Lecture #16 Nonlinear Supervisory Control João P. Hespanha University of California at Santa Barbara Hybrid Control and Switched Systems
Summary Estimator-based nonlinear supervisory control Examples
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
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
Estimator-based nonlinear supervisory control process multi- controller multi- estimator decision logic u measured output control signal y switching signal w NON LINEAR
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
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)
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
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)
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”
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
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)
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:
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)
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…
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 – yy v y
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
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
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
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
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
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 – yy v y = p 2 = q 2
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
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
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.,
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?!
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:
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:
Outline Supervisory control overview Estimator-based linear supervisory control Estimator-based nonlinear supervisory control Examples
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:
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 ))
Example #4: System in strict-feedback form a b u reference
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
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
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
Example #4: System in strict-feedback form a b u reference
Example #4: System in strict-feedback form a b u a b u pointwise min-norm designfeedback linearization design
Example #5: Unstable-zero dynamics (stabilization) estimate output y unknown parameter
Example #5: Unstable-zero dynamics (stabilization with noise) estimate unknown parameter output y
Example #5: Unstable-zero dynamics (set-point with noise) output y reference unknown parameter
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
Example #6: Kinematic unicycle robot
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