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Re: control 2008

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Re: control 2008

Modelling and Control
of Dynamical Systems:
Numerical Implementation
in a Behavioral Framework


Ricardo Zavala Yoe

TOC

Contents
1 Motivating the Behavioral Approach 1
1.1 Suitable Modelling and Control of Systems . . . . . . . . . . . . . . . . . . 1
1.2 Paradigms in Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Closed dynamical systems . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Open dynamical systems and the input/output approach . . . . . . 4
1.2.3 More about the input/output approach . . . . . . . . . . . . . . . . 9
1.2.4 The behavior of the systemis the key . . . . . . . . . . . . . . . . . 10
1.2.5 Some other frameworks for systems and control . . . . . . . . . . . 10
2 Behavioral framework 13
2.1 Modelling by Tearing and Zooming . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Constitutivemodels . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Linear Differential Systems . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Latent variables and elimination . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Equivalent representations of behaviors . . . . . . . . . . . . . . . . . . . . 23
2.5 Observability and detectability . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 Controllability and stabilizability . . . . . . . . . . . . . . . . . . . . . . . 24
2.7 Autonomous behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.8 Defining inputs and outputs . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.9 Controllable part of a behavior . . . . . . . . . . . . . . . . . . . . . . . . 29
2.10 Interconnection of dynamical systems . . . . . . . . . . . . . . . . . . . . . 30
2.10.1 Control as interconnection . . . . . . . . . . . . . . . . . . . . . . . 30
3 Full Interconnection Issues 35
3.1 Implementability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.1 Minimal Annihilators of a Polynomial Matrix . . . . . . . . . . . . 37
3.2 Stabilization and pole placement by regular full interconnection . . . . . . 45
3.3 All regularly implementing controllers . . . . . . . . . . . . . . . . . . . . 49
3.4 All stabilizing controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
VII
VIII CONTENTS
4 Partial Interconnection Issues 59
4.1 Regular implementability by partial interconnection . . . . . . . . . . . . . 59
4.2 Pole placement and stabilization by regular partial interconnection . . . . . 60
4.2.1 Pole placement by regular partial interconnection . . . . . . . . . . 60
4.2.2 Stabilization by regular partial interconnection . . . . . . . . . . . . 67
4.3 All regularly implementing controllers: the observable case . . . . . . . . . 71
4.4 All regularly implementing controllers: the nonobservable case . . . . . . . 77
4.4.1 Reduction to the case that R2 has full column rank . . . . . . . . . 78
4.4.2 Reduction to the observable case . . . . . . . . . . . . . . . . . . . 79
4.5 All stabilizing controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.6 Examples for the nonobservable case . . . . . . . . . . . . . . . . . . . . . 88
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5 Embedding Algorithms 95
5.1 Problemformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2.1 Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3 Pencils andMatrix Pencils . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3.1 Canonical forms of pencils . . . . . . . . . . . . . . . . . . . . . . . 101
5.3.2 A little bit deeper intomatrix pencils . . . . . . . . . . . . . . . . . 101
5.4 The state space representation . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.5 Embedding for a pencil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.6 Transforming the pencil . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.7 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.7.1 QR Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.7.2 Staircase form of ξE − A . . . . . . . . . . . . . . . . . . . . . . . . 109
5.7.3 Algorithm: Embedding P(ξ) . . . . . . . . . . . . . . . . . . . . . 111
5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6 Numerical Implementation 115
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2 Analysis of an example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.3 The geometry of the orbit of a pencil . . . . . . . . . . . . . . . . . . . . . 117
6.4 Matrix pencils asmathematical relations . . . . . . . . . . . . . . . . . . . 120
6.5 Conditioning of the pencil . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.6 Modelling polynomially and assessing numerically . . . . . . . . . . . . . . 128
6.7 Computing the determinant of a polynomial matrix . . . . . . . . . . . . . 131
6.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
CONTENTS IX
7 A new algorithm for embedding problems 135
7.1 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2 Inside the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.3 Numerical computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Conclusions and further research 141
Bibliography 143
Summary 151
Index 153