − Y 0 −X 0 −2S ⎡ ⎢ ⎢ ⎢ ⎢ M2 = ⎢ ⎢ ⎢ ⎣

W CL

⎤

⎡

0 0 0 0

0 0 −Sy S

0

0 0 0

0 0

0

0

0 0 0

0 0

0

0

0 0 0

0 0

0

0

0 0 0

0 0

0

0 ⎥ ⎢ ⎢ 0 0⎥ ⎥ ⎢ ⎥ ⎢ ; M = ⎢ 0 0⎥ 3 ⎥ ⎢ ⎥ ⎢ −S 0⎦ y ⎣ 0 S

⎤

⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎦ 0

The following proposition is stated based upon the use of Lemmas 1 and 2. Proposition 2. If there exist a symmetric positive deﬁnite matrix W , matrices Yu , Yv , Yy , Y , ZE , ZF , ZG , ZH , ZJ , ZK , Xu , Xv , Xy , X , XL , F21 , F1∗ , F2∗ , four diagonal matrices Su , Sv , Sy , S of appropriate dimensions satisfying: M1 + N1 + N1 M2 − F1 + N2 <0 M3 − F2 − F2

(23)

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Sophie Tarbouriech, Isabelle Queinnec, and Germain Garcia

and conditions (19), (20), (21) and (22) ⎡ 0 ⎢ F21 F1 = ⎢ ⎣ 1 0 ⎡ 0 ⎢ XL RL N1 = ⎢ ⎣ 1 0

with

⎤ F1∗ ⎥ ⎥ ; F2 = ⎦ F1∗∗ ⎤

F1∗ ⎥ ⎥ ; N2 = ⎦ F1∗∗

F21 F2∗

(24)

0 F2∗∗

XL RL F2∗ 0

F2∗∗

(25)

then the anti-windup gains Ec = ZE Sv−1 , Fc = ZF Su−1 , Gc = ZG S−1 , Hc = −1 ZH Sv−1 , Jc = ZJ Su−1 , Kc = ZK S−1 , the observer gain L = XL (F21 ) and the

set S0 = {ξ ∈ 2n+m+nc ; ξ W −1 ξ ≤ 1} are solutions to Problem 2. Proof. The four inclusion conditions of S0 in S(u0 ), S(y0 ), S (y0 ), S(v0 ) previously stated in Proposition 1 as conditions (19) - (22) are unchanged. Consider now the Lyapunov function V (ξ) = ξ W −1 ξ and denote M the left-hand side of condition (18). M may be equivalently rewritten as: M = M1 + NM2 + M2 N + NM3 N M M 1 1 2 = 1 N N M2 M3 ⎤

⎡

RL L ⎢ ⎢ 0 ⎢ ⎢ where N = ⎢ 0 ⎢ ⎢ 0 ⎣ 0

00 0 0

⎥ 1 0 0 0⎥ ⎥ ⎥ 0 1 0 0 ⎥ ∈ ((2n+m+nc )+2m+2p)×(p+2m+2p) . ⎥ 0 0 1 0⎥ ⎦ 0 0 0 1

By applying Lemma 2, the satisfaction of M < 0 is equivalent to ﬁnd multipliers F1 and F2 such that M1 M2 M2 M3

+

F1 F2

N −1 +

N −1

F1

F2

<0

By choosing matrices F1 and F2 as in (24), then matrices N1 = F1 N and N2 = F2 N as in (25), with the change of variables LF21 = XL , the satisfaction of relation (23) means that for all ξ ∈ S0 , one gets V˙ (x) < 0, and, as a consequence, that the set

Anti-Windup Strategy for Systems Subject to Actuator and Sensor Saturations

187

S0 is a set of asymptotic stability for the closed-loop system (15). Therefore, one can conclude that the conditions of Proposition 2 allows to exhibit solutions to Problem 2

2.

4.3 Global Stability Conditions Propositions 1 and 2 provide solutions to Problem 2 in a local context. Provided that some open-loop stability assumptions are veriﬁed for matrix A, the global asymptotic stability can be addressed. The following two corollaries give solutions to the global asymptotic stability problems in the case without using Finsler’s Lemma (Corollary 1) then using it (Corollary 2). Corollary 1. If there exist a symmetric positive deﬁnite matrix W , matrices ZE , ZF , ZG , ZH , ZJ , ZK , L, and four diagonal matrices Su , Sv , Sy , S satisfying: ⎡ W (A + RL LCL ) + (A + RL LCL )W ⎢ ⎢ Z R + Z R + Su B − Ku W −2Su u0 F 2 J 1 ⎢ ⎢ ⎢ −2Sv ⎢ −Tyc ZJ ⎢ ⎢ ZE R2 + ZH R1 + Sv Bv0 − Kv W −Tyc ZH ⎢ −Tyc Su ⎢ −ZH Ty c ⎢ ⎢ ⎢ Sy By0 − Sy L RL − Ky W 0 −Sy Dc Ty c −2Sy ⎣ R2 + ZK R1 + S L RL − K W ZG

0

−ZK Ty c

0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

−2S (26)

then the anti-windup gains Ec = ZE Sv−1 , Fc = ZF Su−1 , Gc = ZG S−1 , Hc = ZH Sv−1 , Jc = ZJ Su−1 , Kc = ZK S−1 and the observer gain L are such that the global asymptotic stability of the closed-loop system is ensured. Proof. From the proof of Proposition 1, by considering Gu = Ku , Gy = Ky , G = K , Gv = Kv , Gvu = Tyc Jc −Tyc , Gvv = Tyc Hc , Gvy = Tyc Dc and Gv = Tyc Kc , one gets S(u0 ) = S(v0 ) = S(y0 ) = S (y0 ) = 2n+m+nc . It follows that the sector conditions relative to the nonlinearities φu0 , φv0 , φy0 and φy0 are globally satisﬁed, that is, for all ξ ∈ 2n+m+nc .

2

Corollary 2. If there exist a symmetric positive deﬁnite matrix W , matrices ZE , ZF , ZG , ZH , ZJ , ZK , XL , F21 , F1∗ , F2∗ , and four diagonal matrices Su , Sv , Sy , S satisfying relation (23) but with

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⎡

W A + AW

⎢ ⎢ Z R + Z R + Su B − Ku W −2Su u0 J 1 ⎢ F 2 ⎢ ⎢ −2Sv ⎢ −Tyc ZJ ⎢ M1 = ⎢ ZE R2 + ZH R1 + Sv Bv0 − Kv W −Tyc ZH ⎢ −T S y u c ⎢ −ZH Ty c ⎢ ⎢ ⎢ Sy By0 − Ky W 0 −Sy Dc Ty c −2Sy ⎣ R2 + ZK R1 − K W ZG

0

−ZK Ty c

0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

−2S

then the anti-windup gains Ec = ZE Sv−1 , Fc = ZF Su−1 , Gc = ZG S−1 , Hc = −1 ZH Sv−1 , Jc = ZJ Su−1 , Kc = ZK S−1 and the observer gain L = XL (F21 ) are

such that the global asymptotic stability of the closed-loop system is ensured. Proof. The proof directly follows from the combination of arguments used to prove Proposition 2 and Corollary 1.

2

5 Discussion For a given observer gain L, it is worth to notice that the conditions in Proposition 1 are under LMI form in the decision variables. This fact comes from using model (15) with Lemma 1. The use of a classical sector condition as in [12] or of a polytopic model as in [4] does not lead to LMI conditions but to BMI conditions even in the case of a mere actuator saturation. When the full synthesis problem of ﬁnding simultaneously the observer gain, the anti-windup gains and the estimate of the region of stability given by the Lyapunov matrix W is to be done, the exhibition of a solution through Proposition 1 should be done by means of iterative schemes. Such solutions are however very sensitive to the initial considered guess and only local sub-optimality can be guaranteed. Another option could be to add structural block-diagonal constraints on matrix W to cope with the bilinear nature of the problem [8]. One can then prefer in that case the use of Proposition 2, which, thanks to the use of Finsler’s multipliers, is stated under LMI form, and does not require initial guesses neither iterative schemes. Another important feature of the proposed conditions resides in the separation between the Lyapunov matrix W −1 , and the decision variable

Anti-Windup Strategy for Systems Subject to Actuator and Sensor Saturations

189

L. Furthermore the matrix F21 is a general (nonsingular) matrix, that is it does not present any structural constraint such that the symmetry. Note however that this is done to the detriment of the number of unknown variables which is increased in this case. Moreover, the extension of our approach to provide a global asymptotic stability condition (in the case where the open-loop system is stable) is direct as shown in Corollaries 1 and 2. When a polytopic model is considered it is not possible to exhibit a condition ensuring global stability. This is due to the local character of such a model for representing the saturated system. Another point of interest is to evaluate the inﬂuence of the observation block on the anti-windup strategy efﬁciency. Hence, if the state x is assumed to be accessible, one can achieve a solution to Problem 3 by slightly modify the condition of Proposition 1. The closed-loop saturated system with anti-windup strategy writes: ξ˙r = Ar ξ r + (Brv0 + Rr1 Hc + Rr2 Ec )φv0 + (Bru0 + Rr1 Jc + Rr2 Fc )φu0

(27) +(Bry0 + Rr1 Kc + Rr2 Gc )φy0 ⎡ ⎤ x ⎥ ⎢ n+m+nc ⎥ and the matrices identiﬁed by r directly follow from with ξ r = ⎢ ⎣v⎦ ∈ η the matrices deﬁned in (14) by removing the last line (and the last column in the case of Ar ). Considering also matrices Kr∗ derived from K∗ by removing the last column, one directly obtains a solution to Problem 3 given by the following corollary. Corollary 3. If there exist a symmetric positive deﬁnite matrix W r , matrices Yur , Yvr , Yyr , ZE , ZF , ZG , ZH , ZJ , ZK , Xu , Xv , Xy , and three diagonal matrices Su , Sv , Sy , all of appropriate dimensions, satisfying: ⎡ ⎤ W r Ar + Ar W r ⎢ ⎥ ⎢ r ⎥ r r r ⎥ ⎢ ZF R2 + ZJ R1 + Su Bu0 − Yu −2Su ⎢ ⎥ ⎢ ⎥<0 −2Sv ⎢ r ⎥ r r r ⎢ ZE R2 + ZH R1 + Sv Bv0 − Yv −Xu ⎥ ⎢ ⎥ −Xv − Xv ⎣ ⎦ r r r r 0 −Xy −2Sy ZG R2 + ZK R1 + Sy By0 − Yy ⎡ ⎤ Wr ⎣ ⎦ ≥ 0, i = 1, ..., m r u20(i) Kru(i) W r − Yu(i)

(28)

(29)

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Sophie Tarbouriech, Isabelle Queinnec, and Germain Garcia

⎡ ⎣ ⎡

Yur

⎤

Wr Kry(i) W r

Yvr

−

Yyr

r Yy(i)

2 y0(i)

⎦ ≥ 0, i = 1, ..., p Krv(i)

r Yv(i)

(30) ⎤

W W − ⎢ ⎥ ⎢ ⎥ 2S X 0 Z T − S T − X ⎢ u u yc (i) u J yc (i) u(i) ⎥ ⎥ ⎢ ⎥ ⎢ 2Sv ⎥ ⎢ ⎥ ≥ 0, ⎢ X Z T − X y H y v(i) (i) c ⎥ ⎢ +X + X v v ⎥ ⎢ ⎥ ⎢ ⎢ 2Sy Sy Dc Tyc (i) + ZK Tyc (i) − Xy(i) ⎥ ⎦ ⎣ 2 v0(i) r

r

(31)

i = 1, ..., m then the anti-windup gains Ec = ZE Sv−1 , Fc = ZF Su−1 , Gc = ZG Sy−1 , Hc =

ZH Sv−1 , Jc = ZJ Su−1 , Kc = ZK Sy−1 and the set S0 = {ξ r ∈ n+m+nc ; ξ r W r

−1

ξr ≤

1} are solutions to Problem 3. Proof. It follows the same lines than the one of Proposition 1, applied to the reduced 2

system (27).

6 Optimization Issues For design purposes, according to Propositions 1 and 2, a set Ξ0 deﬁned by its vertices: Ξ0 = co{vj , j = 1, ..., r, vj ∈ 2n+nc } may be used as a shape set to serve for optimizing the size of the set S0 . Two cases can be addressed if we consider the set Ξ0 and a scaling factor β. Thus, we want to satisfy β Ξ0 ⊂ S0 . In case 1, this problem reduces to a feasibility problem with β = 1. In case 2, the objective consists in maximizing β, which corresponds to deﬁne through Ξ0 the directions in which we want to maximize S0 . Thus, the problem of maximizing β can be accomplished by solving the following convex optimization problems: • Optimization problem related to Proposition 1

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191

min μ subject to relations (18), (19), (20), (21), (22) μ vj ≥ 0, j = 1, . . . , r vj W

(32)

• Optimization problem related to Proposition 2 min μ subject to relations (23), (19), (20), (21), (22) μ vj ≥ 0, j = 1, . . . , r vj W

(33)

√ Considering β = 1/ μ, the minimization of μ implies the maximization of β. Other criteria associated to the size of S0 , (e.g. the volume or the size of the minor axis) can be adopted in order to maximize the stability region. Moreover, some structural or norm constraints on the anti-windup gains Ec , and Fc could be considered (see [10]).

7 Numerical Illustrations 7.1 Academical Example of Order 1 Consider system (1) described by the following data: A = 0.1; B = 1; C = 1 with the simple PI controller (2) deﬁned by: Ac = 0; Bc = −0.2; Cc = 1; Dc = −2 The actuator dynamics are given by: Tyc = 1; Tu = −25; Tv = 5 with bounds on the saturation elements: u0 = 1; v0 = 10; y0 = 2

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Sophie Tarbouriech, Isabelle Queinnec, and Germain Garcia

In this ﬁrst example, the state x is directly accessible, and one can directly applies Corollary 3. Then, by solving the optimization problem (32), but with the conditions stated in Corollary 3, and where Ξ0 is the hypercube centered in 0 and with vertices components equal to 1 or −1, one gets Ec = 0.0109 ;

Fc = 2.1133 ; Gc = 0.2000

Hc = −1.0060 ; Jc = 6.3278 ; Kc = 2.0000 for which β = 0.6951. When studying the analysis case, i.e. without anti-windup loops, one gets β = 0.3187. Hence, thanks to the anti-windup strategy, the scaling factor β has been increased of more than 100%. Figure 4 shows the projection in two-dimensional spaces of the ellipsoidal sets of stability S0 in the case without anti-windup (dashed line) and with anti-windup and accessible x (dotted line).

Let us consider the initial state ξ(0) = 1.2 0 0 which belongs to the ellipsoidal set solution of the case with anti-windup strategy, but which does not belong to the 4

v

2 0 −2 −4 −4

−3

−2

−1

−3

−2

−1

0

1

2

3

4

0

1

2

3

4

x

10

η

5 0 −5 −10 −4

x

10

η

5 0 −5 −10 −2.5

−2

−1.5

−1

−0.5

0

v

0.5

1

1.5

2

2.5

Fig. 4. Example 1 - Projection in two-dimensional spaces of ellipsoidal sets S0 . Case without anti-windup (dashed line) and case with anti-windup and accessible x (dotted line)

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y = satu0 (Cx)

2

1

0

−1

−2 0

2

4

6

8

0

2

4

6

8

10

12

14

16

18

20

10

12

14

16

18

20

u = satv0 (v)

1 0.5 0

−0.5 −1

time(s)

Fig. 5. Example 1 - Time-evolution of the saturated closed-loop system initiated in ξ(0) = [1.2 0 0] . Case without anti-windup (dashed line) and case with anti-windup and accessible x (dotted line)

ellipsoidal set obtained without anti-windup. One can check on Figure 5 that the trajectories of the closed-loop system with anti-windup converges towards the origin although the trajectories without anti-windup is unstable (in fact, it remains bounded thanks to the saturation elements).

7.2 F/A-18 HARV Aircraft Example Let us now consider the linearised model of F/A-18 HARV aircraft lateral dynamics [23], in horizontal stabilised ﬂight at Mach 0.7 and altitude 20000ft. The system (1) is described by the following data: ⎤ ⎡ −2.3142 0.5305 −15.5763 0 ⎥ ⎢ ⎥ ⎢ −0.0160 −0.1287 3.0081 0 ⎥ A=⎢ ⎢ 0.0490 −0.9980 −0.1703 0.0440 ⎥ ⎦ ⎣ 1.0000 0.0491 0 0

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Sophie Tarbouriech, Isabelle Queinnec, and Germain Garcia

⎡

23.3987 21.4333

3.2993

⎤

⎥ ⎢ ⎢ −0.1644 0.3313 −1.9836 ⎥ 0 0 1 0 ⎥ B=⎢ ⎢ −0.0069 −0.0153 0.0380 ⎥ ; C = 0 0 0 1 ⎦ ⎣ 0 0 0 and the dynamic compensator (2) evaluated in [22] such that the closed-loop poles are placed in a pre-speciﬁed region of the complex plane is deﬁned by: ⎤ ⎤ ⎡ ⎡ −0.98 0.05 −0.03 −1.84 0.24 −0.03 ⎥ ⎥ ⎢ ⎢ ⎢ 32.55 −4.09 0.42 −16.22 ⎥ ⎢ 0.205 −0.2897 ⎥ ⎥ ⎥ ⎢ Ac = ⎢ ⎥ ⎢ 65.56 −2.90 −6.85 −9.77 ⎥ ; Bc = ⎢ −46.23 0.89 ⎦ ⎦ ⎣ ⎣ 10.91 0.20 −0.05 −9.92 1.59 −0.14 ⎤ ⎤ ⎡ ⎡ 32.55 −0.00 −0.63 −10.57 −2.77 −0.1 ⎥ ⎥ ⎢ ⎢ ⎥ ; Dc = ⎢ −0.64 −0.11 ⎥ Cc = ⎢ 20.11 0.18 −0.26 −7.73 ⎦ ⎦ ⎣ ⎣ −1.61 −0.73 −0.47

−4.22

5.40

The actuator dynamics are given by: ⎤ ⎡ ⎡ 100 −25 0 ⎥ ⎢ ⎢ ⎥ ⎢ T yc = ⎢ ⎣ 0 1 0 ⎦ ; Tu = ⎣ 0 −25 001

0

⎤

0.19

⎤

⎡

0

500

⎥ ⎢ ⎢0 5 ; T = 0⎥ v ⎦ ⎣ 0 −25 00

⎥ 0⎥ ⎦ 5

with bounds on the control surfaces and outputs: ⎤ ⎤ ⎡ ⎡ ⎡ ⎤ 25 100 30 ⎥ ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ u0 = ⎢ ⎣ 10.5 ⎦ ; v0 = ⎣ 40 ⎦ ; y0 = ⎣ 20 ⎦ 30 82 35 Consider a polyhedral set of desired initial condition Ξ0 given by its vertices with components 1 or 0 of the ﬁrst four components corresponding to the aircraft plane. The results obtained in the different cases are summarized as follows: • C1 - Analysis, obtained via the use of optimization problem (32), but with conditions stated in Corollary 3, and without all the variables related to the anti-windup gains: β = 13.080 • C2 - Anti-windup when x is accessible, obtained via the use of optimization problem (32), but with conditions stated in Corollary 3:

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β = 18.494 The anti-windup gains are: ⎤ ⎡ ⎤ ⎡ −0.038 0.001 0.002 −1.000 −0.000 −0.000 ⎥ ⎢ ⎥ ⎢ 0.074 0.004 −0.617 ⎥ ⎢ −3 ⎢ ⎥ ; Hc = ⎢ 0.000 −1.000 −0.000 ⎥ Ec = 10 ∗ ⎢ ⎥ ⎦ ⎣ ⎣ −0.565 0.015 0.339 ⎦ 0.000 −0.000 −1.000 −0.071 0.003 −0.024 ⎤ ⎡ ⎤ ⎡ −3.635 −5.452 −4.187 5.937 0.058 0.044 ⎥ ⎢ ⎢ −98.219 −145.757 −115.444 ⎥ ⎥ ⎢ ⎥ ; Jc = ⎢ 0.088 6.389 0.035 ⎥ ⎢ Fc = ⎢ ⎦ ⎣ 8.979 −3.223 ⎥ ⎦ ⎣ 3.949 −0.212 −0.256 6.096 −11.860 −17.991 −13.481 ⎤ ⎡ ⎤ ⎡ −0.240 0.030 2.770 0.100 ⎥ ⎢ ⎥ ⎢ −0.205 0.290 ⎥ ⎢ ⎥ ; ⎢ 0.640 0.110 ⎥ K Gc = ⎢ = c ⎢ 46.230 −0.890 ⎥ ⎦ ⎣ ⎦ ⎣ 4.220 −0.190 −1.590 0.140 • C3 - Anti-windup when the observer gain is pre-ﬁxed with a pole-placement of A + LC in {−2, −1, −1 ± i}, obtained via the use of optimization problem (32): β = 16.280 The anti-windup gains are: ⎤ ⎡ ⎤ ⎡ 0.090 0.119 −0.266 −0.606 0.706 0.250 ⎥ ⎢ ⎥ ⎢ 19.264 23.140 14.907 ⎥ ⎢ ⎥ ⎥ ⎢ Ec = ⎢ ⎢ −6.339 −7.401 −8.066 ⎥ ; Hc = ⎣ 0.174 −0.554 0.063 ⎦ ⎦ ⎣ 0.279 0.253 −0.504 1.072 1.386 0.099 ⎤ ⎡ ⎤ ⎡ 0.029 1.262 −5.193 5.814 −0.109 −0.530 ⎥ ⎢ ⎥ ⎢ −34.161 5.485 −205.990 ⎥ ⎢ ⎥ ; Jc = ⎢ 0.096 6.197 0.004 ⎥ Fc = ⎢ ⎢ 23.784 20.392 ⎦ ⎣ 37.382 ⎥ ⎦ ⎣ 0.113 0.208 5.429 −1.579 2.663 −19.855 ⎤ ⎡ ⎤ ⎡ −0.761 0.124 −0.493 0.067 ⎥ ⎢ ⎥ ⎢ −19.961 2.436 ⎥ ⎢ ⎥ ; ⎢ −0.164 0.029 ⎥ K Gc = ⎢ = c ⎢ −5.527 0.542 ⎥ ⎦ ⎣ ⎦ ⎣ −0.482 −0.007 −2.363 0.369

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Sophie Tarbouriech, Isabelle Queinnec, and Germain Garcia

• C4 - Anti-windup with simultaneous determination of the observer (Finsler case), obtained via the use of optimization problem (33): β = 17.467 The observer gain solution of the problem is: ⎤ ⎡ 0.068 −0.218 ⎥ ⎢ ⎢ 0.011 0.195 ⎥ ⎥ L=⎢ ⎢ −0.933 −0.098 ⎥ ⎦ ⎣ −0.057 −1.313 which gives the spectrum {−2.08, −0.85, −0.97 ± 1.81i} for A + LC. The anti-windup gains are: ⎤ ⎡ ⎤ ⎡ 0.038 −0.351 −0.560 −0.691 0.521 0.181 ⎥ ⎢ ⎥ ⎢ 19.737 19.806 14.578 ⎥ ⎢ ⎥ ; Hc = ⎢ 0.156 −0.551 0.055 ⎥ Ec = ⎢ ⎢ −8.062 −14.754 −13.305 ⎥ ⎦ ⎣ ⎦ ⎣ 0.215 0.054 −0.511 1.010 0.369 −0.529 ⎤ ⎡ ⎤ ⎡ 0.418 1.532 −7.331 5.630 −0.186 −0.964 ⎥ ⎢ ⎥ ⎢ −22.083 −3.052 −229.981 ⎥ ⎢ ⎥ ; Jc = ⎢ 0.153 6.255 0.010 ⎥ Fc = ⎢ ⎢ 23.839 28.858 ⎦ ⎣ 16.096 ⎥ ⎦ ⎣ 0.195 0.244 5.384 −0.334 2.694 −25.156 ⎤ ⎡ ⎤ ⎡ −0.033 0.022 −0.017 0.007 ⎥ ⎢ ⎥ ⎢ 0.107 0.353 ⎥ ⎢ −3 ⎢ −3 ⎢ ⎥ ; Kc = 10 ∗ ⎣ 0.000 −0.002 ⎥ Gc = 10 ∗ ⎢ ⎥ ⎦ ⎣ −0.721 0.195 ⎦ −0.009 0.005 −0.055 0.057 The comparison of cases C1 and C2 gives the maximum potential increase of the set of initial admissible states obtained by using the proposed anti-windup strategy, when all the state x of the system is accessible. In the present case, the scaling factor β has been increased of about 41%. The comparison of cases C2 associated to Corollary 3, i.e., in the case where the state x is accessible, and C4 associated to the optimization problem (33) allows to evaluate the inﬂuence of the observation loop on the attainable size of the domain of stability S0 . In fact, the case C4 expresses how the scaling factor is degraded with respect to case C2 when an observer has to be computed to recover the unmeasured state x. In the present case, the anti-windup

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197

strategy involving the observer allows to increase the scaling factor of about 33% instead of 41% obtained when x is accessible. It is also interesting to examine the difference between the solutions of cases C3 and C4, for which, by different ways, the closed-loop spectrum of A + LC is almost the same ({−2, −1, −1±i}). The use of the optimization problem (33) based on the use of Finsler’s multipliers gives additional degrees of freedom which allows to obtain larger set of admissible initial conditions. Moreover, the choice of a pre-calculated observer L has a strong inﬂuence on the results of the optimization problem (32). If it is not correctly chosen, its presence in the condition (18) strongly degrades the solution, even in the case without anti-windup. Figure 6 shows the projection in two-dimensional spaces of the ellipsoidal sets of stability S0 . Only the sub-part relative to the system state x is plotted. The actuator, controller and observation error states are assumed to be initially equal to 0 to

100

20

50

10

x3

x2

simplify the presentation of the ﬁgure, although this does not prevent for initializing

0 −50 −500

0

x1

500

−20 −1000

1000

200

20

100

10

x3

x4

−100 −1000

0 −100 −200 −1000

0

500

1000

0

50

100

0

10

20

x1

0

−500

0

x1

500

−20 −100

1000

200

200

100

100

0 −100 −200 −100

−500

−10

x4

x4

0 −10

−50

x2

0 −100

−50

0

x2

50

100

−200 −20

−10

x3

Fig. 6. Example 2 - Projection in two-dimensional spaces of the ellipsoidal subset associated to S0 , relative to the state x of the open-loop system. Case without anti-windup (dashed line), case with anti-windup and a priori given observer gain (solid line), case with anti-windup and accessible x (dotted line) and case with anti-windup and observer gain computed simultaneously (dashdot line)

198

Sophie Tarbouriech, Isabelle Queinnec, and Germain Garcia

those states to any initial values such that the full vector ξ belongs to the set S0 . The projections are given in the case without anti-windup (dashed line), with antiwindup and a priori given observer gain (solid line), with anti-windup and accessible x (dotted line) and with anti-windup and observation gain simultaneously computed (dashdot line). The inﬂuence of the initialization of the observation error can be examined on Figure 7. This ﬁgure shows the projection in two-dimensional spaces of the sub-part of S0 relative to the observation error . The actuator and controller states are assumed to be initially equal to 0. The initial condition for the system state x is taken into account to plot the two-dimensional projections by using the Schur complement. One can easily verify that the initial admissible error is much larger by using the optimization problem (33) derived from the use of Finsler’s multipliers than the one obtained when solving the problem with a pre-ﬁxed observer. 4

1

x 10

5000

3

2

0.5 0

0

−0.5 −1 −4

−2

0

1

2

−2

x 10

4

4

−5000 −4

4 4

x 10

0

1

2

4 4

x 10

5000

3

4

2 0

0

−2 −4 −4

−2

0

1

2

−5000 −1

x 10

4

4

4

4

0.5

1 4

x 10

x 10

2

4

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4

0

2

4

x 10

0 −2 −4 −1

−0.5

4

0 −2

−0.5

0

2

0.5

1 4

x 10

−4 −5000

0

3

5000

Fig. 7. Example 2 - Projection in two-dimensional spaces of the ellipsoidal subset associated to S0 , relative to the state x of the open-loop system. Case without anti-windup (dashed line), case with anti-windup and a priori given observer gain (solid line), case with anti-windup and accessible x (dotted line) and case with anti-windup and observer gain computed simultaneously (dashdot line)

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Finally, note that the extended state vector is of dimension 15, which does not prevent from numerical evaluation of the anti-windup and observer gains. Moreover, note that the size of the problem could be reduced if a mere reduced-order observer was used instead of a full-order observer.

7.3 Benchmark 1 - Inverted Pendulum Let us now focus on the inverted pendulum around its upright position for small inclination angles. The linearized model with proposed numerical data is given as follows:

⎡

⎤ ⎡ ⎡ ⎤ 100 0 ⎥ ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ 0 1⎥ ⎦; B = ⎣0⎦; C = ⎣0 1 0⎦ 001 1 10

0 −1 0

⎢ A=⎢ ⎣ −1 0

⎤

for which a dynamic feedback controller (2) has been a priori computed, without taking into account the actuator and sensor, such that the closed-loop spectrum is given by {−2.5 ± 0.5i, −2.5, −0.5 ± 0.5i, −0.5} as follows: ⎤ ⎤ ⎡ ⎡ −2.50 0.50 0 2.50 −1.50 0 ⎥ ⎥ ⎢ ⎢ ⎥ ; Bc = ⎢ −0.50 2.50 1.00 ⎥ Ac = ⎢ −0.50 −2.50 0 ⎦ ⎦ ⎣ ⎣ 1.75 −3.00 −4.00 0 1.00 2.50 Cc = 1.75 −3.00 −1.50 ; Dc = 0 0 0 According to the benchmark requirements, the actuator dynamics are given by: Tyc = 10; Tu = 10/Ts = 20; Tv = 1/Ts = 2 with bounds on the saturation elements: u0 = 1.25; v0 = 2 Note that in this benchmark example, there is no saturation of the output signal. The optimization problem to be solved is then derived from the conditions of Corollary 3 where all the variables related to the output saturation are removed. The unknown variables become W r , Yur , Yvr , ZE , ZF , ZH , ZJ , Xu , Xv , Su and Sv , which have to satisfy conditions (28) (where the fourth line and column have been removed), (29) (unchanged) and (30) (where the fourth line and column have to be removed). The anti-windup gains solution to this optimization problem are given by:

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Sophie Tarbouriech, Isabelle Queinnec, and Germain Garcia

⎡

−0.0174

⎤

⎡

−123.6841

⎤

⎥ ⎥ ⎢ ⎢ ⎥ ; Fc = ⎢ 122.1753 ⎥ ; Hc = −0.0999 ; Jc = 0.4488 Ec = ⎢ 0.0168 ⎦ ⎦ ⎣ ⎣ −0.0553 −398.2176 The projection in two-dimensional spaces of the ellipsoidal set S0 in all directions is

50

50

2 0 −2 −50

50

200 0 −200 −50

0

50

100 0 −100 −50

0

2

100 0 −100 −2

0 η1

v

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200 0 −200 −20

0 η1

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x3

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0

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x3

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v

η3

η2

v

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2 0 −2 −50

x3

η2

η1

x3

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0

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x2

x2

η2

η1

x2

20 0 −20 −50

η1 x2

η3

0

0

50

x1

v

x3 η2

x2

100 0 −100 −50

50

20 0 −20 −50

x1

v

0

0

0 x1

η3

50

100 0 −100 −50

x1

50 0 −50 −50

50

200 0 −200 −50

η3

0

0 x1

η2

η1

x1

20 0 −20 −50

50

2 0 −2 −50

η3

0

50 0 −50 −50

0

2

v

η3

50 0 −50 −50

x3

x2

plotted in solid line on ﬁgure 8. It is compared with the ellipsoidal set solution of the

20

200 0 −200 −100

0

100

η2

Fig. 8. Example 3 - Projection in two-dimensional spaces of the ellipsoidal set S0 in all directions. Case without anti-windup (dashed line) and case with anti-windup (solid line)

Anti-Windup Strategy for Systems Subject to Actuator and Sensor Saturations

201

analysis problem without anti-windup (dashed line). One can see that, in all directions, the anti-windup scheme allows to increase the size of admissible components. In order to emphasize the set of admissible initial states for the inverted pendulum system while the other initial states are set equal to zero, the projection in twodimensional spaces of this subset of states is plotted on ﬁgure 9. 2

x2

1 0 −1 −2 −1.5

−1

−0.5

−1

−0.5

−1

−0.5

0

0.5

1

1.5

0

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1

1.5

0

0.5

1

1.5

x1

4

x3

2 0 −2 −4 −1.5

x1

4

x3

2 0 −2 −4 −1.5

x2

Fig. 9. Example 3 - Projection in two-dimensional spaces of the ellipsoidal subset associated to S0 , relative to the state x of the open-loop system. Case without anti-windup (dashed line) and case with anti-windup (solid line)

Simulation results also show that the anti-windup strategy allows to improve the performance of the inverted pendulum control. As an example, the trajectories initiated

in ξ r (0) = 1.63 1.45 −1.15 0.5 1.4 1.4 −1.55 are plotted in ﬁgure 10 in the case with anti-windup and in ﬁgure 11 in the case without anti-windup. This shows that the anti-windup strategy allows to obtain the asymptotic stability of trajectories initiated from states which are not stabilizable without anti-windup. The actuator output u (and actuator state ν) is also plotted both in the case with anti-windup (ﬁgure 12) and without anti-windup (ﬁgure 13).

202

Sophie Tarbouriech, Isabelle Queinnec, and Germain Garcia 3

x1

x2

2.5

x3

2

pendulum state x

1.5

1

0.5

0

−0.5

−1

−1.5 0

2

4

6

8

10 time (s)

12

14

16

18

20

Fig. 10. Example 3 - Trajectories initiated in ξ r (0) in the case with anti-windup 20 x1

x2

15

x3

10

pendulum state x

5

0

−5

−10

−15

−20

0

2

4

6

8

10 time (s)

12

14

16

18

20

Fig. 11. Example 3 - Trajectories initiated in ξ r (0) in the case without anti-windup

8 Concluding Remarks In this chapter, LMI conditions have been proposed to treat the anti-windup gain design for linear systems subject to both actuator and sensor saturation. We have particularly focused on the problem of enlargement of the basin of attraction of the closed-loop system. The anti-windup loops use some measured variables (due to the actuator) and some estimated ones (due to the sensor) issued from an observer step. The obtained conditions are proposed via the use of a modiﬁed sector condition which allows to cope with some drawbacks appearing when using other modelling for the closed-loop system (like classical sector condition or polytopic model). The conditions have also been extended through the use of Finsler’s Lemma which allows

Anti-Windup Strategy for Systems Subject to Actuator and Sensor Saturations

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1.5

1

actuator output u

0.5

0

−0.5

−1

−1.5 0

2

4

6

8

10 time (s)

12

14

16

18

20

Fig. 12. Example 3 - Evolution of the actuator output u (solid line) and actuator state ν (dashed line) for trajectories initiated in ξ r (0) in the case with anti-windup

1.5

1

actuator output u

0.5

0

−0.5

−1

−1.5

0

2

4

6

8

10 time (s)

12

14

16

18

20

Fig. 13. Example 3 - Evolution of the actuator output u for trajectories initiated in ξ r (0) in the case with anti-windup

the simultaneous determination of the anti-windup gains and the observer gain necessary to recover the unmeasured states used in the anti-windup scheme with LMI conditions. Different examples, involving one of the proposed benchmarks have allowed to discuss about the different optimization problems, about the inﬂuence of the observer step on the set of initial admissible states and on the numerical tractability of the conditions. In a future work the input-to-state stabilization problem for linear systems subject to L2 -disturbances and the output tracking problem should be addressed.

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References ˚ om and L. Rundqwist. Integrator windup and how to avoid it. In American 1. K. J. Astr¨ Control Conference, pages 1693–1698, Pittsburgh, PA, 1989. 2. C. Barbu, R. Reginatto, A. R. Teel, and L. Zaccarian. Anti-windup for exponentially unstable linear systems with inputs limited in magnitude and rate. In American Control Conference, Chicago, IL, June 2000. 3. D. S. Bernstein. A plant taxonomy for designing control experiments. IEEE Control Systems Magazine, pages 7–14, 2001. 4. Y-Y. Cao, Z. Lin, and D.G. Ward. An antiwindup approach to enlarging domain of attraction for linear systems subject to actuator saturation. IEEE Trans. Aut. Cont., 47(1):140– 145, 2002. 5. Y.Y. Cao, Z. Lin, and B.M. Chen. An output feedback h∞ controller design for linear systems subject to sensor nonlinearities. IEEE Transactions on Circuits and Systems-I, 50(7):914–921, 2003. 6. M. C. de Oliveira and R. E. Skelton. Stability tests for constrained linear systems. In S. O. Reza Moheimani, editor, Perspectives in Robust Control, Lecture Notes in Control and Information Sciences, pages 241–257. Springer-Verlag, 2001. 7. T. Fliegner, H. Logemann, and E.P. Ryan. Time-varying and adative integral control of linear systems with actuator and sensor nonlinearities. In Proc. of 15th International Symposium on Mathematical Theory of Networks and Systems (MTNS), Perpignan, France, 2000. 8. J. C. Geromel, P. L. D. Peres, and S. R. Souza. Convex analysis of output feedback control problems: robust stability and performance. IEEE Trans. Aut. Cont., 41(7):997– 1003, 1996. 9. A.H. Glattfelder and W. Schaufelberger. Control systems with input and output constraints. Springer-Verlag, London, 2003. 10. J.M. Gomes da Silva Jr. and S. Tarbouriech. Antiwindup design with guaranteed regions of stability: An lmi-based approach. IEEE Trans. Aut. Cont., 50(1):106–110, 2005. 11. J.M. Gomes da Silva Jr., S. Tarbouriech, and G. Garcia. Local stabilization of linear systems under amplitude and rate saturating actuators. IEEE Trans. Aut. Cont., 48:842– 847, 2003. 12. J.M. Gomes da Silva Jr., S. Tarbouriech, and R. Reginatto. Analysis of regions of stability for linear systems with saturating inputs through an anti-windup scheme. In Proceedings of the IEE Conference on Control Applications (CCA), Glasgow, Scotland, 2002. 13. G. Grimm, J. Hatﬁeld, I. Postlethwaite, A.R. Teel, M.C. Turner, and L. Zaccarian. Antiwindup for stable linear systems with input saturation: an lmi based synthesis. IEEE Trans. Aut. Cont., 48(9):1509–1525, 2003.

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14. V. Kapila and K. Grigoriadis. Actuator saturation control. Marcel Dekker, Inc., 2002. 15. H. K. Khalil. Nonlinear Systems. MacMillan, 1992. 16. R. Koplon, M.L.J Hautus, and E.D. Sontag. Observability of linear systems with saturated outputs. Linear Algebra and Applications, 205-206:909–936, 1994. 17. M. V. Kothare and M. Morari. Multiplier theory for stability analisys of anti-windup control systems. Automatica, 35:917–928, 1999. 18. G. Kreisselmeier. Stabilization of linear systems in the presence of output measurement saturation. Systems and Control Letters, 29:27–30, 1996. 19. P. Langou¨et, S. Tarbouriech, and G. Garcia. Pilots evaluation by taking into account both limited actuator and incidence - stability analysis. Number Grant F/20062/SA, report in french - Limited diffusion, 2002. 20. R.M. Murray. Geometric approach to control in the presence of magnitude and rate saturation. Technical Report 99-001, Division of Eng. and Appl. Science, California Institute of Technology, 1999. 21. D. Peaucelle and D. Arzelier. An efﬁcient numerical solution for h2 static output feedback synthesis. In European Control Conference, Porto, Portugal, 2001. 22. C. Pittet. Stabilisation des syst`emes a` commande contrainte: application a` une table d’excitation microdynamique. PhD thesis, Universit´e Paul Sabatier, Toulouse, France, 1998. (in french). 23. J.M. Shewchun and E. Feron. High performance control with position and rate limited actuators. Int. J. Robust and Nonlinear Control, 9(10):617–630, 1999. 24. S. Tarbouriech, C.Prieur, and J.M. Gomes da Silva Jr. Stability analysis and stabilization of systems presenting nested saturations. In Proc. of 43rd Conference on Decision and Control (CDC’04), Paradise Island, Bahamas, 2004. 25. A. R. Teel. Anti-windup for exponentially unstable linear systems. Int. J. Robust and Nonlinear Control, 9(10):701–716, 1999. 26. M.C. Turner and L. Zaccarian (Eds). Special issue on anti-windup. Int. J. Systems Science, 37(2), 2006. 27. M.C. Turner and S. Tarbouriech. Anti-windup for linear systems with sensor saturation: sufﬁcient conditions for global stability and l2 gain. internal report, LAAS-CNRS, 2006.

Sampled-Data Nonlinear Model Predictive Control for Constrained Continuous Time Systems Rolf Findeisen, Tobias Raff, and Frank Allg¨ower Institute for Systems Theory and Automatic Control, University of Stuttgart, Germany {findeise,raff,allgower}@ist.uni-stuttgart.de

Summary. Often one desires to control a nonlinear dynamical system in an optimal way taking constraints on the states and inputs directly into account. Classically this problem falls into the ﬁeld of optimal control. Often, however, it is difﬁcult, if not impossible, to ﬁnd a closed solution of the corresponding Hamilton-Jacobi-Bellman equation. One possible control strategy that overcomes this problem is model predictive control. In model predictive control the solution of the Hamilton-Jacobi-Bellman equation is avoided by repeatedly solving an open-loop optimal control problem for the current state, which is a considerably simpler task, and applying the resulting control open-loop for a short time. The purpose of this paper is to provide an introduction and overview to the ﬁeld of model predictive control for continuous time systems. Speciﬁcally we consider the so called sampled-data nonlinear model predictive control approach. After a short review of the main principles of model predictive control some of the theoretical, computational and implementation aspects of this control strategy are discussed and underlined considering two example systems. Key words. Model predictive control, constrained systems, sampled-data

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1 Introduction Many methods for the control of dynamical systems exist. Besides the question of stability often the achieved performance as well as the satisfaction of constraints on the states and inputs are of paramount importance. One classical approach to take these points into account is the design of an optimal feedback controller. If the system is nonlinear, howver, it is often very hard, if not impossible, to derive a closed solution for the corresponding feedback controller. One possible approach to overcome this problem is the application of model predictive control (MPC), often also referred to as receding horizon control or moving horizon control. Basically in model predictive control the open-loop optimal control problem is solved repeatedly at speciﬁc sampling instants for the current, ﬁxed system state. The ﬁrst part of the resulting open-loop input is applied to the system until the next sampling instant, at which the optimal control problem for the new system state is solved again. Since the optimal control problem is solved at every sampling instant only for one ﬁxed initial condition, the solution is much easier to obtain than to deriveobtain a closed solution of the Hamilton-Jacobi-Bellman partial differential equation (for all possible initial conditions) of the original optimal control problem. In general one distinguishes between linear and nonlinear model predictive control (NMPC). Linear MPC refers to MPC schemes that are based on linear dynamical models of the system and in which linear constraints on the states and inputs and a quadratic cost function are employed. NMPC refers to MPC schemes that use for the prediction of the system behavior nonlinear models and that allow to consider nonquadratic cost functions and nonlinear constraints on the states and inputs. By now linear MPC is widely used in industrial applications [77, 78, 41, 75, 40]. For example [78] reports more than 4500 applications spanning a wide range from chemicals to aerospace industries. Also many theoretical and implementation issues of linear MPC theory have been studied so far [55, 75, 68]. Many systems are, however, inherently nonlinear and the application of linear MPC schemes leads to poor performance of the closed-loop. Driven by this shortcoming and the desire to directly use ﬁrst principles based nonlinear models there is a steadily increasing interest in the theory and application of NMPC. Over the recent years much progress in the area of NMPC (see for example [68, 1, 17, 78]) has been made. However, there remain a series of open questions and hurdles

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that must be overcome in order that theoretically well founded practical application of NMPC is possible. In this paper we focus on an introduction and overview of NMPC for continuous time systems with sampled state information, i.e. we consider the stabilization of continuous time systems by repeatedly applying input trajectories that are obtained from the solution of an open-loop optimal control problem at discrete sampling instants. In the following we shortly refer to this as sampled-data NMPC. In comparison to NMPC for discrete time systems (see e.g. [68, 1, 17]) or instantaneous NMPC [68], the inter sampling behavior of the system with respect to the open-loop input must be taken into account, see e.g. [27, 44, 25, 45, 62]. In Section 2 we review the basic principle of NMPC. Before we focus on the theoretical questions, we shortly outline in Section 2.3 how the resulting open loop optimal control problem can be solved. Section 3 contains a discussion on how stability in sampled-data NMPC can be achieved. Section 4 discusses robustness issues in NMPC and Section 5 considers the output feedback problem for NMPC. Before concluding in Section 8 we consider in Section 6 the sampled-data NMPC control of a simple nonlinear example system and in Section 7 the pendulum benchmark example considered throughout this book.

2 Principle of Sampled-Data Model Predictive Control In model predictive control the input applied to the system is given by the repeated solution of a (ﬁnite) horizon open-loop optimal control problem subject to the system dynamics, state and input constraints: Based on measurements obtained at a sampling time (in the following denoted by ti ), the controller predicts the dynamic behavior of the system over the so called control/prediction horizon Tp and determines the input such that an open-loop performance objective is minimized. Under the assumption that the prediction horizon spans to inﬁnity and that there are no disturbances and no model plant mismatch, one could apply the resulting input openloop to the system and achieve (under certain additional assumptions) convergence to the origin. However, due to external disturbances, model plant mismatch and the use of ﬁnite prediction horizons the predicted state and the true system state differ. Thus, to counteract this deviation and to suppress the disturbances it is necessary to incorporate feedback. In model predictive control this is achieved by applying the obtained optimal open-loop input only until the next sampling instant at which the

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closed-loop

control/prediction horizon Tp

input u

control/prediction horizon Tp

closed-loop input u

optimal open loop input u ¯

closed-loop

optimal open loop input u ¯

closed-loop

state x

state x

predicted state x ¯ ti

predicted state x ¯

ti + Tp

ti+1

ti

sampling time ti

ti+1

ti+2

ti+1 + Tp

sampling time ti+1

Fig. 1. Principle of model predictive control.

whole process – prediction and optimization – is repeated (compare Figure 1), thus moving the prediction horizon forward. The whole procedure can be summarized by the following steps: 1. Obtain estimates of the current state of the system 2. Obtain an admissible optimal input by minimizing the desired cost function over the prediction horizon using the system model and the current state estimate for prediction 3. Implement the obtained optimal input until the next sampling instant 4. Continue with 1. Considering this control strategy various questions such as closed-loop stability, robustness to disturbances/model uncertainties and the efﬁcient solution of the resulting open-loop optimal control problem arise, which will be discussed in the following.

2.1 Mathematical Formulation of Sampled-Data NMPC Throughout the paper we consider the stabilization of time-invariant nonlinear systems of the form x(t) ˙ = f (x(t), u(t))

,

x(0) = x0 ,

(1)

where x ∈ Rn denotes the system state and u ∈ Rm is the control or input to the system. We assume that the vector ﬁeld f : Rn × Rm → Rn is locally Lipschitz

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continuous with f (0, 0) = 0. The objective is to (optimally) stabilize the system subject to the input and state constraints: u(t) ∈ U ⊂ Rm , x(t) ∈ X ⊆ Rn , ∀t ≥ 0, where U ⊂ Rm is assumed to be compact and X ⊆ Rn is assumed to be simply connected with (0, 0) ∈ X ×U. Remark 1. (Rate constraints on the inputs) If rate constraints ˙ ∀t ≥ 0 u(t) ˙ ∈ U,

(2)

on the inputs are desired, they can, for example, be transformed to the given form by adding integrators at the input side to the system, see for example Section 7.Note, however, that this transforms the input constraint u ∈ U to a state constraint on the integrator states. We denote the solution of (1) starting at a time t1 from a state x(t1 ), applying a (piecewise continuous) input u : [t1 , t2 ] → Rm by x(τ; u(·), x(t1 )), τ ∈ [t1 , t2 ]. In sampled-data NMPC an open-loop optimal control problem is solved at the discrete sampling instants ti . We assume that these sampling instants are given by a partition π of the time axis: Deﬁnition 1. (Partition) A partition is a series π = (ti ), i ∈ N of (ﬁnite) positive real numbers such that t0 = 0, ti < ti+1 and ti → ∞ for i → ∞. Furthermore, ¯ := supi∈N (ti+1 −ti ) denotes the upper diameter of π and π := inf i∈N (ti+1 −ti ) π denotes the lower diameter of π. Whenever t and ti occur together, ti should be taken as the closest previous sampling instant with ti < t. The input applied in between the sampling instants in NMPC, i.e. in the interval [ti , ti+1 ), is given by the solution of the open-loop optimal control problem

min

[0,Tp ]

J(x(ti ), u ¯(·))

(3a)

u ¯(·)∈L∞

subject to: x ¯˙ (τ) = f (¯ x(τ), u ¯(τ)),

x ¯(ti ) = x(ti )

(3b)

u ¯(τ) ∈ U, x ¯(τ) ∈ X τ ∈ [ti , ti + Tp ]

(3c)

x ¯(ti + Tp ) ∈ E.

(3d)

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Here the bar denotes predicted variables, i.e. x ¯(·) is the solution of (3b) driven by the input u ¯(·) : [ti , ti + Tp ] → U with the initial condition x(ti ). The distinction between the real system state x of (1) and the predicted state x ¯ in the controller is necessary since due to the moving horizon nature even in the nominal case the predicted states will differ from the real states at least after one sampling instant. As cost functional ¯ > 0 we consider J minimized over the control horizon Tp ≥ π ti +Tp ¯(·)) := F (¯ x(τ), u ¯(τ))dτ + E(¯ x(ti + Tp )), (4) J(x(ti ), u ti

where the stage cost F : X ×U → X is assumed to be continuous, satisﬁes F (0, 0) = 0, and is lower bounded by a positive semideﬁnite function αF : R → R+ 0 , i.e. αF (x) ≤ F (x, u) ∀(x, u) ∈ X × U. We furthermore assume that the autonomous system f (x, 0) is zero state detectable via α(x), i.e. ∀(x0 ) ∈ X , αF (x(τ; x0 )) = 0 ⇒ x(τ; x0 ) as t → ∞, where x(τ; x0 ) denotes the solution of the system x˙ = f (x, 0) starting from x(0) = x0 . The so called terminal region constraint E and the so called terminal penalty term E are typically used to enforce stability or to increase the performance of the closed-loop, see Section 3. The solution of the optimal control problem (3) is denoted by u ¯ (·; x(ti )). It deﬁnes the open-loop input that is applied to the system until the next sampling instant ti+1 : ¯ (t; x(ti )), u(t; x(ti )) = u

t ∈ [ti , ti+1 ) .

(5)

As noted above, the control u(t; x(ti )) is a feedback, since it is recalculated at each sampling instant using the new state measurement. We limit the presentation to input signals that are piecewise continuous and refer to an admissible input as: Deﬁnition 2. (Admissible Input) An input u : [0, Tp ] → Rm for a state x0 is called admissible, if it is: a) piecewise continuous, b) u(τ) ∈ U ∀τ ∈ [0, Tp ], c) x(τ; u(·), x0 ) ∈ X ∀τ ∈ [0, Tp ], d) x(Tp ; u(·), x0 ) ∈ E. We furthermore consider an admissible set of problem (3) as: Deﬁnition 3. (Admissible Set) A set X ⊆ X is called admissible, if for all x0 ∈ X there exists a piecewise continuous input u ˜ : [0, Tp ] → U such that a) x(τ; x0 , u ˜(·)) ∈ X, τ ∈ [0, Tp ] and b) x(Tp ; x0 , u ˜(·)) ∈ E. Without further restrictions it is often not clear if for a given x an admissible input exists nor if the minimum of (3) exists. While the existence of an admissible input

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is related to constrained controllability, the existence of an optimal solution of (3) is in general non easy to answer. For simplicity of presentation we assume in the following that the set R denotes an admissible set which admits an optimal solution of (3), i.e. one obtains the following assumption: Assumption 4 (Set R ) There exists an admissible set R such that (3) admits for all x0 ∈ R an optimal (not necessarily unique) solution. It is possible to derive existence results for (3) considering measurable inputs and imposing certain convexity and compactness conditions; see for example [37, 36, 73] and [4, 35, 82]. However, often it is not possible to check the necessary conditions a priori. The main reason for imposing Assumption 4 is the requirement that an optimal/feasible solution at one sampling instant should guarantee (under certain assumptions) the existence of an optimal/feasible solution at the next sampling instant (see Section 3). The optimal value of the cost functional (4) plays an important role in many considerations. It is typically denoted as value function: Deﬁnition 4. (Value function) The value function V (x) is deﬁned as the minimal value of the cost for the state x: V (x) = J(¯ u (·; x); x). The value function is for example used in the proof of convergence and stability. It often serves as a “Lyapunov function”/decreasing function candidate, see Section 3 and [68, 1]. In comparison to sampled-data NMPC for continuous time systems, in instantaneous NMPC the input is deﬁned by the solution of the optimal control problem (3) at all times: u(x(t)) = u ¯ (t; x(t)), i.e. no open-loop input is applied, see e.g. [68, 67]. Considering that the solution of the open-loop optimal control problem requires an often non negligible time, this approach can not be applied in practice. Besides the continuous time considerations, results for NMPC of discrete time systems are also available (see e.g. [68, 1, 17]). We do not go into further details here. Remark 2. (Hybrid nature of sampled-data predictive control) Note, that in sampleddata NMPC the input applied in between the recalculation instants ti and ti+1 is given by the solution of the open-loop optimal control problem (3) at time ti , i.e. the closed-loop is given by

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x(t) ˙ = f (x(t), u(t; x(ti ))) .

(6)

Thus, strictly speaking, the behavior of the system is not only deﬁned by the current state. Rigorously one has to consider a hybrid system [74, 43, 84, 46] consisting of the “discrete” state x(ti ), the continuous state x(t). This is especially important for the stability considerations in Section 3, since the “discrete memory” x(ti ) must be taken into account.

2.2 Inherent Characteristics and Problems of NMPC One of the key problems in predictive control schemes is that the actual closedloop input and states differ from the predicted open-loop ones, even if no model plant mismatch and no disturbances are present. This stems from the fact that at the next sampling instant the (ﬁnite) prediction horizon moves forward, allowing to consider more information thus leading to a mismatch of the trajectories. The difference between the predicted and the closed-loop trajectories has two immediate consequences. Firstly, the actual goal to compute a feedback such that the performance objective over an often desired inﬁnite horizon of the closed-loop is minimized is not achieved. Secondly there is in general no guarantee that the closed-loop system will be stable. It is indeed easy to construct examples for which the closed-loop becomes unstable if a short ﬁnite horizon is chosen. Hence, when using ﬁnite prediction horizons special attention is required to guarantee stability (see Section 3). Summarizing, the key characteristics and properties of NMPC are: • NMPC allows the direct use of nonlinear models for prediction. • NMPC allows the explicit consideration of state and input constraints. • In NMPC a time domain cost functional/performance criteria is minimized online. • In NMPC the predicted behavior is in general different from the closed- loop behavior. • For the application of NMPC an open-loop optimal control problem must be solved on-line. • To perform the prediction the system states must be measured or estimated.

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Remark 3. In this paper we mainly focus on NMPC for the stabilization of timeinvariant continuous time nonlinear systems. However, note that NMPC is also applicable to a large class of other systems, i.e. discrete time systems, delay systems, time-varying systems, and distributed parameter systems, for more details see for example [68, 17, 1]. Furthermore, NMPC is also well suited for tracking problems or problems where one has to perform a transition between different steady states optimally, see e.g. [58, 70, 28]. Before we summarize the available stability results for sampled-data NMPC, we comment in the next section on the numerical solution of the open-loop optimal control problem.

2.3 Numerical Aspects of Sampled-Data NMPC Predictive control circumvents the solution of the Hamilton-Jacobi-Bellman equation by solving the open-loop optimal control problem at every sampling instant only for the currently measured system state. An often untractable problem is replaced by a tractable one. In linear MPC the solution of the optimal control problem (3) can often be cast as a convex quadratic program, which can be solved efﬁciently. This is one of the main reasons for the practical success of linear MPC. In NMPC, however, at every sampling instant a general nonlinear open-loop optimal control problem (3) must be solved on-line. Thus one important condition for the application of NMPC, is the availability of reliable and efﬁcient numerical dynamic optimization algorithms for the optimal control problem (3). Solving (3) numerically efﬁcient and fast is, however, not a trivial task and has attracted much research interest in recent years (see e.g. [66, 83, 2, 81, 5, 56, 18, 65, 6, 64, 23, 24, 22]). Typically so called direct solution methods [7, 6, 76] are used, i.e. the original inﬁnite dimensional problem is turned into a ﬁnite dimensional one by discretizing the input (and also possibly the state). Basically this is done by parameterizing the input (and possibly the states) ﬁnitely and to solve/approximate the differential equations during the optimization. We do not go into further details and instead refer to [66, 7, 22]. However, we note that recent studies have shown the usage of special dynamic optimizers and tailored NMPC schemes allows to employ NMPC to practically relevant problems (see e.g. [2, 81, 65, 29, 24, 34]), even with todays computational power.

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Remark 4. (Sub optimality and NMPC) Since the optimal control problem (3) is typically non convex, it is questionable if the globally minimizing input can be found at all. While the usage of a non optimal admissible input might lead to an increase in the cost, it is not crucial to ﬁnd the global minima for stability of the closed-loop, as outlined in the next Section.

3 Nominal Stability of Sampled-Data NMPC As outlined one elementary question in NMPC is whether a ﬁnite horizon NMPC strategy does guarantee stability of the closed-loop. While a ﬁnite prediction and control horizon is desirable from an implementation point of view, the difference between the predicted state trajectory and the resulting closed-loop behavior can lead to instability. Here we review some central ideas how stability can be achieved. No attempt is made to cover all existing approaches and methods, especially those which consider instantaneous or discrete time NMPC. We do also only consider the nominal case, i.e. it is assumed that no external disturbances act on the system and that there is no model mismatch between the system model used for prediction and the real system.

Stability for inﬁnite prediction horizons: The most intuitive way to achieve stability/convergence to the origin is to use an inﬁnite horizon cost, i.e. Tp in the optimal control problem (3) is set to ∞. In this case the open-loop input and state trajectories resulting from (3) at a speciﬁc sampling instant coincide with the closed-loop trajectories of the nonlinear system due to Bellman’s principle of optimality [3]. Thus, the remaining parts of the trajectories at the next sampling instant are still optimal (end pieces of optimal trajectories are optimal). Since the ﬁrst part of the optimal trajectory has been already implemented and the cost for the remaining part and thus the value function is decreasing, which implies under mild conditions convergence of the states. Detailed derivations can for example be found in [67, 68, 52, 51]. Stability for ﬁnite prediction horizons: In the case of ﬁnite horizons the stability of the closed-loop is not guaranteed a priori if no precautions are taken. By now a series of approaches exist, that achieve closed-loop stability. In most of these approaches the terminal penalty E and the terminal region constraint E are chosen suitable to

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guarantee stability or the standard NMPC is modiﬁed to achieve stability. The additional terms are not motivated by physical restrictions or performance requirements, they have the sole purpose to enforce stability. Therefore, they are usually called stability constraints. Stability via a zero terminal constraint: One possibility to enforce stability with a ﬁnite prediction horizon is to add the so called zero terminal equality constraint at the end of the prediction horizon, i.e. x ¯(t + Tp ) = 0

(7)

is added to the optimal control problem (3) [9, 67, 52, 69]. This leads to stability of the closed-loop, if the optimal control problem has a solution at t = 0. Similar to the inﬁnite horizon case the feasibility at one sampling instant does imply feasibility at the following sampling instants and a decrease in the value function. One disadvantage of a zero terminal constraint is that the predicted system state is forced to reach the origin in ﬁnite time. This leads to feasibility problems for short prediction/control horizon lengths, i.e. to small regions of attraction and performance limitations. Furthermore, from a computational point of view, an exact satisfaction of a zero terminal equality constraint does require in general an inﬁnite number of iterations in the optimization and is thus not desirable. The main advantages of a zero terminal constraint are the straightforward application and the conceptual simplicity.

Control Lyapunov function approaches: In the case that E is a global control Lyapunov function for the system, the terminal region constraint x ¯(t + Tp ) ∈ E is actual not necessary. Even if the control Lyapunov is not globally valid, convergence to the origin can be achieved [50] and it can be established that for increasing prediction horizon length the region of attraction of the inﬁnite horizon NMPC controller is recovered [50, 48]. Approaches using a control Lyapunov functions as terminal penalty term and no terminal region constraint are typically referred to as control Lyapunov function based NMPC approaches. Uniﬁed conditions for convergence: Besides the outlined approaches there exist a series of approaches [71, 12, 11, 14, 61] that are based on the consideration of an (virtual) local control law that is able to stabilize the system inside of the terminal

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region and where the terminal penalty E provides an upper bound on the optimal inﬁnite horizon cost. The following theorem covers most of the existing stability results. It establishes conditions for the convergence of the closed-loop states under sampled-data NMPC. It is a slight modiﬁcation of the results given in [36, 11, 10]. Theorem 1. (Convergence of sampled-data NMPC) Suppose that (a) the terminal region E ⊆ X is closed with 0 ∈ E and that the terminal penalty E(x) ∈ C 1 is positive semi-deﬁnite ¯] → U such that x(τ) ∈ E and (b) ∀x ∈ E there exists an (admissible) input uE : [0, π ∂E ¯] f (x(τ), uE (τ)) + F (x(τ), uE (τ)) ≤ 0 ∀τ ∈ [0, π ∂x

(8)

(c) x(0) ∈ R Then for the closed-loop system (1), (5) x(t) → 0 for t → ∞. Proof. See [26]. Loosely speaking, E is a F -conform local control Lyapunov function in the terminal set E. The terminal region constraint enforces feasibility at the next sampling instant and allows, similarly to the inﬁnite horizon case, to show that the value function is strictly decreasing. Thus stability can be established. Note that this result is nonlocal in nature, i.e. there exists a region of attraction R which is of at least the size of E. Various ways to determine a suitable terminal penalty term and terminal region exist. Examples are the use of a control Lyapunov function as terminal penalty E [50, 49] or the use of a local nonlinear or linear control law to determine a suitable terminal penalty E and a terminal region E [71, 12, 11, 14, 61]. Remark 5. (Sub optimality) Note that we need the rather strict Assumption 4 on the set R to ensure the existence of a new optimal solution at ti+1 based on the existence of an optimal solution at ti . The existence of an admissible input at ti+1 , i.e. u ˜ is already guaranteed due to existence of local controller, i.e. condition (b). In principle the existence of an optimal solution at the next time instance is not really required for the convergence result. The admissible input, which is a concatenation of the remaining old input and the local control already leads to a decrease in the cost

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function and thus convergence. To increase performance from time instance to time instance one could require that the cost decreases from time instance to time instance more than the decrease resulting from an application of the “old” admissible control, i.e. feasibility implies convergence [12, 79]. Remark 6. (Stabilization of systems that require discontinuous inputs) In principle Theorem 1 allows to consider the stabilization of systems that can only be stabilized by feedback that is discontinuous in the state [36], e.g. nonholonomic mechanical systems. However, for such systems it is in general rather difﬁcult to determine a suitable terminal region and a terminal penalty term. To weaken the assumptions in this case, it is possible to drop the continuous differentiability requirement on E, requiring merely that E is only Lipschitz continuous in E. From Rademacher’s theorem [16], it follows that E is continuously differentiable almost everywhere and that (8) holds for almost all τ and the proof remains nearly unchanged. More details can be found in [37]. Remark 7. (Special input signals) Basically it is also possible to consider only special classes of input signals, e.g. one could require that the input is piecewise continuous in between sampling instants or that the input is parameterized as polynomial in time or as a spline. Modifying Assumption 4, namely that the optimal control problem posses a solution for the considered input class, and that condition (8) holds for the considered inputs, the proof of Theorem 1 remains unchanged. The consideration of such inputs can for example be of interest, if only piecewise constant inputs can be implemented on the real system or if the numerical on-line solution of the optimal control problem allows only the consideration of such inputs. One example of such an expansion are the consideration of piecewise constant inputs as in [61, 62]. So far only conditions for the convergence of the states to the origin where outlined. In many control applications also the question of asymptotic stability in the sense of Lyapunov is of interest. Even so that this is possible for the sampled-data setup considered here, we do not go into further details, see e.g. [37, 26]. Concluding, the nominal stability question of NMPC is by now well understood and a series of NMPC schemes exist, that guarantee closed-loop stability.

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4 Robustness of Sampled-Data NMPC The results reviewed so far are based on the assumption that the real system coincides with the model used for prediction, i.e. no model/plant mismatch or external disturbances are present. Clearly, this is very unrealistic and the development of a NMPC framework to address robustness issues is of paramount importance. In general one distinguishes between the inherent robustness properties of NMPC and the design of NMPC controllers that take the uncertainty/disturbances directly into account. Typically NMPC schemes that take uncertainty that acts on the system directly into account are based on game-theoretic considerations. Practically they often require the on-line solution of a min-max problem. A series of different approaches can be distinguished. We do not go into details here and instead refer to [54, 13, 8, 60, 13, 59, 53, 57, 38]. In contrast to this we are interested in the so called inherent robustness properties of sampled-data NMPC. By inherent robustness we mean the robustness of NMPC to uncertainties/disturbances without taking them directly into account. As shown elsewhere sampled-data NMPC posseses under certain conditions inherent robustness properties. This property stems from the close relation of NMPC to optimal control. Results on the inherent robustness of instantaneous NMPC can for example be found in [63, 9, 68]. Discrete time results are given in [80, 42] and results for sampled-data NMPC are given in [33, 71]. Typically these results consider additive disturbances of the following form: x˙ = f (x, u) + p(x, u, w),

(9)

where p : Rn × Rm × Rl → Rn describes the model uncertainty/disturbance and where w ∈ W ∈ Rl might be an exogenous disturbance acting on the system. However, assuming that f locally Lipschitz in u these results can be simply expanded to the case of input disturbances. This type of disturbances is of special, since it allows to capture the inﬂuence of numerical solution errors of the open-loop optimal control problem. Further examples of input disturbances are neglected fast actuator dynamics, computational delays, or numerical errors in the solution of the underlying optimal control problem. For example, inherent robustness was used in [21, 20] to establish stability of a NMPC scheme that employs approximated solutions of the optimal control problem.

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Summarizing, some preliminary results for the inherent robustness and the robust design of NMPC controllers exist. However, these result are either not implementable since they require a high computational load or they are not directly applicable due to their restrictive assumptions.

5 Output Feedback Sampled-Data NMPC One of the key obstacles for the application of NMPC is that at every sampling instant ti the system state is required for prediction. However, often not all system states are directly accessible, i.e. only an output (10)

y = h(x, u)

is directly available for feedback, where y ∈ Rp are the measured outputs and where h : Rn × Rm → Rp maps the state and input to the output. To overcome this problem one typically employs a state observer for the reconstruction of the states. In principle, instead of the optimal feedback (5) the “disturbed” feedback u(t; x ˆ(ti )) = u ¯ (t; x ˆ(ti )),

t ∈ [ti , ti+1 )

(11)

is applied. Yet, due to the lack of a general nonlinear separation principle, stability is not guaranteed, even if the state observer and the NMPC controller are both stable. Several researchers have addressed this problem (see for example [32] for a review). The approach in [19] derives local uniform asymptotic stability of contractive NMPC in combination with a “sampled” state estimator. In [58], see also [80], asymptotic stability results for observer based discrete-time NMPC for “weakly detectable” systems are given. The results allow, in principle, to estimate a (local) region of attraction of the output feedback controller from Lipschitz constants. In [72] an optimization based moving horizon observer combined with a certain NMPC scheme is shown to lead to (semi-global) closed-loop stability. In [31, 30, 47], where semiglobal stability results for output feedback NMPC using high-gain observers are derived. Furthermore, in [32], based on the inherent robustness properties of NMPC as outlined in Section 4 for a broad class of state feedback nonlinear model predictive controllers, conditions, on the observer that guarantee that the closed-loop is semi-global practically stable.

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Even so that a series of output feedback results for NMPC using observers for state recovery exist, most of these approaches are far away from being implementable. Thus, further research has to address this important question to allow practical application of NMPC.

6 A Simple Nonlinear Example The following example is thought to show some of the inherent properties of sampled-data NMPC and to show how Theorem 1 can be used to design a stabilizing NMPC controller that takes constraints into account. We consider the following second order system [39] x˙ 1 (t) = x2 (t)

(12a)

x˙ 2 (t) = −x1 (t) + x2 (t) sinh(x21 (t) + x22 (t)) + u(t),

(12b)

which should be stabilized with the bounded control u(t) ∈ U := {u ∈ R| |u| ≤ 1} ∀t ≥ 0 where the stage cost is given by F (x, u) = x22 + u2 .

(13)

According to Theorem 1 we achieve stability if we can ﬁnd a terminal region E and a C 1 terminal penalty E(x) such that (8) is satisﬁed. For this we consider the unconstrained inﬁnite horizon optimal control problem for (12). For this simple example one can calculate the explicite solution leading to the control law 2

2

u∞ (x) = −x2 ex1 +x2 which minimizes the corresponding cost ∞ ! x22 (τ) + u2 (τ) dτ, J∞ (x, u(·)) =

(14)

(15)

0

and that the associated value function, which will be used as terminal penalty term, is given by 2

2

E(x) := V∞ (x) = ex1 +x2 − 1.

(16)

It remains to ﬁnd a suitable terminal region. According to Theorem 1 (b) there must exist for all x ∈ E an open-loop input u which satisﬁes the constraints such that (8) is satisﬁed. If we deﬁne E as

Sampled-Data Model Predictive Control for Constrained Systems

E := {x ∈ R2 |E(x) ≤ α}

223

(17)

we know that along solution trajectories of the closed-loop system controlled by u∞ (x), i.e. x˙ = f (x, u∞ ), the following holds ∂E f (x, u∞ (x)) + F (x(τ), u∞ (x)) = 0, ∂x

(18)

however, α must be chosen such that u∞ (x) ∈ U. It can be veriﬁed, that for α = 1 β

− 1, where β satisﬁes 1 − βeβ = 0, u∞ (x) ∈ U ∀x ∈ E. The derived terminal 2

penalty term E(x) and the terminal region E are designed to satisfy the conditions of Theorem 1, thus the resulting NMPC controller should be able to stabilize the closed-loop. The resulting NMPC controller with the prediction horizon set to Tp = 2 is compared to a feedback linearizing controller and the optimal controller (14) (where the input of both is limited to the set U by saturation). The feedback linearizing controller used is given by:

! uFl (x) := −x2 1 + sinh(x21 + x22 ) ,

(19)

which stabilizes the system globally, if the input is unconstrained. The implemented input for the feedback linearizing controller (and the unconstrained optimal controller (14)) is given by u(x) = sign(uFl (x)) min{1, |uFl (x)|},

(20)

where the sign operator is deﬁned as usual, i.e. sign(x) :=

−1, 1,

x<0 x≥0

. For the

NMPC controller the sampling instants are given by an equidistant partition of the time axis, i.e. π = (ti ) with ti+1 = ti + δ and t0 = 0, where the sampling time δ is δ = 0.1. The open-loop optimal control problem (3) is solved by a direct solution method. Speciﬁcally the input signal is parametrized as piecewise constant with a time discretization of 0.05 over the prediction horizon, i.e. at every sampling instant an optimization problem with 40 optimization variables is solved. Figure 2 shows the simulation results in the phase plane for the initial conditions x1 (0) = −1.115 and x2 (0) = −0.2 of all three controllers. Note that the initial conditions are such that for all controllers the input constraints are not active at the beginning. However, after some time the maximum applicable input is reached, i.e. the saturation in (20) is active. As can be seen, both, the optimal controller and the feedback linearizing controller are not able to stabilize the system for the considered initial conditions.

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1

0.8

0.6

x

2

0.4

0.2

0

−0.2

−0.4

−0.6 −1.5

−1

−0.5

0

x

0.5

1

1.5

1

Fig. 2. Phase plot x2 over x1 starting from the initial conditions x(0) = [−1.115, −0.2] for the NMPC controller (black solid), the saturated feedback linearizing controller (dark gray solid) and the saturated optimal controller (gray solid). The inner ellipsoid (gray dashed) is the border of the terminal region E of the NMPC controller, while the outer curve (black dashed) are the points for which the optimal controller u∞ (x) just satisﬁes the input constraint (saturation not active).

In comparison the NMPC controller is able to stabilize the system while meeting the input constraints (see Figure 3). Note that inside the terminal region the NMPC controller and the optimal control law u∞ (x) coincide, since the constraints are not active and since (18) is satisﬁed with equality. Thus, the terminal penalty term E(x) can be seen as an approximation of the cost that appears up to inﬁnity. As can be seen from this example, if a value function/Lyapunov function and a local controller as well as the corresponding region of attraction is known, NMPC can be utilized to increase the overall region of attraction of the closed-loop while satisfying the input and state constraints.

7 Inverted Pendulum Benchmark Example As a second example, underling the achievable performance in the case of input and state constraints, we consider the inverted pendulum on a cart system considered as

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225

x1

0 −0.5 −1

x

2

1 0 −1

u

1 0 −1 0

2

4

6

8

10

12

Time

Fig. 3. Simulation results starting from the initial condition x(0) = [−1.115, −0.2] for the NMPC controller (black solid), the saturated feedback linearizing controller (dark gray solid) and the saturated optimal controller (gray solid).

an benchmark for constrained control in the book ⎡ ⎤ ⎡ ⎤ ⎤ ⎡ 1 0 0 −1 0 ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ x(t) ˙ =⎢ ⎣−1 0 1⎦ x(t) + ⎣0⎦ u(t) + ⎣0⎦ z(t). 0 1 0 1 0

(21)

The variable x1 denotes the horizontal speed of the pendulum, x2 the horizontal displacement of the pendulum and x3 the horizontal speed of the cart. The load z represents a horizontal force on the pendulum which is persistent with unknown but bounded magnitude. Furthermore, u is the force of the actuator acting on the cart which is constrained in the magnitude by |u| ≤ 1.25 and in the slew rate by |du(t)/dt| ≤ 2s−1 . In order to take the slew rate constraint on the control input into account, the system (21) is augmented by an integrator at the control input. Thus, the input constraint |u| ≤ 1.25 is transformed to a state constraint on the new state, e.g. |ξ4 | ≤ 1.25. With the state ξ = [ξ1 , ξ2 , ξ3 , ξ4 ]T = [x1 , x2 , x3 , u]T and the new control input v(t) = du(t)/dt one obtains the augmented system

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⎡ ⎤ ⎡ ⎤ ⎤ 1 0 0 −1 0 0 ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ˙ = ⎢−1 0 1 0⎥ ξ(t) + ⎢0⎥ v(t) + ⎢0⎥ z(t). ξ(t) ⎢0⎥ ⎢0⎥ ⎢ 0 1 0 1⎥ ⎣ ⎦ ⎣ ⎦ ⎦ ⎣ 0 1 0 0 00 ⎡

(22)

Therefore, the input constraints of the system (21) are |ξ4 | < 1.25 and |dv(t)/dt| < 2s−1 for the system (22). Note that the input constraints of the system (22) can be cast in the optimal control problem (3). In the following, two control problems are considered. The control objective of the ﬁrst problem is to track a reference signal r while the control objective of the second problem is to stabilize the system under the inﬂuence of a disturbance z. For both control problems the stage cost is chosen as ⎤ ⎡ 10 0 0 0 ⎥ ⎢ ⎥ ⎢ T ⎢ 0 10 0 0⎥ (ξ − ξs ) + v 2 , F (ξ, v) = (ξ − ξs ) ⎢ ⎥ ⎣ 0 0 10 0⎦ 0 0 0 1 where is a small positive parameter, e.g. = 0.00001, and ξs the set point which depends on the reference signal r and the disturbance z. The parameter in the stage cost is chosen small in order to recover the classical quadratic stage cost on the state x and on the input u. To guarantee closed-loop stability, the terminal cost E and the terminal region E are calculated off-line by a procedure as in the quasiinﬁnite horizon model predictive control scheme described in [12, 15]. The resulting terminal cost E is given by

Fig. 4. Inverted pendulum on a cart.

Sampled-Data Model Predictive Control for Constrained Systems

⎡

142.6 −148.8 −67.0 −17.3

⎢ ⎢−148.8 169.0 E(ξ) = (ξ − ξs ) ⎢ ⎢ −67.0 79.2 ⎣ T

−17.3

21.1

79.2 44.5 12.1

227

⎤

⎥ 21.1 ⎥ ⎥ (ξ − ξs ) 12.1 ⎥ ⎦ 5.3

and the terminal region E is given by E = {ξ ∈ R4 |E(ξ) ≤ 3.2}.

(23)

Note, that the design of the terminal penalty term and terminal region constraint is rather easy, since the system itself is linear. Furthermore, the control and prediction horizon is chosen to TP = 6 and the sampling time is chosen to δ = 0.1.

7.1 Tracking In the following the tracking problem is studied. The control objective is that the state variable x1 tracks asymptotically the reference signal r. However, the tracking problem cannot solved directly via the NMPC controller with the optimal control problem (3). Therefore, the tracking problem was considered as a sequence of set point changes. The set points of the system (22) depend on the reference signal r, i.e. ξs = [r 0 r 0]T . Figure 5 shows the closed-loop system states x, the control input u and the reference signal r. Figure 5 shows that the reference signal r is tracked sufﬁciently well while satisfying the constraints. Note that no pre-information about the reference signal is considered in the prediction, which would improve the performance of predictive control further.

7.2 Disturbance Attenuation In the following the task is to stabilize the state x1 under a persistent disturbance z with unknown but bounded magnitude. It is assumed that the full state ξ can be measured but not the disturbance z. Also in this control problem the NMPC controller with the optimal control problem (3) cannot directly be applied to stabilize the state x1 under the disturbance z. A typical approach to solve such kind of disturbance attenuation problem in model predictive control is to estimate the disturbance z via an observer and to use the estimated disturbance zˆ in the prediction of the model predictive controller. The disturbance z can be estimated via the observer

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1

x ,r

2

1

0

0

5

10

15

20

25

30

35

40

45

0

5

10

15

20

25

30

35

40

45

0

5

10

15

20

25

30

35

40

45

1

2

x ,x

3

2.5 2

0 −1 2

u

1.25 0

−1.25 −2

Time

Fig. 5. Simulation results of r (gray solid), x1 (black solid), x2 (gray solid), x3 (black solid), and u (black solid) for the tracking problem.

⎡ ⎤ ⎤ 0 0 −1 0 0 1 ⎢ ⎥ ⎥ ⎢ ⎢0⎥ ⎢−1 0 1 0 0⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ˆ ˆ˙ = ⎢ + ⎢0⎥ v(t) + L(y(t) − yˆ(t)) φ(t) ⎢ 0 1 0 1 0⎥ φ(t) ⎢ ⎥ ⎥ ⎢ ⎢1⎥ ⎢ 0 0 0 0 0⎥ ⎣ ⎦ ⎦ ⎣ 0 0 0 000 ⎤ ⎡ 10000 ⎥ ⎢ ⎢0 1 0 0 0⎥ ⎥ˆ yˆ(t) = ⎢ ⎢0 0 1 0 0⎥ φ(t), ⎦ ⎣ 00010 ⎡

(24)

where φˆ = [ξˆ zˆ]T is the augmented state. The observer gain L was chosen such that the disturbance z is estimated sufﬁciently fast in order to obtain a good performance. Figure 6 shows the closed-loop system states x, the control input u and the disturbance z. As can be seen the state x1 is asymptotically stabilized under the disturbance z while satisfying the constraints. Summarizing, NMPC shows good performance while satisfying the constraints.

Sampled-Data Model Predictive Control for Constrained Systems

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1

1

x ,z

2

0 −1 0

5

10

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5

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10

15

20 Time

25

30

35

40

2

2

x ,x

3

1 0 −1 0 2

u

1.25 0

−1.25 −2 0

Fig. 6. Simulation results of z (gray solid), x1 (black solid), x2 (gray solid), x3 (black solid), and u (black solid) for the disturbance attenuation problem.

8 Conclusions Model predictive control, especially linear model predictive control, is by now widely applied in practice. However, increasing productivity demands, tighter environmental regulations, higher quality speciﬁcations and demanding economical considerations require to operate processes over a wide region of operating conditions, for which linear models are often not adequate. This inadequacy has lead in recent years to an increased theoretical and practical interest in NMPC. In this paper d the main principles and the existing results of sampled-data NMPC for continuous time systems subject to constraints where reviewed. As outlined, in NMPC an open-loop optimal control problem is solved repeatedly at ﬁxed sampling instant considering the current system state and the resulting control is applied openloop for a short time. Since NMPC is based on an open-loop optimal control problem, it allows the direct consideration of a nonlinear system model and the inclusion of constraints on states and inputs. As outlined a series of questions for NMPC, such as

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the stability of the closed-loop, are by now well understood, even so that there still remain some open issues.

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Explicit Model Predictive Control Urban Maeder1 , Raphael Cagienard2 , and Manfred Morari3 1

ETH Zurich [email protected]

2

ETH Zurich [email protected]

3

ETH Zurich [email protected]

Summary. In this contribution, explicit model predictive control is applied to the benchmark problems. Emphasis is put on the reduction of complexity both for the off-line and on-line computation. For the pendulum, a control-invariant set is computed ﬁrst. By using this set in the subsequent optimal control problem, a simple solution can be attained which guarantees invariance of the resulting closed-loop system by construction. The computation of such invariant sets for systems with a partially constrained state-space is discussed. For the reactor, a different approach was chosen. First, a simple model predictive control problem is solved explicitly which results in a low-complexity controller. Invariance is established a posteriori by analysing the explicit control law. Both approaches employ an analysis method to prove stability.

1 Introduction This section gives an introduction to explicit model predictive control, emphasising the concepts and methods applied to the benchmark problems. Model Predictive Control (MPC) has become an accepted and widely used tool for the control of constrained, multivariable plants. In MPC, an open-loop optimal control problem is solved at every time step. The control signal optimising a performance measure is then applied to the plant; the procedure is repeated at every time step. The strengths of MPC lie mainly in its handling of constraints and its relative simplicity. Commonly, a quadratic performance measure over a ﬁnite prediction horizon is employed. Problems of this type are easily translated into a quadratic program

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(QP). Many optimisation software packages include a QP solver which is efﬁcient enough to perform these optimisations on-line. However, the complexity can still be prohibitive for fast plants and low-end embedded computers. A remedy for this is to compute the closed-form solution to the optimisation problem off-line, which leads to a reduction of the complexity of the on-line algorithm. For constrained linear and quadratic optimisation problems – and hence for quadratic constrained ﬁnite horizon control – this turns out to be a tractable method, yielding powerful new abilities and possibilities for MPC. Furthermore, the closed-form solution allows the analysis of the behaviour of the system under control. For example, stability and feasibility of the closed-loop may be established by using standard analysis tools.

1.1 Model Predictive Control We will use linear, time-invariant, discrete-time models x(k + 1) = Ax(k) + Bu(k),

(1)

with A ∈ n×n and B ∈ m×n . State- and input variables are subject to the constraints x(k) ∈ X ⊆ n ,

u(k) ∈ U ⊆ m .

(2)

U = {u ∈ m ; Hu u Ku },

(3)

with X and U being polyhedral X = {x ∈ n ; Hx x Kx },

containing the origin in their interior. The constraints may either be by design or actual physical constraints. The control problem with prediction horizon Np is deﬁned by Np −1 ∗ JN (x(k)) p

:=

min

u0 ,...,uNp −1

(ui Rui + xi Qxi ) + xNp QNp xNp

(4a)

i=0

subj. to xi ∈ X, ui−1 ∈ U, ∀ i ∈ {1, . . . , N },

(4b)

xNp ∈ Tset ,

(4c)

xi+1 = Axi + Bui , x0 = x(k),

(4d)

R > 0, Q ≥ 0, QNp ≥ 0.

(4e)

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The terminal state xNp is constrained to lie in Tset . By properly choosing Tset and QNp , stability of the closed-loop can be guaranteed [14]. Solving (4) yields the optimal open-loop sequence [u∗0 . . . u∗Np −1 ], of which only the ﬁrst element u∗0 is applied to the plant. The optimisation is repeated at every time step (Receding Horizon Control policy). The optimisation problem (4) can be rewritten as the QP ∗ T JNp (x0 ) = x0 Y x0 + min UNp HUNp + x0 F UNp UNp

subj. to GUNp W + Ex0 , H > 0,

(5a) (5b) (5c)

where the column vector UNp = [u0 , . . . , uNp −1 ] ∈ Np m is the optimisation vector and H, F, Y, G, W, E are easily obtained from Q, R, QNp and by substituting i−1 xi = Ai x0 + j=0 Aj Bui−1−j . See, for example [13].

1.2 Multi-parametric Quadratic Programming In parametric programming, the goal is to ﬁnd the characterisation of the solution of (5) for a range of parameter values. Speciﬁcally, we want to obtain the closed ∗ form solution UN (x0 ) of (5) for any feasible x0 . We term this type of optimisation p

problem multi-parametric QP, where the preﬁx ’multi’ stresses the vector nature of x0 . We will brieﬂy state some deﬁnitions and properties of the mp-QP [3], [19]. Deﬁnition 1 (Feasible Set XNp ). We deﬁne the feasible set XNp ⊆ Rn as the set of states x for which the optimization problem (5) is feasible, i.e. XNp = {x ∈ n ; ∃UNp ∈ n , GUNp W + Ex}.

(6)

Deﬁnition 2 (Polyhedral Partition). A collection of polyhedral sets {Pr }R r=1 = {P1 , . . . , PR } is a polyhedral partition of a polyhedral set Θ if (i)

-R r=1

Pr = Θ

(ii) (Pr \ ∂Pr ) ∩ (Pq \ ∂Pq ) = ∅, ∀r = q, where ∂ denotes the boundary.

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Deﬁnition 3 (Region). Each polyhedron Pr of the polyhedral partition {Pr }R r=1 is referred to as a region. Theorem 1 (Properties mp-QP). Consider the multi-parametric Quadratic Pro∗ gram (5). Then, the set of feasible parameters XNp is convex, the optimiser UN : p

XNp → Np m is continuous and piecewise afﬁne (PWA) over polyhedra, i.e. ∗ UN (x) = Fr x + Gr , if x ∈ Pr = {x ∈ n ; Hr x Kr }, r = 1, . . . , R p

(7)

and the optimal value function J ∗ : XNp → is continuous, convex and piecewise quadratic (PWQ). For the proof to Theorem 1, refer to [3], [4]. Theorem 1 gives rise to an ef∗ ﬁcient on-line implementation of the controller, since the optimiser UN (x) = p

[u∗0 , . . . , u∗Np −1 ] and hence the optimal control input u∗0 = fr x + gr can easily be determined, where fr and gr are the ﬁrst m rows of the matrices Fr and Gr , respectively. Thus, to obtain the optimal control input for a given state x(k), the corresponding region Pr has to be identiﬁed; the optimal control law is afﬁne in x within this region. An in-depth discussion of multi-parametric programs is given in [19], [3], [4], [18]. A Matlab toolbox is available for download [15].

1.3 Sets In the following, we will introduce some useful facts and deﬁnitions about invariantand controllable sets commonly used in MPC. For an in-depth treatment of this topic, we refer the reader to [11]. Deﬁnition 4 (Maximum LQR Admissible Set). Let XLQR denote the maximum admissible set for the system (1) subject to the optimal unconstrained LQR feedback law KLQR : XLQR = { x ∈ n ; (A + BKLQR )k x ∈ X, KLQR (A + BKLQR )k x ∈ U, ∀k ≥ 0}.

(8)

XLQR is positive invariant for the closed-loop system. A method for computing XLQR can be found in [7]. For LTI systems, XLQR is polyhedral..

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241

Deﬁnition 5 (Control Invariant Set). A set C ⊆ X is said to be a control invariant set for the system (1) subject to the constraints (2), if for every x ∈ C , there exists a u ∈ U such that Ax + Bu ∈ C . Deﬁnition 6 (Ns -Step Reachable Set KNs (S)). Let KNs (S) be the largest set of states for which there exists a feasible control sequence which drives the state into the target set S in at most Ns steps. A particular choice of the target set is S = XLQR . It is easy to see that KNs (XLQR ) is both a control-invariant set and a stabilisable set for the system, since for any initial state x0 ∈ KNs (XLQR ), there exists a ﬁnite control sequence which drives the system into XLQR . In the following, KNs (XLQR ) will be abbreviated by KNs . The computation of KNs is usually performed iteratively [11] KNs +1 = K1 (KNs ),

K0 = XLQR .

(9)

It is easy to see that KNs ⊆ KNs +1 . Furthermore, (9) will eventually produce the maximum stabilisable set K∞ , though ﬁnite termination cannot be guaranteed [11]. Invariant- and control invariant sets are widely used in control; two applications speciﬁc to MPC shall be introduced here brieﬂy. • Stable MPC. Stability of the MPC algorithm in (4) can be guaranteed by choosing Tset = XLQR and QNp = PARE , where PARE is the solution to the discrete-time Algebraic Riccati Equation (e.g. [14]). While this method yields a stabilising controller by design, the region of attraction depends on the prediction horizon Np . Since the complexity of mp-QP is exponential in Np in the worst case, selecting a large enough prediction horizon such that the region of attraction is sufﬁciently large may be prohibitive. On the other hand, if the terminal set constraint is dropped, the region where the control law is deﬁned is enlarged, but stability and all-time constraint satisfaction (feasibility) are no longer guaranteed. • N-Step MPC. In [9], it is proposed to drop the terminal set constraint, but instead to conﬁne the state to stay within a control-invariant set. The controller is deﬁned by

242

Urban Maeder, Raphael Cagienard, and Manfred Morari Np −1 ∗ JN (x0 ) p

=

s.t.

min

u0 ,...,uNp −1

xNp p QNp xNp p

+

(xi Qxi + ui Rui )

i=0

(10)

xi+1 = Axi + Bui ui−1 ∈ U,

xi ∈ KNs , ∀ i ∈ {1, . . . , Np }.

Note the distinction made between the prediction horizon Np and the number of iterations for computing the invariant set Ns . This controller guarantees all time feasibility, but not asymptotic stability, which must be checked a posteriori by analysis. The feasible region for this controller depends only on the controlinvariant set employed, not on the prediction horizon. Therefore, Np can in general be much smaller than in standard MPC; in fact, it is often sufﬁcient to set Np = 1 for the controller to be stabilising. The decrease in complexity – in terms of number of regions – achieved by this method is substantial.

1.4 Stability Analysis A major advantage of explicit MPC is the possibility of analysing the behaviour of the closed-loop system. The equations for system (1), combined with the controller partition form a PWA dynamic system x(k + 1) = fP W A (x(k)) (11) = (A + Bfr )x(k) + Bgr , if x(k) ∈ Pr ,

r = 1, . . . , R.

The system equations are thus deﬁned over the polyhedral partition SP W A = -R r=1 Pr . Assumption 5 We will make the following assumptions on PWA systems: • Pi ∩ Pj (i = j, Pi , Pj ∈ n ) is empty or lower dimensional, • SP W A is positive invariant. Note that since we assume the autonomous PWA dynamics (5) to stem from an optimal control problem of a constrained LTI system, (5) is deﬁned over the whole feasible set XNs . Both the stable- and N-step MPC controllers satisfy Assumption 5 by design.

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243

In order to check stability, we identify a piece-wise quadratic Lyapunov function over the partition. In each region Pr , the function PWQ(x) is deﬁned by PWQr (x) = x Qr x + x Lr + Cr ,

if x ∈ Pr ,

(12)

r = 1, . . . , R.

The following constraints are imposed on the function PWQ(x) βx x ≥ PWQr (x) ≥ αx x, α, β > 0, ∀x ∈ Pr , ∀r ∈ R, PWQj (fP W A (x)) − PWQi (x) ≤ −ρx x, ρ > 0, ∀x ∈ Pij ,

(13a) (13b)

where Pij is deﬁned as the subset of a region Pi from where the system state evolves into region Pj in the next step Pij = {x ∈ n ;

x ∈ Pi , fP W Q (x) ∈ Pj }.

(14)

The decay rate for moving from one region Pi to Pj is thus ΔP W Qij (x) = PWQj (fP W A (x)) − PWQi (x). Note that this also applies for i = j, that is, when the trajectory evolves within a region. Theorem 2. The origin x = 0 is exponentially stable on SP W A if there exists a Lyapunov function PWQ(x) as in (12)-(13). For the proof of Theorem 2, refer to [5]. Since the constraints (13) are not convex, they are relaxed by employing the S-procedure. In this manner, the constraints (13) are transformed into (convex) linear matrix inequalities (LMI), such that the problem can be handled by standard SDP (semi-deﬁnite programming) software [5], [8]. Remark 1 (Consvervatism). There is some conservatism in the procedure described above. First, there might not exist a PWQ Lyapunov function over the given partition at all, even though the system is exponentially stable. In such a case, more complex function types – such as piece-wise polynomial – can be applied. Second, the constraint relaxation which is necessary to make (13) convex is not lossless. Hence, even if a PWQ Lyapunov function over the partition exists, the solver might still not ﬁnd it.

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2 Inverted Pendulum 2.1 Model The benchmark system is sampled employing a zero-order hold. The sampling interval is chosen as Ts = 0.05s, such that the bandwidth- and disturbance rejection requirements can be fulﬁlled. The system equations are x(k + 1) = Ax(k) + Bu(k) + Bd d(k) y(k) = Cx(k), ⎤ ⎡ 1.001 −0.05 −0.001 ⎥ ⎢ ⎥, A=⎢ −0.05 1.003 0.05 ⎦ ⎣ −0.001 0.05 1.001 ⎤ ⎡ ⎤ ⎡ 0 0.05 ⎥ ⎢ ⎥ ⎢ ⎥ , Bd = ⎢ −0.001 ⎥ , B=⎢ 0.001 ⎦ ⎣ ⎦ ⎣ 0.05

(15)

0

C = [1 0 0], with the constraints |u(k)| ≤ 1.25,

|u(k) − u(k − 1)| ≤ 2Ts .

(16)

2.2 Target Equilibrium The system equilibria depend on the disturbance and the reference input. They are identiﬁed by solving the steady-state equation (I − A)x∞ − Bu∞ = Bd d∞ Cx∞ = r∞ .

(17a) (17b)

Equation (17a) ensures the solution is an equilibrium, while (17b) selects the equilibrium which matches the requested reference r∞ . Deﬁnition 7 (Target Equilibrium). The quadruple (x∞ , u∞ , d∞ , r∞ ) is termed ‘target equilibrium’ for the system (A, B, Bd ), if it is a solution to (17).

Explicit Model Predictive Control

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The solution to (17) is readily found ⎡

x∞ u∞

=

I − A −B C

−1 ·

0

Bd 0

0 I

·

d∞

r∞

0

1

⎤

⎥ 0⎥ d ∞ ⎥ . 1⎥ ⎦ r∞ −1 0

⎢ ⎢1 =⎢ ⎢0 ⎣

(18)

2.3 Augmented State To account for slew rate constraints on the input, the system is put into input velocity form by applying u(k) = u(k − 1) + Δu(k).

(19)

The input u(k) enters the system equations as an additional state variable, whereas Δu(k) is considered the plant’s input variable. In order to be able to reject an asymptotically constant load, a disturbance model is incorporated assuming integral (step) disturbance dynamics. The actual disturbance magnitude d(k) is estimated by an observer, as will be laid out later. The estimate ˆ enters the augmented state vector. The resulting system equations are d(k) ⎤

⎡ ⎢ ⎢ ⎣

x(k + 1)

⎡

⎤

⎤⎡ A B Bd

x(k)

⎤

⎡ B

⎥ ⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎥ = ⎢ 0 1 0 ⎥ ⎢ u(k − 1) ⎥ + ⎢ 1 ⎥ Δu(k) ⎦ ⎣ ⎦ ⎣ ⎦ ⎦⎣ ˆ ˆ 0 d(k + 1) d(k) 0 0 1 u(k)

(20)

We denote the augmented state vector by x ˜(k), the associated system matrices by A˜ ˜ The constraints are modiﬁed accordingly and B. ˜ xx H ˜(k) < 1.25,

|Δu(k)| < 2Ts .

(21)

˜ x = [0 0 0 1 0]. The augmented system (20) is employed in the optimisation with H model.

2.4 Feasible State Space The constraints (21) do not bound x ˜(k) in every direction; the set of feasible states is unbounded. The following deﬁnition is introduced.

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Deﬁnition 8 (Partially Constrained State-Space). Consider the LTI system in (1) subject to polyhedral constraints (3). We say the state space of the system is partially constrained, if (Hx , A) is not completely observable. The unobservable subspace is termed ‘unconstrained subspace’. Consider the observability matrix ⎡

⎤

Hx

⎢ ⎥ ⎢ Hx A ⎥ ⎢ ⎥ O=⎢ ⎥. . .. ⎢ ⎥ ⎣ ⎦ n−1 Hx A

(22)

Assume rank(O) = m < n. Deﬁnition 8 suggests the transformation x = T z,

T =

Tu

(23)

,

Tc

where Tc ∈ m×n contains m linearly independent rows of O and the rows of Tu ∈ (n−m)×n are selected such that T is invertible. Then, the transformed system acquires the structure z(k + 1) =

A11 A12 0 A22

z(k) +

B1 B2

u(k),

(24)

with the constraints [0 Hz ] z(k) Kz ,

Hu u(k) Ku .

(25)

Thus, any generic partially constrained system (1) subject to (3) can be decomposed into a constrained- and unconstrained part. This structure can be exploited for the computation of control-invariant sets, as will be shown below.

2.5 Cost Function In this section, an LQ cost function is derived for the augmented system (20). A cost function for the case with acting disturbances, but no reference tracking is derived.

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Then, we show how reference tracking can easily be incorporated by extending any stabilising control law obtained for the former case. First, the LQ cost function for a single time step which penalises the deviation from the target equilibrium is considered l(˜ x−x ˜∞ , Δu) = (˜ x−x ˜∞ ) Q(˜ x−x ˜∞ ) + Δu RΔu.

(26)

By (18), the steady-state target is x ˜∞ = [x∞ u∞ d∞ ] . The steady-state disturbance ˆ d∞ is not known, it is replaced by the current disturbance estimate d(k), such that ˆ ˆ =x x ˜∞ = [0 1 0 -1] d(k). By inserting d(k) ˜(5) (k), we can rewrite (26) ˜x ˜ + Δu RΔu, l(˜ x, Δu) = x ˜ Q

(27)

˜ can easily be obtained from (18) and (26). where the weight matrix Q

Reference Tracking A controller obtained from the cost function (27) does not consider tracking of reference velocity, it always regulates the pendulum to zero velocity. In principle, the same scheme as employed for the disturbance – state vector- and cost function augmentation – can be applied as well. However, it is easy to extend any given controller to the reference tracking case without adding on the number of states, thus avoiding unnecessary complexity. Consider the tracking error for a given time step e(k) = x ˜(k) − [1 0 1 0 0] r∞ (k).

(28)

The reference tracking problem amounts to constructing a controller which drives e(k) to the origin. The state constraints in (21) are reformulated ˜ x [1 0 1 0 0] r∞ | ≤ 1.25. ˜ x e(k) − H |H . /0 1

(29)

=0

The constraints are not affected for any r∞ . Suppose K(˜ x) is a feasible and stabilising control law. Assume its region of attraction covers all feasible equilibria. Then, the extended control law

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Urban Maeder, Raphael Cagienard, and Manfred Morari

Δu(k) = K(˜ x(k) − [1 0 1 0 0] r∞ ).

(30)

is feasible and solves the reference tracking problem. Clearly, this is anticipated by the physical properties of the pendulum.

2.6 Estimator Since the disturbance is not measurable, an estimator for (20) is to be devised. By assuming noise-free measurements of x(k), a simple dead-beat disturbance estimator is employed ˆ x ˆ(k + 1|k) = Ax(k) + Bu(k) + Bd d(k|k − 1) ˆ + 1|k) = d(k|k ˆ d(k − 1) + L(x(k) − x ˆ(k|k − 1)),

(31)

where x ˆ, dˆ are estimates of the state and disturbance, respectively. The estimator gain is chosen to be L = (Bd Bd )−1 Bd , the pseudo inverse of Bd . When using this estimator, the disturbance estimate is error free, but delayed by one time step ˆ d(k|k − 1) = d(k − 1).

(32)

2.7 Control Invariant Set In this section, we propose a method for computing the maximum control-invariant set for partially constrained systems. This set, or a subset of it, can subsequently be used in an N-step MPC scheme (10). Control-invariant sets have been used widely in the past for safely controlling unstable plants with saturating actuators. In [1], a control-invariant set is employed for controlling an inverted pendulum. However, the computation method is speciﬁc to the problem, and no rate constraints were considered. In [20], a method for approximating control-invariant sets for continuous-time plants is given. The inverted pendulum closely relates to a class of control problems in aviation, where saturating actuators and pilot-induced oscillations led to crashes of several prototype ﬁghter aircraft in the past [10], [16]. For instance, a control-invariant set is used in an anti-windup scheme in [2] to guarantee feasibility and safe control of

Explicit Model Predictive Control

249

an unstable tail-less airplane. Compared to these speciﬁc methods, our approach is general; it can be used for plants with arbitrary constraints on both the state and the input. Consider the LTI with the following structure z(k + 1) =

A11 A12

0 A22

z(k) +

B1 B2

u(k),

(33)

with A11 ∈ ns and A22 ∈ nu , n = ns + nu and |λi (A11 )| ≤ 1,

(34)

subject to the polyhedral state- and input constraints [0 Hz ] z(k) Kz ,

Hu u(k) Ku .

(35)

Remark 2 (Decomposition Method). An arbitrary system can be decomposed into (33) by ﬁrst applying the transform (23). If any |λi (A11 )| > 1, another transform T2∗ 0 T2 = (36) 0 I can be constructed such that T2∗ A11 T2∗−1 =

As11 A∗12

0 Au11

(37)

with Au11 strictly unstable and the Eigenvalues of As11 inside or on the unit disc. By applying T2 , the decomposition (33) is ﬁnally obtained.

Theorem 3 (Stabilisable Set for Partially Constrained Systems). Consider the system (33) subject to (34) and (35). Assume (A, B) stabilisable. Let KNus be a stabilisable set for the subsystem (A22 , B22 ) subject to Hz z2 (k) Kz and Hu u(k) Ku . Then, KNs = ns × KNus is a stabilisable set for the complete system.

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Urban Maeder, Raphael Cagienard, and Manfred Morari

Proof. The proof is performed by constructing a control sequence which drives the state to the origin. By deﬁnition, for every z = [z1 z2 ] ∈ K∞ , there exists an admissible input sequence which drives z2 to the origin. We therefore consider an arbitrary state z(0) = [z1 (0) 0] and show how to drive the state to the origin from z(0). For the unconstrained (both state and input) case, we consider the class of input sequences U = [u(0) , . . . u(N − 1) ] with Horizon N = n, such that z(N ) = 0 (i.e. terminal state is the origin) with z(N ) = AN z(0) + [AN −1 B, . . . , B]U . Such a sequence exists due to controllability of the system and it is given by U = −[AN −1 B, . . . , B]−1 AN z(0). For the constrained case, we note that the sequence U0 = [0, . . . , 0] is strictly feasible and the state evolves strictly inside the feasible set because the feasible sets X and U both contain the origin in their interior. Consider the sequence αU with 1 ≥ α ≥ 0. The sequence is an interpolation between U0 and U . By convexity, there exists an α > 0 such that αU is a feasible sequence, both for state- and input constraints. Applying this sequence, z(N ) = AN z(0) + α[AN −1 B, . . . , B]U . Insertion yields z(N ) = (1 − α)AN z(0) and hence z1 (N ) = (1 − α)AN 11 z1 (0) for the upper subsystem, since z2 (0) = z2 (N ) = 0. By repeatedly choosing an αi for every N 2k steps, the evolution of the state thus is z1 (kN ) = AkN 11 i=1 (1 − αi ), and hence z1 (kN ) → 0 as k → ∞. Remark 3 (Stability Margin). For unstable systems, the border of the maximum control-invariant set is not necessarily stabilisable. The nearer the state is to the border, the more of the available control authority is spent on merely keeping the state inside the set, but no authority remains for regulating the system away from the border. The system, though never leaving the set, might have very poor performance. Therefore, it is sensible to employ a stability margin for unstable systems, to avoid stickiness. If KNs is computed by the procedure (9), the margin can be increased by selecting a lower Ns , since KNs ⊆ KNs +1 .

Application to the Pendulum For the benchmark system (20), the similarity transform z = T x ˜ with

Explicit Model Predictive Control

⎡

⎤

−0.787 −0.599 0.122

0

−0.358 0.843

0

⎢ ⎢ 0.394 ⎢ ⎢ T = ⎢ −0.062 ⎢ ⎢ −0.350 ⎣ 0.316

251

0.086

⎥ −0.080 ⎥ ⎥ ⎥ 0.261 0.229 0 0.936 ⎥ ⎥ 0.495 0.350 0.670 −0.247 ⎥ ⎦ −0.447 −0.316 0.742 0.223

(38)

leads to decomposition (33) with nu = 2, which means the dimension of the set computation is reduced from 5 to 2. As pointed out above, the choice of the number of iterations deﬁnes the margin of control authority; the number of iterations Ns = 25 was chosen by trial. In Figure 1, cuts of the invariant set through x(1) = x(3) = u = 0 and x(2) = d = u = 0 are shown. It has 52 facets and is unbounded, containing the set of equilibrium states (18). Most of the facets only appear in the

1.5

1.5

1

1

0.5

0.5

x(3)

d

u/d cut, which is not very interesting otherwise and is thus not reproduced here.

0

0

−0.5

−0.5

−1

−1

−1.5 −1.5

−1

−0.5

0

x(2)

0.5

1

1.5

−1.5 −1.5

−1

−0.5

0

x(1)

0.5

1

1.5

Fig. 1. K25 , cuts with x(1) = x(3) = u = 0 (left) and x(2) = d = u = 0 (right)

2.8 Controller For the pendulum, the N-step MPC scheme (10) is applied Np −1 ∗ JN (˜ x0 ) p

s.t.

=

min

Δu0 ,...,ΔuNp −1

˜N x x ˜Np Q ˜ p Np

+

˜ xi + Δui RΔui ) (˜ xi Q˜

i=0

˜xi + BΔu ˜ i x ˜i+1 = A˜ ˜i ∈ KNs , ∀ i ∈ {1, . . . , Np }. |Δui−1 | < 2Ts , x

(39)

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Urban Maeder, Raphael Cagienard, and Manfred Morari

The cost function is constructed as in (27) with the weights

Q = diag [1 1 1 0.1 0] ,

R = 0.1,

(40)

yielding ⎤

⎡ 1 0 0 0

⎢ ⎢0 1 ⎢ ⎢ ˜ Q = ⎢0 0 ⎢ ⎢0 0 ⎣ 0 -1

0

⎥ 0 0 -1 ⎥ ⎥ ⎥ 1 0 0 ⎥. ⎥ 0 0.1 0.1 ⎥ ⎦ 0 0.1 1.1

(41)

The weights are chosen by trial to obtain satisfying large-signal (constrained) behaviour, i.e. when the benchmark test sequences are applied. The small-signal bandwidth requirement is separately treated below. ˜ N , the algebraic Riccati equation is For the computation of the terminal weight Q p solved iteratively, because the plant is technically not controllable, which results in some standard solvers failing. ˜ ˜ PN B) ˜ −1 (B ˜ + Q, ˜ ˜ PN A) PN +1 = A˜ PN A˜ − (A˜ PN B)(R +B

(42)

˜ Iteration (42) is applied until PN converges. The value of PN is then with P1 = Q. applied as terminal weight. The optimisation problem (39) is solved for Np = 25, employing multi-parametric quadratic programming. The piece-wise afﬁne control law is obtained ˜ x(k)). Δu(k) = K(˜

(43)

˜ consists of 55 polyhedrons; cuts through x(1) = x(3) = u = 0 The partition of K(·) and x(2) = d = u = 0 are shown in Figure 2.

Bandwidth The controller (30) is tuned for robust large-signal behaviour, it does not meet the bandwidth requirement in the linear range. In order to improve this behaviour, a fast local controller is created which overrides the global controller near the origin.

1.5

1.5

1

1

0.5

0.5

x(3)

d

Explicit Model Predictive Control

0

0

−0.5

−0.5

−1

−1

−1.5 −1.5

−1

−0.5

0

x(2)

0.5

1

1.5

253

−1.5 −1.5

−1

−0.5

0

x(1)

0.5

1

1.5

Fig. 2. Controller partition, cuts through x(1) = x(3) = u = 0 (left) and x(2) = d = u = 0 (right)

First, a linear state feedback controller is computed as in (42). The cost function is constructed with a strong emphasis on the pendulum velocity Q = diag [50 1 1 0.1 0] < big),

R = 0.1.

(44)

For the resulting linear controller, ˜ + R) ˜ −1 (B ˜ ˜ PN B ˜ PN A), Kloc = (B

(45)

the maximum admissible set Xloc is computed as in (8). The local controller is then employed when the state is in Xloc , whereas the global controller is active otherwise. Since the system will stay within Xloc once it enters it, there are no switching issues; the system will be stable if the global controller stabilises it since the state eventually enters the linear region. This controller achieves a tracking magnitude of −2.6db at 2rad/s, thus fulﬁlling the requirement.

2.9 Stability Analysis Since the explicit controller computed above does not guarantee asymptotic stability by design, it has to be checked a posteriori. In [17] and [12], it is argued that a system with saturating input cannot be globally exponentially stabilised if it has poles on or outside the unit circle.

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Urban Maeder, Raphael Cagienard, and Manfred Morari

For the pendulum, similar arguments apply. Asymptotic stability can be achieved by deﬁnition for the maximum stabilisable set K∞ . Since in this particular case, the set is unbounded in the direction of an integrator mode, exponential stability can not be achieved over it. Clearly, the analysis algorithm in Section 1.4 can therefore not be applied directly. The following algorithm is devised to remove this limitation by choosing a subset of the state space where exponential stability is attained. Algorithm 5 (Local Stability Analysis of Unbounded Partitions) -R Assume an invariant partition XN = r Pr 1. Select large μ > 0. n 2. Calculate new partition {Pr }R r=1 such that Pr = Pr ∩ {x ∈ ; |x|∞ ≤ μ}.

Store indices of affected regions I = {i ∈ {1, . . . , R}; Pi = Pi } 3. Find Lyapunov function as in (12) subject to (13) on partition {Pr }R r=1 . 4. If I = {}, then V = ∞. Break. 5. Solve V =

min

x∈Pi , i∈I

P W Qi (x),

s.t. |x|∞ = μ.

(46)

In Step 2, the partition is bounded by a hypercube. The parameter μ should be chosen such that only unbounded regions are affected. In step 5, the smallest level of the Lyapunov function is determined for which there is a x that leaves the hypercube. Solving for V in Step 5 amounts to solving 2n QPs for every region in I . The complexity of the algorithm is negligible compared to the complexity of ﬁnding the Lyapunov funtion in Step 3. Theorem 4 (Stability). If a Lyapunov function is found for the PWA system (11) by Algorithm 5, then the system is exponentially stable for x = 0 in the region XN = XN ∩ {x ∈ n ; P W Q(x) ≤ V }. Proof. By intersecting the partition with a hypercube of size μ, its invariance property is lost. Invariance is re-established by Step 4 after the Lyapunov function is computed. Trajectories can only exit the partition through μ-hyperplanes due to the invariance of the original partition {Pr }R r=1 . Since V is the smallest value for which

Explicit Model Predictive Control

255

P W Q(x) touches such a μ-hyperplane, and since P W Q(x) < V , ∀x ∈ XN and P W Q(fP W A (x)) − P W Q(x) < 0, x ∈ XN , trajectories starting in XN cannot leave it, hence XN is invariant. Therefore, Theorem 2 applies.

Application to the Inverted Pendulum Stability analysis was performed for the case without disturbance. Not every combination of the number of iterations for computing the invariant set Ns , and the prediction horizon Np actually yield a stabilising controller. Moreover, because of the conservatism discussed in Remark 1, it is not possible to ﬁnd a Lyapunov function for every stabilising controller. By selecting a larger Ns – and thus a larger set KNs – the system is allowed to move closer to the stability border, where it might get stuck [16]. In such a case, the prediction horizon may be increased appropriately, which increases the likelihood of the controller to be stable – at the cost of greater complexity in terms of number of regions. Table 1 shows the minimum prediction horizon Np necessary to obtain a stabilising controller for various Ns . Lyapunov analysis was restricted to horizons no larger than 10 due to computational complexity. Additionally, stability was checked by simulating trajectories for a series of initial conditions. The starting points where generated by taking some extreme vertices of the controller regions, and by additionally limiting |x0 |∞ < 10. If convergence was encountered, the system was considered ‘trajectory stable’. It is interesting to note that for some choices of parameters, the system is stable in simulation, but a Lyapunov function cannot be found. For example, choosing Np = 1, Lyapunov stability can be proved for up to Ns = 10. On the other hand, Ns = 10 leads to infeasibility when the disturbance benchmark trajectory is applied, since the controller does not react decisively enough. A larger Ns is required. The pair Ns = 25 and Np = 1 was found to yield satisfying behaviour. However, a Lyapunov function could not be obtained for this controller. Table 2 shows the number of regions resulting from varying prediction horizons. For larger prediction horizons (Np > 8), controller computation takes several hours, the Lyapunov stability check days to complete, which seriously limits application of these controllers.

256

Urban Maeder, Raphael Cagienard, and Manfred Morari Table 1. Minimum prediction horizon for stability Number of

Min. Pred.

Iterations for

Horizon Np for

Invariant Set,

Lyapunov

Ns

Stability

5

1

1

10

1

1

15

5

1

25

> 10

1

40

> 10

3

Min. Pred. Horizon Np for Stable Trajectory

Table 2. Prediction horizon vs. number of regions for Ns = 25 Prediction

Number of

Horizon Np

Regions

1

55

2

171

3

471

5

1509

7

3432

9

6977

10

9585

To sum up, the quick growth in complexity for ﬁnding the Lyapunov function, and the discrepancy between stable simulation behaviour, but unprovable stability limit the potential of the method in practice. One often is interested in ﬁnding the least complex controller satisfying the design speciﬁcations.

2.10 Simulation Results In Figures 3 and 4, the benchmark reference and disturbance trajectories are applied. The parameters for computing the controller were Ns = 25 and Np = 1. It comprises 55 regions.

Explicit Model Predictive Control

257

2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 0

5

10

15

20

25

30

35

40

45

50

Fig. 3. System response to velocity reference test sequence. y(t) (-), r(t) (--), u(t) (-.),

d dt

u(t)

(..). Controller with Ns = 25, Np = 1; 55 regions.

1.5

1

0.5

0

−0.5

−1

−1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 4. System response to disturbance test sequence. y(t) (-), d(t) (--), u(t) (-.),

d dt

u(t) (..).

Controller with Ns = 25, Np = 1; 55 regions.

In Figure 5, trajectories are shown for varying Ns , while keeping Np = 1. Increasing Ns results in quicker control action. On the other hand, the controller eventually enters limit cycles, e.g. for Ns = 40.

258

Urban Maeder, Raphael Cagienard, and Manfred Morari

4.5

1

4 0.8

3.5 0.6

y(t)

y(t)

3

2.5

2

0.4

0.2

1.5 0

1 −0.2

0.5

0

0

2

4

6

8

10

12

14

16

18

20

−0.4

0

2

4

6

8

10

12

14

16

Fig. 5. Step responses for varying Ns , and Np = 1 for all trajectories. Left reference, right disturbance. Ns = 10, 29 regions (-), Ns = 25, 55 regions (-.), Ns = 40, 55 regions (--).

2.11 Conclusion The concepts of control-invariant sets and MPC have been applied to the benchmark problem. An extension to the usual techniques for computing control-invariant sets has been shown to be efﬁcient for partially constrained systems. The explicit solution to the N-step low-complexity MPC scheme has been computed and shown to work well for the reference and disturbance trajectories. The main features of this type of controller are its simplicity and efﬁciency (less than 100 regions to be evaluated at runtime) and its large feasible region.

2.12 Relation to Furuta Pendulum The Furuta pendulum [6] is a particular and popular instance of the inverted pendulum, which was originally developed for educational purposes, but has received quite some interest by researchers. In the following, we will apply the method used for the benchmark problem to the linearised model; due to the similarity to the benchmark pendulum, this is a straightforward task. The results obtained are compared to those in [1]. The system equations of the Furuta pendulum are ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ˙θ 0 θ 010 ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ θ¨ ⎥ = ⎢ α 0 0 ⎥ ⎢ θ˙ ⎥ + ⎢ β ⎥ u ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ δ φ˙ γ 00 φ¨

(47)

Explicit Model Predictive Control

259

with α = 31.3, β = −71.2, γ = −0.5888 and δ = 191 and the input constraint |u| ≤ 0.3.

(48)

√ The system’s eigenvalues are λ1 = 0, λ2,3 = ± α. Like the benchmark example, the Furuta pendulum has an integrator mode and a stable/unstable pair of real poles. The same algorithm used for the benchmark is applied to the Furuta pendulum. Computation of the invariant set is reduced to dimension 1, since no slew rate constraints are present. The resulting set has two facets; it is identical to the set obtained by adhoc methods in [1]. The resulting controller partition comprises 5 regions. Closedloop behaviour under reference tracking is shown in Figure 6. It is essentially identical to the plots in [1]. The strength of our method lies in its generality, since it can be applied both to higher dimensional systems and also when additional state constraints are present.

0.6 0.4 0.2 0 −0.2 −0.4 0

5

10

15

20

0

5

10

15

20

0

5

10

15

20

20 10 0 −10 −20

0.3 0.2 0.1 0 −0.1 −0.2 −0.3

˙ (..); middle: φ(t) ˙ Fig. 6. Step responses of Furuta pendulum. Top: θ(t) (-), θ(t) (-), r(t) (..); bottom: u(t)

260

Urban Maeder, Raphael Cagienard, and Manfred Morari

3 Continuous Stirred Tank Reactor (CSTR) with a strong exothermal reaction 3.1 Modelling the CSTR The nonlinear, continuous-time model of the plant is transformed into a description, which consists of two linear, time-invariant, discrete-time models so that our techniques can be applied. When the plant is heated up and cooled down, no reactant is inside the vessel, i.e. the reactant concentration x1 as well as the reactant feed inﬂow u1 are zero. The concentration dynamics can be omitted and the nonlinearity f vanishes because x1 = 0. Therefore, for x1 = 0, u1 = 0, a model of the plant with only 2 states and 1 input −1 1 0 xh (t) + uh (t), (49) x˙ h (t) = 5 −10 5 is considered, where xh (t) = [x2 (t) x3 (t)] and uh (t) = u2 (t). After heating up the plant to the nominal temperature, reactant is fed into the vessel and the exothermal reaction starts. The nonlinearity cannot be omitted. We linearize the equations around the nominal operating point x ¯p = [1 1 0] , u ¯p = [1 −1] . The content temperature is assumed to stay above the ignition temperature, i.e. x2 > x2s , as long as reactant is contained inside the vessel. For x1 = 0, x2 > x2s the linearized model with 3 states and 2 inputs is given by ⎤ ⎡ ⎡ −5 −50 0 5 ⎥ ⎢ ⎢ ⎢ x˙ p (t) = ⎢ 1 ⎥ ⎦ xp (t) + ⎣0 ⎣1 9 5 −10

0

⎤ 0 ⎥ 0⎥ ⎦ up (t), 05

(50)

¯p , u ¯p , i.e. xp (t) = where xp (t), up (t) denote the deviations from the operating point x [x1 (t) x2 (t) x3 (t)] − x ¯p and up (t) = [u1 (t) u2 (t)] − u ¯p . In the following we will refer to the areas of operation assigned to the two different models of the plant as Modes, i.e. Mode 1:

x1 = 0, u1 = 0 : model (49),

(51a)

Mode 2:

x1 = 0, x2 > x2s : model (50).

(51b)

Explicit Model Predictive Control

261

We will consider the more stringent slew rate constraints on the input u2 as given in case b) of the problem formulation, i.e. |du2 (t)/dt| ≤ 5s−1 . The continuous time models are converted to discrete-time models with sampling time Ts = 0.04s and a zero-order hold. The sampling period is chosen 10 times faster than the required closed loop bandwidth for the temperature control, which is speciﬁed at 15 rad/s. The model for Mode 1 in the discrete-time state space description is given as xh (k + 1) =

0.9642 0.0323 0.1616 0.6734 /0 1 . Ah

xh (k) +

0.0035 0.1651 . /0 1

uh (k),

(52)

Bh

where xh (k) = [x2 (k) x3 (k)] and uh (k) = u2 (k). The discrete-time model for Mode 2 is

⎤ ⎤ ⎡ ⎡ 0.7792 −2.1688 −0.0372 0.1786 −0.0025 ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ xp (k+1) = ⎣0.0434 1.3901 0.0397 ⎦ xp (k)+⎣0.0042 0.0040 ⎥ ⎦ up (k), (53) 0.0037 0.1983 0.6738 0.0003 0.1651 . . /0 1 /0 1 Ap

Bp

¯p and up (k) = [u1 (k) u2 (k)] − u ¯p . where xp (k) = [x1 (k) x2 (k) x3 (k)] − x

3.2 Prediction Model The models derived in the previous section are augmented to be able to cope with slew rate constraints on inputs, to apply time varying reference tracking and to implement control with integral action.

Mode 1 To account for slew rate constraints on the cooling/heating ﬂuid entry temperature, the input u2 is formulated in the form u2 (k) = u2 (k − 1) + Δu2 (k) and the rate of change Δu2 (k) is considered as the new input to the system. Model (52) is augmented by the state u2 (k − 1) to keep track of the previous input, i.e. u2 (k) = u2 (k − 1) + Δu2 (k).

(54)

262

Urban Maeder, Raphael Cagienard, and Manfred Morari

Time varying reference tracking is implemented by augmenting the prediction model with an additional state for each state to be tracked. For Mode 1 the content temperature x2 has to be steered to the reference value r2 . The additional state r2 is kept constant during the predicted behavior of the plant: (55)

r2 (k + 1) = r2 (k). Thus the augmented state vector x ˜h (k) is x ˜h (k) = [x2 (k) x3 (k) u2 (k − 1) r2 (k)] , and the update equations for the prediction model are given by ⎞ ⎛ ⎛ ⎞ Ah Bh 0 B ⎟ ⎜ ⎜ h⎟ ⎟ ˜h (k) + ⎜ 1 ⎟ u x ˜h (k + 1) = ⎜ ⎝ 0 1 0⎠ x ⎝ ⎠ ˜h (k), .

0

0 1 /0 1 ˜h A

(56)

(57)

0 . /0 1 ˜h B

where u ˜h (k) = Δu2 (k) and 0 is the null matrix of appropriate dimensions.

Mode 2 As for Mode 1, input u2 is formulated in Δu-form to account for slew rate constraints: u2 (k) = u2 (k − 1) + Δu2 (k).

(58a)

There are no constraints on the rate of change of the reactant feed inﬂow u1 . Additionally to the content temperature x2 , in Mode 2 the reactant concentration x1 is steered to the reference r1 , i.e. the plant (53) is augmented by the two reference states r1 , r2 : r1 (k + 1) = r1 (k),

(59a)

r2 (k + 1) = r2 (k).

(59b)

The content temperature x2 is required to be tracked with zero steady state error, i.e. e2 = r2 − x2 → 0, this is considered by including an integrator state i2 in the problem formulation:

Explicit Model Predictive Control

i2 (k + 1) = i2 (k) + (r2 (k) − x2 (k)).

263

(60)

The augmented state vector xp (k) is referred to the nominal operating point x ¯p = [1 1 0 −1 1 1 0]T , i.e. x ˜p (k) = [x1 (k) x2 (k) x3 (k) u2 (k − 1) r1 (k) r2 (k) i2 (k)]T − x ¯p . The update equations for the linearized prediction model around x ¯2 are ⎞ ⎛ ⎛ ⎞ Ap Bp2 0 0 Bp ⎟ ⎜ ⎜ ⎟ ⎜ 0 1 0 0⎟ ⎜Π1 ⎟ ⎟ x ⎜ ⎟u ˜ (k) + x ˜p (k + 1) = ⎜ p ⎜ 0 0 I 0⎟ ⎜ 0 ⎟ ˜p (k) ⎝ ⎝ ⎠ 2 ⎠ Π2 0 Π3 1 0 . /0 1 . /0 1 ˜p A

(61)

(62)

˜p B

where u ˜p (k) = [u1 (k) Δu2 (k)] − u ¯p and u ¯p = [1 0] , Bp2 is the second column of Bp , i.e. Bp2 = Bp ·[0 1] , Π1 = [0 1], Π2 = [0 −1 0] and Π3 = [0 1]. I2 is the identity matrix of dimensions 2 × 2 and 0 is the null matrix of appropriate dimensions.

3.3 Cost function We ﬁrst formulate quadratic costs l1 (˜ xh , u ˜h ) and l2 (˜ xp , u ˜p ) for Mode 1 and for Mode 2 for one single time step, respectively. Summation over the prediction horizon Np yields the cost functions J1,Np (˜ xh (k)) and J2,Np (˜ xp (k)) to be minimized in the optimization problem. For Mode 1, the state x2 is steered to the reference value r2 , which is realized by penalizing (x2 −r2 ) with the weight Qh1 . The input u1 required at the given reference point is not known in advance and different from zero. Therefore, in the Δu(k)formulation the previous value u2 (k − 1) is not penalized, but only its deviation Δu2 (k − 1). Furthermore, we do not penalize the jacket temperature x3 , which will allow it to take a value yielding a steady state point corresponding to the reference r2 . The quadratic cost l1 (˜ xh , u ˜h ) for one single time step is l1 (˜ xh , u ˜h ) = (x2 − r2 ) Qh1 (x2 − r2 ) + Δu2 Rh1 Δu2 .

(63)

Summing up the cost l1 (˜ xh , u ˜h ) over each single time step i in the prediction horizon N and appending a terminal cost QhNp = Qh1 yields the cost function J1,N (˜ xh (k)) for Mode 1:

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Urban Maeder, Raphael Cagienard, and Manfred Morari Np −1

J1,Np (˜ xh (k)) :=

! l1 (˜ xh,i , u ˜h,i ) + x ˜h,Np QhNp x ˜h,Np ,

(64)

i=0

where x ˜h,i denotes the predicted state at time k + i obtained by starting from the measured state x ˜h,0 = x ˜h (k) at time k and by applying to the system (57) the input sequence {˜ uh,0 , . . . , u ˜h,i−1 }. In Mode 2, beside the state x2 , the state x1 is steered to the reference value r1 , i.e. (x1 − r1 ) and (x2 − r2 ) are weighted with Qp1 and Qp2 , respectively. Penalizing x3 is omitted. The inputs required at the steady state are not known in advance, therefore the state u2 (k − 1) and the input u1 are not penalized. Note that for the weight Rp1 belonging to u1 a small value has to be chosen instead of zero to fulﬁll Rp > 0 (see (4e)). The integrator state i2 is weighted with Qp3 . The quadratic cost l2 (˜ xp , u ˜p ) for one single time step is given by ⎡ ⎤ ⎡ ⎤⎡ ⎤ (x1 − r1 ) (x1 − r1 ) Qp1 0 0 ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ xp , u ˜p ) = ⎢ l2 (˜ (x 0 Q (x − r ) 0 − r ) 2 ⎦ ⎣ p2 2 ⎦ ⎣ 2 ⎦⎣ 2 i2 0 0 Qp3 i2 u1 Rp1 0 u1 + . 0 Rp2 Δu2 Δu2

(65a)

(65b)

xp (k)) is and the resulting cost function J2,Np (˜ Np −1

xp (k)) := J2,Np (˜

! l2 (˜ xp,i , u ˜p,i ) + x ˜p,Np QpNp x ˜p,Np .

(66)

i=0

3.4 Constrained Optimal Control Formulation Given the prediction model and the cost function we formulate an optimal control problem, where the cost function is minimized subject to physical and design constraints. Physical constraints are given on the magnitude and the rate of change of input u2 . The prediction model is considered as a design constraint, which deﬁnes the state update equations for the predicted behavior of the plant. For Mode 1, we formulate the optimization problem

Explicit Model Predictive Control ∗ J1,N (˜ xh (k)) := p

min

u ˜h,i ,...,˜ uh,Np −1

(67a)

J1,Np (˜ xh (k))

subj. to − 1.1 ≤ x ˜h(3),i ≤ 1.1, − 0.5 ≤ u ˜h,i ≤ 0.5,

265

i = 0, . . . , Np ,

(67b)

i = 0, . . . , Np − 1,

(67c)

˜h u ˜h,i + B ˜h,i , x ˜h,i+1 = A˜h x

x ˜h,0 = x ˜h (k),

(67d)

˜h,i . For Mode 2, the where x ˜h(3),i denotes the third element of the column vector x physical constraint that reactant can only be fed into the plant and not extracted has to be considered additionally in the optimization problem: ∗ J2,N (˜ xp (k)) := p

min

u ˜p,i ,...,˜ up,Np −1

subj. to − 1.1 ≤ x ˜p(4),i ≤ 1.1, 0≤u ˜p(1),i

(68a)

J2,Np (˜ xp (k)) i = 0, . . . , Np ,

i = 0, . . . , Np − 1,

− 0.5 ≤ u ˜p(2),i ≤ 0.5,

i = 0, . . . , Np − 1,

˜p u ˜p,i + B ˜p,i , x ˜p,i+1 = A˜p x

x ˜p,0 = x ˜p (k).

(68b) (68c) (68d) (68e)

Although problems (67) and (68) look similar, problem (68) is more difﬁcult to solve than (67) because problem (68) comprises 7 states and 2 inputs compared to (67) with 4 states and 1 input. Note that in this formulation no terminal set with an appropriate terminal weight is included as in problem (4). Thus feasibility and stability are not guaranteed.

3.5 Explicit Feedback Control Law As described in Section 1.2, problems (67) and (68) can be considered as multiparametric Quadratic Programs and solved for all x ˜h (k), x ˜p (k) within a polyhedral set of values. Because the state space in (67), (68) is only partially constrained, the exploration of the solution is restricted to the space where the plant is operating. For example, the content temperature x2 has to be steered to the reference values r2 = 1 and r2 = 0.98, therefore the solution is derived in a neighborhood of x2 = 1, i.e. for 0.9 ≤ x2 ≤ 1.1. Otherwise, the controller complexity is increased unnecessarily. The parameters to be tuned are the prediction horizon Np , the weights Qh1 , Rh1 for Mode 1 and Qp1 , Qp2 , Qp3 , Rh2 for Mode 2. As mentioned in section 3.3, Rp1 is set to 1e−6 to fulﬁll Rp > 0, and not considered as tuning parameter. The parameter

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values are a trade off between controller performance and its complexity in terms of number of regions resulting from the state feedback solution. The closed loop bandwidth for the temperature control in the linear range is designed to be 15 rad/s. The explicit state feedback solution for the optimal control problems (67), (68) are derived for the following sets of parameters Mode 1 : Np = 2, Qh1 = 10, Rh1 = 1,

(69a)

Mode 2 : Np = 2, Qp1 = Qp2 = 7.5, Qp3 = 3, Rh1 = 1.

(69b)

Table 3 summarizes the controllers complexity. Table 3. Number of regions of controllers. controller states inputs number of regions R Mode 1

4

1

14

Mode 2

7

2

82

3.6 Stability Analysis In Section 3.4 conditions resulting in a controller with feasibility and stability guaranties were omitted on purpose in order to obtain a controller of low complexity. Because the explicit solution is at our disposal, these properties can be analyzed a posteriori. We will investigate the state feedback solution around the nominal operating point, i.e. the Mode 2 controller and ﬁxed reference values r1 = 1, r2 = 1. Reference tracking is omitted, and the reduced state space consists of x ˘p (k) = [x1 (k) x2 (k) x3 (k) u2 (k − 1) i2 (k)] − x ¯p and u ˘p (k) = [u1 (k) Δu2 (k)] − u ¯p , where x ¯p = [1 1 0 −1 0] and u ¯p = [1 0] . Our aim is to show feasibility and stability of the origin, which corresponds to the nominal operating point due to the coordinate transformation. The explicit solution is derived in the reduced state space x ˘p (k), u ˘p (k) with the parameters (69b). The resulting controller is identical to the controller in the previous section and r1 = 1, r2 = 1. The state feedback solution is used to close the loop to an autonomous PWA dynamic system of the form

1.1

1.08

1.08

1.06

1.06

1.04

1.04

1.02

1.02 2

1.1

1

x

x

2

Explicit Model Predictive Control

1

0.98

0.98

0.96

0.96

0.94

0.94

0.92

0.92

0.9

0

0.1

0.2

0.3

0.4

0.5

0.6 x

0.7

0.8

0.9

1

(a) Cut x3 = 1, u2 = 1, i2 = 0.

1

1.1

267

0.9

0

0.1

0.2

0.3

0.4

0.5

0.6 x

0.7

0.8

0.9

1

1.1

1

(b) Cut x3 = 0, u2 = −1, i2 = 0.

Fig. 7. Polyhedral partition of the invariant set XI intersected with the x3 -, u2 - and i2 -planes: (a) plant heated up at nominal temperature without reactant inside, (b) nominal operating point.

˘r , if x x ˘2 (k + 1) = A˘r x ˘2 (k) + B ˘2 (k) ∈ Pr ,

r = 1, . . . , R,

(70)

˘p f˘r , B ˘r = B ˘p g˘r and f˘r , g˘r deﬁne the afﬁne control law on the where A˘r = A˘p + B feasible set of states XN . We ﬁrst identify the invariant subset XI of XN , since positive invariance of the set XN is not fulﬁlled by design, and afterwards search for a PWQ Lyapunov function on XI . Then, Theorem 2 states that the origin of the closed loop system (70) is exponentially stable on XI . In [8] an algorithm is reported for computing the invariant set XI , where trajectories which exit the feasible set XN are removed iteratively until no such states can be found. The invariant set found covers the state space from where production starts up to the nominal operating point. In Figure 7, cuts of the invariant set XI trough x3 = 1, u2 = 1, i2 = 0 and x3 = 0, u2 = −1, i2 = 0 are shown, which correspond to the plant heated up at nominal temperature without reactant inside and to the nominal operating point, respectively. For any initial state within XI , all constraints will be met at all future time-instants, but no guarantee is given that the state vector will ever reach the origin. Therefore, a PWQ Lyapunov function has to be identiﬁed on XI . Using the MPT Toolbox [15] an appropriate Lyapunov function was found. Consequently, the closed-loop system is feasible and stable on the set XI around the nominal operating point.

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3.7 Simulation Results The given test sequence is applied to the system. The plant is heated up using Mode 1, operates in Mode 2 when reactant is in the vessel and is cooled down in Mode 1, again. We switch between the controllers depending on the conditions (51). The inputs are obtained by identifying the region which contains the actual state vector xh (k), xp (k) and subsequent evaluation of the corresponding afﬁne function. They are applied to the nonlinear, continuous time plant with a zero-order hold element. The evaluation is repeated with a sampling interval of Ts = 0.04s. Figure 8 depicts the evolution of the states and the inputs by applying the reference sequence. The upper plot of Figure 8 shows the three states x1 (t), x2 (t), x3 (t) of the nonlinear continuous time plant, the lower plot of Figure 8 shows the inputs

1

0.5

0

−0.5

0

2

4

6

8

10

12

14

16

18

10

12

14

16

18

t

1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0

2

4

6

8 t

Fig. 8. Trajectory evolution of the full test sequence. Upper plot: reactant concentration x1 (t) (--), content temperature x2 (t) (-), jacket temperature x3 (t) (-.). Lower plot: reactant feed ﬂow u1 (t) (--), temperature of heating/cooling ﬂuid u2 (t) (-), magnitude constraints on u2 (t) (..).

Explicit Model Predictive Control

269

1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 0.94 0.92 0.9

6

7

8

9

10

11

12

13

14

15

t

Fig. 9. Trajectory evolution around nominal operating point. Reactant concentration x1 (t) (-), content temperature x2 (t) (-), jacket temperature x3 (t) (-.), reference values r1 (t), r2 (t) (..).

u1 (t), u2 (t) and in dotted lines the magnitude constraints on u2 (t). Note that the input u1 (t) is neither constrained in magnitude nor in rate of change, except that reactant can only be fed into and not extracted from the plant (u1 (t) ≥ 0). In Figure 9 the trajectories x1 (t), x2 (t), x3 (t) around the nominal operating point are displayed. Additionally, the references r1 (t), r2 (t) are plotted in dotted lines. We see that the content temperature x2 (t) is controlled with zero steady state error, contrary to the reactant concentration x1 (t). 3.8 Conclusion For the CSTR benchmark example a linearized description has been derived, which consists of two LTI models. The two models have been extended to cover time varying reference tracking, slew rate constraints on inputs and control with integral action. For each augmented model a controller has been derived with the emphasis to keep the controller complexity low, but with the drawback that no stability and feasibility guarantees are given by construction. Because the explicit solution was at disposal, an a posteriori analysis veriﬁed that the controller is stable and feasible.

References ˚ ˚ om K. J., Safe Manual Control of the Furuta Pendulum. Proceedings of 1. Akesson J., Astr¨ the 2001 IEEE International Conference on Control Applications, Mexico City, Mexico, (2001)

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2. Barbu C., Galeani S., Teel A. R., Zaccarian L., Non-linear anti-windup for manual ﬂight control. International Journal of Control, 78(14):1111–1129 (2005) 3. Bemporad M., Morari M., Dua V., Pistikopoulos E. N., The explicit linear quadratic regulator for constrained systems. Automatica, 38(1):3–20 (2002) 4. Borrelli F., Constrained Optimal Control Of Linear And Hybrid Systems. Lecture Notes in Control and Information Sciences, Springer (2003) 5. Ferrari-Trecate G., Cuzzola F. A., Mignone D. and Morari M., Analysis of discrete-time piecewise afﬁne and hybrid systems. Automatica, 38(12):2139–2146 (2002) 6. Furuta K., Yamakita M., Kobayashi S., Nishimura M., A new Inverted Pendulum Apparatus for Education, IFAC Symposium on Advances in Control Education, Boston, USA, 191–196 (1991) 7. Gilbert E.G. and Tan K.T., Linear systems with state and control constraints: the theory and applications of maximal output admissible sets. IEEE Trans. Automatic Control, 36(9):1008–1020 (1991) 8. Grieder P. , L¨uthi M., Parillo P. and Morari M., Stability & Feasibility of Receding Horizon Control. European Control Conference, Cambridge, UK (2003) 9. Grieder P., Morari M, Complexity Reduction of Receding Horizon Control. IEEE Conference on Decision and Control, Maui, Hawaii, 3179–3184 (2003) 10. Hanley J. G., A Comparison of Nonlinear Algorithms to Prevent Pilot-Induced Oscillations caused by Actuator Rate Limiting. Air Force Institute of Technology (2003) 11. Kerrigan E.C., Robust Constraints Satisfaction: Invariant Sets and Predictive Control. PhD thesis, Department of Engineering, The University of Cambridge, Cambridge, England (2000) 12. Ling Z., Saberi A., Semi-global exponential stabilization of linear discrete-time systems subject to input saturation via linear feedbacks System and Control Letters 24, 125–132 (1995) 13. Maciejowski J.M., Predictive Control with Constrains. Prentice Hall, 2002. 14. Mayne D.Q., Rawlings J.B., Rao C.V., Scokaert P.O.M., Constrained model predictive control: Stability and optimality. Automatica, 36(6):789–814 (2000) 15. Kvasnica

M.,

Grieder

M.,

Baoti´c

M.,

Multi-Parametric

Toolbox

(MPT).

http://control.ee.ethz.ch/ mpt/ (2004) 16. Pachter M., Miller R.B., Manual Flight Control with Saturating Actuators, IEEE Control Systems Magazine, 18(1):10–19 (1998) 17. Sontag E.D., Sussmann E.J., Nonlinear Output Feedback Design for Linear Systems with Saturating Controls. Proc. 29th IEEE Conference Decision and Control, Honolulu, Hawaii, 3414–3416 (1990) 18. Tøndel P., Constrained Optimal Control via Multiparametric Quadratic Programming. PhD thesis, Department of Engineering Cybernetics, NTNU ,Trondheim, Norway (2000)

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19. Tøndel P., Johansen T.A. and Bemporad A., An algorithm for multiparametric quadratic programming and explicit MPC solutions. Automatica, 39(3) (2003) 20. Yakubovich V.A., Nakaura S., Furuta K., Tracking Domains for Unstable Plants with Saturating-Like Actuators. Asian Journal of Control, 1(4):229–244 (2000)

Constrained Control Using Model Predictive Control J.M. Maciejowski1 , P.J. Goulart1 , and E.C. Kerrigan2 1

Cambridge University Engineering Department, Cambridge CB2 1PZ, England {jmm,pjg42}@eng.cam.ac.uk

2

Imperial College London, Dept. of Electrical and Electronic Engineering and Dept. of Aeronautics, London SW7 2AZ, England [email protected]

1 Introduction The most common way of dealing with constraints in control systems is to ignore them, pretend that the system is linear, and ﬁx things up in a more-or-less ad-hoc fashion after performing a linear design. There are some rather systematic ways of ‘ﬁxing things up’, including certain anti-windup techniques, but logically they are an afterthought, a way of dealing with the nuisance of constraints after the central work has been done. Until recently, things could not be done differently, because almost all the control design techniques available to us were linear techniques.3 This situation has been changed by the emergence of Model Predictive Control (MPC) as a successful technique for controlling systems subject to constraints — several books are already available [6, 7, 10, 11, 17, 19, 20, 21]. MPC allows constraints to be included as part of the problem formulation, and thus faced up to from the outset. Furthermore, this is true for output or state constraints (arising from safety or product quality considerations, etc), as well as input constraints (arising from actuator limitations). It can be applied to typical control problems in which there are set-points as well as constraints, but also to problems in which there are only constraints, and no set-points (such as level control in a buffer tank). A very nice aspect of MPC is that there is no particular difﬁculty in dealing with multivariable systems 3

Although quite a lot of nonlinear methods have been developed, the proportion of problems to which they can be applied is very small.

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in which some output variables have set-points, some have constraints, and some have both set-points and constraints. The technological foundation of MPC is repeated online optimisation. It is therefore usually considered to be a technique suitable only for low-bandwidth applications with low sampling and update rates, such as those found in process control. But modern computing power is so great that in fact MPC can be used in applications usually considered to be relatively high-bandwidth, such as aerospace, automotive, and robotics. This will be illustrated in this chapter by a (simulated) application to the control of an inverted pendulum.

2 Outline of MPC 2.1 The Basic Idea Figure 1 shows the basic idea of predictive control, for the simplest case of a singleinput, single-output plant. We assume a discrete-time setting, and that the current time is labelled as time step k. At the current time the plant output is yk , and the ﬁgure shows the previous history of the output trajectory. Also shown is a set-point trajectory, which is the trajectory that the output should follow, ideally. The value of the set-point trajectory at any time t is denoted by st . Optionally, there may be a reference trajectory rk+i|k which is distinct from the setpoint trajectory. This starts at the current output yk , and deﬁnes an ideal trajectory along which the plant should return to the set-point trajectory, for instance after a disturbance occurs. The reference trajectory therefore deﬁnes an important aspect of the closed-loop behaviour of the controlled plant. It is not necessary to insist that the plant should be driven back to the set-point trajectory as fast as possible, although that choice remains open. It is frequently assumed that the reference trajectory approaches the set-point exponentially from the current output value, with the ‘time constant’ of the exponential. The notation rk+i|k indicates that the reference trajectory depends on the conditions at time k, in general. A predictive controller has an internal model which is used to predict the behaviour of the plant, starting at the current time, over a future prediction horizon. This predicted behaviour depends on the assumed input trajectory u ˆk+i|k (i =

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275

st rt|k N yt yˆt|k f yˆt|k

k

k+N

Time

k

k+N

Time

Input

Fig. 1. MPC: The Basic Idea

0, 1, . . . , N − 1) that is to be applied over the prediction horizon, and the idea is to select that input which promises the best predicted behaviour. We shall assume that the internal model is linear; this makes the calculation of the best input relatively straightforward. The notation u ˆ rather than u here indicates that at time k we only have a prediction of what the input at time k + i may be; the actual input at that time, uk+i , will probably be different from u ˆk+i|k . Note that we assume that we have the output measurement yk available when deciding the value of the input uk . This implies that our internal model must be strictly proper. In the simplest case we can try to choose the input trajectory such as to bring the plant output at the end of the prediction horizon, namely at time k + N , to the required value rk+N . In this case we say, using the terminology of Richalet [20], that we have a single coincidence point at time k + N . There are several input trajectories {ˆ uk|k , u ˆk+1|k , . . . , u ˆk+N −1|k } which achieve this, and we could choose one of them, for example the one which requires the smallest input energy. But it is usually adequate, and in fact preferable, to impose some simple structure on the input trajectory,

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parametrised by a smaller number of variables. Figure 1 shows the input assumed to vary over the ﬁrst three steps of the prediction horizon, but to remain constant thereafter: u ˆk+2|k = u ˆk+3|k = · · · , u ˆk+N −1|k , so that there are three ‘parameters’ to choose: u ˆk|k , u ˆk+1|k , u ˆk+2|k . These parameters (or other similar ones) are chosen so as to minimise some cost function. Once a future input trajectory has been chosen, only the ﬁrst element of that trajectory is applied as the input signal to the plant. That is, we set uk = u ˆk|k , where uk denotes the actual signal applied. Then the whole cycle of output measurement, prediction, and input trajectory determination is repeated, one sampling interval later: a new output measurement yk+1 is obtained; a new reference trajectory rk+i|k+1 (i = 2, 3, . . .) may be deﬁned; predictions are made over the horizon k + 1 + i, with i = 1, 2, . . . , N ; a new input trajectory u ˆk+1+i|k+1 , with i = 0, 1, . . . , N − 1 is chosen; and ﬁnally the next input is applied to the plant: uk+1 = u ˆk+1|k+1 . Since the prediction horizon remains of the same length as before, but slides along by one sampling interval at each step, this way of controlling a plant is often called a receding horizon strategy. This idea remains valid if the plant being controlled is multivariable, that is, if there are several inputs and/or several outputs. In such cases the variables u and/or y become vectors, and we shall henceforth assume that this is the case. The only other major ingredient of MPC that we need is constraints. The choice of future trajectory u ˆk+i|k , (i = 0, 1, . . . , N − 1) should be such that the input signals and their rates of change remain within allowed constraints, and such that the outputs — and possibly inferred variables, namely values which are not measured directly — also remain within allowed constraints.

2.2 Computing the Inputs If we assume that the plant model is linear, and that the cost function is quadratic, then the problem which has to be solved is a standard ﬁnite-horizon linear quadratic (or ‘LQ’) problem, for which a complete theory exists [5]. If linear inequality constraints on inputs, states or outputs are added, ﬁnding the optimal input trajectory becomes a quadratic programming (or ‘QP’) problem. If the cost function is changed from being quadratic to being ‘1-norm’ or ‘∞-norm’, namely one which penalises

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277

absolute values of errors, or the peak error, respectively, then ﬁnding the optimal solution becomes a linear programming (or ‘LP’) problem. It is standard to assume that the plant is linear and time-invariant, and that after discretisation of time at a single sampling rate it can be modelled as a strictly-proper state-space system: xk+1 = Axk + Buk + Ewk yk = Cxk + vk

(1) (2)

where x is a state vector, u and y are vectors of control inputs and measured outputs, respectively, and w and v are vectors of unknown state disturbances and measurement errors, respectively. A, B, C, E are constant matrices of appropriate dimensions. Other linear model forms are often assumed for MPC, but these can all be obtained from a state-space model, so no loss of generality is involved in assuming this form. (In particular, a conversion of MPC problems from the ‘GPC’ variant of MPC to a state-space form is described in [17].) Plants with time delays can also be modelled by this form, providing that all delays are multiples of the sampling interval. Given a future input trajectory u ˆk+i|k (i = 0, 1, 2, . . .) predicted state and/or output trajectories x ˆk+i|k , yˆk+i|k can be calculated using the model (1), (2) together with any assumptions that are made about the disturbances wk+i and measurement errors vk+i . The future input trajectory is chosen as one that minimises a cost function of the form J=

Hp

(ˆ yk+i|k , rk+i , u ˆk+i−1|k ) + F (ˆ xk+Hp +1|k )

(3)

i=1

in which the function (·, ·, ·) is a stage cost on the output, reference, and input signals during a prediction horizon of length Hp , and the function F (·) is a terminal cost on the state. This minimisation is performed subject to satisfying constraints on the inputs and states, typically in the form of linear inequalities: ⎡ ⎤ x ˆ ⎢ k+i|k ⎥ M⎢ ˆk+i|k ⎥ ⎣ u ⎦≤μ

(4)

Δˆ uk+i|k where Δˆ uk+i|k = u ˆk+i|k − u ˆk+i−1|k , M is a matrix and μ is a vector. In ‘robust MPC’ the minimisation of the cost is sometimes minimisation of the worst case

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cost, for a range of allowable disturbances and/or plant uncertainties. Thus a straight minimisation problem is replaced by a min-max problem, which usually requires much more computational effort to solve. Common choices of stage cost are: Quadratic: Absolute values: Peak values:

¯) 22 (y, r, u) = Q1/2 (y − r) 22 + R1/2 (u − u

(5)

¯) 1 (y, r, u) = Q(y − r) 1 + R(u − u

(6)

¯) ∞ (y, r, u) = Q(y − r) ∞ + R(u − u

(7)

¯ is an equilibrium value where Q = QT ≥ 0 and R = RT > 0 typically, and u of u that is compatible with the set-point r. (The precise conditions on Q and R may vary.) The ﬁrst of these leads to a quadratic programming problem to be solved, while the second and third lead to a linear programming problem. (In the ‘robust’ case, if min-max problems are to be solved, several QP or LP problems may need to be solved at each step.) Note that the computation of the optimal control input is not much more complicated if the model, the constraints, and the cost function vary with time, although the analysis of closed-loop properties becomes much more difﬁcult in such cases. (The only additional computational complexity arises because the data for the optimisation problem has to be re-assembled at each step. This is usually a minimal overhead in comparison with the solution of the optimisation problem.)

2.3 Closed-Loop Stability MPC has a reputation for usually giving closed-loop stability. Certainly if one chooses a problem speciﬁcation at random, one is much more likely to obtain a stable closed loop with an MPC controller than one would ﬁnd if one chose the parameters of a ‘classical’ (even simple PI) controller at random. But it is possible to construct examples in which the MPC solution gives an unstable control loop, so the stability property should not be taken for granted. The reputation of MPC in this regard was established in industrial applications, in particular in the petrochemical industry. Most such applications involve a benign stable plant, and the control requirements are relatively undemanding, allowing a considerable degree of ‘de-tuning’, which tends to give closed-loop stability when used with stable plants. As MPC becomes

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used for a wider range of applications, with less stable plants, more aggressive controller requirements, and more complex patterns of active constraints, closed-loop stability is likely to be an increasing problem. Fortunately, there is a well-developed theory regarding stability of MPC, which leads to a number of ways of formulating the MPC problem which guarantee closed-loop stability. The two basic ideas are: • Impose a terminal constraint at the end of the prediction horizon, or • Use an inﬁnite prediction horizon. Suppose that the set-point is constant, and that we change coordinates such that the set-point corresponds to an equilibrium at x = 0. Imposing a terminal constraint of the form x ˆk+Hp |k = 0 always gives closed-loop stability, providing that the resulting optimization problem is always feasible, and that the global optimal solution is found and applied at each step [13]. But this is an unnecessarily strong constraint, and it can be relaxed to a constraint of the form x ˆk+Hp |k ∈ XF , providing that the terminal set XF has suitable properties. For example, if a conventional control exists which stabilises 0 and for which XF is contained in the domain of attraction of 0, and all the constraints are satisﬁed everywhere on XF , then that is enough to guarantee closedloop stability, if one postulates a dual-mode control, which applies the MPC solution for x ∈ XF and the conventional (typically linear) control for x ∈ XF . Allowing the terminal state x ˆk+Hp |k to belong to a set rather than a point makes it more likely that the MPC problem will have a feasible solution at each step. Using an inﬁnite horizon is an alternative way of ensuring closed-loop stability. For if the closed-loop were unstable then the cost function would be inﬁnite, and hence the controller would not be optimal. This argument assumes that the plant is stabilisable, and that the MPC problem is feasible, and that the global optimum is found at each step. An alternative argument can be made invoking Bellman’s principle of optimality, and using the value function (that is, the optimal value of J — see (3)) as a Lyapunov function [17, Chapter 6]. But the difﬁculty with using an inﬁnite horizon is that there are inﬁnitely many decision variables {ˆ uk+i|k : i = 0, 1, 2, . . .} to be chosen. This difﬁculty can be overcome by reparametrising the MPC problem in such a way as to reduce the number of decision variables to a ﬁnite number. The usual way of doing this is to assume that there is a horizon length N , such that the constraints can be assumed to be inactive for i ≥ N . Then one only

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chooses {ˆ uk+i|k : i = 0, 1, 2, . . . , N − 1}, and assumes some ﬁxed feedback law uk+i = μ(xk+i ) for i ≥ N . In particular, if the stage cost in (3) is taken to be quadratic ˆk+i|k ) = Q1/2 x ˆk+i|k 22 + R1/2 u ˆk+i|k 22 (ˆ xk+i|k , rk+i , u

(8)

(assuming rk+i = 0 for all i) and the horizon is taken to be inﬁnite, then the optimal value of the cost is J 0 = xTk P xk

(9)

where P is the symmetric positive semi-deﬁnite solution of the algebraic Riccati equation P = AT P A − AT P B(B T P B + R)−1 B T P A + Q

(10)

(assuming the linear plant model (1)–(2)). Consequently, the following cost function has been suggested [9]: J=

N "

Q

1/2

x ˆk+i|k 22

+ R

1/2

u ˆk+i|k 22

#

+x ˆTk+N +1|k P x ˆk+N +1|k

(11)

i=1

A thorough survey of techniques for ensuring stability of MPC controllers is given in [18].

2.4 Robust Feasibility and Stability Solutions to MPC problems are relatively straightforward if model uncertainty is ignored. But they become signiﬁcantly more complicated if uncertainty is taken into account, and an effort is made to ensure that constraints are satisﬁed despite the uncertainty. This is of course a requirement in any practical application of MPC. In general model uncertainty can be very ‘unstructured’ — the real system is likely to be nonlinear, time-varying, of unknown state dimension, etc. But useful results can be obtained by assuming more restricted forms of uncertainty, such as uncertain parameter values in a model of the form (1) – (2). In this chapter we shall assume an even more restricted form of uncertainty, namely that everything in the model is known in advance, except for the values of the state disturbance sequence {wk }. We shall also assume that the full state vector can be measured, with negligible measurement noise. The state disturbance values will be assumed to belong to a set wk ∈ W , with

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281

W being a known convex polytope. Note that a consequence of these assumptions is that it is possible to infer the value of wk−1 from a measurement of xk . The strategy for ensuring robustness is to establish that a feasible trajectory {uk+i , xk+i+1 : i = 0, 1, . . . , N } can be found, from the current state xk , to a terminal set xk+N +1 ∈ XF , whatever disturbance trajectory is encountered, and that the terminal set has an appropriate robust invariance property. Of course if xk is such that this is impossible, then the constraints in the problem speciﬁcation cannot be satisﬁed — but the aim is to ensure that this can never arise. An essential idea is that this robust feasibility problem typically has no solution if one tries to determine in advance what the solution trajectory should be, because there is usually no single trajectory which is appropriate for all possible disturbance realizations. But it may have a solution providing that one allows the trajectory from each time onwards to depend on the particular disturbances that have actually occurred up to that time — in other words, if one decides on a feedback policy rather than a speciﬁc trajectory [18]. But optimization over feedback policies is in general very difﬁcult if constraints have to be taken into account. Most proposals for achieving this using ﬁnite-dimensional optimization are computationally intractable, because the size of the optimization problem grows exponentially with the size of the problem data (ie with the horizon length, the number of constraints, etc) [22, 17]. One possibility is to pre-compute (off-line) the solution to a suitable robust constrained ﬁnite-horizon problem [2, 6, 14]. This solution turns out to be a time-varying piecewise-afﬁne control law; once computed, it can be stored in a table and the appropriate “piece” deployed, depending on the current state. Unfortunately this ‘explicit’ approach is currently feasible only for small problems, because the computation of the solution can grow exponentially with the problem data. A popular approach to overcoming the computational complexity of optimization over feedback policies is to somehow ﬁx a stabilising linear state-feedback law, and then to use on-line optimization to ﬁnd perturbations to the resulting control sequence in order to satisfy constraints [1, 8, 15]. This approach attempts to reduce the effects of uncertainty by means of the state feedback, so that optimization of a single trajectory becomes feasible despite the huge variety of possible disturbance sequences that might be encountered. But the choice of the linear state-feedback law is essentially ad hoc, and it is not clear how best to choose it. Attempts to improve on this by simultaneously optimising the state-feedback law and the perturbations to

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it required for feasibility usually result in a non-convex problem, because state and input predictions depend nonlinearly on the state-feedback law. We have recently proposed a way of overcoming this non-convexity [12]. The key to this is a re-parametrization of the control policy as an afﬁne function of past disturbances: ui =

i−1

Mij wj + vi ,

for i = 0, 1, . . . , N − 1.

(12)

j=0

where {Mij } is a set of matrix gains which are to be chosen, and {vi } is a set of perturbations which are also to be chosen. A similar parametrization was also proposed in [23, 16, 4]. It is shown in [12, 3] that this parametrization of the control policy is equivalent to the more conventional state-feedback parametrization: ui =

i

Lij xj + gi

for i = 0, 1, . . . , N − 1.

(13)

j=0

But the crucial difference is that, given xk (and hence w0 , . . . , wk−1 ), the set of admissible Mij and vi in (12) (for i = k, k+1, . . . , k+N −1 and j = 0, 1, . . . , k−1) is convex, whereas the set of admissible Lij and gi in (13) is not.

3 Application to Inverted Pendulum We apply MPC to the inverted pendulum benchmark problem. In this case we found that we could easily meet the performance requirements and satisfy the constraints, without needing any of the machinery from section 2.4. We ran MPC in discrete time, using a time-step of 0.15 sec, and simulated its performance on a continuoustime model of the pendulum. Figure 2 shows a Simulink block diagram of the inverted pendulum, with both an MPC controller and a basic conventional controller available as options. The output of the MPC controller at each update instant is a change in the required force, as required by the rate-commanded actuator. A cost function of the form (11) was used, but modiﬁed to: J=

N "

# xk+i|k − x ¯) 22 + R1/2 Δˆ uk+i−1|k 22 + Q1/2 (ˆ

i=1

¯)T P (ˆ xk+N +1|k − x ¯) (14) (ˆ xk+N +1|k − x

rref

rref

y

MPC Controller

Rate Commanded Actuator u_dot

y

DT linear controller

u_dot

pendulum rref

u

u

Constrained Control Using Model Predictive Control

z y

u_dot

283

time MPC controller. A conventional linear controller is also shown as an alternative option.

Fig. 2. A Simulink model of the continuous-time invered pendulum controlled by a discrete-

zload

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where x ¯ is an equilibrium value of the state vector that corresponds to the current set-point (see equations (18)), and Δˆ uk+i|k = u ˆk+i|k − u ˆk+i−1|k is the predicted change in the control input signal. We used a horizon N = 35 steps (ie 5.25 sec), with weighting matrices ⎡

100 0

0

0

⎤

⎥ ⎢ ⎢ 0 500 0 0 ⎥ ⎥ Q=⎢ ⎢ 0 0 10 0 ⎥ ⎦ ⎣ 0 0 0 0.01

and

R = 0.01

(15)

where the ﬁrst 3 states are as deﬁned in the benchmark deﬁnition, namely the pendulum speed, the displacement between the pendulum and the slider, and the slider speed. The fourth state variable was the last input applied to the system: x(4)k = uk−1 , which is needed in order to keep track of the predicted values u ˆk+i|k during the prediction horizon — these are needed to ensure that actuator limits are not violated. The corresponding terminal weight P that appears in (11) (and is the solution of (10)) was ⎡

2211.6 −1018.3 −179.12 −15.432

⎢ ⎢ −1018.3 1928.5 P =⎢ ⎢ −179.12 271.34 ⎣ −15.432 23.043

271.34 77.455 7.0224

⎤

⎥ 23.043 ⎥ ⎥ 7.0224 ⎥ ⎦ 1.0311

(16)

Of course the model has to be modiﬁed appropriately, to be consistent with the newly-deﬁned state vector, and the use of Δu instead of u as the input. But this is straightforward and standard [7, 17]. Figure 3 shows the internal structure of the MPC controller. At each step a simple disturbance estimator is run, assuming that the state disturbance is given by the difference between the latest measured state and the previously expected state: dˆk = xk − x ˆk|k−1

(17)

(Recall that it is assumed that all the states are measured.) This is then used to estimate the equilibrium value of the force u ¯ required from the actuator, assuming that the disturbance will remain constant at this estimated value: d¯ = dˆk . x ¯ and u ¯ are found as the solutions to the simultaneous equations: ¯ x ¯ = A¯ x + Bu ¯ + d,

Cx ¯ = rk

(18)

xbar

sfunsetpt

NOT MPC controller

Stop Simulation STOP

Constrained Control Using Model Predictive Control

set point calculator disturbance estimation

ubar du

dbar

dbar

1 u_dot dbar

sfunmpc y

2 y

rref

1 rref

equilibrium set-point calculator.

285

Fig. 3. Internal structure of the MPC controller, showing the disturbance estimator and the

x

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J.M. Maciejowski, P.J. Goulart, and E.C. Kerrigan

where rk is the current set-point for the pendulum speed. The resulting value u ¯ then becomes the fourth element of the augmented x ¯. Note that this results in offsetfree control in the presence of unknown but constant disturbance forces, even if the steady-state gains in the model are not accurate. To demonstrate that this design meets the requirements, we simulated a single run, over 120 sec of simulated time. The speciﬁed reference trajectory for the pendulum speed was applied over the ﬁrst 60 seconds, and the speciﬁed disturbance force was applied over the last 60 sec, as shown in Figure 4. 2.5

2

1.5

1

0.5

0

0

20

40

60 Time (sec)

80

100

120

Fig. 4. Reference trajectory for pendulum speed (solid line) and disturbance force (dashed line) during simulations.

With the higher limit on the actuator rate, namely |u| ˙ ≤ 10, the results of the simulation are shown in Figure 5. The top and middle graphs show the speeds of the pendulum (with set-point) and the slider, respectively, and it can be seen that the pendulum speed follows the set-point closely, with no steady-state offset. The slider’s speed trajectory is similar to that of the pendulum, except for brief accelerations/decelerations in order to produce accelerations/decelerations of the pendulum. The second graph shows the distance between the slider and the pendulum. This is held to zero (except

Constrained Control Using Model Predictive Control

287

x1 4 2 0 −2 x

2

1.5 1 0.5 0 −0.5 x

3

4 2 0 −2 −4

u 2 1 0 −1 −2

du/dt 10 5 0 −5 −10

0

20

40

60 Time

80

100

120

Fig. 5. Results of the simulation with |u| ˙ ≤ 10. From the top, the graphs show (1) x1 , the pendulum speed (with set-point), (2) x2 , the distance between the pendulum and the slider, (3) x3 , the slider speed, (4) u, the force applied by the actuator, (5) u, ˙ the rate of change of the force.

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J.M. Maciejowski, P.J. Goulart, and E.C. Kerrigan

during transients) when a constant speed set-point is being followed, since it is assumed that there is no friction. But when the constant external disturbance is present, then the slider and the pendulum are a constant distance apart, in order to produce a counteracting force on the pendulum. The fourth graph shows the force applied by the actuator on the slider. It will be seen that this is limited at ±1.25, as required. The bottom graph shows that the rate limits are also respected. It is assumed that the controller does not anticipate set-point changes. If it did have advance awareness of set-point changes then the delay in tracking the controller speed could be eliminated. (Note that this feature is not unique to MPC, despite common belief [5].) If the slew rate of the actuator force is reduced to |u| ˙ ≤ 2 the results are shown in Figure 6. It can be seen that the results are essentially the same as in Figure 5, so that the reduction of the slew rate limit is not of great signiﬁcance in this problem. Note, however, that the control signals generated by the MPC controller in the two cases are not the same; Figure 5 shows that u˙ = 10 at t ≈ 108 sec, so that the full range of allowed rates is used when the higher limit is available. The quadratic programming problem involved in implementing the MPC solution was solved using an interior point method, implementing the speciﬁc ‘banded matrix’ scheme described in [17, section 3.3.1]. The time taken to run the simulation in Simulink was about 15 sec, on a Pentium 4 with 3GHz clock speed and 1 Gbyte of RAM, running Linux. That is, the simulation runs more than 10 times faster than real time. Note that our running speed was limited by network communications, by writing the simulation results to a ‘scope’ while running the simulation, and by simulating the plant as well as the controller. Furthermore the simulation was run within Matlab version 7, making much use of the object-oriented mechanisms, thus incurring considerable run-time overhead. Thus an MPC controller could easily be made to run much faster than this for this problem. A signiﬁcant feature of MPC is that it can often be made to work quite easily. This was the case for the inverted pendulum benchmark. Once the form of the cost function and the structure of the MPC controller had been chosen, the only tuning that had to be done was the choice of the horizon N and of the weights Q and R (see (14)). Finding a suitable combination of these was straightforward and did not take much time.

Constrained Control Using Model Predictive Control

289

x1 4 2 0 −2 x

2

1.5 1 0.5 0 −0.5 x

3

4 2 0 −2

u 2 1 0 −1 −2

du/dt 2 1 0 −1 −2

0

20

40

60 Time

80

100

120

Fig. 6. Results of the simulation with |u| ˙ ≤ 2. From the top, the graphs show (1) x1 , the pendulum speed (with set-point), (2) x2 , the distance between the pendulum and the slider, (3) x3 , the slider speed, (4) u, the force applied by the actuator, (5) u, ˙ the rate of change of the force.

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4 Conclusions Model Predictive Control (MPC) is particularly well-suited to the control of constrained systems. It allows constraints to be included in the problem formulation from the start. The basic form of MPC is easy to understand and to implement, and is no longer conﬁned to application to slow processes. Formulations which ensure stability, at least from some region of attraction about the desired equilibrium, are standard. Formulations which ensure feasibility, and hence stability, from some prescribed set of initial conditions are also available, but typically require much more computation to implement them. However, as we have shown on the inverted pendulum benchmark problem, such advanced formulations are frequently not needed for many problems.

References 1. A. Bemporad. Reducing conservativeness in predictive control of constrained systems with disturbances. In Proc. 37th IEEE Conference on Decision and Control, pages 1384– 1391, Tampa, FL, December 1998. 2. A. Bemporad, F. Borrelli, and M. Morari. Min-max control of constrained uncertain discrete-time linear systems. IEEE Transactions on Automatic Control, 48(9):1600–1606, September 2003. 3. A. Ben-Tal, S. Boyd, and A. Nemirovski. Extending scope of robust optimization: Comprehensive robust counterparts of uncertain problems. Mathematical Programming, 107(1–2):63–89, June 2006. 4. A. Ben-Tal, A. Goryashko, E. Guslitzer, and A Nemirovski. Adjustable robust solutions of uncertain linear programs. Mathematical Programming, 99(2):351–376, March 2004. 5. R.R. Bitmead, M. Gevers, and V. Wertz. Applied Optimal Control: The Thinking Man’s GPC. Prentice-Hall, Englewood Cliffs, NJ, 1990. 6. F. Borrelli. Constrained Optimal Control of Linear and Hybrid Systems. Lecture Notes in Control and Information Sciences. Springer, 2004. 7. E.F. Camacho and C.Bordons. Model Predictive Control. Advanced Textbooks in Control and Signal Processing. Springer, London, 1999. 8. L. Chisci, J.A. Rossiter, and G. Zappa. Systems with persistent state disturbances: predictive control with restricted constraints. Automatica, 37:1019–028, 2001. 9. D. Chmielewski and V. Manousiouthakis. On constrained inﬁnite-time linear quadratic optimal control. Systems and Control Letters, 29:121–129, 1996.

Constrained Control Using Model Predictive Control 10. R. Dittmar and B.-M. Pfeiffer.

Modellbasierte pr¨adiktive Regelung.

291

Oldenbourg,

M¨unchen, 2004. 11. G.C. Goodwin, M.M. Seron, and J.A. De Dona. Constrained Control and Estimation: an Optimisation Approach. Springer, London, 2005. 12. P.J. Goulart, E.C. Kerrigan, and J.M. Maciejowski. Optimization over state feedback policies for robust control with constraints. Automatica, 42(4):523–533, April 2006. 13. S.S. Keerthi and E.G. Gilbert. Optimal inﬁnite-horizon feedback laws for a general class of constrained discrete-time systems: stability and moving-horizon approximations. Journal of Optimization Theory and Applications, 57:265–293, 1988. 14. E.C. Kerrigan and J.M. Maciejowski. Feedback min-max model predictive control using a single linear program; robust stability and the explicit solution. International Journal of Robust and Nonlinear Control, 14:395–413, 2004. 15. Y.I. Lee and B. Kouvaritakis. Constrained receding horizon predictive control for systems with disturbances. European Journal of Control, pages 1027–1032, August 1999. 16. J. L¨ofberg. Minimax Approaches to Robust Model Predictive Control. PhD thesis, Link¨oping University, April 2003. 17. J.M. Maciejowski. Predictive Control with Constraints. Prentice-Hall, Harlow UK, 2002. (Japanese translation by S. Adachi and M. Kanno published by Tokyo Denki University Press, 2005.). 18. D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokaert. Constrained model predictive control: stability and optimality. Automatica, 36:789–814, 2000. 19. D.M. Prett and C.E. Garcia. Fundamental Process Control. Butterworths, Boston, 1988. 20. J. Richalet, G. Lavielle, and J. Mallet. La Commande Pr´edictive. Eyrolles, Paris, 2004. 21. J.A. Rossiter. Model-based Predictive Control: A Practical Approach. CRC Press, Boca Raton, 2003. 22. P.O.M. Scokaert and D.Q. Mayne. Min-max feedback model predictive control for constrained linear systems. IEEE Transactions on Automatic Control, 43(8):1136–1142, 1998. 23. D.H. van Hessem and O.H. Bosgra. A conic reformulation of model predictive control including bounded and stochastic disturbances under state and input constraints. In Proc. 41st IEEE Conference on Decision and Control. December 2002.

Risk Adjusted Receding Horizon Control of Constrained Linear Parameter Varying Systems M. Sznaier1 , C. M. Lagoa2 , X. Li3 , and A. A. Stoorvogel4 1

EE Dept., The Pennsylvania State University. University Park, PA 16802, USA, [email protected]

2

EE Dept., The Pennsylvania State University. University Park, PA 16802, USA, [email protected]

3

EE Dept., The Pennsylvania State University. University Park, PA 16802, USA, [email protected]

4

Department of Information Technology and Systems Delf University of Technology P.O. Box 5031, 2600 GA Delft, The Netherlands. email: [email protected]

Summary. In the past few years, control of Linear Parameter Varying Systems (LPV) has been the object of considerable attention, as a way of formalizing the intuitively appealing idea of gain scheduling control for nonlinear systems. However, currently available LPV techniques are both computationally demanding and (potentially) very conservative. In this chapter we propose to address these difﬁculties by combining Receding Horizon and Control Lyapunov Functions techniques in a risk–adjusted framework. The resulting controllers are guaranteed to stabilize the plant and have computational complexity that increases polynomially, rather than exponentially, with the prediction horizon.

Key words: Model Predictive Control, LPV Systems, Risk Adjusted Control.

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M. Sznaier, C. M. Lagoa, X. Li, and A. A. Stoorvogel

1 Introduction A widely used practice to handle nonlinear dynamics is to linearize the plant around several operating points and then use gain–scheduled linear controllers. However, while intuitively appealing, this idea has several pitfalls [15]. Motivated by these shortcomings, during the past few years considerably attention has been devoted to the problem of synthesizing controllers for Linear Parameter Varying Systems, where the state–space matrices of the plant depend on time–varying parameters whose values are not known a priori, but can be measured by the controller. This research has resulted in controller synthesis methods guaranteeing worst case performance bounds (for instance in an H2 or H∞ sense, see e.g. [1] and references therein). While successful in many situations, these techniques are potentially very conservative in others, since they are based on sufﬁcient conditions. In addition, these methods are computationally very demanding, requiring the solution of a set of functional matrix inequalities. Obtaining computationally tractable problems requires using both ﬁnite expansion approximations as well as a gridding of the parameter space, leading to further conservatism. The present chapter seeks to reduce both the computational complexity and conservatism entailed in currently available LPV synthesis methods by combining risk– adjusted (see eg. [5, 6, 13]) and Receding Horizon (see [12] and references therein) ideas. Our main result shows that, by searching over a set of closed–loop strategies the problem can be reduced to ﬁnding a solution to a ﬁnite set of Linear Matrix Inequalities (LMIs). Finding the exact solution to this problem has computational complexity that grows exponentially with the horizon length. To circumvent this difﬁculty, we propose to use a stochastic approximation algorithm that is guaranteed to converge to the solution with probability one, and whose computational complexity grows only polynomially with the horizon. The chapter draws inspiration, in addition to [16] and [9], from [14], [7] and [3]. The main difference with [14] and [7] is the use of Receding Horizon techniques and parameter dependent Lyapunov functions. Compared with [3], we consider the case of case of LPV dynamics and we obtain a controller that minimizes the worst case performance, rather than its expected value, over all trajectories compatible with the current parameter value. Finally, the use of closed–loop strategies, based on the solution of a set of LMIs, results in substantial reduction of computational complexity.

Risk Adjusted Receding Horizon Control

295

In this sense the work presented here is related to the earlier work in [18] advocating the use of Riccati based receding horizon controllers for H∞ control of Linear Time Varying systems, and to the work in [11], proposing an LMI–based optimization of closed–loop control strategies.

2 Preliminaries 2.1 The LPV Quadratic Regulator Problem In this chapter we consider the following class of LPV systems: x(t + 1) = A [ρ(t)] x(t) + B2 [ρ(t)] u(t) z(t) = C1 [ρ(t)] x(t) + D12 [ρ(t)] u(t)

(1)

where x ∈ Rnx , u ∈ Su ⊆ Rnu , and z ∈ Rmz represent the state, control, and regulated variables respectively, Su is a convex set containing the origin in its interior, ρ denotes a vector of time–varying parameters that can be measured in real time, and where all matrices involved are continuous functions of ρ. Further, we will assume that the set of parameter trajectories is of the form: FΘ = {ρ : ρ(t + 1) ∈ Θ [ρ(t)] , t = 0, 1, . . .}

(2)

where P ⊂ Rnρ is a compact set and Θ : P → P is a given set valued map . Our goal is, given an initial condition xo , and an initial value of the parameter ρo , to ﬁnd an admissible parameter dependent state–feedback control law u[x(t), ρ(t)] ∈ Su that minimizes the index: J(xo , ρo , u) =

sup

∞

z T (k)z(k), x(0) = xo

(3)

ρ∈FΘ ,ρ(0)=ρo k=0 T D12 = In the sequel, for simplicity, we make the following standard assumptions: D12

I, C1T D12 = 0. In addition, the explicit dependence of matrices on ρ will be omitted, when it is clear from the context. Deﬁnition 1 A function Ψ : Rnx × P → R+ that satisﬁes the following condition: ,, + + <0 (4) max min max Ψ[A(ρ)x + B2 (ρ)u, θ] − Ψ(x, ρ) ρ∈P

u∈Su

θ∈Θ(ρ)

for all x = 0 is said to be a parameter dependent Control Lyapunov Function (CLF) for system (1).

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M. Sznaier, C. M. Lagoa, X. Li, and A. A. Stoorvogel

The following result characterizes a family of CLFs in terms of the solution to a (functional) Afﬁne Matrix Inequality. Lemma 1 Assume that the pair {A(.), C(.)} is uniformly observable for all parameter trajectories. If there exist a continuous matrix function Y (ρ) > 0 such that: ⎡ ⎤ −Y (ρ) + B2 (ρ)B2T (ρ) Y (ρ)AT (ρ) − B2 (ρ)B2T (ρ) Y (ρ)C1T (ρ)

. ⎢ M (ρ) = ⎣A(ρ)Y (ρ) − B2 (ρ)B2T (ρ)

−Y (θ)

0

0

−I

C1 (ρ)Y (ρ)

⎥ ⎦≤0 (5)

for all ρ ∈ P and θ ∈ Θ(ρ), then V (x, ρ) = xT Y −1 (ρ)x is a parameter dependent CLF for system (1), with associated control action given by u(x, ρ) = −B2T (ρ)Y −1 (ρ)x

(6)

Moreover, the corresponding trajectory satisﬁes: sup

∞

z T (t)z(t) ≤ xTo Y −1 (ρo )xo

(7)

ρ∈FΘ ,ρ(0)=ρo t=0

Proof. Given any admissible parameter trajectory ρ(.) ∈ FΘ , let T T −1 T −1 T T v = x(t) Y [ρ(t)] x(t + 1) Y [ρ(t + 1)] x(t) C1 [ρ(t)]

(8)

Pre/postmultiplying M [ρ(t)] by v T and v yields: xT (t + 1)Y −1 [ρ(t + 1)]x(t + 1) − xT (t)Y −1 [ρ(t)]x(t) ≤ −z T (t)z(t)

(9)

Summing this inequality along the trajectories of the closed–loop system yields ∞ T T −1 [ρ(0)]xo . Since Y > 0 is a continuous function of 0 z (t)z(t) ≤ xo Y ρ(t) ∈ P compact, it follows that there exist some constant c such that Y −1 (ρ) < ∞ cI. Thus 0 z T (k)z(k) is bounded, which implies that z T (t)z(t) → 0 as t → ∞. Finally, it is easy to show that uniform observability of {A, C}, combined with continuity of all the matrices involved and the fact that ρ ∈ P compact, implies uniform observability of the closed loop system {A − B2 B2T Y −1 , C1 − D12 B2T Y −1 }. Thus z(t) → 0 implies that x(t) → 0.

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2.2 Motivation: A Conceptual Algorithm for Receding Horizon Control of Constrained LPV Systems In this section, motivated by the work in [16] we introduce a conceptual receding horizon control law for constrained LPV systems. This law provides the motivation for the risk–adjusted approach pursued latter in the chapter. Let Ψ : Rnx × P → R+ be a CLF for system (1) such that it satisﬁes the additional condition: ,, + + T max min max Ψ[A(ρ)x + B2 (ρ)u, θ] + z z − Ψ(x, ρ) ≤0 ρ∈P

u∈Su

θ∈Θ(ρ)

(10)

and, given a horizon N , deﬁne (recursively) the following function J(x, ρ, n, N ): J(x, ρ, N, N ) = Ψ(x, ρ) , + T J(x, ρ, i, N ) = min z (i)z(i)+ max J[A(ρ)x + B(ρ)u, θ, i + 1, N ] , i < N u∈Su

θ∈Θ(ρ)

(11) Finally, let x(t), ρ(t) denote the present state and parameter values, and consider the following Receding Horizon control law: uRH [x(t), ρ(t)] = argmin J[x(t), ρ(t), t, t + N ]

(12)

u∈Su

Theorem 1 Assume that the pair {A(.), C(.)} is uniformly observable for all admissible parameter trajectories ρ(t) ∈ FΘ . Then (i) the control law uRH renders the origin an asymptotically stable equilibrium point of (1), and (ii), as N → ∞, its performance approaches optimality monotonically. Proof: Begin by noting that (10) implies that J(x, ρ, t + N, t + N + 1) , + T = min z (t + N )z(t + N ) + max J(Ax + Bu, θ, t + N + 1, t + N + 1) u∈Su θ∈Θ(ρ) , + = min z T (t + N )z(t + N ) + max Ψ(Ax + Bu, θ) u∈Su

θ∈Θ(ρ)

≤ Ψ(x, ρ) = J(x, ρ, t + N, t + N ) It follows that

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J(x, ρ, t + N − 1, t + N + 1) , + T = min z (t + N − 1)z(t + N − 1) + max J(Ax + Bu, θ, t + N, t + N + 1) u∈Su θ∈Θ(ρ) , + T ≤ min z (t + N − 1)z(t + N − 1) + max J(Ax + Bu, θ, t + N, t + N ) u∈Su

θ∈Θ(ρ)

= J(x, ρ, t + N − 1, t + N ) A simple induction argument shows now that J(x, ρ, t, t + N + 1) ≤ J(x, ρ, t, t + N )

(13)

For a given initial condition and parameter trajectory, let V (t) = J[xcl (t), ρ(t), t, t + N ] where xcl (t) denotes the closed–loop trajectory corresponding to the control law uRH . From the deﬁnition of J it follows that V (t) = z T (t)z(t) + max J[xcl (t + 1), θ, t + 1, t + N ] θ∈Θ[ρ(t)]

≥ z T (t)z(t) + J[xcl (t + 1), ρ(t + 1), t + 1, t + N ]

(14)

≥ z T (t)z(t) + V (t + 1) where the last inequality follows from (13). As before, asymptotic stability of the origin follows from a combination of standard Lyapunov and observability arguments. Property (ii) follows from (13) combined with Bellman’s optimality principle [4]. Remark 1 In the unconstrained case (e.g. Su = Rnu ) a CLF is readily available from Lemma 1 since it can be easily shown that Ψ = xT Y −1 (x, ρ)x, where Y satisﬁes the AMIs (5), satisﬁes the inequality (10). In addition, by summing ( 14) along the trajectories and using (13) it follows that this CLF choice yields the following worst–case performance bound: sup ρ∈FΘ

∞

z T (k)z(k) ≤ V (0) ≤ xTo Y −1 (ρo )xo

k=0

3 Risk Adjusted Receding Horizon As shown in Theorem 1, the receding horizon control law uRH is guaranteed to stabilize the system while optimizing a (monotonically improving with N ) approximation

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299

to the performance index. However, solving the min–max optimization problem (11) is far from trivial, even when making several approximations such as assuming memoryless, arbitrarily fast time–varying parameters5 . To avoid this difﬁculty, motivated by the work in [18], in the sequel we will search over closed–loop strategies, rather than control actions. For simplicity, we consider ﬁrst the unconstrained control case and defer the treatment of constraints until section 3.3

3.1 The Unconstrained Control Case In the sequel, we will denote by XN : Θ × {n, n + 1, . . . , n + N } → P the set of all bounded matrix functions that map N –length admissible parameter trajectories to P , the class of all symmetric positive deﬁnite matrices. With this deﬁnition, consider now the following receding horizon type control law: Algorithm 6 0.- Data: A horizon N , a CLF Y (ρ) that satisﬁes (5). 1.- Let ρ(n) denote the measured value of the parameter at time n and solve the following LMI optimization problem in X ∈ XN : min γ

subject to:

X ∈XN

−γ xT (n)

x(n) −X(n)

⎡

≤0

A11

A12

A13

AT13

0

−I

⎤

. ⎢ ⎥ M (ρ) = ⎣AT12 −X(n + i + 1) 0 ⎦ ≤ 0

(15)

A11 = −X(n + i) + B2 [ρ(n + i)]B2T [ρ(n + i)] A12 = X(n + i)AT [ρ(n + i)] − B2 [ρ(n + i)]B2T [ρ(n + i)]) A13 = X(n + i)C1T [ρ(n + i)] for all ρ(n + i + 1) ∈ Θ[ρ(n + i)], i = 0, 1, . . . , N − 1

with boundary condition X(n + N ) = Y [ρ(n + N )], and where, with a slight notational abuse, we denote X[ρ(t), t] simply as X(t). 5

In this case the problem becomes essentially equivalent to synthesizing a controller for a system subject to parametric uncertainty, and it is well known that these problems are generically NP hard.

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2.- At time n use as control action u[ρ(n), x(n), N ] = −B2T [ρ(n)]X −1 [ρ(n), n]x(n)

(16)

3.- Set n = n + 1 and go to step 1. Theorem 2 Assume that the pair {A(ρ), C(ρ)} is uniformly observable for all admissible parameter trajectories. Then, the control law (16) renders the origin an asymptotically stable point of the closed loop system for all ρ(.) ∈ FΘ . Moreover, performance improves monotonically with the horizon length N . Proof To establish stability, let x(n), ρ(n) and X(n + i), i = 1, . . . , N , denote the present state and parameter values and the corresponding solution to (15). Since {X(n + 1), X(n + 2), . . . , Y [ρ(n + N )], Y [ρ(n + N + 1)} is a feasible solution for (15) starting from any ρ(n + 1) ∈ θ[ρ(n)], we have that xT (n + 1)X −1 [ρ(n + 1)]x(n + 1) ≤ xT (n)X −1 [ρ(n)]x(n) − z T (n)z(n)

(17)

From an observability argument it can be easily shown that this, coupled with the fact that X > 0, implies that xT (n)X −1 (n)x(n) → 0 as n → ∞. Combined with the fact that X −1 (n) is bounded away from zero (since X is bounded), this implies that x(n) → 0. Finally, the second property follows from the fact that if {X(n), X(n + 1), . . . , Y [ρ(n + N )]} is a feasible solution for (15) with horizon N , then {X(n), X(n + 1), . . . , Y [ρ(n + N )], Y [ρ(n + N + 1)]} is feasible for N + 1. Thus γ(N + 1) ≤ γ(N ). Remark 2 Note that, for any N , the sequence: {Y [ρ(n)], Y [ρ(n + 1)], . . . , Y [ρ(n + N )]}

(18)

is a feasible solution for (15). Thus, if there exist a sequence of matrices X(ρ, i) that yields a lower value of the bound xT (n)X −1 [ρ(n), n]x(n) the optimization (15) will ﬁnd it. Hence the proposed controller will outperforms the standard LPV control

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law (6). However, its computational complexity is comparable (or worse), since it requires ﬁnding feasible solutions to both (5) and (15). However, since the solution to (5) is used only as a boundary condition for (15) at t = n+N , even if a conservative solution is used here6 , performance degradation is considerably less severe than the one incurred by using this conservative solution to compute the control action, for instance via (6). Further, as we show in the sequel, the computational complexity of solving (15) can be substantially reduced using a risk–adjusted approach.

3.2 Stochastic Approximation In principle, inﬁnite–dimensional optimization problems of the form (15) can be (approximately) converted to a ﬁnite dimensional optimization by using a ﬁnite exm pansion X(ρ, t) = i=1 Xi (t)hi (ρ), where hi (.) are known continuous functions ([1]). However, the computational complexity of the resulting problem grows exponentially with the horizon length N . In this chapter, we propose to avoid this difﬁculty by pursuing a stochastic approximation approach, whose complexity grows polynomially with N . To this effect, assume that ρˆ = [ρ(n + 1)ρ(n + 2) · · · ρ(n + N )] has a non–zero probability density for all ρ ∈ Fθ . Note that, to compute u(n), one only needs X[ρ(n)]. This observation allows for reformulating the optimization problem (15) to eliminate the need to explicitly compute X[ρ(n+1)]...X[ρ(n+N )] as follows: Given a ﬁxed instant n, ρ(n) and ρˆ ∈ Fθ , deﬁne

⎡

−γ xT (n)

⎤

⎢ ⎢ x(n) −X(n) ⎢ . ⎢ M [ρ(n)] M (n) = ⎢ ⎢ .. ⎢ . ⎣

⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦ M [ρ(n + N − 1)]

In terms of M , the constraints in the optimization problem (15) can be expressed as: . λ (γ, X(n), ρ(n), ρˆ) = 6

min

X(n+1),...X(n+N )

5 6 λmax M (n) < 0.

Such a solution can be found using risk–adjusted methods following the approach used in [7].

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Thus, in this context, one does not need to compute the explicit value of X(i), i = n + 1, . . . , n + N , as long as the minimum above can be computed. Note in passing, that this reformulation preserves convexity, since the function λ(·) is a convex function of its arguments. To complete the approximation of the original optimization problem by a stochastic one, given ζ > 0, deﬁne g˜(x) =

eζx − 1 ζ

and collect all the optimization variables (e.g., γ and the entries of X(n)) in a vector xn . Deﬁne the following functions . f0 (xn , ρ(n), ρˆ) = γ; . f1 (xn , ρ(n), ρˆ) = g˜ [λ (γ, X(n), ρ(n), ρˆ)] . Consider now the following convex problem min Eρˆ [f0 (xn , ρ(n), ρˆ)] = min γ s.t. Eρˆ [f1 (xn , ρ(n), ρˆ)] ≤ 0;

(19)

where Eρˆ[·] denotes the expected value with respect to the random variable ρˆ. It can be easily shown that the solution to this problem tends to the solution of the problem (15) as ζ → ∞. We are now ready to provide the main result of this section: an algorithm for solving problem (19) in polynomial time. For technical reasons, in the sequel we will assume that the solution to this problem is known to belong to a given compact convex set X (where the matrix X(n) has bounded entries and is positive deﬁnite). Let π(·) denote the projection onto X ; i.e., ˜ 2 . π(x) = arg min x − x ˜ ∈X x

and consider the following algorithm: Algorithm 7 1. Generate a feasible solution Y (ρ) to (5), using for instance the procedure proposed in [7]. 2. Initialization: Determine x0n , yi0 ; i = 0, 1 and z10 . Let k = 0. 3. Generate a sample ρˆk = [ρk (n + 1), . . . , ρk (n + N − 1)]

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4. Let If z1k ≤ −γ0 /k γ then 6 5 k k = π x − b y xk+1 k n n 0 . 6 5 k k = π x − b y Otherwise, xk+1 k n n 1 .. $

5. Let yik+1 = yik + ak

% ∂fi (xn , ρ(n), ρˆk ) − yik ∂xn xn =xk n

i = 0, 1 and

! z1k+1 = z1k + ak f1 (xn , ρ(n), ρˆk ) − z1k ;

6. If z1l < 0 and |γk−1 − γk | < for l = k − Ngood + 1, . . . , k + 1 stop. Otherwise, let k = k + 1 and go to step 3. Theorem 3 Let ak =

α0 kα ;

bk =

β0 , kβ

where α0 , α, β0 and β are positive constants.

Furthermore, assume that γ0 and γ are also positive. Then, if ∞

ak =

k=0

∞

bk = ∞;

k=0

∞

bk =0 k→∞ ak

a2k < ∞; lim

k=0

and 2β − α − 2γ > 1 the sequence xkn converges with probability one to the solution of the problem (19). Proof: Direct application of Theorem 1 in [8].

3.3 Adding Control Constraints In this section we brieﬂy indicate how to modify the algorithm presented above to handle constraints in the control action. A difﬁculty here is that, while it is relatively easy to incorporate these constraints when searching for a control action in the minimization of (3), this is not the case when dealing instead with closed–loop control strategies of the form (16). Following [17], we propose to accomplished this indirectly, by suitably scaling some of the terms in (15). Speciﬁcally, assume that the control constraint set is of the form Su {u : u ∞ ≤ 1} and consider the following control law:

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Algorithm 8 0.- Data: A horizon N , an initial condition x(0) = xo 0.1- Find a CLF Y (ρ, τ, x) that satisﬁes the following (functional) LMI, parametric in ρ, x and the scalar τ: ⎡

A12

AT12

−Y (θ, τ, x)

0

AT13

0

−τ · I

B2T (ρ)

0

0

⎢ ⎢ ⎢ ⎣

max ρ∈P

⎤

−Y (ρ, τ, x)

θ ∈ Θ(ρ)

A13 B2 (ρ)

⎥ ⎥ ⎥≤0 0 ⎦ 0

(20)

−τ · I

A12 = Y (ρ, τ, x)AT (ρ) − B2 (ρ)B2T (ρ) A13 = Y (ρ, τ, x)C1T (ρ) 1 xT 1 bTi ≥ 0, ≥ 0, i = 1, nu x Y (ρ, τ, x) bi Y (ρ, τ, x)

(21)

where bi denotes the ith column of B2 (ρ). 0.2- Let τmin (ρ, xo ) = inf {τ : (20)–(21) are feasible} with x = xo and deﬁne τo = maxρ∈P τmin (ρ, xo ). 1.- Let ρ(n) denote the measured value of the parameter at time n and solve the following LMI optimization problem in X ∈ XN and the scalar γ: min γ, subject to:

X ∈XN

−γ xT (n) x(n) −X(n)

1

≤0

(22)

τ(n + i) ≤ τo (xo )

T

x (n)

≥ 0,

x(n) X(n)

bTj (n

1

+ i)

≥ 0;

bj (n + i) X(n + i)

(23)

j = 1, . . . , nu , i = 0, 1, . . . , N − 1 ⎡

−X(n + i)

A12

A13

B2 (n + i)

AT12

−X(n + i + 1)

0

0

AT13

0

−τ(n + i) · I

0

B2T (n + i)

0

0

−τ(n + i) · I

⎢ ⎢ ⎢ ⎣

⎤ ⎥ ⎥ ⎥≤0 ⎦

(24)

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for all ρ(n + i + 1) ∈ Θ[ρ(n + i)], with boundary condition X(n + N ) = Y [ρ(n + N ), τ(n + N ), x(n + N )], where A12 = X(n + i)AT (n + i) − B2 (n + i)B2T (n + i) A13 = X(n + i)C1T (n + i) and i = 1, 2, . . . , N and where Y (.) satisﬁes the AMI (20). 2.- At time n use as control action u(n, N ) = −B2T [ρ(n)]X −1 [ρ(n), n]x(n)

(25)

3.- Set n = n + 1 and go to step 1. Theorem 4 Assume that the origin is an exponentially stable equilibrium point of A[ρ(.)] for all ρ ∈ FΘ and the pair {A(ρ), C(ρ)} is uniformly observable for all admissible parameter trajectories. Then, the control law (25) (i) is admissible, in the sense that it satisﬁes the control constraints, and (ii) it renders the origin a globally exponentially stable point of the closed loop system for all admissible parameter trajectories. Proof. See the Appendix. Remark 3 If the open loop system is not exponentially stable but the pairs (A, B2 ) and (A, C1 ) are uniformly controllable and observable respectively, the algorithm above will locally stabilize the system in some neighborhood of the origin, which 7 can be estimated by computing the set S = ρ∈P S(ρ), where S(ρ) = x : lim xT X −1 (ρ, τ)x < ∞ τ→∞

Remark 4 As before, the computational complexity of the optimization problem above grows exponentially with the horizon, even when approximating X by a ﬁnite expansion. However, the stochastic approach used in Algorithm 7 can also be used here, with minimal modiﬁcations, to obtain polynomial growth approximations.

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4 Illustrative Example Consider the following discrete time LPV system: 1 0.1 A(ρ(t)) = , B2 (ρ(t)) = 0 0.0787 0 1 − 0.1ρ(k) C1 (ρ(t)) =

1 , D12 (ρ(t)) = √ 1 1 2 −1 0 1 0

with admissible parameter set FΘ = {ρ(t) : ρ(t + 1) ∈ [−1, 1], t = 0, 1, . . .} .

(26)

It can be veriﬁed that the matrix function: Y (ρk ) = Y0 + Y1 ρk + Y2 ρk 2 0.0296 −0.0196 Y0 = −0.0196 0.0290 0.0003 −0.0016 Y1 = −0.0016 0.0044 −0.0691 0.0302 Y2 = 10−3 0.0302 0.2542 satisﬁes(5). Thus V (x, ρ) = xT Y −1 (ρ)x is a parameter dependent CLF for the plant. Figure 1 shows a typical parameter trajectory and the corresponding cost evolution.

In this case the initial condition was x0 = [0.1 0.1] , and the following values were used for the risk adjusted RH controller: N = 10, ζ = 30, α = 0.6, α0 = 1, β = 1, β0 = 10−3 , γ = 0.15 and γ0 = 10−6 . At each time instant, an approximate solution to (15) was obtained using 100 iterations of the stochastic approximation. On a Intel Pentium III 1.6GHz, the total CPU time to perform this iteration is 81.88 seconds. As reference, solving the parametric LMI (5) takes 474.44 seconds, although this needs to be done only once, off–line. The overall cost of the trajectory is J = 1.0886. For comparison, the standard LPV controller (6) results in a cost J = 1.2805. Thus, in this case the risk adjusted controller yields roughly a 20% performance improvement. Similar results were obtained for other initial conditions and parameter trajectories.

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Fig. 1. Top: Parameter Trajectory. Bottom: Cost Evolution

5 Conclusions Practical tools for synthesizing controllers for LPV systems have emerged relatively recently and are still far from complete. Among others, issues not completely solved yet include non–conservative handling of performance speciﬁcations and overall computational complexity.

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In this chapter we take some steps towards removing these limitations by combining Receding Horizon and Control Lyapunov Function ideas in a risk–adjusted framework. Motivated by some earlier results on regulation of LTV and LPV systems ([18, 16]) the main idea of the chapter is to recast the inﬁnite horizon regulation problem into a (approximately) equivalent ﬁnite horizon form, by searching over a suitable set of closed–loop strategies. As shown here, this leads to a globally stabilizing control law that is guaranteed to outperform techniques currently used to deal with LPV systems. However, in principle this is achieved at the expense of computational complexity, since this law requires the on–line solution of a set of functional LMIs. We propose to address this difﬁculty by using a risk–adjusted approach, where in exchange for a slight probability of constraint violation, one obtains a substantial reduction in computational complexity. Moreover, this approach scales polynomially, rather than exponentially, with system size ([10, 19]). These results were illustrated with a simple example where a risk-adjusted receding horizon controller was used to control a second order LPV plant. As shown there the proposed risk-adjusted receding horizon controller improves performance vis-a-vis a conventional LPV controller, while substantially reducing the computational effort required by a comparable Receding Horizon controller. Research is currently under way seeking to extend these results to the output feedback case and to address the issue of model uncertainty, both parametric and dynamic.

Acknowledgements This research was supported in part by NSF under grants ANI-0125653, ECS-9984260, ECS-0115946, ECS-0221562, and IIS-0312558.

References 1. G. BALAS et al. (1997). Lecture Notes, Workshop on Theory and Application of Linear Parameter Varying Techniques, 1997 ACC. 2. B. R. Barmish and C. M. Lagoa (1997). “The Uniform Distribution: A Rigorous Justiﬁcation for its Use in Robustness Analysis”, Mathematics of Control, Signals and Systems,10, 203–222.

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3. I. Batina, A. A. Stoorvogel and S. Weiland (2001). “Stochastic Disturbance Rejection in Model Predictive Control by Randomized Algorithms,” Proc. 2001 ACC, Arlington, Va, pp. 732–737. 4. R. E. Bellman and S. E. Dreyfus, (1962). Applied Dynamic Programming, Princeton University Press. 5. G. Calaﬁore, F. Dabbene and R. Tempo (2000). “Randomized Algorithms for Probabilistic Robustness with Real and Complex Structured Uncertainty,” IEEE Trans. on Autom. Contr., 45,12, 2218–2235. 6. X. Chen and K. Zhou (1997). “ A Probabilistic Approach to Robust Control”, Proc. 36 IEEE CDC, pp. 4894–4895. 7. Y. Fujisaki, F. Dabbene and R. Tempo (2001) “Probabilistic Robust Design of LPV Control Systems,” Proc. 2001 IEEE CDC, Orlando, FL, pp. 2019–2024. 8. A. M. Gupal (1974). “A Certain Stochastic Programming Problem with Constraints of a Probabilistic Nature,” Kibernetika (Kiev), n. 6, pp. 94–100. 9. A. Jadbabaie, J. Yu and J. Hauser (1999). “Receding horizon control of the Caltech ducted fan: a control Lyapunov function approach,” Proc. 1999 IEEE CCA, pp. 51–56,. 10. P. Khargonekar and A. Tikku (1996). “Randomized Algorithms for Robust Control Analysis and Synthesis Have Polynomial Complexity”, Proceedings of the 35th IEEE CDC, pp. 3470–3475. 11. M. Kothare, V. Balakrishnan and M. Morari (1996). “Robust Constrained Model Predictive Control Using Linear Matrix Inequalities,” Automatica, 32, 10, pp. 1361–1379. 12. D. Q. Mayne, J. B. Rawlings, C.V. Rao and P. O. M. Scokaert (2000). “Constrained Model Predictive Control: Stability and Optimality,” Automatica, 36, 6, pp. 789–814. 13. L. Ray and R. Stengel (1993). “A Monte Carlo Approach to the Analysis of Control System Robustness”, Automatica,29,1, 229–236. 14. B. T. Polyak and R. Tempo (2000). “Probabilistic Robust Design with Linear Quadratic Regulators,” Proc. 39 IEEE CDC, pp. 1037–1042. 15. J. S. Shamma and M. Athans (1992). “Gain Scheduling: Potential Hazards and Possible Remedies,” IEEE. Contr. Sys. Mag., 12, 1, pp. 101–17. 16. M. Sznaier (1999). “Receding Horizon: An easy way to improve performance in LPV systems,” Proc. 1999 ACC,” pp. 2257–2261. 17. M. Sznaier and R. Suarez (2000). “Control of Constrained Systems using Receding Horizon Control Lyapunov Functions,” Workshop on Systems with Time–Domain Constraints, Eindhoven University of Technology, Eindoven, The Netherlands. 18. G. Tadmor (1992). “Receding horizon revisited: and easy way to robustly stabilize an LTV system,” Systems and Control Letters, 18, pp. 285–294. 19. R. Tempo, E. W. Bai, and F. Dabbene, F. (1996). “Probabilistic Robustness Analysis: Explicit Bounds for the Minimum Number of Samples,” Proc. 35th IEEE CDC, pp. 3424– 3428.

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A Computing Subgradient of λmax (M (x) ) In this section we describe how to compute a subgradient of λmax (M (x) ). To com˜, pute such a subgradient ﬁrst note that, given x λmax (M (˜ x) ) = max y T M (˜ x)y = y ∗ T M (˜ x)y ∗

y 2 =1

where y ∗ is an eigenvector of euclidean norm one associated with maximum eigenvalue of M (˜ x). Given that the maximum above is achieved by y ∗ then ∗T ∗ ∂λmax (M (x) ) ∂y M (x)y = . ∂x ∂x x=˜ x x=˜ x

This can be easily computed since, in the case addressed in this chapter, M (x) is an afﬁne function of x.

B Proof of Theorem 4 In order to prove this theorem we need the following preliminary result: Lemma 2 Assume that the origin is an exponentially stable equilibrium point of the open loop system A [ρ(t)] for all admissible parameter trajectories and the pair {A, C} is uniformly observable. Let Y [ρ(t), τ] denote a feasible solution to (20) corresponding to a given τ. Then, given τ1 > τ2 , (20) admits a solution Y [ρ(t), τ1 ] > Y [ρ(t), τ2 ]. Proof: Let δ = τ2 − τ1 . From the hypotesis it follows that for all admissible parameter trajectories ρ ∈ Fθ there exists some Δ [ρ(t)] > 0 that satisﬁes the inequality: ⎤ ⎡ −Δ [ρ(t)] Δ [ρ(t)] AT [ρ(t)] Δ [ρ(t)] C1T [ρ(t)] 0 ⎥ ⎢ ⎥ ⎢ A [ρ(t)] Δ [ρ(t)] −Δ[ρ(t + 1)] 0 0 ⎥<0 ⎢ ⎢C [ρ(t)] Δ [ρ(t)] 0 −δ · I 0 ⎥ ⎦ ⎣ 1 0 0 0 −δ · I

(27)

. Consider now any solution of (20) Y [ρ(t), τ2 ] and let Y [ρ(t), τ1 ] = Y [ρ(t), τ2 ]+ Δ [ρ(t)] > Y [ρ(t), τ2 ]. Adding (27) and (20) we have that Y [ρ(t), τ1 ] satisﬁes (20) for the value τ = τ1 .

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Proof of Theorem 4 Controllability of the pair{A(.), B(.)} implies that for some τ > 0 large enough, (20) has a solution Y > 0. From Lemma 2 it follows then that as τ → ∞, the LMI (20) always admits a solution Y → ∞, which in turn implies that, for any initial condition xo , there always exist some ﬁnite τ such that (21) is feasible. Since P is compact, τo in step [0.2] is well deﬁned. Consider now a ﬁxed value of τ ≤ τo and let v = x(t)T Y −1 [ρ(t)] x(t + 1)T Y −1 [ρ(t + 1)]

x(t)T C1T [ρ(t)] B2T [ρ(t)]Y −1 [ρ(t)]x(t) τ τ

(28) Pre/postmultiplying (20) by v and v T yields, after some algebra: T

x (t + 1)Y

−1

[ρ(t + 1), τ]x(t + 1) − x (t)Y T

−1

z(t) 2 [ρ(t)] x(t) ≤ − τ

(29)

From the deﬁnition of τo it follows that Y (ρ, τo ) satisﬁes (20) for all admissible parameter trajectories ρ ∈ FΘ . This fact, together with (29) implies that the LMIs(22)– (24) are feasible for all parameter trajectories, since X = Y is a feasible solution. A compactness argument shows that along the trajectories of the closed loop system c1 I > X(t) > c2 I for some constants c1 , c2. Finally, pre/postmultiplying (24) by x(t)T C1T (t) B2T (t)X −1 (t)x(t) T −1 T −1 v = x(t) X (t) x(t + 1) X (t + 1) τ τ and v T yields: xT (t + 1)X −1 (t + 1)x(t + 1) − xT (t)X −1 (t)x(t) ≤ −

z(t) 2 τ(t)

(30)

From this point on, the proof of asymptotic stability proceeds along the lines of the proof of Theorem 4 using (30) instead of (17). To show that the control law (25) satisﬁes the constraints |ui (.)| ≤ 1, note that |ui (n)| = | − bTi (n)X −1 (n)x(n)| ≤ bTi (n)X − 2 (n) 2 X − 2 (n)x(n) 2 ≤ 1 1

1

where the last inequality follows from the LMIs (23), coupled with the fact that, along the closed loop trajectories xT (n + i)X −1 x(n + i) ≤ xT (n)X −1 (n)x(n) ≤ 1 .

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Note in passing that from the proof above it follows that ∞ zT z 1 T ≤ xTo X −1 (xo , ρo , τo )xo z (t)z(t) ≤ τo t=0 τ(t) t

This gives an intuitive explanation of τ: it is a parameter used to scale the performance index so that the control constraints are satisﬁed.

Case Studies on the Control of Input-Constrained Linear Plants Via Output Feedback Containing an Internal Deadzone Loop Dan Dai1 , Tingshu Hu2 , Andrew R. Teel1 , and Luca Zaccarian3 1

Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA diane|[email protected]

2

Department of Electrical and Computer Engineering, University of Massachusetts Lowell, MA 01854, USA [email protected]

3

Dipartimento di Informatica, Sistemi e Produzione, University of Rome, Tor Vergata, 00133 Rome, Italy [email protected]

Summary. In this chapter we address several case studies using LMI optimization methods for designing output feedback control laws to achieve regional performance and stability of linear control systems with input saturation. Algorithms are developed for minimizing the upper bound on the regional L2 gain for exogenous inputs with L2 norm bounded by a given value, and for minimizing this upper bound with a guaranteed reachable set or domain of attraction. Based on the structure of the optimization problems, using the projection lemma, the output feedback controller synthesis is cast as a convex optimization over linear matrix inequalities. The problems are studied in a general setting where the only requirement on the linear plant is detectability and stabilizability.

keywords: Output feedback control, input saturation, L2 gain, reachable set, domain of attraction, LMIs

1 Introduction The behavior of linear, time-invariant (LTI) systems subject to actuator saturation has been extensively studied for several decades. More recently, some systematic

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design procedures based on rigorous theoretical analysis have been proposed through various frameworks. Most of the research efforts geared toward constructive linear or nonlinear control for saturated plants can be divided into two main strands. In the ﬁrst one, called anti-windup design, a pre-designed controller is given, so that its closed-loop with the plant without input saturation is well behaved (at least asymptotically stable but possibly inducing desirable unconstrained closed-loop performance). Given the pre-designed controller, anti-windup design addresses the controller augmentation problem aimed at maintaining the pre-designed controller behavior before saturation and introducing suitable modiﬁcations after saturation so that global (or regional) asymptotic stability is guaranteed (local asymptotic stability already holds by the properties of the saturation nonlinearity). Anti-windup research has been largely discussed and many constructive design algorithms have been formally proved to induce suitable stability and performance properties. Many of these constructive approaches (see, e.g., [4, 5, 8, 9, 10, 11, 12, 20, 24, 31]) rely on convex optimization techniques and provide Linear Matrix Inequalities (LMIs) [2] for the anti-windup compensator design. The second research strand, can be called “direct design”, to resemble the fact that saturation is directly accounted for in the controller design and that no speciﬁcation or constraint is imposed on the behavior of the closed-loop for small signals. Direct designs for saturated systems range from the well-known Model Predictive Control (MPC) techniques [21], especially suitable for discrete-time systems) to sophisticated nonlinear control laws which are able to guarantee global asymptotic stability for all linear saturated and globally stabilizable plants (see, e.g., the scheduled Riccati approach in [22] and the nested saturations of [26, 28]). Several LMI-based methods for direct controller design for linear plants with input saturation have also been proposed (see, e.g., [6, 19, 23, 25]). It is not our scope to mention here all the extensive literature on direct design for saturated systems, but it is worth mentioning that several constructive methods are available that differ in simplicity, effectiveness and formality. Compared to anti-windup control, direct design is a simpler task to accomplish, because there’s no constraint on the closed-loop behavior for small signals. Therefore, the designer has full freedom in the selection of the controller dynamics. Antiwindup design, on the other hand, allows to guarantee that a certain prescribed unconstrained performance is met by the closed-loop as long as the saturation limits are

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not exceeded (this performance often consists in some linear performance measure when a linear plant+controller pair is under consideration) and that this performance is gradually deteriorated as signals grow larger and larger outside the saturation limits. In this chapter, we will discuss and illustrate on several examples a synthesis method for the construction of output feedback controllers with an internal deadzone loop. This type of structure corresponds to the typical framework used since the 1980’s for the design of control systems for saturated plants. See for example the work in [29, 3, 30, 13, 27, 6] and other references in [1]. In our approach we will use the same tools used in our recent papers [17] for static anti-windup design, and we will recast the underlying optimization problem for the selection of all the controller matrices (whereas in [17] only the static anti-windup gain was selected and the underlying linear controller matrices were ﬁxed). This approach parallels the approach proposed in [23] where classical sector conditions were used and extra assumptions on the direct input-output link of the plant were enforced. A similar assumption was also made in the recent paper [7] which uses similar tools to ours to address both magnitude and rate saturation problems in a compensation scheme with lesser degrees of freedom than ours. Here we use the regional analysis tool adopted in [17], and we extend the general output feedback synthesis to characterize the regional L2 gain and reachable set for a class of norm bounded disturbance inputs, as well as the estimate of domain of attraction. The overall synthesis is cast as an optimization over LMIs, and under a detectability and stabilizability condition on the plant, the proposed design procedure will always lead to regionally stabilizing controllers if the plant is exponentially unstable, to semi-global results if the plant is non-exponentially unstable, and to global results if the plant is already exponentially stable. An interesting advantage of the approach proposed here is that due to the type of transformation that we use, it is possible to derive system theoretic interpretation of the feasibility conditions for the controller design (such as stabilizability and detectability of the pant). This result is novel and was not previously observed in [23]. This chapter is organized as follows: In Section 2 we formulate three problems that will be addressed in the chapter; in Section 3 we state the LMI-based main conditions for output feedback controller synthesis and the procedure for the controller construction; in Section 4 we give the feasible solutions for the problems we presented

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in Section 2; in Section 5 we illustrate the proposed constructions on several case studies. Notation For compact presentation of matrices, given a square matrix X we denote HeX := X + X T . For P = P T > 0, we denote E(P ) := {x : xT P x ≤ 1}.

2 Problem Statement Consider a linear saturated plant, ⎧ ⎪ ⎪ ⎨ x˙ p = Ap xp + Bp,u sat(yc ) + Bp,w w P y = Cp,y xp + Dp,yu sat(yc ) + Dp,yw w ⎪ ⎪ ⎩z = C x + D sat(y ) + D w p,z p

p,zu

c

(1)

p,zw

where xp ∈ np is the plant state, yc ∈ nu is the control input subject to saturation, w ∈ nw is the exogenous input (possibly containing disturbance, reference and measurement noise), y ∈ ny is the measurement output and z ∈ nz is the performance output. The goal of this chapter is the synthesis of a plant-order linear output feedback controller with internal deadzone loops: x˙ c = Ac xc + Bc y + E1 dz(yc ) C yc = Cc xc + Dc y + E2 dz(yc ),

(2)

where xc ∈ nc (with nc = np ) is the controller state, yc ∈ nu is the controller output and dz(·) : nu → nu is the deadzone function deﬁned as dz(yc ) := yc − sat(yc ) for all yc ∈ nu with sat(·) : nu → nu being the symmetric saturation function having saturation levels u ¯1 , . . . , u ¯nu with its i-th component satu¯i (·) depending on the i-th input component yci as follows: ⎧ ⎪ ¯i , if yci ≥ u ¯i , ⎪ ⎨u satu¯i (yci ) = yci , if − u ¯i ≤ yci ≤ u ¯i , ⎪ ⎪ ⎩ −¯ u , if y ≤ −¯ u. i

ci

(3)

i

The resulting nonlinear closed-loop (1), (2), is depicted in Figure 1. The same output feedback controller structure was considered in [23], where convex synthesis methods for global (rather than regional, as we consider here) stability and

Output Feedback for Input-Constrained Linear Plants w C

yc

317

z P

y

Enhanced Controller

Fig. 1. The linear output feedback control system with deadzone loops.

performance were developed. In [23], it was assumed for simplicity that Dp,yu = 0 (we will remove this assumption here). It is well known that linear saturated plants are characterized by weak stabilizability conditions. In particular, since by linearity the controller authority becomes almost zero for arbitrarily large signals, then global asymptotic stability can only be guaranteed for plants that are not exponentially unstable, while global exponential stability can only be guaranteed if the plant is already exponentially stable. Due to this fact, global results are never achievable (not even with a nonlinear controller) when wanting to exponentially stabilize plants that are not already exponentially stable. On the other hand, local and regional results are always achievable and semiglobal ones are achievable with non-exponentially unstable plants. The following three regional properties are then relevant for the controller design addressed here: Property 1. Given a set Sp ⊂ np , the plant (1) is Sp -regionally exponentially stabilized by controller (2) if the origin of the closed-loop system (1), (2) is exponentially stable with domain of attraction including Sp × Sc (where Sc ⊂ nc is a suitable set including the origin). Property 2. Given a set Rp ⊂ np and a number s > 0, controller (2) guarantees (s, Rp )-reachability for the plant (1) if the response (xp (t), xc (t)), t ≥ 0 of the closed-loop system (1), (2) starting from the equilibrium point (xp (0), xc (0)) = (0, 0) and with w 2 < s, satisﬁes xp (t) ∈ Rp for all t ≥ 0. Property 3. Given two numbers s, γ > 0, controller (2) guarantees (s, γ)-regional ﬁnite L2 gain for the plant (1) if the performance output response z(t), t ≥ 0 of the closed-loop system (1), (2) starting from the equilibrium point (xp (0), xc (0)) = (0, 0) and with w 2 < s, satisﬁes z 2 < γ w 2 .

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Based on the three properties introduced above, in this paper we are interested in providing LMI-based design tools for the synthesis of an output feedback controller of the form (2) guaranteeing suitable stability, reachability and L2 gain properties on the corresponding closed-loop. In particular, we will address the following problems: Problem 1. Consider the linear plant (1), a bound s on w 2 , a desired reachability region Rp and a bound γ on the desired regional L2 gain. Design a linear output feedback controller (2) guaranteeing (s, Rp ) reachability, (s, γ)-regional ﬁnite L2 gain and which maximizes the exponential stability region Sp of the closed-loop (1), (2). Problem 2. Consider the linear plant (1), a bound s on w 2 , a desired stability region Sp and a bound γ on the desired regional L2 gain. Design a linear output feedback controller (2) guaranteeing Sp regional exponential stability, (s, γ)-regional ﬁnite L2 gain and which minimizes the (s, Rp ) reachability region of the closed-loop (1), (2). Problem 3. Consider the linear plant (1), a bound s on w 2 , a desired stability region Sp and a desired reachability region Rp . Design a linear output feedback controller (2) guaranteeing Sp regional exponential stability, (s, Rp ) reachability and which minimizes the (s, γ)-regional ﬁnite L2 gain of the closed-loop (1), (2).

3 LMI-Based Design In this section, a set of main feasibility conditions for solving Problems 1 to 3 will be presented in addition to giving a constructive procedure to design a state space representation of the linear output feedback controller (2).

3.1 Main Feasibility Theorem The results that we will derive are based on the sector description of the deadzone originally developed in [15, 14]. The main idea of the description is as follows. For a scalar saturation function satu¯ (·), if |v| ≤ u ¯ then satu¯ (u) is between u and v for all u ∈ . Applying this description to deal with the saturating actuator in Figure 1, we will have satu¯i (yci ) between yci and Hi x as long as |Hi x| ≤ u ¯i . Here x is the

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319

combined state in Figure 1 and Hi can be any row vector of appropriate dimensions. It turns out that the choice of Hi can be incorporated into LMI optimization problems. It should be noted that [6] and [9] exploit the same idea to deal with saturations and deadzones, however in our approach yc and x are related to each other in a more general way, rather than yc = F x, as in [6, 9]. The following theorem will be used in the following sections to provide solutions to the problem statements given in Section 2. Theorem 1. Consider the linear plant (1). If the following LMI conditions in the T unknowns Q11 = QT11 > 0, P11 = P11 > 0, γ2 > 0, Yp ∈ nu ×np , K1 ∈ np ×npy ,

K2 ∈ npy ×npy , K3 ∈ npz ×npy are feasible: ⎡ ⎤ A Q + Bpu Yp 0 Bpw ⎢ p 11 ⎥ γ2 I ⎥<0 He⎢ C Q + D Y − D pz 11 p,zu p p,zw ⎣ ⎦ 2 ⎡ ⎢ He⎢ ⎣

0

0

− I2

P11 Ap + K1 Cpy P11 Bpw + K1 Dp,yw K2 Cpy

Cpz + K3 Cpy Q11 I > 0. I P11 2 2 u ¯i /s Ypi ≥ 0, YpiT Q11

(4a)

K2 Dp,yw − I/2

⎤ 0 0

Dp,zw + K3 Dp,yw −γ2 I/2

⎥ ⎥<0 ⎦

(4b)

(4c)

i = 1, . . . , nu ,

(4d)

(where Ypi denotes the ith row of Yp ), then there exists an output feedback controller of the form (2) and of oder np which guarantees the following three properties for the closed-loop: 1. (s, γ) regional ﬁnite L2 gain; 2. Sp regional exponential stability with Sp = E((s2 Q11 )−1 ); 3. (s, Rp ) reachability region with Rp = E((s2 Q11 )−1 ). Moreover, given any feasible solution to the LMI constraints (4), a state-space representation of a controller guaranteeing these properties can be determined based on the matrices (Q11 , P11 ) by way of the Procedure 1 reported next.

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Remark 1. The LMI conditions in (4) have a system theoretic interpretation: 1) The ﬁrst condition (4a) corresponds to a strengthened stabilizability condition for the plant (1). Indeed, substituting Yp = Kp Q11 , (4a) corresponds to the LMI formulation of the bounded real lemma (see, e.g., [10]) characterizing the L2 gain from w to z for the plant controlled by a state feedback law u = Kp xp . Therefore, (4a) constrains γ to be not smaller than the L2 gain of the plant stabilized by static state feedback. Note however that the corresponding state feedback gain Kp is further constrained by (4d), so that the open-loop plant L2 gain may be only reduced to a certain extent when larger values of s make the constraint (4d) tighter. As s approaches +∞, Yp will approach 0 and the constraint on γ enforced by (4a) will approach the global constraint corresponding to the L2 gain of the open-loop plant. 2) The second condition (4b) corresponds to a strengthened detectability condition for the plant. Indeed, if we had K2 = 0 in the condition above, then the corre−1 sponding equation would mean that (if we take L = P11 K1 ) there exist L and

K3 such that the observer with unknown input ˜ + L(y − y˜) x ˜˙ = Ap x ˜ y˜ = Cyp x ˜ − K3 (y − y˜), z˜ = Cpz x for the system x˙ = Ap x + Bpw w y = Cyp x + Dp,yw w z = Cpz x + Dp,zw w guarantees gain γ from w to the output observation error (z − z˜). The fact above can be checked writing down the error dynamics e = x − x ˜ and imposing that there exists a disturbance attenuation Lyapunov function V = eT P11 e. In particular, the LMI (4b) corresponds to imposing that: 2eT P11 (Ap + LCpy )e + 2eT P11 (Bp,w + LDp,yw )w + where ze = z − z˜.

1 T ze ze − wT w < 0, 2 γ

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321

3.2 Controller Construction We provide next a constructive algorithm for determining the matrices of the linear controller whose existence is established in Theorem 1. Procedure 1 (Output feedback construction) Step 1. Solve the feasibility LMIs. Find a solution (Q11 , P11 , K1 , K2 , K3 , Yp , γ) to the feasibility LMIs conditions (4). Step 2. Construct the matrix Q. (see also [11, 16].) Deﬁne the matrices Q11 ∈ np ×np and Q12 ∈ np ×nc , with nc = np as a solution to the following equation: Q11 P11 Q11 − Q11 = Q12 QT12 .

(5)

Since Q11 and P11 are invertible and Q−1 11 < P11 by the feasibility conditions, then Q11 P11 Q11 − Q11 is positive deﬁnite. Hence there always exists 4 a matrix Q12 satisfying equation (5). Deﬁne the matrix Q22 ∈ nc ×nc as Q22 := I + QT12 Q−1 11 Q12 .

(6)

Finally, deﬁne the matrix Q ∈ n×n (n = np + nc ) as Q11 Q12 Q := . QT12 Q22

(7)

Step 3. Controller synthesis LMI. Construct the matrices Ψ ∈ n¯ ×¯n , H

∈

(nc +nu )×¯n , G ∈ (nc +nu +ny )×¯n , and T ∈ n¯ ×¯n (¯ n = np +nc +nu +nw +nz ) as follows: ⎡

Ap Q11 Ap Q12 −Bpu U

⎤ Bpw

0

⎢ ⎥ ⎢ ⎥ 0 0 0 0 0 ⎢ ⎥ ⎢ ⎥ Ψ = ⎢ −Yp −Yc −U 0 0 ⎥, ⎥ ⎢ ⎥ ⎢ 0 0 0 −I/2 0 ⎦ ⎣ 2 Cpz Q11 Cpz Q12 −Dp,zu U Dp,zw −γ I/2 4

Note that equation (5) always admits inﬁnite solutions, parametrizing an inﬁnite number of compensators inducing the same closed-loop performance. Understanding how to exploit this degree of freedom for the selection of a most desirable compensator is subject of future research.

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⎡

⎤T

0 Bpu ⎢ ⎥ ⎢I 0 ⎥ ⎢ ⎥ ⎢ ⎥ H = ⎢0 I ⎥ , ⎢ ⎥ ⎢0 0 ⎥ ⎣ ⎦ 0 Dp,zu

⎡

⎤

0 I0 0 0 ⎢ ⎥ ⎥, G=⎢ C 0 0 D 0 py p,yw ⎣ ⎦ 0 0I 0 0

T = diag{Q, I, I, I} where Yc ∈ nu ×np is deﬁned as Yc := Yp (Q11 )−1 Q12 . Then, deﬁne the un¯ U ∈ (nc +nu )×(nc +nu +ny ) as: known variable Ω ¯ U := Ω

A¯c C¯c

¯c B ¯c D

¯1 U E ¯2 U E

(8)

Finally, solve the output feedback controller LMI: ¯ U GT ) < 0 M = He(Ψ + Ψ1 ) = He(Ψ + H T Ω

(9)

¯ U , and U ∈ nu ×nu , U > 0 diagonal. in the unknowns Ω ¯ U in Step 4. Computation of the controller matrices. From the matrices U and Ω ¯ as Step 3, compute the matrix Ω ¯ := Ω

A¯c C¯c

¯c B ¯c D

¯1 E ¯2 E

¯ U diag(I, I, U −1 ). := Ω Finally, the controller parameters in Ω can be determined applying the following transformation: Ω :=

Ac Bc E1

Cc Dc E2 ⎡ ⎤ I 0 0 ⎢ ⎥ ¯ ⎢ −XD ¯ p,yu C¯c X ¯ XD ¯ p,yu (I − E ¯2 ) ⎥ = Ω ⎣ ⎦ 0

¯ c )−1 . ¯ := (I + Dp,yu D where X

0

I

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4 Feasibility of the Output Feedback Synthesis Problems In this section, the feasibility conditions established in Theorem 1 will be used to provide conditions for the solvability of Problems 1 to 3.

4.1 Feasibility of and Solution to Problem 1 We ﬁrst use the result of Theorem 1 to give a solution to Problem 1. To this aim, we use the guaranteed L2 performance and reachability region of items 1 and 3 of Theorem 1 and maximize the size of the guaranteed stability region by maximizing the size of the ellipsoid E((s2 Q11 )−1 ) which, according to item 2 of Theorem 1 is an estimate of the domain of attraction. We state the corollary below for a generic measure αR (·) of the size of the ellipsoid E((s2 Q11 )−1 ). This is typically done with respect to some shape reference of the desired stability region Sp . Corollary 1. Given s, Rp and γ, a solution to Problem 1 is given (whenever feasible) by applying Procedure 1 to the optimal solution of the following maximization problem: sup Q11 ,P11 ,K1 ,K2 ,K3 ,Yp

,γ2

αR (E((s2 Q11 )−1 )), subject to

(4a), (4b), (4c), (4d),

(10a)

2 E(Q−1 11 /s ) ⊂ Rp

(10b)

The formulation in Corollary 1 can be easily particularized to the problem of maxi2 2 −1 mizing the volume of E(Q−1 )) = det(s2 Q11 ). 11 /s ) by selecting αR (E((s Q11 )

Alternative easier selections of αR can correspond to maximizing the size of a region which has a predeﬁned shape. For example, when focusing on ellipsoids, one can seek for stability regions of the type Sp = E(Sp−1 ) = {xp : xTp (αSp )−1 xp ≤ 1},

(11)

where α is a positive scalar such that larger values of α correspond to larger sets Sp . Then the optimization problem (10) can be cast as

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α, subject to

sup Q11 ,P11 ,K1 ,K2 ,K3 ,Yp ,γ

(4a), (4b), (4c), (4d), (10b)

(12a)

αSp ≤ s2 Q11 .

(12b)

Similarly, if one takes a polyhedral reference region: Sp = αco{x1 , x2 , ..., xnp },

(13)

then constraint (12b) can be replaced by 2 xpi T Q−1 11 xpi ≤ αs ,

or

αs2 xTpi xpi Q11

(14)

≥ 0,

i = 1, . . . , np .

(15)

We ﬁnally note that the constraint (10b) on the guaranteed reachability region can be expressed by way of different convex (possibly LMI) conditions depending on the shape of the set Rp . Guidelines in this direction are given in the following section. Remark 2. Based on Corollary 1, reduced LMI conditions can be written to only maximize the estimate of the domain of attraction without any constraint on the other performance measures: sup

αR (E((s2 Q11 )−1 )), subject to

Q11 ,P11 ,K1 ,Yp

He[Ap Q11 + Bpu Yp ] < 0

(16a)

He[P11 Ap + K1 Cpy ] < 0

(16b)

(4c), (4d)

(16c)

From (16) it is straightforward to conclude that if the plant is exponentially stable, then global exponential stability and ﬁnite L2 gain can be achieved by the proposed output feedback controller (Yp = 0 and K1 = 0 are sufﬁcient). If the plant is not exponentially unstable, semiglobal results are obtainable. Regional results can always be obtained in the general case and the size of the maximal feasible domain of attraction depends on the particular problem.

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4.2 Feasibility of and Solution to Problem 2 We now use the result of Theorem 1 to give a solution to Problem 2 following similar steps as the ones in the previous section. When focusing on reachable sets, smaller estimates are desirable, so that there’s a guaranteed bound on the size of the state when the system is disturbed by external inputs. Since by Theorem 1 the reachability region estimate coincides with the estimate of the domain of attraction, the goal addressed in Problem 2 is in contrast with the goal addressed in the previous section. The corresponding equivalent to Corollary 1 is the following (where αR (·) is a measure of the size of the ellipsoid E((s2 Q11 )−1 )): Corollary 2. Given s, Sp and γ, a solution to optimal stability region Problem 2 is given (whenever feasible) by applying Procedure 1 to the optimal solution to the following minimization problem: min

Q11 ,P11 ,K1 ,K2 ,K3 ,Yp ,γ2

αR (E((s2 Q11 )−1 ), subject to

(4a), (4b), (4c), (4d),

(17a)

Sp ⊂ E((s2 Q11 )−1 )

(17b)

Similar to the previous section, the volume of the reachability set can be minimized by selecting αR (E((s2 Q11 )−1 )) = det(s2 Q11 ) in (17). Alternative easier selections of αR correspond to focusing on ellipsoids when choosing Rp = E(αRp−1 ) = {xp : xTp (αRp )−1 xp ≤ 1},

(18)

where Rp = RpT > 0, then the optimization problem (17) becomes min

Q11 ,P11 ,K1 ,K2 ,K3 ,Yp ,γ2

α, subject to

(4a), (4b), (4c), (4d), (17b)

(19a)

s2 Q11 ≤ αRp

(19b)

Similarly, Rp can be selected as the following unbounded set: Rp (α) = {xp : |Cxp | ≤ α},

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where C ∈ 1×np is a given row vector. Then E((s2 Q11 )−1 ) ⊂ Rp (α) if and only if CQ11 C T ≤ α2 /s2 . If both (19a) and CQ11 C T ≤ α2 /s2 are enforced to hold in the LMI optimization, then it follows that |Cxp (t)| ≤ α for all t if w 2 ≤ s. Therefore, if our objective is to minimize the size of a particular output Cxp , we may formulate the following optimization problem: min

Q11 ,P11 ,K1 ,K2 ,K3 ,Yp ,α

α, subject to

(4a), (4b), (4c), (4d), (17b)

(20a)

CQ11 C T < α2 /s2 ,

(20b)

where the guaranteed L2 gain is be incorporated in (4a) and (4b) and the guaranteed stability set is incorporated in (17b). Remark 3. If there is no interest for a guaranteed L2 gain, then the problem to minimize the desirable reachable set Rp can be simpliﬁed to the feasibility of the LMIs in (4) and (17b), by removing the second block row and the second block column of the matrices in (4a) and the third block ones in (4b).

4.3 Feasibility of and Solution to Problem 3 Similar to what has been done in the previous two sections with reference to Problems 1 and 2, we use here Theorem 1 to give a solution to Problem 3. Corollary 3. Given s, Rp and Sp , a solution to Problem 3 is given (whenever feasible) by applying Procedure 1 to the optimal solution to the following minimization problem: min

Q11 ,P11 ,K1 ,K2 ,K3 ,Yp

,γ2

γ2 , subject to

(4a), (4b), (4c), (4d),

(21a)

Sp ⊂ E((s2 Q11 )−1 ) ⊂ Rp

(21b)

If we only focus on ellipsoidal reachability and stability sets so that for two given matrices Sp = SpT > 0 and Rp = RpT > 0 Sp := E(Sp−1 ) and Rp := E(Rp−1 ), then the optimization problem (21) can be cast as the following convex formulation:

Output Feedback for Input-Constrained Linear Plants

min

Q11 ,P11 ,K1 ,K2 ,K3 ,Yp ,γ2

327

γ2 , subject to

(4a), (4b), (4c), (4d),

(22a)

Sp ≤ s2 Q11 ≤ Rp

(22b)

Alternative shapes for the guaranteed reachability set Rp and for the guaranteed stability region Sp can be selected by following the indications given in the previous two sections.

5 Examples In this section we illustrate the proposed controller constructions on several case studies. For the different control problems, several examples are organized as follows: in Section 5.1 we address the disturbance rejection problem; in Section 5.2 some cases of set point regulation are studied; in Section 5.3 both the disturbance rejection and the reference tracking problems will be addressed in an exponentially unstable example. 5.1 Disturbance Rejection Example 1. We consider the system used in [10], which has one control input, one disturbance input, four states and one measurement output. The plant state is xp = T ˙ p p˙ θ θ , where p is the horizontal displacement of the cart and θ is the angle of the pendulum. The plant parameters are given by: ⎡ ⎤ 0 1 0 0 0 0 ⎢ ⎥ ⎢ −330.46 −12.15 −2.44 0 2.71762 0 ⎥ ⎥ ⎡ ⎤ ⎢ ⎢ ⎥ 0 0 0 1 0 0 ⎥ Ap Bp,u Bp,w ⎢ ⎥ ⎥ ⎢ ⎢ ⎢ Cp,y Dp,yu Dp,yw ⎥ = ⎢ −812.61 −29.87 −30.10 0 6.68268 15.61 ⎥ ⎥ ⎣ ⎦ ⎢ ⎢ ⎥ ⎢ Cp,z Dp,zu Dp,zw 1 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 1 0 0 0 ⎣ ⎦ 0 0 1 0 0 0 For each s > 0, the achievable L2 gain by a plant order output feedback can be determined with the algorithm based on Theorem 1. If choosing different values of s over (0, ∞), the achievable performance can be obtained as a function of s.

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The solid line in Figure 2 reports the achievable L2 gain by a suitable plant order output feedback controller, as a function of s. For comparison purposes, we report in the same ﬁgure (dashed line) the L2 gain achievable by a dynamic anti-windup compensation when using a speciﬁc unconstrained controller (see [16] for details).

200 180 160 140

γ

120 100 80 60 40 20 0 −3 10

−2

10

−1

10

0

10 ||w||

1

10

2

10

3

10

2

Fig. 2. Achievable nonlinear L2 gains. Proposed output feedback (thin solid); Dynamic antiwindup (dashed)

In [10], when the cart-spring-pendulum system is subject to the larger pendulum tap, which is modeled as a constant force of 7.94N with duration 0.01s, the closed-loop response with an LQG controller exhibits undesirable oscillations if the control input is constrained in the range of the D/A converter: [-5, +5] Volts. In [10] a dynamic anti-windup compensator is used to preserve the local LQG behavior and improve the response after saturation. We compare that result to our direct design. In particular, we use Procedure 1 to construct an output feedback controller by ﬁxing s = 0.14. The corresponding optimal L2 gain is γ = 2.26 and the controller matrices are

Output Feedback for Input-Constrained Linear Plants

⎡

Ac Bc E1

329

Cc Dc E2 −215.1

5.7

⎢ ⎢ −1781.3 −1154.1 ⎢ ⎢ = ⎢ −93.8 −53.9 ⎢ ⎢ −4914.0 −2441.5 ⎣ −74.4474 −38.2744

−6.0

137.7

3558.8 −3149.6 −0.6334

⎤

⎥ 360.2 −4122.6 −3202.5 −48.7601 ⎥ ⎥ ⎥ −159.9 21.2 −1047.7 3788.7 −4.4397 ⎥ . ⎥ −416.8 735.1 1491.6 1290.1 −131.6461 ⎥ ⎦ −5.9264 11.7716 120.6582 3.7499 −0.9998 −62.5

θ (radians)

0.5

0

−0.5

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4 5 6 time (seconds)

7

8

9

10

p (meters)

0.1

0

−0.1

u (volts)

5

0

−5

Fig. 3. Simulated response to the large pendulum tap. Constrained response with LQG controller (dashed); Response with the linear output feedback controller (thin solid); Response with the LQG controller and dynamic anti-windup [10] (dash-dotted).

The thin solid curve in Figure 3 represents the response of the closed-loop system with our output feedback controller to the same disturbance that generates the undesirable response arising from the saturated LQG controller (taken from [10]), which is represented by the dashed curve in the same ﬁgure. Moreover, for comparison purposes, the dash-dotted curve represents the response when using the dynamic compensator proposed in [10] on top of the LQG controller. Note that the proposed controller guarantees more desirable large signal responses as compared to the anti-windup closed-loop of [10], indeed our controller is not constrained to satisfy the small signal speciﬁcation as in the anti-windup approach of [10]. On the other hand, it must be recognized that synthesizing the controller by

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direct design reduces the small signal performance (before saturation) of the overall closed-loop, which can be imposed as an arbitrary linear performance when using the anti-windup tools as in [10] (or the more advanced techniques recently proposed in [17]).

5.2 Reference Tracking In this section, we will include two examples to show how effectively the proposed output feedback controller deals with the reference tracking problem. Before the examples, we introduce a plant augmentation that constrains the arising control system to guarantee zero tracking error. The modiﬁed problem corresponds to augmenting Ap with extra integrators that are inserted at the output y according to the block diagram reported in Figure 4. Plant Pnew Reference input

w=r

−1 y−w znew

Control input

u

P

1/s

ynew

Original Palnt

Fig. 4. The augmented plant for set point regulation.

According to the block diagram of Figure 4, not only should the output of the integrator(s) be sent to the controller, but also its input. Namely, the signal ynew is formed by the integrators output and an extra set of signals corresponding to the tracking error y − r. Suppose that the original plant equations are: x˙ p = Axp + Bu u P y = Cxp + Du u

(23)

then, the augmented plant equations, which correspond to the form in (1), are given by the following parameters:

Output Feedback for Input-Constrained Linear Plants

⎡

Ap

Bp,u

Bp,w

Cp,y/z Dp,y/zu Dp,y/zw

A 0 Bu 0

331

⎤

⎥ ⎢ ⎢ C 0 Du −I ⎥ ⎥ ⎢ =⎢ ⎥ ⎣0 I 0 0 ⎦ C 0 Du −I

(24)

As another design criterion, suppose that we wish to recover as much of the linear performance as possible for one output at the expense of the performance in the other output. For this problem, we add a new diagonal matrix δ to the LMI conditions in Theorem 1 (so that the 2 by 2 entry involving γ in (4a) and the corresponding entry in (4b) will be both changed from −γ2 I/2 to −δ−1 γ2 I/2). This extra matrix allows to guarantee the regional ﬁnite L2 gain in the form of z T δz ≤ γ2 wT w, thus providing extra degrees of freedom in the performance optimization. If the exogenous input w contains not only references to be tracked but also disturbances, the augmented plant will be different from the form (24). We will discuss this case in the next section. Example 2. Consider the damped mass-spring system used in [16], [31]. The motion equations for this system are given by 0 1 0 x˙ p = xp + u −k/m −f /m 1/m

y = 1 0 xp

T

where xp :=

represents position and speed of the body connected to the q q˙ spring, m is the mass of the body, k is the elastic constant of the spring, f is the damping coefﬁcient, u represents a force exerted on the mass. According to the parameters chosen in [16], [31]: m = 0.12kg, k = 0.8kg/s2 , f = 0.0008kg/s, the plant

matrices are given by: ⎡

Ap

Bp,u

Bp,w

Cp,y/z Dp,y/zu Dp,y/zw

0

1

0

0

⎢ ⎢ −6.667 −0.007 0 8.333 ⎢ ⎢ =⎢ 1 0 0 0 ⎢ ⎢ 0 0 1 0 ⎣ 1 0 0 0

0

⎤

⎥ 0 ⎥ ⎥ ⎥ −1 ⎥ ⎥ 0 ⎥ ⎦ −1

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Dan Dai, Tingshu Hu, Andrew R. Teel, and Luca Zaccarian

For our simulation, the system is starting from the rest position and with the reference switching between ±1 meters every 5s and going back to zero permanently after 15 s. We apply Procedure 1 with the output weight δ−1 = diag(0.42 , 1), and set s = 0.25, such that the optimization problem gives an upper bound of 1.5841 on the weighted L2 gain and the resulting controller matrices are: ⎡ −5.5080 0.1603 −10.7442 1.5543 ⎢ ⎢ −27.1931 −11.3053 −45.9518 −8.5028 Ac Bc E1 =⎢ ⎢ −1.3964 −0.2274 −10.2038 56.6145 Cc Dc E2 ⎣ −0.9526 −0.4371 −1.7830 −0.3329

−5.6132 −1.5255

⎤

⎥ −29.2707 −7.3341 ⎥ ⎥ 6.8613 −0.0527 ⎥ ⎦ −1.1344 0.7160

The corresponding closed-loop response is represented by the thin curve in Figure 5. The mass position is tracking the reference well.

Control input u

Performance output z

Mass position y

The response of damped mass−spring system 2

0

−2

0

5

10

15

20

25

0

5

10

15

20

25

0

5

10

15

20

25

5

0

−5 1

0

−1

Time (s)

Fig. 5. Response to the system in Example 2. Response with linear output feedback controller(thin solid); Reference input (dotted).

Example 3. Consider a fourth-order linear model of the longitudinal dynamics of the F8 aircraft introduced in [18], [10]. The state-space equations are given by

Output Feedback for Input-Constrained Linear Plants

⎡

−0.8 −0.0006 −12

0

⎤

⎡

−19

−3

333

⎤

⎥ ⎥ ⎢ ⎢ ⎢ 0 −0.014 −16.64 −32.2 ⎥ ⎢ 0.66 −0.5 ⎥ ⎥ ⎥u ⎢ ⎢ xp + ⎢ x˙ p = ⎢ ⎥ ⎥ 0 ⎦ ⎣ 1 −0.0001 −1.5 ⎣ −0.16 −0.5 ⎦ 1 0 0 0 0 0 00 0 1 y= xp 0 0 −1 1 The two inputs of the plant are the elevator angle and ﬂaperon angle, each one limited between ±25 degrees in the simulation. The two outputs of the plant are the pitch angle and ﬂight path angle, which are supposed to track the reference in T put w = 10 10 . Similar to the previous example, the original model of the F8 aircraft is ﬁrst augmented by extra integrators to guarantee zero steady-state error. Therefore, the new output consists in the pitch and ﬂight path angles and the corresponding rates. In the LMI computation of Procedure 1, we note that the output weight helps to balance the performance of the two sets of outputs and select δ−1 = diag(0.1, 0.1, 0.92 , 1.152 ) and s = 1. The upper bound on the weighted L2 gain is 1.3056, and the controller matrices are given by: ⎡

−47.7 −0.9 −86.3 −587.6 −1585.3 221.2

⎤

⎥ ⎢ ⎢ −27.0 −0.3 23.3 −395.6 −875.3 −65.3 ⎥ ⎥ ⎢ ⎢ 6.5 −0.0 −579.4 556.6 −56.7 1532.4 ⎥ ⎥ ⎢ Ac = ⎢ ⎥ ⎥ ⎢ −0.3 0.7 −39.3 6.5 −77.2 104.7 ⎥ ⎢ ⎥ ⎢ −0.6 0.2 2.8 −12.6 −28.1 −7.5 ⎦ ⎣ 0.2 −0.1 −16.6 14.9 −4.3 40.2 ⎤ ⎡ 34.8 −19.7 −226.5 −14.3 ⎥ ⎢ ⎢ −48.3 35.5 −149.0 −2.3 ⎥ ⎥ ⎢ ⎥ ⎢ Bc = ⎢ 16.4 −44.9 43.4 41.2 ⎥ ⎥ ⎢ ⎢ −256.2 −15.5 −77.8 −1.2 ⎥ ⎦ ⎣ 1692.4 171.15 7.2 448.4 0.0217 −0.0004 −0.2902 0.5522 0.6161 0.7687 Cc = −0.0277 0.0039 2.2378 −2.3055 −0.1292 −5.9286

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Dan Dai, Tingshu Hu, Andrew R. Teel, and Luca Zaccarian

Dc = E1 = E2 =

−0.0033 −0.0012 0.1159

0.0331

−0.0159 0.0046 −0.1995 −0.2073 0.0659 0.0488 −0.0981 −0.0029 0.0017 −0.0027

T

0.0595 −0.0025 0.0234

0.2079 0.0163 0.8089 0.9999 0.0004 0.0004 0.9969

The time histories in Figure 6 correspond to the closed-loop response to the reference r = [10 10]T . From the ﬁgure it appears that the controller makes wide use of the input authority (the second input is kept into saturation until the ﬂight angle reaches the desired value in a bang-bang type of control action) guaranteeing a fast

12

12

10

10

8 6 4 2 0

0

1

2

3

4

6 4 2 0

1

2

3

4

5

0

1

2 3 time (s)

4

5

30

0

−10

−20

−30

8

0

5

flaperon angle (deg)

elevator angle (deg)

flight angle (deg)

pitch angle (deg)

and desirable output response.

0

1

2 3 time (s)

4

5

20 10 0 −10

Fig. 6. Response of the F8 aircraft with reference tracking.

5.3 Reference Tracking and Disturbance Rejection In the previous examples, we have shown the desirable performance of the proposed output feedback construction. In particular, we have shown its ability to effectively solve both the disturbance rejection and the reference tracking problems. In this section we study an exponentially unstable example and consider the disturbance

Output Feedback for Input-Constrained Linear Plants

335

Plant Pnew r

−1

w

y−r

d

Reference and disturbance input

znew P

u

ynew

1/s

Control input

Original Palnt

Fig. 7. The plant augmented for reference tracking and disturbance rejection.

rejection and reference tracking problems together. Before the example, the block diagram represented in Figure 4 is modiﬁed to include both references and disturbances. The counterpart of the original plant model (23) and of the augmented plant parameters (24), correspond to: P

x˙ p = Axp + Bu u + Bd d

with

⎡

Ap

(25)

y = Cxp + Du u + Dd d

Bp,u

Bp,w

Cp,y/z Dp,y/zu Dp,y/zw

A 0 Bu 0 Bd

⎤

⎥ ⎢ ⎢ C 0 Du −I Dd ⎥ ⎥ =⎢ ⎢ ⎥ 0 I 0 0 0 ⎣ ⎦ C 0 Du −I Dd

(26)

r . Using equations (25) and (26) as an intermediate step, can use with w = d Procedure 1 for the following example. Example 4. Consider the inverted pendulum around its upright position for small inclination angles. The pendulum mass m1 shall be concentrated at its center of gravity (cg), and the connecting link of length L to the slider at its bottom is mass-free and rigid. The slider mass m3 may move horizontally without friction. A horizontal force is applied by the actuator to the slider, in order to control the speed error to zero. The linearized model in appropriately scaled variables is given by:

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Dan Dai, Tingshu Hu, Andrew R. Teel, and Luca Zaccarian

τ1 x˙ 1 = −x2 + d; τ2 x˙ 2 = −x1 + x3 ;

(27)

τ3 x˙ 3 = x2 + u; y = x3

with parameter values τ1 = τ2 = τ3 = 1.0s. In (27), the state variables correspond to the horizontal speed of the pendulum cg (represented by x1 ), the horizontal displacement between the cg’s of the pendulum and of the slider (represented by x2 ) and the horizontal speed of the slider (represented by x3 ). The load d represents a horizontal force exerted on the pendulum cg, which is persistent and with unknown but bounded magnitude. The input u represents the force exerted by the actuator, which is constrained in magnitude (stroke) between ±1. To deal with the persistent disturbance while tracking the reference, we ﬁrst augment the plant model matrices as follows: ⎤ ⎡ ⎡ ⎤ 0 1 0 ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ 0 0⎥ r ⎢ −1 0 1 0 ⎥ ⎢0⎥ ⎥ ⎢ ⎥ ⎢ ⎥ x˙ p = ⎢ ⎢ 0 1 0 0 ⎥ xp + ⎢ 1 ⎥ u + ⎢ 0 0 ⎥ d ⎦ ⎣ ⎦ ⎣ ⎣ ⎦ −1 0 0 0 10 0 ⎡

0 −1 0 0

y=

⎤

0001 0010

xp +

0 0 −1 0

r d

The plant has one pole at the origin and one on the positive real axis: it is exponentially unstable. The test sequence is designed as follows: 1) Start in closed-loop operation from the initial conditions x(0) = 0, reference r = 0 and disturbance d = 0. 2) Apply a step reference of r = 0.08 to the system for 30 s and then reset to zero. 3) At time t = 60s, apply a load step disturbance d = 0.2 to the system. In Procedure 1 we choose s = 0.012 (small region), and solve the LMIs to get the minimized L2 gain γ = 37.78. The controller matrices are given next and the simulation results are shown in Figure 8.

Output Feedback for Input-Constrained Linear Plants

⎡

Ac Bc E1

337

Cc Dc E2 −90.3862 67.9126

1.6317

⎤

⎥ 2.4002 ⎥ ⎥ ⎥ −123.9217 35.4146 44.7308 −13.8450 −116.1458 −5.0009 ⎥ ⎥ −11.3325 7.9340 −13.8678 280.5706 −0.6618 −0.6745 ⎥ ⎦ −47.4669 14.9467 16.8633 0.8502 −46.4682 −1.0000 25.8891 −24.7031 −15.6606 58.9859

Control input u

Performance output z

Horizontal speed y

⎢ ⎢ −61.4551 ⎢ ⎢ = ⎢ 193.4138 ⎢ ⎢ 18.6906 ⎣ 74.7057

−5.3992 −20.5277 −22.1765 35.6479 60.7153

1 0 −1

0

20

40

60

80

100

0

20

40

60

80

100

0

20

40

60

80

100

2 1 0 −1 1 0 −1

time (s)

Fig. 8. Response of the inverted pendulum.

The thin solid curve in Figure 8 represents the response of the closed-loop system with input saturation. The dash-dotted curve represents the reference input and the load disturbance. The upper plot shows that the speed of the slider succeeds in tracking the reference and rejecting the disturbance while exhibiting desirable performance.

6 Conclusions In this chapter we illustrated on several example studies a novel synthesis method for the construction of a linear output feedback controller with an internal deadzone loop. By using regional analysis tools developed in [16], a systematic method for the

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synthesis of the proposed controller is cast as an optimization problem over LMIs and a variety of regional stability and performance goals can be achieved. For all the proposed examples, the simulations show desirable closed-loop performance.

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13. W.M. Haddad and V. Kapila. Anti-windup and guaranteed relative stability margin controllers for systems with input nonlinearities. In Proceedings of the 13th IFAC World Congress, volume H, pages 143–148, San Francisco (CA), USA, June 1996. 14. T. Hu, Z. Lin, and B.M. Chen. Analysis and design for linear discrete-time systems subject to actuator saturation. Systems and Control Letters, 45(2):97–112, 2002. 15. T. Hu, Z. Lin, and B.M. Chen. An analysis and design method for linear systems subject to actuator saturation and disturbance. Automatica, 38(2):351–359, 2002. 16. T. Hu, A.R. Teel, and L. Zaccarian. Nonlinear L2 gain and regional analysis for linear systems with anti-windup compensation. In American Control Conference, pages 3391– 3396, Portland (OR), USA, June 2005. 17. T. Hu, A.R. Teel, and L. Zaccarian. Regional anti-windup compensation for linear systems with input saturation. In American Control Conference, pages 3397–3402, Portland (OR), USA, June 2005. 18. P. Kapasouris, M. Athans, and G. Stein. Design of feedback control systems for stable plants with saturating actuators. In Proceedings of the Conference on Decision and Control, pages 469–479, Austin (TX), USA, December 1988. 19. T.A. Kendi, M.L. Brockman, M. Corless, and F.J. Doyle. Controller synthesis for inputconstrained nonlinear systems subject to unmeasured disturbances. In ACC, pages 3088– 3092, Albuquerque (NM), USA, May 1997. 20. V.M. Marcopoli and S.M. Phillips. Analysis and synthesis tools for a class of actuatorlimited multivariable control systems: A linear matrix inequality approach. Int. J. Robust and Nonlinear Control, 6:1045–1063, 1996. 21. D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokaert. Constrained model predictive control: Stability and optimality. Automatica, 36(6):789–814, 2000. 22. A. Megretski. L2 BIBO output feedback stabilization with saturated control. In 13th Triennial IFAC World Congress, pages 435–440, San Francisco, USA, 1996. 23. E.F. Mulder and M.V. Kothare. Static anti-windup controller synthesis using simultaneous convex design. In ACC, pages 651–656, Anchorage (AK), USA, June 2002. 24. E.F. Mulder, M.V. Kothare, and M. Morari. Multivariable anti-windup controller synthesis using linear matrix inequalities. Automatica, 37(9):1407–1416, September 2001. 25. C.W. Scherer, P. Gahinet, and M. Chilali. Multi-objective output-feedback control via lmi optimization. IEEE Trans. Aut. Cont., 42(7):896–911, July 1997. 26. H.J. Sussmann, E.D. Sontag, and Y. Yang. A general result on the stabilization of linear systems using bounded controls. IEEE Trans. Aut. Cont., 39(12):2411–2424, 1994. 27. K. Takaba. Anti-windup control system design based on Youla parametrization. In Proceedings of the American Control Conference, volume 3, pages 2012–2017, San Diego (CA), USA, June 1999. 28. A.R. Teel. Global stabilization and restricted tracking for multiple integrators with bounded controls. Systems and Control Letters, 18:165–171, 1992.

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29. F. Tyan and D.S. Bernstein. Anti-windup compensator synthesis for systems with saturating actuators. Int. J. Robust and Nonlinear Control, 5(5):521–537, 1995. 30. F. Wu, K.M. Grigoriadis, and A. Packard. Anti-windup controller design using linear parameter-varying control methods. Int. Journal of Control, 73(12):1104–14, August 2000. 31. L. Zaccarian and A.R. Teel. A common framework for anti-windup, bumpless transfer and reliable designs. Automatica (B), 38(10):1735–1744, 2002.

Set Based Control Synthesis for State and Velocity Constrained Systems Franco Blanchini1 , Stefano Miani2 , and Carlo Savorgnan2 1

University of Udine, Dept. of Mathematics and Computer Science, Via delle Scienze 208, 33100, Udine, Italy. email [email protected]

2

University of Udine, Dept. of Electrical, Mechanical and Managerial Eng., Via delle Scienze 208, 33100, Udine, Italy. email [email protected], [email protected]

Summary. In the present contribution, the control synthesis for state and input constrained systems for linear systems via set-valued techniques will be presented. The results will be developed for both uncertain and gain scheduled constrained control problems, where the difference between the two is the knowledge of the uncertain parameter entering the system at each time instant in the gain scheduled case. It will be shown how slew-rate constrained control problems can be recast into standard constrained control problems by means of an extended system. To present such technique, an overview of some of the existing results with some numerical algorithms to derive the stability conditions and the control will be given.

1 Introduction Controlling systems in the presence of constraints is one of the areas which has received great attention in the recent literature [17, 18, 15, 21, 11, 13]. The main ingredients of such control problems are rather standard: the system dynamics, the input limitations, both on the amplitude and the slew–rate, the output or, in general, the state variables constraints and, possibly, the disturbances acting on the system. The ﬁnal goal of constrained control problems is optimizing some performance measure and, possibly, the requirement of an as simple as possible control law.

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Franco Blanchini, Stefano Miani, and Carlo Savorgnan

Even for linear systems, constrained control problems are rather difﬁcult to solve. Several techniques have been proposed to face constrained synthesis. Such techniques evolved from the adaptation of analysis results to synthesis problems (e.g. the circle criterion), when the designer was basically verifying the extent of the effects of the nonlinearity in the closed loop system for a linear controller, to a more mature stage when ﬁrst basic problems on the determination of globally stabilizing control laws for constrained systems were considered [12, 22], and later more complex control laws (e.g. receding horizon and nonlinear controllers) were seeked, keeping in mind from the beginning the presence of constraints and performance requirements [10, 1]. It is worth stressing that most of the recent results have been made available on one side by recent new ideas, on the other side by the availability of reliable numerical algorithms which have made tractable problems which were far from being solvable in the 90’s. In this chapter we will present some results which allow the designer to cope with constrained problems for a speciﬁc class of systems. Precisely, the main focus of this chapter will be time-varying systems with structured uncertainty affecting the system matrices, and for these it will be shown how to solve input and slew-rate limitations design problems. Although here, for the sake of simplicity, only the case of symmetric constraints will be considered, it is worth saying that all the results given in the sequel can be extended to the nonsymmetric case.

2 Deﬁnitions and Existing Results As already mentioned, the aim of the present work is to show how assessed techniques in the area of constrained control problems can be applied to deal with slewrate constrained problems as well. To this aim, in the present section some of the basic results in the area of constrained control (excluding the presence of rate constraints) are reported. In this section constrained control problems for n dimensional linear time-varying systems of the form x+ (t) = A(w(t))x(t) + B(w(t))u(t) y(t) = C(w(t))x(t) + D(w(t))u(t)

(1)

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are considered. The term x+ denotes the time derivative x(t) ˙ in the continuous-time case or the one step forward delay x(t + 1) in the discrete-time case. In either cases, the system matrices belong to the sets whose expression are A= C=

s

i=1 Ai wi (t), s i=1 Ci wi (t),

B= D=

s i=1

s

Bi wi (t)

i=1 Di wi (t)

(2)

where the matrices Ai ∈ IRn×n , Bi ∈ IRn×m , Ci ∈ IRp×n and Di ∈ IRp×m are given and w(t) is a piecewise constant function which attains its values in W = {w : wi ≥ 0,

s

wi = 1}

(3)

i=1

For the above systems, several control problems can be deﬁned, depending on the availability of information of the state variables, the presence of state and/or input limitations (both on the amplitude and the slew-rate) and the knowledge of the parameter w(t) at each time-instant. In the present context, it will be assumed that the state information is available, and thus all the design problem faced here will eventually lead to state-feedback strategies. In other words, in this work the variable y(t) will be essentially used to characterize the performance measure. It is assumed that the following constraints are given ⎧ ⎪ ⎪ ⎨ u(t) ∈ U x(t) ∈ X ⎪ ⎪ ⎩ y(t) ∈ Y

(4)

where U, X and Y are polyhedral compact sets containing the origin as an interior point. Note that, in principle, the constraint y(t) ∈ Y includes all the mentioned constraints provided that matrices C(·) and D(·) are suitably chosen. In particular we can face state speed constraints x˙ = Ax + Bu ∈ S. Note also that constraints of the form y = C(w)x(t) + D(w)u ∈ Y are equivalent to a ﬁnite number or constraints, precisely Ci x + Di u ∈ Y , ∀i We consider the following problems: • when the value of w(t) is not available to the controller, we will talk about robust synthesis;

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• when such information is available for control purposes, we will talk about gain scheduled synthesis; • when the value of w(t) is known and, moreover, w(t) attains its value only on one of the vertices of the set W deﬁned in (3), say w(t) ∈ Ws = {w : wi = 1, wj = 0 if j = i f or some i}, we will talk about switched gain scheduled synthesis. Remark 1. We let the reader note that, even in the continuous–time case, in view of the fact that the signal w(t) is piecewise-constant, the gain scheduled and switched gain scheduled synthesis problems are rather different. For example, the continuous– time system x+ (t) = x(t) + (w1 (t) − w2 (t))u(t)

(5)

is clearly switched gain scheduling stabilizable (even in the presence of constraints) by the control law u(t) = −2(w1 (t) − w2 (t))x(t), but there is obviously no gain scheduling stabilizing control law due to to the lack of reachability when w1 (t) = w2 (t) = 1/2. To recap, the aim of the present work is that of determining state-feedback gain scheduled or robust controllers for either continuous or discrete-time constrained control problems of the form (1), with performance measure on the variable y(t) (also deﬁned in (1)). Before getting into these issues, some deﬁnitions which will be useful in the sequel are reported. We refer the reader to one of the many books on convex analysis [16] for a more detailed exposition on polyhedron, polytopes and their representations. Deﬁnition 1. An r × r matrix H belongs to the set H if and only if there exists τ > 0 and a matrix P , whose 1-norm is smaller than 1, such that H = τ−1 (P − I). Deﬁnition 2. P ∈ IRn is a polyhedral set or polyhedron if it can be written as follows:

P = x ∈ IRn : fiT x ≤ 1, i = 1, . . . , m

(6)

where fi are vectors in IRn and f T denotes the transpose vector of f . A more compact notation can be obtained by stacking the vectors fiT in the matrix F as to obtain

P = x ∈ IRn : F x ≤ 1

(7)

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where 1 represents a column vector of appropriate dimensions and whose entries are all equal to 1. A bounded polyhedron P ∈ IRn is a polytopic set or polytope and it can be also represented in terms of its r vertices xj . Indeed, denoting by X ∈ IRn×r the full row rank matrix X = [x1 , . . . , xr ], it is possible to write P as the convex hull of its

vertices: P = Co(X) =

x ∈ IRn : x = Xα,

r

8 αi , αi ≥ 0 .

(8)

i=1

The above is also known as vertex representation. By applying the above deﬁnitions it is immediate to see that the sets A and B where A(w) and B(w) attain their values are polytopic matrix sets. The vertices of A and B are respectively Ai and Bi . The reason why polytopic sets have been introduced is that they play a key role in the next fundamental result, which is the core result on which the control of slew–rate constrained systems will be built on in the next sections. Theorem 1. Given the linear time–varying system (1) and the constraints ⎧ ⎪ ⎪ ⎨ x(t) ∈ X u(t) ∈ U ⎪ ⎪ ⎩ y(t) ∈ Y

(9)

where X , U, Y are symmetric polyhedra, the following statements hold. • Continuous–time, robust synthesis [5]. There exists a state feedback controller u(x) = φ(x) such that the closed loop evolution complies with constraints (9) and such that the closed loop state evolution asymptotically converges to 0 if and only if there exist a matrix X ∈ IRn×r , a matrix U ∈ IRp×r and s matrices Hi ∈ H such that, for any i, Ai X + Bi U = XHi

(10)

and each of the state vectors xj corresponding to the columns of the matrix X and each of the input vectors uj corresponding to the columns of the matrix U are such that the following state and input conditions are satisﬁed

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⎧ j ⎪ ⎪ ⎨x ∈ X

∀ j = 1, . . . , r

uj ∈ U ∀ j = 1, . . . , r ⎪ ⎪ ⎩ C xj + D uj ∈ Y ∀ j = 1, . . . , r ∀ i = 1, . . . , s i i

(11)

• Continuous–time, switched gain scheduled synthesis [9]. There exists a state feedback controller u(x) = φ(x, i) such that the closed loop evolution complies with constraints (9) and such that the closed loop state evolution asymptotically converges to 0 if and only if there exist a matrix X ∈ IRn×r , s matrices Ui ∈ IRp×r and s matrices Hi ∈ H such that, for any i = 1, . . . s, Ai X + Bi Ui = XHi

(12)

and each of the state vectors xj corresponding to the columns of the matrix X and each of the input vectors uji corresponding to the columns of the matrix Ui are such that the following state and input conditions are satisﬁed ⎧ j ⎪ ∀ j = 1, . . . , r ⎪ ⎨x ∈ X ⎪ ⎪ ⎩

uji ∈ U ∀ j = 1, . . . , r ∀ i = 1, . . . , s Ci xj +

Di uji

∈Y

(13) ∀ j = 1, . . . , r ∀ i = 1, . . . , s

• Continuous–time gain scheduled synthesis [6]. The existence a state feedback controller u(x) = φ(x, w) such that the closed loop system complies with constraints (9) and such that the closed loop state evolution asymptotically converges to 0 is equivalent to the existence of a control u(x) = φ(x) which achieves the same goal. In simple words, gain-scheduling and robust stabilization are equivalent and they are both equivalent to condition (12) with a single matrix U . Note that according to Remark 1 this equivalence does not hold between robust and switched gain scheduling stabilization unless B is a certain matrix (see also [9] for a detailed analysis of the relations between the different cases). • Discrete–time, robust synthesis [5]. There exists a state feedback controller u(x) = φ(x) complying with constraints (9) and such that the closed loop state evolution asymptotically converges to 0 if and only if there exist a matrix X ∈ IRn×r , a matrix U ∈ IRp×r and s non-negative matrices Pi such that, for any i, Ai X + Bi U = XPi , Pi 1 < 1

(14)

and each of the state vectors xj corresponding to the columns of the matrix X and each of the input vectors uj corresponding to the columns of the matrix U satisfy conditions the conditions (11)

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• Discrete–time, switched gain scheduled synthesis [9]. There exists a state feedback controller u(x) = φ(x, i) complying with constraints (9) and such that the closed loop state evolution asymptotically converges to 0 if and only if there exist a matrix X ∈ IRn×r , s matrices Ui ∈ IRp×r and s non-negative matrices Pi such that, for any i, Ai X + Bi Ui = XPi , Pi 1 < 1

(15)

and each of the state vectors xj corresponding the columns of the matrix X and each of the input vectors uji corresponding the columns of the matrix Ui satisfy conditions (13). • Discrete–time, gain scheduled synthesis [9]. Differently form the continuous– time case there is no equivalence between the gain scheduled and robust stabilizability. However, when B(w) = B, an if and only if condition for the existence of a state feedback gain scheduled control law complying with the imposed requirements is given by Ai X + BUi = XPi , Pi 1 < 1

(16)

(note that it is slightly different from equation (15)) together with the conditions on the state, input and output limitations. To see the role played by polytopic sets in Theorem 1, we will ﬁrst brieﬂy concentrate on the discrete-time conditions for robust constrained stabilization, since such case is the easiest to be interpreted. Indeed, by deﬁning the symmetric polytope P = Co(−X, X) (so that the columns of X and −X represent the vertices of P ) and by considering each equation given by the columns of the matrix equation (14), one gets Ai xj + Bi uj = XPij

(17)

where Pij denotes the j-th column of matrix Pi . Now, such condition corresponds to say that for any system vertex i, whenever the state variable corresponds to one vertex of P (x(t) = xj ), there exists a value of the input that does not depend on i (u(t) = uj ) and such that any value that the next step state xj+ can attain (and i which clearly depends on the system vertex i) is strictly included in P . The latter assertion can be immediately veriﬁed by rewriting the equation corresponding to the jth column of (14) as

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Franco Blanchini, Stefano Miani, and Carlo Savorgnan j+

x

j

j

= Ai x + Bi u =

Xpji

=

r

j pk,j i x

(18)

k=1

Since pji is a column of a matrix whose 1-norm is smaller than one, then r k,j j+ ∈ Co(−X, X). Though the above condition k=1 |Pi | < 1 and therefore x holds only on each of the vertices of the set P , it can be shown that, in view of the linearity of the system and the convexity of P , the inclusion x+ ∈ Co(−X, X) is guaranteed for every value of w(t) and for every point of P , say it is possible to derive a state feedback stabilizing control law u = φ(x). Moreover, since the state and input values xj and uj satisfy (11), it can be shown that the derived control law complies with the state, input and output constraints. In the continuous–time case the main condition to be checked, apart from the conditions (11) which are needed for the constraints, is (10). From the deﬁnition of Hi , such condition can be written as Ai X + Bi U = Xτ−1 (Pi − I)

(19)

for a proper value of τ, then (I + τAi )X + τBi U = XPi

(20)

where Pi is such that Pi 1 < 1. Thus, equation (20) corresponds to the robust condition for the discrete-time system x+ (t) =

s i=1

wi (t)(I + τAi )x(t) +

s

wi (t)τBi u(t)

(21)

i=1

also known as Euler Approximating System (EAS) of (1) (with time constant τ). Therefore, an interpretation of Theorem 1 is the following: a constrained continuoustime system is robustly stabilizable if and only if there exists τ > 0 such that the corresponding discrete-time EAS is robustly stabilizable system. Similar considerations can be made for the gain scheduling and switched gain scheduling conditions. In this case the matrix U is substituted by Ui . This fact simply implies that the value of u that stabilizes the system depends on the value of the index i.

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2.1 Algorithms for the Calculation of Contractive Sets So far polytopic sets useful in ascertaining the existence of constrained control laws have been presented as the focal point. The usefulness of the results in Theorem 1 would be anyhow limited if no constructive algorithm for the determination of such sets were available. Fortunately, numerical algorithms for the calculation of such matrices are available [20]. Such numerical algorithms rely on the basic ideas by Bertsekas and Rhodes [3] and essentially consist in the backward calculation of reachability sets. Before getting into this issue, the deﬁnition of contractive set in the robust and gain scheduling case as well as that of pre–image set are introduced. Deﬁnition 3. P is a robust (resp. gain scheduling) λ-contractive set , with 0 < λ < 1, with respect to the discrete system (1) if for every x ∈ P there exist a value of the system input u (resp. u(w)) such that A(w)x + B(w)u ∈ λP (resp. A(w)x + B(w)u(w) ∈ λP ) for any w ∈ W , where λP = {x : x/λ ∈ P }. To keep things simple and avoid introducing subdifferentials and other mathematichal tools (see [5] for a formal deﬁnition), we will say that a set P is β < 0 contractive with respect to the continuous–time system (1) if it is λ = 1 + τβ contractive for the corresponding EAS for a certain τ > 0. The above deﬁnition clearly extends to the case of switched gain scheduled systems when w is not allowed to live in the entire W , but just on its vertices. It is rather assessed that the existence of a λ (resp. β in the continuous–time case) contractive set P is equivalent to the fact that, for every x(0) ∈ P , x(t) ≤ γ1 λt x(0) , (resp. x(t) ≤ γ2 eβt x(0) )

(22)

To relate the just introduced deﬁnition with what previously presented in Theorem 1, it is worth observing that the sets P introduced in the theorem and described by means of the matrices X (say P = Co(−X, X)), are contractive, but with an unspeciﬁed value of λ < 1 (resp. β). The just introduced parameter λ gives a measure of the contractivity and therefore of the convergence speed of the system. A concept on which the algorithm to be introduced is based is the following: Deﬁnition 4. Given a polyhedral set P = {x ∈ IRn : F x ≤ 1} and the discrete-time system (1), the one step pre-image of P (referred to as P (1) ) is the set of states which

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can be brought to P in one step by a proper control action. Such set can be computed as [4, 8]: • in the robust case P (1) = {x ∈ IRn : ∃u such that F (Ai x + Bi u) ≤ 1, ∀i};

(23)

• in the switched gain scheduling case P (1) = {x ∈ IRn : ∀i, ∃ui such that F (Ai x + Bi ui ) ≤ 1}.

(24)

The relation between preimage sets and λ-contractive ones is clariﬁed by the following result. Theorem 2. The set P is λ-contractive with respect to the discrete-time system (1) if P ⊆ preimage(λP ). In [4] it has been shown that a λ-contractive set can be computed starting from any arbitrary P (0) and recursively calculating the pre-image set for the dynamic system x(t + 1) =

B(w(t)) A(w(t)) x(t) + u(t) λ λ

as P (∞) =

9

P (k)

(25)

(26)

k=0,...,∞

It turns out that P (∞) is the largest λ-contractive set in P (0) , say condition (22) cannot be satisﬁed for any x(0) ∈ P . With this background, we are now ready to present the algorithm to calculate λcontractive sets for robust synthesis problems. We consider the discrete-time system (1) with constraints on input, state and output variables as in Theorem 1. In the algorithm a tolerance is introduced to cope with the possibility of an inﬁnite number of iterations and the intersection (26) is carried out at each set to avoid storing all the pre-image P (k) . Algorithm 9 Set 0 < λ < 1, 0 < < (1 − λ), k = 0. Assign the initial polytope P (0) = X = {x : F (0) x ≤ 1}.

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351

1. Calculate the expanded set E (k) ∈ IRn+m E (k) = {(x, u) ∈ IR(n+m) : F (k) [Ai x + Bi u] ≤ λ1, ∀i};

(27)

2. Compute the projected set P r(k) ∈ IRn P r(k) = {x ∈ IRn : ∃u : (x, u) ∈ E (k) , (x, u) ∈ Y , u ∈ U};

(28)

3. Calculate the intersection P (k+1) ∈ IRn P (k+1) = P (k)

9

P r(k)

(29)

(P (k+1) = {x : F (k+1) x ≤ 1}); 4. If P (k+1) is (λ + )-contractive stop, else k = k + 1 and go to step 1. Without getting into the details of the numerical methods needed to properly implement the algorithm, we just focus on the most interesting issues in the algorithm, whose implementation is freely available [20]. • By the assignment P (0) = X , the set calculated when the stopping criterion is met is the largest λ-contractive set compatible with the constraints. • Step 2 is crucial for the algorithm. The projection of E (k) bypasses the problem of calculating simultaneously the contractive set and a proper value of the input that guarantees the contractivity. Indeed, at each step no information about the controller is considered. • The constraints u ∈ U and y ∈ Y are taken into account in (28). Some simple changes are needed to accomplish the computation of a λ-contractive set when the switched gain scheduling synthesis problem is considered. As in the previous algorithm the discrete-time system (1) is considered and the constraints are as in Theorem 1. Algorithm 10 Set 0 < λ < 1, 0 < < (1 − λ), k = 0. Assign the initial polytope P (0) = X = {x : F (0) x ≤ 1}. (k)

1. Calculate the expanded sets Ei (k)

Ei

∈ IRn+m

= {(x, u) ∈ IR(n+m) : F (k) [Ai x + Bi u] ≤ λ1};

(30)

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Franco Blanchini, Stefano Miani, and Carlo Savorgnan (k)

2. Calculate the projected sets P ri (k)

P ri

∈ IRn (k)

= {x ∈ IRn : ∃u : (x, u) ∈ Ei , (x, u) ∈ Y , u ∈ U};

(31)

3. Calculate the intersection P (k+1) ∈ IRn P (k+1) = P (k)

9

(k)

P ri

(32)

i=1,...,s

(P (k+1) = {x : F (k+1) x ≤ 1}); 4. If P (k+1) is (λ + )-contractive stop, else k = k + 1 and go to step 1. The proposed algorithms (9 and 10) can be extended to consider systems with an unknown input disturbance d(t) ∈ D ∈ IRq , say to consider systems of the form x+ (t) = A(w(t))x(t) + B(w(t))u(t) + Ed(t) y(t) = C(w(t))x(t) + D(w(t))u(t)

(33)

Indeed, since we are in a worst-case setting, it is only needed to change the expansion steps resulting in, for the robust case E (k) = {(x, u) ∈ IR(n+m) : F (k) [Ai x + Bi u] ≤ λ(1 − max F (k) Ed), ∀i} (34) d∈D

and, for the switched gain scheduling case (k)

Ei

= {(x, u) ∈ IR(n+m) : F (k) [Ai x + Bi u] ≤ λ(1 − max F (k) Ed)}. d∈D

(35)

Remark 2. The presented algorithms deal only with discrete-time systems. When a continuous-time system is considered, it is still possible to use the algorithms 9 and 10 by considering the EAS with a proper value of τ > 0. So far, the algorithms presented only give a method to calculate the matrices X in Theorem 1. The calculation of the matrices U , Ui , Hi and Pi can be carried out by using linear programming techniques. In the next section it will be shown how such matrices come in hand to derive stabilizing feedback control laws.

2.2 State Feedback Variable Structure Linear Controllers Once a contractive set is computed it is important to ﬁnd a control law. Such law is a pure state feedback variable structure linear control of the form u = Φ(x) in the

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353

uncertain case and a state feedback variable structure gain scheduling linear control law of the form u = Φ(x, w) in the case of gain scheduled problems. To see how this law is derived, we need to recall that the coupled matrices X and U (or Ui ) in Theorem 1 deﬁne the vertices and the corresponding control values to be applied in order to have the inclusion x+ ∈ P . When the state x of the system is not one of the vertices, but belongs anyway to the set P , then such value can be expressed as a (non unique) convex combination of the vertices of P , say x = Xα. The control u(x) = U α can be shown [14, 5] to guarantee the satisfaction of the imposed constraints. A clever way to determine, given x, the vector of multipliers α in a unique way passes through the partition of a polytope in simplicial sectors, as per the next deﬁnition: Deﬁnition 5. Given a polytopic set P in IRn deﬁned by the convex hull of the vertices xi , we can denote by Xh the full rank matrix obtained by stacking a set of (linearly independent) n vertices: Xh = [xh1 xh2 . . . xhn ] The set

(36)

⎧ ⎫ n n ⎨ ⎬ xhj αj , αj ≤ 1, αj ≥ 0 Sh = x = ⎩ ⎭ j=1

(37)

j=1

is called a simplicial sector of P . It is immediate to see that if a state x belongs to a simplicial sector Sh , then the value α can be computed as

α= x

h1

h2

x

··· x

hn

−1

x

(38)

Now, it can be shown that any polytopic set P can be partitioned in simplicial sectors. More precisely, a particular selection of the vertices is the one given by choosing {h1 h2 . . . hn } such that all the vertices lie on the same face of P , Sh has nonempty 7 interior, Sh Sk = ∅ if h = k and h Sh = P . Once such partition has been determined, it is possible to derive the following variable structure linear controller u = Φ(x) = Kh x

(39)

Kh = Uh Xh−1

(40)

with

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where the index h corresponds to the simplicial sector Sh to which the state x belongs and the matrix Uh is given by stacking the columns uhj of U . It can be shown [5] that the controller given by equation (39) is robustly stabilizing for the system considered and locally Lipschitz. Moreover, if the matrix X and U are such that the conditions (11) are satisﬁed, then the proposed control law satisﬁes the state limitations as well as the input and output constraints. A procedure similar to the one just presented can be used also for the derivation of stabilizing controllers for the other problems considered in Theorem 1. For example, to cope with discrete-time switched system, it is possible to use the equation Ai X + Bi Ui = XPi

(41)

u = Φ(x) = Khi x

(42)

Khi = Uhi Xh−1

(43)

which provides the control

with where h is such that x ∈ Sh and Uhi is the matrix obtained by stacking the columns h

ui j of Ui . Similarly, for discrete–time gain scheduled systems with B certain, it is possible to use the following control law u(x, w) =

s

wi Khi x

(44)

i=1

3 Amplitude and Slew–Rate Constrained Control Problems So far the basic results concerning constrained control systems have been introduced. In this section, it will be shown how the presented results can be applied to the solution of stabilization problems when slew-rate constraints are also present. To this aim, and extended system will be deﬁned and it will be shown how the determination of a constrained control law, as per the results of the previous section, allows to solve the slew-rate constrained control problem. In the present section it will be assumed that the constraints on the state, input and output are given by (9) and that the slewrate constraints are also represented by a polyhedral set u(t) ˙ ∈V

(45)

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3.1 The Rate Bounding Operator A typical way to face the bounded variation problem is that of inserting in the input channel a suitable operator. In the case of a bounded amplitude system, the typical operator is the saturation one. It is usually assumed that a “desired” input ω is achieved by some feedback. This “desired” input ω then enters an operator which provides the “actual control”. For instance in the scalar case the (symmetric) saturation operator is ⎧ ⎪ ¯ if u ≥ u ¯ ⎪ ⎨ u Satu¯ (u) = u if −¯ u≤u≤u ¯ ⎪ ⎪ ⎩ −¯ u if u ≤ −¯ u

(46)

The saturation operator has an obvious multi–input version. Conversely if the problem is rate–bounded, a more sophisticated model is necessary. An operator model to this purpose is proposed in [19] and another approach has been proposed by [2]. The corresponding operator is now a dynamic one. If w is the desired signal and u the actual signal, assuming |u| ˙ ≤ v¯, the operator depicted in Figure 1 can be expressed as u(t) ˙ = satv¯ [ω − u]

r

ω

(47)

u

v v

F(s)

y

Fig. 1. Rate bounding operator

Given these operators, the problem normally consists on the determination of a controller which produces the “desired control value” ω in such a way that there are no bad consequences once this is saturated. Here, a different approach is used. We produce a control law which satisﬁes both amplitude and rate constraints. In other words, we do not make use of any saturation

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operator, but we will derive state feedback control laws which respect the given constraints.

3.2 The Continuous-Time Case To cope with slew–rate limitations and formulate the problem in the settings of state constrained systems previously introduced, we use the following extended system: x(t) ˙ A(w(t)) B(w(t)) x(t) 0 = + v(t) (48) u(t) ˙ 0 0 u(t) I This extended system is simply obtained from the original one by adding an integrator on the input of the original system. Therefore, the control u(t) becomes a new state variable and a new input vector v(t) is deﬁned. By doing so, the original constraint u ∈ U has become a state constraint for the new system and the slew-rate constraint for the original system, u(t) ˙ ∈ V , has now become an input constraint, say v(t) ∈ V . It is clear that the stabilization of the original constrained system (1) with bounded input and slew–rate constraints requires the stabilization of the system without slew– rate constraints, which, to the light of the results of the previous section, is equivalent to the existence of a polytopic contractive set of the form P = {x = Xα, α 1 ≤ 1}

(49)

that doesn’t take into account the slew-rate constraints. In view of the equivalence between the robust and gain scheduling stabilization (and the switched gain scheduling control in the case of uncertain B is certain), we assume constrained robust stabilization and, precisely, we assume the existence of a contractive P , namely the existence of a matrix U ∈ IRp×r , whose columns are in U, and s matrices Hi ∈ H such that the following equation Ai X + Bi U = XHi

(50)

is satisﬁed as well as the conditions (11). Now, we extend equation (50) to check what happens by considering the slew–rate constraints. To this aim, assuming for brevity that the matrix

Set Based Control Synthesis for State and Velocity Constrained Systems

M=

X

357

(51)

U

is full row rank3 , it sufﬁces to take s matrices Vi ∈ IRp×r such that, for any i, X Ai Bi X 0 Hi (52) + Vi = U 0 0 U I It is straightforward that the above is simply obtained by choosing (53)

Vi = U Hi

Now, equation (52) implies that the extended system admits a linear variable structure gain scheduled control law of the form (54)

v(t) = φ(x, u, w)

which depends on the parameter w and satisﬁes the constraints on the state, input and output variables. Because u(t) is nothing but the integral of v(t), this means that by applying u(t) ˙ = v(t), v(t) = φ(x, u, w)

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it is possible to achieve the same closed loop we would achieve by applying the static controller (54) to the augmented system. The above is summarized in the next proposition: Proposition 1. Assume the gain scheduled synthesis problem for system (1), has a solution with a single matrix U (i.e. the robust problem) and that the matrix M is full row rank. Assume furthermore that (52) holds, that the columns of X and U satisfy the state, input and output constraints (9) and that each column vi of V is such that vi ∈ V . Then, for every x(0) ∈ Co(−X, X) there exists u(0) ∈ U such that the closed loop evolution, when the gain scheduled linear variable structure control law (44) is applied: v = Φ(x, u, w) =

s i=1

wi Khi

x u

is such that v(t) ∈ V , x(t) ∈ X , u(t) ∈ U and y(t) ∈ Y , for any w(t) ∈ W and t ≥ 0. 3

If this is not the case, then it is possible to augment such equation, similarly to what has been done in a different setting in [9]

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The technique proposed has the following shortcomings: • The control is not robust anymore since the knowledge of w is required. Indeed, also if we have a single matrix U satisfying equation (10), the extended system admits an extended contractive set Q represented by the matrix M which satisﬁes the appropriate equation with matrices Vi depending on i. • When calculating the set P , slew-rate constraints are neglected and therefore there is no guarantee that u˙ ∈ V since this is checked a-posteriori. Indeed, depending on the problem, the columns of V might not belong to V . • For a given x(0), a proper initialization value for u(0) has to be found. The ﬁrst issue can be solved, in the robust case, by resorting to the results in [6] (and reported in the previous section in Theorem 1) concerning the equivalence between robust and gain scheduling synthesis. Such equivalence relies on the approximation of the extended polytopic region P by a smoothed version P˜ which allows to use a gradient based control law (see also [7]) which is based on the state only (which, in the present case, is composed by the actual state, x, and by the input u) and not on the parameter w. Note that the extended input matrix is [0 I]T which is independent of w and therefore the gradient based controller proposed in [7] can be applied. A more appropriate approach, which takes into account also the second issue, is to start directly by considering the extended system. Precisely we apply the iterative procedure to compute matrices satisfying the equations Ai Bi X X 0 Hi + Vi = U 0 0 U I We assume Vi = V if we are interested in the robust case. The above conditions are equivalent to the solution of the gain scheduled stabilization problem for the extended system and, in view of the equivalence, also to the robust stabilization problem of the extended system, say it is possible to derive a control law v (though not continuous). For a more in depth analysis of the relations between the extended and original system the reader is referred to [9]. Concerning the last point, we postpone the discussion to Section 3.4. Summarizing the presented technique, we have the following synthesis procedure. 1. expand the original system to obtain system (52);

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359

2. calculate the largest polytopic contractive set Q (for a given contractivity β) for the system obtained in the previous step by means of the procedures presented Section 2.1; 3. use the controller (56)

u(t) ˙ = v(t) v(t) = φ(x, u)

(or φ(x, u, w)), where φ is a variable structure linear controller for the expanded system. The so derived control law is such that for every initial (extended) state [xT uT ]T the state evolution satisﬁes the given amplitude and slew–rate constraints and converges to zero as in (22). For a given β, since the so computed contractive is the largest β contractive set, there are no other initial states outside Q starting from which the same convergence is assured while respecting the constraints. 3.3 The Discrete-Time Case In the discrete-time case, the extension which allows to derive a stabilizing controller is

x(t + 1) u(t + 1)

=

A(w(t)) B(w(t)) 0

I

x(t) u(t)

+

0 I

v(t)

(57)

Once again, slew-rate constraints have been transformed into the new input constraints and the original input constraints have been transformed into state constraints and thus it is possible to proceed exactly as in the continuous-time case. Unfortunately, introducing a one step delay in the actual system input is not a payless operation. In fact, the actual input value u(t) can be set only by proper choice of the value of the (fake) input variable at the previous time instant, v(t − 1). The next theorem holds. Theorem 3. Assume the extended system (57) can be stabilized by means of a constrained control law v(t) = φ(x(t), u(t), w(t)) (or in particular v(t) = φ(x(t), u(t))). Then, the original system can be stabilized by a control law complying with the constraints (45). The set of all initial control–states for which the constrained convergence is assured is 8 x X Q = = β, β 1 ≤ 1 . u U

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3.4 The Initialization Problem We now wish to go back to the problem of the initial control value. It is indeed quite clear that if control rate are present, the possibility of initializing the control value is a key point in the determination of the set of states that can be brought to the origin without constraints violation (either state, input, output or rate). More precisely, in applying the previous results, we have that convergence is guaranteed under bounds for all states of the extended systems namely for all pair (xT , uT )T ∈ Q . Therefore the convergence to the origin with constraints satisfaction does not depend only on the initial state but also on the initial control value. Therefore we can distinguish two cases: • the control value can be initialized; • the control value is initially assigned. The ﬁrst cases, which seems more unrealistic, can be indeed justiﬁed if one assumes that the “rate bound” is not a hard constrains but rather a soft performance speciﬁcation which can be instantaneously broken. In this way a “high” rate initial control variation is admitted (a step) to consequently assure a bounded variation. In the second case the control value is given u(0) = u ¯. The two different cases reﬂect in the set of states which can be brought to the origin without constraint violation, according to the next result: Theorem 4. Assume that for a slew–rate constrained synthesis problem the largest λ (resp. β in the continuous–time case) contractive set Q for the extended system has been found. Then the set X {0} of initial values x(0) starting from which the state evolution converges to the origin at least as in (22) without constraints violation is given by • the projection of Q on IRn : X {0} = {x : [xT uT ]T ∈ Q for some u} • the intersection of Q with the plane u = u ¯. 9 X {0} = X7 = Q {[xT uT ]T : u = u ¯}

(59)

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361

Proof. The ﬁrst item is rather obvious to the light of what has been said in the previous section, and we focus on the second one only. If u(0) = u ¯ is ﬁxed, then since the computed set is the largest λ contractive (resp. β) set, constraint satisfaction and convergence to the origin of the state evolution for the extended system as in (22) is satisﬁed only for the states [xT uT ]T ∈ Q . Since the u component is ﬁxed, this in turn requires x ∈ X7 .

4 Examples The next very simple example shows how the amplitude–constrained domain of attraction shrinks under slew–rate contraints. Example 1. Consider the continuous–time system (60)

x(t) ˙ = x(t) + u(t) with |u(t)| ≤ 1. A domain of attraction is the set P = {x : |x| ≤ a}

(61)

where a < 1. Let us now investigate on what happens if we impose |u(t)| ˙ ≤ v¯. We need to add the equation (62)

u(t) ˙ = v(t) obtaining the system

x(t) ˙ u(t) ˙

=

11 00

x(t) u(t)

+

0 1

v(t)

(63)

where |v(t)| ≤ v¯. In Figure 2 we can see the domains of attraction of system (63) for v¯ = 2, 1, 0.5, 0.1. Such sets are computed by choosing τ = 0.1 and setting λ = 0.9999 and = 0.00001. The lines resemble continuous curves since the vertices of the polytopes are really close (when v¯ = 0.1 there are more than 350 vertices). If there were no constraints on v(t) (¯ v = ∞) we would have aspected a squared domain of attraction {(x, u) : |x| ≤ 1, |u| ≤ 1}. By choosing a ﬁnite value of v¯ and decreasing it we can observe how the domain of attraction shrinks.

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1 0.8 0.6 0.4

u

0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

−0.8

−0.6

−0.4

−0.2

0 x

0.2

0.4

0.6

0.8

1

Fig. 2. Domains of attraction for system (63) with the given constraints.

In the next example, the behavior of the bounded slew–rate control law derived in the previous sections is applied to a cart-pendulum system. Example 2. Consider the the classical example of the inverted pendulum depicted in Figure 3. The problem that we analyze is stabilizing the position of the pendulum to the top vertical position. The pendulum has a concentrated mass m = 1 Kg and a connecting member length L = g (where g represents the acceleration of gravity). A disturbance z acts on the center of gravity of the pendulum. The cart is characterized by a mass M = 1 Kg. A force u is applied on the cart to control the system. The speciﬁcation considered are the following • actuator magnitude saturation |u(t)| ≤ 1.25; • actuator rate saturation u(t) ˙ ≤ 10. By choosing the state variables to be the pendulum horizontal speed x1 , the horizontal displacement between the cart and the pendulum x2 and the horizontal speed of the cart x3 , we can linearize the system equations at the origin obtaining the equations

Set Based Control Synthesis for State and Velocity Constrained Systems

363

Fig. 3. Pendulum state variables.

x˙ 1 (t) = −x2 (t) + z(t)

(64)

x˙ 2 (t) = −x1 (t) + x3 (t)

(65)

x˙ 3 (t) = x2 (t) + u(t)

(66)

To consider the actuator rate saturation we augment the system obtaining ⎡

x˙ 1 (t)

⎤

⎡

⎡ ⎤ ⎡ ⎤ 1 0 ⎢ ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ 0 1 0⎥ ⎥ ⎢ x2 (t) ⎥ + ⎢ 0 ⎥ v(t) + ⎢ 0 ⎥ z(t) ⎢ ⎢0⎥ ⎥ ⎢ ⎥ 1 0 1⎥ ⎣ ⎦ ⎦ ⎣ x3 (t) ⎦ ⎣ 0 ⎦ 0 000 1 u(t)

0 −1 0 0

⎥ ⎢ ⎢ ⎢ x˙ 2 (t) ⎥ ⎢ −1 ⎥ ⎢ ⎢ ⎢ x˙ (t) ⎥ = ⎢ 0 ⎣ 3 ⎦ ⎣ 0 u(t) ˙

⎤⎡

x1 (t)

⎤

(67)

where u(t) becomes a state variable and v(t) is the new input. By using the techniques previously shown, we ﬁrst investigate on which is a proper value of the bandwith that can be achieved with a variable structure control. For this purpose we ﬁrst neglect the disturbance input and we impose a bandwidth of B = 2 rad/s. It can be shown that a worst case bound on the bandwidth is given by B=

1−λ τ

(this bound is really conservative). We choose τ = 0.1 and λ = 0.8. The

resulting contractive set is really complex (it is deﬁned by more than 2000) so we lower the requirement on B.

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Franco Blanchini, Stefano Miani, and Carlo Savorgnan

For B = 1 rad/s, by choosing τ = 0.1 and λ = 0.9, the resulting contractive set is simpler (it counts 144 inequalities and 618 vertices). To test the control obtained for the calculated set, it can be interesting simulate the system starting from an initial state corresponding to the steady condition where x2 attains the maximal value such that the pendulum reaches the vertical position. This value can be therefore calculated by choosing the greatest value of x2 such that the point (0, 0, x2 , 0) is contained in the contractive set. The value obtained is x2 = 0.168. In Figure 4, the intersection of the contractive set with the hyperplanes x3 = 0 and u = 0 is shown. 1 0.8 0.6 0.4

x2

0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

−0.8

−0.6

−0.4

−0.2

0 x1

0.2

0.4

0.6

0.8

1

Fig. 4. Intersection between the hyperplanes x3 = 0, u = 0 and the domain of attraction (B = 1 rad/s).

Figure 5 shows the simulation with the initial condition (x1 , x2 , x3 , u)0

=

(0, 0.168, 0, 0). It can be observed how the value of v(t) (the solid line in the bottom graph) reaches the minimum allowed. By reintroducing the disturbance we can now investigate on which is the maximal magnitude of z(t) (referred to as zmax ) that is allowed to achieve the stability of the system without breaking the constraints. Using the bisection method, the maximal value is found to be in the open interval (0.09, 0.1). The calculated domain of

Set Based Control Synthesis for State and Velocity Constrained Systems

365

0.2 0.15

x2

0.1 0.05 0 −0.05

0

1

2

3

4

5 t

6

7

8

9

10

0

1

2

3

4

5 t

6

7

8

9

10

5

u,v

0

−5

−10

Fig. 5. Simulation of system (67) obtained for (x1 , x2 , x3 , u)0 = (0, 0.168, 0, 0) and z(t) = 0. In the top graph x2 (t) is represented by a solid line. In the bottom graph u(t) and v(t) correspond respectively to the dashed and solid line.

attraction (τ = 0.1 and λ = 0.8) is a polytope deﬁned by 102 inequalities and 428 vertices. Figure 6 shows the evolution of x2 (t), z(t), u(t) and v(t) starting form a null initial condition when a persistent disturb z(t ≥ 1) = 0.09 is applied. We can observe that the constraints are far from being broken. This follows from the fact that only certain signals z(t) lead the control input to act close to the constraints. The control we obtained by applying the technique explained in the previous sections guarantees indeed that for every signal |z(t)| ≤ zmax the system is stable and satisﬁes the constraints. Allowing z(t) to be any signal involves a drawback: when the designer is aware that the disturbance has certain properties, these information can’t be taken into account and the results obtained become quite conservative. To remark this fact, the simulation illustrated in Figure 7 shows that also when |z(t)| ≤ zmax is not veriﬁed the control calculated may work well. For example, when 0 f or t < 1 z(t) = 0.5 f or t ≥ 1 the constraints are not broken.

(68)

366

Franco Blanchini, Stefano Miani, and Carlo Savorgnan 0.1 0.08

x2, z

0.06 0.04 0.02 0 −0.02

0

1

2

3

4

5 t

6

7

8

9

10

0

1

2

3

4

5 t

6

7

8

9

10

1

u, v

0.5

0

−0.5

Fig. 6. Simulation of system (67) applying a persistent disturb z(t ≥ 1) = 0.09. In the top graph x2 (t) and z(t) are represented respectively by a dashed and a solid line. In the bottom graph u(t) and v(t) correspond to the dashed and solid line. 0.6

x2, z

0.4

0.2

0

−0.2

0

1

2

3

4

5 t

6

7

8

9

10

0

1

2

3

4

5 t

6

7

8

9

10

6

u, v

4 2 0 −2 −4

Fig. 7. Simulation of system (67) applying a persistent disturb z(t ≥ 1) = 0.5. In the top graph x2 (t) and z(t) are represented respectively by a dashed and a solid line. In the bottom graph u(t) and v(t) correspond to the dashed and solid line.

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References 1. A. Bemporad, M. Morari, V. Dua, and E. Pistikopoulos. The explicit linear quadratic regulator for constrained systems. Automatica J. IFAC, 38(1):3–20, 2002. 2. J. M. Berg, Hammet K. D., Schwartz C. A., and S. Banda. An analisys of the destabilizing effect of daisy chained rate-limited actuators. Trans. Control Systems Technology, 4(2), 1996. 3. D. P. Bertsekas and I. B. Rhodes. On the minmax reachability of target set and target tubes. Automatica J. IFAC, 7:233–247, 1971. 4. F. Blanchini. Non-quadratic lyapunov function for robust control. Automatica J. IFAC, 31(3):451–461, 1995. 5. F. Blanchini. Set invariance in control – a survey. Automatica J. IFAC, 35(11):1747–1767, 1999. 6. F. Blanchini. The gain scheduling and the robust state feedback stabilization problems. IEEE Trans. Automat Control, 45(11):2061–2070, 2000. 7. F. Blanchini and S. Miani. Constrained stabilization via smooth lyapunov functions. Systems Control Lett., 35:155–163, 1998. 8. F. Blanchini and S. Miani. Stabilization of LPV systems: state feedback, state estimation and duality. SIAM J. Control Optim., 42(1):76–97, 2003. 9. F. Blanchini, S. Miani, and C. Savorgnan. Stability results for continuous and discrete time linear parameter varying systems. In Proc. of the 2005 IFAC conference, page CDROM, Prague, 2005. 10. Y. Cao, Z. Lin, and Y. Shamash. Set invariance analysis and gain-scheduling control for LPV systems subject to actuator saturation. Systems Control Lett., 46(2):137–151, 2002. 11. R. Freeman and L. Praly. Integrator backstepping for bounded controls and control rates. IEEE Trans. Automat Control, 43(2):256–262, 1998. 12. E. G. Gilbert and K. K. Tan. Linear systems with state and control constraints: the theory and the applications of the maximal output admissible sets. IEEE Trans. Automat Control, 36(9):1008–1020, 1991. 13. J. M. Gomes da Silva Jr., S. Tarbouriech, and Garcia G. Local stabilization of linear systems under amplitude and rate saturating actuators. IEEE Trans. Automat Control, 48(5):842–847, 2003. 14. P. Gutman and M. Cwikel. Admissible sets and feedback control for discrete-time linear dynamical systems with bounded controls and states. IEEE Trans. Automat Control, 31(4):373–376, 1986. 15. D. Henrion, S. Tarbouriech, and V. Kucera. Control of linear systems subject to input constraints: a polynomial approach. Automatica, 37(4):597–604, 2001.

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16. J-B. Hiriart-Urruty and C. Lemarc´echal. Convex analysis and minimization Algorithms I and II. Springer-Verlag, 1993. 17. T. Hu and Z. Lin. Control Systems with Actuator Saturation: Analysis and Design. Birkhauser, Boston, 2001. 18. F. Mesquine, F. Tadeo, and A. Benzaouia. Regulator problem for linear systems with constraints on control and its increment or rate. Automatica, 40(8):1387–1395, 2004. 19. A. Saberi, A. Stoorvogel, and P. Sannuti. Communication and control engineering series. In Control of linear systems with regulation and input constraints. Springer-Verlag London, Ltd., London, 2000. 20. C. Savorgnan.

Maxis-g software package.

http://www.diegm.uniud.it/savorgnan or

http://www.diegm.uniud.it/smiani/Research.html, 2005. 21. J. Solis-Daun, R. Suarez, and J. Alvarez-Ramirez. Global stabilization of nonlinear systems with inputs subject to magnitude and rate bounds: a parametric optimization approach. SIAM J. Control Optim., 39(3):682–706, 2000. 22. H. J. Sussmann, E. D. Sontag, and Y. Yang. A general result on the stabilization of linear systems using bounded controls. IEEE Trans. Automat Control, 39:2411–2425, 1994.

Output Feedback for Discrete-Time Systems with Amplitude and Rate Constrained Actuators J.M. Gomes da Silva Jr.1 , D. Limon2 , T. Alamo2 , and E.F. Camacho2 1

Department of Electrical Engineering, UFRGS, Av. Osvaldo Aranha 103, 90035-190 Porto Alegre-RS, Brazil [email protected]

2

Dept. Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Camino de los Descubrimentos s/n, 41092 Sevilla, Spain limon,alamo,[email protected]

Summary. The aim of this chapter is the presentation of a technique for the design of stabilizing dynamic output feedback controllers for discrete-time linear systems with rate and amplitude saturating actuators. Two synthesis objectives are considered: the maximization of the domain of attraction of the closed-loop system under some time-domain performance constraints; and the guarantee that the trajectories of the system are bounded in the presence of L2 -bounded disturbances, while ensuring an overbound for the L2 gain from the disturbance to the regulated output. The nonlinear effects introduced by the saturations in the closed-loop system are taken into account by using a generalized sector condition, which allows to propose theoretical conditions to solve the synthesis problems directly in the form of linear matrix inequalities (LMIs). From these conditions, convex optimization problems are proposed in order to compute the controller aiming at the maximization of the basin attraction of the closed-loop system, as well as aiming at ensuring a level of L2 disturbance tolerance and rejection. Numerical examples are provided to illustrate the application of the proposed methodology. Keywords: constrained control, control saturation, output feedback, stabilization, discretetime systems.

1 Introduction The physical impossibility of applying unlimited control signals makes the actuator saturation an ubiquitous problem in control systems. In particular, it is well known

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J.M. Gomes da Silva Jr., D. Limon, T. Alamo, and E.F. Camacho

that the input saturation is source of performance degeneration, limit cycles, multiple equilibrium points, and even instability. Hence, it was great the interest in studying these negative effects and also in proposing control design procedures, in global, semiglobal and local contexts of stability, taking directly into account the control bounds: see for instance [20], [10], [11], and references therein. It should be pointed out that most of these works consider only input amplitude saturation and state feedback control strategies. Although the proposition of state feedback methods allow to have a good insight into the problem, the practical applicability of these methods is limited. On the other hand the works proposing output feedback strategies consider, in general, observer-based control laws ensuring global or semi-global stabilization. However, when the open-loop system is not null controllable or additional performance and robustness requirements have to be satisﬁed, local (or regional) stabilization approaches are needed and an implicit additional objective is the enlargement of the basin of attraction of the closed-loop system. In this context, the amount of works proposing output feedback control strategies is even smaller. Works formally addressing the stabilization in the presence of both amplitude and rate saturation started to appear late 90s. Global and semi-global stabilization results using both state feedback and observer-based control laws were proposed in [14], [18] and [19]. Concerning a local stabilizing context, we can cite the results presented in [8], [1], [21], where the synthesis of state feedback control laws are proposed. On the other hand, the synthesis of dynamic output feedback controllers ensuring local stability is considered in [22] and [15]. In [22], a method for designing dynamic output controllers using of the Positive Real Lemma is proposed. The main objective pursued in that paper is the minimization of an LQG criterion. A region of stability is associated to the closed-loop system. However, it should be pointed out that the size and the shape of this region are not taken into account in the design procedure, which can lead to very conservative domains of stability. Furthermore, the controller is computed from the solution of strong coupled Riccati equations which, in general, are not simple to solve. A time-varying dynamic controller is proposed in [15]. Considering amplitude bounded disturbances, the main focus of that paper is the disturbance rejection problem. The problem of characterization of estimates of the region of attraction is not explicitly discussed. The stabilizing conditions are obtained from a polytopic differential inclusion representation of the saturation terms. Hence, considering the parameter that deﬁnes the region of validity of the polytopic

Output Feedback with Amplitude and Rate Constrained Actuators

371

representation, the derived conditions are BMIs, which implies the use of iterative LMI relaxation schemes for computing the controller. On the other hand, it should be pointed out that all the references above are concerned only with continuoustime systems and the rate limitation is considered in the modelling of the actuator, i.e. a position-feedback-type model [22] is considered. In this case, the rate saturation is modelled, in fact, as a saturation of the actuator state. Hence, the plant plus the actuator appears as a nonlinear system which renders the formal analysis in the sampled-data control case quite involved. An alternative approach consists therefore in designing a discrete-time nonlinear controller (i.e. considering saturations in the controller) in order to prevent that control signal (to be sent to the actuator) violates the rate and amplitude bounds. The aim of this chapter is the proposition of a technique for the design of stabilizing dynamic output feedback controllers for discrete-time linear systems with rate and amplitude constrained actuators. Two implicit design objectives are particularly considered. The ﬁrst one concerns the synthesis of the controller aiming at the maximization of the region of attraction of the closed-loop system, while guaranteeing a certain degree of time-domain performance for the system operation in a neighborhood of the origin (equilibrium point). The second one concerns the disturbance tolerance and rejection problems. Since the control signals are bounded, the proposed conditions allows to ensure that the system trajectories are bounded for given L2 -bounded disturbances acting on the system. Furthermore, the synthesis can be carried out in order to minimize the L2 gain from the disturbance to the regulated output. The results apply to both stable and unstable open-loop systems. In order to deal with the rate limitation, we propose the synthesis of a nonlinear dynamic controller which is composed by a classical linear dynamic controller in cascade with input saturating integrators and two static anti-windup loops. It should be pointed out that, differently from the anti-windup approaches (see for instance [12], [9],[7], and references therein), where the controller is supposed to be given, here the idea consists in computing simultaneously the controller and the anti-windup gains. The anti-windup gains appear therefore as extra degrees of freedom in the synthesis problem. The theoretical conditions for solving the synthesis problem are based on a generalized sector condition proposed in [7]. This condition encompasses the classical sector condition, used for instance in [13], and allows (differently from the classical

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J.M. Gomes da Silva Jr., D. Limon, T. Alamo, and E.F. Camacho

one) the formulation of local stability conditions directly in LMI form. Using then the classical variables transformations as proposed in [17] and [3], it is possible to formulate conditions that allow to compute a dynamic controller that stabilizes the closed-loop system. Convex optimization problems are therefore proposed in order to compute the controller aiming at the maximization of the basin attraction of the closed-loop system, as well as aiming at ensuring a level of L2 disturbance tolerance and rejection. Numerical examples are provided to illustrate the application of the proposed method.

2 Problem Statement Consider the discrete-time linear system x(t + 1) = Ax(t) + Bu(t) + Bw w(t) y(t) = Cx(t)

(1)

z(t) = Cz x(t) + Dz,u u(t) + Dz,w w(t) where x(t) ∈ n , u(t) ∈ m , y(t) ∈ p , z(t) ∈ r , w(t) ∈ q are the state, the input, the measured output, the regulated output and the disturbance vectors, respectively, and t ∈ N . Matrices A, B, Bw , C, Cz Dz,u and Dz,w are real constant matrices of appropriate dimensions. Pairs (A, B) and (C, A) are assumed to be controllable and observable respectively. The input vector u is subject to limitations deﬁned as follows. • Amplitude constraints: |u(i) (t)| ≤ ρa(i) , i = 1, ..., m

(2)

where ρa(i) > 0, i = 1, ..., m denote the control amplitude bounds. • Rate constraints: |Δu(i) (t)| = |u(i) (t) − u(i) (t − 1)| ≤ ρr(i) , i = 1, ..., m

(3)

where ρr(i) > 0, i = 1, ..., m denote the control rate bounds. The disturbance vector w(t) is assumed to be limited in energy, that is w(t) ∈ L2 and for some scalar δ, 0 <

1 δ

< ∞, one has:

Output Feedback with Amplitude and Rate Constrained Actuators

w

22 =

∞

w(t) w(t) ≤

t=0

1 δ

373

(4)

We suppose that only the output y(t) is available for measurement. Hence our aim is to compute a stabilizing dynamic compensator. In order to cope with the rate constraints, in the sequel we consider a controller composed by an n + m order dynamic compensator in cascade with m input saturating integrators and two anti-windup loops, as follows: v(t + 1) = Im v(t) + satρr (yc (t)) xc (t + 1) = Ac xc (t) + Bc [y(t) v(t) ] + Ec (satρa (v(t)) − v(t)) +Fc (satρr (yc (t)) − yc (t))

(5)

yc (t) = Cc xc (t) + Dc [y(t) v(t) ] where xc (t) ∈ n+m is the dynamic compensator state, yc (t) ∈ m is the compensator output, matrices Ac , Bc , Cc , Dc , Ec and Fc have appropriate dimensions. Ec and Fc are anti-windup gains. satρ (·) : m → m is a componentwise saturation map deﬁned as follows: (satρ (v))(i) = satρ(i) (v(i) ) = sign(v(i) ) min(ρ(i) , |v(i) |), ∀i = 1, ..., m, where ρ(i) , denotes the ith bound of the saturation function. As a consequence of the amplitude control bounds, the effective control signal applied to system (1) is a saturated one: u(t) = satρa (v(t))

(6)

The whole closed-loop system is depicted in Figure 1. The objective of this chapter is therefore to provide a methodology for computing the controller matrices Ac , Bc , Cc , Dc , Ec and Fc in order to address the following problems. • Maximization of the basin of attraction: maximize the domain of attraction of the closed-loop system under some time-domain performance constraints. • Tolerance and disturbance rejection: ensure that the trajectories of the system are bounded for any disturbance satisfying (4) and, in addition, ensure an overbound for the L2 gain from the disturbance to the regulated output.

374

J.M. Gomes da Silva Jr., D. Limon, T. Alamo, and E.F. Camacho v(t) m-integrators

−

− +

Plant

+

Ec

Fc yc (t)

y(t)

u(t)

Compensator

Dynamic Controller

Fig. 1. Closed-loop system

3 Preliminaries For a given vector α ∈ m deﬁne function ψη (α) : m → m as

ψη (α) = α − satη (α). Note that ψη (α) corresponds to a decentralized deadzone nonlinearity ψη (α) = (ψη (α))(1) ... (ψη (α))(m) with

(ψη (α))(i)

⎧ ⎪ ⎪ ⎨ α(i) − η(i) if α(i) > η(i) = 0 if −η(i) ≤ α(i) ≤ η(i) i = 1, . . . , m ⎪ ⎪ ⎩α + η (i) (i) if α(i) < −η(i)

(7)

Deﬁne now the following vectors and matrices x x ˜ x ˜= ; ξ= v xc A=

A B

B

Bw

0

C 0

; Bw = ; B1 = ; C= ; B= 0 0 Im 0 Im 0 Im L = 0 Im ; Cz = Cz Dz,u ; Dz,u = Dz,u ; Dz,w = Dz,w

Output Feedback with Amplitude and Rate Constrained Actuators

A=

A + BDc C BCc

B1

Bw

; B1 = ; Bw = ; B= Ac Ec 0 L = L 0 ; K = Dc C Cc ; C = Cz 0

Bc C

B

375

Fc

Dz,u = Dz,u ; Dz,w = Dz,w From the deﬁnitions above, the closed-loop system can be re-written as ξ(t + 1) = Aξ(t) − B1 ψρa (Lξ(t)) − Bψρr (K ξ(t)) + Bw w(t) z(t) = Cz ξ(t) − Dz,u ψρa (Lξ(t)) + Dz,w w(t)

(8)

Considering the generic nonlinearity ψη (α) and deﬁning the set S(η) = {α ∈ m , β ∈ m ; |α(i) − β(i) | ≤ η(i) , i = 1, . . . , m}

(9)

the following lemma can be stated [7]. Lemma 1. If α and β are elements of S(η) then the nonlinearity ψη (α) satisﬁes the inequality: ψη (α) T (ψη (α) − β) ≤ 0

(10)

for any diagonal positive deﬁnite matrix T ∈ m×m . Lemma 2. Consider the following system composed by m-integrators: v(t + 1) = Im v(t) + q(t) u(t) = satρa (v(t)) If |q(t)(i) | ≤ ρr(i) , i = 1, ..., m, it follows that |Δu(i) (t + 1)| = |u(i) (t + 1) − u(i) (t)| ≤ ρr(i) Proof: Considering that the Lipschitz constant of the sat(·) function is equal to 1, it follows directly that |Δu(i) (t+1)| = |satρa(i) (v(i) (t)+q(i) (t))−satρa(i) (v(i) (t))| ≤ |q(t)(i) | ≤ ρr(i) 2

376

J.M. Gomes da Silva Jr., D. Limon, T. Alamo, and E.F. Camacho

4 Stabilization In this section we focus on the synthesis of the controller (5) in order to guarantee the asymptotic stability of the closed-loop system (8). With this aim we consider the disturbance free case, i.e., w(t) = 0, ∀t ≥ 0. A result concerning the local (or regional) stabilization is presented. This result provides LMI conditions that can be used in order to compute the controller matrices and an associated region of stability for the closed-loop system. It can therefore be applied to either stable and unstable open-loop systems. Finally, it is shown how the derived LMI conditions can be cast in optimization problems for computing the controller in order to maximize an ellipsoidal estimate of the region of attraction of the closed-loop system, while ensuring a certain timedomain performance in the linear region of the closed-loop system.

4.1 Theoretical Result Theorem 1. If there exist symmetric positive deﬁnite matrices X, Y

(n+m)×(n+m)

ces Aˆ ∈

∈

, positive deﬁnite diagonal matrices Sr , Sa ∈ , and matrim×(n+m) (n+m)×(p+m) ˆ ∈ ˆ ∈ m×(p+m) , , Cˆ ∈ ,B ,D m×m

(n+m)×(n+m)

Zr1 , Zr2 , Za1 , Za2 ∈ m×(n+m) , Qr , Qa ∈ (n+m)×m such that the following linear matrix inequalities are veriﬁed 3 ⎡ X ⎢ ⎢ In+m Y ⎢ ⎢ Zr1 Zr2 ⎢ ⎢ ⎢ Za1 Za2 ⎢ ⎢ ˆ ⎣ AX + BCˆ A + BDC ˆ Aˆ Y A + BC

2Sr

0

2Sa

BSr B1 Sa Qr

X

Qa In+m

⎡ ⎢ ⎢ ⎣

3

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥>0 ⎥ ⎥ ⎥ ⎦ Y

(11)

⎤ X

In+m

Y

⎥ ⎥ ⎦≥0

(12)

ˆ (i) C − Zr2(i) ρ2 Cˆ(i) − Zr1(i) D r(i)

stands for symmetric blocks; • stands for an element that has no inﬂuence on the development

Output Feedback with Amplitude and Rate Constrained Actuators

⎡ ⎢ ⎢ ⎣

377

⎤ X

In+m

Y

⎥ ⎥ ⎦≥0

(13)

X(n+i) − Za1(i) [0 Im ](i) − Za2(i) ρ2a(i)

for all i = 1, ..., m, then the dynamic controller (5) with Fc = N −1 (Qr Sr−1 − Y B) Ec = N −1 (Qa Sa−1 − Y B1 ) ˆ Dc = D Cc = (Cˆ − Dc CX)(M )−1 ˆ − Y BDc ) Bc = N −1 (B 5 6 Ac = N −1 Aˆ − (Y AX + Y BDc CX + N Bc CX + Y BCc M ) (M )−1

(14)

where matrices M and N verify N M = In+m − Y X, guarantees that the region

(15) E(P ) = {ξ ∈ 2(n+m) ; ξ P ξ ≤ 1} Y N X M with P = and P −1 = , is a region of asymptotic stability for N • M • the closed-loop system (8). Proof: Consider the closed loop system (8) and the candidate Lyapunov function V (ξ(t)) = ξ(t) P ξ(t), P = P > 0. Assuming w(t) = 0, ∀t ≥ 0, the variation of V (ξ(t)) along the trajectories of system (8) is given by ΔV (ξ(t)) = V (ξ(t + 1)) − V (ξ(t)) = −ξ(t) P ξ(t) + ξ(t) A P Aξ(t) − 2ξ(t) A P B1 ψρa (Lξ(t)) −2ξ(t) A P Bψρr (K ξ(t)) + ψρr (K ξ(t)) B P Bψρr (K ξ(t)) +ψρa (Lξ(t)) B1 P B1 ψρa (Lξ(t)) + 2ψρa (Lξ(t)) B1 P Bψρr (K ξ(t)) (16) Consider matrices Ga , Gr ∈ m×(2(n+m)) and deﬁne now the following sets:

Ξ(ρa ) = {ξ ∈ 2(n+m) ; |L(i) ξ − Ga(i) ξ| ≤ ρa(i) , i = 1, ..., m}

Ξ(ρr ) = {ξ ∈ 2(n+m) ; |K(i) ξ − Gr(i) ξ| ≤ ρr(i) , i = 1, ..., m} From Lemma 1, provided that ξ(t) ∈ Ξ(ρa ) ∩ Ξ(ρr ), it follows that ΔV (ξ(t)) ≤ ΔV (ξ(t)) − 2ψρr (K ξ(t)) Tr [ψρr (K ξ(t)) − Gr ξ(t)] −2ψρa (Lξ(t)) Ta [ψρa (Lξ(t)) − Ga ξ(t)]

(17)

378

J.M. Gomes da Silva Jr., D. Limon, T. Alamo, and E.F. Camacho

For ease of notation in the sequel we denote ψr (t) = ψρr (K ξ(t)) and ψa (t) = ψρa (Lξ(t)). Therefore, expression (17) can be put in matrix form, as follows: ⎡ ⎤ ⎡ ⎤ −A P ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ B ⎥ P −A B B1 )θ(t) (18) − ΔV (ξ(t)) ≤ −θ(t) (⎣ −Tr Gr 2Tr ⎥ ⎦ ⎣ ⎦ B1

−Ta Ga 0 2Ta

where θ(t) = [ξ(t) ψr (t) ψa (t)]. From Schur’s complement it follows that ΔV (ξ(t)) < 0 if ξ(t) ∈ Ξ(ρa ) ∩ Ξ(ρr ) and

⎡

P

⎢ ⎢ −Tr Gr 2Tr ⎢ ⎢ −T G 0 2T ⎣ a a a −A Deﬁne now a matrix Π = lows that X − Y −1

⎤

⎥ ⎥ ⎥>0 ⎥ ⎦

(19)

B1 P −1

B

X In+m

[17],[3]. Note now that, from (11), it folM 0 > 0. Therefore, In+m − Y X is nonsingular. Thus it is always

possible to compute square and nonsingular matrices N and M verifying the equation N M = In+m − Y X. From the nonsigularity of M it is inferred that Π is nonsingular. Pre and post-multiplying (19) respectively by ⎡ ⎤ ⎡ −Π 0 −Π 0 0 0 ⎢ ⎥ ⎢ ⎢ ⎢ 0 Sr 0 0 ⎥ ⎥ and ⎢ 0 Sr ⎢ ⎢ 0 0 ⎢ 0 0 S 0 ⎥ ⎣ ⎦ ⎣ a 0 0 0 ΠP 0 0

0

0

⎤

⎥ 0 ⎥ ⎥ Sa 0 ⎥ ⎦ 0 PΠ 0

with Sa = Ta−1 and Sr = Tr−1 one gets: ⎡

Π P Π

⎤

⎢ ⎥ ⎢ Gr Π ⎥ 2S r ⎢ ⎥>0 ⎢ G Π ⎥ 0 2S ⎣ ⎦ a a Π P AΠ Π P BSr Π P B1 Sa Π P Π From the deﬁnition of Π, it follows that X I BS n+m r Π P Π = ; Π P BSr = In+m Y Y BSr + N Fc Sr

(20)

Output Feedback with Amplitude and Rate Constrained Actuators

Π P B1 Sa =

B 1 Sa Y B1 Sa + N E c Sa

; Π P AΠ =

Γ11 Γ12

379

Γ21 Γ22

with Γ11 = (A + BDc C)X + BCc M Γ12 = A + BDc C Γ21 = Y AX + Y BDc CX + N Bc CX + Y BCc M + N Ac M Γ22 = Y (A + BDc C) + N Bc C ˆ Cc M + Dc CX = Consider now the following change of variables Dc = D, ˆ Y AX+Y BDc CX+N Bc CX+Y BCc M +N Ac M = A, ˆ Y BDc +N Bc = B, ˆ C, Y BSr + N Fc Sr = Qr , Y B1 Sa + N Ec Sa = Qa , Gr Π = [Zr1 Zr2 ] and Ga Π = [Za1 Za2 ]. Hence, since Π, Sr and Sa are nonsingular, it follows that if (11) is veriﬁed, (19) holds with the matrices Ac , Bc , Cc , Dc , Ec and Dc deﬁned as in (14). On the other hand, considering that ˆ K Π = [Dc CX + Cc M Dc C] = [Cˆ DC] ˜ I] ˜ LΠ = [[0 Im ] 0]Π = [X

(21)

˜ and I˜ correspond respectively to the matrices composed by the last m lines where X pre and post-multiplying inequalities (12) and (13) of matrices X and In+m . Hence −1 (Π ) 0 respectively by and its transpose, it is easy to see that (12) and (13) 0 1 ensures respectively that E(P ) ⊂ Ξ(ρr ) and E(P ) ⊂ Ξ(ρa ) [2]. Thus, if relation (11), (12) and (13) are satisﬁed, one obtains ΔV (ξ(t)) < 0, ∀ξ(t) ∈ E(P ), ξ(t) = 0, which means that E(P ) is a contractive region for system (8), i.e., if ξ(0) ∈ E(P ), then the corresponding trajectory converges asymptotically to the origin.2

4.2 Optimization Problems According to theorem 1, any feasible solution of the set of LMIs (11), (12) and (13) provides a stabilizing, and probably different, dynamic controller. Between all these solutions, it is interesting to chose one that optimizes some particular objective.

380

J.M. Gomes da Silva Jr., D. Limon, T. Alamo, and E.F. Camacho

The objective considered in this section concerns the maximization of an ellipsoidal estimate of the closed-loop system domain of attraction. In particular, it can be of interest the maximization of the projection of E(P ) onto the states of the plant (i.e., x). This set is denoted as Ex (P ) and is given by Ex (P ) = {x ∈ n ; ∃v ∈ m , xc ∈ n+m ; [x v xc ] ∈ E(P )} −1 = {x ∈ n ; x X11 x ≤ 1}

where X11 ∈ n×n is obtained from X =

X11 X21 X22

.

Note that for any initial state x(0) ∈ Ex (P ), initial values of the states of the dynamic controller v(0) and xc (0) can be found such that ξ(0) ∈ E(P ), i.e. such that the asymptotically stability of the closed-loop system is ensured. In order to maximize the size of Ex (P ), we can for instance adopt one of the following criteria: −1 • Minimization of the trace of X11 . This can be indirectly accomplished by con-

sidering an auxiliary matrix variable R and the following optimization problem: min trace(R) s.t. (11), (12), (13),

R In In X11

>0

(22)

• Minor axis maximization. It can be accomplished by maximizing the smallest eigenvalue of X11 (which corresponds to minimize the largest eigenvalue of −1 X11 ) , as follows:

max λ s.t. (11), (12), (13), X11 ≥ λIn

(23)

• Maximization along certain directions. This can be accomplished by considering a shape set Ξ0 , which describes the directions in which the ellipsoid should be maximized and a scaling factor η > 0 [5],[6]. Considering a polyhedral shape set described by the convex hull of its vertices (i.e. directions in which Ex (P )

should be maximized) Ξ0 = Co{vr ∈ n ; r = 1, . . . , nv }, the optimization problem to be solved in this case is the following:

Output Feedback with Amplitude and Rate Constrained Actuators

max η

s.t. (11), (12), (13),

1 ηvr ηvr X11

≥ 0 , r = 1, . . . , nv

381

(24)

• Volume maximization. This can be accomplished from the following convex optimization problem: max log(det(X11 ))) s.t. (11), (12), (13)

(25)

Instead of the maximization of Ex (P ), another option consists in considering the maximization of the cut of E(P ) by the hyperplane plane x, denoted by Ec (P ) and deﬁned as

Ec (P ) = {x ∈ n ; [x 0 0] ∈ E(P )} = {x ∈ n ; x Y x ≤ 1}

Similar optimization problems to (22), (23), (24) and (25), can thus be formulate −1 . considering Y instead of X11

It is worth noticing that performance requirements such as contraction rate, pole placement of the closed-loop system or quadratic cost minimization can be added to the problem. In this case, the resulting optimization problem can be also written as an LMI optimization problem. For instance the controller can be designed in order to ensure some degree of timedomain performance in a neighborhood of the origin. In general we consider this neighborhood as the region of linear behavior of the closed-loop system [8], i.e., the region where saturation does not occur: RL = {ξ ∈ 2(n+m) ; |K(i) ξ| ≤ ρa(i) , |L(i) ξ| ≤ ρr(i) , i = 1, . . . , m} In this case the time-domain performance can be achieved if we consider the pole placement of matrix A in a suitable region inside the unit circle. Considering an LMI framework, the results stated in [17] can be used to place the poles in a called LMI region in the complex plane. For example, if we verify the following LMI, ⎡ ⎤ −σX −σIn+m ⎢ ⎥ ⎢ −σIn+m ⎥ −σY ⎢ ⎥<0 ⎢ AX + BCˆ A + BDC ˆ −σX −σIn+m ⎥ ⎣ ⎦ Aˆ

ˆ −σIn+m Y A + BC

−σY

(26)

382

J.M. Gomes da Silva Jr., D. Limon, T. Alamo, and E.F. Camacho

it is easy to show that the poles of A will be placed in a disk centered in zero and with ray σ < 1. In this case, smaller is σ faster will be the decay rate of the timeresponse inside the linearity region. Hence a constraint like (26), with a given σ < 1, can be added to the optimization problems (22), (23), (24) and (25). Note that all these optimization problems are eigenvalues problems (EVPs) [2]. On the other hand, considering speciﬁcally (26), another problem of interest can be the minimization of σ, while ensuring the stability for a given set of admissible initial conditions Ξ0 . In this case, the following generalized eigenvalue problem (GEVP) can be formulated [6]: max σ s.t. (11), (12), (13), (26),

1 vr

vr X11

≥ 0 , r = 1, . . . , nv

(27)

Remark 1. In [13], it was shown that, at least in some cases, the use of saturating control laws does not help in obtaining larger regions of stability. It is, however, very important to highlight that no constraints on the control rate, neither on the performance, nor on the robustness, were taken into account in this analysis. In this case, although the optimal region of stability is obtained with a linear control law, the closed-loop poles associated to this solution can be very close to the imaginary axis, which implies a very slow behavior, as shown in [8]. In that paper, a clear trade-off between performance, effective saturation and the size of the region of stability of the closed-loop system is discussed.

5 Disturbance Tolerance and Rejection The problem we aim to address in this section regards the synthesis of the controller (5) in order to minimize the L2 norm of the regulated output, z (disturbance rejection problem). Since the inputs are bounded, as pointed in [16] and [4], a ﬁrst concern in this case consists in ensuring that the system trajectories will be bounded for any disturbance satisfying (4) (tolerance disturbance problem). Similarly to the procedure adopted in section 4, we derive conditions, in LMI form, which allow to compute the controller in order to ensure that trajectories of the closed-loop system remain bounded, provided that a certain L2 -bound on the disturbances is respected and that the system initial condition belongs to a certain

Output Feedback with Amplitude and Rate Constrained Actuators

383

admissible set. The same conditions, allow to determine an upper bound to the L2 norm of the regulated output, z. Based on these conditions, convex optimization problems are proposed to compute a controller aiming at maximizing the tolerance to the disturbances (i.e. maximize the L2 bound on the admissible disturbances) or, given the bound on the admissible disturbance, compute the controller aiming at maximize the disturbance rejection (i.e. minimize an upper bound on the L2 gain between the disturbance and the regulated output).

5.1 Theoretical Result The following theorem provides a theoretical condition to address both the disturbance tolerance and rejection problems. Theorem 2. If there exist symmetric positive deﬁnite matrices X, Y ∈ (n+m)×(n+m) , positive deﬁnite diagonal matrices Sr , Sa ∈ m×m , matrices Aˆ ∈ (n+m)×(n+m) , ˆ ∈ (n+m)×(p+m) , D ˆ ∈ m×(p+m) , Zr1 , Zr2 , Za1 , Za2 ∈ Cˆ ∈ m×(n+m) , B m×(n+m) , Qr , Qa ∈ (n+m)×m and positive scalars μ and γ such that the following inequalities are veriﬁed ⎡ X ⎢ ⎢ In+m Y ⎢ ⎢ Zr1 Zr2 ⎢ ⎢ ⎢ Za1 Za2 ⎢ ⎢ 0 0 ⎢ ⎢ ⎢ AX + BCˆ A + BDC ˆ ⎢ ⎢ ˆ Aˆ Y A + BC ⎣ −Cz X

−Cz

2Sr

0

2Sa

0

0

Iq

BSr

B1 Sa

−Bw

X

Qr

Qa

−Y Bw In+m

0 −Dz,u Sa Dz,w

0 ⎤

⎡ ⎢ ⎢ ⎣

X

In+m

Y

ˆ (i) C − Zr2(i) μρ2 Cˆ(i) − Zr1(i) D r(i)

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥>0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ Y ⎦

(28)

0 γIr

⎥ ⎥≥0 ⎦

⎡ ⎢ ⎢ ⎣

(29)

⎤ X

In+m

Y

X(n+i) − Za1(i) [0 Im ](i) − Za2(i) μρ2a(i)

⎥ ⎥≥0 ⎦

(30)

384

J.M. Gomes da Silva Jr., D. Limon, T. Alamo, and E.F. Camacho

for all i = 1, ..., m, then the dynamic controller (5) with the matrices as deﬁned in (14), where matrices M and N verify N M = In+m − Y X, is such that 1. When w = 0: a) the closed-loop trajectories remain bounded in the set E1 = {ξ ∈ 2(n+m) ; ξ P ξ ≤ μ−1 } for any ξ(0) ∈ E0 = {ξ ∈ 2(n+m) ; ξ P ξ ≤ β}, with Y N X M P = , P −1 = N • M • and β = μ−1 − δ−1 ≥ 0, and for any disturbance satisfying (4). b) ||z||22 < γ||w||22 + γξ(0) P ξ(0).

2. When w = 0, the set E1 is included in the basin of attraction of the closed-loop system (8) and it is contractive.

Proof: Consider the closed-loop system (8) and the candidate Lyapunov function V (ξ(t)) =

ξ(t) P ξ(t), P = P > 0, P ∈ 2(n+m)×2(n+m) . Deﬁne J (t) = ΔV (ξ(t)) − w(t) w(t) + γ−1 z(t) z(t) with ΔV (ξ(t)) = V (ξ(t + 1)) − V (ξ(t)). If J (t) < 0 along the system trajectories and ξ(0) ∈ E0 , it follows that T −1 t=0

J (t) = V (T ) − V (0) −

T −1

w(t) w(t) + γ−1

t=0

T −1

z(t) z(t) < 0, ∀T,

t=0

whence: • V (T ) < V (0) + ||w||22 ≤ β + δ−1 = μ−1 , ∀T , i.e. the trajectories of the system do not leave the ellipsoid E1 , for ξ(0) ∈ E0 and w(t) satisfying (4); • considering T → ∞, ||z||22 < γ||w||22 + γξ(0) P ξ(0); • if w = 0 then ΔV (ξ(t)) < −γ−1 z(t) z(t) < 0.

Output Feedback with Amplitude and Rate Constrained Actuators

385

Computing therefore J (t) along the system trajectories (8) one obtains: J (t) = V (ξ(t + 1)) − V (ξ(t)) = −ξ(t) P ξ(t) + ξ(t) A P Aξ(t) − 2ξ(t) A P B1 ψρa (Lξ(t)) −2ξ(t) A P Bψρr (K ξ(t)) + ψρr (K ξ(t)) B P Bψρr (K ξ(t)) +ψρa (Lξ(t)) B1 P B1 ψρa (Lξ(t)) + 2ψρa (Lξ(t)) B1 P Bψρr (K ξ(t)) +2ξ(t) A P Bw w(t) − 2ψρa (Lξ(t)) B1 P Bw w(t) −2ψρr (K ξ(t)) B P Bw w(t) + w(t) (Bw P Bw − Iq )w(t)

+γ−1 (Cz ξ(t) − Dz,u ψρa (Lξ(t)) + Dz,w w(t)) (Cz ξ(t)− Dz,u ψρa (Lξ(t)) + Dz,w w(t)) Provided that ξ(t) ∈ Ξ(ρa ) ∩ Ξ(ρr ), it follows that J (t) ≤ J (t) − 2ψρr (K ξ(t)) Tr [ψρr (K ξ(t)) − Gr ξ(t)] −2ψρa (Lξ(t)) Ta [ψρa (Lξ(t)) − Ga ξ(t)]

(31)

From here, writing (31) in a matrix form and following the same steps as in the proof of Theorem 1, we conclude that if (28) is satisﬁed, J (t) < 0 holds provided that ξ(t) ∈ Ξ(ρr ) ∩ Ξ(ρa ). On the other hand, (29) and (30) ensures that E1 ⊂ Ξ(ρr ) and E1 ⊂ Ξ(ρa ). Hence, provided (28), (29) and (30) hold, ξ(0) ∈ E0 and ||w||22 ≤ δ−1 , it follows that ξ(t) never leaves E1 and therefore J (t) < 0, which concludes the proof of item 1. Suppose now that w = 0 and ξ(0) = ξ ∈ E1 , therefore it follows that ξ(0) ∈ Ξ(ρr )∩Ξ(ρa ). In this case, the satisfaction of (28) ensures that ΔV (ξ(t)) < 0, ∀t ≥ 0, ξ(t) = 0. Since this reasoning is valid for all ξ ∈ E1 it follows that this set is a region where the asymptotic stability of the closed-loop system (8) is ensured. This completes the proof of item 2. 2

The idea behind the result stated in Theorem 2 is depicted in Figure 2. It follows that once the initial state is in E0 , provided ||w|| ≤ δ−1 , the corresponding trajectories remain conﬁned in the region E1 . This last set can therefore be seen as an estimate of the reachable domain of the systems trajectories. Note that β, μ and δ are related by the equation β = μ−1 − δ−1 . Hence a trade-off between the size of admissible initial conditions E0 , which is related to β, and the tolerance to the disturbance, related to δ−1 appears: if we want to consider the possibility of larger admissible initial conditions, smaller will be the L2 bound on the tolerated disturbances, and vice-versa.

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J.M. Gomes da Silva Jr., D. Limon, T. Alamo, and E.F. Camacho

E1

o

E0

ξ(0)

ξ(T )

Fig. 2. Domain of admissible initial states (E0 ) and the reachable set of trajectories (E1 )

5.2 Optimization Problems The conditions in Theorem 2 are LMIs. Hence, if a linear criterion is used, convex optimization problems can be formulated in order to compute the dynamic controller (5). In the sequel, we focus on the design based on two criteria of particular interest: the maximization of the admissible disturbance bound (tolerance disturbance maximization) and the minimization of the L2 gain between the disturbance and the regulated output (disturbance rejection). For simplicity, in both cases we consider that ξ(0) = 0, i.e., the system is supposed to be in equilibrium at t = 0.

Tolerance Disturbance Since we suppose that ξ(0) = 0, in this case μ−1 = δ−1 . The idea is therefore to minimize μ (i.e. maximize the bound on the disturbance) for which a controller can be computed in order to ensure that the closed-loop trajectories are bounded. This can be accomplished by the following optimization problem. min μ s.t. (28), (29) and (30)

(32)

In this case, the value of γ obtained does not matter.

Disturbance Rejection Considering ξ(0) = 0, an upper bound to the L2 -gain between w and z is given by √ γ. Hence, assuming that the admissible disturbance level δ−1 = μ−1 is given, the

Output Feedback with Amplitude and Rate Constrained Actuators

387

following problem can be considered. min γ

(33)

s.t. (28), (29) and (30)

6 Numerical Examples 6.1 Example 1 Consider the discrete-time linear system given by 0.8 0.5 0 A= , B= , C= 01 −0.4 1.2 1 This system must be controlled with the following saturating limits |u(t)| ≤ 1,

|Δu(t)| ≤ 0.3

We consider also that, in order to ensure a certain time-domain performance when the system is not saturated, the poles of matrix A must be placed inside a disk of radius 0.9 The stabilizing dynamic controller has been computed solving the optimization problem (22). The obtained controller is given by ⎤ ⎤ ⎡ ⎡ 0.6326 −0.0482 0.0166 14.8846 8.1483 ⎥ ⎥ ⎢ ⎢ ⎥ ; Bc = ⎢ 39.1828 24.2906 ⎥ Ac = ⎢ −0.8236 0.0628 −0.0216 ⎦ ⎦ ⎣ ⎣ 1.9616 −0.1494 0.0515 −42.6207 −142.5520

Cc = 0.0079 −0.0006 0.0002 ; Dc = −0.0582 −0.5910 ⎤ ⎤ ⎡ ⎡ 6.5054 4.0208 ⎥ ⎥ ⎢ ⎢ ⎥ ; Fc = ⎢ −16.2020 ⎥ Ec = ⎢ 30.9151 ⎦ ⎦ ⎣ ⎣ −60.5314

−200.5261

The projection of the stability region E(P ) onto the plant states is given by: 0.0472 −0.0223 Ex (P ) = {x ∈ 2 : x x ≤ 1} −0.0223 0.1742

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J.M. Gomes da Silva Jr., D. Limon, T. Alamo, and E.F. Camacho 2.5

2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5 −5

−4

−3

−2

−1

0

1

2

3

4

5

Fig. 3. Region Ex (P ) and state portrait of the controlled system.

In Figure 3 this contractive ellipsoid as well as the trajectories of the controlled system for several initial states are shown. For a given initial state, the initial controller states i.e v(0) and xc (0), are chosen in such a way that ξ(0) = [x(0) v(0) xc (0) ] is in E(P ). In Figure 4 the evolution of the system output, y(t), the control action, u(t), and increment of the control action, Δu(t), are depicted when the system starts from x(0) = [2.8 − 1.5] , v(0) = 1.2966, xc (0) = [449.4707 − 709.7774 − 156.1419] . Notice that the limit requirements in u(t) and Δu(t) are satisﬁed thanks to the proposed saturating dynamic output feedback. Indeed, note that both u(t) and Δu(t) are effectively saturated in the ﬁrst samples. In Table 1, it is illustrated the trade-off between saturation, the size of the stability region and the time-domain performance in terms of the pole placement of matrix A −1 in a disk of radius σ. Considering different values of σ, the values of trace(X11 ) ' and det(X11 ), obtained from the solution of (22), are shown in two situations: the

ﬁrst one regards the application of the results of Theorem 1, i.e. the saturation and the nonlinear behavior of the closed-loop system are effectively considered; the second one concerns the linear solution, i.e. the stability region is forced to be contained in the region of linear behavior of the closed-loop system (RL ) in order to avoid saturation. Note that, as expected, smaller is σ (i.e. more stringent is the performance requirement), smaller is the obtained stability region. On the other hand, the solutions considering the saturation and the nonlinear behavior lead to larger domains of stability.

Output Feedback with Amplitude and Rate Constrained Actuators

389

y(t)

0 −1 −2 0

5

10

15

20

25

30

0

5

10

15

20

25

30

0

5

10

15 samples

20

25

30

1

u(t)

0.5 0 −0.5

Δ u(t)

0.1 0 −0.1 −0.2 −0.3

Fig. 4. Evolution of y(t), u(t) and Δu(t). Table 1. Trade-off: saturation × region of stability × performance saturated solution linear solution ' ' −1 −1 σ trace(X11 ) det(X11 ) trace(X11 ) det(X11 ) 0.99

0.0715

31.8254

0.0798

28.0904

0.95

0.1193

19.8778

0.1622

14.1177

0.9

0.2214

11.3793

0.3178

7.5111

0.85

0.4161

6.2837

0.5748

4.3165

0.8

0.7846

3.3509

1.0328

2.4725

6.2 Example 2 Consider the pendulum continuous time model presented in the preface. It has been discretized with a sampling time Ts = 0.05 s, obtaining the following discrete-time model matrices ⎡

⎤

1.0013 −0.0500 −0.0013 ⎥ ⎢ ⎥, A=⎢ −0.0500 1.0025 0.0500 ⎦ ⎣ −0.0013 0.0500 1.0013 ⎤ ⎤ ⎡ ⎡ −0.0021 5.0021 ⎥ ⎥ ⎢ ⎢ ⎥ , Bw = 10−2 · ⎢ −0.1251 ⎥ , B = 10−2 · ⎢ 0.1251 ⎦ ⎦ ⎣ ⎣ 5.0021

−0.0021

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C=

100

010

,

Cz = 1 0 0 ,

Dz,u = 0,

Dz,w = 0.

The amplitude and rate control limits are respectively: |u(t)| ≤ 1.25 , |Δ(u(t))| ≤ 2.0 Regarding the problem of maximizing the region of attraction of the closed-loop system, in Table 2 it is shown the results obtained from the solution of optimization problem (22). As in the previous example, it can be noticed a trade-off between the saturation allowance, the size of the region of stability and the time-performance requirement, expressed as a pole placement in disk of radius σ. Table 2. Trade-off: saturation × region of stability × performance

σ

saturated solution linear solution ' ' −1 −1 trace(X11 ) det(X11 ) trace(X11 ) det(X11 )

0.99

10.0454

591.1106

12.5708

479.7308

0.975

23.1943

184.1258

36.1758

156.9039

0.95

70.7120

94.3923

135.0452

49.1979

0.925 229.6418

0.0576

473.9738

0.0202

0.0049

2345.4

0.0018

0.9

1169.8

Considering, for instance σ = 0.975, the solution of (22) gives the following controller matrices: ⎡

0.3933

0.0002 −0.0006 −0.0001

⎤

⎥ ⎢ ⎢ 806.5808 0.3946 −1.2876 −0.2052 ⎥ ⎥ Ac = ⎢ ⎢ −30.0468 −0.0146 0.0481 0.0070 ⎥ ⎦ ⎣ −523.8979 −0.2556 0.8369 0.1304 ⎤ ⎡ 0.5754 0.6402 −0.0291 ⎥ ⎢ ⎢ 1796.5662 −2838.9892 −26.8975 ⎥ ⎥ Bc = ⎢ ⎢ 1181.7759 −777.2075 9.0466 ⎥ ⎦ ⎣ −1538.1978 2106.5383 50.7020 Cc = 4.8971 0.0024 −0.0078 −0.0012 ; Dc = 10.9580 −17.2724 −1.1629

Output Feedback with Amplitude and Rate Constrained Actuators

⎡

−0.0291

⎤

⎡

−0.000001

391

⎤

⎥ ⎥ ⎢ ⎢ ⎢ −0.1133 ⎥ ⎢ 164.382729 ⎥ ⎥ ⎥ ⎢ ⎢ ; Fc = ⎢ Ec = ⎢ ⎥ ⎥ ⎣ 14.8337 ⎦ ⎣ 35.515640 ⎦ 65.9817 93.770018 Considering the controller given by the matrices above, the projection of the stability region E(P ) onto the plant states is given by: ⎤ ⎡ 10.1414 −10.1832 −4.2598 ⎥ ⎢ 3 ⎢ Ex (P ) = {x ∈ : x ⎣ −10.1832 10.5713 4.7669 ⎥ ⎦ x ≤ 1} −4.2598 4.7669 2.4816 In Figure 5 the evolution of the states of the pendulum is shown for the initial condition x(0) = [0.0050, 0.2428, 0.1447]T , v(0) = −3.4855 and xc (0) = [0.1769, −572.9046, −157.2404, 471.0735]T . The corresponding evolutions of u(t) and Δ(u(t)) are depicted in Figure 6. 0.1

x1

0 −0.1 −0.2 0

50

100

150

0

50

100

150

0

50

100

150

0.3

x2

0.2 0.1 0 −0.1

0.2

x3

0 −0.2 −0.4 −0.6 samples

Fig. 5. Evolution of state of the pendulum.

Regarding now the disturbance tolerance problem, from (32), it follows that the maximal value of the bound on ||w||22 , for which it is possible to ensure that the trajectories are bounded is given by μ−1 = δ−1 = 10.2736. In this case, the obtained controller matrices are

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u

0

−0.5

−1

0

50

0

50

100

150

100

150

0.6 0.5

Δu

0.4 0.3 0.2 0.1 0 −0.1 samples

Fig. 6. Evolution of u(t) and Δu(t).

⎡

0.4493

−0.0402 −0.0000 0.0000

⎤

⎥ ⎢ ⎢ −6.0391 0.5410 0.0000 −0.0000 ⎥ ⎥ ⎢ Ac = ⎢ ⎥ ⎣ 3876.4632 −347.1736 −0.0020 0.0102 ⎦ 97.9460 −8.7760 −0.0001 −0.0003 ⎤ ⎡ −0.0253 0.6773 −0.0326 ⎥ ⎢ ⎥ ⎢ −0.1301 8.3458 0.4154 ⎥ Bc = ⎢ ⎢ 5406.6797 −9120.6486 −135.8883 ⎥ ⎦ ⎣ 136.6425 −230.4732 −10.1011 Cc = 5.4145 −0.4849 −0.0000 0.0000 ; Dc = 7.5517 −12.7393 −1.1898 ⎤ ⎤ ⎡ ⎡ −0.0326 −0.0000 ⎥ ⎥ ⎢ ⎢ ⎢ 0.4146 ⎥ ⎢ −0.0045 ⎥ ⎥ ⎥ ⎢ Ec = ⎢ ⎢ 0.0052 ⎥ ; Fc = ⎢ 715.9553 ⎥ ⎦ ⎦ ⎣ ⎣ −14.8473 −20.6986 Table 3 shows the different values of μ−1 , obtained from (32), with an additional constraint of constraint of pole placement in a disk of radius σ. It can be noticed trade off between time-domain performance and the tolerance to disturbances. Finally, in Table 4, it is shown the values of γ (related to the minimization of the L2 gain) obtained from the solution of optimization problem (33), considering different values of δ. As expected, the larger is δ (i.e. the lower is the bound on the disturbances), smaller is the value of γ, i.e. better is the disturbance rejection.

Output Feedback with Amplitude and Rate Constrained Actuators

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Table 3. Trade-off: σ × maximal bound on the disturbance energy σ

μ−1 = δ−1

1.00

10.2737

0.99

5.3997

0.975

2.5087

0.95

0.9449

0.925

0.3356

0.9

0.0741

Table 4. Trade-off: bound on the disturbance energy × disturbance rejection δ−1

γ

10 17094.7 8 211.8103 6

45.4023

4

14.5536

2

4.7053

1

2.2508

0.5

1.2816

0.1

0.4768

0.01 0.1689

7 Conclusions In this chapter a technique for the design of stabilizing dynamic output feedback controllers for discrete-time linear systems with rate and amplitude constrained actuators was proposed. Two problems have been considered: the stabilization problem in the absence of disturbances; and the tolerance and disturbance problem. In order to cope with the rate saturation problem, a controller structure consisting of a classical linear dynamic compensator in cascade with m input-saturating integrator system and two static anti-windup loops was adopted. The obtained closed-loop system presents in this case two amplitude saturation terms. With this particular structure, it is ensured that the control signal delivered to the actuator will always respect the rate constraints.

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In order to obtain theoretical conditions allowing to compute a local stabilizing controller, the nonlinear effects of the saturation terms are taken into account by considering a generalized sector condition. The main advantages of such approach are twofold: it leads to conditions directly in LMI form; it allows the simultaneous synthesis of a dynamic compensator and the anti-windup loops. This last fact links, someway, the anti-windup approach (which adds corrections in a pre-computed controller, designed disregarding the saturation effects) with the paradigm of synthesizing a dynamic controller directly taking into account the saturation nonlinearities. Based on the derived conditions, convex optimization problems can be formulated in order to compute the controller considering a synthesis criterion. Three problems have been formulated in this sense. The ﬁrst one regards the synthesis of the controller with the aim of maximizing an associated region for which the asymptotically stability can be ensured. This problem indirectly addresses the more complex problem of computing a controller in order to maximize the region of attraction of the closed-loop system. Additional performance constraints for the behavior of the system inside the linearity region of the system (where no saturation is active), in terms of pole placement in LMI regions, are considered in the optimization problem. It has been noticed, by means of examples, a clear trade-off between the size of the stability region and the achieved performance in the linearity region: more stringent performances lead to smaller regions of stability. The second formulated problem concerns the computation of the controller aiming at the maximization of an upper bound on the L2 norm of the disturbances for which the corresponding trajectories are guaranteed bounded. Finally, given an L2 bound on the admissible disturbances acting on the system, the problem of determining the controller which minimizes an upper bound on the L2 gain between the disturbance input and the regulated output, while ensuring bounded trajectories and internal stability, is formulated. Another trade-off is observed in this case: smaller are the L2 -norm of the admissible disturbances, smaller is the achieved L2 gain, that is, more effective is the disturbance rejection. The best L2 gain tends to be obtained for small enough disturbances, which do not lead the trajectories outside the linearity region. In this case, the obtained gain is the one of the linear (unsaturated) closed-loop system. On the other hand, by allowing the saturation and take into account the nonlinear behavior of the system it is possible to ”tolerate”, larger disturbances, while ensuring the trajectories are bounded.

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References 1. A. Bateman and Z. Lin. An analysis and design method for linear systems under nested saturation. System & Control Letters, 48(1):41–52, 2002. 2. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics, 1994. 3. M. C. de Oliveira, J. C. Geromel, and J. Bernussou. Design of dynamic output feedback decentralized controllers via a separation procedure. Int. Journal of Control, 73(5):371– 381, 2000. 4. H. Fang, Z. Lin, and T. Hu. Analysis of linear systems in the presence of actuator saturation and L 2 -disturbances. Automatica, 40(7):1229–1238, 2004. 5. J. M. Gomes da Silva Jr. and S. Tarbouriech. Stability regions for linear systems with saturating controls. In Proc. of European Control Conference 1999 (ECC’99), Karlsrhue, Germany, 1999. 6. J. M. Gomes da Silva Jr. and S. Tarbouriech. Local stabilization of discrete-time linear systems with saturating controls: an LMI-based approach. IEEE Trans. on Automatic Control, 46:119–125, 2001. 7. J. M. Gomes da Silva Jr. and S. Tarbouriech. Anti-windup design with guaranteed regions of stability: an lmi-based approach. IEEE Trans. on Automatic Control, 50(1):106–111, 2005. 8. J. M. Gomes da Silva Jr., S. Tarbouriech, and G. Garcia. Local stabilization of linear systems under amplitude and rate saturating actuators. IEEE Trans. on Automatic Control, 48(5):842–847, 2003. 9. G. Grimm, J. Hatﬁeld, I. Postlethwaite, A. Teel, M. Turner, and L. Zaccarian. Antiwindup for stable systems with input saturation: an lmi-based synthesis. IEEE Trans. on Automatic Control, 48(9):1500–1525, 2003. 10. T. Hu and Z. Lin.

Control systems with actuator saturation: analisys and design.

Birkhauser, 2001. 11. V. Kapila and K. Grigoriadis (Editors). Actuator Saturation Control. Marcel Dekker, Inc., 2002. 12. N. Kapoor, A. R. Teel, and P. Daoutidis. An anti-windup design for linear systems with input saturation. Automatica, 34(5):559–574, 1998. 13. T. Kiyama and T. Iwasaki. On the use of multi-loop circle for saturating control synthesis. Systems & Control Letters, 41:105–114, 2000. 14. Z. Lin. Semi-global stabilization of linear systems with position and rate-limited actuators. System & Control Letters, 30:1–11, 1997. 15. T. Nguyen and F. Jabbari. Output feedback controllers for disturbance attenuation with actuator amplitude and rate saturation. Automatica, 36:1339–1346, 2000.

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16. C. Paim, S. Tarbouriech, J. M. Gomes da Silva Jr., and E. B. Castelan. Control design for linear systems with saturating actuators and L 2 -bounded disturbances. In Proc. of the 41st IEEE Conference on Decision and Control, Las Vegas - USA, 2002. 17. C. Scherer, P. Gahinet, and M. Chilali. Multiobjective output-feedback control via lmi optimization. IEEE Trans. on Automatic Control, 42(7):896–911, 1997 1997. 18. G. Shi, A. Saberi, and A. A. Stoorvogel. On Lp performance with global internal stability for linear systems with actuators subject to amplitude and rate saturations. In Proceedings of the 19th American Control Conference (ACC’2000), pages 730–734, Chicago, USA, 2000. 19. A. A. Stoorvogel and A. Saberi. Output regulation for linear systems to amplitude plus rate saturating actuators. Int. J. of Robust and Nonlinear Control, 9(10):631–657, 1999. 20. S. Tarbouriech and G. Garcia (Editors). Control of uncertain systems with bounded inputs. Springer Verlag, 1997. 21. S. Tarbouriech, C. Prieur, and J. M. Gomes da Silva Jr. Stability analysis and stabilization of systems presenting nested saturations. In Proc. of the 43rd IEEE Conference on Decision and Control, Paradise Island , Bahamas, 2004. 22. F. Tyan and D. S. Bernstein. Dynamic output feedback compensation for linear systems with independent amplitude and rate saturation. Int. Journal of Control, 67(1):89–116, 1997.

Decentralized Stabilization of Linear Time Invariant Systems Subject to Actuator Saturation Anton A. Stoorvogel1,2 , Ali Saberi3 , Ciprian Deliu1,2 , and Peddapullaiah Sannuti4 1

Department of Mathematics, and Computing Science, Eindhoven Univ. of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands , [email protected],[email protected].

2

Department of Electrical Engineering, Mathematics and Computer Science, Delft Univ. of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands [email protected],[email protected].

3

School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99164-2752, U.S.A. [email protected].

4

Department of Electrical and Computer Engineering, Rutgers University, 94 Brett Road, Piscataway, NJ 08854-8058, U.S.A. [email protected].

Summary. We are concerned here with the stabilization of a linear time invariant system subject to actuator saturation via decentralized control while using linear time invariant dynamic controllers. When there exists no actuator saturation, i.e. when we consider just linear time invariant systems, it is known that global stabilization can be done via decentralized control while using linear time invariant dynamic controllers only if the so-called decentralized ﬁxed modes of it are all in the open left half complex plane. On the other hand, it is known that for linear time invariant systems subject to actuator saturation, semi-global stabilization can be done via centralized control while using linear time invariant dynamic controllers if and only if the open-loop poles of the linearized model of the given system are in the closed left half complex plane. This chapter establishes that the necessary conditions for semi-global stabilization of linear time invariant systems subject to actuator saturation via decentralized control while using linear time invariant dynamic controllers, are indeed the above two conditions, namely (a) the decentralized ﬁxed modes of the linearized model of the given system are in the open left half complex plane, and (b) the open-loop poles of the linearized model of the

Support for this work was provided by the Ofﬁce of Naval Research under grant number N000140310848.

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given system are in the closed left half complex plane. We conjecture that these two conditions are also sufﬁcient in general. We prove the sufﬁciency for the case when the linearized model of the given system is open-loop conditionally stable with eigenvalues on the imaginary axis being distinct. Proving the sufﬁciency is still an open problem for the case when the linearized model of the given system has repeated eigenvalues on the imaginary axis.

1 Introduction Non-classical information and control structure are two essential and distinguishing characteristics of large-scale systems. The research on decentralized control was formally initiated by Wang and Davison in their seminal paper [17] in 1973, and has been the subject of intense study during the 70’s and 80’s. Most recently there has been a renewed interest in decentralized control because of its fundamental role in the problem of coordinating the motion of multiple autonomous agents which by itself has attracted signiﬁcant attention. Coordinating the motion of autonomous agents has many engineering applications besides having links to problems in biology, social behavior, statistical physics, and computer graphics. The engineering applications include unmanned aerial vehicles (UAVs), autonomous underwater vehicles (AUVs) and automated highway systems (AHS). A fundamental concept in the study of stabilization using decentralized feedback controllers is that of ﬁxed modes. These are the poles of the system which cannot be shifted by just using any type of decentralized feedback controllers. The idea of ﬁxed modes was introduced by Wang and Davison [17] who also show that decentralized stabilization is possible if and only if the ﬁxed modes are stable. More deﬁnitive results are obtained by Corfmat and Morse [4] who present necessary and sufﬁcient conditions under which spectrum assignment is possible in terms of the remnant polynomial of complementary subsystems. Since ﬁxed modes constitute such an important concept in decentralized control, their characterization and determination has been the subject of many papers in the literature. The majority of existing research in decentralized control makes a critical assumption that the interconnections between the subsystems of a given system are unknown but have known bounds. In this regard, tools borrowed from robust control theory and Lyapunov theory are used for the purpose of either synthesis or analysis of decentralized controllers [10, 14, 13]. For the case when the interconnections between

Decentralized Control with Input Saturation

399

the subsystems are known, the existing research is very sparse. In fact, in any case, beyond the decentralized stabilization, no results are yet available dealing with the fundamental control issues such as exact or almost disturbance decoupling, control for various performance objectives, etc. From a different perspective, input saturation in any control scheme is a common phenomenon. Every physically conceivable actuator has bounds on its output. Valves can only be operated between fully open and fully closed states, pumps and compressors have a ﬁnite throughput capacity and tanks can only hold a certain volume. Ignoring such saturation effects in any control system design can be detrimental to the stability and performance of controlled systems. A classical example for the detrimental effect of neglecting actuator constraints is the Chernobyl unit 4 nuclear power plant disaster in 1986 [16]. During the last decade and the present one, there has been an intense research activity in the area of control of linear plants with saturating actuators. Such intense research activity has been chronicled in special issues of journals and edited books (e.g. for recent literature see [8, 15, 2]). Fundamental fuel behind such a research activity has been to accentuate the industrial and thus the practical engineering relevance of modern control theory. In this regard, the primary focus of the research activity has been to take into account a priori the presence of saturation nonlinearities in any control system analysis and design. A number of control issues have been considered so far including internal, external, or internal plus external stabilization and output regulation among others. Although not all aspects of these issues have been completely resolved, it is fair to say that a good understanding of these issues exists at present. However, issues related to performance, robustness etc., are very poorly understood and still remain as challenging and complex problems for future research. Having been involved deeply in the past with research on linear systems subject to constraints on its input and state variables, we are now ready to open up a new front line of research in decentralized control by bringing into picture the constraints of actuators. The focus of this chapter is to determine the necessary and sufﬁcient conditions for decentralized stabilization of linear systems subject to constraints on actuators. Obviously, this is related to the seminal work of Wang and Davison [17] but goes beyond it by bringing into picture the input constraints on the top of decentralized constraint.

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2 Problem Formulation and Preliminaries Consider the linear time invariant systems subject to actuator saturation, ⎧ ν ⎪ ⎨ x˙ = Ax + Bi sat ui Σ: i=1 ⎪ ⎩ yi = Ci x, i = 1, . . . , ν,

(1)

where x ∈ Rn is a state, ui ∈ Rmi , i = 1, . . . , ν are control inputs, yi ∈ Rpi , i = 1, . . . , ν are measured outputs, and ‘sat’ denotes the standard saturation element with the property that for any vector u of arbitrary dimension, sat(u) is a vector of the same dimension as u, and moreover for any positive integer j less than or equal to the dimension of u, the j’th component of sat u, denoted by (sat u)j , compared to the j’th component of u, denoted by (u)j , has the property, ⎧ ⎪ ⎪ 1 if 1 < (u)j , ⎪ ⎨ (sat u)j = (u)j if − 1 ≤ (u)j ≤ 1, ⎪ ⎪ ⎪ ⎩−1 if (u) < −1. j

Here we are looking for ν controllers of the form, z˙i = Ki zi + Li yi , zi ∈ Rsi Σi : u i = Mi zi + Ni y i .

(2)

The controller Σi is said to be i-th channel controller. Before we state the problem we study in this chapter, we would like to recall the concept of semi-global stabilization via decentralized control. Deﬁnition 1. Consider a system Σ of the form (1). Then, we say that Σ is semiglobally stabilizable via decentralized control if there exists nonnegative integers s1 , . . . , sν such that for any given collection of compact sets W ⊂ Rn and Si ⊂ Rsi , i = 1, . . . ν, there exist a decentralized set of controllers ν controllers Σi , i = 1, . . . ν, of the form (2) such that the origin of the resulting closed-loop system is asymptotically stable and the domain of attraction includes W × S1 × · · · × Sν . The problem we would like to study in this chapter can be stated as follows: Problem 1. Consider a system Σ of the form (1). Develop the necessary and sufﬁcient conditions such that Σ is semi-globally stabilizable via decentralized control.

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Remark 1. For the case when ν = 1, the above decentralized control problem retorts to centralized semi-global stabilization of linear time invariant systems subject to actuator saturation. Such a problem has been studied in depth by Saberi and his coworkers. By now it is well known that such a centralized semi-global stabilization problem is solvable by a linear time invariant dynamic controller if and only if the linearized model of the given system is stabilizable and detectable and all the openloop poles of linearized model are in the closed left half complex plane.

3 Review of Decentralized Stabilization of Linear Time Invariant Systems Before we proceed to consider the conditions for the solvability of Problem 1, it is prudent to review the necessary and sufﬁcient conditions for the global decentralized stabilization of linearized model of the given system Σ. To do so, we ﬁrst write the linearized model of the given system Σ of (1) as, ⎧ ν ⎪ ⎨ x˙ = Ax + Bi ui ¯: Σ i=1 ⎪ ⎩ yi = Ci x, i = 1, . . . , ν.

(3)

The classical decentralized global stabilization problem or more general decentral¯ can be stated as follows: ized pole placement problem for the linearized model Σ Find linear time invariant dynamic controllers Σi , i = 1, . . . ν, of the form (2) ¯ and the controllers Σi , such that the poles of the closed-loop system comprising Σ i = 1, . . . ν, has pre-speciﬁed poles in the open left half complex plane. It is easy to observe that, if (A, Bi ) and (A, Ci ) are respectively controllable and observable pairs for some i, the above decentralized pole placement problem can be solved trivially. Wang and Davison in [17] considered the general decentralized pole placement prob¯ Before we state their result, we need to recall the lem for the linearized model Σ. important concept of decentralized ﬁxed modes as was introduced by Wang and Davison: ¯ of the form (3). Then, λ is called a decentralized Deﬁnition 2. Consider a system Σ ¯ if for all matrices K1 , . . . , Kν we have that λ is an ﬁxed mode of the system Σ

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eigenvalue of AK := A +

ν

Bi Ki Ci .

i=1

Wang and Davison proved in [17] that there exist dynamic controllers Σi , i = 1, . . . ν, of the form (2) such that the poles of the closed-loop system comprising ¯ and the controllers Σi , i = 1, . . . ν are at pre-speciﬁed locations in the open left Σ ¯ are themselves half complex plane provided that the decentralized ﬁxed modes of Σ in the open left half complex plane and the set of pre-speciﬁed locations in the open ¯ This obvileft half complex plane includes the set of decentralized ﬁxed modes of Σ. ously implies that the decentralized stabilization of the linear time invariant system ¯ is possible if and only if the decentralized ﬁxed modes of it are all in open left half Σ complex plane. ¯ play a crucial role The above result implies that the decentralized ﬁxed modes of Σ in decentralized stabilization of linear time invariant systems. As such it is important to know how to compute such ﬁxed modes. One of the easiest procedure to do so is as follows: Since Ki = 0, i = 1, . . . ν, are admissible, in this case AK retorts to A, and hence in view of Deﬁnition 2, the decentralized ﬁxed modes are naturally a subset of the eigenvalues of A. Thus the ﬁrst step is to compute the eigenvalues of A. Second, it can be shown that if Ki , i = 1, . . . ν, are randomly chosen, then with probability one the decentralized ﬁxed modes are common eigenvalues of A and AK . Since algorithms are well developed to determine the eigenvalues of a matrix, the computation of decentralized ﬁxed modes is quite straightforward. After the introduction of the concept of decentralized ﬁxed modes, there has been quite some research on interpretations of this concept. The crucial step in understanding the decentralized ﬁxed modes was its connection to complementary systems as introduced by Corfmat and Morse in the paper [4]. The paper [1] by Anderson and Clements used the ideas of Corfmat and Morse to yield the following characterization of decentralized ﬁxed modes: ¯ of (3). We deﬁne, Lemma 1. Consider the system Σ ⎛

" # B = B1 · · · Bν ,

⎞ C1 ⎜ . ⎟ . ⎟ C=⎜ ⎝ . ⎠. Cν

Decentralized Control with Input Saturation

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Then λ is a decentralized ﬁxed mode if and only if at least one of the following three conditions is satisﬁed: • λ is an uncontrollable eigenvalue of (A, B). • λ is an unobservable eigenvalue of (C, A). • There exists a partition of the integers {1, 2, . . . , ν} into two disjoint sets {i1 , . . . , iα } and {j1 , . . . , jν−α } where 0 < α < ν for which we have ⎛ ⎞ λI − A Bi1 · · · Biα ⎜ ⎟ ⎜ Cj1 ⎟ 0 · · · 0 ⎜ ⎟ < n. rank ⎜ . .. . . .. ⎟ ⎜ .. ⎟ . . . ⎠ ⎝ Cjν−α 0 · · · 0 Basically the decentralized ﬁxed modes are therefore common blocking zeros of all complementary systems which are, moreover, either unobservable or uncontrollable for each complementary system. For a detailed investigation of blocking zeros we refer to the paper [3]. Other attempts to characterize the decentralized ﬁxed modes can be found in for instance [12, 6, 7]. The above discussion focuses on developing the necessary and sufﬁcient condition under which stabilization of a linear time invariant system by a set of decentralized controllers is possible. The next issue that needs to be discussed pertains to how does one construct systematically the set of decentralized controllers that stabilize a given system assuming that it is possible to do so. In this regard, it is important to recognize that implicit in the proof of pole placement result of Wang and Davison [17] is a constructive algorithm. This algorithm requires as a ﬁrst step the (possibly random) selection of Ki , i = 1, . . . ν, such that all the eigenvalues of AK = A +

ν

Bi Ki Ci

i=1

are distinct from those of A except for the decentralized ﬁxed modes. Then, dynamic feedback is successively employed to arrive at a dynamic controller Σi , i = 1, . . . ν, placing the poles of resulting closed-loop system that are both controllable and observable eigenvalues of the pairs (A, Bi ) and (A, Ci ) respectively. Also, Corfmat and Morse [4] have studied the decentralized feedback control problem from the point of view of determining a more complete characterization of conditions for stabilizability and pole placement as well as constructing a set of stabilizing decentralized

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¯ controllers. Their basic approach is to determine conditions under which a system Σ of the form (3) can be made controllable and observable from the input and output variables of a given controller by static feedback applied by the other controllers. Then dynamic compensation can be employed at this controller in a standard way to place the poles of the closed-loop system. ¯ controllable and observable It is not hard to see that a necessary condition to make Σ from a single controller is that none of the transfer functions Ci (sI − A)−1 Bj vanish identically for all i = 1, . . . ν, and j = 1, . . . ν. A system satisfying this condition is termed strongly connected. If a system is not strongly connected, the given system can be decomposed into strongly connected subsystems and each subsystem can be made then controllable and observable from one of its controllers. As outlined in an early survey paper by Sandell et al [11], as a practical design method, the Corfmat and Morse method suffers some defects. At ﬁrst, it can be noted that even if all the modes of a large scale system can be made controllable and observable from a single controller (or a few controllers if the given system is not strongly connected), some of the modes may be very weakly controllable and observable. Thus, impractically large gains may be required to place all the poles from a single controller. Second, it is unclear that the approach uses the designer’s available degrees of freedom in the best way. Essentially, the approach seems to require that all the disturbances in the system propagate to a single output, where they can be observed and compensated for by the control signals at an adjacent input. Finally, concentration of all the complexity of the control structure at a single (or few) controllers may be undesirable. As pointed out once again in [11], the constructive approach of Wang and Davison also suffers similar drawbacks as mentioned above. Although there is no explicit attempt in their approach to make all of the strongly connected subsystems controllable and observable from a single controller, the generic outcome of the ﬁrst step of their approach will be precisely this situation. After the early ﬁrst phase of work of Wang and Davison [17] as well as Corfmat and Morse [4], there has been a lot of second phase of work (see [14, 13] and references there in) on how to construct the set of decentralized controllers for a large ¯ of (3) scale system. These researchers view the given large scale system such as Σ

Decentralized Control with Input Saturation

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as consisting of ν interconnected subsystems, the i-th subsystem being controlled by the i-th controller Σi . Then, the research in decentralized control is dominated by the point of view of considering the interconnections between the subsystems essentially as disturbances, and then using robust control theory to design strongly robust subsystems in such a way that the effect of such disturbances is minimal. Essentially, the framework of viewing the interconnections as disturbances is fundamentally ﬂawed. Such work belongs to the ﬁeld of centralized robust control theory. In our opinion, the decentralized control is still in its infancy, and is a very complex and open ﬁeld.

4 Main Results In this section, we will present the necessary and sufﬁcient conditions for semi-global stabilizability of linear time invariant systems with actuator saturation by utilizing a set of decentralized linear time invariant dynamic controllers. We have the following theorem that pertains to necessary conditions, the proof of which is given in Section 5. Theorem 1. Consider the system Σ given by (1). There exists nonnegative integers s1 , . . . , sν such that for any given collection of compact sets W ⊂ Rn and Si ⊂ Rsi , i = 1, . . . ν, there exist ν controllers of the form (2) such that the origin of the resulting closed loop system is asymptotically stable and the domain of attraction includes W × S1 × · · · × Sν only if ¯ given by (3) are in the open left half complex • All decentralized ﬁxed modes of Σ plane, and • All eigenvalues of A are in the closed left half plane. The following theorem says that besides decentralized ﬁxed modes being in the open left half complex plane, a sufﬁcient condition for semi-global stabilizability of (1) when the set of controllers given by (2) are utilized is that all the eigenvalues of A be in the closed left half plane with those eigenvalues on the imaginary axis having algebraic multiplicity equal to one.

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Theorem 2. Consider the system Σ given by (1). There exists nonnegative integers s1 , . . . , sν such that for any given collection of compact sets W ⊂ Rn and Si ⊂ Rsi , i = 1, . . . ν, there exist ν controllers of the form (2) such that the origin of the resulting closed loop system is asymptotically stable and the domain of attraction includes W × S1 × · · · × Sν if ¯ given by (3) are in the open left half complex • All decentralized ﬁxed modes of Σ plane, and • All eigenvalues of A are in the closed left half plane with those eigenvalues on the imaginary axis having algebraic multiplicity equal to one. The above theorem is proved in Section 6. Our work done up to now convinces us to state the following conjecture that the necessary conditions given in Theorem 1 are also sufﬁcient for semi-global stabilizability of decentralized linear systems with actuator saturation. Conjecture 1. Consider the system Σ given by (1). There exists nonnegative integers s1 , . . . , sν such that for any given collection of compact sets W ⊂ Rn and Si ⊂ Rsi , i = 1, . . . ν, there exist ν controllers of the form (2) such that the origin of the resulting closed loop system is asymptotically stable and the domain of attraction includes W × S1 × · · · × Sν if and only if ¯ given by (3) are in the open left half complex • All decentralized ﬁxed modes of Σ plane, and • All eigenvalues of A are in the closed left half plane.

5 Proof of Theorem 1 We prove Theorem 1 in this section. Assume that decentralized semi-global stabilization of the given system Σ of (1) is possible. Then, the decentralized stabilization ¯ of Σ as given in (3) is possible. By the result of Wang of the linearized model Σ and Davison [17], this implies that it is necessary to have all the decentralized ﬁxed ¯ in the open left half complex plane. However, we have a simple alternate modes of Σ proof of this fact as follows: Since the linearized model needs to be asymptotically

Decentralized Control with Input Saturation

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stable, there exists ν linear controllers achieving locally an asymptotically stable system. We deﬁne the following matrices for these ν controllers of the form (2): ⎛ ⎞ ⎞ ⎛ K1 0 · · · 0 L1 0 · · · 0 ⎜ ⎟ ⎟ ⎜ ⎜ 0 L . . . ... ⎟ ⎜ 0 K . . . ... ⎟ 2 2 ⎜ ⎜ ⎟ ⎟ K=⎜ . ⎟, L = ⎜ . ⎟, ⎜ . .. .. ⎜ . .. .. ⎟ ⎟ . . 0⎠ . . 0⎠ ⎝ . ⎝ . 0 · · · 0 Kν

⎛ M1 0 · · · 0 ⎜ ⎜ 0 M . . . ... 2 ⎜ M =⎜ . ⎜ . .. .. . . 0 ⎝ .

⎞

⎛

⎟ ⎟ ⎟ ⎟, ⎟ ⎠

0 · · · 0 Mν

0 · · · 0 Lν

N1 0 · · · ⎜ ⎜ 0 N ... 2 ⎜ N =⎜ . ⎜ . .. .. . . ⎝ .

⎞ 0 .. ⎟ . ⎟ ⎟ ⎟. ⎟ 0⎠

0 · · · 0 Nν

For any λ with Re λ ≥ 0 there exists a δ such that (λ + δ)I − K is invertible and the closed loop system when replacing K by K − δI is still asymptotically stable. But then the linearized model of the closed loop system cannot have a pole at λ which implies that we must have that 5 6 ! det λI − A − B M (λI − (K − δI))−1 L + N C = 0. Hence the block diagonal matrix S = M (λI − (K − δI))−1 L + N has the property that det (λI − A − BSC) = 0, and thus λ is not a ﬁxed mode of the system. Since this argument is valid for any λ in the closed right half plane this implies that all the ﬁxed modes must be in the open left half plane. This proves the necessity of the ﬁrst item of Theorem 1. To prove the necessity of the second item of Theorem 1, assume that λ is an eigenvalue of A in the open right half plane with corresponding left eigenvector p, i.e. pA = λp. Then we have d px(t) = λpx(t) + v(t) dt where v(t) :=

ν i=1

pBi sat ui (t).

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Anton A. Stoorvogel, , Ali Saberi, Ciprian Deliu, , and Peddapullaiah Sannuti

˜ > 0 such that v(t) ≤ M ˜ for all t > 0. But then There clearly exists an M $ % ˜ ˜ λt M M + |px(t)| > e |px(0)| − Re λ Re λ which does not converge to zero since Re λ ≥ 0, provided the initial condition is such that

˜ M . Re λ Note that this is valid for all controllers and therefore we can clearly not achieve |px(0)| >

semi-global stability.

6 Preliminary Lemmas and Proof of Theorem 2 We will use two lemmas. The ﬁrst lemma given below is a well-known classical result from Lyapunov theory. Lemma 2. Consider a matrix A ∈ Rn×n , and assume that it has all its eigenvalues in the closed left half plane with those eigenvalues on the imaginary axis having a geometric multiplicity equal to the algebraic multiplicity. Then, there exists a matrix P > 0 such that A P + P A ≤ 0.

(4)

Another useful tool is the following continuity result related to (4). Lemma 3. Assume that we have a sequence of matrices Aδ ∈ Rn×n parameterized by δ and a matrix A ∈ Rn×n such that Aδ → A as δ → 0. Assume that A has all its eigenvalues in the closed left half plane, and that there are p distinct eigenvalues of A on the imaginary axis (i.e. there are p eigenvalues of A on the imaginary axis each with algebraic multiplicity equal to 1). Moreover, assume that Aδ also has all its eigenvalues in the closed left half plane. Let P > 0 be such that (4) is satisﬁed. Then there exists for small δ > 0 a family of matrices Pδ > 0 such that Aδ Pδ + Pδ Aδ ≤ 0 and Pδ → P as δ → 0.

Decentralized Control with Input Saturation

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Proof. We ﬁrst observe that there exists a matrix S such that $ % A 0 11 , S −1 AS = 0 A22 and such that all the eigenvalues of A11 are on the imaginary axis while A22 has all its eigenvalues in the open left half plane. Since A11 and A22 have no common eigenvalues and Aδ → A, there exists a parameterized matrix Sδ such that for δ sufﬁciently small Sδ−1 Aδ Sδ =

$ A11,δ 0

0

%

A22,δ

where Sδ → S, A11,δ → A11 and A22,δ → A22 as δ → 0. This follows from classical results on the sensitivity of invariant subspaces (see for instance [9, 5]). Given is a matrix P > 0 such that A P + P A ≤ 0. Let us deﬁne $ % ¯11 P¯12 P P¯ = S P S = . ¯ P¯12 P22 Obviously, we note that $ % % $ A11 0 A11 0 P¯ + P¯ ≤ 0. 0 A22 0 A22

(5)

Next, given an eigenvector x1 such that A11 x1 = λx1 with Re λ = 0, we have $ % % $ % $ %∗ $ A A11 0 x1 0 x1 11 ¯ + P¯ P = 0. 0 A22 0 A22 0 0 Using (5), the above implies that $ % % $ % $ A x1 0 A11 0 11 P¯ + P¯ = 0. 0 A22 0 A22 0 Since all the eigenvalues on the imaginary axis of A11 ∈ Rv×v are distinct we ﬁnd that the eigenvectors of A11 span Rv and hence $ % % $ % $ A11 0 I A11 0 P¯ + P¯ = 0. 0 0 A22 0 A22 This leads to

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$ $ % % $ % A A11 0 0 0 0 11 P¯ + P¯ = ≤ 0. 0V 0 A22 0 A22 This immediately implies that A11 P¯12 + P¯12 A22 = 0 and since A11 and A22 have no eigenvalues in common we ﬁnd that P¯12 = 0. Thus, we have A11 P¯11 + P¯11 A11 = 0 and A22 P¯22 + P¯22 A22 = V ≤ 0. Next, since A22 has all its eigenvalues in the open left half plane, there exists a parameterized matrix P22,δ for δ small enough such that A22,δ P¯22,δ + P¯22,δ A22,δ = V ≤ 0 while P¯22,δ → P22 as δ → 0. Let A11 = W ΛA W −1 with ΛA a diagonal matrix. Because the eigenvectors of A11 are distinct and A11,δ → A11 , for δ small enough the eigenvectors of A11,δ depend continuously on δ and hence there exists a parameterized matrix Wδ such that Wδ → W while A11,δ = Wδ ΛA W −1 with ΛA diagonal. The matrix P¯11 δ

δ

δ

satisﬁes A∗11 P¯11 + P¯11 A11 = 0 This implies that ΛP = W ∗ P¯11 W satisﬁes Λ∗A ΛP + ΛP ΛA = 0. The above equation then shows that ΛP is a diagonal matrix. We know that Λ Aδ → Λ A . We know that ΛAδ is a diagonal matrix whose diagonal elements have real part less than or equal to zero while ΛP is a positive-deﬁnite diagonal matrix. Using this, it can be veriﬁed that we have Λ∗Aδ ΛP + ΛP ΛAδ ≤ 0. We choose P¯11,δ as P¯11,δ = (Wδ∗ )−1 ΛP Wδ−1 . Obviously, our choice of P¯11,δ satisﬁes A∗11,δ P¯11,δ + P¯11,δ A11,δ ≤ 0.

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We observe that P¯11,δ → P¯11 as δ → 0. But then $ % ¯11,δ 0 P Pδ = (Sδ−1 ) Sδ−1 ¯ 0 P22,δ satisﬁes the conditions of the lemma. This completes the proof of Lemma 3. We proceed now with the proof of Theorem 2. Our proof is constructive and involves a sequential design. We present a recursive algorithm which at each step applies a decentralized feedback law which stabilizes at least one eigenvalue on the imaginary axis while preserving the stability of the stable modes of the system in such a way that the magnitude of each decentralized feedback control is guaranteed never to exceed 1/n. Therefore, after at most n steps the combination of these decentralized feedback laws will asymptotically stabilize the system without ever violating the magnitude constraints of each of the inputs. The basic steps of the algorithm are as formalized below: Algorithm: • Step 0 (Initialization): We ﬁrst initialize our algorithm at step 0. To do so, let 0 A0 := A, B0,i := Bi , C0,i := Ci , ni,0 := 0, Ni,ε := 0, i = 1, . . . , ν and

x0 := x. Moreover, deﬁne P0ε := εP , where P > 0 is a matrix such that A P + P A ≤ 0. Since all the eigenvalues of A on the imaginary axis have multiplicity 1, we know from Lemma 2 that such a matrix P exists. • Step k: For the system Σ given by (3), we have to design ν parameterized decentralized feedback control laws, k k p˙ki = Ki,ε pi + Lki,ε yi , k,ε Σi : k ui = Mi,ε pki + Ni,ε yi + vik in case ni,k > 0, and otherwise Σk,ε : i

pki ∈ Rni,k

k yi + vik , ui = Ni,ε

(6)

(7)

for i = 1, . . . , ν. The closed-loop system comprising the above decentralized feedback control laws and the system Σ of (1) can be written as

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Σk,ε cl

⎧ ν ⎪ ⎨ x˙ k = Aε xk + Bk,i vik k : i=1 ⎪ ⎩ yi = Ck,i xk , i = 1, . . . , ν,

where xk ∈ Rnk with nk = n +

(8)

ν

ni,k is given by ⎛ ⎞ x ⎜ k⎟ ⎜ p1 ⎟ ⎜ ⎟ xk = ⎜ . ⎟ . ⎜ .. ⎟ ⎝ ⎠ pkν i=1

(9)

In view of (9), we can rewrite ui as k xk + vik ui = Fi,ε k for some appropriate matrix Fi,ε .

The above decentralized feedback control laws given by either (6) or (7) are to be designed in such a way that they satisfy the following properties: 1) The matrix Aεk has all its eigenvalues in the closed-left half plane, and those eigenvalues of Aεk which are on the imaginary axis are distinct. 2) The number of eigenvalues of Aεk on the imaginary axis must at least be one less than the number of eigenvalues of Aεk−1 on the imaginary axis (i.e. at each step of our recursive algorithm we design a decentralized feedback law which stabilizes at least one more eigenvalue on the imaginary axis while preserving the stability of the stable modes of the system designed until then). 3) There exists a family of matrices Pkε such that Pkε → 0 as ε → 0 while for vik = 0, i = 1, . . . , ν, the closed-loop system Σk,ε cl of (8) is such that xk (t) Pkε xk (t) is non-increasing in t for all initial conditions, i.e. (Aεk ) Pkε + Pkε Aεk ≤ 0.

(10)

Moreover, there exists an ε∗ such that for all ε ∈ (0, ε∗ ] we have ui (t) ≤

k n

for all states with xk (t) Pkε xk (t) ≤ n − k + 1.

(11)

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It is easy to verify that all of the above conditions are true for k = 0. • Terminal step: There exists a value for k, say ≤ n, such that the matrix Aε has all its eigenvalues in the open-left half plane. We set vi = 0 for i = 1, . . . , . The decentralized control laws Σ ,ε i , i = 1, . . . , as given by (6) or (7), all together, represent a decentralized semi-global state feedback law for the given system Σ of (1). More precisely, for any given compact sets W ⊂ Rn , and Si ⊂ Rni, for i = 1, . . . ν, there exists an ε∗ such that the origin of the closed-loop system comprising the given system Σ of (1) and the decentralized control laws Σ ,ε i , i = 1, . . . , as given by (6) or (7) is exponentially stable for any 0 < ε < ε∗ , and the compact set W × S1 × · · · × Sν is within the domain of attraction. Moreover, for all the initial conditions within W × S1 × · · · × Sν , the said closed-loop system behaves like a linear dynamic system, that is the saturation is not activated implying that ui < 1 for all i = 1, . . . , ν. The fact that the decentralized control laws Σ ,ε i , i = 1, . . . , as given by (6) or (7) are semi-globally stabilizing follows from the property 3) as given in step k of the above algorithm. To be explicit, we observe that, for an ε sufﬁciently small, the set Ωε1 := {x ∈ Rn |x P ε x ≤ 1} is inside the domain of attraction of the equilibrium point of the closed-loop system comprising the given system Σ of (1) and the decentralized control laws Σ ,ε i , i = 1, . . . , as given by (6) or (7). This follows from the fact that for all the initial conditions within Ωε1 , it is obvious from (11) that ui ≤ 1 for all i = 1, . . . , ν. This implies that the said closed-loop system behaves like a linear dynamic system, that is the saturation is not activated. Moreover, this linear dynamic system is asymptotically stable since Aε has all its eigenvalues in the open left half plane, and hence the state converges to zero asymptotically. Next, since P ε → 0 as ε → 0, for an ε sufﬁciently small, we have that the compact set W × S1 × · · · × Sν is inside Ωε1 . This concludes that the decentralized control laws Σ ,ε i , i = 1, . . . , as given by (6) or (7) are semi-globally stabilizing. This completes the description of our recursive algorithm to design the decentralized feedback control laws having the properties as given in Theorem 2.

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It remains to prove that the above recursive algorithm succeeds in designing the decentralized feedback control laws having the properties as given in Theorem 2. In order to do so, we assume that the design of decentralized feedback control laws as described in step k can be done, and then prove that the corresponding design in step k + 1 can be done. We proceed now to prove this. After step k we have for the system Σ of (1), ν feedback control laws of the form (6) or (7) such that the system (8) obtained after applying these feedbacks has the properties 1), 2) and 3). We consider the closed-loop system Σk,ε cl of (8). Let λ be an eigenvalue on the imaginary axis of Aεk . We know that decentralized feedback laws do not change the ﬁxed modes and therefore, since λ was not a ﬁxed mode of the original system (1), it is not a ﬁxed mode of the system (8) obtained after applying ν ¯ i such that feedback laws either. Hence there exists a K Aεk

+

ν

¯ i Ck,i Bk,i K

i=1

has no eigenvalue at λ. Therefore, Aεk

+δ

ν

¯ i Ck,i Bk,i K

i=1

has no eigenvalue at λ for almost all δ > 0 (the determinant of λI − Aεk − ν ¯ i Ck,i is a polynomial in δ and is nonzero for δ = 1 and therefore δ i=1 Bk,i K the determinant has a ﬁnite number of zeros). Let j be the largest integer such that Aε,δ k

=

Aεk

+δ

j

¯ i Ck,i Bk,i K

i=1

has the same number of eigenvalues on the imaginary axis as Aεk for δ > 0 small enough. This implies that Aε,δ k still has all its eigenvalues in the closed left half plane for δ small enough. We know that (10) is satisﬁed and hence using Lemma 3 we ﬁnd that there exists a P¯ ε,δ such that k

¯ ε,δ ¯ ε,δ ε,δ (Aε,δ k ) Pk + Pk Ak ≤ 0

while P¯kε,δ → Pkε as δ → 0. Hence for δ small enough xk P¯kε,δ xk ≤ n − k +

1 2

=⇒ xk Pkε xk ≤ n − k + 1

(12)

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415

and for δ small enough we have that ¯ i xk ≤ δK

1 2n

for all xk with xk Pkε xk ≤ n − k + 1.

(13)

For each ε choose δ = δε small enough such that the above two properties (12) and ¯ i , P¯ ε = P¯ ε,δε and (13) are satisﬁed. We deﬁne K ε = δε K i

k

A¯εk := Aεk +

j

k

Bk,i Kiε Ck,i .

i=1

By the deﬁnition of j, we know that Aεk

+

j+1

Bk,i Kiε Ck,i

i=1

has less eigenvalues on the imaginary axis than A¯εk . Hence (A¯εk , Bk,j+1 , Ck,j+1 ) has a stabilizable and detectable eigenvalue on the imaginary axis. Choose V such that VV =I

and

ker V = ker Ck,j+1 | A¯εk .

We choose the following decentralized feedback law, vik = Kiε xk + vik+1 ,

(14)

i = 1, . . . , j ,

k p˙ = Aεs p + V Bk,j+1 vj+1 + K(Ck,j+1 V p − yj+1 ) k+1 k vj+1 = Fρ p + vj+1

i = j + 2, . . . , ν.

vik = vik+1 ,

(15) (16)

Equations (14), (15), and (16) together represent our decentralized feedback control laws at step k + 1. Here p ∈ Rs and Aε is such that Aε V = V A¯ε while K is chosen s

such that

Aεs

s

k

+ KCk,j+1 V has all its eigenvalues in the open left half plane while Aεs + KCk,j+1 V and A¯εk have no eigenvalues in common. Moreover, for all ρ the matrix A¯ε + Bk,j+1 Fρ V has at least one eigenvalue less on the imaginary axis than k

Aεk

does, and still has all its eigenvalues in the closed left half plane while Fρ → 0

as ρ ↓ 0. Rewriting the resulting system in a new basis consisting of xk and p − V xk results in

Anton A. Stoorvogel, , Ali Saberi, Ciprian Deliu, , and Peddapullaiah Sannuti

416

x ¯˙ k+1 =

$ A¯εk + Bk,j+1 Fρ V

%

Bk,j+1 Fρ Aεs

0

+ KCk,j+1 V

x ¯k+1 +

ν

¯k+1,i v k+1 B i

i=1

(17)

i = 1, . . . , ν,

yi = C¯k+1,i x ¯k+1 ,

$

where ¯k+1,i = B

%

Bk,i

,

−V Bk,i

# " ¯ Ck+1,i = Ck,i 0

for i = j + 1 while ¯k+1,j+1 = B

$ % Bk,j+1 0

$ C¯k+1,j+1 =

, $

and x ¯k+1 =

xk

p − V xk

Ck,j+1 0 V

%

I

% .

Obviously, the above feedback laws (14), (15), and (16) satisfy at step k + 1 the properties 1), and 2) as mentioned in step k. What remains to show is that they also satisfy property 3). Moreover, we need to write the control laws (14), (15), and (16) in the form of (6) or (7) for step k + 1. In what follows we focus on these aspects. For any ε there exists a Rkε > 0 with (Aεs + KCk,j+1 V ) Rkε + Rkε (Aεs + KCk,j+1 V ) < 0 such that Rkε → 0 as ε ↓ 0. Since Fρ → 0 as ρ → 0, for each ε we have for ρ small enough Fρ e <

1 2n

for all e such that e Rkε e ≤ n − k +

1 2

(18)

where e = p − V xk . We have that A¯εk + Bk,j+1 Fρ V is a perturbation of A¯εk , but it has at least one eigenvalue less on the imaginary axis than A¯ε while having all its eigenvalues in the closed k

left half plane. But then, using Lemma 3, for ρ small there exists a matrix P¯ρε > 0 such that (A¯εk + Bk,j+1 Fρ V ) P¯ρε + P¯ρε (A¯εk + Bk,j+1 Fρ V ) ≤ 0 with P¯ρε → P¯kε as ρ → 0. Finally because A¯εk and Aεs + KCk,j+1 V have disjoint eigenvalues we note that for ρ small enough we get that A¯ε + Bk,j+1 Fρ V and Aε + KCk,j+1 V have disjoint k

s

eigenvalues since Fρ → 0 as ρ ↓ 0. But then there exists a Wε,ρ such that

Decentralized Control with Input Saturation

417

Bk,j+1 Fρ + (A¯εk + Bk,j+1 Fρ V )Wε,ρ − Wε,ρ (Aεs + KCk,j+1 V ) = 0 while Wε,ρ → 0 as ρ ↓ 0. Note that this implies that $ %$ %$ % ε ¯ I 0 I −W 0 P ε,ρ ε,ρ ρ P¯k+1 = ε −Wε,ρ I 0 Rk 0 I has the property that: ¯ ε,ρ ¯ ε,ρ ¯ε,ρ (A¯ε,ρ k+1 ) Pk+1 + Pk+1 Ak+1 ≤ 0

for A¯ε,ρ k+1

=

$ A¯εk + Bk,j+1 Fρ V

ε,ρ = lim P¯k+1 ρ↓0

%

Bk,j+1 Fρ Aεs + KCk,j+1 V

0

and

(19)

$ % P¯kε 0 0 Rkε

.

We consider x ¯k+1 such that ε,ρ x ¯k+1 ≤ n − k. x ¯k+1 P¯k+1

(20)

Then we can choose ρ small enough such that xk P¯kε xk ≤ n − k +

1 2

and (p − V xk ) Rkε (p − V xk ) ≤ n − k + 12 .

(21)

We choose for each ε a ρ = ρε such that (18) is satisﬁed while (20) implies that (21) is satisﬁed and ﬁnally Fρ V xk <

1 2n

for all xk such that xk P¯kε xk ≤ n − k + 12 .

In order to establish that the bounds on the inputs are satisﬁed in step k + 1, we set vik+1 = 0 for i = 1, . . . ν. Then, we obtain for i = 1, . . . , j, k xk + Kiε xk ≤ ui (t) = Fi,ε

k n

+

1 2n

≤

k+1 n

for all x ¯k+1 such that (20) is satisﬁed. For i = j + 1 we obtain, k xk + Fρε p ui (t) = Fi,ε k xk − Fρε V xk + Fρε (p − V xk ) ≤ = Fi,ε

Finally for i = j + 2, . . . , ν, we obtain

k n

+

1 2n

+

1 2n

=

k+1 n .

418

Anton A. Stoorvogel, , Ali Saberi, Ciprian Deliu, , and Peddapullaiah Sannuti k ui (t) = Fi,ε xk ≤

k n

≤

k+1 n .

We now focus on rewriting the decentralized control laws (14), (15), and (16) in the form of (6) or (7) for step k + 1. We set ni,k+1 = ni,k for i = j + 1 while for i = j + 1 we set ni,k+1 = ni,k + s and $ % pki k+1 pi = =p or pk+1 i p in case ni,k > 0 or ni,k = 0 respectively. We can then rewrite the system (17) in terms of the state xk+1 (deﬁned by (9)) instead of x ¯k+1 which requires a basis transformation Tk+1 , i.e. x ¯k+1 = Tk+1 xk+1 . We deﬁne ε,ρε ε = Tk+1 Tk+1 P¯k+1 Pk+1

and obviously, for i = 1, . . . , ν, we can write the relationship between yi , vik+1 and ui in the form (6) or (7) depending on whether ni,k+1 = 0 or not. We can now rewrite the control laws (14), (15), and (16) in the form k+1 k+1 p˙k+1 = Ki,ε pi + Lk+1 i i,ε yi , Σk+1,ε : i k+1 ui = Mi,ε pk+1 + Ni,ε yi + vik+1 i

pk+1 ∈ Rni,k+1 i

in case ni,k+1 > 0, and otherwise k+1 : Σk+1,ε yi + vik+1 , ui = Ni,ε i

(22)

(23)

for i = 1, . . . , ν. It is then clear that properties 1), 2) and 3) are satisﬁed in step k +1. This concludes the proof of Theorem 2.

References 1. B.D.O. A NDERSON AND D.J. C LEMENTS, “Algebraic characterization of ﬁxed modes in decentralized control”, Automatica, 17(5), 1981, pp. 703–712. 2. D.S. B ERNSTEIN AND A.N. M ICHEL, Guest Eds., Special Issue on saturating actuators, Int. J. Robust & Nonlinear Control, 5(5), 1995, pp. 375–540. 3. B.M. C HEN , A. S ABERI , AND P. S ANNUTI, “On blocking zeros and strong stabilizability of linear multivariable systems”, Automatica, 28(5), 1992, pp. 1051–1055. 4. J.P. C ORFMAT AND A.S. M ORSE, “Decentralized control of linear multivariable systems”, Automatica, 12(5), 1976, pp. 479–495.

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5. G.H. G OLUB AND C.F. VAN L OAN, Matrix Computations, John Hopkins Univ. Press, 2nd Ed., 1989. 6. Z. G ONG AND M. A LDEEN, “On the characterization of ﬁxed modes in decentralized control”, IEEE Trans. Aut. Contr., 37(7), 1992, pp. 1046–1050. ¨ 7. N. G UNDES ¸ AND C.A. D ESOER, Algebraic theory of linear feedback systems with full and decentralized compensators, vol. 142 of Lecture Notes in Control and Information Sciences, Springer Verlag, 1990. 8. V. K APILA AND G. G RIGORIADIS, Eds., Actuator saturation control, Marcel Dekker, 2002. 9. T. K ATO, Perturbation theory for linear operators, Springer Verlag, Berlin, Second Ed., 1976. 10. A.N. M ICHEL AND R.K. M ILLER, Qualitative analysis of large scale dynamical systems, Academic Press, 1977. 11. J R . N. R. S ANDELL , P. VARAIYA , M. ATHANS , AND M.G. S AFONOV, “Survey of decentralized control methods for large scale systems”, IEEE Trans. Aut. Contr., 23(2), 1978, pp. 103–128. 12. H. S ERAJI, “On ﬁxed modes in decentralized control systems”, Int. J. Contr., 35(5), 1982, pp. 775–784. 13. D.D. S ILJAK, Large-scale dynamic systems : stability and structure, North Holland, Amsterdam, 1978. 14.

, Decentralized control of complex systems, Academic Press, London, 1991.

15. A. S ABERI AND A.A. S TOORVOGEL, Guest Eds., Special Issue on control problems with constraints, Int. J. Robust & Nonlinear Control, 9(10), 1999, pp. 583–734. 16. G. S TEIN, “Respect the Unstable”, in IEEE Conference of Decision and Control, Tampa, FL, 1989. Bode prize lecture. 17. S.H. WANG AND E.J. DAVISON, “On the stabilization of decentralized control systems”, IEEE Trans. Aut. Contr., 18(5), 1973, pp. 473–478.

On the Stabilization of Linear Discrete-Time Delay Systems Subject to Input Saturation Karim Yakoubi and Yacine Chitour Laboratoire des signaux et syst`emes, Univ. Paris-Sud, CNRS, Sup´elec, 91192 Gif-sur-Yvette cedex, France. [email protected] [email protected] Summary. This chapter deals with two problems on stabilization of linear discrete systems by static feedbacks which are bounded and time-delayed, namely global asymptotic stabilization and ﬁnite gain lp −stabilizability, p ∈ [1, ∞]. Furthermore, an explicit construction of the corresponding feedback laws is given. The feedback laws constructed also result in a closedloop system that is globally asymptotically stable. The results presented here are parallel to our earlier results on the continuous-time delay case, and show that, for the ﬁrst issue, such stabilization is possible if and only if the system is stabilizable and the transition matrix has spectral radius inside or on the unit circle. For the ﬁnite-gain lp −stabilization issue, we assume that the system is neutrally stable. We provide upper bounds for the corresponding lp −gain which is delay-independent.

Key words: Saturated feedback, global stabilization, Lyapunov functions, Linear discrete-time delay systems, Finite-gain stability.

1 Introduction In this chapter we address two issues relative to the stabilization of a linear discrete system subject to input saturation and time-delayed, of the type (S) : x(k + 1) = Ax(k) − rBσ(u(k − h)),

(1)

422

Karim Yakoubi and Yacine Chitour

where (i) x(k) ∈ Rn and u(k) ∈ Rm , with n the dimension of the system and m the number of inputs; (ii) the input bound r ∈ (0, 1] only depend on (S) and there is an arbitrary constant delay h ∈ N∗ appearing in the input; (iii) σ : Rm → Rm represents the actuator saturation (deﬁnitions are given in section 2). The results presented here are an analog of those established for the continuoustime counterpart, in the papers [9] and [10]. Although the organization of the current work is tightly patterned after that of [9] and [10], and many of the arguments given here are speciﬁc of the discrete-time case and cannot be reduced to corresponding arguments in the continuous-time case. In the zero-delay case, the above problems have been widely investigated in the last years: see for example [11] and reference therein for stabilization and [1], [2] and reference therein for ﬁnite-gain stabilizability. It was shown in [1] that, for neutrally stable discrete-time linear systems, ﬁnite-gain lp −stabilization can be achieved by linear output feedback, for every p ∈ (1, ∞] except p = 1. In [2], the authors complete the results by showing that they also hold true for the case p = 1. A result due to Yang, Sontag and Sussmann ([11]) shows that such stabilization is possible if and only if the system (S) is stabilizable and the transition matrix has spectral radius less or equal to one. It is trivial to see that these conditions are also necessary in the case of non-zero delay and it seems natural to expect them to be also sufﬁcient. The objective of this chapter is to show that the results of [9] and [10], however, do carry over to linear discrete-time delay systems. More precisely, for the problem of global asymptotic stabilization we show that, a linear discrete-time delay system subject to input saturation can be globally asymptotically stabilized via feedback if and only if the system is stabilizable and all its poles are located inside or on the unit circle. The argument is an extension to the non-zero delay case of that of [11]. Concerning the problem of ﬁnite gain lp −stabilizability, p ∈ [1, ∞], we should also mention that, for neutrally stable linear discrete time delay systems subject to actuator saturation, ﬁnite gain lp −stabilization can be achieved by linear output feedback for all p ∈ [1, ∞]. We determine a suitable “storage function” Vp and establish for it a “dissipation inequality” of the form ΔVp (k) ≤ −λ1p xu (k) p + λ2p u(k) p , for some constants λ1p > 0 and λ2p > 0 possibly depending on the input bound r and the delay h, with λ1p 0 λ2p (deﬁnition is given in section 2). For more discussion on passivity, see [8] for instance. In addition, by choosing carefully the factor r and

Stabilization and Finite-Gain Stabilizability

423

the linear feedback inside the saturation, we are able to provide upper bounds for the lp −gains of (S) which are independent of h > 0. Since the delay h is a constant positive integer, the standard trick to get rid of the delay consists of extending the state space to Xk := (x(k), x(k − 1), · · · , x(k − h)), k ≥ h, and to rewrite the dynamics for Xk . As concerns the stabilization question for the X−system, one may be able to derive similar results as ours even though the corresponding linear algebraic problem does not seem easy. However, as regards the lp −stabilization problem for the X−system, the lp −gain will depend on the delay since the latter appears in the dimension of the state space of X. This is the reason why using the X−system is not relevant for our studies. Organization of the chapter. This chapter is organized as follows. In sections 2 and 3, we provide the main notations used in the chapter and state the Krasovskii Stability Theorem. Section 4 present the ﬁnite-gain lp −stabilizability problem. Solution to the delay-independent stabilization based on feedbacks of saturation type is described in section 5. Section 6 contains our result of output problem. The chapter closes with section 7, which draws the conclusions of our current work.

2 Notations and Deﬁnition of Saturation Functions For x ∈ Rn , x and xT denote respectively the Euclidean norm of x and the transpose of x. Similarly, for any n × m matrix K, K T and K denote respectively the transpose of K and the induced 2−norm of K. Moreover, λmin (K) and λmax (K) denote the minimal and the maximal singular values of the matrix K. If f (.) and g(.) are two real-valued functions, we mean by f (r) 0 g(r), that there are positive constants ξ1 and ξ2 , independent of r small enough, with ξ1 ≤ ξ2 such that ξ1 g(r) ≤ f (r) ≤ ξ2 g(r). For h > 0, let xk (θ) := x(k + θ), −h ≤ θ ≤ 0. Initial conditions for timedelay systems are elements of Ch := C([−h, 0], Rn ), the Banach space of continuous Rn − valued functions deﬁned on [−h, 0], equipped with the supremum norm, xk h = sup−h≤θ≤0 x(k + θ) . Then, for xk ∈ l∞ ([−h, ∞), Rn ), we have xk l∞ = sup−h≤s<∞ x(k + s) ). Initial conditions for time-delay systems " #T are x0 = x(0), x(−1), . . . , x(−h) .

424

Karim Yakoubi and Yacine Chitour

We deﬁne a class of saturation functions to be considered in this chapter. Deﬁnition 1. (Saturation functions) We call σ : R −→ R a saturation function (“Sfunction” for short) if there exist two real numbers 0 < a ≤ Kσ such that for all t, t ∈ R, (i) |σ(t) − σ(t )| ≤ Kσ inf(1, |t − t |), (ii) |σ(t) − at| ≤ Kσ tσ(t), It is assumed here that the function is normalized at the origin, i.e. a = σ (0) = 1. A constant Kσ deﬁned as above is called an S-bound for σ. When the context is clear, we simply use K to denote an S-bound. Note that (i) is equivalent to the fact that σ is bounded and globally lipschitz, while (ii) is equivalent to lim inf |σ(t)| > 0,

tσ(t) > 0 for t = 0,

|t|→∞

lim sup t→0

σ(t) − t < ∞. t2

For an m−tuple l = (l1 , . . . , lm ) of nonnegative integers, deﬁne |l| = l1 + . . . + lm . We say that σ is an R|l| − valued S-function if σ = (σ1 , . . . , σ|l| ) = (σ11 , . . . , σl11 , . . . , σ1m , . . . , σlmm ) ! = (σi1 )1≤i≤l1 , (σi2 )1≤i≤l2 , . . . , (σim )1≤i≤lm ,

j

where, for 1 ≤ j ≤ m, (σij )1≤i≤lj is an Rl -valued S-function (i.e : (σij )1≤i≤lj = (σ1j , . . . , σljj ) where each component σij , 1 ≤ i ≤ lj is an S-function and " # (σij )1≤i≤lj (x) = σ1j (x1 ), . . . , σljj (xlj ) , j

for x = (x1 , . . . , xlj )T ∈ Rl .)

3 Krasovskii Stability Theorem We consider some stability properties of the solutions of the discrete time-delay linear systems. They are deﬁned in a similar fashion as in the linear delay-free system.

Stabilization and Finite-Gain Stabilizability

425

For stability analysis of systems with time-delay, the Krasovskii Theorem is used extensively. Consider the discrete-time functional differential equation of retarded type ⎧ ⎪ ⎪ ⎨ x(k + 1) = f (xk ), k ≥ k0 ; (Σ)h : ⎪ ⎪ ⎩ x (θ) = Ψ(θ), ∀θ ∈ [−h, 0]. k0

It is assumed that Ψ ∈ Ch and the map f is continuous and Lipschitz in Ψ and f (0) = 0. Deﬁnition 2. We say that the trivial solution x(k) ≡ 0 of (Σ)h is globally asymptotically stable (GAS for short) if the following conditions holds: (i) for every ε > 0, there exists a δ > 0 such that, for any Ψ ∈ Ch , with ||Ψ||Ch ≤ δ, there exists k0 ≥ 0, such that the solution xΨ of (Σ)h satisﬁes ||(xΨ )k ||h ≤ ε, for all k ≥ k0 ; (ii) for all Ψ ∈ Ch , the trajectory of (Σ)h with initial condition Ψ converges to zero as k → ∞. The Krasovskii Theorem gives conditions under which x(k) = 0 is asymptotically stable. As in the continuous time-delay case, the Krasovskii stability theorem is given by: Theorem 1. (Krasovskii Stability Theorem): Suppose that the function f : Ch → Rn takes bounded sets of Ch in bounded sets of Rn and suppose that u(s), v(s) and w(s) are scalar, continuous, positive and nondecreasing functions. If there is a continuous function V : Ch → R+ and a positive number ρ such that for all xt ∈ LV (ρ) := {ψ ∈ Ch : V (ψ) ≤ ρ}, the following conditions hold. 1. u( xk (0) ) ≤ V (xk ) ≤ v( xk h ). 2. ΔV (xk ) ≤ −w( xk (0) ). Then, the solution x(k) ≡ 0 of (Σ)h is asymptotically stable. Moreover, the set LV (ρ) is an invariant set inside the domain of attraction. If items 1 and 2 hold for all x ∈ Rn and u(s) → ∞ as s → ∞, then, the solution x(k) ≡ 0 is GAS.

426

Karim Yakoubi and Yacine Chitour

4 On Finite-Gain Stabilizability 4.1 Statement of the Results In this section, we establish ﬁnite-gain lp −stabilizability in the various p− norms, using linear state feedbacks stabilizers. we start by giving some deﬁnitions. For p ∈ [1, ∞] and 0 ≤ h(≤ T ), we use lp to denote lp (−h, ∞) and we let y lp ( y lp ([−h, T ]) respectively) denote the lp −norm : $ y lp =

∞

$

% p1 y(k) p

, ( y lp ([−h,T ]) =

k=−h

T

% p1 y(k) p

respectively),

k=−h

if p < ∞ and y l∞ =

sup

−h≤k<∞

{ y(k) } , ( y l∞ ([−h,T ]) =

sup

−h≤k≤T

{ y(k) } , respectively).

Consider a control discrete-time delay input system of the form (Σ)h : x(k + 1) = f (x(k), u(k − h)) , for k ∈ Z+ ,

(2)

where the state x and the control u take respectively values in Rn and Rm and f : Rn × Rm → Rn , is locally Lipschitz in (x, u), with f (0, 0) = 0. Trajectories of (Σ)h starting at an initial condition x0 ∈ Ch and corresponding to an input u ∈ lp are deﬁned for a time interval I of Z+ (which may depend on x0 and u) and verify (2) for almost every k ∈ I. Let ¯ 0 be the zero function in Ch , h ≥ 0. Deﬁnition 3. (lp −stability): Given p ∈ [1, ∞], the discrete-time delay system (Σ)h is said to be lp −stable if, for every u ∈ lp , we have xu ∈ lp , where xu denotes the solution of (Σ)h corresponding to u with initial condition x0 = ¯ 0. Deﬁnition 4. (Finite gain lp − stability): Given p ∈ [1, ∞], the discrete-time delay system (Σ)h is said to be ﬁnite-gain lp −stable if it is lp −stable, and there exists a positive constant mp such that, for every u ∈ lp , xu lp ≤ mp u lp . Furthermore, the inﬁmum over such numbers mp will be called the lp −gain of the system.

Stabilization and Finite-Gain Stabilizability

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In the next theorem, we summarize our results. Theorem 2.. Let A, B be n×n, n×m matrices respectively. Let σ be an Rm −valued S−function. Assume that A be neutrally stable and (A, B) is stabilizable. Then, for every h > 0 there exists an n × m matrix Fh such that, the following delayed system ! (S)rh : x(k+1) = Ax(k)−rBσ FhT x(k − h) + u1 (k − h) +ru2 (k−h), for k ∈ Z+ , is unrestricted ﬁnite-gain lp -stable, for every p ∈ [1, ∞]. Moreover, there exist a constant C0 > 0 and, for every 1 ≤ p ≤ ∞, a real mp > 0 such that, for every h > 0 there is an r∗ (h) ∈ (0, 1], for which the trajectories xu1 ,u2 of (S)rh , r ∈ (0, r∗ (h)], starting at ¯ 0 and corresponding to u1 , u2 ∈ lp with u2 l∞ ≤ C0 , verify xu1 ,u2 lp ≤ mp ( u1 lp + u2 lp ). Remark 1. In the absence of u1 and u2 , the equilibrium point ¯ 0 is globally asymptot! ically stable for the delayed system x(k + 1) = Ax(k) − rBσ FhT x(k − h) . Remark 2. It will be clear from our argument that we can in fact obtain the following stronger ISS-like property ([6] and references there): xψ u1 ,u2 lp ≤ θp ( ψ h ) + Mp ( u1 lp + u2 lp ), where ψ ∈ Ch is the initial condition for the trajectory xψ u1 ,u2 corresponding to u1 , u2 and θp is a K -function (i.e. θp : R+ → R+ , is continuous, strictly increasing and satisﬁes θp (0) = 0.).

4.2 Proof of Theorem 2 From elementary linear algebra, a neutrally stable matrix A is similar to a matrix $ % A1 0 , (3) 0 A2 where A1 is a q × q convergent or discrete-time Hurwitz (i.e. all its eigenvalues have magnitude less than 1) matrix and A2 is an (n − q) × (n − q) orthogonal matrix. So, up to a change of coordinates, we may assume that A is already in the form (3). In these coordinates, we write

428

Karim Yakoubi and Yacine Chitour

$ B=

B1 B2

% ,

where B2 is an (n − q) × m matrix and we write vectors as x = (xT1 , xT2 )T , and u2 = (uT21 , uT22 )T . For h > 0, consider the feedback law (0, FhT ). Then system (S)rh , with this choice of FhT , can be written as ⎧ T ⎪ ⎪ ⎨ x1 (k + 1) = A1 x1 (k) − rB1 σ[Fh x2 (k − h) + u1 (k − h)] + ru21 (k − h), ⎪ ⎪ ⎩ x (k + 1) = A x (k) − rB σ F T x (k − h) + u (k − h)! + ru (k − h). 2 2 2 2 1 22 h 2 Since A1 is discrete-time Hurwitz, it will be sufﬁcient to show that there exists an r∗ (h) ∈ (0, 1], such that the x2 − subsystem is ﬁnite-gain lp −stable, for all r ∈ (0, r∗ (h)]. The controllability assumption on (A, B) implies that the pair (A2 , B2 ) is also controllable. Therefore, the theorem is a consequence of the following proposition. Proposition 1. Let σ, u1 , u2 be as in Theorem 2. Let (A, B) a controllable pair with A orthogonal (i.e., AT A = Idn ). Then, for every h > 0, there is an r∗ (h) ∈ (0, 1], such that for every r ∈ (0, r∗ (h)], the system ! (S)rh : x(k + 1) = Ax(k) − rBσ B T Ah+1 x(k − h) + u1 (k − h) + ru2 (k − h), for k ∈ Z+ , veriﬁes the conclusion of Theorem 2. To prove this proposition, ﬁrst we need to establish a few lemmas carried out in [1]. Lemma 1. For any p > l > 0, there exist two scalars M1 , M2 > 0 such that, for any two positive scalars ξ and ζ, ξ p−l ζ l ≤ M1 ξ p + M2 ζ p . 2 Lemma 2. For any l > 0, there exists a constant Cl > 0 such that (a + b)l ≤ Cl (al + bl ), ∀a, b ≥ 0, Cl := 2l−1 . Consequently, if l ≤ 1, we have (a + b)l ≤ al + Cl bl , ∀a, b ≥ 0.

2

Stabilization and Finite-Gain Stabilizability

429

Lemma 3. Let A be orthogonal, and suppose that the pair (A, B) is controllable. Then, for every r > 0 such that rB T B < 2In , set Ar := A − rBB T A and Pr the unique symmetric positive-deﬁnite solution of the Lyapunov equation ATr Pr Ar − Pr = −In . Then, there exists a r∗ > 0 and χ1 , χ2 > 0 independent of r, such that for every r ∈ (0, r∗ ],

χ1 χ In ≤ Pr ≤ 2 In . r r

2

Remark 3. By Lemma 3, there exists r1∗ ∈ (0, r∗ ] such that λmax (Pr ) 0 λmin (Pr ) 0

1 , r

∀r ∈ (0, r1∗ ].

(4)

Lemma 4. Let Ar and Pr be as given in Lemma 3, then for any p ∈ (1, ∞), there exists a r∗ > 0 such that p

p

[z T ATr Pr Ar z] 2 − [z T Pr z] 2 ≤ −r where ζ > 0 is some constant independent of r.

2−p 2

ζ|z|p , r ∈ (0, r∗ ],

2

Lemma 5. Let A and B be as given in Lemma 3. For any l ∈ [1, ∞) and any r ∈ (0, 1], |σ(B T Az + u)|l ≤ 2l−1 |B|l |z|l + 2l−1 |u|l .

2

Lemma 6. Let A and B be as given in Lemma 3. Pick any z ∈ Rn and d˜ ∈ Rm , any ˜ number η ≥ 3, and any nonnegative real number l. Denote z˜(k) = B T Az(k)+ d(k). Then, provided |z| > rη|Bσ(˜ z )|, we have z ) + M r2 |z|l−2 |Bσ(˜ z )|2 , |Az − rBσ(˜ z )|l ≤ |z|l − lr|z|l−2 z T AT Bσ(˜ for some constant M > 0 which is independent of r.

2

We are now ready to prove Proposition 1. Proof of Proposition 1. Let h > 0 and y be the solution of ⎧ ! T ⎪ y(k + 1) = A − rBB A y(k) + ru2 (k − h), for k ∈ Z+ , ⎪ ⎨ ⎪ ⎪ ⎩y = ¯ 0, on [−h, 0], 0

(5)

430

Karim Yakoubi and Yacine Chitour

with u2 ∈ lp and u2 l∞ ≤ C0 , where C0 is a positive constant to be determined later. (By an obvious abuse of notation, lp denotes here lp ([−h, ∞), Rn ).) Let r1∗ > 0 be such that rBB T < 2In . Then, Ar := A−rBB T A is asymptotically stable for all r ∈ (0, r1∗ ], (see [3]). So, for every r ∈ (0, r1∗ ], (5) is lp −stable for any 1 ≤ p ≤ ∞. Set γp denotes its lp −gain, then y lp ≤ γp u2 lp .

(6)

We need the following lemma. Lemma 7. Let γp be the lp −gain of (6). So γp 0 O(1). Proof of Lemma 7: We separate the proof for p ∈ [1, ∞) and for p = ∞. Proof for p ∈ [1, ∞) : The equation (5) may also be written as y(k + 1) = Ar y(k) + ru2 (k − h), y ∈ Rn , u2 ∈ Rn , for k ∈ Z+ . For this system, deﬁne the function Vy as " # p2 T Vy (y) = y Pr y ,

(7)

(8)

where Pr is as given in Lemma 3. We next evaluate the increments Vy (y(k + 1)) − Vy (y(k)), which we denote as ΔVy , along any given trajectory of (7). Thus, " #p " #p 2 2 ΔVy = Vy (y(k + 1)) − Vy (y(k)) = y T (k + 1)Pr y(k + 1) − y T (k)Pr y(k) =

y

T

(k)ATr

+

ruT2 (k

p p ! 2 2 T − h) Pr (Ar y(k) + ru2 (k − h)) − y (k)Pr y(k)

p 2 = y T (k)ATr Pr Ar y(k) + 2ry T (k)ATr Pr u2 (k − h) + r2 uT2 (k − h)Pr u2 (k − h) p 2 − y T (k)Pr y(k) ≤ y T (k)ATr Pr Ar y(k) + 2ry T (k)ATr Pr u2 (k − h) + r2 uT2 (k − h)Pr u2 (k − h) p − y T (k)Pr y(k) p2 ν 2 −1 , (9)

for some ν > 0 between ν1 := y T (k)ATr Pr Ar y(k) + 2ry T (k)ATr Pr u2 (k − h) + r2 uT2 (k − h)Pr u2 (k − h) and ν2 := y T (k)Pr y(k) . The last inequality follows p

from ﬁnite increments theorem applied to the function s 2 :

Stabilization and Finite-Gain Stabilizability

431

p p p p s12 − s22 = (s1 − s2 ) ν 2 −1 , 2

for ν between s1 and s2 . By Lemmas 3 and 1, there exist r2∗ ∈ (0, r1∗ ] and a constant π1 > 0 independent of r, such that for any r ∈ (0, r2∗ ], we have # p " ΔVy ≤ − 12 y(k) 2 + π1 u2 (k − h) 2 p2 ν 2 −1

≤

# ⎧" p p 1 2 2 p 2 −1 2 −1 ⎪ − y(k) + π u (k − h) max{ν , ν }, if 1 2 ⎪ 1 2 2 2 ⎨ " # ⎪ ⎪ ⎩ − 1 y(k) 2 + π u (k − h) 2 p min{ν p2 −1 , ν p2 −1 }, if 1 2 1 2 2 2

p 2

− 1 > 0,

p 2

− 1 ≤ 0. (10)

Case 1:

0. By using Lemmas 1-3, there exists r3∗ ∈ (0, r2∗ ] such that for any

r∈

is bounded as follows:

p 2 −1 > (0, r3∗ ], ΔVy

ΔVy ≤

p 2

"

−

1 2 2 y(k)

× ≤

p p 2 −1 2 (3χ2 )

"

"

+ π1 u2 (k − h)

2

χ2 2 r y

−

#

+ 2χ2 y u2 + rχ2 u2 2

1 2 2 y(k)

+ π1 u2 (k − h)

2

#"

# p2 −1

2 1 y 2 r

+ r u2

2

# p2 −1

" # p p ≤ p2 2 2 −2 (3χ2 ) 2 −1 − 12 y(k) 2 + π1 u2 (k − h) 2 " # p−2 p p × 21− 2 y p −1 + r 2 −1 u2 p−2 r2

≤ −Cp1 ≤ −Cp4

y p

p r 2 −1

y p

p r 2 −1

+ Cp2 y p −1 u2 (k − h) 2 + Cp3 r 2 −1 u2 (k − h) p p−2

p

r2

+ Cp5 u2 (k−h) , p −1 p

r2

(11)

where Cpi > 0, 1 ≤ i ≤ 4, are constants independent of r. In deriving (11), the ﬁrst inequality by Lemma 3, the second and last inequalities by Lemma 1 and The third inequality by Lemma 2. p 2 − (0, r4∗ ],

Case 2: r∈

1 ≤ 0. By Lemmas 9 and 4, there exists r4∗ ∈ (0, r3∗ ] such that for any

432

Karim Yakoubi and Yacine Chitour

ΔVy = y T (k)ATr Pr Ar y(k) + 2ry T (k)ATr Pr u2 (k − h)

+ "

≤ y

T

ATr Pr Ar y

≤ −r

2−p 2

≤ −r

2−p 2

≤ −r

2−p 2

p

# p2

ζ y + C

r2 uT2 (k

"

− y Pr y " p 2

T

p2 p2 T − h)Pr u2 (k − h) − y (k)Pr y(k)

# p2

p2 T T 2 T + C p2 2ry Ar Pr u2 + r u2 Pr u2

2χ2 y u2 + rχ2 u2

2

# p2

" # p p p p p ζ y p + C p2 (2χ2 ) 2 y 2 u2 2 + C p2 r 2 χ22 u2 p p

p

p

p

p

ζ y p + C p2 (2χ2 ) 2 y 2 u2 2 + (C p2 )2 r 2 χ22 u2 p

≤ −Cp6 y p + Cp7 u2 p , (12) where Cp6 > 0 and Cp7 > 0 are constants independent of r. Combining case 1 with case 2, we have, for any r ∈ (0, r4∗ ], ΔVy ≤ −ε1 (r) y(k) p + ε2 (r) u2 (k − h) p , where ε1 (r) 0 ε2 (r) 0

⎧ ⎪ ⎪ ⎨

1

p r 2 −1

, if

⎪ ⎪ ⎩ O(1), if

p 2

(13)

− 1 > 0, (14)

p 2

− 1 ≤ 0.

For every k ∈ Z+ , summing both sides of (14) from 0 to k, we have V (y(k)) + ε1 (r) y p(lp )[0,k] ≤ ε2 (r) u2 p(lp )[0,k] .

(15)

Using the fact that V ≥ 0, we conclude that y lp ≤ γp u2 lp ,

(16)

" ε (r) # P1 2 γp := 0 O(1). ε1 (r)

(17)

where, according to (14)

Proof for p = ∞ : From (13) we get for p = 2,

Stabilization and Finite-Gain Stabilizability

ΔVy ≤ −ε1 y(k) 2 + ε2 u2 2l∞ ,

433

(18)

where ε1 > 0 and ε2 > 0 are constants independent of r. Thus, ΔVy is negative outside the ball centered at the origin, of radius

&

ε2 ε1 u2 l∞ .

It follows that χ1 y(k) 2 ≤ y T (k)Pr y(k) := Vy (y(k)) ≤ sup & r ε2 |ξ|≤

u ε1

Vy (ξ) ≤

2 l∞

χ2 ε2 u2 2l∞ . r ε1 (19)

Thus implies that y l∞ ≤ γ∞ u2 l∞ , where γ∞

" χ ε # 12 2 2 := 0 O(1). χ1 ε1

(20) 2

Let x be the solution of (S)rh starting at ¯ 0 ∈ Ch and corresponding to u1 , u2 . Set z = x − y. Then z satisﬁes ⎧ ! T h+1 T h+1 ⎪ z(k + 1) = Az(k)−rB[σ B A z(k−h)+B A y(k−h)+u (k−h) − ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ − B T Ay(k)], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩z = ¯ 0, on [−h, 0]. 0

(21) ˜2 (k) = B T Ay(k). Choosing Let u ˜1 (k − h) = B T Ah+1 y(k − h) + u1 (k − h) and u the constant C0 > 0 such that γ∞ B C0 ≤ γ, with γ = lim|ξ|→∞ inf σ(ξ) > 0, we get ˜ u2 l∞ ≤ B y l∞ ≤ γ∞ B C0 ≤ γ.

(22)

Now (21) can be written as ⎧ 5 ! 6 T h+1 ⎪ z(k + 1) = Az(k) − rB σ B A z(k − h) + u ˜ (k − h) − u ˜ (k) , ⎪ 1 2 ⎨ (23)

⎪ ⎪ ⎩z = ¯ 0, on [−h, 0]. 0 From (23), we have z(k) = Ah z(k − h) − r −u ˜2 (i)] = Ah z(k − h) − r

! k−i−1 T h+1 A B[σ B A z(i − h) + u ˜ (i − h) − 1 i=k−h

k−1

h−1 i=0

Ak−i−1 B[σ(B T Ah+1 z(k − 2h + i)+

+u ˜1 (k − 2h + i)) − u ˜2 (k − h + i)].

434

Karim Yakoubi and Yacine Chitour

Then, ˜ B T Ah+1 z(k − h) + u ˜1 (k − h) = B T Az(k) + d(k), where h−1 ˜ =u d(k) ˜1 (k − h) + r i=0 B T Ah−i B[σ(B T Ah+1 z(k − 2h + i)+ +u ˜1 (k − 2h + i)) − u ˜2 (k − h + i)]. According to (23), we obtain " # ⎧ T ˜ ⎪ z(k + 1) = Az(k) − rB σ B Az(k) + d(k) − u ˜ (k) 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

:= Az(k) − rB [σ (˜ z (k)) − u ˜2 (k)] , for k ∈ Z+ ,

(24)

0, on [−h, 0], z0 = ¯

˜ We separate the proof for p ∈ (1, ∞), p = 1 and where z˜(k) = B T Az(k) + d(k). for p = ∞. We consider the Lyapunov function V2,p used in [1] (or in [4] and [10]) and given by (25)

V2,p (z) := p,r V0,p (z) + V1,p (z), where p,r > 0 will be chosen later and V0,p (z) := z p+1 ,

p

V1,p (z) = (z T Pr z) 2 ,

(26)

with Pr , the positive deﬁnite symmetric matrix, satisfying ATr Pr Ar − Pr = −In , where the Hurwitz matrix Ar deﬁned by A − rBB T A. If W : Rn → R, then ΔW denote the increment W (x(k + 1)) − W (x(k)) along any given trajectory of (24). In the sequel Ci , i = 1, 2, . . . denote constants that are independent of r. Proof for p ∈ (1, ∞) : Along the trajectories of (24) and as in [1], we separate each of the cases z > rη ( Bσ(˜ z ) + B u ˜2 ) and z ≤ rη ( Bσ(˜ z ) + B u ˜2 ) , where η is any number ≥ 3 independent of r. Case 1: z ≤ rη ( Bσ(˜ z ) + B u ˜2 ) . By Lemma 5, for any r ∈ (0, r1∗ ],

Stabilization and Finite-Gain Stabilizability

435

ΔV0,p := V0,p (z(k + 1)) − V0,p (z(k)) = z(k + 1) p+1 − z(k) p+1 = Az(k) − rB (σ(˜ z) − u ˜2 ) p+1 − z(k) p+1 p+1

≤ Az(k) − rB (σ(˜ z) − u ˜2 ) p+1 ≤ ( z + r ( Bσ(˜ z ) + B u ˜2 )) p+1

≤ [r(η + 1) ( Bσ(˜ z ) + B u ˜2 )] p+1

≤ 2p [r(η + 1) Bσ(˜ z ) ]

p+1

+ 2p [r(η + 1) B u ˜2 ]

˜ p + ˜ ≤ rC1 z p + rC2 ( d u2 p ). (27) In deriving (27), we have used the fact that σ, r and u ˜2 are bounded. case 2: z > rη ( Bσ(˜ z ) + B u ˜2 ) . By Lemmas 6, 5 and 1, there exists r2∗ ∈ (0, r1∗ ] such that for any r ∈ (0, r2∗ ], ΔV0,p = Az − rB(σ(˜ z) − u ˜2 ) p+1 − z p+1 ≤ z p+1 − (p + 1)r z p−1 z T AT B (σ(˜ z) − u ˜2 ) + ! z ) 2 + B u ˜2 2 − z p+1 + 2r2 M z p−1 Bσ(˜ (28) ≤ −(p + 1)r z

p−1 T

p−1 ˜T

z˜ σ(˜ z ) + (p + 1)r z

z )+ d σ(˜

z ) 2 + B u ˜2 + 2M r2 z p−1 Bσ(˜ ˜2 2 + (p + 1)r z p B u

!

" # ˜ p + ˜ ≤ −(p + 1)r z p−1 z˜T σ(˜ z ) + rC3 z p + rC4 d u2 p , where M is deﬁned in Lemma 6. In deriving (28), the ﬁrst inequality by Lemma 6, the second and third inequalities are the consequence of the fact that σ, r, u ˜2 are bounded and Lemmas 1 and 5. Combining case 1 with case 2, we have, for any r ∈ (0, r2∗ ],

436

Karim Yakoubi and Yacine Chitour

ΔV0,p ≤

" # ⎧ p−1 T p p p ˜ ⎪ −(p + 1)r z d , z ˜ σ(˜ z ) + rC z + rC + ˜ u ⎪ 5 6 2 ⎪ ⎪ ⎪ ⎨ if z > rη ( Bσ(˜ z ) + Bσ(˜ z ) ) , ⎪ ⎪ ⎪ " # ⎪ ⎪ p ⎩ rC5 z p +rC6 d ˜ p + ˜ z ) + Bσ(˜ z ) ) , u2 , if z ≤ rη ( Bσ(˜ (29)

where C5 = max{C1 , C3 } and C6 = max{C2 , C4 }. Rewrite the system equation (24) as ⎧ ⎪ z + σ(˜ z ) + d˜ − u ˜2 ], for k ∈ Z+ , ⎪ ⎨ z(k + 1) = Ar z(k) − rB[−˜ (30)

⎪ ⎪ ⎩z = ¯ 0, on [−h, 0], 0

and separate each of the cases z > rη ( Bσ(˜ z ) + B u ˜2 ) and z ≤ rη ( Bσ(˜ z ) + B u ˜2 ) . The increment ΔV1,p along the trajectories of (30) veriﬁes case 1: z ≤ rη ( Bσ(˜ z ) + B u ˜2 ) . By using Lemmas 3, 4, 5 and Jensen’s inequality, ΔV1,p along the trajectories of (30) is bounded as follows: !p !p ΔV1,p = z(k + 1)T Pr z(k + 1) 2 − z(k)T Pr z(k) 2 !p p ≤ Pr 2 Az(k) − rB (σ(˜ z (k)) − u ˜2 ) p − z T (k)Pr z(k) 2 + + z T ATr Pr Ar z

! p2 p

p

p

p

≤ −r

2−p 2

ζ z p + 2p−1 χ22 (η + 1)p r 2 ( Bσ(˜ z ) p + B u ˜2 p )

≤ −r

2−p 2

ζ z p + 2p−1 χ22 (η + 1)p r 2 (2p−1 B 2p z p +

˜ p + B u ˜2 p ) + 2p−1 B p d ≤ −r

2−p 2

p 2

p 2

ζ z + r C7 z + r C8 p

p

"

# p p ˜ d + ˜ u2 ,

r ∈ (0, r2∗ ],

where χ2 and ζ > 0 are as deﬁned in Lemmas 3 and 4, respectively. Case 2: z > rη ( Bσ(˜ z ) + B u ˜2 ) . We next evaluate the increments ΔV1,p along the trajectories of (30). We follow the computations leading to equation (44) of [1] and obtain using ν := 2rz T AT Pr BB T Az − 2rz T AT Pr B (σ(˜ z) − u ˜2 ) +

Stabilization and Finite-Gain Stabilizability

437

r2 (σ(˜ z) − u ˜2 ) B T Pr B (σ(˜ z) − u ˜2 ) , the existence of r3∗ ∈ (0, r2∗ ] such that for T

every r ∈ (0, r3∗ ], ΔV1,p ≤ −r

2−p 2

+ C11 r

ζ z p + C9 r

2−p 2

"

4−p 2

Pr B z p−1 z˜T σ(˜ z ) + C10 r

2−p 2

z p +

#

(31)

˜ p + ˜ u2 p . d

Summarizing, we may combine case 1 with case 2, to obtain for every r ∈ (0, r3∗ ], ⎧ 2−p 4−p 2−p ⎪ −r 2 ζ z p + C12 r 2 Pr B z p−1 z˜T σ(˜ z ) + C13 r 2 z p + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ " # ⎪ 2−p ⎪ ⎪ p p ˜ 2 ⎪ u2 , if z > rη ( Bσ(˜ +C14 r d + ˜ z ) + B u ˜2 ) , ⎪ ⎨ ΔV1,p ≤ " # ⎪ ⎪ 2−p 2−p 2−p ⎪ p p ˜ ⎪ 2 ξ z p + C 2 z p + C 2 r r + ˜ u −r d , 13 14 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ if z ≤ rη ( Bσ(˜ z ) + B u ˜2 ) , (32) where C12 = C9 , C13 = max{C10 , rp−1 C7 } and C14 = max{C11 , rp−1 C8 }. We choose p,r as p,r :=

C12 2−p r 2 Pr B . p+1

(33)

Using eqs. (29) and (32), we deduce that there exists a r4∗ ∈ (0, r3∗ ] such that for every r ∈ (0, r4∗ ] the estimation of ΔV2,p along the trajectories of (24) veriﬁes " # 2−p 2−p ˜ p + ˜ u2 p . ΔV2,p ≤ −r 2 C15 z p + r 2 C16 d (34) Proof for p = 1 : For z ∈ Rn , consider V1,1 (z) = z T Pr z and V0,1 (z) = z 2 . For clarity, let us repeat here the system equation (24) " # ˜ z(k + 1) = Ar z(k) − rB −˜ z (k) + σ(˜ z (k)) + d(k) − u ˜2 (k) (35) = Az(k) − rB (σ(˜ z (k)) − u˜2 (k)) , ˜ Along trajectories of (35), we obtain where z˜ = B T Az + d. ˜ u2 + ΔV0,1 ≤ −2r˜ z T σ(˜ z ) + 2rd˜T σ(˜ z ) + 2r ˜ z ˜ u2 + 2r d ˜ (36) z ) 2 + 2r2 B u ˜2 2 . + 2r2 Bσ(˜

438

Karim Yakoubi and Yacine Chitour

Moreover, for every z ∈ Rn , d˜ ∈ Rm and u ˜2 ∈ Rn we have V1,1 (z(k + 1)) = z(k + 1)T Pr z(k + 1) T

z (k)) − u˜2 (k)) B T ]Pr × = [z T (k)AT − r (σ(˜ (37) [Az(k) − rB (σ(˜ z (k)) − u˜2 (k))] ≤

C17 2 r z

# 2 2 ˜ + C18 r d + ˜ u2 , "

and also, T

V0,1 (z(k + 1)) = [z T (k)AT − r (σ(˜ z (k)) − u˜2 (k)) B T ]× [Az(k) − rB (σ(˜ z (k)) − u˜2 (k))]

(38)

" # ˜ 2 + ˜ u2 2 . ≤ C19 z 2 + d Equation (37) clearly implies " # √ C20 ˜ V1,1 (z(k + 1)) ≤ √ z + rC21 d + ˜ u2 . r 1 2

(39)

˜ + ˜ It is convenient to treat separately two cases namely whether z ≤ r( d u2 ) ˜ + ˜ or not. Assume ﬁrst that z ≤ r( d u2 ). We clearly have from (39) and (36), respectively that # 1 1 √ " 2 2 ˜ + ˜ ΔV1,1 ≤ V1,1 (z(k + 1)) ≤ C22 r d u2 ,

(40)

˜ + ˜ ΔV0,1 ≤ rC23 ( d u2 ).

(41)

and

For the second inequality, we have used the fact that σ and u ˜2 are bounded. Adding (40) and (41), we get

√ ˜ + ˜ ΔV2,1 ≤ C24 r( d u2 ).

(42)

We deduce that

# √ " √ ˜ + ˜ ˜ + ˜ u2 ) + r r( d u2 ) − z ΔV2,1 ≤ C24 r( d √ √ ˜ + ˜ u2 ). ≤ − r z + C25 r( d

(43)

Stabilization and Finite-Gain Stabilizability

439

˜ + ˜ We now treat the case where z > r( d u2 ). From (39), it follows that 1 C26 2 V1,1 (z(k + 1)) ≤ √ z , r

(44)

1 1 1 C27 C28 2 2 2 √ z ≤ V1,1 (z(k)) ≤ V1,1 (z(k)) + V1,1 (z(k + 1)) ≤ √ z . r r

(45)

which implies that

After elementary calculus, we get ˜ + ˜ ΔV0,1 ≤ −2r˜ z T σ(˜ z ) + C29 r z + C30 r( d u2 ).

(46)

1 2 ) as As in ([2]), we write Δ(V1,1

ΔV1,1 (z(k))

1 2 Δ(V1,1 )(z(k)) =

1 2

(47)

.

1

2 V1,1 (z(k)) + V1,1 (z(k + 1))

We follow the computations leading to equation (II. 23) of ([2]) and obtain, there is an r5∗ ∈ (0, r4∗ ] such that " #2 1 ˜ + ˜ z T σ(˜ z ) + C32 r d u2 , ΔV1,1 ≤ − z 2 + C31 z ˜ 2

∀r ∈ (0, r5∗ ], (48)

˜

u2 According to (47) and using (45) and the fact that r d + ˜ ≤ 1, we get,

˜ z

# √ √ T √ " ˜ u2 . Δ(V1,1 ) ≤ −C33 r z + C34 r˜ z σ(˜ z ) + C35 r d + ˜ 1 2

∀r ∈ (0, r5∗ ], (49)

Now, consider the following function 1

where

2 V2,1 = 1,r V0,1 + V1,1 ,

(50)

C34 1,r := √ . 2 r

(51)

It follows that there exists r∗ ∈ (0, r5∗ ] such that # √ √ " ˜ + ˜ ΔV2,1 ≤ −C36 r z + C37 r d u2 ,

∀r ∈ (0, r∗ ].

(52)

So, in either case, the Lyapunov function V2,p , 1 ≤ p < ∞, verify # " 1 p 2 p p ˜ u2 , ∀r ∈ (0, r∗ ], ΔV2,p ≤ −θp (r) z + θp (r) d + ˜

(53)

where

440

Karim Yakoubi and Yacine Chitour

θp1 (r) 0 θp2 (r) 0 r Now let Hμ (k) = μ

h−1

2−p 2

p ∈ [1, ∞).

,

k−1

(54)

z(θ) p ,

j=0 θ=k−2h+j

where p ∈ [1, ∞) and μ > 0 will be chosen later. Its increments along the trajectories of (24) satisﬁes, ΔHμ (k) = Hμ (k + 1) − Hμ (k) =μ

h−1 k

p θ=k+1−2h+j z(θ) −

j=0

= μh z(k) p − μ

h−1 j=0

k−1

p θ=k−2h+j z(θ)

(55)

z(k − 2h + j) p .

Finally, consider the following function V (z(k)) = V2,p (z(k)) + Hμ (k). Along the trajectories of (24), from (53) and (55), we have: # " 1 p 2 p p ˜ u2 (k) − ΔV (z(k)) ≤ −(θp (r) − μh) z(k) + θp (r) d(k) + ˜ −μ

h−1 j=0

(56) z(k − 2h + j) p .

˜ it follows that: From the deﬁnition of d, ˜ d(k) ≤ ˜ u1 (k − h) + r B 2 max{1, K}

h−1 j=0

+ ˜ u2 (k − h + j) ) + r2 max{1, K} B 3

( ˜ u1 (k − 2h + j) + h−1 j=0

z(k − 2h + j) ,

so, p ˜ ≤ C38 ˜ u1 (k − h) p + rp C39 2(h−1)(p−1) d(k)

h−1

+ ˜ u2 (k − h + j) p ) + r2p C40 2(h−1)(p−1) From (56) and (57), we have:

u1 (k j=0 ( ˜

h−1 j=0

− 2h + j) p +

z(k − 2h + j) p , (57)

Stabilization and Finite-Gain Stabilizability

441

ΔV (z(k)) ≤ −(θp1 (r) − μh) z(k) p + θp2 (r) max{1, C38 }( ˜ u1 (k − h) p + + ˜ u2 (k) p ) + rp θp2 (r)C39 2(h−1)(p−1)

h−1

u1 (k j=0 ( ˜

− 2h + j) p +

! + ˜ u2 (k − h + j) p ) − μ − r2p θp2 (r)C40 2(h−1)(p−1) × h−1 j=0

z(k − 2h + j) p .

Now, if we choose μ∗ (h) and r∗ (h) ∈ (0, r∗ ] such that, ⎧ 1 θp (r) ∗ (h−1)(p−1) 2 ⎪ ⎪ μ (h) = min , C 2 θ (r) , 40 p ⎪ 2h ⎪ ⎨ +" , ⎪ #1 ⎪ θp2 (r) 2p ⎪ 1 ∗ ∗ ⎪ ,r , ⎩ r (h) = min 2C40 h2(h−1)(p−1) θp1 (r)

(58)

we get for μ ≤ μ∗ (h) and r ≤ r∗ (h), ΔV (z(k)) ≤ − 12 θp1 (r) z(k) p + θp2 (r)[max{1, C38 } ( ˜ u1 (k − h) p + ˜ u2 (k) p ) + rp C39 2(h−1)(p−1)

h−1 j=0

( ˜ u1 (k − 2h + j) p + ˜ u2 (k − h + j) p )]

u1 (k − h) p + ˜ u2 (k) p )+ := −12 θp1 (r) z(k) p +θp2 (r)[max{1, C38 } ( ˜ p ˜ + ˜ n(k) p )], + rp C39 2(h−1)(p−1) ( m(k)

p where m(k) ˜ =

h−1 j=0

˜ u1 (k−2h+j) p and ˜ n(k) p =

h−1 j=0

(59) ˜ u2 (k−h+j) p .

A simple computation shows that 1

1

m ˜ lp ≤ h p ˜ u1 lp and ˜ n lp ≤ h p ˜ u2 lp .

(60)

For every k ∈ Z+ , summing both sides of (59) from 0 to k, we have V (z(k))+ 12 θp1 (r)

k j=0

z(j) p ≤ θp2 (r)[max{1, C38 }

k

u1 (j j=0 ( ˜

− h) p +

+ ˜ u2 (j) p ) + rp C39 2(h−1)(p−1) × k j=0

p ( m(j) ˜ + ˜ n(j) p )].

(61)

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Karim Yakoubi and Yacine Chitour

We conclude that, for every 1 ≤ p < ∞ and k ∈ Z+ , # " V (z(k))+ 12 θp1 (r) z p(lp )[0,k] ≤ θp2 (r)[max{1, C38 } ˜ u1 p(lp )[0,k] + ˜ u2 p(lp )[0,k] + p

(h−1)(p−1)

+r C39 2

"

m ˜ p(lp )[0,k]

+

˜ n2 p(lp )[0,k]

#

]. (62)

Using the fact that V ≥ 0 and the inequality (60), we conclude that ! u1 lp + ˜ u2 lp , z lp ≤ θp3 (r) ˜

(63)

where, according to (54)

# θp2 (r) " 3 p (h−1)(p−1) θp (r) := 2 1 max{1, C38 } + hr C39 2 θp (r)

p1 0 O(1).

(64)

˜2 = B T Ay, we have Since z = x − y, u ˜1 = B T Ah+1 y + u1 and u ⎧ ⎪ ˜ u2 lp ≤ B y lp ≤ B γp u2 lp , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ˜ u1 lp ≤ B γp u2 lp + u1 lp , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z ≥ x − y ≥ x − γ u . lp lp lp lp 2 lp p

(65)

Combining this with (63), we get ! x lp ≤ θp3 (r) u1 lp + γp 1 + 2 B θp3 (r) u2 lp !

(66)

≤ mp u1 lp + u2 lp , where

!

mp = max θp3 (r), γp 1 + 2 B θp3 (r) .

(67)

By using (64) and (6) it is clear that mp 0 O(1),

∀p ∈ [1, ∞).

(68)

So, by choosing rh ≤ 1, the ﬁnite-gain mp , 1 ≤ p < ∞ is delay-independent. More precisely, there exists a positive constant C41 independent of r and h such that ! x lp ≤ C41 u1 lp + u2 lp . Proof for p = ∞. Letting p = 2, from (34) and (59) we have

Stabilization and Finite-Gain Stabilizability

443

2

ΔV (z(k)) ≤ −C42 z(k) 2 + C43 ( ˜ u1 l∞ + ˜ u2 l∞ ) , √ 5 6 ˜ l∞ ≤ h ˜ where C43 := C16 max{1, C38 } + hr2 C39 2h−1 , since m u1 l∞ and √ ˜ n l∞ ≤ h ˜ u& 2 l∞ . Hence, ΔV (z) is negative outside the ball ¢ered at the C43 C43 origin of radius u1 l∞ + ˜ u2 l∞ ) := λR where λ = C42 ( ˜ C42 and R = ( ˜ u1 l∞ + ˜ u2 l∞ ) , from which it follows that, for any state z(k) in the trajectory: V (z(k)) ≤ sup V (ξ) ≤ 2,r λ3 R3 + λ2 R2 |ξ|≤λR

First assume that R ≤ 1. Then, we have χ1 z(k) 2 ≤ z T (k)Pr z(k) ≤ V (z(k)) ≤ r

3h + 1 2 2 χ2 + μh λ R . r 2

(69)

3h + 1 2 + μh λ R2 . 2,r λ + λ r 2 (70) 3

2 χ2

This implies that z l∞ ≤ Kr1 R, where $ Kr1 :=

2 r2,r λ3 + χ2 λ2 + rμh 3h+1 2 λ χ1

If R ≥ 1, we have

2,r z(k) ≤ V (z(k)) ≤ 3

% 12 .

3h + 1 2 + μh λ R3 , 2,r λ + λ r 2 3

2 χ2

(71)

(72)

and we get z l∞ ≤ Kr2 R, where $ Kr2 :=

2 r2,r λ3 + χ2 λ2 + rμh 3h+1 2 λ r2,r

% 13 .

(73)

Let Kr∞ = max{Kr1 , Kr2 }. Then, by using (33), it is clear that Kr∞ 0 O(1).

(74)

So, by choosing rh ≤ 1, the ﬁnite-gain Kr∞ is delay-independent. More precisely, there exists a positive constant C44 independent of r and h such that z l∞ ≤ C44 ( ˜ u1 l∞ + ˜ u2 l∞ ) . It is implies that z l∞ ≥ x l∞ − y l∞ ≥ x l∞ − γ∞ u2 L∞ . We conclude that

(75)

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Karim Yakoubi and Yacine Chitour

x l∞ ≤ C44 ( ˜ u1 l∞ + ˜ u2 l∞ ) + γ∞ u2 l∞ ≤ C44 ( u1 l∞ + 2γ∞ B u2 l∞ ) + γ∞ u2 l∞

(76)

≤ m∞ ( u1 l∞ + u2 l∞ ) , where m∞ = max {C44 , γ∞ (1 + 2C44 B )} 0 O(1).

2

5 Delay-Independent GAS using a Stabilizing Feedback of Nested Saturation Type 5.1 Statements of the Results We determine two explicit expressions of globally asymptotically stabilizing feedbacks for general linear discrete-time delay systems, both of nested saturation type, according to the results of the stabilization of delay free-system and continuous-time delay linear systems. The above problem was ﬁrst studied for delay-free continuoustime systems. It was shown in [7] and [9] that, under standard necessary condition (no eigenvalue of the uncontrolled system have positive real part and that the standard stabilizability rank-condition hold), there exists explicit expressions of globally asymptotically stabilizing feedbacks. Then, it is natural to investigate whether this technique can be extended to the case of linear discrete-time systems where there is a delay in the input. We start by giving some deﬁnitions, ﬁrst introduced in [7] and adapted here to the delay case. Deﬁnition 5. (cf. [11]) Given an n−dimensional system (Σ)h : x(k + 1) = f (x(k − h)), k ∈ Z, we say that (Σ)h is IICS (integrable-input converging-state) if, whenever {e(k)}∞ 0 ∈ l1 , every"solution k → x(k) of x(k +1) = f (x(k −h))+e(k) converges to zero as k → ∞. Such a concept is needed in order to state Theorem 3 and an

intermediary result (Lemma 9 given below), which is useful for the induction step in # the proof of Theorem 3 . For a system x(k + 1) = f (x(k), u(k − h)), x ∈ Rn , u ∈ Rm , we say that a feedback u(k − h) := u ¯(x(k − h)) is stabilizing if 0 is a globally asymptotically stable equilibrium of the closed-loop system x(k + 1) = f (x(k), u ¯(x(k − h))) . If, in addition, this closed-loop system is IICS, then we will say that u ¯ is IICS-stabilizing.

Stabilization and Finite-Gain Stabilizability

445

Deﬁnition 6.. (cf. [11]) For an n×n matrix A, let N (A) be the number of eigenvalues z of A such that |z| = 1 and m z ≥ 0, counting multiplicity. In the next theorem, we summarize our results. Theorem 3. Let (S)rh be a linear system x(k + 1) = Ax(k) + Bu(k − h) with state space Rn and input space Rm . Assume that (S)rh is stabilizable and A has no unstable eigenvalues. Let N = N (A). Then, for every ε > 0 and h > 0, there exist a sequence σ = (σ1 , . . . , σN ) of S-functions with σ ≤ ε, an m−tuple l = (l1 , . . . , lm ) of nonnegative integers such that |l| = l1 + . . . + lm = N, a number j j r∗ (h) ∈ (0, 1] and for each 1 ≤ j ≤ m, linear functions fh,i , gh,i : Rn −→ R, 1 ≤

i ≤ lj , such that for all r ∈ (0, r∗ (h)], there are IICS-stabilizing feedbacks j j j j (∗) uj (.) = −rσljj fh,l (x(.)) + α σ j lj −1 lj −1 fh,lj −1 (x(.))+ j j j + . . . + α1 σ1 (fh,1 (x(.))) ,

(77)

where αji ≥ 0, for all i ∈ [1, lj − 1], and " " # # j j j j (∗∗) uj (.) = −r βjlj σljj gh,l (x(.)) + β σ (x(.)) + g j lj −1 lj −1 h,lj −1 + ... −

βj1 σ1j

"

(78)

#

j (x(.)) gh,1

,

where βj1 , . . . , βjlj are nonnegative constants such that βj1 + . . . + βjlj ≤ 1. 5.2 Proof of Theorem 3 The proof of Theorem 3 is based on two lemmas, exactly as in the argument of [11], [7] and [9]. More precisely, Lemma 8 below is Lemma 3.1 of [11] and Lemma 9, which is actually the main technical point, in the nonzero delay version of Lemma 4.1 of [11]. In order to facilitate the analysis of the stabilizability properties by bounded feedback of (S)rh , a linear transformation is carried out in [11]. Lemma 8. (cf. [11]) Consider an n-dimensional linear single-input system (S1 )rh : x(k + 1) = Ax(k) + bu(k − h).

446

Karim Yakoubi and Yacine Chitour

Suppose that (A, b) is a controllable pair and all eigenvalues of A have magnitude 1. (i) If λ = 1 or λ = −1 is an eigenvalue of A, then there is a linear change of coordinates T x = (y1 , . . . , yn )T = (¯ y T , yn−1 , yn )T of Rn which transforms (S1 )rh into the form ⎧ ⎪ ⎪ ⎨ y¯(k + 1) = A1 y¯(k) + b1 (yn (k) + u(k − h)) , ⎪ ⎪ ⎩ y (k + 1) = λ (y (k) + u(k − h)) , n n

(79)

where the pair (A1 , b1 ) is controllable and yn is a scalar variable. (ii) If A has an eigenvalue of the form α+iβ, with β = 0, then there is a linear change y T , yn−1 , yn )T of Rn that puts (S1 )rh in the of coordinates T x = (y1 , . . . , yn )T = (¯ form

⎧ ⎪ y¯(k + 1) = A1 y¯(k) + b1 (yn (k) + u(k − h)) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

yn−1 (k + 1) = αyn−1 (k) − β (yn (k) + u(k − h)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y (k + 1) = βy (k) + α (y (k) + u(k − h)) , n

n−1

(80)

n

where the pair (A1 , b1 ) is controllable and yn−1 , yn are scalar variables. The following lemma is the key technical point of the proof. Though its conclusion are similar to Lemma 4 in [9], the proof that we provide is quite different and not totally obvious. Lemma 9. Let a, b be two real constant such that a2 + b2 = 1 and b = 0. Let ej = (ej (0), ej (1), ej (2), . . .), j = 1, 2, be two element of l1 . Then, there exist a constant C0 > 0, a real m∞ > 0 and, for every h > 0 an r∗ (h) ∈ (0, 1] and an 2 × 1 matrix Fh such that, for all r ∈ (0, r∗ (h)], if x = (x1 , x2 )T : R≥0 → R2 is any solution of the control system

Stabilization and Finite-Gain Stabilizability

447

⎧ ! T ⎪ x (k + 1) = ax (k) − bx (k) + rb[σ F x(k − h) + u(k − h) + 1 1 2 ⎪ h ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + v(k − h)] + re1 (k), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ! r (S2 )h : x2 (k + 1) = bx1 (k) + ax2 (k) − ra[σ FhT x(k − h) + u(k − h) + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + v(k − h)] + re2 (k), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T ¯ on [−h, 0], x0 = ((x1 )0 , (x2 )0 ) = 0, (81) with ¯0 the zero function in Ch , and u, v ∈ l∞ ([−h, ∞), R), with v l∞ ≤ C0 veriﬁes: (i) There exists a ﬁnite constant m∞ > 0 independent of r, such that lim sup x(k) ≤ m∞ ( u l∞ + v l∞ + e l∞ ).

(82)

k−→∞

where e = (e1 , e2 )T . (ii) In the absence of u, v and e, the equilibrium (x, y) = (0, 0) is globally asymptotically stable. Remark 4. We will in fact actually obtain the following stronger ISS-like property (see [6] and references there): " # lim sup x(k) ≤ θ∞ ( Ψ h ) + m∞ u L∞ + v L∞ + e L∞ , k−→∞

where Ψ is the initial condition for x and θ∞ is a class-K function (i.e. θ∞ : R≥0 → R≥0 is continuous, strictly increasing and satisﬁes θ∞ (0) = 0.) Proof of lemma 9. The proof is an immediate consequence of Theorem 2 with an explicit for all The feedback law FhT = B T J h+1 , where $ expression % $ parameters. % a −b −b J= and B = . The Hurwitz matrix Jr is given by b a a ⎛ ⎜ Jr = J − rBB T J = ⎜ ⎝

a −(1 − r)b b (1 − r)a

⎞ ⎟ ⎟, ⎠

448

Karim Yakoubi and Yacine Chitour

and the unique symmetric positive-deﬁnite solution of the Lyapunov equation JrT Pr Jr − Pr = −Id2 , is ⎛ ⎜ Pr = ⎜ ⎝

2b2 +a2 r 2 b2 r(2−r)

1−r − ab 2−r

1−r − ab 2−r

1+(1−r)2 r(2−r)

⎞ ⎟ ⎟. ⎠

Corollary 1. For n = 1, 2, let J be an n×n matrix, equal to either 1 or −1 if n = 1, or of the form

⎛ ⎜ ⎜ ⎝

α −β β α

⎞ ⎟ ⎟, ⎠

in the case n = 2, with α2 +β2 = 1 and β = 0. Let b = 1 if n = 1, and b = (0, 1)T if n = 2. Then, there exist a constant v0 > 0, independent of r and, for every ε > 0 and h > 0, there is an r∗ (h) ∈ (0, 1], and an n × 1 matrix Fh , such that for any functions v : Z≥−h → R, with v l∞ ≤ v0 , and e : Z≥0 → Rn , e ∈ l1 , if ψ : R≥0 → Rn , is any solution of the system " # x(k + 1) = J x(k) − rσ FhT x(k − h) − ξv(k − h) b + rηv(k − h)b + re(k), deﬁned for all k ≥ 0, where ξ + η = 1, ξη = 0, it follows that there exists T ≥ 0 such that, for r ∈ (0, r∗ (h)], lim sup |ψ(k)| ≤ ε,

∀t ≥ T.

k→∞

Proof of Corollary 1: When n = 2, the conclusion follows from lemma 9. Assume n = 1. Let V (x(k)) = x(k)2 . Let x(.) : [−h, +∞) −→ R be a solution of x(k + 1) = λ [x(k) − rσ (fh x(k − h) − ξv(k − h)) + rηv(k − h)] + re(k), with fh = λh , where λ = ±1. Then, if |e(k)| ≤ e0 , it follows that

Stabilization and Finite-Gain Stabilizability

449

ΔV (x(k)) = x(k + 1)2 − x(k)2 ! 2 ≤ −2rx(k)σ λh x(k − h) − ξv(k − h) + r2 (K + ηv0 + e0 ) + + 2r|x(k)| (e0 + ηv0 ) ≤ −2rx(k)σ(x(k)) + 2r|x(k)||[σ(x(k)) − σ(λh x(k − h)− 2

− ξv(k − h))]| + r2 (K + ηv0 + e0 ) + 2r|x(k)| (e0 + ηv0 ) .

Since |σ (.)| = 1, we have ΔV (x(k)) ≤ −2rx(k)σ(x(k)) + 2r|x(k)| x(k) − λh x(k − h) + ξv(k − h) + 2

+ r2 (K + ηv0 + e0 ) + 2r|x(k)| (e0 + ηv0 ) ≤ −2rx(k)σ(x(k)) + 2r|x(k)| (rh(K + ηv0 + e0 ) + e0 + v0 ) + 2

+ r2 (K + ηv0 + e0 ) , according to h

x(k) = λ x(k−h)−r

k−1

5 ! 6 λk−j σ λh x(j −h)−ξv(j −h) −ηv(j −h)−λe(j) .

j=k−h

Pick ε > 0, choose constants v0 and r∗ (h) such that hr∗ (h) (K + ηv0 + e0 ) + v0 + e0 ≤

ε , 8

2

(K + ηv0 + e0 ) ≤

ε . 4

Then, if |x(t)| ≥ 4ε , it follows that ΔV (x(k)) ≤ −rx(t)σ(x(t)) + r2 Therefore, |x(k)| ≤

ε 2

ε ε ε ε ≤ −r σ( ) + r2 < 0. 4 4 4 4

for sufﬁciently large k. We conclude that lim sup x(k) < ε.

2

k−→∞

Proof of Theorem 3. Without loss of generality, making a change of coordinates if necessary, we may assume that the system (S)rh has the following partitioned form:

450

Karim Yakoubi and Yacine Chitour

(S)rh :

⎧ ⎪ ⎪ ⎨ x1 (k + 1) = A1 x1 (k) + B1 u(k − h),

x1 (k) ∈ Rn1 ,

⎪ ⎪ ⎩ x (k + 1) = A x (k) + B u(k − h), 2 2 2 2

x2 ∈ Rn2 ,

where n1 + n2 = n, all the eigenvalues of A1 have magnitude 1 and all the eigenvalues of A2 have magnitude less than 1. Here, we take % $ % $ B1 A1 0 ,B = . A= 0 A2 B2 The controllability assumption on (A, B) implies that the pair (A1 , B1 ) is also controllable. Since A2 is asymptotically stable, it will be sufﬁcient to show that if we ﬁnd an IICS-stabilizing feedback of type (*) or type (**) for the system x1 (k + 1) = A1 x1 (k) + B1 u(k − h) then the same feedback will stabilize (S)rh as well because the second equation, x2 (k + 1) = A2 x2 (k) + B2 u(k − h), can be seen as an asymptotically stable linear system forced by a function that converges to zero. Thus, in order to stabilize (S)rh , it is enough to stabilize the subsystem x1 (k + 1) = A1 x1 (k) + B1 u(k − h). Without loss of generality, in our proof of Theorem we will suppose that (S)rh is already in this form, that is, we assume that all the eigenvalues of A have magnitude 1 and that the pair (A, B) is controllable. i) The single-input case: We start with the single-input case and prove the theorem by induction on the dimension of the system. For dimension zero there is nothing to prove. Consider now a single-input n−dimensional system, n ≥ 1, and suppose that Theorem 3 has been established for all single-input systems of dimensions less than or equal to n − 1. Write N = N (A), and pick any ε > 0. If 1 or −1 is an eigenvalue of A, we apply the ﬁrst part of Lemma 1 and rewrite our system in the form ⎧ ⎪ ⎪ ⎨ y¯(k + 1) = A1 y¯(k) + (yn (k) + u(k − h)) b1 , ⎪ ⎪ ⎩ y (k + 1) = λ (y (k) + u(k − h)) , n n

(83)

where y¯ = (y1 , . . . , yn−1 )T . In the case when neither 1 nor −1 is an eigenvalue of A which has a pair of eigenvalues of the form α + iβ, with β = 0. So, we apply the second part of Lemma 1. Then we take

Stabilization and Finite-Gain Stabilizability

⎧ ⎪ y¯(k + 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

451

= A1 y¯(k) + (yn (k) + u(k − h)) b1 ,

yn−1 (k + 1) = αyn−1 (k) − β (yn (k) + u(k − h)) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y (k + 1) = βy n n−1 (k) + α (yn (k) + u(k − h)) ,

(84)

where y¯ = (y1 , . . . , yn−2 )T . So, in either case, we can rewrite our system in the ⎧ ⎪ ⎪ ⎨ y¯(k + 1) = A1 y¯(k) + (yn (k) + u(k − h)) b1 ,

form

(85)

⎪ ⎪ ⎩ y˜(k + 1) = J (˜ y (k) + u(k − h)b0 ) ,

where J is as in Corollary 1, (J, b0 ) is a controllable pair, with y˜ = yn , b0 = 1 in the ﬁrst case and y˜ = (yn−1 , yn )T , b0 = (0, 1)T in the second case. To consider the problem of integrable-input converging-state (IICS), we must study solutions of the following system: ⎧ ⎪ e(k), ⎪ ⎨ y¯(k + 1) = A1 y¯(k) + (yn (k) + u(k − h)) b1 + r¯ (86)

⎪ ⎪ ⎩ y˜(k + 1) = J (˜ e(k), y (k) + u(k − h)b0 ) + r˜

where e˜, e¯ are arbitrary elements of l1 , bounded by e˜0 and e¯0 , and have the same dimensions as y¯, y˜, respectively. We will design a feedback of the form # " T ˜ u(k − h) = r σN Fh y˜(k − h) − ξv(k − h) − ηv(k − h) ,

F˜hT = bT0 J h+1 ,

where ξ and η are constants such that ξη = 0, ξ + η = 1, and v is to be chosen later. From Corollary 1, there exists r1∗ (h) ∈ (0, 1] and for σ = σN , we may pick 0 < v0 <

ε 2

such that, if v l∞ ≤ v0 , then all trajectories of (86) satisfy ˜ y l∞ <

ε 2

for

r ∈ (0, r1∗ (h)]. Consider one such trajectory. According to (ii) of Def. 1, one has # " σN F˜hT y˜(k − h) − ξv(k − h) = F˜hT y˜(k − h) − ξv(k − h)+ (87) + O( F˜hT y˜(k − h) − ξv(k − h) 2 ). So, the ﬁrst block equation in (86) becomes

452

Karim Yakoubi and Yacine Chitour

y¯(k + 1) = A1 y¯(k) − v(k − h)b1 + # T T 2 ˜ ˜ e(k) + yn (k) + rFh y˜(k−h)+O( Fh y˜(k − h)−ξv(k − h) ) b1 + r¯ "

= A1 y¯(k) − rv(k − h)b1 + e¯(k), for all large k, where e¯(k) = [yn (k) + rF˜hT y˜(k − h) + O( F˜hT y˜(k − h) − ξv(k − h) 2 )]b1 + r¯ e(k). Then for every ε > 0 there exists r2∗ (h) ∈ (0, r1∗ (h)], such that lim sup e¯(k) < ε, ∀r ∈ (0, r2∗ (h)]. k→∞

Note that (A1 , b1 ) is controllable and all eigenvalues of A1 have magnitude 1. Applying the inductive hypothesis to the single-input system of dimension less than or equal to n − 1, y¯(k + 1) = A1 y¯(k) − rv(k − h)b1 + e¯(k), we know that there is a feedback v(k − h) = u ¯(¯ y (k − h)) having (∗) or (∗∗) form (cases ξ = 1, η = 0, and ξ = 0, η = 1, respectively), such that all the trajectories of y¯(k + 1) = A1 y¯(k) − rv(k − h)b1 go to zero as k → ∞, for every vector function e¯ ∈ l1 . The proof for the single-input case is completed. 2 ii) The general case: Next, we deal with the general case of m ≥ 1 inputs and prove Theorem 3 by induction on m. First, we know from the proof above that the theorem is true if m = 1. Assume that Theorem 3 has been established for all κ−input systems, for all κ ≤ m − 1, m ≥ 2, and consider an m−input system (S)rh : x(k + 1) = Ax(k) + Bu(k − h) + e(k) where e(k) ∈ l1 is any decaying vector function. Assume without loss of generality that the ﬁrst column b1 of B is nonzero and consider the Kalman controllability decomposition of the system (S 1 )rh : x(k + 1) = Ax(k) + b1 u(k − h) (see [5], Lemma 3.3.3). We conclude that, after a change of −1 coordinates y = Syx x, (S 1 )rh has the form ⎧ ⎪ ⎪ y1 (k + 1) = A1 y1 (k) + A2 y2 (k) + ¯b1 u(k − h), ⎨

⎪ ⎪ ⎩ y (k + 1) = A y (k), 2 3 2 Where (A1 , ¯b1 ) is a controllable pair. In these coordinates, (S)rh has the form ⎧ ¯ ⎪ ¯ ¯(k − h) + e1 (k), ⎪ ⎨ y1 (k + 1) = A1 y1 (k) + A2 y2 (k) + b1 u1 (k − h) + B1 u ⎪ ⎪ ⎩ y (k + 1) = A y (k) + B ¯2 u ¯(k − h) + e2 (k), 2 3 2

(88)

Stabilization and Finite-Gain Stabilizability

453

¯1 , B ¯2 are matrices of appropriate dimensions. So it where u ¯ = (u2 , . . . , um )T and B sufﬁces to show the conclusion for (88). Let n1 , n2 denote the dimensions of y1 , y2 respectively. Recall that N = N (A). Let σ = (σ1 , · · · , σN ) be any ﬁnite sequence of saturation functions. Then, for the single-input controllable system y1 (k + 1) = A1 y1 (k) + ¯b1 u1 (k − h), there is a feedback u1 (k − h) = v1 (y1 (k − h)), such that (i)

(89)

v1 is N1 = N (A1 )(∗)−form (or N1 = N (A1 )(∗∗)−form); (ii) the

resulting closed-loop system is IICS. Since (88) is controllable, we conclude that the (m−1)−input subsystem y2 (k+1) = ¯2 u A3 y2 (k) + B ¯(k − h) is controllable as well. By the inductive hypothesis, this subsystem can be stabilized by a feedback T

u ¯(k − h) = v¯2 (y2 (k − h)) = [v2 (y2 (k − h)), . . . , vm (y2 (k − h))] ,

(90)

such that (i) there exists an (m − 1)−tuple k¯ = (N2 , . . . , Nm ) of nonnegative in¯ = N − N1 , such that v¯2 is |k|(∗)−form ¯ ¯ tegers and |k| (or |k|(∗∗)−form); (ii) the resulting closed-loop system is IICS. ¯1 [v2 (y2 (k − h)), . . . , vm (y2 (k − h))]T + e1 (k) still converges Since A2 y2 (k) + B to zero, we conclude that the coordinate y1 in equation (88), with u1 , u ¯ given by (89) and (90), converges to zero. Therefore, the feedback given in (89) and (90) globally stabilizes the system x(k + 1) = Ax(k) + Bu(k − h) and the resulting closed-loop system is IICS.

2

6 Output Feedback Design This section deals with the problem of dynamic output feedback stabilization via linear feedbacks of detectable and stabilizable systems subject to time-delayed and saturated input. Our main result is given in the following theorem. Theorem 4. Consider a linear discrete-time delay input system of the form x(k + 1) = Ax(k) + Bu(k − h) with an Rp −valued output y(k) = Ex(k), p ∈ N, such that all the eigenvalues of A are located inside or on the unit circle, (A, B) is

454

Karim Yakoubi and Yacine Chitour

stabilizable and (A, E) is detectable. Then, for any observer system x ˆ(k) = Aˆ x(k)+ Bu(k − h) + L(y(k) − E x ˆ(k)), (L is any matrix such that all the eigenvalues of A − LE are strictly inside the unit circle) and for any bounded IICS-stabilizing feedback u(k) := u ¯(ˆ x(k)) is as given in Theorem 3, the composite system x(k+1) = Ax(k) + B u ¯(ˆ x(k − h)), x ˆ(k) = Aˆ x(k) + B u ¯(ˆ x(k − h)) + L(Ex(k) − E x ˆ(k)) has the origin as a globally asymptotically stable equilibrium. Proof: Set e(k) = x ˆ(k) − x(k). Then, the error e satisﬁes e(k + 1) = (A − LE)e(k) and e(k) → 0 as k → ∞, since the matrix A − LE is Hurwitz. The equation for x becomes

" # x(k + 1) = Ax(k) + B u ¯ x(k − h) + e(k − h) .

(91)

We know that the trajectories of x(k + 1) = Ax(k) + B u ¯(x(k − h)) go to zero, the problem is whether the same is true for (91). To do this for the feedback deﬁned in Theorem 3, we recall that this feedback is globally Lipschitz. Therefore, we can rewrite (91) as x(k + 1) = Ax(k) + B u ¯ (x(k − h)) + e˜(k − h),

(92)

where e˜(k − h) = B [¯ u (x(k − h) + e(k − h)) − u ¯(x(k − h))] is bounded by a constant multiple of e(k − h). Thus e˜(k − h) → 0 as t → ∞. Then Theorem 3 implies that all solutions of (91) converge to zero which completes our proof.

2

7 Conclusions In this chapter we have dealt with the ﬁnite-gain lp −stabilizability, p ∈ [1, ∞] and global asymptotic stabilization of linear discrete systems subject to input saturation and time-delayed. We have shown that these systems under the stabilizability, neutrally stability assumption and the transaction matrix has spectral radius inside or on the unit circle are globally asymptotically stable and ﬁnite-gain lp −stabilizable. Moreover, the corresponding lp −gain is delay-independent. The explicit construction of such control laws was also given.

Stabilization and Finite-Gain Stabilizability

455

References 1. X. Bao, Z. Lin and E. Sontag (2000) “Finite gain stabilization of discrete-time linear systems subject to actuator saturation”, Automatica Vol 36 pp. 269–277. 2. Y. Chitour and Z. Lin (December 2003) “Finite gain lp stabilization of discrete-time linear systems subject to actuator saturation: the case p = 1”, Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA. 3. J. Choi (1999) “On the stabilization of linear discrete-time systems subject to input saturation“, Systems and Control Letters, Vol 36, pp. 241-244. 4. W. Liu, Y. Chitour and E.D. Sontag (1996) “On ﬁnite gain stabilizability of linear systems subject to input saturation”, Siam J. Control Opt., Vol 34, NO 4, pp. 1190-1219. 5. E.D. Sontag “Mathematical control theory: Deterministic Finite Dimensional systems” Springer-Verlag, New York. 6. E. D. Sontag (1998) “Comments on integral variants of ISS”, Systems and Control Letters 43, 93-100. 7. H.J. Sussmann, E.D. Sontag and Y. Yang (1994) “A general result on the stabilization of linear systems using bounded controls”, IEEE Trans. Automat. Contr. Vol. 39 pp. 24112425. 8. A.J. Van Der Schaft (1996) “L2 -gain and passivity techniques in nonlinear control”, Lecture Notes in Control and Inform. Springer, London. 9. K. Yakoubi and Y. Chitour, “Linear systems subject to input saturation and time delay: Global asymptotic stabilization”, submitted. 10. K. Yakoubi and Y. Chitour, “Linear systems subject to input saturation and time delay: Finite-gain Lp −stabilization”, to appear in Siam J. Control and Optimization.. 11. Y. Yang, E. D. Sontag and H. J. Sussmann (1997) “Global stabilization of Linear Discrete-Time Systems with Bounded Feedback”, Systems and Control Letters, Vol 30, pp. 273-281.

List of Contributors

Teodoro Alamo (Universidad de Sevilla, Spain) - Chapter 13 Frank Allgower (University of Stuttgart, Germany) - Chapter 7 Franco Blanchini (University of Udine, Italy) - Chapter 12 Raphael Cagienard (ETH, Zurich, Switzerland) - Chapter 8 Eduardo Camacho (Universidad de Sevilla, Spain) - Chapter 13 Yacine Chitour (Supelec, Gif-sur-Yvette, France) - Chapter 15 Dan Dai (University of California, USA) - Chapter 11 Ciprian Deliu (Eindhoven Univ. of Technology, The Netherlands) - Chapter 14 Haijun Fang (University of Virginia, USA) - Chapter 3 Rolf Findeisen (University of Stuttgart, Germany) - Chapter 7 Germain Garcia (LAAS-CNRS, Toulouse, France) - Chapter 6 Adolf H. Glattfelder (ETH, Zurich, Switzerland) - Chapter 4 Jo˜ao Manoel Gomes da Silva Jr. (UFRGS, Porto Alegre, Brazil) - Chapter 13 Paul J. Goulart (Cambridge University, UK) - Chapter 9 Peter Hippe (Universitaet Erlangen-Nuernberg, Germany) - Chapter 2 Guido Herrmann (University of Leicester, UK) - Chapter 5 Tingshu Hu (University of Massachusetts, Lowell, USA) - Chapter 11 David Humphreys (General Atomics, USA) - Chapter 1 Eric C. Kerrigan (Imperial College London, UK) - Chapter 9 Miroslav Krstic (University of California, San Diego, USA) - Chapter 1 Constantino M. Lagoa (The Pennsylvania State University, USA) - Chapter 10 Daniel Limon (Universidad de Sevilla, Spain) - Chapter 13 X. Li (The Pennsylvania State University, USA) - Chapter 10

458

List of Contributors

Zongli Lin (University of Virginia, USA) - Chapter 3 Jan M. Maciejowski (Cambridge University, UK) - Chapter 9 Urban M¨ader (ETH, Zurich Switzerland) - Chapter 8 Stefano Miani (University of Udine, Italy) - Chapter 12 Manfred Morari (ETH, Zurich Switzerland) - Chapter 8 Ian Postlethwaite (University of Leicester, UK) - Chapter 5 Isabelle Queinnec (LAAS-CNRS, Toulouse, France) - Chapter 6 Tobias Raff (University of Stuttgart, Germany) - Chapter 7 Walter Schaufelberger (ETH, Zurich, Switzerland) - Chapter 4 Ali Saberi (Washington State University, USA) - Chapter 14 Peddapullaiah Sannuti (Rutgers University, USA) - Chapter 14 Carlo Savorgnan (University of Udine, Italy) - Chapter 12 Eugenio Schuster (Lehigh University, USA) - Chapter 1 Anton A. Stoorvogel (Eindhoven Univ. of Technology, and Delft Univ. of Technology, The Netherlands) - Chapters 10, 14 Mario Sznaier (The Pennsylvania State University, USA) - Chapter 10 Sophie Tarbouriech (LAAS-CNRS, Toulouse, France) - Chapter 6 Andrew R. Teel (University of California, USA) - Chapter 11 Matthew C. Turner (University of Leicester, UK) - Chapter 5 Michael Walker (General Atomics, USA) - Chapter 1 Karim Yacoubi (Supelec, Gif-sur-Yvette, France) - Chapter 15 Luca Zaccarian (Universita’ di Roma, Italy) - Chapter 11

Index

amplitude and rate restriction ersatz model

37

37

amplitude and rate saturation 52

antiwindup

52, 335

38

simulation model example

inverted pendulum

Kalman-Yakubovich lemma linear programming L1 -norm

92, 101, 119

augmented system

277, 278

42, 47, 51

Lyapunov function, PWQ

41, 48, 92, 95, 96, 117

coincidence point

275

control invariant set controller windup cost function

Min-Max-selector

331 103

model predictive control

240 31

MPC

273

multi-parametric programming optimal reachability set

316

optimal regional L2 gain

detectability condition

320

directionality problem

48, 52

disturbance model disturbance rejection

239

326

output feedback construction override

321

92, 101, 103–106, 114–116, 119

327, 334

plant windup

31, 37, 41

pole assignment

94, 109, 111

polyhedral regions 332

280

Popov criterion

324

111

prediction horizon

feasibility conditions

323

44

279

F8 aircraft dynamics feasibility

273

276

deadzone loop

dual-mode

242

39 mass-spring system

circle criterion

50

274, 279

319 quadratic programming

integral action

95, 104, 109, 111

internal model

274

receding horizon

276

276, 278

460

Index

reference shaping ﬁlter reference tracking

330, 334

reference trajectory regional L2 gain

saturation

state observer

317

regional reachability

317

317

343, 345, 346

terminal constraint terminal set

344, 346, 347

279

279, 281

357 windup prevention

360

spring-cart-pendulum system 249

344, 346, 347

switched gain scheduled

114, 116 initial input value

43

gain scheduled robust

280

slew–rate constraints

320

synthesis

91, 95, 96, 99, 100, 103, 107,

stabilisable set

278

stabilizability condition

274

regional exponential stability Riccati equation

stability

45

327

stable and unstable systems stable systems

41

40

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