30
lag lag p s z s K s C ) ( Lag Controller Design 0 0 K p z lag lag K p z K p j z j K j C lag lag lag lag 0 ) ( ing destabiliz p z j C lag lag , 0 ) ( tan ) ( tan ) ( 1 1
lag
lag
ps
zsKsC
)(
Lag Controller Design
00 Kpz laglag
Kp
zK
pj
zjKjC
lag
lag
lag
lag
0)(
ingdestabilizpz
jClaglag
,0)(tan)(tan)( 11
0
10
20
30
40M
agn
itu
de
(dB
)
10-4
10-3
10-2
10-1
100
101
-90
-60
-30
0
Ph
ase
(deg
)Bode Diagram
Frequency (rad/sec)
gcd
zgcd/5
Kill PM by 10 to 12 deg
z/p=5
z/p=10
z/p=20
z/p=40
0
10
20
30
40M
agn
itu
de
(dB
)
10-4
10-3
10-2
10-1
100
101
-90
-60
-30
0
Ph
ase
(deg
)Bode Diagram
Frequency (rad/sec)
gcd
zgcd/10
Kill PM by 5 to 7 deg
z/p=5
z/p=10
z/p=20
z/p=40
0
5
10
15
20
25
30
35M
agn
itu
de
(dB
)
10-5
10-4
10-3
10-2
10-1
100
-90
-60
-30
0
Ph
ase
(deg
)Bode Diagram
Frequency (rad/sec)
gcd
zgcd/20
Kill PM by 2 to 3 deg
z/p=5
z/p=10
z/p=20
z/p=40
Want these:DC gain boosting
Don’t want these:PM reduction!
Lag and lead-lag Design Steps• From plant, draw Bode plot
• From specs => PMd and gcd
– If there is speed or BW req, gcd, • In this case, if PM not enough, design PD or lead
– Otherwise, choose gcd to have PM>PMd
• Find K to enforce gcd:
• Find Kp,v,a-have with K and C above
• Find Kp,v,a-des from ess specs
• zlag/plag = Kp,v,a-des/Kp,v,a-have
• Let zlag= gcd/5~20, depending on PM room
• Compute plag
1gcd )(
jCGK
Lag design example• Plant transfer function is given by:
• Desired design specifications are:– Step response overshoot <= 10%– Steady state tracking error to ramp input is
<=0.01
)1)(1(
1)(
501
51
sss
sG
Note: no speed or BW requirement, so just choose wgcd to have enough PM
n=[1]; d=[1/5/50 1/5+1/50 1 0];figure(1); clf; margin(n,d);%proportional control design:figure(1); hold on; grid; V=axis;Mp = 10/100;zeta = sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));PMd = zeta * 100 + 3;semilogx(V(1:2), [PMd-180 PMd-180],':r');%get desired w_gcx=ginput(1); w_gcd = x(1);K = 1/abs(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));Kva = K*n(end)/d(end-1); ess=0.01; Kvd=1/ess;z = w_gcd/5; p = z/(Kvd/Kva);ngc = conv(n, Kp*[1 z]); dgc = conv(d, [1 p]);figure(1); hold on; margin(ngc,dgc);[ncl,dcl]=feedback(ngc,dgc,1,1);figure(2); step(ncl,dcl); grid;figure(3); margin(ncl*1.414,dcl); grid;
-100
-50
0
Mag
nit
ud
e (d
B)
10-1
100
101
102
103
-270
-225
-180
-135
-90
Ph
ase
(deg
)
Bode DiagramGm = 34.8 dB (at 15.8 rad/sec) , Pm = 77.8 deg (at 0.981 rad/sec)
Frequency (rad/sec)
-200
-100
0
100
200M
agn
itu
de
(dB
)
10-4
10-2
100
102
-270
-180
-90
Ph
ase
(deg
)
Bode DiagramGm = 26 dB (at 15 rad/sec) , Pm = 51.7 deg (at 2.31 rad/sec)
Frequency (rad/sec)
0 2 4 6 8 10 120
0.2
0.4
0.6
0.8
1
1.2
1.4Step Response
Time (sec)
Am
plit
ud
e
n=[1]; d=[1/5/50 1/5+1/50 1 0];figure(1); clf; margin(n,d); hold on; grid; V=axis;Mp = 10/100; zeta = sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));PMd = zeta * 100 + 3;semilogx(V(1:2), [PMd-180 PMd-180],':r');x=ginput(1); w_gcd = x(1); %get desired w_gcK = 1/abs(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));Kva = K*n(end)/d(end-1); ess=0.01; Kvd=1/ess;z = w_gcd/10; p = z/(Kvd/Kva);ngc = conv(n, K*[1 z]); dgc = conv(d, [1 p]);figure(1); hold on; margin(ngc,dgc);[ncl,dcl]=feedback(ngc,dgc,1,1);figure(2);step(ncl,dcl); grid;figure(3); margin(ncl*1.414,dcl); grid;
-150
-100
-50
0
50
100
150M
agn
itu
de
(dB
)
10-4
10-2
100
102
-270
-225
-180
-135
-90
Ph
ase
(deg
)
Bode DiagramGm = 27.7 dB (at 15.5 rad/sec) , Pm = 60.2 deg (at 2.01 rad/sec)
Frequency (rad/sec)
0 5 10 15 20 250
0.2
0.4
0.6
0.8
1
1.2
1.4Step Response
Time (sec)
Am
plit
ud
e
clear all; n=[1]; d=[1/5/50 1/5+1/50 1 0];figure(1); clf; margin(n,d); grid; V=axis; hold on; Mp = 10/100; zeta =sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));PMd = zeta * 100 + 7;semilogx(V(1:2), [PMd-180 PMd-180],':r');x=ginput(1); w_gcd = x(1); %get desired w_gcK = 1/abs(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));Kva = K*n(end)/d(end-1); ess=0.01; Kvd=1/ess;z = w_gcd/10; p = z/(Kvd/Kva);ngc = conv(n, K*[1 z]); dgc = conv(d, [1 p]);figure(1); hold on; margin(ngc,dgc);[ncl,dcl]=feedback(ngc,dgc,1,1);figure(2);step(ncl,dcl); grid;figure(3); margin(ncl*1.414,dcl); grid;
-100
0
100
Mag
nit
ud
e (d
B)
10-4
10-2
100
102
-270
-225
-180
-135
-90
Ph
ase
(deg
)
Bode DiagramGm = 31.2 dB (at 15.6 rad/sec) , Pm = 66.9 deg (at 1.42 rad/sec)
Frequency (rad/sec)
0 5 10 15 20 250
0.2
0.4
0.6
0.8
1
1.2
1.4Step Response
Time (sec)
Am
plit
ud
e
-150
-100
-50
0
50M
agn
itu
de
(dB
)
10-2
10-1
100
101
102
103
-270
-180
-90
0
Ph
ase
(deg
)
Bode DiagramGm = 27.9 dB (at 15.6 rad/sec) , Pm = 98.2 deg (at 2.18 rad/sec)
Frequency (rad/sec)
Lead-Lag design example• Plant transfer function is given by:
• Desired design specifications are:– Step response overshoot <= 16%– Step response rise time <= 2 sec– Steady state tracking error to unit acceleration
input is <=1
)5(
1)(
2
sssG
Note: we have all three types of specs: speed, relative stability, and tracking
Strategy
• First do a lead design to fix speed and overshoot requirement
• Then do a lag design to fix the ess.
n=[1]; d=[1 5 0 0];figure(1); clf; margin(n,d); grid; hold on;Mp=16/100; zeta = sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));PMd = zeta * 100 + 5;tr = 2; wn = 1.8/tr; w_gcd = wn*0.8;PM = pi+angle(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));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(evalfr(tf(n,d),j*w_gcd)));ngc = conv(n, K*[1 zlead]); dgc = conv(d, [1 plead]);figure(1); margin(ngc,dgc);[ncl,dcl]=feedback(ngc,dgc,1,1);figure(2); step(ncl,dcl); grid;figure(3); margin(ncl*1.414,dcl); grid;
-200
-100
0
100M
agn
itu
de
(dB
)
10-2
10-1
100
101
102
103
-270
-225
-180
-135
-90
Ph
ase
(deg
)
Bode DiagramGm = 20.1 dB (at 3.74 rad/sec) , Pm = 55.4 deg (at 0.72 rad/sec)
Frequency (rad/sec)
0 5 10 15 20 25 30 35 400
0.2
0.4
0.6
0.8
1
1.2
1.4Step Response
Time (sec)
Am
plit
ud
e
Overshoot is too large.
Plus, we know the lag controller will further deteriorate Mp.
So, redesign for better Mp.
-100
0
100
Mag
nit
ud
e (d
B)
10-3
10-2
10-1
100
101
102
-270
-225
-180
-135
-90
Ph
ase
(deg
)
Bode DiagramGm = 22.4 dB (at 4.89 rad/sec) , Pm = 65.4 deg (at 0.72 rad/sec)
Frequency (rad/sec)
-150
-100
-50
0
50M
agn
itu
de
(dB
)
10-2
10-1
100
101
102
103
-270
-180
-90
0
Ph
ase
(deg
)
Bode DiagramGm = 18.7 dB (at 4.89 rad/sec) , Pm = 96.5 deg (at 1.14 rad/sec)
Frequency (rad/sec)
0 5 10 15 20 25 30 35 400
0.2
0.4
0.6
0.8
1
1.2
1.4 Step Response
Time (sec)
Am
plit
ud
e
About 12% overshoot.So, let’s go ahead with lag design.
Kaa = ngc(end)/dgc(end-2); ess=1; Kad=1/ess;zlag = w_gcd/20; plag = zlag/(Kad/Kaa);ngcc = conv(ngc, [1 zlag]); dgcc = conv(dgc, [1 plag]);figure(1); margin(ngcc,dgcc);[ncl,dcl]=feedback(ngcc,dgcc,1,1);figure(4); step(ncl,dcl); grid;figure(5); margin(ncl*1.414,dcl); grid;
We don’t have too much room to spare for Mp, so choose 20 so that the lag controller only kills about 3 degrees of PM.
-200
-100
0
100
200M
agn
itu
de
(dB
)
10-4
10-2
100
102
-270
-180
-90
Ph
ase
(deg
)
Bode DiagramGm = 22.3 dB (at 4.86 rad/sec) , Pm = 62.7 deg (at 0.721 rad/sec)
Frequency (rad/sec)
-150
-100
-50
0
50M
agn
itu
de
(dB
)
10-2
10-1
100
101
102
103
-270
-180
-90
0
90
Ph
ase
(deg
)
Bode DiagramGm = 18.6 dB (at 4.86 rad/sec) , Pm = 93.4 deg (at 1.17 rad/sec)
Frequency (rad/sec)
0 5 10 15 20 250
0.2
0.4
0.6
0.8
1
1.2
1.4Step Response
Time (sec)
Am
plit
ud
e
