1 Modern Control Theory Digital 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 , rise time t r, and overshoot M p (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  i. Def: The angle to the test point from pole number i is  i. Therefore, In def. 2, notice,

8 Root locus of a motor position control (example)

9 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 s 0  Sketch by hand (6 rules)  Matlab s0s0 p1p1 

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

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

18 Root Locus characteristics

19 Root Locus characteristics

20 Root Locus characteristics = Starting point for the asymptotes

21 Root Locus characteristics n-m=1n-m=2n-m=3n-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 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 at 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 Root Locus characteristics 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 => 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=r 1 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 j  -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)