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EE 4314
Examples on Compensator Design
Spring 2011
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Stability Margins
Gain margin: (GM): Factorby which the gain can beraised before instability
results Phase margin (PM):
Amount by which the
phase of G(jω) exceeds-180° when |KG(jω)|=1
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Stability Margins
GM and PM are measuresof how close the Nyquistplot comes to encircling
the -1 point
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Lead Compensator Design Summary
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Lag Compensator Design Summary
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Design Example – Root Locus (p. 261)
G s s e s
160 s 2.5 s 0.7
s2 5 s 40 s2 0.03 s 0.06
The transfer function between theelevator input and the pitch attitude
is
where
is the pitch angle* (degrees)
is the elevator angle (degrees)
e
Design the controller so that
1. Rise time of 1 sec or less2. Overshoot less than 10%
* see sec 10.3 for details
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Design Example – Root Locus
According to the specifications:
From (3.60) p.116:
t r
1.8
n
From (3.64) p.118:
M p e / 1 2
n 1.8 rad /sec
0.6
As a result, desired closed-loop pole should be at:
p j d
where
n
d n 1 2 (3.56) p.112
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Design Example – Root Locus
Let’s look at open loop response of the system, by using Matlab:
>> num = 160 .* conv([1 2.5], [1 0.7]);>> den = conv([1 5 40], [1 0.03 0.06]);>> sysOl = tf(num, den);>> step(sysOl)
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Design Example – Root Locus
Now, look at the root locus of the system:
>> Wn = 1.8;
>> DR = 0.6;>> sigma = DR * Wn;>> Wd = Wn * sqrt(1 - DR^2);>> p = -sigma + j*Wd;>> q = -sigma - j*Wd;>> rlocus(sysOl)>> hold on
>> plot(p, '*')>> plot(q, '*')>> grid on
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Design Example – Root Locus
Root Locus Plot
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Design Example – Root Locus
Using “rlocfind” command to find the gain K that yield the closed-looppoles that locate as close to the desired poles location as possible:
>> rlocfind(sysOl)Select a point in the graphics window
selected_point =
-0.9755 + 0.4105i
ans =
0.3260
Use K = 0.3260
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Design Example – Root Locus
Use proportional gain K = 0.326 to see if it gives satisfactory close-
loop response:
>> sysClP = feedback(0.326*sysOl, 1)
Transfer function:52.16 s^2 + 166.9 s + 91.28
--------------------------------------------s^4 + 5.03 s^3 + 92.37 s^2 + 168.4 s + 93.68
>> step(sysClP)
The overshoot is greater than10 %, rise time is okay.
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Design Example – Root Locus
Since proportional control is not enough, we can use leadcompensator – as a lead compensator would shift the root locusfurther to the left (lowering rise time and decreasing the transientovershoot (p. 249).
From p. 250, the zero of the lead compensator should be placed in
the neighborhood of the desired closed-loop Wn, and the poleof the lead compensator should be 5 to 20 times of the value of thezero. Let’s pick z = 2 (as Wn = 1.8) and p = 2 * 10 = 20.
D s K s 2
s 20
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Design Example – Root Locus
Now pick gain K of the lead compensator:
>> lead = tf([1 2], [1 20]);>> sysLead = lead * sysOl;>> rlocus(sysLead)>> hold on>> plot(p, '*')>> plot(q, '*')>> grid on>> hold off
>> rlocfind(sysLead)Select a point in the graphics window
selected_point =
-8.7708 +14.3523i
ans =
1.7599
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Design Example – Root Locus
Simulate the step response of the new closed-loop system:
>> sysClLead = feedback(1.7599 * sysLead, 1)
Transfer function:281.6 s^3 + 1464 s^2 + 2295 s + 985.5
------------------------------------------------------s^5 + 25.03 s^4 + 422.4 s^3 + 2270 s^2 + 2327 s + 1034
>> step(sysClLead)
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Design Example – Root Locus
D s 1.7599 s 2
s 20
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Design Example – Compensator Design from Freq. Response
Example 6.17 (p. 358) – a Type 1 Servomechanism System
KG s K 10 s s /2.5 1 s /6 1
Design a lead compensator so that the PM = 45° and Kv = 10.
From p. 356:
Step 1: Pick K to satisfy error constant
from p. 181
K v lim s0
sK G s K 100 1 0 1
K 1
D i E l C t D i f F R
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Design Example – Compensator Design from Freq. Response
Step 2: Evaluate the PM of the uncompensated system using Kobtained from step 1
>> Gs = tf(10, conv([1 0], conv([1/2.5 1], [1/6 1])))
Transfer function:10
----------------------------0.06667 s^3 + 0.5667 s^2 + s
>> [mag, phase, w] = bode(Gs);>> [GM, PM, Wcg, Wcp] = margin(mag, phase, w)
GM =
0.8540
PM =
-4.1158
Wcg =
3.8730
Wcp =
4.1910
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Design Example – Compensator Design from Freq. Response
Step 3: Add extra margin (about 5-10°) and determine the phase lead
max 45o
5o
(4o
)
54o
Step 4: Determine from (6.40) p. 349:
1 sin max
1 sin max
0.1
Step 5: Pick at the crossover frequency: ma x
1/T m ax
T 0.5
D s Ts 1
Ts 1
0.5 s 1
0.05 s 1
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Design Example – Compensator Design from Freq. Response
Step 6: Draw the compensated frequency response and check the PM:
>> lead = tf([0.5 1], [0.05 1]);>> sysLead = lead * Gs;>> [mag, phase, w] = bode(sysLead);>> [GM, PM, Wcg, Wcp] = margin(mag, phase, w)
GM =
2.3204
PM =
22.9367
Wcg =
11.5132
Wcp =
7.2994
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Design Example – Compensator Design from Freq. Response
We can use “sisotool” command to find GM, PM, Wcp, Wcg:
>> sisotool(sysLead)
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Design Example – Compensator Design from Freq. Response
Step 6: Iterate on the design until all specifications are met.Add an additional lead compensator if necessary.
We can see that with only on lead compensator, we can only get thePM of 23°.
So, we need to repeat Step 1 – Step 5 again to add additional leadcompensator.
Assuming the second iteration gives:
D s Ts 1
Ts 1
0.25 s 1
0.025 s 1
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Design Example – Compensator Design from Freq. Response
Let’s check the PM new compensated system:
>> secondLead = tf([0.25 1], [0.025 1]);
>> sysDoubleLead = secondLead * sysLead;>> [mag, phase, w] = bode(sysDoubleLead);>> [GM, PM, Wcg, Wcp] = margin(mag, phase, w)
GM =
3.8669
PM =
45.6362
Wcg =
30.7887
Wcp =
13.7805
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Design Example – Compensator Design from Freq. Response
The overall compensator becomes:
D s 0.5 s 1(0.05 s 1)
0.25 s 1(0.025 s 1)
Let’s take a look at the step responses of the compensated system:
>> sysLeadCl = feedback(sysLead, 1);>> sysDoubleLeadCl = feedback(sysDoubleLead, 1);>> step(sysLeadCl)>> hold on>> step(sysDoubleLeadCl)>> hold off>> grid on>> legend('Single Lead Compensator', 'Double Lead Compensator')
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Design Example – Compensator Design from Freq. Response
Step responses of the single lead compensator system vs doublelead compensator system
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Compensator Design Conclusion
Lead Compensator => PD Controller
• Speed up system response• Lowering the rise time•
Decreasing the overshoot
Lag Compensator => PI Controller
• Improve steady-state accuracy