1 / 95

Switched Supervisory Control

Tutorial on Logic-based Control. Switched Supervisory Control. Jo ã o 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.

roana
Download Presentation

Switched Supervisory Control

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


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

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

  3. 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

  4. 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

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

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

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

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

  9. Types of supervision Pre-routed supervision Performance-based supervision (direct) s = 2 • 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 s = 1 s = 3 • try one controllers after another in a pre-defined sequence • stop when the performance seems acceptable Estimator-based supervision (indirect) • estimate process model from observed data • select controller based on current estimate – Certainty Equivalence not effective when the number of controllers is large

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

  11. Estimator-based supervision’s setup: Example #1 process control signal u y measured output Process is assumed to be unknown parameters p*=(a*, b*) 2P› [–1,1] £ {–1,1} we consider three candidate controllers: controller C1: u = 0 to be used when a*· –.1 controller C2: u = 1.1y to be used when a* > –.1 & b* = –1 controller C3: u = – 1.1y to be used when a* > –.1 & b* = +1 process parameter p* is equal to p›(a,b) 2P use controller Cq with controller selection function

  12. Estimator-based supervision’s setup 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 for each process in a family Mp, at least one candidate controller Cq, q2Q provides adequate performance. controller selection function controller Cq with q =c(p)provides adequate performance process in Mp, p2P

  13. Estimator-based supervisor: Example #1 y measured output – y multi-estimator + – u control signal + Process is assumed to be unknown parameters p*=(a*, b*) 2P› [–1,1] £ {–1,1} Multi-estimator 8p=(a, b) 2P› [–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)

  14. Estimator-based supervisor: Example #1 y measured output – y switching signal decisionlogic multi-estimator + s – u control signal + Process is assumed to be unknown parameters p*=(a*, b*) 2P› [–1,1] £ {–1,1} Multi-estimator 8p=(a, b) 2P› [–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: processlikely in Mp should useCq, q = c(p) epsmall set s = c(p) Certainty equivalence inspired

  15. Estimator-based supervisor y measured output – y switching signal decisionlogic multi-estimator + s – u control signal + Process is assumed to be in family process inMp,p2P 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: processlikely in Mp should useCq, q = c(p) epsmall set s = c(p) Certainty equivalence inspired

  16. Estimator-based supervisor y measured output – y switching signal decisionlogic multi-estimator + s – u control signal + A stability argument cannot be based on this because typically process in Mp)ep small but not the converse Process is assumed to be in family process inMp, p2P 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: processlikely in Mp should useCq, q = c(p) epsmall set s = c(p) Certainty equivalence inspired

  17. Estimator-based supervisor y measured output – y switching signal decisionlogic multi-estimator + s – u control signal + Process is assumed to be in family process inMp, p2P 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 “small ep) small state” Decision logic: overall system is detectable through ep sets = c(p) overall state is small epsmall Certainty equivalence inspired, but formally justified by detectability

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

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

  20. Abstract supervision Estimator and performance-based architectures share the same common architecture multi-estimator decision logic switching signal w s multi- controller u process y measured output control signal switched system

  21. The four basic properties (1-2) : Example #1 Multi-estimator is Process is Matching property: At least one of the ep is “small” Why? decision logic ep* is“small” essentially a requirement on the multi-estimator switching signal detectable means “small ep) small state” s Detectability property: For each p2P, the switched system is detectable through ep when s = c(p) index of controller that stabilizes processes in Mp

  22. Detectability a system is detectable if for every pair eigenvalue/eigenvector (li,vi) of A <[li] ¸ 0 )C vi¹ 0 for short: pair (A,C) is detectable From solution to linear ODEs… ¹ 0 (assuming Adiagonalizable, otherwise terms in tk eli t appear) 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

  23. Detectability system is detectable if for every pair eigenvalue/eigenvector (li,vi) of A <[li] ¸ 0 )C vi¹ 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

  24. The four basic properties (1-2) : Example #1 detectable means “small ep) small state” Detectability property: For each p2P, 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 C3: u=– 1.1y (3 = s = c(p)) .5 – 1.1 = –.6 < 0 Thus, ep small ) yp small )y = yp– ep small )u small etc. • Questions: • Where we just lucky in getting –.6 < 0 ? NO (why?) • Does detectability hold if u=– 1.1y does not stabilize the process (e.g., a* = .5, b*=-1)? YES (why?)

  25. The four basic properties (1-2) process in Matching property: At least one of the ep is “small” Why? decision logic 9p*2P: processin Mp* ep* is“small” essentially a requirement on the multi-estimator switching signal s Detectability property: For each p2P, 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…

  26. The four basic properties (3-4) Small error property: 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) r(t) can be viewed as current parameter “estimate” decision logic 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) Typically requires some form of “slow switching” Both are essentially (conflicting) properties of the decision logic

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

  28. Dwell-time switching monitoring signals start p2P measure of the size of ep over a “window” of length 1/l forgetting factor wait tD seconds Non-destabilizing property:The minimum interval between consecutive discontinuities of s is tD> 0. (by construction)

  29. Small error property (e.g., L2 noise and no unmodeled dynamics) Assume P finite and 9p*2P :   when we select r = p at time t we must have  • Two possible cases: • Switching will stop in finite time T at some p2P: < 1 ·C*

  30. Small error property (e.g., L2 noise and no unmodeled dynamics) Assume P finite and 9p*2P :   when we select r = p at time t we must have  • Two possible cases: • 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*

  31. Small error property (e.g., L2 noise and no unmodeled dynamics) Assume P finite and 9p*2P :   when we select r = p at time t we must have  Small error property: (L2case)Assume thatPis a finite set. If 9p*2P for which then at least one error L2 “switched” error will be L2

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

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

  34. 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 2. model is linearly parameterized on p (leads to quadratic optimization) 3. there are closed form solutions (e.g., mp polynomial on p) 4. mp is convex on p wait tD seconds usual requirement in adaptive control results still hold if there exists a computational delay tC in performing the optimization, i.e.

  35. The four basic properties r(t) can be viewed as current parameter “estimate” Small error property: 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) decision logic controller consistent with parameter “estimate” Non-destabilization property: Detectability is preserved for the time-varying switched system (not just for constant s) switching signal s Matching property: At least one of the ep is “small” Detectability property: For each p2P, the switched system is detectable through ep when s = c(p) index of controller that stabilizes processes in Mp

  36. Analysis outline(linear case, w = 0) 1st by the Matching property: 9p*2Psuch that ep* is “small” 2nd by the Small error property: 9r such that s = c(r) and er is “small”(when compared with ep*) 3rd by the Detectability property: there exist matrices Kp such that the matrices Aq– Kp Cp, q = c(p) are asymptotically stable 4th the switched system can be written as decision logic switching signal s injected system asymptotically stable by non-destabilization property “small” by 2nd step )x is small (and converges to zero if, e.g., er2L2)

  37. 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 IH T axis’s inertia (.023) PDE (small bending): Boundary conditions: beam’s length(113 cm) transversalslice’s inertia beam’s elasticity beam’smass density(.68Kg total mass)

  38. 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 IH T axis’s inertia (.023) Series expansion and truncation: eigenfunctions of the beam

  39. Example #2: One-link flexible manipulator mass at the tip x mt y(x, t) q torque applied at the base Assumed not known a priori: mt2 [0, .1Kg] xsg2 [40cm, 60cm] IH T axis’s inertia (.023) Control measurements: ´ base angle ´ base angular velocity ´ tip position ´ bending at position xsg(measured by a strain gauge attached to the beam at position xsg)

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

  41. Example #2: One-link flexible manipulator u torque Class of admissible processes: Mp´ family around a nominal transfer function corresponding to parameters p› (mt, xsg) parameter set: grid of 18 points in [0, .1]£[40, 60] unknown parameter p› (mt, xsg) 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

  42. Example #2: One-link flexible manipulator 0.7 0.6 3 0.5 0.4 mt 2 0.3 xsg 0.2 0.1 1 0 0 5 10 15 20 25 t 0 s -1 18 16 -2 tip position 14 torque 12 10 set point 8 -3 0 5 10 15 20 25 6 t 4 2 0 0 5 10 15 20 25 30 t

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

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

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

  46. 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

  47. Example #3: Uncertain gain + - multi- controller 1 ·k < 40

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

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

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

More Related