1. Routh Hurwitz Stability Test & Analysis of Closed-loop System
Root Locus Analysis

2. Given Transfer Function of a system
Characteristic polynomial of the system:
ROUTH HURWITZ TABLE

4. Characteristic Polynomial:
EXAMPLE:
Characteristic Polynomial:
ROUTH HURWITZ TABLE:
2 SIGN CHANGES IN FIRST COLUMN. THEREFORE TWO UNSTABLE POLES

5. ONE SIGN CHANGE IN FIRST COLUMN.
THEREFORE ONE UNSTABLE POLE

6. Y(s) = KP G(s) [R(s) – H(s)Y(s)]
CLOSED-LOOP SYSTEM
R(s)
Y(s)
E(s)
U(s)
E(s) = R(s) – H(s)Y(s)
U(s) = KP E(s)
= KP [R(s) – H(s)Y(s)]
Y(s) = G(s) U(s)
Y(s) = KP G(s) [R(s) – H(s)Y(s)]

7. ROOTS (poles) OF THE CLOSED-LOOP SYSTEM
The poles of the closed-loop system now depend on the gain Kp and the compensator H(s) as well as the poles of the original plant i.e. G(s)
We want a quick way to determine what will happen to the closed-loop poles when the gain Kp is increased from zero to infinity.
A ROOT LOCUS sketch enables us to see the migration of the closed-loop poles.

8. THE ROOT-LOCUS Let Since
Equation 1
Kp = 0, Equation 1 becomes D(s) = 0, The poles (roots) of the closed-loop system are the poles of the transmittance GH
Kp = infinity, Equation 1 becomes N(s) = 0, The poles (roots) of the closed-loop system are the ZEROS of the transmittance GH

9. Example to illustrate that the roots of the closed-loop go from the POLES of GH to the ZEROS of GH
Poles of GH : s =0, s = -1, s = -4
ZEROS of GH: s = j, s = j

10. The root locus on the previous slide is reproduced here
K = 0
Real axis segment
Real axis segment
A real axis segment of the root locus exists if the total number of poles + zeros to the right of a point are an ODD number

11. Centroid =
n = number of poles of GH = 3
m = number of zeros of GH = 2
Asymptotic angles
If n – m < 2 then asymptotic angles are meaningless

12. Poles of GH at s = 0, s = -1, s = -4
Zeros of GH at s = -6, s = j, s = j

13. Poles of GH at s = -5, s = -1 + j, s = -1 - j
Zeros at infinity
System remains stable for a range of gain Kp from zero to 72

14. Centroid =
n = number of poles of GH = 3
m = number of zeros of GH = 0
Asymptotic angles
Draw straight lines starting from the centroid at making angles of 60, -60 and 180. The locii are asymptotic to these straight lines

15. What is G(s)H(s) from the root-locus?

16. ROOT LOCUS SKETCHING 1. Form the product G(s)H(s)
2. Draw the POLES and ZEROS of G(s)H(s)
3. Draw the REAL-AXIS SEGMENTS of the root locus
4. Find the CENTROID
5. Find the ASYMPTOTIC ANGLES (when locii go to zeros at infinity)
i is an integer 0, 1, 2, 3, -----

17. 6. Find the range of gain Kp for which the system remains stable, since stability is determined by the closed-loop poles
Use the Routh Hurwitz table to find the gain Kp that gives a table row of all zeros, (boundary of stability)

18. Closed-loop Poles
Kcritical = 74
The third row can be made all zero

19. Kcritical = 74 s = jω Therefore, at Kcriticical the poles are
To find the value of the poles at cross over, let
Therefore, at Kcriticical the poles are

20. System remains stable for a range of gain Kp from zero to 72

21. G(s) H(s) Kp CLOSED-LOOP SYSTEM
Exercise to see how the poles of this closed-loop system move with increasing K
G(s)
H(s) Kp
CLOSED-LOOP SYSTEM
R(s)
Y(s)
E(s)
U(s)
Perform Root Locus Analysis on the product GH

22. 4 -6 -4 -2 -4