1. Lecture #16 Nonlinear Supervisory Control João P. Hespanha University of California at Santa Barbara Hybrid Control and Switched Systems

2. Summary Estimator-based nonlinear supervisory control Examples

3. Supervisory control supervisor process  controller 1 controller n y u w  Motivation: in the control of complex and highly uncertain systems, traditional methodologies based on a single controller do not provide satisfactory performance. bank of candidate controllers measured output control signal exogenous disturbance/ noise switching signal Key ideas: 1.Build a bank of alternative controllers 2.Switch among them online based on measurements For simplicity we assume a stabilization problem, otherwise controllers should have a reference input r

4. Supervisory control Supervisor: places in the feedback loop the controller that seems more promising based on the available measurements typically logic-based/hybrid system supervisor process  controller 1 controller n y u w  Motivation: in the control of complex and highly uncertain systems, traditional methodologies based on a single controller do not provide satisfactory performance. measured output control signal exogenous disturbance/ noise switching signal bank of candidate controllers

5. Estimator-based nonlinear supervisory control process multi- controller multi- estimator decision logic u measured output control signal y  switching signal w NON LINEAR

6. Class of admissible processes: Example #4 process u control signal y measured output state accessible p=(a, b) 2  › [–1,1] £ {–1,1} (a *, b * ) ´ unknown parameters Process is assumed to be in the family

7. Class of admissible processes process u control signal y measured output w exogenous disturbance/ noise Typically Process is assumed to be in a family parametric uncertainty unmodeled dynamics  p ´ small family of systems around a nominal process model N p metric on set of state-space model (?) Most results presented here: independent of metric d (e.g., detectability) or restricted to case  p = 0 (e.g., matching)

8. now the control law stabilizes the system Candidate controllers: Example #4 To facilitate the controller design, one can first “back-step” the system to simplify its stabilization: after the coordinate transformation the new state is no longer accessible p=(a, b) 2  › [–1,1] £ {–1,1} Process is assumed to be in the family virtual input state accessible Candidate controllers

9. Class of admissible processes Assume given a family of candidate controllers  p ´ small family of systems around a nominal process model N p u  measured output control signal switching signal y Multi-controller: (without loss of generality all with same dimension)

10. Multi-estimator multi- estimator u measured output control signal y – + – + we want:Matching property: there exist some p* 2  such that e p* is “small” Typically obtained by: How to design a multi-estimator? process in 9 p * 2  : process in  p* e p* is “small” when process “matches”  p* the corresponding error must be “small”

11. Candidate controllers: Example #4 p=(a, b) 2  › [–1,1] £ {–1,1} Process is assumed to be state not accessible Multi-estimator: for p = p * … ))  for p = p * we can do state-sharing to generate all the errors with a finite-dimensional system exponentially Matching property

12. Designing multi-estimators - I Suppose nominal models N p, p 2  are of the form no exogenous input w state accessible Multi-estimator: asymptotically stable A When process matches the nominal model N p*  exponentially Matching property: Assume  = { N p : p 2  } 9 p* 2 , c 0, * >0 :|| e p* (t) || · c 0 e - * t t ¸ 0 (state accessible)

13. Designing multi-estimators - II Suppose nominal models N p, p 2  are of the form Multi-estimator: When process matches the nominal model N p*  Matching property: Assume  = { N p : p 2  } 9 p* 2 , c 0, c w, * >0 :|| e p* (t) || · c 0 e - * t + c w t ¸ 0 with c w = 0 in case w(t) = 0, 8 t ¸ 0 (output-injection away from stable linear system) asymptotically stable A p nonlinear output injection (generalization of case I) State-sharing is possible when all A p are equal an H p ( y, u ) is separable:

14. Designing multi-estimators - III Suppose nominal models N p, p 2  are of the form Matching property: Assume  = { N p : p 2  } 9 p* 2 , c 0, c w, * >0 :|| e p* (t) || · c 0 e - * t + c w t ¸ 0 with c w = 0 in case w(t) = 0, 8 t ¸ 0 (output-inj. and coord. transf. away from stable linear system) asymptotically stable A p (generalization of case I & II) The Matching property is an input/output property so the same multi-estimator can be used:  p   p ‘ ±  p -1 ´ cont. diff. coordinate transformation with continuous inverse  p -1 (may depend on unknown parameter p)

15. Switched system process multi- controller multi- estimator decision logic u y  switched system w detectability property? The switched system can be seen as the interconnection of the process with the “injected system” essentially the multi-controller & multi-estimator but now quite…

16. Constructing the injected system 1stTake a parameter estimate signal  : [0, 1 ) ! . 2ndDefine the signal v  e  = y   y 3rdReplace y in the equations of the multi-estimator and multi-controller by y   v. multi- controller multi- estimator u – yy v y 

17. Switched system = process + injected system injected system u v – + – + y process w   Q:How to get “detectability” on the switched system ? A:“Stability” of the injected system

18. Stability & detectability of nonlinear systems Notation:  :[0, 1 ) ! [0, 1 ) is class  ´ continuous, strictly increasing,  (0) = 0 is class  1 ´ class  and unbounded  :[0, 1 ) £ [0, 1 ) ! [0, 1 ) is class  ´  ( ¢,t) 2  for fixed t & lim t !1  (s,t) = 0 (monotonically) for fixed s Stability:input u “small”  state x “small” Input-to-state stable (ISS) if 9  2 ,  2  Integral input-to-state stable (iISS) if 9  2  1,  2 ,  2  strictly weaker

19. Stability & detectability of nonlinear systems Stability:input u “small”  state x “small” Input-to-state stable (ISS) if 9  2 ,  2  Integral input-to-state stable (iISS) if 9  2  1,  2 ,  2  strictly weaker One can show: 1.for ISS systems:u ! 0 ) solution exist globally & x ! 0 2.for iISS systems: s 0 1  (||u||) < 1) solution exist globally & x ! 0

20. Stability & detectability of nonlinear systems Detectability:input u & output y “small”  state x “small” Detectability (or input/output-to-state stability IOSS) if 9  2 ,  u  y 2  Integral detectable (iIOSS) if 9  2  1,  2 ,  u  y 2  strictly weaker One can show: 1.for IOSS systems:u, y ! 0 ) x ! 0 2.for iIOSS systems: s 0 1  u (||u||), s 0 1  y (||y||) < 1) x ! 0

21. Certainty Equivalence Stabilization Theorem Theorem: (Certainty Equivalence Stabilization Theorem) Suppose the process is detectable and take fixed  = p 2  and  = q 2  1.injected system ISS  switched system detectable. 2.injected system integral ISS  switched system integral detectable Stability of the injected system is not the only mechanism to achieve detectability: e.g., injected system i/o stable + process “min. phase” ) detectability of switched system (Nonlinear Certainty Equivalence Output Stabilization Theorem) injected system u v – + – + y process   switched system

22. Achieving ISS for the injected system We want to design candidate controllers that make the injected system (at least) integral ISS with respect to the “disturbance” input v Theorem: (Certainty Equivalence Stabilization Theorem) Suppose the process is detectable and take fixed  = p 2  and  = q 2  1.injected system ISS  switched system detectable. 2.injected system integral ISS  switched system integral detectable Nonlinear robust control design problem, but… “disturbance” input v can be measured (v = e  = y  – y) the whole state of the injected system is measurable (x C, x E ) injected system multi- controller multi- estimator u – yy v y   = p 2   = q 2 

23. Designing candidate controllers: Example #4 p=(a, b) 2  › [–1,1] £ {–1,1} Multi-estimator: To obtain the injected system, we use epep Candidate controller q =  ( p ) : injected system is exponentially stable linear system (ISS) Detectablity property

24. Decision logic decision logic  switching signal switched system estimation errors injected system process For nonlinear systems dwell-time logics do not work because of finite escape

25. Scale-independent hysteresis switching monitoring signals p 2 p 2  measure of the size of e p over a “window” of length 1/ forgetting factor start wait until current monitoring signal becomes significantly larger than some other one n y hysteresis constant class  function from detectability property All the  p can be generated by a system with small dimension if  p (||e p ||) is separable. i.e.,

26. Scale-independent hysteresis switching number of switchings in [ , t ) Theorem: Let  be finite with m elements. For every p 2  maximum interval of existence of solution  uniformly bounded on [0, T max )  Assume  is finite, the  p are locally Lipschitz and and FIX not bounded?!

27. Scale-independent hysteresis switching number of switchings in [ , t ) Theorem: Let  be finite with m elements. For every p 2  and Assume  is finite, the  p are locally Lipschitz and maximum interval of existence of solution Non-destabilizing property: Switching will stop at some finite time T * 2 [0, T max ) Small error property:

28. Analysis 1stby the Matching property: 9 p * 2   such that || e p* (t) || · c 0 e - * t t ¸ 0 2ndby the Non-destabilization property: switching stops at a finite time T * 2 [0, T max ) )  (t) = p &  (t) =  (p) 8 t 2 [T *,T max ) 3rdby the Small error property: (w = 0, no unmodeled dynamics) Theorem: Assume that  is finite and all the  p are locally Lipschitz. The state of the process, multi-estimator, multi-controller, and all other signals converge to zero as t ! 1.  solution exists globally T max = 1 & x ! 0 as t ! 1 the state x of the switched system is bounded on [T *,T max ) 4thby the Detectability property:

29. Outline Supervisory control overview Estimator-based linear supervisory control Estimator-based nonlinear supervisory control  Examples

30. now the control law stabilizes the system Example #4: System in strict-feedback form Suppose nominal models N p, p 2  are of the form state accessible To facilitate the controller design, one can first “back-step” the system to simplify its stabilization:

31. Example #4: System in strict-feedback form Suppose nominal models N p, p 2  are of the form Multi-estimator: When process matches the nominal model N p* it is separable so we can do state-sharing exponentially Matching property Candidate controller q =  (p): makes injected system ISS  Detectability property ))

32. Example #4: System in strict-feedback form a b u   reference

33. Example #4: System in strict-feedback form Suppose nominal models N p, p 2  are of the form state accessible In the previous back-stepping procedure: the controller drives  ! 0  nonlinearity is cancelled (even when a < 0 and it introduces damping) One could instead make still leads to exponential decrease of  (without canceling nonlinearity when a < 0 ) pointwise min-norm design

34. Example #4: System in strict-feedback form Suppose nominal models N p, p 2  are of the form state accessible A different recursive procedure:  pointwise min-norm recursive design  ! 0  ! 0   exponentially In this case

35. Example #4: System in strict-feedback form Suppose nominal models N p, p 2  are of the form Multi-estimator ( option III ): When process matches the nominal model N p*  exponentially  Matching property Candidate controller q =  (p):  Detectability property

36. Example #4: System in strict-feedback form a b u   reference

37. Example #4: System in strict-feedback form a b u  a b u  pointwise min-norm designfeedback linearization design

38. Example #5: Unstable-zero dynamics (stabilization) estimate output y unknown parameter

39. Example #5: Unstable-zero dynamics (stabilization with noise) estimate unknown parameter output y

40. Example #5: Unstable-zero dynamics (set-point with noise) output y reference unknown parameter

41. Example #6: Kinematic unicycle robot x2x2 x1x1  u 1 ´ forward velocity u 2 ´ angular velocity p 1, p 2 ´ unknown parameters determined by the radius of the driving wheel and the distance between them This system cannot be stabilized by a continuous time-invariant controller. The candidate controllers were themselves hybrid

42. Example #6: Kinematic unicycle robot

43. Example #7: Induction motor in current-fed mode 2  2 ´ rotor flux u 2  2 ´ stator currents  ´ rotor angular velocity  ´ torque generated Unknown parameters:  L 2 [  min,  max ] ´ load torque R 2 [R min, R max ] ´ rotor resistance “Off-the-shelf” field-oriented candidate controllers:  is the only measurable output