Reset Control Systems

Advances in Industrial Control

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Alfonso Baños r Antonio Barreiro

Reset Control Systems

Alfonso Baños Fac. Informática Depto. Informática y Sistemas Grupo de Informática Industrial Universidad de Murcia Murcia 30071 Spain [email protected]

Antonio Barreiro Depto. Ingeniería de Sistemas y Automática ETSII Universidad de Vigo Campus de Lagoas-Marcosende Vigo 36310 Spain [email protected]

ISSN 1430-9491 Advances in Industrial Control ISBN 978-1-4471-2216-6 e-ISBN 978-1-4471-2250-0 DOI 10.1007/978-1-4471-2250-0 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2011940675 © Springer-Verlag London Limited 2012 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: VTeX UAB, Lithuania Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To Angeles and Alfonso (A. Baños) To Rosa, María, and Marta (A. Barreiro)

Series Editors’ Foreword

The series Advances in Industrial Control aims to report and encourage technology transfer in control engineering. The rapid development of control technology has an impact on all areas of the control discipline. New theory, new controllers, actuators, sensors, new industrial processes, computer methods, new applications, new philosophies all lead to new challenges. Much of this development work resides in industrial reports, feasibility study papers and the reports of advanced collaborative projects. The series offers an opportunity for researchers to present an extended exposition of such new work in all aspects of industrial control for wider and rapid dissemination. The name of Isaac Horowitz is usually associated with quantitative feedback theory (QFT), but his first-order reset element (FORE) is probably not so well known. This Horowitz FORE compensator was a generalisation of the seminal Clegg Integrator (1958) in which integral control was coupled with a reset mechanism to inculcate additional performance and design benefits. These two compensators are the original inspiration and source for the results and applications reported in this Advances in Industrial Control monograph, Reset Control Systems, by Alfonso Baños and Antonio Barreiro. In fact, the authors’ whole reset control research project seems to have been much motivated by original meetings in Spain with Professor Horowitz. The monograph is a comprehensive presentation of reset control systems, covering the fundamental theoretical foundations, design rules and applications experience. The monograph has four chapters on theoretical fundamentals. Opening these is an invaluable introductory chapter (Chap. 1) that guides the reader through the historical development of reset control. This particular chapter also sets out the theoretical context of reset systems and identifies their relationship with other related theoretical fields, for example impulsive and hybrid systems. Chapters. 2 to 4 contain the core theoretical results of the monograph and set out, possibly for the first time, a complete systems analysis framework for reset control. Design for reset control is treated in Chap. 5, where the focus is on identifying the type of systems for which reset control offers good design and performance benefits. Alongside this; sets of useful compensator tuning rules are presented for vii

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different compensator and system types. The chapter also investigates the properties of the authors’ own novel PI + CI reset compensator. In control engineering, the assessment of usefulness, effectiveness, design ease and performance gain is through real applications studies. In this respect, the case studies of Chap. 6 provide a trio of demanding “benchmark” applications. The authors expose all the steps of trying to accomplish successful reset control system designs. The applications are from process control (a pilot heat exchanger plant), control by tele-operation (a remote controlled process application that uses the Internet for communication) and utility process operations (a solar collector field control problem). This chapter allows both the practitioner and the theoretician to see how reset control works in practice, and the insights provided will undoubtedly inspire new developments for the reset control field. This invaluable and comprehensive study of reset control systems by Professors Baños and Barreiro joins several other recent Advances in Industrial Control monographs that give more fundamental presentations of new potential applicable topics. One such monograph is Process Control by Jie Bao and Peter L. Lee (ISBN 978-184628-892-0, 2007) that was received enthusiastically; another is Fractional-order Systems and Controls by Concepción A. Monje, YangQuan Chen, Blas M. Vinagre, Dingyü Xue and Vicente Feliu (ISBN 978-1-84996-334-3, 2010), and a third is, Internet-based Control Systems by Shuang-Hua Yang (ISBN 978-1-84996-358-9, 2011). The industrial control engineer, the control academic or control postgraduate researcher may find these three “fundamentals with applications” texts along with, Reset Control Systems essential volumes for inclusion in a complete control library. Industrial Control Centre, Glasgow, Scotland, UK

M.J. Grimble M.A. Johnson

Foreword

As part of the small community of control researchers, often referred to as the QFTers, I first met Alfonso Baños in 1997 at an international QFT conference in Glasgow, Scotland. We kept in touch throughout the years via emails, and in 2008 he arranged for his PhD student, Joaquín Carrasco, to spend a few months in my research lab. Joaquín’s thesis focused on reset control, and he wanted to learn about our activities in that area. Since then, I continued to follow their work and have come to fully appreciate their contributions to the area of reset control. My foray into reset control research was somewhat accidental. While working on challenging industrial robust control problems, we ‘stumbled’ upon a series of papers from the 1970s written by the late professor I. Horowitz, the founder of QFT, and his students. I recall that right away we noticed something unique about these papers. Horowitz presented nontraditional solutions to overcome the limitations of linear, time-invariant (LTI) control. Horowitz was already familiar with Cleggintegrator and similar concepts that were introduced a few decades earlier on purely empirical grounds. Horowitz’s results on Clegg-integrator and First-Order Reset Element (FORE) offered the practicing control engineer a systematic and transparent procedure for using such nonlinear elements in feedback control design. In our own work, we initially attempted to use these ideas in experiments, and motivated by successful results, we decided to investigate if one can explain such results theoretically. Orhan Beker, a PhD student of CV Hollot and I, was able to prove for the first time that reset control can overcome classic LTI performance limitations in a feedback system comprising a plant with an integrator. I believe that these encouraging results may have spurned other groups, including Baños’s and Barreiro’s, to probe deeper into this area which falls under the more general subjects of hybrid control systems and impulsive control. This book presents a compilation of numerous results derived by the authors and colleagues. The material is written in self-contained manner and offers the interested reader an exposition of current state-of-the-art results. The authors present stability theory in progression starting from reset-times independent results and reset-times dependent results, to the more general passivity-based results. A whole chapter is devoted to time-delay systems where LTI limitations are especially harsh and stabilix

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Foreword

ity conditions take the form of linear matrix inequalities. Numerical examples are sprinkled throughout the presentation so to bring life into the theory. In addition to the book’s theoretical focus, the authors appear to have made deliberate effort to remain true to the original motivation behind Horowitz’s original work to provide the control engineer with a practical technique for design of reset control systems in applications. They achieve this goal using describing function analysis as a design tool, extend the notion of reset to a partial reset compensator where only some of the compensator states are reset, and introduce a new reset compensator that consists of a PI compensator whose integrator is reset only at a fraction of Clegg integrator reset times. An entire chapter presents experimental applications of reset control systems in a temperature control of a heat exchanger, a bilateral teleoperation, and a temperature control of a solar collector field. In closing, I would like to thank the authors for asking me to write this forward. This well written book strikes a fine balance between sophisticated theory and engineering design and therefore should be accessible to most readers. In writing this book, the authors have succeeded in continuing Horowitz’s philosophy for feedback control research. Amherst, USA

Yossi Chait

Preface

Although the origin of reset control systems goes back to 1958 with the founding work of Clegg, the subject is still in its infancy. The seminal works of Horowitz and coworkers in the 1970s were the first attempts to build a synthesis theory for basic reset compensators such as the Clegg integrator or the FORE (first-order reset element), but the lack of a control theory, approaching basic questions such as wellposedness and stability, for several decades has put aside of the mainstream further developments in reset control, both in theory and applications. It is not until the late 1990s that reset control starts to develop with the works of Chait, Hollot and coworkers, giving a significative impetus to the field. In the meantime, the area of hybrid control systems and impulsive control also give numerous results, having clear connections with reset control. For the last decade, a number of international groups have been working actively on reset control, and fortunately, reset control has started to be seen as an attractive control design technique with a significant potential for practical control applications. The first direct contact of the authors with reset control was in September 2000, when Isaac Horowitz was visiting our group in Spain. We were actively working in nonlinear QFT at that time, that uses linear compensation, and in several fruitful discussions he gave us clear arguments for the many benefits of using nonlinear/reset compensation to attack the “nonlinear tiger” and also to overcome fundamental limitations of LTI systems. We started to seriously work on reset control in 2006, and since then our goal has been to develop a formal theory to cover basic theoretical aspects, and also to define simple compensator structures and tuning rules for them, with the focus on some practical problems, including applications in process control and teleoperation, and in general systems with time-delays. The first chapter of the book is an introduction to reset control systems, pursuing two objectives. The first objective is to give a quick and simple description of what a reset control is, and to provide basic explanations on why and when it is convenient to use this strategy. This objective is covered by the first two sections and is summarized in this key idea: a reset control is a simple nonlinear control technique very effective for linear plants subject to fundamental design limitations. The second objective of the chapter is to give a brief survey on the literature on analysis xi

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and design of reset control systems. The historical perspective begins with the early ideas on reset control, including the popular Clegg integrator and the first-order reset element (FORE) introduced by Horowitz and coworkers. In addition, the first series of rigorous results on analysis and design of reset controllers using a statespace description are given, including full reset and partial reset compensators. In addition, relationships between reset control and the wider field of impulsive and hybrid control systems will be analyzed from different points of view. In Chap. 2, a definition of a reset control system, or a reset system in general, is given. In general, as it is common in impulsive systems, reset systems may exhibit different types of solutions, in particular having complex patterns such as beating, deadlock, and Zenoness. In control practice, this type of behavior is considered pathological and thus several conditions will be given for reset control systems to be well-posed. On the other hand, important properties of reset systems may be derived by analyzing the reset instants that correspond to a given initial condition. These patterns will be also analyzed, and their relationship with the observability and reachability of the base linear system will be shown. Chapter 3 is devoted to the stability problem of reset control systems with finitedimensional base systems. The stability problem is addressed from different, complementary points of view: (i) internal or Lyapunov stability, (ii) external or input– output stability with passivity analysis, and (iii) stability by the describing function method. Internal stability techniques are subdivided into techniques giving rise to stability conditions that do not depend directly on the reset instants (reset-times independent), or alternatively, are reset-times dependent. The first case is obtained directly using continuous time Lyapunov functions (that gives rise to the so-called Hβ condition), while the second case (reset-times dependent) requires a discretization at the after-reset values and a subsequent discrete-time Lyapunov analysis. Then, the input–output L2 stability is studied, and a number of results are presented in connection with passivity and dissipativity properties of reset feedback loops. Finally, the standard describing function tool is used for approximately predicting the appearance or absence of oscillations. Stability of time-delay systems under reset control is approached in Chap. 4. Since reset control is able to overcome fundamental limitations, and time-delay is one source of such limitations, then it is of great interest to study the problem of delayed reset systems. The stability is addressed by choosing an appropriate Lyapunov–Krasovskii functional, and by imposing that the functional should decrease in the continuous and reset modes. The resulting conditions take the form of linear matrix inequalities, and, depending on the chosen functional, these LMIs can be delay-dependent or delay-independent. In both cases, those LMIs, derived from time-domain stability conditions, are translated into equivalent frequency-domain conditions by means of adequate tools, like the Kalman–Yakubovich–Popov lemma, or passivity techniques. From the latter frequency-domain conditions, useful interpretations are exhibited regarding the achieved robustness, in terms of scaled smallgain or positive realness of certain subsystems. Finally, several examples illustrate the application of the stability conditions, showing the potentials of reset control when applied to time-delay systems.

Preface

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In Chap. 5, reset compensation has been used to overcome limitations of LTI compensation. In this chapter, a new reset compensator, referred to as PI + CI, is introduced. It basically consists of adding a Clegg integrator to a PI compensator, with the goal of improving the closed loop response by using the nonlinear characteristic of this element. It turns out that by resetting a percentage of the integral term in a PI compensator, a significant improvement can be obtained over a well-tuned PI compensator in some relevant practical cases, such as systems with dominant lag and integrating systems. The main goal is the development of PI + CI tuning rules for basic dynamic systems in a wide range of applications, including first- and higher-order plus dead time systems. In addition, a number of design improvements such us the use of a fixed or variable reset band, the integration with QFT, and the use of a variable reset percentage are discussed. Finally, in Chap. 6, several practical applications of reset control systems will be developed, all based on the PI + CI compensator: a temperature control system of a heat exchanger, a bilateral teleoperation control system, and finally, a temperature control of a solar collector field. The first two applications have been tested by means of experiments in plants, while the third has been tested by using a (wellproven) simulator of the field. This book is a compendium of the several works developed by our groups in the last five years. These works have been performed in collaboration with Joaquín Carrasco, Angel Vidal, Alejandro Fernández, Juan Ignacio Mulero, Sebastián Dormido, José Carlos Moreno, Manuel Berenguel, and Arjan van der Schaft. In fact, they are responsible for many parts of this book. The authors also acknowledge the support of ‘Ministerio de Ciencia e Innovación’ (Spanish government) under the joint projects DPI2004-07670, DPI2007-66455, and DPI2010-20466. Thus, the book has been planned as a means to systematize and make available to a wider audience a number of publications that are disseminated over several journals and conference proceedings. It is intended for control researchers interested in a solid introduction to reset control, including the several approaches available in the literature, and also for control engineers interested in application of simple and efficient control techniques that may overcome fundamental limitations of the universally used PI/PID compensators. Murcia, Spain Vigo, Spain

Alfonso Baños Antonio Barreiro

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 What Is Reset Control? . . . . . . . . . . . . . . . . . . . . . . . 1.2 Why Reset Control? . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Overcoming Fundamental Limitations in the Time Domain 1.2.2 Overcoming Limitations in the Frequency Domain . . . . . 1.3 Early Ideas on Reset Control . . . . . . . . . . . . . . . . . . . . 1.3.1 Reset Control with the Clegg Integrator . . . . . . . . . . . 1.3.2 Reset Control with the First-Order-Reset-Element (FORE) 1.4 First General Approaches to Analysis of Reset Control Systems . . 1.4.1 General Setup and Asymptotic Stability . . . . . . . . . . . 1.5 Reset Systems as Impulsive Systems . . . . . . . . . . . . . . . . 1.5.1 Theory of Impulsive Differential Equations . . . . . . . . . 1.5.2 Impulsive Control Theory . . . . . . . . . . . . . . . . . . 1.5.3 Impulsive Dynamical Systems . . . . . . . . . . . . . . . . 1.6 Reset Systems as Hybrid Systems . . . . . . . . . . . . . . . . . . 1.7 Other Recent Results on Reset Control . . . . . . . . . . . . . . . 1.7.1 Design Based on L2 and H2 Performance . . . . . . . . . 1.7.2 Design Based on L2 Gain and Nonzero References . . . . 1.7.3 Design Based on Fixed Reset Instants tk . . . . . . . . . . 1.8 Preview of the Chapters . . . . . . . . . . . . . . . . . . . . . . .

1 1 8 9 11 20 20 23 25 25 27 29 31 34 37 43 44 47 48 50

2

Definition of Reset Control System and Basic Results . . 2.1 Preliminaries and Problem Setup . . . . . . . . . . . 2.1.1 Reset Control System Solutions . . . . . . . . 2.1.2 Characterization of Reset Intervals . . . . . . 2.2 Zenoness, Beating, and Deadlock . . . . . . . . . . . 2.2.1 Well-posedness: Beating and Deadlock . . . . 2.2.2 Zeno Solutions . . . . . . . . . . . . . . . . . 2.3 Reset Instants and the After-Reset Surface Dimension 2.3.1 dim(MR ) = 1 . . . . . . . . . . . . . . . . . 2.3.2 dim(MR ) = 2 . . . . . . . . . . . . . . . . .

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2.3.3 dim(MR ) ≥ 3 . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Reset Control Systems with Exogenous Inputs . . . . . . . . . . . 2.4.1 A Well-posed Reset Control System with Exogenous Input 2.4.2 A Reset Control System with Zeno Solutions . . . . . . . . . . . . . . . . . . . . . . . .

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Stability of Reset Control Systems . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.2 Lyapunov Stability . . . . . . . . . . . . . . . . . . . 3.2.1 Reset-Times Independent Conditions . . . . . 3.2.2 Reset Times-Dependent Stability Criteria . . . 3.2.3 Stabilization of Reset Control Systems . . . . 3.3 Reset Control Systems with Inputs/Passivity Analysis 3.3.1 Full Reset Compensators . . . . . . . . . . . 3.3.2 Partial Reset Compensators . . . . . . . . . . 3.3.3 L2 -stability of the Reset Control System . . . 3.3.4 Example . . . . . . . . . . . . . . . . . . . . 3.4 Describing Function Analysis . . . . . . . . . . . . . 3.4.1 FORE and Clegg Integrator . . . . . . . . . . 3.4.2 Reset Compensators with Reset Band . . . . . 3.4.3 Limit Cycle Analysis . . . . . . . . . . . . . 3.4.4 Justification of the Describing Function . . . .

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93 93 94 94 99 105 113 117 120 123 125 127 128 129 136 139

4

Stability of Time-Delay Reset Control Systems . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Problem Motivation and Statement . . . . . . . . . . . . 4.3 Delay-Independent Conditions in the Time Domain . . . 4.4 Delay-Independent Conditions in the Frequency Domain . 4.4.1 The Hβ -condition . . . . . . . . . . . . . . . . . 4.4.2 The Generalized Hβ -condition . . . . . . . . . . 4.4.3 Interpretation of the Stability Conditions . . . . . 4.5 Example: Delay-Independent Stability . . . . . . . . . . 4.6 Delay-Dependent Conditions in the Time Domain . . . . 4.7 Delay-Dependent Conditions in the Frequency Domain . 4.8 Example: Delay-Dependent Stability . . . . . . . . . . .

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147 147 149 153 155 156 158 159 162 164 168 175

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Design of Reset Control Systems . . . . . . . . . . . . . 5.1 The PI + CI Compensator . . . . . . . . . . . . . . . 5.1.1 PI + CI Tuning for First Order Plants . . . . . 5.1.2 First Order Plus Deadtime (FOPDT) Systems . 5.1.3 High Order Systems . . . . . . . . . . . . . . 5.1.4 Integrating Systems . . . . . . . . . . . . . . 5.1.5 Summary of Tuning Rules . . . . . . . . . . . 5.2 Design Improvements . . . . . . . . . . . . . . . . . 5.2.1 Fixed Reset Band . . . . . . . . . . . . . . . 5.2.2 Variable Reset Band/Advanced Reset . . . . . 5.2.3 Variable Reset Percentage . . . . . . . . . . . 5.2.4 Robust Control Design Based on QFT . . . .

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181 181 184 188 195 198 202 202 203 205 206 208

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Contents

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Application Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Temperature Control in a Heat Exchanger . . . . . . . . . . . 6.1.1 Process Model . . . . . . . . . . . . . . . . . . . . . 6.1.2 PI + CI Design . . . . . . . . . . . . . . . . . . . . . 6.2 Teleoperation of a Gantry Crane . . . . . . . . . . . . . . . . 6.2.1 Passive Teleoperation and System Overview . . . . . 6.2.2 Reset Procedure . . . . . . . . . . . . . . . . . . . . 6.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Control of Solar Collector Fields . . . . . . . . . . . . . . . 6.3.1 The Solar Collector Field . . . . . . . . . . . . . . . 6.3.2 PI + CI Design . . . . . . . . . . . . . . . . . . . . . 6.3.3 PI + CI Compensator with Variable Reset Percentage

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211 211 212 216 224 225 228 231 237 238 239 242

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Chapter 1

Introduction

1.1 What Is Reset Control? A reset controller, as the name suggests, is no more than a standard controller endowed with a reset mechanism, that is, a strategy that resets to zero the controller state (or part of it) when some condition holds. The reset condition (the event that triggers the reset action) is typically the zero crossing of the controller input, but other choices are possible as well. To explain it with a simple example, consider the control system in Fig. 1.1. The plant is a standard linear time invariant (LTI) system with a transfer function P (s). For the reset controller, represented customarily with a broken box, its underlying continuous mode of operation is given as well by a LTI transfer function C(s). But the broken box indicates that it is a reset controller, so a reset mechanism has to be specified. For the sake of concreteness, suppose that the base linear system C(s) is given by a first order transfer function K , (1.1) s +a and thus the base linear dynamics are described by u˙ = −au + Ke. Suppose that the reset condition is, as usual, the zero crossing of the input (e = 0) and that the reset action is the zero reset of the controller state. Thus, the equations that define this reset controller (a first-order-reset-element or FORE [31]) are:  u(t) ˙ = −au(t) + Ke(t) when e(t) = 0, (1.2) u(t + ) = 0 when e(t) = 0. C(s) =

The first equation describes the continuous dynamics or the flow mode. The second equation defines the discrete or impulsive dynamics, also called jump mode because at the time instants tk (k = 1, 2, . . .) when the error crosses zero (e(tk ) = 0) the controller state jumps from u(tk− ) to u(tk+ ) = 0. Now let us see what happens in this reset control system, and compare its behavior with the purely linear base system. Let us take the same values as in [18], K = a = 1 for the controller and P (s) = (s + 1)/(s(s + 0.2)) for the plant. The A. Baños, A. Barreiro, Reset Control Systems, Advances in Industrial Control, DOI 10.1007/978-1-4471-2250-0_1, © Springer-Verlag London Limited 2012

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Introduction

Fig. 1.1 Standard reset control system formed by a linear plant P (s) and a reset controller with base linear system C(s)

responses y(t) to a unit step reference r(t) applied at t = 1 are plotted in Fig. 1.2. This figure shows that the base linear system is very underdamped, with a large overshoot, greater than 70%. The reset system response (dotted line in Fig. 1.2) is much more damped, the main overshoot decreases to 40%, and the remaining oscillations disappear very quickly. Figure 1.3 plots the corresponding control signals, showing that the convergence to zero is much faster for the reset system than for the linear system. This figure illustrates more clearly the discontinuities or jumps of the reset system from u(tk− ) to u(tk+ ) = 0, where tk are the reset times. The reset times are the instants tk when e(tk ) = 0 or, looking at Fig. 1.2, when y(tk ) = r(tk ) = 1. This figure shows the improvement achieved with reset control: an acceptable, bounded overshoot, but with the same speed of response (and small rise time) than the linear system, much more underdamped. In this way, reset control

Fig. 1.2 Step response y(t) of the reset system (solid) and base linear system (dotted)

1.1 What Is Reset Control?

3

Fig. 1.3 Control action u(t) of the reset system (solid) and base linear system (dotted)

appears as a solution for overcoming the typical linear design trade-offs between bandwidth and robustness, or other competing objectives. We will discuss in the next section the subject of fundamental limitations and reset control. An intuitive explanation of the benefits of reset control can be understood looking at Fig. 1.2. The initial part of the step response of both the base linear system and the reset system is very fast, which is a good and desirable property. But after the first time the output reaches the reference, that is y(t1 ) = r(t1 ) = 1, and the behavior of the linear and reset systems is completely different. The linear system has a lot of “inertia”, the linear states are far from the steady state values y = 1, y˙ = 0, u = 0 and consequently a large overshoot appears. On the other hand, the reset controller performs a strong resetting action from u(t1− ) = 0 to u(t1+ ) = 0. This has the effect of “discharging” or “emptying” the controller state so that the step response produces a much smaller overshoot. A similar favorable effect appears at the second reset time t2 , and so on. The reset system reaches steady state after the fourth reset time t4 (with a very small error) but the linear system is still very oscillatory at t = t4 and its settling time is much worse. Although the previous example shows more advantages in favor of reset control, several skeptical doubts could arise: (i) It might be that there exists another linear controller, different from C(s) = 1/(s + 1), that performs equally well or better than

4

1

Introduction

Fig. 1.4 Basic reset integrator with reset condition input c(t) and after-reset value input a(t)

the reset controller, and (ii) It might be that, starting from a well-tuned linear control system, the inclusion of a reset provokes instability. Actually, these two grounds for concern are justified. Regarding (ii), one can find examples of stable linear control systems that become unstable if some reset is applied. This tells us that reset should be designed with great care, if we want to improve system performance and robustness. Regarding (i), it is true that there exist powerful design methods for linear controllers that easily find solutions C(s) that satisfy the given design objectives. In control practice, reset control may perform better, but it also may perform worse than a well designed linear compensator, and thus in general reset should be used carefully. The key point is that there exist some classes of linear plants P (s) that are very difficult to control with linear controllers and are subject to linearly unsolvable trade-offs between competing design objectives. These problematic classes of systems include plants containing integrators, right half-plane poles or zeros, or time delays. The corresponding unsolvable trade-offs are called linear fundamental limitations. These fundamental limitations are well known in the literature (see, e.g., [2], [27], [37]) and impose hard restrictions on the system performance. For example, if a system has an open-loop positive zero s = c > 0, then for any linear controller the closed-loop step response exhibits an undershoot Mu , and (see [27]) Mu ≥ (1 − δ)/(exp(cts ) − 1) where ts is the settling time into the band [1 − δ, 1 + δ]. So we cannot make ts and Mu (good speed and robustness) simultaneously small, and things get worse as c → 0. These limitations are derived assuming linear plants and controllers, thus in principle it might be possible that a nonlinear controller (like a reset controller) overcomes the limitation and outperforms any linear solution. This is the main point that justifies the use of reset control. In the following section, the relations between fundamental limitations and reset control will be presented in more detail. Before that, let us look at some other possible reset controllers in order to gain insight into certain dynamical aspects of reset systems. This presentation will be intuitive and informal; the formal treatment of well-posedness, analysis and design will be addressed later in this book. The basic block for simulation or block diagram realization of reset systems is presented in Fig. 1.4. In the same way as standard integrators are basic dynamic blocks for realization of finite-dimensional systems, the reset systems require two extra inputs apart from the usual input e(t), the signal that is being integrated. As Fig. 1.4 shows, there is a second input c(t), the reset condition, that has the effect that the zero crossings of a signal c(t), that is, c(tk ) = 0, define the reset instants t = tk (k = 1, 2, . . .) and trigger the reset action. The third input a(t) defines the after-reset value to be applied at the output: u(tk+ ) = a(tk ).

1.1 What Is Reset Control?

5

Fig. 1.5 Simulation of a reset integrator: e(t) is the standard input, c(t) gives the reset condition, and a(t) the after-reset value for the output u(t)

This is a basic block that can be built, for example, in MATLAB-SIMULINK, or can be programmed easily in any other environment. Formally, the equations of this basic reset integrator are: ⎧ ˙ = e(t) when c(t) = 0, ⎨ u(t) + (1.3) u(t ) = a(t) when c(t) = 0, ⎩ u(0) = u0 . If the input signals are sufficiently regular (forget for the moment possible difficulties arising from discontinuities or pathologies in them) then the simulation of this reset integrator is very easy. To illustrate the role of each signal, Fig. 1.5 plots a simulation example: the output u(t) integrates the main input (a slow sinusoidal), but resets the output value at the zero crossings of c(t), to the after-reset values given at those moments by a(t). The initial condition is u(0) = 0. Normally, the after-reset value a(t) is 0. Only in special cases is this third input required, for example, in partial reset systems, that is, when u(tk+ ) = p · u(tk− ) for some number p, called the percentage of reset. The realization of this strategy requires the use of the third input, and some easy arrangement for circumventing the algebraic loop a = pu. When the after-reset input is omitted, the default value a = 0 is assumed and the reset becomes zero reset. Furthermore, if we identify the two in-

6

1

Introduction

Fig. 1.6 Clegg integrator

puts c(t) = e(t), that is, the reset instants appear at the zero crossings of the same signal that is integrated, then we reach the well-known reset integrator or Clegg integrator [22]. Figure 1.6 shows the block representing the Clegg integrator. The Clegg integrator is simply given by  u(t) ˙ = e(t) when e(t) = 0, (1.4) u(t + ) = 0 when e(t) = 0 with u(0) = u0 . Figure 1.7 shows the response of the integrator to a sinusoidal input, and with u0 = 0. Using Clegg integrators, every finite-dimensional reset system can be built in the form of a block diagram realization based on this primitive block. For example, the FORE system in (1.2) can be represented by the block diagram in Fig. 1.8.

Fig. 1.7 Simulation of a Clegg integrator for a sinusoidal input

1.1 What Is Reset Control?

7

Fig. 1.8 Block diagram of the FORE (first-order-reset-element) with base linear system K/(s + a)

Clegg integrators and FOREs are very simple, low-order, particular cases of reset controllers. A general reset controller with linear base dynamics can be described by the set of equations: ⎧ x˙ (t) = Ax(t) + Be(t) when e(t) = 0, ⎪ ⎪ ⎨ x(t + ) = A x(t) when e(t) = 0, ρ (1.5) ⎪ u(t) = Cx(t) + De(t), ⎪ ⎩ x(0) = x0 where e(t), u(t) are the input and output signals, and x(t) ∈ Rn is the state vector. The base linear system of (1.5) is given by the transfer function C(s) = C(sI − A)−1 B + D, defined by the first and third equation, describing the flow mode and holding almost all the time, when e(t) = 0. The impulsive dynamics, or the jump mode, is triggered by e(t) = 0. If some of the states are affected by reset and some are not, we can, without loss of generality, reorder states, for example, non-reset states first and reset states last. In this way, the reset matrix Aρ takes the form Aρ = diag(1, . . . , 1, 0, . . . , 0)    

n1

(1.6)

n2

with n1 + n2 = n. If n2 = 0, we have a linear controller. If 0 < n2 < n, we have a so-called partial reset controller and only part of the states are reset states. If n2 = n, the controller is called full reset and x(tk+ ) = 0 ∈ Rn . The FORE in Fig. 1.8 is a particular case of a full reset system with scalar parameters (A, B, C, D, Aρ ) = (−a, K, 1, 0, 0). In summary, to conclude this section, a reset controller, or a reset system, is a dynamical system (typically linear and finite-dimensional) endowed with a reset mechanism. The reset mechanism has to be carefully designed by answering two questions: • When to apply reset?—Reset condition This amounts to defining the reset condition input c(t) in Fig. 1.4 so that c(tk ) = 0 defines the reset instants tk (k = 1, 2, . . .). Usually, it is made equal to the tracking error of the control system c(t) = e(t), but other modifications are possible (combinations of e(t) and e(t), ˙ deadbands or thresholds for e(t), etc.). • How to apply reset?—Reset action This amounts to defining the after-reset value signal a(t) in Fig. 1.4 so that u(tk+ ) = a(t). Normally, reset is applied as zero reset, a(t) = 0. Sometimes it is useful to perform only a percentage of reset, a(t) = pu(t) so that u(tk+ ) = p(tk )u(tk ) with 0 < p(t) < 1. The designer has to choose judiciously the reset conditions and reset actions in order to improve the system performance, and also in order to ensure wellposedness. The well-posedness could be lost through several causes: algebraic

8

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Introduction

loops, accumulation of reset instants tk → t∞ < ∞, etc., giving rise to different pathologies (deadlock, livelock, Zenoness). There are simple recipes (called regularizations) to eliminate these anomalies. These well-posedness aspects will be discussed in the following chapter. Now that reset systems are introduced, let us see in the following section the main reasons why these type of systems are useful in control.

1.2 Why Reset Control? The main reason for using reset controllers is that, just by including the mechanism of resetting, they are able to overcome fundamental limitations in linear systems. This idea was present from the very beginning: Clegg’s integrator [22] was introduced as a solution for improving feedback performance, due to its ability to provide the same modulus attenuation as a linear integrator (−20 dB/dec) but with a phase (≈ −38°) much more favorable in terms of phase margins and robustness. This ability for overcoming linear limitations was also implicit in the subsequent proposals of reset control: from the first-order-reset-element (FORE) by Horowitz and coworkers (see [31], [32]) to the many recent generalizations of reset controllers. In spite of being the key reason for using this class of controllers, the literature that specifically studies the overcoming of design limitations by reset controllers is very scarce: only a few works show explicitly certain linear limitation and how the reset controller outperforms it. On the other hand, the topic of linear fundamental limitations has been widely treated in the literature, and there is a large number of references that quantify linear limitations in the time and frequency domains; see, for example, [2], [37], [27]. It is out of the scope of this book to give a detailed presentation on the topic of fundamental limitations, the interested reader can find more information in the aforementioned references. The purpose of this section is to give a brief glimpse of the problem, by presenting a representative choice of some time domain and frequency domain linear trade-offs, and then to motivate the use of reset controllers by showing how they can solve the trade-offs and outperform the linear solutions. A complete study, quantitative and systematic, on how reset control outperforms linear limitations deserves further research. We provide here only a presentation based on examples, to give the reader the main idea and to justify the interest on reset control. Before proceeding with the section, two observations have to be made. First, notice that, to overcome linear limitations, other hybrid approaches apart from reset are possible [25]. We only claim here that reset control is an efficient and simple choice among other hybrid control strategies. Second, as concluded in [17], the fact that certain reset controllers overcome some linear limitations could introduce as a drawback a new different set of limitations, specific to reset control. This is true if we consider standard reset controllers (that reset to zero when tracking error is zero). But if we consider general reset controllers (that can be implemented with the reset integrator in Fig. 1.4) the freedom in the choice of the reset condition and reset action enlarges the design possibilities. Actually, the set of all ‘generic’ reset

1.2 Why Reset Control?

9

Fig. 1.9 Standard linear control system

controllers contains as particular case the subset of linear ones (obtained putting, for example, c(t) = 1 ∀t, so that reset condition is never met). In this way, reset controllers form a superset, or extension, of linear ones, so the set of achievable design objectives is obviously larger. A recent rigorous study on performance improvement by means of reset can be found in [1].

1.2.1 Overcoming Fundamental Limitations in the Time Domain The linear fundamental limitations of control systems are classified into time domain and frequency domain limitations. The systems that are ‘hard to control’ (having right half-plane poles or zeros, or time delays) present simultaneously both types of limitations, but it is customary to address the limitations separately, studying the restrictions on the achievable time responses or frequency responses. Let us start with time domain. First, consider the standard (one degree of freedom) feedback control system in Fig. 1.9, with the sensitivity S(s) and complementary sensitivity T (s) closed-loop transfer functions given by S(s) =

1 , 1 + C(s)P (s)

T (s) =

C(s)P (s) , 1 + C(s)P (s)

(1.7)

and satisfying S(s) + T (s) = 1. It follows that −S(s) is the closed-loop transfer function from noise n(t) to tracking error e(t), and T (s) is the closed-loop transfer function from reference r(t) to plant output y(t). Leaving aside the restrictions arising from actual limitations in the real physical components (for example, precision and range of sensors or actuators), there are structural limitations affecting the closed-loop in Fig. 1.9, appearing even in ideal or nominal conditions. A detailed exposition of linear limitations can be found in [27], [37]. We present here a selection of results taken from [27, Sect. 8.6]. These restrictions arise from interpolation constraints, that is, from transmission of poles or zeros from the open-loop transfer functions P (s) or C(s) to the closedloop ones. These roots will be called uncanceled poles or uncanceled zeros. It can be seen that certain types of uncanceled poles or zeros transmitted to closed-loop impose severe restriction on certain closed-loop time responses. Notice that the attempt to cancel a plant pole or zero (with a controller zero or pole, respectively), when lying in the right complex half-plane, is impossible with perfect accuracy, and hence totally unacceptable as it leads to instability in practice. On the other hand, although such approximate cancellation in the left half-plane

10

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Introduction

does not produce instability, in [35] it is shown that there is a trade-off between input disturbance rejection (better when not canceling but shifting the slow undesired poles) and robustness (which is better when canceling and not shifting the poles). According to [27] (Sect. 8.6), we have the following interpolation constraints in the sensitivity functions: (i) The complementary sensitivity T (s) has a zero at all uncanceled zeros of P (s). (ii) The sensitivity S(s) has a zero at all uncanceled poles of P (s). These two basic properties can be particularized to open-loop integrators, openloop general poles or zeros, or imaginary-axis poles and zeros. For concreteness, let us see what happens with open-loop general poles and zeros, reproducing part of Lemma 8.3 in [27]: Proposition 1.1 Consider the feedback system in Fig. 1.9, with all the closed-loop poles to the left of −α < 0. Assume that the controller has at least one pole at the origin. For an uncanceled plant zero z0 or plant pole η0 satisfying Re(z0 ) > −α or Re(η0 ) > −α, and for a unit reference step, we have the following: ∞ (i) 0 e(t)e−z0 t dt = z10 ; ∞ (ii) 0 e(t)e−η0 t dt = 0; ∞ (iii) 0 y(t)e−z0 t dt = 0. The following observations are consequences of the previous constraints [27]. First, if −α < z0 < 0, that is, z0 is an uncanceled minimum phase zero, then the error must change sign because the initial value of the integral (i) is positive (since e(0) = 1) and the total integral (from 0 to ∞) is negative. In other words, we have the following linear limitation: L1 Any uncanceled minimum phase zero z0 < 0, to the right of the closed-loop poles, implies overshoot in the step response. A similar discussion in the case z0 > 0 leads to the following limitation: L2 Any uncanceled non-minimum phase zero gives rise to undershoot in the step response and, if z0 > 0 is small, to a positive large value of the error integral (i). Finally, from (ii): L3 Any uncanceled pole η0 , to the right of the closed-loop poles, must produce overshoot in the step response. Using the previous limitations,  ∞introducing  t  ∞a suitably defined settling time ts , partitioning the integral in (i) as 0 = 0s + ts , and manipulating bounds, more quantitative results are reached (see Lemmas 4.2 and 4.3 in [27] or Proposition 3 in [35]) showing that there are restrictions affecting the overshoot Mp (or undershoot Mu ) and the settling time ts . In other words, if we make good ts (small ts and fast response) then the overshoot Mp (or undershoot Mu ) becomes bad, and vice versa. It can be shown that reset control systems provide a very simple mechanism for overcoming the previous linear limitations. The results that follow are based

1.2 Why Reset Control?

11

on [17], where a simple example is presented, showing that reset control does not suffer from the time-domain limitations of linear controllers. So, consider as in [17] a linear feedback system of Fig. 1.9 with a plant: 1 P (s) = s and a controller C(s) that stabilizes the closed loop. It is known that the tracking error e(t) to a unit step reference r(t) = 1 ∀t ≥ 0 satisfies: ∞ 1 e(t) dt = , K v 0 where the velocity gain is given by Kv = lims→0 sC(s)P (s). Now, consider the definition of rise time tr given by

 t tr = sup τ : y(t) ≤ ∀t ∈ [0, τ ] . τ τ Then [17] proves that 2 ⇒ the unit step response y(t) overshoots. (1.8) tr > Kv From this result, [17] derives a set of design objectives impossible to meet with linear controllers: Design problem: Given the plant P (s) = 1/s, obtain a controller that stabilizes the closed loop and has (i) steady-state error to unit ramp no greater than 1, (ii) rise time of y(t), in response to unit step, greater than 2, and (iii) no overshoot in the step response of y(t). From (1.8) it is clear that no stabilizing linear controller C(s) exists that fulfills all these specifications [17]. At the same time, [17] presents a very simple reset controller (FORE) that achieves these specifications:  u(t) ˙ = −u(t) + e(t) when e(t) = 0, (1.9) when e(t) = 0. u(t + ) = 0 Figure 1.10 plots the unit step response of the reset control system (top), showing how rise-time and overshoot specifications are met (ramp steady state error is met as well). For comparison, the bottom plot in Fig. 1.10 shows the response of the base linear control system, with C(s) = 1/(s + 1), that, according to (1.8), must overshoot and is subject to this linear limitation.

1.2.2 Overcoming Limitations in the Frequency Domain The previous section has briefly presented some fundamental limitations in the time domain. They imply constraints in the achievable time domain specifications (restrictions on overshoot, rise time and settling time, etc.) that cannot be solved with linear controllers.

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Introduction

Fig. 1.10 Example in [17]. Unit step responses: (top) reset control and (bottom) linear control

Likewise, this section introduces their frequency domain counterparts, and shows how reset controllers can overcome these limitations. The presentation is based on [2] and [3]. The reader can also consult [27] and, for a more detailed treatment, [37]. The frequency domain limitations, for the typical control system in Fig. 1.9, arise from the structure of the sensitivities S(s) and T (s) (1.7) that impose certain restrictions on the modulus or phase of these complex-valued functions. Some restrictions are consequences of the Bode’s integral formulas, and some others are related to the Bode’s gain–phase relation. Let us see first the Bode’s integral formulas and their effect on performance limitations in the frequency domain [3]. Proposition 1.2 Consider the feedback system in Fig. 1.9. Assume that the loop transfer function L(s) = C(s)P (s) satisfies lim sL(s) = 0. s→0

1 Then the sensitivity function S(s) = 1+L(s) satisfies the integral identity ∞  log |S(iω)| dω = π pk , 0

where the summation is over all right half-plane poles of L(s).

(1.10)

1.2 Why Reset Control?

13

Equation (1.10) implies, following [3], that the sensitivity function S(iω) has to distribute is attenuation over different frequencies. Recall that |S(iω)| is the closedloop frequency attenuation from noise n(t) to error e(t) (or from disturbance d(t) to plant input) in Fig. 1.9. So if S(iω) is made smaller for some frequencies, then necessarily is has to become larger at other frequencies, in order that the total integral in (1.10) remains constant. This means [3] that if disturbance attenuation is improved in one frequency range, then it will be worse at another. This property is known as the waterbed effect. The total integral in (1.10) is greater (worse attenuation ability) as the number of right half-plane poles pk increases, or as their values pk increase, so that fast unstable poles are worse than slow ones, as intuitively predicted. In the best case, when L(s) has no poles in the right half-plane, we have:

∞

log |S(iω)| dω = 0.

(1.11)

0

The Bode’s integral limitation can be given a nice geometrical interpretation [3] as shown in Fig. 1.11 for the case L(s) = 10/((s + 2)(s − 1)). The figure plots log |S(iω)| versus ω in linear scale. The horizontal line log |S(iω)| = 0 is also plotted as a reference. Since L(s) has a right half-plane pole at p = 1, then the total  integral in (1.10) is equal to π pk = π . Thus, the area of the curve above the horizontal line log |S(iω)| = 0 (positive contribution to the integral) minus the area  below the horizontal line (negative contribution) must be equal to π pk = π . This means that if we want to make |S(iω)| < 1 at lower frequencies (which is typically desired for disturbance attenuation) then the appearance of disturbance amplification |S(iω)| > 1 is unavoidable at higher frequencies in order to balance the integral constraint (1.10) (waterbed effect). Even in the best case (1.11) with no poles of L(s) in the right half-plane, the negative value of the integral at low frequencies has to be compensated by a positive value at high frequencies. If we consider the complementary sensitivity function T (s) = L(s)/(1 + L(s)), there exists as well a similar Bode’s integral formula [3]: 0

∞

1 log |T (iω)| dω = π , zi ω2

(1.12)

where the summation is taken over all the right half-plane zeros of L(s). Now, an alternative to the Bode’s integral constraints, in order to make evident the fundamental limitations of linear design in the frequency domain, is to consider a well-known restriction affecting minimum phase systems: the Bode’s gain–phase relation. In what follows, an exposition on fundamental limitations derived from the Bode’s gain–phase relation is presented based on the results in [2] (see also [3]). Subsequently, some examples will be given, showing that reset control systems are able to overcome these limitations and outperform linear controllers.

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Introduction

Fig. 1.11 Waterbed effect: plot of log |S(iω)| versus ω for L(s) = 10/((s + 2)(s − 1))

Proposition 1.3 (Bode’s gain–phase relation) Assume that the transfer function G(s) is minimum phase, that is, it does not have poles and zeros in the right halfplane. Then the phase is uniquely given by the gain curve by means of the relation π ∞ d log |G(iω)| f (ω, ω0 ) arg G(iω0 ) = d log ω, (1.13) 2 0 d log ω where the weighting kernel is f (ω, ω0 ) =

   ω + ω0  2 .  log  ω − ω0  π2

Notice that the kernel weights the frequencies close to ω, for if ω → ω0 then f (ω, ω0 ) → ∞, and if ω → ±∞ then f (ω, ω0 ) → 0. From this interpretation of the weight, it can be seen that the relation (1.13) gives rise to the approximate Bode’s gain–phase relation, for minimum phase systems: arg G(iω0 ) ≈

π d log |G(iω)| . 2 d log ω

(1.14)

d log |G(iω)| d log ω

(1.15)

If we define n(ω) :=

1.2 Why Reset Control?

15

as the slope of the gain curve, in logarithmic scale, then the approximate Bode’s gain–phase relation, for minimum phase systems, says: π (1.16) arg G(iω0 ) ≈ n(ω). 2 In other words, if G(iω) at a given point ω has −20 dB/dec (respectively, −30, −40, . . .dB/dec), then we can predict an approximate phase of −π/2 rad (respectively, −3π/4, −π, . . . rad), a property that is well-known when sketching Bode plots by hand.

1.2.2.1 Limitations Derived from the Approximate Gain–Phase Relation In this subsection, we present a brief exposition, based on [2] and [3], on frequency domain limitations specifically derived from the approximate relation (1.16). After that, it will be shown, following [24], that reset control systems are able to overcome these limitations. If a plant P (s) has poles or zeros in the right half-plane, or time-delays, it is called a non-minimum phase system and it is subject to some fundamental limitations. Following [2], let us factor the loop transfer function L(s) = C(s)P (s) of one such system as L(s) = Lm (s)La (s), where Lm (s) is the minimum phase part and La (s) the non-minimum phase part. We normalize this factorization so that |La (iω)| = 1 (all-pass). If the non-minimum phase factor comes from a pure time delay h, we can identify La (s) = e−sh . If the plant has one non-minimum phase zero, L(s) = L1 (s)(z − s) with z > 0, then we can factor L(s) = Lm (s)La (s) with Lm (s) = L1 (s)(z + s) and the all-pass factor z−s . z+s Likewise, for an open-loop pole in the right half-plane s = p > 0, the all-pass factor is La (s) = (s + p)/(s − p). If the open-loop has several poles or zeros in the right half-plane, or time delays, then the all-pass factor La (s) contains several all-pass factors as the previously presented ones. Following [2], we characterize the bandwidth by the gain crossover frequency ωc as given by La (s) =

|L(iωc )| = |Lm (iωc )| = 1 (since La is all-pass). Let us introduce here n(ω) =

d log |L(iω)| d log |Lm (iω)| = log ω log ω

and let nc = n(ωc ) be the ‘crossover slope’. Two measures of robust stability are given by the guaranteed phase margin φm and by the crossover slope nc . The negative feedback loop formed around L(s) is more robustly stable as φm is larger (60° better than 45°) and nc is more negative (−20 dB/dec better than −10 dB/dec). Two relations hold:

16

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Introduction

arg L(iωc ) ≥ −π + φm , (1.17) π (1.18) arg Lm (iω) ≈ n(ω) . 2 The first inequality ensures the guaranteed phase margin φm , and the second one is the Bode’s relation. If we use arg L = arg Lm + arg La and Bode’s relation (with an equality = in place of ≈), we reach π (1.19) − arg La (iωc ) ≤ π − φm + arg Lm (iωc ) = π − φm + nc . 2 This last relation, π − arg La (iωc ) ≤ π − φm + nc =: c(φm , nc ), (1.20) 2 is called the design inequality in [2]. If we particularize La (iωc ), we typically obtain restrictions on the crossover frequency ωc that limit in some way the achievable bandwidth of the linear control system. For example, if the plant has a non-minimum phase zero and the all-pass factor is La (s) = (z − s)/(z + s) then, from arg La (iωc ) = −2 arctan(ωc /z), we reach     c(φm , nc ) π − φm + nc π/2 = z tan . (1.21) ωc ≤ z tan 2 2 Notice that the limitation imposed by the coefficient c(·,·) becomes more restrictive as the robustness requirements (φm , nc ) are stronger. For example, if (φm , nc ) = (45°, −10 dB/dec), then c = π/2 ≈ 1.57, but if (φm , nc ) = (60°, −20 dB/dec) then c = π/6 ≈ 0.52. The restriction (1.21) also becomes stronger as the zero approaches the origin (slow non-minimum phase zeros are bad) [2]. Similar limitations could be derived from unstable open-loop poles or from time delays, see [2]. Therefore, the fact that a plant contains right half-plane poles or zeros, or time delays, makes it very difficult to control these types of systems, at least if linear controllers are used.

1.2.2.2 Overcoming Frequency Domain Limitations with Reset How can a reset control contribute to solving this problem? In this subsection, based on [24] and [42], we present briefly some examples of reset systems that are not subject to the limitations for linear controllers. Checking the previous steps of the inequalities (1.19) and (1.20), it follows that the limitation could be alleviated if we could replace arg Lm (iωc ) by a quantity less negative (larger) than nc π/2. Unfortunately, no linear system is able to solve this, minimum phase linear systems Lm (s) are restricted to the approximate Bode relation (1.16). So the only way to achieve a significative reduction of the limitation is by using some new type of nonlinear system Lr (the reset controller). It will be useful to define an approximate frequency response Lr (ω) of Lr (describing function analysis could provide this approximation of the frequency response, a detailed analysis of reset control system with the describing function will be given in Chap. 3).

1.2 Why Reset Control?

17

The reset system Lr can be used in place of Lm , or combined in series with it Lr (ω)Lm (iω). Since the angles and decibel slopes nr , n of the systems Lr (ω), Lm (iω) behave additively, it follows that the Bode’s relation (1.19) is drastically relaxed (and the limitation outperformed) if π arg Lr (ω)  nr (ω) , 2

(1.22)

at least at the frequency ω = ωc . Fortunately, it is well documented that a variety of reset controllers (FORE, Clegg, etc.) satisfy the previous relation. For example, the Clegg integrator (1.4) LCI (an integrator with reset) has an approximate describing function LCI (ω) =

1.62 −i38° , e ω

(1.23)

that is, arg LCI (iω) = −38° and nCI (ω) = −20 dB/dec, which means a valuable improvement of about 52° in phase lag, compared to the linear integrator, with phase lag −90°. Consider a very simple example: Let us compare two negative feedback loops, the first one containing a standard integrator Lm (s) = 1/s and a pure delay La (s) = e−sh ; and the second one with a Clegg integrator Lr = LCI and a delay La (s) = e−sh as well. In both cases, the modulus slope has to be nc = −20 dB/dec. Let us particularize (1.19), with φm = π/3 = 60°, having in mind that arg La (iω) = arg e−iωh = −ωh. Thus we reach π π − ≈ 0.52 with linear integrator, 3 2 π π ≈ 1.43 with Clegg integrator. ωc h ≤ π − − 38° 3 180° ωc h ≤ π −

(1.24) (1.25)

Thus, for the same phase margin and delay, the reset system outperforms the linear one. The bandwidth limitation ωc h ≤ 0.52 is improved to ωc h ≤ 1.43 by a factor 1.43/0.52 = 2.75 (almost three times larger) provided by reset control. Going further with this idea, one can think if it is possible to improve the Clegg integrator, providing also −20 dB/dec but with a phase lag still less negative than −38°. One simple idea for achieving this is replacing the reset condition e(t) = 0 by a new condition u(t) = 0 where u(t) is obtained passing e(t) through some block Fc (s) providing phase advance. Thus, consider now the Clegg Integrator with advanced reset [24] (or with variable reset band [42]): x˙ (t) = e, +

x = 0,

u(t) = 0, u(t) = 0,

(1.26)

u(t) = L −1 [Fc (s)E(s)] , where L , L −1 are the direct and inverse Laplace transforms, and Fc (s) is an anticipative block, for example, Fc (s) = Kp + Kd s, which introduces a phase lead or phase advance on e(t) (if Fc (s) ≡ 1 then u = e, and we recover the standard Clegg

18

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Introduction

Fig. 1.12 Pure time-delay plant P (s) stabilized by: Advanced Clegg integrator (top), Clegg integrator (center), and standard integrator (bottom)

Integrator). The describing function of an Advanced Clegg Integrator with a phase advance of φ = φ(ω) = arg Fc (iω) is:    1 4 cos2 φ 4 cos φ sin φ −i 1− . (1.27) LACI (ω) = ω π π The particular case when φ = 0 stays constant recovers the standard Clegg integrator (1.23). If φ = 0 is held approximately constant over a frequency range then |LACI | = k/ω, and we obtain −20 dB/dec as the standard integrator. The key feature is the behavior of the phase, the plot of arg LACI versus φ reveals that the less negative phase is obtained for φ ≈ +30° giving arg LACI ≈ −25°. Thus, by designing the phase lead block Fc (s) so that it provides φ ≈ +30° along the frequency range of interest (frequency where gain margin is defined), we will obtain LACI ≈ ω1 e−i25° , which is the same attenuation as that of an integrator but with 90° − 25° = 65° of phase improvement, a very important advantage, regarding fundamental limitations. Example 1.1 To conclude this section, let us translate the ideas above into a simple but quantitative design problem. Let us consider as shown in Fig. 1.12 a one-second time-delay plant P (s) = e−s and three possible controllers with adjustable gain: an integrator, a Clegg integrator, and an advanced Clegg integrator. These are not examples of real reset control systems that work well in practice because among other reasons the reset action destroys the steady state property for tracking external references. Other real examples of useful practical application of reset will be presented later in the book. So, consider that we are only interested in the stability of the zero solution, and how restrictive the trade-off between speed (ωc ) and robustness (φm ) is in each of the three cases. If the design objective is chosen to be a gain crossover frequency ωc of, say, 4π/9 rad/s = 80°/s, then it is easy to see that the corresponding gains for the integrator and Clegg are respectively k = kI = 4π/9, k = kCI = (4π/9)/1.62, and the resulting phase margins φm are respectively 10° and 62°.

1.2 Why Reset Control?

19

Fig. 1.13 Bode plots of the delay and three integrators example

On the other hand, to take full advantage of the advanced Clegg, one has to choose an adequate lead filter Fc (s), for example, a PD: Fc (s) = Kp + Kd s. As Fc (s) defines the signal so that its zero crossings induce resetting, Fc (s) can be normalized as Fc (s) = 1 + sT . The phase advance is φ(ω) = arg(1 + j ωT ). As the maximum advance is at φ = 30° = π/6, and we would like to exploit this advance at ωc (for maximizing the phase margin), the best choice is φ(ωc ) = 30° which is √ √ solved by ωc T = 1/ 3 implying T = 9/(4 3π). Having fixed the lead compensator Fc (s), it is easy to obtain the gain kACI = 1.32 achieving ωc = 4π/9 crossover frequency. Figure 1.13 plots the resulting open-loop responses of the solutions with a standard integrator, a Clegg, and an advanced Clegg. All the three achieve the same crossover frequency ωc = 4π/9 = 1.4 rad/s, but the corresponding margins are respectively 10°, 62° and 75°, showing the advantages provided by (advanced) resetting for overcoming fundamental trade-offs. The corresponding time plots, with a zero reference and x(0) = 1, are shown in Fig. 1.14.

20

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Introduction

Fig. 1.14 Time plots of the delay and three integrators example

1.3 Early Ideas on Reset Control In this section, we present a brief glimpse of the early ideas on reset control. It is based on [21] that reviews the contributions to reset control of Prof. I. Horowitz, and it is based also on [32] and [31] by Horowitz and coworkers. It is recognized [21] that reset controllers began with the Clegg integrator (Clegg, 1959) [22]; however, it was not until the work of Horowitz that a quantitative design procedure was developed, first around the Clegg integrator in Krishnan and Horowitz (1974) [32], and then around the first-order-reset-element (FORE) in Horowitz and Rosenbaum (1975) [31]. Since the final design procedures in [32] and [31] are strongly based on Quantitative Feedback Theory (QFT), which is beyond the scope of this book, this section is focused on the ideas by Horowitz and coworkers regarding the Clegg and FORE controllers: its behavior, interpretation, analysis, and features.

1.3.1 Reset Control with the Clegg Integrator In [32], Krishnan and Horowitz proposed for the first time a complete and quantitative control design procedure based on the Clegg integrator. The block diagram of

1.3 Early Ideas on Reset Control

21

Fig. 1.15 (Top) The two degree of freedom structure in Krishnan and Horowitz [32]; (bottom) structure for G with linear integrator in parallel with Clegg integrator (CI)

the considered control system is shown in Fig. 1.15 (top), where we can see a twodegree-of-freedom structure with prefilter F (s) acting on the reference r(t), output disturbance d(t), and measurement noise n(t). The plant (possibly including linear factors from the controller) is given by P (s), and G is the controller that could be linear or nonlinear. The justification of the use of reset control is presented in [32] as a simple mechanism to alleviate the limitations of linear design. More precisely, the effect of sensor noise n(t) at the plant input u(t) in Fig. 1.15 (top) is given, in the linear case, by B(s) :=

U (s) G(s) =− , N(s) 1 + L(s)

where L(s) = G(s)P (s). Typically, L(s) is designed such that |L(j ω)|  1 over a sufficiently large frequency range. This may require |G(j ω)|  1 over a larger frequency range. As a consequence, large G in the range where L is small (and 1 + L ≈ 1) causes, from the above equation, large B and amplification of noise tending to saturate the plant. Hence, it is desirable to attenuate L as fast as possible outside the frequency range where specifications require large L. However, from Bode gain–phase relation, a fast attenuation of |L| is accompanied by a very negative phase, which compromises stability. These linear limitations justify the use of a nonlinear element whose describing function has a smaller phase than that of a linear element with the same magnitude characteristic, and such a nonlinear element is the Clegg integrator. More precisely, Krishnan and Horowitz propose as nonlinear controller G in Fig. 1.15 (top), the configuration in Fig. 1.15 (bottom), formed by a linear integrator b/s in parallel with the Clegg integrator (CI), and a linear compensator G(s). It was well known that the describing function of the CI is (see Sect. 3.4): CI (j ω) :=

1.62e−j 38° 1.62 j 52° , = e ω jω

that is, the same magnitude slope as an integrator, but with phase −38°, which means 52° less than that of the linear integrator. One simple example shows the benefit of a Clegg integrator [32]: Consider an unforced typical negative feedback loop with two linear integrators L(s) = (1/s)(1/s). Then, the autonomous response

22

1

Introduction

is oscillating and does not converge to zero. On the other hand, if either of the integrators is replaced by a Clegg integrator, that is, L = (1/s) ◦ (CI ) or L = (CI ) ◦ (1/s), then the system is stable and all initial conditions tend to zero in finite time. Notice that in order that the sinusoidal describing function CI (j ω) is representative of the approximate response of the CI, the input should have a mean value of zero. For this reason, [32] proposes the parallel configuration in Fig. 1.15 (bottom), containing a linear integrator b/s. Another interesting interpretation of the dynamic response of the Clegg integrator is given by Krishnan and Horowitz in [32], and consists of noticing that it can be understood as a linear integrator which is affected by an adequate train of corrective Dirac impulses, as an additional input. More precisely, let 0 ≤ t1 < t2 < · · · < tn be the zero crossings of e(t), the input to the CI. Then x(t), its output, is t t n tk+1  x(t) = e(τ ) dτ = e(τ ) dτ − e(τ ) dτ. 0

tn

Hence, if we define xk := x(tk ) =  t

x(t) = 0

 tk+1

e(τ ) −

tk n 

k=1 tk

e(τ ) dτ , we reach  xk δ(t − tk ) dτ.

(1.28)

k=1

Thus the Clegg integrator can be replaced by a linear integrator corrected by a train of Dirac impulses as additional input, the weight of the impulse xk being the value of the CI output at tk . This is a very interesting interpretation proposed in [32]. This idea will be exploited later in this book, in Chap. 4 on stability analysis of time-delay systems under reset control. Another important contribution is the analysis of the noise response of the Clegg integrator and its application to the design procedure. The motivation for this noise analysis is derived from the formulation of the design problem: it consists of a deterministic and a stochastic objective. The deterministic objective is: Given the plant and the range of parameter uncertainty, find feasible controllers that satisfy specifications in the form of bounds on the responses to command and disturbance inputs. The stochastic objective is in an optimal sense: Among all previous feasible controllers, find the one which minimizes the rms value of the noise appearing at the input of the plant in Fig. 1.15 (top), originating from sensor noise n(t). Let x(t), y(t) be the input and output of the Clegg integrator, and let v(t) = t − tk the interval between t and the immediate previous zero crossing tk : t x(τ ) dτ. y(t) = t−v(t)

If x(·) is a stationary random process, the interval v(t) is itself a stationary random process. The computations of the statistics of v(·) as a function of the statistics of x(·) have been reported only for a few processes. If x(·) is a zero mean Gaussian Markov process with autocorrelation Rxx (τ ) = ae−α|τ | ,

1.3 Early Ideas on Reset Control

23

then the probability density function of v is p(v) =

2αe−αv , √ π 1 − e−2αv

implying the mean value v¯ = log 2/α and variance σv2 = π/(12α 2 ). Thus for highly uncorrelated processes (α → ∞) the variance tends to zero, and it seems reasonable to replace v(t) by v¯ in the integral defining the CI output which leads, among other results, to Syx (ω) 1 − e−j ωv¯ = , Sxx (ω) jω

(1.29)

the right-hand side being equal to the random describing function of the CI. These results on noise analysis are extended to the parallel of CI with an integrator in Fig. 1.15 (bottom), and finally to the noise response of the whole loop. In this way, Krishnan and Horowitz present a complete design procedure which clearly reveals that the optimum nonlinear solution with a Clegg integrator (the feasible controller that minimizes the rms value of noise at the plant input) outperforms the optimum solution among only linear controllers.

1.3.2 Reset Control with the First-Order-Reset-Element (FORE) The analysis and design procedures were subsequently extended by Horowitz and Rosenbaum in [31] to the analysis and design of reset control systems with a firstorder-reset-element (FORE). Since the approach is similar, regarding general justifications and ideas, and these aspects have already been summarized in the previous subsection, we will describe here only the basic design procedure. To facilitate this description we will follow the report by Chait and Hollot in [21] on the Horowitz design procedure, and to avoid technical details related to the QFT theory, we will illustrate the procedure by means of one of the application examples. The design of the FORE proceeds in two steps: • First, design a linear controller C(s) so that the linear closed-loop response satisfies disturbance rejection and noise suppression specifications (at the expense of violating the required stability margins). 1 • Second, introduce the factors s+b (s +b)C(s), leaving unaltered the linear system. Apply the linear control law given by (s + b)C(s) at the plant input. Convert the 1 preceding block s+b into a FORE (introducing resetting). Choose the pole s = −b in the FORE to reduce overshoot and improve stability margins. The second step is crucial and it is important to understand that performance improvement does not come with blind reset: a badly tuned reset controller results in worse performance than that of the underlying LTI system. To get an idea of what this procedure is able to do, let us briefly describe an application example on control of a flexible mechanical shaft. The plant consists of

24

1

Introduction

three inertias J3 , J2 , J1 connected with flexible shafts. The servo motor drives the inertia J3 and a tachometer measures the speed of the inertia J1 , resulting in the fifth-order linear plant: 46083950 . (s + 1.524)(s 2 + 3.1s + 2820)(s 2 + 3.62s + 9846) The following specifications were proposed to highlight the trade-offs in linear design and the improvement achieved with reset control: P (s) =

1. Bandwidth: The gain crossover frequency, defined by |C(j ωc )P (j ωc )| = 1, must satisfy ωc > 3π . 2. Disturbance rejection: For low frequencies, ω ≤ π ,    y(j ω)     d(j ω)  ≤ 0.2, where y(j ω), d(j ω) denote the Fourier transforms of the plant output and disturbance, respectively, of the feedback loop (Fig. 1.15). 3. Noise suppression: For high frequencies, ω ≥ 10π ,    y(j ω)     n(j ω)  ≤ 0.3. 4. Asymptotic performance: Zero steady-state tracking error to constant references and disturbances. 5. Overshoot: Overshoot in output to a reference step should be less than 20% (no prefilter is permitted). Using classical loop shaping techniques, it is impossible to meet all of the above specifications with linear and time-invariant controllers. To illustrate this difficulty, two seventh order controllers are proposed, C1 (s) and C2 (s). C1 (s) only fails to satisfy the noise suppression specification and, when trying to arrange this with C2 (s), this arrangement comes at the expense of the phase margin, and hence the overshoot specification is violated. The ultimate reason for this infeasibility is related to the Bode’s gain–phase relationship, that imposes a restriction on the linear designs. Turning to the reset control design and using the previously described design procedure, a valid choice for the first step is the controller C2 (s) since it satisfies all the design specifications except the overshoot constraint. Consequently, in the second step, the direct path in the feedback loop is reorganized as 1 [(s + b)C2 (s)]P (s) s +b so that the base-linear loop transfer function remains unaltered: L(s) = C2 (s)P (s). The actual linear control law implemented at the plant input is given by (s +b)C2 (s), 1 is endowed with reset and thus transformed into a and the first order system s+b FORE. In [21], an analysis of the influence of the FORE base linear pole at s = −b is performed, and the optimum value b = 14 is obtained. The final solution is proven to be satisfactory and all the specifications are met.

1.4 First General Approaches to Analysis of Reset Control Systems

25

To conclude this section, let us quote some statements in [21] on the pioneering work of Horowitz on reset controllers in the 1970s: ‘History shows that Prof. Isaac Horowitz was often ahead of the curve in his feedback control research . . . Horowitz motivated their use [reset controllers] by showing that with qualitative design, they can exhibit better performance trade-offs than those in linear, time-invariant systems’.

1.4 First General Approaches to Analysis of Reset Control Systems Although the works by Horowitz on reset control were not based on describing function or any other kind of approximate analysis, they considered only feedback systems with either Clegg or FORE elements, subject to step inputs, and computed overshoot assuming second-order dominance. Furthermore, the main obstacle for widespread dissemination of reset control was, at that time, the lack of a general theory, not restricted to Clegg and FORE, over basic questions such as well-posedness and stability of reset control systems. This theoretical framework for general reset control was subsequently developed in a number of works by Y. Chait, C.V. Hollot, and coworkers in the University of Massachusetts at Amherst. The developed framework is contained in a large number of journal papers and conference proceedings, and in several Ph.D. theses. These researchers considered for the first time higher order controllers and partial state resetting, and achieved a number of original results on stability and performance of reset systems. The approach by Chait, Hollot, and coworkers addresses reset control systems, and by this we mean here feedback control systems where the zero crossing of the error signal triggers the resetting of some of the controller states. This is different (although highly related) to the approach in the field of impulsive dynamical systems where reset instants may arise from other independent mechanisms, not necessarily related to zero crossing of feedback errors. This approach to reset systems as special cases of impulsive systems will be treated in the following section. In this section, we summarize the main results in the framework elaborated by Chait, Hollot, and coworkers. For brevity, we cannot report here a detailed account of the results in all the publications; see [18], [21], and references therein for details. For clarity, we choose [18] as the main reference for the following exposition that addresses sequentially the topics of setup, stability, steady-state, and transient performance of reset control systems.

1.4.1 General Setup and Asymptotic Stability The framework elaborated by Chait and Hollot [21] can be considered as an extension of the works by Horowitz where for the first time general reset controllers (with

26

1

Introduction

Fig. 1.16 Reset control system studied by Beker, Hollot, Chait, and Han in [18]

arbitrary order and partial reset) were addressed. To deal with these general cases, the authors in [18] and [21] adopt a modeling setup based on state space instead of transfer functions. They showed that a reset control system like the one in Fig. 1.16 can be represented by a closed-loop model based on state space variables in the form: / M (t), x˙ (t) = Acl x(t) + Bcl w(t) if x(t) ∈  + x t = AR x(t) if x(t) ∈ M (t),

(1.30)

yp (t) = Ccl x(t). This is an impulsive-differential equation (IDE) where the first line is the ordinary differential equation describing the continuous mode, the second line defines the reset actions, or instantaneous state jumps, and the third line simply gives the plant output yp . The closed-loop matrices (Acl , Bcl , Ccl ) are easily obtained from the state-space matrices of the plant and of the base linear controller. The exogenous signal is w(t) = r(t) − n(t) − d(t). The complete state vector x(t) is formed by stacking the states of the plant and of the controller. The number of plant and controller states are np and nr , respectively, and n = np + nr is the total state dimension. Within the controller, it may happen that only part of the state is affected by reset. Let nρ ≤ nr be the number of controller states to be reset. In this way, Chait, Hollot, and coworkers introduced the concept of a partial reset: when nρ < nr , not all the controller states are reset. Thus, the framework in [18] generalizes the analysis by Horowitz to controllers with arbitrary dimension nr and to reset laws that can be either ‘full reset’ (nρ = nr ) or ‘partial reset’ (nρ < nr ). By an appropriate reordering of states, the reset states can be stacked in the last positions, so that the reset matrix AR takes the form: AR = diag(1, . . . , 1, 0, . . . , 0)

(1.31)

with nρ 0s. Another interesting proposal in [18] is the definition of the so-called reset surface M (t):   (1.32) M (t) = ξ ∈ Rn : e(t) = 0, (I − AR )ξ = 0 where the feedback tracking error is e(t) = r(t) − n(t) − d(t) − yp (t) = w(t) − Ccl x(t). The first condition in the definition of M (t) is e(t) = 0, which shows the dependence on exogenous signals via e(t) or w(t) and also shows the time-varying nature of M (t) when the exogenous signals are not constant. The second condition, (I − AR )ξ = 0 is imposed for well-posedness, to avoid the so-called re-resetting,

1.5 Reset Systems as Impulsive Systems

27

that is, the fact that immediately after a reset, the state vector could satisfy again the reset condition, implying formally an infinite sequence of resettings to be performed instantaneously, also referred to as beating. Thus, to avoid beating, the term (I − AR )ξ = 0 is added in the definition of M (t). As a consequence of this definition,   / M (t). x(t) ∈ M (t) ⇒ x t + ∈ In addition, the stability of reset control systems is approached, firstly from an internal or asymptotic point of view, and secondly from an external or input–output point of view. Regarding internal stability, the unforced or autonomous version of the reset system (1.30) is considered: / M, x˙ (t) = Acl x(t) if x(t) ∈  + x t = AR x(t) if x(t) ∈ M , where

(1.33)

  M = ξ ∈ Rn : Ccl ξ = 0, (I − AR )ξ = 0 .

As a tool for addressing internal stability, a general theorem for arbitrary Lyapunov functions is given, followed by a particularization to quadratic stability. This condition is called the Hβ condition and is easily testable by using linear matrix inequalities, or directly by frequency domain plots. When reset affects one single state (nρ = 1), the problem is reduced to the search for the single variable β ∈ R. A property that follows immediately from the Hβ condition is that the class of quadratically stable reset control systems requires their base-linear systems to be stable, that is, Acl stable. On the other hand, the authors pointed out that reset can stabilize an unstable base-linear system, and showed it with an example. Some other examples show that there exist reset control systems that are stable but not quadratically stable. In spite of these examples, it is noticed that the class of quadratically stable systems is rich and contains many common and practical cases of reset control applications. Furthermore, it can be seen that the Hβ condition can be easily incorporated into the Horowitz design scheme introduced in the previous subsection. The Hβ condition has been also proven to be useful in establishing other interesting properties of reset control systems, for example regarding some type of input-state stability. There are an important number of results by Chait, Hollot, and coworkers regarding other aspects of performance in the steady state and transient responses, that cannot be included here, for brevity. The interested reader can find this information in [18], [21], and in the references therein.

1.5 Reset Systems as Impulsive Systems In the two previous Sects. 1.3 and 1.4, we have reviewed, for the approximate period 1974–2004, the field that we call here ‘reset control systems’, and by this we mean

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1

Introduction

feedback control systems with resetting applied at the zero-crossings of the tracking error signal. At the same time, the field of ‘reset control systems’, as previously defined, could be classified as a subarea of the much wider field of ‘impulsive systems’. This term of ‘impulsive systems’ is used in different areas of applied mathematics and dynamical systems theory, and there is an important body of results and a large literature on the subject. It is out of the scope of this section to present a detailed survey of results on impulsive systems. ‘Reset control systems’ can be considered as a particular class of ‘impulsive systems’ characterized by the fact that the reset condition is triggered by the zero crossing of the tracking error signal, in a typical feedback loop. As the error signal is a function of the state for unforced systems (and the forcing external signals can be described by ‘exosystems’) we can say that in ‘reset control systems’ the reset events are time-independent and only state-dependent. On the other hand, the research in impulsive theory considers impulses applied at instants depending only on time or both time-dependent and state-dependent. More generally, as reported in Bainov and Simeonov (1989) [4], impulsive systems may be classified as (i) systems with impulses at fixed instants, (ii) systems with impulse effect at variable instants, and (iii) autonomous systems with impulse effects. Hence, the class (i) has timedependent impulses, the class (ii) has both time- and state-dependent impulses, and class (iii) which contains ‘reset control systems’ has only state-dependent impulses. Traditionally, most of the research effort has been dedicated to cases (i) and (ii), and thus case (iii), which includes reset control, is less developed. Summarizing, ‘reset control systems’ form a particular class of ‘impulsive systems’ with a specific state-dependent mechanism that triggers reset. Furthermore, the field of reset control considers implicitly as the main motivation the idea that the reset action is an effective way for overcoming fundamental limitations in linear feedback. This motivation regarding control limitations is not present in the general literature on impulsive control. In spite of the previous differences between the points of view of ‘reset control’ and ‘impulsive systems’, there are strong and deep connections between the concrete problems addressed by both fields. Thus, the research on reset control can profit from employing results on impulsive systems as addressed in the corresponding areas of applied mathematics and mathematical control theory. Having this is mind, we include here three subsections based on three relevant monographs in the field: Theory of impulsive differential equations by Lakshmikantham, Bainov, and Simeonov [33], Impulsive control theory by Yang [43], and Impulsive and hybrid dynamical systems by Haddad, Chellaboina, and Nersesov [30]. The following comments and discussions on these three books are focused only on aspects regarding specifically the class of ‘reset control systems’, that is, we discuss how reset control can be treated within the framework presented in the books, and how general results could be particularized, if possible, to reset control.

1.5 Reset Systems as Impulsive Systems

29

Fig. 1.17 Impulses at fixed times

Fig. 1.18 Impulses at variable times

1.5.1 Theory of Impulsive Differential Equations In the book Theory of Impulsive Differential Equations by Lakshmikantham, Bainov and Simeonov [33], the impulsive differential systems are classified in the three following classes: (i) Systems with impulses at fixed times. This type of system is described by  x˙ = f (t, x) if t = tk , (1.34) Δx = Ik (x) if t = tk , where tk , for k = 1, 2, 3, . . ., is a sequence of prescribed reset times and Δx(tk ) = x(tk+ ) − x(tk ) are the state ‘jumps’ correspondingly applied. Figure 1.17 illustrates a typical time response of a system in this class. (ii) Systems with impulses at variable times. These systems are given formally by  x˙ = f (t, x) if t = τk (x), (1.35) Δx = Ik (x) if t = τk (x), where τk (x) is a sequence of state-dependent reset times having the property that τk (x) < τk+1 (x),

lim τk (x) = ∞

k→∞

for all x. Figure 1.18 shows a possible time response of one such a system. Notice that solutions may hit the same surface τk (x) several times and also present complex behaviors called ‘pulse phenomenon’ and ‘confluence’. In general, a simplifying assumption is to consider functions τk to be continuous over Rn .

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Introduction

Fig. 1.19 Impulses at a general reset surface h(t, x) = 0

Fig. 1.20 Autonomous systems with impulses at a state-dependent surface

The class (ii) can be extended to a more general class in the form:  x˙ = f (t, x) if h(t, x) = 0, Δx = Ik (x) if h(t, x) = 0,

(1.36)

where h(t, x) = 0 might represent a more general, arbitrary time-dependent reset surface. This situation is illustrated by Fig. 1.19. Finally, we have (iii) Autonomous systems with impulses. These systems are described by  x˙ = f (t, x), if x ∈ / M, Δx = Ik (x), if x ∈ M,

(1.37)

where M is an arbitrary subset in the domain of definition of x ∈ Rn . Figure 1.20 plots a possible trajectory, in the one-dimensional case, n = 1 and x ∈ R. Although the analysis of an autonomous system with impulse effects may appear to be easier than the other cases, in general it has been proven harder since in general the continuity assumption over functions τk cannot be achieved. In summary, the classification proposed by [33] is based on the resetting mechanism, and considers three possibilities: only time-dependent reset (class (i), t = tk ), only state-depending reset (class (iii), x ∈ M) and both, time- and state-reset (class (ii), t = τk (x)). The reset control systems that we are interested in, as depicted in Fig. 1.16 or defined by (1.30), fit well into the third class (iii). The unforced system (1.30) without exogenous input (w = 0) is directly in the form (1.37). The same is true in the case of constant (step) exogenous inputs. Furthermore, for the typical control inputs (step, ramp, sinusoid, etc.) it is always possible to express them as the output of an appropriately defined system (the so-called ‘exosystem’). As a consequence, the

1.5 Reset Systems as Impulsive Systems

31

zero crossing of the error that defines the surface M (t) in (1.32) can be expressed as a condition on the system states and on the states of the exosystem. In this way, the reset condition can be made time-independent so that reset systems belong to the class (iii) of autonomous systems with impulses. Most of the results in [33] apply to the cases (i) and (ii). Regarding stability, which is a basic property required for a control system, the results in [33] for class (i) systems cannot be applied to reset control since the reset times in reset control are not prescribed, but are state-dependent. In addition, most of the literature about state-dependent impulsive systems has been developed for systems which can be described by functions τk with some specific properties, for example, it is usual to assume that they are continuous over their domain. As it will be seen in detail in Chap. 2, in general this is not the case for reset control systems. To conclude this section, the book [33] presents a systematic treatment of impulsive systems, addressing well-posedness and continuity of solutions, stability criteria, comparison results, vector Lyapunov functions, and many other aspects. The interested reader can find a detailed treatment of these topics in [33], [4], and in the references therein.

1.5.2 Impulsive Control Theory Another very interesting monograph on impulsive systems is Impulsive Control Theory (2001) by T. Yang [43]. This book presents an approach in many parts similar to the one in the previously discussed reference [33], that is, focused on impulsive systems of classes (i) and (ii). But [43] has a point of view much closer to the field of control, than that of [33], with a more mathematical emphasis. In fact, as T. Yang notices in the preface to [43], there are several reasons why the field of control has not fully had the benefit from the results in the field on impulsive systems. One reason is that for many years impulsive control has been restricted to very specific problems, like mechanical systems with impacts or spacecraft control. Another reason is that the early research activities on impulsive control were reported as Russian literature, not well-known to the English speaking community. Even after publication of many English monographs on impulsive control, they did not attract at the beginning the control community because the examples were too limited and they targeted mainly mathematicians as potential readers. But nowadays, the ideas on impulsive control are much more disseminated into the control, community and the number of control applications has increased, including problems such as chaotic communication systems or nanoelectronic devices. Impulsive systems are classified into the same classes that [33] uses: (i) impulses at fixed times, (ii) impulses at variable times, and (iii) autonomous systems with impulses (which are called discontinuous dynamical systems in [43]). But, at the same time, an interesting parallel classification is presented in [43] based on the way the control action enters the system. In the case of impulses at variable times

32

1

Introduction

t = τk (x), the classification is as follows. A type-I impulsive control system is given by: ⎧ if t = τk (x), ⎨ x˙ = f (t, x) Δx = U (k, y) if t = τk (x), (1.38) (I) ⎩ y = g(t, x), where the control input is implemented as sudden jumps in the state, given by the impulsive control law U (k, y). A type-II impulsive control system is given by: ⎧ ˜ if t = τk (x), ⎨ x˙ = f (t, x, u) (1.39) (II) Δx = U (k, y) if t = τk (x), ⎩ y = g(t, x, u) ˜ if u˜ = γ (t, x), so that there are actually two control laws, the impulsive control law U (k, y) and the continuous control law u˜ = γ (t, y). A type-III impulsive control system is given by: ⎧ ˜ if t = τk (x), ⎨ x˙ = f (t, x, u) (1.40) (III) if t = τk (x), Δx = jk (x) ⎩ y = g(t, x, u) ˜ if u˜ = γ (t, x), where for this type of system the control u˜ is continuous and what is impulsive is the plant itself. Most of the results are presented for systems with impulses at fixed t = tk or variable t = τk (x) times with some regularity condition over functions τk , but not for autonomous systems with impulses, hence they are not applicable to the reset control systems as in Fig. 1.16 or defined by (1.30). On the other hand, it is out the scope of this section to report a detailed account of the results in [43] but, just to give an idea of one among the many approaches presented, let us see a simple example. The example is taken from Chap. 6 and illustrates the concept of practical stability and the comparison system. Given a system of type-II (1.39) and given the pair (μ, ν) with 0 < μ < ν, the system is said to be practically stable when x0  < μ

⇒

x(t, t0 , x0 ) < ν,

where x(t, t0 , x0 ) is the system solution for t ≥ t0 starting at x(t0 ) = x0 . Hence, the concept of practical stability is weaker than that of asymptotic stability and only imposes boundedness on the state evolution. Consider the following two-dimensional chaotic system [43]:  x˙ = y − f (x), (1.41) y˙ = −εx − ζy + η sin ωt, where 1 f (x) = bx + (a − b)(|x + 1| − |x − 1|), 2 with a = −1.02, b = −0.55, ε = 1, ζ = 1.015, η = 0.15, and ω = 0.75. This is a chaotic system, with the chaotic attractor shown in Fig. 1.21.

1.5 Reset Systems as Impulsive Systems

33

Fig. 1.21 Chaotic attractor of an uncontroller chaotic system in Yang [43]

The system is controlled in an impulsive fashion, as a type-II system in the form:  x˙ = y − f (x) , if t = τi , y˙ = −εx − ζy + ηsin ωt, (1.42) x(τi+ ) = di x(τi ) , if t = τi . y(τi+ ) = di y(τi ), Using the Lyapunov function V = |x| + |y|, the stability of the two-dimensional system can be derived from the stability of a certain one-dimensional comparison system. The practical stability of the controlled system (1.42) with respect to (μ, ν) is implied by the practical stability of the following comparison system:  w˙ = φw + θ if t = τi , (1.43) + w(τi ) = di w(τi ) if t = τi , where φ = max{|ε + |a||, |1 − ζ |} and θ = |η|. Sufficient stability conditions are derived for the impulsive practical stabilization of the chaotic attractor. A simplification is obtained letting the resetting coefficients constant di = d and the reset instants to be fixed and equally spaced, i.e., τi+1 − τi = δ for all i. It can be seen that a valid stabilizing parameter choice (d, δ) is given, for any μ < ∞ and for some ν, by the conditions

34

1

Introduction

Fig. 1.22 Partial stability of the impulsively controlled chaotic system in [43]

d < e−φδ , θ |1 − eφδ | < v. φ 1 − deφδ A valid solution is ν = 1 and (d, δ) = (0.06, 1.01). So we can conclude that no matter how large the initial condition is, (x0 , y0 ) < μ < ∞, the resetting controls in (1.42) achieve the practical stability (boundedness) (x(t), y(t)) < ν = 1 around the origin. The simulation results in Fig. 1.22 show how the uncontrolled attractor is destroyed and the controlled solution remains within the specified bounds.

1.5.3 Impulsive Dynamical Systems An important number of results on stability and the control of impulsive systems are contained in the book Impulsive and Hybrid Dynamical Systems by Haddad, Chellaboina, and Nersesov [30], and in many other references by the authors and coworkers (see [29], [30] and the references therein). One of the main distinctive features in [30], compared to the monographs discussed in the previous subsections, is the strong emphasis put on energy-based modeling and the control of impulsive systems. The book contains an in-depth presentation of a dissipativity theory for

1.5 Reset Systems as Impulsive Systems

35

analysis and design of impulsive systems. The book also addresses impulsive systems from the point of view of nonnegative and compartmental systems, optimal control, disturbance rejection, robustness against uncertainties, periodic solutions, and so on. In spite of this, again the developed results are not directly applicable to general reset control systems (the approach is based on a number of regularity conditions that give some restrictions over the functions τk ). Some brief comments about dissipativity theory (modeling, definitions, and basic principles) will be given in the following. It is out of scope of this subsection to give a detailed account of the many aspects treated in [30]. However, to give an idea of the powerful technique of energy-based control, we will include a simple but illustrative example. An impulsive system is described, in its most general form, by the equations x˙ (t) = fc (x(t))

if (t, x(t)) ∈ / S,

Δx(t) = fd (x(t))

if (t, x(t)) ∈ S ,

(1.44)

where the resetting set S can be both time- and state-dependent. In order to have well-posedness of the resetting times, two assumptions are imposed. The first assumption A1 imposes that if a trajectory reaches the closure of S , at a point not in S , then the trajectory must be directed away from S . The second assumption A2 imposes that when a trajectory intersects S , it jumps out of S . The general system (1.44) contains as a particular case the state-dependent reset systems, obtained replacing the reset condition (t, x(t)) ∈ S by certain x(t) ∈ Z , where Z is the subset of the state-space where reset takes place. In any case, these are autonomous or unforced systems, and in order to address dissipativity, it is necessary to include input and output signals, giving rise to models in the form: x˙ (t) = fc (x(t)) + Gc (x(t))uc (t)

if (x(t), uc (t)) ∈ /Z,

Δx(t) = fd (x(t)) + Gd (x(t))ud (t)

if (x(t), uc (t)) ∈ Z ,

yc (t) = hc (x(t)) + Jc (x(t))uc (t)

if (x(t), uc (t)) ∈ /Z,

yd (t) = hd (x(t)) + jd (x(t))ud (t)

if (x(t), uc (t)) ∈ Z .

(1.45)

The impulsive system (1.45) is said to be dissipative with respect to continuous and discrete supply rates sc and sd if T  0≤ sc (uc (t), yc (t)) dt + sd (uc (tk ), yc (tk )), (1.46) t0

k

for all T ≥ t0 and for all input and output, continuous and discrete, signals uc , ud , yc , yd , in their corresponding signal spaces. Basically, it is an extension of the usual dissipation inequality in which the power into the system contains the usual continuous term, typically sc (uc (t), yc (t)) = yc (t) uc (t), plus discrete energy packets sd (ud (tk ), yd (tk )) applied at the reset instants tk . A complete framework for dissipativity of impulsive systems is given, including generalizations of the available and required storage functions, Kalman– Yakubovich–Popov conditions, stability of feedback interconnections, port-

36

1

Introduction

Fig. 1.23 Mechanical equivalence of an example in [30]

controlled Hamiltonian systems, and so on. For brevity, it is not possible to include more details but, before concluding this subsection, we will include an insightful example, just to give an idea of how powerful and intuitive this approach is. Consider a second order Lienard plant: q¨ + f (q(t)) = u(t), ∈ Rn

: Rn

(1.47)

→ Rn

with q and f being infinitely differentiable, f (q) = 0 if and only if ∂fj ∂fi q = 0, and ∂qj = ∂qi for all i, j . The equations represent a mechanical system with n particles of unit mass coupled by elastic forces given by f (·). Defining p = q, ˙ the Hamiltonian is obtained as the sum of the kinetic and potential energies q 1 H (p, q) = p p + f  (σ ) dσ, (1.48) 2 0 where the integral is independent of the path, from the assumptions on f . The proposed feedback controller is a copy or replica of the plant, but endowed with a reset mechanism. More precisely, /Z, q¨c + g(qc (t) − q(t)) = 0 if (q(t), p(t), qc (t), pc (t)) ∈ Δqc (t) = q(t) − qc (t)

if (q(t), p(t), qc (t), pc (t)) ∈ Z ,

Δpc (t) = −pc (t)

if (q(t), p(t), qc (t), pc (t)) ∈ Z ,

(1.49)

u(t) = g(qc (t) − q(t)). To gain insight into the plant–controller feedback system resulting, notice that in the scalar case, the continuous dynamics, without the impulsive equations, represent two unit masses with positions q, qc and velocities p, pc connected as in Fig. 1.23, by means of two possibly nonlinear springs with restoring forces f (q), g(qc − q). Notice that the nonimpulsive feedback system (without resetting) is a Hamiltonian lossless system and the total energy (plant energy + controller energy) remains constant. Hence along the solutions of the system, the four states q, p = q, ˙ qc , and pc = q˙c change in such a way that the total energy is constant, so that sometimes the energy flows from the plant to the controller, and sometimes from the controller back to the plant. In order to achieve stabilization of the physical plant to the equilibrium (q, p) = (0, 0), is necessary to implement in the virtual controller some kind of dissipation of the virtual energy stored in it. It can be seen that resetting is an efficient way of doing so. Two things have to be decided: when and how to reset. Since we want to destroy the virtual energy in the controller, once the instant tk is decided, the

1.6 Reset Systems as Hybrid Systems

37

maximum dissipation is achieved by fully emptying the controller energy (kinetic plus potential), in other words, the appropriate resettings are:     pc (t) → pc t + = 0. (1.50) qc (t) → qc t + = q(t), These resettings are given in the form of incremental impulses in (1.49), it remains to decide when to apply reset. Since the energy may flow in both directions between the plant and the controller, it makes little sense to apply reset at instants when the continuous dynamics are extracting energy from the plant. In fact, it is proposed to start the controller with zero energy (qc (0) = q(0), pc (0) = 0, and thus the energy at the beginning goes out of the plant) and reset the controller at each q −q moment when its energy Hc (qc , pc ) = pc pc /2 + 0 c g  (σ ) dσ achieves a maximum, that is, when   d 0 = Hc (qc , pc ) = − g(qc − q) p. (1.51) dt This gives rise to the resetting set Z     Z = (q, p, qc , pc ) : g(qc − q) p = 0, (q − qc , pc ) = (0, 0) , (1.52) which contains the maximum condition 0 = dH /dt plus the condition to avoid re-resetting (beating). To see how these ideas work, let us particularize to a twomass system with nonlinear plant spring f (q) = q + q 3 and a linear virtual spring g(qc − q) = 3(qc − q). The results of the simulations are shown in Figs. 1.24, 1.25, 1.26, 1.27.

1.6 Reset Systems as Hybrid Systems In the tutorial by Goebel et al. [26], a modeling of hybrid systems is given that, in the words of the authors, emphasizes the robustness of asymptotic stability to data perturbation, disturbances, and measurement error. In general, a hybrid system has the form  x˙ ∈ F (x) if x ∈ C, (1.53) + x ∈ G(x) if x ∈ D, where the dynamics is given by both a constrained differential inclusion and a constrained difference inclusion, with F and G being the set-valued maps, and C and D, the flow and the jump set, respectively, subsets of Rn . Among other basic assumptions, it is assumed that C and D are closed sets in Rn , and in general D may overlap with the interior of C, thus the jump set D enables rather than forces jumps. The general model (1.53) is representative of a large class of hybrid systems as it has been shown in [26]. In particular, the Clegg integrator and the FORE are first modeled in this framework in [44], where the FORE compensator, with input e and output xr , is given by  x˙r = λr xr + e if exr ≥ 0, (1.54) x+ = 0 if exr ≤ 0.

38

1

Introduction

Fig. 1.24 Example in Haddad, Chellaboina, and Nersesov [30]. Plant position and velocity

In addition, a plant with state xp ∈ Rn−1 is considered and described by  x˙ p = Ap xp + Bpu u + Bdu d, (1.55) y = Cp xp , and finally, a closed-loop reset system is given by the hybrid system ⎧ ⎨ x˙ = Ax + Bd d if x ∈ C, x+ = Ar x if x ∈ D, ⎩ y = Cx,

(1.56)

where d is an input disturbance to the plant, the flow set C is given by C = {x ∈ Rn |xT Mx ≥ 0}, and the jump set by D = {x ∈ Rn |xT Mx ≤ 0} with   0 −CpT M= . (1.57) −Cp 0 This representation of a reset system gives solutions compatible with previous definition of reset systems, as long as the initial condition of the FORE (or Clegg integrator) is zero. The advantage is that, without loss of generality, the dynamics of the compensator is refined to be in the sector {(xr , e) ∈ R2 |xr e ≥ 0} instead of the whole plane R2 . The implication of this is a less conservative approach to stability, as developed in the mentioned works. In addition, temporal regularization has been

1.6 Reset Systems as Hybrid Systems

39

Fig. 1.25 Example in Haddad, Chellaboina, and Nersesov [30]. Controller position and velocity

used in order to the system solutions to be well defined for forward time. As a result, the reset system is given by ⎧ ⎨ τ˙ = 1, x˙ = Ax + Bd d if x ∈ C or τ ≤ ρ, if x ∈ D and τ ≥ ρ, τ + = 0, x+ = Ar x (1.58) ⎩ y = Cx. For this formulation of reset control systems, several results about bounded-input bounded-state and bounded-input bounded-output stability are developed. In particular, it is shown that if the following pair of LMIs  ⎧ T A P +P A+τF M P B C T ⎪ ⎨ T < 0, B P −γ I 0 (1.59) C 0 −γ I ⎪ ⎩ T AR P AR − P − τR M ≤ 0 is feasible for some P = P T > 0, τF , τR ≥ 0, and γ > 0, then there exists some ρ > 0 small enough such as the reset control system (1.58) is finite gain L2 -stable from d to y. It has been also shown that even for simple examples the above statement needs some refinement by using piecewise quadratic Lyapunov functions, and that in general it may be necessary to solve a large number of LMIs to obtain a tight bound of the gain.

40

1

Introduction

Fig. 1.26 Example in Haddad, Chellaboina, and Nersesov [30]. Control signal

For example, consider an integrator controlled by a FORE (Example 2 in [44]): ⎧ ⎨ τ˙ = 1, x˙1 = x2 + d, x˙2 = −x1 + βx2 if x1 x2 ≤ 0 or τ ≤ ρ, if x1 x2 ≥ 0 and τ ≥ ρ, τ + = 0, x1 + = x1 , x2 + = 0 (1.60) ⎩ y = x1 ,  0 −1 . Note that FORE has where the flow and jump sets are obtained for M = −1 0 1 a base linear system with transfer function s−β , and that the closed-loop base sys1 tem is given by the transfer function s 2 −βs+1 , thus the base closed-loop system is unstable for β ≥ 0. In addition, for β ≥ 2 the dominant poles are real. It has been shown that for any value of β ∈ R the L2 -gain from d to y is welldefined and that it tends to zero as β tends to infinity (for positive values of β, the base closed-loop system is unstable, and thus it does not have a well-defined L2 -gain). Figures 1.28–1.29 show the time evolution of the closed-loop state of the reset control system for a rectangular pulse as disturbance, for β = 2. In this case, the obtained L2 -gain from d to y is 0.58. As a result, the reset control system can beat the base linear control system, having less L2 -gain from d to y and thus having a better disturbance rejection.

1.6 Reset Systems as Hybrid Systems

41

Fig. 1.27 Example in Haddad, Chellaboina, and Nersesov [30]. Plant, controller and total energy

It should be pointed out that for higher order systems, and especially partial reset compensators, the reset condition based on the sign change in the product of the input and output signals results in solutions of the reset control systems that are different from the reset condition based on the zero crossing of the input (which is the classical condition used by Clegg and Horowitz), even though the initial condition of the compensator state is zero. Consider, for example, (Fig. 1.30) a reset compensator given by a parallel connection of an integrator and a Clegg integrator (this is a compensator structure that has been used, for example, in [32]). This is a simple partial reset compensator, and may be described by (with reset condition e(t) = 0) ⎧  1−pr  ⎪ ⎨ x˙ (t) = pr e(t) if e(t) = 0,   x(t + ) = 10 00 x(t) if e(t) = 0, ⎪ ⎩ u = ( 1 1 ) x,

(1.61)

where the base linear compensator is an integrator, and pr is a parameter with value 0 ≤ pr ≤ 1. Note that for pr = 0 the reset compensator corresponds to a linear

42

1

Introduction

Fig. 1.28 Example 2 in Nesic, Zaccarian, and Teel [44]. Time evolution of states x1 and x2 for a rectangular pulse disturbance

integrator, and that for pr = 1 it is a CI. On the other hand, using the alternative reset condition, the reset compensator is given by ⎧  1−pr  ⎪ if e(t)u(t) ≥ 0 or τ ≤ ρ, ⎨ τ˙ (t) = 1, x˙ (t) = pr e(t)   1 0 τ (t + ) = 0, x(t + ) = 0 0 x(t) if e(t)u(t) ≤ 0 and τ ≥ ρ, ⎪ ⎩ u = ( 1 1 ) x.

(1.62)

Figure 1.31 shows a time response of both implementations for a sinusoidal input and zero initial conditions. Both responses are identical up to the first reset, then the compensator (1.62) performs several resets every ρ time units, and thus both responses diverge up to next common reset. Note that due to the partial reset the signs of the input and output are different for a finite time interval after the first reset, and in fact there would be deadlock for ρ = 0 with the reset condition based on the sign change.

1.7 Other Recent Results on Reset Control

43

Fig. 1.29 Example 2 in Nesic, Zaccarian, and Teel [44]. Time evolution of disturbance d and closed-loop output y = x1 Fig. 1.30 I + CI compensator

1.7 Other Recent Results on Reset Control To conclude the historical survey of reset control, this section presents several recent results to the analysis and design of reset control systems. Although the number of recent publications on reset control is not very large, it is not possible, for brevity reasons, to give a complete account of all of them. For the sake of concreteness, a review of three works is given in the following: the first two works are somehow extensions and are based on the reset system description as a hybrid system outlined in Sect. 1.6, while the third work introduces reset control systems with a simple resetting law (reset is performed at fixed instants). More specifically, the first work [1] provides a general framework for reset systems design based on L2 and H2 gain concepts; the second work [34] also approaches the design from the L2 sense and specializes to constant or varying, a priori known, references; finally, the design technique in [28] adopts fixed reset instants tk and finds the controller parameters from the optimization of a quadratic performance index.

44

1

Introduction

Fig. 1.31 Sinusoidal response of I + CI with different resetting law

1.7.1 Design Based on L 2 and H 2 Performance This subsection presents a summary of some relevant ideas in [1]. This work contributes new design techniques for reset controllers that generalize previous results in [36, 44], using the same resetting law. The approach is also based on certain LMIs that provide computable upper bounds of the L2 and H2 norms. The plant–controller configuration is based on the typical H∞ framework shown in Fig. 1.32. The block K denotes the SISO controller, with input y(t) ∈ R and output u(t) ∈ R. The block P denotes the MIMO augmented plant that includes the system to be controlled and possible input and output weighting filters. The vector of exogenous signals w(t) ∈ Rnw typically includes references, disturbances, and sensor noise, or their pre-weighted signals. Finally, z ∈ Rnz is the vector of controlled outputs that collects signals (tracking error, actuation, plant output, etc.) that are to be minimized in some sense (L2 or H2 ). The augmented plant P is modeled as a MIMO linear system (with the common assumption that there is no direct feedthrough from u to y) and the reset controller K is described by a linear base system combined with the reset to zero of part of the controller states. The resetting law is based on the approach reported in Sect. 1.6,

1.7 Other Recent Results on Reset Control

45

Fig. 1.32 Plant–Controller configuration in Aangenent et al. [1]

Fig. 1.33 Partitioning of the (y, u)-space. The gray area is the reset set. The white area is the flow set, partitioned in angular regions C1 , . . . , CN , for PWQ Lyapunov analysis

that is, the resets are applied when input and output have opposite signs. Figure 1.33 shows the reset set (in gray) or the set R  in the (y, u) plane where reset is applied, D :

y ≤ 0,

u ≥ 0,

or

y ≥ 0,

u ≤ 0,

and the flow set (white area) C  or the set for the continuous mode of operation, C :

y ≤ 0,

u ≤ 0,

or y ≥ 0,

u ≥ 0.

Both sets intersect on the lines u = 0 and y = 0 where reset takes precedence over flow. Analogously to the developments given in [36, 44] and outlined in Sect. 1.6, the flow set is divided into a number of angular regions C  = C1 ∪ · · · ∪ CN in order to apply a piecewise quadratic (PWQ) Lyapunov analysis that will be commented on later. Combining the plant and controller equations as in Fig. 1.32, the model of the closed-loop system Σ is obtained. This system Σ is described by a base linear closed loop x˙ = Ax + Bw, z = Cx + Dw and a resetting law x + = AR x, where x contains the plant and controller states and (A, B, C, D, AR ) are derived from the plant and controller models. The reset set D and flow set C for the closed-loop system Σ can be expressed in terms of the state x and the input w D:

TR (x, w) ≥ 0,

or TR (x, w) ≤ 0,

C:

Tf (x, w) ≥ 0,

or Tf (x, w) ≤ 0,

for some 2 × (nx + nw ) matrices TR , Tf . Well-posedness is addressed considering inputs w(t) belonging to the set of real Bohl functions, that is, signals that can be expressed in the form w(t) = H eF t v for some (H, F, v) of adequate dimensions. Then a condition based on x(0) and on a finite number of derivatives w(0), w(0), ˙ w(0), ¨ . . . is presented that guarantees local existence of solutions. After these basic results, several stability results are

46

1

Introduction

developed using L2 and H2 analysis. For the L2 analysis, recall that the L2 gain Σ∞ of the system Σ is given by Σ∞ =

z2 0<w2 <∞ w2 sup

for any input–output pair (w, z) of the system Σ, with initial state x(0) = 0. Thus, it represents the input–output amplification in the sense of the L2 norms w2 , z2 . Stabilization of the closed-loop reset system Σ in Fig. 1.32 requires a finite gain Σ∞ < ∞, and optimal performance requires minimizing Σ∞ < ∞ with respect to the free parameters in the controller K. Unfortunately, the computation of the gain of a nonlinear system is extremely difficult, in general. However, a technique for estimating upper bounds γ of Σ∞ is proposed based on dissipativity with respect to quadratic Lyapunov storage functions. To this end, quadratic Lyapunov functions V (x) = x  P x with P > 0 are postulated, resulting in two LMIs (continuous and discrete). These two LMIs are easily computable in the two parameters γ , P . From a bisection search on γ , checking feasibility w.r.t. P > 0, a procedure for computing an upper bound of Σ∞ is obtained. In order to improve accuracy in the bounding Σ∞ < γ and to avoid conservativeness, more general storage functions V (x) can be used. A detailed treatment for piecewise quadratic (PWQ) Lyapunov functions is given. In this way, each angular sector in the flow set in Fig. 1.33 (and the corresponding regions in the (w, x)-space) are assigned to a Lyapunov function Vi (x) = x  Pi x with i = 1, . . . , N . The treatment is more involved and the reduction of conservatism requires in some cases to include strictly proper input weighting filters, but finally, a set of LMIs is obtained that are applied to representative examples suggesting that are quite accurate for the L2 gain characterization. Regarding the H2 gain analysis, the first thing that had to be defined is the interpretation of the H2 norm for nonlinear reset systems. One of the possible interpretations of the H2 norm of linear SISO systems, given by stable strictly proper transfer functions G(s) = C(sI − A)−1 B, is  ∞ 1/2 2 G2 = |g(t)| dt , 0

that is, the total energy of g(t), the SISO impulse response of G(s). This impulsive response can also be written as g(t) = Ce At B, so it can be identified with g(t) = CeAt x0 , that is, the autonomous response with initial condition x0 = B ∈ Rn . In the MIMO case, a similar interpretation holds based on initial conditions x0 equal to the columns Bi of B, or equal to sums of columns Bi , Bj , . . . of B. Inspired by this interpretation, the following definition is proposed: the H2 norm of the reset system Σ in Fig. 1.32 corresponding to an initial value x0 ∈ Rn is the total output energy defined as  ∞ 1/2 z (t)z(t) dt , Σ2,x0 = 0

1.7 Other Recent Results on Reset Control

47

Fig. 1.34 Reset control system in [34]

where z is the output trajectory corresponding to zero input (w = 0) and initial state x0 . In this way, the H2 analysis of reset systems can follow a treatment very similar to the L2 analysis. Typically, the H2 problems are posed with a diagram like the one in Fig. 1.32, were P is the augmented plant and the input w is a vector containing the reference, the disturbance, and the sensor noise. Inputs are assumed to be known a priori and this knowledge is captured in the input filters. Possible input filters include: 1 , • unit step W (s) = s+ε 1 • unit ramp W (s) = (s+ε)2 , • sine wave W (s) = (s+ε)ω2 +ω2 ,

with ε > 0 a small offset. A complete and detailed H2 analysis is developed based on standard Lyapunov and PWQ Lyapunov functions and providing computable LMIs so that, given the initial state x0 and the bound γ , the feasibility of the LMIs implies the bound Σ2,x0 ≤ γ . Iterating with various initial conditions and/or controller parameters, and by a bisection search on γ , these LMIs provide accurate, powerful tools for performance optimization on reset control systems.

1.7.2 Design Based on L 2 Gain and Nonzero References The approach in [34] is in many parts similar to that in [1], and also uses the framework outlined in Sect. 1.6. Both works share the same resetting law (based on the sign of input and output), the L2 point of view, the Lyapunov and PWQ Lyapunov analysis, and the formalism leading finally to LMI analysis conditions. What makes it different is some specific treatment of references and disturbances, and a different L2 tracking criterion. This subsection presents briefly some details of the interesting work in [34], in order to complement the section with additional, useful tools, for L2 analysis of reset systems. The reset control system setup is shown in Fig. 1.34. The strictly proper plant is given by (Ap , Bp , Cp ), and the base linear model of the controller is given by (Ar , Br , Cr , Dr ). The flow or continuous mode is given by e · yr ≥ 0, and the jump, or reset, mode by e · yr ≤ 0. With zero reference (or constant reference after coordinate change)   n these sets are translated to the closed-loop state-space x = (x p , xr ) ∈ R , and thus   the flow set F is given by x Mx ≥ 0 and the jump set J is given by x Mx ≤ 0.

48

1

Introduction

For well-posedness, it is assumed that x ∈ J ⇒ x+ ∈ F , and it is shown that Dr ≥ 0 is a sufficient condition for that. Also, it is assumed that the disturbances are L2 -bounded: ∞ 1 d22 = d  (t)d(t) dt ≤ , δ 0 for some finite 1/δ. The target L2 tracking criterion is a bound in the form ˙ 22 ≤ η, e22 + e

(1.63)

which is shown to imply asymptotic tracking, that is, limt→∞ e(t) = 0. Since r(t) is assumed to be known a priori, it is directly included in the model that in this way only considers the error response e(t) with respect to bounded disturbances d(t). Let us briefly see the case of constant r = r0 (the case r(t) = r0 e−εt is also considered). The closed-loop flow and jump modes are respectively given by x˙ = Ax + Br0 ,

x+ = AJ x,

and thus the equilibrium state is xe = −A−1 Br0 . Introducing the coordinate change x˜ = x − xe and considering disturbances d(t), the closed-loop equations take the form (flow and jump): x˜ + = AJ x˜ , x˜˙ = A˜x + Bd d, e˜ = −C x˜ . This is a hybrid system from d to e, and it is analyzed by means of the Lyapunov function V (˜x) = x˜  P x˜ and the quantity  1  L = V (˜x) + e e + e˙ e˙ − γ d  d. γ It is required that L < 0 along the continuous trajectories of the system (state in the flow set) and non-increasing variation ΔV (˜x) = V (˜x+ ) − V (˜x) ≤ 0 along resets (state in the jump set). By processing the inequalities for these two conditions, two computable LMIs are obtained and thus their feasibility is a sufficient condition: • If d = 0, the reset control system is globally asymptotically stable. • If d = 0, then a relation like (1.63) holds in the form ˜ 0 + γ 2 d22 , e22 + e ˙ 22 ≤ γ x˜  0 Px

(1.64)

which in particular implies asymptotic tracking limt→∞ e(t) = 0. To conclude this subsection, we refer the reader to [34] for the treatment with PWQ Lyapunov functions (to reduce conservativeness), and for time-varying references in the form r(t) = r0 e−εt , and for illustrative numerical examples and simulations.

1.7.3 Design Based on Fixed Reset Instants tk In the following, a brief summary of the results given in [28] and [45] will be provided. The main differences with the previous design approaches to reset control

1.7 Other Recent Results on Reset Control

49

are that the reset instants tk are a priori fixed and known, and that the after reset values of the controller state xr (tk+ ) are not zero, but calculated online for optimal performance. For the sake of concreteness, let us follow the presentation in [45]. The system considered is the typical reset control system like the one in Fig. 1.34. No disturbances are considered (d = 0), and in addition the reference input is assumed to be constant r(t) = r. The closed-loop reset system is then described by: x˙ = Ax + Br  + x tk = Mk x + Nk r

if t = tk , if t = tk ,

with pre-specified reset times t1 < t2 < · · ·. Note that in this case the reset control system analysis is much simpler, indeed it is a particular type of an impulsive system with impulses at fixed times (see Sect. 1.5.1), and thus a number of results including well-posedness and stability can be directly applied. For example, internal stability, with zero reference r = 0, can be derived by using an auxiliary discrete-time (referred to as induced difference system):     +  = Mk+1 eA(tk+1 −tk ) x tk+ =: Lk+1 x tk+ . x tk+1 If Δtk := tk − tk−1 < ΔT for some fixed ΔT < ∞ and for all k, then the reset control system is (asymptotically) stable if and only if the induced difference system is (asymptotically) stable. As a corollary, in the simplest case in which both Δtk = δ and Mk = M are constant, the system is asymptotically stable if and only if all the eigenvalues of MeAδ =: L have modulus strictly smaller than one, that is, L is Schur stable. The main idea in [45] and [28] is to let free the controller after-reset state xr (tk+ ) (not necessarily equal to zero), and to compute xr (tk+ ) in order to minimize a quadratic performance index in the form: tk+1 ˙ k+1 ) + e (s)P1 e(s) ds, Jk = e (tk+1 )P0 e(tk+1 ) + e˙ (tk+1 )Q0 e(t tk

where the tracking error is e(t) = r(t) − y(t) = r(t) − Cx(t). The index Jk can be seen as a moving index for the kth time interval [tk , tk+1 ] containing two terminal contributions (weighted by P0 , Q0 ) and a distributed contribution (the one weighted by P1 ). The choice of xr (tk+ ) at the beginning of the interval is the one that minimizes Jk along the interval [tk , tk+1 ], in a moving horizon optimization that resembles in part the idea in model-based predictive control (MPC). For the solution, it is assumed that for any reference values r ∈ R there exists a steady state value xss such that Axss + Br = 0,

Cxss − r = 0,

which, in the case of stable A, amounts to G(0) = 1, that is, unit dc gain of the closed loop G(s) = C(sI − A)−1 B. The solution for xr (tk+ ) that minimizes Jk has the form    −1 xr tk+ = − Γk22 Γk21 (xp − xpss ) + xrss ,

50

1

Introduction

where the subscripts p, r, ss stand for the plant, reset controller, and steady-state, respectively, and Γk21 , Γk21 are matrices that can be obtained from the problem data (A, B, C, P0 , Q0 , P1 ). To conclude this section, some simulation results that correspond to a real experimental application (a PZT microactuator positioning stage) are shown. The dynamics of the microactuator plant can be described by the state space model: ⎧ ⎨ x˙1 (t) = x2 (t), x˙2 (t) = −a1 x1 (t) − a2 x2 (t) + bu(t), ΣP : ⎩ y(t) = x1 (t), where x1 , x2 are position and velocity of the moving stage and u is the control input, and the parameters are a1 = 106 , a2 = 1810, and b = 3 × 106 . On the other hand, the reset controller has a PI as base linear controller, and a periodic reset action: ⎧ if t = tk , ⎨ x˙r (t) = e(t) + xr (tk ) = E1 x1 (t) + E2 x2 (t) + Gr if t = tk , ΣR : ⎩ u(t) = ki xr (t) + kp e(t). The PI gains were tuned to kp = 0.08 and ki = 300. The optimization method in [45] was applied to an index Jk with weights P0 = 2.1, Q0 = 10−6 , and P1 = 0. The reset time interval was fixed constant to Δtk = tk − tk−1 = 1 ms. Then, the optimal solution is given by the constant matrices E1 = −2.8 × 10−4 , E2 = −6.8 × 10−7 , and G = 0.0014. Figures 1.35 and 1.36 show the simulation results for the position output and control signal, respectively, in three cases: the linear control, the standard reset control, and the reset control with reset at fixed instants and variable reset. It can be seen that the linear controller (the PI controller used as base linear system in the reset controllers) is fast, but very underdamped. On the other hand, the standard reset controller (that resets to zero at the zero crossings of the error) is completely inadequate, producing sustained oscillations with underdamping around the steady state values. Finally, the solution presented by [45] (with Δtk = 1 ms and the weighting matrices given above), called an ‘improved reset controller’, gives rise to clearly the best response. It is even faster than the linear solution, but with zero overshoot. To be completely fair, it should be noted that the standard reset controller is not the best possible design; note that the steady-state oscillation is due to the fact that there is no integrator in the loop, thus it is not a good idea to reset the integrator to zero, a potentially better design would be a PI + CI controller (see Chap. 5). In addition, the proposed compensator uses state feedback, which means that, in principle, it is more sensitive to plant uncertainty if an observer is used.

1.8 Preview of the Chapters In Chap. 2, the definition of a reset control system, or reset system in general, is given. In general, as it is common to impulsive systems, reset systems may exhibit different types of solutions, in particular having complex patterns such as beating,

1.8 Preview of the Chapters

51

Fig. 1.35 Step responses of the example in Zheng et al. [45]

deadlock, and Zenoness. In control practice, this type of behavior is considered pathological and thus several conditions will be given for reset control systems to be well-posed. On the other hand, important properties of reset systems may be derived by analyzing the reset instants that correspond to a given initial condition. These patterns will be also analyzed, and their relationship with the observability and reachability of the base linear system will be shown. This material is based on the work [15]. Chapter 3 is devoted to the stability problem of reset control systems with finitedimensional base systems. The stability problem is addressed from different, complementary points of view: (i) internal or Lyapunov stability, (ii) external or input– output stability with passivity analysis, and (iii) stability by the describing function method. Internal stability techniques are subdivided into techniques giving rise to stability conditions that do not depend directly on the reset instants (reset-times independent) or, alternatively, are reset-times dependent. The first case is obtained directly using continuous time Lyapunov functions (giving rise to the so-called Hβ condition), while the second case (reset-times dependent) requires a discretization at the after-reset values and a subsequent discrete-time Lyapunov analysis. Then, the input–output L2 stability is studied, and a number or results are presented in connection with passivity and dissipativity properties of reset feedback loops. Finally, the standard describing function tool is used for approximately predicting the appearance or absence of oscillations, and as a good practical tool to evaluate the

52

1

Introduction

Fig. 1.36 Control inputs for the step responses of the example in Zheng et al. [45]

phase lead obtained by reset compensation. This material is based on several published works [7, 9, 11, 13, 14, 19, 20]. Stability of time-delay systems under reset control is approached in Chap. 4. Since reset control is able to overcome fundamental limitations, and time-delay is one source of such limitations, it is of great interest to study the problem of delayed reset systems. The stability is addressed by choosing an appropriate Lyapunov– Krasovskii functional, and by imposing that the functional should decrease in the continuous and reset modes. The resulting conditions take the form of linear matrix inequalities, and, depending on the chosen functional, these LMIs can be delaydependent or delay-independent. In both cases, those LMIs, derived from timedomain stability conditions, are translated into equivalent frequency-domain conditions by means of adequate tools, like the Kalman–Yakubovich–Popov lemma, or passivity techniques. From the latter frequency-domain conditions, useful interpretations are exhibited regarding the achieved robustness in terms of scaled small-gain or positive realness of certain subsystems. Finally, several examples illustrate the application of the stability conditions, showing the potentials of reset control when applied to time-delay systems. The chapter is based on [5, 10, 12, 16]. In Chap. 5, reset compensation has been used to overcome limitations of LTI compensation. In this chapter, a new reset compensator, referred to as PI + CI, is introduced. It basically consists of adding a Clegg integrator to a PI compensator with the goal of improving the closed-loop response by using the nonlinear charac-

References

53

teristic of this element. It turns out that by resetting a percentage of the integral term in a PI compensator, a significant improvement can be obtained over a well-tuned PI compensator in some relevant practical cases, such as systems with dominant lag and integrating systems. The main goal is the development of PI + CI tuning rules for basic dynamic systems in a wide range of applications, including first and higher order plus dead time systems. In addition, a number of design improvements such us the use of a fixed or variable reset band, the integration with QFT, and the use of a variable reset percentage are discussed. This material is based on [6, 8, 24, 38–41]. Finally, in Chap. 6, several practical applications of reset control systems will be developed, all based on the PI + CI compensator: a temperature control system of a heat exchanger, a bilateral teleoperation control system, and finally, a temperature control of a solar collector field. The first two applications have been tested by means of experiments in plants, while the third has been tested by using a (wellproven) simulator of the field. The chapter is based on [23, 24, 38, 39, 41, 42].

References 1. Aangenent, W.H.T.M., Witvoet, G., Heemels, W.P.M.H., van de Molengraft, M.J.G., Steinbuch, M.: Performance analysis of reset control systems. Int. J. Robust Nonlinear Control 20(11), 1213–1233 (2009) 2. Åström, K.J.: Limitations on control system performance. Eur. J. Control 6, 2–20 (2000) 3. Åström, K.J., Murray, R.M.: Feedback Systems. An Introduction for Scientists and Engineers. Princeton University Press, Princeton (2008) 4. Bainov, D.D., Simeonov, P.S.: Systems with Impulse Effect: Stability, Theory and Applications. Ellis Horwood, Chichester (1989) 5. Baños, A., Barreiro, A.: Delay-independent stability of reset control systems. In: 32nd Annual Conference of IEEE Industrial Electronics Society, Paris, France (2006) 6. Baños, A., Vidal, A.: Definition and tuning of PI + CI reset controller. In: European Control Conference, Kos, Grecia (2007) 7. Baños, A., Carrasco, J., Barreiro, A.: Reset times-dependent stability of reset control system. In: European Control Conference, Kos, Grecia (2007) 8. Baños, A., Vidal, A.: Design of PC + CI reset compensators for second order plants. In: IEEE International Symposium on Industrial Electronics, Vigo, España (2007) 9. Baños, A., Carrasco, J., Barreiro, A.: Reset times-dependent stability of reset control with unstable base systems. In: IEEE International Symposium on Industrial Electronics, Vigo, España (2007) 10. Baños, A., Barreiro, A.: Delay dependent stability of reset control systems. In: American Control Conference, New York, EE.UU (2007) 11. Baños, A., Dormido, S., Barreiro, A.: Stability analysis of reset control systems with reset band. In: 3rd IFAC Conference on Analysis and Design of Hybrid Systems, Zaragoza, España (2009) 12. Baños, A., Barreiro, A.: Delay-independent stability of reset systems. IEEE Trans. Autom. Control 54(2), 341–346 (2009) 13. Baños, A., Barreiro, A.: Limit cycles analysis in reset systems with reset band. Nonlinear Anal. Hybrid Syst. (2010). doi:10.1016/j.nahs.2010.07.004 14. Baños, A., Carrasco, J., Barreiro, A.: Reset times-dependent stability of reset control systems. IEEE Trans. Autom. Control 56(1), pp. 217–223 (2011) 15. Baños, A., Mulero, J.I.: On the well-posedness of reset control systems. Technical Report TR-DIS-1-2011, University of Murcia (2011)

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16. Barreiro, A., Baños, A.: Delay-dependent stability of reset systems. Automatica 46(1), 216– 221 (2010) 17. Beker, O., Hollot, C.V., Chait, Y., Han, H.: Plant with integrator: an example of reset control overcoming limitations of linear systems. IEEE Trans. Autom. Control 46(11), 1797–1799 (2001) 18. Beker, O., Hollot, C.V., Chait, Y., Han, H.: Fundamental properties of reset control systems. Automatica 40, 905–915 (2004) 19. Carrasco, J., Baños, A., van der Schaft, A.: A passivity approach to reset control of nonlinear systems. In: 34th Annual Conference of the IEEE Industrial Electronics Society, Orlando, EE.UU (2008) 20. Carrasco, J., Baños, A., Barreiro, A.: Stability of reset control systems with inputs. In: 16th IEEE Mediterranean Conference on Control and Automation, Ajaccio, Francia (2008) 21. Chait, Y., Hollot, C.V.: On Horowitz’s contributions to reset control. Int. J. Robust Nonlinear Control 12, 335–355 (2002) 22. Clegg, J.C.: A nonlinear integrator for servomechanism. Trans. AIEE, Part II 77, 41–42 (1958) 23. Fernández, A.F., Barreiro, A., Baños, A., Carrasco, J.: Reset control for passive teleoperation applications in process control. In: 34th Annual Conference of the IEEE Industrial Electronics Society, Orlando, EE.UU (2008) 24. Fernández, A.F., Barreiro, A., Baños, A., Carrasco, J.: Reset control for passive bilateral teleoperation. IEEE Trans. Ind. Electron. (2010). doi:10.1109/TIE.2010.2077610 25. Feuer, A., Goodwin, G.C., Salgado, M.: Potential benefits of hybrid control for linear time invariant plants. In: Proc. Amer. Control Conf., Alburquerque, pp. 2790–2794 (1997) 26. Goebel, R., Sanfelice, R., Teel, A.R.: Hybrid dynamical systems. IEEE Control Syst. Mag. (2009) 27. Goodwin, G.C., Graebe, S.F., Salgado, M.E.: Control System Design. Prentice-Hall, Upper Saddle River (2001) 28. Guo, Y., Wang, Y., Zheng, J., Xie, L.: Stability analysis, design and application of reset control systems. In: Proceedings of the IEEE Int. Conf. on Control and Automation, Guangzhou, China, May 30–June 1 (2007) 29. Haddad, W.M., Nersesov, S.G., Chellaboina, V.S.: Energy-based control for hybrid portcontrolled Hamiltonian systems. Automatica 39, 1425–1435 (2003) 30. Haddad, W.M., Chellaboina, V.S., Nersesov, S.G.: Impulsive and Hybrid Dynamical Systems. Stability, Dissipativity and Control. Princeton University Press, Princeton (2006) 31. Horowitz, I.M., Rosenbaum, P.: Nonlinear design for cost of feedback reduction in systems with large parameter uncertainty. Int. J. Control 24(6), 977–1001 (1975) 32. Krishnan, K.R., Horowitz, I.M.: Synthesis of a nonlinear feedback system with significant plant-ignorance for prescribed system tolerances. Int. J. Control 19(4), 689–706 (1974) 33. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) 34. Loquen, T., Tarbouriech, S., Prieur, Ch.: Stability of reset control systems with nonzero reference. In: Proceedings of the 47th IEEE Conf. on Decision and Control, Cancun, Mexico (2008) 35. Middleton, R.H., Graebe, S.F.: Slow stable open loop poles: to cancel or not to cancel. Automatica 35, 877–886 (1999) 36. Neši´c, D., Zaccarian, L., Teel, A.R.: Stability properties of reset systems. Automatica 44(8), 2019–2026 (2008) 37. Seron, M., Braslavsky, J.H., Goodwin, G.C.: Fundamental Limitations in Filtering and Control. Springer, London (1997) 38. Vidal, A., Baños, A., Moreno, J.C., Berenguel, M.: PI + CI compensation with variable reset: application on solar collector fields. In: 34th Annual Conference of the IEEE Industrial Electronics Society, Orlando, EE.UU (2008) 39. Vidal, A., Baños, A.: QFT-based design of PI + CI reset compensator: applications in process control. In: 16th IEEE Mediterranean Conference on Control and Automation, Ajaccio, Francia (2008)

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40. Vidal, A., Baños, A.: Stability of reset control systems with variable reset: application to PI + CI compensation. In: European Control Conference, Budapest, Hungary (2009) 41. Vidal, A., Baños, A.: Reset compensation applied on industrial heat exchangers. In: 14th IEEE International Conference on Emerging Technologies and Factory Automation, Mallorca, Spain (2009) 42. Vidal, A., Baños, A.: Reset compensation for temperature control: experimental applications on heat exchangers. Chem. Eng. J. 159(1–3), 170–181 (2010) 43. Yang, T.: Impulsive Control Theory. Lecture Notes in Control and Information Sciences, vol. 272. Springer, Berlin (2001) 44. Zaccarian, L., Nesic, D., Teel, A.R.: First order reset elements and the Clegg integrator revisited. Proc. Am. Control Conf. 1, 563–568 (2005) 45. Zheng, J., Guo, Y., Fu, M., Wang, Y., Xie, L.: Improved reset control design for a PZT positioning stage. In: Proceedings of the 16th IEEE Int. Conf. on Control Applications, Singapore (2007)

Chapter 2

Definition of Reset Control System and Basic Results

2.1 Preliminaries and Problem Setup The main focus of the book will be on the use of single-input single-output reset compensators having as reset condition the classical condition of zero input. Therefore, the main goal will be to analyze and design reset control systems with linear and time invariant base systems. This chapter will be devoted to the definition of a reset control system, developing basic conditions for a system to be well-posed. In addition, an analysis of the dependence of zero crossing instants with respect to initial conditions will be given, showing the complex patterns that may result depending on the dimension of the after-reset surface. Consider the feedback control system of Fig. 2.1, where the system P , with state xp , is described by  (P )

x˙ p (t) = Ap xp (t) + Bp u(t),

xp (0) = xp0 ,

y(t) = Cp xp (t)

(2.1)

and the reset compensator R, with state xr , is modeled in principle by the impulsive differential equation  x˙ r (t) = Ar xr (t) + Br e(t) if e(t) = 0, (R) (2.2) xr (t + ) = Aρ xr (t) if e(t) = 0, where xr (0) = xr0 and v(t) = Cr xr (t). Here np is the dimension of the state xp , and nr is the dimension of the state xr . In addition, xr (t + ), or simply x+ r , is the value xr (t + ε) with ε → 0+ . Aρ is a diagonal matrix with diagonal elements (Aρ )ii = 0 if the compensator state component (xr )i is to be reset, and (Aρ )ii = 1 otherwise, i = 1, . . . , nr . In general, it is assumed that the first nρ¯ compensator state components are not reset, and the last nρ compensator states are reset or set to zero at the reset instants t in which the compensator input e(t) is zero. Thus, nr = nρ¯ + nρ and Aρ is given by Aρ = diag(Inρ¯ , Onρ ). For the particular case corresponding to nρ¯ = 0 A. Baños, A. Barreiro, Reset Control Systems, Advances in Industrial Control, DOI 10.1007/978-1-4471-2250-0_2, © Springer-Verlag London Limited 2012

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2

Definition of Reset Control System and Basic Results

Fig. 2.1 Reset controller R applied to an LTI plant

and nr = nρ , that is, if Aρ is the zero matrix, the reset compensator will be referred to as full reset. Otherwise, it will be referred to as partial reset. The closed-loop autonomous unforced system is given by e(t) = −y(t), u(t) = v(t), where by definition the closed-loop state of dimension n = np + nr is   x (2.3) x= p . xr The result is the closed-loop system (without exogenous inputs)  x˙ (t) = Ax(t) if x(t) ∈ / M, x(t + ) = AR x(t)

if x(t) ∈ M

(2.4)

 A B C x  , A = −BrpCp Ap r r , AR = with x(0) = x0 and y(t) = Cx(t), and where x0 = xp0 r0 diag(Inp , Aρ ) = diag(Inp , (Inρ¯ , Onρ )), and C = (Cp 0). Therefore, to complete the closed-loop system equations, the set M , which will be referred to later as the reset surface, needs to be defined. Another set, the afterreset surface MR , also plays an important role in the definition of closed-loop system solutions. Note that reset actions occur when the state x(t) contacts the reset surface M at some instant t, that is, x(t) ∈ M , and then the state jumps to AR x(t) ∈ MR . In general, the set MR will be defined as MR = R(AR ) ∩ N (C),

(2.5)

where R(X) and N (X) stand for the image and the null subspace of the linear operator given by the matrix X, respectively. Thus, MR is the set of states x that belong both to the null space of C (and then the output is y = Cx = 0) and to the image space of AR (they are the after-reset states). In addition, the set M is defined as M = N (C) \ MR

(2.6)

where the after-reset states are removed from the reset surface. Otherwise, an infinite number of resets would be produced after a reset action; in general, the reset system (2.4) can exhibit rather complex behaviors that have been referred to as beating or pulse phenomena in the literature on impulsive systems [1, 10]. They are related to the fact that reset instants may not be well defined or may not be distinct; in addition, even if the reset instants are well defined and are distinct, they may converge to a finite number and then the reset system exhibits Zenoness. This topic will be analyzed in detail in Sect. 2.2.2; however, a common solution in practice to avoid these behaviors is to use time regularization.

2.1 Preliminaries and Problem Setup

59

A time regularization solution can be simply constructed adopting the scheme proposed in [12]: the system (2.4) is modified including a temporal restriction over the reset instants by simply avoiding resets if some minimum time between resets Δm has not passed. Thus, the closed-loop system will be described by  ˙ = 1, Δ(t) x˙ (t) = Ax(t), (x(t) ∈ / M ) ∨ (Δ ≤ Δm ), (2.7) Δ(t + ) = 0, x(t + ) = AR x(t), (x(t) ∈ M ) ∧ (Δ > Δm ) with Δ(0) = 0, x(0) = x0 , and y(t) = Cx(t), and where the reset action is only performed if the state x(t) contacts the reset surface M at some reset instant tj , j = 1, 2, . . . , and in addition every reset interval Δk = tk − tk−1 satisfies Δk > Δm , k = 1, 2, . . . , where Δm > 0 is some given constant.

2.1.1 Reset Control System Solutions The reset control system given by (2.4) or (2.7) is a special class of a system with impulse effects, or an impulsive system. There is a large literature on impulsive systems [1, 5, 8, 10, 11, 14]. Most of the work has been done in the area of systems with impulses at fixed instants, or systems with impulses dependent on the state. The case in which reset and after-reset surfaces are time independent, which is usually referred to as autonomous impulsive systems, has attracted considerably less attention in spite of being relevant for engineering applications including control systems, this being the case of reset control systems. The framework developed in [8] will be (partly) adopted here. The LTI system described by the first equation in (2.4) will be referred to as the linear base system, or simply the base system, while the second equation in (2.4) will be referred to as the resetting law. Let D ⊂ Rn be an initial conditions set, and Ix0 ⊆ [0, ∞), with x0 ∈ Rn , a dense subset of [0, ∞) such that Jx0 = [0, ∞) \ Ix0 is a countable set with a finite or infinite number of elements; in fact, this set will be the set of reset times corresponding to the initial condition x0 . In general, an initial condition x0 may produce a finite or infinite number of resets, depending on whether the set Jx0 is finite or infinite. A function x : [0, ∞) × D → Rn is a solution to the reset control system (2.4) if the following conditions are satisfied: 1. x(·, x0 ) is left-continuous in t, that is, limτ →t − x(τ, x0 ) = x(t, x0 ) for all x0 ∈ D and t ∈ (0, ∞). 0) 2. x(·, x0 ) is differentiable in t, and dx(t,x = Ax(t, x0 ), for all t ∈ Ix0 . dt + 3. x(t , x0 ) = AR x(t, x0 ), for all t ∈ Jx0 . In addition, functions τk : D → [0, ∞), k = 1, 2, . . . , are defined such that τk (x0 ) is the kth reset instant of the solution x(·, x0 ). Therefore, functions Δk : Rn → [0, ∞), k = 1, 2, . . . , are defined such that Δk (x0 ) = τk (x0 ) − τk−1 (x0 ) is the kth reset interval, with τ0 (x0 ) = 0 for any x0 ∈ Rn . The following result is directly adapted from [1] with minimal effort.

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Definition of Reset Control System and Basic Results

Proposition 2.1 For any initial condition x0 ∈ D , assume that τ1 (x0 ) < τ2 (x0 ) < · · · < τk (x0 ) < · · · , and τk (x0 ) → ∞ as k → ∞, then there exists a unique solution x(·, x0 ) (x(·) in short) to the reset control system (2.4) that can be written in the form x(t, x0 ) = W (t, x0 )x0 for t > 0, where the transition matrix W (t, x0 ) is given by W (t, x0 ) = eA(t−τk (x0 )) AR eA(τk (x0 )−τk−1 (x0 )) AR · · · AR eAτ1 (x0 )

(2.8)

for t ∈ (τk (x0 ), τk+1 (x0 )]. Proof Since reset instants are directly determined by the initial condition, for a given initial condition x0 ∈ D and its reset instants tk = τk (x0 ), k = 1, 2, . . . , the system (2.4) is an impulsive system with impulses at fixed instants t1 , t2 , . . . , where by assumption 0 = t0 < t1 < t2 < · · · . A direct application of Theorem 3.6 and Corollary 3.2 in [1] gives the result.  Note that in general if the formulation (2.7) for reset system is used then the reset intervals are lower-bounded by the time regularization constant Δm , that is, Δk (x0 ) > Δm , k = 1, 2, . . . for any x0 ∈ R, and thus the assumptions of Proposition 2.1 are clearly satisfied. If time regularization is not performed, deadlock and beating could be present if τk+1 (x0 ) = τk (x0 ) for some initial condition x0 and some k; or even a Zeno solution may be obtained if τk (x0 ) < ∞ for k → ∞. Alternatively, if the initial instant is t0 = 0 and the initial condition is x(t0 ) = x0 , then a solution x(t, t0 , x0 ) is defined in a similar way. In Sect. 2.2, a detailed analysis of beating, deadlock, and Zeno solutions in the reset system (2.4) will be given. It will be shown that under mild conditions it does not have these behaviors and that time regularization is not necessary to have welldefined solutions for forward time.

2.1.2 Characterization of Reset Intervals An important question that has not been previously approached in the reset control literature is the analysis of reset instants for a given reset control system. For example, given the base system of (2.4), it is not evident if every initial condition will cross the reset surface M at some finite instant. Usually, since at least a reset action is wanted for initial conditions in some set D , a common practice is to design the base closed-loop system to have a pair of dominant complex poles, and thus to force crossings after some arbitrarily large time. In the following, it will be shown that this practice is, in fact, theoretically supported, and in addition an upper bound over resets intervals will be computed. In general, the problem of computing crossings of the solution of a linear system governed by x˙ (t) = Ax(t), with an initial condition x0 ∈ R, with a given hyperplane, is a particular instance of the reachability problem for linear systems. In general, for arbitrary values of the state matrix A, it has been shown to be an open problem, referred to as the continuous Skolem–Pisot problem [4, 9]. As discussed above,

2.1 Preliminaries and Problem Setup

61

this is a central problem in reset control where a base system has to be designed for the reset system to perform crossings with a reset surface. For the analysis of the crossings with the hyperplane defined by the row vector C, that is, the instants t > 0 at which CeAt x0 = 0, it is convenient to use the equation Ce(A+λI )t x0 = 0 for some λ ∈ R, obtaining the same results [4]. This is a simple, but key simplifying result because all the eigenvalues of A can be assumed to have non-positive real parts without loss of generality, for some λ properly chosen. In addition, it has also been shown [4] that if for some initial condition x0 the matrix A does not have dominant real eigenvalues and (A, C, x0 ) is reduced, the continuous Skolem–Pisot problem always has a solution for that initial condition. By definition, an eigenvalue of A is dominant if it is the rightmost placed eigenvalue of A in the complex plane. In addition, the term CeAt x0 can always be split as CeAt x0 = y1 (t) + y2 (t), where by definition the dominant term y1 (t) is not identically zero if (C, A, x0 ) is reduced, and in addition y2 (t) tends to zero exponentially fast as t increases. This means that a reset control system with such a matrix A will always produce crossings for the initial condition x0 . Note that the set of states x0 for which (A, C, x0 ) is not reduced has a linear subspace structure, and will be re¯ Note that if the modes corresponding to dominant eigenvalues are ferred to as R. unobservable then (A, C, x0 ) is not reduced for any x0 ∈ Rn , and thus R¯ = Rn , but in the case they are observable, (A, C, x0 ) may not be reduced for some initial condition x0 ∈ Rn , and in general R¯ is not the empty set. In general, if the matrix A ∈ Rn×n has eigenvalues λ1 , λ2 , . . . , λs such that ki = index(λi ), then it can be expressed as ⎛ ⎞ Q1 s ⎜ ⎟ A = (P1 . . . Ps )J ⎝ ... ⎠ = Pi J (λi )Qi , (2.9) i=0 Qs where J is the Jordan form, and J (λi ) the Jordan segment associated to the eigenvalue λi . In addition, for a function f : R → R such that f (λi ), f (λi ), . . . , f (ki −1) (λi ) exist for each i = 1, . . . , s, the value of f (A) can be determined by using a generalization of the spectral theorem for non-diagonalizable matrices f (A) =

s k i −1

f (j ) (λi ) i=1 j =0

j!

(A − λi I )j Gi ,

(2.10)

where Gi = Pi Qi is the spectral projector corresponding to the eigenvalue λi . In the case in which the matrix A is diagonalizable, the expression of f (A) is much simpler since index(λi ) = 1, for i = 1, . . . , s. In this case, f (A) =

s

f (λi )Gi ,

(2.11)

i=1

where in addition the spectral projectors are simply given by Gi = vi wTi , with vi and vj being the right-hand eigenvector and the left-hand eigenvector corresponding to the eigenvalue λi , respectively.

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In the following, it will be shown how a reset control system produces an infinite number of resets, and that the reset intervals are upper-bounded, if the base system does not have real dominant eigenvalues. Proposition 2.2 Consider the reset control system (2.4), where the matrix A does not have real dominant eigenvalues, and a nonempty closed set of initial conditions ¯ then reset intervals are uniformly upper-bounded, that is, Δk (x0 ) = D ⊂ Rn \ R, τk+1 (x0 ) − τk (x0 ) < ΔM , k = 1, 2, . . . , for any x0 ∈ D and some finite constant ΔM > 0. Proof It is assumed, without loss of generality, that A has dominant imaginary eigenvalues. The case in which A has a pair of dominant eigenvalues will be considered in the following (with multiplicity not necessarily equal to one), the more general case of multiple dominant eigenvalues is a bit more involved but follows a similar reasoning. The eigenvalues of A are ordered in such a way that its spectrum is σ (A) = {λ1 , λ2 , . . . , λs } = {iβ1 , −iβ1 , α3 + iβ3 , . . . , αs + iβs }, and αl < 0, l = 3, . . . , s. In addition, let kl , l = 1, . . . , s, be the index of each eigenvalue. Then, the output y(t) of the reset control system can be expressed (using (2.9) and (2.10) for computing eAt ) as y(t) =

s k l −1 j λl t

t e l=1 j =0

j!

C(A − λl I )j Gl x0 ,

(2.12)

where Gl is the spectral projector associated to the eigenvalue λl . It can be split in two parts as y(t) = y1 (t) + y2 (t), where y1 (t) =

2 k l −1 j iβl t

t e l=1 j =0

j!

C(A − iβl I )j Gl x0

(2.13)

and y2 (t) =

s k l −1 j αl t iβl t

t e e l=3 j =0

j!

C(A − (αl + iβl )I )j Gl x0 .

(2.14)

Here y2 (t) tends to zero exponentially fast as t increases since αl < 0, l = n + 1, . . . , s, and thus it is clear that for any ε2 > 0 it is always possible to choose an instant t2 large enough such that |y2 (t)| < ε2 x0 , for t ≥ t2 . k −1 j In addition, defining constant matrices Zlj = j l=0 (A−iβjl!I ) Gl ∈ C and matrix  −1 polynomials Pl (t) = ( jkl=0 Zlj t j ), the dominant term of the output y1 (t) can be expressed as   y1 (t) = C P1 (t)eiβl t + P2 (t)e−iβl t x0 = C(P1,R (t) cos(β1 t) + P1,I (t) sin(β1 t))x0 ,

(2.15)

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63

where P2 (t) = P¯1 (t), P1,R = 2 Real{P1 }, and P1,I = 2 Imag{P1 }. Now, for instants tm = m 2π β1 , with m ∈ N, the value of the dominant term is given simply by y1 (tm ) = CP1,R (tm )x0 , and since (C, A, x0 ) is reduced for any x0 ∈ D , and D is closed, there must exist some lower bound ε1 > 0 such that |y1 (tm )| > ε1 x0 , for any m and x0 ∈ D . This can be shown by contradiction, if it were false then it would exist a sequence of states {x01 , x02 , . . . } → x0 such as for any t > 0, CP1,R (t)x0n → 0 as n → ∞. Thus, since D is closed it must contain a state x0 such as (C, A, x0 ) is not reduced, which is a contradiction. Also note that for an m1 large enough, the sign of y1 (tm ) is constant for m > m1 (for example, choosing tm1 as the Cauchy’s bound of the roots of the polynomial CP1,R (t)x0 ). It will be assumed that y1 (tm1 ) > 0, otherwise tm = m πβ , m ∈ N, may be chosen. Therefore, it is possible to find a large enough instant t2 such as ε2 < ε1 and thus | yy21 (t(tmm )) | < y2 (tm ) y1 (tm ) ) > 0,

ε2 x0 ε1 x0

=

ε2 ε1

< 1, and then it is

true that y(tm ) = y1 (tm )(1 + provided that tm > max{tm1 , t2 }. A similar reasoning may be applied to show that y(tn ) < 0, with tn = n βπ1 , for some tn > max{tn1 , t2 }, where n1 is computed in a similar way. As a result, the output of the reset system y(t) = CeAt x0 , for some initial condition x0 ∈ Rn , is equal to 0 for t < tm , with the upper bound tm being independent of the initial condition x0 . Then, since the reset action is only active for t > Δm , Δ1 (x0 ) = τ1 (x0 ) < Δm + tm =: ΔM . Obviously, the rest of the reset intervals are also upper-bounded by ΔM , and thus the proof is complete. 

2.2 Zenoness, Beating, and Deadlock In control practice, it is required that the reset control system solutions x(t, x0 ) be well defined in the sense that they exist and are unique for any t > 0, as given in Proposition 2.1. As it has been previously discussed, this may be complicated by the fact that these solutions may exhibit complex phenomena such as non-continuability of solutions or deadlock, beating or livelock, and Zenoness. These concepts will be used as defined in [8]. Deadlock occurs when the state x(t) cannot evolve in time because no continuation, continuous or discrete, is possible. To avoid deadlock, a typical assumption (adapted from [8]) is that / M, (A1) ∀x(t) ∈ MR , ∃ε > 0 such that ∀δ ∈ (0, ε) x(t + δ) ∈ that is, the after-reset states evolve with the continuous base dynamics for some finite time interval. Beating appears when the system solution encounters the reset surface after resetting, which in our case is simply avoided by assuming that (A2) MR ∩ M = ∅, that is, assuming that the after-reset states are not elements of the reset surface. Finally, a Zeno solution exists if a system solution has infinitely many reset actions in a finite time. By definition, the reset control system (2.4), with sets MR and M , is wellposed if it satisfies conditions (A1) and (A2) (and thus it does not exhibit beating or

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deadlock, but Zeno solutions may exist in principle). Note that in the reset control system (2.4), the reset action occurs at the instants t at which the output y(t) is zero. The set MR is defined as in (2.5), that is, as the set of states that belong both to the null space of C (and then the output is y = Cx = 0) and to the image space of AR (they are the after reset states). In addition, the set M will be defined as in (2.6), where the points that are the after-reset states are removed from the reset surface to satisfy condition (A2), otherwise an infinite number of resets may be produced after a reset action, and beating would be present.

2.2.1 Well-posedness: Beating and Deadlock To have well-defined solutions to the reset systems as given in last section, reset instants have to be well defined and distinct. In general, two phenomena that have to be avoided are deadlock (non-continuability of solutions), and beating or livelock. Another important type of solutions like Zeno solutions will be treated in next section. In general, a reset control system as defined by (2.4)–(2.6) does not exhibit beating, once the surfaces MR and M are defined according to (2.5) and (2.6), respectively. On the other hand, additional assumptions have to be made to avoid deadlock. Proposition 2.3 The reset control system (2.4)–(2.6) is well-posed if the after-reset surface MR is a subset of the observable subspace of the linear base system, that is, MR ∩ N (Obase ) = {0} where

⎛ ⎜ ⎜ Obase = ⎜ ⎝

C CA .. .

(2.16)

⎞ ⎟ ⎟ ⎟. ⎠

CAn−1 Proof Beating is avoided by defining the after-reset surface MR by (2.6). Thus, the proof is centered around deadlock. In general, given any initial condition x0 ∈ D , the reset surface M is first contacted at the instant t1 = τ1 (x0 ). Then the reset instant t1 is simply given by t1 = inf{t > 0|x(t, x0 ) ∈ M }, and thus is a solution of the equation CeAt1 x0 = 0. In addition, it must be true that eAt1 x0 ∈ / R(AR ) according to the definitions of the after-reset and reset surfaces as given by (2.5) and (2.6), respectively. For simplicity, consider first that the closed-loop state matrix A has distinct eigenvalues, then the matrix exponential may be computed by using the Caley–Hamilton method, that is, e At1 = α0 I + α1 A + · · · + αn−1 An−1 ,

(2.17)

2.2 Zenoness, Beating, and Deadlock

65

where αi , i = 0, . . . , n − 1, are given by ⎧ λt e 1 1 = α0 + α1 λ1 + · · · + αn−1 λn−1 ⎪ 1 , ⎪ ⎪ ⎪ ⎪ ⎨ eλ2 t1 = α0 + α1 λ2 + · · · + αn−1 λn−1 , 2 ⎪ ··· ⎪ ⎪ ⎪ ⎪ ⎩ λn t1 = α0 + α1 λn + · · · + αn−1 λn−1 e n .

(2.18)

Using the vector notation λT = (λ1 λ2 . . . λn ), α T = (α0 α1 . . . αn−1 ) and eλt1 =  n λi ti e , where e stands for the unit vector (0 . . . 0 1 0 . . . 0)T in which the i i i=1 e ith component is 1, (2.18) can be compactly written as eλt1 = V (λ)T α,

(2.19)

where V (λ) is a (nonsingular) Vandermonde matrix. Now, by eliminating α from (2.17) and (2.19) the equation 0 = CeAt1 x0 is transformed into ⎛ ⎜ ⎜ 0 = αT ⎜ ⎝

C CA .. .

⎞ ⎟ T ⎟ ⎟ x0 = eλt1 U (λ)Obase x0 , ⎠

(2.20)

CAn−1 where U (λ) = V (λ)−1 . Now, if condition (2.16) is satisfied, then the right-hand side of (2.20) is an analytical function (in fact, a sum of exponentials) that is not zero for all t ≥ 0. As a result, it has no isolated zeros, and then the reset instant t1 is lower-bounded. In other words, solutions of (2.20) may not be obtained for arbitrarily small values of the reset instant t1 , and thus deadlock does not occur if condition (2.16) is satisfied. In the case in which the eigenvalues of A may be repeated, a similar argument may be applied. Note that eAt1 may be written as the infinite series  ∞ λi t1i Ai t1i D(A) = ∞ i=0 i! . Thus the polynomial D(λ) = i=0 i! can be factorized as D(λ) = Q(λ)P (λ) + R(λ), with R(λ) = 0, or deg(R) < deg(P  ) = n. jIn addition, R has degree no greater than n − 1, and thus R(λ) = n−1 j =0 αj λ . Since the characteristic polynomial is zero for the eigenvalues of A, D(λk ) = R(λk ) for  λik t1i λk t1 = R(λ ) = n−1 α λj k = 0, 1, . . . , n − 1. And then D(λk ) = ∞ k i=0 i! = e j =0 j k T λt 1 for k = 0, 1, . . . , n − 1. This can be compactly expressed as V (λ)α = e , and the expression (2.19) is obtained. Now, if A has r different eigenvalues with respective  multiplicity order ni , and as a consequence the characteristic polynomial is p(λ) = ri=1 (λ − λi )ni , then again there exist unique polynomials Q and R such as D(λ) = Q(λ)P (λ) + R(λ) where λt1 D(λ) n−1= e i and R = 0 or deg(R) < deg(P ). Here R can be expressed as R(λ) = i=0 αi λ , where the coefficients are unique. Since p and its derivatives up to order

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nr are zero at λi , d j D(λi ) d j R(λi ) = dλj dλj

∀i = 1, 2, . . . , r, ∀j = 0, 1, . . . , ni − 1.

(2.21)

This can be expressed by μ = W α, where μ=

r n i −1

j

ei eλi t1 ⊗ ej λi ,

(2.22)

i=1 j =0

⎞ ⎛ r n i −1 j

∂ V (λ) ⎠ . ei ⊗ ej eTi W =⎝ j ∂λi i=1 j =0

(2.23)

By using arguments based on the Lagrange–Hermite interpolation problem, it can be shown that, in fact, the matrix W is invertible. And then an expression similar to (2.20) may be obtained. Using similar arguments as those after (2.20), the proposition is proved for the general case of repeated eigenvalues.  Obviously, a reset control system will be well-posed if the base linear system is observable. But some unobservable base linear systems can also define well-posed reset control systems. Therefore, note that in the proof of Proposition 2.3 no particular structure of the matrices A, C, and AR has been used. Thus, the result is in general valid for any reset system given by (2.4) with arbitrary values of those matrices. In the following, two examples corresponding to an ill-posed (not well-posed) reset system and a well-posed reset control system are given.

2.2.1.1 Example (Ill-posed Reset System) This example is used in [12] for analyzing some weak points in the definition of reset systems given in [3]. Consider a reset system (2.4) with the following system matrices ⎛ ⎞ ⎛ ⎞ 1 0 0 −1 0 0   C = 1 0 0 , (2.24) A = ⎝ 0 −1 −1 ⎠ , AR = ⎝ 0 1 0 ⎠ , 0 0 0 0 1 −1 where in addition the sets MR and M are defined according to (2.5) and (2.6) as   MR = R(AR ) ∩ N (C) = span (0, 1, 0)T , and     M = N (C) \ MR = span (0, 1, 0)T , (0, 0, 1)T span (0, 1, 0)T . Note that this reset system cannot be realized as a control reset system with the structure of Fig. 2.1.

2.2 Zenoness, Beating, and Deadlock

67

In [12], it is correctly argued that for any initial condition x0 = (0, a, 0)T ∈ MR , the solution is ill-defined since the continuous dynamics makes the system instantly evolve to the set MR and then the system instantly resets once the reset surface is reached, making infinitely many resets without leaving the reset surface; in fact, there is deadlock. Note that this is due to the fact that the after-reset surface MR is a subset of the unobservable subspace of the linear base system, which is given in this case by ⎛⎛ ⎞⎞ ⎛⎛ ⎞⎞ C 1 0 0 (2.25) N ⎝⎝ CA ⎠⎠ = N ⎝⎝ −1 0 0 ⎠⎠ = N (C) ⊃ MR . 1 0 0 CA2

2.2.1.2 Example (Well-posed Reset Control System) This example, adapted from [3], shows how an unobservable linear base system may define a well-posed reset system, as long as the unobservable subspace does not contain after-reset states. Consider a reset control system (2.4)–(2.6) with ⎛ ⎞ ⎛ ⎞ 0 0 1 1 0 0   A = ⎝ 1 −0.2 1 ⎠ , AR = ⎝ 0 1 0 ⎠ , C= 010 (2.26) 0 −1 −1 0 0 0 that has an unobservable mode corresponding to a stable pole–zero cancellation in s+1 , and the linear base system, where the plant has a transfer function P (s) = s(s+0.2) 1 the base compensator is C(s) = s+1 (corresponding to a first order reset element— FORE). In addition, the after-reset and reset surfaces are given by MR = R(AR ) ∩ N (C) = span{(1, 0, 0)T } and M = N (C) \ MR = span{(1, 0, 0)T , (0, 0, 1)T } \ span{(1, 0, 0)T }, respectively. In this case, the set MR is not a subset of the linear base system unobservable subspace given by ⎛⎛ ⎞⎞ ⎛⎛ ⎞⎞ C 0 1 0   −0.2 1 ⎠⎠ = span (1, 0, −1)T . N ⎝⎝ CA ⎠⎠ = N ⎝⎝ 1 −0.2 −0.96 −0.2 CA2 (2.27) As a result, Proposition 2.3 may be used to ensure that the system is well-posed. Figure 2.2 shows the system solutions corresponding to two initial conditions.

2.2.2 Zeno Solutions In principle, the reset control system (2.4) may exhibit Zeno solutions even in the case where it is well-posed (assuming that condition (2.16) is satisfied). Zeno solutions are solutions to (2.4) that have an infinite number of jumps in a compact time interval. However, as it will be shown in the following, condition (2.16) is sufficient to avoid the existence of Zeno solutions.

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Fig. 2.2 System solution for the well-posed reset system example

Proposition 2.4 The reset system (2.4)–(2.6) does not have Zeno solutions if MR ∩ N (Obase ) = {0}. Proof The basic idea of the proof consists of showing that the reset system (2.4)– (2.6), with an initial condition in MR , can only have finite sequences of reset intervals Δk , k = 1, 2, . . . , m − 1 such as Δm−1 < Δm−2 < · · · < Δ1 = ε, for some ε > 0 arbitrarily small but fixed, and some finite positive integer m. In fact, in the following it will be shown that at most there will be sequences of length m − 1, with m being the dimension of the after-reset surface MR . Without loss of generality, it is considered that the plant state equations (2.1) are given in an observer canonical form, that is, ⎞ ⎞ ⎛ ⎛ 0 0 ··· 0 −a0 b0 ⎜1 0 ··· 0 ⎜ b1 ⎟ −a1 ⎟   ⎟ ⎟ ⎜ ⎜ Bp = ⎜ . ⎟ , Cp = 0 0 . . . 1 , Ap = ⎜ .. .. . . ⎟, .. .. . ⎠ ⎝. . ⎝ . ⎠ . . . 0 0 · · · 1 −anp −1 bnp −1 (2.28) then C = (0, 0, . . . , 1, 0, . . . , 0) and thus ⎞ ⎞ ⎛ ⎛ 0 0 ··· 0 0 1 0 ... 0 C ⎜ CA ⎟ ⎜ 0 0 · · · 0 1 −anp −1 X . . . X ⎟ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ .. ⎜ .. .. .. .. .. .. . . .. ⎟ .. ⎜ ⎟ ⎜ . . . .⎟ . . . . . Obase = ⎜ ⎟, ⎟=⎜. ⎟ ⎜ CAnp −1 ⎟ ⎜ 1 −an −1 · · · X X X X · · · X p ⎠ ⎠ ⎝ ⎝ .. .. . . . . . .. .. .. .. .. . . . .. . .. . . . (2.29) where X stands for a non (necessarily) zero term.

2.2 Zenoness, Beating, and Deadlock

69

For simplicity, the case of full reset is approached at first. Thus, an after-reset state x ∈ MR is given by x = (x1 , x2 , . . . , xnp −2 , xnp −1 , 0, 0, . . . , 0)T

(2.30)

for some values x1 , . . . , xnp −1 ∈ R, with np being the number of plant states. Thus, m = np − 1 in the case of full reset. Let us start with the case m = 1, which corresponds to second order plants and full reset compensators of arbitrary order. In this case, if the reset control system is well-posed, it is well known that the reset is periodic after reaching the set MR from any initial condition; thus starting from MR , the reset will be periodic and Zeno solutions are not possible. In fact, there exists no initial condition in the set MR that contacts M in an arbitrarily small time since reset intervals are constant. The case m = 2 is analyzed in the following. Consider an initial condition x0 = x1 ∈ MR , that is, x1 = (x1 , x2 , 0, 0, . . . , 0)T . If the solution x(t, 0, x1 ) contacts the reset surface MR at time t1 = ε1 , thus Δ1 = ε1 , for some ε1 > 0 arbitrarily small, then ε12 (2.31) CA2 x1 + · · · . 2 Since the control system (2.4)–(2.6) is well-posed, the right-hand side of (2.31) is not identically zero for any x1 ∈ MR . Now using the special structure given in (2.29), one obtains   ε1 (2.32) 0 = x2 + x1 + O ε12 , 2 0 = CeAε1 x1 = Cx1 + ε1 CAx1 +

where the terms of order ε12 and higher maybe be neglected, in principle. Note that for (2.32) to be satisfied for an arbitrarily small ε1 > 0 it must be true that x1 = 0 and x2 = 0. In addition, the following after-reset state x2 is given by x2 = AR x(t1 , 0, x1 ) = (x1 + O(ε12 ), x2 + ε1 x1 + O(ε12 ), 0, 0, . . . , 0)T . Repeating the argument, the solution x(t, t1 , x2 ) will again contact M at the instant t2 = t1 + Δ2 . If Δ2 = ε2 ≤ ε1 for some ε2 > 0, then it is verified that 0 = x2 + ε1 x1 +

  ε2 x1 + O ε12 , 2

(2.33)

where the properties O(ε22 ) = O(ε12 ) for ε2 ≤ ε1 and O(kε) = O(ε), for a real constant k, have been used. Now, using (2.32) and (2.33), the result is that given some ε1 > 0 arbitrarily small, ε2 = −ε1 + O(ε12 ) < 0, which is absurd. Thus, by contradiction it is true that ε2 > ε1 , and thus any initial condition in the set MR that produces a first reset interval ε1 > 0 arbitrarily small, gives a larger second reset interval ε2 > 0. Thus Zeno solutions do not exist in this case either. In the rest of the proof, the terms O(ε1m ) are directly neglected for simplicity. For the case m = 3, consider an initial condition x0 = x1 = (x1 , x2 , x3 , 0, 0, . . . , 0)T ∈ MR . Applying a similar argument, the result is now that if a sequence of decreasing

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reset intervals {ε1 , ε2 , ε3 } exists, with ε1 ≥ ε2 ≥ ε3 > 0 and ε1 being arbitrarily small, then ε2 ε1 x2 + 1 x1 = 0, 2 6   2   2 ε1 ε1 ε2 ε2 ε2 x3 + ε1 + + + x2 + x1 = 0, 2 2 2 6     (ε1 + ε2 )2 (ε1 + ε2 )ε3 ε32 ε3 + + x2 + x1 = 0, x3 + ε1 + ε2 + 2 2 2 6 x3 +

(2.34)

and now, eliminating x1 , x2 , and x3 , after some computation, ε3 is given as a function of ε1 and ε2 by the second order equation (ε1 + ε2 )ε2 + (ε1 + 2ε2 )ε3 + ε32 = 0

(2.35)

having the solutions ε3 = −ε2 < 0 and ε3 = −(ε1 + ε2 ) < 0, which is a contradiction. Thus, no initial condition in MR can produce a sequence of resets intervals that converge to zero, and again Zeno solutions do not exist for the case m = 3. For the general case in which the dimension of MR is m, with initial state x0 = x1 = (x1 , x2 , . . . , xm , 0, 0, . . . , 0)T , a similar reasoning results in the set of equations m

k=1 i m

i=1 k=1

ε1m−1 xk = 0, (m + 1 − k)!

ε2m−i ε1i−k xk = 0, (m + 1 − i)!(i − k)!

(2.36)

... m i

ε m−i (ε1 + · · · + εm−1 )i−k m

i=1 k=1

(m + 1 − i)!(i − k)!

xk = 0,

which results in an algebraic equation of order m in εm , with the solutions εm = −εm−1 , εm = −(εm−1 + εm−2 ), . . . , εm = −(εm−1 + εm−2 + · · · + ε1 ). And again, a sequence of reset intervals {ε1 , ε2 , . . . , εm } with ε1 ≥ ε2 ≥ · · · ≥ εm > 0 and ε1 arbitrarily small cannot exist, showing that a Zeno solution is not possible in the full-reset case. The case of partial-reset can be conveniently transformed into the full-reset form by a change of coordinates, by a simple resorting of coordinates so that the bijectivity is guaranteed. We will consider the system structure decomposition by writing the states as xT = (xTp , xρ¯ T , xTρ ) where xp ∈ Rnp stands for the states of the plant, xρ¯ ∈ Rnρ¯ for the non-resetting compensator states, and xρ ∈ Rnρ for the resetting

2.2 Zenoness, Beating, and Deadlock

71

compensator states. Define the linear transformation T from Rn to Rn such that ⎛

⎞ ⎛ ⎞ xp xρ¯ T x = T ⎝ xρ¯ ⎠ = ⎝ xp ⎠ = z, xρ xρ that is,

⎛

0nρ¯ ×np ⎝ T = Inp ×np 0nρ ×np

Inρ¯ ×nρ¯ 0np ×nρ¯ 0nρ ×nρ¯

⎞ 0nρ¯ ×nρ 0np ×nρ ⎠ . Inρ ×nρ

(2.37)

(2.38)

Note that T is a square matrix, all of whose entries are 0 or 1, and in each row and column of T there is precisely one 1. This means that T is a permutation matrix. Clearly, such a matrix is unitary, hence orthogonal, so T T = T −1 . The nonsingular matrix T allows us to rewrite the dynamical system via a similarity transformation (congruence transformation): if z(t) ∈ / M˜,

¯ z˙ (t) = Az(t)   z t + = A¯ R z(t)

if z(t) ∈ M˜,

(2.39)

¯ y(t) = CT z(t) = Cz(t), T

where A¯ = T AT T , A¯ R = T AR T T , and C¯ = CT T , and in addition the reset surface is transformed into M˜ = {z ∈Rn : T T z ∈ M }. Note that C¯ = CT T = enp +nρ¯ so that the output is not changed by the transformation, i.e., y(t) = znp +nρ¯ (t) as expected. Henceforth, (2.39) is in full-reset form. Since observability is invariant under similarity transformations, it is clear that (2.4)–(2.6) is well-posed if and only if (2.39) is well-posed. Finally, to complete the proof it is necessary to show that the observability matrix has the structure given in (2.29) (using state transformations x  if needed). This is simply done by considering the substate z1 = xpρ¯ . In general, w  z  there always exists a state transformation of z = xρ1 to w = xρ1 such that the state submatrix corresponding to z1 is in the observability staircase form, and thus the observability matrix has the structure given in (2.29) once unobservable states are eliminated. This concludes the proof. 

2.2.2.1 Example: Well-posed Reset Control System with Partial Reset Consider a reset control system (2.4)–(2.6) where the plant, with state xp = given by the state space model  Ap =

 0 1 , 1 −1

Bp =

  1 , 0

  Cp = 0 1 ,

x1  x2

, is

(2.40)

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Definition of Reset Control System and Basic Results

and the reset compensator, with state xr = 

Br =

  1 , 1

x4

, is given by

 1 0 , 0 0 (2.41) that is, the reset control system has a partial reset compensator: it is a parallel connection of an integrator and a Clegg integrator, where only the state x4 is set to zero at the reset instants. The closed-loop system is defined by the matrices ⎛ ⎛ ⎞ ⎞ 0 1 1 1 1 0 0 0 ⎜ 1 −1 0 0 ⎟ ⎜0 1 0 0⎟   ⎟ ⎟ A=⎜ AR = ⎜ C= 0100 , ⎝ 0 −1 0 0 ⎠ , ⎝0 0 1 0⎠, 0 −1 0 0 0 0 0 0 (2.42) and the closed-loop state x = ( x1 x2 x3 x4 )T . This reset control system is well-posed, since ⎧⎛ ⎞ ⎛ ⎞⎫ ⎧⎛ ⎞⎫ 1 0 ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨⎜ ⎟ ⎜ ⎟⎬ ⎨⎜ 0⎟ ⎜0⎟ 0 ⎟ ⎜ ⎟ MR = span ⎜ , , (2.43) , N (O base ) = span ⎝ ⎝ ⎝ ⎠ ⎠ ⎠ 0 −1 ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ⎩ ⎭ 0 1 0 Ar =

 0 0 , 0 0

x3 

  Cr = 1 1 ,



Aρ =

and then MR ∩ N (Obase ) = {0}. Following the reasoning given in the proof of Proposition 2.4, the closed-loop state x can be transformed into a state z in which the observability matrix has the form (2.29). In this case, this is obtained with z = ( x3 x1 x2 x4 )T . Thus, the initial conditions that produce a crossing in an arbitrarily small time ε > 0 are of the form z1 = ( 1 − 2ε 0 0 )T , or equivalently, T  ε x1 = − 0 1 0 . 2

(2.44)

Now, the second after-reset state is given by T  ε T  ε 010 , x2 = AR eAε − 0 1 0 = 2 2

(2.45)

and, according to Proposition 2.4, x2 cannot produce a new crossing in a time less than or equal to ε. This fact can be verified by computing solutions to the implicit equation 0 = CeAt ( α 0 1 0 )T for t, given α ∈ R. The solution is shown in Fig. 2.3, where t = τ1 (( α 0 1 0 )T ) is given. Note that for t to have an arbitrarily small value, an initial condition x1 in the after-reset surface must have the form (2.44), that is, α = −ε/2 in Fig. 2.3. Then, as a result the state after the first reset x2 has the form (2.45), that is, α = +ε/2 (see also Fig. 2.3). And then the value of the second reset instant can be obtained from Fig. 2.3. The result is that if the first reset instant is arbitrarily small, then the second reset instant is arbitrarily close to 3.15.

2.3 Reset Instants and the After-Reset Surface Dimension

73

Fig. 2.3 First reset instant as a function of α

2.3 Reset Instants and the After-Reset Surface Dimension In general, reset instants tk = τk (x0 ), k = 1, 2, . . . , can take different and complex patterns for different initial conditions x0 ∈ D , and in fact this is a key property of reset control systems since it determines the way in which the reset instants evolve and also some important properties. For example, the fact that τ1 (x0 ) has a discontinuity in the example of the last section (Fig. 2.3) is directly related with the non-existence of Zeno solutions. A useful property is that functions τk (x0 ) are homogeneous (of degree 0) since τk (αx0 ) = τk (x0 ), for any α > 0 and k = 1, 2, . . . . If the set of initial conditions is D , this means that the computation of reset intervals can be simplified, for example, to those states that are elements of the unit ball (centered at the origin). On the other ¯ as defined before Proposition 2.2, can be hand, the set of not reduced states R, obtained by using the spectral projectors of the corresponding eigenvalues. In the rest of this section, it is assumed that the reset control systems are wellposed and that their base systems have a state matrix A with a simple pair of complex dominant eigenvalues λ1 = α1 + iβ1 and λ2 = α1 − iβ1 (with index 1). As far as the computation of the set of not reduced states R¯ is concerned, it may be assumed that α1 = 0 without loss of generality. Thus, for an initial condition x0 the output dominant term before the first reset instant is given by (using (2.15))   y1 (t) = C Re{G1 } cos(β1 t) + Im{G1 } sin(β1 t) x0 ,

(2.46)

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Definition of Reset Control System and Basic Results

where G1 is the spectral projector given by G1 = v1 wT1 , and v1 and w1 are the right and the left eigenvectors, respectively. Thus, the set R¯ can be simply computed as   C Re{G1 } , (2.47) R¯ = N C Im{G1 } where Re{G1 } and Im{G1 } stands for the real and imaginary parts of G1 , respec¯ tively. And thus the set of reduced states is given simply by Rn \ R. The order of the reset control system, and in particular the dimension of the after-reset surface, is key in the analysis of the reset instants, thus in the following different cases corresponding to dimensions of the after-reset surface 1, 2, and ≥ 3 are treated separately.

M R) = 1 2.3.1 dim(M In this case, since the after-reset surface has dimension 1, any initial condition x0 ∈ MR can be generated by one vector u, that is, x0 = αu for some α ∈ R. Thus, simply by using the homogeneity property of the functions tk = τk (x0 ), k = 1, 2, . . . , it is clear that τk (x0 ) = τk (αu) = τk (u), that is, they are all constant functions over the set MR , as long as (C, A, u) is reduced (otherwise there are no crossings). In other words, starting from an initial condition in the after-reset surface, the reset instants are periodic. For initial conditions that are not elements of the after-reset surface, the first reset instant is in general different. Using Proposition 2.2, if the reset system has a pair of dominant complex eigenvalues then the first reset instant is uniformly bounded, that is, τ1 (x0 ) < Δ, for some upper bound Δ > 0 and for any initial condition x0 such that (C, A, x0 ) is reduced. In the case in which (C, A, x0 ) is not reduced, τ1 (x0 ) is not necessarily bounded, that is, the initial condition may not cross the reset surface.

2.3.1.1 Example   Consider the reset control system (2.4)–(2.6) where the plant, with state xp = xx12 , is given by the state space model       0 0 1 Ap = , Bp = , Cp = 0 1 , (2.48) 1 −1 0 and the reset compensator, with state xr = x3 , is given by Ar = −1,

Br = 1,

Cr = 1,

Aρ = 0,

(2.49)

that is, the reset compensator is FORE, and the state x3 is set to zero at the reset instants.

2.3 Reset Instants and the After-Reset Surface Dimension

75

The closed-loop system is given by the matrices ⎛

0 A = ⎝1 0

⎛

⎞ 1 0 0 AR = ⎝ 0 1 0 ⎠ , 0 0 0

⎞ 0 1 −1 0 ⎠ , −1 −1

  C= 010 ,

(2.50)

and the closed-loop state is x = ( x1 x2 x3 )T . It can be easily checked that the system is well-posed since the base linear system is observable. By definition, the after-reset surface is given by ⎧⎛ ⎞⎫ ⎨ 1 ⎬ MR = span ⎝ 0 ⎠ . ⎩ ⎭ 0

(2.51)

Now, the subspace of not reduced states R¯ is computed. The closed-loop state matrix A has the eigenvalues λ1 = −0.12 + j 0.74, λ2 = −0.12 − j 0.74, and λ3 = −1.75. Thus, the reset control system has two complex dominant eigenvalues, with their spectral projectors being ⎛

0.41 − j 0.28 0.15 + j 0.34 G1 = ⎝ 0.12 − 0.j 41 0.29 + j 0.14 0.16 + j 0.34 −0.27 + j 0.07

⎞ 0.12 − j 0.41 −0.16 − j 0.34 ⎠ , 0.29 + j 0.14

The subspace of not reduced states is R¯ = N has dimension 1 and is given by

C Re{G1 } C Im{G1 }

⎧⎛ ⎞⎫ ⎨ −0.41 ⎬ R¯ = span ⎝ 0.55 ⎠ . ⎩ ⎭ 0.72

G2 = G 1 .

(2.52) , which in this example

(2.53)

Since, according to (2.51) and (2.53), every nonzero after-reset state is reduced, it results in that the reset instants are periodic as discussed above. However, note that if the initial condition is not an after-reset state, two types of solutions may occur (see Figs. 2.4–2.5): • solutions with no crossings if ⎛

⎞ −0.41 x0 = α ⎝ 0.55 ⎠ , 0.72 for some real number α, and • solutions with a first crossing at a finite time and then an infinite number of periodic crossings.

76

2

Definition of Reset Control System and Basic Results

Fig. 2.4 Closed-loop output for three different initial conditions

Fig. 2.5 Control input for three different initial conditions

M R) = 2 2.3.2 dim(M The case of dim(MR ) = 2 is slightly more involved since, as it will be described below, the resets instants are not periodic in general, and a large variety of solutions may appear regarding the number and structure of crossings associated to a given initial condition. By simplicity, the particular case of a third order plant and a full reset compensator is considered. Without loss of generality, consider an observer

2.3 Reset Instants and the After-Reset Surface Dimension

canonical form realization of the plant, that is, ⎞ ⎛ ⎛ ⎞ 0 0 −a0 b0 Ap = ⎝ 1 0 −a1 ⎠ , Bp = ⎝ b1 ⎠ , 0 1 −a2 b2

77

  Cp = 0 0 1 .

(2.54)

Thus an after-reset state x ∈ MR is given by x = (x1 , x2 , 0, 0, . . . , 0)T

(2.55)

  for some values xx12 ∈ R2 . Using again the fact that tk = τk (x0 ), k = 1, 2, . . . , are homogeneous, for the systems solutions with an initial condition in the after-reset surface it is enough to check the reset instants some subset of R2 , for example, x1  in ρ cos θ  the unit circle. Using polar coordinates x2 = ρ sin θ , for ρ ∈ [0, ∞), θ ∈ [0, 2π), it is true that ⎞⎞ ⎞⎞ ⎛⎛ ⎛⎛ ρ cos θ cos θ ⎜⎜ ρ sin θ ⎟⎟ ⎜⎜ sin θ ⎟⎟ ⎟⎟ ⎟⎟ ⎜⎜ ⎜⎜ ⎟⎟ ⎜⎜ 0 ⎟⎟ ⎜⎜ (2.56) τk ⎜⎜ ⎟⎟ = τk ⎜⎜ 0 ⎟⎟ ⎜⎜ .. ⎟⎟ ⎜⎜ .. ⎟⎟ ⎝⎝ . ⎠⎠ ⎝⎝ . ⎠⎠ 0 0 for k = 1, 2, . . . . As a result the reset instants are functions of the single parameter θ . In general, the system’s solutions may exhibit no crossings, a finite number of crossings, or infinitely many crossings. If the initial condition is an after-reset state, that is, x0 ∈ MR , no crossings or a finite number of crossings may occur if some of the after-reset states are not reduced. Otherwise, an infinite number of crossings is produced.

2.3.2.1 Example Consider again the reset control system with a FORE compensator, with parameters Ar = −1, Br = 1, and Cr = 1, and a plant given by the state space model ⎛ ⎛ ⎞ ⎞ 0 0 −0.35 3   Ap = ⎝ 1 0 −2.40 ⎠ , (2.57) Bp = ⎝ 1 ⎠ , Cp = 0 0 1 . 0 1 −4.35 0 The reset control system is well-posed since the base linear system is observable. Using the parameter θ as in (2.56), the function τ1 can be computed by solving the implicit equation ⎛ ⎞ cos θ ⎜ sin θ ⎟ ⎟ CeAt1 ⎜ (2.58) ⎝ 0 ⎠=0 0

78

2

Definition of Reset Control System and Basic Results

Fig. 2.6 Reset instants corresponding to after-reset states as a function of the parameter θ

for t1 = τ1 (θ ). The result is given in Fig. 2.6, where it can be seen that the mapping τ1 has a discontinuity at θ = π , and in addition it results in the reset instants being uniformly bounded for initial conditions in the after-reset surface. This is due to the fact that all the after-reset states are reduced, as it will be seen below. The after-reset surface is given by ⎧⎛ ⎞ ⎛ ⎞⎫ 1 0 ⎪ ⎪ ⎪ ⎨⎜ ⎟ ⎜ ⎟⎪ ⎬ 0 ⎟,⎜1⎟ MR = span ⎜ (2.59) ⎝ 0 ⎠ ⎝ 0 ⎠⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0 0 and the subspace of not reduced states by R¯ = N



C Re{G1 } C Im{G1 }



⎧⎛ ⎞ ⎛ ⎞⎫ 0.20 −0.81 ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎜ ⎟⎬ 0.31 ⎟ ⎟ , ⎜ 0.47 ⎟ , = span ⎜ ⎝ 0.92 ⎠ ⎝ −0.05 ⎠⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0.16 0.35

(2.60)

thus it can be easily checked that MR ∩ R¯ = {∅}, which means that all the after-reset states are reduced. In Figs. 2.7 and 2.8, several simulations corresponding to different initial conditions in the after-reset surface MR are shown, including closed-loop outputs and control inputs. Note that since the after-reset states are reduced, there are an infinite number of resets corresponding to each initial condition, and as indicated above, a bound over the reset intervals may be found. In the case in which the initial condition is not an after-reset surface state, it may occur that no crossings are produced. This is, in fact, the case for initial conditions in the set R¯ given by (2.60). Otherwise, an infinite number of crossings are produced

2.3 Reset Instants and the After-Reset Surface Dimension

79

Fig. 2.7 Closed-loop output for three initial conditions in the after-reset surface

Fig. 2.8 Control input for three initial conditions in the after-reset surface

in this example. Different simulations for not reduced initial conditions are shown in Figs. 2.9–2.10

80

2

Definition of Reset Control System and Basic Results

Fig. 2.9 Closed-loop output for three initial conditions (not reduced)

Fig. 2.10 Control input for three initial conditions (not reduced)

M R) ≥ 3 2.3.3 dim(M For higher order reset control systems, the functions τk , k = 1, 2, . . . , depend on more than one parameter, and in general they can be described by the values of τ1 over the unit ball in Rm , where m is the number of the after-reset states. In general, the function τ1 can exhibit an infinite number of discontinuities over that domain

2.3 Reset Instants and the After-Reset Surface Dimension

81

even in the case m = 2, resulting in very complex patterns of the resets instants as a function of the initial condition. A detailed analysis of these patterns is still an open issue. In the following, an example corresponding to a second order partial reset compensator and a third order plant is analyzed.

2.3.3.1 Example In this example, a PI + CI compensator is used. It consists of a parallel connection of a PI compensator and a Clegg integrator, and it is a partial reset compensator. Consider the PI + CI compensator with a state space realization given by     0 0 1 , Br = , Ar = 0 0 1     1 0 Cr = 0.2 0.2 , Dr = 0.2, Aρ = (2.61) 0 0 and a third order plant given by ⎛ ⎞ −2.20 −1.32 −0.72 0 0 ⎠, Ap = ⎝ 2 0 1 0

⎛ ⎞ 1 Bp = ⎝ 0 ⎠ , 0

  Cp = 0 0 1.33 . (2.62)

In addition, closed-loop state matrices are ⎛ ⎞ −2.20 −1.32 −0.99 0.20 0.2 ⎜ 2 0 0 0 0 ⎟ ⎜ ⎟ ⎜ 1 0 0 0 ⎟ A=⎜ 0 ⎟, ⎝ 0 0 −1.33 0 0 ⎠ 0 0 −1.33 0 0 and

⎛

1 ⎜0 ⎜ AR = ⎜ ⎜0 ⎝0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

  C = 0 0 1.33 0 0 , (2.63)

⎞ 0 0⎟ ⎟ 0⎟ ⎟. 0⎠ 0

(2.64)

In this case, the base system is unobservable, with the unobservable subspace in this case being given by ⎧⎛ ⎞⎫ 0 ⎪ ⎪ ⎪ ⎪ ⎪⎜ ⎪ ⎪ ⎪ ⎨⎜ 0 ⎟ ⎟⎬ ⎜ ⎟ N (Obase ) = span ⎜ 0 ⎟ ; (2.65) ⎪ ⎪ ⎪ ⎝ −1 ⎠⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 1

82

2

Definition of Reset Control System and Basic Results

Fig. 2.11 Closed-loop output for three initial conditions in the after reset surface

however, the reset control system is well-posed since the after-reset surface is ⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎫ 1 0 0 ⎪ ⎪ ⎪ ⎪⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎪ ⎪ ⎪ ⎨⎜ 0 ⎟ ⎜ 1 ⎟ ⎜ 0 ⎟⎪ ⎬ ⎜ ⎜ ⎟ ⎜ ⎟ ⎟ MR = span ⎜ 0 ⎟ , ⎜ 0 ⎟ , ⎜ 0 ⎟ (2.66) ⎪ ⎪ ⎪ ⎝ 0 ⎠ ⎝ 0 ⎠ ⎝ 1 ⎠⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0 0 0 and MR ∩ N (Obase ) = {0}. In addition, it can be checked that the after reset states are not reduced, in other words, assuming initial conditions in the after-reset surface there are always an infinite number of crossings. In Fig. 2.11, closed-loop outputs corresponding to different initial conditions are obtained. Also in Fig. 2.12 the different control inputs are shown. Note that in general, as the dimension of the after-reset surface increases, the structure of the reset instants becomes not that simple, for example, in the interval [0, 15]s one initial condition does not produce a crossing while the others produce five reset actions. As a conclusion, as the dimension of the after reset surface increases, the reset instants occur with more complex patterns.

2.4 Reset Control Systems with Exogenous Inputs As it is usual in control practice, reset control systems are driven by external or exogenous inputs such as reference or disturbance signals. In this case, well-posedness of the reset control system can be analyzed using the arguments given in previous sections, if the exogenous inputs are generated by an exosystem.

2.4 Reset Control Systems with Exogenous Inputs

83

Fig. 2.12 Control input for three initial conditions in the after-reset surface

Consider the reset control system of Fig. 2.1, where the plant and the reset compensator are given by (2.1) and (2.2), respectively, and with a reference input r and a disturbance input d generated by exosystems, with the state space models 

˙ 1 (t) = A1 w1 (t), w

w1 (0) = w10 ,

r(t) = C1 w1 (t),

(2.67)

with w1 ∈ Rm1 , and 

˙ 2 (t) = A2 w2 (t), w

w2 (0) = w20 ,

d(t) = C2 w2 (t),

(2.68)

with w2 ∈ Rm2 . These exosystems allow the generation of signals like steps, ramps, sinusoids, . . . . Now, the feedback connection is given by e = r − y and u = v + d, and the base linear closed-loop system may be described by  x˙ (t) = Ax(t) +

0 Br



 r(t) +

0 Bp

 d(t)

(2.69)

  where x = xxpr , and xp and xr are the plant and compensator states. In addition, the reset instants are defined as those instants t at which the closed-loop output y(t) = Cx(t) is equal to the reference signal r(t) = C1 w1 (t). Define the augmented state z as z = ( wT1 wT2 xT )T . Then, the reset map can be defined in the augmented

84

2

state space by

Definition of Reset Control System and Basic Results

⎛

I ⎜0 A¯ R = ⎜ ⎝0 0

0 I 0 0

0 0 I 0

⎞ 0 0 ⎟ ⎟ 0 ⎠ Aρ

(2.70)

and the after-surface M¯R and the reset surface M¯ as ¯ M¯R = R(A¯ R ) ∩ N (C), ¯ \ M¯R M¯ = N (C)

(2.71)

with C¯ = ( C1 0 −Cp 0 ). Finally, the closed-loop system in the augmented state space is given by  ¯ z˙ (t) = Az(t) if z(t) ∈ / M, (2.72) z(t + ) = A¯ R z(t) if z(t) ∈ M with the state space matrix ⎛

A1 ⎜ 0 A¯ = ⎜ ⎝ 0 Br C1

0 A2 Bp C2 0

0 0 Ap −Br Cp

⎞ 0 0 ⎟ ⎟. Bp Cr ⎠ Ar

(2.73)

In the augmented space state representation, Propositions 2.3 and 2.4 can be directly applied, giving the next result. Proposition 2.5 Consider the reset control system of Fig. 2.1, with the plant and reset compensator given by (2.1) and (2.2), respectively, and with inputs r and d generated by the exosystems (2.67)–(2.68). If M¯R ∩ N (O¯base ) = {0}, where O¯base is the observability matrix ⎛ ⎞ C¯ ⎜ ⎟ C¯ A¯ ⎜ ⎟ O¯base = ⎜ (2.74) ⎟ .. ⎝ ⎠ . C¯ A¯ n+m1 +m2 −1 then 1. The reset system is well-posed. 2. The reset system does not have Zeno solutions.   Proof The augmented state can be partitioned as z = xz1r , and thus the state matrix (2.73) is partitioned as   B¯ p Cr A¯ p , (2.75) A¯ = Br C¯ p Ar

2.4 Reset Control Systems with Exogenous Inputs

85

where ⎛

A1 A¯ p = ⎝ 0 0

⎞ 0 u0 A2 u 0 ⎠ , Bp C2 uAp

⎞ 0 B¯ p = ⎝ 0 ⎠ , Bp ⎛

  C¯ p = C1 0 −Cp . (2.76)

Part 1 of the proposition is a direct application of Proposition 2.3 to the system (2.72) ¯ On the other hand, since the state matrix A¯ ¯ A¯ R , and C. defined by the matrices A, has the structure given in (2.75), it always possible to make a state transformation of z1 in such a way that corresponding principal minor be in the observer form. Thus, the reasoning used in the proof of Proposition 2.4 may be used to prove Part 2.  In general, a simple sufficient condition for the well-posedness of a reset control system with exogenous inputs it that the augmented closed-loop system (2.72) be observable, that is, the matrix O¯base be full rank.

2.4.1 A Well-posed Reset Control System with Exogenous Input A simple example is shown here to illustrate Proposition 2.5. Consider the reset control system of Fig. 2.1, consisting of the feedback interconnection of a Clegg integrator and an integrator. The Clegg integrator has the state equations ⎧ x˙r (t) = r(t), xr (t) − r(t) = 0, ⎪ ⎪ ⎨ xr (t + ) = 0, xr (t) − r(t) = 0, ⎪ ⎪ ⎩ v(t) = xr (t).

(2.77)

In addition, consider a sinusoidal reference input r(t) = a sin(ωt + φ), for some given constants a, ω > 0, and φ. It is given by the exosystem 

˙ 1 (t) = w



0 −ω

ω w1 (t), 0

w1 (0) =

 a sin φ  a cos φ ,

r(t) = ( 1 0 )w1 .

(2.78)

Since a disturbance signal is not considered in this example, Proposition 2.5 can be used by eliminating the row and column blocks corresponding to the disturbance ¯ The result is ¯ A¯ R , and C. exosystem in the matrices A, ⎛

0 ⎜ −ω A¯ = ⎜ ⎝ 0 1

ω 0 0 0

⎞ 0 0 0 0⎟ ⎟, 0 1⎠ −1 0

  C¯ = 1 0 −1 0 .

(2.79)

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Fig. 2.13 Reference signal and closed-loop output corresponding to a well-posed reset control system

Now, the observability matrix of the (augmented) base system is ⎞ ⎛ 1 0 −1 0 ⎜ 0 ω 0 −1 ⎟ ⎟ O¯base = ⎜ ⎝ −(1 + ω2 ) 0 1 0 ⎠ 1 0 −ω(1 + ω2 ) 0

(2.80)

which is full rank for any ω > 0 (and does not depend on a), and thus the system is well-posed for any sinusoidal reference input. Figures 2.13 and 2.14 show a simulation of the reset control system for ω = 0.5 rad/s, with an initial condition w1 (0) = (1 − 1)T for the exosystem, and with zero initial condition for the Clegg integrator and the integrator.

2.4.2 A Reset Control System with Zeno Solutions In general, Proposition 2.5 gives a simple and checkable condition for wellposedness of the reset control system of Fig. 2.1 with exogenous inputs, for the plant and compensator given by (2.1) and (2.2), respectively. For well-posedness, the result may be also used for any reset system that can be expressed as (2.72) with ¯ A¯R , and C, ¯ since no particular structure of these matrices is arbitrary values of A, used to prove the result. However, for avoiding Zeno solutions the structure of these matrices, related with (2.1) and (2.2), is a central part of the result. Thus, a reset control system with a different structure may have Zeno solutions, this is the case, for example, corresponding to a non-strictly proper plant in Fig. 2.1.

2.4 Reset Control Systems with Exogenous Inputs

87

Fig. 2.14 Control input corresponding to a well-posed reset control system

In the following, an example developed in [6] is shown here to illustrate the existence of Zeno solutions in reset control systems with external inputs, where the plant is simply P (s) = 1. The resulting feedback system is a Clegg integrator with unity feedback. In addition, a sinusoidal reference input is also considered. The state space models (2.77) and (2.78) are again used, as a result the augmented state matrices are in this case (the row and column block corresponding to the plant are simply removed) ⎛ ⎛ ⎞ ⎞ 0 ω 0 1 0 0   A¯ = ⎝ −ω 0 0 ⎠ , A¯ R = ⎝ 0 1 0 ⎠ , C¯ = 1 0 −1 , (2.81) 1 0 −1 0 0 0 and the observability matrix of the augmented base system is given by ⎛ ⎞ 1 0 −1 ω 1 ⎠ O¯base = ⎝ −1 2 1 − ω −ω −1

(2.82)

which is full rank for any ω > 0. Although Proposition 2.5 cannot be used in this case because the plant does not fit the model (2.2), the well-posedness can be assured since O¯base is full rank. However, the well-posedness argument cannot be used for assessing the existence of Zeno solutions. In fact, this system has a Zeno solution with sequences of reset instants {tk }, k = 1, 2, . . . , converging to t ∗ = nπ for every integer n ≥ 1. Figure 2.15 (top) shows the output response y(t) of the closed loop (and output of the Clegg integrator) to the sinusoidal reference w(t) = sin ωt. It can be shown that a Zeno behavior appears to the left of every t = kπ , k = 1, 2, 3, . . . . The solution

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Fig. 2.15 (Top) Response y(t) to a sinusoidal reference of a feedback loop with a Clegg integrator; (Bottom) detailed zoom of the three first visible resets close to t = π

starting from the right of every y(kπ) = 0 escapes the Zeno behavior until it again reaches another Zeno point at t = (k + 1)π . Figure 2.15 (bottom) plots a detailed zoom of the first three visible resets close to t = π . These reset times are t1 ≈ 2.28, t2 ≈ 2.92, and t3 ≈ 3.12. In fact, there exists an infinite sequence of resets {tk } converging to t∞ = π . It is illustrative to see in Fig. 2.16 the time response of this system in a threedimensional plot that shows the time evolution of the state (w11 , w12 , xp ) = (r, r˙ , y) . The first two coordinates are from the exosystem, and the first one is the reference r(t) to the closed loop, in this case r(t) = sin ωt and r˙ (t) = cos ωt, with ω = 1. Thus the trajectory lies entirely within the cylinder r 2 + r˙ 2 = 1. The three-dimensional plot reveals clearly that, in the state space (r, r˙ , y), there are actually two Zeno accumulation points, namely Z1 = (0, 1, 0) and Z2 = (0, −1, 0). The first one corresponds in the time domain (Fig. 2.15) to the zero crossings of the reference for t = 2kπ and the second one for t = (2k + 1)π , with k integer. Figure 2.16 also shows two relevant planes: the null space N (C) of C = (1, 0, −1), that is, the plane y = r that triggers the reset action, and the plane MR given by y = 0 where the state is projected immediately after a reset y(tk+ ) = 0.

2.4 Reset Control Systems with Exogenous Inputs

89

Fig. 2.16 Trajectory (r(t), r˙ (t), y(t)) with Zeno behavior

The analysis of Zeno behavior in dynamical systems is an involved topic, and we will not address it here rigorously. Notice from the previous proposition in this section that the reset control systems that we are interested in (well-posed) do not have Zeno behavior. However, in order to give an idea of the asymptotic behavior close to the Zeno limit, let us complete the example with a simplified study based on the Poincaré map. We refer the interested reader to the literature, for example, [8], [7], and [13] for a full exploration of these features. From Fig. 2.16 it is clear that from any initial condition the state evolves until it reaches the reset condition at (rk , r˙k , yk ) with yk = rk and then resets to the point xk = (rk , r˙k , 0). Then, it flows again until a new reset condition holds and a new reset moves the state to xk+1 = (rk+1 , r˙k+1 , 0). The sequence of after-reset points {xk } defines a discrete-time map, or iteration xk → xk+1 , called the Poincaré map of the reset system. Note that, since rk2 + r˙k2 = 1 for all k, the Poincaré sequence {xk } ∈ P is here defined on a one-dimensional manifold P given by r 2 + r˙ 2 = 1 and y = 0. Since P is one-dimensional, the Poincaré map can be determined also from the time domain plots of r(t) and y(t) in Fig. 2.15, for if we determine the sequence of reset times {tk }, then the Poincaré sequence is {xk } = {(sin tk , cos tk , 0)}. In Fig. 2.15 (bottom), we see the first three reset times t1 ≈ 2.28, t2 ≈ 2.92, and t3 ≈ 3.12. These tk tend from the left to t∞ = π . Let us suppose that we start at t = tk with y(tk ) = 0.

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The evolution of the Clegg integrator state is governed by: y(tk ) = 0.

y˙ = r(t) − y(t) = sin t − y(t),

Since tk tends to π , a key simplifying assumption is that, in the limit, we can approximate sin t by −(t − π). This simplification is also helpful because the Poincaré maps give rise to implicit equations not solvable analytically. In this way, close to t = π , we have: y˙ = −(t − π) − y(t),

y(tk ) = 0,

having the solution y(t) = (tk − π − 1)e−(t−tk ) − t + π + 1,

t ≥ tk .

To determine the next reset time tk+1 , we have to impose the condition y(tk+1 ) = r(tk+1 ). Again we replace r(tk+1 ) = sin tk+1 by −(tk+1 −π). Thus the Poincaré map tk → tk+1 is given implicitly by the condition −(tk+1 − π) = (tk − π − 1)e−(tk+1 −tk ) − tk+1 + π + 1. Introduce the change of variables tk − π = dk , with dk → 0− . Using the asymptotic approximation e−(tk+1 −tk ) = e−(dk+1 −dk ) ≈ 1 − (dk+1 − dk ) gives rise to −dk+1 = (dk − 1)(1 − dk+1 + dk ) − dk+1 + 1, from which we can solve explicitly dk+1 = dk2 /(dk − 1). This law for dk → 0− approaches dk+1 = −dk2 , or equivalently, approaches the limit relation |tk+1 − π| = |tk − π|2 which proves that there exists an infinite sequence of reset times tk tending to π and governed, in the limit, by a quadratic recurrence law. This explains why in Fig. 2.15 there are only a few visible resets: to detect the reset at tk = π + dk , we should implement a simulation step size smaller than |dk |, that, from the quadratic law |dk+1 | = |dk |2 , decreases very fast with k, implying a strong computational cost.

References 1. Bainov, D.D., Simeonov, P.S.: Systems with Impulse Effect: Stability, Theory and Applications. Ellis Horwood Limited, Chichester (1989) 2. Baños, A., Mulero, J.I.: On the well-posedness of reset control systems. Technical Report TR-DIS-1-2011, University of Murcia (2011) 3. Beker, O., Hollot, C.V., Chait, Y., Han, H.: Fundamental properties of reset control systems. Automatica 40, 905–915 (2004) 4. Bell, P.C., Delvenne, J.-C., Jungers, R.M., Blondel, V.D.: The continuous Skolem–Pisot problem: on the complexity of reachability for linear ordinary differential equations (2009). arXiv:0809.2189v2 [maths.DS]

References

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5. Branicky, M.S., Borkar, V.S., Mitter, S.K.: A unified framework for hybrid control: model and optimal control theory. IEEE Trans. Autom. Control 43(1), 31–45 (1998) 6. Carrasco, J.: Stability of reset control systems. Ph.D. Thesis (2009) (in Spanish) 7. Grizzle, J.W., Abba, G., Plestan, F.: Asymptotically stable walking for biped robots. Analysis via systems with impulse effects. IEEE Trans. Autom. Control 46, 51–64 (2001) 8. Haddad, W.M., Chellaboina, V., Nersesov, S.G.: Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control. Princeton University Press, Princeton (2006) 9. Hainry, E.: Reachability in linear dynamical systems. In: Computability in Europe 2008. Lectures Notes in Computer Science, vol. 5028. Springer, Berlin (2008) 10. Lakshmikanthan, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) 11. Michel, A., Hu, B.: Towards a stability theory of general hybrid dynamical systems. Automatica 35, 371–384 (1999) 12. Neši´c, D., Zaccarian, L., Teel, A.R.: Stability properties of reset systems. In: Proc. of 16th IFAC World Congress, Prague, Czech Republic (2005) 13. Schumacher, J.M.: Time-scaling symmetry and Zeno solutions. Automatica 45, 1237–1242 (2009) 14. Yang, T.: Impulsive Control Theory. Lecture Notes in Control and Information Sciences, vol. 272. Springer, Berlin (2001)

Chapter 3

Stability of Reset Control Systems

3.1 Introduction Stability is a main concern in control systems, and thus stability of reset control systems is a key issue both from an analysis and a design point of view. In the classical works of Clegg and Horowitz, the main stability tool was the describing function that, although giving only approximate results, provides information about the phase lead characteristic of reset compensators. In the seminal work [18], the stability problem is explicitly formulated, concluding that the design problem “will be solved more satisfactorily when stability criteria are developed for feedback loops containing nonlinear elements such as FORE ( first order reset element)”. In the last decade, the stability problem has been approached from different perspectives, and nowadays a number of results are available for reset control systems, including systems with time delays. A basic question is about the relationship between the stability of the reset control system and its base linear system. It is important to emphasize that the stability of a reset control system is not always guaranteed by the stability of the base system; in other words, reset actions can destabilize a stable base linear system, thus reset should be carefully used in practice. Conversely, a proper reset control design may stabilize an unstable base linear system, thus stability of the base linear system is neither sufficient nor necessary for the stability of the reset control system. This chapter addresses the stability of reset control systems with finite-dimensional base systems. Section 3.2.1 deals with reset-times independent conditions, that is, stability conditions that are not explicitly based on the reset instants, including the Hβ condition developed in [9]. In Sect. 3.2.2, we study stability conditions depending on the reset time intervals and based on the induced discrete-time system obtained by sampling the reset control system at the after-reset instants. This approach is further extended in Sect. 3.2.3 to the stabilization problem, imposing stabilizing upper bounds (resp., lower bounds) on the reset intervals, that is, forcing (resp., avoiding) reset actions. Section 3.3 considers input–output L2 stability by using passivity-based techniques and providing a number of useful results on the passivity and dissipativity of different reset systems. Finally, the chapter concludes with Sect. 3.4 where an analysis by the describing function is presented, A. Baños, A. Barreiro, Reset Control Systems, Advances in Industrial Control, DOI 10.1007/978-1-4471-2250-0_3, © Springer-Verlag London Limited 2012

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applied to different types of reset compensators, including fixed/variable reset band and variable reset. The describing function is especially useful for evaluating in the frequency domain the overcoming of linear modulus-phase limitations, showing the phase lead property of reset compensators.

3.2 Lyapunov Stability In this section, the stability of the autonomous reset control system (2.4)–(2.6) will be analyzed, assuming that it is well-posed. For any equilibrium point, both the (continuous) linear base dynamics and the (discrete) resetting dynamics need to be at an equilibrium, that is, for any equilibrium point xe of (2.4), both x˙ e = 0 and xe = AR xe are satisfied. In general, the origin xe = 0 will be the equilibrium point of the reset control system (2.4). Note that xe = 0 is an element of the after reset surface MR , and thus 0 ∈ / M. The following stability definitions are standard, with the notation adopted from [9] and [27]. The equilibrium point xe = 0 of the reset system, or simply the reset system, is stable if for any ε > 0 there exists a δ(ε) such that if x0  < δ then x(t, x0 ) < ε for all t ≥ 0, where x(t, x0 ) is the solution of (2.4) at instant t and for the initial condition x(0) = x0 . On the other hand, the equilibrium point xe = 0 of the reset system, or again simply the reset system, is asymptotically stable if it is stable and, in addition, there exists some number η such that if x0  < η then x(t, x0 ) → 0 as t → ∞. Since the origin xe = 0 is the only equilibrium point, stability is equivalent to global stability. The stability of a discrete-time linear system given by x(k + 1) = A(k)x(k), x(k0 ) = x0 is considered following [27]. It is uniformly stable if there exists a finite positive constant γ such that for any k ≥ k0 and any initial condition x0 the corresponding solution satisfies x(k) ≤ γ x0 . It is uniformly asymptotically stable if, in addition, there exists a constant 0 ≤ λ < 1 such that for any k0 and x0 the solution satisfies x(k) ≤ γ λk−k0 x0 , for any k ≥ k0 .

3.2.1 Reset-Times Independent Conditions Recently, the reset control systems stability problem has been successfully addressed in [9], [10] for general reset compensators, allowing full or partial state reset. This work is significant since stability can be checked by a simple condition: the (strictly) positive realness of a transfer functions matrix Hβ , referred to as the Hβ -condition. The Hβ -condition is based on a quadratic Lyapunov function that must be decreasing over all the state space along the system trajectories and non-increasing on the reset jumps. In [24], the condition has been relaxed by using a different definition of reset and piecewise quadratic Lyapunov functions. However, this definition of reset system is not equivalent to the original (see Sect. 1.6) and it will not be developed here.

3.2 Lyapunov Stability

95

3.2.1.1 The Hβ -condition The Hβ stability condition is applied to the reset control system (2.4), where in principle the reset surface M is defined by   (3.1) M = x ∈ Rn : Cx = 0, (I − AR )x = 0 , to avoid beating. According to this definition, M is the set of closed-loop states in the null space N (C) that does not contain fixed points of AR . Note that this definition is equivalent to (2.6). In addition, it will be assumed that the system is well-posed, that is, MR ∩ Obase = {0}, to avoid deadlock and also the existence of Zeno solutions. The basic approach for addressing stability is a Lyapunov theorem that requires the Lyapunov function to be decreasing in the continuous and jump modes, but with no explicit assumption on the reset times [9]. In this way, the derived results are reset-times independent: Proposition 3.1 Let V : Rn → R be a continuously differentiable, positive-definite, radially unbounded function such that V˙ (x) :=



∂V ∂x



Ax < 0,

for x = 0,

(3.2)

and ΔV (x) := V (AR x) − V (x) ≤ 0,

for x ∈ M .

(3.3)

Then the reset control system (2.4)–(2.6) is asymptotically stable. This basic result can be specialized to the quadratic Lyapunov case. The reset control system (2.4)–(2.6) is said to be quadratically stable if it satisfies (3.2)–(3.3) for some V (x) = x P x with P > 0. Recall that the row C has the structure C = (Cp , 01×nρ¯ , 01×nρ ),

(3.4)

where Cp is 1 × np and np is the number of plant states. The number of controller states is nρ¯ + nρ , where nρ is the number of controller reset states (note that nρ¯ > 0 in the partial reset case). Thus, the reset matrix takes the form AR = diag(0np ×np , 0nρ¯ ×nρ¯ , Inρ ×nρ ),

(3.5)

and the following stability result is obtained, where the matrices C0 and B0 are defined by ⎞ ⎛ 0np ×nρ (3.6) B0 = ⎝ 0nρ¯ ×nρ ⎠ . C0 = (βCp , 0nρ ×nρ¯ Pρ ), Inρ ×nρ

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3 Stability of Reset Control Systems

Proposition 3.2 The following conditions are equivalent: (i) The reset system (2.4)–(2.6) is quadratically stable. (ii) There exists a constant β ∈ Rnρ ×1 such that the restricted Lyapunov equation P > 0,

A P + P A < 0, B0 P = C0

(3.7) (3.8)

has a solution for P . (iii) (Hβ -condition) There exist a positive-definite matrix Pρ ∈ Rnρ ×nρ and β ∈ Rnρ ×1 such that Hβ (s) := C0 (sI − A)−1 B0

(3.9)

is a strictly positive real (SPR) transfer function, where (A, B0 ) is controllable and (A, C0 ) is observable. Proof The complete proof can be found in [9], but the main steps are as follows. First, the equivalence between (3.2) and (3.7) is immediate. Second, (3.3) can be expressed in the form x (A

R P AR − P )x ≤ 0 for all x ∈ M , which is equivalent, by continuity, to the same inequality for all x ∈ N (C). Processing this inequality gives rise to (3.8). Finally, the equivalence between (3.7), (3.8), and (3.9) follows from the wellknown Kalman–Yakubovich–Popov lemma, or positive real lemma (see, for example, [19]), which is a standard tool for passing between time-domain (Lyapunov) and frequency domain conditions.  Recall that the transfer function matrix Hβ (s), with dimension nρ × nρ , is strictly positive real when Hβ (s −ε) is positive real for some ε > 0. In our case, since Hβ (s) is strictly proper, the Hβ -condition amounts to [19]: • Hβ (s) is Hurwitz, that is, all poles of elements of Hβ (s) have negative real parts; • Hβ (j ω) + Hβ (−j ω) > 0 for all finite ω ∈ R; • limω→∞ ω2 (Hβ (j ω) + Hβ (−j ω)) > 0. The availability of frequency domain stability conditions, like the SPR condition on Hβ above, is important for many well-known reasons: integration of analysis and design, treatment of uncertainty, etc. Note that the Hβ -condition gives quadratic stability by checking if Hβ is SPR for some β and some Pρ . Since the Hβ -condition is easily checkable especially in the scalar case corresponding to the reset of only one state (nρ = 1), it has been proved to be an efficient method for stability analysis of reset control systems. In general, Hβ (s) will be a transfer function matrix with dimension nρ × nρ , that is, with size equal to the number of reset states. The Hβ -condition results in that the quadratic Lyapunov function V (x) = xT P x is decreasing over the time intervals (tk , tk+1 ) and non-increasing at the reset instants tk , k = 1, 2, . . . . Similar stability conditions have also been previously used in [23] and the seminal work [2], in the context of impulsive systems with impulses at fixed instants.

3.2 Lyapunov Stability

97

In spite of its usefulness, the Hβ -condition has some limitations. Firstly, since the quadratic function V (x) has a continuous derivative over Rn , it is clear that (3.2) is equivalent to xT (AT P + P A)x < 0 for all x ∈ Rn and x = 0, and thus the base linear system must be always asymptotically stable. Secondly, note that even in the case in which the base linear system and the reset system are asymptotically stable, it might happen that the Hβ condition does not hold. This is, for example, the case if the reset action induces small positive increments ΔV (x) > 0 that are compensated by large decrements between reset instants, due to V˙ (x) < 0, so that the net effect is a net decrement of V (x), and thus the reset system is stable. As a result, the Hβ -condition has two basic limitations: (i) it does not apply to reset systems with unstable base system, and (ii) it is only a sufficient condition, and thus there exist stable reset systems for which an SPR Hβ does not exist. In the following, two examples are developed: an example in which the Hβ -condition gives stability of a reset system; and a second example of a stable reset system with a stable base system in which the Hβ -condition does not apply. Example: Hβ is SPR Consider Example 6 in [10], which uses a reset control system with plant P (s) = 1/s and base linear controller Rb (s) = 1/(s + 1), connected in negative feedback, without exogenous inputs. If the state vector is x = (x1 , x2 )

with x1 the plant state and x2 the (reset) controller state, then it results in a reset system like that in (2.4)–(2.6) with    

 1 0 0 1 , C= 10 . (3.10) , AR = A= 0 0 −1 −1 Since the base linear system is observable, the reset control system is well-posed. In addition, C0 and B0 are given by  

 0 B0 = , C0 = β 1 , (3.11) 1 and thus it is easily obtained that (A, B0 ) is controllable and (A, C0 ) is observable. In addition, from (3.9), Hβ is simply given by (for this case nρ = 1 and then Pρ = 1 without loss of generality):  

 s +1−β −1 0 , = 2 Hβ (s) = β 1 (sI − A) 1 s +s +1 and its frequency response is Hβ (j ω) =

ω2 ω(β − ω2 ) + j . (1 − ω2 )2 + ω2 (1 − ω2 )2 + ω2

Finally, since Hβ (s) is Hurwitz, Re{Hβ (j ω)} > 0 for all finite ω, and furthermore, limω→∞ ω2 Re{Hβ (j ω)} = 1 > 0 for any value of β ∈ R, we can conclude that Hβ is SPR, and as a consequence, the control system is quadratically stable and thus asymptotically stable.

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In this example, the after-reset surface MR and reset surface M are given by MR = N (C) ∩ R(AR ) = {0},    1 {0}. M = N (C) \ MR = span 0

(3.12)

Note that since the after-reset surface is the origin 0, the state evolution of this system is very simple, the equilibrium point is reached after the first reset if a reset occurs. In addition, since√the matrix A does not have real dominant eigenvalues, its eigenvalues are − 12 ± j 23 , and (A, C, x0 ) is reduced for any initial condition x0 , so the reset surface is always reached in a finite time. This is a very simple example of a reset control system that is asymptotically stable, and this stability is perfectly captured by the Hβ -condition. But it might happen that the Hβ -condition fails in detecting stability, as seen in the following example. Example: Hβ is not SPR This example shows a case of a reset control system with a stable base system in which the Hβ -condition does not apply, and thus the stability of the reset system cannot be asserted. As it will be shown in Sect. 3.2.2.1 (see the second example), this case corresponds indeed to a stable reset system. Consider a reset control system with a base compensator Rb (again corresponding to a FORE compensator) and a plant P as given by the transfer functions: Rb (s) =

3.2 , s −1

P (s) =

s + 0.25 . s 2 + 2s

(3.13)

A state-space realization for the closed-loop system is given by matrices A, AR , and C: ⎛ ⎞ ⎛ ⎞ 1 0 0 −2 0 1.6

 0 0 ⎠, C = 1 0.25 0 , A=⎝ 1 AR = ⎝ 0 1 0 ⎠ , 0 0 0 −2 −0.5 1 (3.14) where A is Hurwitz (all its eigenvalues have strictly negative real part). In this example, the compensator is full reset, and so nρ¯ = 0. In addition, np = 2 and nρ = 1. Thus β is a real number, and 



(3.15) C0 = βCp Pρ = β 0.25β 1 , where Pρ = 1 without loss of generality. In addition, 



B0 = 0np 0nρ¯ Inρ = 0 0 1 .

(3.16)

The transfer function Hβ (s) after a simple computation is Hβ (s) =

5s 2 + (8β + 10)s + 2β . 5s 3 + 5s 2 + 6s + 4

(3.17)

3.2 Lyapunov Stability

99

Fig. 3.1 Minimum of the real part of Hβ (j ω) for ω ∈ [0, ∞) vs β

In Fig. 3.1, it is shown that although the base system is stable (A is Hurwitz) the Hβ condition is not satisfied since there does not exist any β ∈ R such as Re{Hβ (j ω)} > 0 for any ω > 0.

3.2.2 Reset Times-Dependent Stability Criteria The results in the previous section are limited by the requirement that the Lyapunov function has to be decreasing both in the continuous flow and at the reset changes. Although in many practical cases this is the natural and expected behavior, there are stable reset systems such that the Lyapunov function sometimes increases in the continuous flow or at the reset instants. On the other hand, the results in the previous section do not depend explicitly on the reset instants, so that in many cases the important influence of the reset intervals on the system stability cannot be adequately captured. For the two mentioned reasons, this section presents an approach to reset systems stability that explicitly takes into account the reset instants and reset intervals. The resulting conditions are thus called reset times-dependent stability criteria. To this end, stability of the reset control system is based on the stability of a time varying discrete time system representing the dynamics of the reset system between consecutive after-resets instants. Related work has been recently developed in [16], but there reset is simplified to be performed at fixed instants, and thus earlier results developed in [2] can be directly used.

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In this section, stability conditions will be developed by using an auxiliary timevariant discrete time system, where the dynamics of the auxiliary system will explicitly depend on reset time intervals. This approach is further extended in Sect. 3.2.3 to the stabilization problem, imposing restrictions over reset intervals to be in some designed set. A simple observation of the reset control system equation (2.4), and its solution as given in Proposition 2.1, shows that if it is sampled at the after-reset instants tk+ , k = 0, 1, 2, . . . , the sampled system is given by

 

+ , x0 = AR eA(τk+1 (x0 )−τk (x0 )) x tk+ , x0 x tk+1

(3.18)

since x(t, x0 ) = eA(t−tk ) x(tk+ , x0 ), for t ∈ (tk , tk+1 ]. In addition, if the assumptions of Proposition 2.2 are satisfied, more specifically the base system has no real dominant modes and (A, C, x0 ) is reduced for any initial condition x0 ∈ Rn \ {0}, there exist a well defined sequence of reset intervals (Δ1 (x0 ), Δ2 (x0 ), . . . ), in the sense that they are upper-bounded by some constant ΔM . Strictly speaking, it is only necessary to introduce some upper bound over reset intervals if the base system is unstable, otherwise the reset system will be unstable. Therefore, since in general the problem of determining if an initial condition x0 crosses the surface M is an open issue, in this work it will be assumed that if the base system is unstable then it has the property that for any initial condition x0 ∈ MR , its solution x(t, x0 ) = CeAt x0 is equal to zero at some instant tc ≤ ΔM , for some constant ΔM > 0. A reset control system with either a stable base system or an unstable base system with that property will be referred to as regular. Note that in particular reset control systems having an unstable base linear systems with complex dominant modes, with (A, C, x0 ) reduced for any x0 ∈ Rn \{0}, are regular. In the case of low order reset control systems in which the dimension of the after reset surface MR is m = 0 or m = 1, the sampled system (3.18) results in an auxiliary discrete time system with particularly simple dynamics. The case in which the dimension of the reset surface is m = 0, that is, MR = {0}, corresponds to a first order plant and a full reset compensator; in this case, the equilibrium point 0 is reached at the first reset instant if there is a crossing, thus the stability problem reduces to the reachability of the reset surface and the following result easily follows. Proposition 3.3 A regular reset control system with MR = {0} is asymptotically stable. In addition, a reset control system with MR = {0} which is not regular is not stable. The case m = 1 has been analyzed in Sect. 2.3.1. In this case, for any initial condition x0 , reset intervals are periodic with period Δ after the first reset interval Δ1 (x0 ) = τ1 (x0 ), and thus the auxiliary discrete-time system associated to the sampled system (3.18) is time invariant. It is given by xd (k + 1) = AR eAΔ xd (k),

xd (0) = AR eAτ1 (x0 ) x0 .

(3.19)

As a result the stability of the reset control system is given by the stability of the auxiliary time-invariant discrete time system, taking into account that the reset sur-

3.2 Lyapunov Stability

101

face is reached from any nonzero initial condition in the case in which the base system is not stable. The following result easily follows. Proposition 3.4 A regular reset control system with m = 1 is asymptotically stable if the matrix AR eAΔ is Schur stable, that is, all its eigenvalues are strictly inside the unit circle. In addition, a reset control system with m = 1 is not stable if it is not regular, or is regular and AR eAΔ is not Schur stable. In general, for the case m > 1, the reset intervals sequences may exhibit complex patterns as has been discussed in Sects. 2.3.2 and 2.3.3. However, the stability problem of the reset control system can be reduced to solutions with initial conditions in the after reset surface MR . In addition, if the base system is not stable, the regularity condition is needed to assure that MR is uniformly reachable in a finite time; otherwise, the reset control system is not stable. On the other hand, the auxiliary time-varying discrete time system will be given by xd (k + 1) = AR eAΔk+1 xd (k),

xd (0) = x0

(3.20)

where (Δ1 , Δ2 , . . . ) is a sequence of reset intervals corresponding to some initial condition. In the following, the stability of the reset control system will be analyzed in terms of the stability of a family of auxiliary discrete-time systems in which all the possible sequences of reset intervals are included. Since in general for the stability of the auxiliary systems it will be necessary that reset intervals be lower-bounded, the original reset control system will be modified to include a dwell-time constant Δm . In this way, the reset control system to be considered in the following is  ˙ = 1, Δ(t) x˙ (t) = Ax(t) (x(t) ∈ / M ) ∨ (Δ ≤ Δm ), (3.21) + + Δ(t ) = 0, x(t ) = AR x(t) (x(t) ∈ M ) ∧ (Δ > Δm ). Proposition 3.5 A regular reset control system (3.21) is (asymptotically) stable if the time-varying discrete time system (3.20) is uniformly (asymptotic) stable for any sequence (Δ1 , Δ2 , . . . ), where Δk > Δm , k = 1, 2, . . . . Proof If the discrete time system given by (3.20) is uniformly stable for any sequence (Δ1 , Δ2 , . . . ), with Δk > Δm , k = 1, 2, . . . then it is true that xd (k) ≤ γ xd (k0 ), k ≥ k0 , for any k0 and xd (k0 ) = x0 , and some γ > 0. In particular, for any x0 ∈ Rn and k0 = 0, reset intervals will be given by Δk = Δk (x0 ) = tk − tk−1 = τk (x0 ) − τk−1 (x0 ), k = 1, 2, . . . , making a sequence (Δ1 , Δ2 , . . . ) with Δk > Δm by assumption. Then, the solution of the system (2.4) at the after-reset instants satisfies x(tk+ , x0 ) = xd (k) ≤ γ x0 , k ≥ 0. Now, using standard arguments based on the Gronwall inequality (see, for example, [32]) it is possible to bound the solution between two consecutive reset instants. Since the solution x(t, x0 ) at some instant t ∈ (tk , tk+1 ] is given by  t

 Ax(s, x0 ) ds, (3.22) x(t, x0 ) = x tk+ , x0 + tk

102

3 Stability of Reset Control Systems

by taking norms, we get 

 x(t, x0 ) ≤ x tk+ , x0  + α



t

x(s, x0 ) ds,

(3.23)

tk

where the induced norm A is always bounded by some real number α, that is, A ≤ α. Finally, using Gronwall inequality, x(t, x0 ) ≤ x(tk+ , x0 )eα(t−tk ) , showing that the solution x(t) is always bounded in every interval t ∈ (tk , tk+1 ], resulting in x(t, x0 ) ≤ γ eα(t−tk ) x0 , where γ and α do not depend on the initial condition x0 ∈ M . In addition, since the reset control system is regular, if the base linear system is unstable, that is, A is not Hurwitz, the solution x(t, x0 ) satisfies x(t, x0 ) ≤ γ eαΔk x0  ≤ γ eαΔM x0 

(3.24)

for t ≥ 0, including reset instants. Thus, it is obtained that the system (2.4) is stable. Note that in the case in which A is Hurwitz, reset intervals do not need to be upperbounded since the exponential is always bounded, and stability of (2.4) directly follows. Asymptotic stability follows by using similar arguments and the fact that x(tk+ , x0 ) ≤ γ λk x0 , k ≥ 0 for any x0 ∈ Rn , with 0 ≤ λ < 1, by uniformly asymptotic stability of the discrete time system (3.19). Then, as t → ∞, it is true that k → ∞ since t ∈ (tk , tk+1 ] and tk+1 − tk = Δk+1 (x0 ) > Δm for any x0 ∈ Rn , and in addition since the reset control system is regular, x(t, x0 ) ≤ γ λk eαΔk x0  ≤ γ λk eα·ΔM x0 , thus x(t, x0 ) → 0 for any x0 ∈ Rn as t → ∞.

(3.25) 

Proposition 3.6 A regular reset control system is asymptotically stable if there exist a positive definite matrices sequence Pk , k = 1, . . . , such that ηI ≤ Pk ≤ ρI and

Δ

eA

k

AR Pk+1 AR eAΔk − Pk ≤ −εI

(3.26)

for some finite positive constants η, ρ, and ε, and for any reset interval sequence (Δ1 , Δ2 , . . . ), where Δk > Δm , k = 1, 2, . . . . Proof It is a direct application of Proposition 3.5 and discrete time-varying system’s stability results as given, for example, in [27, Theorem 23.3].  In comparison with the reset times-independent results in the previous section, notice that there the stability conditions are:  T T if x ∈ / M \ {0}, x (A P + P A)x < 0 (3.27) xT (ATR P AR − P )x ≤ 0 if x ∈ M , which implies that the quadratic Lyapunov function V (x) = xT P x is decreasing over the time intervals (tk , tk+1 ) and non-increasing at the reset instants tk , k =

3.2 Lyapunov Stability

103

1, 2, . . . . These stability conditions have also been previously published in [23] and the seminal work [2], in the context of impulsive systems with impulses at fixed instants. It is clear that from the first condition of (3.27) the base linear system must always be asymptotically stable. Note that even in the case in which the base linear system is asymptotically stable, the second condition of (3.27) may be not compatible in general with the solutions of the first condition. In addition, in the previous section it has been shown that (3.27) is equivalent to the Hβ -condition. Thus, the main contribution of Proposition 3.5 (and Proposition 3.6) is to circumvent the Hβ -condition by allowing that the continuous time evolution between two consecutive reset instants, in (tk , tk+1 ), k = 1, 2, . . . , is not necessarily associated with a Lyapunov function (quadratic or not) in those intervals, thus results are less conservative: (i) the proposition may be applied to unstable base linear systems, (ii) it may be applied to stable base linear systems in which condition (3.27) cannot be applied, as we will see in the two following examples.

3.2.2.1 Examples Unstable Base System Consider a reset control system as given by (2.4), where A, AR , and C are given by ⎛

−2 A=⎝ 1 0

⎞ 0 1 0 0⎠, −1 1

⎛

1 AR = ⎝ 0 0

0 1 0

⎞ 0 0⎠, 0

 C= 001 .

This corresponds to the feedback system of Fig. 2.1 with P (s) =

1 s(s+2)

(3.28)

and a FORE

In this case, the reset control system is regular with base compensator Rb (s) = since although A is not Hurwitz it has a pair of dominant complex eigenvalues, its spectrum is σ (A) = {0.57 + 0.37i, 0.57 − 0.37i, −2.15}; and, in addition, (C, A, x0 ) is reduced for x0 = [x01 0 0]T ∈ MR and x01 = 0, since in this case equation (2.7) is given (after some computation) by 1 s−1 .

 y(t) = 0.42e0.57t cos(0.37t) + 0.37 e0.57t sin(0.37t) − 0.42 e−2.15t x01 . Thus, after the first reset, reset instants are periodic with period Δ = 2.3 (the first zero of y(t) that is bigger than Δm = 0.1) for any initial condition. Note that AR eAΔ is Schur stable (its eigenvalues are strictly inside the unit circle), but that the reset control system is not regular since the after-reset surface is not reached for initial conditions x0 ∈ span{(−0.898, 0.418, 0.133)T }. Thus, Proposition 3.4 can be used to state that the reset control system is unstable. If the initial condition set were restricted to MR [7] then the reset control system would be stable since AR eAΔ is Schur stable.

104

Fig. 3.2 Max. eigenvalue of eA

3 Stability of Reset Control Systems

Δ

AR P AR eAΔ − P + εI vs Δ

Stable Base System This example is given to show how the Hβ -condition can be overtaken for reset control systems with stable base system. Consider a reset control system as given by (3.21) with a dwell-time constant Δm = 0.1 and matrices ⎛ ⎞ ⎛ ⎞ 1 0 0 −2 0 1.6

 0 ⎠, C= 001 (3.29) A = ⎝ 0.5 0 AR = ⎝ 0 1 0 ⎠ , 0 0 0 −2 −1 1 3.2 corresponding to Rb (s) = s−1 and P (s) = s+0.25 . It can be easily checked that, als 2 +2s though A is Hurwitz (and thus the reset control system is regular), the Hβ -condition is not satisfied, that is, there does not exist any β ∈ R (β is scalar in this case) such that Re{Hβ (j ω)} = Re[ β 0 1 ](j ωI − A)−1 [ 0 0 1 ]T > 0 (see the second example in Sect. 3.2.1). Now, it has been found that (see the search procedure given in next section) (3.26) is feasible for constant matrices Pk = P , where P is given by ⎛ ⎞ 22.07 7.91 −7.01 P = ⎝ 7.91 41.96 1.22 ⎠ (3.30) −7.01 1.22 46.86

and for any Δk ≥ 0.03, k = 1, 2, . . . . Figure 3.2 shows the maximum eigenvalue of

eA Δ AR P AR eAΔ − P + εI vs Δ, for ε = 0.001. It is always negative for Δ ≥ 0.03. As a result, Proposition 3.6 guarantees that the reset control system is asymptotically stable. As it will be developed in next section, for the case of stable base

3.2 Lyapunov Stability

105

Fig. 3.3 Base system (dashed line), and reset system (solid line)

systems this method can be used to compute the values of the dwell-time constant Δm that stabilizes the reset control system. In this example, stability is assured for any Δm ≥ 0.03. Figure 3.3 shows a time simulation of the closed-loop output for the base and the reset control system.

3.2.3 Stabilization of Reset Control Systems In this section, Proposition 3.5 and Proposition 3.6 will be used to adjust the dwelltime constant Δm in such a way that, by avoiding reset actions at some crossings with the reset surface, stability of the reset control system is achieved. The case in which the base system is stable is first approached, note that for the unstable case a separate treatment is necessary, since it is not enough to avoid reset actions, it is also necessary to force resets at some instants even if the reset surface is not crossed.

3.2.3.1 Stable Base System Proposition 3.7 For the reset control system (3.21) with A Hurwitz, there always exists a value Δm > 0 of the dwell-time constant such as for any Δ > Δm the inequality

Δ

eA

AR P AR eAΔ − P ≤ −εI

(3.31)

106

3 Stability of Reset Control Systems

is satisfied for some P > 0 and ε > 0. Therefore, the reset control system (3.21) is asymptotically stable for that Δm . Proof Since A is Hurwitz and then the reset control system is regular, stability directly follows from Proposition 3.6, by taking Pk = P , for k = 1, 2, . . . . Note that since A is Hurwitz, the matrix eAΔ can be made arbitrarily small in norm, by choosing Δ large enough. Thus, for any ε > 0 there always exists a Δm such for any Δ > Δm it is true that eAΔ  ≤ ε. Some valid explicit relations Δ = f (ε) could be derived using some canonical form of A and eAΔ (diagonal, Jordan form, etc.) and bounding |eλi Δ |, where λi are the (stable) eigenvalues of A. Then, for the sequence P1 = P2 = · · · = P for some P > 0, a Δ large enough can be chosen (Δ > Δm ) so that the left side of the inequality (3.31) approaches 0−P = −P , which is obviously smaller than −εI for small ε > 0.  A Search Procedure for Δm Proposition 3.7 gives a condition that is easy to check in practice by using available algorithms for solving LMIs. However, some care has to be taken because as it is well known the norm eAΔ  is not necessarily monotonic with respect to Δ. For A Hurwitz, when Δ → ∞ it is true that eAΔ → 0, but this does not imply monotonicity of eAΔ  as a function of Δ. Before the search procedure, a lower bound of the solution (ΔLB ) can be found by the map of the feasibility of the LMI (3.31) vs Δ. This lower bound is defined by ΔLB ∈ R such that for all Δ ≥ ΔLB the LMI (3.31) is feasible for some P (Δ). A search procedure for Δm is proposed as follows: 1. Initial value: Let us start with a large initial value ΔL so that the LMI (3.31) is feasible for the single value Δ = ΔL with respect to some P = PL > 0, and also for all Δ > ΔL the LMI (3.31) is feasible. 2. Feasible values of P : A set of feasible values of P is obtained simply by sampling the interval (0, ΔL ] and solving the LMI (3.31) for P with a given sampled value of Δ. More precisely, with some step size Δ1 , a set of reset intervals {Δ1,1 = Δ1 , Δ2,1 = 2Δ1 , Δ3,1 = 3Δ1 , . . . } ⊂ (0, ΔL ] is chosen. Now, for every Δk,1 a matrix Pk > 0 is computed by solving the LMI (3.31). As a result, a set of feasible pairs (Δk,1 , Pk ), k = 1, 2, . . . , is obtained. Let P = {P1 , P2 , . . .}. 3. Feasible values of Δ: The next step is to compute the minimum value Δm,k such that for any Δ > Δm,k the LMI (3.31) is feasible for some Pk ∈ P. In order to obtain that result, every matrix Pk ∈ P generates a matrix Qk (Δ) =

eA Δ AR Pk AR eAΔ − Pk + εI . Then, the interval (0, ΔL ] is sampled with another step size Δ2 such that the maximum eigenvalue of Qk (ΔL − iΔ2 ), where i = 0, 1, . . . , is computed and i is increased until the maximum is non-negative for some value of i = ik + 1; therefore, for all Pk ∈ P, a Δm,k = ΔL − (ik )Δ2 is found. Again the LMI has to be checked with some accuracy Δ2 (not necessarily equal to Δ1 ), for Δ ∈ {Δ1,2 = Δ2 , Δ2,2 = 2Δ2 , Δ3, = 3Δ2 , . . . }. As a result, a set of pairs (Δk,m , Pk ), k = 1, 2, . . . , is now obtained. 4. Finally, the desired Δm is the minimum value of Δm,k for k = 1, 2, . . . , obtained for some km , and Pkm is the associated matrix.

3.2 Lyapunov Stability

107

The number of LMIs to be checked is of order Δ2L /(Δ1 Δ2 ). This procedure guaranties that a minimum value Δm and a matrix P = Pkm exist such that the LMI (3.31) is feasible for Δ > Δm . On the other hand, note that Proposition 3.7 gives a simple and practical method for designing stabilizing reset control systems. Once the bound Δm has been computed, the reset control system should only reset when reset intervals are larger than Δm . Stabilization by Removing Reset Actions at Some Crossings As it is well known, there exist cases in which reset actions can destabilize a stable base closedloop system. This is the case of the example to be analyzed in the following, which is borrowed from [9]. It will be shown how stability of the reset system can be assured by simply avoiding those reset intervals that are not large enough. Consider a standard reset control system where the base compensator Rb (corre1 sponding to a FORE compensator) and the plant P are given by Rb (s) = s+1 and 2

(s+1)(s+10) P (s) = s 4 +4s 3 +6s 2 +5s−98 , respectively, corresponding to a state-space realization with matrices A and AR : ⎛ ⎞ −4 −0.750 −0.312 1.531 4 ⎜ 8 0 0 0 0 ⎟ ⎜ ⎟ ⎜ 2 0 0 0 ⎟ A=⎜ 0 ⎟, ⎝ 0 0 4 0 0 ⎠ ⎛−0.25 −0.656 ⎞−1.875 −0.391 −1 (3.32) 1 0 0 0 0 ⎜0 1 0 0 0⎟ ⎜ ⎟ ⎟ AR = ⎜ ⎜0 0 1 0 0⎟. ⎝0 0 0 1 0⎠ 0 0 0 0 0

Note that the reset control system is regular since A is Hurwitz. It has been found that the LMI (3.31) is feasible for Δ ≥ 27.02 and some P > 0. As a result of the search procedure a matrix Pm has been found: ⎛ ⎞ 0.4680 0.0594 0.0395 0.0296 0.0556 ⎜ 0.0594 0.3630 0.0034 0.119 −0.0161 ⎟ ⎜ ⎟ 4⎜ (3.33) Pm = 10 ⎜ 0.0395 0.0034 0.4300 0.0597 0.1110 ⎟ ⎟ ⎝ 0.0296 0.1190 0.0597 0.170 0.2810 ⎠ 0.0556 −0.0161 0.1110 0.2810 1.0600 making the LMI (3.31) feasible for this Pm and Δ ≥ Δm = 27.02. Figure 3.4 shows

the maximum eigenvalue of eA Δ AR P AR eAΔ − P , which is always negative for values of Δ ≥ 27.02. Since the reset control system produces crossings with the reset surface with intervals that are smaller than 27.02, it is not guaranteed that the reset control system is stable in this case. Figure 3.5 shows a simulation for this system, comparing the closed loop outputs both for the base and the reset control system, for Δm = 0.1. Using again Proposition 3.7, it can be concluded that the closed-loop reset system is

108

Fig. 3.4 Max. eigenvalue of eA

3 Stability of Reset Control Systems

Δ

AR P AR eAΔ − P + εI vs Δ

Fig. 3.5 Closed loop output for Δm = 0.1

asymptotically stable if the reset intervals Δk , k = 0, 1, . . . , are always greater than Δm = 27.1. Figure 3.6 shows a simulation of the reset control system avoiding resets when the reset interval is smaller than Δm = 27.1. Note that the system, although stable,

3.2 Lyapunov Stability

109

Fig. 3.6 Reset control system output for Δm = 27.1

Fig. 3.7 Reset control system output for Δm = 30

is very oscillatory since the reset intervals are close to the stability limit. Figure 3.7 shows a simulation for Δm = 30. In this case, the system behavior is much less oscillatory due to the fact that reset intervals are far away from the stability limit.

110

3 Stability of Reset Control Systems

3.2.3.2 Unstable Base System In the previous subsection, stability of reset control systems has been considered for stable base systems, that is, for the case in which the matrix A is Hurwitz. In this subsection, Proposition 3.5 is used to elaborate a sufficient condition for stability of reset control systems in the case in which the closed loop matrix A is not Hurwitz, and thus the base linear system is unstable. It should be noted that when A is not Hurwitz the Hβ -condition, developed in [9], is not applicable. In addition, in the case in which A is not Hurwitz and thus the base system is not stable, solutions of (3.31) are typically given for some bounded set (usually a union of intervals) I ⊂ R for a given definite positive matrix P . First, the case in which the reset control system is regular is investigated. In this case, the reset control system (3.21) is redefined to allow resets only at the instants in which Δ ∈ I :  ˙ = 1, Δ(t) x˙ (t) = Ax(t) (x(t) ∈ / M ) ∨ (Δ ∈ / I ), (3.34) Δ(t + ) = 0, x(t + ) = AR x(t) (x(t) ∈ M ) ∧ (Δ ∈ I ), and thus the stability of reset control system easily follows. Proposition 3.8 For a regular reset control system (3.34), assume that I ⊂ R is a closed bounded set such that for any scalar Δ ∈ I the inequality (3.31) is satisfied for some ε > 0, and a positive-definite matrix P . Then, the reset control system is asymptotically stable. In the case in which the base system is not regular, that is, when there exist initial conditions that do not reach the reset surface and thus the reset control system is directly unstable since the base system is not stable, some reset may be forced in order to stabilize the system. In this case, let ΔI be the upper bound of I , that is, ΔI = max{I }, then the reset control system may be modified, resulting in  ˙ = 1, Δ(t) x˙ (t) = Ax(t) (x(t) ∈ / M ) ∨ (Δ ∈ / I ) ∧ (Δ = ΔI ), (3.35) + + Δ(t ) = 0, x(t ) = AR x(t) (x(t) ∈ M ) ∧ (Δ ∈ I ) ∨ (Δ = ΔI ). Proposition 3.9 For a reset control system (3.35), assume that I ⊂ R is a closed bounded set such that for any scalar Δ ∈ I the inequality (3.31) is satisfied for some ε > 0, and a positive-definite matrix P . Then the reset control system is asymptotically stable. A search procedure similar to that developed in the previous section can be used to obtain an interval I , although there are some particularities (see [4]). In the following, two examples are developed. The first example consists of a lower order base system (with m = 1), in particular the plant is a second order model with an integrator and a real pole. In the second example, a higher order base system (m = 4) is considered.

3.2 Lyapunov Stability

111

A Low Order Base System Consider a reset control system with the base compensator Rb and the plant P given by Rb (s) =

8 , s +1

P (s) =

1 . (s + 2)s

(3.36)

The (unstable) closed-loop base system is given by the transfer function T (s) =

s3

8 . + 2s + 8

(3.37)

+ 3s 2

A state-space realization of the base system transfer is obtained as (Rb )  (P )

Ap =

Ar = −1,

 −2 0 , 1 0

Br = 4,

Cr = 2,

  1 Bp = , 0

Dr = 0,

 Cp = 0 1 ,

from where the closed-loop state matrices are given by ⎛ ⎛ ⎞ ⎞ −2 0 2 1 0 0 0 0 ⎠, A=⎝ 1 AR = ⎝ 0 1 0 ⎠ , 0 −4 −1 0 0 0 For the positive-definite matrix P given by ⎛ 29.18 10.56 P = ⎝ 10.56 176.23 1.49 −19.39

(3.38)

Dp = 0, (3.39)

 C= 010 .

⎞ 1.49 −19.39 ⎠ , 93.07

(3.40)

(3.41)

it has been found that the resulting set I is given by I = [0.92, 1.72] ∪ [3.16, 3.51]

(3.42)

as shown in Fig. 3.8. In this example, there are initial conditions that do not produce crossings with the surface reset, and thus the reset control system is not regular. As a result, Proposition 3.9 can be used to show that the reset control system (3.35) is stable with I given by (3.42) and ΔI = 3.51. Figure 3.9 shows a time simulation comparing closed-loop outputs for the reset control system and its base system. A High Order Base System Consider the reset control system with base compensator Rb and plant P given by Rb (s) =

350 , s +1

P (s) =

s5

+ 13s 4

s +3 + 87s 3 + 305s 2 + 350s

(3.43)

that results in a unstable base control system, for example, the (unstable) closedloop transfer function from the reference input to the output is given by

112

3 Stability of Reset Control Systems

Fig. 3.8 Maximum eigenvalue of eA

TΔ

AR P AR eAΔ − P + εI vs Δ

Fig. 3.9 Closed-loop output for the linear base control system (dashed line), and the reset control system (solid line)

T (s) =

s6

+ 14s 5

+ 100s 4

350(s + 3) . + 392s 3 + 655s 2 + 700s + 1050

(3.44)

3.3 Reset Control Systems with Inputs/Passivity Analysis

Fig. 3.10 Maximum eigenvalue of eA

TΔ

113

AR P AR eAΔ − P + εI vs Δ

In this case, it has been found that for the positive-definite matrix P ⎛ ⎞ 16.22 1.16 2.14 2.42 0.79 3.08 ⎜ 1.16 14.68 1.46 5.44 1.93 1.74 ⎟ ⎜ ⎟ ⎜ 2.14 ⎟ 1.46 9.77 2.92 10.78 2.26 ⎜ ⎟ P =⎜ ⎟ 2.42 5.44 2.92 9.22 10.45 −0.68 ⎜ ⎟ ⎝ 0.79 1.93 10.78 10.45 28.11 −9.55 ⎠ 3.08 1.74 2.26 −0.68 −9.55 27.35

(3.45)

the resulting set I is I = [1.12, 1.92] ∪ [3.37, 3.96] ∪ [5.72, 6.01] ∪ [8.05, 8.08]

(3.46)

as shown in Fig. 3.10. On the other hand, this reset control system is not regular since the base system is unstable and in addition there exist initial conditions that do not reach the surface reset. Using again Proposition 3.9, the reset control system (3.35) is stable with I given by (3.46) and ΔI = 8.08. Figure 3.11 shows a time simulation comparing closed-loop outputs for the reset control system and its base system.

3.3 Reset Control Systems with Inputs/Passivity Analysis Regarding L2 -stability of reset systems with inputs, a number of works have been developed that give results for particular cases of reset compensators and/or inputs.

114

3 Stability of Reset Control Systems

Fig. 3.11 Closed-loop output for the linear base control system (dashed line), and the reset control system (solid line)

The work [24] approaches the problem for compensators in which its output has the same sign as its input, and the zero reference case is considered. In addition, in [1], L2 -stability conditions for the case of nonzero references are given. The conservatism given by the Hβ -condition is improved for that kind of system. On the other hand, dissipative systems theory was developed in [33], where the passive system concept which was developed in electric circuit and mechanical systems was extended to abstract systems. That theory is widely used in interconnected systems due to the fact that the feedback interconnection of two passive systems is a passive system. In [29], passivity techniques are shown to be a powerful tool for nonlinear control. Dissipative systems theory has been developed for a hybrid system in [17], where dissipative hybrid system’s definitions are developed for classic concepts such as supply rate. A few works have dealt with passive systems theory and hybrid systems [35], [11], [25], [8]. In spite of the fact that a single input–output approach to passive systems theory cannot be sufficient for more general hybrid systems, this section will show how several passive properties can be obtained for reset compensators. The goals of this section are: (a) to obtain stability conditions that are applicable to feedback interconnections of reset compensators and nonlinear plants and (b) to find passive reset compensators that can be used in passive control techniques. The passivity approach will be used for these purposes. Easily checkable passivity conditions on the base linear compensator will be presented. In general, the feedback system given by Fig. 3.12 will be considered, where w and d are the reference and disturbance inputs, respectively. R is a single-input single-output (SISO) reset compensator to be defined later on, and P is an SISO plant.

3.3 Reset Control Systems with Inputs/Passivity Analysis

115

Fig. 3.12 Reset controller R applied to an LTI plant

By definition, the linear  ∞ space L2 consists of all measurable functions f (·) : R+ → R such that 0 |f (t)|2 dt < ∞, with the L2 -norm  ·  : L2 → R+ ∞ 1 given by f  = ( 0 |f (t)|2 dt) 2 . Let ΠT be a projector operator such that ΠT (f )(t) = f (t), for 0 ≤ t ≤ T , and ΠT (f )(t) = 0, for t > T . Then the extended space L2,e is given by all functions f (·) : R+ → R such that ΠT (f ) belongs to L2 . The feedback interconnection is given simply by e(t) = w(t) − y(t),

u(t) = v(t) + d(t).

(3.47)

The feedback system of Fig. 3.12 is L2 -stable (with respect to inputs w and d) if for every input signals w ∈ L2 and d ∈ L2 , the outputs u ∈ L2 and y ∈ L2 . In addition, it is finite-gain stable if there exists a positive constant γ > 0 such that y2 + u2 ≤ γ (w2 + d2 ). Here the plant P is represented by the state space model  x˙p = f (xp , u) if u ∈ R, (P ) (3.48) if y ∈ R, y = g(xp , u) where np is the dimension of the state xp , f : Rnp × R → Rnp is locally Lipschitz, g : Rnp × R → R is continuous, f (0, 0) = 0, and g(0, 0) = 0. In addition, following the notation given in [17], the reset compensator dynamic is described by three elements: (a) a continuous-time dynamical equation, (b) a difference equation, and (c) a resetting law. Whenever the resetting law is not held, the system flows continuously; otherwise, when the resetting law is satisfied, the system jumps in a discrete way. An LTI base compensator will be considered in this work, then the reset compensator R is given by the impulsive differential equation (IDE): ⎧ x˙r = Ar xr + Br e if e = 0, ⎪ ⎪ ⎨ xr+ = Aρ xr if e = 0, (3.49) (R) ⎪ ⎪ ⎩ u = Cr xr + Dr e, where nr is the dimension of the state xr , Aρ is a diagonal matrix with zeros in the states to be reset and ones in the rest of compensator states, nρ is defined as

116

3 Stability of Reset Control Systems

the dimension of the reset subspace, and nρ is defined as the dimension of the nonreset subspace (nρ + nρ = nr ). When Aρ = 0, R will be referred to as a full reset compensator; otherwise, it will be referred to as a partial reset compensator. The first equation in (3.49) describes the continuous compensator dynamic at the non-resetting instants, and the second equation gives the reset operation as a jump of the compensator state at the reset instants. Note that reset instants are produced when the compensator input is zero. The base compensator is simply obtained by elimination of the reset actions in (3.49) and thus has the transfer function Rbl (s) = Cr (sI − Ar )−1 Br + Dr , where it is assumed that (Ar , Br , Cr , Dr ) is a minimal realization. It is assumed that the feedback system is well-posed and thus Zeno solutions are avoided. Alternatively, time regularization may be imposed (see (2.7)), where the resetting law is deactivated during a time Δm after every reset action. As a result, it is assumed that there exists a finite number of resets over every finite time interval, thus the reset actions will be given by the countable set {t1 , t2 , . . . , tk , . . . } where tk+1 − tk > Δm for all k = 1, 2, . . . and limt→∞ tk = ∞. A system H : L2,e → L2,e , with input u and output y = H u is said to be passive if there exists a constant β ≤ 0 such that 

T

u (t)y(t) dt ≥ β,

∀T ≥ 0, ∀u ∈ L2 .

(3.50)

0

If there are constants δ ≥ 0 and ε ≥ 0 such that 

T 0





T

u (t)y(t) dt ≥ β + δ 0





T

u (t)u(t) dt + ε

y (t)y(t) dt,

(3.51)

0

for all functions u, and all T ≥ 0, then the system is input strictly passive (ISP) if δ > 0, output strictly passive (OSP) if ε > 0, and very strictly passive (VSP) if δ > 0 and ε > 0. In particular, for a linear time-invariant system with a transfer function given by H (s) = C(sI − A)−1 + D, with A Hurwitz and the pair (A, B) controllable, it is true that [21] 1. The system is passive if and only if Re[H (j ω)] ≥ 0 for all ω. 2. The system is ISP if and only if there is a δ > 0 such that Re[H (j ω)] ≥ δ > 0 for all ω. 3. The system is OSP if and only if there is an ε such that Re[H (j ω)] ≥ ε|H (j ω)|2 for all ω. Note that these three properties are verifiable in a Nyquist diagram: if H (j ω) is in the closed right half-plane then the system is passive; if H (j ω) is in Re[H (j ω)] ≥ 1 δ > 0, the system is ISP; and if H (j ω) is inside the circle with center at s = 2ε and 1 radius r = 2ε , then the system is OSP. When the system is in state space representation, the passivity of the input–output maps as before is replaced by the following notions from dissipative systems theory.

3.3 Reset Control Systems with Inputs/Passivity Analysis

Consider the system H : L2,e → L2,e given by  x˙ = f (x, u) if u ∈ R, (H ) y = g(x, u) if y ∈ R,

117

(3.52)

where f and g have the same properties as in (3.48). H is said to be dissipative respect to a supply rate w(t) = w(u(t), y(t)) if there exists a storage function V (x) ≥ 0 such that the following dissipation inequality holds  T V (x) ≤ V (x(0)) + w(u(t), y(t)) dt; ∀u, ∀x(0), ∀T ≥ 0. (3.53) 0

Following [29], the relationship between dissipative and passive systems is given by the choice of a particular supply rate: 1. The system H is passive if it is dissipative with respect to supply rate wp = u y and V (0) = 0. 2. The system H is input strictly passive (ISP) if it is dissipative with respect to supply rate wi = u y − εu u, for some ε > 0. 3. The system H is output strictly passive (OSP) if it is dissipative with respect to supply rate wo = u y − δy y, for some δ > 0. 4. The system H is very strictly passive (VSP) if it is dissipative with respect to supply rate wv = u y − δu u − εy y, for some ε > 0 and δ > 0. Comparing (3.50) and (3.53), since V (x) ≥ 0, the relationship between β and the initial condition is clear, namely, β = −V (x(0)). An important result is the Passivity Theorem (see [29]): the feedback interconnection of Fig. 3.12 is finite-gain stable if εR + δP > 0, εP + δR > 0, where εR , δR , εP , and δP are such that they satisfy (3.51) for the systems R and P , respectively.

3.3.1 Full Reset Compensators The main goal of this section will be to analyze the passivity properties of reset compensators as described by (3.49). Firstly, conditions on reset compensators are derived in order to show when they are (strictly) passive systems. Full reset compensators and partial reset compensators are studied in different sections. Finally, the passivity theorem will be used to show stability properties of the feedback system given in Fig. 3.12. Notice that plants can be nonlinear in general, and will be described by (3.48). In spite of the fact that passivity theory (as an input–output theory) is not appropriate in general for hybrid systems, it can be successfully used for full reset

118

3 Stability of Reset Control Systems

compensators given by (3.49). This is due to the fact that the system loses all of its memory at every reset time, i.e., the base compensator is restarted with zero initial condition. Therefore, a space states description is not necessary because the output compensator just depends on the input since the last reset time. This is the key point in the next proposition. Proposition 3.10 A full reset compensator R (given by (3.49) with Aρ = 0) is passive, ISP, OSP, or VSP if the base compensator is passive, ISP, OSP, or VSP, respectively. Proof The result is proved for the VSP case, the rest of the cases follow similar arguments. If the base compensator is VSP, then there exist δ > 0 and ε > 0 such that (3.51) is satisfied for all T ≥ 0 and for all u ∈ L2 . Since, by assumption, in every finite time interval there are a finite number of resets instants {0 < t1 , t2 , . . . } with 0 < t1 < t2 < · · · and limk→∞ tk = ∞, there exists a finite integer k such that for every T ∈ (tk , tk+1 ] the integration interval [0, T ] can be divided into a finite number of subintervals 

T

u (t)y(t) dt =



t1

u (t)y(t) dt +

i=k−1   ti+1

0

0

i=1



T

+

u (t)y(t) dt

ti

u (t)y(t) dt.

(3.54)

tk

In addition, since the base system is time invariant and VSP by assumption, it is true that  t1  t1  t1



u (t)y(t) dt ≥ β + δ u (t)u(t) dt + ε y (t)y(t) dt, (3.55) 

0 ti+1

u (t)y(t) dt ≥ δ

ti

and

0



ti+1

u (t)u(t) dt + ε

ti



T

tk

u (t)y(t) dt ≥ δ



0



ti+1

y (t)y(t) dt,

(3.56)

ti

T tk

u (t)u(t) dt + ε



T

y (t)y(t) dt

(3.57)

tk

for any u ∈ L2 and for i = 1, 2, . . . , k − 1. In (3.56) and (3.57), β = 0 has been taken due to the system having null initial conditions at ti+ for all i = 1, 2, . . . , k. In addition, the integral limits ti+ can be taken as ti by integral properties. Then, after direct substitution of (3.55), (3.56), and (3.57) into (3.54), we at once conclude that  T  T  T



u (t)y(t) dt ≥ β + δ u (t)u(t) dt + ε y (t)y(t) dt. (3.58) 0

0

0

Thus, the reset compensator is a VSP system with the same constants β, δ, and ε as its base compensator. 

3.3 Reset Control Systems with Inputs/Passivity Analysis

119

Remark 3.1 Note that the formulation of passivity, ISP, OSP, and VSP given in [29] (Sects. 2.2 and 3.1) is completely general, and thus does not involve any assumption regarding the continuity of system trajectories of the reset system R as given by (3.49). In particular, the state trajectories may be discontinuous at the reset instants, and passivity definitions and the passivity theorem can be directly applied to reset systems in order to obtain input–output stability properties. Remark 3.2 Note that in the proof of the previous proposition, linearity of the base compensator has been used simply to state that it has null initial conditions at ti+ for all i = 1, 2, . . . . Thus, it holds for any full-reset compensator, not necessarily linear, as long as β = 0 or, using the dissipativity framework to be elaborated in the next subsection, the storage function of the compensator satisfies V (0) = 0. Some examples of reset compensators are analyzed in the following.

3.3.1.1 The First Order Reset Element (FORE) A typical reset compensator is the first order reset element (FORE) introduced in [18, 20]. The transfer function of its base compensator is FORE(s) =

k s+b

(3.59)

where it will be assumed that k, b > 0. The real part of its frequency response is given by kb , b2 + ω2 and thus, applying Proposition 3.10, it is directly obtained that Re[FORE(j ω)] =

(3.60)

• FORE is passive since Re[FORE(j ω)] > 0 for all ω, • FORE is not ISP because limω→∞ Re[FORE(j ω)] = 0, • FORE is OSP, since for ε = bk b Re[FORE(j ω)] = |FORE(j ω)|2 . k 3.3.1.2 A Reset Lag Compensator (Reset-PI) In order to obtain VSP reset compensators, it is necessary that their base compensators have the same number of poles as zeros. A possible choice is a base lag compensator PI(s) = k

1+Ts 1+γTs

(3.61)

120

3 Stability of Reset Control Systems

where γ ∈ (1, ∞). Its frequency response real part is given by Re[PI(j ω)] = k

1 + γ (T ω)2 , 1 + (γ T ω)2

(3.62)

then it is easy to check that Re[PI(j ω)] > min(1, γk ), and finally, | PI(j ω)|2 = k 2

1 + (T ω)2 . 1 + (γ T ω)2

(3.63)

Applying Proposition 3.10, it is now obtained that • Reset-PI is passive due to Re[PI(j ω)] > 0 for all w, • Reset-PI is ISP with δ = min(1, γk ), • Reset-PI is OSP with ε = γk . 3.3.1.3 A Reset-PID Compensator As example of higher order reset element, consider as base a PID compensator with the transfer function PID(s) = kp γ

1 + Ti s 1 + Td s 1 + γ Ti s 1 + αTd s

(3.64)

where 0 ≤ Td < Ti , 1 ≤ β < ∞, and 0 < α ≤ 1. This base compensators is VSP, then the full reset compensator based on it, referred to as reset-PID, is VSP. Several other cases are possible for particular combination of parameters, a particular study of the Nyquist plot is necessary in each case.

3.3.2 Partial Reset Compensators In the partial reset compensator case, where only some of the compensator states are reset, dissipativity theory needs to be used (in contrast with the full reset case). Note that after each reset action, partial reset results in a possibly nonzero initial condition for the compensator state in comparison (as opposed to full reset, which produces a zero initial condition). Thus, for every initial condition after reset, a different input– output system has to be taken into consideration. As a result, the compensator state has to be included in the formulation and the result of Proposition 3.10 is not applicable. Dissipative systems theory leads to a solution of this problem due to the fact that the dissipativity property holds for every initial condition. Proposition 3.11 A partial reset compensator R (given by (3.3) with the reset matrix Aρ = diag(Inρ , 0nρ )) is passive, ISP, OSP, or VSP if its base compensator is

3.3 Reset Control Systems with Inputs/Passivity Analysis

121

dissipative with respect to supply rate wp , wi , wo , or wv , respectively, and a storage function V (x) that satisfies V (Aρ x) ≤ V (x)

(3.65)

for every x ∈ Rnr . Proof Again the case VSP is considered, the remaining cases follow similar arguments. Since, by assumption, the base compensator is dissipative with respect the storage function V and the supply rate wv = u y − δu u − εy y for some δ > 0 and ε > 0, the following inequality is satisfied  T wv (u(t), y(t)) dt, ∀u ∈ L2,e , ∀x(0) ∈ Rnr , ∀T ≥ 0. V (x(T )) ≤ V (x(0)) + 0

(3.66) For a given input u(t), by assumption there is a finite number of reset times ti , i = 1, 2, . . . , k, in the interval [0, T ], with T ∈ (tk , tk+1 ]. Thus, it is true that  T

 wv (t) dt (3.67) V (x(T )) ≤ V x tk+ + tk

since in the interval [tk+ , T ] the reset compensator behaves like its base compensator with x(tk+ ) the initial condition. Applying a similar argument, it is true that  ti

+  V (x(ti )) ≤ V x ti−1 + wv (t) dt (3.68) ti−1

for i = 2, . . . , k, and also



V (x(t1 )) ≤ V (x(0)) +

ti

(3.69)

wv (t) dt. 0

Then, applying condition (3.65) at the reset instants, it is obtained that V (x(ti+ )) = V (Aρ x(ti )), i = 1, . . . , k. Using this fact and (3.67)–(3.69), the result is



+  V x(T ) ≤ V x tk−1 +

+  = V x tk−1 +

+  ≤ V x tk−2 +



tk

 wv (t) ds +

tk−1



T

wv (t) dt tk

T

wv (t) dt tk−1



T

wv (t) dt ≤ · · · ,

tk−2

and then it is concluded that the reset compensator satisfies  T wv (t) dt. 0 ≤ V (x(T )) ≤ V (x(0)) + 0

(3.70)

122

3 Stability of Reset Control Systems

As a result, it is true that  0

T

u (t)y(t) dt ≥ β + δ



T

u (t)u(t) dt + ε

0



T

y (t)y(t) dt,

(3.71)

0

where β = −V (x(0)), and thus the reset compensator is VSP.



Remark 3.3 A necessary and sufficient condition for the satisfaction of (3.65) can be given as follows for a convex storage function V . Note that if V is convex then V satisfies (3.65) for all x ∈ Rnr if and only if ∂V ∂x (Aρ x) ⊥ ker Aρ (see [26]). In the linear case (quadratic V ) and Aρ being given as the projection on the first vector component, this is equivalent to the decoupled quadratic function V given by:   0 Q11 x (3.72) V (x) = x

0 Q22 where Q11 and Q22 are positive-definite matrices such that Q11 ∈ Rnρ ×nρ and Q22 ∈ Rnρ ×nρ . It is clear that V satisfies (3.65) since   0

0 V (Aρ x) − V (x) = −x x ≤ 0, ∀x ∈ Rnr . 0 Q22 Remark 3.4 A well-known condition for an LTI system with state-space representation (A, B, C, D) to be SPR is that   −QA − A Q QB − C

>0 (3.73) B Q − C D + D

for some Q > 0. In addition, the system (A, B, C, D) is dissipative with respect to the storage function V (x) = x Qx (see [21]). Note that if the LMI (3.73) is   satisfied for a block diagonal Q = Q011 Q022 with the structure given by (3.72), then Proposition 2 can be applied and as a result the partial reset compensator with base system (A, B, C, D) will be VSP. Some examples of passive partial reset compensators are given in the following.

3.3.2.1 Partial Reset PIIR Compensator Consider the high order reset elements with base compensator PII R defined as PII R (s) = k

1 + γ1 Ti s 1 + γ2 Tr s 1 + Ti s 1 + Tr s

(3.74)

where γ1 ∈ (0, 1) and γ2 ∈ (0, 1). Now, the transfer function is not enough to define the reset compensator because, with the same Ti , Tr , γ1 , and γ2 , infinitely many

3.3 Reset Control Systems with Inputs/Passivity Analysis

123

Fig. 3.13 Reset choice in PII R (s)

reset compensators can be defined. Thus, a state space description has to be given. In particular, consider the base compensator PII R (s) =

1 + 2s 1 + 0.012s 1 + 3s 1 + s

(3.75)

given by the block diagram of Fig. 3.13, where the left block gives the state x1 = u1 to be reset. Then, the compensator is defined by its state space matrices given by     −0.333 0.247 0.003 Ar = , Br = , 0 −1 1   Dr = 0.008, Cr = 0.444 0.658 , and Aρ =

 1 0

 0 . 0

(3.76)

Note that by simply checking the Nyquist plot of the system (Ar , Br , Cr , Dr ) it is straightforward to show that it is SPR (see Fig. 3.14). In addition, a decoupled storage function has been found that satisfies (3.72), which is given by   1.20 0 V (x) = x

x. (3.77) 0 8.94 As a result, an application of Proposition 3.11 (see Remark 3.4) guarantees that the partial reset compensator is VSP.

3.3.3 L 2 -stability of the Reset Control System In this subsection, L2 -stability will be studied for the feedback system of Fig. 3.12, by directly using Propositions 3.10 and 3.11, and the passivity theorem. Both the full reset and the partial reset cases will be investigated. Proposition 3.12 The reset feedback control system of Fig. 3.12, with R being a full reset compensator with base compensator Rb , is finite-gain stable if some of the following conditions are satisfied: • Rb is ISP and P is ISP. • Rb is OSP and P is OSP.

124

3 Stability of Reset Control Systems

Fig. 3.14 Nyquist plot of PII R (s)

• Rb is VSP and P is passive. • Rb is passive and P is VSP. In contrast to the full reset case, for the partial reset compensator a state space description is necessary, the input–output map is not enough. In this case, as it was shown in Proposition 3.11, a condition over the storage function has to be added. Proposition 3.13 The reset feedback control system of Fig. 3.12, with R being a partial reset compensator with base linear compensator Rb , is finite-gain L2 stable if 1. Rb is dissipative with respect to the supply rate wi and a storage function V that satisfies (3.65), and P is ISP. 2. Rb is dissipative with respect to the supply rate wo and a storage function that satisfies (3.65), and P is OSP. 3. Rb is dissipative with respect to the supply rate wv and a storage function that satisfies (3.65) and P is passive. 4. Rb is dissipative with respect to the supply rate wp and a storage function holds (3.65) and P is VSP. Note that if a system G is OSP (with parameters δG = 0 and εG > 0), then G is finite-gain L2 stable with gain ≤ ε1G (see, for example, Theorem 2.2.14 in [29]). Let us apply this to Propositions 3.12 and 3.13, for example, in the last case where P is VSP (with parameters εP and δP ), and Rbl is passive (and thus δR = εR = 0). Then, after simple manipulations we get that the closed-loop system (with inputs

3.3 Reset Control Systems with Inputs/Passivity Analysis

125

Fig. 3.15 Online pH process

w, d, and outputs u, y) is finite-gain L2 stable with gain less than or equal to 1 1 1 min{εP ,δP } = max{ εP , δP }. The other cases of Propositions 3.12 and 3.13 are similar. As a result, the gain of the reset control system is less than or equal to the gain of the base control system (while the precise computation of the gain of the closed-loop gain should be performed by other means).

3.3.4 Example In this example, Propositions 3.11–3.12 will be used to investigate the input–output stability of a reset control system in which the plant is nonlinear; in particular, a reset compensator for a pH control system will be designed based on the above described passivity properties. Firstly, consider a stream (with flow rate F ) of distilled water that is mixed with a titrating stream (with flow rate u) of a strong acid in a continuous stirred tank reactor (CSTR) [34] with the following assumptions (see Fig. 3.15): (a) the mixture is perfect and instantaneous, (b) the pH sensor in placed very near to acid injection, and (c) the flow rate of titrating stream is negligible compared with the flow rate of the process stream, i.e., F + u  F . Under these restrictions, the process can be modeled as a linear plant in cascade with a nonlinear passive memoryless system (see Fig. 3.16). The tank will be considered as the part of the pipe where acid is injected. Due to conservation of charge, the concentration of the charge sum (W ) is taken as the state of the linear subsystem, i.e., W = [H+ ] − [OH− ]. The dynamics of the CSTR is given by V W˙ = −F (W − Wdw ) + uWa

(3.78)

where V is the volume of the tank, F is the flow rate of the process stream, u is the flow rate of the titrating stream and Wa is the concentration of charge in the titrating stream. In particular, for distilled water Wdw = 0, but also a stream process of strong base (Wb < 0) can be studied if −Wb  Wa . Furthermore, the output of the linear system equals the state, yL = W . The relationship between this concentration and

126

3 Stability of Reset Control Systems

Fig. 3.16 Final model of pH process under some considerations

the pH is given by  pH(W ) = − log

W+

 √ W 2 + 4 × 10−14 . 2

(3.79)

Taking an operation point, with a flow rate constant uss , an equilibrium point is obtained as −F Wss + uss Wa = 0. Therefore, Wss = uFss Wa . Working with increments with respect to the equilibrium point, ΔW = W − Wss , the new dynamics are given by ˙ = −FΔW + ΔuWa V ΔW

(3.80)

where Δu = u − uss . The nonlinear memoryless function is defined by ΔpH = pH (Wss ) − pH (W ). The system was inverted in order for an increment in the input to result in an increment in the output. Finally, the model was recast as the block interconnection of Fig. 3.16, where L is an LTI system given by:  x˙ = −ax + bu, (L) (3.81) y = x, where a =

F V

and b =

Wa V .

The static nonlinearity is given by φ(x) = pH(Wss ) − pH(x),

(3.82)

from which it follows that xφ(x) > 0 for all x = 0. It can be easily shown (see, for example, [19] or [21]) that the system of Fig. 3.16 with L and φ given by (3.81)– (3.82) is passive. A numerical example is developed for the next data: V = 0.1 l, F = 5 × 10−2 l/s, Wa = 1 mol/l, uss = 0.5 × 10−4 l/s. The linear plant from (3.81) is L(s) = and the static nonlinearity is given by   φ(x) = 3 − − log10

20 , 2s + 1

x+

 √ x 2 + 4 × 10−14 . 2

(3.83)

(3.84)

A reset-PII R compensator has been also designed for tracking some pH references, consisting in deviations about the equilibrium point. By using Proposition 3.12, finite-gain L2 -stability of the feedback system can be directly guaranteed since reset-PII R is VSP (see Sect. 3.3.2.1). A simulation of the control system

3.4 Describing Function Analysis

127

Fig. 3.17 Simulation of pH process with a partial reset compensator

is shown in Fig. 3.17, where in addition an important characteristic of reset compensation is clearly shown: the possibility of obtaining very fast responses without overshooting, overcoming fundamental limitations of LTI compensation.

3.4 Describing Function Analysis The analysis of reset control systems with the describing function (DF) will follow standard methods, and in the following it will be assumed that the reader is familiar with the DF analysis (see, for example, [32] for a good introduction to the subject). Although it is well-known that the describing function may fail in some cases, it has proved to be an efficient method and has been formally justified in many practical cases (see [14, 22]), including some types of switched systems (see [28]). As it has been emphasized in this book, the main motivation for the use of reset compensation is to improve the performance of a previously designed LTI control system, with the goal to reset some states of a base LTI compensator to improve control system performance both in terms of speed of response and relative stability. In general, these specifications will be impossible to achieve by means of LTI compensation. It turns out that the describing function technique, in spite of being approximate, is a good practical tool to evaluate the phase lead obtained by reset compensation. As it will be seen below, the fact that for reset compensators (with

128

3 Stability of Reset Control Systems

Fig. 3.18 FORE sinusoidal response

zero error crossing as the resetting law) the describing function is only a function of the input frequency (it does not depend on the input amplitude) makes the analysis very simple, and thus is a good tool for a first stability analysis. In the following, the describing functions of several basic reset compensators, such as the Clegg integrator and FORE, are obtained.

3.4.1 FORE and Clegg Integrator The FORE describing function is computed in this section, the Clegg integrator describing function is then obtained as a particular case. FORE is a simple reset K compensator with a first order base compensator given by FOREbase (s) = s+a . As it is well known (see, for example, [32]), for a given system its DF is calculated as the ratio between the fundamental component of its sinusoidal response and the sinusoidal input. The response v of the reset compensator FORE to the sinusoidal input e(t) = E sin(ωt) is given in the s-domain simply by V (s) =

Eω K , s + a s 2 + ω2

(3.85)

and the time response v(t) will consist of two terms: one corresponding to the transient response given by the compensator modes, and the other corresponding to the steady response given by the sinusoidal input modes. Since the reset instants will be given by the crossings of the sinusoidal input with zero, and thus they will be periodic with fundamental period 2π/ω (see Fig. 3.18), it turns out that v(t) will also be periodic with that period. To compute the FORE describing function, a simple procedure is to add to the base frequency response the contribution of the transitory terms vt (t) in v(t) over a period, for example [0, 2π ω ]. By symmetry (see Fig. 3.18), it will be computed by

3.4 Describing Function Analysis

129

using the interval [0, πω ], where vt (t) =

KEω −at e . a 2 + ω2

(3.86)

The result is 1 π

 π/ω 0

vt (t)e−j ωt ω dt

=

E 2j

π j 2Kω2 1 + e−a ω , 2 2 π(a + j ω)(a + ω )

(3.87)

and finally adding to the frequency response of the base system, the describing function of FORE results as   π K 2ω2 (1 + e−a ω ) . FORE(ω) = 1+j a + jω π(a 2 + ω2 )

(3.88)

Note the simple form of the describing function that only depends on the frequency. This allows us to make direct comparisons of the FORE describing function and the frequency response of its base system. The Clegg integrator describing function can be simply obtained by setting K = 1 and a = 0 in (3.88):   4 1 1+j , CI(ω) = jω π

(3.89)

◦

and since 1 + j π4 = 1.62ej 51.85 , it is clear that Clegg integrator gives a phase lead of almost 52° over an integrator (it also increases the gain with a factor of 1.62 approximately). Figure 3.19 shows this fundamental property of the Clegg integrator and FORE, that is, the achievement of a phase lead up to 52◦ with respect their base linear compensator.

3.4.2 Reset Compensators with Reset Band In control practice, the compensator implementation is usually done by using a reset band. In addition, it has been noted that the use of a reset band may improve stability and performance in systems with time-delays [30] due to the fact that phase lead can be even improved by using a reset band; see Sect. 5.2 for a more detailed justification of how reset band can improve reset control system stability and performance. In this section, the focus will be on the computation of the DF for a number of basic reset compensators with a reset band. In these cases, the resetting law is given by the crossing of the error signal when entering some (fixed or variable) band.

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3 Stability of Reset Control Systems

Fig. 3.19 Nichols plot of FORE and CI describing functions

Reset control systems with a reset band are feedback control systems (see Fig. 2.1) where the compensator R is given by the impulsive differential equation ⎧ x˙r (t) = Ar xr (t) + Br e(t) if (e(t), e(t)) ˙ ∈ / Bδ , ⎪ ⎪ ⎨ (R) (3.90) xr (t + ) = Aρ xr (t) if (e(t), e(t)) ˙ ∈ Bδ , ⎪ ⎪ ⎩ v(t) = Cr xr (t) + Dr e(t), where the reset band surface Bδ is given by Bδ = {(x, y) ∈ R2 |(x = −δ ∧ y > 0) ∨ (x = δ ∧ y < 0)}, with δ some non-negative real number. In this way, the compensator states are reset at the instants at which its input is entering into the reset band. In general, the reset band surface Bδ will consist of two reset lines Bδ+ and Bδ− in the plane, as shown in Fig. 3.20. In the particular case δ = 0, the standard reset compensator is obtained. On the other hand, if δ is big enough in relation to the error amplitude, then no reset action is produced, and the reset compensator reduces to its base compensator given by the transfer function Rb (s) = Cr (sI − Ar )−1 Br + Dr . The following notation will be used to distinguish between reset compensators with and without reset band: Rδ will denote a reset compensator with reset band Bδ ,

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131

Fig. 3.20 Reset band

Fig. 3.21 FORE with reset band sinusoidal response

whereas R0 (or simply R) is the standard reset compensator without reset band. Note that Rb stands for the base reset compensator. For simplicity, the describing function (DF) of a FOREδ compensator is considered first.

3.4.2.1 FOREδ To obtain the describing function with a reset band, again the sinusoidal response needs to be calculated. In this case, note that since FOREδ is time-invariant, the FOREδ response to e(t) = E sin(ωt) over the period [−tδ , 2π ω − tδ ] is the same (except for a translation) as its response to e(t − tδ ) over the period [0, 2π ω ] (see Fig. 3.21). Thus, FOREδ DF can be obtained as the contribution of the transitory terms in the response to e(t − tδ ) = E cos(ωtδ ) sin(ωt) − E sin(ωtδ ) cos(ωt), plus the base frequency response. But note that the transitory terms consist of two terms: one term v1t corresponding to the response to E cos(ωtδ ) sin(ωt), and a second term vt2 with the response to −E sin(ωtδ ) cos(ωt). δ ) −at By using (3.86), the term vt1 is directly given by vt1 = KEa 2cos(ωt e , and thus +ω2 using again symmetry arguments, its harmonic balance (corresponding to the fun-

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3 Stability of Reset Control Systems

damental component) is given by  π/ω

vt1 (t)e−j ωt ω dt 0  2π/ω 1 e(t − tδ )e−j ωt ω dt 2π 0 1 π

=

π KEω2 cos(ωtδ ) (1 + e−a ω ) π(a+j ω)(a 2 +ω2 ) . E −j ωtδ 2j e

(3.91)

sin(ωtδ ) −at e , and its On the other hand, the second transitory term is vt2 = − KEa a 2 +ω 2 harmonic balance is now π  KEaω sin(ωtδ ) 1 π/ω (1 + e−a ω ) vt2 (t)e−j ωt ω dt π(a+j ω)(a 2 +ω2 ) π 0 = . (3.92)  2π/ω E −j ωtδ 1 e(t − tδ )e−j ωt ω dt 2j e 2π 0

Finally, the FOREδ describing function is obtained by adding to (3.91) and (3.92) the base frequency response. After some simple manipulations the result is  2  π  FOREδ (E, ω) δ δ j sin−1 ( δ ) j 2ω(1 + e−a ω ) E , (3.93) ω 1− +a =1+ e FOREb (j ω) π(a 2 + ω2 ) E E K , and the identities sin(ωtδ ) = Eδ and cos(ωtδ ) = where FOREb (j ω) = j ω+a ! 2 1 − Eδ have been used. Note that in contrast to (3.89), FOREδ (E, ω) depends both on the magnitude and the frequency of the input. In general, expression (3.93) is valid for 0 ≤ Eδ ≤ 1. Obviously, for δ > E, no reset action is performed, and thus the describing function will directly be the base frequency response. Note that the FOREδ introduces extra phase lead for frequencies over a1 , in comparison with FORE for some values of δ (see Fig. 3.22). Since in addition for increasing values of δ a significant phase lag is introduced, it seems that a proper selection of δ should be in the range 0 < δ/E < 0.8.

3.4.2.2 Clegg Integrator with a Reset Band The describing function of the Clegg integrator with a reset band, that will be referred to as CI δ , is obtained as a particular case of FOREδ for K = 1 and a = 0. The result is !   1 − ( Eδ )2 ) j 4( 1 −1 δ CI δ (E, ω) = ej sin ( E ) . 1+ (3.94) jω π Again, note that the reset band introduces extra phase lead for values 0 < δ/E < 0.8 and at every frequency (see Fig 3.23).

3.4.2.3 General Reset Compensator with Reset Band In the case in which the general reset compensator (3.90) has a state matrix Ar with real and distinct eigenvalues, the compensator may be implemented as a sum of

3.4 Describing Function Analysis

133

Fig. 3.22 FOREδ (E, ω). Phase for different values of δ/E

Fig. 3.23 CI d (E, ω). Phase for different values of δ/E

different FORE compensators, and thus the calculation of its describing function is straightforward. In addition, partial reset may be implemented by considering the base frequency response of the non-reset element. On the other hand, although the derivation is somehow involved (see Appendix A of [6]), a closed-form expression of the describing function can be obtained for the

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3 Stability of Reset Control Systems

general case of full reset with a reset band. The result is: Rδ (E, ω) = Rbase (j ω) +

 

Ar 1 δ π 1 −1 δ δ Cr e ω Ar sin −1( E ) G , ω I + e−Ar ω · e( ω −j I )(π−sin ( E )) Br , π E (3.95)

where G is given by   −1 −1 G(x, ω) = (j ωI − Ar )−1 e−j sin (x) + (j ωI + Ar )−1 ej sin (x) · ω(j ωI − Ar )−1 . (3.96) Using (3.95)–(3.96) with δ = 0, a general closed-form expression can also be obtained for the describing function R0 (j ω) of a full reset compensator without a reset band. It is given by R0 (E, ω) = Rb (j ω) +

Ar 1 π 1 Cr e ω Ar π F (0, ω) I + e−Ar ω · e( ω −j I )(π−π) Br (3.97) π

and after simplification results in (note that it does not depend on E) ω  Cr (j ωI − Ar )−1 π  

π + (j ωI + Ar )−1 · (j ωI − Ar )−1 eAr ω + I Br .

R0 (j ω) = Rb (j ω) +

(3.98)

Note that in general R0 (j ω) will depend only on frequency ω, and Rδ (E, ω) both on the amplitude E and the frequency ω. Obviously, the expression for FOREδ as K , Ar = −a, Br = 1, given in (3.93) is a particular case of (3.95) for Rb (j ω) = j ω+a Cr = K. In addition, the CI δ FD expression in (3.94) is a particular case for K = 1 and a = 0.

3.4.2.4 Reset Systems with a Variable Reset Band A variable reset band is considered as an alternative to overcome the influence of, for example, dominant time delays over the reset action. In practical applications (see [15, 31]), this has been shown to be a good strategy to achieve stability and performance design specifications (see Chap. 5 for a detailed exposition). In the fixed band case, the reset instants are determined when the error signal is approaching zero at a fixed distance given by the reset band. A variable reset band means that at every reset instant the band value may be different, and in general this value will be determined by a combination of the error value and its derivative. The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. In this way, the reset instants will be predicted by using the error signal derivative approximation. This approximation states that the error signal derivative at a point is equal to the slope of the tangent

3.4 Describing Function Analysis

135

Fig. 3.24 Variable reset band surface

line to the graph of the error signal at that point. Therefore, the following expression is obtained tan α =

de(t) −e(t) = , dt h

(3.99)

where h is the time delay process. From this equation, the new variable reset condition is expressed as h

de(t) + e(t) = 0. dt

(3.100)

As it can be seen, no parameter has to be fixed in this new condition, since the time delay h can be estimated from plant tests (usually a worst-case estimate is a good option). With this new reset condition (3.100), the reset controller (3.90) is expressed now in the state-space as ⎧ x˙r (t) = Ar xr (t) + Br e(t) if (e(t), e(t)) ˙ ∈ / Bhv , ⎪ ⎪ ⎨ xr (t + ) = Aρ xr (t) if (e(t), e(t)) ˙ ∈ Bhv , (R) (3.101) ⎪ ⎪ ⎩ v(t) = Cr xr (t) + Dr e(t), where the variable reset band surface Bhv is given by   ˙ ∈ R2 | he(t) ˙ + e(t) = 0 . Bhv = (e(t), e(t))

(3.102)

In this case, the reset surface is a continuous function of the error signal, as it can be seen in Fig. 3.24. When there is no time delay, h = 0, the reset band surface (3.102) is reduced to the reset compensator without reset band. In the frequency domain, the describing function of the Clegg integrator with a variable reset band may be computed, resulting in    j2

1 −1 CI h (ω) = 1+ 1 + ej 2 tan (ωh) , (3.103) jω π

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3 Stability of Reset Control Systems

Fig. 3.25 Phase of CI h

where h is the time delay of the process. Note that in the case of the variable reset band the describing function only depends on the input frequency and that, in contrast to CI, CI h gives a frequency dependent phase lead. Figure 3.25 shows the phase of CI h for different values of h. It is clear from this figure that CI h gives an 1 ◦ extra phase lead over a CI up to 27◦ in the frequency interval [ 0.1 h , h ] (and up to 75 over an integrator) that is especially useful for systems with time delays.

3.4.3 Limit Cycle Analysis Consider the feedback system of Fig. 2.1. The question now is to analyze the existence of limit cycles of reset control systems with a reset band by using the describing function of the reset compensator obtained above. A well-known method is to compute possible values of E and ω such that the describing function of the reset compensator Rδ (E, j ω) satisfies 1 + Rδ (E, j ω)P (j ω) = 0, which is usually referred to as the harmonic balance principle. This is usually made by analyzing the crossings of the Nyquist plot of the plant P with plots of −1/Rδ (E, j ω) vs ω for different values of E. It is well known that this method only gives approximate results, and it works well only if the plant P (s) filters higher order harmonics of the periodic signal v in Fig. 2.1 (low pass condition). Alternatively, the above procedure is equivalent to finding the crossings of the critical locus −1/Rδ (E, ω) with the plant Nyquist plot P (j ω), and then obtaining crossings for values of E and ω that will correspond to the amplitude and frequency of a limit cycle. In the following, two examples are developed for the reset compensators FOREδ and CI δ , respectively. Note that since, in general, their describing

3.4 Describing Function Analysis

137

Fig. 3.26 Time simulation of feedback system with FOREδ and first order plant (note that e = −y)

functions depend on the parameter δ/E, where δ/E ∈ (0, 1) defines the reset band in relation to the compensator input amplitude, if a limit cycle exist for some δ then there always exist limit cycles for any value of E such that δ/E ∈ (0, 1). Thus, a normalized reset band δ/E and a describing function with amplitude E = 1 will be used. As a result, the critical locus −1/Rδ/E (1, ω) will be used to evaluate the crossings, and then values of δ/E and ω will be obtained instead. It turns out that for each given frequency ω, critical loci −1/Rδ/E (1, ω) are circle segments in the Nyquist plane as δ/E goes from 0 to 1. 3.4.3.1 FOREδ with a First Order Plant A simple analysis shows that there are no limit cycles in the case corresponding to a feedback control system (see Fig. 2.1) with a FOREδ reset compensator having a stable base compensator and a first order stable plant P (s). The reason is that the after-reset state of the closed-loop system will always correspond to a zero compensator output, and an alternating value of the error δ, −δ, . . . is obtained. The only case in which a limit cycle may exist is the case in which the error signal reaches the reset band from each one of the two values δ and −δ, and this is only possible if the base feedback system is unstable. In this example, a FOREδ compensator with an unstable base compensator 2 1 , δ = 0.1, and a first order plant P (s) = s+0.5 is considered. A simFOREb (s) = s−1 ulation of the feedback system reveals (see Fig. 3.26) that a limit cycle exists with amplitude E = 0.20 and frequency ω = 1 rad/s.

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3 Stability of Reset Control Systems

Fig. 3.27 Crossings of critical loci −1/FOREδ/E (1, ω) for ω = 0.5, 1, 1.5 rad/s with Nyquist plot of the first order plant

In Fig. 3.27, the crossings of critical loci −1/FOREδ/E (1, ω) for ω = 0.5, 1, 1.5 rad/s with P (j ω) are shown. Then, it is possible to evaluate numerically those crossings corresponding to a limit cycle, and in addition values of δ/E and ω. The result is that there exist limit cycles for any value of E such as δ/E = 0.68 and for ω = 1 rad/s. In this example, the amplitude will be E = δ/0.68 = 0.1/0.68 = 0.15. Note that the describing function method gives a very accurate prediction of the limit cycle. In addition, a perturbation method can be used to analyze the (local) stability of the limit cycle. It can be shown that for the limit cycle to be stable a sufficient condition is ∂R(E, ω) ∂I (E, ω) ∂R(E, ω) ∂I (E, ω) − >0 ∂E ∂ω ∂ω ∂E

(3.104)

at (E0 , ω0 ) = (0.15, 1), where R(E, ω) = Re(1 + FOREδ (E, ω)P (j ω)) and I (E, ω) = Im(1 + FOREδ (E, ω)P (j ω)). In this case, condition (3.104) can be checked numerically, and as a result the limit cycle is stable.

3.4.3.2 CIδ with a Second Order Plant Consider again the feedback system of Fig. 2.1, in this case with a Clegg integrator CI δ as reset compensator with δ = 0.25 and a second order plant P (s) = s+1 . s2 A simulation has been made first to analyze the existence of limit cycles, the result is that there exists a limit cycle as shown in Fig. 3.28 with amplitude E = 0.5 and frequency ω = 0.92 rad/s. Let us consider now the use of the describing function method. Different critical loci −1/CI δ/E (1, ω) for ω = 0.1, 0.3, 0.5, 0.92 rad/s are shown in Fig. 3.29. After a numerical evaluation of the different crossings, the result is that there exists a limit

3.4 Describing Function Analysis

139

Fig. 3.28 Crossings of critical locus −1/CI b (E, ω) with Nyquist plot of the plant

cycle for any value of E such as δ/E = 0.82 and ω = 0.92 rad/s. For this example, the amplitude is E = δ/0.82 = 0.3. Again, the prediction of the limit cycle is very accurate. The deviation of the amplitude is consistent with the fact that signal v in Fig. 3.28 may have significant higher order harmonics. In addition, condition (3.104) is satisfied for (E0 , ω0 ) = (0.3, 0.92), and thus the limit cycle is also stable.

3.4.4 Justification of the Describing Function As it is well known, in spite of the fact that the describing function technique gives only approximate results, it has been found useful in many practical applications. Although a formal justification is clearly desirable, it turns out that the several analytical justifications that can be found in the literature are of limited use. The first work in this direction is [14], where it is shown that a limit cycle exists, using Fig. 2.1, if R is bounded and P is able to filter enough high order harmonics (low pass condition). Also, the works [22] and [28] develop formal results for nonlinear systems that are sector-bounded or exhibit a passivity condition, respectively. These results are not applicable to reset control systems, and in general an analytical treatment seems to be a hard open problem. In the following, the low pass condition will be explored for some particular types of reset control systems with a reset band. In addition, the harmonics structure of the sinusoidal response of FOREδ and CI δ will be investigated. In general, the response of FOREδ to a sinusoidal input e(t) = E sin(ω(t − tδ )) is periodic with

140

3 Stability of Reset Control Systems

Fig. 3.29 Crossings of critical locus −1/CI b (E, ω) with Nyquist plot of the second order plant

fundamental period 2π/ω (see Fig. 3.21). In addition, it is also symmetric and can be expressed as

v(t) =

⎧ KE(ω cos(ωt )+a sin(ωt )) δ δ e−at + M(ω)E sin(ω(t − tδ ) + Φ(ω)) ⎪ ⎪ a 2 +ω2 ⎪ ⎪ ⎪ ⎨ if t ∈ [0, π ], ω ⎪ −y(t − πω ) ⎪ ⎪ ⎪ ⎪ ⎩ if t ∈ [ πω , 2π ω ],

(3.105)

K where M(ω) and Φ(ω) are the magnitude and phase of | j ω+a |. Now, computation of high order harmonics is relatively simple in this case since by symmetry only odds harmonics are different from zero. Defining C(E, ω, δ) = KEω(ω cos(ωtδ )+a sin(ωtδ )) (1 + e−aπ/ω ), higher harmonics have a particularly simple π(a 2 +ω 2 ) form. computation, spectral coefficients of the Fourier series v(t) = "∞ After jsome kωt are given by (see Fig. 3.30) a e k=−∞ k

 ak =

C(E,ω,δ) a+j kω

0

for k = ±3, ±5, . . . , for k = ±2, ±4, . . . .

(3.106)

3.4 Describing Function Analysis

141

Fig. 3.30 Sinusoidal response of FOREδ for ω = 1 rad/s, K = 1, and a = −1, and approximations with the first plus third harmonics, and with 20 harmonics

Therefore, by using the first n odd harmonics of FOREδ response as an approximation, a measure of the approximation can be defined as the L2 [0, 2π ω ]-norm of the residual term. Then, v(t) as given by (3.105) can be expressed by v(t) = " vn (t)+ vˆn (t) where vn (t) = nk=−n a2k−1 ej (2k−1)ωt , and vˆn (t) = v(t)−vn (t). Then vˆn 2 = 2

∞  k=n+1

|a2k−1 |2 = 2

∞  k=n+1

a2

C , + (2k − 1)2 ω2

(3.107)

where C stands for C(E, ω, δ) (since C does not depend on k, the arguments have been dropped for simplicity). By using the ψ function (also known as the digamma function), the norm can be computed in a closed-form as        C ω + ia ω + ia ω − ia C vˆ n  = Im Ψ n+ Ψ n+ −Ψ n+ = i2aω 2ω 2ω aω 2ω (3.108) for a = 0. Figure 3.31 shows a measure of how low order harmonics represent the vˆ n  only depends on the sinusoidal FOREδ response. Note that in general the ratio v 2

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3 Stability of Reset Control Systems

Fig. 3.31 Ratio (1 − | ωa | = 0.25

vˆ n  v )

vs n (number of odds harmonics) for | ωa | = 1, | ωa | = 0.5, and

parameters a and ω, and is given by vˆ n  = v



a )} Im{Ψ (n + 12 + i 2ω a Im{Ψ ( 12 + i 2ω )}

1 2

.

(3.109)

For the Clegg integrator with a reset band CI δ , the norm is        1−i C C 1+i 1+i − Ψ n + = , vˆ n 2 = Im Ψ n + Ψ n + 2 2 2 i2ω2 ω2 (3.110) vˆn  and in this case the ratio v is constant (independent of the frequency ω), and is given by (it has the same values as that of FOREδ for |a/ω| = 1, see Fig. 3.32) 

1

    1 2 1 = 0.8331 Im Ψ n + . (1 + i) 1 2 Im{Ψ ( 2 (1 + i))} (3.111) For the example of FOREδ plus a first order plant, the low-pass condition can be now evaluated by computing how the plant filters the harmonics of the FOREδ sinusoidal response. In Fig. 3.33, it is shown how for different values of |a/ω| the relative weight of the first harmonic is much more significant, and it is a good approximation of the filtered FOREδ response. For the example of CI δ with a second order plant, the results are analogous. Note that the harmonics structure of the FOREδ or the CI δ response is always given as shown in Fig. 3.31, then the low-pass condition can be evaluated for a vˆn  = v

Im{Ψ (n + 12 (1 + i))}

2

3.4 Describing Function Analysis

143

Fig. 3.32 Sinusoidal response of CI δ for ω = 0.92 rad/s, and approximations with fundamental component, with the first plus third harmonics, and with 20 harmonics

Fig. 3.33 Ratio (1 − | ωa | = 0.25

vˆ n  v )

vs n (number of odds harmonics) for | ωa | = 1, | ωa | = 0.5, and

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3 Stability of Reset Control Systems

given plant by computing how the harmonics are filtered, and then evaluating how the plant modifies the harmonics structure, for example, as shown in Fig. 3.33. This figure shows that the relative error norm of the first harmonic approximating the FOREδ response is around 10%, and that the relative error considering both the first and third harmonics is around 5%, thus it may be expected that the describing function method works well in many cases.

References 1. Aangenent, W.H.T.M., Witvoet, G., Heemels, W.P.M.H., van de Molengraft, M.J.G., Steinbuch, M.: An LMI-based L2 gain performance analysis for reset control systems. In: Proceedings of the American Control Conference (2008) 2. Bainov, D.D., Simeonov, P.S.: Systems with Impulse Effect: Stability, Theory and Applications. Ellis Horwood, Chichester (1989) 3. Baños, A., Carrasco, J., Barreiro, A.: Reset times-dependent stability of reset control system. In: European Control Conference, Kos, Greece (2007) 4. Baños, A., Carrasco, A., Barreiro, A.: Reset times dependent stability of reset systems with unstable base system. In: Proc. IEEE International Symposium on Industrial Electronics, Spain (2007) 5. Baños, A., Dormido, S., Barreiro, A.: Stability analysis of reset control systems with reset band. In: 3rd IFAC Conference on Analysis and Design of Hybrid Systems, Zaragoza, Spain (2009) 6. Baños, A., Dormido, S., Barreiro, A.: Limit cycles analysis in reset systems with reset band. Nonlinear Anal. Hybrid Syst. (2010). doi:10.1016/j.nahs.2010.07.004 7. Baños, A.: Carrasco, J., Barreiro, A.: Reset times-dependent stability of reset control systems. IEEE Trans. Autom. Control 56(1), 217–223 (2011) 8. Bemporad, A., Bianchini, G., Brogi, F.: Passivity analysis and passification of discrete-time hybrid systems. IEEE Trans. Autom. Control 53(4), 1004–1009 (2008) 9. Beker, O.: Analysis of reset control systems. Ph.D. Thesis, University of Massachusetts, Amherst (2001) 10. Beker, O., Hollot, C.V., Chait, Y., Han, H.: Fundamental properties of reset control systems. Automatica 40, 905–915 (2004) 11. Camlibel, M.K., Heemels, W.P.M.H., Schumacher, J.M.: On linear passive complementary systems. Eur. J. Control 8(3), 220–237 (2002) 12. Carrasco, J., Baños, A., van der Schaft, A.: A passivity approach to reset control of nonlinear systems. In: 34th Annual Conference of the IEEE Industrial Electronics Society, Orlando, Florida, USA (2008) 13. Carrasco, J., Baños, A., van der Schaft, A.: A passivity-based approach to reset control systems stability. Syst. Control Lett. 59(1), 18–24 (2010) 14. Bergen, A.R., Franks, R.L.: Justification of the describing function method. SIAM J. Control 9, 568–589 (1971) 15. Fernández, A., Barreiro, A., Baños, A., Carrasco, J.: Reset control for passive bilateral teleoperation. IEEE Trans. Ind. Electron. (2010). doi:10.1109/TIE.2010.2077610 16. Guo, Y., Wang, Y., Xie, L., Zheng, J.: Stability analysis and design of reset systems: theory and an application. Automatica 45(2), 492–497 (2009) 17. Haddad, W.M., Chellaboina, V., Nersesov, S.G.: Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control. Princeton University Press, Princeton (2006) 18. Horowitz, I.M., Rosenbaum, P.: Nonlinear design for cost of feedback reduction in systems with large parameter uncertainty. Int. J. Control 24(6), 977–1001 (1975) 19. Khalil, H.K.: Nonlinear Systems, 2nd edn. Prentice Hall, New York (2002)

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20. Krishman, K.R., Horowitz, I.M.: Synthesis of a nonlinear feedback system with significant plant-ignorance for prescribed system tolerances. Int. J. Control 19(4), 689–706 (1974) 21. Lozano, R., Brogliato, B., Egeland, O., Maschke, B.: Dissipative Systems Analysis and Control. Springer, London (2000) 22. Mees, A., Bergen, A.: Describing functions revisited. IEEE Trans. Autom. Control 20(4), 473–478 (1975) 23. Michel, A., Hu, B.: Towards a stability theory of general hybrid dynamical systems. Automatica 35, 371–384 (1999) 24. Neši´c, D., Zaccarian, L., Teel, A.R.: Stability properties of reset systems. Automatica 44(8), 2019–2026 (2008) 25. Pogromski, A., Jirstrand, M., Spangeus, P.: On stability and passivity of a class of hybrid systems. In: Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, Florida, USA (1998) 26. Rockafellar, R.T., Wets, J.-B.: Variational Analysis. Series of Comprehensive Studies in Mathematics, vol. 317. Springer, Berlin (1998) 27. Rugh, W.J.: Linear System Theory, 2nd edn. Prentice Hall, New York (1996) 28. Sanders, S.R.: On limit cycles and the describing function method in periodically switched circuits. IEEE Trans. Circuits Syst. I 20(4), 473–478 (1975) 29. Van der Schaft, A.: L2 -Gain and Passivity Techniques in Nonlinear Control. Springer, Berlin (2000) 30. Vidal, A., Baños, A.: QFT-based design of PI + CI compensators: application in process control. In: IEEE Mediterranean Conference on Control and Automation, pp. 806–811 (2008) 31. Vidal, A., Baños, A.: Reset compensation for temperature control: experimental application on heat exchangers. Chem. Eng. J. 159(1–3), 170–181 (2010) 32. Vidyasagar, M.: Nonlinear Systems Stability. Prentice-Hall, London (1993) 33. Willems, J.C.: Dissipative dynamical systems—Part I: General theory. Arch. Ration. Mech. Anal. 45, 321–351 (1972) 34. Wright, R.A., Kravaris, C.: On-line identification and nonlinear control of an industrial pH process. J. Process Control 11, 361–374 (2001) 35. Zefran, M., Bullo, F., Stein, M.: A notion of passivity for hybrid systems. In: Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, USA (2001)

Chapter 4

Stability of Time-Delay Reset Control Systems

4.1 Introduction Since the early proposals for reset control, including the Clegg integrator [13] and the first-order reset element (FORE) [20], [15], the underlying idea was to achieve fast and robust control solutions for problems under linear limitations. The simple nonlinear effect of resetting was proved to be useful for overcoming linear fundamental limitations. As it is well known in the literature on fundamental limitations [1] (see also Chap. 1 of this monograph), linear plants that have time-delays (and also plants having right half-plane poles or zeros) are specially hard to control: if the delay is significant, it is impossible to obtain a linear controller providing simultaneously a fast and a robust behavior. One has to choose between a fast response but with poor stability margin, or a robust stability margin, but with slow response. As a consequence, since reset control is a simple nonlinear method for overcoming fundamental limitations, and since time-delay is one of the main sources of such limitations, it is of great interest to study the problem of delayed reset systems. A design idea to obtain a reset controller for a plant with time-delay can be taken from the design approach proposed in [15] and [20]: first, tune the base LTI controller so that the base LTI feedback loop becomes stable and satisfies part of the performance specifications (typically, fast transient and large bandwidth). Then, in a second step, a reset action is included to achieve the other part of the desired objectives (typically, bounded overshoot and large stability margins). However, this method should be carefully applied since it is well known [8] that the reset action may destabilize a well designed LTI base control system, and this fact is one of the main drawbacks for the practical application of reset control. A solution to this problem has been reported in [8, 9] for finite-dimensional LTI plants. The result is a frequency domain condition, the Hβ -condition (see the previous Sect. 3.2.1.1 of this monograph), that guarantees closed-loop stability of the reset system. Since then, a number of useful results have appeared addressing the analysis and design of reset systems [6], [17], [18], [25]. However, none of them considered the A. Baños, A. Barreiro, Reset Control Systems, Advances in Industrial Control, DOI 10.1007/978-1-4471-2250-0_4, © Springer-Verlag London Limited 2012

147

148

4

Stability of Time-Delay Reset Control Systems

case of time-delay plants, so it is then of great interest to study delayed reset systems. It should be emphasized that this is extremely difficult: the combination of reset and delay moves the problem to the field of hybrid and distributed systems. Even the issue of existence and uniqueness of solutions may cause trouble unless special care is taken, for example, by means of the time-regularization procedure [25]. It should be noted that stability of delay impulsive systems is a very active research area in different areas of mathematics and systems theory, two classical monographs are [2] and [21]. In general, as reported in [2] (see the previous Sect. 1.5.1 of Chap. 1 in this monograph), impulsive systems may be classified into (i) systems with impulses at fixed instants, (ii) systems with impulse effect at variable instants, and (iii) autonomous systems with impulse effects. Systems (ii) and (iii) have state-dependent impulses, but there is an important difference: the reset surface is time independent in the case (iii). Traditionally, most of the research effort has been dedicated to cases (i) and (ii), and thus case (iii) that includes reset control systems is less developed. In addition, in the control field, the main effort has been directed at systems without time delays [28], [18]. For the case of impulsive delay systems, available results in the literature are concentrated in the case (i) (see [22], [14], [24], and the references therein), and case (ii) (see [23]). The material in this chapter is based on the journal papers [5] and [7], and on the preliminary conference papers [3] and [4]. Some of the results in these references can be regarded as generalizations of the Hβ -condition for LTI plants with time-delays. In [5], the case of delay-independent stability was considered, a standard Lyapunov–Krasovskii functional was used, and the instrumental technique for passing between time and frequency domains was the Kalman–Yakubovich–Popov (KYP) lemma. In [7], the delay-dependent case was addressed. As the combination of delay-dependent LMIs with the KYP lemma may give rise to extremely large matrix inequalities, which do not show much insight into the results, special care was taken in the choice of appropriate Lyapunov functionals and passivity/frequency domain techniques. There are several Lyapunov–Krasovskii (LK) functionals that can be used, so that the delay h appears explicitly in the stability conditions. Among these delaydependent results, the complete LK functional (Theorem 5.19 in [16]) has the advantage that it gives a sufficient but also necessary condition for stability; however, it is a distributed functional and requires some kind of discretization for practical computation. Another useful result is given by Proposition 5.17 in [16], with good LMI computability and which is not too conservative, in general. This functional was studied in [4] for application to reset systems. Finally, for interpretability in terms of uncertainty, the LK functional in [26] has the advantage that it allows for an easy interpretability (detailed in [29]) so that the stability conditions can be understood in terms of scaled small-gain for an uncertain plant. In the delay-dependent case, for passing from the time-domain to the frequency-domain, in place of the involved KYP techniques in [5], a novel interpretation was presented in [7] based on passivity and impulsive control. In this way, all the stability conditions can also be interpreted in terms of robust stability against certain uncertainties.

4.2 Problem Motivation and Statement

149

Fig. 4.1 Reset control loop for a time-delay plant

The organization of the chapter is as follows. After introducing formally the problem and basic results in Sect. 4.2, the time-domain delay-independent stability of reset systems is studied in Sect. 4.3 with the equivalent frequency domain conditions and interpretations presented in Sect. 4.4, followed by an example in Sect. 4.5. Then, the delay-dependent stability results are considered in Sect. 4.6 with the frequency domain counterpart in Sect. 4.7, including a passivity analysis of reset systems and robustness interpretations. The chapter concludes with an illustrative example of the delay-dependent case in Sect. 4.8.

4.2 Problem Motivation and Statement Consider a control system formed by a plant P0 and a reset controller R (both singleinput single-output):  x˙ p = Ap xp + Bp u, (P0 ) (4.1) y = C p xp , ⎧ x˙ r = Ar xr + Br e, ⎪ ⎪ ⎨ (R) xr (t + ) = Aρ xr (t), (4.2) ⎪ ⎪ ⎩ v = Cr xr + Dr e, where the second equation of the controller represents the impulsive or reset actions, applied when a certain reset condition, to be specified later, holds. If the connection from P0 to R (or from R to P0 ) is affected by a delay h1 (or h2 ), then it is easy to see that the total delay h = h1 + h2 can be moved to the plant input or output. For instance, move the delay to the plant input, as shown in Fig 4.1. As the main interest is stability, let us suppose that the exogenous loop signals are zero (r = d = n = 0), so that the closed loop connections are: u(t) = v(t − h),

e(t) = − y(t),

and the closed loop system is easily described by       x (t) x˙ p (t) xp (t − h) =A p + Ad , x˙ r (t) xr (t) xr (t − h) and by impulsive dynamics

xr t + = Aρ xr (t).

(4.3)

(4.4)

150

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Stability of Time-Delay Reset Control Systems

With this is mind, we generalize to the system given by the impulsive-delayed differential equation (IDDE)  x˙ (t) = Ax(t) + Ad x(t − h) if x(t) ∈ / M, (IDDE) (4.5) if x(t) ∈ M , x(t + ) = AR x(t)   where x = (x p , xr ) is the global state and, from (4.1)–(4.2), the matrices A, Ad take the form     0 Bp Cr −Bp Dr Cp Ap , Ad = . A= −Br Cp Ar 0 0

Since the results in this chapter are independent of the precise structure of A, Ad , let us assume in what follows, for the sake of generality, that A, Ad are arbitrary matrices, not necessarily coming from a control loop like the one in Fig. 4.1. Now, to complete the system equations, it remains to introduce AR and M . Let us suppose that only the second part of the controller states are reset to zero, that is, Aρ = diag(1, . . . , 1, 0, . . . , 0). Then AR takes the form [9]: AR = diag(In12 , 0n3 ),

(4.6)

so, the reset action acts over the last n3 states of the state x ∈ Rn . These dimensions are defined by: n = n1 + n2 + n3 = n1 + n2 +n3 ,     n23

(4.7)

n12

where n1 (n23 ) is the number of states of the plant (controller). Accordingly, partition the state vector:

  x = x (4.8) 1 , x2 , x3 .     x p

x r

In principle, it is considered that the reset is applied at the time t when x(t) hits the hypersurface   (4.9) M = x ∈ Rn : Cx = 0 . As it is desired that 0 = Cx defines a hypersurface, C is a row vector, C ∈ R1,n . From the state partition (4.8), the reset condition matrix C can be partitioned as: C = (C1 , C2 , C3 ). There are several possibilities for C. If the IDDE system comes from a control loop like the one in Fig. 4.1, and the reset condition is activated when the error crosses zero, then C = (Cp , 0, 0). The most general case is an arbitrary row matrix C, but it causes some difficulties if C3 = 0, that is, if the part x3 of the controller states affected by reset have influence via C3 on the reset condition. For this reason, several results in this chapter assume the structure C = (C1 , C2 , 0) for C.

4.2 Problem Motivation and Statement

151

In this way, (4.5), (4.6), and (4.9) define, in principle, the system that we are going to study. However, it is known that the IDDE (4.5) may exhibit beating and have Zeno solutions as has been remarked in [25] for the case h = 0. A certain strategy for avoiding these pathological solutions has been analyzed in Chap. 2 of this monograph. This problem of well-posedness of solutions affects all hybrid systems, and in particular reset systems. Note that the well-posedness issue has been addressed in Chap. 2 for finite-dimensional reset systems. However, the existence of the time-delay complicates the situation further because it makes the system infinitedimensional. In this chapter, the useful idea of time regularization (TR) will be used, and thus the IDDE (4.5) is recast in the form (IDDE-TR): x˙ (t) = Ax(t) + Ad x(t − h)

τ + = 0, x t + = AR x(t)

τ˙ = 1,

if (x(t) ∈ / M ) ∨ (τ ≤ ρ), if (x(t) ∈ M ) ∧ (τ > ρ),

(4.10)

where ρ is a finite positive scalar, arbitrarily small. In this way, every reset in (4.10) will take place at least ρ seconds after the previous reset instant. That is, the reset instants tk (k = 1, 2, . . .) satisfy tk+1 − tk > ρ (k = 1, 2, . . .). Choosing a sufficiently small time-step ρ the behavior the IDDE and IDDE-TR systems (4.5) and (4.10) is almost equal for most practical systems, and in this way the regularized system is formally free of dynamic pathologies. Now, it is important to distinguish between the (lumped) instantaneous state x(t) ∈ Rn and the true (distributed) state given by some piecewise continuous function xt : [−h, 0] → Rn , where xt (θ ) = x(t + θ ), θ ∈ [−h, 0]. In the following, C([−h, 0], Rn ) will stand for the set of piecewise continuous functions mapping [−h, 0] to Rn , with a finite number of discontinuity points. For the sake of concreteness, the piecewise continuous functions in C([−h, 0], Rn ) will be assumed continuous from the left at the discontinuity points, that is, x(tk ) = x(tk− ) and, after a reset, x(tk+ ) = AR x(tk ). Then, the existence and uniqueness of solutions to (4.10) is simply guaranteed in a constructive way. First, as the counter τ starts at τ = 0, there exists a unique solution to the following initial value problem:  x˙ = Ax(t) + Ad x(t − h) for t ≥ 0, (4.11) x(θ ) = x0 (θ ) for θ ∈ [−h, 0], where x0 (·) is the initial condition. The starting trajectory of the linear system (4.11) is then well defined and is not affected by the first reset until some instant t1 > ρ. Now, in a recursive way, if the trajectory is well defined until some reset instant t = tk , then introduce the reset jump x(tk+ ) = AR x(tk ) and after the reset, solve a new linear problem similar to (4.11). In a constructive way, the solution is  x(t) = e

A(t−tk )

AR x(tk ) +

h

eA(t−tk −σ ) Ad x(tk + σ − h) dσ,

(4.12)

0

and this solution is valid for all t ∈ (tk , tk+1 ], up to reaching the new reset time tk+1 > tk + ρ.

152

4

Stability of Time-Delay Reset Control Systems

For the stability analysis of (4.10), let us rewrite the system in a form that makes explicit only the state x and implicit the time counter τ :  x˙ (t) = Ax(t) + Ad x(t − h), (4.13) (IDDE-TRx ) x(t + ) = AR x(t) only if x(t) ∈ M . This should be understood as equivalent to (4.10) but we only pay attention to x(t), driven by the continuous dynamics most of the time, and subject to reset only if x(t) ∈ M (note that the ‘iff’ condition is not used, due to the fact that τ < ρ may disable the reset). Now, let us introduce a basic result to address the stability of time-delay reset systems. It is common practice in stability analysis of time-delay systems [16] to start with a Lyapunov–Krasovskii functional V (xt ) and force its derivative to be negative, V˙ (xt ) < 0, along the solutions of the system. If the system also contains jumps or resettings, then it is natural to impose that the variation of the functional at the reset instants t = tk has to be nonpositive, ΔV (t) = V (xt + ) − V (xt ) ≤ 0. There are other possibilities, different from imposing V˙ < 0 and ΔV ≤ 0 as basic stability conditions. These requirements may be conservative, in general, and within the area of impulsive systems there is a vast amount of literature offering alternative techniques. But, at the same time, notice that many reset design approaches (see, e.g., [15], [20], [9], [18]) are based on the useful idea of tuning the linear system to be fast transient, underdamped but stable, and introducing reset for adding extra damping and decreasing overshoot. Within this idea, both the continuous and the impulsive mode must be stable, and so the basic stability principle (V˙ < 0 and ΔV ≤ 0) is very well suited and adequate. This principle is formalized in the following proposition taken from [7] and based on the results in [9] for systems without time-delays. Proposition 4.1 Let the functional V (xt ) : C([−h, 0], Rn ) → R be continuously differentiable, positive-definite, radially unbounded and such that along the solutions of the reset system d V (xt ) < 0, dt

xt = 0,

ΔV = V (xt + ) − V (xt ) ≤ 0,

x(t) ∈ M .

(4.14) (4.15)

Then, the zero equilibrium xt = 0 is globally asymptotically stable. Proof The detailed proof is omitted for brevity, but it is quite similar to the proof of Theorem 1 in [9], and to the proof of Proposition 3.1 in [5]. The basic idea is that, from the decreasing behavior, V (xt ) → c ≥ 0 as t → ∞. But if we suppose that c > 0, then xt would always lie outside a ball around the origin 0, so that necessarily V˙ < −γ , by continuity, for some γ > 0. But hence V would become negative, and so by contradiction c = 0. 

4.3 Delay-Independent Conditions in the Time Domain

153

4.3 Delay-Independent Conditions in the Time Domain In this section, we study the stability of reset control systems, the results are valid for any of the formulations: the nonregularized IDDE (4.5), the explicit regularized IDDE-TR (4.10), and the implicit regularized IDDE-TRx (4.13). The results will be stated for some of the models, (4.5), (4.10), or (4.13), but they are equally valid for all of them because the regularization only affects well-posedness, and the stability treatment is formally the same. According to Chap. 5 in [16], the time-domain approach to stability of timedelayed systems can be based on the Razumikhin Theorem for Lyapunov functions or on Lyapunov–Krasovskii functionals. In any case, the resulting conditions can be delay-independent (valid for all h > 0) or delay-dependent. Since the reset actions complicate the treatment, compared to time-delay systems without reset, then for the sake of simplicity let us consider Lyapunov–Krasovskii functionals and let us start with the delay-independent case. From Sect. 5.4 in [16], perhaps the simplest stability criterion can be obtained from the simplest Lyapunov– Krasovskii functional:  0  x (4.16) V (t) = V (xt ) = xt (0) P xt (0) + t (θ )Qxt (θ ) dθ, −h

for some P , Q > 0. It is clear that V (t) ≥ 0 and that V (t) = 0 if and only if xt (θ ) = 0 ∈ Rn , for each θ ∈ [−h, 0]. Thus the functional is positive-definite, and also differentiable and radially unbounded. It is clear as well that the zero state xt = 0 is a fixed point, or equilibrium point, of the reset system. The question now is whether this equilibrium is stable or not. First, let us formalize a little the sense of stability to be used in relation to the aforementioned quadratic functional. Definition 4.1 The IDDE reset control system (4.5) is said to be quadratically (asymptotically) stable when there exist P , Q > 0 such that the functional V in (4.16) satisfies the conditions (4.14) and (4.15). These two inequalities ensure that there exists a common Lyapunov function for the system (4.5) without reset (the base LTI system), using (4.14), and for the reset system xt + (0) = AR xt (0) using (4.15). Proposition 4.2 The system (4.5) is quadratically stable if and only if there exist matrices P , Q > 0, of size n × n, satisfying the following conditions:    A P + P A + Q P Ad < 0, (4.17) A −Q dP and

Θ  A R P AR − P Θ ≤ 0

for any matrix Θ with Im Θ = Ker C.

(4.18)

154

4

Stability of Time-Delay Reset Control Systems

Proof Condition (4.17) is commonly used in delay-independent stability of timedelay systems [16], and easily follows from the fact that (4.16) can be written as V (xt ) = x(t) P x(t) +



0

−h

x (t + θ )Qx(t + θ ) dθ,

and thus its derivative easily gives d V (xt ) = x (t)P x˙ (t) + x˙  (t)P x(t) + x (t)Qx(t) dt − x (t − h)Qx(t − h), so, using (4.5), the result is V˙ =



x(t) x(t − h)

 

A P + P A + Q A dP

P Ad −Q



 x(t) , x(t − h)

from which (4.17) is finally derived. It remains to prove that condition (4.18) follows from (4.15). Since the reset action is only active when xt (0) ∈ M , and thus does not affect the delay buffer x(t + θ ), for any θ ∈ (−h, 0), the integral part of the Lyapunov functional (4.16) does not contribute to the jump, thus the reset jump in the Lyapunov functional ΔV in (4.15) results in that 

ΔV = x t (0) AR P AR − P xt (0) ≤ 0, for every xt (0) ∈ M = Ker C, from which (4.18) directly follows.



Note that the reset action adds condition (4.18) to the standard condition of delayindependent stability given by the linear matrix inequality (LMI) (4.17). Since the matrix AR has a very particular structure, this new condition can be considered as an extra LMI, or simply interpreted as restriction over the structure of the matrix P that is a solution of the LMI. Let us study the structure of condition (4.18). First, notice that from (4.6) AR has the form AR = diag(Inρ¯ , Onρ ),

(4.19)

where, for convenience, the dimensions in (4.7) have been renamed nρ = n3 ,

nρ¯ = n − nρ = n1 + n2 ,

so that nρ is the number of (controller) states affected by reset and nρ¯ is the number of (plant and controller) states not affected by reset. Now, partitioning P according to (4.19), we have:     P1 P2 P1 0  A , (4.20) A P A = A = R R R R P P 0 0 3 2

4.4 Delay-Independent Conditions in the Frequency Domain

with P1 > 0 square of size nρ¯ = n − nρ , thus (4.18) becomes       

θ1    0 −P2 θ1 = θ1 θ2 ≤ 0, θ1 θ2 AR P AR − P θ2 θ2 −P2 −P3

155

(4.21)

where Θ  = (θ1 θ2 ). Now, assume here that C has the form C = (C1 , 0) with C1 of size (1 × nρ¯ ), as typically the last nρ states do not influence the reset condition.   Notice that in this case x = (x 1 , x2 ) belongs to Ker C = Ker(C1 , 0) if and only if C1 x1 = 0. Furthermore, (4.21) is nonpositive when x 1 P2 x2 = 0 for all x1 , x2 such that C1 x1 = 0. This amounts to P2 = C1 Mβ for some matrix Mβ , and hence the next proposition easily follows. Proposition 4.3 The system (4.5), with C = (C1 , 0) and C1 a row of length n − nρ , is quadratically stable if and only if there exist matrices P , Q > 0, with   C1 Mβ P1 (4.22) P= Mβ C1 P3 that satisfy (4.17), where Mβ is some (column) matrix of size nρ × 1.

4.4 Delay-Independent Conditions in the Frequency Domain The availability of frequency domain stability conditions is important for many wellknown reasons: integration of analysis and design, treatment of uncertainty, etc. In addition, previous stability conditions developed in reset systems [9] and [8] are in the frequency domain, and give usually easier checkable conditions than its LMIs counterpart. They are obtained by use of the Kalman–Yakubovich–Meyer (KYM) lemma. In addition, there exist frequency domain approaches to the stability of delay systems (see [12]) based on generalizations of the Kalman–Yakubovich–Popov (KYP) lemma. In this section, frequency domain conditions for time-delay reset systems will be derived. Again the move from time to frequency domain will require some generalization of the KYP lemma as well. A nice, general version of KYP can be found in [27]. We recall here this result: Proposition 4.4 (Generalized KYP lemma) Given (A, B) controllable with (sI − A)−1 B having no poles on the imaginary axis, and a matrix M = M  of adequate size, the following statements are equivalent:  ∗   (j ωI − A)−1 B (j ωI − A)−1 B M (i) ≤ 0, for all ω ∈ R ∪ {∞}. I I (4.23)    A P +PA PB ≤ 0. (4.24) (ii) There exists P = P  such that M + 0 B P The equivalence for strict inequalities holds even if (A, B) is not controllable.

156

4

Stability of Time-Delay Reset Control Systems

4.4.1 The Hβ -condition In order to find how to use the generalized KYP lemma to obtain frequency conditions for (4.17) and (4.18), it will be useful to recast the results in [8] using this lemma. In the following, the Hβ -condition for stability of reset systems without time-delay will be proved by using a different proof to that in [8]. It is included here since it will reveal a basic procedure to be followed afterwards in the time-delay case. Proposition 4.5 Consider the IDDE system (4.5) with Ad = 0 and reset surface   M = x(t) ∈ Rn : Cx(t) = 0 , where the row vector C has the form C = (C1 , 0), with C1 of length nρ¯ = n − nρ (nρ is the number of reset states). Assume also that A is Hurwitz and (A, B0 ) is controllable, where the input matrix is B0 = [0nρ¯ Inρ ]. Then, the IDDE system (4.5) is quadratically stable if and only if Hβ (j ω) = C0 (j ωI − A)−1 B0 is strictly positive real (SPR) for some Mβ and some P3 > 0, where B0 = [0nρ¯ Inρ ] and C0 = [Mβ C1 P3 ]. Proof The necessary and sufficient conditions for quadratic stability in this case are as follows: First, ∃P > 0 A P + P A < 0,

(4.25)

V (x(t)) = x (t)P x(t)

to be negafor the time derivative of the Lyapunov function tive along the trajectories of the system except for the reset times, and, second,

x (t) A (4.26) R P AR − P x(t) ≤ 0, ∀x(t) ∈ M for the jump ΔV to be ≤ 0 in the reset surface. Now, condition (4.26) yields, following (4.20) and (4.21),   0 −P2  x(t) ≤ 0, ∀x(t) ∈ M , (4.27) x (t) −P2 −P3 which gives rise, by Proposition 4.3, to a structure for P in the form   P1 C1 Mβ ≥0 Mβ C1 P3

(4.28)

for some column matrix Mβ of dimensions nρ ×1, and some P3 > 0. This restriction can be finally recast as B0 P = C0 for B0 = [0nρ¯ Inρ ] and C0 = [Mβ C1 P3 ].

(4.29)

4.4 Delay-Independent Conditions in the Frequency Domain

Notice that (4.25) and (4.29) are equivalent to ∃P > 0, ∃ε > 0    P B0 − C0 A P + P A + 2εP ≤ 0. B0 P − C0 0 Now, (4.30) can be rewritten in the form (ii) in KYP lemma:      Aε P + P Aε P B0 0 −C0 + ≤ 0, ∃P > 0, ∃ε > 0, −C0 0 B0 P 0  

157

(4.30)

(4.31)

M0

where Aε = A + εI . So, using KYP, since (A, B0 ) is controllable, ∃ε > 0 such that   ∗  (j ωI − Aε )−1 B0 (j ωI − Aε )−1 B0 M0 ≤ 0 for all ω ∈ R ∪ {∞}. I I (4.32) Notice that from the last equation (statement (i) in KYP) we recover (4.31) (statement (ii) in KYP), but with ∃P = P  in place of ∃P > 0. However, since A is Hurwitz, the existence of a symmetric P such that A P + P A = Q < 0 implies [19] that such P is positive-definite, i.e., P > 0. Hence, the last two equations (4.31) and (4.32) are completely equivalent. From the last condition, putting Hβ (j ω) = C0 (j ωI − A)−1 B0 and noticing that j ωI − Aε = j ωI − (A + εI ) = (j ω − ε)I − A, the positive real condition required for Hβ (s − ε), for some small ε > 0, is obtained. In other words (see [19]), it amounts to the strictly positive real (SPR) requirement on the system Hβ (j ω) given by the triple (A, B0 , C0 ), which is precisely the so-called Hβ -condition in [9].  Remark 4.1 It can be seen that the previous SPR condition, characterized in [19], is equivalent to the Output Strictly Passive (OSP) condition as defined in [10]. More precisely, since A is Hurwitz and Hβ is strictly proper, the SPR condition amounts, following [19], to Hβ (j ω) + Hβ (j ω) > 0 for all finite ω, and for ω → ∞ to

lim ω2 Hβ (j ω) + Hβ (j ω) > 0.

ω→∞

On the other hand, [10] characterizes the OSP condition, with A Hurwitz and (A, B0 ) controllable, by

Hβ (j ω) + Hβ (j ω) ≥ ε|Hβ (j ω)|2 for some ε > 0 small enough. By expanding Hβ (s) = H1 /s + H2 /s 2 + · · · and putting s = j ω with ω → ∞, the equivalence between the two previous conditions, that is, the equivalence between the SPR and OSP requirements for a strictly proper Hβ (s) can be verified. We will find again the OSP condition later, in Sect. 4.7, where the passage from time to frequency domain, instead of using KYP, will be solved by an approach based on approximating the resets by suitable Dirac delta pulses.

158

4

Stability of Time-Delay Reset Control Systems

Note that the Hβ -condition gives quadratic stability by checking if Hβ is SPR for some Mβ and some P3 . In the original formulation [8], P3 was simplified to the identity P3 = I , but this is not the general case. This was rectified in the subsequent work [9]. Since the SPR condition is easily checkable specially in the scalar case corresponding to the reset of only one state (nρ = 1), it has been proved to be an efficient method for stability analysis of reset control systems. In the following, application of the above procedure is extended to cope with the time-delay case.

4.4.2 The Generalized Hβ -condition Proposition 4.6 Consider the IDDE system (4.5) under the same conditions as in Proposition 4.5, that is, with M given by Cx = 0 and with C = (C1 , 0), but with Ad = 0. Assume also that A is Hurwitz and (A, B1 ) controllable, with B1 = (B0 , Ad ) and B0 = [0nρ¯ Inρ ]. Then, this IDDE system is quadratically stable if and only if for some P3 , Q > 0, for some Mβ and for some ε > 0, the KYP lemma is satisfied with the matrices A, M, B in the KYP frequency condition (4.23) given by A → Aε = A + εI, B → B1 = (B0 Ad ), and

⎛

Q M → Mε = ⎝ −C0 0

−C0 0 0

⎞ 0 ⎠ 0 −Q + εI

(4.33)

where C0 = [Mβ C1 P3 ]. Proof The treatment of the reset-delay conditions in (4.17) and (4.18) can follow a completely parallel manipulation to conditions in Proposition 4.5. First, notice from Proposition 4.3 that (4.18) is equivalent to (4.22). This condition has already been reformulated in the form (0 I ) P = (Mβ C1 P3 ),     B0

(4.34)

C0

where I and P3 inside B0 , C0 have size nρ × nρ . Now, let us try to reproduce manipulations similar to those from (4.30) up to (4.32). The starting point is: Quadratic stability amounts to the existence of P , Q > 0 satisfying (4.17) and (4.34), rephrased as    A P + P A + Q + 2εP P Ad (4.35) ≤ 0, B0 P = C0 , A P −Q + εI d

4.4 Delay-Independent Conditions in the Frequency Domain

159

for some ε > 0, which, imitating (4.25)–(4.30) is equivalent to ⎛  ⎞ P Ad P B0 − C0 Aε P + P Aε + Q ⎠ ≤ 0, (4.36) ∃P > 0, ∃ε > 0, ⎝ A −Q + εI 0 dP  B0 P − C 0 0 0 where Aε = A + εI . Now, reproducing (4.30) → (4.31), and simultaneously interchanging the second and third row-blocks and column blocks, we obtain that ∃ε > 0: ⎞ ⎛ B1 ⎛ ⎞

  0 Q −C0 ⎜ A P + P A P (B , A ) ⎟ ε 0 d ⎟ ε ⎝ −C0 ⎠+⎜ (4.37) 0 0 ⎟ ≤ 0. ⎜   B ⎠ ⎝ 0 0 0 −Q + εI P 0   A d Mε

The last equation is ready for KYP, reaching to the frequency condition for quadratic stability: ∃ε > 0, ∗    (j ωI − Aε )−1 B1 (j ωI − Aε )−1 B1 ≤ 0 for all ω ∈ R ∪ {∞}, Mε I I (4.38) where Mε , B1 are given in (4.37). This concludes the proof.  The matrix Mε could be diagonalized in the form Mε = F  J F , with J diagonal with entries ±1 or 0. Thus, depending on the signature of Mε (the number of positive, zero, and negative eigenvalues), condition (4.38) might be presented more suitably as a small-gain frequency condition, or as a positive-real frequency condition.

4.4.3 Interpretation of the Stability Conditions The previous conditions (4.37) and (4.38) are very useful for a quick implementation of the stability tests, however, they are based on big matrices that do not really help in understanding the nature of the stability conditions. To clarify this, an interpretation will be given in terms of the structural properties of the dynamics. This will be achieved in three steps. In the first step, the IDDE system (4.5) is equivalently represented by the block diagram in Fig. 4.2. The upper loop realizes the continuous dynamics x˙ = Ax + Ad Dh x (the first equation in (4.5)), where Dh ≡ e−j ωh is the delay operator, by assigning Cd = I n ,

Bd = Ad ,

Δd → Dh ,

(4.39)

where the delay Dh ≡ e−j ωh is treated as a particular case of an uncertainty Δd satisfying small-gain: Δd (j ω) ≤ 1

for all ω.

(4.40)

160

4

Stability of Time-Delay Reset Control Systems

Fig. 4.2 Block diagram representation of the IDDE system

The impulsive or reset action is implemented by the lower loop which is zero most of the time. It only works at the reset instants when the reset states (the last nρ states) are switched to zero. This resetting is easily shown to be equivalent to the application of Dirac impulses δ(·) in the following form. Consider the reset system Rr from yr to ur defined by:  δ(t − tk )P3−1 yr (t), (4.41) (Rr ) ur (t) = − k

with Br = (0, Inρ ) ,

Cr = (Mβ C1 , P3 ),

where tk are the reset instants characterized by Cx(tk ) = 0 and by tk ≥ tk−1 + ρ. The equivalence stems from the fact that, around the reset instant tk , the signal ur contains the delta function with weight: ur (t) = −δ(t − tk )P3−1 yr (t),

t ∈ (tk − ε, tk + ε), ε < ρ.

Thus, the instantaneous response of x˙ = Ax + Bd ud + Br ur = Ax + Ad x(t − h) + Br ur to a Dirac input in ur is given by a state jump of value



Δx = x tk+ − x tk− = Br



tk+ tk−

ur = −Br P3−1 yr (tk )

  0 =− P3−1 (Mβ C1 , P3 )x(tk ). I Now, since at the reset instants  0 = Cx(tk ) = (C1 , 0)

x12 (tk ) x3 (tk )

 = C1 x12 (tk )

where the state has been partitioned making explicit the last nρ coordinates x3 (the coordinates affected by reset),    



0 0 , = − Δx = x tk+ − x tk− = x3 (tk ) −P3−1 P3 x3 (tk )

4.4 Delay-Independent Conditions in the Frequency Domain

161

which is the precise change that produces the required reset. Thus so far we have proved that the IDDE system can be realized in the block diagram form given by Fig. 4.2. To complete the first step, notice that the resetting subsystem Rr : yr → −ur satisfies  ∞  −u yr (tk )P3−1 yr (tk ) ≥ 0, (4.42) r (τ )yr (τ ) dτ = 0

k

so, as shown in Fig. 4.2, the resetting block can be treated as a particular case, Δr → Rr of a general uncertainty Δr : yr → −ur satisfying the positivity condition:  ∞ −ur = Δr yr ⇐⇒ −u (4.43) r (τ )yr (τ ) dτ ≥ 0. 0

In the second step, the interpretation of the upper loop condition is provided. The quadratic stability of the continuous dynamics amounts to (4.14), V˙ < 0, and to the first LMI in (4.35). This LMI can be treated within the KYP lemma, with M = diag(Q, −Qε ), for Qε = Q − εI , and it is easy to show that it is equivalent to Gd (j ω)

     1/2 −1 Q (j ωI − Aε ) Ad Qε−1/2  < 1 for all ω,

(4.44)

where Gd (s) is the open-loop nominal (Δd = 0) transfer function of the upper loop in Fig. 4.2 (from ud to yd ). In this way, (4.44) is a Q-scaled strict small-gain condition on the uncertain upper loop. The small-gain condition is strict due to the presence of the small ε > 0 in Aε = A + εI and Qε = Q − εI . It is important to notice that stability is achieved for all values of the delay 0 ≤ h < ∞. This means, in particular, stability for h = 0 (so that A + Ad is stable) and for h → ∞ (so that A stable). Notice also that, from (4.35), A P + P A ≤ −Q − 2εP < 0, so that x P x is a Lyapunov function for the system x˙ = Ax. In the third and final step, an interpretation of the lower loop condition is given. As the lower loop performs the resettings, the required condition (4.15) is ΔV ≤ 0 at the reset instants. It is equivalent to the second condition in (4.35), rewritten now as Br P = Cr . Furthermore, it amounts (given A P + P A < 0) to:    P Br − Cr Aε P + P A ε ≤ 0. (4.45) ∃ε > 0, Br P − Cr 0 Recasting the previous LMI via the KYP lemma, it easily follows ∃ε > 0,

0 ≤ G∗r (j ω − ε) + Gr (j ω − ε),

(4.46)

where Gr (s) = Cr (j ωI − A)−1 Br is the open-loop nominal (Δr = 0) transfer function of the lower loop in Fig. 4.2 (from ur to yr ). In this way, (4.45) is a strictly positive real (SPR) condition on the uncertain lower loop. Summarizing, the LMIs (4.37) and (4.38), expressed in terms of large matrices, have been interpreted here in terms of structural properties of the IDDE system.

162

4

Stability of Time-Delay Reset Control Systems

Basically, the mechanized procedure of KYP and LMIs (computationally oriented) corresponds to the underlying structure in Fig. 4.2, in which the delay is treated as a small-gain uncertainty and the resetting is viewed as a positive uncertainty. Using this framework, different conditions might be deduced, by encapsulating the delay or the reset action within alternative uncertainty structures. Furthermore, the reset action could be changed freely and redesigned, provided it remains within the required sector bounds. In the delay-dependent case, Remark 4.6 (that can be found in Sect. 4.7) will present an interpretation in terms of robustness for some uncertain loop (see Fig. 4.6) very similar to the one in this section. The main difference is that the delaydependent case will be addressed by passivity techniques applied to an equivalent system where reset is approximated by Dirac impulses.

4.5 Example: Delay-Independent Stability To illustrate the stability results, let us consider an example based on Example 5.7 in [16]. Consider a time-delay reset system with state x(t) = (x1 (t) x2 (t)) given by x˙ (t) = Ax(t) + Ad x(t − h), with

 A=

 −2 0 , 0 −0.9

 Ad =

 −0.25 0 , −0.25 −0.25

(4.47)

endowed with the reset law

x2 t + = 0 when x1 (t) = 0. This system is stable irrespective of the delay in the case of no reset [16]. In the following, it will be also shown that it is also stable if we reset the state x2 (t). Following Proposition 4.6, in this case B0 = (0 1), and C0 = (Mβ p3 ), where p3 is simply a positive real number, p3 = 0.1, and Mβ = 0 is a possible choice, then C0 = (0 0.1). In addition, a positive-definite matrix Q is chosen as  Q=

 0.1 −0.05 . −0.05 0.1

Thus, using Proposition 4.6, the matrix Mε is given by (for ε = 0.01) ⎛

Q Mε = ⎝ −C0 0

−C0 0 0

⎞ 0 ⎠ 0 −Q + εI

4.5 Example: Delay-Independent Stability

163

Fig. 4.3 The largest eigenvalue of H (j ω) vs ω

⎛

0.1 −0.05 0 ⎜ −0.05 0.1 −0.1 ⎜ −0.1 0 =⎜ ⎜ 0 ⎝ 0 0 0 0 0 0

⎞ 0 0 0 0 ⎟ ⎟ 0 0 ⎟ ⎟, −0.09 0.05 ⎠ 0.05 −0.09

and finally, the frequency response matrix H (j ω) resulting from the KYP lemma (4.38) is H (j ω) = G (j ω)MG(j ω), where

 G(s) =

(j ωI − Aε )−1 B1 I



with Aε = A + εI and B1 = (B0 , Ad ). The matrix H (j ω) is easily shown to be negative semidefinite for every frequency ω > 0, thus it satisfies the KYP condition (4.38). Figure 4.3 shows the largest eigenvalue of H (j ω) versus ω (using a logarithmic ω axis). Since these values are always below zero, H (j ω) ≤ 0 ∀ω > 0, and thus the reset system given by (4.47) is stable independently of the delay.

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4

Stability of Time-Delay Reset Control Systems

Fig. 4.4 Simulation of the delay-independent example: x1 (t) (dashed) and x2 (t) (solid)

Notice the behavior as ω → ∞ in Fig. 4.3: G(j ω) tends to (02×3 , I3×3 ) , and thus H (j ω) tends to the 3 × 3 submatrix of Mε formed by its last three rows and columns. This submatrix has a zero eigenvalue, and this explains the high frequency limit in Fig. 4.3. Figure 4.4 shows a simulation of the example with a delay h = 2 s, and with the initial conditions x1 (θ ) = x2 (θ ) = 0 for θ ∈ [−2, 0) and x1 (0) = 10, x2 (0) = 0. It can be seen that initially (t < h = 2) the state x1 (t) tends to zero with the stable dynamics given by A = diag(−2, −0.9). But when t = h = 2 s, the influence of Ad x(t − h) appears and the trajectories x(t) are affected, giving rise to several resets at the times t1 ≈ 2.1, t2 ≈ 4.6, t3 ≈ 7.3, . . .. These reset times are defined by x1 (tk ) = 0. The simulation of this reset system for h = 2 s shows a stable behavior. Since we have proven that this system is stable independently of the delay h, we can as well predict stability for all fixed h ≥ 0.

4.6 Delay-Dependent Conditions in the Time Domain This section addresses delay-dependent stability of reset systems, that is, we will search for adequate Lyapunov–Krasovskii functionals such that, from an application

4.6 Delay-Dependent Conditions in the Time Domain

165

of the basic Proposition 4.1, the derived stability conditions depend explicitly on the delay h. In this way, the stability conditions may hold only for some values of h. This is more natural since many practical systems are typically stable for small values of the delay, but unstable for large delays. In order to gain generality, we assume here for C the structure C = (C1 , C2 , 0),    based on the state partition (4.8) x = (x 1 , x2 , x3 ) and the corresponding dimensions (4.7) n = n1 +n2 +n3 . This structure is equivalent to the one used in the delayindependent case, with the notation equivalence nρ = n3 and nρ¯ = n−nρ = n1 +n2 . The proposed structure C = (C1 , C2 , 0) is more general than C = (C˜ 1 , 0, 0), the one adopted in [9]. For comparison with this last case, let us state the fact that there exists an invertible transformation T , T −1 ∈ Rn×n such that C˜

C T

   (C1 , C2 , 0) = (C˜ 1 , 0, 0) diag(T12 , In3 ) .

(4.48)

One possible solution is C˜ 1 = (1, 0, . . . , 0) and T12 with its first row equal to (C1 , C2 ) and the remaining rows completed for full rank. Now, the key point to obtain delay-dependent stability conditions is to choose an appropriate Lyapunov–Krasovskii (LK) functional. There are several LK functionals that can be used for generating delay-dependent conditions. The complete LK functional (Theorem 5.19 in [16]) has the advantage that it gives a sufficient but also necessary condition for stability, however, it is a distributed functional and requires discretization for practical computation. Another useful result is given by Proposition 5.17 in [16], with good LMI computability and which is generally not too conservative. Finally, the LK functional in [26] has the advantage that the stability conditions are interpretable in a robust stability framework, in terms of a certain small-gain loop [29]. This LK functional takes the integral-quadratic form given by: V (xt ) = V1 (xt ) + V2 (xt ) + V3 (xt ),

(4.49)

with 1 V1 (xt ) = x (t)P x(t), 2  0 V2 (xt ) = x (t + θ )Qx(t + θ ) dθ, −h

and

 V3 (xt ) =

0



−h θ

0

G(xt+ξ ) A d XAd G(xt+ξ ) dξ dθ,

(4.50) (4.51)

(4.52)

where G(xt+ξ ) = Ax(t + ξ ) + Ad x(t + ξ − h).

(4.53)

In this way, by applying Proposition 4.1 to V (xt ), we will arrive at conditions ensuring global and asymptotical stability. The requirement that d[V (xt )]/dt < 0

166

4

Stability of Time-Delay Reset Control Systems

has been already treated in [26], and it is finally expressed in an LMI form. The requirement that ΔV (xt ) ≤ 0 when x(t) ∈ M can be addressed in a special form, related to the one in Theorem 5 of [9]. By combining these two approaches, we obtain the following proposition. Proposition 4.7 The IDDE reset system (4.5) is globally and asymptotically stable, if there exist Q > 0, V > 0, W , and P = P  = (Pij )i,j =1,2,3 > 0,

(4.54)

(0, 0, In3 )P = (Mβ (C1 , C2 ), P33 ),

(4.55)

with

for some column matrix Mβ of appropriate size, and such that ⎛

N

⎜ ⎜ ⎝

−W Ad −Q 0

A A dV  A d Ad V −V 0

h(W + 12 P ) 0 0 −V

⎞ ⎟ ⎟ < 0, ⎠

(4.56)

where N=

1  (A + Ad ) P + P (A + Ad ) + W Ad + A d W + Q, 2

and each block “ ” is the transpose of the corresponding symmetric block. Proof From Proposition 4.1, what has to be proved is that (i) dV /dt < 0 and (ii) ΔV ≤ 0 for x ∈ M . It can be seen that (4.56) implies (i), this has been already proved in [26]. So, it only remains to prove that (4.55) implies (ii). Actually, it will be shown that (4.55) and (ii) are equivalent, that is, (4.55) holds iff (ii) is true. This equivalence will be made explicit by expressing ΔV ≤ 0 for x ∈ M as a quadratic ˜ that M = Ker C = T −1 Ker C, ˜ that form. First, observe from (4.48) with C = CT is, M = Ker C = T −1 Ker C˜ = T −1 Imag diag(K1 , In2 , In3 ),  

(4.57)

K

with K1 such that Imag K1 = Ker C˜ 1 .

(4.58)

Then, it can be seen that, after a reset jump x(t) → AR x(t), the LK functional V = V1 + V2 + V3 only changes in its first contribution V1 because the integrals in V2 and V3 are not affected. In this way, the condition ΔV = ΔV1 ≤ 0 for x ∈ M = T −1 Imag K amounts to the negative semidefiniteness, P¯ ≤ 0, of P¯ = K  T − (AR P AR − P ) T −1 K,   Π (P )

(4.59)

4.6 Delay-Dependent Conditions in the Time Domain

167

which defines Π(P ). But, it can be seen that

P¯ = K  Π T − P T −1 K = K  Π(P˜ )K.  

(4.60)

P˜

This is true because AR and T −1 commute, due to their block-diagonal structure. Now, partitioning P˜ = (P˜ij ) for i, j = 1, 2, 3, it holds that ⎛

0 Π(P˜ ) = − ⎝ 0 P˜ 

13

0 0 P˜ 

23

⎞ P˜13 P˜23 ⎠ . P˜33

(4.61)

Introducing K in (4.57), we get ⎛

0 P¯ = K  Π(P˜ )K = − ⎝ 0 K P˜13 1

0 0  P˜23

⎞ K1 P˜13 P˜23 ⎠ . P˜33

(4.62)

Recall that (ii) ΔV ≤ 0 for x ∈ M is true iff P¯ ≤ 0. And this is equivalent, imposing that the off-diagonal terms are zero, to −P˜33 ≤ 0,

P˜23 = 0,

and

∃Mβ :

 P˜13 = Mβ C˜ 1 ,

(4.63)

as Imag K1 = Ker C˜ 1 , for some column matrix Mβ of appropriate size. Then, using T  P˜ T = P , we can say that  



(0, 0, In3 )P = P˜13 , P˜23 T12 , P˜33 = (Mβ C˜ 1 , 0)T12 , P˜33 , (4.64) and using (4.48) and P˜33 = P33 , one immediately obtains (4.55) = (4.64), proving the fact that (4.55) holds iff P¯ ≤ 0 iff ΔV ≤ 0 for x ∈ M , which concludes the proof.  ˜ T = In , and P = P˜ , Remark 4.2 Notice that when C2 = 0, one can take C = C, which is the case in [9]. In this way, the result here extends the previous results not only to the delayed case, but also to a more general form of the resetting hypersurface M , with C2 = 0. Remark 4.3 Usually, the term V3 in (4.52) is expressed by replacing Ax(t + ξ ) + Ad x(t + ξ − h) by x˙ (t + ξ ) [26]. Here, this is possible only in the continuous mode of operation. At the reset instants, the relation is no longer valid, so that the correct expression for V3 is the one in (4.52). Remark 4.4 Delay margin. If the delay h is not given, it is important to estimate the set of values h ∈ Ireset for which (4.54) and (4.56) are feasible. Notice that stability of the reset system is more restrictive than stability of the linear system, so that Ireset ⊂ Ilin . Determining those h ∈ Ilin for which x˙ = Ax + Ad x(t − h) is stable

168

4

Stability of Time-Delay Reset Control Systems

is well known. It is known that h can have an effect of conditional stability: Ilin = [0, η1 ) ∪ (η2 , η3 ) ∪ · · · ∪ (ηm−1 , ηm ), with the ηk easily computable [16]. With this is mind, as Ireset is bounded by Ilin , one can always define a fine mesh of values {hk }N k=1 covering this set and check feasibility of the N related LMIs. Furthermore, (4.56) can be written as F + hG < 0, and one can always force negative definiteness by checking Fk + hk Gk < −αk I , for some αk > 0. Then, it is easy to show that we have a stable  range (hk − δk , hk + δk ) =: Ik provided that δk = αk /Gk . Thus, the union k Ik =: Inum provides a numerically guaranteed stable range Inum ⊂ Ireset that can be made arbitrarily close to Ireset by increasing the number of LMIs tested.

4.7 Delay-Dependent Conditions in the Frequency Domain This section presents frequency domain conditions related to (4.55) and (4.56) to verify stability. One standard tool for moving from time-domain LMIs to frequency-domain conditions is the Kalman–Yakubovich–Popov (KYP) lemma used in Sect. 4.4. The KYP lemma could have been used here as well, however, we prefer to present here an approach based on passivity, with useful connections between resetting and control by impulses. The approach is based on the notion of passivity of a system with respect to certain storage function (the Lyapunov functional). The IDDE system will be interpreted as equivalent to the base-linear subsystem (BLS) in feedback connection with the resetting subsystem. It will be shown that the reset subsystem is passive, so that the global system will be stable if the BLS (with certain inputs and outputs) is strictly passive. This last condition has an easy frequency domain interpretation. Thus, to give a frequency-domain interpretation of Proposition 4.7, note that condition (4.55) can be rewritten as B  P = H or H  = P B,

(4.65)

with B  = (O, O, In3 ),

H = (Mβ C1 , Mβ C2 , P33 ).

(4.66)

After this, consider the base linear system (BLS) x˙ (t) = Ax(t) + Ad x(t − h), and the base linear system with inputs and outputs  x˙ (t) = Ax(t) + Ad x(t − h) + Bu(t), (BLSio ) y(t) = H x(t).

(4.67)

(4.68)

Proposition 4.8 If the conditions (4.55) and (4.56) hold, then the base-linear system BLSio is strictly passive with respect to the storage functional V (xt ) (4.49).

4.7 Delay-Dependent Conditions in the Frequency Domain

169

Proof The arguments are sketched as follows. Recall that the system BLSio is strictly passive with respect to V (xt ), by definition, when y  (t)u(t) > V˙BLSio (xt ),

xt = 0,

(4.69)

for all u(t), y(t), xt solutions to the system. By rebuilding the time derivative of the Lyapunov–Krasovskii functional, it can be seen that 1 1 V˙BLSio = V˙BLS + x (t)P Bu(t) + u (t)B  P x(t), 2 2

(4.70)

where the subindex of the V˙ denotes the system where the time derivative of V is computed. The last expression is easy to deduce and shows that the only influence of u in BLSio affecting V˙ = V˙1 + V˙2 + V˙3 is through the term V˙1 . The derivatives of the remaining integral terms V2 , V3 can be written [26] as: V˙2 = x (t)Qx(t) − x (t − h)Qx(t − h) and V˙3 = hF (t) −



0

−h

F (t + θ ) dθ,

with F (t) = (Ax(t) + Ad x(t − h)) A d XAd (Ax(t) + Ad x(t − h)) . And hence V˙2 , V˙3 depend only on past values x(t + θ ), θ ≤ 0 of the state and not on the input u(t), so that (4.70) is true. Then, the passivity condition amounts to y  (t)u(t) = x (t)H  u(t) > V˙BLS (xt ) + x (t)P Bu(t).

(4.71)

Now, the only possibility for the previous relation to hold is (setting u = 0) that V˙BLS (xt ) < 0 (ensured by (4.56)) and (as u(t) is completely free) that (4.65) is satisfied (equivalent to (4.55)). Notice that (4.56) is only a sufficient condition for V˙BLS (xt ) < 0, according to [26]. This is the reason why the ‘only if’ direction cannot be proved here.  Now, it is useful to show that the overall IDDE (4.5) is passive (and stable) by proving that it is the negative feedback connection between BLSio and certain suitable subsystem R that performs the resetting actions. To show this, consider the ideal resetting system R : y(t) → −u(t) =: v(t) with input y(t) = H x(t) and output −u(t) = v(t) given by  −1 δ(t − tk )P33 y(t), (4.72) (R) v(t) = k

where δ(t − tk ) is a Dirac impulse applied at t = tk , the instants when the reset condition holds. This resetting system has already been used in Sect. 4.4.3. Now, we will see more formally how R plays an important role also in the delay-dependent case.

170

4

Stability of Time-Delay Reset Control Systems

Proposition 4.9 The IDDE system (4.5) is equivalent to the negative feedback interconnection between the base linear system with inputs and outputs BLSio and the ideal resetting subsystem R. Proof During the continuous mode, both configurations share the same autonomous dynamics. At the reset times tk , the IDDE system resets to zero the component x3   of the controller state x 23 = (x2 , x3 ) (the last n3 states). At the same instants tk , the system R behaves as follows. Notice that y(tk ) = H x(tk ) = Mβ C1 x1 (tk ) + Mβ C2 x2 (tk ) + P33 x3 (tk ),

(4.73)

and, by the reset condition Cx = C1 x1 + C2 x2 = 0, y(tk ) = P33 x3 (tk ),

(4.74)

so from (4.72) u(t) = −v(t) contains, locally around t = tk , a Dirac delta with weight −1 u(t) ≈ −δ(t − tk )P33 P33 x3 (tk ) = −δ(t − tk )x3 (tk ).

This is the precise input u(t) that, applied to the BLS with input matrix B, performs the zero reset of x3 , the last n3 components of the state.  Figure 4.5 illustrates the main idea of the previous proposition. The evolution of certain reset state x3 (t) is shown in Fig. 4.5 (top). It obeys the equation x˙3 = f3 (x(t), x(t − h)) and resets to x3 (tk+ ) = 0 at reset instants given by Cx(tk ) = 0. Then, this behavior is equivalently modeled by x˙ 3 = f3 (x(t), x(t − h)) + u(t) with  the feedback law u = − k δ(t − tk )x3 (t) with corrective ideal Dirac impulses. Indeed, this effect can be approximated by rectangular pulses δε (t) with width ε  and height 1/ε in the form u = − k δε (t − tk )x3 (tk ). Figure 4.5 (bottom) shows the required train of finite pulses, for ε = 0.1. These ideas are exploited and formalized in the following. Now the question is: Could the stability of the IDDE system be proved with passivity techniques? From Proposition 4.7 (internal stability), one can always derive an input–output frequency domain result, using the KYP lemma, as in Sect. 4.4. However, it would be useful to find an independent approach based only on passivity. The main problem on the I/O stability of the loop {BLSio , −R} is that the Dirac impulses are not signals in L2 . Then, the only possibility is to consider L2 approximations of ideal impulses. So, consider the system Rε , y(t) → −u(t) =: v(t), with input y(t) = H x(t) and output −u(t) = v(t) given by  −1 δε (t − tk )P33 |y(tk )| ◦ sign(y(t)), (4.75) (Rε ) v(t) = k

where δε (t − tk ) is a rectangular pulse starting at t = tk , with width ε and height 1/ε, and “◦” stands for componentwise vector product. Thus Rε acts by producing pulse trains with height proportional to the value |y(tk )| of its inputs at the reset instants. The function ‘sign’ is included for passivity

4.7 Delay-Dependent Conditions in the Frequency Domain

171

Fig.  4.5 (Top) A reset state x3 (t). (Bottom) Equivalent resetting impulses u = − k δε (t − tk )x3 (tk ) (ε = 0.1)

because y(t) may vary fast enough in the interval [tk , tk + ε), changing its sign. Connecting Rε to BLSio , from (4.66) and (4.68), and for t ∈ [tk , tk + ε), we have y(t) ˙ = H x˙ (t) = H Ax(t) + H Ad x(t − h) +H Bu(t),   =:y˙X (t)

thus, from H B = P33 and u(t) = −v(t) in (4.75), for t ∈ [tk , tk + ε): y(t) ˙ = H Ax(t) + H Ad x(t − h) −   =:y˙X (t)

|y(tk )| ◦ sign(y(t)). ε

(4.76)

To better see how Rε works, consider the scalar case with y(tk ) =: yk > 0, then y(t) ˙ = y˙X (t) −

yk sign(y(t)). ε

(4.77)

So, if ε  min(yk )/ max(|y˙X |), then the second term in (4.77) dominates the first one, which is the objective. Thus, we make here two assumptions:

172

4

Stability of Time-Delay Reset Control Systems

(i) the state x(t) is restricted to evolve in {x : x(t) < Xmax }, and (ii) the resetting is applied only when |y(tk )| > ymin . For the first assumption, the bound Xmax can be taken arbitrarily large, covering the physically meaningful values of the states. This assumption also implies that the stability results will have a local character, and ensures that y˙X (t) is bounded. For the second assumption, ymin can be taken arbitrarily small, equal to the resolution level of the quantization in digital implementation. Both assumptions are necessary, otherwise the dominance of yk /ε and the resetting effect will be lost. If assumptions (i) and (ii) hold, then (4.77) for yk > 0 and ε → 0 approaches y(t) ˙ ≈−

yk ε

for t ∈ [tk , tk + ε).

Integrating this equation from y(tk ) = yk gives rise to y(tk + ε) = 0, that is, we reset the output y(t) at the end of the pulse width. Since we are arbitrarily close to the reset instant t ∈ [tk , tk + ε), with ε → 0, from (4.73) and (4.74) it follows that y(t) ≈ P33 x3 (t) for t ∈ [tk , tk + ε). Thus we are resetting as well the required part of the state x3 at the end of the pulse: x3 (tk + ε) = 0 when ε → 0. In this way, Rε approaches the ideal reset R (4.72) when ε → 0, and the resetting action in Proposition 4.9 is correctly applied because from (4.77) when ε → 0 the value yk = P33 x3 (tk ) decreases linearly to y(tk + ε) = 0 in an arbitrarily fast way. Now, the system Rε given by (4.75) maps L2e to L2e . However, Rε is not bounded in L2 since it is possible to find signals in L2 such that their images under Rε are not in L2 . A similar problem has been discussed in [11] in the framework of sampled-data systems. The solution adopted there to overcome the problem will be also used here. Basically, the system Rε is restricted to operate in the space F L2 of filtered L2 signals, F being an LTI system with a strictly proper transfer function and arbitrarily large bandwidth. More precisely, Rε is restricted to ˙ ≤ M} where the bound M on y˙ can always operate on F L2 = {y ∈ F L2 : |y| be imposed, for any fixed ε, from the discussion after (4.77). Thus, the system ⊂ L → L is bounded, that is, R (F L ) ⊂ L . This is a necessary Rε : F L2e 2e 2e ε 2 2 condition for Rε to be output strictly passive. ⊂ L → L Proposition 4.10 The system Rε : F L2e 2e 2e (that approximates the ideal reset R by finite rectangular pulses, and approaches R when ε → 0) is an output strictly passive (OSP) system for all ε > 0.

Proof Recall that a system Rε : y → v, satisfying  0

T

y  (t)v(t) dt ≥ β + δi



T 0

y  (t)y(t) dt + δo



T

v  (t)v(t) dt

(4.78)

0

for all y, v, T and some β ≤ 0, is passive when δi , δo ≥ 0, input strictly passive (ISP) when δi > 0, output strictly passive (OSP) when δo > 0, and strictly passive (SP) when δi , δo > 0 (for details, see Sect. 3.3).

4.7 Delay-Dependent Conditions in the Frequency Domain

173

Let us suppose here, for simplicity, the scalar case n3 = 1 and P33 = 1. The multivariable case requires a more careful treatment and will be reported elsewhere. First, the L2 norm of the output is Rε y = 2

 

2 δε (t − tk )|y(tk )| sign(y(t)) dt,

k

and after some manipulation, Rε y2 = (1/ε)



|y(tk )|2 .

k

Then, the inner product v, u = Rε y, y is, by definition,    Rε y, y = δε (t − tk )|y(tk )| sign(y(t)) y(t) dt, k

where after some work (using |y| = sign(y)y) Rε y, y = (1/ε)

 k

 |y(tk )|

tk +ε

|y(t)| dt.

tk

, |y| Now, since y is a filtered L2 signal, that is, y ∈ F L2e ˙ is bounded, and one gets  tk +ε |y(t)| dt ≥ εα(ε)|y(tk )|, tk

where the coefficient α(ε) > 0 can be obtained from (4.77) by taking the worst  case evolution of y(t) (least integral value |y|) as a function of the corresponding bounds |yk | ≥ ymin and |y˙X | ≤ (H A + H Ad )Xmax . It becomes also clear that when ε → 0 the evolution for (t, y(t)) in (4.77) is a straight line from (tk , yk ) to (tk + ε, 0) so that |y| = εyk /2, meaning that α(ε) → 1/2 when ε → 0. Finally, comparing < Rε y, y > and Rε y2 ,  Rε y, y ≥ α |y(tk )|2 = αεRε y2 ≥ δRε y2 , k

and thus, from (4.78), Rε is output strictly passive (OSP) for some δ such that 0 < δ ≤ αε.  Remark 4.5 An important consequence of the last result is that the stability condition for the feedback system {BLSio , −Rε } can be translated into a frequency condition on BLSio . To see this, recall that a negative feedback system {G1 , −G2 } is finite-gain stable if any of (i)–(iii) holds [10]: (i) G1 , G2 are ISP, (ii) G1 , G2 are OSP, (iii) G1 is SP and G2 is passive (see also Sect. 3.3 for details).

174

4

Stability of Time-Delay Reset Control Systems

Fig. 4.6 Robust stability interpretation described in Remark 4.6

Clearly, we are interested in case (ii), so if BLSio is OSP, then closed-loop stability (finite-gain) is guaranteed. Recall also that a stable and controllable LTI system G(s) is OSP if there exists some δo > 0 such that G(j ω) + G (j ω) ≥ δo G(j ω)2 (see [10]). In the scalar case, this means that G(j ω) lies inside a circle with center s = 1/(2δo ) and radius r = 1/(2δo ). In summary, the required stability condition is Hβ (j ω) + Hβ (j ω) ≥ δHβ (j ω)2 , δ > 0, that is, an OSP condition on

−1 Hβ (j ω) = H j ωI − A − Ad e−j ωh B.

(4.79)

So this is a generalization of the results in [9]. In the particular case when Ad → 0 (without time-delay), and when H = (Mβ C1 , Mβ C2 , P33 ) → (Mβ C1 , O, P33 ), one gets the so-called Hβ -SPR (positive realness) condition in [9]. Strictly speaking, this passivity analysis ensures stability of the loop {Hβ (s), R} in the I/O sense. In general, recovering internal stability would require some mild assumptions (controllability, detectability, etc.) However, note that the passivity result here is presented as an extra interpretation of an original LK analysis which directly ensures internal asymptotic stability. Remark 4.6 (Robustness in terms of small-gain and positive uncertainties) A similar interpretation of stability to the one given for the delay-independent case in Sect. 4.4.3 can be given now. One nice feature of the delay-dependent LK functional (4.49) is that it allows an interpretation within a scaled small-gain framework [26]. In the same way, the previous passivity analysis can be recast in a robust stability setting. Combining the two parts, it can be said that the IDDE system (4.13) satisfying the conditions of Proposition 4.7 can be interpreted as shown in Fig. 4.6 (compare with Fig. 4.2).

4.8 Example: Delay-Dependent Stability

175

This figure contains two sets of blocks: the white fixed blocks and the gray uncertain blocks. The set of fixed blocks is derived from the base linear system and from some matrices appearing in the LMI stability conditions, including B, H in (4.66). There are three uncertain blocks, the block Δ is the delay system e−hs . The block Z is the zero-order holder system (e −hs − 1)/(hs). The third block R is the impulsive resetting subsystem introduced above. The proof of the equivalence is omitted for brevity, but it can be checked starting from (13) in [29], and with some mild notation changes, as M = I + X−1 W  , with X = 12 P . The interesting point of the relation between (4.13) and Fig. 4.6 is not only the equivalence when Z, Δ and R are as described above, but also the important conclusion that stability is preserved when the first two blocks are replaced by any pair of small-gain blocks (Δ1 , Δ2 ) satisfying Δi ∞ ≤ 1, i = 1, 2. And the third bock R can be replaced as well by any other OSP block with a stable strictly proper transfer function Δ3 with Re[Δ3 (iω − ε)] ≥ 0 for some ε > 0. Thus, Fig. 4.6 shows explicitly the degree of robustness achieved by the LK stability conditions, that is, which amount of uncertainty, and in what form, can be tolerated. This offers useful insights into theoretical and practical aspects of the problem. Remark 4.7 (Using alternative LK functionals and LMIs) Within the field of linear time-delay stability there is a large literature, and thus, there are many other LK functionals alternative to (4.49). In [4], a functional taken from [16] was used. Similarly, many other LK functionals could be combined to the passivity analysis presented here, although it is not always possible to derive robust interpretations as in the previous remark.

4.8 Example: Delay-Dependent Stability Consider a feedback system according to the setup of Fig. 4.1. Here the plant has a transfer function e−hs P0 (s) given by (the example is adapted from [8]) e−hs P0 (s) = e−hs

s +1 , s(s + 0.2)

and the feedback compensator R is a FORE compensator with base LTI system given by 1 . s +1 It can be shown that the base LTI control system is stable for delays h < 0.2. In the following, the passivity result derived in Sect. 4.7 will be used to show that the reset control system is also stable for that range of delays. Thus, the reset action does not destabilize the control system. This is simply made by checking if the system given by (4.79) is OSP. A delay h = 0.15 will be chosen. FOREb (s) =

176

4

Stability of Time-Delay Reset Control Systems

Fig. 4.7 minω Re{(H (j ω))} vs β for h = 0.15

Fig. 4.8 Re{(H (j ω))} vs ω for h = 0.15 and β = 0.4

In this simple case, the plant has two states and the reset compensator only one state, thus n1 = 2, n2 = 0, and n3 = 1. Using the realizations of P0 (s) and FORE b (s):    

−0.2 0 1 , Bp = , Cp = 1 1 , Ap = 1 0 0 and Ar = −1, Br = 1, Cr = 1, Dr = 0; the matrices A, Ad ⎛ ⎛ ⎞ −0.2 0 0 0 0 0 ⎠, A=⎝ 1 Ad = ⎝ 0 −1 −1 −1 0

are ⎞ 0 1 0 0⎠. 0 0

4.8 Example: Delay-Dependent Stability

177

Fig. 4.9 Time response y(t) (top) and u(t) (bottom) with reset (solid) and without reset (dotted)

In addition, here Mβ = (β), P33 are scalar, in particular P33 = 1 may be chosen without lost of generality. As a result, matrices H and B are given by (4.66):



B = 0 0 1 . H = βCp P33 = β β 1 , With the above values, system (4.79) is checked to be OSP. A simple exploration (see Fig. 4.7) shows that in fact it is OSP for values of β between 0.29 and 0.56. Figure 4.7 shows the minimum value of Re{H (j ω)} over the ω-axis for different values of β. In particular, Fig. 4.8 shows the positive value of Re(H (j ω)) (logarithmic scale) vs ω for β = 0.4. As a result, the reset control system is guaranteed to be also stable, in this case for h = 0.15. An identical procedure could be used to show that the reset action never destabilizes a stable base LTI control system, and its stability limit is h = 0.2, the same limit that the one for the base LTI control system. Figure 4.9 plots the step response, showing the plant output y(t) and plant input u(t). Notice that the reset instants appear actually when the error is zero, r(t) − y(t) = 0, but the corrective effects (discontinuities in u(t) and derivative discontinuities in y(t)) appear in the plant h = 0.15 later, due to the delay between the controller and the plant (see Fig. 4.1). Notice also the damping provided by reset that improves the trade-off between fast rise time and small overshoot.

178

4

Stability of Time-Delay Reset Control Systems

An important question is whether the reset control system might be stable even in the case in which the base LTI system is unstable, in this case for h > 0.2. The proposed method cannot answer this question because the stability of the base LTI system is a necessary condition. A way to overcome this problem could be to develop stability conditions based on the reset-times, as in Sect. 3.2.2. But those ideas should be extended to time-delay, infinite-dimensional reset systems. This is a more involved problem that requires further work. Notice also from the time response in Fig. 4.9 that if we want to avoid the delay of h in the reset corrective effect on the plant output y(t), then we could arrange it by anticipating the reset actions in some way. This anticipative reset can be implemented, for example, by using the concept of reset band introduced in Sect. 3.4. Reset band is proposed in the next chapter as a design improvement and applied in different cases as reported in Chap. 6.

References 1. Åström, K.J.: Limitations on control system performance. Eur. J. Control 6, 2–20 (2000) 2. Bainov, D.D., Simeonov, P.S.: Systems with Impulse Effect: Stability, Theory and Applications. Ellis Horwood, Chichester (1989) 3. Baños, A., Barreiro, A.: Delay-independent stability of reset systems. In: 32nd IECON Conference, Paris, France (2006) 4. Baños, A., Barreiro, A.: Delay-dependent stability of reset control systems. In: Proceedings of the American Control Conference. ACC, New York (2007) 5. Baños, A., Barreiro, A.: Delay-independent stability of reset systems. IEEE Trans. Autom. Control 54(2), 341–346 (2009) 6. Baños, A., Carrasco, J., Barreiro, A.: Reset-times dependent stability of reset control with unstable base systems. In: Proceedings of the IEEE Int. Symp. on Industrial Electronics (ISIE), Vigo, Spain (2007) 7. Barreiro, A., Baños, A.: Delay-dependent stability of reset systems. Automatica 46, 216–221 (2010) 8. Beker, O.: Analysis of reset control systems. Ph.D. Thesis, University of Massachusetts, Amherst (2001) 9. Beker, O., Hollot, C.V., Chait, Y., Han, H.: Fundamental properties of reset control systems. Automatica 40, 905–915 (2004) 10. Carrasco, J., Baños, A., van der Schaft, A.: A passivity based approach to reset control systems stability. Syst. Control Lett. 59, 18–24 (2010) 11. Chen, T., Francis, B.A.: Input-output stability of sampled data systems. IEEE Trans. Autom. Control 36, 1 (1991) 12. Chen, J., Latchman, H.A.: Frequency sweeping tests for stability independent of the delay. IEEE Trans. Autom. Control 40, 1640–1645 (1995) 13. Clegg, J.C.: A nonlinear integrator for servomechanism. Trans. AIEE, Part II 77, 41–42 (1958) 14. De la Sen, M., Luo, N.: A note on the stability of linear time-delay systems with impulsive inputs. IEEE Trans. Circuits Syst. 50(1), 149–152 (2003) 15. Horowitz, I.M., Rosenbaum, P.: Nonlinear design for cost of feedback reduction in systems with large parameter uncertainty. Int. J. Control 24(6), 977–1001 (1975) 16. Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-Delay Systems. Birkhäuser, Boston (2003) 17. Guo, Y., Wang, Y., Zheng, J., Xie, L.: Stability analysis, design and application of reset control systems. In: Proceedings of the IEEE Int. Conf. on Control and Automation, Guangzhou, China (2007)

References

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18. Haddad, W.M., Nersesov, S.G., Chellaboina, V.S.: Energy-based control for hybrid portcontrolled Hamiltonian systems. Automatica 39, 1425–1435 (2003) 19. Khalil, H.J.: Nonlinear Systems, 2nd edn. Prentice Hall, Upper Saddle River (1996) 20. Krishman, K.R., Horowitz, I.M.: Synthesis of a nonlinear feedback system with significant plant-ignorance for prescribed system tolerances. Int. J. Control 19(4), 689–706 (1974) 21. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) 22. Liu, X., Ballinger, G.: Uniform asymptotic stability of impulsive delay differential equations. Comput. Math. Appl. 41, 903–915 (2001) 23. Liu, X., Wang, Q.: Stability of nontrivial solution of delay differential equations with statedependent impulses. Appl. Math. Comput. 174, 271–288 (2006) 24. Liu, X., Shen, X., Zhang, Y., Wang, Q.: Stability criteria for impulsive systems with time delay and unstable system matrices. IEEE Trans. Circuits Syst. 54, 10 (2007) 25. Neši´c, D., Zaccarian, L., Teel, A.R.: Stability properties of reset systems. Automatica 44, 8 (2008) 26. Park, P.: A delay-dependent stability criterion for systems with uncertain time-invariant delays. IEEE Trans. Autom. Control 44, 876–877 (1999) 27. Rantzer, A.: On the Kalman–Yakubovich–Popov lemma. Syst. Control Lett. 28, 7–10 (1996) 28. Yang, T.: Impulsive Control Theory. Lecture Notes in Control and Information Sciences, vol. 272. Springer, Berlin (2001) 29. Zhang, J., Knospe, C.R., Tsiotras, P.: Stability of time-delay systems: equivalence between Lyapunov and scaled small-gain conditions. IEEE Trans. Autom. Control 46, 482–486 (2001)

Chapter 5

Design of Reset Control Systems

5.1 The PI + CI Compensator It is well known that PID (proportional-integral-derivative) compensators are by far the most applied form of feedback in use. As it is noted in [1], more than 90% of all control loops are PID, in fact, most of them are PI since the derivative action in not used very often. It is generally acknowledged that this predominance of PI/PID compensation is due to its simplicity (three parameters to tune), and its effectiveness in a wide range of applications: motor drives, automotive industry, flight control, process control, etc. In this chapter, a new type of reset compensator will be defined using as base system a PI compensator. The idea will be to introduce a new nonlinear/hybrid PI compensator, that will be referred to as PI + CI compensator, having in mind both the simplicity of tuning (only a new parameter, the reset percentage, is added), and the effectiveness from the application point of view. It will be mandatory to obtain simple tuning rules that can improve the performance of PI + CI over PI compensation. Thus, one basic goal is to analyze the potentials of PI + CI compensation, and identify the class of systems in which its performance is superior compared with a well-tuned PI compensator. An important point is that although, in principle, using the derivative term in a PID compensator may offer a better performance than a PI + CI compensator, this has an important price in terms of cost of feedback, even if the D term is filtered. This is the reason why the PI + CI compensator is compared with a PI compensator, they both have a similar cost of feedback and the PI + CI compensator, being a nonlinear/hybrid compensator, can potentially overcome PI compensator fundamental limitations. From the applications side, a basic goal will be to develop PI + CI tuning rules for basic plants including first order plus deadtime (FOPDT) systems, second and higher order plus deadtime systems, and systems with integrators. Related work about experimental demonstrations of reset control has been developed recently. In [22], a FORE compensator is designed and demonstrated for a tape-drive system, exploiting the advantages of reset control to overcome fundamental limitations of the LTI base system. The works [2] and [20] use some form of PI/PID compensation A. Baños, A. Barreiro, Reset Control Systems, Advances in Industrial Control, DOI 10.1007/978-1-4471-2250-0_5, © Springer-Verlag London Limited 2012

181

182

5 Design of Reset Control Systems

Fig. 5.1 PI + CI controller structure

in the base system that, in contrast to PI + CI compensation, performs a full reset of the compensator integrator term. In addition, in [23], a type of reset compensation with reset actions at fixed instants has been developed. The PI + CI compensator is defined by means of a base PI compensator, that will be referred to as PIbase , in which the integral term is partly reset. Then, the starting point is a PIbase compensator that has a transfer function depending on its proportional gain, kp , and its integral time constant, τi , given by   1 . (5.1) PIbase (s) = kp 1 + τi s In the state space, a realization of the PIbase compensator is ⎧ ⎨ x(t) ˙ = e(t), (PIbase ) ⎩ v(t) = kp x(t) + kp e(t). τi

(5.2)

Finally, the PI + CI compensator is defined by adding in parallel a Clegg integrator (CI) to a PI controller (see Fig. 5.1). As a result, the PI + CI compensator will have three terms: a proportional term P , an integral term I, and a Clegg integrator (CI) term. Note that a CI term by itself does not have the characteristic of eliminating the steady-state error in response to step disturbances by itself, thus the integral term I is used for that purpose. The main benefit of using the CI term will be to improve the transient response of the system, being possible in principle to get a better response without excessive overshooting in comparison with its base PIbase . As it will be seen later, if a well-tuned PIbase does not have a closed-loop response with enough overshoot then the compensator PIbase may be detuned to obtain a faster response, and then reset is performed to improve settling time and reduce overshooting. In addition, it is expected that a PI + CI compensator will obtain better performance in comparison with a well-tuned PI compensator in some cases. It may be argued that the comparison should be also done with a PID compensator, but it should be noted that in general a PID compensation would have a significatively higher cost of feedback (sensitivity with respect to sensor noise) even in the case

5.1 The PI + CI Compensator

183

in which the term D is filtered, thus for a fair comparison only a PI compensator is considered. The block diagram structure of a PI + CI controller is shown in Fig. 5.1, where the input e is the error signal, the output v is the control signal, kp is the proportional gain, and τi is the integral time. The new parameter preset , the reset ratio, is a dimensionless constant with values preset ∈ [0, 1] that accounts for the relative weight of the CI term over the I term. Note that a PI + CI compensator is reduced to a PI compensator if reset is eliminated, that is, preset = 0. And on the other hand, a P + CI compensator would be obtained if preset = 1. However, as it was discussed above, in general the reset should not be applied on the whole of the integral term because the fundamental asymptotic property of the integral term would be lost, for example, the steady-state error in the closed-loop step response could not disappear. In the state space, a PI + CI compensator can be expressed by using two states, one corresponding to the I term and the other corresponding to the CI term. Thus the state will be xr = (xi , xir ) , where xi will correspond to the I-term state and xir to the CI-term state. A PI + CI state space model is given simply by (4.2), where the state space matrices are given by   0 0 Ar = , (5.3) 0 0   1 , (5.4) Br = 1  kp  1 − preset preset , Cr = (5.5) τi D r = kp , and the matrix Aρ ∈ R2×2 selects the state to be reset, xir , and is given by   1 0 . Aρ = 0 0

(5.6)

(5.7)

Alternatively, a (more general) PI + CI compensator can be realized by using one state, but then the reset matrix Aρ , in this case is a scalar, will be a nonzero number α, and then the matrix AR will be a diagonal matrix with ones corresponding to the plant states and α = 0 corresponding to the PI + CI state. A realization of the PI + CI with one state is: ⎧ x˙pi+ci (t) = e(t) if e(t) = 0, ⎪ ⎪ ⎨ xpi+ci (t + ) = (1 − rk )xpi+ci (t) if e(t) = 0, (5.8) ⎪ ⎪ ⎩ kp v(t) = τi xpi+ci (t) + kp e(t), where the previous model is obtained in the particular case of rk =

preset xir (tk ) , (1 − preset )xi (tk ) + preset xir (tk )

184

5 Design of Reset Control Systems

with tk , k = 1, 2, . . . , being the reset instants, and xi and xir the states of the integrator and the Clegg integrator, respectively, in the two states model. In general, several different values of rk may be used. As a result, this representation gives a particular type of reset control system with variable reset that need a specific treatment ([17] is an initial work in this direction). On the other hand, as it is well known, the describing function gives good results in practice if some low-pass filtering is assumed in the plant. A very important characteristic of the CI term is that its describing function, previously given by (3.89), does not depend on the input amplitude, only on the frequency. This property is also inherited by the PI + CI compensator whose describing function is simply given by   1 − preset preset 1.62 −j 38.1◦ e (PI + CI)(j ω) = kp 1 + + j ωτi τi ω   j (ωτi + π4 preset ) + 1 . (5.9) = kp j ωτi Note that the PI + CI describing function reduces to the PI frequency response function whenever reset is eliminated, that is, preset = 0. In Fig. 5.2, the describing function of a PI + CI compensator is shown for several values of the reset ratio preset . Note that the reset action may introduce a phase lead of up to 50 degrees and a relatively small increment of the magnitude for frequencies lower than 1/τi . This is a basic property of the PI + CI compensator that, in comparison with its base PI compensator, allows achieving both a bigger phase margin and a crossover gain frequency. This means that a better performance both in terms of speed of response and relative stability may be potentially obtained by means of PI + CI compensation, overcoming fundamental limitations of PI compensation. In addition, the simple form of the describing function makes it very appropriate from a design perspective, and in fact it will serve as a basis of the tuning rules to be developed in next sections.

5.1.1 PI + CI Tuning for First Order Plants The simplest plant case is considered first, a first order plant, to elaborate a PI + CI design procedure based on easy-to-apply tuning rules. The main goal is to exploit the reset action by using the parameter preset , to obtain a significant improvement in terms of closed-loop performance. Basically, a PI compensator is tuned in the first place, and then a proper value of preset is chosen to obtain the desired performance specifications. Obviously, the design of the PI compensator has to be done in such a way that its performance can be improved by partly resetting its state. As it will be seen below, a fast response with significant overshoot will be usually chosen for the base PI compensation. Afterwards, CI will be used to reduce overshooting without sacrificing speed of response.

5.1 The PI + CI Compensator

185

Fig. 5.2 PI + CI describing function for several values of preset (kp = τi = 1)

Consider the standard feedback reset system of Fig. 2.1, where the reset controller R is a PI + CI compensator, and now the plant P is described by

(P )

x˙p (t) = − τ1 xp (t) + τk u(t),

xp (t0 ) = xp0 ,

y(t) = xp (t),

(5.10)

where xp is the plant state and xp0 the plant initial state. In this case, the time constant τ and the gain k are scalars. After some simple computation, the closedloop system (2.4) is given by the matrices ⎛ 1+kp k − τ A = ⎝ −1 −1

kp k τ τi (1 − preset )

0 0

0 0

  C= 100 , ⎛ 1 AR = ⎝0 0

⎞ kp k τ τi preset

0 1 0

⎞ 0 0⎠ . 0

⎠,

(5.11)

(5.12)

(5.13)

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5 Design of Reset Control Systems

5.1.1.1 Autonomous Closed-Loop System Using (2.4), the time evolution of the autonomous closed-loop state is given simply by x(t) = eA(t−tk ) x(tk )

(5.14)

for t ∈ (tk , tk+1 ], where the state is now x = (xp , xi , xir ) , and A is given by (5.11). In addition, the closed-loop output y is given by y = Cx = (1, 0, 0)x = xp , and by definition after each reset time both the plant state xp (t) and the reset integral state xir (t) are zero. Thus, taking a state after reset x = (0, xik , 0) at t = tk+ , it can be shown after some computation that the output y(t) in the interval (tk , tk+1 ] is given by y(t) =

α 2 + β 2 −α(t−tk ) sin(β(t − tk ))(1 − preset )xik , e β

(5.15)

where −α ± jβ are the nonzero eigenvalues of the matrix A (4.2), with α and β being respectively α= and

1 + kp k 2τ

   kp k  . β = + α 2 − τ τi 

(5.16)

(5.17)

k k

Note that α 2 < τpτi is assumed, otherwise the output y would not be oscillatory. Using (5.15), the reset instants tk+1 , k = 1, 2, . . . , are given by 0 = y(tk+1 ) =

α 2 + β 2 −α(tk+1 −tk ) e sin(β(tk+1 − tk ))(1 − preset )xik , β

(5.18)

and then it is not difficult to show that reset instants {t1 , t2 , . . . } are periodic with fundamental period Δreset given by Δreset =

π . β

(5.19)

On the other hand, the integral term state, xi (t), is not necessarily zero after the reset instants. Its time evolution for t ∈ (tk , tk+1 ], k = 1, 2, . . . , is given by       1 − preset −α(t−tk )  xi (t) = preset + α sin β(t − tk ) + β cos β(t − tk ) xik , e β (5.20) and thus the values of xi (t) at the reset instants are simply given by  − απ  xi,k+1 = x(tk+1 ) = preset − (1 − preset )e β xik .

(5.21)

5.1 The PI + CI Compensator

187

5.1.1.2 Closed-Loop Stability As it has been shown above, the closed-loop system consisting of a PI + CI compensator and a first order plant performs reset actions periodically, with a period given by (5.19). This fact considerably simplifies the closed-loop stability analysis since it can be shown that a necessary and sufficient condition for internal stability of the closed-loop system (see [5] and [21]) is simply that the matrix AR eAΔreset be Schur stable, that is, that its eigenvalues be strictly inside the unit circle. In this case, the matrix AR eAΔreset can be computed in closed form with some relative effort (details are omitted here for brevity). The result is: ⎞ ⎛ − απ 0 0 −e β ⎟ ⎜ − απ (5.22) AR eAΔreset = ⎝ 0 1 − (1 − preset )(1 + e β ) ∗ ⎠ , 0 0 0 where “∗” stands for a nonzero term. Note that the nonzero eigenvalues of − απ AR eAΔreset are λ1 = −e β and λ2 = 1 − (1 − preset )(1 − λ1 ). Thus, AR eAΔreset is Schur if and only if the nonzero eigenvalues of A are strictly in the left half-plane (and thus a minimal realization of the base system is Hurwitz) independently of the reset ratio preset , since |λ1 | < 1 if and only if α > 0, and λ2 ∈ [λ1 , 1) for values preset ∈ [0, 1). Thus, the closed-loop reset system is stable if and only if the closed-loop base linear system is stable, which is a very convenient result. Unfortunately, this result is no longer valid for high order plants and plants with deadtime, and some of the stability results given in Chaps. 3 and 4 could be used.

5.1.1.3 PI + CI Tuning Rules for First Order Plants Following the goal of designing a PI + CI controller to improve the performance of a previously designed PI controller, the additional degree of freedom given by the parameter preset of the Clegg integrator will be used to reduce the overshooting of the response. In this simple case, overshooting can be eliminated after the second reset time if the integral state is forced to be zero at t2 , the second reset time, reaching the closed-loop system steady state at that instant. Therefore, making xi (t2 ) = 0 in (5.21), the result is preset =

e

− απ β

1+e

− απ β

,

(5.23)

which will relate the reset ratio, preset , with the integral time constant of the PI controller, τi , and its proportional gain, kp , through the parameters α and β. Note that in general the integral term state cannot be forced to be zero at the first reset instant t1 since (5.21) is only valid when xik , k = 1, 2, . . . , is an after-reset state. It would be only possible if the initial state were an after-reset state, but this is a

188

5 Design of Reset Control Systems

trivial case since it corresponds to a zero initial state, and thus the (autonomous) reset system state would not depart from the origin. For a closed-loop step response that reaches the steady state at the second crossing, the tuning rules of the PI + CI controller for a first order plant can be summarized as: 1. Design the PIbase compensator to obtain kp and τi for a desired oscillatory system 1+kk kk response (( 2τ p )2 < τ τpi ) by using any tuning rules. 2. Compute α and β by using (5.16) and (5.17). 3. Finally, calculate the reset ratio, preset , by using (5.23).

5.1.1.4 Example Consider the following plant with a unit step as a reference signal:

x˙p (t) = −0.5xp (t) + 1.5u(t), (P ) y(t) = xp (t).

(5.24)

The following tuning rules are applied: 1. The chosen constant values for PIbase are kp = 2 and τi = 0.15. This gives a fast response with significant overshoot (see Fig. 5.3). 2. From (5.16) and (5.17), α = 1.75 and β = 4.12. 3. Finally, by using (5.23), the reset ratio is preset = 0.21 in order to reach the steady-state just after the second reset time. As a result, overshoot and undershoot of the PI response are decreased or even eliminated, without sacrificing the speed of response. Figure 5.3 shows step responses and control signals for both compensators PIbase and PI + CI, respectively. Note how responses are identical up to the first reset instant, and that the undershooting/overshooting is eliminated by PI + CI compensation after the second reset instant; in fact, the steady-state is reached at that instant. This is impossible to achieve by LTI compensation. In addition, it can be seen how this improvement can be done by simply resetting the control signal twice (with a slight reset of the control signal, the reset ratio is 21%). Finally, note that not only is the settling time considerably reduced, but also the first overshoot is decreased.

5.1.2 First Order Plus Deadtime (FOPDT) Systems The focus of this section is on the use of PI + CI compensation for first order plus deadtime systems. In this case, the presence of delays makes it impossible to obtain explicit solutions to state space equations. Thus, a state-space approach similar to that in the above section is problematical.

Fig. 5.3 System responses and control signals for PI and PI + CI controllers

5.1 The PI + CI Compensator 189

190

5 Design of Reset Control Systems

Internal model control (IMC) will be used to tune the PIbase compensator. The IMC method consists of considering a known model of the system in order to get a desired closed-loop response. For that purpose, the compensator parameters will depend on the system parameters, and on the time constant of the desired closedloop response, λ. Specifically, when the system is modeled as an FOPDT system P (s) =

k e−hs , τs + 1

(5.25)

the PI compensator is tuned by using the following rules (see [15]): kp =

τ , k(λ + h)

τi = τ.

(5.26) (5.27)

The value of the desired closed-loop time constant λ can be chosen freely, but from (5.27) it has to satisfy −h < λ < ∞ to get a positive and nonzero controller gain. In general, the optimal value of λ is determined by a trade-off between: 1. Fast response and good disturbance rejection (small value of λ), and 2. Stability and robustness (large value of λ). Among the different ways of choosing the value of λ that have been studied in the literature (see [12–14]), the one given by S. Skogestad [14] will be used in the following. Thus, the value λ is chosen to be equal to the time delay λ = h. This gives a reasonably fast response with good robustness margins. In this way, the PI proportional gain (5.26) is given by kp =

τ . 2kh

(5.28)

In the following, this method will be used as a basis to tune the PI + CI compensator, but the tuning procedure will depend on whether the delay is dominant or not.

5.1.2.1 Lag Dominant Systems A lag dominant system is a system in which its time delay is much smaller than its time constant, τ  h, and thus time delay is not dominant over the system response. In this case, tuning the integral time τi by using (5.27) gives a base compensator with a small integral term, and thus it can be expected that the reset action does not improve the linear response. On the other hand, this choice results in a long settling time for input load disturbances. To improve the load disturbance response, a modification of the IMC method proposed by Skogestad will be used here, where τi is tuned by the rule [14]: τi = 8h.

(5.29)

5.1 The PI + CI Compensator

191

Table 5.1 Performance indexes Reference IAE (s)

Disturbance ITAE

(s2 )

Stability margins

IAE (s)

ITAE

(s2 )

ϕm (°)

Am (dB)

PI-IMC

2.20

5.05

5.85

237.45

61.3

9.92

PI-SIMC

3.09

14.07

2.33

67.2

52.5

9.64

PI + CI-SIMC

2.30

6.47

2.33

67.2

51

9.19

This expression, together with (5.28), is used in the Skogestad modification of internal model control method (it will be referred to as SIMC) to tune a PI controller. This rule produces a higher value of the integral term, and thus it may be expected that reset can have a bigger influence. As an example, consider the following system: P (s) =

1 e−s . 20s + 1

(5.30)

Direct application of (5.26) and (5.27) gives a PI-IMC compensator with parameters kp = 10 and τi = 20 s. On the other hand, by applying the SIMC method, a PISIMC compensator is obtained with the same proportional gain, kp = 10, but with a smaller integral time constant, τi = 8 s. Finally, these parameters will be used to tune the PI + CI controller, PI + CI-SIMC, with reset ratio value preset = 0.55. In Fig. 5.4, closed-loop responses and control signals are compared for the PIIMC, PI-SIMC, and PI + CI-SIMC compensators. It can be seen that the PI-SIMC gives a disturbance response faster than the one given by the PI-IMC controller. On the contrary, when the reference signal is changed, the PI-SIMC controller increases the overshoot in comparison with the PI controller tuned by IMC, making its transitory response longer. The PI + CI controller will be used to reduce this transitory response by decreasing the overshoot, and without modifying its disturbance response, as it is shown in Fig. 5.4. As a result, it can be concluded that the PI + CI compensator will give a good trade-off between reference tracking and disturbance rejection, without increasing the cost of feedback (as a general rule, the cost of feedback for a PI + CI compensator is similar to the cost of its base compensator). In order to get a better comparison between the different compensators, the values of performance indexes such as the integral of absolute error (IAE) and the integral time absolute error (ITAE) are presented in Table 5.1. From these indexes, it can be concluded that the PI + CI controller makes the disturbance response faster, and therefore much better, in comparison with the PIMC (IAEPI+CI-SIMC = 2.33 s versus IAEPI-IMC = 5.85 s), just by making the reference response slightly worse (IAEPI+CI-SIMC = 2.30 s versus IAEPI-IMC = 2.20 s). In addition, the robustness indexes, phase and gain margins, are also computed in Table 5.1 (by using the describing function in the case of PI + CI), and it can be expected that the robustness properties of PI + CI are close to the values obtained by its base compensator. As a result, for FOPDT systems with dominant lag the PI + CI compensator gives a good balance between disturbance rejection and reference tracking without

Fig. 5.4 System responses and control signals for PI-IMC, PI-SIMC, and PI + CI-SIMC controllers

192 5 Design of Reset Control Systems

5.1 The PI + CI Compensator

193

sacrificing robustness. After extensive simulation, it has been found that a value of preset ∈ [0.3, 0.6] gives a good response having better rejection/worse reference tracking characteristic for increasing values of preset . Rules for tuning a PI + CI compensator for a lag dominant system are now summarized: 1. Tune the base PI controller, PIbase , by using kp =

τ 2kh

and τi = 8h.

2. Choose the reset ratio preset in the interval [0.3, 0.6], balancing between performance and robustness.

5.1.2.2 Time Delay Dominant Systems A system is said to be time-delay dominant when its time constant τ is at least of the same order as its delay h, usually h ∈ [0.5, 1]τ . An example of this kind of system is the following model: P (s) =

1 e−s . 2s + 1

(5.31)

For this system, a PI-IMC compensator is tuned by using the IMC method (see (5.27)–(5.28)). The following parameters are obtained: kp = 1 and τi = 2 s. At first, a PI + CI-IMC is tuned by using as base compensator the PI-IMC and a small value of preset , for example, preset = 0.1. In Fig. 5.5, the closed-loop responses and the control signals are shown. It can be seen that the reset action makes the PI + CI behavior worse for a reference change since overshoot does not decrease and undershoot increases considerably. On the other hand, when an input load disturbance is considered, the PI + CI controller behaves in the same way as the PI one since the response does not cross the reference signal, and thus there are no resets. In general, reset action does not improve the linear response due to the fact that the base PI compensator, tuned by the IMC method, gives a response with a low overshoot; in such a way, the control signal reaches the steady state very fast and with practically no oscillations. Then, when the reset is performed, the control signal is close to its steady-state value and any reset ratio, even small valued, makes the control signal move away from its steady state-value giving a worse response. To solve this problem, a detuning of the PIbase compensator is performed. This detuning consists of making the closed-loop response faster, and thus more oscillatory, by fixing a smaller value of the time constant of the desired closed-loop response λ. Note that the λ value had been fixed previously to be equal to the time delay, h. Thus, in order to make the response faster, now λ will be fixed to be two thirds of the time delay, that is, λ = 23 h. With this new λ value, the integral time constant of the base PI controller (5.27) does not vary, but the proportional gain (5.28) does. By replacing λ = 23 h in (5.26),

Fig. 5.5 System responses and control signals for PI-IMC and PI + CI-IMC controllers

194 5 Design of Reset Control Systems

5.1 The PI + CI Compensator

195

Table 5.2 Performance indexes Reference

Disturbance

IAE (s)

ITAE

PI-IMC

2.17

PI-des

2.10

PI + CI-des

2.15

(s2 )

Stability margins

IAE (s)

ITAE

5.05

2.00

5.01

1.67

5.03

1.67

(s2 )

ϕm (°)

Am (dB)

47.89

61.3

9.94

39.46

55.6

8.36

39.46

56

8.23

this new value of the proportional gain is expressed as kp =

3τ . 2kh

(5.32)

With this modification, the detuned PI (PI-des) base parameters are now kp = 1.2 and τi = 2 s. In addition, the detuned compensator is the base for the reset compensator PI + CI-des, where a small reset ratio is also considered, for example, preset = 0.1. In Fig. 5.6, closed-loop responses and the control signals are compared for the PI-IMC, PI-des, and PI + CI-des compensators. In addition, for a better comparison, the values of the performance indexes are presented in Table 5.2. It can be seen that PI-des, in spite of having bigger overshoots and undershoots, by sacrificing some robustness, behaves better than the PI-IMC since, when the reference signal changes, the system response is faster, and smaller IAE and ITAE values are obtained. When input disturbances are considered, it is seen that PI-des also makes the system response better, by obtaining a faster response with lower performance indexes. However, comparing PI + CI with its base compensator PI-des, it is deduced that the PI + CI response is slightly worse with similar robustness values. Thus, it has to be concluded that without any modification PI + CI compensation is not appropriate for FOPDT systems with dominant delay. This worse behavior of the reset system is due to the detrimental effect of the dominant time delay over the reset action. In the next section, we will discuss how the use of a reset band can improve this behavior.

5.1.3 High Order Systems In order to be able to get tuning rules for more general systems, a simple approach is to approximate them by an FOPDT model. For this purpose, in this subsection the approximation given in [14] is used. This approximation was developed by using the simple half-rule which considers that the largest neglected (denominator) time constant (lag) is distributed evenly to the effective delay and the smallest retained time constant. It will be briefly explained in the following for the sake of completeness. A system P (s) given by  j (Tj 0 s + 1) −h0 s P (s) =  , (5.33) e i (τi0 s + 1)

Fig. 5.6 System responses and control signals for PI-IMC, PI-des and PI + CI-des controllers

196 5 Design of Reset Control Systems

5.1 The PI + CI Compensator

197

where τi0 > τi+1,0 , and Tj 0 > Tj +1,0 is approximated by an FOPDT system P˜ (s) =

k e−hs , τs + 1

(5.34)

where the model time constant τ is calculated with the following expression: τ = τ10 +

τ20 , 2

(5.35)

and the effective time delay h is computed by using h = h0 +

τ20  τi0 . + 2

(5.36)

i3

Finally, if the general system (5.33) has zeros, the numerator term (T0 s + 1) is canceled by a neighboring denominator term (τ0 s + 1), where both T0 and τ0 are positive and real, using the following approximations ⎧ T0 ⎪ if T0  τ0  h, ⎪ τ0 ⎪ ⎪ ⎪ ⎪ T0 T0 s + 1 ⎨ h if T0  h  τ0 , (5.37) τ0 s + 1 ⎪ 1 if h  T0  τ0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T0 if τ  T  5h. τ0

0

0

If there is more than one positive numerator time constant, then one T0 should be approximated at a time, starting with the largest T0 . Once the general system is approximated to an FOPDT model, the tuning rules developed in the above section can be directly applied in order to tune the PI + CI compensator. For example, consider the following system: P (s) =

(15s + 1) e−s . (25s + 1)(20s + 1)(s + 1)

(5.38)

Firstly, its effective time delay is computed by using (5.36), and the value h = 1.5 s is obtained. Secondly, the zero (15s + 1) is canceled against the neighboring pole (20s + 1) with (5.37). Since the condition τ0  T0  5h is fulfilled, the zero and pole are approximated as: 15s + 1 15 ≈ = 0.75. 20s + 1 20

(5.39)

Finally, the approximated time constant is calculated with (5.35), so τ = 25.5 s is obtained. Therefore, the system (5.38) is approximated by the following FOPDT model: 0.75 (5.40) e−1.5s . P˜ (s) = 25.5s + 1

198

5 Design of Reset Control Systems

Table 5.3 Performance indexes Reference

Disturbance

IAE (s)

ITAE

PI-IMC

4.52

PI-des

5.55

PI + CI-des

4.81

(s2 )

Stability margins

IAE (s)

ITAE

25.42

6.29

38.00

3.18

26.70

3.18

(s2 )

ϕm (°)

Am (dB)

457.98

50.6

10.31

200.30

42.6

9.77

200.30

40.8

9.25

From this model it is seen that its time constant, τ = 25.5 s, is much bigger than its time delay, h = 1.5 s, thus the model (5.40) is a lag dominant system. As a result, it may be expected that reset compensation will be effective in this case to improve performance without sacrificing robustness and cost of feedback. Firstly, a base PISIMC compensator is tuned by using (5.28) and (5.29); the result is kp = 11.3 and τi = 12 s. In addition, for the PI + CI-SIMC compensator, an initial value of the reset ratio preset = 0.55 is chosen. For comparison purposes, a PI-IMC compensator will be also used. Using (5.27) and (5.28), the parameters are kp = 11.3 and τi = 25.5 s. In Fig. 5.7, closed-loop responses and control signals are shown for PI-IMC, PI-SIMC, and PI + CI-SIMC compensators. Both PI-SIMC and PI + CI-SIMC give a faster response than PIIMC, overall when input disturbances are considered. However, when the reference signal changes, the system responses with PI-SIMC and PI + CI-SIMC exhibit more overshoot, and therefore worse performance indexes, as it can be seen in Table 5.3. When PI + CI is used, the PI-SIMC overshoot is reduced and better performance indexes for references are obtained. In spite of this overshoot reduction, the PI + CI response is slightly worse than the PI-IMC response when reference changes are considered (IAEPI+CI-SIMC = 4.81 s versus IAEPI-IMC = 4.52 s), but it is much better when input disturbances are considered (IAEPI+CI-SIMC = 3.18 s versus IAEPI-IMC = 6.29 s). As a conclusion, and without surprise, it can be said that PI + CI compensation gives the best balance between performance and robustness for higher order systems with deadtime that can be well approximated by FOPDT systems with dominant lag.

5.1.4 Integrating Systems Integrating systems, that is, systems with an integrator, are known to be systems for which reset control may give significant improvement over linear control. This property, already mentioned in the seminal work of Clegg [7], has been explicitly shown, for example, in [6]. In the following, it will be shown how PI + CI compensation can beat a well tuned PI compensator for this type of systems, and in addition, tuning rules will be given. In the previous systems, the reset ratio could never be fixed at its maximum value since this value would make the system response oscillate indefinitely and the steady-state would not be reached. In the case in which the system has one

Fig. 5.7 System responses and control signals for PI-IMC, PI-SIMC, and PI + CI-SIMC controllers

5.1 The PI + CI Compensator 199

200

5 Design of Reset Control Systems

Table 5.4 Performance indexes Reference

Disturbance

IAE (s)

ITAE

PI

6.43

PI + CI

4.17

(s2 )

Stability margins

IAE (s)

ITAE

62.80

4.48

20.44

4.48

(s2 )

ϕm (°)

Am (dB)

328.3

48.6

24.28

328.3

48.4

23.9

integrator, this is no longer true, and thus the reset ratio can be set at its maximum value preset = 1. An FOPDT system can be approximated to an integrating system when its time constant is really large, τ → ∞: P (s) =

k k k e−hs e−hs = e−hs . τs + 1 τs s

(5.41)

Obviously, this integrating system behaves as a lag dominant system, τ  h, in such a way that the PI + CI controller is tuned by using the tuning rules developed for those systems. However, in order to be able to apply these tuning rules, the proportional gain has to be modified by using k = τk . As a result, PI + CI tuning rules for integrating systems like (5.41) are simply: 1. Tune the base PI controller, PIbase , by using kp =

τ 2kh

and τi = 8h.

2. Set the reset ratio to its maximum value, preset = 1. The following example is considered to demonstrate the effectiveness of these tuning rules: P (s) =

(0.17s + 1)2 , s(s + 1)2 (0.028s + 1)

(5.42)

which can be approximated with the following model: 1 P˜ (s) = e−1.69s . s

(5.43)

With this model, the PI + CI compensator is tuned by using the tuning rules given above. Firstly, the PIbase parameters are computed, and the following values are obtained: kp = 0.3 and τi = 13.5 s. Obviously, these parameters are used in the PI + CI compensator together with a reset ratio, in this case, preset = 1. In Fig. 5.8, the closed-loop responses and control signals are drawn for the PI and PI + CI compensators (with the plant given by (5.41)). In addition, the values of the performance indexes and stability margins are shown in Table 5.4. From Fig. 5.8 and Table 5.4, it can be deduced that the reset action improves considerably the linear system behavior since the PI + CI controller not only gives a response with less overshoot but also reduces the settling time without modifying

Fig. 5.8 System responses and control signals for PI and PI + CI controllers

5.1 The PI + CI Compensator 201

202

5 Design of Reset Control Systems

the response speed. On the other hand, when input disturbances are considered, the reset control system behaves in the same way as the linear one since the response does not cross the reference and thus the reset action does not occur. It is worthy to note that in this case the stability margins, see Table 5.4, are hardly modified with the reset action. Therefore, it can be concluded that the PI + CI controller makes the response much better without modifying its robustness properties.

5.1.5 Summary of Tuning Rules The PI + CI reset compensator simply consists of a parallel combination of a PI compensator and a Clegg integrator (CI). Both terms may be combined in such a way that the PI + CI behaves like a PI base compensator until the first reset action, thus tuning a proper base PI compensator is a natural way to tune the PI + CI compensator, that will only need an extra parameter preset . In principle, PI + CI compensation may have advantages for controlling systems usually found in a wide range of applications. In particular, it has been found that PI + CI compensation gives significant improvements in first order plus deadtime (FOPDT) systems with dominant lag, and also in integrating systems. On the other hand, higher order systems that can be well approximated by lag dominant FOPDT and integrating systems can also benefit from PI + CI compensation. Tuning rules have been developed that basically consist of using a previously tuned PI compensator and resetting some percentage of the integral term. In general, this partial reset of the integral term results in an improvement on the closed-loop transient response, reducing the overshoot percentage and overall the settling time corresponding to the design without reset, and without degrading the speed of response as well as robustness and cost of feedback. On the one hand, the first order case has been developed by using directly the state-space equations, and the tuning of the PI + CI compensator results in a step response that reaches the steady-state in a finite time (exactly at the second reset time). A summary of the tuning rules may be found in Table 5.5. Note that, without modification, PI + CI compensation is not recommended for systems with dominant delays. In this case, there are several modifications that will be introduced in next section, including the use of a reset band (with fixed or variable size) and a variable reset ratio.

5.2 Design Improvements In practice, a number of simple modifications of the original reset compensator definition have been found useful in improving performance and robustness of the closed-loop system. In this section, several modifications will be analyzed, including both changes over the reset law that regulates the instants at which reset actions are performed and in the compensator design.

5.2 Design Improvements

203

Table 5.5 Tuning rules System

kp

First order without delay

< τ τpi  1+kk kk α = 2τ p , β = τ τpi − α 2

preset =

First order with dominant lag

τ 2kh

8h

[0.3, 0.6]

First order with dominant delay

3τ 2kh

τ

–

(

Second and higher order Integrating systems

τi 1+kkp 2 2τ )

kk

preset e

− απ β

1+e

− απ β

Approximate by a first order plus time delay 1 2kh

8h

1

Regarding the reset law, a first simple modification will consist of using a reset band, as introduced in Sect. 3.4.2, that is, reset actions are produced when the error signal is entering a predefined reset band. In a second stage, this reset band is considered to be time varying depending on the error signal and its time derivative; in this way, reset actions may be more effective, for example, if different setpoints are defined in the control problem, and moreover, not only in the transitory but also in the steady state, adapting the bandwidth to the magnitude of the error and its time derivative. In general, these changes over the reset law are specially effective when significant time-delays are present, for example, in applications like process control or teleoperation. On the other hand, to improve robustness especially in the case in which large plant parametric uncertainty exists, a frequency domain robust control technique like Quantitative Feedback Theory (QFT) is useful for guaranteeing some prescribed degree of robustness in stability and performance. Thus, the base linear compensator can be designed using QFT with some robustness specifications. Finally, considering the case of PI + CI design, it has been found useful in some cases to use a time varying reset percentage; in general, this is related with the use of partial reset compensators with a time varying reset dynamics.

5.2.1 Fixed Reset Band A reset compensator with a fixed reset band has been defined in (3.90). For a reset compensator R with input e and output v, the reset action is performed at the instant t when the pair (e(t), e) ˙ is in the set Bδ . In general, the use of a fixed reset band is expected to be a good design choice for control of systems with significative delay h, and with a constant reference r; for example, this is the case of regulating a process around a fixed setpoint r. In this way, the bandwidth δ is directly related with h and r since the goal is to anticipate the crossing for the reset action to be performed. Thus, for a closed-loop step response

204

5 Design of Reset Control Systems

Fig. 5.9 System responses and control signals for PI and PI + CI controllers

with a rise time tr , a good approximate value of δ is simply given by δ 0.8h . ≈ r tr

(5.44)

For example, consider the integrating system given in Sect. 5.1.4. In this case, the rise time has a value tr ≈ 2.5 s and the delay is h = 1.69 s. Thus, the value of the reset band is δ = 0.5. Figures 5.9 and 5.10 show the step response and the control signal for the PI + CI compensator with a reset band of δ = 0.5, respectively. Note that in comparison with the PI + CI compensator, the step response has improved in terms of the overshoot and undershoot. In addition, using the describing function analysis of the Clegg integrator with reset band given in Sect. 3.4.2, it also possible to come up with a value of δ/r for obtaining a maximum phase lead around the frequency 1/ h. In Chap. 3, the describing function of the Clegg integrator with a fixed reset band was computed and given by    j 4 1 − ( Eδ )2 1 δ j sin−1 ( Eδ ) 1+ e DCI (E, ω) = , (5.45) jω π where E is the error amplitude. Thus, the PI + CI describing function is simply given by     δ 2 j 4p reset 1 − ( E ) δ 1 −1 δ ej sin ( E ) . (5.46) 1+ DPI+CI (E, ω) = kp 1 + j ωτi π

5.2 Design Improvements

205

Fig. 5.10 System responses and control signals for PI and PI + CI controllers

For comparison purposes, in Fig. 5.11, the Bode plot is shown for a PI compensator (kp = τi = 1) and for a PI + CI compensator (kp = τi = 1 and preset = 0.50) with several fixed reset bands ( Eδ = 0.25, 0.50, 0.75, and 1.00). It can be seen that the controller phase is increased with the fixed reset band for some values. Specifically, in this case, when preset = 0.50, the higher controller phase is obtained in a range Eδ ∈ [0.25, 0.50]. This phase increment should make the delay effect less influential over the responses, by giving smaller overshoots and undershoots, and hence, smaller IAE and ITAE values.

5.2.2 Variable Reset Band/Advanced Reset The main drawback of a fixed reset band is that the compensator is specifically tailored for a fixed reference (or disturbance) value. In practice, if different levels of references and/or disturbances are present then a variable bandwidth is necessary. For example, in Fig. 5.12, the example considered above in Fig. 5.9 is used with different step levels; note that the PI + CI with fixed band (dashed line) adjusted for a unit step gives poor results for the different step levels at t = 15 and t = 30 s. A solution to this problem is to consider reset compensators with a variable reset band, as given in Sect. 3.4.2.4. Both Figs. 5.12 and 5.13 give a comparison of fixed and variable reset compensators. It turns out that the variable reset band is a more efficient design choice in control practice where different levels of references and/or disturbances may occur.

206

5 Design of Reset Control Systems

Fig. 5.11 Bode plots for PI + CI controller with several fixed reset bands

Fig. 5.12 System responses and control signals for PI and PI + CI controllers

5.2.3 Variable Reset Percentage Until now, the reset percentage in the PI + CI controller has been given by a fixed value, preset , which means that the PI + CI controller will always reset the same amount of the integral term. In the following, the reset percentage is going to be modified at each reset time so that the PI + CI controller improves the system response. This modification consists of making the reset percentage value variable at the reset times through an element

5.2 Design Improvements

207

Fig. 5.13 System responses and control signals for PI and PI + CI controllers

of prediction, such as the rate of change of the error signal. Specifically, the reset percentage will be changed as a function of the error derivative through the tuning of a new parameter, the derivative time constant, τd . In addition, a low-pass filter will be used with the pure derivative term to avoid the amplification of the sensor noise, F (s) =

1 , τf s + 1

(5.47)

where the filter time constant, τf , will be another parameter to be tuned. In this way, the variable reset percentage, pˆ reset , can be given through the following expression: pˆreset (t) = p¯ reset − τd

deF , dt

(5.48)

where p¯ reset is a constant base value of reset percentage and eF is the error signal filtered by using F . As it can be seen, from (5.47)–(5.48), three parameters must be fixed to tune a PI + CI controller with variable reset percentage from a PI + CI one with fixed reset percentage. To choose the values of these parameters, it is important to observe that the variable reset percentage must make the reset action act strongly over the transitory response, but weakly when the system reaches the steady-state. In this way, the base reset percentage, p¯ reset , should have a varying value in order to avoid a high influence on the system response when the steady-state is reached. On the other hand, the reset action should have strong influence over the transitory response, so the derivative time constant, τd , is fixed for that purpose. Finally, the filter time constant, τf , should be tuned to avoid the undesirable noise amplification. In Chap. 6, an application example of the variable reset percentage technique will be developed (see Sect. 6.3)

208

5 Design of Reset Control Systems

5.2.4 Robust Control Design Based on QFT The control of uncertain plants is of considerable importance from both theoretical and practical points of view. Although most feedback control systems provide some robustness with respect to uncertainty, the development of techniques that explicitly consider some uncertainty in the design process is a relatively recent development. One exception to this statement is found in the work of Isaac Horowitz [9]. Since the late 1950s, his work has focused on the control of systems with parametric uncertainty. The linear case was first considered, and later work in the 1970s [10, 11] also considered robust control of some nonlinear and/or time-varying systems with parametric uncertainty. In those works, Horowitz developed a technique for translating a nonlinear and/or time-varying problem to an equivalent linear problem that, at least in some cases, can be solved using his earlier linear robust design techniques. All of these early ideas were then expanded and integrated by Horowitz and coworkers, leading to a set of control design methods including scalar/multivariable, linear/nonlinear, time-invariant/time-varying, single-loop/multiple-loop systems, now referred to as quantitative feedback theory (QFT) [9]. QFT is a robust control technique that is especially well suited for control problems with large plant uncertainty, and that uses frequency domain specifications. A key step in QFT is the mapping of these specifications into allowed regions of the Nichols plane by using plant uncertainty regions referred to as templates. These allowed regions are relative to the open-loop gain function, and their borders are usually referred to as boundaries. The final step is loop shaping and aimed to shape the open-loop function by minimizing its high frequency gain. QFT works in the frequency domain, thus plant models can be derived from transfer functions (usually with parametric uncertainty) or directly by sets of frequency responses. QFT basically consists of several design steps: 1. Computation of templates. A template represents, at a given frequency, the uncertainty of the plant. It is a region of the Nichols plane, with each template point being given by the phase and magnitude of a plants set element. For a set of working frequencies, the first design step consists of computing the templates set. 2. Computation of boundaries. Given (robust) stability and performance specifications, each template generates a boundary. If the nominal open-loop gain avoids the boundaries, one boundary at every working frequency, then closed-loop specifications are satisfied for all the plants considered in the template. 3. Nominal open loop shaping. Once the boundaries are computed, the next design step is to compute (shape) the open-loop gain that fits them in some optimal way. In general, this is a hard computational problem that usually has been solved heuristically. Once the open-loop gain is obtained, the feedback compensator is directly computed. In addition, a precompensator can be added to the feedback structure if tracking specifications are to be satisfied. A detailed description of these design steps may be found in [9]. For the case of reset control, and in particular for PI + CI design, the novelty is simply that the

References

209

open loop gain function is described by L(s) = G(s)P (s), where the frequency response of the feedback compensator is directly given by the describing function of the PI + CI compensator (5.9), that is, G(j ω) = (PI + CI)(j ω). Alternatively, it can be used to design the base compensator, guaranteeing that the response has the desired oscillatory characteristic for the reset to be effective, and in the presence of large plant uncertainty. In Chap. 6, a number of examples and case studies will be developed by using QFT.

References 1. Åström, K.J., Hägglund, T.: The future of PID control. Control Eng. Pract. 9, 1163–1175 (2001) 2. Bakkeheim, J., Smogeli, O.N., Johansen, T.A., Sorensen, A.J.: Improved transient performance by Lyapunov-based reset of PI thruster control in extreme seas. In: Proc. IEEE Conference on Decision and Control, pp. 4052–4057 (2006) 3. Baños, A., Vidal, A.: Definition and tuning of a PI + CI reset controller. In: Proc. European Control Conference, Kos, Greece (2007) 4. Baños, A., Vidal, A.: Design of PI + CI reset compensators for second order plants. In: Proc. IEEE International Symposium on Industrial Electronic, Vigo, Spain (2007) 5. Baños, A., Carrasco, J., Barreiro, A.: Reset-times dependent stability of reset systems. In: Proc. European Control Conference, Kos, Greece (2007) 6. Beker, O., Hollot, C.V., Chait, Y.: Plant with integrator: an example of reset control overcoming limitations of feedback. IEEE Trans. Autom. Control 46(11), 1797–1799 (2001) 7. Clegg, J.C.: A nonlinear integrator for servomechanisms. Trans. AIEE, Part II 77, 41–42 (1958) 8. Fernández, A.F., Barreiro, A., Baños, A., Carrasco, J.: Reset control for passive bilateral teleoperation. IEEE Trans. Ind. Electron. (2010). doi:10.1109/TIE.2010.2077610 9. Horowitz, I.M.: Quantitative Feedback Theory. QFT Press, Boulder (1992) 10. Horowitz, I.M., Rosenbaum, P.: Nonlinear design for cost of feedback reduction in systems with large parameter uncertainty. Int. J. Control 24(6), 977–1001 (1975) 11. Krishman, K.R., Horowitz, I.M.: Synthesis of a nonlinear feedback system with significant plant-ignorance for prescribed system tolerances. Int. J. Control 19(4), 689–706 (1976) 12. Lee, Y., Park, S., Lee, M., Brosilow, C.: PID controller tuning for desired closed-loop responses for SI/SO systems. AIChE J. 44(1), 106–115 (1998) 13. Rivera, D.E., Morari, M., Skogestad, S.: Internal model control 4, PID controller design. Ind. Eng. Chem. Process Des. Dev. 25(1), 252–265 (1986) 14. Skogestad, S.: Simple analytic rules for model reduction and PID controller tuning. J. Process Control 13(4), 291–309 (2003) 15. Smith, C.L., Corripio, A.B., Martin, J.: Controller tuning from simple process models. Instrum. Technol. 22(12), 39–44 (1975) 16. Vidal, A., Baños, A.: QFT-based design of PI + CI reset compensator: applications in process control. In: 16th IEEE Mediterranean Conference on Control and Automation, Ajaccio, France (2008) 17. Vidal, A., Baños, A.: Stability of reset control systems with variable reset: application to PI + CI compensation. In: European Control Conference, Budapest, Hungary (2009) 18. Vidal, A., Baños, A.: Reset compensation applied on industrial heat exchangers. In: 14th IEEE International Conference on Emerging Technologies and Factory Automation, Mallorca, Spain (2009)

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19. Vidal, A., Baños, A., Moreno, J.C., Berenguel, M.: PI + CI compensation with variable reset: application on solar collector fields. In: 34th Annual Conference of the IEEE Industrial Electronics Society, Orlando, Florida, USA (2008) 20. Wu, D., Guo, G., Wang, Y.: Reset Integral-Derivative Control for HDD servo systems. IEEE Trans. Control Syst. Technol. 15(1), 161–167 (2007) 21. Yang, T.: Impulsive Control Theory. Lecture Notes in Control and Information Sciences, vol. 272. Springer, Berlin (2001) 22. Zheng, Y., Chait, Y., Hollot, C.V., Steinbuch, M., Norg, M.: Experimental demonstration of reset control design. Control Eng. Pract. 8(2), 113–120 (2000) 23. Zheng, J., Guo, Y., Fu, M., Wang, Y., Xie, L.: Development of an extended reset controller and its experimental demonstration. IET Control Theory Appl. 2, 866–874 (2008)

Chapter 6

Application Cases

6.1 Temperature Control in a Heat Exchanger Temperature control is a common problem in the process industry, such as power generation, chemical plants, refrigeration, and food processing among others. This difficult temperature control problem arises from the large process uncertainty and the different operation conditions which are present during the whole process. For example, in batch reactors the temperature has to be adjusted to an optimal temperature profile in spite of the drastic changes at setpoints with the goal of obtaining a specific product with desired properties [3, 15, 32]. In other applications, such as thermal treatments in heat exchangers, the outlet temperature has to be maintained at a particular setpoint for a wide variety of products and different operation points in order to destroy harmful microorganisms and optimize food quality factors [5, 23, 33]. Because of these particularities, thermal control is difficult to achieve with conventional PID controllers. Since variable operating conditions occur, the parameters of the PID controller require frequent retuning to adjust them to these different operating conditions. Several works deal with the temperature control problem by using adaptive model predictive control [1–3, 27, 32]. Other works are focused on the use of the Quantitative Feedback Theory (QFT) to tune robust controllers which take into account the uncertainty of the thermal process [9, 16]. QFT is a robust control technique that is especially well suited for control problems with large plant uncertainty (see Sect. 5.2.4 for a brief introduction). A family of linear time invariant (LTI) plants obtained by local linearization may be used to model the heat exchanger nonlinear dynamics around different operation points, and then a compensator can be designed by using QFT [8]. The result is a robust compensator that guarantees a specified worst-case behavior for all the considered operation points. In general, the robustness and performance have to be balanced due to fundamental limitations of the resulting LTI compensator, this being the main motivation for using reset compensation, in which this type of nonlinear/hybrid control is investigated with the goal of improving performance without sacrificing robustness. In Chap. 5, tuning rules of the PI + CI compensator for first and higher order systems with time delay have been developed. In this section, this reset controller A. Baños, A. Barreiro, Reset Control Systems, Advances in Industrial Control, DOI 10.1007/978-1-4471-2250-0_6, © Springer-Verlag London Limited 2012

211

212

6 Application Cases

Fig. 6.1 Pilot plant at the University of Murcia

is tuned by using QFT and applied to the temperature control of an industrial heat exchanger. In Sect. 6.1.1, a dynamic model of the heat exchanger is obtained for different operation points based on experimental data. In Sect. 6.1.2, several PI + CI compensators are designed. First, a PI + CI compensator is tuned by using QFT, and then several design improvements such as the use of a fixed reset band and a variable reset band, are analyzed. The different reset compensators are experimentally tested and compared with a well designed PI compensator.

6.1.1 Process Model A temperature control problem will be studied by using an industrial heat exchanger, commonly used in the food industries for thermal treatments. For this purpose, the temperature process will be modeled through pseudorandom binary sequence (PRBS) experiments and step tests application. PI + CI compensation will be used to fix the outlet temperature at a desired value with minimum performance indexes, such as the integral absolute error (IAE). The heat exchanger used in this study is part of a pilot plant which has been designed for food industry applications, and it is a part of the pilot plant of the Computer and Control Engineering group at the University of Murcia (see Fig. 6.1). It contains two HRS UNICUS double tube heat exchangers, consisting of a 3 meter long tube within another one in parallel flow. That is to say, the product flows through the inner tube in parallel to the serviced fluid which flows through the annulus between the inner and the outer tube. In addition, the shell and tube heat exchanger have scraper bars fitted in each interior tube whose movement mixes the fluid and cleans the heat exchange surface. This also keeps heat transfer high and makes cleaning tasks easier in food industries. The physical details of this heat exchanger are shown in Table 6.1. The product is pumped from a storage tank up to the heat exchanger by an industrial helicoidal impeller pump. Two resistance thermometers used as thermocouples

6.1 Temperature Control in a Heat Exchanger Table 6.1 Some parameters of the industrial heat exchanger

213

Number of inner tubes

1

Shell diameter

104 mm

Inner tube diameter

76.2 mm

Type of flow

parallel

Material

stainless steel

Tubes length

3000 mm

Fig. 6.2 Electropneumatic globe valve

are placed at both the inlet and the outlet of the inner tube of the heat exchanger for temperature measurements. The product is heated in the inner tube from the saturated steam condensation at the shell side. The rate of heat transfer depends on the temperature difference between the cold and hot fluids in such a way that the higher this difference is, the faster the heat is transferred. Therefore, the rate of heat transfer, q in kcal/h, is expressed through the following equation: q = U AΔT ,

(6.1)

where A is the area for heat transfer in m2 , ΔT is the temperature difference in °C and U is the overall heat transfer coefficient, kcal/°C m2 . The saturated steam is generated in an electrical boiler of 12 kW of power which generates 16 kg of steam per hour at a pressure which oscillates between 4.3 and 5 bar. These pressure drops will be taken into account as a perturbation, d, in the control scheme. The control signal will be precisely the steam flow rate, which will be controlled through the aperture (expressed in %) of an electropneumatic globe valve placed just before the shell inlet in the heat exchanger, as seen in Fig. 6.2. The thermocouples and the control valve are connected through a PLC to a data acquisition system, where a SCADA is installed for monitoring and control purposes, as can be seen in Fig. 6.3. The SCADA system reads both the inlet and outlet temperatures, Tin and Tout , and from the temperature difference, ΔT = Tout − Tin ,

214

6 Application Cases

Fig. 6.3 Industrial heat exchanger control

the control system provides the corresponding electropneumatic globe valve aperture, u in %, in order to reach the desired reference temperature. The industrial heat exchanger is modeled by using water as the product with a flow rate of 250 L/h. The identification is done through pseudorandom binary sequence (PRBS) experiments and step test application. On the one hand, the heat exchanger models are obtained by first order approximation describing the response of the temperature along the heat exchanger to several step changes of valve aperture. Specifically, this identification was done by opening the control valve from 10 up to 25% in 5% steps and then by closing the control valve up to 10% by using again 5% steps as can be seen in Fig. 6.4. In addition, the PRBS experiments were also done to model the heat exchanger. These experiments [30] consist of doing different tests by using the control signal

Fig. 6.4 Heat exchanger identification

6.1 Temperature Control in a Heat Exchanger

215

Fig. 6.5 PRBS experiment Table 6.2 Heat exchanger uncertainty Plant Pa –b%

Gain ka –b%

Time constant τa –b% (s)

Time delay ha –b% (s)

P10–15%

0.32

87

88

P15–20%

0.35

82

83

P20–25%

0.40

80

65

P25–20%

0.40

89

50

P20–15%

0.35

95

76

P15–10%

0.31

114

84

as the input, which will have a spectrum with significant components in some regions of interest. Then, the different response data are used for getting an uncertain parametric model. The range [10, 25]% for the control valve aperture is identified as the working range for this study. One example of a PRBS experiment is shown in Fig. 6.5 for a valve aperture between 10 and 15%. With these modeling experiments, the heat exchanger can be represented by a set of first order plus time delay (FOPTD) transfer functions given by Pa−b% (s) =

ka−b% e−ha−b% s . τa−b% s + 1

(6.2)

For each valve aperture change, from a% to b%, a heat exchanger model is obtained, as given in Table 6.2. The heat exchanger uncertainty is expressed in parametric form as ka−b% ∈ [0.31, 0.40], τa−b% ∈ [80, 114], and ha−b% ∈ [50, 88]. Note that every parameter combination is not possible, in fact, the gain is increasing for increasing valve aperture values, while the time lag and the delay are decreasing.

216

6 Application Cases

6.1.2 PI + CI Design In the following, a PI + CI compensator is designed by using the results given in Chap. 5. Different control experiments using the industrial heat exchanger will be carried out with different control systems including a base PI compensator and different reset PI + CI compensators (with/without reset band, . . . ). The experiment will consist of setting a fixed product flow of 250 L/h, as considered in the modeling procedure, and opening the control valve at 10%. When the steady-state temperature is reached at this control valve aperture, the reference temperature is increased and decreased by step changes with a magnitude of 2 °C.

6.1.2.1 PI Compensator Design For comparison purposes, a linear PI controller is (well-)tuned by using the Internal Model Control (IMC) technique as described in Chap. 5. This method consists of considering a known model of the process in order to get a desired closed-loop response. For this purpose, the controller parameters will depend on the system parameters and on the time constant of the desired closed-loop response, λ. Specifically, when the process is modeled as an FOPDT system given by (6.2), the controller takes the structure of the PI controller with the following parameters [38]: kp =

τ , k(λ + h)

τi = τ.

(6.3) (6.4)

The value of the desired closed-loop time constant λ can be chosen freely, but due to (6.3) it has to satisfy −h < λ < ∞ to get a positive and nonzero controller gain. In general, the optimal value of λ should be determined by a trade-off between: 1. Fast response and good disturbance rejection (small value of λ) and 2. Stability and robustness (large value of λ). Between the different ways of fixing the value of λ which have been studied in the literature [31, 35, 36], the one given in [36] is used in this work. So, the value λ is chosen to be equal to the time delay λ = h. This gives a reasonably fast response with good robustness margins. In this way, the PI proportional gain (6.3) is given as kp =

τ . 2kh

(6.5)

For the industrial heat exchanger modeled by (6.2) and with parameters as in Table 6.2, a nominal model has to be considered to tune the PI controller via IMC. In this work, the chosen model is the one in which the control valve is opened from 15% up to 20% P15−20% =

0.35 −83s . e 82s + 1

(6.6)

6.1 Temperature Control in a Heat Exchanger

217

Fig. 6.6 Responses and control signals of PI controller tuned by IMC

For this model, by using (6.4) and (6.5), the following PI parameters are obtained: kp = 1.41 and τi = 82 s. From now on, this controller is going to be named as PIIMC. The experiment explained previously has been carried out with this controller. That is, the control valve is opened at 10% until the steady-state is reached. The difference between the outlet and inlet temperatures, ΔT = Tout − Tin , is taken as 0 °C. Then 2 °C step changes are introduced into the reference temperature from 0 °C up to 6 °C and from this value down to 0 °C again. In Fig. 6.6, the temperature increments in the heat exchanger are shown together with the control signal, that is, the control valve aperture. In both signals, the noise introduced by the thermocouples and the perturbations due to the pressure drops occurring at the electric boiler are visible. Note that the higher the control valve is opened, the more oscillatory the response at the steady-state is. Regarding the control system, it can be seen that the PI-IMC controller is robust for all the reference changes. In addition, fast response is obtained (rise time between 98 and 233 seconds) with a significant overshoot in some cases (between 4.5 and 26%).

6.1.2.2 Reset Compensation Design Here the goal is to design a reset control system that guarantees both more robust and better performance than the PI-IMC compensator for the industrial heat exchanger modeled by (6.2) and Table 6.2. This better performance will consist of reducing the overshoot without decreasing its speed of response. For this purpose, a PI + CI compensator is going to be used. As was explained in Chap. 5, this compensator is designed from a PIbase just by adding a proper reset ratio preset . Since the PIIMC hardly has overshoot, a faster PIbase compensator with more overshoot has to be tuned for the PI + CI to be effective. First, Quantitative Feedback Theory

218

6 Application Cases

(QFT) will be used to design PIbase , by using design specifications about closedloop stability and tracking performance. In a second stage, this PI + CI compensator will be designed choosing a proper reset ratio preset . PIbase Tuning The PIbase compensator will be tuned with QFT by following the design steps (briefly) explained in Chap. 5. First of all, plant templates are computed taking into account parametric uncertainty as given in (6.2) and Table 6.2, with the nominal plant being P15−20% (s) =

0.35 −83s e . 82s + 1

(6.7)

In this case, the templates are obtained for a wide set of working frequencies, from 0.002 up to 0.02 rad/s. Secondly, the boundaries are computed for each frequency from design specifications by using templates. Design specifications will be a minimum (robust) phase margin of 24◦ and desired tracking performance, such as a rise time between 45 and 180 seconds and a maximum overshoot between 7 and 75%. Note that a relatively small phase margin has been specified for the base PI control system. In addition, note also that the response is specified to be faster and with more overshoot than the response obtained with the PI-IMC controller. These design specifications are considered due to the fact that the reset compensation will improve the minimum phase margin and will reduce the overshoot of the PI compensator without reducing the response speed. The next step consists of shaping the nominal open loop. Here a PI controller is tuned by making the open-loop gain fit both the stability and tracking boundaries. A shaping is shown in Fig. 6.7, corresponding to a proportional gain of kp = 1.6 and an integral time constant of τi = 60 s. Finally, once the base PI compensator is tuned, a closed-loop system analysis is performed to ensure that the system fits the given stability and tracking specifications. This is usually referred as to compensator validation in QFT. This analysis has to be done for a wider set of frequencies than the one used in the templates computation, and in this case, this set goes from 0.00001 up to 100 rad/s with a much bigger grid. In Fig. 6.8, the stability analysis is shown, whereas the tracking analysis is presented in Fig. 6.9. As it can be seen, the designed closed-loop system satisfies both stability and tracking specifications for every frequency. Therefore, the base PI compensator design is finished. PI + CI Tuning Once PIbase has been designed, the obtained parameters are chosen as a part of the PI + CI parameters. But in addition, one more parameter, the reset ratio preset , must be tuned. The expected result is an increment of the phase margin, and thus a better transient response for the industrial heat exchanger. Several values of preset are considered: 0.1, 0.2, 0.3, and 0.4. These PIbase and PI + CI compensators are used to control the already described heat exchanger. For this purpose, the same experiment carried out with the PI-IMC

6.1 Temperature Control in a Heat Exchanger

219

Fig. 6.7 Base PI open-loop design

Fig. 6.8 Stability analysis

controller is repeated. Figure 6.10 shows the temperature increment and the valve aperture when the PIbase and PI + CI compensations are used.

220

6 Application Cases

Fig. 6.9 Tracking analysis

Fig. 6.10 Responses and control signals of base PI and PI + CI controllers

As it can be seen, the PIbase and the PI + CI responses are equal until the first reset action is performed. After this reset, the PI + CI response has the same or smaller first overshoot while the other overshoots and undershoots are increased with respect to the base PI system, even for small values of preset . This behavior is compatible with the results given in Chap. 5, where PI + CI is shown to behave worse than its base compensator for delay dominant systems.

6.1 Temperature Control in a Heat Exchanger

221

Table 6.3 Performance indexes for the PI and PI + CI control systems Controller

IAE (s)

ITAE (s2 )

Phase margin (°)

Base PI

3.31 × 103

12.45 × 106

41

PI + CI preset = 0.1

3.75 × 103

14.14 × 106

44

PI + CI preset = 0.2

3.28 × 103

12.09 × 106

46

PI + CI preset = 0.3

3.62 × 103

13.12 × 106

47

PI + CI preset = 0.4

3.87 × 103

14.54 × 106

47

To evaluate the tracking performance, the integral absolute error (IAE) and the integral time absolute error (ITAE) are computed:  ∞ IAE = |e(t)| dt, (6.8) 0

 ITAE =

∞

t|e(t)| dt

(6.9)

0

which should be as small as possible. These performance indexes confirm that the reset action makes the error increase, and therefore the response gets worse. In addition, for comparison of results, phase margins are also computed in Table 6.3 for these compensators. These values show the extra phase lead given by the PI + CI compensator: for the PI controller the phase margin is 41° whereas for the PI + CI reset one the phase margin can be increased up to 47°. The worse response under reset conditions, as shown in Fig. 6.9, is due to the fact that the effect of the reset action takes place after the crossing of the output with zero, due to the existence of dominant time delays in the system. In the next section, a simple modification of the PI + CI compensator will be investigated to significantly improve control system performance.

6.1.2.3 Reset Compensation with Reset Band The reset action is a good choice to improve the transient response of a closed-loop system. But there are systems whose response improvement is not as good as can be expected. This is due to the presence of a dominant time delay. The problem in these dominant time delay systems is that the reset action is done at reset times tk when the error is equal to zero, but the system suffers the reset action at another time, specifically at tk + h, where h is the time delay of the plant. This lack of coordination between the reset times and the times when the system undergoes the reset action can be overcome by modifying the reset condition. The reset condition in the PI + CI controller has been usually given by the fact that the error signal had to be equal to zero. In this section, the reset condition is modified with the goal of doing reset some time before the crossing between the

222

6 Application Cases

Fig. 6.11 Responses and control signals of PI-IMC and PI + CI controllers with a fixed reset band Table 6.4 Performance indexes for the PI and PI + CI control systems with a fixed reset band Controller

IAE (s) 2.59 × 103

9.56 × 106

= 0.6

2.16 × 103

8.17 × 106

= 0.7

2.82 × 103

10.86 × 106

PI-IMC PI + CI preset = 0.10 and PI + CI preset = 0.35 and

δ E δ E

ITAE (s 2 )

error signal and zero. To specify when the reset action must exactly be done, two techniques are considered in this study: a fixed reset band and a variable reset band. Fixed Reset Band The experiment is carried out again, but now for two PI + CI compensators with fixed reset band. In this case, two different values of the reset ratio are set and their corresponding fixed reset bands are computed. In one controller, a low reset ratio is chosen and for this value, preset = 0.1, a fixed reset band has been computed with the goal of having the maximum phase margin in the control system. In this case, the maximum phase margin of 47° has been obtained for a fixed reset band of Eδ = 0.6. On the other hand, for a second PI + CI controller with a fixed reset band, a higher reset ratio value is chosen, preset = 0.35. In this case, the control system reaches the maximum phase margin of 60° when Eδ = 0.7. In Fig. 6.11, the heat exchanger responses and the control signals for the PI + CI compensator and for the PI-IMC compensator are shown. In addition, their performance indexes are compared in Table 6.4. From Fig. 6.11 and Table 6.4, it can be said that the PI + CI compensator with preset = 0.35 and Eδ = 0.7, in spite of having a higher phase margin, has a response that it is not as good as that given by the PI + CI compensator with preset = 0.1 and δ E = 0.6. This last PI + CI controller not only has lower IAE and ITAE values than

6.1 Temperature Control in a Heat Exchanger

223

Table 6.5 Performance indexes for the PI and PI + CI control systems with a variable reset band ITAE (s 2 )

Controller

IAE (s)

PI-IMC

2.59 × 103

9.56 × 106

PI + CI preset = 0.1 and h = 50 s

2.13 × 103

8.09 × 106

PI + CI preset = 0.4 and h = 50 s

2.92 × 103

10.67 × 106

the other PI + CI controller, but also has less overshoots and undershoots in such a way that it gives the better response. In fact, this PI + CI controller response is also much better than the PI-IMC response since it is faster and it has similar overshoots and undershoots with lower performance indexes. However, as it can be seen from the control signals, the control system just undergoes the reset action once in each reference. Therefore, after the first and the only reset instant, the control system behaves as the lineal base PI controller. In order to improve further the system performance, a variable reset band will be considered in the next section. Variable Reset Band Here a variable reset band is considered to overcome the influence of the dominant time delay over the reset action. For this purpose, the approach described in Chap. 5 will be followed. In this case, the reset conditions are also given when the error signal is approaching zero, but now the value of the error when the reset action is done will change constantly from one reset instant to another. For a variable reset band to be applied, only a nominal value of the time delay has to be fixed in the variable reset band. For the industrial heat exchanger studied in this work, the time delay is uncertain, and several values of time delay can be chosen. Theoretically, the reset instants are going to take place h time units before the time the error signal crosses to zero. Therefore, in order not to degrade the response to any reference change, the nominal time delay is taken as the minimum one, that is, h = 50 s. In addition, due to the presence of sensor noise, the derivative part of this reset band will be filtered by a low-pass filter with a constant time 10 times smaller than the nominal time delay. For comparison purposes, the same experiment explained previously is carried out under the same operation conditions. In this case, the PI-IMC controller response is compared to the PI + CI controller with variable reset band and two different reset ratios, preset = 0.1 and preset = 0.4. In Fig. 6.12, both the responses and the control signals are drawn for these three controllers. In addition, the IAE and ITAE values are reported in Table 6.5. From Fig. 6.12 and Table 6.5, it is deduced that the PI + CI controller with a variable reset band and reset ratio equal to preset = 0.1 is the best option to control the temperature increments in an industrial heat exchanger since, in comparison with the PI controller tuned with IMC, a faster response is obtained with a similar overshoot and undershoot. In this case, the PI + CI controller does several resets rather than just one as with the fixed reset band. Therefore, it can be concluded that

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6 Application Cases

Fig. 6.12 Responses and control signals of PI-IMC and PI + CI controllers with a variable reset band

the variable reset band can be used to significantly improve the reset performance and the linear IMC response when dominant time delays are present in the system dynamics.

6.2 Teleoperation of a Gantry Crane This section reports on the application of reset control to a real laboratory teleoperated system consisting of a scale gantry crane teleoperated through the Internet by means of a haptic device. The section is based on the works [21] and [22] where a novel approach to passive teleoperation is presented that considers reset actions appropriately applied. The motivation of these studies is to alleviate the limitations arising from the existence of communication delays between the master and slave sides. Communication delays are problematic for networked controlled systems and in particular for teleoperated systems. They give rise to a trade-off between speed and robustness, which cannot be overcome by means of linear controllers. In order to solve this problem, in this section we report a novel approach that combines passivity-based techniques and reset control principles. In this way, it is possible to obtain simultaneously the robust stability properties of passive control and the performance improvement enabled by reset strategies. The experimental and simulation results presented confirm the good behavior achieved with this method. Bilateral teleoperation is an area that has motivated an enormous amount of research in the last two decades [24]. Modern teleoperation is usually based on applying passivity-based control techniques [39] to teleoperated systems. Using passivity

6.2 Teleoperation of a Gantry Crane

225

tools has a number of advantages: they provide a multi-domain and modular approach to modeling, and transparent control principles based on energy and power concepts. The passivity ideas can be adapted to the teleoperation framework, where a human operator (actuating a master device) manipulates a remote machine (slave device) which is possibly interacting with the environment. If some time delay exists, a communication channel transmitting velocity and force (which provides the operator with a feeling of telepresence that is essential in applications [26, 29]) is no longer passive. As an alternative, wave variables—which are obtained from velocities and forces via the scattering transformation, see [4, 34]—should be transmitted, thus making the delayed line behave like an analog lossless LC line. This is called line-passivation. It turns out that after line-passivation, and with passive controllers at the master and slave sides, the overall teleoperation system becomes passive. Since it is assumed that the external elements—human operator and environment—behave in a passive way, this means that the system is stable for all possible constant values of the time delay. This delay-independent stability property can also be extended to time-varying delays [18], and therefore it is possible to use the Internet as the communication channel, as has already been shown in several applications [37, 44]. Delay-independent stability can be interpreted partly as an advantage and partly as a drawback: on the one hand, it is an advantage because the designer does not have to worry about the delay. On the other hand, it is a drawback because, if the passive controllers are stable for very large delay values, they tend to provide worse performance than other (non-passive) controllers, which are unstable for large delays but stable for small nominal delays. Realizing that the source of the problem is the time-delay, which introduces a linear trade-off between speed and robustness, it becomes clear that using the resetcontrol principle in teleoperation might enable us to combine the robustness of the passive solutions with the fast position tracking performance provided by a proper design of the reset action. The section is organized as follows. Section 6.2.1 provides an overview of the system, and the principles of passive teleoperation. Then, Sect. 6.2.2 shows how the use of reset is capable of overcoming the fundamental limitations of linear controllers, namely when time delays are present. The proposed reset strategy is presented and justified and the stability is proven from passivity properties. Finally, two examples are presented in Sect. 6.2.3: a simulated simple robot and a real application with a laboratory gantry crane teleoperated with a haptic device.

6.2.1 Passive Teleoperation and System Overview The block diagram that describes the overall system is given in Fig. 6.13. It consists of the following subsystems: a master device commanded by an operator, a master impedance controller (represented as the parallel system of a spring Km and a damper Bm ), the scattering transformations corresponding to master and slave sides,

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6 Application Cases

Fig. 6.13 Teleoperation scheme

the communication channel, a slave impedance controller (Ks , Bs ) and a slave device in contact with the environment. The interchanged variables are either power variables (forces fi and velocities x˙i , with i ∈ {h, m, s, e} and with the additional subindex d for delayed variables); or wave variables (um , us , wm , ws ) which result from the scattering transformation of the former. The communication channel contains the time delays which will be considered constant and known throughout this study. This is a frequent assumption when deriving theoretical results for the first time; these may be extended for time-varying delays later. Here we restrict ourselves to constant time delays; only in the last example of Sect. 6.2.3, and as a preliminary experimental validation, we apply the method to unknown, time-varying delays. It is well-known that a time delay provokes a loss of passivity, which can be arranged using the scattering theory [4] or the equivalent wave-variables [34] formulation given by the equations:    √ −1/2   2B −I um f m √ = (6.10) x˙md wm − 2B −1/2 B −1 and



ws x˙sd

√

 =

2B −1/2 −B −1

√ −I−1/2 2B



fs us

 ,

(6.11)

where the matrix B > 0 is the line impedance. The actually transmitted variables are the wave variables: us (t) = um (t − ΔT ),

wm (t) = ws (t − ΔT ),

(6.12)

where ΔT ≥ 0 is the communication delay, unknown but considered constant. This configuration behaves as a lossless system with respect to the storage (Hamiltonian) function   1 t  Hc (t) = um (τ )2 + ws (τ )2 dτ, (6.13) 2 t−ΔT that is, the integral of the power of the waves for the duration of the transmission. It easily follows that −fs (t)x˙sd (t) + fm (t)x˙md (t) = H˙ c (t).

(6.14)

6.2 Teleoperation of a Gantry Crane

227

This relation is true for all ΔT , so, from the point of view of passivity, we do not have to be concerned about its actual value. The interconnection of passive subsystems leads to an overall system that is itself passive [39], and hence stable behavior is guaranteed. As wave reflections can occur at both sides of a teleoperation scheme, there is a need for impedance matching [34], which can be achieved if both sites are placed under velocity control. This is one of the motivations for using the two symmetric “impedance controllers” mentioned before, which are placed at each end of the transmission channel. Thus, each side is receiving force information and providing velocity signals. The master side of the setup consists of a master device, its impedance controller, and the corresponding scattering transformation. The master device interacts with the human operator, allowing him to specify the desired movement of the plant (velocity command), while receiving some force feedback representing information about the operating conditions. Similarly, the slave side consists of the slave device, its impedance controller, and the scattering transformation. The master and slave devices will be considered as mechanical systems whose equations can be written in the port-Hamiltonian notation as: z˙ = [J (z) − R(z)] x˙ = g T (z)

∂H (z) , ∂z

∂H (z) + g(z)f, ∂z

(6.15)

where z are the states, x˙ the output (velocities), f the input (forces), H the Hamiltonian function, and J, R the interconnection and dissipation structures, respectively. Reproducing the previous equations with subindices m, s, we obtain the description of the blocks ‘Master device’ and ‘Slave device’ in Fig. 6.13, additively including the forces fh , fe . Finally, the equations of the impedance controllers can be written as  fm = Km (x˙m − x˙md ) dt + Bm (x˙m − x˙md ),  (6.16) fs = Ks (x˙sd − x˙s ) dt + Bs (x˙sd − x˙s ), which are two Proportional-Integral (PI) controllers with transfer functions   Km fm = + Bm em , em = x˙m − x˙md , s (6.17)   Ks fs = es = x˙sd − x˙s . + Bs es , s In this way, we are not only transmitting energy through the strings Km , Ks , but also dissipating some of it in the dampers Bm , Bs . Consequently, artificial damping is injected, which is a requisite to ensure a passive behavior when it is possible that the plant interacts with the environment. The parameters of this controller are the

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6 Application Cases

stiffness constants of the springs, Km and Ks , and the viscous frictions, Bm and Bs . If the latter are chosen to be equal to the scattering parameter of the transmission line, B, no wave reflections will take place [34].

6.2.2 Reset Procedure In this section, a reset strategy for improving the performance of the impedance controllers is presented. Recall that their transfer functions are given by (6.17), thus defining them as Proportional-Integral (PI) controllers with respect to the velocity error. The proposed reset procedure entails, to begin with, partially resetting the integral value of the master PI, that is, the reset compensator is a PI + CI (see Chap. 5 for a detailed description). Thus, the overall master controller can be then written in the form ⎧ x˙I = x˙CI = x˙ m − x˙md if c = 0, ⎪ ⎪ ⎨  x+  x I I = 0 if c = 0, (6.18) + xCI ⎪  xI ⎪ ⎩ fm = Km [(1 − p) p] xCI + Bm em , with c the condition determining the reset instants (discussed later in (6.24)), (xI , xCI ) the states of the linear and Clegg integrators, respectively, and (1 − p, p) the parameters that multiply the integrator states. The PI + CI controller can be implemented as two parallel integrators (see Fig. 5.1, with kp = Bm , Km = kp /τi ), one of them being a Clegg integrator and the other a normal one. The outputs of these two integrators are multiplied by p and 1 − p respectively, for 0 < p < 1. Thus, the modification of parameter p provides a simple way of tuning the controller. The reason for performing only “partial” reset is that a totally reset integrator loses the fundamental property of eliminating the steady-state error. Since feedback control is performed on the velocity signal, full reset leads to a loss of information about the slave position that cannot be recovered. When this happens, position drift appears, a well-known problem in bilateral teleoperation. However, when partial reset is used, position information is being kept in the linear integrator (the nonreset integrator placed in parallel to the reset one). This can be more formally explained as follows. A steady state is achieved if all the forces and velocities in Fig. 6.13 are zero. According to [17] and references therein, the steady-state tracking-position error, referred to as “position drift”, is zero if:  t Δxm = (x˙m − x˙md ) = 0, (6.19) 0

where t is some time at steady-state. Let us assume that the error is initially zero, that is, Δxm (0) = 0. Now consider the master impedance controller in Fig. 6.13 replaced

6.2 Teleoperation of a Gantry Crane

by a PI + CI as in Fig. 5.1, with the equivalence kp = Bm , τi = xir = xCI . The general equation for 0 < p < 1 is

229 Bm Km ,

xi = xI , and

y = fm = Bm (x˙m − x˙md ) + Km (xI (1 − p) + xCI p), (6.20)

t where xI = 0 (x˙m − x˙md ) dt and xCI = tk (x˙m − x˙md ) dt, with tk being the last reset time. Now, when in steady state, the controller has zero input (velocities) and zero output (force). The problem is whether there may exist nonzero steady-state values xI , xCI compatible with zero input and output. We will discuss three possible cases: PI, CI, and PI + CI. When a PI is used as a matching controller, the fact that the output is zero under zero input yields (6.19) directly because Δxm corresponds to the state of the PI controller. On the other hand, if a CI controller (this is, a PI + CI with p = 1) is used, Δxm can be non-null under zero output and zero input due to the fact

t that every reset action is deleting the controller’s memory; null reset state entails tk (x˙m − x˙md ) = 0, where tk is the instant of the last reset action. The advantage of using a PI + CI with p < 1 is that (6.19) is guaranteed. This is based on the fact that the reset state (xCI ) cannot maintain a non-null value under zero input. In principle, there may exist a combination (xCI , xI ) of non-null states in (6.18) that yield zero output (fm ) with zero input. However, a sustained zero input entails that the reset law holds and that the reset state is made zero. Therefore, at steady-state, the linear state (xI ) has to be zero, and since Δxm = xI , (6.19) holds and the steady-state tracking-position error is zero, as in the linear (PI) case. Hence, the PI + CI manages to improve the transient response while retaining the zero steady-state error property. This entails a trade-off between transient and steady state performance, and the parameter p offers an intuitive way of adjusting it. This can be done by a simple trial-and-error procedure, decreasing its value from p = 1 until the steady state performance is good enough, that is, with an overshoot and settling time which are both small enough to be acceptable. The option of resetting the master side controller and not the slave one is supported by several sets of simulations and experiments. In them, the effect of resetting the master has been found to be much more important, while reset of the slave has had little influence. The explanation is related to the fact that the signal round trip (from the velocity specified by the operator x˙m to force feedback fm in Fig. 6.13) is closed by the master controller forming an external loop around the plant (slave) that contains the time-delays. On the other hand, the slave PI forms an inner loop not affected by delay. As the expected performance improvement due to reset is more effective under fundamental limitations (time-delays), the master controller, and not the slave, is the one that should be reset. The value of p should be chosen so that it is close enough to 1 to reduce overshoot, but not so high as to affect the capability of eliminating the steady state position error.

t

6.2.2.1 Passivity of the PI + CI We make use of the result that the parallel connection of two passive systems is itself passive [39]. Since a PI + CI is the parallel connection of a PI compensator

230

6 Application Cases

and a CI compensator (see Fig. 5.1), its passivity results from the passivity of both subsystems as follows. Without loss of generality, gains have been restructured in order to show a simpler and easier to understand proof. First, passivity of a PI can be proven with the classical result for linear systems [28]: a system with a transfer function H (j ω) is passive if Real[H (j ω)] ≥ 0 for all ω ∈ R. The PI transfer function is given by:   1 HPI (j ω) = k 1 +

, (6.21) T jω where T =

T 1−p

and T is the integration time constant. Its real part is given by Re[HPI (j ω)] = k ≥ 0.

(6.22)

Therefore, the PI compensator is passive. On the other hand, using Proposition 3.10 (see also [14]), passivity of a full reset compensator results from the passivity of its base linear compensator. The CI base linear compensator is a linear inblc blc tegrator and its transfer function is HCI (j ω) = j1ω . Since Real[HCI (j ω)] = 0 ≥ 0 for all ω ∈ R, the linear integrator is passive, and therefore, using the above mentioned result, the full reset compensator CI is also passive. Hence, it is immediately seen that the PI + CI compensator defined by (6.18) is passive. 6.2.2.2 Reset Instants The aforementioned passivity property is independent of the reset instants, provided that there are a finite number of resets in any finite interval. This enables the designer to choose the reset instants in the most convenient way for each application. In the case of teleoperation with a known, constant delay, it is possible to profit from this freedom by anticipating to this delay, in the following sense: the reset instants can be chosen so that they correspond to the times when the velocity error is approaching zero. The resulting strategy can be visualized in Fig. 6.14, where the evolution of the velocity error, ev (t) = x˙m (t) − x˙md (t), is plotted. By monitoring the velocity error and its variation, the reset instants can be chosen as those satisfying the following condition: ev (t1 ) (6.23) ⇔ ev (t1 ) + e˙v (t1 )ΔT = 0. e˙v (t1 ) = − ΔT Hence, (6.23) defines the reset condition c = 0 in (6.18), which amounts to applying a PD control on the error signal ev (t). That is, with a slight abuse of notation, the signal c(t) that triggers resetting in (6.18) is fixed to c(t) = (1 + sΔT ) · ev (t).

(6.24)

Thus, instead of waiting for the error to be zero, this situation is predicted and reset is carried out in advance. This mechanism also holds if the slave interacts with the environment: in case of a contact, the slave’s velocity is modified and the reset instants are changed accordingly.

6.2 Teleoperation of a Gantry Crane

231

Fig. 6.14 Choosing the reset instants

6.2.3 Examples 6.2.3.1 Example 1: Simple Robot In order to show that the reset control action is useful in different applications, two different teleoperated systems with two different approaches are chosen. First, we choose a simple model of a one-dimensional robot used in many simulations of bilateral teleoperation, consisting of a mass with friction. The “Master Device” and “Slave Device” blocks depicted in Fig. 6.13 are then given by: 1 (fh − fm ), mr s + b r 1 x˙s = (fs − fe ). mr s + b r

x˙m =

(6.25)

The force exerted by the human operator, fh , is simulated as a sinusoidal force command applied to the master device. The force fm produced by the master side impedance controller is fed back to the master device, influencing its movement in the same way as the operator command. Symmetrically, the force acting on the slave device is the output of the impedance controller at the slave side. Firstly, we will show simulation results that illustrate the potential of reset control for eliminating overshoot. Two identical “robots” with mass mr = 1 kg and friction br = 1 kg/s are considered at the master and slave sides. A constant time delay of 0.5 s is present in both directions of the communications channel (round trip delay is 1 s). The coefficients of the impedance controllers are Km = Ks = 10 kg/s2 , Bm = Bs = 10 kg/s, and the scattering factor is also set to b = 10 kg/s. This is an aggressive tuning of the controllers, intended to provide fast transient response; hence the high-gain values Km and Ks . In Fig. 6.15, the result of a maneuver in free space is plotted, with the master trajectory as a solid line and the slave as a dashed line. It can be noticed that the non-reset controller indeed manages to obtain this fast transient, but at the cost of a large overshoot. A partial reset controller with p = 0.5 yields the same fast transient while achieving a 50% reduction in overshoot. Finally, full reset (p = 1) eliminates overshoot almost completely (90–100% reduction). However, the steady state behavior is not shown in these simulations since the setpoint is always varying. We consider now a different maneuver, including contact with the environment, where this steady state behavior can be evaluated. We use the same master and slave devices as before, and place them initially at the same

232

6 Application Cases

Fig. 6.15 Teleoperation in free space: (solid lines) master device, (dashed) slave; (top) without reset, (middle) with partial reset (p = 0.5), (bottom) full reset

6.2 Teleoperation of a Gantry Crane

233

position, xm (0) = xs (0) = 0 m; but now there is a wall in the slave location at xs = 0.5 m. A sinusoidal force with a frequency of 1 rad/s is applied by the simulated operator at the master side (fh ), with an amplitude of 2 N during the first cycle and of 5 N during the second and third. After 15 seconds, this force is made zero, simulating that the operator stops holding the master device. The coefficients of the impedance controllers are now Km = Ks = 15 kg/s2 , Bm = Bs = 20 kg/s, and the scattering factor is also set to b = 20 kg/s. The evolution of the system is shown in Fig. 6.16, for the no-reset, full-reset, and partial-reset cases (from top to bottom). If no reset is performed, the plant exhibits considerable overshoot. A reset strategy can be applied to solve this issue. Since the overall delay is ΔT = 1, the reset instants satisfy ev + e˙v = 0. In this case, a full reset strategy (see Fig. 6.16, middle) and a partial reset strategy with p = 0.85 (Fig. 6.16, bottom) achieve similar results as far as overshoot reduction is concerned: at the time of ≈ 3 seconds, the overshoot is eliminated; at the time of ≈ 6 seconds, it is reduced by more than 50%; and at the time of ≈ 13 seconds, by more than 60%. In both cases, when moving in free space, the overshoot is reduced and, due to the fact that we are anticipating the time delay, the slave position tracks the master position more accurately, so the shapes of the trajectories are more similar. The difference in performance between both strategies is that with full reset there is an error in steady state, which does not appear with partial reset: in this latter case, after the master device is released by the operator, this is pushed by the feedback force until it matches the position of the slave device, thus eliminating the steady state position error. Obtaining simultaneously these two features (good tracking with reduced overshoot, and no error in steady state) is possible due to the use of partial reset. In Fig. 6.17, the force feedback to the user (fm ) is plotted for the no-reset (dashed line) and reset cases (solid line). As expected, the use of reset decreases the magnitude of the force at the reset instants, resembling the physical equivalent of a charged spring whose deformation is suddenly reduced.

6.2.3.2 Example 2: Gantry Crane Now we apply the reset control scheme to the teleoperation of a gantry crane. A real plant, Inteco’s 3DCrane model, was used as a slave device in the experiments. The output to be controlled is the position of the cart moving along the x axis. Its movement is affected by the oscillations of a payload hanging from a cable with constant length. Unlike in the previous case, the master device is not similar to the slave plant; instead, the desired velocity is directly indicated by the operator. Two different experiments will be carried out: in the first one, the operator is simulated in order to obtain a completely repeatable maneuver; in the second one, the maneuver is actually specified by a human operator. Let us first examine the case where the operator’s command is simulated as a velocity pulse with an amplitude of 0.2 m/s and which is 1.5 s long. The feedback force has thus no effect on the velocity setpoint. A constant time delay of 0.25 s is considered (round trip delay of 0.5 s). The controllers are aggressively tuned with

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6 Application Cases

Fig. 6.16 Manoeuvre with contact: (solid lines) master device, (dashed) slave; (top) no reset, (middle) full reset, (bottom) partial reset (p = 0.85)

6.2 Teleoperation of a Gantry Crane

235

Fig. 6.17 Feedback forces with partial reset and without reset

Fig. 6.18 Gantry crane, with and without partial reset (p = 0.95)

coefficients Km = Ks = 60 kg/s2 , Bm = Bs = b = 6 kg/s. The desired and achieved positions are pictured in Fig. 6.18. It can be noticed that the reset action eliminates the overshoot by 80%, while providing a fast rise time. Next we perform a human-commanded teleoperation maneuver. The operator handles again a SensAble PhanTom Omni haptic device, and the overall teleoperation setup is depicted in Fig. 6.19. The controllers are tuned as before, but this time the time delay is no longer constant. Instead, the use of a packet-switched network such as the Internet is simulated, applying the passivity-maintaining solution for signal reconstruction presented in [18] that was applied to crane teleoperation in [20]. The delay follows a normal distribution, with an average value of 0.15 s for each channel (master-to-slave and slave-to-master), a standard deviation of 0.01, and a 5% packet loss. The reset controller is hence tuned to anticipate an expected round trip delay of 0.3 s. In Fig. 6.20, it is shown that the overshoot is completely eliminated. In conclusion, this section has reported on the advantages of reset control for overcoming fundamental limitations, namely time delays, in passive teleoperation.

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6 Application Cases

Fig. 6.19 Setup for crane teleoperation

Fig. 6.20 Teleoperating a crane over the Internet: (solid lines) master device, (dashed) slave (gantry crane); (top) no reset, (bottom) partial reset (p = 0.95)

The proposed method resets the state of the master impedance controller and benefits from choosing the reset instants so that the system anticipates the communications channel time delay.

6.3 Control of Solar Collector Fields

237

The simulated and experimental results presented demonstrate the good performance of the proposed scheme for different applications. The solution has shown its usefulness even in the Internet-based teleoperation with time-varying delays. An interesting research direction for future work is to consider the possibility of using a time-varying parameter p, and its implications for passivity. An extension for unknown, time-varying delays, is another natural line of research.

6.3 Control of Solar Collector Fields This case study is motivated by a temperature control problem that is common to distributed collector fields in solar plants [12, 13]. In this kind of system, it is important to maintain a constant outlet temperature of the thermal fluid which flows through the collector field, in spite of the fact that the solar radiation varies during the day. The way to achieve a constant outlet temperature is by the adjustment of the fluid flow which will have to change substantially during operation, making operation conditions change constantly. Due to this variation, the temperature control problem is hard to solve in solar plants by conventional means, for example, using PID controllers with simple rules for tuning, or even autotuning. The reason is simple: the controller has to be designed for fixed operation conditions in the presence of potentially large uncertainty with respect to them. Since the dynamics is highly nonlinear and uncertain, a PID simply designed to work efficiently under fixed operation conditions usually degrades considerably in its performance when these conditions are changed [11, 19]. In [19], QFT (Quantitative Feedback Theory) is used for the control of a distributed collector field, ACUREX field, located at the Plataforma Solar de Almeria, Spain. QFT is a robust control technique that is especially well suited for control problems with large plant uncertainty. A (very) brief summary is given in Chap. 5, see [25] for a detailed exposition by Prof. Horowitz. The basic idea of [19] consists of using a linear time invariant (LTI) model with parametric uncertainty which was obtained by local linearization of the distributed collector field nonlinear dynamics around different operation conditions, following the method developed in [8]. The result is a robust PID compensator that guarantees a specified worst-case behavior for all the operation conditions considered. In general, the robustness and performance has to be balanced due to fundamental limitations of the resulting PID compensator, this being the main motivation of the present case study, in which reset control is investigated with the goal of improving performance without sacrificing robustness. In this section, the potentials of PI + CI compensation in improving robustness and performance of QFT-based designed base compensators will be investigated. In particular, the tuning rules developed in Chap. 5 for systems with time delays will be used (see also [6, 7]). First, the uncertain model for the solar field is defined through the uncertainty of two parameters, and then the QFT design method is applied using the uncertain model in order to get a well-tuned PI + CI compensator. Finally, the response of the ACUREX field is improved just by modifying the controller using a variable reset percentage.

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6 Application Cases

Fig. 6.21 ACUREX field diagram

Fig. 6.22 Closed-loop system

6.3.1 The Solar Collector Field The ACUREX field is a distributed collector system located at the Plataforma Solar de Almeria, Spain. This distributed solar collector field consists of a series of parabolic mirrors that reflect solar radiation onto a pipe where thermal oil is pumped from the bottom of a storage tank. This oil flow gets heated while circulating, and is returned to the top of the storage tank to be used in other industrial applications such as in thermal electrical generators and in desalination plants [11]. The diagram of this collector field can be seen in Fig. 6.21. In a distributed collector field, the objective of the control system is to maintain the outlet oil temperature of the field at a desired level in spite of disturbances, such as the irregular solar irradiance levels and inlet oil temperature changes. For control purposes, let us consider the feedback system given by Fig. 6.22, where P is the uncertain collector field and R is the reset compensator. In addition, a series feedforward controller is added to the control structure in order to reject measurable disturbances, solar irradiation I , and inlet oil temperature Tin , as much as possible. The uncertainty of the ACUREX field has been studied by taking the feedforward controller as part of the system model. For different operating conditions, that is, for different ranges of oil flow, the frequency response of this system (feedforward and plant) was obtained. From these frequency response data, it can be stated that the field exhibits a number of antiresonance modes (frequencies at which the magnitude of the frequency response changes abruptly) within the control bandwidth (the main

6.3 Control of Solar Collector Fields

239

one located around 0.02 rad/s) [11]. Thus, the controller tuning should be careful to avoid exciting these antiresonance modes.

6.3.2 PI + CI Design Since the main goal of this work is to design a robust control system that guarantees both stable and optimal operation for the ACUREX field, QFT will be used to design the base LTI control system. Closed-loop stability and tracking specifications are considered. In a second stage, this base compensator will be used to design the final reset control system.

6.3.2.1 Base PI Tuning In the design of the base LTI compensator, a PI structure will be used to satisfy some design specifications. After that, the PI parameters will be used to design the PI + CI compensator, a key issue being the election of a proper reset percentage. First of all, plant templates are computed from the sets of frequency responses obtained in [11]. In this case, the templates are obtained for a wide set of working frequencies, from 0.001 up to 0.1 rad/s. Secondly, boundaries are computed for each frequency from design specifications by using templates. Design specifications will be a (robust) phase margin of 9°, a settling time of 15 minutes (trying to avoid the excitation of the resonance frequencies) and a maximum overshoot of 40%. Note that a relatively small phase margin has been specified for the base LTI control system. This is due to the fact that the reset PI + CI compensator will be designed to improve this minimum phase margin. The next step consists of shaping the nominal open loop. For this purpose, the nominal plant Gn considered is the one obtained from the frequency response at low flow conditions, the worst operating condition, when the field is more difficult to control: Gn (s) =

0.296 s2 0.0084 s + 0.00842 ) e−20s . s2 0.82 s2 )(1 + 0.01 s + 0.012 ) 0.00452

0.83(1 + (1 +

1.6 0.0045 s

+

(6.26)

Here the PI controller is tuned by making the open-loop gain fit both the stability and tracking boundaries. A shaping is shown in Fig. 6.23, corresponding to a proportional gain of kp = 1.80 and an integral time constant of τi = 150 s. Finally, once the base PI compensator is tuned, a closed-loop system analysis is performed to ensure that the system fits the given stability and tracking specifications. This is usually referred as to compensator validation in QFT. This analysis has to be done for a wider set of frequencies than the one used in the templates computation, and in this case, this set goes from 0.0001 up to 0.1 rad/s with a larger

240

6 Application Cases

Fig. 6.23 Boundaries and loop gain shaping

grid. In Fig. 6.24, the stability analysis is shown, whereas the tracking analysis is depicted in Fig. 6.25. As it can be seen, the designed closed-loop system satisfies both stability and tracking specifications, for every frequency, thus the base PI compensator design is finished.

6.3.2.2 PI + CI Tuning Once the base PI compensator has been designed, the obtained PI parameters are chosen as part of the PI + CI parameters. But, in addition, another parameter, the reset percentage (preset ), must be tuned. For a proper selection of this parameter, it is important to note that for high values of preset a worse closed-loop response can be expected. Here, the tuning rules given in Chap. 5 (see also [6, 7]) are used for tuning this parameter for the nominal loop. The result is an increment of the phase margin and thus a better transient response for the nominal loop, and also for rest of possible loops derived from the uncertainty due to the extra phase lead given by reset compensation. A value preset = 0.35 is obtained. For comparison purposes, Bode plots of the nominal loops with the plant (6.26) and both compensators: the base PI compensator (kp = 1.80, τi = 150), and the PI + CI compensator (kp = 1.80, τi = 150, and preset = 0.35) are shown in Fig. 6.26. The extra phase lead given by

6.3 Control of Solar Collector Fields

241

Fig. 6.24 Validation of stability specifications

Fig. 6.25 Validation of tracking specifications

the PI + CI compensator without significant change in the magnitude, in relation to the base PI design can be seen in the figure. Note that, although reset compensation by PI + CI can considerably improve the phase margin of a PI controller, these improvements need to be carefully interpreted since in this case the closed-loop system is nonlinear, and the results have been

242

6 Application Cases

Fig. 6.26 Bode plot of both PI and PI + CI controllers

obtained by using the describing function of the PI + CI system. To simulate the ACUREX field dynamics, a simulation software package [10] and input data from 30 November 2007 are used. In Fig. 6.27, the reset effect over the distributed collector field can be seen. It shows that both systems (for PI and PI + CI controllers) are equal until the reset action is performed. On the other hand, the control signal is shown in Fig. 6.28. This figure shows that the distributed collector field operates at medium-high oil flows for both compensators, in such a way that the antiresonance modes are not excited as they lie at high frequencies. Although the responses given in Fig. 6.27 fulfill the design specifications, it can be seen that the effect of reset over the ACUREX field leads the system to have less overshoot than in the PI case, but more undershoot when the reference temperature is increased. Also when the reference temperature is decreased, the reset produces more overshoot. In what follows, a modification of the reset PI + CI compensator will be investigated to improve the control system performance.

6.3.3 PI + CI Compensator with Variable Reset Percentage Reset action may be a good choice to improve the transient response of a closedloop system, but there are systems whose response improvement is not better than the one with the linear PI controller, in fact, the response is actually worse. This is the case of distributed collector fields. This is due to the fact that, in this kind of systems, the dynamic characteristics of the field are constantly changing, and the

6.3 Control of Solar Collector Fields

243

Fig. 6.27 Field responses of both PI and PI + CI controllers

Fig. 6.28 Oil flow of both PI and PI + CI controllers

reset action can be very sensitive to these changes. In fact, in the PI + CI design, the reset percentage has been tuned by fixing a low value, preset = 0.2, but from

244

6 Application Cases

Fig. 6.29 Field responses of PI + CI with both fixed and variable reset

Fig. 6.27, it can be seen that this fixed value of reset percentage has only improved the field response at some reset times. To extend this response improvement over all the reset times, the reset percentage should be variable. In the following, the reset percentage is modified at each reset time using (5.47) and (5.48). Consider the system responses given in Fig. 6.27, that is, the reference temperature will experience the same step changes along the simulation. The control will be done by the PI + CI controller tuned in previous section, where its parameters were kp = 0.77, τi = 120, and preset = 0.2. In addition, another PI + CI controller is used with the same parameters, but with a variable reset percentage as in (5.47)–(5.48), where the chosen values are p¯ reset = 0.05, τd = 0.55, and τf = 2. The systems responses are shown in Fig. 6.29. Comparing closed-loop responses in Fig. 6.29, it can be said that closed-loop response is considerably improved by making the reset percentage variable since both overshoots and undershoots are strongly reduced without decreasing the speed of response. Finally, for comparison purposes, in Fig. 6.30, closed-loop responses are plotted for the base PI controller and the already tuned PI + CI controller with variable reset percentage. As can be seen in Fig. 6.30, the PI + CI controller with a variable reset percentage makes the response to have less overshoot in comparison to the PI, without modifying the response speed. Therefore, it can be concluded that a variable reset percentage improves considerably the PI response.

References

245

Fig. 6.30 Field responses of PI and PI + CI with variable reset controllers

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12. Camacho, E.F., Rubio, F.R., Berenguel, M., Valenzuela, L.: A survey on control schemes for distributed solar collector fields. Part I: Modeling and basic control approaches. Sol. Energy 81, 1240–1251 (2007) 13. Camacho, E.F., Rubio, F.R., Berenguel, M., Valenzuela, L.: A survey on control schemes for distributed solar collector fields. Part II: Advanced control approaches. Sol. Energy 81, 1252– 1272 (2007) 14. Carrasco, J., Baños, A., van der Schaft, A.J.: A passivity-based approach to reset control systems stability. Syst. Control Lett. 59(1), 18–24 (2010) 15. Carrizales-Martínez, G., Fermat, R., González-Alvarez, V.: Temperature control via robust compensation of heat generation: Isoparaffin/olefin alkylation. Chem. Eng. J. 125, 89–98 (2006) 16. Cervera, J.: Ajuste automático de controladores en QFT mediante estructuras fraccionales. Ph.D. Thesis, University of Murcia, Spain (2006) 17. Chopra, N., Spong, M.W., Ortega, R., Barabanov, N.E.: On tracking performance in bilateral teleoperation. IEEE Trans. Robot. 22(4), 861–866 (2006) 18. Chopra, N., Berestesky, P., Spong, M.W.: Bilateral teleoperation over unreliable communication networks. IEEE Trans. Control Syst. Technol. 16(2), 304–313 (2008) 19. Cirre, C.M., Moreno, J.C., Berenguel, M.: Robust QFT control of a solar collectors field. In: Martínez, D. (ed.) Improving Human Potential Programme—Research Results at PSA Within the Year 2002 Access Campaign. Serie Ponencias CIEMAT, pp. 27–33 (2003) 20. Fernández, A., Barreiro, A., Raimúndez, C.: Digital passive teleoperation of a gantry crane. In: Proc IEEE Int. Symp. Ind. Electron., Vigo, Spain (2007) 21. Fernández, A.F., Barreiro, A., Baños, A., Carrasco, J.: Reset control for passive teleoperation applications in process control. In: 34th Annual Conference of the IEEE Industrial Electronics Society, Orlando, Florida, USA (2008) 22. Fernández, A.F., Barreiro, A., Baños, A., Carrasco, J.: Reset control for passive bilateral teleoperation. IEEE Trans. Ind. Electron. (2010). doi:10.1109/TIE.2010.2077610 23. Fischer, M., Nelles, O., Isermann, R.: Adaptive predictive control of a heat exchanger based on a fuzzy model. Control Eng. Pract. 6, 259–269 (1998) 24. Hokayem, P., Spong, M.W.: Bilateral teleoperation: an historical survey. Automatica 42(12), 2035–2057 (2006) 25. Horowitz, I.M.: Quantitative Feedback Theory. QFT Press, Boulder (1992) 26. Hyodo, S., Soeda, Y., Ohnishi, K.: Verification of flexible actuator from position and force transfer characteristic and its application to bilateral teleoperation system. IEEE Trans. Ind. Electron. 56(1), 36–42 (2009) 27. Karacan, S., Hapoˇglu, H., Alpbaz, M.: Application of optimal adaptive generalized predictive control to a packed distillation column. Chem. Eng. J. 84, 389–396 (2001) 28. Khalil, H.K.: Nonlinear Systems. Prentice Hall, New York (2002) 29. Khan, S., Sabanovic, A., Nergiz, A.O.: Scaled bilateral teleoperation using discrete-time sliding-mode control. IEEE Trans. Ind. Electron. 56(9), 3609–3618 (2009) 30. Landau, I.D.: System Identification and Control Design. Prentice Hall, New York (1990) 31. Lee, Y., Park, S., Lee, M., Brosilow, C.: PID controller tuning for desired closed-loop responses for SI/SO systems. AIChE J. 44(1), 106–115 (2004) 32. Loulch, Z., Cabassud, M., Le Lann, M.V.: A new strategy for temperature control of batch reactors: experimental application. Chem. Eng. J. 75, 11–20 (1999) 33. Maidi, A., Diaf, M., Corriou, J.P.: Optimal linear PI fuzzy controller design of a heat exchanger. Chem. Eng. Process. 47, 938–945 (2008) 34. Niemeyer, G., Slotine, J.-J.: Stable adaptive teleoperation. Int. J. Ocean Eng. 16(1), 152–162 (1991) 35. Rivera, D.E., Morari, M., Skogestad, S.: Internal model control. 4. PID controller design. Ind. Eng. Chem. Process Des. Dev. 25(1), 252–265 (1986) 36. Skogestad, S.: Simple analytic rules for model reduction and PID controller tuning. J. Process Control 13(4), 291–309 (2003)

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Index

A ACUREX field, 238 After-reset surface, 58, 73 Anticipating reset, 178, 230 B Beating, 63 C Clegg integrator definition, 6 describing function, 17, 129 Horowitz’s design procedure, 20 Clegg integrator with advanced reset/variable reset band definition, 17 describing function, 18, 135 Clegg integrator with reset band describing function, 132 Crane, 233 D Deadlock, 63 Delay margin, 167 Delay-dependent stability frequency domain, 168 time domain, 164 Delay-independent stability frequency domain, 155 time domain, 153 Delayed differential equation, 150 Delta (Dirac) function, 160 Detuned PI (PI-des), 195 Dirac impulse (delta), 169 Dominant eigenvalue, 61 Dwell time, 105

E Existence and uniqueness of solutions, 151 F Feedback force, 233 Feedforward, 238 First order plants, 184 First order plus deadtime (FOPDT), 188, 215 lag dominant, 190 time-delay dominant, 193 FORE (first order reset element) definition, 1 describing function, 129 Horowitz’s design procedure, 23 passivity, 119 FORE with reset band, 131 describing function, 132 G General reset control system, 26 full reset, 26, 117 partial reset, 26, 120 with reset band, 130 General reset controller, 7 describing function, 134 with reset band describing function, 132 H H-beta condition, 27, 156 generalized, 158 Half-rule, 195 Haptic device, 235 Heat exchanger, 212 High-order systems, 195 Hybrid systems, 37

A. Baños, A. Barreiro, Reset Control Systems, Advances in Industrial Control, DOI 10.1007/978-1-4471-2250-0, © Springer-Verlag London Limited 2012

249

250 I IAE (integral absolute error), 221 Impedance controllers, 227 Impulsive control system, 32 Impulsive dissipative system, 35 Impulsive systems, 27 autonomous systems with impulses, 30 impulses at fixed times, 29 impulses at variable times, 29 Input strictly passive (ISP), 116 Input–output stability, 113 Integrating systems, 198 Internal model control (IMC), 190, 216 Skogestad IMC (SIMC), 190 ITAE (integral time absolute error), 221 J Jordan form, 61 K Kalman–Yakubovich–Popov (KYP) lemma, 155 L Limit cycles, 136 Linear base system, 59 Linear fundamental limitations, 4 frequency domain, 11 time domain, 9 Linear matrix inequality (LMI), 154, 166 Lyapunov stability definitions, 94 H-beta condition, 95 reset-times dependent conditions, 99 reset-times independent conditions, 94 Lyapunov–Krasovskii (LK) functionals, 153, 165, 175 O Output strictly passive (OSP), 116, 157, 174 P Passive system, 116 pH control, 125 PI+CI, 181 design for a heat exchanger, 216 design for solar collector field, 239 design for teleoperation, 228 summary of tuning rules, 202 variable reset band/advanced reset, 205, 223 with fixed reset band, 203, 222 Port-Hamiltonian systems, 227

Index Position drift, 228 PRBS (pseudo random binary signal), 215 Q Quantitative Feedback Theory (QFT), 208, 218, 237 templates, boundaries, and loop shaping, 208 templates, boundaries and loop shaping, 239 R Rectangular pulses, 172 Reduced initial condition, 61, 100 Regular reset system, 100 Reset band, 129 Reset lag compensator, 119 Reset percentage (ratio), 183, 240 variable, 206, 242 Reset PID compensator, 120 Reset PII R compensator, 122 Reset surface, 58 Reset times, 2 Robustness, 174 S SCADA, 213 Scattering, 226 Small gain, 159, 174 Solar collector fields, 237 Spectral projector, 61 Stabilization by reset, 105 stable base system, 105 unstable base system, 110 Storage functional, 168 Strictly positive real (SPR), 157, 174 T Teleoperation, 224 passive, 225 Temperature control, 211 Time regularization, 59, 151 V Variable reset, 184 Variable reset band, 134 Very strictly passive (VSP), 116 W Well-posedness, 63 Z Zeno solutions, 63, 67

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