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Sliding Mode Control in Engineering


SLIDING MODE CONTROL IN ENGINEERING

edited by Wilfrid Perruquetti
Ecole Central de Lille Villeneuve d'Ascq, France

Jean Pierre Barbot
Ecole Nationale Superieure d'Electronique et de ses Applications Cergy-Pontoise, France

MARCEL

MARCEL DEKKER, INC.
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Library of Congress Cataloging-in-Publication Data Perruquetti, Wilfrid Sliding mode control in engineering. Wilfrid Perruquetti, Jean Pierre Barbot. p. cm. — (control engineering) Includes bibliographical references and index. ISBN 0-8247-0671-4 (alk. paper) 1. Automatic Control 2. Sliding Mode Control I. Barbot, Jean Pierre, II. Title. III. Control Engineering (Marcel Dekker Inc.) TJ213 .P415 2002 629.8—dc21 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above. Copyright ? 2002 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA

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

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

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

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SERIES

INTRODUCTION

in Engineering is another outstanding entry to Dekker's Control Engineering series. Neil Munro

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Preface
Many physical systems naturally require the use of discontinuous terms in their dynamics. This is, for instance, the case of mechanical systems with friction. This fact was recognized and advantageously exploited since the very beginning of the 20th century for the regulation of a large variety of dynamical systems. The keystone of this new approach was the theory of differential equations with discontinuous right-hand sides pioneered by academic groups of the former Soviet Union. On this basis, discontinuous feedback control strategies appeared in the middle of the 20th century under the name of theory of variable-structure systems. Within this viewpoint, the control inputs typically take values from a discrete set, such as the extreme limits of a relay, or from a limited collection of prespecified feedback control functions. The switching logic is designed in such a way that a contracting property dominates the closedloop dynamics of the system thus leading to a stabilization on a switching manifold, which induces desirable trajectories. Based on these principles, one of the most popular techniques was created, developed since the 1950s and popularized by the seminal paper by Utkin (see [30] in chapter 7): the sliding mode control. The essential feature of this technique is the choice of a switching surface of the state space according to the desired dynamical specifications of the closed-loop system. The switching logic, and thus the control law, are designed so that the state trajectories reach the surface and remain on it. The main advantages of this method are: ? its robustness against a large class of perturbations or model uncertainties ? the need for a reduced amount of information in comparison to classical control techniques ? the possibility of stabilizing some nonlinear systems which are not stabilizable by continuous state feedback laws

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The first implementations had an important drawback: the actuators had to cope with the high frequency bang-bang type of control actions that could produce premature wear, or even breaking. This phenomenon was the main obstacle to the success of these techniques in the industrial community. However, this main disadvantage, called chattering, could be reduced, or even suppressed, using techniques such as nonlinear gains, dynamic extensions, or by using more recent strategies, such as higher-order sliding mode control (see Chapter 3). Once the constraint sliding function (CSF) was chosen according to some design specifications (stabilizing dynamics or tracking), then two difficulties may appear: Dl) the CSF should be of relative degree one (differentiating once for this function with respect to time: the control should appear) in order to provide the existence of a sliding motion; and D2) the CSF may depend on the whole state (and not only on the measured outputs). To circumvent Dl) one may use a new CSF of relative degree one (see the introduction of Chapter 3 and the choice of the CSF in subsection 13.3.1). Another promising alternative to this difficulty is based on higherorder sliding mode controller design (see Chapter 3). Concerning D2) when the CSF depends on other variables than the measured outputs, a natural solution is provided by observer design. This approach has one advantage which concerns the natural filtering of the measurements (see Chapter 4 p. 121). But the drawback is that the class of admissible perturbations is reduced, since the perturbation should match two conditions: one for the control (see Chapter 1, p. 20) and the other for the observer (see Section 4.5). We are currently living in an important time for these types of techniques. Now they may become more popular in the industrial community: they are relatively simple to implement, they show a great robustness, and they are also applicable to complex problems. Finally, many applications have been developed (see the Table of Contents): ? Control of electrical motors, DTC ? Observers and signal reconstruction ? Mechanical systems ? Control of robots and manipulators ? Magnetic bearings

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Based on these facts, several active researchers in this field combined their efforts, thanks to the support of many French institutions1, to present new trends in sliding mode control. In order to clearly present new trends, it is necessary to first give an historical overview of classical sliding mode (Chapter 1). In the same manner of thinking, it is important to recall and introduce, from a very clear educational standpoint, a mathematical background for discontinuous differential equations, which is done in Chapter 2. Next, a new concept in variable structure systems is introduced in Chapter 3 : the higher-order sliding mode. Such control design is naturally motivated by the limits of classical sliding mode (see Chapter 1) and completely validated by the mathematical background (see Chapter 2). On the basis of these chapters, some control domains and methods are discussed with a sliding mode point of view: ? Chapter 4 deals with observer design for a large class of nonlinear systems. ? Chapter 5 presents a complementary point of view concerning the design of dynamical output controllers, instead of observer and state controllers. ? Chapter 6 presents the link between three of the most popular nonlinear control methods (i.e., sliding mode, passivity, and flatness) illustrated through power converter examples. ? Chapter 7 is dedicated to stability and stabilization. The domain of sliding mode motion is particularly investigated and the usefulness of the regular form is pointed out. ? Chapter 8 recalls some problems due to the discretization of the sliding mode controller. Some solutions are recalled and the usefulness under sampling of the higher-order sliding mode is highlighted. ? Chapter 9 deals with adaptive control design. Here, some basic features of control algorithms derived from a suitable combination of sliding mode and adaptive control theory are presented. ? Chapters 10 and 11 are dedicated to time delay effects. They deal, respectively, with relay control systems and with changes of behavior due to the delay presence.
, GdR Automatique, GRAISyHM, LAIL-UPRESA CNRS 8021, ECE-ENSEA and Ecole Centrale de Lille.

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? Chapter 12 is dedicated to the control of infinite-dimensional systems. A disturbance rejection for such systems is particularly presented. In order to increase interest in the proposed methods, the book ends with two applicative fields. Chapter 13 is dedicated to robotic applications and Chapter 14 deals with sliding mode control for induction motors.

Wilfrid PERRUQUETTI Jean-Pierre BARBOT FRANCE

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Contents
Preface Contributors 1 Introduction: An Overview of Classical Sliding Mode Control A.J. Fossard and T. Floquet 1.1 Introduction and historical account 1.2 An introductory example 1.3 Dynamics in the sliding mode 1.3.1 Linear systems 1.3.2 Nonlinear systems 1.3.3 The chattering phenomenon 1.4 Sliding mode control design 1.4.1 Reachability condition 1.4.2 Robustness properties 1.5 Trajectory and model following 1.5.1 Trajectory following 1.5.2 Model following 1.6 Conclusion References 2 Differential Inclusions and Sliding Mode Control T. Zolezzi 2.1 Introduction 2.2 Discontinuous differential equations and differential inclusions 2.3 Differential inclusions and Filippov solutions 2.4 Viability and equivalent control 2.5 Robustness and discontinuous control 2.6 Numerical treatment

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2.7 Mathematical appendix 2.8 Bibliographical comments References

3 Higher-Order Sliding Modes L. Fridman and A. Levant
3.1 Introduction 3.2 Definitions of higher order sliding modes 3.2.1 Sliding modes on manifolds 3.2.2 Sliding modes with respect to constraint functions 3.3 Higher order sliding modes in control systems 3.3.1 Ideal sliding 3.3.2 Real sliding and finite time convergence 3.4 Higher order sliding stability in relay systems 3.4.1 2-sliding stability in relay systems 3.4.2 Relay system instability with sliding order more than 2 3.5 Sliding order and dynamic actuators 3.5.1 Stability of 2-sliding modes in systems with fast actuators 3.5.2 Systems with fast actuators of relative degree 3 and higher 3.6 2-sliding controller 3.6.1 2-sliding dynamics 3.6.2 Twisting algorithm 3.6.3 Sub-optimal algorithm 3.6.4 Super-twisting algorithm 3.6.5 Drift algorithm 3.6.6 Algorithm with a prescribed convergence law 3.6.7 Examples 3.7 Arbitrary-order sliding controllers 3.7.1 The problem statement 3.7.2 Controller construction 3.7.3 Examples 3.8 Conclusions References

4 Sliding Mode Observers J-P. Barbot, M. Djemai, and T. Boukhobza
4.1 Introduction 4.2 Preliminary example 4.3 Output and output derivative injection form

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4.4

4.5 4.6 4.7

Nonlinear observer Sliding observer for output and output derivative nonlinear injection form Triangular input observer form 4.4.1 Sliding mode observer design for triangular input observer form 4.4.2 Observer matching condition Simulations and comments Conclusion Appendix 4.7.1 Proof of Proposition 39 4.7.2 Proof of Theorem 41 4.7.3 Proof of Theorem 49

4.3.1 4.3.2

References 5 Dynamic Sliding Mode Control and Output Feedback C. Edwards and S.K. Spurgeon
5.1 Introduction 5.2 Static output feedback of uncertain systems 5.3 Output feedback sliding mode control for uncertain systems via dynamic compensation 5.3.1 Dynamic compensation (observer based) 5.3.2 Control law construction 5.3.3 Design example 5.4 Dynamic sliding mode control for nonlinear systems 5.4.1 Design example 5.5 Conclusions References

6 Sliding Modes, Passivity, and Flatness H. Sira- Ramirez
6.1 6.2 Introduction The permanent magnet stepper motor 6.2.1 The simpler D-Q nonlinear model of the PM stepper motor 6.2.2 The control problem 6.2.3 A passivity canonical model of the PM stepper motor 6.2.4 A controller based on "energy shaping plus damping injection" 6.2.5 Differential flatness of the system 6.2.6 A dynamic passivity plus flatness based controller 6.2.7 Simulation results

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6.2.8 A pulse width modulation implementation 6.3 The "boost" DC-to-DC power converter 6.3.1 Flatness of the "boost" converter 6.3.2 Passivity properties through 6.3.3 A passivity-based sliding mode controller 6.3.4 Non-minimum phase output stabilization 6.3.5 Trajectory planning 6.3.6 Simulation results 6.3.7 Dc-to-ac power conversion 6.3.8 An iterative procedure for generating a suitable inductor current reference 6.3.9 Simulation results 6.4 Conclusions References 7 Stability and Stabilization W. Perruquetti 7.1 Introduction 7.2 Notation 7.3 Generalized regular form 7.3.1 Obtention of the regular form 7.3.2 Effect of perturbations on the regular form 7.4 Estimation of initial sliding domain 7.4.1 Problem formulation 7.4.2 Sliding domain and initial domain of sliding motion 7.4.3 Application 7.5 Stabilization 7.5.1 Stabilization in the case d = m 7.5.2 Stabilization in the case d > m 7.6 Conclusion References 8 Discretization Issues J-P. Barbot and T. Boukhobza 8.1 Introduction 8.2 Mathematical recalls 8.3 Classical sliding modes in discrete time 8.4 Second-order sliding mode under sampling 8.5 The sampled "twisting algorithm" References

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9 Adaptive and Sliding Mode Control G. Bartolini 9.1 Introduction 9.2 Identification of continuous linear systems in I/O form 9.3 MRAC model reference adaptive control 9.3.1 MRAC with accessible states 9.3.2 Adaptive control for SISO plant in I/O form: an introductory example with relative degree equal to one 9.3.3 Generalization to system of relative degree greater than one 9.4 Sliding mode and adaptive control 9.5 Combining sliding mode with adapt 9.6 Conclusions References

10 Steady Modes in Relay Systems with Delay L. Fridman, E. Fridman, and E. Shustin 10.1 Introduction 10.2 Steady modes and stability 10.2.1 Steady modes 10.2.2 Stability 10.3 Singular perturbation in relay systems with time delay 10.3.1 Existence of stable zero frequency periodic steady modes for a singularly perturbed multidimensional system 10.3.2 Existence of stable zero frequency steady modes in systems of arbitrary order 10.4 Design of delay controllers of relay type 10.4.1 Stabilization of the simplest unstable system 10.4.2 Stable systems with bounded perturbation and relay controllers with delay 10.4.3 Statement of the adaptive control problem 10.4.4 The case of definite systems 10.4.5 The case of indefinite systems 10.5 Generalizations and open problems 10.5.1 The case when \F(x)\ > 1 for some x 10.5.2 Systems and steady modes of the second order 10.5.3 Stability and instability of steady modes for multidimensional case 10.6 Conclusions 10.7 Appendix: proofs References

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11 Sliding Mode Control for Systems with Time Delay F. Gouaisbaut, W. Perruquetti, and J.-P. Richard 11.1 Introduction 11.2 SMC under delay effect: a case study 11.2.1 Problem formulation 11.2.2 A case study 11.2.3 An example with simulation 11.3 A SMC design for linear time delay systems 11.3.1 Regular form 11.3.2 Asymptotic st mall delays 11.3.3 Sliding mode controller synthesis 11.3.4 Example: delay in the state 11.4 Conclusion References 12 Sliding Mode Control of Infinite-Dimensional Systems Y. Orlov 12.1 Introduction 12.2 Motivation: disturbance rejection in Hilbert space 12.3 Mathematical description of sliding modes in Hilbe 12.3.1 Semilinear differential equation 12.3.2 Discontinuous control input and sliding mode equation 12.4 Unit control synthesis for uncertain systems with a finitedimensional unstable part 12.4.1 Disturbance rejection in exponentially stabilizable systems 12.4.2 Disturbance rejection in minimum phase systems 12.5 Conclusions References 13 Application of Sliding Mode Control to Robotic Systems N. M'Sirdi and N. Nadjar-Gauthier 13.1 Introduction 13.2 Modeling and properties of robotic systems 13.2.1 Dynamics of mechanical systems 13.2.2 Control design approach 13.2.3 Examples 13.3 Sliding mode for robot control 13.3.1 Sliding mode control for a pneum stem 13.3.2 Sliding mode control of a hydraulic robot 13.3.3 Simulation results

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13.4 SM observers based control 13.4.1 Observer design 13.4.2 Tracking error e : observer and control 13.4.3 Stability of observer based control 13.4.4 Simulation results 13.4.5 Conclusion 13.5 Appendix 13.5.1 Pneumatic actuators model 13.5.2 Hydraulic manipulator model 13.5.3 Proof of lemma References 14 Sliding Modes Control of the Induction Motor: a Benchmark Experimental Test A. Glumineau, L. C. De Souza Marques, and R. Boislive 14.1 Introduction 14.2 Sliding modes control 14.3 Application to the indu 14.4 Benchmark "horizontal handling" 14.4.1 Speed and flux references and load disturbance 14.4.2 Induction motor parameters (squirrel cage rotor) 14.4.3 Variations of the parameters for robustness test 14.5 Simulation and experimentation results 14.5.1 Results of simulations 14.5.2 Experimental results 14.6 Conclusion References

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Contributors
J-P. BARBOT, Equipe Commande des Systemes (ECS), ENSEA. 6 av. du Ponceau, 95014 Cergy - FRANCE.

G. BARTOLINI, Institute di Elettrotecnica, University of Cagliari, Piazza d'Armi, 09123 Cagliari - ITALY.

R. BOISLIVEAU, IRCCYN: Institut de Recherche en Communications et Cybernetique de Nantes, UMR CNRS 6597, Ecole Centrale de Nantes, BP 92101, 44 321 Nantes cedex 03 - FRANCE.

T. BOUKHOBZA, IUT de Kourou, Universite des Antilles Guyane, Av. Bois Chaudat, B.P. 725, 97 387 Kourou Cedex - FRANCE.

L.C. DE SOUZA MARQUES, DAS: Departamento de Automate e Sistemas, Universidade Federal de Santa Catarina, CP 476, 88040-900, FLORIANOPOLIS - BRAZIL.

M. DJEMAI, Equipe Commande des Systemes (ECS), ENSEA, 6 av. du Ponceau, 95014 Cergy Cedex - FRANCE.

C. EDWARDS, Control Systems Research Group, University of Leicester, University Road, Leicester LEI 7RH, England - UNITED KINGDOM.

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T. FLOQUET, Equipe Commande des Systemes (ECS), ENSEA, 6 av. du Ponceau, 95014 Cergy - FRANCE and LAIL UPRES A CNRS 8021, Ecole Centrale de LILLE, BP 48, 59651 Villeneuve d'Ascq CEDEX - FRANCE.

A.J. FOSSARD, DERA/CERT/ONERA, Complexe scientifique de Rangueil, 2 av. Edouard Belin, 31 4000 Toulouse - FRANCE.

E. FRIDMAN, Dept. of Electrical Engineering, Tel Aviv University, Ramat-Aviv, 69978 Tel Aviv - ISRAEL.

L. FRIDMAN, Division of Postgraduate Study and Investigations, Chihuahua Institute of Technology, av. Tecnologico 2909, 31310 Chihuahua MEXICO.

A, GLUMINEAU, IRCCyN: Institut de Recherche en Communications et Cybernetique de Nantes, UMR CNRS 6597, Ecole Centrale de Nantes, BP 92101, 44 321 Nantes cedex 03 - FRANCE.

F. GOUAISBAUT, Ecole Centrale de Lille, LAIL-UPRES A CNRS 8021, BP 48, 59651 Villeneuve d'Ascq CEDEX - FRANCE.

A. LEVANT, Institute for Industrial Mathematics, 4/24 Yehuda HaNachtom St., Beer-Sheva 84311 - ISRAEL.

N. M'SIRDI, LRP: Laboratoire de Robotique de Paris, University of Versailles Saint-Quentin en Yvelines, 10 avenue de 1'Europe, 78140 Velizy - FRANCE.

N. NADJAR-GAUTHIER, LRP: Laboratoire de Robotique de Paris, University of Versailles Saint-Quentin en Yvelines, 10 avenue de 1'Europe, 78140 Velizy - FRANCE.

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Y. ORLOV, Electronics Department, CICESE, P.O.Box 434944, San Diego, CA 92143-4944 - USA.

W. PERRUQUETTI, LAIL UPRES A CNRS 8021, Ecole Centrale de LILLE, BP 48, 59651 Villeneuve d'Ascq CEDEX - FRANCE.

J.P. RICHARD, LAIL-UPRESA CNRS 8021, Ecole Centrale de Lille, BP 48, 59651 Villeneuve d'Ascq CEDEX - FRANCE.

H. SIRA-RAMIREZ, CINVESTAV-IPN, Depto. Ing. Electrica, Programa Mecatronica, Avenida IPN, No. 2508, Colonia San Pedro Zacatenco, A.P. 14-740 Mexico D.F. - MEXICO.

S.K. SPURGEON, Control Systems Research Group University of Leicester, University Road, Leicester LEI 7RH, England - UNITED KINGDOM.

E. SHUSTIN, School of Mathematical Sciences, Tel Aviv University, Ramat-Aviv, 69978 Tel Aviv - ISRAEL.

T. ZOLEZZI, Dipartimento di Matematica, via Dodecanese 35, 16 146 Geneva - ITALY.

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Chapter 1

Introduction: An Overview of Classical Sliding Mode Control
A.J. FOSSARD* and T. FLOQUET**
* DERA/CERT/ONERA, Toulouse, France ** EN SEA, Cergy-Pontoise, France

1.1

Introduction and historical account

Sliding mode control has long proved its interests. Among them, relative simplicity of design, control of independent motion (as long as sliding conditions are maintained), invariance to process dynamics characteristics and external perturbations, wide variety of operational modes such as regulation, trajectory control [14], model following [30] and observation [24]. Although the subject has already been treated in many papers [5, 6, 13, 20], surveys [3], or books [7, 17, 28], it remains the object of many studies (theoretical or related to various applications). The main purpose of this chapter is to introduce the most basic and elementary concepts such as attractivity, equivalent control and dynamics in sliding mode, which will be illustrated by examples and applications. Sliding mode control is fundamentally a consequence of discontinuous control. In the early sixties, discontinuous control (at least in its simplest form of bang-bang control) was a subject of study for mechanics and control engineers. Just recall, as an example, Hamel's work [15] in France, or Cypkin's [27] and Emelyanov's [9] in the USSR, solving in a rigorous way the

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problem of oscillations appearing in bang-bang control systems. These first studies, more concerned with analysis and where the phenomena appeared rather as nuisances to be avoided, turned rapidly to synthesis problems in various ways. One of them was related to (time) optimal control, another to linearization and robustness. In the first case, discontinuities in the control, occurring at given times, resulted from the solution of a variational problem. In the second, which is of interest here, the use of a discontinuous control was an a priori choice. The more or less high frequency of the commutations used depended on the goal pursued (linearization), as produced by the beating spoilers used in the early sixties to control the lift of a wing, conception of corrective nonlinear networks enabling them to bypass the Bode's law limitations and, of course, generation of sliding modes. Although both approaches and objectives were at the beginning quite different, it is interesting to note that they turned out to have much in common. In fact, it was when looking for ways to design what we now call robust control laws that sliding mode was discovered at the beginning of the sixties. For the needs of military aeronautics, and even before the term of robustness was used, control engineers were looking for control laws insensitive to the variations of the system to be controlled. The linear networks used at these time did not bring enough compensation to use high gains required to get parametric insensitivity: they match the Bode's law according to which phase and amplitude effects are coupled and antagonist. At the beginning of 1962, on B. Hamel's idea, studies of nonlinear compensators were initiated, whose aim was to overcome previous limitations. Typically, these networks, acting on the error signal x of the feedback system, were defined by the relation u = |Fi(x,i,...,)|sgn(F 2 (a:,i,...,)) where | | denotes the absolute value and F\ and F<2 are appropriate linear filters. Hence the output was discontinuous but modulated by a function of x and its derivatives. Under the simplest form, one had, for instance, u = — \x\ sgn(x + kx) instead of the classical PD corrector. It is easy to see that, under the approximation of the first harmonic: ? the equivalent gain of such a network (for a sinusoidal input x — XQ sin ujt) is independent of the amplitude XQ and only depends on the pulsation a; (as a linear network), hence the denomination of pseudo linear network; (1-1)

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? it produces a lead phase without any increase (and even decrease) of the dynamics amplitude. For instance, in the previous case, if </?(u;) is the phase of 1 + kp (p denoting the Laplace operator) at w, the real Re and imaginary parts Im of the equivalent gain are given by
Re = I — ^ ((p — sin (p cos ip)

Im = ^ sin2 (p leading to the generalized transfer locus of Figure 1.1 where, for comparison, the loci for a simple PD (dotted line) and a classical lead phase network (thin line) are given. This shows that a lead phase can be obtained (theoretically till |), not only without increase of the dynamics rate but also with a small reduction (from 1 to ?).

l + kp

Figure 1.1: Nyquist plots In fact, it appeared simultaneously in France and in the former USSR, that these laws presented two different aspects: ? pseudo linear compensation: astute combinations of linear and nonlinear signals, including commutations, can lead to appreciable advantages while being freed from the disadvantages specific to purely linear systems;

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? they generate a sliding motion by controlling the evolution of the system through commutations. This mode is certainly nonoptimal but exhibits a rather interesting sensitivity.

1.2

An introductory example

By way of illustration, let us take the simple example of a variable inertia 2 §5- [1], as shown in Figure 1.2.

>

,

X

orrni'T 1 —L A/X )i l?nr D g l l l X T^

u

a2
p2

y

Figure 1.2: Variable inertia Taking as state variables x\ = x, x2 = x, the system can be put in the following state space representation
xi = x2
X2 — 0?

(1.2)

where the control law u is designed as in (1.1) and is given by

u =-

kx2]

(1.3)

In the following, a — x\ + kx2 = 0 will be called the switching surface. The term switching illustrates the fact that the control law u commutes while crossing the line a = 0. Then, one can easily see that (Figure 1.3): ? the phase plane is divided into four regions; ? in regions I and III (where xi sgn(xi + kx2) > 0), trajectories are ellipses given by aLx\ + x\ — cst; ? in regions II and IV (where x\ sgn(xi + kx2] < 0), trajectories are hyperbolas with asymptotes x2 = the control only commutes on the boundary surface #1 + kx2 = 0;

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? by a suitable choice of fc, all trajectories are directed toward this surface (regardless of which side of the surface they are). Consequently, once it is reached, a new phenomenon appears: the trajectories are "sliding" along this surface.

Figure 1.3: Trajectories in the portrait phase The classical theory of ordinary differential equations however is unable to explain what occurs here (the solution of the system (1.2) is known to exist and be unique if u is a Lipschitz function, and so continuous). Consequently, the design of appropriate mathematical tools appears necessary and alternative approaches and construction of solutions can be found in Filippov's work [11] and in other's using the theory of differential inclusions [2]. Those results are not developed here since they are the subject of the chapter Differential Inclusions and Sliding Mode Control. To understand more "physically" what is happening, a very simple interpretation can be given just by introducing some kind of imperfections in the switching devices, for instance a time delay T. Under such an assumption, the motion proceeds along a succession of small arcs (sequentially ellipsoi'dal and hyperbolic) between the lines x\ +k x^ = 0 and x\ +k x-z = 0,

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crossing the origin, with

k-r 1 + a2kr k-r k = I — a2kr
A; =

When T tends to zero, the amplitude of these oscillations tends to zero, whereas the frequency increases indefinitely and the representative point "slides" along the line x + kx = 0 (Figure 1.4).

kx

—0

Figure 1.4: Trajectories with time delay Further important remarks must be made: In the sliding motion, a = 0, which implies that the dynamics is now defined by 1 x — —x k Therefore, the second-order system behaves then like a first-order system, with time constant k and independent of the inertia a, and the trajectory will slide along a = 0 to the origin (thus a = 0 is also called the sliding surface). Note also that, with the discontinuous control, the system is equivalent to a proportional-derivative feedback associated with an infinite gain. As a — 0, x2 + ka?u = 0. On the sliding surface, the motion is consequently the same as if, instead of the discontinuous control, an "equivalent"

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continuous control defined by

had been used. This equivalent control can be considered as the mean value of the discontinuous control u on the sliding surface, modulated in width and amplitude. Yet, in sliding motion, the control switches with a high frequency between the values — |#i| and |xi|. This phenomenon is known as chattering and is a drawback of sliding modes (see section 1.3.3). The latter dynamical behavior is called the ideal sliding mode, that is to say that there exists a finite time te such that for all t > te, s(x(t)) = 0 Of course, the ideal sliding mode along x + kx = 0 only exists for a timecontinuous system and without delay, which is not the case in real system. Attention is drawn to the fact that, under sampling, the situation is much more complicated. The problem is beyond the scope of this introductory chapter and the interested reader will find developments in subsequent chapters, for instance Discretization Issues or Sliding Mode Control for Systems with Time Delay. This simple example allowed us to enhance some characteristics of the sliding phenomenon and it has been shown that the sliding mode was initiated at the first switching. Of course, this is not always the case unless some precautions are taken. For instance, if the discontinuous control u = — sgn(xi + ?#2) is used instead of (1.3), the sliding mode only occurs in the layer as can be seen in Figure 1.5. This comes from the fact that the switching surface is known to be attractive if the condition ss < 0 is fulfilled. This will be detailed in the following sections, as well as the dynamics in sliding motion, the notion of equivalent control, the chattering phenomenon and the robustness properties of the sliding mode.

1.3
1.3.1

Dynamics in the sliding mode
Linear systems

Let us consider a linear process, eventually a multi-input system, defined by x = Ax + Bu (1.5)

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

Figure 1.5: Portrait phase and sliding mode domain where x e H n , u B,m and rank 5 = m. Let us also define the sliding surface as the intersection of m linear hyperplanes where C is a full rank (m x n) matrix and let us assume that a sliding motion occurs on S. In sliding mode, s = 0 and s = CAx + CBu = 0. Assuming that CB is invertible (which is reasonable since B is assumed to be full rank and s is a chosen function), the sliding motion is affected by the so-called equivalent control 'lCAx Consequently, the equivalent dynamics, in the sliding phase, is defined by xe = \I-B(CB)~1C\ Axe = Aexe (1.6)

The physical meaning of the equivalent control can be interpreted as follows. The discontinuous control u consists of a high frequency component (uhf) and a low frequency one (us): u — Uhf + us. Uhf is filtered out by the bandwidth of the system and the sliding motion is only affected by us, which can be viewed as the output of the low pass filter
TUS

Us — U, T

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This means that ue ~ us and represents the mean value of the discontinuous control u. C being full rank, Cx = 0 implies that m states of the system can be expressed as a linear combination of the remaining (n — m) states. Thus, in sliding motion, the dynamics of the system evolves on a reduced order state space (whose dimension is (n — m)). It is easy to verify that Ae is independent of the control and has at most (n — m) nonzero eigenvalues, depending on the chosen switching surface, while the associated eigenvectors belong to ker(C). As B is full rank, there exists a basis where it is equivalent to the matrix

where B2 is an invertible (m x m) matrix. Let us decompose the state as x = [xf,x^] T , where xi & JR n ~ m , x2 6 H m . Thus, the system (1.5) becomes
x\ =
X2 — Ai2Xi + A22X2 + B2U

and

C=[d

C2]

the (m x m) matrix C2 being assumed invertible (which is the necessary and sufficient condition for CB to be invertible since det(CJ5) = Then one can compute Ae as following Au n-^n *V21 ^1^
I

Ai2

1

— W2

— C2 6*1^22 J
Ai2

L

2

— 1 f~*

°1

0 1 I" AU -Ai2C2lC 1 7 \[ o

7
~t— 1 X~t

0
v_/1

0

yo

J

7"

Under this form, the characteristic polynomial of Ae clearly appears to be = A m P(A 11 -A 12 C 2 - 1 Ci) Thus Ae has at least m null eigenvalues and the sliding dynamics is defined by

x2 = —C2 C\x\

These last equations are interesting since they show that:

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

? designing C is analogous to design a state feedback matrix ensuring the desired behavior for the reduced order system (An, AH], provided that the pair (An,A\2] is controllable (which is the case if and only if the original pair (A,B] is controllable). Then the problem is a classical one which can be solved by the usual control techniques of direct eigenvalue and eigenvector placement or quadratic minimization [4], [28]; ? the dynamics only depends on the matrix AH, A\-z, and not on A^\, A^. For a single-input system, this means in particular, that if the system is written under the canonical controllability form,
/ 0

1

0

0

\

0 \

x=
0 -a0 \

\

then the sliding dynamics is independent from the parameters a^ of the system. Note that this remark can be generalized to multi-input systems. However, observe that, for this kind of system, the design of the control law is more complex than in the single-input case as the required sliding motion must take place at the intersection of the ra switching surfaces. Broadly speaking, at least three strategies can be considered: ? the first one uses a hierarchical procedure where the system is gradually brought to the intersection of all the surfaces. Denoting <Si,..., Sm 771 the m linear hyperplanes such that <S = H Si, and starting from an i=l arbitrary initial condition, the control MI is designed to induce a sliding mode on the surface ?Si, for any control u^, ???, um. This done, the second control u-2 (while the system is still sliding on S\ = 0) leads to Si fl $2 and generates a sliding mode on this surface, and so on till a sliding motion takes place at the intersection of the m switching surfaces (Figure 1.6); ? another solution lies in reducing the system in m single-input subsystems such that every surface Si only depends on the ith component of the discontinuous part of the control. These first two policies lead to a rather simple procedure. However this implies a high prompting and wear of the actuators of the system since

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Figure 1.6: A sliding mode motion with two control functions the control commutes at many more points of the state space than those constituting the sliding surface S. Situations where one control drives the state away from the required intersection by imposing a sliding motion on a subset of surfaces can also occur. A way to face these problems is to make the sliding motion appear only at the intersection of all the manifolds. The control is continuous at the crossing of any separate surface and discontinuous only at the intersection of all of them. For this, the following control laws were proposed (see [7]) [called the unit vector approach],

u = uP —
or

pCx

u = UP —

pMx

where the matrix M and N are such that
ker M = ker N = ker C

1.3.2

Nonlinear systems
x = f(x)+g(x)u(t) (1.7)

Let us now consider the following nonlinear system affine in the control:

and a set of ra switching surfaces S = {x H n : s(x) = [si(x), . . . , sm(x)]T = 0} (1.8)

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

An extension of the previous results leads to: ? the associated equivalent control
ds , J ds ?, ,

obtained by writing that s(x) = |p [f(x] + g(x)u(t)} = 0; dx the resulting dynamics, in sliding mode
9(xe)

i \

f(Xe]

Note that a must be designed such that ||g(x) is regular. However, it is clear that, outside specific cases, the determination of the switching surfaces, in order to get a prescribed dynamics, is not as easy as in the linear case. One of these specific cases is when the system (1.7) can be transform into the so-called regular form [18], [19]: xi = /i(a?i,? 2 ) xi = /20i, x2) + g z ( x i , x2)u

(1.9)

with x\ e K n ~ m , x-2 R m and g-2 regular. Suppose that the control problem is to stabilize the system at a prescribed point with the following dynamics x\ = f ( x i , h ( x i ) ) Defining s(x) = x% — h(x\) and a control u such that a sliding mode occurs on s = 0 solves the problem, and the resulting sliding motion then evolves on a reduced order manifold of dimension (n — m) (#2 can be viewed as the input of the subsystem whose state is x\). This can be illustrated by the example of the two-arm manipulators which can be found in [25]. Yet, the transformation of the system into the regular form can induce complex diffeomorphisms. An alternative is to proceed by pseudo linearization as in [21].

1.3.3

The chattering phenomenon

An ideal sliding mode does not exist in practice since it would imply that the control commutes at an infinite frequency. In the presence of switching imperfections, such as switching time delays and small time constants in the actuators, the discontinuity in the feedback control produces a particular

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chattering

Sliding surface

Figure 1.7: The chattering phenomenon dynamic behavior in the vicinity of the surface, which is commonly referred to as chattering (Figure 1.7). This phenomenon is a drawback as, even if it is filtered at the output of the process, it may excite unmodeled high frequency modes, which degrades the performance of the system and may even lead to unstability [16]. Chattering also leads to high wear of moving mechanical parts and high heat losses in electrical power circuits. That is why many procedures have been designed to reduce or eliminate this chattering. One of them consists in a regulation scheme in some neighborhood of the switching surface which, in the simplest case, merely consists of replacing the signum function by a continuous approximation with a high gain in the boundary layer: for instance, sigmoid functions (see [23]) or saturation functions as shown in Figure 1.8. However, although the chattering can be removed, the robustness of sliding mode is also compromised. Another solution to cope with chattering is based on the recent theory of higher-order sliding modes (see Chapter 3).

sat

Figure 1.8: Saturation function sat(s)

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

The real motion near the surface can be seen as the superposition of a "slow" movement, along the surface, and a "fast" one, perpendicular to this surface (the chattering phenomenon). To put in a prominent position these two movements, let us consider again our introductive example and let us approximate, in an ^-neighborhood of the surface, the signum function by a saturation function whose slope is -. Taking ? as a (small) perturbation parameter, the behavior in the boundary layer can be described, under the standard singularly perturbed form, by
X\ — X2

The slow motion is defined by setting e = 0, hence
1
and

s

-j k

with X\Q being the value of x\ at point M\ (see Figure 1.9). As it has been seen in section 1.2, this corresponds to the dynamics in the sliding motion. In the time scale ~ , the fast motion is defined by

that is and the global motion is approximated by

X2 = X2fi + X2f - X2QS = -TXi0e

-k

k

+ (X2Q + -r

k

which gives the trajectories in Figure 1.9.

1.4
1.4.1

Sliding mode control design
Reachability condition

It has been said that, in the sliding, the motion was independent from the control. Nonetheless, it is obvious that the control must be designed such

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

Figure 1.9: tion

a) Singular perturbed motion e = 0 ; b) Real mo-

that it drives the trajectories to the switching surface and maintains it on this surface once it has been reached. The local attractivity of the sliding surface can be expressed by the condition
a—>0"i"

lim %i(f + gu)<Q

and

lim ff (/ + gu) > 0

or, in a more concise way,
ss < 0

(1.10)

which is called the reachability condition [17].

Example 1 In a way of illustration, let us consider a de-motor modeled by the following transfer function

Y(p) =

1

-U(p)

that is, in a state-space representation:
x\ = X2 X2 = -X2 + u II — Ti y — ?t'l

(1.11)

Let us assume that the sliding surface is designed as

s = X2 + axi = 0, a > 0

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Thus s = (a-l)x2 + u (1.12) Using the control law u — —ksgn(s), k > 0, the reachability condition is satisfied in the domain
A;}

snce
ss
-A:) < 0

One should note that condition (1.10) is not sufficient to ensure a finite time convergence to the surface. Indeed, in the latter example, the control

u = (1 — a)x2 — ks
provides s = —ks, but the convergence to s = 0 is only asymptotic since s(t) = s(0)e- fct where s(0) is the initial value of s. Condition (1.10) is often replaced by the so-called r/ -reachability condition ss<-r,\s\ (1.13)

which ensures a finite time convergence to s = 0, since by integration \s(t}\ - s(0)| < -rjt showing that the time required to reach the surface, starting from initial condition s(0), is bounded by

In a practical way, the control law is generally displayed as u = ue+Ud where ue is the equivalent control (allowing us to cancel the known terms on the right hand side of (1.12)) and where Ud is the discontinuous part, ensuring a finite time convergence to the chosen surface. The example (1.11) was simulated using the following control law
u = (1 — a}x<2 — fcsgns

where the term (1 — a)x2 represents the equivalent control (since s = 0 implies u + (a — I}x2 — 0)- One can also note that the 77 -reachability condition is satisfied. Figures 1.10 and 1.12 show obviously that the sliding

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

motion takes place after about 1.3 sec. Indeed, after this time, the dynamics of the system is represented by the reduced order system given by the chosen surface, i.e.: x\ = —Q.XI = #2 and the control switches at high frequency. In Figure 1.12 one can see that the equivalent control, in sliding motion, represents the mean value of the control u. The portrait phase, in Figure 1.11, illustrates the two steps of the dynamics behavior: first, a parabolic trajectory before the surface is reached (which is called the reaching phase) and then the sliding along the designed line s = 0 (x^ = —ax\) to the origin.

O

O.5

Figure 1.10: Evolution of the states versus time x\ (dotted) and x2 (solid)

1.4.2

Robustness properties

An important feature of sliding mode control is its robustness properties with respect to uncertainties. In the case of invariant and nonperturbed systems, recall first that the use of a continuous component, equal to ue, allows the use of a discontinuous component as small as desired. Indeed, for the sake of simplicity, consider the linear system (1.5) and choose the following controller u = ue — k (CB}~ sgn(s)

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

0.5 I

Figure 1.11: Portrait phase of the sliding motion

=3

0

0

0.5

1

1.5

r
=3 0

-1 i
-3

/?^-r ? r ii ii . v--\
!
0.5

, i ~j——-1 ^-~L~——_
r

I

I

P i i r 1V
2.5

r

0

1

1.5

2

3

3.5

4

4.5

5

t i m e , sec

Figure 1.12: Discontinuous and equivalent control

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

with ue = - (CB]

l

CAx. This implies

ss = sCx = s [CAx + CBue - fcsgn(s)] = -k \s\ < 0 and consequently k might be taken high enough when the trajectory is far from the switching surface (so that the reaching time is short) and then as small as desired in order to limit the chattering. Actually the use of a large enough discontinuous signal is necessary to complete the reachability condition despite parametric uncertainties and exogenous perturbations. Still, to be as simple as possible, consider the system under the canonical controllable form but with parametric uncertainties Aa;
0

1

0

0

\

0 \

X —

1
0

0

\ — a0

1 -a n _i - Aa n _i

0 1 )

where the Aa^ are all supposed to be bounded such that

a~ < |Aai| < oil
Let the switching surface be

s = [CQ ci c n _ 2 1] x = 0
(corresponding to the sliding dynamics pn~l + cn-2pn~2 + . . . + CQ = 0). The control law is chosen as follows

u=

- kn sgn(s)

The ^-reachability condition, (1.13), can be satisfied by two ways, and thus despite the uncertainties: ? if constant gains are set as ko = ao, ki = ai — Cj_i, i — 1,..., n — 1,
n

one gets ss = — ^ A aj_iXjS — kn \s\ and thus setting

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

is sufficient to satisfy (1.13). The magnitude of the discontinuity in the control is a function of the state and of the uncertainties on the process. The control law is easy to design but the discontinuity can be important (and consequently the chattering). ? another solution relies on using commuting gains. Taking &o — ^o + ao, ki = ki + ai - Cj_i, i — 1 , . . . , n - 1 leads to

and the condition ss < —77 \s can be satisfied by choosing kn — rj as a small positive scalar and

The structure of the control law is a little more complex but the amplitude of the discontinuity in the control is reduced. Sliding modes are also known to be insensitive to exogenous perturbations satisfying the so-called matching condition (originally stated and proved by Drazenovic in [6]), that is to say that these perturbations act exactly in the input channels. Considering the perturbed linear system

x = Ax + Bu + A(:r,t)
where A is an unknown but bounded function, the matching condition means that the sliding mode is insensitive to the uncertain function A if it is in the range space of the input matrix B: that is, there exists a known matrix D and an unknown function 5 such that A = DS and rank[B D] = rank B. Indeed, it is easy to show that, in that case, (l-B(CB}~lC\ A = since and thus the dynamics in sliding motion remains independent of the exogenous input A (xe = \I - B (CB}~1 C\Ax — Aex). It is important to note that the system only becomes insensitive to those perturbations during sliding mode but remains affected by the perturbations during the reaching phase (that is to say before the sliding surface has been reached).

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1.5

Trajectory and model following

In the previous sections, variable structure control and sliding modes have been designed for regulation purposes but they can also be used for trajectory and model following.

1.5.1

Trajectory following

Without going into the details, and with the aim of outlining the interest of sliding mode controls in trajectory following, let us consider a simple linear single-input system
y(n)

=u

written in the canonical controllable representation
/ 0 1 ??? 0 0 ^ i

( 0\

x=

n
\ -a0 ???

n

1

i i
-On-l /

Vi

where x = [y, y , . . . , y(n-l)]T. Assume that the control problem is to constrain the output y to follow a prescribed trajectory yd(t) and set

Defining the sliding surface to be s(t) = C(x — Xd) and designing a control law leading to a sliding motion on this surface gives x = Xd- It should be noted, in comparison with the regulation case, that here, the surface is time-varying and that the dynamics of the response is imposed by the desired trajectory (and not by the coefficients of the surface). It should also be noted that this idea can be enlarged to nonlinear multi-input systems. Consider for instance the system
X\ = 3Xi + X-2 + XiX<2 COS 2^2 + #2 = X\ — X2 COS X\ + U2

whose outputs are
2/1
X-2

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The control problem is to constrain these outputs to follow trajectories corresponding to second order responses with respect to step inputs. It is sufficient to take the sliding surfaces
°i — CjCj T~ Cj , 1 —= 1) ^

with &i = xl — Xid , and to generate controls u^ such that SiSi < 0. Taking ui = kuei + anXi + ai 2 X2 + 0:13X1X2 + x\d - k\ sgn BI gives Si = (GI + fcn)ei+(3 -I- aii)xi + (l + 012) x\+(cosx-2 + 0:1 so that with ku — — ci, an — —3, 0:12 = —1
?Ml = (COSX2 + 0:13) Xi±2S\ - k\ \Si\

Thus, taking implies < 0, Vfci > 0 The control u<2 can be designed similarly such that 52*2 < 0. Then each output follows the predefined trajectories.

1.5.2

Model following

Variable structure control and sliding mode can also be used for model following, that is to control the process in such a way that it behaves like a given model (of the same order). The idea is to force a sliding motion on the surfaces
O —— J\f>[?fYi *? ) "~ -t-^e^e ~~: ^

where x and x m are respectively, the process and model state vectors. It is easy to see that, in sliding motion, the error dynamics is given by ±e = 1 - QAKe

with 6 - B(KeB}~lKe.
Except for the case of perfect matching, which supposes that rank [B, Bm] — rank [??, Am - A] — rank B there exists a steady-state error which can be computed by the equation [(1 - 9) A]T \ _(-[(!- 6) A]T A^BmUr,

~

0

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

where [(1 — 0) A]T denotes the matrix constituted by the (n — m) independent lines of (1 — 0) A. In the general case, when the conditions can not be met, one will only focus on the outputs and integrators to be added on the error ym — y so that the steady state error is null (Figure 1.13).
u

x = Ax + Bu y = Cx
X

y

y-xe +

V b ^X

m

?
71

?>

u

m

ym — ^m-Em Model

ym

K-i

Figure 1.13: Model following By way of illustration, let us consider the following case of a process given by the transfer function
v

'

2 S

+ 4(5s + 4

where p and 6 are parameters which may vary. The control problem is to follow a model corresponding to
rov

'

s2 +1.45 + 1

The following figure shows the results of simulations enhancing the fact that the model following scheme is able to cope with important parametric variations. In Figure 1.14, continuous variations of S and p have been assumed such that 6 = | and p = 2 4- |t (that is to say, for the span time of 9 seconds, 6 is varying from 0 to 1 and p from 2 to 14). As far as the problem of model following is concerned, variable structure control laws using sliding modes can also be found in [12], [29] or [30].

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

Figure 1.14: Example of model following

1.6

Conclusion

In this introductory chapter, the basic properties and interests of sliding modes have been enhanced. Since this technique involves differential equations with discontinuous right-hand sides, the concept of solution needs to be redefined and alternative approaches to the classical ordinary differential equation theory must be developed. One concerns differential inclusions and is presented in Chapter 2. The main benefits of sliding mode control are the invariance properties and the ability to decouple high dimensional problems into sub-tasks of lower dimensionality. However, it has been shown that imperfections in switching devices and delays were inducing a high-frequency motion called chattering (the states are repeatedly crossing the surface rather than remaining on it), so that no ideal sliding mode can occur in practice. Yet, solutions have been developed to reduce the chattering and so that the trajectories remain in a small neighborhood of the surface, like the higher-order sliding modes developed in Chapter 3. The continuous case has been considered in this introduction, but the problems induced by sliding modes under sampling and in the presence of delays are treated in Chapters 8, 10, 11. The control problem given here was a regulation one and the illustrative examples were quite simple. However, sliding modes find their application in many other area such as observers (Chapter 4), output feedback (Chapter 5) or trajectory following (Chapter 6), and in practical applications such

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

as robotics (Chapter 13) and control of induction motors (Chapter 14).

References
[1] C. Bigot and A.J. Fossard, "Compensation et auto adaptation passive par lois pseudo lineaires", Automatisme, Mai 1963. [2] F.H. Clarke, Y.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth analysis and control theory, Graduate Texts in Mathematics 178, Springer Verlag, New-York, 1998. [3] R.A. De Carlo, S.H. Zak, G.P. Matthews, "Variable structure control of nonlinear variable systems: a tutorial", Proceedings of IEEE, Vol. 76, pp. 212-232, 1988. [4] C.M. Dorling and A.S.I. Zinober, "Robust hyperplane design in multivariable structure control systems", Int. J. Control, Vol. 43, No. 5, pp. 2043-2054, 1988. [5] S. Drakunov and V. Utkin, "Sliding mode observers. Tutorial", Proceedings of the 34th Conference on Decision and Control, NewOrleans, LA, December 1995. [6] B. Drazenovic, "The Invariance Conditions in Variable Structure Systems", Automatica, Vol. 5, No. 3, pp. 287-295, 1969. [7] C. Edwards and S. Spurgeon, Sliding mode control: theory and applications, Taylor and Francis, 1998. [8] O.M.E. El-Ghezawi, A.S.I. Zinober and S.A. Billings, "Analysis and Design of Variable Structure Systems using a Geometric Approach", Int. J. Control, Vol. 38, No. 3, pp. 657-671, 1983. [9] S.V. Emelyanov, "On pecularities of variable structure control systems with discontinuous switching functions", Doklady ANSSR, Vol. 153, pp. 776-778, 1963. [10] F. Esfandiari and H.K. Khalil, "Stability Analysis of a Continuous Implementation of Variable Structure Control", IEEE TAG, Vol. 36, No. 5, pp. 616-620, 1991. [11] A.F. Filippov, Differential equations with discontinuous right handsides, Mathematics and its applications, Kluwer Ac. Pub, 1983.

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[12] A.J. Fossard, "Commande a structure variable: poursuite approchee du modele. Application a 1'helicoptere", (in French) Rapport Technique DERA 167/91, Decembre 1991. [13] K. Furuta, "Sliding mode control of a discrete system", Systems and Control Letters, Vol. 14, pp. 145-152, 1990. [14] J. Guldner and V.I. Utkin, "Tracking the gradient of artificial potential fields: sliding mode control for mobile robots", Int. J. Control, Vol. 63 (1996) No. 3, pp. 417-432. [15] B. Hamel, "Contribution a 1'etude mathematique des systemes de reglage par tout ou rien", (in French) 1949. [16] B. Heck, "Sliding mode control for singularly perturbed systems", Int. J. Control, Vol. 53, pp. 985-1001, 1991. [17] U. Itkis, Control systems of variable structure, Wiley, New-York, 1976. [18] A.G. Lukyanov and V.I. Utkin, "Methods of Reducing Equations for Dynamic Systems to a Regular Form", Automation and Remote Control, Vol. 42, No. 4, pp. 413-420, 1981. [19] W. Perruquetti, J.P. Richard and P. Borne, "A Generalized Regular Form for Sliding Mode Stabilization of MIMO Systems", Proceedings of the 36th Conference on Decision and Control, December 1997. [20] H. Sira-Ramirez, "Differential geometric methods in variable-structure control", Int. J. Control, Vol. 48, No. 4, pp. 1359-1390, 1988. [21] H. Sira-Ramirez, "On the sliding mode control of nonlinear systems", Systems and Control Letters, Vol. 19, pp. 303-312, 1992. [22] H. Sira-Ramirez and S. K. Spurgeon, "Robust Sliding Mode Control Using Measured Outputs", J. of Math. Systems, Estimation and Control, Vol. 6, No. 3, pp. 359-362, 1996. [23] J.J.E. Slotine, "Sliding controller design for nonlinear systems", Int. J. Control, Vol. 40, No. 2, pp. 421-434, 1984. [24] J.J.E. Slotine, J.K. Hedrick and E.A. Misawa, "On sliding observers for nonlinear systems", Transactions of the ASME: Journal of Dynamic Systems Measurement and Control, Vol. 109, pp. 245-252, 1987. [25] J.J.E. Slotine and W. Li, Applied nonlinear control, Prentice Hall, Englewood Cliffs, NJ, 1991.

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[26] W.-C. Su, S. Drakunov and U. Ozguner, "Implementation of variable structure control for sampled data systems", Proceedings of the IEEE Workshop on Robust Control via Variable Structure and Lyapunov Techniques, Benevento, Italy, pp. 166-173, 1994. [27] Y. Z. Tzypkin, Theory of control relay systems, Moscow: Gostekhizdat, 1955 (in Russian). [28] V.I. Utkin, Sliding Modes in Control Optimization, Communication and Control Engineering Series, Springer-Verlag, 1992. [29] K.K.D. Young, "Asymptotic stability of model reference systems with variable structure control", IEEE TAG, Vol. AC-20, No. 2, pp. 279281, 1977. [30] A.S.I. Zinober, O.M.E. El-Ghezawi and S.A. Billings, "Multivariablestructure adaptative model-following control systems", Proceedings of IEE, Vol. 129, pp. 6-12, 1982.

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Chapter 2

Differential Inclusions and Sliding Mode Control
T. ZOLEZZI Dipartimento di Matematica, Genova, Italy

2.1

Introduction

A basic problem in the field of variable structure control is the following. We are given a controlled system of ordinary differential equations with prescribed initial value x = f(t,x,u),x(Q) =a where the dynamics are denned by a given function
/ : [0, +00) x Q x U -> RN

(2.1)

and the fixed initial condition a fi which is an open set in M N and U is a closed set in E M . The M-dimensional control variable ueU

(2.2)

is constrained to belong to the given control region ?7; the TV-dimensional state variable x is required to fulfill a given sliding condition s[x(t)] = 0 for all t
P

(2.3)

where s : R^ —>? IR is a fixed mapping which defines the sliding manifold

s(z) = 0

(2.4)

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

The overall problem is to select an admissible control law u = u(t,x), usually in the feedback form, such that through the corresponding state x issued from a at time 0 sends in finite time the initial position a to some point x(t*} fulfilling (2.3) and keeps the state vector x ( t ) on the sliding manifold (2.4) for all t > t* (in a prescribed time interval). For simplicity we treat the case that s does not depend on time, even if what follows can be extended to the more general sliding condition s(t,x) = 0 In these notes we deal only with the mathematical description of the sliding motion, assuming that a suitable control law has been found which solves the attainability problem, to reach in finite time the sliding manifold (hence we assume t* = 0). At least three methods are known to control the given system in order to fulfill the state constraint (2.3). Componentwise sliding control. Let P = M, then a suitably defined pair of feedback control laws uf = u ~ i ( t , x } , u ~ — u~(t,x),i = 1,... ,M are used for each component of s to obtain the control law u*(t,x) = u^(t,x) if Si(x) > 0

u* (t, x) — u~ (t, x) if Si(x) < 0
Here Si(x) denotes the i—th component of the vector s ( x ) . Proper choice of u^ and of u~ allows us to keep x(t] on the sliding manifold (2.3).

sl(x) < 0
Unit control. Let

f ( t , x, u] = A(t, x) + B(t, x)u

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and denote by Ds the Jacobian matrix of s. Let E(t,x) = Ds(x)B(t,x). Then, under suitable nonsingularity assumptions, the control law u(t,x) = -a(t,x)E(t,x)'s(x)/\E(t,xYs(x)\

with a proper choice of the gain a allows us to reach the sliding manifold (2.3) and to keep the state vector on it. Sliding mode simplex method. For every ?,?, points u i ( ? , x ) , . . in U are found such that the vectors gi(t,x) = D s ( x ) f [ t , x , U i ( t , x ) } form a simplex in R .
p

s(x)
92

For every x, s(x) belongs to some cone generated by the edges gi(t, x), i ^ /i, for the smallest index h. Then the choice of the control law u*(t,x) - Uh(t,x) guarantees the sliding mode condition under suitable assumptions about the shape of the simplex.

2.2

Discontinuous differential equations and differential inclusions

All the above control methods share the following basic feature: the corresponding control law u* undergoes discontinuities as a function of x. More precisely, u* is (quite often) a piecewise continuous function of x. By inserting u* into (2.1) we are forced to consider states x of the control system such that x(t) = f ( t , x ( t ) , u * [ t , x ( t ) ] ) and the corresponding dynamics <?(*,*) = /[t,zX(i,z)] (2.5)

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is a discontinuous function of x. A basic issue of the mathematical description of the sliding mode control method is then the following. Which is the meaning of the solution concept of the differential equation x —g(t,x),x(Q) = a with a discontinuous g ( t , ?)? Example 2 There exists (almost everywhere) no solution of the scalar equation x = - sgn x,x(Q} = 0 (2.7) Here sgn x — x/\x\,x ^ 0, sgn (0) = 1. —sgn a;
1
(2.6)

-1

The previous example shows that, in general, discontinuous initial value problems (2.6) fail to possess classical (i.e., almost everywhere) solutions. A generalization of the concept of solution is required. A natural way of modifying the solution concept to (2.7) is to enlarge the right-hand side at 0, taking into account the behavior of g(x) — — sgn x when x ^ 0. This leads us to consider the multifunction G : K —> R defined by G(x) = {g(x}} = {- sgn x}, x + 0; G(0) = [-1,1] and the initial-value problem for the differential inclusion
(2.8)

which has the constant solution y(t) — 0 for every t. The set-valued function G agrees with the singleton {g(x}} whenever g is continuous; at 0, G(0) is obtained by taking the set of all values of g(x) as \x\ is sufficiently small and > 0, that is { —1,1}, then its convex hull [—1,1] and finally the intersection when x\ —> 0 (which here has no effect). In this way we restore existence without loosing contact with the original equation (2.7).

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

A

G(x)

-1

Let us remark that the existence behavior of (2.7) is very sensitive to changes of the initial value. Example 3 The scalar equation
x — — sgn x,

=a

(2.9)

has (everywhere) local solutions for each a ^ 0 given by x(t) = a — t, 0 < t < a or x(t] = t + a,Q <t < —a. If we consider
g(x) — 0, x = 0; g(x] = — sgn x, x ^ 0

then (2.9) has (almost everywhere) global solutions (i.e., on the whole time interval [0, -foo) for every initial value a, namely x(t) = (a — t)+ if a > 0, x(t) = (a + t)- if a < 0, x(t) - 0 if a = 0.

2.3

Differential inclusions and Filippov solutions

We consider first initial value problems for differential inclusions and briefly review some existence theorems.

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

We are given a multifunction (set-valued mapping)
C< ? ii _ > M N (JT . O —, 1B>

where Q is an open set of RN , which takes on nonempty values G(x) C RN . Existence of classical (i.e., almost everywhere) solutions to the initial value problem xG(x),x(Q) = a (2.10) is related to continuity properties of G, as follows. G is called ? upper semicontinuous at x0 6 Q if for every open set A such that G(x0) C A, we have G(x] C A for all x sufficiently close to x0; ? lower semicontinuous at x0 e Q, if for every open set A such that G(x0) fl -A 7^ 0 we have (7(a:) D ^4 / 0 for all x sufficiently close to x0. Example 4 Consider

Then G\ is upper semicontinuous, not lower semicontinuous at 0. Consider

Then G^ is lower semicontinuous, not upper semicontinuous at 0.

1

'

— *, v

/ ; 1

I Gi(x)

X

X

-1
-1

A solution to the initial- value problem (2.10) is a function

for some positive T < +00 such that its derivative exists for almost all t G (0, T) and it is locally integrable, fa ydt = y(b] — y(a] for every pair a, b in (0,T), and y(t] e G[y(t)} for almost all t e (0, T) (2.11)

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The conditions imposed on y in the previous definition [except (2.11)] amount to local absolute continuity of y. Control problems quite often require examining the behavior over a prescribed time interval, for example [0, +00) if asymptotic stability is the main issue. For this reason, global existence theorems are most significant. Theorem 5 (Existence) Let G be nonempty compact convex valued and upper semicontinuous. Suppose there exist constants A, B such that sup {\u\ : u G(x)} < A\x\ + B for every x Then problem (2.10) has solutions on [0, +00) for every a G f2. Example 6 Let G(x] = {- sgn x} if x ^ 0,G(0) = {-1,1}. Then G is upper semicontinuous, compact valued, convex valued except at 0 (and G fails to be lower semicontinuous at 0). The initial value problem zeG(z),z(0) = 0 lacks existence.

G(x}

-I

The previous example shows that convexity of G(x) for every x cannot be omitted in the existence theorem 5. The previous theorem can be extended to time-varying right-hand sides xeG(t,x),x(Q} = a under suitable measurability and growth properties of G. Theorem 7 (Existence) Let the multifunction G: [0,+oo) x R ^ ^ E ^ be nonempty closed convex valued and such that G(t, ?) is upper semicontinuous for all t, for every x there exists a measurable function h such (2.12)

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

that h(t] 6 G(t,x) for almost all t > 0, and there exist locally integrable functions b, c such that sup {\u : u e G(t,x)} < &(?)|a;| + c(t] for almost all t > 0 and every x. Then (2.12) has solutions on [0, +00). Both existence theorems 5 and 7 require upper semicontinuity of the right-hand side. If the right-hand side is lower semicontinuous (a case of less interest in the next developments) with respect to the state variable, then an existence theorem similar to theorem 7 holds without requiring convexity of the values. We come back to initial-value problems for discontinuous differential equations x = g(t,x),x(0) = a (2.13) We have seen that the concept of solution to (2.6) needs to be properly redefined in order to guarantee existence. The basic definition we are going to review is due to Filippov, as follows. Let

g : [0, +00) x 0 -? RN
be measurable and such that for every A there exists B = B(t) locally integrable such that almost everywhere
sup {\g(t, x)\ : t + \x\ < A} < B(t)

We associate to g the multifunction G, as follows. Denote by B(x,e] the ball in R^ of center x and radius e. Consider the set
{g(t, y):ye B(x, e)} = g[t, B(x, e)], t > 0, x G ft

Then let G(t, x) = r\c\ co {g[t, B(x, e) \ L] : e > 0, meas L = 0} (2.14)

where cl co A denotes the closed convex hull of A, i.e., the intersection of all closed convex sets containing the set A. Definition 8 A Filippov solution y to (2.6) is a locally absolutely continuous function y : [0, T) —? R^ such that

y(t)?G[t,y(t)]
for almost every t E (0, T).

(2.15)

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Thus the Filippov definition replaces the discontinuous differential equation (2.6) by the differential inclusion (2.15). The construction of G from g generalizes what we have seen after example 2. Removing sets of measure 0 from the values taken by g corresponds, roughly speaking, to purposedly ignoring possible misbehavior of the right-hand side in (2.6) on small sets. For every t and x,G(t,x) defined by (2.14) turns out to be a nonempty closed convex set and the multifunction G(t, ?) is upper semicontinuous, moreover if g(t, ?) is continuous at z then G(t,z) = {g(t, z}}. It follows that, if g ( - , x ) is a measurable function, and g(t, ?) is everywhere continuous, then y is a classical solution to (2.6) if and only if y is a Filippov solution. For control systems (2.1) with discontinuous feedback control u* = u*(t,x) we obtain as a particular case the notion of Filippov solutions to (2.5). Hence x = f(t,x(t},u*[t,x(t)]) if x is a Filippov solution to (2.1) with u — u*,/ is smooth and u*(t, ?) is continuous at x(t}. Quite often in applications, / is a smooth function and the discontinuous behavior is due to the insertion of the discontinuous control feedback u* inside /. The properties of the multivalued function G given by (2.14) allows us to apply the existence theorems for initial value problems of differential inclusions. Theorem 9 (Existence) There exist Filippov solutions to (2.1) with u = w*(?, x) on [0,+00) provided: ? SI = R w , /(-,#, u) is measurable for every x and u, f ( t , ?, ?) is continuous for almost every t>Q; ? there exist locally integrable functions 6, c such that \f(t,x,u)\<b(t)\x\+c(t) for almost all t > 0, every x and every u G U; ? u* is measurable. We refer the reader to [2] as far as the physical meaning of Filippov solutions is involved, showing that this notion has not only a proper mathematical meaning but, as documented in [2], also a physical significance, which is relevant for control applications. (However there are stabilization problems in nonlinear control via discontinuous feedback in which Filippov solutions are not adequate, and something different must be used.) Using directly the Filippov definition based on (2.14) is often rather complicated. In the following we describe an explicit formula which allows

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

us to obtain, in a simple yet useful case in practice, an explicit expression for the Filippov dynamics. We shall consider scalar control, smooth sliding manifold of codimension one, and piecewise smooth dynamics. More precisely, suppose that

s is continuously differentiate, its gradient Ds(x) ^ 0 whenever s(x) = 0, and the smooth surface 5 defined by (2.4) partitions fi in two disjoint open sets G~,G+ (with common boundary S). Assume that g given by (2.5) is bounded and its restriction to both G+ , G~ converges as x —? x0 G S to limiting values g+(t, x0),g~(t,x0}, respectively, for all x0 e 5. Denote by SW'SW the projections of g ,g~ on the unit normal vector N to S at each point, oriented from G~ to G+. Let y be absolutely continuous in a given time interval such that, for every t,
+

Then y is a Filippov solution to (2.6) if and only if for almost every t

where
9~N

or explicitly

' ^ ? ^ '
with everything on the right being evaluated at t , y ( t ) . Thus, in this case, the Filippov dynamics is obtained explicitly as a convex combination of the vectors g+ ,g~ .
N.

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2.4

Viability and equivalent control

The basic problem we started with, was to find a feedback control law u* such that the solution to (2.1) corresponding to u* fulfilled the sliding condition (2.3). The appropriate meaning of state variable corresponding to the possibly discontinuous feedback u* through (2.1) was obtained via the Filippov definition discussed in the previous section. In this section we take into account the state constraint (2.3) and consider a more general version of the resulting problem, to find solutions to the following problem x e F(x), x ( t ) e K for all t where the multifunction and the closed set K C R are fixed (fi being an open set of In order to avoid technical points we consider only autonomous differential equations (2.17), i.e., we assume that F does not depend upon t. Our control problem (2.1), (2.2), (2.3) obtains as a particular case provided / = /(#, w), i.e., the given dynamics is time-invariant, by taking F(x) = f ( x , U} = {/(x, u) : u <= U}, x e 0 (2.18) (2.17)

which is the set of all admissible velocities (so to speak) of the given control system, and K = {zeSl: s(z) = 0} (2.19) Let us pause to remark that (by a known result), if / is continuous and U is compact, then the set of all trajectories of the control system

x(t} = f[t,x(t},u(t}],u(t}

U almost everywhere

corresponding to open loop (measurable) control laws u(-), coincides with the set of all solutions to the differential inclusion x(t)f(t,x(t),U) Even if this result has no relevance for us here, it shows that differential inclusions can provide a convenient mathematical framework for the study of certain control problems. A solution y to the differential inclusion

x e F(x)

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is called viable for K if y fulfils (2.17), i.e., y(t) K for all t. So we are interested in those special solutions (if any) of the differential inclusion in (2.17) which fulfill the sliding condition defined by the constraint set K. More precisely we are given the initial value a = x(Q) G K and we look for conditions guaranteeing that there exists at least one viable solution, i.e., a solution to (2.17) issued from a at 0. If F is single-valued, i.e., we are considering a system of ordinary differential equations x = g ( x ) , it is natural to impose, as a sufficient condition to viability, that the dynamics be tangent to the set K. This will force a solution starting on K to remain there forever.

K

Then we are led to consider the tangent cone to K at a given point x K, T(K,x) which is the set of all points w = lim n _ >00 (x n — x)/tn where the sequence xn 6 K, xn —> x and the sequence of positive numbers tn —>? 0. If K is a smooth surface 5 obtained by (2.19), then the tangent cone T(K,x) turns out to be the tangent space of S at x. We denote by
Ds(x)

the P x N Jacobian matrix of s at x, whose (j, h) element is given by

where Sj is the j-th component of s. We have Proposition 10 Let K be given by (2.19). Let s e Cl(RN,Rp) be with Ds(x) of maximum rank if s(x) = 0. Then T(K, x} = {w e R^ : Ds(x)w = 0} Now we consider the autonomous control problem x 6 F ( x ) , x ( Q ) = a, x ( t ) E K (2.20)

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where F is given by (2.18) and K is a given closed set. The differential inclusion (2.20) models (as a particular case) the control problem. Indeed, all states from (2.1) (if time-invariant) corresponding to arbitrary control laws fulfilling (2.2) obey (2.20) for almost all t. Moreover, all Filippov dynamics x corresponding to discontinuous feedback control laws from U fulfill (2.20) provided U is compact, / is continuous, and f ( x , U] is convex for all x e fi. (2.21) The main point is the following. The tangency condition we discussed before turns out to be a necessary and sufficient viability condition, which shows (in principle) how to control the system in order to fulfill the sliding condition (2.3). Theorem 11 (Viability) Suppose that (2.21) is verified and assume that |/(x, u)\ < a\x\ + b for suitable constants a,b and for every x ? fi,u G U. Then the following are equivalent for every a ? K there exists a solution x to (2.20) on [0, +00); F(x) n T(K, z) ^ 0 for every xeK (2.22)

Let us write down the viability condition (2.22) in the case of interest, i.e., K is defined by (2.19) and s is as in Proposition 10. Then (2.22) is true if and only if for every x ? K there exists some point u = u(x) ? U such that Ds(x}f(x,u)=Q (2.23) Condition (2.23) can be obtained formally by differentiating the sliding condition s[x(t)] = 0 and working as x were a classical solution of (2.10), which could be false as we know, since Filippov solutions are not pointwise solutions. Then an equivalent control u for (2.1), (2.2) and (2.3) (in the time invariant case we are discussing), is any feedback control law u such that (2.23) holds and the classical solution to (2.1) corresponding to u verifies the sliding condition (2.3). This last requirement is automatically true provided s[rc(0)] = 0 since for almost every t d/dts[x(t)] = Ds[x(t)]f[x(t),u(x(t})] =0

hence s[z(?)] is constant. If U is compact, / is continuous and in addition we assume that the mapping

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for every x ? Q is one-to-one on a neighborhood of ?7 to Mp and its range contains 0, then the equivalent control u — u(x) defined by (2.23) is uniquely defined and is a continuous function of x in i7. Unfortunately it is not true (even if it is tempting to admit) that the sliding dynamics corresponding to the equivalent control agree with those obtained via Filippov's concept of solution.
co/(x,t/)

Example 12 The control system is (N = 2) Xi = 0.80:2 + ux\,x-2 = — O.Txi + 4u3xi the sliding manifold (P = 1) is defined by

s(x) = Xi + x-2
and the scalar control u e [—1,1]. The discontinuous feedback control we consider is given by u*(x] = — sgn (s(x)xi) which can be shown to guarantee the sliding condition. Thus the control where s(x) > 0 is given by u+ = — sgn x\, while u~ — sgn x\ is the control law where s(x) < 0. Here the equivalent control is the constant u = 0.5 obtained as the unique real root of u + 4-u3 = 1, giving rise to the dynamics Xi = 0.2a:i on x\ + x-2 = 0 By applying (2.16) we get the Filippov dynamics Xi = —O.lxi on x\ + X2 = 0 which is different from that corresponding to the equivalent control.

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u* = 1

X-2

u* = -1
u* = -1

u* = -1

u* = 1

\

u* = -1

x\ + x-2 = 0

However, the sliding dynamics obtained by using the equivalent control agree with Filippov's dynamics in the L— particular case of control systems cj J. A u o VQ orrin^i in ^-Vio r?/~vnl-Y?/~vl 01 i m ol i a whichi are affine in the control signal, i.e.,
/(t, x, u) = A(t, x) + B(t, x)u
(2.24)

where A, B are matrices of the appropriate dimensions.

A + BU+

s(x) = 0

Example 13 / / / is given by (2.23) with scalar control u,M = P = l,A and B being measurable with respect to t and continuous with respect to x, and s is continuously differentiate, then a sufficient condition to existence and uniqueness of the equivalent control is that Ds(x)-B(t,x) 7^0 for almost all t and every x. Then the equivalent control is given by u(t, x) = -Ds(x) ? A(t, x}/Ds(x) ? B(t, x) and is again measurable in t and continuous in x. More generally, for multivariable control systems (2.24) with M = P, a sufficient condition for the existence and uniqueness of the equivalent control is that the M x M matrix Ds(x)B(t,x) is everywhere nonsingular. In this case the equivalent control is u(t,x) = -[Ds(x}B(t,x)]-lDs(x)A(t,x)

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The equivalence between Filippov and equivalent control states deals with the following situation (componentwise sliding mode control described in section 2.1). Let (2.24) hold, M = P, and each Si be continuously differentiable, i = 1, . . . , M . Then for each x with s(x) = 0, every sufficiently small neighborhood of x turns out to be a disjoint union of open regions GI, . . . , Gq and points of the sliding surface. We are given q = 1M feedback control laws U i ( t , x ) , which are measurable in t and continuous in x. Let y be absolutely continuous in the given time interval [0, T] such that = 0 for every t. Theorem 14 (Equivalence) y is a Filippov solution to (2.1) corresponding to the feedback u* defined by u\ on G\, . . . , uq on Gq if and only if y is a classical solution to (2.1), corresponding to the equivalent control, provided U is closed convex and Ds(x}B(t,x) is nonsingular for every t and x close to S . Proof of a particular case of Theorem 14. Let M = P = I and suppose that the conditions leading to (2.16) are met. Then the dynamics corresponding to the equivalent control are as in Example 13, namely

y = A - B(Ds ? A)/Ds ? B
In order to compare this with (2.16) we write w* = u+ if s(x) > 0, u* = u~ if s(x) < 0, and compute (Ds ? g-}g+ - (Ds ? g+}g- = [(Ds ? A}B - (Ds ? B}A](u+ - u~] thus, by (2.16) , the conclusion. The practical value of Theorem 14 is obvious. For control systems (2.24) (under the above conditions), all calculations involving Filippov sliding mode controls can be correctly performed by formally differentiating the sliding condition (2.3) and working with states corresponding in the pointwise (classical) sense to the equivalent control; no discontinuous differential equation is involved at this stage. An interesting property of the equivalent control, assuming (2.24) and suitable smoothness properties, involves the convergence of states, fulfilling only approximately the sliding condition, to the sliding state corresponding to the equivalent control, when the boundary layer width tends to disappear (regularization procedure). This fact will be discussed from a more general point of view in the next section.

2.5

Robustness and discontinuous control

Feedback control is important, among other reasons, mainly because of its robustness properties. In this section we briefly summarize a mathematical

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

interpretation of a form of robustness which deals with the dynamic behavior of sliding mode control systems under discontinuous feedback, and lies at the roots of practical control methods. Given the variable structure control system (2.1), (2.2) and (2.3) we distinguish between ? real states which are solutions to (2.1) fulfilling only approximately the sliding condition and ? ideal states which solve (2.1) and fulfill exactly condition (2.3). The following problem is relevant in this connection. Find conditions on the variable structure control system (2.1), (2.2) and (2.3) such that the following two properties hold: ? for every sequence of real states, whenever their initial values converge to the sliding manifold, then they converge towards a well-defined ideal state; ? one can approximate any ideal sliding state by real states fulfilling only approximately the sliding condition as the sliding error tends to zero. We would like to obtain such robustness properties, no matter what the reasons are of violating the sliding condition (like disturbances, control errors, uncertainties, delays, etc.). Taking into account the discussion of Section 2.4 we assume that s is continuously differentiate and the mapping Dsf(t,x,-) takes on the value 0, and is one-to-one on U for all x in some neighborhood V of the sliding manifold (2.4) and almost every t. We denote by u(t,x,w) the unique solution u 6 U of D s ( x } f ( t , x , u ] = w for a given iu, hence the equivalent control is now w(t,o:,0). Given p > 1,T > 0 and m(t) > 0 such that JQ [m(t}]pdt is finite, let H denote the set of all parametrized functions ae(t}, > 0, such that < m(t) and sup {| / a e (s)ds|;0 < t < JQ Given ae 6 H suppose that xe solves almost everywhere (2.1) with u = ?u[i,x,o e (t)]. Then d/dts[x?(t)] = Ds[x?(t)]f[t,Xt(t),u?(t)] = a?(t) (2.25)

where ue = u[t, x ? ( t } , a e ( t } } . Integrating (2.25) between 0 and t we get s[x?(t)] —> 0 uniformly on [0,T] as e —* 0

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

The parameter e describes the amount of violation of the sliding condition (2.3) due to some imperfection (whatever they be). The sliding error is measured by a e . Let y be a classical solution on [0,T] of (2.1) corresponding to the equivalent control -u(-,-,0) such that s[y(0)] = 0, hence s[y(t)} — 0 for all t 6 [0,T] (because of the definition of u). The required robustness conditions are then satisfied provided the control system fulfills the following approximability property in (0,T): for every at in H such that u [ t , x , a e ( t ) } exists for almost every t and x S F, if we have s[z e (0)] ->0 as e - + 0 then x e (0) —* y(0) implies xe —> y uniformly on [0,T]. Thus we have the following behavior provided (2.1), (2.2) and (2.3) satisfies the approximability property. If the control law we are employing yields small sliding errors, then reduction of the sliding error at the initial time implies uniformly small deviations from the desired (sliding) dynamical behavior (described by the equivalent control). Thus all real states converge to a well defined (uniquely determined) sliding state of the control system as the disturbances disappear, provided the initial values tend to the sliding manifold. Therefore approximability holds if and only if we can uniformly approximate any ideal sliding state by real states, disregarding the particular nature of the disturbances which are responsible for the sliding errors.

Example 15 The control system is

N = 3,M = P = 2, with control constraint \ui\ < 1, \u-2 < I ; the sliding manifold is given by si(x) — x\,s<2,(x) = x^j the initial condition is x(0) = 0. Here the equivalent control u = 0 gives rise to the motion y(t} = 0 for all t. Partition the time interval in 2n equal subintervals and consider the control laws uin(t) = U2n(t) = — 1 or +1 alternatively. Then for the corresponding states, as n —> +oo,xin(t) —?> 0, X2n(t) —>? 0, however x^n(t) = t for every n. Approximability fails (and the sliding state z(i] = (0,0, t)' does not correspond to the equivalent control). The dynamic behavior of the system on the sliding manifold is in some sense ambiguous, and lacks robustness: by reducing the sliding error the corresponding real states do not converge to y.

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

?3

z(t]

X-2

It can be proved that, under suitable smoothness and nonsingularity conditions, approximability is verified in each of the following cases:

f ( t , x, u) = A(t, x] + B(t, x)u
f(t,x,u) = (x2,x3,...,xN,g)

(2.26)

where g = g(t,xi,X2,... ,XN,U) is strictly monotone with respect to the scalar control variable u. Approximability is a theoretical basis to justify on rigorous grounds several sliding mode control procedures as far as their robustness properties are involved.

2.6

Numerical treatment
x 0 = a 0 < ? <T

The simplest way to solve numerically the initial value problem

is to look for a suitable extension of the classical Euler method, as follows. Choose a uniform grid
0 < ti < t2 < . . . < tn = T

with step size h = T/n, n a given positive integer, hence

Let x0 = a and for j = 0, 1, . . . , n — 1 compute any point

such that

e Xj +

(T/n)G(tj,Xj)

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Consider the corresponding piecewise affine continuous function yn(t) = Xj + (n/T}(t - tj}(xj+i - Xj),tj < t < tj+i,j = 0 , 1 , . . . ,n - 1

Then yn can be considered as an approximate solution to (2.12) on [0,T]. Theorem 16 (Convergence) As n —> +00, yn converges uniformly on [0,T], up to subsequences, to some solution of (2.12) provided G is upper semicontinuous with nonempty compact convex values and sup {\z\ : z e G(t,x)} < k\x\ -f h for every t,x and some constants k, h. Thus convergence of the Euler method is guaranteed for discontinuous feedback control systems (under the previous assumptions). More refined methods, known to have better convergence properties when applied to smooth differential equations, cannot be guaranteed to converge when extended more or less directly to apply to, say, (2.15). Indeed, smoothness properties under which convergence is guaranteed for differential inclusions, are usually not satisfied for piecewise continuous differential equations. If applicable, such methods require special care to handle discontinuous differential equations. See also Chapter 8 of this book (Discretization Issues, by J-P. Barbot et al.).

2.7

Mathematical appendix

We collect here a few mathematical definitions which have been used in these notes. A bounded subset A of the real numbers has Lebesgue measure zero if for every E > 0 there exists a countable collection B of disjoint intervals Bn, n — 1, 2, ? ? ? , such that A c U{Sn : n = 1,2, ? ? ? } and the total length of B, i.e. Z^^ (length Bn) is < ?. Any finite set, the set of points of any sequence, the set of all decimal numbers in a given bounded interval are all examples of sets of measure 0 in R. Almost everywhere means except of a set of measure 0. Hence (Section 2.2) if x is an almost everywhere solution of the differential equation x = g(t, x) on some bounded interval, then x ( t ) — g[t,x(t)} for all t except those in a set of measure 0 (which could be empty of course). The family of all Lebesgue measurable subsets of RN contains all compact and all open sets, all subsets of sets of measure 0 (which are de-

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fined similarly as the case N = 1), and it is invariant under complementation, countable unions and intersections. A given real-valued function / : R^ —> R is Lebesgue measurable if and only if all sublevel sets {x RN : f ( x ) < c] are measurable (for all real c). A vector-valued function is measurable if and only if its components are. Very roughly speaking, most of the functions we encounter in the control sciences are indeed measurable. A function y : [p, q] —> RN is absolutely continuous if and only if y has a derivative y(t) at almost every point t of the interval [p, g],y is integrable there and for all pairs of points a, b in [p, q] we have Jaydt = y(b) — y(a). Hence y(t) = y(p) + f y(s) ds,p < t < q, which allows us to represent the absolutely continuous function y via its derivative. Of course every continuously differentiable function is absolutely continuous (e.g., any classical solution of (2.6) with a continuous g). A set C C ^N is convex if and only if for every pair of points w, v G C we have that all points cm + (1 — a)v,Q < a < I belong to C as well: i.e., if u, v are in C then the whole segment with ends u, v belongs to C.

2.8

Bibliographical comments

Section 2.1. A comprehensive treatment of the whole subject of sliding mode control with several applications can be found in [2]. Basic points of design of variable structure control are described in [8], see also [13]. The simplex method was discovered by Bajda-Isozimov (Automation Remote Control 46, 1985) and further developed by Bartolini-Parodi-Utkin-Zolezzi (to appear in Mathematical Problems in Engineering). Sections 2.2, 2.3. The basic definition and the mathematical properties of Filippov solutions are in [3], see also the treatise [6]. Further definitions are compared in [7]. In [1] we find an exposition of the basic mathematical results about differential inclusions, see also [11]. An interesting discussion about the very beginning of relating the theory of discontinuous differential equations with control problems is in [9]. The physical meaning of Filippov solutions is discussed in [2]. Stabilization of control systems via discontinuous control require notions of solution of control systems which are different from Filippov's, see Clarke-Ledyaev-Sontag in IEEE Trans. Autom. Control 42 (1997), and Bressan, preprint SISSA (Trieste) n. 144 ( 1998). Section 2.4. The theory of viability is discussed in [1] (and at a greater length in [10]). The concept of equivalent control and its physical meaning can be found in [2]. A survey of several concepts related to viability is in [5].

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Section 2.5. Approximability was introduced in [4], see Bartolini-Zolezzi in [13] for further developments. Section 2.6 See the survey [12], which among other things presents some computer plots of numerical solutions to a discontinuous differential equation.

Acknowledgements
Work partially supported by MURST. We thank L. Levaggi for editorial assistance and for drawing the diagrams.

References
[1] J.P. Aubin, A. Cellina, Differential inclusions, Springer, 1984. [2] V.I. Utkin, Sliding modes in control optimization. Springer, 1992. [3] A.F. Filippov, "Differential equations with discontinuous right-hand side", Amer. Math. Soc. Translations, 42, pp. 199-231, 1964. [4] G. Bartolini, T. Zolezzi, "Control of nonlinear variable structure systems.", J. Math. Anal. Appl, 118, pp. 42-62, 1986. [5] F.H. Clarke, Y.S. Ledyaev, R.J. Stern, P.R. Wolenski, "Qualitative properties of trajectories of control systems: a survey", J. Dynamical Control Systems, 1 , pp. 1-48, 1995. [6] A.F. Filippov, "Differential equations with discontinuous righthand side", Kluwer, 1988. [7] O. Hajek, "Discontinuous differential equations", I,II. J. Equations, 32, pp. 149-185, 1979. Differential

[8] R. Decarlo, S. Zak, G. Matthews, "Variable structure control of nonlinear multivariable systems: a tutorial", Proceedings IEEE 76, pp. 212-232, 1988. [9] A.F. Filippov, "Application of the theory of differential equations with discontinuous right-hand sides to non-linear problems in automatic control", Proceedings Ist IFAC International Congress, Vol.11. Butterworths, pp. 923-927, 1961. [10] J.P. Aubin, Viability theory, Birkhauser, 1991. [11] K. Deimling, Multivalued differential equations, De Gruyter, 1992.

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[12] A. Dontchev, F. Lempio, "Difference methods for differential inclusions: a survey", SIAM Review, 34, pp. 263-294, 1992. [13] A. Zinober (ed.), Deterministic control of uncertain systems, P. Peregrinus, 1990.

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Chapter 3

Higher-Order Sliding Modes
L. FRIDMAN* and A. LEVANT**
* Chihuahua Institute of Technology, Chihuahua, Mexico ** Institute for Industrial Mathematics, Beer-Sheva, Israel

3.1

Introduction

One of the most important control problems is control under heavy uncertainty conditions. While there are a number of sophisticated methods like adaptation based on identification and observation, or absolute stability methods, the most obvious way to withstand the uncertainty is to keep some constraints by "brutal force". Indeed any strictly kept equality removes one " uncertainty dimension". The most simple way to keep a constraint is to react immediately to any deviation of the system stirring it back to the constraint by a sufficiently energetic effort. Implemented directly, the approach leads to so-called sliding modes, which become main operation modes in the variable structure systems (VSS) [55]. Having proved their high accuracy and robustness with respect to various internal and external disturbances, they also reveal their main drawback: the so-called chattering effect, i.e., dangerous high-frequency vibrations of the controlled system. Such an effect was considered as an obvious intrinsic feature of the very idea of immediate powerful reaction to the minutest deviation from the chosen constraint. Another important feature is proportionality of the maximal deviation from the constraint to the time interval between the measurements (or to the switching delay).

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To avoid chattering some approaches were proposed [15, 51]. The main idea was to change the dynamics in a small vicinity of the discontinuity surface in order to avoid real discontinuity and at the same time to preserve the main properties of the whole system. However, the ultimate accuracy and robustness of the sliding mode were partially lost. Recently invented higher order sliding modes (HOSM) generalize the basic sliding mode idea, acting on the higher order time derivatives of the system deviation from the constraint instead of influencing the first deviation derivative as it happens in standard sliding modes. Along with keeping the main advantages of the original approach, at the same time they totally remove the chattering effect and provide for even higher accuracy in realization. A number of such controllers were described in the literature [16, 34, 35, 38, 3, 5]. HOSM is actually a movement on a discontinuity set of a dynamic system understood in Filippov's sense [22]. The sliding order characterizes the dynamics smoothness degree in the vicinity of the mode. If the task is to provide for keeping a constraint given by equality of a smooth function s to zero, the sliding order is a number of continuous total derivatives of s (including the zero one) in the vicinity of the sliding mode. Hence, the rth order sliding mode is determined by the equalities
s=

s - 5 = ... = 5(r-1) = 0

(3.1)

forming an r-dimensional condition on the state of the dynamic system. The words "rth order sliding" are often abridged to "r-sliding". The standard sliding mode on which most variable structure systems (VSS) are based is of the first order (s is discontinuous). While the standard modes feature finite time convergence, convergence to HOSM may be asymptotic as well, r-sliding mode realization can provide for up to the rth order of sliding precision with respect to the measurement interval [35, 38, 41]. In that sense r-sliding modes play the same role in sliding mode control theory as Runge-Kutta methods in numerical integration. Note that such utmost accuracy is observed only for HOSM with finitetime convergence. Trivial cases of asymptotically stable HOSM are easily found in many classic VSSs. For example there is an asymptotically stable 2-sliding mode with respect to the constraint x = 0 at the origin x = x = 0 (at the one point only) of a 2-dimensional VSS keeping the constraint x + x = 0 in a standard 1-sliding mode. Asymptotically stable or unstable HOSMs inevitably appear in VSSs with fast actuators [23, 25, 26, 27, 30]. Stable HOSM reveals itself in that case by spontaneous disappearance of the chattering effect. Thus, examples of asymptotically stable or unstable sliding modes of any order are well known [16, 14, 50, 35, 30]. On the contrary, examples of r-sliding modes attracting in finite time are known for r = I

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(which is trivial), for r = 2 [34, 16, 17, 35, 4, 5] and for r = 3 [30]. Arbitrary order sliding controllers with finite-time convergence were only recently presented [38, 41]. Any new type of higher-order sliding controller with finite-time convergence is unique and requires thorough investigation. The main problem in implementation of HOSMs is increasing information demand. Generally speaking, any r-sliding controller keeping s = 0 needs s, s,..., s^r~1^ to be available. The only known exclusion is a so-called "super-twisting" 2-sliding controller [35, 37], which needs only measurements of s. First differences of s(r~2^ having been used, measurements of s,5, ...,s( r ~ 2 ) turned out to be sufficient, which solves the problem only partially. A recently published robust exact differentiator with finite-time convergence [37] allows that problem to be solved in a theoretical way. In practice, however, the differentiation error proves to be proportional to e^2 \ where k < r is the differentiation order and ? is the maximal measurement error of s. Yet the optimal one is proportional to e( r ~ fc )/ r (s(r) is supposed to be discontinuous, but bounded [37]). Nevertheless, there is another way to approach HOSMs. It was mentioned above that r-sliding mode realization provides for up to the rth order of sliding precision with respect to the switching delay T, but the opposite is also true [35]: keeping \s\ = O(TT) implies \s^\ ==? O(rr~l},i = 0,1,...,r — 1, to be kept, if s^ is bounded. Thus, keeping \s\ = O(TT] corresponds to approximate r-sliding. An algorithm providing for fulfillment of such relation in finite time, independent on r, is called rth order real-sliding algorithm [35]. Few second order real sliding algorithms [35, 52] differ from 2-sliding controllers with discrete measurements. Almost all rth order real sliding algorithms known to date require measurements of 5, s,..., s( r ~ 2 ) with r > 2. The only known exceptions are two real-sliding algorithms of the third order [7, 39], which require only measurements of s. Definitions of higher order sliding modes (HOSM) and order of sliding are introduced in Section 3.2 and compared with other known control theory notions in Section 3.3. Stability of relay control systems with higher sliding orders is discussed in Section 3.4. The behavior of sliding mode systems with dynamic actuators is analyzed from the sliding-order viewpoint in Section 3.5. A number of main 2-sliding controllers with finite time convergence are listed in Section 3.6. A family of arbitrary-order sliding controllers with finite time convergence is presented in Section 3.7. The main notions are illustrated by simulation results.

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

3.2

Definitions of higher order sliding modes

Regular sliding mode features few special properties. It is reached in finite time, which means that a number of trajectories meet at any sliding point. In other words, the shift operator along the phase trajectory exists, but is not invertible in time at any sliding point. Other important features are that the manifold of sliding motions has a nonzero codimension and that any sliding motion is performed on a system discontinuity surface and may be understood only as a limit of motions when switching imperfections vanish and switching frequency tends to infinity. Any generalization of the sliding mode notion must inherit some of these properties. First let us recall what Filippov's solutions [21, 22] are of a discontinuous differential equation

x = v(x)
where x G IR , v is a locally bounded measurable (Lebesgue) vector function. In that case, the equation is replaced by an equivalent differential inclusion
n

x e V(x]
In the particular case when the vector-field v is continuous almost everywhere, the set-valued function V(x) is the convex closure of the set of all possible limits of v(y) as y —> x, while {y} are continuity points of v. Any solution of the equation is denned as an absolutely continuous function x(t), satisfying the differential inclusion almost everywhere. The following Definitions are based on [34, 16, 17, 19, 35, 30]. Note that the word combinations "rth order sliding" and "r-sliding" are equivalent.

3.2.1

Sliding modes on manifolds

Let S be a smooth manifold. Set S itself is called the 1-sliding set with respect to S. The 2-sliding set is defined as the set of points x G ?, where V(x] lies entirely in tangential space Tx to manifold S at point x (Figure Definition 17 It is said that there exists a first (or second) order sliding mode on manifold S in a vicinity of a first (or second) order sliding point x, if in this vicinity of point x the first (or second) order sliding set is an integral set, i.e., it consists of Filippov's sense trajectories. Let ?Si = <S. Denote by to manifold <S. Assume that smooth manifold. Then the respect to 82 . Denote by <S3 <$2 the set of 2-sliding points with respect t$2 may itself be considered as a sufficiently same construction may be considered with the corresponding 2-sliding set with respect

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to e>2- ?3 is called the ^-sliding set with respect to manifold S. Continuing the process, we can achieve sliding sets of any order. Definition 18 It is said that there exists an r-sliding mode on manifold S in a vicinity of an r-sliding point x G <Sr, if in this vicinity of point x the r-sliding set Sr is an integral set, i.e., it consists of Filippov's sense trajectories.

3.2.2

Sliding modes with respect to constraint functions

Let a constraint be given by an equation s(x) = 0, where s : Rn —> R is a sufficiently smooth constraint function. It is also supposed that total time derivatives along the trajectories s, s , s , . . . , s^r~1^ exist and are single-valued functions of x, which is not trivial for discontinuous dynamic systems. In other words, this means that discontinuity does not appear in the first r — I total time derivatives of the constraint function s. Then the rth order sliding set is determined by the equalities
s=s

= S = ... = s(r~l) = 0

(3.2)

Here (3.2) is an r-dimensional condition on the state of the dynamic system. Definition 19 Let the r-sliding set (3.2) be non-empty and assume that it is locally an integral set in Filippov's sense (i.e., it consists of Filippov's trajectories of the discontinuous dynamic system). Then the corresponding motion satisfying (3.2) is called an r-sliding mode with respect to the constraint function s (Figure 3.1). To exhibit the relation with the previous Definitions, consider a manifold S given by the equation s(x] = 0. Suppose that s, s, s,..., s^r~2^ are differentiate functions of x and that
ran k{Vs,Vs,...,Vs ( r - 2 ) } = r - l

(3.3)

holds locally (here rank V is a notation for the rank of vector set V). Then Sr is determined by (3.2) and all Si, i — 1,..., r — I are smooth manifolds. If in its turn Sr is required to be a differentiate manifold, then the latter condition is extended to
ran k{Vs,Vs,...,Vs (r - 1} } = r

(3.4)

Equality (3.4) together with the requirement for the corresponding derivatives of s to be differentiate functions of x will be referred to as the sliding

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s=0

s=s=0

s=0

Figure 3.1: Second order sliding mode trajectory regularity condition, whereas condition (3.3) will be called the weak sliding regularity condition. With the weak regularity condition satisfied and S given by equation 5 = 0, Definition 19 is equivalent to Definition 18. If regularity condition (3.4) holds, then new local coordinates may be taken. In these coordinates the system will take the form
= J/25
2/r-l = Mr

Proposition 20 Let regularity condition (3-4) be fulfilled and r-sliding manifold (3.2) be non-empty. Then an r-sliding mode with respect to the constraint function s exists if and only if the intersection of the Filippov vector-set field with the tangential space to manifold (3.2) is not empty for any r-sliding point. Proof. The intersection of the Filippov set of admissible velocities with the tangential space to the sliding manifold (3.2), mentioned in the Proposition, induces a differential inclusion on this manifold. This inclusion satisfies all the conditions by Filippov [21, 22] for solution existence. Therefore manifold (3.2) is an integral one. Let s now be a smooth vector function, s : Rn —> M m , s — ( s i , . . . , s m ), and also r — (r\,... ,r TO ), where TI are natural numbers.

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Definition 21 Assume that the first r; successive full time derivatives of Si are smooth functions, and a set given by the equalities Si — si = s'i = ... = slrl~1' =0, i = 1 , . . . , m is locally an integral set in Filippov's sense. Then the motion mode existing on this set is called a sliding mode with vector sliding order r with respect to the vector constraint function s. The corresponding sliding regularity condition has the form rank{Vs i ,...,VsJ r i ~ 1 ) |i = 1,... ,m} = n + . . . + rm Definition 21 corresponds to Definition 18 in the case when r\ = ... = rm and the appropriate weak regularity condition holds. A sliding mode is called stable if the corresponding integral sliding set is stable. Remarks 1. These definitions also include trivial cases of an integral manifold in a smooth system. To exclude them we may, for example, call a sliding mode "not trivial" if the corresponding Filippov set of admissible velocities V(x] consists of more than one vector. 2. The above definitions are easily extended to include non-autonomous differential equations by introduction of the fictitious equation i = I. Note that this differs slightly from the Filippov definition considering time and space coordinates separately.

3.3

Higher order sliding modes in control systems

Single out two cases: ideal sliding occurring when the constraint is ideally kept and real sliding taking place when switching imperfections are taken into account and the constraint is kept only approximately.

3.3.1

Ideal sliding

All the previous considerations are translated literally to the case of a process controlled

x = f ( t , x , u ) , s = s(t,x) e R, u = U(t,x) e R

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where x G M n , t is time, u is control, and / and s are smooth functions. Control u is determined here by a feedback u — ?/(?,#), where U is a discontinuous function. For simplicity we restrict ourselves to the case when s and u are scalars. Nevertheless, all statements below may also be formulated for the case of vector sliding order. Standard sliding modes satisfy the condition that the set of possible velocities V does not lie in tangential vector space T to the manifold s = 0, but intersects with it, and therefore a trajectory exists on the manifold with the velocity vector lying in T. Such modes are the main operation modes in variable structure systems [54, 55, 12, 57] and according to the above definitions they are of the first order. When a switching error is present the trajectory leaves the manifold at a certain angle. On the other hand, in the case of second order sliding all possible velocities lie in the tangential space to the manifold, and even when a switching error is present, the state trajectory is tangential to the manifold at the time of leaving. To see connections with some well-known results of control theory, consider at first the case when x = a(x) + b(x)u, s = s(x) G E, u G R

where a, 6, s are smooth vector functions. Let the system have a relative degree r with respect to the output variable s [31] which means that Lie derivatives Lf,s, LbLas,..., L(}L1a~2s equal zero identically in a vicinity of a given point and L^L1a~1s is not zero at the point. The equality of the relative degree to r means, in a simplified way, that u first appears explicitly only in the rth total time derivative of 5. It is known that in that case s^ = Llas for i = 1 , . . . , r — 1, regularity condition (3.4) is satisfied automatically and also -j^s^ = LbL1a~1s ^ 0. There is a direct analogy between the relative degree notion and the sliding regularity condition. Loosely speaking, it may be said that the sliding regularity condition (3.4) means that the "relative degree with respect to discontinuity" is not less than r. Similarly, the rth order sliding mode notion is analogous to the zero-dynamics notion [31]. The relative degree notion was originally introduced for the autonomous case only. Nevertheless, we will apply this notion to the non-autonomous case as well. As was done above, we will introduce for the purpose a fictitious variable xn+\ — t, x n +i = 1. It should be mentioned that some results by Isidori will not be correct in this case, but the facts listed in the previous paragraph will still be true. Consider a dynamic system of the form

x = a(t, x) + b(t, x)u, s = s(t, x}, u~ U(t, x] G R

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Theorem 22 Let the system have relative degree r with respect to the output function s at some r -sliding point (to,xo). Let, also, the discontinuous function U take on values from sets [K, oo) and (—00, —K] on some sets of non zero measure in any vicinity of any r-sliding point near point (IQ^XQ). Then it provides, with sufficiently large K, for the existence of r-sliding mode in some vicinity of point (?o,a?o)- r-sliding motion satisfies the zerodynamics equations. Proof. This Theorem is an immediate consequence of Proposition 20, nevertheless, we will detail the proof. Consider some new local coordinates y — (yii ? ? ? i Vn)-, where y\ = s, y% = s, . . . , yr = s^r~l\ In these coordinates manifold Lr is given by the equalities y\ = y-2 = . . . = yr — 0 and the dynamics of the system is as follows:
2/1 =1/2, . - . , 2/r-i =Vr yr = h(t,y) + g(t,y)u, g ( t , y ) /=0

(3.5)

Denote Ueq = — h ( t , y } / g ( t , y ) . It is obvious that with initial conditions being on the r-th order sliding manifold Sr equivalent control u = Ueq(t, y) provides for keeping the system within manifold Sr. It is also easy to see that the substitution of all possible values from [-K, K] for u gives us a subset of values from Filippov's set of the possible velocities. Let \Ueq\ be less than KQ, then with K > KQ the substitution u = Ueq determines Filippov's solution of the discontinuous system which proves the Theorem. The trivial control algorithm u = — Ksign s satisfies Theorem 22. Usually, however, such a mode will not be stable. It follows from the proof above that the equivalent control method [54] is applicable to r-sliding mode and produces equations coinciding with the zero-dynamics equations for the corresponding system. The sliding mode order notion [11, 14] seems to be understood in a very close sense (the authors had no opportunity to acquaint themselves with the work by Chang). A number of papers approach the higher order sliding mode technique in a very general way from the differential-algebraic point of view [48, 49, 50, 43]. In these papers so-called "dynamic sliding modes" are not distinguished from the algorithms generating them. Consider that approach. Let the following equality be fulfilled identically as a consequence of the dynamic system equations [50] : P(s(r) , . . . , s, s, x, u(k} , . . . , u, u) = 0 (3.6)

Equation (3.6) is supposed to be solvable with respect to s^ and u^k\ Function s may itself depend on u. The rth order sliding mode is considered

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as a steady state s = 0 to be achieved by a controller satisfying (3.6). In order to achieve for s some stable dynamics a = s(r~l} + a lS (r - 2) + . . . + a r _is = 0 the discontinuous dynamic a = —signer (3-7)

is provided. For this purpose the corresponding value of s^ is evaluated from (3.7) and substituted into (3.6). The obtained equation is solved for u^. Thus, a dynamic controller is constituted by the obtained differential equation for u which has a discontinuous right hand side. With this controller successive derivatives s , . . . , s^r~^ will be smooth functions of closed system state space variables. The steady state of the resulting system will satisfy at least (3.2) and under some relevant conditions also the regularity requirement (3.4), and therefore Definition 19 will hold. Hence, it may be said that the relation between our approach and the approach by Sira-Ramirez is a classical relation between geometric and algebraic approaches in mathematics. Note that there are two different sliding modes in system (3.6) and (3.7): a standard sliding mode of the first order which is kept on the manifold a = 0, and an asymptotically stable r-sliding mode with respect to the constraint s = 0 which is kept in the points of the r-sliding manifold s = s = s = ... = s^r~1^ — 0.

3.3.2

Real sliding and finite time convergence

Recall that the objective is synthesis of such a control u that the constraint s(t, x] = 0 holds. The quality of the control design is closely related to the sliding accuracy. In reality, no approach to this design problem provides for ideal keeping of the prescribed constraint. Therefore, there is a need to introduce some means in order to provide a capability for comparison of different controllers. Any ideal sliding mode should be understood as a limit of motions when switching imperfections vanish and the switching frequency tends to infinity (Filippov [21, 22]). Let e be some measure of these switching imperfections. Then sliding precision of any sliding mode technique may be featured by a sliding precision asymptotics with E —> 0 [35]: Definition 23 Let ( t , x ( t , e ) ) be a family of trajectories, indexed by ? 6 R M ; with common initial condition (to^x(to)), and let t > to (or t 6 [tQ,T]). Assume that there exists t\ > to (ort\ E [to,T]) such that on every segment

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[?',?"], where t' > t\ (or on [t\ , T}), the function s(t,x(t,e)) tends uniformly to zero with e tending to zero. In that case we call such a family a realsliding family on the constraint s = 0. We call the motion on the interval [to, ti] a transient process, and the motion on the interval [ti, oo) (or [ti,T]) a steady state process. Definition 24 A control algorithm, dependent on a parameter e G R M , is called a real-sliding algorithm on the constraint s — 0 if, with e —? 0, it forms a real-sliding family for any initial condition. Definition 25 Let 7(5) be a real-valued function such that 7(2) —-> 0 as ? —> 0. A real- sliding algorithm on the constraint s = 0 is said to be of order r (r > 0) with respect to 7(5) if for any compact set of initial conditions and for any time interval [Ti,T2J there exists a constant C, such that the steady state process for t G [Ti , T2] satisfies

In the particular case when 7(5) is the smallest time interval of control smoothness, the words "with respect to 7" may be omitted. This is the case when real sliding appears as a result of switching discretization. As follows from [35], with the r-sliding regularity condition satisfied, in order to get the rth order of real sliding with discrete switching it is necessary to get at least the rth order in ideal sliding (provided by infinite switching frequency). Thus, the real sliding order does not exceed the corresponding sliding mode order. The standard sliding modes provide, therefore, for the first-order real sliding only. The second order of real sliding was really achieved by discrete switching modifications of the second-order sliding algorithms [34, 16, 17, 18, 19, 35]. Any arbitrary order of real sliding can be achieved by discretization of the same order sliding algorithms from [38, 39, 41] (see Section 3.7). Real sliding may also be achieved in a way different from the discrete switching realization of sliding mode. For example, high gain feedback systems [47] constitute real sliding algorithms of the first order with respect to A:"1, where A; is a large gain. A special discrete-switching algorithm providing for the second order real sliding were constructed in [52] . Another example of a second order real sliding controller is the drift algorithm [18, 35] . A third order real-sliding controller exploiting only measurements of s was recently presented [7]. It is true that in practice the final sliding accuracy is always achieved in finite time. Nevertheless, besides the pure theoretical interest there are also some practical reasons to search for sliding modes attracting in finite time. Consider a system with an r-sliding mode. Assume that with minimal

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switching interval r the maximal r-th order of real sliding is provided. That means that the corresponding sliding precision \s\ ~ rr is kept, if the r-th order sliding condition holds at the initial moment. Suppose that the r-sliding mode in the continuous switching system is asymptotically stable and does not attract the trajectories in finite time. It is reasonable to conclude in that case that with r —? 0 the transient process time for fixed general case initial conditions will tend to infinity. If, for example, the sliding mode were exponentially stable, the transient process time would be proportional to rln(r~ 1 ). Therefore, it is impossible to observe such an accuracy in practice, if the sliding mode is only asymptotically stable. At the same time, the time of the transient process will not change drastically if it was finite from the very beginning. It should be mentioned, also, that the authors are not aware of a case when a higher real-sliding order is achieved with infinite-time convergence.

3.4

Higher order sliding stability in relay systems

In this section we present classical results by Tsypkin [53] (published in Russian in 1956) and Anosov (1959) [1]. They investigated the stability of relay control systems of the form

y\ = 2/2, ? a - , yi-ik = yi ? si n
yi = Y%=i i,jyj +
yi = ]Cj=i a,itjyj,

S 2/i

(3-8)

i = / + 1,..., n

where a^j = const, k / 0, and y\ = y^ = ... = yi = 0 is the Z-th order sliding set. The main result is as follows: ? for stability of equilibrium point of relay control system (3.8 ) with second order sliding (Z = 2), three main cases are singled out: exponentially stable, stable, and unstable; ? it is shown that the equilibrium point of the system (3.8) is always unstable with Z > 3. Consequently, all higher order sliding modes arriving in the relay control systems are unstable with an order of sliding more than 2. Consider the ideas of the proof.

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

3.4.1

2-sliding stability in relay systems
yi=V2, m = ay i +by2 + k sign yi (3.9)

Consider a simple example of a second-order dynamic system

The 2-sliding set is given here by y\=yi — 0. At first, let k < 0. Consider the Lyapunov function (3.10) Function E is an energy integral of system (3.9) Computing the derivative of function E, we achieve

E = 2^2 + g l
It is obvious that for some positive oti < ot2i 01 < fa <*i\yi\ + Piy% <E< a2\yi\+fay% Thus, the inequalities —^E < E < —j\E or ^\E < E < ^2E hold for b < 0 or b > 0, respectively, in a small vicinity of the origin with some 72 > 7i > 0. Now let k > 0. It is easy to see in that case that the trajectories cannot leave the set y\ > 0, y2 = y\ > 0 if a > 0. The same is true with y\ < k/\a\ if a < 0. Starting with an infinitisimally small y\ > 0, y2 > 0, any trajectory inevitably leaves some fixed origin vicinity. It allows three main cases to be singled out for investigation of the stability of the system (3.9). ? Exponentially stable case. Under the conditions 6<0, fc<0 (3.11)

the equilibrium point y\ = 7/2 = 0 is exponentially stable. ? Unstable case. Under the condition

k > 0 or b > 0
the equilibrium point yi = y2 = 0 is unstable. ? Critical case.

k < 0, 6 < 0, bk = 0 With 6 = 0, k < 0 the equilibrium point j/i = 7/2 = 0 is stable.
It is easy to show that if the matrix A consisting of ajj, i,j > 2 is Hurvitz and conditions 02,2 < 0, k < 0 are true, then the equilibrium point of system (3.8) is exponentially stable.

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

3.4.2

Relay system instability with sliding order more than 2

Let us illustrate the idea of the proof on an example of a simple third-order system 2/i = 2/2, 2/2 = 2/3, 2/3 = asi2/i + ^322/2 + a33y3 - fcsignj/i, k > 0 (3.12) Consider the Lyapunov function1 V = 2/12/3 - ijyl Thus, V = -%i| +2/i(a 3 i2/i +a 32 y 2 +0332/3) and V" is negative at least in a small neighborhood of origin (0,0,0). That means that the zero solution of system (3.12) is unstable. On the other hand, in relay control systems with an order of sliding more than 2, a stable periodic solution can occur [46, 32].

3.5

Sliding order and dynamic actuators

Let the constraint be given by the equality of some constraint function s to zero and let the sliding mode s = 0 be provided by a relay control. Taking into account an actuator conducting a control signal to the process controlled, we achieve more complicated dynamics. In that case the relay control u enters the actuator and continuous output variables of the actuator z are transmitted to the plant input (Figure 3.2). As a result, discontinuous switching is hidden now in the higher derivatives of the constraint function [55, 23, 24, 25, 26, 27, 9].

3.5.1

Stability of 2-sliding modes in systems with fast actuators

Condition (3.11) is used in [25, 26, 27, 9] for analysis of sliding mode systems with fast dynamic actuators. Here is a simple outline of these reasonings. One of the actuator output variables is formally replaced by s after application of some coordinate transformation. Let the system under consideration be rewritten in the following form:

fj,z = Az + Bf] + D\x iri] = Cz + brj + D2x + k sign s s — r\ x = F(z,rj,s,x]
1

(1 i"*}

This function was suggested by V.I. Utkin in private communications.

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

Plant

Relay Controller

Figure 3.2: Control system with actuator where z <E Em, x R n , 77 and s G R. With (3.11) fulfilled and Re Spec A < 0, system (3.13) has an exponentially stable integral manifold of slow motions which is a subset of the second-order sliding manifold and given by the equations z = H(fj.t x) = -A~lDix +
s = T = 0.

Function H may be evaluated with any desired precision with respect to the small parameter /i. Therefore, according to [25, 26, 27, 9], under the conditions ReSpecA<0, b < 0, k < 0 (3.14)

the motions in such a system with a fast actuator of relative degree 1 consists of fast oscillations, vanishing exponentially, and slow motions on a submanifold of the second-order sliding manifold. Thus, if conditions (3.14) of chattering absence hold, the presence of a fast actuator of relative degree 1 does not lead to chattering in sliding mode control systems. Remark The stability of the fast actuator and of the second-order sliding mode in (3.13) still does not guarantee absence of chattering if dimz > 0 and §j T^ 0, for in that case fast oscillations may still remain in the 2-sliding mode itself. Indeed, the stability of a fast actuator corresponds to the stability of the fast actuator matrix

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

ReSpec ( ^ f
\\ O

u

) <0

Consider the system

^if] = 24^i — 6022 — 9?7 + D^x + k sign s

s = r) x = F(zi,z2,rj,s,x)
where zi,z2,rj,s are scalars. It is easy to check that the spectrum of the matrix is { — 1, —2, —3} and condition (3.11) holds for this system. On the other hand the motions in the second-order sliding mode are described by the system IJLZ\ = z\ + Z2 + DIX
/JiZ2 -— ^Z"2 r JLJ2X

The fast motions in this system are unstable and the absence of chattering in the original system cannot be guaranteed. Example Without loss of generality we illustrate the approach by some simple examples. Consider, for instance, sliding mode usage for the tracking purpose. Let the process be described by the equation x = u, x,u K, and the sliding variable be s = x-f(t), / : R - > R so that the problem is to track a signal f ( t ) given in real time, where I / I ) I / I ) I/I < 0.5. Only values of :r, /, u are available. The problem is successfully solved by the controller u — —signs, keeping s = 0 in a 1-sliding mode. In practice, however, there is always some actuator between the plant and the controller, which inserts some additional dynamics and removes the discontinuity from the real system. With respect to Figure 3.2 let the system be described by the equation

x —v
where v E R is an output of some dynamic actuator. Assume that the actuator has some fast first order dynamics. For example LJLV — u — v

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

The input u of the actuator is the relay control u = —sign s where n is a small positive number. The second order sliding manifold 8-2 is given here by the equations

s = x- f ( t ) = 0, s = v - f ( t ) = 0
The equality

shows that the relative degree here equals 2 and, according to Theorem 22, a 2-sliding mode exists, provided \i < I . The motion in this mode is described by the equivalent control method or by zero- dynamics, which is the same: from s = s — s = 0 we achieve u = pf(t} + v, v = f ( t ) and therefore It is easy to prove that the 2-sliding mode is stable here with yu small enough. Note that the latter equality describes the equivalent control [54, 55] and is kept actually only in the average, while the former two are kept accurately in the 2-sliding mode. Let

f ( t ) = 0.08 sin t + 0.12 cos 0.3* , z(0) = 0, u(0) = 0
The plots of x(t) and f ( t ) with p, = 0.2 are shown in Figure 3.3, whereas the plot of v(t) is demonstrated in Figure 3.4.

3.5.2

Systems with fast actuators of relative degree 3 and higher

The equilibrium point of any relay system with relative degree > 3 is always unstable [1, 53] (Section 3.4.2). That leads to an important conclusion: even being stable, higher order actuators do not suppress chattering in the closed-loop relay systems. For investigation of chattering phenomena in such systems, the averaging technique was used [25, 29]. Higher-order actuators may give rise to high-frequency periodic solutions. The general model of sliding mode control systems with fast actuators has the form [10] x = h(x,s,r), z,u(s}}, s = 77 (3.15)

W = 9z(x, s, 77, z),fj.z = gi [x, s, 77, z, u(s)})

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

1.897384E-01

n 1 1
: {

r

\k \ \ \ \
N

0

r

v\

\\ v
5.988000

-6.961547E-02 O.OOOOOOE+OO Figure 3.3: Asymptotically stable second-order sliding mode in a system with a fast actuator. Tracking: x(t) and 7.630808E-01

-4.040195E-01 O.OOOOOOE+OO 5.988000

Figure 3.4: Asymptotically stable second-order sliding mode in a system with a fast actuator: actuator output v(t)

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

where z R m , 77, s e R, x 6 X C R n , w(s) = signs, and g\,gi,h are smooth functions of their arguments. Variables s and x may be considered as the state coordinates of the plant. 77, z are the fast-actuator coordinates, and fj, being the actuator time constant. Suppose that following conditions are true: 1. The fast-motion system
ds — =77,

ar

dr) — =?2 (2,0,77,2),

ar

dz . ... — =0i z, 0,17,2, u(s)

ar

, . (3.16)

has a T(x)-periodic solution (SQ(T, x), T)Q(T, x), ZQ(T, x)) for any x ? X. System (3.16) generates a point mapping ^(x,rj,z) of the switching surface s = 0 into itself which has a fixed point (17* (x) , z* (x)} , \I>(z,r7*(z),2*(z)) = ( r f ( x } , z * ( x ) } . Moreover, the Frechet derivative of \?(z, 77, z) with respect to variables (77,2) calculated at (rj* (x) , z* (x)) is a contractive matrix for any x G X. 2. The averaged system

1 rTW =1 x ——l ( ) Jo

(3.17)

has an unique equilibrium point x = XQ. This equilibrium point is exponentially stable. Theorem 26 [29]. Under conditions 1 and 2 system (3.15) has an isolated orbitally asymptotically stable periodic solution with the period /z(T(xo) + O(/^)) near the closed curve
(XQ, //So(t//Z, XQ), 77o(t//Z, XQ), Zo(t/fl, XQ))

Example
Consider a mathematical model of a control system with actuator and the overall relative degree 3 x = —x — u, s = 21
fj,zi = 22, i*>z<2 = —1z\ — 3^2 — u

(3.18)
(3.19)

Here zi,zz,s,x 6 R, u(s) = signs, // is the actuator time constant. The fast motions taking place in (3.18),(3.19) are described by the system

dr

= -22! -3z2-u, w = sign^

(3.20)

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

Then the solution of system (3.20) for ? > 0 with initial condition ?(0) = 0, 2i(0) = 210, 22(0) = 220 is as follows

3 1 1 1 ?( r ) = 2 Z l ° ~ 22 ioe~ T + -zwe~2T + -z20 - z20e~T + -z20e~2r
1 3

2r

ZI(T) = 22i 0 e- T - 2i 0 e" 2r + 2 20 e~ T - 2 20 e~ 2r - - + e~T - -e"2r 2 2 (r) = 2zwe~2T - 2zwe~T - z20e"r + 2^ 2 oe~ 2r - e~T + e~2r Consider the point mapping ?(2:1,2:2) °f the domain 2:1 > 0, 2:2 > 0 on the switching surface ? = 0 into the domain z\ < 0, z2 < 0 with sign? > 0 made by system (3.20). Then —(21,22) = (— 1(21, 22), ?2(21, 22)) ?1(21,22) = 22ie- T -2ie- 2 T + 2 2 e- T -2 2 e- 2 T - - + e~ T - -e"2T H 2 (2i, 22 ) - 22ie"2T - 22ie"T - 22e"T + 222e"2T - e"T + e"2T where T(2i, 22) is the smallest root of equation ?(T(2!, 2 2 )) = -21 - 22ie- T + 2?ie~ 2T + ifi ~ ^~T
1 ryrrt 1 O 71 1 _ OT1 ^

+ 2 22C

~2T+4-6

+

4E



System (3.20) is symmetric with respect to the point ? = 21 = 22 = 0. Thus, the initial condition (0,2^,22) and the semi-period T* = T(z*,Z2) for the periodic solution of (3.20) are determined by the equations ?(2^,22) — — (2*, z2) and 4(^(21,22)) = 0, and consequently -2^-22*6
O .,, ^ ,*, T T
1

*

+-2*e

- * - *

_2r 1* OT

+-z*-z2e

- L ^

^

_ T*

+ -z2e

- L *

_ OT^*

__1T* + - - 6 T* +4- -e~2T* -U + e~ 0 2 4 46 "

22ie- 2T * - 22*e~ T * - z2e~T* + 222e~ 2T * - e"T* + e"2T* = -z2 (3.21) Expressing 2^,23 from the latter two equations of (3.21), we achieve
T*(eT' + e3T* + l + eVT*)

_ 5eT* _ 3e3T*

+

3 + ^IT*

=Q

(3 ^

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

Equations (3.22) and (3.21) have positive solution T* ? 2.2755, z\ ? 0.3241, z2 ? 0.1654 corresponding to the existence of a 2T*-periodic solution in system (3.20). mi Thus

dT dT
V i s '

Q_

/

3 _ Op-T , 1--2T 2 ' 2

1 _ Cp-T , \C -IT P

(2*! + z2 + l}e~T - (z,
and

2

^ 2

= 1e~T - e~

+ [e-Clz, + 2z2 + 1) - e'(2Zl + z2
I)]'

= 2e -2T

Calculating the value of Frechet derivative § at (zl,z%}, using the found | value of T*, achieve dE( * ? . _ [ -0.4686 ? l*!.^;-''03954
L

-0.1133
00979

The eigenvalues of matrix J are —0.3736 and 0.0029. That implies existence and asymptotic stability of the periodic solution of (3.20). The averaged equation for system (3.18) and (3.19) is
x = —x

and it has the asymptotically stable equilibrium point x = 0. Hence, system (3.18) and (3.19) has an orbitally asymptotically stable periodic solution which lies in the O(/^-neighborhood of the switching surface.

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

3.6

2-sliding controllers

We follow here [36, 35, 6].

3.6.1

2-sliding dynamics
x = f ( t , x , u ) , s = s ( t , x ] e R, u = U(t, x) e R
n

Return to the system (3.23)

where x E R , ? is time, u is control, and /, s are smooth functions. The control task is to keep output s = 0. Differentiating successively the output variable s, we achieve functions s, s,... Depending on the relative degree [31] of the system, different cases should be considered a) relative degree r = 1, i.e., -j^s ^ 0 b) relative degree r > 2, i.e., ?s^ = 0 (i = 1 , 2 , . . . , r - 1), ?s^ r ) ^ 0 In case a) the classical VSS approach solves the control problem by means of 1-sliding mode control, nevertheless 2-sliding mode control can also be used in order to avoid chattering. For that purpose u will become an output of some first-order dynamic system [35]. For example, the time derivative of the plant control u(t) may be considered as the actual control variable. A discontinuous control u steers the sliding variable s to zero, keeping s = 0 in a 2-sliding mode, so that the plant control u is continuous and the chattering is avoided [35, 5]. In case b) the p-sliding mode approach (with p > r) is the control technique of choice. Chattering avoidance: the generalized constraint fulfillment problem When considering classical VSS the control variable u(t) is a feedbackdesigned relay output. The most direct application of 2-sliding mode control is that of attaining sliding motion on the sliding manifold by means of a continuous bounded input u(t) being a continuous output of a suitable first-order dynamic system driven by a proper discontinuous signal. Such first-order dynamics can be either inherent to the control device or specially introduced for chattering elimination purposes. Assume that / and s are respectively Cl and C2 functions, and that the only available current information consists of the current values of ?, w(t), s(t,x) and, possibly, of the sign of the time derivative of s. Differentiating the sliding variable s twice, the following relations are derived:
r\ r\

s = —s(t, x) + —s(t, x ) f ( t , x, u} ot ox

(3.24)

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

r\

r\

r\

s(t) = -?-s(t, x, u) + -z-s(t, x, u)f(t, x, u) + -?-s(t, x, u)u(t) ot ox ou

(3.25)

The control goal for a 2-sliding mode controller is that of steering s to zero in finite time by means of control u(t) continuously dependent on time. In order to state a rigorous control problem, the following conditions are assumed: 1) Control values belong to the set U — {u : |u| < UM}, where UM > 1 is a real constant; furthermore the solution of the system is well defined for all ?, provided u(t] is continuous and Vt u(t] e U. 2) There exists HI 6 (0,1) such that for any continuous function u(t) with |w(i)| > wi, there is ti, such that s(t}u(t) > 0 for each t > ti. Hence, the control u(t] = — sign(s(to)), where to is the initial value of time, provides hitting the manifold s = 0 in finite time. 3) Let s(t, x, u) be the total time derivative of the sliding variable s(i, x). There are positive constants SQ, UQ < 1, Tm, TM such that if \s(t, x)\ < SQ then
o

0 < T m < — s(t,x,u) <TM ou

,\lu^U,x^X

(3.26)

and the inequality \u\ > UQ entails su > 0. 4) There is a positive constant $ such that within the region \s\ < SQ the following inequality holds W, x e X, u U
r\ r\

— s(t,x,u) + —s(t,x,u)f(t,x,u)

(3.27)

The above condition 2 means that starting from any point of the state space it is possible to define a proper control u(t] steering the sliding variable within a set such that the boundedness conditions on the sliding dynamics defined by conditions 3 and 4 are satisfied. In particular they state that the second time derivative of the sliding variable s, evaluated with fixed values of the control w, is uniformly bounded in a bounded domain. It follows from the theorem on implicit function that there is a function ueq(t,x} which can be considered as equivalent control [55], satisfying the equation s — 0. Once s = 0 is attained, the control u = ueq(t,x) would provide for the exact constraint fulfillment. Conditions 3 and 4 mean that \s\ < SQ implies \ueq\ < UQ < 1, and that the velocity of the ueq changing is bounded. This provides for a possibility to approximate ueq by a Lipschitzian control.

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

The unit upper bound for UQ and u\ is actually a scaling factor. Note also that linear dependence on control u is not required here. The usual form of the uncertain systems dealt with by the VSS theory, i.e., systems affine in u and possibly in x, are a special case of the considered system and the corresponding constraint fulfillment problem may be reduced to the considered one [35, 20]. Relative degree two. In case of relative degree two the control problem statement could be derived from the above by considering the variable u as a state variable and u as the actual control. Indeed, let the controlled system be f ( t , x, u) = a(t, x) + b(t, x ) u ( t ) (3.28) where a : R n+1 -> Rn and 6 : R n+1 -> Rn are sufficiently smooth uncertain vector functions, [^s(i,x)]6(t,x) = 0. Calculating, we find that s = <f>(t,x)+i(t,x)u (3.29)

It is assumed that \(p\ < $, 0 < Fm < 7 < F M , $ > 0 Thus in a small vicinity of the manifold s = 0 the system is described by (3.28), (3.29) if the relative degree is 2, or by (3.23) and s = (f>(t,x) + ~f(t,x)u if the relative degree is 1. (3.30)

3.6.2

Twisting algorithm

Let relative degree be 1. Consider local coordinates y\ = s and 7/2 = s, then after a proper initialization phase, the second order sliding mode control problem is equivalent to the finite time stabilization problem for the uncertain second-order system with )</? < <i>, 0 < Fm < 7 < FM, $ > 0.
1
2

= yi = <p(t,x) 4-7(t,x)u

/ 3 31)

with 1/2(0 immeasurable but with a possibly known sign, and y>(t,x) and 7(t,x) uncertain functions with $>0,|</> < ^ > , 0 < r m < 7 < F M (3.32)

Being historically the first known 2-sliding controller [34], that algorithm features twisting around the origin of the 2-sliding plane yiOy? (Figure 3.5). The trajectories perform an infinite number of rotations while converging in finite time to the origin. The vibration magnitudes along the axes as well as the rotation times decrease in geometric progression. The control

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

Figure 3.5: Twisting algorithm phase trajectory derivative value commutes at each axis crossing, which requires availability of the sign of the sliding-variable time-derivative y-2. The control algorithm is defined by the following control law [34, 35, 17, 20], in which the condition on \u\ provides for \u\ < I :
—u u(i) = { -V r TO sign(yi) if \u\ > I if yiyz < 0; \u\ < I if yiy2 > 0; \u\ < 1
(3.33)

The corresponding sufficient conditions for the finite time convergence to the sliding manifold are [35]

VM>Vr Vm : v771 ^ p* ?' ^>
The similar controller

(3.34)

TMv
if yi2/2 < 0

u(t} =

if y\m > o

is to be used in order to control system (3.28) when the relative degree is 2. By taking into account the different limit trajectories arising from the uncertain dynamics of (3.29) and evaluating time intervals between successive crossings of the abscissa axis, it is possible to define the following

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

upper bound for the convergence time [6]
1 - Vtw Here y\M is the value of the y\ variable at the first abscissa crossing in the y \Oy-2 plane, tjwl is the corresponding time instant and
ttwoo < *M! + Qtw--.-^—v\\y\M, I (3-35)

In practice when y% is immeasurable, its sign can be estimated by the sign of the first difference of the available sliding variable y\ in a time interval r, i.e., signal*)) is estimated by sign(yi(i) — y\ (t — r)). In that case the 2-sliding precision with respect to the measurement time interval is provided, and the size of the boundary layer of the sliding manifold is A ~ O(r2} [35]. Recall that it is the best possible accuracy asymptotics with discontinuous y? = s.

3.6.3

Sub-optimal algorithm

That 2-sliding controller was developed as a sub-optimal feedback implementation of a classical time-optimal control for a double integrator. Let the relative degree be 2. The auxiliary system is 2/2 =(pt,x + it>xu

The trajectories on the y\Oy^ plane are confined within limit parabolic arcs which include the origin, so that both twisting and leaping (when y\ and y<2 do not change sign) behaviors are possible (Figure3.6). Also here the coordinates of the trajectory intersections with axis y\ decrease in geometric progression. After an initialization phase the algorithm is defined by the following control law [4, 5, 6]:

v(t) = -a(t)VMsign(yi(t)
a

- \yiM]
(3.37)

*

I

*/ W*) - 5Z/iMHl/iM - J/i(*)l > 0 if [ i / i ( t ) - i y i M ] [ j / i M - j / i ( t ) ] < 0

where y\M is the latter singular value of the function y i ( t ) , i.e. the latter value corresponding to the zero value of y% — y\ . The corresponding sufficient conditions for the finite-time convergence to the sliding manifold are as follows [4]: a* e ( 0 , J ] n ( 0r M ) v ' l v ,^ '
VM> max
T7
(

$

* ,

4$

4*

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

Figure 3.6: Sub-optimal algorithm phase trajectories Also in that case an upper bound for the convergence time can be determined [4] 1 / +0pt _i +*, 4- ^opt+ ?, + < M\ I Pi /In, I ("\ "1Q\ n \ li/lMi I \O.dVJ 1- 'opt
I L x

Here y\M and IMI are defined as for the twisting algorithm, and
^opt
3

'opt

_ na*

The effectiveness of the above algorithm was extended to larger classes of uncertain systems [6] . It was proved [5] that in case of unit gain function the control law (3.37) can be simplified by setting a — 1 and choosing VM > 2$. The sub-optimal algorithm requires some device in order to detect the singular values of the available sliding variable y\ = s. In the most practical case can be estimated by checking the sign of the quantity D(t) = [y\(t —r] — yi(t}] Vi(t) m which | is the estimation delay. In that case the control amplitude VM must belong to an interval instead of a half-line:
/ / $ \ \ VM e max —— ,VMl(T,yiM) ,^M 2 (r;yi M ) L
\ \"

(3.40)

m

/

J

Here VM\ < ^M2

are

the solutions of the second-order algebraic equation

V

+ 1=0

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

In the case of approximated evaluation of y\M the second order real sliding mode is achieved, and the size of the boundary layer of the sliding manifold is A ~ O(r2}. It can be minimized by choosing VM as follows [6]:
3Fm -

1 + 3Fm -

An extension of the sub-optimal 2-sliding controller to a class of sampled data systems such that the gain function in (3.29 ) is constant, i.e., 7(-) = 1, was recently presented [6].

3.6.4

Super-twisting algorithm

This algorithm has been developed to control systems with relative degree one in order to avoid chattering in VSC. Also in this case the trajectories on the 2-sliding plane are characterized by twisting around the origin (Figure 3.7), but the continuous control law u(t} is constituted by two terms. The first is defined by means of its discontinuous time derivative, while the other is a continuous function of the available sliding variable.

Figure 3.7: Super-twisting algorithm phase trajectory The control algorithm is defined by the following control law [35]:
-u if -Wsign(yi) if -A|s 0 | p sign(yi) -X\yi\psign(yi)

\u\ > 1 \u\ < 1 if \yi\ > SQ if |yi| < s0

(3.41)

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

and the corresponding sufficient conditions for the finite time convergence to the sliding manifold are [35]
W>

0 < p < 0.5

That controller may be simplified when controlled systems (3.28) are linearly dependent on control, u does not need to be bounded and SQ — oo: u= -\\s\psign(yi)+ui iii = — W^sign(yi) The super-twisting algorithm does not need any information on the time derivative of the sliding variable. An exponentially stable 2-sliding mode arrives if the control law (3.41) with p = I is used. The choice p = 0.5 ensures that the maximal possible for 2-sliding realization real-sliding order 2 is achieved. Being extremely robust, that controller is successfully used for real-time robust exact differentiation [37] ( see further).

3.6.5

Drift algorithm

The idea of the controller is to steer the trajectory to the 2-sliding mode s — 0 while keeping s relatively small, i.e., to cause "drift" towards the origin along axis y\. When using the drift algorithm, the phase trajectories on the 2-sliding plane are characterized by loops with constant sign of the sliding variable y\ (Figure 3.8). That controller intentionally yields real 2-sliding and uses sample values of the available signal y\ with sampling period r. The control algorithm is defined by the following control law [35, 16, 18] (relative degree is 1):
f —u if \u\ > I u = < -ymsign(A7/li) */ y^yi, < 0; \u\ < I [ -VMsign(A7/ii) if yiAyi, > 0; \u\ < 1

(3.43)

where Vm and VM are proper positive constants such that Vm < VM and yA*- is sufficiently large, and l^y\i = yi(ti)—yi(ti—r), t [?j,?j + i). The corresponding sufficient conditions for the convergence to the sliding manifold are rather cumbersome [18] and are omitted here for the sake of simplicity. Also here a similar controller corresponds to relative degree 2: ? _ / -Krn sign(Ayi i ) T — ' /A ^
m

if yiAyi 4 < 0 f A \n

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Figure 3.8: Drift algorithm phase trajectories After substituting y2 for Ayi, a first order sliding mode on y2 = 0 would be achieved. That implies y\ — const, but since an artificial switching time delay appears, we ensure a real sliding on y2 with most of the time spent in the region y\y^ < 0. Therefore, y\ —> 0. The accuracy of the real sliding on y2 = 0 is proportional to the sampling time interval r\ hence, the duration of the transient process is proportional to r~l. Such an algorithm does not satisfy the definition of a real sliding algorithm (Section 3.3) requiring the convergence time to be uniformly bounded with respect to T. Consider a variable sampling time Ti+i[yi(ti)} = ti+\ — ?;, i = 0 , 1 , 2 , . . . with T = max(rM,min(r m , 77(2/1 (?j)| p )), where 0.5 < p < 1 , TM > rm > 0, 77 > 0. Then with 77, ^ sufficiently small and Vm sufficiently large, the drift algorithm constitutes a second-order real sliding algorithm with respect to T —> 0. That algorithm has no overshoot if the parameters are chosen properly [18].

3.6.6
This sient of s. is as

Algorithm with a prescribed convergence law

class of sliding controllers features the possibility of choosing a tranprocess trajectory: the switching of u depends on a suitable function The general formulation of such a class of 2-sliding control algorithms follows:
—u -VMsign(y2 - g(yi)}
if \u\ > I if \u\ < I
(3.44)

Here VM is a positive constant and the continuous function g(y\) is smooth everywhere but in y\ — 0. A controller for the relative degree 2 is formed

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

Figure 3.9: Phase trajectories for the algorithm with prescribed law of variation of s in an obvious way: u = -V-sign [7/2 - 9(yi)] Function g must be chosen in such a way that all solutions of the equation 7/1 = g(y\) vanish in finite time and the function g' ? g be bounded. For example, the following function can be used 0(2/i) = -Ai|j/i p sign(yi), A > 0, 0.5 < p < I The sufficient condition for the finite time convergence to the sliding manifold is defined by the following inequality

VM>

sup [?'(7/1)2(7/1)]

(3.45)

and the convergence time depends on the function g [16, 35, 56]. That algorithm needs 7/2 to be available, which is not always the case. The substitution of the first difference of y\ for 7/2 i-e., sign[A7/ii — Tig(yi)] instead of sign [ y i - g ( y i } } (t [ti,ti+i), TI = ij-tj_i), turns the algorithm into a real sliding algorithm. The real sliding order equals two if g(-) is chosen as in the above example with p = 0.5 [35]. Important remark. All the above-listed discretized 2-sliding controllers, except for the super-twisting one, are sensitive to the choice of the measurement interval T. Indeed, given any measurement error magnitude, any information of significance of the first difference AT/^ is eliminated with sufficiently small T, and the algorithm convergence is disturbed. That problem was shown to be solved [40] by a special feedback determinating

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r as a function of the real-time measured value of y\ . In particular, it was shown that the feedback r = max(rM , min(r m , 77(3/1 (?i) p )), 0.5 < p < 1, TM > Tm > 0, TJ > 0, makes the twisting controller robust with respect to measurement errors. Moreover, the choice p = 1/2 is proved to be the best one. It provides for keeping the second-order real-sliding accuracy s = O(r2) in the absence of measurement errors and for sliding accuracy proportional to the maximal error magnitude otherwise. Note that the super-twisting controller is robust due to its own nature and does not need such auxiliary constructions.

3.6.7

Examples

Practical implementation of 2-sliding controllers is described in [42] . Continue the example series 3.5.1 and 3.5.2. The process is given by x = u. x,weM, s = x-f(t), /:R->R

so that the problem is to track a signal /(?) given in real time, where I / U / U / I < 0-5- Only values of x,/, u are available. Following is the appropriate discretized twisting controller: u(ti)\ > I u = < -5signs(ti), s(ti}Asi > 0, \u(ti)\ < 1 -signs(ti), s(ti)Asi < 0, \u(ti)\ < I Here t{ < t < ?;+i. Let function / be chosen as in examples 3.5.1 and 3.5.2:

f ( t ) = 0.08 sin t + 0.12 cos 0.3* , x(0) = 0, u(0) = 0 .
The corresponding simulation results are shown in Figure 3.10 and 3.11. The discretized super-twisting controller [19, 35, 37] serving the same goal is the algorithm

^ j"[*;| < I
Its simulation results are shown in Figures 3.12 and 3.13.

3.7

Arbitrary-order sliding controllers

We follow here [38, 39, 41].

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1.903470E-01

-6.939553E-02 O.OOOOOOE+00 5.988000 Figure 3.10: Twisting 2-sliding algorithm. Tracking: x(t) and f ( t )

3.919948E-01

-1.077855E-01 O.OOOOOOE+00 5.988000 Figure 3.11: Twisting 2-sliding algorithm. Control u(t)

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1.884142E-01

-6.939553E-02 O.OOOOOOE+00 5.988000 Figure 3.12: Super-twisting 2-sliding controller. Tracking: x(t) and

3.732793E-01

\
-1.077609E-01 O.OOOOOOE+00 5.988000 Figure 3.13: Super-twisting 2-sliding controller. Control u(t).

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3.7.1

The problem statement
x = a(t,x) + b(t,x)u,s = s(t,x) (3.46)

Consider a dynamic system of the form

where x G M n , a, b, s are smooth functions, u ? R. The relative degree r of the system is assumed to be constant and known. That means, in a simplified way, that u first appears explicitly only in the r-th total derivative of s and -^s^ ^ 0 at the given point. The task is to fulfill the constraint s(t,x) = 0 in finite time and to keep it exactly by discontinuous feedback control. Since s, s, ..., s^r~1^ are continuous functions of t and x, the corresponding motion will correspond to an r-sliding mode. Introduce new local coordinates y = (j/i,..., j/ n ), where yi — s,y 2 = s, ...,y r = s^"1). Then

? = (2/r+i, ???, 2/n) (3.47) Let a trivial controller w = — K signs be chosen with K > sup|w e9 |, weq = — h ( t , y ) / g ( t , y ) [55]. Then the substitution u — ueq defines a differential equation on the r-sliding manifold of ( 3.46). Its solution provides for the r-sliding motion. Usually, however, such a mode is not stable. It is easy to check that g = LbLra~ls — -^s^ . Obviously, h = LTas is the rth total time derivative of s calculated with u = 0. In other words, functions h and g may be defined in terms of input-output relations. Therefore, dynamic system (3.46) may be considered as a "black box". The problem is to find a discontinuous feedback u = U(t,x) causing finite-time convergence to an r-sliding mode. That controller must generalize the 1-sliding relay controller u = —K signs. Hence, g ( t , y ) and h(t,y) in (3.47) are to be bounded, h > 0. Thus, we require that for some Km,KM,C>Q < KM: \Lras\ < C (3.48)

? = T)(t, s, s, ..., s(r~l\?) + 7(?, s, s, ..., s (r ~ 1} , f )w,

3.7.2

Controller construction
Nijr = \S\(r-

Let p be a positive number. Denote

Ni,r = (\S\P/r + \S\P/(r~V + ... + | s (i-l)|

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=s

where /?i, ...,/3 r _i are positive numbers. Theorem 27 Let system (3.46) have relative degree r with respect to the output function s and (3.48) be fulfilled. Then with properly chosen positive parameters (3\,..., fa_ i controller u= -asign ^ T ._i ) T .(s,s,...,s ( r " 1 ) )J . (3.49)

provides for the appearance of r-sliding mode s = 0 attracting trajectories in finite time. The positive parameters fli,...,fa-\ are to be chosen sufficiently large in the index order. Each choice determines a controller family applicable to all systems (3.46) of relative degree r. Parameter a > 0 is to be chosen specifically for any fixed C, Km, KM- The proposed controller is easily generalized: coefficients of JV;;r may be any positive numbers, etc. Obviously, a is to be negative with -j^s^ < 0. Certainly, the number of choices of fa is infinite. Here are a few examples with fa tested for r < 4, p being the least common multiple of 1,2, ...,r. The first is the relay controller and the second is listed in Section 3.6. l.u = —a signs 2.u = — a sign (s + |s|1//2signs), 3.u = -asign(s + 2(|s| 3 + |s|2)1/6sign(s + |s|2/3signs), 4.w - -asign{s(3) + 3(s6 + s4 + |s|3)1/12sign[s+ (54 + s |3)i/6 sign (s + 0.5|s 3/4signs)]}, 5.u = -asign(s( 4 ) + fa(\s\12 + |s|15 + |s|20+ +/32(|s
12

The idea of the controller is that a 1-sliding mode is established on the smooth parts of the discontinuity set F of (3.49) (Figure3.14). That sliding mode is described by the differential equation T/V-I^ = 0 providing in its turn for the existence of a 1-sliding mode ?/V-i,r — 0- But the primary sliding mode disappears at the moment when the secondary one is to appear. The resulting movement takes place in some vicinity of the subset of F satisfying V ; r-2,r — 0, transfers in finite time into some vicinity of the

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s, s ,..., s

Figure 3.14: The idea of r-sliding controller subset satisfying t/V-3,r — 0 and so on. While the trajectory approaches the r-sliding set, set F retracts to the origin in the coordinates s, s, ...,s(r~l\ Set F with r = 3 is shown in Figure 3.15. An interesting controller, so-called "terminal sliding mode controller", was proposed by [56]. In the 2-dimensional case it coincides with a particular case of the 2-sliding controller with given convergence law (Section 3.6). In the r-dimensional case a mode is produced at the origin similar to the r-sliding mode. The problem is that a closed-loop system with terminal sliding mode does not satisfy the Filippov conditions [22] for the solution existence with r > 2. Indeed, the control influence is unbounded in vicinities of a number of hyper-surfaces intersecting at the origin. The corresponding Filippov velocity sets are unbounded as well. Thus, some special solution definition is to be elaborated, the stability of the corresponding quasi-sliding mode at the origin and the very existence of solutions are to be shown. Controller (3.49) requires the availability of s, s, ...,s^r~1^. The needed information may be reduced if the measurements are carried out at times ti with constant step r > Q. Consider the controller

u(t) = -asi
? sign
)xl

')J

(3-50)

Theorem 28 Under conditions of Theorem 27 with discrete measurements both algorithms (3.49) and (3.50) provide in finite time for some positive constants QQ, ai, ??-, ar-\ for fulfillment of inequalities
--1

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?3.03

,.-3.03

Figure 3.15: The discontinuity set of the 3-sliding controller

That is the best possible accuracy attainable with discontinuous s^. Convergence time may be reduced by changing coefficients (3j. Another way is to substitute \~^s^ for s^\\ra for a and ar for r in (3.49) and (3.50), A > 0, causing convergence time to be diminished approximately by A times. Implementation of r-sliding controller when the relative degree is less than r. Introducing successive time derivatives u, w,..., u^r~k~1^ as new auxiliary variables and u^r~k^ as a new control, achieve different modifications of each r-sliding controller intended to control systems with relative degrees k — 1,2, ...,r. The resulting control is (r — k — l)-smooth function of time with k < r, a Lipschitz function with k = r — 1 and a bounded " infinite-frequency switching" function with k = r. Chattering removal. The same trick removes the chattering effect. For example, substituting u^r~^ for u in (3.50), receive a local r-sliding controller to be used instead of the relay controller u = —signs and attain rth order sliding precision with respect to r by means of (r — 2)-smooth control with Lipschitz (r — 2)th time derivative. It must be modified for global usage.

Controlling systems nonlinear on control. Consider a system x = f(t,x,u)

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nonlinear in the control variable u. Let ^js^^(t,x,u) = 0 for i = 1, ...,r — 1, -j^s(r\t,x,u} > 0. It is easy to check that
5^ +1

) = A;+IS + I-^A,

A.(-) = |(-) + ?-(.)f(t,x,u)

The problem is now reduced to that considered above with relative degree r + 1 by introducing a new auxiliary variable u and a new control v = u. Discontinuity regularization. The complicated discontinuity structure of the above-listed controllers may be smoothed by replacing the discontinuities under the sign-function with their finite-slope approximations. As a result, the transient process becomes smoother. Consider, for example, the above-listed 3-sliding controller. The function sign(s+ |s|2/3signs) may be replaced by the function max[—l,min(l, |s|~ 2 / 3 (s + |s|2/3signs)/?)] for some sufficiently small e > 0. For e = 0.1 the resulting tested controller is u = -asign(s + 2(|s|3 + |s| 2 )^ max[-l,min(l, 10|s|~5~(s + [assigns))]) (3.51) Controller (3.51) provides for the existence of a standard 1-sliding mode on the corresponding continuous piece-wise smooth surface. Theorem 29 Theorems 27 and 28 remain valid for controller (3.51). Real-time control of output variables The implementation of the above-listed r-sliding controllers requires realtime observation of the successive derivatives s, s,..., s^T~^. Thus, theoretically no model of the controlled process needs to be known. Only the relative degree and 3 constants are needed in order to adjust the controller. Unfortunately, the problem of successive real-time exact differentiation is usually considered to be practically unsolvable. Nevertheless, under some assumptions the real-time exact robust differentiation is possible. Indeed, let input signal rj(t) be a Lebesgue-measurable locally bounded function defined on [0,oo) and let it consist of a base signal r]o(t) having a derivative with Lipschitz's constant C > 0 and a bounded measurable noise N(t). Then the following system realizes a real-time differentiator [37]:
v=

= 1/1- \\v - ??(?)|1/2sign [v - j](t)\, z>i = -//sign [i/ - r/(t)]

where //, A > 0. Here v(t) is the output of the differentiator. Solutions of the system are understood in the Filippov sense. Parameters may be chosen in the form n = 1.1C, A = 1.5C1/2, for example (it is only one of possible choices). That differentiator provides for finite time convergence to the exact derivative of rjo(t) if N ( t ) = 0. Otherwise, if sup N(t) = e

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

it provides for accuracy proportional to C fl / 2 ? 1 / 2 . Therefore, having been implemented k times successively, that differentiator will provide for kth order differentiation accuracy of the order of e^2 ). Thus, full local real-time robust control of output variables is possible, using only output variable measurements and knowledge of the relative degree [41]. When the base signal r]o(t) has (r-l)th derivative with Lipschitz's constant C > 0, the best possible kth order differentiation accuracy is dk Cklr ? (r-fc)/r^ w h ere dk > 1 may be estimated (the asymptotics may be improved with additional restrictions on ?7o(?))- Moreover, it is proved that such a robust exact differentiator really exists [37]. The corresponding differentiator has been submitted by A. Levant for possible presentation at the European Control Conference in Portugal (2001). Theorem 30 An optimal k-th order differentiator having been applied, rsliding controller (3-49) provides locally for the sliding accuracy s^\ < Ci?^r~"l^r, i = 0,1, ...,r — 1, where e is the maximal possible error of realtime measurements of s and Q are some positive constants. Theorem 30 probably determines the best sliding asymptotics attainable when only s is available.

3.7.3 Examples Car control
Consider a simple kinematic model of car control [45]
x — v cos </?, y = v sin (p v tp = - tan d 6 =u

where x and y are Cartesian coordinates of the rear-axle middle point, ip is the orientation angle, v is the longitudinal velocity, I is the length between the two axles, and 6 is the steering angle. The task is to steer the car from a given initial position to the trajectory y — g ( x ) , while x,y, and </? are assumed to be measured in real time. Define s = y-g(x) Let v — const = lOra/s, / — 5m, g(x) — 10sin0.05x+5, x — y — ip = d = Q at t = 0. The relative degree of the system is 3 and both 3-sliding controller No. 3 and its regularized form (3.51) may be applied here. It was taken a = 20. The corresponding trajectories are the same, but the performance

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

GO

TO \

M

90

Figure 3.16: Car trajectory tracking
S, S, S

Figure 3.17: Regularized 3-sliding controller is different. The trajectory and function y = g(x) with measurement step T = 2 ? 10~~4 are shown in Figure3.16. Graphs of s, s, s are shown in Figure 3.16 and 3.17 for regularized and not regularized controllers, respectively. 4-sliding control Consider a model example of a tracking system. Let input z(t) and the control system satisfy equations
2 (4) + 32 + 22 = 0

=u

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s, s, s

Figure 3.18: Standard 3-sliding controller

Figure 3.19: 4-sliding tracking The task is to track z by x, s = x — z, thus the 4th controller with a = 40 is used. Initial conditions for z and x at time t = 0 are z(O) - 0, 2(0) = 0, 2(0) = 2, 2 (3) (0) - 0 = l,ar(0) =
(3)

(0) -

A mutual graph of x and z with r — 0.01 is shown in Figure 3.19. A mutual graph of x^ and z^ with T = 0.001 is shown in Figure 3.20. Mutual graphs of s, s, s, s^ with r = 0.001 are demonstrated in Figure 3.21. The attained accuracies are \s\ < 1.33 ? 10~4 with T = 0.01 and \s\ < 1.49 ? 10~12 with r = 0.0001.

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x, z

Figure 3.20: Third derivative tracking

Figure 3.21: Tracking deviation and its three derivatives

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

The authors are grateful to A. Stotsky for helpful discussions on VSS car control.

3.8

Conclusions

? A general review of the current state of the higher order sliding theory, its main notions and results were presented. ? It was demonstrated that higher order sliding modes are natural phenomena for relay control systems if the relative degree of the system is more than 1 or a dynamic actuator is present. ? Stability was studied of second order sliding modes in relay systems with fast stable dynamic actuators of relative degree 1. ? Instability of higher order sliding modes was shown in relay systems with dynamic actuators of relative degree 2 and more. ? A number of the most popular 2-sliding controllers were listed and compared. ? A family of arbitrary order sliding controllers with finite time convergence was presented. ? The discrete switching modification of presented sliding controllers provided for the sliding precision of their order with respect to the measurement time interval. ? A robust exact differentiator was presented allowing for full control of output variables using only measurements of their current values. ? A number of simulation examples were presented.

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[3] G. Bartolini, A. Ferrara , E. Usai, "Applications of a sub-optimal discontinuous control algorithm for uncertain second order systems", Int. J. of Robust and Nonlinear Control, Vol. 7, No.4, pp. 299-310, 1997. [4] G. Bartolini, M. Coccoli and A. Ferrara, "Vibration damping and second order sliding modes in the control of a single finger of the AMADEUS gripper", International J. of Systems Science, Vol. 29, No. 5, pp. 497-512, 1998. [5] G. Bartolini, A. Ferrara and E. Usai, "Chattering avoidance by secondorder sliding mode control", IEEE Trans. Automat. Control, Vol. 43, No. 2, pp. 241-246, 1998. [6] G. Bartolini, A. Ferrara, A. Levant and E. Usai On second order sliding mode controllers. Proc.of 5th Int. Workshop on VSS, Longboat Key, Florida, 1998. [7] G. Bartolini, A. Levant, A. Pisano, E. Usai, "2-sliding mode with adaptation" , Proc. of the 7th IEEE Mediterranean Conference on Control and Systems, Haifa, Israel, 1999. [8] J. Z. Ben-Asher, R. Gitizadeh, A. Levant, A. Pridor, I. Yaesh, "2sliding mode implementation in aircraft pitch control", Proc. of 5th European Control Conference, Karlsruhe, Germany, 1999. [9] S.V. Bogatyrev, Fridman L.M., "Singular correction of the equivalent control method. Differentialnye uravnenija (Differential equations)", Vol. 28, No. 6, pp. 740-751, 1992. [10] A.G. Bondarev, S.A. Bondarev, N.Y. Kostylyeva and V.I. Utkin, "Sliding Modes in Systems with Asymptotic State Observers", Automatica i telemechanica (Automation and Remote Control), Vol. 46, No. 5, pp. 679-684, 1985. [11] L.W. Chang, "A MIMO sliding control with a second order sliding condition", ASME WAM, paper no. 90-WA/DSC-5, Dallas, Texas, 1990. [12] R.A. DeCarlo, S.H. Zak, G.P. Matthews, "Variable structure control of nonlinear multivariable systems: a tutorial", Proceedings of the Institute of Electrical and Electronics Engineers, Vol. 76, pp. 212-232, 1988. [13] S.V. Drakunov, V.I. Utkin, "Sliding mode control in dynamic systems", Int. J. Control, Vol. 55, No. 4, pp. 1029-1037, 1992.

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[14] H. Elmali, N. Olgac, "Robust output tracking control of nonlinear MIMO systems via sliding mode technique", Automatica, Vol. 28, No. 1, pp. 145-151, 1992. [15] S.V. Emelyanov, S.K. Korovin, "Applying the principle of control by deviation to extend the set of possible feedback types", Soviet Physics, Doklady, Vol. 26, No. 6, pp. 562-564, 1981. [16] S.V. Emelyanov, S.K. Korovin and L.V. Levantovsky, "Higher order sliding modes in the binary control systems", Soviet Physics, Doklady, Vol. 31, No. 4, pp. 291-293, 1986. [17] S.V. Emelyanov, S.K. Korovin and L.V. Levantovsky, "Second order sliding modes in controlling uncertain systems", Soviet Journal of Computer and System Science, Vol. 24, No. 4, pp. 63-68, 1986. [18] S.V. Emelyanov, S.K. Korovin and L.V. Levantovsky, "Drift algorithm in control of uncertain processes", Problems of Control and Information Theory, Vol. 15, No. 6, pp. 425-438, 1986. [19] S.V. Emelyanov, S.K. Korovin and L.V. Levantovsky, "A new class of second order sliding algorithms", Mathematical Modeling, Vol. 2, No. 3, pp. 89-100, (in Russian), 1990. [20] S.V. Emelyanov, S.K. Korovin and A. Levant, "Higher-order sliding modes in control systems", Differential Equations, Vol. 29, No. 11, pp. 1627-1647, 1993. [21] A.F. Filippov, "Differential Equations with Discontinuous Right-Hand Side", Mathematical Sbornik, Vol. 51, No. 1, pp. 99-12 (in Russian), 1960. [22] A.F. Filippov, "Differential Equations with Discontinuous Right-Hand Side", Kluwer, Dordrecht, the Netherlands (1988). [23] L.M. Fridman, "On robustness of sliding mode systems with discontinuous control function", Automatica i Telemechanica (Automation & Remote Control), Vol. 46, No. 5, pp. 172-175 (in Russian), 1985. [24] L.M. Fridman, "Singular extension of the definition of discontinuous systems", Differentiate uravnenija (Differential equations), Vol. 8, pp. 1461-1463 (in Russian), 1986. [25] L.M. Fridman, "Singular extension of the definition of discontinuous systems and stability", Differential equations, Vol. No.26 (1990) (10), pp. 1307-1312.

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[26] L.M. Fridman, "Sliding Mode Control System Decomposition", Proceedings of the First European Control Conference, Grenoble, Vol. 1, pp. 13-17,1991. [27] L.M. Fridman, "Stability of motions in singularly perturbed discontinuous control systems", in Prepr. of XII World Congress IFAC, Sydney, Vol. 4, pp. 367-370, 1993. [28] L. Fridman, E. Fridman, E. Shustin, "Steady modes in an autonomous system with break and delay", Differential Equations (Moscow), Vol. 29, No. 8, 1161-1166, 1993. [29] L.M. Fridman, "The problem of chattering: an averaging approach", In Young K.D., Ozguner U.(Eds.) Variable Structure Systems, Sliding Mode and Nonlinear Control, Lecture Notes in Control and Information Sciences 247, Springer Verlag, pp. 361-385, 1999. [30] L. Fridman, A. Levant, "Sliding modes of higher order as a natural phenomenon in control theory", In Garofalo F., Glielmo L. (Eds.) Robust Control via Variable Structure and Lyapunov Techniques, Lecture Notes in Control and Information Sciences 217, Springer Verlag, pp. 107-133, 1996. [31] A. Isidori, "Nonlinear Control Systems", Second edition, Springer Verlag, New York, 1989. [32] K.H. Johanson, K.J. Astrom, A. Rantzer, "Fast switches in relay feedback systems", Automatica, Vol. 35, pp. 539-552, 1999. [33] U. Itkis, "Control Systems of Variable Structure", Wiley, New York, 1976. [34] L.V. Levantovsky, "Second order sliding algorithms: their realization", In "Dynamics of Heterogeneous Systems", Institute for System Studies, Moscow, pp. 32-43 (in Russian), 1985. [35] A. Levant (Levantovsky, L.V.), "Sliding order and sliding accuracy in sliding mode control", International Journal of Control, Vol. 58, No. 6, pp. 1247-1263, 1993. [36] A. Levant, "Higher order sliding: collection of design tools", Proc. of the 4th European Control Conference, Brussels, Belgium, 1997. [37] A. Levant, "Robust exact differentiation via sliding mode technique", Automatica, Vol. 34, No. 3, pp. 379-384, 1998.

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[38] A. Levant, "Arbitrary-order sliding modes with finite time convergence" , Proc. of the 6th IEEE Mediterranean Conference on Control and Systems, June 9-11, Alghero, Sardinia, Italy, 1998. [39] A. Levant, "Controlling output variables via higher order sliding modes", Proc. of the 5th European Control Conference, Karlsruhe, Germany, 1999. [40] A. Levant, "Variable measurement step in 2-sliding control", Kybernetika, Vol. 36, No. 1, pp. 77-93, 2000. [41] A. Levant, "Universal SISO sliding-mode controllers with finite-time convergence", to appear in IEEE Trans. Automat. Control (2001). [42] A. Levant, A. Pridor, R. Gitizadeh, I. Yaesh , J. Z. Ben-Asher, "Aircraft pitch control via second-order sliding technique", AIAA Journal of Guidance, Control and Dynamics, Vol. 23, No. 4, pp. 586-594, 2000. [43] X.-Y. Lu, S.K. Spurgeon, "Output feedback stabilization of SISO nonlinear systems via dynamic sliding modes", International Journal of Control, Vol. 70, No. 5, pp. 735-759, 1998. [44] Z. Man, A.P. Paplinski, H.R. Wu, "A robust MIMO terminal sliding mode control for rigid robotic manipulators", IEEE Trans. Automat. Control, Vol. 39, No. 12, pp. 2464-2468, 1994. [45] R. Murray, S. Sastry, "Nonholonomic motion planning: steering using sinusoids", IEEE Trans. Automat. Control, Vol. 38, No. 5, pp. 700716, 1993. [46] Y.I. Neimark, "The point mapping method in the theory of nonlinear oscillations", Nauka, Moscow (in Russian), 1972. [47] V.R. Saksena, J. O'Reilly, P.V. Kokotovic, "Singular perturbations and time-scale methods in control theory", Survey 1976-1983, Automatica, Vol. 20, No. 3, pp. 273-293, 1984. [48] H. Sira-Ramirez, "On the sliding mode control of nonlinear systems", Systems & Control Letters, Vol. 19, pp. 303-312, 1992. [49] H. Sira-Ramirez, "Dynamical sliding mode control strategies in the regulation of nonlinear chemical processes", International Journal of Control, Vol. 56, No. 1, pp. 1-21, 1992. [50] H. Sira-Ramirez, "On the dynamical sliding mode control of nonlinear systems", International Journal of Control, Vol. 57, No.5, pp. 10391061, 1993.

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[51] J.-J. E. Slotine, "Sliding controller design for nonlinear systems", Int. J. of Control, Vol. 40, No.2, 1984. [52] W.-C. Su, S. Drakunov, U. Ozguner, "Implementation of variable structure control for sampled-data systems", Proceedings of IEEE Workshop on Robust Control via Variable Structure & Lyapunov Techniques, Benevento, Italy, pp. 166-173, 1994. [53] Y.Z. Tsypkin, "Relay control systems", Cambridge University Press, Cambridge, 1984. [54] V.I. Utkin, "Variable structure systems with sliding modes: a survey", IEEE Transactions on Automatic Control, 22, pp. 212-222, 1977. [55] V.I. Utkin, "Sliding Modes in Optimization and Control Problems", Springer Verlag, New York, 1992. [56] Y. Wu, X. Yu, Z. Man, "Terminal sliding mode control design for uncertain dynamic systems", Systems & Control Letters, Vol. 34, pp. 281-287, 1998. [57] A.S.I. Zinober, (editor), "Deterministic Control of Uncertain Systems", Peter Peregrinus, London, 1990.

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Chapter 4

Sliding Mode Observers
J-P. BARBOT*, M. DJEMAF, and T. BOUKHOBZA** - EN SEA, Cergy, France Universite des Antilles, Kourou, France

4.1

Introduction

Sliding mode techniques have been widely studied and developed for the control problem and observation in the occidental countries1 since the works of Utkin [43]. As discussed by many authors [22, 40, 21, 37, 49, 50, 20, 4, 31, 24, 33], this methodology has several drawbacks in control design, adaptive control and observation. More particularly, several authors have used sliding observer for linear and nonlinear systems, and in many applications such as robotics [41, 12, 13, 28], mobile robots [5], AC motors [16, 17, 18] and converters [36]. This kind of observer is very useful and was developed for many reasons: - to work with reduced observation error dynamics - for the possibility of obtaining a step-by-step design - for a finite time convergence for all the observables states - to design, under some conditions, an observer for nonsmooth systems, and - robustness under parameter variations is possible, if the condition (dual of the well-known matching condition) is verified.
It is important to highlight the paternity and the major contribution of the Russian school in the sliding mode domain.
1

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Here, we highlight a few advantages of the sliding observer. One advantage is the possibility to design an observer for a system with an undetermined but bounded specific variable structure, however, throughout this chapter we choose to focus our attention on widening the class of considered systems in the design of the observer. Historically, in nonlinear control theories, the problem of a nonlinear observer design with linearization of the observation error dynamics for a class of nonlinear systems, called the input injection form, has been investigated ([29, 45, 46]...). Some necessary and sufficient conditions to obtain such a form are given in [46]. From this form, it is "easy" to design an observer. Unfortunately, the geometric conditions to obtain this form are very often too restrictive with respect to the system considered. Thus, in [11] we have given an extension of the results obtained in [29, 30, 35, 45, 46], for systems that can be written in an output injection form to systems which can be written in the form of the output and the output's derivative injection. We first recall this result and then we deal with a more general case, which is the triangular observer form [1]. Here, aiming for simplicity, we only present the case of single output system. The multi-output case may be found in [6], where the implicit triangular observer form is introduced in order to take into account the fa'ct that the information quantity given by one output and its derivatives may change along the state space. Roughly speaking, in the nonlinear case, in the neighborhood of XQ, information about the state can be given by the output y\ (one component of the output) and its derivative, and in another neighborhood of xi, information can be given by 2/2 (another component of the output) and its derivative. In both forms considered in this presentation, input derivatives are prohibited. Indeed, if they are allowed it is possible to use the observer form proposed in [25] and in that case a sliding observer is also widely used (see for example [34]). As in other chapters, some recall on high order sliding mode are given [31], then for the sake of clarity we do not present the high order sliding observer [7, 3, 7]. Moreover, we deferred some technical proofs to the appendix. We find that it is important to end this introduction with the following warning: in this chapter we omit many interesting aspects, for example, the observer design without coordinate change [14], high gain [10], and noise sensibility [47]. The subject is too large and open, to be able to squeeze it in an introductory presentation. The main purpose of this chapter is to highlight the utilities and difficulties of sliding mode technique for the observer design.

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

4.2

Preliminary example

In this section, the sliding observer is introduced based on a simple academic example. Let E be the system:
xi
X2

=
=

x-2
f(xi,X2)

(4.1)

where x ? R2 and y 6 M is the output and the function /(zi, ?2) is bounded ( \ f ( x i , X 2 } \ < B] but not necessary smooth, thus (4.1) is a particular case of variable structure dynamics. One wants to observe the state x with the additional constraint to obtain the real value of #2 in finite time. To do this, one uses a classical sliding mode observer, but completed with a new component 52. xi x-2 y x-2 = x-2 + \\sgn(x\ — xi) = /(xi,x 2 ) + Ei\-2sgn(x-2 — x 2 ) = xi =? x-2 + Ei\isgn(x\ — x\) (4.2)

where x represents the estimated value of x and E\ = \ \i x\ — x\ else ?"1=0 and sign denote the usual sign function. From (4.1) and (4.2), the error observation (e = x — x} dynamics are: ei = e2-\isgn(ei) (4.3)

Considering the nonempty manifold S = {e/e\ = 0} and the Lyapunov function V — \e\, one proves the attractivity of S as follows. One gets: V = e\e-2 — \\eisgn(e\), which verifies the inequality V < 0 when AI is chosen such that AI > |e2|max (where |e|max denotes the maximal value of e, V t e [0, oo]). As one uses a sgn function and as the Lyapunov function V is decreasing, one obtains the convergence to the sliding surface S = 0 in finite time to (and moreover, we have |e|max = |e|^ax and |e|^aa, is the maximal value of e, V i e [0,to])- Thus, for AI > |e2| ma x ? ^i converges to :TI in finite time and remains equal to x\ for t > to. Moreover, one also has that e\ = 0 V t > to, so that from (4.3), e2 = Xisgn(ei) (4.4)

Therefore, the observer output, #2 = X2 + Xisgn(ei) is equal to x% V t > to-

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Remark 31 This is obviously only true without any noise measurement, but this difficulty may be partially overcome by a sgn function modification (see [47] for analysis and design of observer with respect to noise) or by high order sliding mode [31]. Up to now, we proved for the system (4.1) that the observer (4.2) is suitable to give all the values of the state in finite time. The condition AI > 1 62] max can only be verified if e2 has stable dynamics, which is fulfilled after to for A2 > 0, where we have ?2 = f(xi,x2) - f(xi,x2) - Ei\2sgn(x2 - x2) with x2 = x2 and EI = 1 then e2 = -X2sgn(e2) Therefore, one gets |e2| max , which is bounded by the way that t0 and f(xi,x2) are bounded. The observer (4.2) with assumptions AI > |e2|^ax and A2 > 0 ensures a finite time convergence of (e\, e2) to (0, 0).

Remark 32 The time to can be very short because it is natural to initialize x\ — x\.

4.3

Output and output derivative injection form

Following, we recall some classical results on nonlinear observer theory.

4.3.1

Nonlinear observer

First of all, we recall the definition of observability indices. Definition 33 [29] Let the system

which is observable at XQ if there exists a neighborhood U. of XQ and p-tuple of integers (/zi, ...,/z p ) such that

1

i > 2 > ??? >

> 0 and

=n

-

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2) After suitable reordering of the hi at each x 6 U, the n row vectors \ L?r (dhi) : i = 1, ...,p; j = 1,..., /^ > are linearly independent. 3) ///i, ...,/ p satisfies (i) and after suitable reordering the n row vectors {L^~l(dhi) : i = l,...,p; j = 1, ...,/*} are linearly independent at some x 6 U then (/i,..., lp) > (/^i,..., fj,p) in the lexicographic ordering [(/i > ^i) or (/i = Hi and 1-2 > 1^2) or... or (/i = /zi, ...,/ p = /zp)]. The integers (/zi, ...,/z p ) are ca//ed i/ie observability indices at XQ. Remark 34 In the nonlinear case, the previous notion of observability index is local. In the linear case, this notion is global. As it is shown in [29, 30, 45], an interesting nonlinear systems is the output injection form without forced terms:

x = Ax + y = Cx
where:
A!

(4.6)

0 0

0 0 Ap

A=

0 0

sa

matrix

=
0 0 0

(4.7)

Ci
and

0 0

0

0 0

cp

0

Ci is a line vector E Mi , such that : Ci = (1,0, ...,0). This is interesting because for such a class, one can design an observer that allows us to obtain an observation error with stable linear dynamics. In fact, for the nonlinear observable system:

y = MO

=

(4.8)

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where / and h are smooth functions, necessary and sufficient conditions for the existence of a diffeomorphism x = $>(?) to transform the system (4.8) into (4.6) are given in [46]. Theorem 35 [46] There exists a change of coordinates transforming (4-8) into (4-6) only if there exists a p— tuple of integers (//i, ...,// p ), p,\ > M2 > ??? > Up such that we have the following: 1) If one denotes (with a possible reordering of the hi)

then dim span Q = n in a neighborhood of ?° . 2) If one denotes for j = 1, ...,p,

f

J

i ? . _ -,i , . . . , [I j K —

then span Q3 = span Q fl Qj for j — I,..., p. Theorem 36 [46] There exists a change of coordinates transforming (4-8) to (4-6) if and only if 1. and 2. in the previous Theorem hold and, moreover, if there exists vector fields gl, ...,gp satisfying: L g l L l f ~ l ( h j ) = Sijdi^, i,j = l,...,p, / = l,...,//j such that: (ad^_r-,gl,adl,r-,g^\ = 0/or i,j = l,...,p; k = 0, ...,/^i — 1; / = 0, ...,//j - 1. Thus, it immediately follows: Corollary 37 The conditions of Theorem 36 are sufficient to construct an observer that is asymptotically locally stable.

4.3.2

Sliding observer for output and derivative nonlinear injection form

output

In this section, one first constructs an asymptotically stable observer for the following class of systems called output and output derivative nonlinear injection form: ±i Vi
Vi

= AiX + <l>i(y,y) = Xi,i
=

f o r i = l,...,p

(4.9)

Xi,2

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with

and all AI matrix are of appropriated dimensions. Secondly, one exhibits the necessary and sufficient conditions under which the system (4.8) may be rewritten as (4.9). For the sake of simplicity, one introduces the following notations:
/ \T

*Li ~
JL
A

V^z,!} ^i^i ?"; ^i^Hi ) /~ ~ ~ \T
— I ?-6 J , ^2 5 ? ? ? } * ~ p )
/ / v / s
A

\

T

f

1

where Xi — Xi^ + Ei\i,\sgn(yi — Xi^) and E\ — 1 if (x\,\ — ?^1)= ???=(XI,P — XI, P )=O, else Ei=Q. Let us construct for the system (4.9) the sliding observer:

=

-&) ?i,3 + 02(y, y) + Ei\i^sgn(yi - fa]
(4.10)

for i — 1 ^ * ? ? ^ T) whprp* ?"/? — IT- o -Iiwi t — JL /-^ w iivyJi \-f* yi — "-"i 2t 1^

From this, one deduces a part of the error's observation dynamic (yi -Xi,i) and e;,2 = fa - xii2):

-yO

Therefore, using the same method as in the previous section one obtains: Theorem 38 Under the conditions: 1) Ai,i > \e2,i\max for i = 1, ...,p. AlltheXij i = l,...,p, j = 2,...,Hi aresuchthatsl — (Ai + -^u Where u\ = (1,0, ...,0)T and Ai is the

is a Hurwitz polynomial.

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(Hi — 1) x (fa — 1) matrix defined by A, is a elR^- 1 )*^- 1 ) matrix =
/ 0 1 0 Q '-. l \ 0 0 0

The observer (4-10) gives, infinite time to, the convergence ofy (respectively y) to y (respectively to y), and an asymptotic linear stable observation error dynamics on the sliding surface fe^i = 0). Proof The dynamics of the observation error are 6i = AiCi + (f>(y, y) — (j)(y, y) — Aj for i = l,...,p. It is clear that, after a finite time to, one has y — y, so 0(?/> y) — 4>(y, y) — 0- So that, for V t > to the error dynamics will be on the reduced manifold (e^i = 0), V

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