Linear, time-varying approximations to nonlinear dynamical systems: with applications in control and optimization

302 Pages·2010·1.44 MB·English

by  María Tomás-Rodríguez

Lecture Notes
in Control and Information Sciences 400
Editors:M.Thoma,F.Allgöwer,M.Morari
María Tomás-Rodríguez and Stephen P. Banks
Linear, Time-varying
Approximations to Nonlinear
Dynamical Systems
with Applications in Control and Optimization
ABC
SeriesAdvisoryBoard
P.Fleming,P.Kokotovic,
A.B.Kurzhanski,H.Kwakernaak,
A.Rantzer,J.N.Tsitsiklis
Authors
Dr.MaríaTomás-Rodríguez Prof.StephenP.Banks
CityUniversityLondon UniversityofSheffield
SchoolofEngineering& Dept.AutomaticControl&
MathematicalSciences SystemsEngineering
NorthamptonSquare MappinStreet
London Sheffield
UnitedKingdom UnitedKingdom
E-mail:[email protected] E-mail:s.banks@sheffield.ac.uk
ISBN978-1-84996-100-4 e-ISBN978-1-84996-101-1
DOI 10.1007/978-1-84996-101-1
LectureNotesinControlandInformationSciences ISSN0170-8643
LibraryofCongressControlNumber:2009942768
(cid:2)c2010Springer-VerlagBerlinHeidelberg
(cid:2) (cid:2)
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To my parents and brother, (M T-R).
To David and Xu, (S P B).
Preface
This book is the culmination ofseveralyears’researchon nonlinearsystems.
In contrast to the case of linear systems, where a coherent and well-defined
theory has existed for many years (indeed, in many respects, we may regard
linear systems theory as ‘complete’), nonlinear systems theory has tended to
beasetofdisparateresultsonfairlyspecifickindsofsystems.Ofcourse,there
arecoherenttheories ofnonlinearsystems using differentialand/oralgebraic
geometric methods, but, in many cases, these have very strong conditions
attached which are not satisfied in general. In an attempt to build a theory
which has great generality we have been led to consider systems with the
structure
x˙ = A(x;u)x+B(x;u)u
(possibly also with a measurement equation).
This appearstobe quite restrictive,but, asweshallsee,almosteverysys-
tem canbe put in this form,so that the theory is, infact, almostcompletely
general. We shall show that systems of this form can be approximated arbi-
trarilyclosely(onanyfinitetimeinterval-nomatterhowlarge)byasequence
of linear, time-varying systems. This opens up the prospect of using existing
lineartheoryinthe(global)solutionofnonlinearproblemsanditisthiswith
whichthebookisconcerned.Itisaresearchmonograph,butitcouldbeused
asagraduate-leveltext;wehavetriedtokeepthenotationstandard,sothat,
for the most part, the mathematical language is well-known. In some parts
of the book, some previous knowledge of Lie algebras, differential geometry
and functional analysis is necessary. Since there are, of course, many excel-
lent(classical)textsonthesesubjects,wehavemerelygivenreferencestothe
requisite mathematical ideas.
Now we outline the detailed contents of the book. In Chapter 1 we intro-
duce the systems with which we shall be interested and show how they are
relatedto the mostgeneralnonlinearsystems. Chapter 2 begins the detailed
analysisofthesesystems,andinparticular,we discussthe‘iterationscheme’
VIII Preface
whichis the maintechnicaltoolofthe approach.Since the method givesrise
to sequences of linear, time-varying systems, Chapter 3 is a detailed analy-
sis of such systems; in particular, we determine some explicit solutions and
study the stability and spectral theory of these systems. In Chapter 4 we
showthatmuchofthelinearspectraltheoryofsystemscanbegeneralisedto
nonlinear systems and in Chapter 5 we give a main application of the ideas
to the spectral assignmentproblem in nonlinear systems. Optimal controlof
linear systems is a major part of ‘classical’ control theory and in Chapter
6 we show how to use the iteration method to extend the theory to nonlin-
earsystems.WealsodiscusstheoptimalityviatheHamilton-Jacobi-Bellman
equation.The needfor morerobustcontrollersled to the discoveryofsliding
controllers, which again are generalised to nonlinear systems (and nonlinear
sliding surfaces)inChapter7.InChapter 8we showhowthe method relates
to fixed-point theory and how it can be used inductively to derive certain
conditions on nonlinear systems. The generalisationof the technique to par-
tial differential equations and systems is given in Chapter 9, together with
examples from moving boundary problems and solitons (nonlinear waves).
Lie algebraic methods have significant impact on linear systems theory and
in Chapter 10 we see that it can also give a powerful structure theory for
nonlinear systems. The global theory of nonlinear systems on manifolds is
outlined in Chapter 11 where we show how to piece together a number of
local systems into a global one by use of the theory of connections. Low-
dimensional systems on manifolds are considered in the cases of 2, 3 and 4
dimensions. Finally, in Chapter 12, we speculate on the future possibilities
of the iteration method and show that it is likely to be applicable in many
other circumstances in nonlinear systems theory. The appendices give some
background on linear algebra, Lie algebras, manifold theory and functional
analysis.
Finally, we should acknowledge the influence of many of our students and
colleagues who have been associated with this work over the years and, in
particular,MetinSalamci,TayfunCimen,DavidMcCaffrey,ClaudiaNavarro-
Hernandez, Oscar Hugues-Salas, Zahra Sangelaji, Sherif Fahmy, Yi Song,
Evren Gurkan Covasoglu, Xianhua Zheng, Chunyan Du, Wei Chen, Xu Xu,
Salman Khalid, Serdar Tombul and Mehmet Itik. They have all contributed
in various ways to the evolution of this technique.
Sheffield, London, Mar´ıa Tom´as-Rodr´ıguez
January 2010 Stephen P. Banks
Contents
1 Introduction to Nonlinear Systems ...................... 1
1.1 Overview............................................. 1
1.2 Existence and Uniqueness .............................. 2
1.3 Logistic Systems ...................................... 3
1.4 Control of Nonlinear Systems ........................... 4
1.5 Vector Fields on Manifolds ............................. 5
1.6 Nonlinear PartialDifferential Equations .................. 6
1.7 Conclusions and Outline of the Book..................... 8
References ................................................ 9
2 Linear Approximations to Nonlinear Dynamical
Systems ................................................. 11
2.1 Introduction .......................................... 11
2.2 Linear, Time-varying Approximations.................... 12
2.3 The Lorenz Attractor .................................. 16
2.4 Convergence Rate ..................................... 17
2.5 Influence of the Initial Conditions on the Convergence...... 20
2.6 Notes on Different Configurations ....................... 22
2.7 Comparison with the Classical Linearisation Method....... 23
2.8 Conclusions........................................... 26
References ................................................ 27
3 The Structure and Stability of Linear, Time-varying
Systems ................................................. 29
3.1 Introduction .......................................... 29
3.2 Existence and Uniqueness .............................. 29
3.3 Explicit Solutions ..................................... 32
3.4 Stability Theory....................................... 46
3.5 Lyapunov Exponents and Oseledec’s Theorem............. 51
X Contents
3.6 Exponential Dichotomy and the Sacker-SellSpectrum ...... 57
3.7 Conclusions........................................... 59
References ................................................ 60
4 General Spectral Theory of Nonlinear Systems .......... 61
4.1 Introduction .......................................... 61
4.2 A Frequency-domain Theory of Nonlinear Systems......... 61
4.3 Exponential Dichotomies ............................... 70
4.4 Conclusions........................................... 73
References ................................................ 74
5 Spectral Assignment in Linear, Time-varying Systems ... 75
5.1 Introduction .......................................... 75
5.2 Pole Placement for Linear, Time-invariant Systems ........ 77
5.3 Pole Placement for Linear, Time-varying Systems.......... 79
5.4 Generalisation to Nonlinear Systems ..................... 89
5.5 Application to F-8 Crusader Aircraft..................... 94
5.6 Conclusions........................................... 97
References ................................................ 98
6 Optimal Control......................................... 101
6.1 Introduction .......................................... 101
6.2 Calculus of Variations and Classical Linear
Quadratic Control ..................................... 101
6.3 Nonlinear Control Problems ............................ 106
6.4 Examples............................................. 109
6.5 The Hamilton-Jacobi-Bellman Equation, Viscosity
Solutions and Optimality ............................... 114
6.6 Characteristics of the Hamilton-Jacobi Equation........... 117
6.7 Conclusions........................................... 120
References ................................................ 121
7 Sliding Mode Control for Nonlinear Systems............. 123
7.1 Introduction .......................................... 123
7.2 Sliding Mode Control for Linear Time-invariant Systems ... 124
7.3 Sliding Mode Control for Linear Time-varying Systems..... 125
7.4 Generalisation to Nonlinear Systems ..................... 129
7.5 Conclusions........................................... 137
References ................................................ 139
Contents XI
8 Fixed Point Theory and Induction....................... 141
8.1 Introduction .......................................... 141
8.2 Fixed Point Theory.................................... 141
8.3 Stability of Systems.................................... 145
8.4 Periodic Solutions ..................................... 147
8.5 Conclusions........................................... 149
References ................................................ 150
9 Nonlinear Partial Differential Equations ................. 151
9.1 Introduction .......................................... 151
9.2 A Moving Boundary Problem ........................... 152
9.3 Solution of the Unforced System......................... 153
9.4 The Control Problem .................................. 155
9.5 Solitons and Boundary Control.......................... 161
9.6 Conclusions........................................... 167
References ................................................ 167
10 Lie Algebraic Methods .................................. 169
10.1 Introduction .......................................... 169
10.2 The Lie Algebra of a Differential Equation................ 170
10.3 Lie Groups and the Solution of the System ............... 174
10.4 Solvable Systems ...................................... 177
10.5 The Killing Form and Invariant Spaces................... 179
10.6 Compact Lie Algebras ................................. 185
10.7 Modal Control ........................................ 190
10.8 Conclusions........................................... 194
References ................................................ 194
11 Global Analysis on Manifolds............................ 195
11.1 Introduction .......................................... 195
11.2 Dynamical Systems on Manifolds........................ 196
11.3 Local Reconstruction of Systems ........................ 197
11.4 Smooth Transition Between Operating Conditions ......... 199
11.5 From Local to Global .................................. 201
11.6 Smale Theory......................................... 203
11.7 Two-dimensional Manifolds ............................. 205
11.8 Three-dimensional Manifolds............................ 208
11.9 Four-dimensional Manifolds............................. 212
11.10Conclusions .......................................... 215
References ................................................ 216
12 Summary, Conclusions and Prospects for
Development ............................................ 219
12.1 Introduction .......................................... 219
12.2 TravellingWaveSolutionsinNonlinearLattice Differential
Equations ............................................ 219
XII Contents
12.3 Travelling Waves ...................................... 220
12.4 An Approach to the Solution ........................... 221
12.5 A Separation Theorem for Nonlinear Systems ............. 222
12.6 Conclusions........................................... 227
References ................................................ 227
A Linear Algebra .......................................... 229
A.1 Vector Spaces......................................... 229
A.2 Linear Dependence and Bases........................... 231
A.3 Subspaces and Quotient Spaces ......................... 233
A.4 Eigenspaces and the Jordan Form ....................... 234
References ................................................ 237
B Lie Algebras............................................. 239
B.1 Elementary Theory .................................... 239
B.2 Cartan Decompositions of Semi-simple Lie Algebras ....... 241
B.3 Root Systems and Classification of Simple Lie Algebras .... 244
B.4 Compact Lie Algebras ................................. 254
References ................................................ 255
C Differential Geometry ................................... 257
C.1 Differentiable Manifolds ................................ 257
C.2 Tangent Spaces ....................................... 258
C.3 Vector Bundles........................................ 259
C.4 Exterior Algebra and de Rham Cohomology .............. 260
C.5 Degree and Index...................................... 261
C.6 Connections and Curvature ............................. 264
C.7 Characteristic Classes.................................. 267
References ................................................ 269
D Functional Analysis...................................... 271
D.1 Banach and Hilbert Spaces ............................. 271
D.2 Examples............................................. 274
D.3 Theory of Operators ................................... 275
D.4 Spectral Theory....................................... 277
D.5 Distribution Theory ................................... 280
D.6 Sobolev Spaces........................................ 285
D.7 Partial Differential Equations ........................... 287
D.8 Semigroup Theory..................................... 289
D.9 The Contraction Mapping and Implicit Function
Theorems ............................................ 292
References ................................................ 293
Index........................................................ 295