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= -20 1o91o a
(5.141)
(5.142)
(5.143)
The gain margin expressed in decibels is positive if a <1 (Fig. 5.21a) and is
negative if a > 1 as shown in Fig, 5.21b. A positive gain marg:in (in decibels)
means that the system is stable, and a negative gain margin (in decibels) means
that the system is unstable. For a stable minimum phase system, the value of the
gain margin indicates how much the open-loop gain can be increased before the
closed-loop system becomes unstable. For example, a gain margin of30 db implies
that the open-loop gain can be increased by a factor of 31.6228 before the closed-
loop system becomes unstable. On the other hand, if the gain margin is -30 db,
then the closed-loop system is already unstable, and the gain has to be reduced by
a factor of 31.6228 to make the closed-loop system stable.
The phase margin is defined as the amount of additional phase lag at the gain
crossover frequency that can be introduced in the open-loop system to make the
closed-loop system unstable. The gain crossover frequency 002 iS that frequency
when the magnitude of the open-loop transfer function G( jeo) is uruty.
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW 483
The phase margin is usually denoted by #M and is expressed in degrees and is
given by
4M - 180 + 4
(5.144)
M(db)
+
z
Phasc(deg) -90
-180
Positive Gain Margin
~}
┏━━━━━━━━━┓
┃ ~ ~o ┃
┣━━┳━━━━━━┫
┃-~- ┃ Pi ┃
┃ }┃ oa ┃
┣━━┻━━━━━━┫
┃Positive Phase ┃
┃Margin ┃
┗━━━━━━━━━┛
M(db)
+
Z
(deg) -90
-180
~ / ~:y:e'ai'
┏━┳━━━━━━━━━━┓
┃ ┃ ┃
┣━╋━━━━━━━━━━┫
┃~ ┃.\~ o,. ┃
┃ ┃ .b . ┃
┃~-﹢Fig. 5.22 Gain and phase margins using Bode plots.
Margiri
484 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
5.8 Relations Between Time-Domain and Frequency-Domain
Parameters
Generally, the performance requirements for control systems are specified in
terms of time-domain parameters like rise time Tr, settling time Ts, time for peak
amplitude Tp, and percent overshoot Os. In the following, we present some rela-
tions between these time domain parameters and frequency-domain parameters.
These relations will be useful in the analyses and design of control systems using
frequency-domain methods.
Consider a second-order system whose open-loop and unity feedback closed-
loop transfer functions are given by
G(s)=- to~ (5.145)
= s~s + 2-l ton)
T(s)= +2 .. + (5.146)
Let M denote the magnitude of the closed-loop frequency response. rfhen, '
60~
(5.147)
M-.IT(/w)l=~ r+4C2tO~C02
A typical plot of M vs c.o is shown in Fig. 5.23.
201o91o
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