2.1  INTRODUCTION

The design of linear, continuous, feedback control systems is dependent on mathematical techniques such as the Laplace transformation, the signal-flow graph, and the state-variable concept. In addition to these techniques, the design of linear, discrete, feedback control systems requires a knowledge of the z and w transforms, the Fourier transform, and some aspects of information theory. The design of nonlinear, continuous, feedback control systems is dependent on mathematical techniques such as the Fourier transform, and the state-variable concept. The scope of this book does not permit a detailed discussion of all these mathematical devices. The philosophy followed here is to review the theory of those techniques necessary for understanding the design of linear continuous and discrete control systems, and nonlinear continuous control systems, and to focus attention on the specific application of these mathematical tools to these classes of control systems.

This chapter logically develops the many mathematical tools used by the control-system engineer. Starting with a review of the complex variable, complex functions, and the s plane, the presentation follows with the trigonometric and complex forms of the Fourier series. The Fourier integral is next presented, from which the Fourier transform is developed. The limitation of the Fourier transform to control systems is illustrated, and the presentation then develops the Laplace transform. Besides being a logical road to the development of the Laplace transform, the purpose of presenting all of these concepts is that they will be used in the ensuing discussion. For example, the trigonometric form of the Fourier series is used for the describing function analysis of nonlinear control systems. The Laplace transform is the fundamental tool used in the classical transfer function, signal-flow graph, and block-diagram approach for analyzing linear systems which is used extensively throughout the book.

The presentation in this chapter then proceeds to modern control theory aspects which are based on the foundation of matrix theory. After a brief review of those aspects of matrix theory used by the control-system engineer, the state variable is then defined, and the generation of the state and output equations (phase-variable canonical form) of a control system is illustrated. The chapter then concludes with the definition of the state transition matrix and its application. The z and w transforms needed to analyze digital control systems are presented in Chapter 9, where digital control systems are analyzed and designed.

The duality of using the classical Laplace transform/transfer function/block diagram and the modem state-variable approaches is pursued throughout the book. The message of this book is that they enhance each other, and the control-system engineer can benefit by looking at many problems from both viewpoints.