Gain and Phase Margins
Gain and Phase Margins
Gain and Phase Margins
Ghose/2012 205
Why do we need to do all this work and obtain just stability results from Nyquist plots?
Why not just use the Routh-Hurwitz criteria?
The real value of the Nyquist plot lies in the fact that it shows how close the system is
to instability. For example, let us consider the rst example we had.
k
G(j) =
j(j + 1)2
For which the Nyquist plot was such that As k increases, the curve approaches 1.
When k is such that (k > 2), encircles the point 1, and the closed-loop system
becomes unstable. Look at the gure below:
Gain Margin: The gain margin is the factor by which the gain |G(j)| needs to be
increased for the closed-loop system to be neutrally stable. The gain margin (GM) is
dened as:
GM = |G(j180 )|1
where, 180 is such that G(j180 ) = 180 The gain margin is normally expressed in
dB.
Phase Margin: Phase margin (PM) is the amount by which the phase of G(j) exceeds
180 when |G(j)| = 1. It is dened as,
PM = G(jc ) 180
where, the frequency c is such that |G(jc )| = 1. It is also called the phase cross-over
frequency.
Application to Design
Figure 15.25: The gain and phase margins from Bode plot
GM and PM tells you how much uncertainty one can tolerate in the open loop system
before the closed loop system goes to instability.
Phase margin is related to closed loop damping ratio and so to the overshoot. To show
this, consider an open loop system,
n2 n2
GoL (s) = =
s2 + 2n s s(s + 2n )
which produces the closed loop system (with unity feedback and unity gain),
n2
GcL (s) =
s2 + 2n s + n2
n2 n2
G(j) = = 2
j(j + 2n ) j2n
2
n
|G(j)| =
4 + 4 2 n2 w2
|G(jc )| = 1
n4 = c4 + 4 2 n2 c2
1/2
2
c = n 1+ 4 4 2
Lecture Notes on Control Systems/D. Ghose/2012 208
Now,
n2 n2 ( 2 + j2n )
G(j) = =
2 j2n 4 2 2 n2 2
So,
2n c 2n
PM = tan1 = tan1
c2 c
1 2
= tan
[ 1 + 4 4 2 2 ]1/2
So,
Small PM small large overshoot (but fast response).
Large PM large small overshoot (but slow response).
One can come up with design procedure in the frequency domain too. We omit the
details.
PROBLEM SET 9
1. Sketch the Nyquist plots for the following loop transfer functions. Find out N, P ,
and Z and determine if the closed loop system is stable. If yes, then for what
values of K(s) = k is the system stable?
(a)
20
G(s) =
s(1 + 0.1s)(1 + 0.5s)
(b)
3(s + 2)
G(s) =
s3+ 3s + 1
(c)
100
G(s) =
s(s + 1)(s2 + 2)
2. Let the open loop transfer function be given by
k
G(s) =
(s + 1)n
Consider n = 2, 3, 4 and nd out the range of k for which the closed loop system
is stable, using Nyquist plot.
4. For all the systems above determine the phase and gain margins if they are relevant.