19
Contents
• Compensation– Lead Compensation– Lag Compensation– Lag-Lead Compensation
Compensation • Used to Improve the performance of stable or unstable
systems.
PlantG(S)
CompensatorGc(S)
H(S)
-
Lead Compensation
• Used to Improve the Transient behavior of the system.
)1(
)1()(
TS
TS
KSG cc
PlantG(S)
CompensatorGc(S)
H(S)
-
1 where
Lead Compensation
• Consider the Following Unity feedback system
k=4p=[0 -2 ]z=[];usys=zpk(z,p,k)rlocus(usys)csys=feedback(usys,1)[wn b]=damp(csys)sgrid(b,wn)axis([-3 0 -3 3 ])
Lead Compensation contd…
)( 24SS
)(SR )(SC
Design Requirements
• It is desired to increase ωn to 4 rad/sec without changing damping ratio of closed loop poles.
k=4p=[0 -2 ]z=[];usys=zpk(z,p,k)rlocus(usys)wn=[2 4]; b=[0.5 0.5];sgrid(b,wn)axis([-5 0 -5 5 ])
Lead Compensation contd…
)( 24SS
)(SR )(SC
Lead Compensator (Follow the Class Work)
)4.5()9.2(68.4)(
SSSGc
kc=4.68pc=-5.4zc=-2.9lc=zpk(zc,pc,kc)ltiview(lc)
Lead Compensation contd…
Pole-Zero Map
Real Axis
Imag
Axi
s
-5 -4 -3 -2 -1 0-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
5 4 3 2 1
0.998
0.991
0.98 0.96 0.92 0.85 0.7 0.45
0.998
0.991
0.98 0.96 0.92 0.85 0.7 0.45
Pzmap of Lead Compensator
Lead Compensation contd…
Bode Diagram
Frequency (rad/sec)
Phas
e (d
eg)
Mag
nitu
de (d
B)
8
9
10
11
12
13
14
10-1
100
101
102
0
5
10
15
20
Bode plot of Lead Compensator
Lead Compensation contd…
Root locus of Compensated & uncompensated Systems
)4.5)(2()9.2(7.18)()(
SSSSSGSG c
b=0.5; wn=[2 4];ku=4pu=[0 -2 ]zu=[ ];usys=zpk(zu,pu,ku)k=18.7p=[0 -2 -5.4]z=-2.9csys=zpk(z,p,k)rlocus(usys, ‘g’,csys,’b’)Sgrid(b,wn)
)2(4)(
SS
SG
Compensated Vs Uncompensated System
Lead Compensation contd…
Root Locus
Real Axis
Imag
Axi
s
-5 -4 -3 -2 -1 0
-10
-8
-6
-4
-2
0
2
4
6
8
10
4
2
4
2
0.5
0.5
Root locus of Compensated & uncompensated Systems Compensated Vs Uncompensated System
Uncompensatedcompensated
Lead Compensation contd…
Response of compensated & Uncompensated system
23.545.294.723.547.18
)()(1)()(
23
SSS
SSGSGSGSG
c
c
numu=4;denu=[1 2 4];numc=[18.7 54.23];denc=[1 7.4 29.5 54.23];csys=tf(numc,denc);usys=tf(numu,denu);ltiview(usys, ‘g:’,csys,’b’)
424
)(1)(
2
SSSGSG
Compensated Vs Uncompensated System
Lead Compensation contd…
Pole-Zero Map
Real Axis
Imag
Axi
s
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-4
-3
-2
-1
0
1
2
3
4
Pzmap of Compensated & Uncompensated System
Compensated Vs Uncompensated System
Uncompensatedcompensated
Lead Compensation contd…
Step Response
Time (sec)
Ampl
itude
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
1.2
1.4
Step response of Compensated & Uncompensated System
Compensated Vs Uncompensated System
Uncompensatedcompensated
Lead Compensation contd…
Bode Diagram
Frequency (rad/sec)
Phas
e (d
eg)
Mag
nitu
de (d
B)
-80
-60
-40
-20
0
20
10-1
100
101
102
-180
-135
-90
-45
0
Bode plot of Compensated & Uncompensated System
Compensated Vs Uncompensated System
Uncompensatedcompensated
Lead Compensation contd…
Steady State Error of Compensated & Uncompensated System
Compensated Vs Uncompensated System
According to final value theorem
)(lim0
SSGKSv
Steady state error is given by
vss Ke 1
In MATLAB
limit(f(x),x,0)Returns the limit of a function f(x) as x 0.
Lead Compensation contd…
Steady State Error of Compensated & Uncompensated System
Compensated Vs Uncompensated System
)4.5)(2()9.2(7.18)()(
SSSSSGSG c
)2(4)(
SS
SGclearsyms snumu=4denu=s^2+2*sGu=numu/denunumc=18.7*s+54.23denc=s^3+7.4*s^2+10.8*sGc=numc/denckvu=limit(s*Gu,s,0)kvc=limit(s*Gc,s,0)essu=1/kvuessc=1/kvc
Lead Compensation contd…
Exercise#1Repeat the same tutorial using SISO Design tool.
Exercise#2Design another lead compensator for the same system and compare the results.
End of tutorial
You can download this tutorial from:
http://imtiazhussainkalwar.weebly.com/control-system-design-and-analysis.html
