Variable Structure Systems

1. Plant + _ Variable Structure Systems • Motivation Aim : To use linear, static, output feedback and stabilize the system

2. Variable Structure Systems (Continued) Combine these two, introduce the controller Result : Spiraling in at 0

3. Variable Structure Systems (Continued) There are other possibilities

4. Alternate Approach In both of these examples, we use the existing trajectories and attach these to original dynamics so that the goal is achieved. There is another approach – to create new trajectories. Consider the case

5. Variable Structure Control Then

6. where Sliding Mode Control Consider the system

7. Sliding Mode Control (Continued) unmatched uncertainty matched uncertainty

8. Sliding Mode Control (Continued) (1) (2) equivalent control is chosen to cancel the known terms

9. From (1),(2) (3) First write (3) as p scalar equations Sliding Mode Control (Continued)

10. Sliding Mode Control (Continued) Then

11. reaching phase during which trajectories starting off the manifold z=0 move toward it and reach it in finite time  sliding phase during which the motion will be confined to manifold z=0 and the dynamics of the system will be represented by the reduced order system  (i) (ii) (iii) Sliding Mode Control (Continued) Thus sliding mode control Thus

12. switching manifold z>0 delay between the time z changes and the time the control switches z<0 a Robustness & Discontinuous Nonlinearity • Robustness to uncertainties During reaching phase : forcing trajectories toward sliding manifold maintaining them on the manifold This task is affected by matched and unmatched uncertainty During sliding phase : This task is affected by unmatched uncertainty • Discontinuous nonlinearity in sliding mode controller Theoretical issue : uniqueness & existence Practical issue : chattering due to imperfect switching devices and delay

13. 1 1 y y -1 -1 Chattering Effects

14. Chattering Effects (Continued)

15. Chattering Effects (Continued)

16. Theorem

17. Example 1 Ex:

18. Example 1 (Continued)

19. Example 2 Ex:

20. Example 2 (Continued)

21. Example 2 (Continued)

22. Example 2 (Continued)