Table Of ContentLecture Notes
in Control and Information Sciences 391
Editor: M. Thoma
Alan S. I. Zinober (Ed.)
Variable Structure dna
Lyapunov Control
galreV-regnirpS
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A. Bensoussan • M.J. Grimble • P. Kokotovic • H. Kwakernaak • J.L. Massey
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Editor
Alan S. I. Zinober, Phi)
Department of Applied and Computational Mathematics, University of Sheffield,
Sheffield SI0 2TN, UK
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Preface
Mathematical models of actual systems contain uncertainty terms which model
the designer's lack of knowledge about parameter values and disturbances. Such
poorly known quantities may be assumed to be constant or time-varying. Un-
certainties arising from imperfect knowledge of system inputs and inaccuracies
in the mathematical modelling itself, contribute to performance degradation of
the feedback control system. In self-tuning and other stochastic adaptive con-
trol systems, the parameter values and disturbances are constantly monitored
using on-line identification algorithms, and appropriate adaptive globally stable
controllers are implemented. These schemes, however, are costly and result in
additional complexity. Simplicity and reliability are not features which should
be sacrificed in a control system.
In direct contrast to these adaptive controllers, the deterministic control
of uncertain time-varying systems proposes the use of straightforward fixed
nonlinear feedback control functions, which operate effectively over a specified
magnitude range of system parameter variations and disturbances, without
any on-line identification of the system parameters. An immediate advantage
of such an approach is that no statistical information of the system variations
is required to yield the desired dynamic behaviour, and robustness is achieved,
not in an average sense, but for all possible values of the underlying uncer-
tainty. The deterministic approach thus contrasts sharply with many other ad-
aptive control schemes, which require global parameter convergence properties.
Furthermore, if the parameter variations satisfy certain matching conditions,
complete insensitivity to system variations can be achieved. The main areas of
application of deterministic control of uncertain systems include electric mo-
tor drives, robotics, flight control, space systems, power electronics, chemical
processes, automotive control systems and magnetic levitation.
The two main approaches to deterministic control of uncertain systems
are the Sliding Mode Control technique using a special behaviour of variable
structure systems called the sliding regime; the second approach is generically
known as the Lyapunov control design technique. The outstanding feature of
these controllers is their excellent robustness and invariance properties.
The essential property of Variable Structure Control (VSC) is that the
discontinuous feedback control switches on one or more manifolds in the state
space. Thus the structure of the feedback system is altered or switched as the
state crosses each discontinuity surface. Sliding motion occurs when the system
state repeatedly crosses and immediately re-crosses a switching surface, because
all motion in the neighbourhood of the manifold is directed inwards towards the
manifold. Following an initial trajectory onto the switching (sliding) surfaces,
the system state is constrained to lie upon these surfaces and is said to be in
the sliding mode. In the sliding mode the system is totally invariant to a class of
matched disturbances and parameter variations with known upper and lower
bounds; the decoupled system dynamics then being wholly described by the
reduced order dynamics of the selected sliding surfaces. In the sliding mode the
viii
control element has high (theoretically infinite) gain, while the control actually
passed onto the plant takes finite values.
The discontinuous controller can be replaced in many practical applications
with continuous nonlinear control which yields a dynamic response arbitrarily
close to the discontinuous controller, but without undesirable chatter motion.
Following an initial trajectory onto the switching (sliding) surfaces, the system
state is constrained to lie in a neighbourhood of these surfaces.
VSC with a sliding mode was first studied intensively in the 1960's by
Russian authors, notably Emel'yanov and Utkin, although early work was also
done by Fliigge-Lotz in the 1950's. In recent years the subject has attracted the
attention of numerous researchers. This is reflected in learned journals, books,
technical sessions at control conferences and workshops. The basic interest in
the technique stems because of its applicability to linear and nonlinear dy-
namical systems as well as to systems with delays and distributed parameters.
VSC is particularly well suited to the deterministic control of uncertain con-
trol systems. Some of the major interests have been the use of VSC and allied
techniques in model-following and model reference adaptive control, tracking
control and observer systems.
In the 1970's research work consolidated the linear scalar case and some
attempts had been made at solving the more complex multivariable control
problem. The introduction of the geometric approach to linear systems the-
ory was rapidly translated into a general technique which allowed the solution
in full generality of the sliding mode control of linear multivariable systems.
However, some basic problems still remained to be solved. Most notably, the
state observation problem for perturbed linear systems, needed a solution from
the viewpoint of deterministic uncertainty. The 1980's witnessed the emergence
of the initial steps of a general theory for nonlinear systems, most notably, the
differential geometric approach for the study of nonlinear systems structure.
The theoretical results for smooth systems were rapidly translated into a
more intuitive theory, while there has been more rigorous formulation of slid-
ing mode control for nonlinear systems. The theory has now been extended
to distributed parameter systems described by linear partial differential equa-
tions and delay differential systems. The sliding mode control of discrete time
systems for linear and nonlinear systems remained largely unexplored until re-
cently. Important contributions in the area of adaptation and identification of
dynamical systems using sliding mode control, were made towards the end of
the last decade. A user-friendly CAD design package is now available in the
MATLAB environment; thus allowing the control designer who is not expert
in VSC to straightforwardly design and simulate sliding mode controllers.
Recent research is beginning to consolidate nonlinear systems theory from
both a geometric and an algebraic viewpoint. The algebraic approach to cast
linear and nonlinear systems in a unified framework has been researched only
recently, and the implications of non-traditional state space representations
for dynamical systems has yielded interesting emerging consequences in sliding
mode control theory.
xi
Lyapunov control follows the approach of early research workers such as
Leitmann, Corless, Gutman, Palmor and Ryan. Using a Lyapunov function and
specified magnitude bounds on the uncertainties, a nonlinear control law is de-
veloped to ensure uniform ultimate boundedness of the closed-loop feedback
trajectory to achieve sufficient accuracy. The resulting controller is a discon-
tinuous control function, with generally continuous control in a boundary layer
in the neighbourhood of the switching surface. The boundary layer control pre-
vents the excitation of high-frequency unmodelled parasitic dynamics. Control-
lers have been devised for numerous types of system for many different classes
of uncertainty. The control is designed using a Lyapunov design approach and
allows for a range of expected system variation.
The chapters in this book cover the whole spectrum of Variable Struc-
ture and Lyapunov Control research and design. After an introductory chapter
on the theoretical and practical design of multivariable VSC systems, there
are chapters covering numerous aspects of VSC including novel mathematical
approaches exploring a differential algebraic approach for the sliding mode con-
trol of nonlinear single-input single-output systems and module theory for the
study of sliding modes for multivariable linear systems; robust control for sys-
tems with matched and unmatched uncertainty; a frequency domain design
approach; discrete-time control; observer-control systems; the control of un-
certain infinite-dimensional systems; and model-following control systems. Us-
ing the Lyapunov approach universal adaptive nonlinear feedback controllers
are developed, and quadratic Lyapunov techniques are reviewed. Applications
presented in detail include automobile fuel-injection control, magnetic levita-
tion and industrial robotics.
The editor wishes to thank all the authors for their diligent cooperation in
preparing their manuscripts in , ISTEX which has allowed efficient and speedy
text processing using the international electronic mail network. In particular
I wish to thank Hebertt Sira-Ramirez and Sarah Spurgeon for their scholarly
advice and careful review of many of the chapters. Numerous other reviewers
have also assisted in the preparation of this monograph. Mike Piff has pa-
tiently allowed unlimited access to his vast personal knowledge of XETSI , while
Madeleine Floy, Judith Smith and Cerys Morgan have provided efficient ad-
ministrative and secretarial assistance. Finally I wish to thank my family for
their encouragement and patience during the preparation of the book.
University of Sheffield Alan Zinober
August 1993
Table of Contents
List cf Contributors ........................... xix
An Introduction to Sliding Mode Variable Structure Control .... 1
Alan S.I. Zinober
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Regulator System .......................... 2
1.3 Model-Following Control System ................. 3
1.4 The Sliding Mode .......................... 3
1.5 Nonlinear Feedback Ccntrol .................... 5
1.6 Second-Order Example ....................... 7
1.7 Quadratic Performance ....................... 10
1.8 Eigenstlucture A~,signment .................... 11
1.9 Sensitivity Reduction ........................ 13
1.10 Eigenvalue A~,signment in a Region ................ 14
1.10.1 Eigenvalue Assignment in a Sector ............ 14
1.10.2 Eigenvalue Assignment in a Disc ............. 15
1.10.3 Eigenvalue Assignment in a Vertical Strip ........ 16
1.11 Example: Remotely Piloted Vehicle ................ 17
1.12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.13 Acknowledgement .......................... 20
xii
An Algebraic Approach to Sliding Mode Control 23
Hebertt Sira-Ram[rez
2.1 Introduction ............................. 23
2.2 Basic Background to Differential Algebra ............ 24
2.2.1 Basic Definitions ...................... 24
2.2.2 Fliess's Generalized Controller Canonical Forms .... 27
2.2.3 Input-Output Systems ................... 28
2.3 A Differential Algebraic Approach to Sliding Mode Control of
Nonlinear Systems ......................... 30
2.3.1 Differential Algebra and Sliding Mode Control of
Nonlinear Dynamical Systems ............... 30
2.3.2 Dynamical Sliding Regimes Based on Fliess's GCCF . . 32
2.3.3 Some Formalizations of Sliding Mode Control for
Input-Output Nonlinear Systems ............. 34
2.3.4 An Alternative Definition of the Equivalent Control
Dynamics .......................... 35
2.3.5 Higher Order Sliding Regimes .............. 36
2.3.6 Sliding Regimes in Controllable Nonlinear Systems . . . 37
2.4 A Module Theoretic Approach to Sliding Modes in Linear
Systems ............................... 38
2.4.1 Quotient Modules ..................... 40
2.4.2 Linear Systems and Modules ............... 41
2.4.3 Unperturbed Linear Dynamics ............... 41
2.4.4 Controllability ....................... 42
2.4.5 Observability ........................ 42
2.4.6 Linear Perturbed Dynamics ................ 43
2.4.7 A Module-Theoretic Characterization of Sliding Regimes 43
2.4.8 The Switching Strategy .................. 44
2.4.9 Relations with Minimum Phase Systems and Dynamical
Feedback .......................... 45
2.4.10 Non-Minimum Phase Case ................. 45
2.4.11 Some Illustrations ..................... 45
2.5 Conclusions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Robust Tracking with a Sliding Mode ................. 51
Raymond Davies, Christopher Edwards and Sarah K. Spurgeon
3.1 Introduction ............................. 51
3.2 Problem Formulation ........................ 52
3.3 Design of the Sliding Manifold .................. 55
3.4 Nonlinear Controller Development and Associated Tracking
Properties .............................. 61
3.5 Design Example: Temperature Control of an Industrial Furnace 68
3.6 Conclusions ............................. 71
3.7 Acknowledgements ......................... 71
iiix
Sliding Surface Design in the Frequency Domain 75
Hideki Hashimoto dna Yusuke Konno
4.1 Introduction ............................. 75
4.2- Sliding Mode using the LQ Approach ............... 76
4.2.1 Linear Quadratic Optimal Sliding Mode ......... 76
4.2.2 Frequency Shaped LQ Approach ............. 77
4.3 H2/H °° approach ......................... 78
4.3.1 H2/H ~° Optimal Control ................. 78
4.3.2 Generalized Plant Structure ................ 79
4.3.3 Controller Solution ...................... 80
4.4 Simulation .............................. 82
4.4.1 Plant Model ......................... 82
4.4.2 Controller Design ...................... 83
4.4.3 Simulation Results ..................... 84
4.5 Conclusions ............................. 84
Sliding Mode Control in Discrete-Time and Difference Systems 87
Vadim L Utkin
5.1 Introduction ............................. 87
5.2 Semi-Group Systems and Sliding Mode Definition ........ 88
5.3 Discrete-time Sliding Mode Control in Linear Systems ..... 93
5.4 Discrete-Time Sliding Modes in Infinite-Dimensional Systems . 96
5.5 Sliding Modes in Systems with Delays .............. 99
5.6 Finite Observers with Sliding Modes ............... 101
5.7 Control of Longitudinal Oscillations of a Flexible Bar ..... 103
5.8 Conclusions ............................. 106
Generalized Sliding Modes for Manifold Control of Distributed
Parameter Systems ........................... 109
Sergey Drakunov dna Omit Ozgiiner
6.1 Introduction ............................. 109
6.2 Manifold Control: Generalization of the Sliding Mode Control
Concept ................................ 110
6.3 Canonical Form of the Distributed Parameter System ..... 112
6.3.1 Problem Statement ..................... 113
6.3.2 Linear Transformation ................... 114
6.3.3 Nonsingularity of the Integral Transform ......... 115
6.4 Manifold Control of Differential-Difference Systems ....... 117
6.5 Wave Equation ........................... 119
6.5.1 Supressing Vibrations of a Flexible Rod ......... 119
6.5.2 Rod with Additional Mass ................. 121
6.5.3 Semi-Infinite Rod with Distributed Control ....... 123
6.5.4 Dispersive Wave Equation ................. 124
6.6 Diffusion Equation ......................... 125
6.7 Fourth Order Equation ....................... 127
6.7.1 The Euler-Bernoulli Beam ................. 127
