Transient effects of linear dynamical systems [Elektronische Ressource] / von Elmar Plischke

Sujets

• Mathematics

Informations

• Publié par: universitat_bremen
• Publié le: 01 janvier 2005
• Nombre de lectures: 40
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Transient Effects
of Linear Dynamical Systems
von Elmar Plischke
Dissertation
zur Erlangung des Grades eines Doktors der Naturwissenschaften
– Dr.rer.nat. –
Vorgelegt im Fachbereich 3 (Mathematik & Informatik)
der Universit¨at Bremen
im Juli 2005Datum des Promotionskolloquiums: 19. August 2005
Gutachter: Prof. Dr. Diederich Hinrichsen (Universitat¨ Bremen)
PD Dr. Fabian Wirth (NUI Maynooth, Irland)Contents
Introduction 1
1 Preliminaries 7
1.1 Matrix Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Properties of Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Spectral Value Sets and Stability Radii . . . . . . . . . . . . . . . . . . . . 10
1.4 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Contractions and Liapunov Norms 15
2.1 One-Parameter Semigroups in Banach Spaces . . . . . . . . . . . . . . . . 15
2.2 Asymptotic Growth Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Initial Growth Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Liapunov Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 Bounds for the Transient Amplification 51
3.1 (M,β)-Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Bounds from the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 from Singular Value Decompositions . . . . . . . . . . . . . . . . . 60
3.4 Bounds via Liapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5 from the Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4 Examples 91
4.1 Explicit Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Construction of Transient Norms . . . . . . . . . . . . . . . . . . . . . . . 93
4.3 Liapunov Matrices of Minimal Condition Number . . . . . . . . . . . . . . 99
4.4 Dissipativity for Polytopic Norms . . . . . . . . . . . . . . . . . . . . . . . 104
4.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5 Positive Systems Techniques 107
5.1 Properties of Metzler Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 Transient Bounds for Metzler Matrices . . . . . . . . . . . . . . . . . . . . 109
iiiiv CONTENTS
5.3 Optimal Liapunov Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.4 Common Liapunov Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.5 The Metzler Part of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.6 Transient Bounds for General Matrices . . . . . . . . . . . . . . . . . . . . 123
5.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6 Differential Delay Systems 131
6.1 Functional Analytic Approach . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.2 Liapunov Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.3 Existence and Uniqueness of Delay Liapunov Matrices. . . . . . . . . . . . 147
6.4 The One-Delay Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.5 Uncertain Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.6 Multiple Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.7 Scalar Differential Delay Equations . . . . . . . . . . . . . . . . . . . . . . 179
6.8 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7 (M,β)-Stabilization 183
7.1 Synthesis of Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.2 Contractibility for Polytopic Norms . . . . . . . . . . . . . . . . . . . . . . 188
7.3 (M,β)-Stabilizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
7.4 Quadratic (M,β)-Stabilizability . . . . . . . . . . . . . . . . . . . . . . . . 191
7.5 Quadratic Programs for (M,β)-Stabilization . . . . . . . . . . . . . . . . . 196
7.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
List of Symbols 201
Bibliography 203
Index 212Introduction
From a practical point of view the concept of stability may be deceiving: Asymptotic
stability allows for arbitrary growth before a decay occurs. These transient effects which
are only of temporary nature have no influence on the asymptotic stability of a dynamical
system. However, these effects might dominate the system’s performance. Hence we are
in need of information which describes the short-time behaviour of a dynamical system.
Motivation
Everyone has noticed that devices need some time to get ready for use, like an old radio
warming up or a computer booting. On a larger scale, plants also need an amount of
time to reach their working point. But in this initial phase, the plant is particularly
vulnerable. The faster one wants to reach the working point, the more stress the plant
has to endure: there may be overshots which carry some parts of the plant to the limit
of their capacity. One may think of chemicals, which advance towards toxic or explosive
concentrations, before reaching the desired concentration of the reagents, or an autopilot
steering the wrong way before getting on track.
One could believe that such a distinctive behaviour in the initial phase can only occur for
complex dynamical systems. However, this behaviour can already be observed for linear
differential equations which provide simple models for dynamical processes.
Toavoidcatastropheslikethoseindicatedabove,onewishestoeliminatethebadinfluences
in the starting phase, or at least, to keep them small. Furthermore, methods are needed
that allow to predict if the system under consideration shows these transient effects, and
if so, to obtain information on the duration and intensity of these excursions.
This work is mainly concerned with questions dealing with the last two issues, namely
finding bounds on the exponential growth of linear systems. Although there are many
results on exponential bounds, there is still no systematic treatment in the literature.
The mathematical model of a plant will in general not yield the accurate description of the
behaviour of the real plant. Hence we are in need of results which are robust under small
perturbations of the mathematical model. Fortunately, these results follow directly from
our systematic treatment of the exponential bounds.
In addition to the general theory, we study two major classes of linear systems, namely
positivesystemsanddelay-differentialsystems,whichareusedfrequentlyineconomicsand
biology.
12 INTRODUCTION
Moreover, we study the influence of state feedback on the transient behaviour. We ob-
tain necessary and sufficient conditions to obtain a closed-loop system without transient
excursions.
Finite-dimensional linear systems are mostly used as an approximation of more complex
dynamical systems. These are obtained by linearization or discretization. Let us now
discuss two possible ways in which the transients of linear systems may influence the
dynamics.
From Transience to Turbulence
Most linear dynamical systems are obtained by linearizing a nonlinear model of a real
process around an equilibrium point. Now, Liapunov’s theorem implies that the nonlinear
system is asymptotically stable if the linearization is asymptotically stable.
Figure 1: Toy model for turbulence.
But if the asymptotically stable linear system has solutions which move far afield before
eventually decaying, these solutions of the linear system may leave the domain for which
the linear system is a valid approximation of the nonlinear system. Hence small perturba-
tions from the equilibrium point may incite nonlinearities. Models of this kind have been
suggested in Baggett and Trefethen [8] to explain why turbulence of certain flows occurs
at Reynolds numbers much smaller than predicted from a spectral analysis. For example,
let us investigate the following nonlinear time-invariant ordinary differential equation

−5 36 0 −1 2x˙ =Ax+B(x) = x+kxk x, x∈R , (1)
0 −20 1 0
where A is an asymptotically stable, but nonnormal matrix and the nonlinearity B(x) is
conservative (energy-preserving), thus B only adds a rotation of the state-space to theINTRODUCTION 3
linear system x˙ =Ax. Figure 1 shows many trajectories of system (1) starting on a circle
of radius 50.
The trajectories which converge to the origin are colored in black, which gives a rough
approximation of the domain of attraction for the origin. One observes that this domain
of attraction is flat, the nonnormality of the linear system x˙ = Ax quickly drives the
state into regions where the nonlinearity has strong effects on the state. Note that in this
examplenonlinearityandnonnormalityformoppositeforceswhichcreateasortofconveyor
belt driving the states away from the stable origin. The picture drastically changes when
replacing B(x) by−B(x).
From Transience to Permanence
Another interesting observation can be made when approximating infinite dimensional
systems by finite dimensional approximants. Now assume that there exists a sequence of
finite dimensional matrices which approximate an infinite dimensional linear operator, and
that this se