Nonlinear Systems and Controls

754 Pages·2022·16.788 MB·English

by  Jürgen Adamy

Jürgen Adamy
Nonlinear
Systems and
Controls
Nonlinear Systems and Controls
Jürgen Adamy
Nonlinear Systems
and Controls
JürgenAdamy
Institut für Automatisierungstechnik
TU Darmstadt
Darmstadt, Germany
ISBN978-3-662-65632-7 ISBN978-3-662-65633-4 (eBook)
https://doi.org/10.1007/978-3-662-65633-4
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For Claudia, Ursula, and Timm
Preface
Nonlinear dynamic systems and controls play an important role in science,
mathematics,economics,andengineering.Theregulationofhumanbodytem-
perature,theregulationofthemoneysupplyinaneconomy,andtheautopilot
of an airplane are examples. A book about these systems can be written from
thepointofviewofthescientist,themathematician,ortheengineer.Abookof
this kind must orient itself towardsthe objective of the discipline in question:
abookfocusedonthenaturalscienceswouldexplainthedynamicbehaviorof
naturalphenomena,suchasplanetarymotionorthebloodpressureregulation
in mammals. A mathematical approach would focus on theorems and their
proofs, and a book addressed to engineers would describe the theory and its
practical application, such as how to model and control nonlinear processes
and systems like power plants and rockets.
KarlPopper’sthree-worldtheoryillustratesthesethreeperspectives:itisa
modelof thescientificprocessthat dividesthe worldintothe worldofmatter,
theworldofthesubjectiveego,andtheworldofobjectiveknowledge,asshown
in Figure P.1. The material world includes all kinds of matter including flora,
fauna, and the human body. The world of the subjective ego includes the
perceptions, thoughts, and ideas of an individual human being. In contrast,
the objective world consists of, among other things, scientific theories, works
ofart,lawsandagreementsthatareavailabletoallormanypeopletoexamine
and change.
Scientistsobservethematerialworldandusethesubjectiveegotodevelop
a theory of how it works: they make discoveries. A discovery is published in
a scientific journal and is thus made part of the objective world. In this way,
a theory is broadly accessible and can be tested by other scientists and either
confirmed or falsified. In contrast to the classical natural sciences (physics,
chemistry, and biology), mathematics is a method, or we could also saya lan-
guage, with which natural and technological processes can be quantitatively
described. This is not possible using our everyday language. Mathematicians
are constantly expanding this language and exploring new ways of describing
facts using formulas and solving problems mathematically: they draft math-
vii
viii Preface
H
H C H
C C
C C @B~
H C H
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Understanding
scientific theories
Publishing theories Mathematician
and
patenting inventions
Natural scientist Engineer
making discoveries creating inventions
Observation Building
of nature of machinery
Fig. P.1: Karl Popper’s three-world ontology
ematical hypotheses and prove them. Engineers make use both of scientific
theories,especiallyphysicaltheories,and mathematics,thelanguagein which
these theories are formulated. They use this knowledge of the objective world
to create, within the world of their subjective ego, the idea for a new ma-
chine or a new technical process: they invent something new. By realizing
these inventions,they changethe materialworld.Theybuild skyscrapers,air-
planes, powerplants, and robots. Construction plans and system descriptions
published in patents and journals become part of the objective world.
Preface ix
Although these three disciplines are sharply delineated above, there are
a large number of fields in which they overlap. These include experimental
physics, in which scientists build special equipment for their research such
as particle accelerators, and control engineering, in which new mathemati-
cal theories, such as that of flat systems, are developed to solve engineering
problems.
This book is primarily oriented towards the engineer’s point of view. The
aimofthebookisthereforetoconveytheknowledgeofnonlinearsystemsand
controlsthatisimportantforanengineer,andtoillustrateitsapplicability.In
doing so, the book describes the current state of the theory of these systems.
Forthispurpose,thetheoremsofthismathematicaltheorywhichhaveproved
useful in engineering practice have been selected. Since these theorems and
their proofs are available in the literature, the proofs have not been shown
here in the classical theorem-proof sequence: the emphasis in this book is on
teaching the theory and its application rather than proving of the theorems.
Literature Recommendations
To learn more about a particular field of research in detail, it is useful to
consult various books which address that field. In the case of the theory of
nonlinear systems and controls, for example, the works of H. K. Khalil, E.
D. Sontag, S. Sastry, H. Nijmeijer and A. van der Schaft, J.-J.E. Slotine and
W. Li, H.J. Marquez, and A. Isidori are recommended. To understand the
theory of nonlinear systems, the theory of linear dynamic systems is useful
or necessary in many cases. Describing this theory would require a similarly
extensive volume, so it cannot be addressed in detail in this book and it
is assumed that the reader is already familiar with it. Suitable works on this
topichavebeenwrittenbyG.F.Franklin,J.D.PowellandA.Emami-Naeini,
R. C. Dorf and R. H. Bishop, among others.
Acknowledgements
This book is the translation of an improved and extended version of the
third edition of a book written in German on nonlinear systems and con-
trols. For both the German and English editions, a large number of scholars
haveproofreadthe bookforcomprehensibility,errors,and mistakes,and have
contributed to its layout. For this English edition, I would like to especially
thank K. Olhofer-Karova for patiently preparing the LATEXsource code and
Matlab simulations, and V. Ansel for preparing the color images. My thanks
gototheAmericantranslatorH.Krehbielforthelinguisticrevisionofthetext.
Jürgen Adamy, Technische Universität Darmstadt, 2022
Contents
Preface ........................................................ vii
Literature Recommendations .................................. ix
Acknowledgements ............................................ ix
1 Fundamentals of Nonlinear Systems ....................... 1
1.1 System Description and System Behavior................... 1
1.1.1 Linear and Nonlinear Systems....................... 1
1.1.2 System Description and Nonlinear Control Loops...... 2
1.1.3 Equilibrium Points of Nonlinear Systems ............. 5
1.1.4 Example: Satellite ................................. 7
1.1.5 Equilibrium Points of Linear Systems ................ 9
1.1.6 Stability and Asymptotic Stability................... 11
1.1.7 Exponential Stability of Equilibrium Points........... 18
1.1.8 Instability of Equilibrium Points .................... 20
1.1.9 Stability in the Case of Variable Input Signals ........ 24
1.1.10 Limit Cycles...................................... 28
1.1.11 Sliding Modes..................................... 30
1.1.12 Chaos............................................ 32
1.1.13 Discrete-Time Systems............................. 35
1.2 Solution of Nonlinear Differential Equations ................ 37
1.2.1 Existence of Solutions.............................. 37
1.2.2 Numerical Solution and Euler Method ............... 41
1.2.3 Accuracy of the Numerical Solution.................. 43
1.2.4 The Modified Euler Method ........................ 44
xi
xii Contents
1.2.5 The Heun and Simpson Methods .................... 45
1.2.6 The Runge-Kutta Methods ......................... 47
1.2.7 Adaptation of the Step Size......................... 49
1.2.8 The Adams-Bashforth Methods ..................... 50
1.2.9 The Adams-Moulton Predictor-Corrector Method ..... 52
1.2.10 Stability of Numerical Integration Methods ........... 53
1.2.11 Stiff Systems and Their Solutions.................... 56
1.3 Exercises ............................................... 58
2 Limit Cycles and Stability Criteria ........................ 69
2.1 The Describing Function Method .......................... 69
2.1.1 Idea behind the Method............................ 69
2.1.2 Illustrative Example ............................... 73
2.1.3 Characteristic Curves and Their Describing Functions.. 75
2.1.4 Stability Analysis of Limit Cycles ................... 82
2.1.5 Example: Power-Assisted Steering System ............ 85
2.2 Absolute Stability ....................................... 89
2.2.1 The Concept of Absolute Stability................... 89
2.2.2 The Popov Criterion and Its Application ............. 91
2.2.3 The Aizerman and Kalman Conjectures .............. 97
2.2.4 Example: Controlling a Ship ........................ 99
2.2.5 The Circle Criterion ...............................103
2.2.6 The Tsypkin Criterion for Discrete-Time Systems .....110
2.3 Lyapunov’s Stability Theory ..............................113
2.3.1 The Concept and the Direct Method.................113
2.3.2 Illustrative Example ...............................118
2.3.3 Quadratic Lyapunov Functions......................119
2.3.4 Example: Mutualism...............................121
2.3.5 The Direct Method for Discrete-Time Systems ........125
2.3.6 The Indirect Method...............................126
2.3.7 Determining Exponential Stability...................127
2.3.8 Example: Underwater Glider........................129
2.3.9 Catchment Regions................................134
2.3.10 LaSalle’s Invariance Principle .......................137
2.3.11 Instability Criterion ...............................142
2.4 Passivity and Stability ...................................143
2.4.1 Passive Systems...................................143
2.4.2 Stability of Passive Systems ........................146
2.4.3 Passivity of Connected Systems .....................148
2.4.4 Passivity of Linear Systems.........................151
2.4.5 Example: Transporting System for Material Webs .....155
2.4.6 Positive Real Transfer Functions ....................158
2.4.7 Equivalence of Positive Realness and Passivity ........163
2.4.8 Lossless Hamiltonian Systems.......................168
2.4.9 Example: Self-Balancing Vehicle.....................174