1. 1 Scheduling in Anti-windup Controllers: State and Output Feedback Cases Faryar Jabbari Mechanical an Aerospace Engineering Department University of California, Irvine (UCI) November 13, 2007

2. 2 Thanks Responsible Party: Solmaz Sajjadi-Kia Collaborators Thanh Nguyen Sharad Sirivastada Emre Kose Support NSF Grants US D. of Ed GAANN Grants

3. 3 Surveys IJRNC: Michele and Bernstein, eds. (1995) IJRNC: Saberi and Stoorvogel, eds. (1999) Franco Blanchini's review article(TAC, 2000) Tarbouriech, et al., Springer, (1999) Kapila and Grigoriadis, Marcel Dekker (2003) IJRNC: Saberi and Stoorvogel, eds. (2004) Much more!

4. 4 Motivation Old Problem: actuator limitation is ubiquitous `Safe' (Low gain) LTI controllers are often excessively conservative Broad approaches: Oldest: Anti-windup  Nominal high performance controller (linear design)  Anti-windup augmentation Relatively new: Explicit account of saturation nonlinearity  Nonlinear design or low gain designs

5. 5 Current Techniques to Deal with Saturation Direct Approach Considers the controllers limitation at the very beginning of the design Anti-windup Augmentation on top of the nominal controller designed without considering controller bound ||W|| 2 <W 2 max

6. 6 Anti-windup Starting in 60's (Sandberg, among many) Huge body or work, at times intuitive or even ad-hoc Many attempts at unifying, interpreting of all techniques New rigorous stability and performance results  Morari group  Teel group  Many others (literally too numerous to review!)  Positivity, small gain, LMI's, etc.

7. 7 Anti windup (continued) High performance when no saturation Ideal for `occasional' saturation Relatively weak performance when in saturation Typically open loop performance -- so open-loop stability `often' needed (exceptions: Tell, et al. ACC- 05, and a few references there) A single controller/augmentations for all saturation levels (even almost zero?), disturbances, tracking signals, etc.

8. 8 Explicit – direct – approach Low-high gain (Saberi and Lin, 199x) Early LPV : Nguyen and Jabbari (1999, 2000), Scorletti, et al (2001) Scheduling: Older work (full state):  Gutman and Hagander (1985)  Wredenhagen and Belanger (1994)  Megretski (1996 IFAC) Scheduling: Recent work}  Lin (1997), a little bit of observer  Teel (1995), Tarboriech, et al (1999, 2000) - state feedback  Wu, Packard and Grigoriadis (2000) - pure LPV  Stoustrup (2005-07)  Kose and Jabbari (2002, 2003)

9. 9 Direct Approach Stability and performance guarantees Performance not strong in small signal operation `Some' have nice properties: A family of controllers (rather than one) Computationally tractable (e.g., a convex search) High actuator utilization Performance guarantees dependent on actuator size and disturbance estimate Approach flexible to incorporate different design approaches, actuator rate limits, state constraints, tracking, etc.

10. 10 Basic Idea 1: Combining with Scheduling Start with a nominal controller (from somewhere!) Keep it as long as possible Once saturated, switch to a new (family of) of controller (s) that can avoid saturation but can provide guaranteed stability and performance Make sure there are no `cracks' or escape routs! Assumptions: Full state or full order controllers (relaxed later) Disturbance attenuation problem (for now) Information of worst case disturbance (e.g. energy or peak) A small number of controllers (for now -- technical detail)

11. 11 System and Controllers Given Nominal Controller Output Feedback State Feedback or Open loop system Disturbance attenuation problem (ACC & CDC 07) Assumption: known w max (Possibly conservatively) Requirement: closed loop stability, boundedness (e.g., ISS), acceptable performance Key: Use of ellipsoids

12. 12 A simple `safe’ controller Objective: -Use K nom (s) as long as possible, -Once K nom (s) saturates, implement K safe (s) that ensure reasonable behavior Steps: - Analysis: What is the largest disturbance the system can tolerate? W nom - Synthesis Constructing the safe controller

13. 13 Analysis W max >W nom 2 x1x1 x2x2 Max β W nom =(1/β) 1/2

14. 14 Synthesis W max >W nom Nom 123 Safe 1 2 3

15. 15 Full State-Feedback Control (ACC 07) Synthesis (W max >W nom ) Key condition MIN gamma or δ F SAFE =XQ -1

16. 16 Safe Switch Condition Ensures Boundedness

17. 17 Scheduling Conservatism 1) 2) Elliptic invariant set is conservative

18. 18 Scheduling Scheduling: Putting Intermediate Controllers

19. 19 Full State-Feedback Control Scheduling W N =W L <W N-1 <…W 2 <W 1 =W max ; Q N =Q nom Min For i=1:N-1 K i =X i Q i -1 i=1,2,..N

20. 20 Output Feedback (CDC 07) WLOG Assume Fact: Switch Condition

21. 21 A Typical result

22. 22 Full State-Feedback Control Example W nom =2.76 Possible to be exposed to W max =15

23. 23 Full State-Feedback Control

24. 24 Full State-Feedback Control W 1 =W max =15; W 2 =10; W 3 =5; W 4 =W nom =2.76

25. 25 Full State-Feedback Control Switch history Sys. res. in scheduled case vs. the original sys. Res.

26. 26 Output Feedback Example Analysis: W nom =1.55 Synthesis: W max =5 Given nominal controller in the form

27. 27 Output Feedback Example (One Safe Controller) iWiWi Peak Gain Estimate 3 (nom)1.550.64 2 (safe)516.14

28. 28 Output Feedback Example (Scheduled Safe Controller) iWiWi Peak Gain Estimate 3 (nom)1.550.64 233.04 1525.29

29. 29 Output Feedback

30. 30 Future Work Continuous (e.g., spline based) family of controllers: messy but straight forward (will place a bound on how fast the gain can be increased) Mismatch in order of controller and plant: augment the order of the controller Tracking Non-ellipsoidal sets Adding scheduling to the traditional anti-windup scheme …….

31. 31 Going the other way around: Start with a basic Ant-windup set up Use Different anti-windups for different levels of saturation Shouldn’t small saturation leave to better performance guarantee than a sever saturation? (Ans: yes!) But first: Something interesting shows up!! Let us review the basic `Static’ anti-windup set up

32. 32 Static Anti-windup P(s) Sat(.) K(s) ur + - y AW q +- d

33. 33 Static Anti-windup Stability and Wellposedness: Small Gain Theorem

34. 34 Static Anti-windup Λ=XM -1 Q>0, M>0 Performance (stability): L2 Gain

35. 35 Example (Static Anti-windup) Grimm, G., Teel, A.R., and Zaccarian, L., “Results on Linear LMI-Based External Anti-windup Design”, IEEE Trans. on Automatic Control, Vol. 48, No. 9, Sep. 2003.

36. 36 Example (Static Anti-windup) System output and input history when anti-windup augmentation applied

37. 37 Over-saturated Anti-windup P(s) Sat(.) K(s) ur + - y d

38. 38 Over-saturated Anti-windup Q>0 Performance of saturated system for G(t)є [g,1]

39. 39 Over-saturated Anti-windup

40. 40 Over-saturated Anti-windup

41. 41 Over-saturated Anti-windup Q>0 Λ=XM -1 Performance (stability) of Over-saturated Anti-windup: L2 Gain

42. 42 Over-saturated Anti-windup System response: Anti-windup, Over-saturated Anti-Windup, Unconstrained Nominal Traditional Anti-windup: Over-saturated Anti-windup:

43. 43 Example (Over-saturated Anti-windup) input Elevator, limited to ±25 degree Flapron, limited to ±25 degree output Pitch angle Flight path angle Simulation example of F8 aircraft Kapasouris, P., Athans, M., and Stein, G., “Design of Feedback Control Systems for Stable Plants with Saturating Actuators”, Proceeding of the 27th IEEE Conf. on Decision and Control, Austin, TX, December 1988.

44. 44 Example (Over-saturated Anti-windup System response: Unconstrained Nominal, Anti-windup, Unconstrained Nominal

45. 45 Example (Over-saturated Anti-windup) System response: Anti-windup, Over-saturated Anti-Windup, Unconstrained Nominal

46. 46 Summary Tradeoff between `matched uncertainty’ vs better performance guarantee Dynamic Anti-windup case: Reasonably straight forward: the uncertainly is of the LPV (self-scheduled) variety – constant Lyapunov functions suffice Combine `over saturation’ and scheduling is next!