Cascade Compensation
R(s) +
Frequency Domain Design
C(s)
Gc(s)
G(s)

M. Sami Fadali
Professor of Electrical Engineering
University of Nevada
• Gc(s) = compensator transfer function
• Choose compensator to meet design specs.
• Phase Margin, Gain Margin, BW, e()%
1
2
Frequency Response Criteria
Determine Phase Margin
• Find the gain crossover frequency
.
• Express closed-loop criteria in terms of
• Consider the unity feedback system
• Find
Magnitude (dB)
40
where
Bode Diagram
Gm = 7.96 dB (at 10 rad/s) , Pm = 22.5 deg (at 6.08 rad/s)
30
20
10
0
Phase (deg)
-10
-90
• Substitute
for
-135
PM=22.5 deg
-180
-1
10
3
0
1
10
Frequency (rad/s)
10
4
Phase Margin
Gain Crossover
2
4 4 1  2 2
5
Phase Margin vs. Damping Ratio
PM 
100
6
Bandwidth
• Obtain from the magnitude plot of the closedloop transfer function
80
60
• Solve for the 3 dB frequency
:
40
20
0
0
0.5
1
1.5
2
2.5
3

MATLAB:
7
>> bandwidth(T)
8
Bandwidth and Gain Crossover
Solve for 3 dB Frequency
 BW  n
(3 dB point)
c n
/
 c  BW
9
Resonance Frequency and Peak
10
Resonance Frequency and Peak
• Differentiate and equate to zero to find the minimum
 p n
11
Mp
12
Proportional Control
Specified GM
• Only one characteristic can be chosen.
• Phase plot cannot be changed.
• Specify one of the following (stable system):
– Gain Margin.
– Phase Margin.
– Steady-state error.
1. Determine the phase
crossover frequency
1. Find the magnitude at the
crossover frequency.
• Determine the remaining characteristics of the
system after the design is completed.
13
14
Example
Specified GM (Cont.)
3. Add an amplifier gain
Bode Diagram
Magnitude (dB)
100
(or slide curve up or down with sisotool)
/
-100
System: g
Gain Margin (dB): 7.96
At frequency (rad/s): 10
Closed loop stable? Yes
System: g
Phase Margin (deg): 22.5
Delay Margin (sec): 0.0647
At frequency (rad/s): 6.08
Closed loop stable? Yes
-200
-90
Phase (deg)
0
15
-180
-270
-1
10
10
0
1
10
Frequency (rad/s)
10
2
10
3
16
GM Example
GM Absolute Units
.
Magnitude (abs)
10
10
10
Bode Diagram
5
/
/
0
-5
System: g
Gain Margin (abs): 2.5
At frequency (rad/s): 10
Closed loop stable? Yes
,
System: g
Phase Margin (deg): 22.5
Delay Margin (sec): 0.0647
At frequency (rad/s): 6.08
Closed loop stable? Yes
-10
10
-90
Phase (deg)
• Desired
/
-180
-270
-1
10
10
0
1
10
Frequency (rad/s)
10
2
10
3
17
GM Example: MATLAB
18
GM Example: MATLAB (cont.)
>> g=zpk([],[0,-5,-20],1000);
>> [m, f]=bode(g,10) % mag, phase at crossover
m=
0.4000
f=
-180
>> k=1/m/sqrt(10) % 1/mag./GM
k=
0.7906
>>[ GM,PM,wpc,wgc]=margin(g*k)
GM =
3.1623
PM =
28.8031
wpc =
10.0000
wgc =
5.2613
19
20
PM Example
Specified PM
Desired PM 50°
Magnitude (dB)
1. Determine the new gain crossover frequency
(has the desired angle).
at
2. Add an amplifier gain
/
.
/
21
Magnitude (abs)
10
10
0
-5
0.355
System: g
Frequency (rad/s): 3
Magnitude (abs): 2.81
-10
System: g
Frequency (rad/s): 2.99
Phase (deg): -130
10
-90
Phase (deg)
1
2.81
Bode Diagram
5
-180
-270
-1
10
10
0
1
10
Frequency (rad/s)
10
2
-50
System: g
Frequency (rad/s): 3
Magnitude (dB): 8.99
-100
System: g
Frequency (rad/s): 2.99
Phase (deg): -130
-180
-270
-1
10
/
10
=20 log magnitude dB
0
1
10
Frequency (rad/s)
22
MATLAB: Bode
New Gain
10
0
-150
-90
Phase (deg)
1. Find the magnitude at the new gain crossover
.
frequency
Bode Diagram
50
10
3
23
>> [mag,ph]=bode(g,3)
mag =
2.8267
ph =
-129.4945
>> K=1/mag
K=
0.3538
24
PM after Gain Change
Specified Steady-State Error
>> margin(0.35*g)
1. Determine the necessary improvement
in the steady-state error.
Magnitude (abs)
Bode Diagram
Gm = 7.14 (at 10 rad/s) , Pm = 50.8 deg (at 2.98 rad/s)
10
0
System: untitled1
Frequency (rad/s): 9.99
Magnitude (abs): 0.14
10
2. Determine the necessary increase in the
error constant.
-5
e.g. for a type I system
Phase (deg)
-90
-135
,
-180
-225
,
-270
-1
10
0
10
1
10
Frequency (rad/s)
2
10
3
10
25
Steady-state Error Example
• Desired
(Type I)
,
,
27
26