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Continuous Time
Dynamical Systems
State Estimation and Optimal Control
with Orthogonal Functions
B. M. Mohan
S. K. Kar
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
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CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2013 by Taylor & Francis Group, LLC
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Version Date: 20120822
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Library of Congress Cataloging‑in‑Publication Data
Mohan, B. M. (Bosukonda Murali)
Continuous time dynamical systems : state estimation and optimal control with
orthogonal functions / B.M. Mohan, S.K. Kar.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-4665-1729-5 (hardback)
1. Functions, Orthogonal. 2. Differentiable dynamical systems--Automatic
control--Mathematics. 3. Mathematical optimization. I. Kar, S. K. II. Title.
QA404.5.M64 2012
515’.55--dc23 2012028470
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
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Dedication
To my wife, Vijaya Lakshmi,
my son, Divyamsh &
my daughter, Tejaswini
B. M. Mohan
To my parents, Satya Ranjan & Manjushree,
my sister, Sanjeeta,
my wife, Swastika &
my daughters, Santika & the late Ayushi
S. K. Kar
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Contents
List of Abbreviations xi
List of Figures xiii
Preface xix
Acknowledgements xxi
About the Authors xxiii
1 Introduction 1
1.1 Optimal Control Problem . . . . . . . . . . . . . . 1
1.2 Historical Perspective . . . . . . . . . . . . . . . . . 3
1.3 Organisation of the Book . . . . . . . . . . . . . . . 8
2 Orthogonal Functions and Their Properties 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Block-Pulse Functions (BPFs) . . . . . . . . . . . 14
2.2.1 Integration of B(t) . . . . . . . . . . . . . . 15
2.2.2 Product of two BPFs . . . . . . . . . . . . 16
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vi
2.2.3 Representation of C(t)f(t) in terms of BPFs 16
2.2.4 Representation of a time-delay vector in
BPFs . . . . . . . . . . . . . . . . . . . . . 17
2.2.5 Representation of reverse time function vec-
tor in BPFs . . . . . . . . . . . . . . . . . . 18
2.3 Legendre Polynomials (LPs) . . . . . . . . . . . . . 19
2.4 Shifted Legendre Polynomials (SLPs) . . . . . . . . 20
2.4.1 Integration of L(t) . . . . . . . . . . . . . . 22
2.4.2 Product of two SLPs . . . . . . . . . . . . . 23
2.4.3 Representation of C(t)f(t) in terms of SLPs 24
2.4.4 Representation of a time-delay vector func-
tion in SLPs . . . . . . . . . . . . . . . . . 25
2.4.5 Derivation of a time-advanced matrix of
SLPs . . . . . . . . . . . . . . . . . . . . . 27
2.4.6 Algorithm for evaluating the integral in
Eq. (2.75) . . . . . . . . . . . . . . . . . . . 28
2.4.7 Representation of a reverse time function
vector in SLPs . . . . . . . . . . . . . . . . 30
2.5 Nonlinear Operational Matrix . . . . . . . . . . . . 30
2.6 Rationale for Choosing BPFs and SLPs . . . . . . . 32
3 State Estimation 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Inherent Filtering Property of OFs . . . . . . . . . 38
3.3 State Estimation . . . . . . . . . . . . . . . . . . . 39
3.3.1 Kronecker product method . . . . . . . . . . 43
3.3.2 Recursive algorithm via BPFs . . . . . . . . 43
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vii
3.3.3 Recursive algorithm via SLPs . . . . . . . . 44
3.3.4 Modification of the recursive algorithm of
Sinha and Qi-Jie . . . . . . . . . . . . . . . 45
3.4 Illustrative Examples . . . . . . . . . . . . . . . . . 47
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 58
4 Linear Optimal Control Systems Incorporating
Observers 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Analysis of Linear Optimal Control Systems Incor-
porating Observers . . . . . . . . . . . . . . . . . . 68
4.2.1 Kronecker product method . . . . . . . . . 69
4.2.2 Recursive algorithm via BPFs . . . . . . . . 70
4.2.3 Recursive algorithm via SLPs . . . . . . . . 71
4.3 Illustrative Example . . . . . . . . . . . . . . . . . 72
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 77
5 Optimal Control of Systems Described by Integro-
Differential Equations 79
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Optimal Control of LTI Systems Described by
Integro-Differential Equations . . . . . . . . . . . . 80
5.3 Illustrative Example . . . . . . . . . . . . . . . . . 84
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 86
6 Linear-Quadratic-Gaussian Control 89
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 89
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viii
6.2 LQG Control Problem . . . . . . . . . . . . . . . . 90
6.3 Unified Approach . . . . . . . . . . . . . . . . . . 93
6.3.1 Illustrative example . . . . . . . . . . . . . . 96
6.4 Recursive Algorithms . . . . . . . . . . . . . . . . . 97
6.4.1 Recursive algorithm via BPFs . . . . . . . . 101
6.4.2 Recursive algorithm via SLPs . . . . . . . . 102
6.4.3 Illustrative example . . . . . . . . . . . . . . 104
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 105
7 Optimal Control of Singular Systems 109
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 109
7.2 Recursive Algorithms . . . . . . . . . . . . . . . . . 111
7.2.1 Recursive algorithm via BPFs . . . . . . . . 113
7.2.2 Recursive algorithm via SLPs . . . . . . . . 114
7.3 Unified Approach . . . . . . . . . . . . . . . . . . 115
7.4 Illustrative Examples . . . . . . . . . . . . . . . . . 117
7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 121
8 Optimal Control of Time-Delay Systems 125
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 126
8.2 Optimal Control of Multi-Delay Systems . . . . . . 129
8.2.1 Using BPFs . . . . . . . . . . . . . . . . . . 133
8.2.2 Using SLPs . . . . . . . . . . . . . . . . . . 135
8.2.3 Time-invariant systems . . . . . . . . . . . . 137
8.2.4 Delay free systems . . . . . . . . . . . . . . 138
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8.2.5 Illustrative examples . . . . . . . . . . . . . 138
8.3 Optimal Control of Delay Systems with Reverse
Time Terms . . . . . . . . . . . . . . . . . . . . . . 146
8.3.1 Using BPFs . . . . . . . . . . . . . . . . . . 151
8.3.2 Using SLPs . . . . . . . . . . . . . . . . . . 153
8.3.3 Illustrative example . . . . . . . . . . . . . . 154
8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 155
9 Optimal Control of Nonlinear Systems 159
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 159
9.2 Computation of the Optimal Control Law . . . . . 160
9.3 Illustrative Examples . . . . . . . . . . . . . . . . 162
9.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 166
10 Hierarchical Control of Linear Systems 169
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 169
10.2 HierarchicalControlofLTISystemswithQuadratic
Cost Functions . . . . . . . . . . . . . . . . . . . . 170
10.2.1 Partial feedback control . . . . . . . . . . . 172
10.2.2 Interaction prediction approach . . . . . . . 173
10.3 Solution of Hierarchical Control Problem via BPFs 174
10.3.1 State transition matrix . . . . . . . . . . . 175
10.3.2 Riccati matrix and open-loop compensation
vector . . . . . . . . . . . . . . . . . . . . . 177
10.3.3 State vector . . . . . . . . . . . . . . . . . . 178
10.3.4 Adjoint vector and local control . . . . . . . 180
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