1. Root Locus by: Nafees Ahmed
1/12/2019
Root Locus by: Nafees Ahmed

2. Root Locus by: Nafees Ahmed
What is Root Locus ?
H(s)
The characteristic equation of the closed-loop system is
1 + K G(s)H(s) = 0
The root locus is essentially the trajectories of roots of the characteristic equation as the parameter K is varied from 0 to infinity.
1/12/2019
Root Locus by: Nafees Ahmed

3. Root Locus by: Nafees Ahmed
A simple example
A camera control system:
How the dynamics of the camera changes as K is varied ?
1/12/2019
Root Locus by: Nafees Ahmed

4. A simple example (cont.) : pole locations
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Root Locus by: Nafees Ahmed

5. A simple example (cont.) : Root Locus
<<<6.2>>>
(a) Pole plots from the table.
(b) Root locus.
1/12/2019
Root Locus by: Nafees Ahmed

6. Root Locus by: Nafees Ahmedk
G(s) ＋ －
H(s)
poles
1/12/2019
Root Locus by: Nafees Ahmed

7. Root Locus by: Nafees Ahmed
Open loop transfer function
Using open loop transfer function + system parameters to analyze the closed-loop system response
Draw the s-plan root locus
1/12/2019
Root Locus by: Nafees Ahmed

8. Root Locus by: Nafees Ahmed
Example: Draw Root locus
Sol:
In Matlab
>>n=[1];
>>d=[ ];
>>rlocus(n,d)
1/12/2019
Root Locus by: Nafees Ahmed

9. Root Locus by: Nafees Ahmed
1/12/2019
Root Locus by: Nafees Ahmed

10. Root Locus by: Nafees Ahmed
Root locus construction
(i) Root Locus is symmetrical about real axis
(ii) Loci Branches:
Each locus starts from an open pole (K=0) and
terminates at either on an open loop zero or
on infinity (K=∞)
Let
P=No of finite poles
Z=No of finite zeros
No of branches in root loci =P if P>Z
=Z if Z>P if
=>Poles are more if
=>Zeros are more
1/12/2019
Root Locus by: Nafees Ahmed

11. Root Locus by: Nafees Ahmed
(iii) Existence of Root loci on real axis
Note: Only real axis Poles + Zeros; not complex
Poles + zeros = odd
Poles + zeros = even
If Poles + Zeros on RHS= Odd=> That portion will be in root locus
If Poles + Zeros on RHS= Even=> That portion will not be in root locus
1/12/2019
Root Locus by: Nafees Ahmed

12. Root Locus by: Nafees Ahmed
(iv) Asymptotic angles
Reference for angle
Total no of asymptotes =P-Z
1/12/2019
Root Locus by: Nafees Ahmed

13. Root Locus by: Nafees Ahmed
(v) Centroid of the asymptotes
example
Zero : 0
Poles: -2, -3+j3, -3-j3
1/12/2019
Root Locus by: Nafees Ahmed

14. Root Locus by: Nafees Ahmed
(vi) Breakaway and entry points
example
The characteristic function of closed loop system
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Root Locus by: Nafees Ahmed

15. Root Locus by: Nafees Ahmed
(vii) Angle of departure and arrival
Angle of departure from a complex pole
Angle of arrival at a complex zero
example
Angle of departure from the pole:
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Root Locus by: Nafees Ahmed

16. Root Locus by: Nafees Ahmed
example
Angle of approach to the zero:
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Root Locus by: Nafees Ahmed

17. Root Locus by: Nafees Ahmed
(viii) Intersection with Img axis: By Routh Hurwitz criterion
example
The characteristic function of closed loop system:
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Root Locus by: Nafees Ahmed

18. Root Locus by: Nafees Ahmed
Example + -
(i)
(iii)
(ii)
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Root Locus by: Nafees Ahmed

19. Root Locus by: Nafees Ahmed
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20. Root Locus by: Nafees Ahmed
Determination of K on Root loci A B C
Root Loci
1/12/2019
Root Locus by: Nafees Ahmed

21. Root Locus by: Nafees Ahmed
MATLAB method
Example 1
gh=zpk([],[0 –2 -10],[1])
rltool(gh)
Example 2
n=[-3 -9]
m=[1 –1 –1 –15 0]
gh=tf(n,m)
rltool(gh)
1/12/2019
Root Locus by: Nafees Ahmed

22. Root Locus by: Nafees Ahmed
Example 3:
>> n=[1];
>> d=[ ];
>> rlocus(n,d) Or
d may be written as
d=conv([1 0],[1 4 8]);
1/12/2019
Root Locus by: Nafees Ahmed

23. Root Locus by: Nafees Ahmed
Thanks ?
1/12/2019
Root Locus by: Nafees Ahmed