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Desired Bode plot shape
Ess requirement
Noise requirement
0
-90
-180
0dB
wgcd
High low-freq-gain for steady state trackingLow high-freq-gain for noise attenuationSufficient PM near wgc for stability
w
w
PMd
Mid frequency
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Desired Bode plot shape
0
-90
-180
0dB
wgc
High low freq gain for steady state trackingLow high freq gain for noise attenuationSufficient PM near wgc for stability
w
w
Low freq
High freq
Want high gain
Want low gain
Mid freq
Want sufficientPhase margin
Use low pass filters
Use PI or lag control
Use lead or PD control
PM+Mp=70
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C(s) Gp(s)
21
21)(psps
zszsKsC
Controller design with Bode
From specs: => desired Bode shape of Gol(s)Make Bode plot of Gp(s) Add C(s) to change Bode shapeGet closed loop systemRun step response, or sinusoidal response
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Mr and BW are widely used
Closed-loop phase resp. rarely used
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Important relationships• Closed-loop BW are very close to wn
• Open-loop gain cross over wgc ≈ (0.65~0.8)* wn,
• When z <= 0.6, wr and wn are close
• When z >= 0.7, no resonance• z determines phase margin and Mp:
z 0.4 0.5 0.6 0.7
PM 44 53 61 67 deg ≈100z Mp 25 16 10 5 %
PM+Mp ≈70
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Mid frequency requirements• wgc is critically important
– It is approximately equal to closed-loop BW– It is approximately equal to wn
• Hence it determines tr, td directly
• PM at wgc controls z– Mp 70 – PM
• PM and wgc together controls s and wd – Determines ts, tp
• Need wgc at the right frequency, and need sufficient PM at wgc
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Low frequency requirements• Low freq gain slope and/or phase
determines system type• Height of at low frequency determine error
constants Kp, Kv, Ka• Which in turn determine ess
• Need low frequency gain plot to have sufficient slope and sufficient height
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High frequency requirements• Noise is always present in any system• Noise is rich in high frequency contents• To have better noise immunity, high
frequency gain of system must be low
• Need loop gain plot to have sufficient slope and sufficiently small value at high frequency
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Overall Loop shaping strategy• Determine mid freq requirements
– Speed/bandwidth wgc
– Overshoot/resonance PMd
• Use PD or lead to achieve PMd@ wgc
• Use overall gain K to enforce wgc
• PI or lag to improve steady state tracking– Use PI if type increase neede – Use lag if ess needs to be reduced
• Use low pass filter to reduce high freq gain
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Proportional controller design• Obtain open loop Bode plot• Convert design specs into Bode plot req.• Select KP based on requirements:
– For improving ess: KP = Kp,v,a,des / Kp,v,a,act
– For fixing Mp: select wgcd to be the freq at which PM is sufficient, and KP = 1/|G(jwgcd)|
– For fixing speed: from td, tr, tp, or ts requirement, find out wn, let wgcd = (0.65~0.8)*wn and KP = 1/|G(jwgcd)|
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clear all;n=[0 0 40]; d=[1 2 0];figure(1); clf; margin(n,d);%proportional control design:figure(1); hold on; grid; V=axis;Mp = 10; %overshoot in percentagePMd = 70-Mp + 3;semilogx(V(1:2), [PMd-180 PMd-180],':r');%get desired w_gcx=ginput(1); w_gcd = x(1);KP = 1/abs(evalfr(tf(n,d),j*w_gcd));figure(2); margin(KP*n,d);figure(3); mystep(KP*n, d+KP*n);
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Bode Diagram
Frequency (rad/sec)
Ph
ase
(deg
)M
agn
itu
de
(dB
)
-10
0
10
20
30
40
50Gm = Inf, Pm = 17.964 deg (at 6.1685 rad/sec)
10-1
100
101
-180
-135
-90
G(s)=40/s(s+2)
Mp=10%
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0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Time (sec)
Am
plit
ud
e
Unit Step Response
ts=3.65 tp=0.508
Mp=60.4%
ess tolerance band: +-2%
td=0.159
tr=0.19
yss=1
ess=0
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Bode Diagram
Frequency (rad/sec)
Ph
ase
(deg
)M
agn
itu
de
(dB
)
-40
-20
0
20
40Gm = Inf, Pm = 63.31 deg (at 1.0055 rad/sec)
10-1
100
101
-180
-135
-90
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1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
Am
plit
ud
e
Unit Step Response
ts=3.98 tp=2.82
Mp=6.03%ess tolerance band: +-2%
td=0.883
tr=1.33
yss=1
ess=0
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2
1
( ) (1 )
( ) (1 )
Gain: 20 log(| ( ) |) 20 log( )
20 log( 1
Phase: ( ) (1 ) tan ( )
DP D P
P
DP D P
P
P
D
P
D D
P P
KC s K K s K s
K
KC j K K j K j
K
C j K
K
K
K KC j j
K K
PD Controller
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20
30
40
50
60
70
Mag
nit
ud
e (d
B)
10-2
10-1
100
101
102
0
45
90
Ph
ase
(deg
)
Bode Diagram
Frequency (rad/sec)
20*log(KP)
KP/KD
Place wgcd here
Bad for noise
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gcd
gcd
gcd
gcd
gcd
gcd
From specs, find and
( )
a few degrees
tan( ) /
1/ (1 ) ( )
; ( )
( ) ( ) ( ) / 1 ( ) ( )
Perform c.l. step response, tune C
d
D
P D s j
D D P P D
cl
PM
PM angle G j
PM PMd PM
T PM
K T s G s
K T K C s K K s
G s C s G s C s G s
(s) as needed
PD control design
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n=[0 0 1]; d=[0.02 0.3 1 0];
figure(1); clf; margin(n,d);
Mp = 10/100;
zeta = sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));
PMd = zeta * 100 + 3;
tr = 0.3; w_n=1.8/tr; w_gcd = w_n;
PM = angle(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));
phi = PMd*pi/180-PM; Td = tan(phi)/w_gcd;
KP = 1/abs(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));
KP = KP/sqrt(1+Td^2*w_gcd^2); KD=KP*Td;
ngc = conv(n, [KD KP]);
figure(2); margin(ngc,d);
figure(3); mystep(ngc, d+ngc);
Could be a little less
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PD control design Variation
• Restricted to using KP = 1
• Meet Mp requirement
• Find wgc and PM
• Find PMd
• Let f = PMd – PM + (a few degrees)
• Compute TD = tan(f)/wgcd
• KP = 1; KD=KPTD
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n=[0 0 5]; d=[0.02 0.3 1 0];
figure(1); clf; margin(n,d);
Mp = 10/100;
zeta = sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));
PMd = zeta * 100 + 18;
[GM,PM,wgc,wpc]=margin(n,d);
phi = (PMd-PM)*pi/180; Td = tan(phi)/wgc;
Kp=1; Kd=Kp*Td;
ngc = conv(n, [Kd Kp]);
figure(2); margin(ngc,d);
figure(3); stepchar(ngc, d+ngc);
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2
200Example: ( )
4 4
When ( ) 1, 16 , 64%
Want: 16%
o
G ss s
C s PM Mp
Mp
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C(s) G(s)
ssssG
23 3.002.0
1)(
Example
Want: maximum overshoot <= 10% rise time <= 0.3 sec
Can use Lead or PD
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Bode Diagram
Frequency (rad/sec)
Ph
ase
(deg
)M
agn
itu
de
(dB
)
-100
-80
-60
-40
-20
0
20Gm = 23.522 dB (at 7.0711 rad/sec), Pm = 73.367 deg (at 0.9768 rad/sec)
10-1
100
101
102
-270
-225
-180
-135
-90
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0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
Am
plit
ud
e
Unit Step Response
ts=2.68 tp=3
Mp=-1.06%
ess tolerance band: +-2%
td=0.841
tr=1.52
yss=1
ess=0
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n=[0 0 1]; d=[0.02 0.3 1 0];
figure(1); clf; margin(n,d);
Mp = 10; %overshoot in percentage
PMd = 70 – Mp + 3;
tr = 0.3; w_n=1.8/tr; w_gcd = w_n;
PM = angle(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));
phi = PMd*pi/180-PM; Td = tan(phi)/w_gcd;
KP = 1/abs(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));
KP = KP/sqrt(1+Td^2*w_gcd^2); KD=KP*Td;
ngc = conv(n, [KD KP]);
figure(2); margin(ngc,d);
figure(3); mystep(ngc, d+ngc);
Could be a little less
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Bode Diagram
Frequency (rad/sec)
Ph
ase
(deg
)M
agn
itu
de
(dB
)
-60
-40
-20
0
20Gm = Inf, Pm = 62.116 deg (at 6 rad/sec)
100
101
102
-180
-135
-90
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0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
Am
plit
ud
e
Unit Step Response
ts=0.655 tp=0.461
Mp=6.67%ess tolerance band: +-2%
td=0.154
tr=0.225
yss=1
ess=0
Less than spec
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Variation
• Restricted to using KP = 1
• Meet Mp requirement
• Find wgc and PM
• Find PMd
• Let f = PMd – PM + (a few degrees)
• Compute TD = tan(f)/wgcd
• KP = 1; KD=KPTD
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KP=5; n=KP*[0 0 1]; d=[0.02 0.3 1 0];
figure(1); clf; margin(n,d);
figure(3); stepchar(n, d+n);
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Bode Diagram
Frequency (rad/sec)
Ph
ase
(deg
)M
agn
itu
de
(dB
)
-80
-60
-40
-20
0
20Gm = 9.5424 dB (at 7.0711 rad/sec), Pm = 32.613 deg (at 3.7468 rad/sec)
100
101
102
-270
-225
-180
-135
-90
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0.5 1 1.5 2 2.5 3 3.5 4 4.5
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (sec)
Am
plit
ud
e
Unit Step Response
ts=3.17 tp=0.814
Mp=38.9%
ess tolerance band: +-2%
td=0.317
tr=0.317
yss=1
ess=0
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KP=5; n=KP*[0 0 1]; d=[0.02 0.3 1 0];
figure(1); clf; margin(n,d);
Mp = 10;
PMd = 70 – Mp + 10;
[GM,PM,wgc,wpc]=margin(n,d);
phi = (PMd-PM)*pi/180; Td = tan(phi)/wgc;
KP=1; KD=KP*Td;
ngc = conv(n, [KD KP]);
figure(2); margin(ngc,d);
figure(3); stepchar(ngc, d+ngc);
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Bode Diagram
Frequency (rad/sec)
Ph
ase
(deg
)M
agn
itu
de
(dB
)
-60
-40
-20
0
20Gm = Inf, Pm = 52.605 deg (at 3.9488 rad/sec)
100
101
102
-180
-150
-120
-90
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0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
Am
plit
ud
e
Unit Step Response
ts=1.56 tp=0.728
Mp=15.6%ess tolerance band: +-2%
td=0.236
tr=0.321
yss=1
ess=0
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KP=5; n=KP*[0 0 1]; d=[0.02 0.3 1 0];
figure(1); clf; margin(n,d);
Mp = 10;
PMd = 70 – Mp + 18;
[GM,PM,wgc,wpc]=margin(n,d);
phi = (PMd-PM)*pi/180; Td = tan(phi)/wgc;
KP=1; KD=KP*Td;
ngc = conv(n, [KD KP]);
figure(2); margin(ngc,d);
figure(3); stepchar(ngc, d+ngc);
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Bode Diagram
Frequency (rad/sec)
Ph
ase
(deg
)M
agn
itu
de
(dB
)
-60
-40
-20
0
20Gm = Inf, Pm = 57.956 deg (at 4.1131 rad/sec)
100
101
102
-180
-135
-90
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0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (sec)
Am
plit
ud
e
Unit Step Response
ts=1.07 tp=0.695
Mp=10.7%ess tolerance band: +-2%
td=0.22
tr=0.305
yss=1
ess=0
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lead
lead
ps
zsKsC
)(
Lead Controller Design
00 Kzp leadlead
lead
lead
pj
zjKjC
)(
0)(tan)(tan
)()()(
11
leadlead
leadlead
pz
pjzjjC
![Page 45: Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise](https://reader035.vdocument.in/reader035/viewer/2022062417/551739a855034603568b60bf/html5/thumbnails/45.jpg)
101520253035404550
Mag
nit
ud
e (d
B)
10-2
10-1
100
101
102
103
0
30
60
90
Ph
ase
(deg
)Bode Diagram
Frequency (rad/sec)
zlead
plead
leadlead zp
20log(Kzlead/plead)lead
leadz
p
)(tan)(tan 11max lea d
lea d
lea d
lea d
pz
zp
Goal: select z and p so that max phase lead is at desired wgc and max phase lead = PM defficiency!
max
max
1 sinLet
1 sin
gcd gcd/ , *lead leadz p
![Page 46: Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise](https://reader035.vdocument.in/reader035/viewer/2022062417/551739a855034603568b60bf/html5/thumbnails/46.jpg)
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Lead Design • From specs => PMd and wgcd
• From plant, draw Bode plot• Find PMhave = 180 + angle(G(jwgcd)
• DPM = PMd - PMhave + a few degrees
• Choose a=plead/zlead so that fmax = DPM and it happens at wgcd
1
gcdgcdgcd
gcdgcd
max
max
)/()()(
*,/
sin1
sin1
leadlead
leadlead
pjjGzjK
pz
![Page 48: Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise](https://reader035.vdocument.in/reader035/viewer/2022062417/551739a855034603568b60bf/html5/thumbnails/48.jpg)
Lead design example• Plant transfer function is given by:
• n=[50000]; d=[1 60 500 0];
• Desired design specifications are:– Step response overshoot <= 16%– Closed-loop system BW>=20;
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n=[50000]; d=[1 60 500 0]; G=tf(n,d);figure(1); margin(G); Mp_d = 16/100; zeta_d =0.5; % or calculate from Mp_dPMd = 100*zeta_d + 3;BW_d=20;w_gcd = BW_d*0.7; Gwgc=evalfr(G, j*w_gcd);PM = pi+angle(Gwgc);phimax= PMd*pi/180-PM;alpha=(1+sin(phimax))/(1-sin(phimax));zlead= w_gcd/sqrt(alpha);plead=w_gcd*sqrt(alpha);K=sqrt(alpha)/abs(Gwgc);ngc = conv(n, K*[1 zlead]);dgc = conv(d, [1 plead]);figure(1); hold on; margin(ngc,dgc); hold off;[ncl,dcl]=feedback(ngc,dgc,1,1);figure(2); step(ncl,dcl);
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-100
-50
0
50M
agn
itu
de
(dB
)
10-1
100
101
102
103
-270
-225
-180
-135
-90
Ph
ase
(deg
)
Bode DiagramGm = 13.8 dB (at 38.3 rad/sec) , Pm = 53 deg (at 14 rad/sec)
Frequency (rad/sec)
Before designAfter design
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-150
-100
-50
0
50M
agn
itu
de
(dB
)
10-1
100
101
102
103
104
-270
-180
-90
0
Ph
ase
(deg
)
Bode DiagramGm = 8.8 dB (at 38.3 rad/sec) , Pm = 40.6 deg (at 25.2 rad/sec)
Frequency (rad/sec)
Closed-loop Bode plot by:
Magnitude plot shifted up 3dBSo, gc is BW
margin(ncl*1.414,dcl);
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.2
0.4
0.6
0.8
1
1.2
1.4Step Response
Time (sec)
Am
plit
ud
e
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Alternative use of lead• Use Lead and Proportional together
– To fix overshoot and ess– No speed requirement
1.Select K so that KG(s) meet ess req.
2.Find wgc and PM, also find PMd
3.Determine phi_max, and alpha
4.Place phi_max a little higher than wgc
maxgc gc
max
1 sin/ , *
1 sin
( )
lead lead
lead lead lead
lead lead lead
z p
p s z s zC s K K
z s p s p
![Page 54: Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise](https://reader035.vdocument.in/reader035/viewer/2022062417/551739a855034603568b60bf/html5/thumbnails/54.jpg)
Alternative lead design example• Plant transfer function is given by:
• n=[50]; d=[1/50 1 0];
• Desired design specifications are:– Step response overshoot <= 20%– Steady state tracking error for ramp input <=
1/200;
– No speed requirements– No settling concern– No bandwidth requirement
![Page 55: Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise](https://reader035.vdocument.in/reader035/viewer/2022062417/551739a855034603568b60bf/html5/thumbnails/55.jpg)
n=[50]; d=[1/5 1 0];figure(1); clf; margin(n,d); grid; hold on;Mp = 20/100; zeta = sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));PMd = zeta * 100 + 10;ess2ramp= 1/200; Kvd=1/ess2ramp;Kva = n(end)/d(end-1); Kzp = Kvd/Kva;figure(2); margin(Kzp*n,d); grid;[GM,PM,wpc,wgc]=margin(Kzp*n,d);w_gcd=wgc; phimax = (PMd-PM)*pi/180;alpha=(1+sin(phimax))/(1-sin(phimax));z=w_gcd/alpha^.25; %sqrt(alpha); %phimax located higherp=w_gcd*alpha^.75; %sqrt(alpha); %than wgcngc = conv(n, alpha*Kzp*[1 z]); dgc = conv(d, [1 p]);figure(3); margin(tf(ngc,dgc)); grid;[ncl,dcl]=feedback(ngc,dgc,1,1);figure(4); step(ncl,dcl); grid;figure(5); margin(ncl*1.414,dcl); grid;
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-40
-20
0
20
40
60M
agn
itu
de
(dB
)
10-1
100
101
102
-180
-135
-90
Ph
ase
(deg
)
Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 18 deg (at 15.4 rad/sec)
Frequency (rad/sec)
![Page 57: Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise](https://reader035.vdocument.in/reader035/viewer/2022062417/551739a855034603568b60bf/html5/thumbnails/57.jpg)
-20
0
20
40
60M
agn
itu
de
(dB
)
10-1
100
101
102
-180
-150
-120
-90
Ph
ase
(deg
)
Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 9.04 deg (at 31.4 rad/sec)
Frequency (rad/sec)
![Page 58: Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise](https://reader035.vdocument.in/reader035/viewer/2022062417/551739a855034603568b60bf/html5/thumbnails/58.jpg)
-50
0
50
Mag
nit
ud
e (d
B)
10-1
100
101
102
103
104
-180
-150
-120
-90
Ph
ase
(deg
)
Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 52.3 deg (at 50.1 rad/sec)
Frequency (rad/sec)
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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.2
0.4
0.6
0.8
1
1.2
1.4Step Response
Time (sec)
Am
plit
ud
e
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-80
-60
-40
-20
0
20M
agn
itu
de
(dB
)
100
101
102
103
104
-180
-135
-90
-45
0
Ph
ase
(deg
)
Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 77.5 deg (at 82.2 rad/sec)
Frequency (rad/sec)
![Page 61: Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise](https://reader035.vdocument.in/reader035/viewer/2022062417/551739a855034603568b60bf/html5/thumbnails/61.jpg)
-100
-50
0
50
Mag
nit
ud
e (d
B)
10-1
100
101
102
103
104
-180
-135
-90
Ph
ase
(deg
)
Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 52.3 deg (at 50.1 rad/sec)
Frequency (rad/sec)
![Page 62: Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise](https://reader035.vdocument.in/reader035/viewer/2022062417/551739a855034603568b60bf/html5/thumbnails/62.jpg)
Lead design tuning example
C(s) G(s)
2
1( )
( 5)G s
s s
Design specifications: rise time <=2 secovershoot <16%
![Page 63: Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise](https://reader035.vdocument.in/reader035/viewer/2022062417/551739a855034603568b60bf/html5/thumbnails/63.jpg)
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4. Go back and take wgcd = 0.6*wn so that tr is not too small
Desired tr < 2 secWe had tr = 1.14 in the previous 4 designs
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n=[1]; d=[1 5 0 0]; G=tf(n,d);Mp_d = 16; %in percentagePMd = 70 - Mp_d + 4; %use Mp + PM =70tr_d = 2; wnd = 1.8/tr_d; w_gcd = 0.6*wnd;Gwgc=evalfr(G, j*w_gcd);PM = pi+angle(Gwgc);phimax= PMd*pi/180-PM;alpha=(1+sin(phimax))/(1-sin(phimax));zlead= w_gcd/sqrt(alpha);plead=w_gcd*sqrt(alpha);K=sqrt(alpha)/abs(Gwgc);ngc = conv(n, K*[1 zlead]);dgc = conv(d, [1 plead]);[ncl,dcl]=feedback(ngc,dgc,1,1);stepchar(ncl,dcl); gridfigure(2); margin(G); hold on; margin(ngc,dgc); hold off; grid
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-200
-100
0
100
200M
ag
nit
ud
e (
dB
)
10-3
10-2
10-1
100
101
102
103
-270
-225
-180
-135
-90
Ph
ase (
deg
)Bode Diagram
Gm = 22.3 dB (at 3.45 rad/sec) , Pm = 60 deg (at 0.54 rad/sec)
Frequency (rad/sec)
![Page 78: Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise](https://reader035.vdocument.in/reader035/viewer/2022062417/551739a855034603568b60bf/html5/thumbnails/78.jpg)
0 5 10 15 20 25 30 350
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (sec)
Am
pli
tud
eUnit Step Response
ts=20.9 tp=5.85
Mp=16.5%ess tolerance band: +-2%
td=1.56
tr=1.95
yss=1
ess=0
![Page 79: Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise](https://reader035.vdocument.in/reader035/viewer/2022062417/551739a855034603568b60bf/html5/thumbnails/79.jpg)
Lead design
C(s) G(s)
40( )
( 2)G s
s s
Design specifications: Keep tr, td, butreduce overshoot to <16%
![Page 80: Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise](https://reader035.vdocument.in/reader035/viewer/2022062417/551739a855034603568b60bf/html5/thumbnails/80.jpg)
Lag design
C(s) G(s)
50( )
(0.02 1)G s
s s
• Desired design specifications are:– Step response overshoot <= 20%– Steady state tracking error for ramp
input <= 1/200;
![Page 81: Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise](https://reader035.vdocument.in/reader035/viewer/2022062417/551739a855034603568b60bf/html5/thumbnails/81.jpg)
Alternative lead design example• Plant transfer function is given by:
• n=[50]; d=[1/50 1 0];
• Desired design specifications are:– Step response overshoot <= 20%– Steady state tracking error for ramp input <=
1/200;
– No speed requirements– No settling concern– No bandwidth requirement