1. 1 Modern Control Theory Digital Control Lecture 4 By Kirsten Mølgaard Nielsen Based on notes from Jesper Sandberg Thomsen

2. 2 Outline The Root Locus Design Method  Introduction Idea, general aspects. A graphical picture of how changes of one system parameter will change the closed loop poles.  Sketching a root locus Definitions 6 rules for positive root locus (6 rules for negative root locus)  Selecting the parameter value

3. 3 Introduction Closed loop transfer function Characteristic equation, roots are poles in T(s)

4. 4 Introduction Dynamic features depend on the pole locations. For example, time constant , rise time t r, and overshoot M p (1st order) (2nd order)

5. 5 Introduction Root locus  Determination of the closed loop pole locations under varying K.  For example, K could be the control gain. The characteristic equation can be written in various ways

6. 6 Introduction Some root loci examples (K from zero to infinity)

7. 7 Introduction Polynomials b(s) and a(s)

8. 8 Introduction Root locus of a motor position control (example)

9. 9 Introduction break-away point

10. 10 Sketching a Root Locus Definition 1  The root locus is the values of s for which 1+KL(s)=0 is satisfied as K varies from 0 to infinity (pos.). Definition 2  The root locus is the points in the s-plane where the phase of L(s) is 180°. Def: The angle to the test point from zero number i is  i. Def: The angle to the test point from pole number i is  i. Therefore, In def. 2, notice,

11. 11 Sketching a Root Locus s0s0 p1p1  What is meant by a test point? For example

12. 12 Sketching a Root Locus s0s0 p  p*  

13. 13 Sketching a Root Locus How do we find the root locus?  We could try a lot of test points s 0  Sketch by hand (6 rules)  Matlab s0s0 p1p1 

14. 14 Sketching a Root Locus Rule 1 (of 6)  The n branches of the locus start at the poles L(s) and m of these branches end on the zeros of L(s).

15. 15 Sketching a Root Locus Rule 2 (of 6)  The loci on the real axis (real-axis part) are to the left of an odd number of poles plus zeros. Notice, if we take a test point s 0 on the real axis : The angle of complex poles cancel each other. Angles from real poles or zeros are 0° if s 0 are to the right. Angles from real poles or zeros are 180° if s 0 are to the left. Total angle = 180° + 360° l

16. 16 Sketching a Root Locus Rule 3 (of 6)  For large s and K, n-m branches of the loci are asymptotic to lines at angles  l radiating out from a point s =  on the real axis. ll  For example

17. 17 Sketching a Root Locus s0s0          °  Example, n-m = 4 Thus,  l   °

18. 18 Sketching a Root Locus (approximation)

19. 19 Sketching a Root Locus (approximation)

20. 20 Sketching a Root Locus

21. 21 Sketching a Root Locus n-m=1n-m=2n-m=3n-m=4 For ex., n-m=3

22. 22 Sketching a Root Locus Rule 4 (of 6)  The angle of departure of a branch of the locus from a pole of multiplicity q is given by (1)  The angle of departure of a branch of the locus from a zero of multiplicity q is given by (2) Notice, the situation is similar to the approximation in Rule 3

23. 23 Sketching a Root Locus s0s0      Departure and arrival? For K increasing from zero to infinity poles go towards zeros or infinity. Thus, A branch corresponding to a pole departs a some angle. A branch corresponding to a zero arrives at some angle. In general, we must find a s 0 such that angle(L(s))=180° Vary s 0 until angle(L(s))=180°

24. 24 Sketching a Root Locus Rule 5 (of 6)  The locus crosses the j  axis at points where the Routh criterion shows a transition from roots in the left half-plane to roots in the right half-plane.  Routh: A system is stable if and only if all the elements in the first column of the Routh array are positive.

25. 25 Sketching a Root Locus Rule 6 (of 6)  The locus will have multiplicative roots of q at points on the locus where (1) applies.  The branches will approach a point of q roots at angles separated by (2) and will depart at angles with the same separation.

26. 26 Sketching a Root Locus Example Root locus for double integrator with P-control Rule 1: The locus has two branches that starts in s=0. There are no zeros. Thus, the branches do not end at zeros. Rule 3: Two branches have asymptotes for s going to infinity. We get

27. 27 Sketching a Root Locus Rule 2: No branches that the real axis. Rule 4: Same argument and conclusion as rule 3. Rule 5: The loci remain on the imaginary axis. Thus, no crossings of the j  -axis. Rule 6: Easy to see, no further multiple poles. Verification:

28. 28 Sketching a Root Locus Example Root locus for satellite attitude control with PD-control

29. 29 Sketching a Root Locus

30. 30 Sketching a Root Locus

31. 31 Sketching a Root Locus

32. 32 Selecting the Parameter Value The (positive) root locus  A plot of all possible locations of roots to the equation 1+KL(s)=0 for some real positive value of K.  The purpose of design is to select a particular value of K that will meet the specifications for static and dynamic characteristics.  For a given root locus we have (1). Thus, for some desired pole locations it is possible to find K.

33. 33 Selecting the Parameter Value Example

34. 34 Selecting the Parameter Value Matlab A root locus can be plotted using Matlab  rlocus(sysL) Selection of K  [K,p]=rlocfind(sysL)