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Discover the concept of estimator-based nonlinear supervisory control in complex systems. Learn to build a bank of controllers and switch them online based on measurements for improved performance. Key ideas and examples included.
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Hybrid Control and Switched Systems Lecture #16Nonlinear Supervisory Control João P. Hespanha University of Californiaat Santa Barbara
Summary • Estimator-based nonlinear supervisory control • Examples
Supervisory control Motivation: in the control of complex and highly uncertain systems, traditional methodologies based on a single controller do not provide satisfactory performance. supervisor exogenous disturbance/ noise s switching signal measured output w controller 1 s y bank of candidate controllers process u controller n control signal • Key ideas: • Build a bank of alternative controllers • Switch among them online based on measurements For simplicity we assume a stabilization problem, otherwise controllers should have a reference input r
Supervisory control Motivation: in the control of complex and highly uncertain systems, traditional methodologies based on a single controller do not provide satisfactory performance. supervisor exogenous disturbance/ noise s switching signal measured output w controller 1 s y bank of candidate controllers process u controller n control signal • Supervisor: • places in the feedback loop the controller that seems more promising based on the available measurements • typicallylogic-based/hybrid system
Estimator-based nonlinear supervisory control multi-estimator decision logic NON LINEAR switching signal w s multi- controller u process y measured output control signal
Class of admissible processes: Example #4 control signal process u y measured output state accessible (a*, b*) ´unknown parameters Process is assumed to be in the family p=(a, b) 2P [–1,1] £ {–1,1}
Class of admissible processes w exogenous disturbance/noise control signal process u y measured output unmodeled dynamics Process is assumed to be in a family Mp´ small family of systems around a nominal process model Np parametric uncertainty Typically metric on set of state-space model (?) • Most results presented here: • independent of metric d (e.g., detectability) • or restricted to case ep = 0 (e.g., matching)
Candidate controllers: Example #4 Process is assumed to be in the family p=(a, b) 2P [–1,1] £ {–1,1} state accessible To facilitate the controller design, one can first “back-step” the system to simplify its stabilization: virtual input now the control law stabilizes the system after the coordinate transformation the new state is no longer accessible Candidate controllers
Candidate controllers Class of admissible processes Mp´ small family of systems around a nominal process model Np Assume given a family of candidate controllers (without loss of generality all with same dimension) s Multi-controller: switching signal y u measured output control signal
Multi-estimator process in measured output – y multi-estimator + – u control signal + How to design a multi-estimator? we want:Matching property: there exist some p*2 P such that ep* is “small” Typically obtained by: 9p*2P: processin Mp* ep* is“small” when process “matches” Mp*the corresponding error must be “small”
Candidate controllers: Example #4 Process is assumed to be state not accessible Multi-estimator: p=(a, b) 2P [–1,1] £ {–1,1} we can do state-sharing to generate all the errors with a finite-dimensional system forp = p* … exponentially ) ) Matching property g for p = p*
Designing multi-estimators - I (state accessible) Suppose nominal models Np, p2P are of the form no exogenous input w state accessible Multi-estimator: asymptotically stable A When process matches the nominal model Np* exponentially Matching property: Assume M = { Np : p2P} 9p*2 P, c0, l* >0 : || ep*(t) || ·c0 e-l* tt¸ 0
Designing multi-estimators - II (output-injection away from stable linear system) Suppose nominal models Np, p2P are of the form asymptotically stable Ap nonlinear output injection (generalization of case I) Multi-estimator: When process matches the nominal model Np* Matching property: Assume M = { Np : p2P} 9p*2 P, c0, cw, l* >0 : || ep*(t) || ·c0 e-l* t + cwt¸ 0 with cw = 0 in case w(t) = 0, 8t¸ 0 State-sharing is possible when all Ap are equal an Hp( y, u ) is separable:
Designing multi-estimators - III (output-inj. and coord. transf. away from stable linear system) Suppose nominal models Np, p2P are of the form asymptotically stable Ap (generalization of case I & II) ´ cont. diff. coordinate transformation with continuous inverse xp-1 (may depend on unknown parameter p) zp xp‘ ± xp-1 The Matching property is an input/output property so the same multi-estimator can be used: Matching property: Assume M = { Np : p2P} 9p*2 P, c0, cw, l* >0 : || ep*(t) || ·c0 e-l* t + cwt¸ 0 with cw = 0 in case w(t) = 0, 8t¸ 0
Switched system detectability property? multi-estimator decision logic w s multi- controller u process y switched system The switched system can be seen as theinterconnection of the process with the “injected system” essentially the multi-controller & multi-estimator but now quite…
Constructing the injected system 1st Take a parameter estimate signal r : [0,1)!P. 2nd Define the signalver = yr y 3rd Replace y in the equations of the multi-estimator and multi-controller byyr v. s v u yr – multi- controller multi-estimator y
Switched system = process + injected system w Q: How to get “detectability” on the switched system ? A: “Stability” of the injected system y process u – r injectedsystem v + – + r s
Stability & detectability of nonlinear systems Stability: input u “small”statex“small” Input-to-state stable (ISS) if 9b2KL, g2K Integral input-to-state stable (iISS) if 9a2K1, b2KL, g2K strictly weaker Notation: a:[0,1) ! [0,1) is class K´ continuous, strictly increasing, a(0) = 0 is class K1´ class K and unbounded b:[0,1)£[0,1) ! [0,1) is class KL´b(¢,t) 2K for fixed t &limt!1b(s,t) = 0 (monotonically) for fixed s
Stability & detectability of nonlinear systems Stability: input u “small”statex“small” Input-to-state stable (ISS) if 9b2KL, g2K Integral input-to-state stable (iISS) if 9a2K1, b2KL, g2K strictly weaker • One can show: • for ISS systems: u! 0) solution exist globally & x! 0 • for iISS systems: s01g(||u||) < 1) solution exist globally & x! 0
Stability & detectability of nonlinear systems Detectability: inputu & output y“small”statex“small” Detectability (or input/output-to-state stability IOSS) if 9b2KL, gu, gy2K strictly weaker Integral detectable (iIOSS) if 9a2K1, b2KL, gu, gy2K • One can show: • for IOSS systems: u, y! 0)x! 0 • for iIOSS systems: s01gu(||u||), s01gy(||y||) < 1)x! 0
Certainty Equivalence Stabilization Theorem y process switched system u – r injectedsystem + v – + r s Theorem: (Certainty Equivalence Stabilization Theorem) Suppose the process is detectable and take fixed r = p2 P and s = q2 Q 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)
Achieving ISS for the injected system Theorem: (Certainty Equivalence Stabilization Theorem) Suppose the process is detectable and take fixed r = p2 P and s = q2 Q 1. injected system ISS switched system detectable. 2. injected system integral ISS switched system integral detectable s injected system v r = p2 Ps = q2 Q u yr – multi- controller multi-estimator y We want to design candidate controllers that make the injected system (at least) integral ISS with respect to the “disturbance” input v • Nonlinear robust control design problem, but… • “disturbance” input v can be measured (v = er = yr – y) • the whole state of the injected system is measurable (xC, xE)
Designing candidate controllers: Example #4 Multi-estimator: p=(a, b) 2P [–1,1] £ {–1,1} ep To obtain the injected system, we use Candidate controllerq = c(p ): Detectablity property injected system is exponentially stable linear system (ISS)
Decision logic process injectedsystem For nonlinear systems dwell-time logics do not work because of finite escape decision logic switching signal s switchedsystem estimation errors
Scale-independent hysteresis switching class K function from detectability property start monitoring signals p2P measure of the size of ep over a “window” of length 1/l hysteresis constant forgetting factor y n All themp can be generated by a system with small dimension if gp(||ep||) is separable. i.e., wait until current monitoring signal becomes significantly larger than some other one
Scale-independent hysteresis switching Theorem: Let P be finite with m elements. For every p2P FIX not bounded?! number of switchings in [t, t ) and Assume P is finite, the gp are locally Lipschitz and maximum interval of existence of solution uniformly bounded on [0, Tmax) uniformly bounded on [0, Tmax)
Scale-independent hysteresis switching Theorem: Let P be finite with m elements. For every p2P number of switchings in [t, t ) and Assume P is finite, the gp are locally Lipschitz and maximum interval of existence of solution Non-destabilizing property: Switching will stop at some finite time T* 2[0, Tmax) Small error property:
Analysis (w = 0, no unmodeled dynamics) 1st by the Matching property: 9p*2Psuch that || ep*(t) || ·c0 e-l* tt¸ 0 2nd by the Non-destabilization property: switching stops at a finite timeT* 2[0, Tmax))r(t) = p & s(t) = c(p) 8t 2 [T*,Tmax) 3rd by the Small error property: 4th by the Detectability property: the state xof the switched system is bounded on [T*,Tmax) solution exists globally Tmax = 1 & x! 0 as t!1 Theorem: Assume that P is finite and all the gp are locally Lipschitz. The state of the process, multi-estimator, multi-controller, and all other signals converge to zero as t! 1.
Outline • Supervisory control overview • Estimator-based linear supervisory control • Estimator-based nonlinear supervisory control • Examples
Example #4: System in strict-feedback form Suppose nominal models Np, p2P are of the form state accessible To facilitate the controller design, one can first “back-step” the system to simplify its stabilization: now the control law stabilizes the system
Example #4: System in strict-feedback form Suppose nominal models Np, p2P are of the form Multi-estimator: it is separable so we can do state-sharing When process matches the nominal model Np* exponentially ) ) Matching property Candidate controller q = c(p): makes injected system ISS Detectability property
Example #4: System in strict-feedback form b a u areference a
Example #4: System in strict-feedback form Suppose nominal models Np, p2P are of the form state accessible In the previous back-stepping procedure: the controller nonlinearity is cancelled (even when a< 0 andit introduces damping) drivesg! 0 One could instead make still leads to exponential decrease of a (without canceling nonlinearity when a< 0 ) pointwise min-norm design
Example #4: System in strict-feedback form Suppose nominal models Np, p2P are of the form state accessible A different recursive procedure: In this case exponentially g! 0 pointwise min-norm recursive design a! 0 exponentially
Example #4: System in strict-feedback form Suppose nominal models Np, p2P are of the form Multi-estimator ( option III ): When process matches the nominal model Np* exponentially Matching property Candidate controller q = c(p): Detectability property
Example #4: System in strict-feedback form b a u areference a
Example #4: System in strict-feedback form b b a a u u a a pointwise min-norm design feedback linearization design
Example #5: Unstable-zero dynamics unknown parameter estimate output y (stabilization)
Example #5: Unstable-zero dynamics unknown parameter estimate output y (stabilization with noise)
Example #5: Unstable-zero dynamics unknown parameter reference output y (set-point with noise)
Example #6: Kinematic unicycle robot x2 q u1´ forward velocity u2´ angular velocity x1 p1, p2´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 #7: Induction motor in current-fed mode l2Ñ2´ rotor flux u2Ñ2´ stator currents w´rotor angular velocity t´ torque generated wis the only measurable output Unknown parameters: tL2 [tmin, tmax] ´load torque R2 [Rmin, Rmax] ´rotor resistance “Off-the-shelf” field-oriented candidate controllers:
