SWITCHING ADAPTIVE CONTROL

1. SWITCHING ADAPTIVE CONTROL Daniel Liberzon Coordinated Science Laboratory and Dept. of Electrical & Computer Eng., Univ. of Illinois at Urbana-Champaign

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

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

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

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

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

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

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

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

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

11. Multi-estimator: => exp fast EXAMPLE

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

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

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

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

16. SUPERVISOR y1 e1 y Monitoring Signals Generator e2 y2 Multi- Estimator . . . Switching Logic ep yp u Basic idea: . . . . . . . . . , controllers: Plant gives stable closed-loop system => => => plant likely in small small Justification? (“certainty equivalence”)

17. SUPERVISOR y1 e1 y Monitoring Signals Generator e2 y2 Multi- Estimator . . . Switching Logic ep yp u . . . . . . . . . only know converse! => Need: small gives stable closed-loop system This is detectability w.r.t. Basic idea: , controllers: Justification? Plant gives stable closed-loop system => => => plant likely in small small

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

19. SUPERVISOR y1 e1 y Monitoring Signals Generator e2 y2 Multi- Estimator . . . Switching Logic ep yp u . . . . . . . . . We know: is small Switching logic (roughly): => Need: small stable closed-loop switched system This (hopefully) guarantees that is small This is switched detectability

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

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

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

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

24. CANDIDATE CONTROLLERS fixed Plant Controller Multi- estimator

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

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

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

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

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

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

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

32. 2 PARKING PROBLEM p 1 p 1 p Unknown parameters correspond to the radius of rear wheels and distance between them