1. Root Locus Plot of Dynamic Systems

2. We will cover........ Root Locus of LTI models
Root locus of a Transfer Function (TF) model
Root Locus of 1st order system
Root Locus of 2nd order system
Root Locus of higher order systems
Root locus of a Zero-pole-gain model (ZPK)
Root locus of a State-Space model (SS)
Gain Adjustment

3. Root Locus of 1st Order System
Consider the following unity feedback system
Matlab Code
num=1;
den=[1 0];
G=tf(num,den);
rlocus(G)
sgrid

4. Root Locus of 1st Order System
Consider the following unity feedback system
num=[1 0];
den=[1 1];
G=tf(num,den);
rlocus(G)
sgrid
Continued…..

5. Plot the root locus of following first order systems.
Exercise#1
Plot the root locus of following first order systems.

6. Root Locus 2nd order systems
Consider the following unity feedback system
num=1;
den1=[1 0];
den2=[1 3];
den=conv(den1,den2);
G=tf(num,den);
rlocus(G)
sgrid
Determine the location of closed loop poles that will modify the damping ratio to 0.8 and natural undapmed frequency to 1.7 r/sec. Also determine the gain K at that point.

7. Plot the root locus of following 2nd order systems.
Exercise#2
Plot the root locus of following 2nd order systems.
(1)
(2)

8. Root Locus of Higher Order Systems
Consider the following unity feedback system
num=1;
den1=[1 0];
den2=[1 1];
den3=[1 2];
den12=conv(den1,den2);
den=conv(den12,den3);
G=tf(num,den);
rlocus(G)
sgrid
Determine the closed loop gain that would make the system marginally stable.

9. Exercise#3
Plot the root locus of following systems.
(1)
(2)

10. Root Locus of a Zero-Pole-Gain Model
k=2;
z=-5;
p=[ ];
G=zpk(z,p,k);
rlocus(G)
sgrid

11. Root Locus of a State-Space Model
B=[1;0];
C=[1 0];
D=0;
sys=ss(A,B,C,D);
rlocus(sys)
sgrid

12. Exercise#4: Plot the Root Locus for following LTI Models
(1)
(2)
(3)

13. Choosing Desired Gain num=[2 3]; den=[3 4 1]; G=tf(num,den);
[kd,poles]=rlocfind(G)
sgrid

14. Exercise#5
(1)
Plot the root Loci for the above ZPK model and find out the location of closed loop poles for =0.505 and n=8.04 r/sec.
b=0.505;
wn=8.04;
sgrid(b, wn)
axis equal
Open File  New  M-File
Save File as H:\ECE4115\Module1\Mod1_1.m

15. Exercise#5: (contd…) Consider the following unity feedback system (2)
Plot the root Loci for the above transfer function
Find the gain when both the roots are equal
Also find the roots at that point
Determine the settling of the system when two roots are equal.
Open File  New  M-File
Save File as H:\ECE4115\Module1\Mod1_1.m

16. Exercise#5: (contd…) Consider the following velocity feedback system
(3)
Plot the root Loci for the above system
Determine the gain K at which the the system produces sustained oscillations with frequency 8 rad/sec.
Open File  New  M-File
Save File as H:\ECE4115\Module1\Mod1_1.m

17. End of tutorial
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