Tutorial on Logicbased Control Switched Supervisory Control Joo

Tutorial on Logic-based Control Switched Supervisory Control João P. Hespanha University of California at Santa Barbara

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

Multi-controller s controller 1 y measured output bank of candidate controllers controller n switching signal s u control signal Conceptual diagram: not efficient for many controllers & not possible for unstable controllers

Multi-controller s switching signal s controller 1 bank of candidate controllers y measured output u control signal controller n Given a family of (n-dimensional) candidate controllers s y measured output switching signal u control signal

Multi-controller switching times s(t) switching signal s=1 s=3 s=2 s=1 t Given a family of (n-dimensional) candidate controllers s y measured output switching signal u control signal

Supervisor s y measured output u control signal switching signal y u Typically an hybrid system: ´ continuous state d ´ discrete state continuous vector field discrete transition function output function

Types of supervision Pre-routed supervision s=2 s=1 s=3 • try one controllers after another in a pre-defined sequence • stop when the performance seems acceptable not effective when the number of controllers is large Performance-based supervision (direct) • keep controller while observed performance is acceptable • when performance of current controller becomes unacceptable, switch to controller that leads to best expected performance based on available data Estimator-based supervision (indirect) • estimate process model from observed data • select controller based on current estimate – Certainty Equivalence

Summary • • Supervisory control overview Estimator-based linear supervisory control Estimator-based nonlinear supervisory control Examples

Estimator-based supervision’s setup: Example #1 process control signal u Process is assumed to be y measured output unknown parameters p*=(a*, b*) 2 P [– 1, 1] £ {– 1, 1} we consider three candidate controllers: controller C 1: u = 0 to be used when a*· –. 1 controller C 2: u = 1. 1 y to be used when a* > –. 1 & b* = – 1 controller C 3: u = – 1. 1 y to be used when a* > –. 1 & b* = +1 process parameter p* is equal to p (a, b) 2 P use controller Cq with controller selection function

Estimator-based supervision’s setup w control signal u process Process is assumed to be in a family parametric uncertainty exogenous disturbance/n oise y measured output unmodeled dynamics Mp ´ small family of systems around a “nominal” process model Np for each process in a family Mp, at least one candidate controller Cq, q 2 Q provides adequate performance. controller selection function process in Mp, p 2 P controller Cq with q =c(p) provides adequate performance

Estimator-based supervisor: Example #1 measured output y control signal u y – multiestimator Process is assumed to be Multi-estimator + – + unknown parameters p*=(a*, b*) 2 P [– 1, 1] £ {– 1, 1} 8 p=(a, b) 2 P [– 1, 1] £ {– 1, 1} yp ´ estimate of the output y that would be correct if the parameter was p=(a, b) ep ´ output estimation error that would be small if the parameter was p=(a, b) consider the yp corresponding to p = p* Since P has infinitely many elements, this multi-estimator would have to be infinite dimensional !? (not a very good multi-estimator)

Estimator-based supervisor: Example #1 measured output y control signal u y – multiestimator + decision logic – s switching signal + Process is assumed to be Multi-estimator unknown parameters p*=(a*, b*) 2 P [– 1, 1] £ {– 1, 1} 8 p=(a, b) 2 P [– 1, 1] £ {– 1, 1} yp ´ estimate of the output y that would be correct if the parameter was p=(a, b) ep ´ output estimation error that would be small if the parameter was p=(a, b) Decision logic: ep small process likely in Mp should use Cq, q = c(p) Certainty equivalence inspired set s = c(p)

Estimator-based supervisor measured output y control signal u y – + multiestimator decision logic – s switching signal + Process is assumed to be in family process in Mp, p 2 P controller Cq, q =c(p) provides adequate performance Multi-estimator yp ´ estimate of the process output y that would be correct if the process was Np ep ´ output estimation error that would be small if the process was Np Decision logic: ep small process likely in Mp should use Cq, q = c(p) Certainty equivalence inspired set s = c(p)

Estimator-based supervisor measured output y control signal u y – multiestimator + decision logic – s switching signal + A stability argument cannot be based on this because typically Process is assumed to be in family process in Mp ) ep small process in controller Cq, q =c(p) but not the converse Mp, p 2 P provides adequate performance Multi-estimator yp ´ estimate of the process output y that would be correct if the process was Np ep ´ output estimation error that would be small if the process was Np Decision logic: ep small process likely in Mp should use Cq, q = c(p) Certainty equivalence inspired set s = c(p)

Estimator-based supervisor measured output y control signal u y – + multiestimator – decision logic s switching signal + Process is assumed to be in family process in Mp, p 2 P controller Cq, q =c(p) provides adequate performance Multi-estimator yp ´ estimate of the process output y that would be correct if the process was Np ep ´ output estimation error that would be small if the process was Np detectable means “smallep ) small state” Decision logic: ep small set s = c(p) overall system is detectable through ep overall state is small Certainty equivalence inspired, but formally justified by detectability

Performance-based supervision measured output y control signal u performance monitor decision logic s switching signal Candidate controllers: Performance monitor: pq ´ measure of the expected performance of controller Cq inferred from past data Decision logic: ps is acceptable keep current controller ps is unacceptable switch to controller Cq corresponding to best pq

Abstract supervision Estimator and performance-based architectures share the same common architecture multi-est. or perf. monitor decision logic s switching signal multicontroller w u control signal process In this talk we will focus mostly on an estimator-based supervisor… y measured output

Abstract supervision Estimator and performance-based architectures share the same common architecture multiestimator decision logic s switching signal multicontroller switched system w u control signal process y measured output

The four basic properties (1 -2) : Example #1 Matching property: At least one of the ep is “small” decision logic switching signal s Why? Process is Multi-estimator is ep* is “small” essentially a requirement on the multi-estimator detectable means “smallep ) small state” Detectability property: For each p 2 P, the switched system is detectable through ep when s = c(p) index of controller that stabilizes processes in Mp

Detectability a system is detectable if for every pair eigenvalue/eigenvector (li, vi) of A <[li] ¸ 0 ) C vi ¹ 0 for short: pair(A, C) is detectable From solution to linear ODEs… (assuming A diagonalizable, otherwise terms intk eli t appear) ¹ 0 y(t) bounded ) ai C vi = 0 (for <[li] ¸ 0) ) ai = 0 (for <[li] ¸ 0) ) y(t) bounded Lemma: For any detectable system: y(t) bounded ) x(t) bounded & y(t) ! 0 ) x(t) ! 0

Detectability system is detectable if for every pair eigenvalue/eigenvector (li, vi) of A <[li] ¸ 0 ) C vi ¹ 0 Lemma: For any detectable system: y(t) bounded ) x(t) bounded & y(t) ! 0 ) x(t) ! 0 Lemma: For any detectable system, there exists a matrix K such that is asymptotically stable (all eigenvalues of A – KC with negative real part) Can re-write system as (output injection) asympt. stable confirms that: y is bounded /!0 ) x is bounded /!0

The four basic properties (1 -2) : Example #1 detectable means “smallep ) small state” Detectability property: For each p 2 P, the switched system is detectable through ep when s = c(p) index of controller that stabilizes processes in Mp Why? consider, e. g. , p=(a, b)= (. 5, 1) ) use controller C 3: u=– 1. 1 y (3 = s = c(p)) . 5 – 1. 1 = –. 6 < 0 Thus, ep small ) y = yp– ep small ) u small etc. Questions: • Where we just lucky in getting –. 6 < 0 ? NO (why? ) • Does detectability hold if u=– 1. 1 y does not stabilize the process (e. g. , a* =. 5, b*=-1)? YES (why? )

The four basic properties (1 -2) Matching property: At least one of the ep is “small” decision logic switching signal s Why? process in 9 p*2 P: process in Mp* ep* is “small” essentially a requirement on the multi-estimator Detectability property: For each p 2 P, the switched system is detectable through ep when s = c(p) index of controller that stabilizes processes in Mp essentially a requirement on the candidate controllers This property justifies using the candidate controller that corresponds to a small estimation error. Why? Certainty equivalence stabilization theorem…

The four basic properties (3 -4) Small error property: There is a parameter “estimate” decision logic r : [0, 1) ! P current parameter “estimate” for which er is “small” compared to any fixed ep and that is consistent with s, i. e. , s = c(r) switching signal s r(t) can be viewed as controller consistent with parameter “estimate” Non-destabilization property: Detectability is preserved for the time-varying switched system (not just for constant s) Typically requires some form of “slow switching” Both are essentially (conflicting) properties of the decision logic

Decision logic decision logic switching signal s 1. For boundedness one wants er small for some parameter estimate r consistent with s (i. e. , s = c (r)) “small error” 2. To recover the “static” detectability of the time -varying switched system one wants slow switching. “non-destabilization” These are conflicting requirements: 1. r should follow smallest ep 2. s = c(r) should not vary

Dwell-time switching start monitoring signals p 2 P measure of the size of ep over a “window” of length 1/l forgetting factor wait t. D seconds Non-destabilizing property: The minimum interval between consecutive discontinuities of s is t. D > 0. (by construction)

Small error property (e. g. , L 2 noise and no unmodeled dynamics) Assume P finite and 9 p* 2 P : when we select r = p at time t we must have Two possible cases: 1. Switching will stop in finite time T at some p 2 P: <1 · C*

Small error property (e. g. , L 2 noise and no unmodeled dynamics) Assume P finite and 9 p* 2 P : when we select r = p at time t we must have Two possible cases: 2. After some finite time T switching will occur only among elements of a subset P* of P, each appearing in r infinitely many times: <1 · C*

Small error property (e. g. , L 2 noise and no unmodeled dynamics) Assume P finite and 9 p* 2 P : when we select r = p at time t we must have Small error property: (L 2 case) Assume that P is a finite set. If 9 p* 2 P for which then at least one error L 2 “switched” error will be. L 2

Implementation issues monitoring signals start How to efficiently compute a large number of monitoring signals? Example #1: Multi-estimator wait t. D seconds input is linear comb. of y and u ß ß by linearity state-sharing we can generate as many errors as we want with a 2 dim. systems

Implementation issues monitoring signals start How to efficiently compute a large number of monitoring signals? Example #1: Multi-estimator wait t. D seconds 1. we can generate as many monitoring signals as we want with a (2+6)-dim. system state-sharing ß 2. finding r is really an optimization

Implementation issues monitoring signals start When P is a continuum (or very large), it may be issues with respect to the optimization for r. Things are easy, e. g. , 1. P has a small number of elements wait t. D seconds 2. model is linearly parameterized on p (leads to quadratic optimization) 3. there are closed form solutions usual requirement (e. g. , mp polynomial on p) in adaptive control 4. mp is convex on p results still hold if there exists a computational delay t. C in performing the optimization, i. e.

The four basic properties decision logic r(t) can be viewed as Small error property: current parameter There is a parameter “estimate” r : [0, 1) ! P for which er is “small” compared to any fixed ep and that is consistent with s, i. e. , s = c(r) controller consistent with parameter “estimate” switching signal s Non-destabilization property: Detectability is preserved for the time-varying switched system (not just for constant s) Matching property: At least one of the ep is “small” Detectability property: For each p 2 P, the switched system is detectable through ep when s = c(p) index of controller that stabilizes processes in Mp

Analysis outline (linear case, w = 0) decision logic switching signal s 1 st by the Matching property: 9 p*2 P such that ep* is “small” 2 nd by the Small error property: 9 r such that s = c(r) and er is “small” (when compared with ep*) 3 rd by the Detectability property: there exist matrices Kp such that the matrices Aq– Kp Cp, q = c(p) are asymptotically stable 4 th the switched system can be written as asymptotically stable by nondestabilization property injected system “small” by 2 nd step ) x is small (and converges to zero if, e. g. , er 2 L 2)

Example #2: One-link flexible manipulator mass at the tip x mt y(x, t) deviation with respect to rigid body q torque applied at the base T PDE (small bending): beam’s elasticity transversal slice’s inertia beam’s mass density (. 68 Kg total mass) IH axis’s inertia (. 023) Boundary conditions: beam’s length (113 cm)

Example #2: One-link flexible manipulator mass at the tip x mt y(x, t) deviation with respect to rigid body q torque applied at the base T IH axis’s inertia (. 023) Series expansion and truncation: eigenfunctions of the beam

Example #2: One-link flexible manipulator mass at the tip x mt y(x, t) q torque applied at the base T IH axis’s inertia (. 023) Control measurements: Assumed not known a priori: mt 2 [0, . 1 Kg] xsg 2 [40 cm, 60 cm] ´ base angle ´ base angular velocity ´ tip position ´ bending at position xsg (measured by a strain gauge attached to the beam at position xsg)

Example #2: One-link flexible manipulator torque u transfer functions asmt ranges over [0, . 1 Kg] and xsg ranges over [40 cm, 60 cm]

Example #2: One-link flexible manipulator torque u Class of admissible processes: Mp ´ family around a nominal transfer function corresponding to parameters p (mt, xsg) unknown parameter p (mt, xsg) parameter set: grid of 18 points in [0, . 1]£[40, 60] For this problem it is not possible to write the coefficients of the nominal transfer functions as a function of the parameters because these coefficients are the solutions to transcendental equations that must be computed numerically. Family of candidate controllers: 18 controllers designed using LQR/LQE, one for each nominal model

Example #2: One-link flexible manipulator 3 0. 7 0. 6 0. 5 2 0. 4 mt xsg 0. 3 0. 2 0. 1 1 0 0 5 10 t 15 20 25 0 -1 18 16 14 12 -2 -3 0 10 tip position torque set point 5 8 6 4 s 2 10 15 t 20 25 0 0 5 10 15 t 20 25 30

Example #2: One-link flexible manipulator (open-loop)

Example #2: One-link flexible manipulator (closed-loop with fixed controller)

Example #2: One-link flexible manipulator (closed-loop with supervisory control)

Example #3: Uncertain gain + C 1·k<4 The maximum gain margin achievable by a single linear time-invariant controller is 4 Doyle, Francis, Tannenbaum, Feedback Control Theory, 1992

Example #3: Uncertain gain + multicontroller 1 · k < 40

Example #3: Uncertain gain output reference true parameter value monitoring signals

Example #3: 2 -dim SISO linear process Class of admissible processes: nominal transfer function nonlinear parameterized onp – 1 unstable zero-pole cancellations +1 Any re-parameterization that makes the coefficients of the transfer function lie in a convex set will introduce an unstable zero-pole cancellation But the multi-estimator is still separable and state-sharing can be used …

Example #3: 2 -dim SISO linear process output reference true parameter value (without noise)

Example #3: 2 -dim SISO linear process output reference true parameter value (with noise)

Example #3: Disturbance Rejection (unknown frequency) disturbance frequency estimate output (rejection of one sinusoid)

Example #3: Disturbance Rejection (unknown frequency) disturbance frequency estimate output (rejection of two sinusoids)

Example #3: Disturbance Rejection (unknown frequency) disturbance frequency estimate output (rejection of a square wave)

Summary • • Supervisory control overview Estimator-based linear supervisory control 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 s controller 1 bank of candidate controllers controller n exogenous disturbance/ noise switching signal measured output w s u process y control 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

Estimator-based nonlinear supervisory control multiestimator decision logic NON LINEAR switching signal w s multicontroller u control signal process y measured output

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

Class of admissible processes w control signal u process Process is assumed to be in a family parametric uncertainty exogenous disturbance/n oise y measured output unmodeled dynamics Mp ´ small family of systems around a nominal process model Np Typically metric on set of statespace 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) 2 P [– 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 after the coordinate transformation the new state is no longer accessible Candidate controllers stabilizes the system

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) Multi-controller: y measured output s switching signal u control signal

Multi-estimator measured output y control signal u – multiestimator + – + How to design a multi-estimator? we want: Matching property: there exist some p*2 P such that ep* is “small” Typically obtained by: process in 9 p*2 P: process in 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) 2 P [– 1, 1] £ {– 1, 1} we can do state-sharing to generate all the errors with a finite-dimensional system for p = p* … exponentially ) g for p = p* ) Matching property

Designing multi-estimators - I (state accessible) Suppose nominal models Np, p 2 P 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 : p 2 P } 9 p*2 P, c 0, l* >0 : || ep*(t) || · c 0 e-l* t t¸ 0

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

Designing multi-estimators - III Suppose nominal models Np, p 2 P are of the form asymptotically stable Ap (output-inj. and coord. transf. away from stable linear system) (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 : p 2 P } 9 p*2 P, c 0, cw, l* >0 : || ep*(t) || · c 0 e-l* t + cw with cw = 0 in case w(t) = 0, 8 t ¸ 0 t¸ 0

Switched system detectability property? multiestimator decision logic w s multicontroller u process y switched system 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 1 st Take a parameter estimate signal r : [0, 1)! P. 2 nd Define the signal v er = yr y 3 rd Replace y in the equations of the multi-estimator and multi-controller by yr v. s multicontroller u v multiestimator yr – y

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

Stability & detectability of nonlinear systems Stability: input u “small” state x “small” Input-to-state stable (ISS) if 9 b 2 KL, g 2 K Integral input-to-state stable (i. ISS) if 9 a 2 K 1, b 2 KL, g 2 K strictly weaker Notation: a: [0, 1) ! [0, 1) is class K ´ continuous, strictly increasing, a(0) = 0 is class K 1 ´ class K and unbounded b: [0, 1)£[0, 1) ! [0, 1) is class KL ´ b(¢, t) 2 K for fixed t & limt!1 b(s, t) = 0 (monotonically) for fixed s

Stability & detectability of nonlinear systems Stability: input u “small” state x “small” Input-to-state stable (ISS) if 9 b 2 KL, g 2 K Integral input-to-state stable (i. ISS) if 9 a 2 K 1, b 2 KL, g 2 K strictly weaker One can show: 1. for ISS systems: u!0 2. for i. ISS systems: s 01 g(||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 b 2 KL, gu, gy 2 K Integral detectable (i. IOSS) if 9 a 2 K 1, b 2 KL, gu, gy 2 K One can show: 1. for IOSS systems: u, y ! 0 2. for i. IOSS systems: s 01 gu(||u||), s 01 gy(||y||) < 1 ) x ! 0 )x!0 strictly weaker

Certainty Equivalence Stabilization Theorem process y switched system u v – injected system r + – + r s Theorem: (Certainty Equivalence Stabilization Theorem) Suppose the process is detectable and take fixed r = p 2 P and s = q 2 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 = p 2 P and s = q 2 Q switched system detectable. 1. injected system ISS 2. injected system integral ISS switched system integral detectable s multicontroller injected system u multiestimator v yr – r=p 2 P s=q 2 Q 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 (x. C, x. E)

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

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

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

Scale-independent hysteresis switching Theorem: Let P be finite with m elements. For every p 2 P 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)

Scale-independent hysteresis switching Theorem: Let P be finite with m elements. For every p 2 P 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) 1 st by the Matching property: 9 p*2 P such that || ep*(t) || · c 0 e-l* t t¸ 0 2 nd by the Non-destabilization property: switching stops at a finite time T* 2 [0, Tmax) ) r(t) = p & s(t) = c(p) 8 t 2 [T*, Tmax) 3 rd by the Small error property: 4 th by the Detectability property: the state x of 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.

Summary • • 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, p 2 P 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, p 2 P are of the form Multi-estimator: it is separable so we can do state-sharing When process matches the nominal model Np* ) Candidate controller q = c(p): makes injected system ISS exponentially ) Matching property 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, p 2 P are of the form state accessible In the previous back-stepping procedure: the controller drives g ! 0 One could instead make pointwise min-norm design nonlinearity is cancelled (even when a < 0 and it introduces damping) still leads to exponential decrease of a (without canceling nonlinearity when a < 0 )

Example #4: System in strict-feedback form Suppose nominal models Np, p 2 P are of the form state accessible A different recursive procedure: In this case g!0 exponentially pointwise min-norm recursive design a ! 0 exponentially

Example #4: System in strict-feedback form Suppose nominal models Np, p 2 P are of the form Multi-estimator ( option III ): When process matches the nominal model Np* Candidate controller q = c(p): exponentially Matching property 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 pointwise min-norm design a 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 x 2 q u 1 ´ forward velocity u 2 ´ angular velocity x 1 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 l 2 Ñ 2 ´ rotor flux u 2 Ñ 2 ´ stator currents w ´ rotor angular velocity t ´ torque generated w is the only measurable output Unknown parameters: t. L 2 [tmin, tmax] ´ load torque R 2 [Rmin, Rmax] ´ rotor resistance “Off-the-shelf” field-oriented candidate controllers: