Ali Sayyad Thesis 13th April

221 Pages·2017·1.84 MB·English

by  Neil Stewart

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Citation: Sayyad, Ali (2016). Optimisation of condition number for eigenstructure
problems. (Unpublished Doctoral thesis, City, University of London)
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OPTIMISATION OF CONDITION
NUMBER FOR EIGENSTRUCTURE
PROBLEMS
By
Ali A. Sayyad

A THESIS SUBMITTED FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

CITY, UNIVERSITY OF LONDON
SCHOOL OF MATHEMATICS, COMPUTER SCIENCE AND ENGINEERING
DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING
SYSTEMS & CONTROL RESEARCH CENTRE
NORTHAMPTON SQUARE, LONDON EC1V 0HB
UNITED KINGDOM
SEPTEMBER 2016

© Copyright by Ali A. Sayyad, 2016
To my wife and daughter

i
Abstract
This thesis deals with a new solution for the problem of eigenstructure assignment
in control systems design. A wide range of challenging issues is examined involving
the problem of eigenstructure assignment and associated system properties through
different forms of complexity which are strongly related to control system design. In
this thesis, specific attention has been given to the issue of skewness of the closed-
loop eigenframe of the state matrix. In fact, the aim is to develop a new
methodology for determining the best angle between closed-loop eigenvectors by
optimising the minimal condition number of the closed-loop eigenvector matrix.
This problem is strongly linked to sensitivity of eigenvalues to parameter
uncertainty, perturbations to model parameter uncertainty. The importance of this
methodology can be expressed in terms of results related to the Sensitivity of
eigenvalues, Relative measures of controllability and observability and also
deviations from strong stability to overshooting behaviour. Among this variety of
eigenstructure assignment methods, special consideration has been paid to
Geometric Theory [4], [5], which introduces an alternative solution to the
assignability of spectrum of controllability subspaces (cs) based on an eigenvector
approach and then develops a new pole assignment algorithm based on open-
loop/closed-loop spectra as a practical application of this approach. In order to
tackle the problem of measuring the skewness of angles between closed-loop
eigenvalues, some measures for Eigenframe skewness have been defined in general
and so the necessary and efficient conditions have been derived for the angle
between some subspaces in a direct sum decomposition to be maximized. This has
been done via three metrics; Condition Number, Determinant of Gram Matrix and
Singular Values. The thesis presents the parametrisation of closed-loop eigenframes
result by the method generated in [4]. Within this thesis, a non-smooth algorithm has
been developed in order to select the most orthogonal closed-loop eigenframe and so
the influence of selected closed-loop spectra. Also, the parametrisation of
controllability subspaces in a standard direct sum decomposition using matrix
fraction description (MFD) has been derived. Within this thesis the construction and
ii
the existence of controllability subspaces connected to (A,B)-invariant subspaces, has
also been studied. In addition, an algebraic description of the total system behaviour
which leads to an algebraic characterisation of the total input, state and output
behaviour in an implicit formulation is given based on properties of MFD
descriptions, a topic which remains open for future studies.

iii
Table of contents
Abstract .................................................................................................................................. ii
Table of Contents ................................................................................................................ iv
Acknowledgments ............................................................................................................ viii
1 Introduction ........................................................................................................................ 1
1.1 Introduction ................................................................................................................... 1
1.2 Objectives ...................................................................................................................... 3
1.3 Main achievements....................................................................................................... 5
1.4 Summary of upcoming chapters ................................................................................ 6
2 Mathematics and Control background ........................................................................ 12
2.1 Introduction ................................................................................................................. 12
2.2 Mathematics background .......................................................................................... 12
2.2.1 Vector space .......................................................................................................... 12
2.2.2 Subspace ................................................................................................................ 13
2.2.3 Orthogonal and orthonormal vectors ............................................................... 13
2.2.4 Unitary Matrix ..................................................................................................... 13
2.2.5 Vector Matrix........................................................................................................ 13
2.2.6 Matrix norms ........................................................................................................ 14
2.2.7 Hadamard’s inequality theorem ....................................................................... 15
2.2.8 Gramian (Gram) Matrix ...................................................................................... 15
2.2.9 Condition Number .............................................................................................. 16
2.2.10 Vandermonde matrix ........................................................................................ 16
2.2.11 The Jordan Canonical Form of a Matrix ......................................................... 17
2.3 Control background ................................................................................................... 17
2.3.1 Matrix Fraction Description (MFD) .................................................................. 17
2.3.2 Minimal bases of matrix pencils and coprime matrix descriptions ............. 23
2.3.3 Construction of minimal bases of matrix pencils and coprime fraction
descriptions using Toeplitz matrix ............................................................................ 25
2.4 Conclusion ................................................................................................................... 32
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3 Literature review on Eigenstructure assignment: Basic concepts and background
results .................................................................................................................................... 34
3.1 Introduction ................................................................................................................. 34
3.2 Background on Eigenvalues and Eigenvectors ...................................................... 35
3.2.1 Rectilinear motions .............................................................................................. 35
3.2.2 Summary of spectral characterisation .............................................................. 37
3.2.3 Controllability and Observability issues .......................................................... 38
3.3 Forced Rectilinear motions and closed loop Eigenstructure ............................... 40
3.3.1 Physical problems ................................................................................................ 40
3.3.2 Characterisation of Transmission ...................................................................... 41
3.3.3 Feedback and closed-loop Eigenvalues ............................................................ 43
3.3.4 The problem of Eigenspace assignment ........................................................... 45
3.4 Review of results on Eigenstructure assignment ................................................... 48
3.4.1 Early results .......................................................................................................... 48
3.4.2 State feedback results .......................................................................................... 50
3.4.3 Output feedback results...................................................................................... 51
3.4.4 Combined state and output feedback approach ............................................. 53
3.4.5 Approach that reduces controllers complexity ............................................... 54
3.4.6 Results obtained from a subspace theme ......................................................... 55
3.4.7 Parametric state feedback results ...................................................................... 55
3.4.8 Parametric output feedback results ................................................................... 56
3.4.9 Perturbation Theory ............................................................................................ 57
3.4.10 Other approaches ............................................................................................... 64
3.5 Summary and Conclusion ......................................................................................... 66
4 Basic concepts of Eigenstructure assignment from geometric theory ................... 70
4.1 Introduction ................................................................................................................. 70
4.2 Motion along eigenvectors and the A-invariant subspaces of linear
system ................................................................................................................................. 72
4.3 Rectilinear motion of Non-autonomous system in the input output
state spaces and the concept of (A,B)- invariance ........................................................ 74
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4.4 Simple rectilinear motion and the one-dimensional (A,B)-invariant
subspaces ........................................................................................................................... 75
4.5 Rectilinear motion in r-dimensional (A,B)-invariance subspaces,
characteristic decomposition of (A,B)- invariant subspaces ...................................... 77
4.6 Controllability Subspaces .......................................................................................... 83
4.7 Conclusion ................................................................................................................... 86
5 Decomposition of state space into controllability subspaces ................................. 87
5.1 Introduction ................................................................................................................. 87
5.2 (A,B)- invariant subspaces intersecting with B ............................................... 87
5.3 Algebraic characterization of Controllability Subspaces ..................................... 92
5.3.1 Problem statement ............................................................................................... 93
5.3.2 Minimal dimension Controllability Subspaces ............................................... 96
5.4 Assignability of the spectrum of a Controllability subspace ............................... 99
5.4.1 Problem statement ............................................................................................... 99
5.4.2 Assigning the spectrum of a Controllability Subspace ................................ 100
5.4.3 Eigenvalue placement algorithm based on mobility of open to
closed loop spectra ..................................................................................................... 104
5.4.4 Numerical example ........................................................................................... 107
5.5 Selection of the most orthogonal eigenvectors and sensitivity of
closed-loop system based on the open loop – closed loop spectra method .......... 112
5.5.1 Problem statement ............................................................................................. 112
5.6 Conclusion ................................................................................................................ 115
6 Measurement of angle between subspaces in direct sum decomposition.......... 116
6.1 Introduction .............................................................................................................. 116
6.2 Problem statement and preliminary results ......................................................... 117
6.3 Measuring the degree of orthogonality ................................................................. 120
6.3.1 The Gramian ....................................................................................................... 120
6.3.2 Condition Number ............................................................................................ 124
6.3.3 The Spread of Singular Values ........................................................................ 129
6.4 Numerical example .................................................................................................. 133
6.5 Conclusion ................................................................................................................. 137
vi
7 Parameterisation of ordered minimal bases of Controllability Subspaces ........ 139
7.1 Introduction ............................................................................................................... 139
7.2 Implicit system description .................................................................................... 140
7.2.1 Duality issues and behavior ............................................................................. 143
7.2.2 Computation of Input- State generator pairs Problem statement .............. 147
7.2.3 Closed-loop Eigenvectors and frequency transmission ............................... 148
7.2.4 Pole assignment by output feedback and closed-loop Eigenvectors ......... 149
7.2.5 Poles and Zeros .................................................................................................. 152
7.2.6 Design of state feedback controllers using Eigenvector parametrisation . 154
7.2.7 Selection of an independent Eigenframe and resulting state feedback ..... 156
7.2.8 Parametrisation of ordered minimal bases and the selection of
Eigenframes ................................................................................................................. 159
7.3 Conclusion ................................................................................................................. 170
8 Minimsation of the angle between closed-loop Eigenvectors via non-
smooth Algorithm ............................................................................................................. 172
8.1 Introduction ............................................................................................................... 172
8.2 Non-smooth optimisiation problem definitions .................................................. 174
8.3 Generalized gradient of k (A(x)) ...................................................................... 176
8.3.1 k (A) ................................................................................................................... 176
8.3.2 k (A(x))with A(x)=V(x)TV(x) ........................................................................ 178
8.4 Smoothing approximation ..................................................................................... 180
8.5 Non-smooth Optimization Algorithm ................................................................. 182
8.6 Numerical examples ................................................................................................ 185
8.7 Conclusion ................................................................................................................. 192
9 Conclusion and Future Research ................................................................................ 193
9.1 Achievements, Results ............................................................................................. 193
9.2 Future Research ........................................................................................................ 198
References........................................................................................................................... 201
Appendix_1 ........................................................................................................................ 209

vii
Acknowledgements
I would like to express my utmost gratitude to my supervisors Professor Karcanias
and Professor Halikias.
Professor Karcanias who has been a most inspiring, yet demanding supervisor with
a great insights in the field of Control Design Theory which have been invaluable
during the work of this thesis. His constant support and guidance during my time of
research have been priceless and have made this present thesis possible.
I would like also to thank Professor Halikias for his tremendous support and
assistance in various sections of this thesis. His outstanding knowledge in both
theory and programming aspects enabled me to achieve desired results throughout
the time of research.
Finally, I would like to thank my family, Maryam and Diba, for their unconditional
love, support and encouragement all these years. This thesis is dedicated to them for
helping be what I am today.

London, United Kingdom Ali A. Sayyad
September 2016
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