Modern Control Engineering

462 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method

Depending on whether the data are graphical or analytical and whether nonmini-

mum-phase components are included, an appropriate stability test must be used for

multiple-loop systems. If the data are given in analytical form or if mathematical ex-

pressions for all the components are known, the application of the Nyquist stability cri-

terion to inverse polar plots causes no difficulty, and multiple-loop systems may be

analyzed and designed in the inverse GHplane. (See Problem A–7–15.)

7–7 Relative Stability Analysis

Relative Stability. In designing a control system, we require that the system be

stable. Furthermore, it is necessary that the system have adequate relative stability.

In this section, we shall show that the Nyquist plot indicates not only whether a sys-

tem is stable, but also the degree of stability of a stable system. The Nyquist plot also gives

information as to how stability may be improved, if this is necessary.

In the following discussion, we shall assume that the systems considered have

unity feedback. Note that it is always possible to reduce a system with feedback ele-

ments to a unity-feedback system, as shown in Figure 7–62. Hence, the extension of

relative stability analysis for the unity-feedback system to nonunity-feedback sys-

tems is possible.

We shall also assume that, unless otherwise stated, the systems are minimum-phase

systems; that is, the open-loop transfer function has neither poles nor zeros in the right-

halfsplane.

Relative Stability Analysis by Conformal Mapping. One of the important prob-

lems in analyzing a control system is to find all closed-loop poles or at least those clos-

est to the jvaxis (or the dominant pair of closed-loop poles). If the open-loop

frequency-response characteristics of a system are known, it may be possible to esti-

mate the closed-loop poles closest to the jvaxis. It is noted that the Nyquist locus G(jv)

need not be an analytically known function of v. The entire Nyquist locus may be ex-

perimentally obtained. The technique to be presented here is essentially graphical and

is based on a conformal mapping of the splane into the G(s)plane.

R(s) C(s)

G

H

R(s)

GH

1 C(s)

H

+

+

Figure 7–62

Modification of a

system with feedback

elements to a unity-

feedback system.

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