Haddad W.M. Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach
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NonlinearBook10pt November 20, 2007
Preface
Dynamical system theory provides a paradigm for modeling and studying
phenomena that undergo spatial and temporal evolution. A d ynamical
system consists of three elements—namely, a setting (called the state space)
in which the dynamical behavior takes place, such as a torus, topological
space, manifold, or locally compact metric space; a mathematical rule or
dynamic which specifies the evolution of the sys tem over time; and an initial
condition or state from which the system starts at some initial time. Ever
since its inception, the basic questions concerning dynamical system theory
have involved qualitative solutions for the properties of a dynamical system;
questions such as: For a particular initial system state, does th e dynamical
system have at least one solution? What are the asymptotic properties
of the system solutions? How are the system solutions dependent on the
system initial conditions? How are the sys tem solutions dependent on the
form of the m athematical description of the dynamic of the system? How
do system solutions depend on system parameters? And how do system
solutions depend on the properties of the state space on which the system
is defin ed ?
Even though qualitative properties of solutions to dynamical systems
were first pioneered by Henri Poincar´e, mathematical dynamical system
theory can be traced back to Isaac Newton. Newton was the first to model
the motion of physical systems with differential equations. However, the
development of dynamical system theory is a natural outcome of a much
broader theme which is as old as science itself and can be traced back
to the great cosmic theorists of ancient Greece—namely, the universe is
in a constant state of flux ( ). Ancient Greek astronomers
and mathematicians such as Eudoxus, Ptolemy, and Archimedes were
the first to use abstract mathematical models and attach them to the
physical world. More importantly, using abstract th ought and theoretical
calculations they were able to ded uce something that is true about the
physical world. Newton’s greatest achievement, however, was the discovery
that the motion of the planets and moons of the solar system resulted from
a single fundamental source—the gravitational attraction of the heavenly
NonlinearBook10pt November 20, 2007
xxii PREFACE
bodies. As a consequence, Newtonian physics developed into the first field of
modern science—dynamical systems as a branch of mathematical physics—
wherein the circular, elliptical, and parabolic orbits of the heavenly bodies
of our solar system were no longer fundamental determinants of motion,
but rather approximations of the universal laws of the cosmos specified by
governing d ifferential equations of motion.
In the past century, dyn amical system theory quickly spread from the
study of astronomical bodies to modeling and studying numerous physical
phenomena occurring in nature involving changes over time, and it has
become one of the most important and fundamental fields of modern
science and engineering. The app lication of dynamical systems has crossed
interdisciplinary boundaries from chemistry to biochemistry to chemical
kinetics, from medicine to biology to population genetics, from economics
to sociology to psychology, and from physics to mechanics to engineering.
The increasingly complex nature of engineering systems requiring feedback
control to obtain a desired system behavior also gives rise to dynamical
systems. Feedback control th eory involves the analysis and synthesis of a
feedback controller that manipulates s ystem inputs to obtain a desired effect
on the output of the system in the face of system uncertainty and sys tem
disturbances. Furthermore, since most physical and engineering systems are
inherently nonlinear, the resulting feedback dynamical system can exhibit a
very rich dynamical behavior.
One of the most important concepts in the study of dynamical system
theory and control theory is the concept of stability. Stability theory
concerns the behavior of the system trajectories of a dynamical system
when the system initial state is n ear an equilibrium state. Since exogenous
disturbances and s ystem component uncertainty are always present in every
actual system, stability theory plays a central role in dynamical systems
and control. As in the study of dynamical sy s tem theory, the origins
of stability theory can be traced back to the study of idealizations of
astronomical problems involving persistent small oscillations (librations)
about a s tate of motion. One of the first modern treatments of stability
theory was given by Joseph-Louis Lagrange wherein he concluded that
for a conservative mechanical system an isolated minimum of the system
potential en ergy corresponds to a stable equilibrium system state. The most
complete stability analysis framework for d ynamical systems was developed
by Aleksandr Lyapunov. Lyapunov’s method is based on construction of a
function of the system state co ordinates that s erves as a generalized norm
of the solution of the dynamical system. Its appeal comes from the fact
that conclusions about the behavior of the dynamical system can be drawn
without actually computing the system solution trajectories. As a result,
Lyapunov stability th eory has become one of the cornerstones of systems
NonlinearBook10pt November 20, 2007
PREFACE xxiii
and control theory.
Aleksandr Mikhailovich Lyapunov was born on the 6th of June, 1857
to the family of the pr omin ent astronomer Mikhail Vasilyevich Lyapunov
in Yaroslavl, Russia. After completing the gymnasium (high school) in
Nizhny Novogorod where he was taught Greek, Latin, and basic science and
mathematics, in 1876 he entered the Physics and Mathematics Department
of Saint Petersburg University. Upon graduating from Saint Petersburg
University in 1880, he remained in the department of mechanics at the
university to prepare for his academic career which was greatly influenced
by Chebyshev. His master’s work concentrated on the problem of a
homogeneous, incompressible fluid mass held together by gravitational forces
of its particles and rotating about a fixed axis. Chebyshev posed the
question, Under what conditions will the motion of the fluid be stable and
what possible equilibrium f orms can this stable rotating fluid take? This
famous question of the piriform body, which was of considerable importance
to cosmogony and had been studied by prominent mathematicians such
as Newton, MacLaurin, Jacobi, and Poincar´e, was eventually settled by
Lyapunov in 1905.
After receiving his Master’s degree in applied mathematics in 1884
for his work On the Stability of Ellipsoidal Forms in the Equilibrium of a
Rotating Fluid, Lyapunov moved to Kharkov University as a Privatdozent
(private reader or lecturer) in mechanics where h e continued his research
towards his doctoral thesis. In 1892 the Kharkov Mathematical Society
published Lyapunov’s seminal work on The General Problem of the Stability
of Motion which he defended as his doctoral thesis at Moscow State
University (Moskovskii Gosudarstvennii Universitet) later that year. His
paper “Sur le Probl`eme General de la Stabilit´e du Mouvement” publish ed
in the Annales de la Facult´e des Sciences de l’Universit´e de Toulouse in
1907 marks the beginning of modern stability theory in the West. In 1893
Lyapunov was appointed as a professor at Kharkov Un iversity and in 1902 he
returned to Saint Petersburg University as C hair of Applied Mathematics.
In 1900 he was elected as a corresponding member of the Russian Academy
of Sciences, and in 1901 as a full member of the Academy in the field of
applied mathematics.
Even though Lyapunov is pred omin antly known for his work on the
stability of equilibria and motion of mechanical systems with a finite number
of degrees of freedom in the dynamical systems and control community, his
scientific work in cludes fundamental contributions to stability of equilibrium
figures of rotating fluids; equilibrium figures of uniformly rotating fluids;
mathematical physics; probability theory; and theoretical mechanics. In all
these disciplines Lyapunov established a deep level of accuracy and rigor
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xxiv PREFACE
that resu lted in fundamental and classical mathematical results. Lyapunov
tragically died on the 3rd of November, 1918 in Odessa, Russia (now
Ukraine) as a result of a voluntary act three days after the death of his
wife. His wish, which he had stated in an ante mortem note, was to be
buried with his wife.
The main objective of this book is to present and develop necessary
mathematical tools for stability analysis and control design of nonlinear
dynamical systems, with an emphasis on Lyapunov-based methods. This
book is intended to be useful to applied mathematicians, dynamical system
theorists, control theorists, and engineers. Since dynamical system theory
and Lyapunov stability theory lie at the heart of mathematical sciences
and engineering, researchers and graduate students in these fields who s eek
a fundamental understanding of the rich behavior of nonlinear dynamical
systems and control w ill also find this textbook useful. The appropriate
background for this book is a first course in linear system theory and a fi rst
course in advanced (multivariable) calculus.
After a brief introduction on dynamical systems in Chapter 1, a
systematic development of nonlinear ordinary differential equations is
given in Chapter 2. In Chapters 3 and 4, we present fundamental
stability theory as well as advanced stability theory for nonlinear dynamical
systems. Chapter 5 provides a tr eatment of classical dissipativity theory
and absolute stability theory. A detailed treatment of stability of feedback
interconnections, control Lyapunov functions, feedback linearization, zero
dynamics, and stability margins for nonlinear regulators is given in Chapter
6. Chapter 7 focuses on input-output stability and dissipativity theory.
In Chapter 8, we present the nonlinear optimal control problem, while
stability and optimality results for backstepping control problems are given
in Chapter 9. C hapters 10–12 provide novel extensions to disturbance
rejection and robust control of non linear dynamical systems. Finally,
Chapters 13 and 14 present discrete-time extensions of the aforementioned
topics.
The authors would like to thank Dennis S. Bernstein, Sanj ay P. Bhat,
Tomohisa Hayakawa, V. Lakshmikantham, and Frank L. Lewis for their
constructive comments and feedback. I n some parts of the book we have
relied on work we have done jointly with Chaouki T. Abd allah, Dennis
S. Bernstein, Sanjay P. Bhat, Jerry L . Fausz, Qin g Hu i, Vikram Kapila,
Alexander Leonessa, Sergey G. Nersesov, and David A. Wilson; it is a
pleasure to acknowledge their contributions. In addition, we thank Sanjay
P. Bhat, Tomohisa Hayakawa, Qing Hui, Vikram Kapila, and Sergey G.
Nersesov for graciously agreeing to read chapters of the book for clarity and
technical accuracy. We are also grateful to our students for their helpful
NonlinearBook10pt November 20, 2007
PREFACE xxv
comments, insightful suggestions, and corrections , all of which greatly
improved the exposition of the book.
The authors would also like to thank Paul Katinas for providing a
translation of Anaximander’s, Herakleitos’, and Archimedes’ statements qu-
oted in ancient Greek on page vii. In addition, we thank Paul for several
insightful and enlightening philosophical discus sions on the ramifications
of these statements to cosmology, mathematics, science, and engineering.
Anaximander’s aphorism introduces for the first time the concepts of
beginning ( ) and infinity ( ) which are central to the study
of cosmology and modern mathematics. A common misconception among
scholars has been th at modern mathematicians and not ancient Greek
mathematicians were the first to be able to address notions of infinity in
a rigorous and precise manner. However, as has been recently discovered in
the Archimedes Palimpsest, Archimedes was the first to rigorously address
the science of infinity with infinitely large sets using precise mathematical
proofs in his work on The Method of Mechanical Theorems.
Heraklitos’ profound statements created the foundation for all physics
and metaphysics. His first statement marks the beginning of science and
postulates the big bang theory as the origin of the universe as well as the
heat death of the universe. He further postulates that the universe evolves
in accordance with its own laws which are the only unchangeable th ings
in the universe (i.e., universal conservation and nonconservation laws). His
second and third statements give the earliest perception of irreversibility of
nature and the universe along with time’s arrow. The idea that the universe
is in constant change ( ) and there is an underlying order to
this change—the Logos ( )—postulates the existence of entropy as a
physical property of matter permeating the whole of nature and the universe.
In addition, postulating that the fundamental un iform fact in nature is
constant change and everything both is and is not at the same time (
), as well as that energy is change and substance is change
(
), he arrives at a
precursor to the principle of relativity and the mass-energy equivalence.
Finally, Archimedes’ statement leads to th e foundation of mathemat-
ical mechanics. His law of the lever—involving the earliest treatment of
a constrained mechanical system—associated with this statement led to
the idea of energy as the product of force and distance, to the concept
of the conservation of energy, and to the principle of virtual velocities.
In his treatise on The Method of Mechanical Theorems he established
the foundations of integral calculus using infinitesimals, as well as the
foundations of mathematical mechanics. In addition, in one of his p roblems
NonlinearBook10pt November 20, 2007
xxvi PREFACE
he constructed the tangent at any given point for a spiral, establishing the
origins of differential calculus.
Among the greatest gifts of Greece to humankind and world civiliza-
tion are philosophy, mathematics, science, and engineering. Monumental
scholars in these disciplines inspired a deep love for creative work and an
unparalleled thirst for k nowledge ( ). These modern minds in ancient
bodies sto od out in bold relief as towering talents thr ou ghout the centuries,
paving the way for modern mathematics, science, and engineering. Consider,
for example, how intense Archimedes’ passion for stu dy ( ) must have
been. He was deeply immersed in his mathematics when he delivered his
legendary last words, (Do not d isturb my
circles!), before being murdered by a Roman soldier. The golden age of
Greece, which set the standard for western civilization and whose glory
has transcended time itself, was superseded by the Roman imperialists
who brought nothing more than sterility to science and mathematics. The
derailment of the pursuit of abstract science and mathematics in favor of
practicality obfuscated scientific scholarship. This, along with religious
fundamentalism, plunged civilization into the dark ages. The intellectual
light ( ) that was gifted to the world by the ancient Greeks and provided
the pathway for unlocking the most intriguing mysteries of our world lay
idle for over one thousand years. It was not until the fall of Byzantium
to Ottoman tribes from Mongolia that caused Greek scholars to flee to
Florence, Venice, and Rome, sparkin g a r evival in learning and humanism.
This renaissance or rebirth ( ) led to the scientific revolution
which further led to the marvels of modern-day science and engineering.
Atlanta, Georgia, USA, October 2007, Wassim M. Haddad
Knoxville, Tennessee, USA, October 2007, VijaySekhar Chellaboina
NonlinearBook10pt November 20, 2007
Chapter One
Introduction
A system is a combination of components or parts that is perceived as a
single entity. Th e parts making up th e system may be clearly or vaguely
defined. These parts are related to each other through a particular set of
variables, called the states of the system, that completely determine the
behavior of the system at any given time. A dynamical system is a system
whose state changes with time. Specifically, the state of a dynamical system
can be regarded as an information storage or memory of past system events.
The set of (internal) states of a dynamical system must be sufficiently rich to
completely determine the behavior of the system for any future time. Hence,
the state of a dynamical system at a given time is uniquely determined by
the s tate of the system at the initial time and the present input to the
system. In other words, the state of a dynamical system in general depends
on both the present input to the system and the past history of the sys tem.
Even though it is often assu med that the state of a dynamical system is
the least set of state variables needed to completely p redict the effect of the
past upon the future of the system, th is is often a convenient simplifying
assumption.
We regard a dynamical system G as a mathematical model structur e
involving an input, state, and output that can capture the dynamical
description of a given class of physical systems. S pecifically, at each moment
of time t ∈ T, where T denotes a time-ordered sub set of the reals, the
dynamical system G receives an input u(t) (e.g., matter, energy, information)
and generates an output y(t). The values of the input are taken from
the fi xed set U. Furthermore, over a time segment the inpu t function
u : [t
1
, t
2
) → U is not arbitrary but belongs to the admissible input class U,
that is, for every u(·) ∈ U and t ∈ T, u(t) ∈ U . The input class U depends
on the physical description of the system. In addition, each system output
y(t) belongs to the fixed set Y with y(·) ∈ Y over a given time segment,
where Y denotes an output space. In general, the output of G depends on
both the present inp ut of G and the past history of G. Thus, the state,
and hence the output at some time t ∈ T, depend s on both the initial state
x(t
0
) = x
0
and the input segment u : [t
0
, t) → U. In other words, knowledge
NonlinearBook10pt November 20, 2007
2 CHAPTER 1
of both x
0
and u ∈ U is necessary and sufficient to determine the present
and future state x(t) = s(t, t
0
, x
0
, u) of G.
In light of the above discussion, we view a dynamical system as a
precise mathematical object defined on a time set as a mapping between
vector spaces satisfying a set of axioms. A mathematical dynamical system
thus consists of the space of states D of th e system together with a rule or
dynamic that determines the state of the system at a given future time from
a given present state. This is formalized by the following definition. For this
definition T = R for continuous-time systems and T = Z for discrete-time
systems.
Definition 1.1. A dynamical system G on D is the octuple (D, U, U, Y,
Y, T, s, h), where s : T × T × D × U → D and h : T × D × U → Y are such
that the following axioms hold:
i) (Continuity): For every t
0
∈ T, x
0
∈ D, and u ∈ U, s(·, t
0
, x
0
, u) is
continuous for all t ∈ T.
ii) (Con s istency): For every x
0
∈ D, u ∈ U, and t
0
∈ T, s(t
0
, t
0
, x
0
, u) =
x
0
.
iii) (Determinism): For every t
0
∈ T and x
0
∈ D, s(t, t
0
, x
0
, u
1
) = s(t, t
0
,
x
0
, u
2
) for all t ∈ T and u
1
, u
2
∈ U satisfying u
1
(τ) = u
2
(τ), τ ∈ [t
0
, t].
iv) (Group property): s(t
2
, t
0
, x
0
, u) = s(t
2
, t
1
, s(t
1
, t
0
, x
0
, u), u) for all
t
0
, t
1
, t
2
∈ T, t
0
≤ t
1
≤ t
2
, x
0
∈ D, and u ∈ U.
v) (Read-out map): There exists y ∈ Y such that y(t) = h(t, s(t, t
0
, x
0
, u),
u(t)) for all x
0
∈ D, u ∈ U, t
0
∈ T, and t ∈ T.
We denote the dynamical system (D, U, U, Y, Y, T, s, h) by G and we
refer to the map s(·, t
0
, ·, u) as the flow or trajectory corresponding to x
0
∈ D,
t
0
∈ T, and u ∈ U; and for a given trajectory s(t, t
0
, x
0
, u), t ∈ T, we refer
to t
0
∈ T as an initial time of G, x
0
∈ D as an initial condition of G, and
u ∈ U as an input to G. The dynamical system G is isolated if the input
space consists of one element only, that is, u(t) = u
∗
, and the d ynamical
system is undisturbed if u
∗
= 0. If G is isolated, then G is isolated from any
inputs and the environment is the only input acting on the system. T his, for
example, would correspond to a conservative mechanical system wherein the
only external force acting on the system is gravity. In general, the output of
G depends on both the present input of G and the past history of G. Hence,
the output of th e dynamical system at some time t ∈ T depends on the
state s(t, t
0
, x
0
, u) of G, which effectively serves as an information storage
(memory) of past history. Furth ermore, the determinism axiom ensures that
NonlinearBook10pt November 20, 2007
INTRODUCTION 3
the state, and hence the output, before some time t ∈ T is not influenced by
the values of the output after time t. Thus, future inputs to G do not affect
past and present outpu ts of G. T his is simply a statement of causality that
holds for all p hysical sys tems. The notion of a dynamical system as defined
in Definition 1.1 is far to o general to develop useful p ractical deductions for
dynamical systems. This notion of a dynamical system is introduced here
to develop terminology and to introduce certain key concepts. As we will
see in the next chapter, under additional regularity conditions the flow of
a dynamical system describing the motion of th e system as a function of
time can generate a differential equation on the state space, allowing for the
development of a large arr ay of mathematical results leading to useful and
practical analysis and control synthesis tools.
Determining the rule or dynamic that defines the state of physical
and engineering systems at a given fu ture time from a given present state
is one of the central problems of science and engineering. Once the flow
of a dynamical system describing the motion of the system starting from a
given initial state is given, dynamical system theory can be used to describe
the behavior of the system states over time for different initial conditions.
Throughout the centuries—from the great cosmic theorists of ancient Greece
to the present-day quest for a unified field theory—the most important
dynamical system is our universe. By u sing abstract mathematical m odels
and attaching them to the physical world, astronomers, mathematicians,
and physicists h ave used abstract th ought to deduce something that is true
about the natural system of the cosmos.
The quest by scientists, su ch as Brahe, Kepler, Galileo, Newton,
Huygens, Euler, Lagrange, Laplace, and Maxwell, to understand the
regularities inherent in the distances of the planets from the sun and their
periods and velocities of revolution around the sun led to the science of
dynamical systems as a branch of mathematical physics. One of the most
basic issues in dynamical system theory that was spawned from the study
of mathematical models of our s olar system is the stability of dynamical
systems. S y s tem stability involves the investigation of small deviations
from a system’s steady state of motion. In particular, a dynamical system
is stable if the system is allowed to perform persistent small oscillations
about a system equilibrium, or about a state of motion. Among the first
investigations of the stability of a given state of motion is by Isaac Newton.
In particular, in his Principia Mathematica [335] Newton investigated
whether a small perturbation would make a particle moving in a plane
around a center of attraction continue to move near the circle, or diverge
from it. Newton used his analysis to analyze the motion of the moon orbiting
the Earth .
NonlinearBook10pt November 20, 2007
4 CHAPTER 1
Numerous astronomers and mathematicians who followed made sig-
nificant contributions to dynamical stability theory in an effort to sh ow
that th e observed deviations of planets and satellites f rom fixed elliptical
orbits were in agreement with Newton’s principle of universal gravitation.
Notable contribu tions include th e work of Torricelli [431], Euler [116],
Lagrange [252], Laplace [257], Dirichlet [106], Liouville [283], Maxwell [310],
and Routh [369]. The most complete contribution to the stability analysis
of dynamical systems was introdu ced in the late nineteenth century by the
Russian mathematician Aleksandr Mikhailovich Lyapu nov in his seminal
work entitled The General Problem of the Stability of Motion [293–295].
Lyapunov’s direct method s tates that if a positive-definite f unction (now
called a Lyapunov function) of the state coordinates of a dynamical system
can be constructed for which its time rate of change following small
perturbations from the system equilibr ium is always negative or zero, then
the system equ ilibr ium state is stable. In other words, L yapunov’s method is
based on the construction of a Lyapunov function that serves as a generalized
norm of the s olution of a dynamical system. Its appeal comes from the fact
that stability properties of the system solutions are derived directly from the
governing dynamical system equations; hence the n ame, Lyapunov’s direct
method.
Dynamical system theory grew out of the desire to analyze the
mechanics of heavenly bodies and has become one of the most fundamental
fields of modern science as it provides the foundation for unlocking many
of the mysteries in nature and the universe that involve the evolution
of time. Dynamical system theory is used to study ecological systems,
geological systems, biological systems, economic systems, neural systems,
and physical systems (e.g., mechanics, thermodynamics, fluids, magnetic
fields, galaxies, etc.), to cite but a few examples. Dynamical system theory
has also played a crucial role in the analysis and control design of numerous
complex engineering systems. In particular, advances in feedback control
theory have been intricately coupled to progress in dynamical system th eory,
and conversely, dynamical system theory has been greatly advanced by the
numerous challenges posed in the analysis and control design of increasingly
complex feedback control systems.
Since most physical and engineering systems are inherently nonlin-
ear, with system nonlinearities arising from numerous sources including,
for example, friction (e.g., Coulomb, hysteresis), gyroscopic effects (e.g.,
rotational motion), kin ematic effects (e.g., backlash), inp ut constraints (e.g.,
saturation, deadband), and geometric constraints, system nonlinearities
must be accounted for in system analysis and control design. Nonlinear
systems, however, can exh ibit a very rich dynamical behavior, such as
multiple equ ilibria, limit cycles, bifurcations, jump resonance phenomena,