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

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El que sabe que sabe es un sabio. S' ?guelo. El que no sabe que sabe esta dormido. Despi' ertalo. El que sabe que no sabe es sencillo. Instruyelo. ' El que no sabe que no sabe es un necio. Ap' artate de ' el (He who knows and knows he knows, he is wise. Follow him. He who knows and knows not he knows, he is asleep. Wake him. He who knows not and knows he knows not, he is simple. Teach him. He who knows not and knows not he knows not, he is a fool. Shun him.) Proverbio Arabe. Corrige al sabio y lo har' as m' as sabio, corrige al necio y lo har' as tu enemigo. Sedulo curavi, humanas actiones non ridere, non lugere, neque detestari, sed intelligere. (I have made a ceaseless e ort not to ridicule, not to bewail, not to scorn human actions, but to understand them.) Benedictus de Spinoza (1632-1677).
1 Motivating the Behavioral Approach 1
1.1 Suitable Modelling and Control of Systems
1
1.2 Paradigms in Modelling
2
1.2.1 Ch end 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 system is the key
10
1.2.5 Scone other frameworks fin- systems and control
10
2 Behavioral framework 13
2.1 Modelling by Tearing and Zooming
13
2.1.1 Constitutive models
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 cletectability
23
2.6 Controllability and stabilizability
24
2.7 AMA/Minions 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
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 Polo placement by regular partial interconnection
60
4.2.2 Stabilization by regular paitial 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 fin the nonobservable case
88
4.7 Summary
93
5 Embedding Algorithms 95
5.1 Problem formulation
95
5.2 Preliminaries
97
5.2.1 Historical overview
97
5.2.2 Notation
100
5.3 Pencils and Matrix Plicils
101
5.3.1 Canonical lerins of pencils
101
5.3.2 A little bit deeper into matrix pencils
101
5.4 The state space representatk
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 all example
115
6.3 The geometry of the orbit of a pencil
117
6.4 Matrix pencils as mathematical 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 Onicluding remarks
133
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