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Control Systems Analysis

and Design by the

Frequency-Response Method

7–1 Introduction

By the term frequency response,we mean the steady-state response of a system to a

sinusoidal input. In frequency-response methods, we vary the frequency of the input

signal over a certain range and study the resulting response.

In this chapter we present frequency-response approaches to the analysis and design

of control systems. The information we get from such analysis is different from what we

get from root-locus analysis. In fact, the frequency response and root-locus approaches

complement each other. One advantage of the frequency-response approach is that we

can use the data obtained from measurements on the physical system without deriving

its mathematical model. In many practical designs of control systems both approaches

are employed. Control engineers must be familiar with both.

Frequency-response methods were developed in 1930s and 1940s by Nyquist, Bode,

Nichols, and many others. The frequency-response methods are most powerful in con-

ventional control theory. They are also indispensable to robust control theory.

The Nyquist stability criterion enables us to investigate both the absolute and relative

stabilities of linear closed-loop systems from a knowledge of their open-loop frequency-

response characteristics. An advantage of the frequency-response approach is that

frequency-response tests are, in general, simple and can be made accurately by use of

readily available sinusoidal signal generators and precise measurement equipment. Often

the transfer functions of complicated components can be determined experimentally by

frequency-response tests. In addition, the frequency-response approach has the advan-

tages that a system may be designed so that the effects of undesirable noise are negligible

and that such analysis and design can be extended to certain nonlinear control systems.

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