Modern Control Engineering

Section 7–7 / Relative Stability Analysis 467

v =` v =`

v

Im Im

Re Re

0

v

0

v 1 v 1

v 2

v 3

Phase crossover frequencies (v 1 ,v 2 ,v 3 )

Gain crossover frequencies (v 1 ,v 2 ,v 3 )

Figure 7–68
Polar plots showing
more than two phase
or gain crossover
frequencies.

Figure 7–69 Control system.

closed-loop system near the resonant frequency. For satisfactory performance, the phase

margin should be between 30° and 60°, and the gain margin should be greater than 6 dB.

With these values, a minimum-phase system has guaranteed stability, even if the open-

loop gain and time constants of the components vary to a certain extent. Although the

phase and gain margins give only rough estimates of the effective damping ratio of the

closed-loop system, they do offer a convenient means for designing control systems or

adjusting the gain constants of systems.

For minimum-phase systems, the magnitude and phase characteristics of the open-

loop transfer function are definitely related. The requirement that the phase margin be

between 30° and 60° means that in a Bode diagram the slope of the log-magnitude curve

at the gain crossover frequency should be more gradual than –40dBdecade. In most

practical cases, a slope of –20dBdecade is desirable at the gain crossover frequency for

stability. If it is –40dBdecade, the system could be either stable or unstable. (Even if

the system is stable, however, the phase margin is small.) If the slope at the gain crossover

frequency is –60dBdecade or steeper, the system is most likely unstable.

For nonminimum-phase systems, the correct interpretation of stability margins re-

quires careful study. The best way to determine the stability of nonminimum-phase sys-

tems is to use the Nyquist diagram approach rather than Bode diagram approach.

EXAMPLE 7–20 Obtain the phase and gain margins of the system shown in Figure 7–69 for the two cases where

K=10andK=100.

R(s) K C(s) +– s(s+ 1) (s+ 5)