Modern Control Theory Digital Control

1. Modern Control TheoryDigital Control Lecture 4

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 sketching root locus/root locus characteristics • Selecting the parameter value

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

4. Introduction • Dynamic features depend on the pole locations. • For example, time constant t, rise time tr, and overshoot Mp (1st order) (2nd order)

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

7. Definition of 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 yi. • Def: The angle to the test point from pole number i is fi. • Therefore, • In def. 2, notice,

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

9. Root locus of a motor position control (example) break-away point

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

11. Sketching a Root Locus • How do we find the root locus? • We could try a lot of test points s0 • Sketch by hand (6 rules) • Matlab s0 f1 p1

12. Sketching a Root Locus What is meant by a test point? For example s0 f1 p1

13. Sketching a Root Locus s0 f1 p y1 f2 p* A inconviniet method !

14. Root Locus characteristics(can be used for sketching) • 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. Root Locus characteristics • 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 s0 on the real axis : • The angle of complex poles cancel each other. • Angles from real poles or zeros are 0° if s0 are to the right. • Angles from real poles or zeros are 180° if s0 are to the left. • Total angle = 180° + 360° l

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

17. Root Locus characteristics Example, n-m = 4 s0 a f1 = f2 = f3 = f4 = 45° Thus,fl= 45°

18. Root Locus characteristics

19. Root Locus characteristics

20. Root Locus characteristics = Starting point for the asymptotes

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

22. Root Locus characteristics • 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. Root Locus characteristics In general, we must find a s0 such that angle(L(s))=180° s0 f1 f3 y1 f3 f2 • Departure and arrival? • For K increasing from zero to infinity poles go towards zeros or infinity. Thus, • A branch corresponding to a pole departs at some angle. • A branch corresponding to a zero arrives at some angle. Vary s0 until angle(L(s))=180°

24. Root Locus characteristics • Rule 5 (of 6) • The locus crosses the jw 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 => limits for stable K values.

25. Root Locus characteristics • 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. Root Locus characteristics • The locus will have multiplicative roots of q in the point s=r1 We diffrentiate with respect to s

27. Root Locus sketching 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

28. 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 jw-axis. Rule 6: Easy to see, no further multiple poles. Verification:

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

30. Sketching a Root Locus

31. Sketching a Root Locus

32. Sketching a Root Locus

33. 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.

34. Selecting the Parameter Value Example

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