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 Classical continuous feedback paradigm: u y P C u y P Plant: But logical decisions are often necessary: u y C1C1 C2C2 l o g i c P

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. Plant Controller INFORMATION FLOW in CONTROL SYSTEMS

6. 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 Coarse sensing

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

8. Caveat: This doesn’t work in general, need robustness from controller OUR APPROACH (Goal: treat nonlinear systems; handle quantization, delays, etc.) 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

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

10. QUANTIZATION and ISS

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

12. no longer GAS

13. QUANTIZATION and ISS quantization error Assume class

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

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

16. DYNAMIC QUANTIZATION

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

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

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

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

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

22. QUANTIZATION and DELAY where hence Can write

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

24. FINAL RESULT Need: small gain true

25. FINAL RESULT Need: small gain true

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

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

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

29. 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 ]) EXTERNAL DISTURBANCES [ Nešić–L ] State quantization and completely unknown disturbance After zoom-in:

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

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

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

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

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

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

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

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

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

40. SUPERVISORY CONTROL ARCHITECTURE Plant Supervisor Controller u1u1 u2u2 umum y u...... candidate controllers...... – switching controller – switching signal, takes values in

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

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

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. SUPERVISOR Multi- Estimator y1y1 y2y2 u y............ ypyp estimation errors: epep e2e2 e1e1...... Want to be small Then small indicates likely

46. EXAMPLE Multi-estimator: exp fast =>

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

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

49. SUPERVISOR...... Multi- Estimator y1y1 y2y2 ypyp epep e2e2 e1e1 u y............ Monitoring Signals Generator...... Examples:

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

51. SUPERVISOR Switching Logic...... Multi- Estimator y1y1 y2y2 ypyp epep e2e2 e1e1 u y............ Monitoring Signals Generator...... Basic idea: Justification? small => small => plant likely in => gives stable closed-loop system (“certainty equivalence”) Plant, controllers:

52. SUPERVISOR Switching Logic...... Multi- Estimator y1y1 y2y2 ypyp epep e2e2 e1e1 u y............ Monitoring Signals Generator...... Basic idea: Justification? small => small => plant likely in => gives stable closed-loop system only know converse! Need:small => gives stable closed-loop system This is detectability w.r.t. Plant, controllers:

53. DETECTABILITY Want this system to be detectable “output injection” matrix asympt. stable view as output is Hurwitz Linear case: plant in closed loop with

54. SUPERVISOR Switching Logic...... Multi- Estimator y1y1 y2y2 ypyp epep e2e2 e1e1 u y............ Monitoring Signals Generator...... Switching logic (roughly): This (hopefully) guarantees that is small Need:small => stable closed-loop switched system We know: is small This is switched detectability

55. DETECTABILITY under SWITCHING need this to be asympt. stable plant in closed loop with view as output Assumed detectable for each frozen value of Output injection: slow switching (on the average) switching stops in finite time Thus needs to be “non-destabilizing” : Switched system: Want this system to be detectable:

56. SUMMARY of BASIC PROPERTIES Multi-estimator: 1. At least one estimation error ( ) is small Candidate controllers: 3. is bounded in terms of the smallest : for 3, want to switch to for 4, want to switch slowly or stop conflicting when is bounded for bounded & Switching logic: 2. For each, closed-loop system is detectable w.r.t. 4. Switched closed-loop system is detectable w.r.t. provided this is true for every frozen value of

57. SUMMARY of BASIC PROPERTIES Analysis: 1 + 3 => is small 2 + 4 => detectability w.r.t. => state is small Switching logic: Multi-estimator: Candidate controllers: 3. is bounded in terms of the smallest 4. Switched closed-loop system is detectable w.r.t. provided this is true for every frozen value of 2. For each, closed-loop system is detectable w.r.t. 1. At least one estimation error ( ) is small when is bounded for bounded &

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

59. Controller Plant Multi- estimator fixed CANDIDATE CONTROLLERS

60. Linear: overall system is detectable w.r.t. if i.system inside the box is stable ii.plant is detectable fixed Plant Multi- estimator Controller P C E Need to show: => P C E,, => E C, i P ii =>

61. CANDIDATE CONTROLLERS Linear: overall system is detectable w.r.t. if i.system inside the box is stable ii.plant is detectable fixed Plant Multi- estimator Controller P C E Nonlinear: same result holds if stability and detectability are interpreted in the ISS / OSS sense: external signal

62. CANDIDATE CONTROLLERS Linear: overall system is detectable w.r.t. if i.system inside the box is stable ii.plant is detectable fixed Plant Multi- estimator Controller P C E Nonlinear: same result holds if stability and detectability are interpreted in the integral-ISS / OSS sense:

63. SWITCHING LOGIC: DWELL-TIME Obtaining a bound on in terms of is harder Not suitable for nonlinear systems (finite escape) Initialize Find no ? yes – monitoring signals – dwell time Wait time units Detectability is preserved if is large enough

64. SWITCHING LOGIC: HYSTERESIS Initialize Find no yes – monitoring signals – hysteresis constant ? or (scale-independent)

65. SWITCHING LOGIC: HYSTERESIS This applies to exp fast, finite, bounded switching stops in finite time => Initialize Find no yes ? Linear, bounded average dwell time =>