Transcript Control gain & BW

제어설계기법 실습 #6
Frequency-domain properties
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제어설계기법
실습
실험실습 #6 안내서
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실험실습 번호 : #6
실험실습 일시 : 2011.
담당교수 :
담당조교 :
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1. 제목 : frequency-domain properties
2. 목적 : bode plot을 이용하여 control gain으로 인한
system bandwidth의 변화를 살펴본다.
3. 실습에 필요한 기초지식 : 슬라이드 유인물 참조
4. 실습절차 : 슬라이드 유인물 참조
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제어설계기법
실습
Frequency-domain properties
Specifications of frequency-domain
Resonant peak M r
Maximum value of M ( jw)
Indication on the relative stability of a stable closed-loop system
Large M r corresponds to a large maximum overshoot of the step
response.
Resonant frequency r
Frequency at which the peak resonance M r occurs.
Bandwidth BW
Frequency at which M ( jw) drops to 70.7 percent of, or 3dB down
from, its zero-frequency value.
Indication on the transient-response properties in the time
domain.
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제어설계기법
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Bandwidth properties
Bandwidth properties
Larger BW makes higher-frequency signals are more easily
passed through the system.
Smaller BW makes only signals of relatively low frequencies
are passed, and the time response will be slow and sluggish.
BW and rise time are inversely proportional.
larger the bandwidth, faster the system will respond.
Rising time t r , increasing n increases BW and decreases t r
Increasing  decreases BW and increases t r
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제어설계기법
실습
Control gain & BW
 example) relationship of control gain and BW
>> num = 2500;
>> den = conv([1 5 0],[1 50]);
>> data = tf(num,den)
Transfer function:
2500
-------------------s^3 + 55 s^2 + 250 s
>> bode(data)
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제어설계기법
실습
Control gain & BW
 Find the BW in magnitude graph(gain=1)
Bode Diagram
50
System: sys
Frequency (rad/sec): 6.21
Magnitude (dB): -0.016
System: sys
Frequency (rad/sec): 7.63
Magnitude (dB): -3
Magnitude (dB)
0
BW=7.63
-50
-100
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10
0
10
1
10
2
10
3
10
Frequency (rad/sec)
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제어설계기법
실습
Control gain & BW
 decreased gain is applied (gain=0.1)
Bode Diagram
50
System: sys
Frequency (rad/sec): 0.981
Magnitude (dB): 0.000364
System: sys
Frequency (rad/sec): 1.36
Magnitude (dB): -3
BW=1.36
Magnitude (dB)
0
-50
-100
-150
-1
10
0
1
10
2
10
Frequency
10
3
10
(rad/sec)
 increased gain is applied (gain=10)
Bode Diagram
50
System: sys
Frequency (rad/sec): 21.1
Magnitude (dB): -0.000479
System: sys
Frequency (rad/sec): 24.9
Magnitude (dB): -3
BW=24.9
Magnitude (dB)
0
-50
-100
-150
-1
10
0
10
1
10
Frequency
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2
10
3
10
(rad/sec)
제어설계기법
실습
System stability margin
Phase margin & gain margin
Phase margin
 g  L( j )  1  0dB
PM  L( j g )  180o
Gain margin
 P  L( j )  180o
1
GM  20 log10 L( j p )  20 log10
L ( j p )
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제어설계기법
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Check the stability margin in bode plot
Bode plot example in MATLAB
Transfer ftn. 
2500
s 3  55 s 2  250 s
Bode plot result
Bode Diagram
Magnitude (dB)
50
Gain margin
14.82 dB
0
-50
-100
-150
-90
Phase margin
31.72°
Phase (deg)
-135
-180
-225
-270
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10
0
10
1
10
2
10
3
10
Frequency (rad/sec)
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제어설계기법
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Computing the stability margin
Stability margin (gain and phase margins)
>> [gm, pm, wpc, wgc]=margin(mag, phase, w)
>> margin(mag, phase, w)
*gm have to be changed by dB unit.
>> 20*log10(gm)
Example
>> n=1000;
>> d=[1 105 600 0];
>> bode(n,d)
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제어설계기법
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Computing the stability margin
Bode Diagram
50
Gain margin
35.98 dB
Magnitude (dB)
0
-50
-100
-150
-200
-90
Phase margin
74.19°
Phase (deg)
-135
-180
-225
-270
-1
10
0
10
1
10
2
10
3
10
4
10
Frequency (rad/sec)
>> [mag, phase, w]=bode(num,den)
>> [gm, pm, wpc, wgc]=margin(mag, phase, w)
>> margin(mag, phase, w)
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제어설계기법
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Matlab 실험·실습과제6
1) 다음 G(s)는 unity feedback system의 forward-path
transfer function 이다. k=1일 때의 bode plot을 수행하여
margin 값을 확인하고 시스템이 안정한지 확인하시오.
그리고 BW 값을 구하시오
G( s) 
10k
s(1  s)(1  0.5s)
2) 1번의 시스템이 안정하지 않다면, root-locus를 이용하여
적절한 damping ratio(0.707)값을 갖는 k값을 구하시오.
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제어설계기법
실습
Matlab 실험·실습과제6
3) 1, 2번의 결과를 이용하여 다시 bode plot을 그려 gain
margin, phase margin, 그리고 각각의 crossover
point를 구하시오. 그리고 BW를 구하여 1번과 비교한 후
추정할 수 있는 시스템 특성 변화를 설명하시오.
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제어설계기법
실습