Modern Control System Theory and Design, 2nd Edition

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 ...