1. Design & Compensation via Root Locus

2. Contents Compensation Lead Compensation Lag Compensation
Lag-Lead Compensation

3. Compensation
Used to Improve the performance of stable or unstable systems.
Plant G ( S )
Compensator c H -

4. Lead Compensation
Used to Improve the Transient behavior of the system.
Plant G ( S )
Compensator c H -

5. Lead Compensation Consider the Following Unity feedback system k=4
z=[];
usys=zpk(z,p,k)
rlocus(usys)
csys=feedback(usys,1)
[wn b]=damp(csys)
sgrid(b,wn)
axis([ ])
Lead Compensation contd…

6. Design Requirements
It is desired to increase ωn to 4 rad/sec without changing damping ratio of closed loop poles.
k=4
p=[ ]
z=[];
usys=zpk(z,p,k)
rlocus(usys)
wn=[2 4]; b=[ ];
sgrid(b,wn)
axis([ ])
Lead Compensation contd…

7. Lead Compensator (Follow the Class Work)
kc=4.68
pc=-5.4
zc=-2.9
lc=zpk(zc,pc,kc)
ltiview(lc)
Lead Compensation contd…

8. Pzmap of Lead Compensator
Lead Compensation contd…

9. Bode plot of Lead Compensator
Lead Compensation contd…

10. Compensated Vs Uncompensated System
Root locus of Compensated & uncompensated Systems
b=0.5; wn=[2 4];
ku=4
pu=[ ]
zu=[ ];
usys=zpk(zu,pu,ku)
k=18.7
p=[ ]
z=-2.9
csys=zpk(z,p,k)
rlocus(usys, ‘g’,csys,’b’)
Sgrid(b,wn)
Lead Compensation contd…

11. Root locus of Compensated & uncompensated Systems
Compensated Vs Uncompensated System
Uncompensated
compensated
Lead Compensation contd…

12. Compensated Vs Uncompensated System
Response of compensated & Uncompensated system
numu=4;
denu=[ ];
numc=[ ];
denc=[ ];
csys=tf(numc,denc);
usys=tf(numu,denu);
ltiview(usys, ‘g:’,csys,’b’)
Lead Compensation contd…

13. Compensated Vs Uncompensated System
Pzmap of Compensated & Uncompensated System
Uncompensated
compensated
Lead Compensation contd…

14. Compensated Vs Uncompensated System
Step response of Compensated & Uncompensated System
Uncompensated
compensated
Lead Compensation contd…

15. Compensated Vs Uncompensated System
Bode plot of Compensated & Uncompensated System
Uncompensated
compensated
Lead Compensation contd…

16. limit(f(x),x,0) Compensated Vs Uncompensated System
Steady State Error of Compensated & Uncompensated System
According to final value theorem
Steady state error is given by
In MATLAB
limit(f(x),x,0)
Returns the limit of a function f(x) as x
Lead Compensation contd…

17. clear syms s numu=4 denu=s^2+2*s Gu=numu/denu numc=18.7*s+54.23
Compensated Vs Uncompensated System
Steady State Error of Compensated & Uncompensated System
clear
syms s
numu=4
denu=s^2+2*s
Gu=numu/denu
numc=18.7*s+54.23
denc=s^3+7.4*s^2+10.8*s
Gc=numc/denc
kvu=limit(s*Gu,s,0)
kvc=limit(s*Gc,s,0)
essu=1/kvu
essc=1/kvc
Lead Compensation contd…

18. Exercise#1
Repeat the same tutorial using SISO Design tool.
Exercise#2 Design another lead compensator for the same system and compare the results.

19. End of tutorial
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