Formal computational methods for control theory [Elektronische Ressource] / vorgelegt von Daniel Robertz

Daniel Robertz

Sujets

• Mathematics

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2006
• Nombre de lectures: 10
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Formal Computational Methods
for Control Theory
Von der Fakult¨at fu¨r Mathematik, Informatik und Naturwissenschaften
der Rheinisch-Westf¨alischen Technischen Hochschule Aachen zur
Erlangung des akademischen Grades eines Doktors der Naturwissenschaften
genehmigte Dissertation
vorgelegt von
Diplom-Mathematiker
Daniel Robertz
aus Aachen
Berichter: Universit¨atsprofessor Dr. Wilhelm Plesken
Universit¨atsprofessor Dr. Gerhard Jank
Tag der mu¨ndlichen Pru¨fung: 20.06.2006
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek
online verfu¨gbar.iiContents
1 Introduction 1
2 Janet’s Algorithm 5
2.1 Decomposition of Sets of Monomials into Disjoint Cones . . . . . 7
2.2 Janet’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Combinatorial Tools . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Ore Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Janet Bases for Ore Algebras . . . . . . . . . . . . . . . . . . . . 31
2.6 Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.7 Reducing the Complexity of Janet Basis Computations . . . . . . 36
3 Symbolic Computation with Differential Equations 43
3.1 The Jet Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Linearization of Differential Equations . . . . . . . . . . . . . . . 48
3.3 Janet Bases for Linear Differential Equations with Non-constant
Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 The Generalized Hilbert Series 67
4.1 Module-theoretic Approach to Linear Systems . . . . . . . . . . . 67
4.2 Homological Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3 Injective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4 Injective Cogenerators . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5 The Generalized Hilbert Series for Partial Differential Equations . 88
4.6 The Bernstein Filtration . . . . . . . . . . . . . . . . . . . . . . . 91
5 Algebraic Systems Theory 95
5.1 Structural Properties of Linear Systems . . . . . . . . . . . . . . . 97
i5.2 Computation of ext (M,D) . . . . . . . . . . . . . . . . . . . . . 100D
5.3 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4 Autonomous Observables . . . . . . . . . . . . . . . . . . . . . . . 110
5.5 The Maple package OreModules . . . . . . . . . . . . . . . . . . . 115
iiiiv CONTENTS
6 Parametrizing Linear Systems 117
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2 Parametrizing Linear Systems over Ore Algebras . . . . . . . . . . 121
6.3 Parametrizing Linear Systems with Autonomous Observables . . . 125
6.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.5 Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.6 Computing Bases of Free Modules over the Weyl Algebras . . . . 141
7 A Stirred Tank Model 149
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.2 Controllability, parametrizability, flatness . . . . . . . . . . . . . . 151
7.3 Autonomous observables . . . . . . . . . . . . . . . . . . . . . . . 157
7.4 Observability, input-output behavior . . . . . . . . . . . . . . . . 161
7.5 Motion planning . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.6 Optimal control problems . . . . . . . . . . . . . . . . . . . . . . 170
7.7 A discrete-time model . . . . . . . . . . . . . . . . . . . . . . . . 177
7.8 A differential time-delay model . . . . . . . . . . . . . . . . . . . 182
Bibliography 191
Symbol Table 200
Index 201
Curriculum Vitae 204Chapter 1
Introduction
This thesis treats structural properties of control systems, e.g. controllability or
parametrizability of their behavior, from an algebraic point of view. It is in the
tradition of R. E. Kalman, who gave very important impetus to the structural
analysis of control systems in the 1960s. Since then more and more algebraic
methods have been applied and developed to gain insight into the behavior of
control systems.
With this work I contribute to a few aspects of the links between algebra and
control theory emphasizing formal methods and computational issues, which are
also of independent interest. First I describe the main features of the employed
method.
The control system is assumed to be modeled by a set of equations of a
certain kind. In general, these equations are nonlinear. Then the algebraic ap-
proach which is pursued here considers a linearization of these equations. Now
linear equations define a module over a ring which is chosen in accordance with
the type of the given equations (e.g. ordinary or partial differential equations,
difference equations, retarded differential equations, etc.). All consequences of
the given equations are built into this module so that it represents the relations
among the inspected quantities of the control system in an intrinsic way; in par-
ticular, two equivalent sets of linear equations give rise to the same module. The
space of functions which is expected to contain solutions of the given equations
is also assumed to be a module over the same ring; e.g. for differential equations
the action on the function space is by differentiation, for difference equations
by shifts, etc. Structural properties of the control system (e.g. the possibility
to parametrize the solutions of the system in a formal way) are characterized
by properties of the module which is defined by the given equations. However,
these characterizations require a suitable interplay of the module defined by the
equations and the module of admissible functions. In particular, the outlined
approach is not applicable (at least in the full generality) to all kinds of function
spaces. Themostinterestingpropertiesofthemoduleassociatedwiththesystem
12 CHAPTER 1. INTRODUCTION
equations can be dealt with by using homological algebra. This gives a unified
approach to the structural properties and the possibilities to check them compu-
tationally. Amongthestructuralpropertiesofinterestistheexistenceofso-called
autonomous quantities, which cannot be influenced by a control. Whereas it is
difficult in general to draw conclusions from the structural analysis of the lin-
earized system about the properties of the given nonlinear one, sometimes the
existence of such autonomous quantities for the original system can be confirmed
or denied. If the linearized system is completely controllable, then the given
nonlinear one has no autonomous quantities either. Conversely, if autonomous
quantities are found for the linearized system, then one can try to lift them to
the original system.
Among the developers of this algebraic approach are U. Oberst and E. Zerz
who have been working mostly with equations with constant coefficients which
give rise to commutative rings, and J.-F. Pommaret and A. Quadrat who were
the first to tackle the non-commutative case.
The algebraic approach to control systems described above demands algo-
rithms for symbolic computations at different steps of the strategy. If the given
equations are nonlinear, an appropriate way to linearize them is required. The
ring which is chosen in accordance with the type of the given equations needs
to be dealt with constructively. Moreover, methods to compute with modules
over these rings are fundamental in order to realize the constructions used from
homological algebra.
Addressing the first issue, this thesis presents a method which results in a
linearizationofdifferentialequationsthatisindependentofanychosentrajectory.
In particular, no solution of the set of equations is needed. This defines a generic
linearization of the system (Chapter 3).
Secondly, Janet’s algorithm is used for computing with rings and modules
which arise in the present context. Given a finite generating set for a submod-
ule of a free module of tuples over a commutative polynomial algebra, Janet’s
algorithm constructs another finite set, called Janet basis, which generates the
same submodule, but consists of enough elements such that for any given tuple
it can be decided whether it is an element of the submodule or not. More pre-
cisely, a Janet basis defines normal forms for the elements of the residue class
module, enabling in this way effective computations in this module. As a tool for
commutative algebra, Janet bases serve similar purposes as the more commonly
known Gr¨obner bases do. Whereas Gr¨obner bases were introduced in the 1930s
and Buchberger’s algorithm, which computes Gr¨obner bases, was developed in
the 1960s, Janet’s theory from the 1920s even roots in earlier work of M´eray and
Riquier in the last decades of the 19th century. Janet’s formal approach to sys-
tems of differential equations seemed forgotten for a long time, but was revived
by J.-F. Pommaret.3
In this thesis, Janet’s algorithm is a keystone. In Chapter 2, I generalize
Janet’s algorithm from the well-known cases of commutative polynomial alge-
bras and the Weyl algebra to Ore algebras which are of interest in the following
chapters. For system theoretic applications Ore algebras are sufficiently gen-
eral because the most important types of equations define Ore algebras, whose
elements represent the operators which ar