Root-Locus Technique

1. Root-Locus Technique 8

2. Root Loci • Properties of Root Loci • Root Sensitivity • Root Contours • Root Loci of Discrete-Data Systems

3. R(s) C(s) G(s) H(s) (a) Characteristic equation 1+G(s)H(s)=0 , (1)

4. Where, G(s)H(s) is defined as fallow = (2) eq (2) into eq (1) (3)

5. eq (3) into eq (a) The charicteristic eq from eq (4) The parameters of P(s) Q(s) are all flxed

6. Root-Loci Root-Contours • RL (Root-loci) ; • CRL (complementary RL) ; • RC (Root contours) ; Contours of roots when more than one parameters variable • Root-locus ;

7. From eq (5) k=0 root ; p(s) k= root ; Q(s) if k is varied zero to than Root-locus start from open-loop pole finished to open-loop zero

8. -3 –2 -1 Ex) 의 인 점을 확도 K = 0 인 점 K = 인 점

9. Root-Locus Technique : Developed by W.R Evans 1948 If the system has a variable loop gain as follow fig. Root-Locus [The variable loop gain k the location of the closed-loop poles] The characteristic of the transient response The system stability

10. Design Problem [the approprite gain selection] [addition of compensator] By using the root-locus method the designer can predict the effects on the location of the closed-loop poles of varying the gain value or adding open-loop poles and/or open-loop zeros.

11. Definition Root Loci ; System parameter; fixed ;variable RL ( )+ CRL ( ) RC (Root contours) the Root-Loci of multiple-variable parameter at a timer

12. Root-Locus Plots Condition on Magnitude

13. Let’s K is isolated as a multiplying factor Angle condition G(s)H(s) involves a gain parameter K, and the characteristic equation may be written as

14. for

15. (CRL) for The difference between the sums of the angles of the vectors drawn from the zeros and those from the poles of G(s)H(s) to is an odd multiple of For difference values of K, any points on the CRL must satisfy the condition:

16. The difference between the sums of the angles of the vectors drawn from the zeros and from the poles of G(s)H(s) to is an even multiple of , including zero degrees. Once the root loci are constructed , the value of K along the loci can be determined by

17. Ex)

18. Case of RL Case of CRL

19. Ex) 에서 인점의 크기와 각도는?

21. Ex) Closed-loop characteristic eq

22. , 음의 실근 ,음의 복소 공액근 은(+)실근, 는 (-)실근

23. Properties and Construction of the Root-Loci • K=0 and K= points The k=0 points on the root loci are at the poles of G(s)H(s) The K= point on the root loci are the zeros of G(s)H(s) (근궤적은 극점에서 출발하여 영점에서 끝난다.)

24. 2) Number of Branches on the Root-Loci The Number of branches of the Root-Loci is equal to the order of the polynomial (근궤적의 수는 영점수와 극점수중 큰것과 일치) 3) Symmetry of the Root-Loci The Root loci are Symmetrical with respect to the real axis of the s-plane (근궤적은 실수측에 대해 대칭)

25. 4) Asymptotes of Root-Loci Number of Asymptotes Angles of Asymptotes

26. 5) Intersect of the Asymptotes (Centroid) 6) Angles of Departure and Angles of Arrival of the Root-Loci the angle of departure of arrival of a root-locus at a pole or zero , respectively, of G(s)H(s) denetes the angle of the taryent to the locus near the point

27. 여기서 는 특정극(영점) 에서 다른 개루프 즉 (영점) 이 이루는 각의 합이다. Ex)

28. Ex) for