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Sliding mode control in engineering.pdf
dke602_fm.pdf
SLIDING MODE CONTROL IN ENGINEERING
Series Introduction
Preface
Contributors
Contents
dke602_ch01.pdf
Contents
Chapter 1: Introduction: An Overview of Classical Sliding Mode Control
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
dke602_ch02.pdf
Contents
Chapter 2: Differential Inclusions and Sliding Mode Control
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
2.7 Mathematical appendix
2.8 Bibliographical comments
Acknowledgements
References
dke602_ch03.pdf
Contents
Chapter 3: Higher-Order Sliding Modes
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 controllers
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
dke602_ch04.pdf
Contents
Chapter 4: Sliding Mode Observers
4.1 Introduction
4.2 Preliminary example
4.3 Output and output derivative injection form
4.3.1 Nonlinear observer
4.3.2 Sliding observer for output and output derivative nonlinear injection form
4.4 Triangular input observer form
4.4.1 Sliding mode observer design for triangular input observer form
4.4.2 Observer matching condition
4.5 Simulations and comments
4.6 Conclusion
4.7 Appendix
4.7.1 Proof of Proposition 39
4.7.2 Proof of Theorem 41
4.7.3 Proof of Theorem 49
References
dke602_ch05.pdf
Contents
Chapter 5: Dynamic Sliding Mode Control and Output Feedback
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
dke602_ch06.pdf
Contents
Chapter 6: Sliding Modes, Passivity, and Flatness
6.1 Introduction
6.2 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
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 flatness
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
dke602_ch07.pdf
Contents
Chapter 7: Stability and Stabilization
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
dke602_ch08.pdf
Contents
Chapter 8: Discretization Issues
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
dke602_ch09.pdf
Contents
Chapter 9: Adaptive and Sliding Mode Control
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 adaptive control
9.6 Conclusions
References
dke602_ch10.pdf
Contents
Chapter 10: Steady Modes in Relay Systems with Delay
10.1 Introduction
The simplest example of steady modes
Statement of the problem
Organization of the material
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
dke602_ch11.pdf
Contents
Chapter 11: Sliding Mode Control for Systems with Time Delay
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 stability of systems with small delays
11.3.3 Sliding mode controller synthesis
11.3.4 Example: delay in the state
11.4 Conclusion
References
dke602_ch12.pdf
Contents
Chapter 12: Sliding Mode Control of Infinite-Dimensional Systems
12.1 Introduction
Notation
12.2 Motivation: disturbance rejection in Hilbert space
12.3 Mathematical description of sliding modes in Hilbert space
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 finite-dimensional unstable part
12.4.1 Disturbance rejection in exponentially stabilizable systems
12.4.2 Disturbance rejection in minimum phase systems
12.5 Conclusions
References
dke602_ch13.pdf
Contents
Chapter 13: Application of Sliding Mode Control to Robotic Systems
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 pneumatic system
13.3.2 Sliding mode control of a hydraulic robot
13.3.3 Simulation results
13.4 SM observers based control
13.4.1 Observer design
13.4.2 Tracking error equation: 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
dke602_ch14.pdf
Contents
Chapter 14: Sliding Modes Control of the Induction Motor: a Benchmark Experimental Test
14.1 Introduction
14.2 Sliding modes control
14.3 Application to the induction motor
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
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
M A R C EL
D E K K E R
MARCEL DEKKER, INC.
NEW YORK • BASEL
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
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.)
2001058442
TJ213 .P415 2002
629.8—dc21
This book is printed on acid-free paper.
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The publisher offers discounts on this book when ordered in bulk quantities. For more infor-
mation, 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 21
PRINTED IN THE UNITED STATES OF AMERICA
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
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, Con-
stantine 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.
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,
9. Advanced Process Identification and Control, Enso Ikonen and Kaddour
Enright
Forbes T. Brown
Najim
Jean Pierre Barbot
Grigoriadis
10. Modern Control Engineering, P. N. Paraskevopoulos
11. Sliding Mode Control in Engineering, edited by Wilfrid Perruquetti and
12. Actuator Saturation Control, edited by Vikram Kapila and Karolos M.
Additional Volumes in Preparation
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
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 ever-
increasing complex problems faced by practicing engineers. However,
few of these books fully address the applications aspects of control en-
gineering. It is the intention of this new series to redress this situa-
tion.
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 real-
world 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 auto-
mated 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 prob-
lems that could benefit from wider exposure to the applicability of con-
trol methodologies, and the systematic systems-oriented basis inher-
ent 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
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
in Engineering is another outstanding entry to Dekker's Control Engi-
neering series.
SERIES
INTRODUCTION
Neil Munro
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
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 closed-
loop 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 uncer-
tainties
• the need for a reduced amount of information in comparison to clas-
sical control techniques
• the possibility of stabilizing some nonlinear systems which are not
stabilizable by continuous state feedback laws
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
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, dy-
namic 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 dif-
ficulties 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 higher-
order 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 tech-
niques. 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
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
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 Chap-
ter 3 : the higher-order sliding mode. Such control design is naturally moti-
vated 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 non-
linear 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 slid-
ing 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 fea-
tures 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.
Copyright 2002 by Marcel Dekker, Inc. All Rights Reserved.
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