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Wassim M. Haddad & VijaySekhar Chellaboina

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"An excellent textbook. This is an up-to-date, comprehensive, and extremely well presented exposition of modern methods in nonlinear control systems. It contains every topic I would like to see in such a book. The writing is superb and the style is clear and lucid. This is a highly welcome addition to the literature."--Frank L. Lewis, University of Texas, Arlington

"A significant contribution. This book can be used as a textbook for engineering students and others, and it is useful to researchers because it contains up-to-date methods and information. As far as I know, there is no one book that covers all the material contained in this book."--V. Lakshmikantham, Florida Institute of Technology

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"An excellent textbook. This is an up-to-date, comprehensive, and extremely well presented exposition of modern methods in nonlinear control systems. It contains every topic I would like to see in such a book. The writing is superb and the style is clear and lucid. This is a highly welcome addition to the literature."--Frank L. Lewis, University of Texas, Arlington

"A significant contribution. This book can be used as a textbook for engineering students and others, and it is useful to researchers because it contains up-to-date methods and information. As far as I know, there is no one book that covers all the material contained in this book."--V. Lakshmikantham, Florida Institute of Technology

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Nonlinear Dynamical Systems and Control

Princeton University Press

Chapter One

A system is a combination of components or parts that is perceived as a single entity. The parts making up the 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 state 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 system. Even though it is often assumed that the state of a dynamical system is the least set of state variables needed to completely predict the effect of the past upon the future of the system, this is often a convenient simplifying assumption.

We regard a dynamical system G as a mathematical model structure involving an input, state, and output that can capture the dynamical description of a given class of physical systems. Specifically, at each moment of time t [member of] T, where T denotes a time-ordered subset 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 fixed set U. Furthermore, over a time segment the input function u :[t.sub.1], [t.sub.2]) [right arrow] U is not arbitrary but belongs to the admissible input class U, that is, for every u() [member of] U and t [member of] T, u(t) [member of] 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() [member of] y over a given time segment, where Y denotes an output space. In general, the output of G depends on both the present input of G and the past history of G. Thus, the state, and hence the output at some time t [member of] T, depends on both the initial state x([t.sub.0])= [x.sub.0] and the input segment u :[t.sub.0], t) [right arrow] U. In other words, knowledge of both [x.sub.0] and u [member of] U is necessary and sufficient to determine the present and future state x(t)= s(t, [t.sub.0], [x.sub.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 the 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 x T x D x U [right arrow] D and h: T x D x U [right arrow] Y that the following axioms hold:

i) (Continuity): For every [t.sub.0] [member of] T, [x.sub.0] [member of] D, and u [member of] U, s(, [t.sub.0], [x.sub.0], u) is continuous for all t [member of] T.

ii) (Consistency): For every [x.sub.0] [member of] D, u [member of] U, and [t.sub.0] [member of] T, s([t.sub.0], [t.sub.0], [x.sub.0], u)= [x.sub.0].

iii) (Determinism): For every [t.sub.0] [member of] T and [x.sub.0] [member of] D, s(t, [t.sub.0], [x.sub.0], [u.sub.1])= s([t, [t.sub.0], [x.sub.0], [u.sub.2]) for all t [member of] T and [u.sub.1], [u.sub.2] [member of] U satisfying [u.sub.1]([tau])= [u.sub.2]([tau]), [tau] [member of] [[t.sub.0], t].

iv) (Group property): s([t.sub.2], [t.sub.0], [x.sub.0], u) = s([t.sub.2], [t.sub.1], s([t.sub.1], [t.sub.0], [x.sub.0], u), u) for all [t.sub.0], [t.sub.1], [t.sub.2] [member of] T, [t.sub.0] [less than or equal to] [t.sub.1] [less than or equal to] [t.sub.2], [x.sub.0] [member of] D, and u [member of] U.

v) (Read-outmap): There exists y [member of] Y such that y(t) = h(t, s(t, [t.sub.0], [x.sub.0], u), u(t)) for all [x.sub.0] [member of] D, u [member of] U, [t.sub.0] [member of] T, and t [member of] T.

We denote the dynamical system (D, U, U, Y, Y, T, s, h) by G and we refer to the map s(, [t.sub.0], , u) as the flow or trajectory corresponding to [x.sub.0] [member of] D, [t.sub.0] [member of] T, and u [member of] U; and for a given trajectory s(t, [t.sub.0], [x.sub.0], u), t [member of] T, we refer to [t.sub.0] [member of] T as an initial time of G, [x.sub.0] [member of] D as an initial condition of G, and u [member of] 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.sup.*], and the dynamical system is undisturbed if [u.sup.*] = 0. If G is isolated, then G is isolated from any inputs and the environment is the only input acting on the system. This, 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 the dynamical system at some time t [member of] T depends on the state s(t, [t.sub.0], [x.sub.0], u) of G, which effectively serves as an information storage (memory) of past history. Furthermore, the determinism axiom ensures that the state, and hence the output, before sometime t [member of] T is not influenced by the values of the output after time t. Thus, future inputs to G do not affect past and present outputs of G. This is simply a statement of causality that holds for all physical systems. The notion of...