Control and Estimation of Piecewise Affine Systems

242 Pages·2014·5.27 MB·English

by  J. Xu and L. Xie (Auth.)

Control and estimation of piecewise
affine systems
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Control and estimation of
piecewise affine systems
J X
UN U
AND
L X
IHUA IE
AMSTERDAM • BOSTON • CAMBRIDGE • HEIDELBERG • LONDON
NEW YORK • OXFORD • PARIS • SAN DIEGO
SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
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Copyright © J. Xu and L. Xie, 2014
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Control Number: 2013955437
ISBN 978-1-78242-161-0 (print)
ISBN 978-1-78242-162-7 (online)
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Produced from camera-ready copy supplied by the author.
Printed and bound in the United Kingdom
To Gaoyuan, Zhimei, Grace, and Alexander
- Jun Xu
To Meiyun, Jessica and Julie
- Lihua Xie
v
List of Figures
2.1 Several cases on the vector fields . . . . . . . . . . . . . . 24
2.2 Elastic support . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Non-smooth continuous function . . . . . . . . . . . . . . 27
2.4 Dry friction . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Discontinuous of Filippov-type function . . . . . . . . . . 29
2.6 Membership functions of Example 2.4.4 . . . . . . . . . . 30
2.7 A saturation feedback systems with PWA representation . 31
2.8 A saturation example with simulation . . . . . . . . . . . 31
3.1 Stable and unstable phase portrait . . . . . . . . . . . . . 43
3.2 Stable and unstable system trajectories . . . . . . . . . . 43
3.3 The Lyapunov function of the system (3-1) . . . . . . . . 49
3.4 Some trajectories of the Lyapunov function of the system
(3-1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 System state trajectory ±[01]T and ±[10]T . . . . . . . . 73
4.2 System state trajectory from [0 1]T and [0 -1]T . . . . . . 74
5.1 A PQLF along the state trajectory of a PQ stable system 85
5.2 A conewise linear system with four partitions . . . . . . . 95
7.1 State trajectory and Lyapunov function . . . . . . . . . . 134
7.2 Output y via the H∞ controller (Example 7.2.4) . . . . . 140
7.3 Output y via the H∞ controller (Example 7.2.5) . . . . . 141
8.1 Actual state and estimated state via the H∞ estimator. . 162
8.2 Trajectories of the state estimation error . . . . . . . . . . 163
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Control and estimation of piecewise affine systems
8.3 Trajectories of the state estimation error . . . . . . . . . . 164
9.1 Membership functions of Example 9.2.1 . . . . . . . . . . 174
9.2 Membership functions of Example 9.2.2 . . . . . . . . . . 174
9.3 Global membership functions of Example 9.2.3 . . . . . . 176
9.4 Individual membership functions of Example 9.2.3 . . . . 177
9.5 Trajectoriesfromfourinitialconditionsandregionpartitions181
9.6 Membership functions of Example 9.4.1 . . . . . . . . . . 184
9.7 State trajectories of x with different initial conditions . . 185
1
9.8 System state trajectories via the H∞ controller. . . . . . . 188
10.1 Block diagram of electric motor driven pendulum . . . . . 194
10.2 PWL approximation of sine function . . . . . . . . . . . . 196
10.3 The servo state trajectory . . . . . . . . . . . . . . . . . . 197
10.4 State difference between original and approximated systems198
10.5 Servo control system in Hard Disk Drives. . . . . . . . . . 199
10.6 Servo control system with impulsive disturbance d . . . . 200
i
10.7 The comparisonof corrupted PES and normal PES. . . . 201
10.8 Servo control system with impulsive filter F. . . . . . . . 201
10.9 Impulsive signal identification. . . . . . . . . . . . . . . . 203
10.10Impulsive signal filtering. . . . . . . . . . . . . . . . . . . 204
10.11Comparison of signals with and without filtering. . . . . . 209
B.1 Case 1.1.1-1.1.3 . . . . . . . . . . . . . . . . . . . . . . . . 220
B.2 Case 5.1.1-5.1.2 . . . . . . . . . . . . . . . . . . . . . . . . 224
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List of Tables
6.1 Four cases of Y . . . . . . . . . . . . . . . . . . . . . . . 113
2
9.1 Fuzzy relationship . . . . . . . . . . . . . . . . . . . . . . 175
10.1 Motor parameters . . . . . . . . . . . . . . . . . . . . . . 194
B.1 8 cases when λ >0 &λ =0 & η >0 &δ λ η >0 . . . 219
1 2 1 1
B.2 Four cases when λ (cid:2)=0 & λ (cid:2)=0 & CTx≥0 . . . . . . . 221
1 2
B.3 Four cases when λ (cid:2)=0 & λ (cid:2)=0 & CTx<0 . . . . . . . 222
1 2
B.4 Four cases and sixteen subcases . . . . . . . . . . . . . . . 223
B.5 α δ =β γ =0 & c (cid:2)=0, i=1,2 . . . . . . . . . . . . . 226
2 1 2 1 i
B.6 c =0,i=1 or 2 . . . . . . . . . . . . . . . . . . . . . . . 226
i
B.7 8 subcases of Case 7.3 . . . . . . . . . . . . . . . . . . . . 227
xiii
Preface
Piecewise affine (PWA) system has wide applications in analysis and
synthesis of engineering systems. It is considered as a powerful tool to
study nonlinear systems, switched systems, and hybrid systems. This
book presents our research findings in control and estimation of PWA
systems.
We begin with the examination of the stability of PWA systems by
providing a unified view on the stability based on piecewise quadrat-
ic Lyapunov function for both continuous-time and discrete-time PWA
systems. We then introduce a piecewise homogeneous polynomial Lya-
punov function approachto achieve less conservativestability conditions
by using two types of power transformations. The approach leads to
less conservative stability analysis than existing results. In addition to
these stability results based on halfspace representation, we also find
ways to address the stability of PWA systems based on vertex represen-
tation. Specifically, we incorporate vertex information of local partition
andutilizeparameterizedLyapunovfunctionstodeducelessconservative
conditions for stability.
Next,weaddressthecontrollabilityandreachabilityofaclassofPWA
systems. Basedonageneralclassificationmethod,explicitnecessaryand
sufficientconditionsintermsofsystemparametersforcontrollabilityand
reachabilityarepresented. We showthat controllabilityandreachability
can be asserted in finite steps in some situations. We also briefly discuss
the general case.
We then turn to controller design problems for PWA systems. We
first consider the state feedback control for PWA systems. By incor-
porating partition information of the system and applying S-procedure,
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Control and estimation of piecewise affine systems
linear matrix inequality (LMI) based design methods are proposed. To
further reduce the conservatism, we give a method to convert the con-
troldesign probleminto a bilinear matrix inequality (BMI) problemand
propose several iterative procedures for solving the problem. H∞ and
generalized H control of PWA systems are also investigated.
2
Parallel to the control problem, we consider H∞ and generalized H2
estimation for both continuous-time and discrete-time PWA systems.
The Luenberger type estimators with less conservatism are designed by
introducing proper transformations embedded with the partition infor-
mation.
We also take the Takagi-Sugeno fuzzy system as an application of
PWAsystemsandderivethe correspondingstability conditionsandcon-
troller design approaches.
Finally, two mechanical systems, an electric motor driven pendulum
with third order nonlinearities and a hard disk servosystem, are used to
illustrate the PWA model approximation and design issues.
During the courseof the researchembodied inthe book, we werefor-
tunate tohaveclosecollaborationswithseveralcolleagues. Inparticular,
we are grateful to Professors Carlos de Souza, Gary Feng, Minyue Fu,
HongbinMa,XiaoliCao,ZhengzhongHan,andHuanshuiZhangfortheir
helpful discussions, suggestionsand advices. The help from Ms Tingting
Gao for the hard disk example is greatly appreciated.
We would like to dedicate the book to our families for their constant
support and enduring love. In particular, Jun Xu would like to dedicate
the book to his parents, Gaoyuan and Zhimei, his wife, Grace, and son,
Alexander, Lihua Xie to his wife, Meiyun, and daughters, Jessica and
Julie.
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