actuator saturation control

ACTUATOR SATURATION CONTROL

edited by Vikram KapilaPolytechnic University Brooklyn, New York

Karolos M. GrigoriadisUniversity of Houston Houston, Texas

Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.

ISBN: 0-8247-0751-6 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above. Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA


CONTROL ENGINEERINGA Series of Reference Books and Textbooks Editor NEIL MUNRO, PH.D., D.Sc. Professor Applied Control Engineering University of Manchester Institute of Science and Technology Manchester, United Kingdom

1. Nonlinear Control of Electric Machinery, Darren M. Dawson, Jun Hu, and Timothy C. Burg 2. Computational Intelligence in Control Engineering, Robert E. King 3. Quantitative Feedback Theory: Fundamentals and Applications, Constantine H. Houpis and Steven J. Rasmussen 4. Self-Learning Control of Finite Markov Chains, A. S. Poznyak, K. Najim, and E. Gomez-Ramirez 5. Robust Control and Filtering for Time-Delay Systems, Magdi S. Mahmoud 6. Classical Feedback Control: With MATLAB, Bon's J. Lurie and Paul J. Enright 7. Optimal Control of Singularly Perturbed Linear Systems and Applications: High-Accuracy Techniques, Zoran Gajic and Myo-Taeg Lim 8. Engineering System Dynamics: A Unified Graph-Centered Approach, Forbes T. Brown 9. Advanced Process Identification and Control, Enso Ikonen and Kaddour Najim 10. Modern Control Engineering, P. N. Paraskevopoulos 11. Sliding Mode Control in Engineering, edited by Wilfrid Perruquetti and Jean Pierre Barbot 12. Actuator Saturation Control, edited by Vikram Kapila and Karolos M. Grigoriadis Additional Volumes in Preparation


Series IntroductionMany textbooks have been written on control engineering, describing new techniques for controlling systems, or new and better ways of mathematically formulating existing methods to solve the everincreasing complex problems faced by practicing engineers. However, few of these books fully address the applications aspects of control engineering. It is the intention of this new series to redress this situation. The series will stress applications issues, and not just the mathematics of control engineering. It will provide texts that present not only both new and well-established techniques, but also detailed examples of the application of these methods to the solution of realworld problems. The authors will be drawn from both the academic world and the relevant applications sectors. There are already many exciting examples of the application of control techniques in the established fields of electrical, mechanical (including aerospace), and chemical engineering. We have only to look around in today's highly automated society to see the use of advanced robotics techniques in the manufacturing industries; the use of automated control and navigation systems in air and surface transport systems; the increasing use of intelligent control systems in the many artifacts available to the domestic consumer market; and the reliable supply of water, gas, and electrical power to the domestic consumer and to industry. However, there are currently many challenging problems that could benefit from wider exposure to the applicability of control methodologies, and the systematic systems-oriented basis inherent in the application of control techniques. This series presents books that draw on expertise from both the academic world and the applications domains, and will be useful not only as academically recommended course texts but also as handbooks for practitioners in many applications domains. Actuator Saturation Control is another outstanding entry to Dekker's Control Engineering series. NeilMunro


Preface

All real-world applications of feedback control involve control actuators with amplitude and rate limitations. In particular, any physical electromechanical device can provide only a limited force, torque, stroke, flow capacity, or linear/angular rate. The control design techniques that ignore these actuator limits may cause undesirable transient response, degrade the closed-loop performance, and may even cause closed-loop instability. For example, in advanced tactical fighter aircraft with high maneuverability requirements, actuator amplitude and rate saturation in the control surfaces may cause pilot-induced oscillations leading to degraded flight performance or even catastrophic failure. Thus, actuator saturation constitutes a fundamental limitation of many linear (and even nonlinear) control design techniques and has attracted the attention of numerous researchers, especially in the last decade. In prior research, the control saturation problem has been examined via the extensions of optimal control theory, ariti-windup compensation, supervisory error governor approach, Riccati and Lyapunov-based local and semi-global stabilization, and bounded-real, positive-real, and absolute stabilization frameworks. This prior research literature and the currently developing research directions provide a rich variety of techniques to account for actuator saturation. Furthermore, tremendous strides are currently being made to advance the saturation control design techniques to address important issues of performance degradation, disturbance attenuation, robustness to uncertainty/time delays, domain of attraction estimation, and control rate saturation. The scope of this edited volume includes advanced analysis and synthesis methodologies for systems with actuator saturation, an area of intense current research activity. This volume covers some of the significant research advancements made in this field over the past decade. It emphasizes the issue of rigorous, non-conservative, mathematical formulations of actuator saturation control along with the development of efficient computational algorithms for this class of problems. The volume is intended for researchers and graduate students in engineering and applied mathematics with interest in control systems analysis and design. This edited volume provides a unified forum to address various novel aspects of actuator saturation control. The contributors of this edited volume include some nationally and internationally recognized researchers who have


made or continue to make significant contributions to this important field of research in our discipline. Below we highlight the key issues addressed by each contributor. Chapter 1 by Barbu et al. considers the design of anti-windup control for linear systems with exponentially unstable modes in the presence of input magnitude and rate saturation. The chapter builds on prior work by these authors on uniting local and global controllers. Specifically, the anti-windup design of this chapter enables exponentially unstable saturated linear systems to perform satisfactorily in a large operating region. In addition, the chapter provides sufficient conditions for this class of systems to achieve local performance and global stability. Finally, via a manual flight control example involving an unstable aircraft with saturating actuators, it illustrates the efficacy of the proposed control design methodology in facilitating aggressive maneuvers while preserving stability. Chapter 2 by Eim et al. focuses on selecting the actuator saturation level for small performance degradation in linear designs. A novel application of a general stochastic linearization methodology, which approximates the saturation nonlinearity with a quasi-linear gain, is brought to bear on this problem. Specifically, to determine the allowable actuator saturation level, standard deviations of performance and control in the presence of saturation are obtained using stochastic linearization. The resulting expression for the allowable actuator saturation level is shown to be a function of performance degradation, a positive real number based on the Nyquist plot of the linear part of the system, arid the standard deviation of controller output. Numerical examples show that by choosing performance degradation of 10 percent, the actuator saturation level is a weak function of a system intrinsic parameter, viz.. the positive real number based on the Nyquist plot of the linear part of the system. Chapter 3 by Hu et al. is motivated by the issue of asymmetric actuators, a problem of considerable practical concern. In previous research, the authors studied the problem of null controllable regions and stabilizability of exponentially unstable linear systems in the presence of actuator saturation. However, this earlier attempt was restricted to symmetric actuator saturation and hence excluded a large class of real-world problems with asymmetric actuator saturation. This chapter addresses the characterization of null controllable regions and stabilization on the null controllable region, for linear, exponentially unstable systems with asymmetrically saturating actuators. First, it is shown that the trajectories produced by extremal control inputs of linear low-order systems have explicit reachable boundaries. Next, under certain conditions, a closed-trajectory is demonstrated to be the boundarv of the domain of attraction under saturated


linear state feedback. Finally, it is proven that the domain of attraction of second order anti-stable systems under the influence of linear quadratic control can be enlarged arbitrarily close to the null controllability region by using high gain feedback. Chapter 4 by Iwasaki and Fu is concerned with regional H^ performance synthesis of dynamic output feedback controllers for linear time-invariant systems subject to known bounds on control input magnitude. In order to guarantee closed-loop stability and H2 performance, this chapter utilizes the circle and linear analysis techniques. Whereas the circle analysis is applicable to a state space region in which the actuator may saturate, the linear analysis is restricted to a state space region in which the saturation is not activated. It is shown that the circle criterion based control design does not enhance the domain of performance for a specified performance level vis-a-vis the linear design. Finally, since the performance overbound is inherently conservative, it is illustrated that the circle criterion based control design can indeed lead to improved performance vis-a-vis the linear design. Both fixed-gain and switching control design are addressed. Chapter 5 by Jabbari employs a linear parameter varying (LPV) approach to handle the inevitable limitations in actuator capacity in a disturbance attenuation setting. The chapter begins by converting a saturating control problem to an unconstrained LPV problem. Next, a fixed Lyapuriov function based approach is considered to address an output feedback control design problem for polytopic LPV system. To overcome the conservatism of LPV control designs based on fixed Lyapunov function, a parameterdependent LPV control methodology is presented. It is shown that the LPV control design framework is capable of handling input magnitude and rate saturation. A scheduling control design approach to deal with actuator saturation is also considered. Two numerical examples illustrate the effectiveness of the proposed control methodologies. Chapter 6 by Pan arid Kapila is focused on the control of discrete-time systems with actuator saturation. It is noted that a majority of the previous research effort in the literature has focused on the control of continuous-time systems with control signal saturation. Nevertheless, in actual practical applications of feedback control, it is the overwhelming trend to implement controllers digitally. Thus, this chapter develops linear matrix inequality (LMI) formulations for the state feedback and dynamic, output feedback control designs for discrete-time systems with simultaneous actuator amplitude and rate saturation. Furthermore, it provides a direct methodology to determine the stability multipliers that are essential for reducing the conservatism of the weighted circle criterion-based saturation control design. The chapter closes with two illustrative numerical examples which


demonstrate the efficacy of the proposed control design framework. Chapter 7 by Pare et al. addresses the design of feedback controllers for local stabilization and local performance synthesis of saturated feedback systems. In particular, the chapter formulates optimal control designs for saturated feedback systems by considering three different performance objectives: region of attraction, disturbance rejection, and 2-gain. The Popov stability theory and a sector model of the saturation nonlinearity are brought to bear on these optimal control design problems. The bilinear matrix inequality (BMI) and LMI optimization frameworks are exploited to characterize the resulting optimal control laws. Commercially available LMI software facilitates efficient numerical computation of the controller matrices. A linearized inverted pendulum example illustrates the proposed local 2-gain design. Chapter 8 by Saberi et al. focuses on output regulation of linear systems in the presence of state and input constraints. A recently developed novel nonlinear operator captures the simultaneous amplitude and rate constraints on system states and input. The notion of a constraint output is developed to handle both the state and input constraints. A taxonomy of constraints is developed to characterize conditions under which various constraint output regulation problems are solvable. Low-gain and low-high gain control designs including a scheduled low-gain control design are developed for linear systems with amplitude and rate saturating actuators. Finally, output regulation problems in the presence of right invertible and non-right-invertible constraints are also considered. Chapter 9 by Soroush and Daoutidis begins by surveying the notions of directionality and windup and recent directionality and wiridup compensation schemes that account for and negate the degrading influence of constrained actuators. The principal focus of the chapter is on stability and performance issues for input-constrained multi-input multi-output (MIMO) nonlinear systems subject to directionality and integrator wiridup. In particular, the chapter poses the optimal directionality compensation problem as a finite-time horizon, state dependent, constrained quadratic optimization problem with an objective to minimize the distance between the output of the unsaturated plant with an ideal controller and the output of the saturated plant with directionality compensator. Simulation results for a MIMO linear time invariant system and a nonlinear bioreactor subject to input constraints illustrate that the optimal directionality compensation improves system performance vis-a-vis traditional clipping and direction preservation algorithms. Finally, the chapter proposes an input-output linearizing control algorithm with integral action and optimal directionality compensation to handle input-constrained MIMO nonlinear systems


affected by integrator windup. This windup compensation methodology is illustrated to be effective on a simulated nonlinear chemical reactor. Chapter 10 by Tarbouriech and Garcia develops Riccati- and LMIbased approaches to design robust output feedback controllers for uncertain systems with position and rate bounded actuators. The proposed controllers ensure robust stability and performance in the presence of normbounded time-varying parametric uncertainty. In addition, this control design methodology is applicable to local stabilization of open-loop unstable systems. It is noted that in this chapter, the authors present yet another novel approach, viz., polytopic representation of saturation nonlinearities, to address the actuator saturation problem. Two numerical examples illustrate the efficacy of the proposed saturation control designs. Chapter 11 by Wu and Grigoriadis addresses the problem of feedback control design in the presence of actuator amplitude saturation. Specifically, by exploiting the LPV design framework, this chapter develops a systematic anti-windup control design methodology for systems with actuator saturation. In contrast to the conventional two-step anti-windup design approaches, the proposed scheme involving induced 2 gain control schedules the parameter-varying controller by using a saturation indicator parameter. The LPV control law is characterized via LMIs that can be solved efficiently using interior-point optimization algorithms. The resulting gain-scheduled controller is nonlinear in general and would lead to graceful performance degradation in the presence of actuator saturation nonlinearities and linear performance recovery. An aircraft longitudinal dynamics control problem with two input saturation nonlinearities is used to demonstrate the effectiveness of the proposed LPV anti-windup scheme. We believe that this edited volume is a unique addition to the growing literature on actuator saturation control, in that it provides coverage to competing actuator saturation control methodologies in a single volume. Furthermore, it includes major new control paradigms proposed within the last two to three years for actuator saturation control. Several common themes emerge in these 11 chapters. Specifically, actuator amplitude and rate saturation control is considered in Chapters 1, 5, 6, 8, and 10. LMIbased tools for actuator saturation control are employed in Chapters 4, 5, 6, 7, 10, and 11. Furthermore, an LPV approach is used to handle input saturation in Chapters 5 and 11. Finally, scheduled/switching control designs for saturating systems are treated in Chapters 4, 5, and 8. We thank all the authors who made this volume possible by their contributions and by providing timely revisions. We also thank the anonymous reviewers who reviewed an early version of this manuscript and provided valuable feedback. We thank B. J. Clark, Executive Acquisitions Editor,


Marcel Dekker, Inc., who encouraged this project from its inception. Last but not least, we thank Dana Bigelow. Production Editor, Marcel Dekker. Inc., who patiently worked with us to ensure timely completion of this endeavor. Vikmm Kapila Karolos M. Grigoriadis


Contents

Preface Contributors 1 Anti-windup for Exponentially Unstable Linear Systems with Rate and Magnitude Input Limits C. Barbu, R. Reginatto, A.R. Teel, and L. Zaccarian 1.1. Introduction 1.2. The Anti-windup Construction 1.2.1. Problem Statement 1.2.2. The Anti-windup Compensator 1.2.3. Main Result 1.3. Anti-windup Design for an Unstable Aircraft 1.3.1. Aircraft Model and Design Goals 1.3.2. Selection of the Operating Region 1.3.3. The Nominal Controller 1.4. Simulations 1.5. Conclusions 1.6. Proof of the Main Result References 2 Selecting the Level of Actuator Saturation for Small Performance Degradation of Linear Designs Y. Eun, C. Gok$ek, P.T. Kabamba, and S.M. Meerkov 2.1. Introduction 2.2. Problem Formulation 2.3. Main Result 2.4. Examples 2.5. Conclusions 2.6. Appendix References


3 Null Controllability and Stabilization of Linear Systems Subject to Asymmetric Actuator Saturation T. Hu, A. N. Pitsillides, and Z. Lin 3.1. Introduction 3.2. Preliminaries and Notation 3.3. Null Controllable Regions 3.3.1. General Description of Null Controllable Regions 3.3.2. Systems with Only Real Eigenvalues 3.3.3. Systems with Complex Eigenvalues 3.4. Domain of Attraction under Saturated Linear State Feedback 3.5. Semiglobal Stabilization on the Null Controllable Region 3.5.1. Second Order Anti-stable Systems 3.5.2. Higher Order Systems with Two Exponentially Unstable Poles 3.6. Conclusions References 4 Regional 7Y2 Performance Synthesis T. Iwasaki and M. Fu 4.1. Introduction 4.2. Analysis 4.2.1. A General Framework 4.2.2. ApplicationsLinear and Circle Analyses 4.3. Synthesis 4.3.1. Problem Formulation and a Critical Observation 4.3.2. Proof of Theorem 4.1 4.3.3. Fixed-gain Control 4.3.4. Switching Control 4.4. Design Examples 4.4.1. Switching Control with Linear Analysis 4.4.2. Switching Control writh Circle Analysis 4.4.3. Fixed Gain Control with Accelerated Convergence 4.5. Further Discussion References


5 Disturbance Attenuation with Bounded Actuators: An LPV Approach F. Jabbari 5.1. Introduction 5.2. Preliminaries 5.3. Parameter-independent Lyapunov Functions 5.4. Parameter-dependent Compensators and Lyapunov Function 5.5. Numerical Example 5.6. Rate Bounds 5.7. Scheduled Controllers: State Feedback Case 5.7.1. Obtaining the Controller 5.7.2. Special case: Constant Q 5.7.3. A Simple Example 5.8. Conclusion References 6 LMI-Based Control of Discrete-Time Systems with Actuator Amplitude and Rate Nonlinearities H. Pan and V. Kapila 6.1. Introduction 6.2. State Feedback Control of Discrete-Time Systems with Actuator Amplitude and Rate Nonlinearities 6.3. State Feedback Controller Synthesis for Discrete-Time Systems with Actuator Amplitude and Rate Nonlinearities 6.4. Dynamic Output Feedback Control of Discrete-Time Systems with Actuator Amplitude and Rate Nonlinearities 6.5. Dynamic Output Feedback Controller Synthesis for DiscreteTime Systems with Actuator Amplitude and Rate Nonlinearities 6.6. Illustrative Numerical Examples 6.7. Conclusion References


1 Robust Control Design for Systems with Saturating NonlinearitiesT. Pare, H. Hindi, and J. How 7.1. Introduction 7.2. Problems of Local Control Design 7.3. The Design Approach 7.4. System Model 7.5. Design Algorithms 7.5.1. Stability Region (SR) 7.5.2. Disturbance Rejection (DR) 7.5.3. Local 2-Gain (EG) 7.5.4. Controller Reconstruction 7.5.5. Optimization Algorithms 7.6. /VGain Control Example 7.7. Conclusions 7.8. Appendix 7.8.1. Preliminaries 7.8.2. Region of Convergence Design 7.8.3. Local Disturbance Rejection Design 7.8.4. Local 2-Gain Design References

8 Output Regulation of Linear Plants Subject to State and Input ConstraintsA. Saberi, A.A. Stoorvogel, G. Shi, and P. Sannuti 8.1. Introduction 8.2. System Model and Primary Assumptions 8.3. A Model for Actuator Constraints 8.4. Statements of Problems 8.5. Taxonomy of Constraints 8.6. Low-gain and Low-high Gain Design for Linear Systems with Actuators Subject to Both Amplitude and Rate Constraints 8.6.1. Static Low-gain State Feedback 8.6.2. A New Version of Low-gain Design 8.6.3. A New Low-high Gain Design


8.6.4. Scheduled Low-gain Design 8.7. Main Results for Right-invertible Constraints 8.7.1. Results 8.7.2. Proofs of Theorems 8.8. Output Regulation with Non-right-invertible Constraints 8.9. Tracking Problem with Non-minimum Phase Constraints 8.10. Conclusions References 9 Optimal Windup and Directionality Compensation in InputConstrained Nonlinear Systems M. Soroush and P. Daoutidis 9.1. Introduction 9.2. Directionality and Windup 9.2.1. Directionality 9.2.2. Windup 9.2.3. Organization of this Chapter 9.3. Optimal Directionality Compensation 9.3.1. Scope 9.3.2. Directionality 9.3.3. Optimal Directionality Compensation 9.3.4. Application to Two Plants 9.4. Windup Compensation 9.4.1. Scope 9.5. Nonlinear Controller Design 9.5.1. Application to a Nonlinear Chemical Reactor References 10 Output Feedback Compensators for Linear Systems with Position and Rate Bounded Actuators S. Tarbouriech and G. Garcia 10.1. Introduction 10.2. Problem Statement 10.2.1. Nomenclature 10.2.2. Problem Statement


10.3. Mathematical Preliminaries 10.4. Control Strategy via Riccati Equations 10.5. Control Strategy via Matrix Inequalitie 10.6. Illustrative Examples 10.7. Concluding Remarks References 11 Actuator Saturation Control via Linear Parameter-Varying Control Methods F. Wu and KM. Grigoriadis 11.1. Introduction 11.2. LPV System Analysis and Control Synthesis 11.2.1. Induced 2 Norm Analysis 11.2.2. LPV Controller Synthesis 11.3. LPV Anti-Windup Control Design 11.4. Application to a Flight Control Problem 11.4.1. Single Quadratic Lyapunov Function Case 11.4.2. Parameter-Dependent Lyapunov Function Case 11.5. Conclusions References


Contributors

C. Barbu P. DaoutidisY. Eun M. Fu

University of California, Santa Barbara, California University of Minnesota, Minneapolis, Minnesota University of Michigan, Ann Arbor, Michigan University of Newcastle, Newcastle, Australia Laboratoire d'Analyse et d'Architecture des Systemes du C.N.R.S., Toulouse, France University of Michigan, Ann Arbor, Michigan University of Houston, Houston, Texas Stanford University, Stanford, California Massachusetts Institute of Technology, Cambridge, Massachusetts University of Virginia, Charlottesville, Virginia University of Virginia, Charlottesville, Virginia University of California, Irvine, California University of Michigan, Ann Arbor, Michigan Polytechnic University, Brooklyn, New York University of Virginia, Charlottesville, Virginia University of Michigan, Ann Arbor, Michigan Polytechnic University, Brooklyn, New York Malibu Networks, Campbell, California University of Virginia, Charlottesville, Virginia University of California, Santa Barbara, California Washington State University, Pullman, Washington

G. Garcia C. Gokgek K.M. Grigoriadis H. HindiJ. How T. Hu

T. Iwasaki F. Jabbari P.T. Kabamba V. KapilaZ. Lin

S.M. MeerkovH. Pan

T. Pare A.N. Pitsillides R. Reginatto A. Saberi


P. SannutiG. Shi

Rutgers University, Piscataway, New Jersey Washington State University, Pullman, Washington Drexel University, Philadelphia, Pennsylvania Eindhoven University of Technology, Eindhoven, and Delft University of Technology, Delft, the Netherlands Laboratoire d'Analyse et d'Architecture des Systemes du C.N.R.S., Toulouse, France University of California, Santa Barbara, California North Carolina State University, Raleigh, North Carolina University of California, Santa Barbara, California

M. Soroush A.A. Stoorvogel S. Tarbouriech A.R. TeelF. Wu

L. Zaccarian


Chapter 1 Anti-windup for Exponentially Unstable Linear Systems with Rate and Magnitude Input LimitsC. Barbu, R. Reginatto, A.R. Teel, and L. Zaccarian University of California, Santa Barbara, California

1.1.

Introduction

Virtually all control actuation devices are subject to magnitude and/or rate limits and this typically leads to degradation of the nominal performance and even to instability. Historically, this phenomenon has been called "windup" and it has been addressed since the 1950's (see, e.g., [25]). To deal with the "windup" phenomenon, "anti-windup" constructions correspond to introducing control modifications when the system saturates, aiming to prevent instability and performance degradations. Early developments of anti-windup employed ad-hoc methods (see, e.g., [6,9,18] and surveys in [1,19,30]). In the late 1980's, the increasing complexity of control systems led to the necessity for more rigorous solutions to the anti-windup problem (see, e.g., [8]) and in the last decade new approaches have been proposed with the aim of allowing for general designs with stability and performance guarantees [12,17,29,31,36,39,42,47]. In many applications, actuator magnitude saturation is one of the main sources of performance limitation. On the other hand, rate saturation is


particularly problematic in some applications, such as modern flight control systems, where it has been shown to contribute to the onset of pilot-induced oscillations (PIO) and has been the cause of many airplane crashes [7, 16,33]. The combination of magnitude and rate limits is, in general, a very challenging problem and has been considered less in the anti-windup framework. Some results have been obtained for specific applications [2, 27,40], and some results can be adapted to this problem (see, e.g., [12,15, 36]). Additional results on stabilization of systems with inputs bounded in magnitude and rate can be found in [11,23,24,37,38]. On the other hand, these last results don't directly address the anti-windup problem, where the performance induced by a nominal predesigned controller needs to be recovered by means of the anti-windup design. Additional concerns arise when the plant contains exponentially unstable modes. In this case, the operating region for the closed loop system has to be restricted in the directions of the unstable modes, a fact that is especially important when large state excursion is required as in tracking problems with large reference inputs. This problem has been addressed in the literature, especially in the discrete-time case, in the context of the reference governor approach [12,15,26,27] (see also [14,35]). In [15] and [12], the reference of the closed-loop system is modified to guarantee invariance of output admissible sets, corresponding to the control signal remaining within certain limits (see [13] for details), and set-point regulation for feasible references. Additional results, following a more general approach labeled "measurement governor" are given in [36]. The main drawback of these approaches is that the output admissible sets are controller dependent and the results hold only for initial conditions in these sets; in particular, in most applications, the more aggressive the predesigned controller is, the smaller the operating region becomes. This limitation is even more severe when disturbances are taken into account; for example, nothing can be guaranteed when impulsive disturbances propel the state of the system out of the output admissible set. In the continuous-time setting, invariant sets independent of the nominal design are exploited in [39]; for unstable systems, in [27] and [39], the system is allowed to reach the saturation limits during the transients, but the reference value is constrained to be within the steady-state feasibility limits at all times. More recent results [2,3,26] allow the reference to exceed the steady-state feasibility limits during transients. In recent years, among other approaches, a number of results on antiwindup for linear systems have been achieved by addressing the problem with the aim of blending a local controller that guarantees a certain desired performance, but only local stability, with a global controller that guarantees stability disregarding the local performance. The combination of these


two ingredients (according to the approach first proposed in [43]) is attained by augmenting the local design with extra dynamics in a scheme that retains the local controller when trajectories are small enough and activates the global controller when trajectories become too large, thus requiring its stabilizing action. Such an approach has been specialized for anti-windup designs for linear systems [42,45] and has been shown to be successful in a number of case studies [21,40,41,44,46]. The main advantage in adopting the local/global scheme for anti-windup synthesis is that, by identifying the local design with a (typically linear) controller designed disregarding the input limitation, the corresponding unsaturated closed-loop behavior can be recovered (as long as it is attainable within the input constraints) on the saturated system by means of an extra (typically nonlinear) stabilizing controller (the global controller) designed without any performance requirement. This decoupled design greatly simplifies, in some cases, the synthesis of the nonlinear controller for the saturated plant. In this chapter, the uniting technique introduced in the companion papers [42,43] is revisited to design anti-windup compensation for linear systems with exponentially unstable modes in a non-local way. In particular, we address the problem of guaranteeing a large operating region for linear systems with exponentially unstable modes (thus improving the local design given in [42]) and give sufficient conditions for achieving local performance and global stability with large operating regions. Preliminary results in this direction are published in [3]. As compared to the results in [12,15,36], we want to guarantee stability and performance recovery in a region that is not dependent on the nominal controller design. To this aim, instead of focusing our attention on forward invariant regions for the nominal closed-loop system, we consider the nullcontrollability region of the saturated plant and modify the trajectories of the nominal closed-loop system only when they hit the boundaries of (a conservative estimate of) this last region. The resulting anti-windup design is appealing in the sense that the resulting operating region is typically obtained by shrinking the null-controllability region of the saturated system; l since the null-controllability region is unbounded in the marginally unstable directions, 2 it can be extended to infinity in these directions; whereas, in the exponentially unstable directions it needs to be bounded. Subsequently, as an example, the proposed scheme is applied to the linearized short-period longitudinal dynamics of an unstable fighter airShrinking the null-controllability region is desirable to allow a robustness margin toward disturbances and to avoid the stickiness effect described, e.g., in [27]. 2 Results on null-controllability of linear systems with bounded controls can be found, e.g., in [22,34].


craft subject to rate and magnitude limits on the elevator deflection. The anti-windup design applied to this unstable linear system allows to achieve prototypical military specifications for small to moderate pitch rate pilot commands, while guaranteeing aircraft stability for all pitch rate pilot commands. Due to the large operating region achieved by the anti-windup scheme, the controlled aircraft allows the pilot to maneuver aggressively via large pitch rates during transients.

1.2.1.2.1.

The Anti-windup ConstructionProblem Statement

Consider a linear system with exponentially unstable modes having state x G R n . control input 5 G R m , measurable output y G Rp, and performance output z G R 9 . Let the state x be partitioned as x =:I J the vector xu G Rn" contains all of the exponentially unstable states and xs G R n contains all of the other states. The state space representation of the system, consistent with the partition of x, is:s Xu

G R n , where

where all the eigenvalues of Au have strictly positive real part (As can possibly have eigenvalues on the imaginary axis). For system (1.1), assume a (possibly nonlinear) dynamic controller has been previously designed to achieve certain performance specifications in the case where the input is not limited. Let this controller (called "nominal controller"} be given in the form: Nominal controller \ xc \ yc g(xc, uc, r) = k(xc, uc, r),(

I

r AS Ai 1 r B, i . x=Ax+B8=\* Au2 \\ x+\ [ -R u \5 [ 0 A B \z y = Czx + Dz6 = Cvx + Dy8,

,{ }

}

,-. ~\

where xc G R Ur is the controller state, r G R9 is the reference input, and uc G R p , yc G Rm are its input and output, respectively. For the sake of generality, we allow the nominal controller to be nonlinear, although it frequently turns out to be linear. We assume that the design of the nominal controller (1.2) is such that the closed loop system (1.1), (1.2) with the feedback interconnection

S = yc,

uc = y,

(1.3)


is well-posed (i.e., solutions exist and are unique) and internally stable, and provides asymptotic set-point regulation of the performance output,t-^+oo

lim z(t) = r.

We also assume that, for each constant reference r, there exists an equilibrium (x*, x*) for (1.1), (1.2), (1.3), that is globally asymptotically stable and we define (**,j/*)=:(r), (1.4)

as the corresponding state-input pair. Notice that the internal stability assumption implies that the plant (1.1) is stabilizable and detectable. Throughout the chapter we refer to the closed-loop system (1.1), (1.2), (1.3) as the "nominal closed-loop system". We address the problem that arises when the actuators' response is limited both in magnitude and rate. The rate and magnitude saturation effect can be modeled (similarly as in [40] and [2]) by augmenting the plant dynamics with extra states 6 G Rm satisfying the equation: 6 = Rsgn(Ms&t(} -o] , V \M / / (1.5)

where the functions sgn(-) and sat(-) are the standard decentralized unit sign and saturation functions, M and R are positive numbers, and u e Rm is the input to the actuators before saturation. Since the design of the nominal controller disregards the magnitude and rate limits, instability can arise if that controller is connected in feedback with the actual plant (1.1), (1-5), especially because the plant contains exponentially unstable modes. On the other hand, by assumption, the performance induced by the nominal controller is desired for the actual plant (1.1), (1.2) and should be recovered whenever possible. Thus, our antiwindup design problem is to accommodate the requirements of respecting as much as possible the performance induced by the local controller, while guaranteeing stability of the closed-loop system in the presence of magnitude and rate limits, without restricting the magnitude of the reference signal a priori. In the next sections we recall the state of the art for the particular anti-windup approach initiated in [43] and make further contributions to that design methodology especially suited for MIMO exponentially unstable linear systems subject to magnitude and rate limits.


1.2.2.

The Anti-windup Compensator

In recent years, a number of results on anti-windup design for linear systems have been achieved following the guidelines in [43]. The underlying strategy is to augment the nominal controller with the dynamical system (called anti-windup compensator) Anti-windup compensator

v =

, xu, 6) yc,

(1.6)

where = [j j]T G R" x R U u , v = [vf v^}7 G RTO x Rp, and x u , yc as in equations (1.1), (1.2), and to consider the system resulting from (1.1), (1.2), (1.5), (1.6) with the interconnection conditions, | /-I >-7\

The anti-windup design described in this chapter relies on the availability for measurement of the exponentially unstable modes, although to provide such information full state measurement might be required. Nevertheless, if the state of the plant is not available for measurement and the disturbances are small, a fast observer can be used. Throughout the chapter, we will refer to the system (1.1), (1.2), (1.5), (1.6), (1.7) as the "anti-windup closed-loop system". Figure 1 shows the block diagram of the anti-windup closed-loop system, which can be recognized as a natural extension of [39] for the case when the substate xu is available for measurement and both magnitude and rate limits are present.c Nominal V ^ u _ Magnitude and Controller + j ^ , ~~ Rate Saturation -^ +

5 _

Aircraft

TAWy?9

^

xu

+ 'r

-o

Figure 1: Block diagram of the anti-windup scheme. Some of the critical issues that arise in the design of the anti-windup compensator are briefly discussed in the following.


Exponentially unstable systems. A basic issue arising with exponentially unstable plants is that global asymptotic stability cannot be achieved, because the null-controllability region of the plant is bounded in the directions of the exponentially unstable modes. Hence, the results are non-global and the goal is to obtain a large operating region for the closed-loop system without significantly sacrificing performance. The results in [42] apply to exponentially unstable linear systems but only for the solution of the local anti-windup problem, thus not computing explicitly the operating region and possibly resulting in conservative designs. More recently, based on [43], a more explicit construction for exponentially unstable plants with only magnitude saturation was given in [39]. Magnitude and rate saturation. The early anti-windup developments illustrate the importance of magnitude saturation in control applications. On the other hand, rate saturation plays a similar role in terms of the effects introduced in the system. For instance, in flight control problems, it has been remarked in [4,27,28] how the instabilities and/or performance losses due to windup are generated more frequently by the rate limits than by the magnitude limits. The combination of magnitude and rate limits or even general state constraints is a more challenging problem and has been addressed more in the discrete-time setting (see, for instance, [12, 26,27]) than in the continuous-time case. The case of both magnitude and rate saturation is addressed in continuous time in [43] and applied to asymptotically stable plants in [40] and [41]. Reference values. Usually (see, e.g., [12,39]), when the plant contains exponentially unstable modes, the reference signal is not allowed to take large values, although these would generate (at least for a limited amount of time) feasible trajectories for the saturated system. In [26], the problem of allowing large references during transients has been addressed in the context of the reference governor. As pointed out in [26], by allowing the reference to be arbitrarily large, better transient performance for the closed-loop system may be achieved. In [2], arbitrarily large references are allowed during the transients for a particular exponentially unstable plant. It is shown there that the performance of the saturated system is improved adding this extra degree of freedom (nevertheless, due to boundedness of the null-controllability region, the reference cannot be arbitrarily large at the steady state). The main contribution of the approach described in this chapter is in the fact that the resulting anti-windup compensation allows for arbitrarily large references (at least during the transients) for exponentially unstable linear systems when both rate and magnitude saturation are present at the plant's input.


1.2.3.

Main Result

Given a nominal controller and a plant with input magnitude and rate saturation, in this section we give a design algorithm that, on the basis of a desired operating region for the closed-loop system, and for a given stabilizing static feedback that satisfies certain assumptions, provides an anti-windup compensator that achieves stability and allows arbitrarily large references for the saturated system, guaranteeing restricted regulation for any reference outside the operating region. The following definition will be useful in the rest of the chapter. Definition 1.1. The null-controllability region V for system (1.1), (1-5) is the subset V c R" x R m of the state space such that for any initial condition in V, there exists a measurable function u : R>o * R that drives the state of the system asymptotically to the origin. Remark 1.1. A desirable property of the closed loop system is to have an operating region as large as possible. However, it is not always desirable to get very close to the boundary of the null-controllability region. Indeed, assume that there exists a locally Lipschitz controller that renders the nullcontrollability region forward invariant. Then, necessarily, the boundary of the null-controllability region is an invariant set and, by continuity of solutions with respect to initial conditions on compact time intervals, the closer the plant state gets to this boundary, the longer it will take to move away from it. We refer to this behavior as the "stickiness effect".3 It is desirable then to define an "anti-sticking coefficient" and tune the anti-windup compensator using a conservative estimate of the null-controllability region, which guarantees that the trajectories of the system stay far enough from the boundary of the null-controllability region, thus improving the resulting performance. We first specify a region 4 U. C Rn" x Rm where we want the exponentially unstable modes and the inputs of the closed-loop system to operate (accordingly to anti-sticking requirements and/or performance specifications). \Ve specify this region to be a compact set. Then, we assume that a stabilizing static nonlinear state feedback 7 is given that guarantees the first or both of the following properties to hold: 1. positive invariance of the set U for the plant with input magnitude and rate saturation.This effect has been noticed in a number of applications (see, e.g., [27]). The null-controllability region is bounded only in the subspace of the exponentially unstable modes [22], so we only need to specify the operating region in that subspace.


2. convergence to a set-point in W; After U and 7 are chosen, the last ingredient for the design of the antiwindup compensator is the policy to follow when the closed-loop system is driven by a reference whose steady state value corresponds to an infeasible equilibrium for the saturated system (namely, a value r corresponding to a state-input pair (a:*, *,*) = E(r) such that (#*, 6*) If). To this aim, a function P that maps the infeasible set-point to a feasible one will be denned. A typical choice for P is to "project" the infeasible set-point to a feasible point that is, in some sense, "close" to the infeasible one. However, the reference limiting action achieved by P is not used during the transients but only at the steady state. This strategy allows to completely recover, on the saturated system, the nominal responses (even to infeasible references) for the maximal time interval allowable within the specified operating region and the saturation limits. The following statements formally define the requirements described above. Definition 1.2. Define the equilibrium manifold C Rn x Rm as the set of all the state-input pairs (x, 6) of the linear system (1.1) associated with an equilibrium of the nominal closed-loop system, 5 i.e. (with reference to equation (1.4)), := {(x, 5) R n x Rm :3r R9 s.t. (x, 6) = E(r}} .

(1.8)

Let the pair (3, F be such that T is a compact strict subset of ti and (3 : Hn" x Rm > [0, 1] is a continuous function satisfying 6 ' if ? if * R is defined as OLXU, , ?7 U , , yc := 7'x u , < ,/3(a'U5 0; 2. if the initial conditions satisfy (x u (0), 5(0)) e W, then U, \/t > 0 and all the trajectories are bounded; 3. if the initial conditions satisfy (x u (0), 5(0)) e ^ and ift >oo

lim (xa(t), xu(t), xc(t}} = (x*8, x*, x*),

(1.19)

then:

Proof. See Section 1.6.

D

An interpretation of the three results in Theorem 1.1 is in order. Item 1 states that, if the anti-windup compensator is appropriately initialized, and the reference signal is sufficiently small (namely, if it keeps the system within T and does not cause the input to saturate), the anti-windup closed-loop system will perform identically to the nominal closed-loop system. The statement in item 2 conveys the requirement that the trajectory (x(t), 5(t)) of the anti-windup closed- loop system never leaves the operating region U. This statement is completed with item 3 which gives the desired convergence properties for the anti-windup closed-loop system. IfDenote by int(Jr) the interior of the set


the trajectory of the nominal closed-loop system converges to a point in T , the trajectory of the anti-windup closed-loop system converges to the same point; however, different transient behavior should be expected due to the presence of the actuator limits. On the other hand, if the steady-state value of the nominal closed-loop system is outside J-, the same convergence property is not feasible for the anti-windup closed-loop system. In this case, the anti-windup closed-loop system converges to a point which is close to the nominal steady-state value in the sense of the projection function P. The peculiarity of the general structure given in equation (1.16) is in the fact that, with the coordinate transformation X := x , the antiwindup closed-loop system in the (X, xc, x, 6) coordinates is the cascade of two subsystems: the (X, xc) subsystem, exactly reproducing the dynamics of the nominal closed-loop system, and the (x, 8) subsystem, taking into account the effects of the saturation nonlinearity on the plant dynamics:,y , su'bsys'l

*=

==

9(*X,r)k(xc, X, r)

(1.20)

t u

= Ax + B6 = flsgn(M sat ( ) - ) = a(xu, 6, Xu, x -X, yc).

(1.21)

1.3.

Anti-windup Design for an Unstable Aircraft

Magnitude and rate saturation are two of the most frequently encountered nonlinearities in modern flight control. As an example of the antiwindup design synthesized in Section 1.2, we focus on the short-period, longitudinal dynamics of a prototypical unstable fighter aircraft subject to rate and magnitude limits on the elevator deflection. For this system, large pitch rates requested by the pilot may not be achievable while maintaining stability of the aircraft. Based on the results in Theorem 1.1, an anti-windup compensator is designed for the unstable aircraft that achieves prototypical military specifications for small to moderate pitch rate pilot commands, guarantees aircraft stability for all pitch rate pilot commands and allows the pilot to maneuver aggressively.

1.3.1.

Aircraft Model and Design Goals

According to the experimental data in [32], the linearized short-period longitudinal dynamics of the McDonnell Douglas Tailless Advanced Fighter Aircraft (TAFA) model at a dynamic pressure of 450 psf (corresponding


to a specific trim flight condition), are described by the following linear system:6 -2

=: Az + BS6 = Rsgn [Msat (^-] - 6\ , L V AJ / J (1.23)

where the variable q represents the body axis pitch rate and a and 8 are, respectively, the deviation of the angle of attack and of the elevator deflection angle from the trim flight condition. The magnitude and rate limits of the elevator deflection are quantified by M and J?, respectively. In this example, the maximal (deviation of the) elevator deflection angle is limited between 20 deg (M = 0.35) and the maximal elevator deflection rate is limited between 40 deg/sec (R = 0.7). Note that system (1.22) can be diagonalized via a suitable coordinate transformation to obtain xs + bsd xu+bu6, (1.24a) (1.24b)

where As = 4 corresponds to an exponentially stable mode and \u = I corresponds to an exponentially unstable mode. Our control problem is to design a (dynamic) feedback with inputs (a, q, 5) and pitch rate pilot command qj so that, for any trim flight condition (i.e., for any choice of the dynamic pressure) the closed- loop satisfies the following properties: 1. For small to moderate pitch rate commands, the pitch rate response satisfies a prototypical military specification; here, based on [20], we take q(s)_ qd(s) lAs+l s2 + 1.5s + rl

' '

Moreover, this response is recovered asymptotically after large commands. 2. The aircraft is BIBS stablen

from the pilot command input qd-

3. The aircraft is highly maneuverable; i.e., large pitch rates are attained by the control scheme. A system is BIBS (Bounded Input Bounded State) stable if the state response to any bounded input is bounded as well.


1.3.2.

Selection of the Operating Region

In this section we study the structural limitations of the saturated system (1.22), (1.23) (or, equivalently, of system (1.24), (1.23)). In particular, we define and explicitly compute the maximal stability region achievable within the actuator saturation limits. Based on this, we give a selection for the operating region U. introduced in Section 1.2.3. We first compute the null-controllability region V. In particular, first note that V = R x Vp, where Vp C R2 is the projection of V on the ( x u j 5) plane. Now, consider the limitations due to magnitude saturation and observe that, by equation (1.24b), any initial condition outside the set 5} VM := horizontal dotted lines in Figure 2) generates a nonconverging trajectory because xu xu > 0 for all times and XUQ ^ 0. On the other hand, if there is no limitation on the control input rate (namely, R > oc), a simple proportional controller (8 = Kxu, K sufficiently large) can drive to zero any trajectory with initial conditions (x u (0), 5(0)) VM- It follows that VP C VM.

Null-controllability region

-3-

10

15

20

Figure 2: The sets VM ; Vp, IA, F and 8U.


When the effects of rate saturation are considered, Vp can be defined as the set VP := {(xUQ, 60) e VM : M') 0}, (1.26) where 0(t; u o, Ô) u(t)) denotes the trajectory of system (1.24b), (1.23) starting at (x u (0), 5(0)) = (x u o 5 ô) and with a measurable input function u(-). Although equation (1.26) characterizes the region Vp, this definition is implicit, and thus not of practical utility. However, due to the structure of the model (1.24b), (1.23), the boundaries of Vp can be computed explicitly and they correspond to the dashed lines in Figure 2 (the explicit equations are not included here due to space constraints). Note that, while the interior of Vp is weakly forward invariant (namely, there exists at least one selection of the input u(-) that makes it forward invariant), the complement of Vp is strongly forward invariant (namely, regardless of the input w(-), trajectories never leave this set). Hence, if trajectories leave the interior of Vp, they cannot return to the interior of Vp regardless of the control action through the input u. As already pointed out in Remark 1.1, this fact plays an important role in the control design. Indeed, by continuity of solutions with respect to initial conditions on compact time intervals, if the trajectories get close to the boundary of Vp, they will take a long time to move away from this boundary, thus exhibiting an undesired stickiness effect (see, e.g., [27]). To avoid this phenomenon, we choose an operating region U that is strictly smaller than the null-controllability region. In particular, U is chosen as the region Vp that would be obtained if the magnitude and rate limits were 80% of their actual values (see the light shaded area in Figure 2), union with two rectangular regions in the upper left and lower right corners (corresponding to the dark shaded areas in Figure 2). A natural choice for F is then a contraction of the set U. sufficiently close to U. (corresponding to the dash-dotted lines in Figure 2). The diagonal set corresponding to the "stars" represents the projection u of the set on the (xu , 6) plane and corresponds to the set of all the equilibria that the input u can induce on the open-loop system. Based on this choice for the sets U and F, the following result provides a function 7 that satisfies items 1 and 2 of Property 1, thereby guaranteeing by Theorem 1.1, the effectiveness of the anti-windup construction. Theorem 1.2. Given the sets U and F represented in Figure 2, the function 7(x u , O := - (Xuxu + (xu - -* Cfb(s)-jI

-tc-|~i

^

Unsatur ated Aircriift

^

Linear Controller

A''

[a q]

Hi'

Figure 3: Nominal closed-loop system with nominal controller structure. Design of the Linear Controller The linear controller in Figure 3 is constituted by an inner stabilizing static feedback, an outer dynamic feedback and a dynamic feed-forward action. Based on the fact that the linearized aircraft dynamics are minimum phase, the inner stabilizing feedback is chosen as:-2


and the zero dynamics (that become unobservable) are asymptotically stable. Once the inner loop has been closed, the plant is transformed into an integrator and any desired closed-loop transfer function can be obtained by choosing appropriately the feedback and feed-forward dynamic elements Cfb(s) and C//(s), respectively. In particular, to obtain the closed- loop transfer function (1.25) from qd to g, the two dynamic elements have been chosen as: 1 , := 1.5s+l (L28)}

-

Dynamic Command Limiting On the basis of Theorem 1.1, the BIBS stability of the anti-windup closed- loop system designed in Section 1.2.3 is guaranteed for any nominal controller that stabilizes the unsaturated plant. To this aim, the dynamic command limiting block in Figure 3 is not necessary. However, implementing a linear controller such as the one described above without any command limiting action could lead to poor performance when the pilot command is large enough to drive the unsaturated system outside the operating region U. Indeed, the equilibria corresponding to such commands are infeasible for the anti-windup closed-loop system and limiting the steadystate pilot command is important to avoid steady-state differences between these equilibria and the ones achieved by the nominal closed-loop system in Figure 3. Following the above reasoning, it is straightforward that the dynamic command limiter is to be designed with the goal of keeping the nominal trajectory "close" to the operating region by adding a feed-forward action exclusively when the trajectory is not in F\ in this way, any maneuver that stays within the operating region (namely, any feasible maneuver) is not modified by the command limiter, but infeasible trajectories are not allowed to move too far from the operating region itself. Given the magnitude saturation limit M, define qM = 2M as the maximum feasible steady-state pitch rate command for the saturated plant (1.22). Consider the function @L(XU, 6) : R x R > [0, 1] satisfying equation (1.9) but not necessarily equal to fi(xu, 6}. 12 The dynamic commandIt has been verified that good performance is achieved when PL(ZU, 8) < f3(xu, 8) everywhere. In particular, a possible choice is to define a set F$ such that T C F$ C Uand to select

12

0,

1,

if

if

(zu, R be an absolutely continuous function satisfying rj(t)\ < M, W > 0 and f](t) < R for almost all t in [0, oo). The initial value problem C e R SGN ( M sat ( ) (,) , where (0) = ^(0) (1.30)

( 1, if x > 0 SGN(;r) := ^ -1, if x < 0 [ [-1, 1] if x = 0.has the unique solution ("() = r)(t] on [0, oo).

(1.31)

Proof. Since SGN(x] is an upper semicontinuous set valued map and is nonempty, compact, and convex for each x e R. by [10, Theorem 1, page 77], there exists at least one solution and since the right hand side is uniformly bounded, each solution is maximally denned on [0, oo). Let be an arbitrary solution of (1.30) defined on [0, oo) and define e := C r/. Then, e(0) = 0 and, from equation (1.30), e e -flSGN(e) - T), for almost all t in [0, oo), (1.32)

and, since r)(t}\ < R, then for almost all t 6 [0, oo),1 d(ee) 2 dt

= e(-Rsgn(e)-ri) < -R\e +R\e = 0 , where the single-valued function sgn(x) is the sign of x for x ^ 0 and is arbitrarily defined at x = 0. Thus, e 2 (0) = 0 and ^j~ < 0 for almost all times. By standard comparison theorems, the result follows. D Proof of Theorem 1.1. Define X : x - , Y := y + ^2, z^ := Cz + Dz (5 yc}. Z := z z% and rewrite the anti-windup closed-loop


system equations in the new coordinates (J, x c , x, 6):~y

y A

+

JJy

yc

subsystem

= CzX + Dzyc = g(xc, y, r) = k(xc, y, r)X

(1.33)

Ax + B5u,

subsystem

(x,S)

5, P(XU), x - X, yc)

(1.34)

Notice that the system in this form has a cascade structure where the (X, xc) subsystem feeds the (x, 5) subsystem. Item 1. Let the initial conditions of the anti-windup closed loop system be given by= x(0)5(0) =r\

(1.35a) (1.35b) (1.35c) (1.35d)

Thus, (X(0), xc(Q}} = (x(0), x c (0)) and, since the equations describing the (X, xc) subsystem (1.33) are coincident with the ones describing the nominal closed-loop system, by uniqueness of solutions for the unsaturated closed-loop system it follows that

X(t) = x(t), Z(t) = z(t),yc(t) = yc(t),

\/t > 0, \/t > 0,Vt > 0.

(1.36a) (1.36b) (1.36c)

We only need to prove that, under the assumptions of this item, (x(t), 8(t)) ( X ( t ) , y c ( t ) ) is the unique 13 solution of the (x, 6) subsystem (1.34). From this fact, it immediately follows that z^(t) = 0, Vt > 0, and thus z ( t ) = z ( t ) , V^ > 0. To prove this fact we invoke Lemma 1.1 after deriving bounds on and lit I. First notice that P(Xu(t))=Xu(t)=xu(t}, (1.37)

13, is clear that it is a solution, but we cannot directly assert the uniqueness since It the right hand side of (1.34) is not Lipschitz.


which follows from the assumption ( x u ( t ) , yc(t}} F, W > 0 together with equation (1.36a) and property (1.11) of P(-}. Moreover, if (xu. 8} E T, from compactness of J-, by the state equation of system (1.1) and equation (1.5), there exists L > 0 such that 14

Au xu + Bu 5

(1.38)

Defining A := inf we ^ 0 distd^(w] and the maximal time T := A/L, since jpQ is a compact strict subset of J- , by inequality (1.38) and continuity of solutions with respect to time, we can state that for any t0 > 0,

(xu(t0},8(t0))

, W e [t0,t0

(1.39)

where T does not depend on t0. From (1.35a) we obtain xu(0) = xu(Q), which combined with (1.35c) and the theorem assumption that ( x u ( t ) , u ( t ) ) E TQ, Vt, yields (x u (0), 0, there exists Te > 0, independent of the initial conditions, such that 15maxP3(14r15

,

P2(\x(t}

- X(t)\) +

x(t) ~ x(t)\)\x(t) - x(t)\} < e, w e [o, rc].

The norm | denotes the standard Euclidean norm. Recall that f = a? - X.


Thus, recalling the theorem assumptions \yc(t)\ < MO < M, \yc(t) < RO < R, picking e = min{M-Mo, R-Ro}, equations (1.41) can be bounded by \u(t)\ < M, ii(t)\ 0 as t -> oo, and u can be written as u(t) = l(xu(t), 5(t), P(x*u + i(t))', x(t) - x* + e 2 (t), y* + 3 (*)) + /9(ar t t (t), 0, find the level of saturation of the actuator, a, so that &z < (1 + e)crz.

2.3.

Main Result

Let D(r] denote the closed disk in C with radius r, centered at (-r-1, jO) (see Figure 5). Let /3(e, r) be denned by (2.11) Introduce the following assumptions:


2.8 2.6

-r 2.4JlT ^ 2.2

H05

2 1.80

10

Figure 6: Function (3(e,r). (Al) The closed loop system of Figure 3 with w = 0 is globally asymptotically stable. (A2) Transfer functions g and ffi., proper. where L = PiF2C, are strictly

(A3) Equation (2.7) has a unique solution TV*. Theorem 2.1. Let (A1)-(A3) hold, e be the tolerable level of performance degradation, and r be such that the Nyquist plot of the loop gain L lies entirely outside of D ( r ) . Then, 0 < a* < (1 + e)cr 2if

(2.12)

a > /3(e,r)u u . Proof. See the Appendix.

n

Figure 6 illustrates the behavior of /3(e, r) with e = 0.1 and e = 0.05, for a wide range of r. Obviously, /3(e, r) is not very sensitive with respect to r for r > 1. In particular, it is close to 2 for all r > 1, if e 0.1. This justifies the rule-of-thumb given in the Introduction.

2.4.

Examples

Example 2.1. Consider the feedback system with P-controller shown in Figure 7. Using the Popov criterion, one can easily check that this


w>T-^

r = C|| i

60

u

K u^l+ ^Q * 71asat(

i

yp^

i

z

Figure 7: System of Example 2.1. system is asymptotically stable. If no saturation takes place. az = 1.1238 and (T.u = 1.4142. To select a level of saturation, a. that results in less than 10% performance degradation, we draw the Nyquist plot of L = s / s . +2 t wg +10 \ and determine the largest disk -D(r) such that L(ju] lies entirely in its exterior. It turns out that r = 4.2, as shown in Figure 8. Thus, according to Theorem 2.1. a > /3(0.1,4.2)oru = 1.9 x 1.4142 = 2.687 (2.13)

guarantees that the degradation of performance is at most 10%. With

D(r]

-8

-2

0

Figure 8: Nyquist plot arid D(r) for Example 2.1.


Table 1: System performance and accuracy for Example 2.1. (TJ

(A.ll)

Indeed, rewriting a > fi(e,r}au using (A.11) and (2.11) yields a > /3(e,r)au

(A.12) Then, from Lemma A.I, (A. 11) and (A. 12), we obtain FP2C2r

. (A.13)


Next, we show that the quasi-linear gain TV* of the system of Figure 4 corresponding to a given a > /3(e.r)au exists and. moreover, N* > N\. Indeed TV* is defined by FP2C + TV*L_ For N = 1, due to the fact that erf - 1 (l) = oo, we can write: FP2C (A.14)

a\72erf-Kl)

= 0.

(A.15)

is continuous in TV, from (A. 10) and (A.15), 75 l f we conclude that there exists N* > N satisfying (A.14). Finally, using Lemma A.I. we show that cr? < (1 + e}az:2r

Since

(2r+(2r

(A.16) This completes the proof.

n

References[1] A. Gelb and W. Vander Velde. Multiple-input Describing Functions and Nonlinear System Design. New York: McGraw-Hill, (1968). [2] C. Gokgek. Disturbance Rejection and Reference Tracking in Control Systems with Saturating Actuators. PhD thesis. The University of Michigan, 2000. [3] C. Gokgek. P. Kabamba, and S. Meerkov. Disturbance Rejection in Control Systems with Saturating Actuators, Nonlinear Anal, 40 (2000) 213-226. [4] J. Roberts and P. Spanos. Random Variation and Statistical Linearization, New York: John Wiley and Sons, (1990).


[5] W. Wonham and W. Cashman. A Computational Approach to Optimal Control of Stochastic Saturating Systems, Int. J. Contr., 10(1) (1969) 77-98.


Chapter 3 Null Controllability and Stabilization of Linear Systems Subject to Asymmetric Actuator Saturation1T. Hu, A. N. Pitsillides, and Z. Lin University of Virginia, Charlottesville, Virginia

3.1.

Introduction

We consider the problem of controlling exponentially unstable linear systems subject to asymmetric actuator saturation. This control problem involves basic issues such as characterization of the null controllable region by bounded controls and stabilizability on the null controllable region. These issues have been focuses of study of and are now well-addressed for linear systems that are not exponentially unstable. For example, it is wellknown [10,11] that such systems are globally null controllable with bounded controls as long as they are controllable in the usual linear system sense. In regard to stabilizability, it is shown in [12] that a linear system subject to actuator saturation can be globally asymptotically stabilized by smooth feedback if and only if the system is asymptotically null controllable with bounded controls (ANCBC), which, as shown in [10,11], is equivalent to the system being stabilizable in the usual linear sense and having open loopWork supported in part by the US Office of Naval Research Young Investigator Program under grant NOOO14-99-1-0670.


poles in the closed left-half plane. A nested feedback design technique for designing nonlinear globally asymptotically stabilizing feedback laws was proposed in [14] for a chain of integrators and was fully generalized in [13]. The notion of semiglobal asymptotic stabilization on the null controllable region for linear systems subject to actuator saturation was introduced in [7]. The semiglobal framework for stabilization requires feedback laws that yield a closed-loop system which has an asymptotically stable equilibrium whose domain of attraction includes an a priori given (arbitrarily large) bounded subset of the null controllable region. In [7], it was shown that, for linear ANCBC systems subject to actuator saturation, one can achieve semiglobal asymptotic stabilization on the null controllable region using linear feedback laws. On the other hand, the counterparts of the above mentioned results for exponentially unstable linear systems are less understood. Recently, we made an attempt to systematically study issues related to null controllable regions and the stabilizability on them of exponentially unstable linear systems subject to actuator saturation and gave a rather clear understanding of these issues [4]. Specifically, we gave a simple exact description of the null controllable region for a general anti-stable linear system in terms of a set of extremal trajectories of its time-reversed system. We also constructed feedback laws that semiglobally asymptotically stabilize any linear time invariant system with two exponentially unstable poles on its null controllable region. This is in the sense that, for any a priori given set in the interior of the null controllable region, there exists a linear feedback law that yields a closed-loop system which has an asymptotically stable equilibrium whose domain of attraction includes the given set. One critical assumption made in [4] is that the actuator saturation is symmetric. The symmetry of the saturation function to a large degree simplifies the analysis of the closedloop system, it, however, excludes the application of the results to many practical systems. The goal of this chapter is to generalize the results of [4] to the case where the actuator saturation is asymmetric. We will first characterize the null controllable region and then study the problem of stabilization. We take a similar approach as in [4] to characterize the null controllable region. In studying the problem of stabilization, we found the methods used in [4] to derive the main results not applicable to the asymmetric case, since the methods rely mainly on the symmetric property of the saturation function. For a planar anti-stable system under a given saturated linear feedback, we showed in [4] that the boundary of the domain of attraction is the unique limit cycle of the closed-loop system. The uniqueness of the limit cycle was established on the symmetric property of the vector field


and the trajectories. We further showed that if the gain is increased along the direction of the LQR feedback, then the domain of attraction can be made arbitrarily close to the null controllable region. This result was also obtained by applying the symmetric property of the trajectories. In this chapter, we propose a quite different approach to solving these problems for the case of asymmetric saturation. In particular, we will construct a Lyapunov function from the closed trajectory, and show that under certain condition, the Lyapunov function is decreasing within the closed trajectory, thus verifying that the closed trajectory forms the boundary of the domain of attraction. If the state feedback is obtained from the LQR method, then there is a unique closed trajectory (a limit cycle). We will also show that if the gain is increased along the direction of the LQR feedback, then the domain of attraction can be made arbitrarily close to the null controllable region. This result will be developed by a careful examination of the vector field of the closed-loop system. For higher order systems with two anti-stable modes, we have similar results as in the symmetric case: given any compact subset of the null controllable region, there is a controller (switching between two saturated linear feedback laws) that achieves a domain of attraction which includes the given compact subset of the null controllable region.

3.2.

Preliminaries and Notation

Consider a linear system

(3.1)where x(t) e Rn is the state and u(t) e R is the control. Given real numbers u~ < 0 and u+ > 0, define Ua ' \ u : u is measurable and u~~ < u(i] < u+, W G R > . (3-2)

A control signal u is said to be admissible if u G Ua. In this chapter, we are interested in the control of the system (3.1) by using admissible controls. Our first concern is the set of states that can be steered to the origin by an admissible control. Definition 3.1. A state XQ is said to be null controllable if there exist a T G [0, oo) and an admissible control u such that the state trajectory x(t) of the system satisfies x(0) = XQ and x(T) = 0. The set of all null controllable states is called the null controllable region of the system and is denoted by C.


With the above definition, we haver-T

C=

(J

lx = - I'0

e-ATbu(r}dr : uÛa\ .

(3.3)

Te[0,oo) I

Remark 3.1. For a linear system with multiple inputs, x(t)=Ax(t} + Bu(t], (3.4)

where u RTO, and Ui e [M,W/~], u~ < 0,u^~ > 0, let Ci be the null controllable region of the system

then it is easy to verify that the null controllable region of the system (3.4) isC =4=1

Hence, it is without loss of generality that we consider the single input system (3.1). For simplicity, a linear system and the matrix A are said semistable if all the eigenvalues of A are in the closed left half plane; and anti-stable if all the eigenvalues of A are in the open right half plane. We recall a fundamental result from the literature [2, 10, 11]: Proposition 3.1. Assume that (A,b) is controllable. a) If A is semistable, then C = R n . b) If A is anti-stable, then C is a bounded convex open set containing the origin. c) If A =

L n u

Al

A? j

n with Al e R n i x n i anti-stable and A> e

semistable. and b is partitioned as

,

accordingly, then

C = Ci x R n a , where Ci is the null controllable region of the anti-stable system

i(t) =AlXl +blU(t).


Because of this proposition, we can concentrate on the study of null controllable regions of anti-stable systems. For this kind of systems,oo

C = 1x = - /

-Ar e-ATbu(r}dr : u e Ua } ,

(3.5)

where C denotes the closure of C. We also use " U


where u+ (or u~) denotes a constant control v(t) = u+ (or u~). Here we allow ti ti+i (i 7^ 1) and tn-\ = oo, so the above description of c consists of all bang-bang controls with n 1 or less switches. By setting t\ = 0, we immediately get ,

f l, = sgna (u) : u(t] = { (-!),

*e [-00, ii),teM,+i),

( (-1)"-1, *e[* n _i,oo),

0 = t x < 2 < ' ' ' < * n -l < 00 > U

For each u e , we have i>(t) = u~ (or u + ) for all t < 0. Hence, for t < 0,/* $(^) = - /J oo

e-

A(t T)

- 6t;(t)dT = -A~lbu- ( or - A'1bu+).

And for t > 0, v(t) is a bang-bang control with n 2 or less switches. Denote z+ = A~lbu+ and z~ = A~lbu~ , then from Theorem 3.2 we have, Observation 3.3.1. dH = dC is covered by two bunches of trajectories. The first bunch consists of trajectories of (3.8) when the initial state is z+ and the input is a bang-bang control that starts at t = 0 with v = u~ and has n 2 or less switches. The second bunch consists of the trajectories of (3.8) when the initial state is z~ and the input is a bang-bang control that starts at t = 0 with v = u+ and has n 2 or less switches. Remark 3.2. Since the trajectories of the time-reversed system (3.8) and those of the original system are the same except that their directions are opposite, we can also say that 0. It can be verified thats n (c'eAtb) : c O = sgna(sin(^ + ( ? ) ) : Oe [0, ce


Hence the set of extremal controls is Sc = ( v ( t ) = sgna(sin(/% + (9)), t e R : 6 e [0, 27r)} . It is easy to see that C = {*>(*) =sgn a (sin(/%)), t e contains only one element. Denote Tp = -|. then e~ATp = e~aTpI. Let

and

It can be verified that the extremal trajectory corresponding to v(t] = sgna(sin(/3t)) is periodic with period 2Tp: in the first half period it goes from z~ to z+ under the control v = u+ and in the second half period it goes from z+ to z~ under the control v = u~. That is, OK = (e-Atz~- I e-A(t-T}bu+dr : te [0,Tp]l I Jo ), A(tr)-Lj 0 U (IT . j. c \r\ rp . I t [U, -Lp

(3.16) Case 2. A e R 3x3 has eigenvalues a j(3 and a\, with a, /5, ai > 0. a) a = a\. Then similar to Case 1, c = Iv(t) =sgn a (/c + sin(/3t + ^)), t e R : fc e R, ^ e [0,2?r)} . Since sgn a (fc + sin(/3t + 9}} is the same for all k > 1 (or k < 1), we have, t e R : fc e [-1,

Each v G is periodic with period 2Tp, but the lengths of positive and negative parts vary with k. $(t,v) can be easily determined from simulation. A formula can also be derived for $ t , v .


. t 6 R:

i , 0 e [0,27r)It can be shown that+

~

where w + (or u ) denotes a constant control v(t) = u+ (or u ) and\

(3.17) When Q! < a, for each v ^3, f(i) u + (or u ~ ) for all t < 0, so the corresponding extremal trajectories stay at z^~ = A~1bu+ (or z~ = A~1bu~) before t = 0. And after some time, they go toward a periodic trajectory since as t goes to infinity, v(t] becomes periodic; When ai > a, for each v 8^. v(t) ~ u+ (or u~) for all t > 0, and the corresponding extremal trajectories start from near periodic and go toward 2+ or z~. Plotted in Figure 1 are some extremal trajectories on d'R, of the timereversed system (3.8) with

A=

0.5 0 0 0 0.8 -2 0 2 0.8

B=

- 1.

= -0.5.

For higher order systems, the relative locations of the eigenvalues are more diversified and the analysis will be technically much more involved. It can. however, be expected that in the general case, the number of parameters used to describe 8 is n 1.

3.4.

Domain of Attraction under Saturated Linear State Feedback

Consider the open loop system, x ( t ) = Ax(t) +bu(t), (3.18)

with admissible control u E Ua- A saturated linear state feedback is given by u = sat a (/x). where / R l x n is the feedback gain and sat a (-) is the


-0.5

-1

-1.5

Figure 1: Extremal trajectories on d'R,. ot\ < a. asymmetric saturation functionr,

r >M , r e [u~, r < *u~.

Such a feedback is said to be stabilizing if A + bf is asymptotically stable. With a saturated linear state feedback applied, the closed loop system is x(t) = Ax(t) + bs&ta(fx(t)). (3.19)

Denote the state transition map of (3.19) by 0 : (t, XQ) H- x(t). The domain of attraction S of the equilibrium x = 0 of (3.19) is defined by S : \XQ 6 Rn : lim $(t,xo) =I t>00

Obviously, S must lie within the null controllable region C of the system (3.18). Therefore, a design problem is to choose a state feedback gain so that 0,

b=

and / = [ 0 1 ]. Then f x = u and fx = u+ are two horizontal lines (see Figure 3). Since A + bf is Hurwitz, we have 61 < 01,62 < ^2 and that the trajectories go anticlockwise. Denote the region enclosed by F as SI. Since fi contains the origin in its interior, we can define a Minkowski functionalK(X) := min < 7 > 0 : x

(If J7 is symmetric and convex, K(X) is a norm). Clearly, K(X) = I for all x 0,

,and

V(ax) a2V(x),

dxSince Q* = K(X)follows that

dxexists and is continuous for all x R 2 . It

dV(x] dxn

=a0V(: dx

dx

(3.25)

and

(dV(x}\ . \ dx )

'

0, Vx e F.

(3.26)

We conclude that for all x 6 $7, along the trajectory of the system (3.19),fx}} 0

- -

(3 27)

'


This will be proved in the following. With the special form of A. b and /, the trajectory F goes anti-clockwise. Suppose that it starts at the righthand side intersection x(0) with f x u+ and intersects fx u+ and fx = u~ at x(ti}.x(t2) and #(3). We partition the curve F into four parts, I\ : #(3) > x(0); F2 : x(G) > x(i); F3 : x(ti) > x(t 2 ) and F4 : #(2) > #(3) (see Figure 3). Here we note that it might happen that F only intersects one of the two straight lines, s a y f x ,+ In this case, F] and FS are merged into one connected curve and we don't have F We will see that the following proof can be easily adapted for this case.

-0.5-

x(U

x(U

Figure 3: Illustration for the proof of Theorem 3.4. Let fij.i 1,2,3,4 be the region enclosed by Fj and the two straight lines that connect the origin and the two end points of Fj. We now consider V(x) for x in the interior of fi. In other words, we consider V(axr) for a G (0, 1) and xr G F. In the sequal, we use to denote for simplicity.


1) If xr e Ti U T3, then axr e fii U ft3. By (3.25), (3.26) and (3.27), V(axr) (dV(x

v

'

dx

.

rO! = 0.

2) If xr e F2 U F4, then axr ^2 U ^4 and we have V(axr) V ; a fdV(xr)\' (aAxr + bs^fx V dx J (OV(xr}\ ' (dV(xr}\ , f 1 AXr + 6sata a/x r N V dx ) V dx ) g(a,xr).X

dv x Since for a fixed :rr, (9VM\'A, r r , (1 ( ,,, ^\h*nd f^ r are all constants, ( ox ox I'D U 0 for all a e (0, 1) or g(a, xr) < 0 for all a e (0, 1). Denote

Axr + bfx^

k,(xr}=

Axr.

Since g^^ is continuous in x r , both k$xr} and k-2(xr} are continuous functions of x r . For a fixed x r , fci(xr) is the slope of g(a,xr) for small a when sat a (a/x r ) = &fxr, and k 0 for all a (0, 1); If k 0. It follows that

Similarly, on F4, there is a point xr such that xr = ,0 -c2 "

> 0 and

c2

In particular,-a\x-2 + b\u~Xr i ,

1~ \

F d-2I 0 I'

_l_ h


So we have x\ + a^x? bû" < 0. Note that 62 < 0 and u~ < 0. It follows that

k2(Xr) = [ o -ca ] r ~* ] > o.v L J

ix2

[ xi + azX2 J

These show that there exist one point on F2 and one point on F4 such that k2(xr) > 0. In summary of the the above analysis, we have V(x) < 0, for all x in the interior of ^2 and f^, and V(x) = 0 for all x fî U ^3. It follows that no trajectory starting from within f will approach F = oc. ti > Tp. To prove this claim, we recall some simple facts about a secondorder linear system with a pair of complex eigenvalues.v = Av.

(3.38)

For this system, suppose that v(0) / 0, then Lv(t] is monotonically increasing (or decreasing). Consider v(t$ e~Atlv(0). If the trajectory {e~Atv(Q) : t [0,i]} can be separated from the origin with a straight line, then ti < Tp. Now suppose 0 < t\ < Tp. If v(ti) and v(0) are aligned, Tp: If v ( t i ) and v(0) tend to be aligned, then t\ then we must have t\ will approach Tp. then the From z(0) to z(t\) z Az bu~~ . If we let v z part of F from z(0) to z(t\) is a trajectory of (3.38). From Lemma 3.1 we know that z~ does not belong to the half plane kfoz < u~ , so this part of trajectory is below z~ (the origin in the v coordinate). Hence we must have 0 < t\ < Tp. Since z(0) must be the right of fô, we have z\ (0) > 0 for sufficiently large k. It follows that is greater

than a constant. So ||f(i) is also greater than a constant. Note that as k > oo, t'2(0),t>2(i) > 0. Therefore, v(0) and v ( t i ] tend to be aligned, so we get linifcôo ti = Tp. Similarly, Iim fc ^ 00 (t 3 - t 2 ) = Tp. Now we havelim= limfc >oo

-AT

It follows that, as A1 > oo,

~Atland

0,

(3.39)

(3.40)

Recall from (3.36) that0,

the property in (3.40) implies that (3.41)


Let

It can be solved that0(6) + 0(6)-

From (3.39) and (3.41), we know that lirmôo 1 = hm/e-+oo2 = 0, so we havek >oo

lim

A~lb = z~ .

Similarly, we have linifcôo z(0) = z+. It follows from (3.36) thatlim zito) zkK3O

lim z ( t ' i ) z .fe>OO

D

Figure 5: The domains of attraction under different feedback gains. Example 3.1. Consider the open-loop system (3.1) with0.6 0.8

-0.8 0.6

2 4 I'


u~ = 0.5 and u+ = 1. Then we have/ o = [ 0.12

-0.66 ].

In Figure 5, the boundaries of the domains of attraction corresponding to different / = fc/ 0 , k= 0.50005,0.6,0.7,1,2, are plotted from the inner to the outer. It is clear from the figure that the domain of attraction becomes larger as k is increased. The outermost dashed closed curve is dC. 3.5.2. Higher Order Systems with Two Exponentially Unstable Poles

Consider the following open-loop system x ( t ) = Ax(t] + bu(t) =A

*

1 x(t) +

bl

u(t),

(3.42)

where x [ x( x'2 ] , x\ 6 R 2 ,2?2 R n , A\ G R 2 x 2 is anti-stable and A2 e R n is semistable. Assume that (A. b] is controllable. Denote the null controllable region of the subsystem

as Ci, then the null controllable region of (3.42) is C\ xR n . Given 71, 72 > 0, denoteand

When 71 1, ^1(71) = C\ and when 71 < 1, ^1(71) lies in the interior of C\. In this section, we will show that given any 71 < 1 and 72 > 0. a state feedback can be designed such that fh(7i) x ^2(72) is contained in the domain of attraction of the equilibrium x = 0 of the closed-loop system. For e > 0, let P(e) = [ ^1 n^l 1 e R( 2 + r i ) x ( 2 + n > be the unique L ^(t) ^siej J positive definite solution to the ARE A'P + PA - Pbb'P + e 2 / = 0. Clearly, as e [ 0, P(e) decreases. Hence lim ô P(t) exists. Let PI be the unique positive definite solution to the ARE (3.43)

A(Pl +PiAi -PI^^


Then by the continuity property of the solution of the Riccati equation [15],Pi 0 0 0

Let /(e) := b'P(e}. First, consider the domain of attraction of the equilibrium x 0 of the following closed-loop system x(t) = Ax(t) + b s a t a ( f ( c ) x ( t ) } . Let um = min{u~,u + }. It is easy to see that D(e) := {x E R 2+n : x'P(c)x < (3.44)

is contained in the domain of attraction of the equilibrium x = 0 of (3.44) and is an invariant set. Note that if XQ E D(e), then x(t) E D(e) and \f(e)x(t)\ < um for all t > 0. That is, x(t) will stay in the linear region of the closed-loop system, and in D ( e ) . Theorem 3.6. Let /o = b^Pi. For any 71 < 1 and 72 > 0, there exist k > 0.5 and e > 0 such that ^1(71) x 72 (72) is contained in the domain of attraction of the equilibrium x = 0 of the closed-loop system