linearSystemTheory - 1 E LEMENTS O F LINEAR SYSTEM THEORY...

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1 ELEMENTS OF LINEAR SYSTEM THEORY 1.1 INTRODUCTION This book deals with the analysis and design of linear control systems. A prerequisite for studying linear control systems is a knowledge of linear system theory. We therefore devote this first chapter to a review of the most important ingredients of linear system theory. The introduction of control problems is postponed until Chapter 2. The main purpose of this chapter is to establish a conceptual framework, introduce notational convenlions, and give a survey of the basic facts of linear system theory. The starting point is the state space description of linear systems. We then proceed to discussions of the solution of linear state differential equations, the stability of linear systems, and the transform analysis of such systems. The topics next dealt with are of a more advanced nature; they concern controllability. reconstructibility, duality, and phase- variable canonical forms of linear systems. The chapter concludes with a discussion of vector stochastic processes and the response of linear systems to white noise. These topics play an imporlant role in the development of the theory. Since the reader of this chapter is assumed to have had an inlroduclion to linear system theory, the proofs of several well-known theorems are omitted. References to relevant texlbooks are provided, however. Some topics are treated in sections marked with an aslerisk, nolably controllability, recon- struclibility, duality and phase-variable canonical forms. The asterisk indicates that these notions are of a more advanced nature, and needed only in the sections similarly marked in the remainder of the book. 1.2 STATE DESCRIPTION OF LINEAR SYSTEMS 1.2.1 State Description of Nonlinear and Linear Differential Systems Many systems can be described by a set of simultaneous differential equalions of the form w = f [~(f), u(t), fl. 1-1 1
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2 El~mcnts of Lincnr System Theory Here t is the time variable, x(t) is a real n-dimensional time-varying column vector which denotes the state of the system, and tr(t) is a real lc-dimensional column vector which indicates the inptrt variable or corttrol variable. The function f is real and vector-valued. For many systems the choice of the state follows naturally from the physical structure, and 1-1, which will be called the state dijererttial equation, usually follows directly from the ele- mentary physical laws that govern the system. Let y(t) be a real I-dimensional system variable that can be observed or through which the system influences its environment. Such a variable we call an oufpt~t variable of the system. It can often be expressed as = g[x(t), do, 11. 1-2 This equation we call the ot~fpltt eqltatiolz of the system. We call a system that is described by 1-1 and a filtite-di~itensioi~al dijerential sjutern or, for short, a dijere~~tial system. Equations together are called the system eqrmfio~ts. If the vector-valued function g contains u explicitly, we say that the system has a direct liitk.
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linearSystemTheory - 1 E LEMENTS O F LINEAR SYSTEM THEORY...

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