Figure 10.35
Nyquist diagram showing gain and phase margins
1. Gain Margin, GM, and Phase Margin, PΜ, indicate the Relative Stability of the closed-loop system.
2. We assume that the system is a Non-minimum Phase system (no
GH zeros in the RHP).
3. If all the poles of GH are in the LHP, then we can just
II
plot the positive jω axis (Part I)&to
determine stability
using the GM and PM; otherwise, stability needs to be
determined first using the Nyquist criterion, Z = P - N.
4. GM 1 where the Angle of GH = ± 180o
GH
|GH| = 1/a
a = 1/ |GH| =GM
Arg(GH)= ± 180o
 1 
=-20log( GH )
GH


for Bode plots, GM=20log 
o
PM +180 + Angle of GH where GH = 1
5. If we multiply GH by GM, the Nyquist plots shifts to
where it crosses –1 on the real axis and the system
becomes marginally stable. That is, as the GM approaches
1, the system becomes more oscillatory. The GM is less
than 1 and positive for stability, i.e., |GH| real-axis
crossing is less than 1 for stability.
6. For stability, the PM must be positive. As the PM
approaches 0 degrees, the system becomes more
oscillatory.
in dB
GM
Ogata, Modern
Control
Engineering, 3rd
Edition
-1
CONDITIONALLY STABLE
MAY BECOME UNSTABLE
WITH A SLIGHT GAIN
CHANGE
α = PM
Highest
Frequency
Figure 10.37
Gain and phase
margins on the Bode
diagrams
IT IS MUCH EASIER TO FIND
THE GM AND PM FROM BODE
PLOTS.
THE GM IS FOUND BY FINDING
THE MAGNITUDE OF THE
COMPOSITE MAGNITUDE
WHERE THE COMPOSIT PHASE
= -180 DEG.
THE PM IS FOUND BY FINDING
THE PHASE OF THE COMPOSITE
PHASE WHERE THE
COMPOSITE MAGNITUDE = 0dB
AND ADDING +180 DEG AS
SHOWN ON THE GRAPH AT
RIGHT.
PM =
+180 in text
GAIN MARGIN & PHASE MARGIN BODE PLOT EXAMPLE
% KGH(s)=10/[s(s+1)(0.5s+1)]
KGHnum=[10]
KGHden=conv([1 1 0],[0.5 1]) % (s^2+s)*(0.5s+1)
Disp(‘KGH = ‘)
KGH=tf(KGHnum,KGHden)
bode(KGH);
grid
KGH =
10
--------------------0.5 s^3 + 1.5 s^2 + s
THE CLOSED-LOOP SYSTEM IS UNSTABLE
GAIN & PHASE MARGINS ARE NEGATIVE
(Note: Only one of them needs to be negative for the
closed-loop system to be unstable.)
CLOSED-LOOP POLES ARE:
-3.8371
0.4186 + 2.2443i
0.4186 - 2.2443i
Figure 10.36
Bode
log-magnitude
and phase
diagrams
for the system
of Example 10.9
Bode phase
plot for G(s) =
40/[(s +2)
(s +4)(s +5)]:
a. components;
b. composite
PM = 180 deg
GM = 20 dB
Figure 10.55
Effect of 1 sec
delay
1. DELAY ONLY EFFECTS
THE PHASE PLOT
2. A T SECOND DELAY IS
REPRESENTED BY e-TS
e-Ts
s=jω
=e
-Tjω
= 1∠ - Tω,θ
 180 
= -Tω 
 deg
Delay
 π 

 -1s
1
GH(s)e-Ts = 
e , T=1 second delay
s(s+1)(s+2)


-Tjω 
 -1jω
1
GH(jω)e
=
e
 jω(jω+1)(jω+2) 
GH(jω)e
-Tjω
=Mag(GH(jω)) with an
Angle(GH(jω)e
-Tjω
)=
[ Angle of GH(jω) deg ] - ω 
180 

 π 
 180 
θDelay = −ω 

 π 
GH Phase
without delay
Composite Phase
GH Phase + θDelay
Figure 10.56 Step response forclosed-loop system with
G(s) = 5/[s(s +1)(s + 10)]:
a. with a 1 second delay;
b. without delay
Figure 10.39
Representative log-magnitude
plot of Eq. (10.51)
Given a closed - loop system :
ω2
C(s)
n
≈
G
(s) =
CL
2
R(s) s +2ζω +ω 2
n n
Bandwidth is defined as the frequency
at which the magnitude of a closed-loop
system is - 3 dB.
ω BW = ωn (1 - 2ζ 2 ) + 4ζ 4 - 4ζ 2 + 2
dM
Peak M, M , when
= 0 yields :
p
dω
n
1
M =
p
2ζ 1-ζ 2
ω =ω 1-2ζ 2
p n


20log  G
(jω) 
 CL

Figure 10.41
Normalized
bandwidth
vs. damping
ratio for:
a. settling
time;
b. peak time;
c. rise time
Closed-loop System is
assumed to approximate a
2nd order system.
Figure 10.48
Phase margin
vs.
damping ratio
Closed-loop System is
assumed to approximate a
2nd order system.
Phase Margin of GH(s)

PM = ΦM


= tan

2
4 

-2ζ
+
1
+
4ζ


-1 
2ζ