CONTROL with LIMITED INFORMATION ; SWITCHING ADAPTIVE CONTROL

1. CONTROL with LIMITED INFORMATION;SWITCHING ADAPTIVE CONTROL Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign IAAC Workshop, Herzliya, Israel, June 1, 2009

2. SWITCHING CONTROL y u Plant: P y u P Classical continuous feedback paradigm: C y u P But logical decisions are often necessary: C1 C2 l o g i c

3. REASONS for SWITCHING • Nature of the control problem • Sensor or actuator limitations • Large modeling uncertainty • Combinations of the above

4. REASONS for SWITCHING • Nature of the control problem • Sensor or actuator limitations • Large modeling uncertainty • Combinations of the above

5. INFORMATION FLOW in CONTROL SYSTEMS Plant Controller

6. INFORMATION FLOW in CONTROL SYSTEMS • Coarse sensing • Limited communication capacity • many control loops share network cable or wireless medium • microsystems with many sensors/actuators on one chip • Need to minimize information transmission (security) • Event-driven actuators

7. BACKGROUND Our goals: • Handle nonlinear dynamics Previous work: [Brockett,Delchamps,Elia,Mitter,Nair,Savkin,Tatikonda,Wong,…] • Deterministic & stochastic models • Tools from information theory • Mostly for linear plant dynamics • Unified framework for • quantization • time delays • disturbances

8. OUR APPROACH • Model these effects via deterministic error signals, • Design a control law ignoring these errors, • “Certainty equivalence”: apply control, • combined with estimation to reduce to zero Technical tools: • Input-to-state stability (ISS) • Lyapunov functions • Small-gain theorems • Hybrid systems (Goal: treat nonlinear systems; handle quantization, delays, etc.) Caveat: This doesn’t work in general, need robustness from controller

9. QUANTIZATION finite subset of is partitioned into quantization regions QUANTIZER Encoder Decoder

10. QUANTIZATION and ISS

11. QUANTIZATION and ISS – assume glob. asymp. stable (GAS)

12. QUANTIZATION and ISS no longer GAS

13. QUANTIZATION and ISS quantization error Assume class

14. QUANTIZATION and ISS quantization error Assume Solutions that start in enter and remain there This is input-to-state stability (ISS) w.r.t. measurement errors In time domain:

15. LINEAR SYSTEMS 9 feedback gain & Lyapunov function Quantized control law: (automatically ISS w.r.t. ) Closed-loop:

16. DYNAMIC QUANTIZATION

17. DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state

18. DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state

19. DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state Zoom out to overcome saturation

20. DYNAMIC QUANTIZATION – zooming variable Hybrid quantized control: is discrete state Proof: ISS from to small-gain condition ISS from to After ultimate bound is achieved, recompute partition for smaller region Can recover global asymptotic stability

21. QUANTIZATION and DELAY QUANTIZER DELAY Architecture-independent approach Delays possibly large Based on the work of Teel

22. QUANTIZATION and DELAY where Can write hence

23. SMALL–GAIN ARGUMENT Assuming ISS w.r.t. actuator errors: In time domain: Small gain: if [Teel ’98] then we recover ISS w.r.t.

24. FINAL RESULT small gain true Need:

25. FINAL RESULT Need: small gain true

26. FINAL RESULT solutions starting in enter and remain there Need: small gain true Can use “zooming” to improve convergence

27. EXTERNAL DISTURBANCES [Nešić–L] State quantization and completelyunknown disturbance

28. EXTERNAL DISTURBANCES [Nešić–L] State quantization and completelyunknown disturbance

29. EXTERNAL DISTURBANCES [Nešić–L] State quantization and completelyunknown disturbance After zoom-in: Issue: disturbance forces the state outside quantizer range Must switch repeatedly between zooming-in and zooming-out Result: for linear plant, can achieve ISS w.r.t. disturbance (ISS gains are nonlinear although plant is linear; cf.[Martins])

30. HYBRID SYSTEMS as FEEDBACK CONNECTIONS continuous discrete • Other decompositions possible • Can also have external signals [Nešić–L, ’05, ’06]

31. SMALL–GAIN THEOREM • Input-to-state stability (ISS) from to : • ISS from to : (small-gain condition) Small-gain theorem [Jiang-Teel-Praly ’94] gives GAS if:

32. SUFFICIENT CONDITIONS for ISS • ISS from to if ISS-Lyapunov function [Sontag ’89]: • ISS from to if: and # of discrete events on is [Hespanha-L-Teel ’08]

33. LYAPUNOV– BASED SMALL–GAIN THEOREM and # of discrete events on is Hybrid system is GAS if:

34. SKETCH of PROOF is nonstrictly decreasing along trajectories Trajectories along which is constant? None! GAS follows by LaSalle principle for hybrid systems [Lygeros et al. ’03, Sanfelice-Goebel-Teel ’07]

35. APPLICATION to DYNAMIC QUANTIZATION quantization error ISS from to with some linear gain Zoom in: where ISS from to with gain small-gain condition!

36. RESEARCH DIRECTIONS http://decision.csl.uiuc.edu/~liberzon • Quantized output feedback • Performance-based design • Disturbances and coarse quantizers (with Y. Sharon) • Modeling uncertainty (with L. Vu) • Avoiding state estimation (with S. LaValle and J. Yu) • Vision-based control (with Y. Ma and Y. Sharon)

37. REASONS for SWITCHING • Nature of the control problem • Sensor or actuator limitations • Large modeling uncertainty • Combinations of the above

38. unmodeled dynamics parametric uncertainty Also, noise and disturbance MODELING UNCERTAINTY Adaptive control (continuous tuning) vs. supervisory control (switching)

39. EXAMPLE stable not implementable Controller family: Could also take controller index set Scalar system: , otherwise unknown (purely parametric uncertainty)

40. Controller – switching signal,takes values in – switching controller SUPERVISORY CONTROL ARCHITECTURE Supervisor candidate controllers u1 Controller y Plant u u2 Controller . . . um . . .

41. Prescheduled (prerouted) • Performance-based (direct) • Estimator-based (indirect) TYPES of SUPERVISION

42. Prescheduled (prerouted) • Performance-based (direct) • Estimator-based (indirect) TYPES of SUPERVISION

43. OUTLINE • Basic components of supervisor • Design objectives and general analysis • Achieving the design objectives (highlights)

44. OUTLINE • Basic components of supervisor • Design objectives and general analysis • Achieving the design objectives (highlights)

45. y1 e1 y e2 y2 Multi- Estimator estimation errors: . . . ep yp u . . . . . . Want to be small Then small indicates likely SUPERVISOR

46. Multi-estimator: => exp fast EXAMPLE

47. Multi-estimator: => exp fast EXAMPLE disturbance

48. STATE SHARING Bad! Not implementable if is infinite The system produces the same signals

49. y1 e1 y Monitoring Signals Generator e2 y2 Multi- Estimator . . . ep yp u . . . . . . . . . Examples: SUPERVISOR

50. EXAMPLE Multi-estimator: – can use state sharing