Control Using Logic & Switching: Part III Supervisory Control

Transcription

1 Control Using Logic & Switching: Part III Spervisor Control Ttorial for the 40th CDC João P. Hespanha Universit of Sothern California Universit of California at Santa Barbara Otline Spervisor control overview Estimator-based linear spervisor control Estimator-based nonlinear spervisor control Examples 1

2 Spervisor control Motivation: in the control of complex and highl ncertain sstems, traditional methodologies based on a single controller do not provide satisfactor performance. spervisor controller 1 bank of candidate controllers controller n switching control exogenos distrbance/ noise w measred otpt Ke ideas: 1. Bild a bank of alternative controllers 2. Switch among them online based on measrements For simplicit we assme a stabilization problem, otherwise controllers shold have a reference inpt r Spervisor control Motivation: in the control of complex and highl ncertain sstems, traditional methodologies based on a single controller do not provide satisfactor performance. spervisor controller 1 bank of candidate controllers controller n switching control exogenos distrbance/ noise w measred otpt Spervisor: places in the feedback loop the controller that seems more promising based on the available measrements tpicall logic-based/hbrid sstem 2

3 Mlti-controller measred otpt controller 1 bank of candidate controllers controller n switching control Conceptal diagram: not efficient for man controllers & not possible for nstable controllers Mlti-controller measred otpt controller 1 bank of candidate controllers controller n switching control Given a famil of (n-dimensional) candidate controllers switching measred otpt control 3

4 (t) switching = 1 Mlti-controller switching times = 1 = 3 = 2 t Given a famil of (n-dimensional) candidate controllers switching measred otpt control Spervisor spervisor switching measred otpt control Tpicall an hbrid sstem: ϕ continos state δ discrete state continos vector field discrete transition fnction otpt fnction 4

5 Tpes of spervision Pre-roted spervision = 1 = 2 = 3 tr one controllers after another in a pre-defined seqence stop when the performance seems acceptable not effective when the nmber of controllers is large Performance-based spervision (direct) keep controller while observed performance is acceptable when performance of crrent controller becomes nacceptable, switch to controller that leads to best expected performance based on available data Estimator-based spervision (indirect) estimate model from observed data select controller based on crrent estimate Certaint Eqivalence Estimator-based spervision s setp w exogenos distrbance/ noise control measred otpt nmodeled Process is assmed to be in a famil dnamics M p small famil of sstems arond a nominal model N p parametric ncertaint for each in a famil M p, at least one candidate controller C q, q Q provides adeqate performance. controller selection fnction in M p, p P controller C q with q =χ(p) provides adeqate performance 5

6 measred otpt control Estimator-based spervisor decision logic switching Process is assmed to be in famil in M p,p P controller C q, q =χ(p) provides adeqate performance Mlti-estimator p estimate of the otpt that wold be correct if the was N p e p otpt estimation error that wold be small if the was N p Decision logic: e p small likel in M p shold se C q, q = χ(p) set = χ(p) Certaint eqivalence inspired measred otpt control Estimator-based spervisor Process is assmed to be in famil in M p, p P Mlti-estimator p estimate of the otpt that wold be correct if the was N p e p otpt estimation error that wold be small if the was N p Decision logic: mltiestimator mltiestimator decision logic switching A stabilit argment cannot be based on this becase tpicall in M controller p e p small C q, q =χ(p) bt not the converse provides adeqate performance e p small likel in M p shold se C q, q = χ(p) set = χ(p) Certaint eqivalence inspired 6

7 measred otpt control Estimator-based spervisor mltiestimator decision logic switching Process is assmed to be in famil in M p, p P controller C q, q =χ(p) provides adeqate performance Mlti-estimator p estimate of the otpt that wold be correct if the was N p e p otpt estimation error that wold be small if the was N p Decision logic: e p small set overall sstem overall state = χ(p) is detectable is small throgh e p Certaint eqivalence inspired, bt formall jstified b detectabilit Performance-based spervision measred otpt control performance monitor decision logic switching Candidate controllers: Performance monitor: π q measre of the expected performance of controller C q inferred from past data Decision logic: π is acceptable π is nacceptable keep crrent controller switch to controller C q corresponding to best π q 7

8 Abstract spervision Estimator and performance-based architectres share the same common architectre decision logic mlti-est. or perf. monitor switching w mlticontroller control measred otpt In this talk we will focs mostl on an estimator-based spervisor Abstract spervision Estimator and performance-based architectres share the same common architectre decision logic mltiestimator switching w mlticontroller control measred otpt switched sstem 8

9 The for basic properties (1-2) decision logic Matching propert: At least one of the e p is small Wh? in p * P: in M p* e p* is small switching essentiall a reqirement on the mlti-estimator Detectabilit propert: For each p P, the switched sstem is detectable throgh e p when = χ(p) index of controller that stabilizes es in M p essentiall a reqirement on the candidate controllers This propert jstifies sing the candidate controller that corresponds to a small estimation error. Wh? Certaint eqivalence stabilization theorem The for basic properties (3-4) switching decision logic Small error propert: There is a switching ρ : [0, ) P for which e ρ is small compared to an fixed e p and that is consistent with, i.e., = χ(ρ) ρ(t) can be viewed as crrent parameter estimate controller consistent with parameter estimate Non-destabilization propert: Detectabilit is preserved for the time-varing switched sstem (not jst for constant ) Tpicall reqires some form of slow switching Both are essentiall (conflicting) properties of the decision logic 9

10 Analsis otline (linear case, w = 0) switching decision logic 1st b the Matching propert: p * P sch that e p* is small 2nd b the Small error propert: ρ sch that = χ(ρ) and e ρ is small (when compared with e p* ) 3rd b the Detectabilit propert: there exist matrices K p sch that the matrices A q K p C p, q = χ(p) are asmptoticall stable injected 4th the switched sstem can be written as sstem asmptoticall stable b non-destabilization propert small b 2nd step x is small (and converges to zero if, e.g., e ρ L 2 ) Otline Spervisor control overview Estimator-based linear spervisor control Estimator-based nonlinear spervisor control Examples 10

11 Estimator-based linear spervisor control decision logic mltiestimator switching LINEAR w mlticontroller control measred otpt Class of admissible es d exogenos distrbance n measrement noise control measred otpt The transfer fnction from to is an nknown element of parametric ncertaint M p small famil of sstems arond a nominal transfer fnction ν p Tpicall mltiplicative nmod. dnamics additive nmod. dnamics or co-prime factorization of ν p (SISO) 11

12 A word on norms Given a and λ 0 Given a transfer fnction ν and λ 0 e λ t weighted L 2 norm of trncated to [0, t ) e λ t weighted H norm of ν (finite if all poles of ν have real part smaller than λ) The e λ t weighted L 2 indced norm of transfer fnction ν is nmericall eqal to the e λ t weighted H norm of ν ν ν,λ = γ de to initial conditions we will sometimes need a stabilit margin λ > 0... Candidate controllers Class of admissible es M p small famil of sstems arond a nominal transfer fnction ν p Assme given a famil of candidate controller transfer fnctions and a controller selection fnction χ : P Q sch that p P controller κ q, q = χ(p) stabilizes all es in M p χ maps parameter vales with the corresponding stabilizing controller No other constrain is posed on the candidate controllers: κ q can be designed sing an (nonadaptive) techniqe (e.g., pole placement, LQG/LQR, H-infinit, etc.) 12

13 measred otpt control Mlti-estimator mltiestimator with A E asmptoticall stable Matching propert: There exist positive constants c 0, c w, c ε, λ and some p* P sch that noise/distrb. initial cond. e p* is L 2 when noise/distrb. and nmodeled dnamics are L 2 recall nmodeled dnamics A simple mlti-estimator Class of admissible es (2 elements) Assming realizations are detectable, we cold make Lenberger observers asmptoticall stable Mlti-estimator: A E x E D E B E c p* (s I - A p* ) -1 b p* e p* = p* - 0 (exp. fast) with noise & nmodeled dnamics 13

14 Mlti-controller Given a famil of candidate controller transfer fnctions Compte (n C -dimensional) stabilizable and detectable realization Mlti-controller switching measred otpt control detectabilit propert? Switched sstem decision logic mltiestimator w mlticontroller switched sstem The switched sstem can be seen as the interconnection of the with the injected sstem essentiall the mlti-controller & mlti-estimator bt now qite 14

15 Constrcting the injected sstem 1st Take an (arbitrar) switching ρ : [0, ) P. 2nd Define the v ú e ρ = ρ 3rd Replace in the eqations of the mlti-estimator and mlti-controller b ρ v. v injected sstem Constrcting the injected sstem 1st Take an (arbitrar) switching ρ : [0, ) P 2nd Define the v ú e ρ = ρ 3rd Replace in the eqations of the mlti-estimator and mlti-controller b ρ v. mlticontroller mltiestimator ρ v v injected sstem 15

16 The injected sstem mlticontroller mltiestimator ρ v B inspection: for each p P, q Q eigenvales of A p q { sbset of eigenvales of A E } U { poles of the feedback interconnection of ν p with κ q } q =χ( p ) κ q stabilizes ν p A p q is asmptoticall stable If we choose ρ sch that =χ(ρ) then A ρ is alwas stable Switched sstem = injected sstem w Wh? becase v ú e ρ = ρ v injected sstem ρ ρ Using this diagram one can prove the detectabilit propert b inspection 16

17 v injected sstem ρ w Detectabilit propert ρ Sppose ρ = p P w = 0 = χ( p ) Q e p = 0 ( e ρ = v = 0 ) stabilit of injected sstem state & otpts, p of injected sstem 0 inpt & otpt, = p e p 0 state 0 whole state of switched sstem 0 detectabilit of The state of the switched sstem converges to zero along an soltion compatible with zero inpt w & otpt e p detectabilit Detectabilit propert w Detectabilit propert: Given an p P, setting ρ = p P and = χ( p ) Q : 1. The injected sstem is asmptoticall stable 2. The switched sstem is detectable throgh e p v injected sstem ρ ρ Also known as the Certaint Eqivalence Stabilization Theorem Stabilit of the injected sstem is not the onl mechanism to achieve detectabilit: e.g.: injected sstem i/o stable min. phase detectabilit of switched sstem (Certaint Eqivalence Otpt Stabilization Theorem) 17

18 Decision logic switching decision logic switched sstem 1. For bondedness one wants e ρ small for some ρ consistent with (i.e., = χ (ρ)) small error 2. To recover the static detectabilit of the time-varing switched sstem one wants slow switching. non-destabilization These are conflicting reqirements: 1. ρ shold follow smallest e p 2. = χ(ρ) shold not var estimation errors injected sstem Dwell-time switching start monitoring s p P measre of the size of e p over a window of length 1/λ forgetting factor wait τ D seconds Non-destabilizing propert: The minimm interval between consective discontinities of is τ D > 0. 18

19 Small error propert Assme P finite and p * P : (e.g., L 2 noise and no nmodeled dnamics) when we select ρ = p at time t we mst have Two possible cases: 1. Switching will stop in finite time T at some p P: < C* Small error propert Assme P finite and p * P : (e.g., L 2 noise and no nmodeled dnamics) Two possible cases: when we select ρ = p at time t we mst have 2. After some finite time T switching will occr onl among elements of a sbset P* of P, each appearing in ρ infinitel man times: < C* 19

20 Small error propert Assme P finite and p * P : (e.g., L 2 noise and no nmodeled dnamics) when we select ρ = p at time t we mst have Small error propert: (L 2 case) Assme that P is a finite set. If p * P for which then at least one error L 2 switched error will be L 2 Small error propert Small error propert: (L 2 case) Assme that P is a finite set. If p * P for which then at least one error L 2 switched error will be L 2 (for switching ρ defined b the logic) Small error propert: (general case) Assme that Q is a finite set with m element. For ever p P, t 0, switching ρ t : [0, t ) P sch that: 1. = χ(ρ t ) except at most on m time intervals of length τ D 2. althogh the bond ma not hold for e ρ, it will hold for another switching ρ t that is almost alwas consistent with The small error propert can still be generalized for the case when Q is not finite (i.e., infinitel man controllers) 20

21 Implementation isses start monitoring s How to efficientl compte a large nmber of monitoring s? It is alwas possible to write: appropriatel defined fnction wait τ D seconds From this and the definition of µ p : So, b linearit, all the µ p can be generated b: dimension is independent of the nmber of element in P Implementation isses start monitoring s When P is a continm (or ver large), it ma be isses with respect to the optimization for ρ. Things are eas, e.g., wait τ D seconds 1. P has a small nmber of elements 2. model is linearl parameterized on p (leads to k(p) qadratic) 3. there are closed form soltions (e.g., k(p) polnomial) 4. k(p) is convex on p sal reqirement in adaptive control reslts still hold if there exists a comptational dela τ C in performing the optimization, i.e. 21

22 Matching propert: p* P sch that Analsis (w = 0, ε = 0) Detectabilit propert: for frozen ρ = p P and = χ( p ) Q the injected sstem is asmptoticall stable Non-destabilizing propert: The minimm interval between consective discontinities of is τ D > 0. Small error propert (L 2 case, P finite): Matching propert: p* P sch that Analsis (w = 0, ε = 0) Detectabilit propert: for frozen ρ = p P and = χ( p ) Q the injected sstem is asmptoticall stable Non-destabilizing propert: The minimm interval between consective discontinities of is τ D > 0. Small error propert (L 2 case, P finite): Assmption (slow switching): τ D is large enogh and λ is small enogh so that the injected sstem is nif. exp. stable state transition matrix of injected sstem an switching with interval between consective discontinities no smaller than τ D 22

23 Analsis (w = 0, ε = 0) Matching propert: p* P sch that Detectabilit propert: for frozen ρ = p P and = χ( p ) Q the injected ρ sstem is asmptoticall stable v injected Non-destabilizing propert: The minimm sstem interval between consective discontinities of is τ D > 0. Small error propert (L 2 case, P finite): Assmption (slow switching): τ D is large enogh and λ is small enogh so that the injected sstem is nif. exp. stable state transition matrix of injected sstem an switching with interval between consective discontinities no smaller than τ D Analsis (w = 0, ε = 0) v injected sstem ρ ρ 1st b the Matching propert: p * P sch that 2nd b the Small error propert: 3rd b the Non-destabilization propert & assmption the injected sstem is nif. exp. stable (state transition matrix decas faster then e λ t ) 23

24 Analsis (w = 0, ε = 0) v injected sstem ρ ρ 2nd b the Small error propert: 3rd b the Non-destabilization propert & assmption the injected sstem is nif. exp. stable (state transition matrix decas faster then e λ t ) 4th b 2nd and 3rd (same for and ρ ) Analsis (w = 0, ε = 0) v injected sstem ρ ρ 4th b 2nd and 3rd (same for and ρ ) 5th b the detectabilit: state the is also L 2 and converges to zero 24

25 Analsis (w = 0, ε = 0) v injected sstem ρ ρ Theorem: Assming that P is finite and in the absence of noise and nmodeled dnamics (i.e., ε = 0, w(t) = 0, t 0) the states of the, the mlti-estimator, and the mlti-controller are all (e λt -weighted) L 2 and converge to zero as t. Matching propert: p* P sch that Analsis (general case) Detectabilit propert: for ρ = p P and = χ( p ) Q the injected sstem is asmptoticall stable Non-destabilizing propert: The minimm interval between consective discontinities of is τ D > 0. Small error propert (Q finite): p P, t 0, ρ t : [0, t ) P sch that: 1. = χ(ρ t ) except at most on m time intervals of length τ D 2. we will start b cheating and assming that = χ(ρ t ) all the time... Assmption (slow switching): τ D is large enogh and λ is small enogh so that state transition matrix of injected sstem an switching with interval between consective discontinities no smaller than τ D 25

26 Analsis (general case) v injected sstem ρ ρ Consider a fixed interval [0, t ) 1st b the Matching propert: p * P sch that 2nd b the Small error propert: ρ t sch that = χ( ρ t ) & 3rd se ρ t from Small error propert to constrct the injected sstem Analsis (general case) v injected sstem ρ t ρ t 2nd b the Small error propert: ρ t sch that = χ( ρ t ) & 3rd se ρ t from Small error propert to constrct the injected sstem since = χ( ρ t ) the injected sstem switches among stabilit matrices 4th b the Non-destabilization propert & assmption the injected sstem is nif. exp. stable (state transition matrix decas faster then e λ t ) 26

27 Analsis (general case) v injected sstem ρ t ρ t 2nd b the Small error propert: ρ t sch that = χ( ρ t ) & 4th b the Non-destabilization propert & assmption the injected sstem is nif. exp. stable (state transition matrix decas faster then e λ t ) finite λ,[0, t ) indced norm from v to : Analsis (general case) v injected sstem ρ t ρ t 2nd b the Small error propert: ρ t sch that = χ( ρ t ) & 4th b the Non-destabilization propert & assmption: 5th b small-gain argment (sing 2nd and 4th) jst as before: v bonded & injected sstem stable all s bonded 27

28 Analsis (general case) v injected sstem ρ t ρ t Where did we se = χ( ρ t )? 3rd se ρ t from Small error propert to constrct the injected sstem since = χ( ρ t ) the injected sstem switches among stabilit matrices 4th b the Non-destabilization propert & assmption the injected sstem is nif. exp. stable (state transition matrix decas faster then e λ t ) Analsis (general case) v injected sstem ρ t ρ t Where did we se = χ( ρ t )? 3rd se ρ t from Small error propert to constrct the injected sstem since = χ(ρ t ) except at most on m time intervals of length τ D, the injected sstem switches among stabilit matrices except at most on m time intervals of length τ D 4th b the Non-destabilization propert & assmption the injected sstem is still nif. exp. stable (state transition matrix decas faster then e λ t ) 28

29 Analsis (general case) v injected sstem ρ t ρ t Theorem: Assming that Q is a finite set with m element and the λ,[0, t ) norm of all s can be bonded b expressions of the form (finite indced norms) Moreover: w(t) is niforml bonded for t [0, ) all s niforml bonded w(t) 0 as t all s converge to zero as t So far Fast switching Assmption (slow switching): τ D is large enogh and λ is small enogh so that state transition matrix of injected sstem an switching with interval between consective discontinities no smaller than τ D Can be relaxed to Assmption (fast switching): λ is small enogh so that all matrices A p χ(p) λ I, p P are asmptoticall stable (an dwell-time τ D will do) 29

30 Fast switching Assmption (fast switching): λ is small enogh so that all matrices A p χ(p) λ I, p P are asmptoticall stable (an dwell-time τ D will do) Theorem: Assming that the is SISO and that the mlti-controller is realized as there exists a constant sch that when (no loss of generalit) the λ,[0, t ) norm of all s can be bonded b expressions of the form (finite indced norms) Moreover: w(t) is niforml bonded for t [0, ) all s niforml bonded w(t) 0 as t all s converge to zero as t proof: tilize internal strctre of injected sstem & inject more errors to boost rate of deca Other logics Dwell-time switching Scale-independent hsteresis switching start start hsteresis constant wait τ D seconds wait fixed amont of time wait ntil crrent monitoring becomes significantl larger than some other one n 30

31 Other logics Scale-independent hsteresis switching Hierarchical hsteresis switching start start wait ntil crrent monitoring becomes significantl larger than some other one n wait ntil crrent monitoring, or another one consistent with, becomes significantl larger than some other one n Other logics For both logics we have: Small error propert: For ever p P, t 0, switching ρ t : [0, t ) P sch that: all the time! Non-destabilizing propert: For ever p P nmber of switchings in the interval [τ, t ) nmber of elements in P (scale-indep.) or in Q (hierarchical) average dwell-time tpe growth either large h (hsteresis constant) or small λ (forgetting factor) leads to stabilit of injected sstem 31

32 Otline Spervisor control overview Estimator-based linear spervisor control Estimator-based nonlinear spervisor control Examples Estimator-based nonlinear spervisor control decision logic mltiestimator NON LINEAR switching w mlticontroller control measred otpt 32

33 Class of admissible es w exogenos distrbance/ noise control measred otpt nmodeled Process is assmed to be in a famil dnamics M p small famil of sstems arond a nominal model N p parametric ncertaint Tpicall metric on set of state-space model (?) Most reslts presented here: independent of metric d (e.g., detectabilit) or restricted to case ε p = 0 (e.g., matching) Candidate controllers Class of admissible es M p small famil of sstems arond a nominal model N p Assme given a famil of candidate controllers (withot loss of generalit all with same dimension) Mlti-controller: switching measred otpt control 33

34 Mlti-estimator measred otpt control mltiestimator How to design a mlti-estimator? we want: Matching propert: there exist some p* P sch that e p* is small Tpicall obtained b: in p * P: in M p* e p* is small when matches M p* the corresponding error mst be small Designing mlti-estimators - I Sppose nominal models N p, p P are of the form (state accessible) no exogenos inpt w state accessible Mlti-estimator: asmptoticall stable A When matches the nominal model N p* Matching propert: Assme M = { N p : p P } exponentiall p* P, c 0, λ * >0 : e p* (t) c 0 e -λ* t t 0 34

35 State-sharing Mlti-estimator: state of mlti-estimator is x E ú { z p : p P } can be large when P has man elements Sppose A p (, ) is separable, i.e., matrix vector B linearit, the p can be generated b: tre, e.g., is is linearl parameterized matrix with the size of M with ever colmn eqal to The dimension of the mlti-estimator is independent of the nmber of elements in P (cold even b infinit) Designing mlti-estimators - II Sppose nominal models N p, p P are of the form (otpt-injection awa from stable linear sstem) asmptoticall stable A p Mlti-estimator: nonlinear otpt injection (generalization of case I) When matches the nominal model N p* Matching propert: Assme M = { N p : p P } p* P, c 0, c w, λ * >0 : e p* (t) c 0 e -λ* t c w t 0 with c w = 0 in case w(t) = 0, t 0 State-sharing is possible when all A p are eqal an H p (, ) is separable: 35

36 Designing mlti-estimators - III Sppose nominal models N p, p P are of the form (otpt-inj. and coord. transf. awa from stable linear sstem) asmptoticall stable A p ζ p ú ξ p ξ p -1 (generalization of case I & II) cont. diff. coordinate transformation with continos inverse ξ p -1 (ma depend on nknown parameter p) The Matching propert is an inpt/otpt propert so the same mlti-estimator can be sed: Matching propert: Assme M = { N p : p P } p* P, c 0, c w, λ * >0 : e p* (t) c 0 e -λ* t c w t 0 with c w = 0 in case w(t) = 0, t 0 detectabilit propert? Switched sstem decision logic mltiestimator w mlticontroller switched sstem Also now the switched sstem can be seen as the interconnection of the with the injected sstem 36

37 Constrcting the injected sstem 1st Take a switching ρ : [0, ) P. 2nd Define the v ú e ρ = ρ 3rd Replace in the eqations of the mlti-estimator and mlti-controller b ρ v. mlticontroller mltiestimator ρ v Switched sstem = injected sstem w Q: How to get detectabilit on the switched sstem? A: Stabilit of the injected sstem v injected sstem ρ ρ 37

38 Stabilit & detectabilit of nonlinear sstems Stabilit: inpt small state x small Inpt-to-state stable (ISS) if β KL, γ K Integral inpt-to-state stable (iiss) if α K, β KL, γ K strictl weaker Notation: α:[0, ) [0, ) is class K continos, strictl increasing, α(0) = 0 is class K class K and nbonded β:[0, ) [0, ) [0, ) is class KL β(,t) K for fixed t & lim t β(s,t) = 0 (monotonicall) for fixed s Stabilit & detectabilit of nonlinear sstems Stabilit: inpt small state x small Inpt-to-state stable (ISS) if β KL, γ K Integral inpt-to-state stable (iiss) if α K, β KL, γ K strictl weaker One can show: 1. for ISS sstems: 0 soltion exist globall & x 0 2. for iiss sstems: 0 γ( ) < soltion exist globall & x 0 38

39 Stabilit & detectabilit of nonlinear sstems Detectabilit: inpt & otpt small state x small Detectabilit (or inpt/otpt-to-state stabilit IOSS) if β KL, γ, γ K Integral detectable (iioss) if α K, β KL, γ, γ K strictl weaker One can show: 1. for IOSS sstems:, 0 x 0 2. for iioss sstems: 0 γ ( ), 0 γ ( ) < x 0 Interconnecting stable & detectable sstems cascade (detectable) sstem 1 (stable) sstem 2 (detectable) Lemma 1 (cascade): i. sstem 1 ISS & sstem 2 detectable cascade detectable ii. sstem 1 integral ISS & sstem 2 detectable cascade integral detectable feedback (detectable) sstem 1 (detectable) Lemma 2 (feedback): i. sstem 1 detectable feedback detectable ii. sstem 1 integral detectable feedback integral detectable 39

40 Certaint Eqivalence Stabilization Theorem switched sstem (version 1) v injected sstem ρ ρ switched sstem (version 2) Σ 1 ρ Σ 2 injected sstem e ρ ρ Certaint Eqivalence Stabilization Theorem 1st detectable sstem Σ 2 detectable 2nd injected sstem ISS & 1st 3rd cascade Σ 1 Σ 2 detectable cascade Σ 1 Σ 2 detectable switched sstem detectable or 2nd injected sstem integral ISS & 1st cascade Σ 1 Σ 2 integral detectable 3rd cascade Σ 1 Σ 2 integral detectable switched sstem integral detectable switched sstem (version 2) Σ 1 ρ Σ 2 injected sstem e ρ ρ 40

41 Certaint Eqivalence Stabilization Theorem switched sstem v injected sstem ρ ρ Theorem: (Certaint Eqivalence Stabilization Theorem) Sppose the is detectable and take fixed ρ = p P and = q Q 1. injected sstem ISS switched sstem detectable. 2. injected sstem integral ISS switched sstem integral detectable Stabilit of the injected sstem is not the onl mechanism to achieve detectabilit: e.g., injected sstem i/o stable min. phase detectabilit of switched sstem (Nonlinear Certaint Eqivalence Otpt Stabilization Theorem) Achieving ISS for the injected sstem Theorem: (Certaint Eqivalence Stabilization Theorem) Sppose the is detectable and take fixed ρ = p P and = q Q 1. injected sstem ISS switched sstem detectable. 2. injected sstem integral ISS switched sstem integral detectable injected sstem mlticontroller mltiestimator ρ v ρ = p P = q Q We want to design candidate controllers that make the injected sstem (at least) integral ISS with respect to the distrbance inpt v Nonlinear robst control design problem, bt distrbance inpt v can be measred (v = e ρ = ρ ) the whole state of the injected sstem is measrable (x C, x E ) 41

42 Designing candidate controllers - I (feedback linearizable) Sppose the mlti-estimator is of the form with ( AD C p, B ) stabilizable To obtain the injected sstem, we make = p v : mst be inpt-to-state stabilized b the q = χ( p ) candidate controller (with respect to distrbance v) Candidate controller q = χ( p ): with AD C p B F p asmptoticall stable injected sstem ISS: Also applicable if the mlti-estimator is a coordinate transformation awa from this form Designing candidate controllers - II (inpt/otpt feedback linearizable) Sppose for each p P we can 1. partition the state of the mlti-estimator as 2. write its dnamics as with ( AD p C p, B p ) stabilizable ISS with respect to inpt ( x p,, ) Also applicable if the mlti-estimator is a coordinate transformation (possibl p-dependent) awa from this form This form is qite common becase tpicall one can linearize the sstem with respect to each individal p bt not with respect to all the p simltaneosl 42

43 Designing candidate controllers - II (inpt/otpt feedback linearizable) Sppose for each p P we can 1. partition the state of the mlti-estimator as 2. write its dnamics as with ( AD p C p, B p ) stabilizable ISS with respect to inpt ( x p,, ) Also applicable if the mlti-estimator is a coordinate transformation (possibl p-dependent) awa from this form Candidate controller q = χ( p ): with AD p C p B p F p asmptoticall stable injected sstem ISS: (cascade of ISS sstems) Detectabilit propert Detectabilit propert: For an of the previos mlti-estimators and candidate controller 1. The injected sstem is ISS (and also integral ISS) 2. The switched sstem is detectable throgh e p (and also integral detectable) Remarks: a. Other controllers also lead to detectabilit, e.g., one cold 1st se the feedback linearizing controller to find an ISS control Lapnov fnction 2nd se the ISS control Lapnov to constrct a more robst controller (e.g., sing an inverse optimal design) b. It is possible to achieve iiss for mch larger classes of sstems (e.g., sstems that cannot even be controlled b smooth time-invariant feedback) 43

44 Decision logic decision logic For nonlinear sstems dwell-time logics do not work becase of finite escape switching switched sstem estimation errors injected sstem Scale-independent hsteresis switching start monitoring s class K fnction from detectabilit propert hsteresis constant measre of the size of e p over a window of length 1/λ p P forgetting factor wait ntil crrent monitoring becomes significantl larger than some other one n All the µ p can be generated b a sstem with small dimension if γ p ( e p ) is separable. i.e., 44

45 Scale-independent hsteresis switching Theorem: Let P be finite with m elements. For ever p P and nmber of switchings in [τ, t ) Assme P is finite, the γ p are locall Lipschitz and maximm interval of existence of soltion niforml bonded on [0, T max ) niforml bonded on [0, T max ) Scale-independent hsteresis switching Theorem: Let P be finite with m elements. For ever p P and nmber of switchings in [τ, t ) Assme P is finite, the γ p are locall Lipschitz and maximm interval of existence of soltion Non-destabilizing propert: Switching will stop at some finite time T * [0, T max ) Small error propert: 45

46 Analsis (w = 0, no nmodeled dnamics) 1st b the Matching propert: p * P sch that e p* (t) c 0 e -λ* t t 0 2nd b the Non-destabilization propert: switching stops at a finite time T * [0, T max ) ρ(t) = p & (t) = χ(p) t [T *,T max ) 3rd b the Small error propert: 4th b the Detectabilit propert: the state x of the switched sstem is bonded on [T *,T max ) soltion exists globall T max = & x 0 as t Theorem: Assme that P is finite and all the γ p are locall Lipschitz. The state of the, mlti-estimator, mlti-controller, and all other s converge to zero as t. Otline Spervisor control overview Estimator-based linear spervisor control Estimator-based nonlinear spervisor control Examples 46

47 Example: One-link flexible maniplator x mass at the tip m t (x, t) deviation with respect to rigid bod torqe applied at the base T I H θ axis s inertia (.023) PDE (small bending): beam s elasticit transversal slice s inertia Bondar conditions: beam s length (113 cm) beam s mass densit (.68Kg total mass) Example: One-link flexible maniplator x mass at the tip m t (x, t) deviation with respect to rigid bod torqe applied at the base T I H θ axis s inertia (.023) Series expansion and trncation: eigenfnctions of the beam 47

48 Example: One-link flexible maniplator x mass at the tip m t (x, t) torqe applied at the base T Control measrements: I H θ axis s inertia (.023) Assmed not known a priori: m t [0,.1Kg] x sg [40cm, 60cm] base angle base anglar velocit tip position bending at position x sg (measred b a strain gage attached to the beam at position x sg ) Example: One-link flexible maniplator torqe Bode Diagrams Bode Diagrams 50 From: torqe 50 From: torqe 0 0 Phase (deg); Magnitde (db) To: base angle Phase (deg); Magnitde (db) To: base anglar velocit Freqenc (rad/sec) Freqenc (rad/sec) transfer fnctions as m t ranges over [0,.1Kg] and x sg ranges over [40cm, 60cm] Bode Diagrams Bode Diagrams 50 From: torqe 40 From: torqe 0 20 Phase (deg); Magnitde (db) To: tip position Phase (deg); Magnitde (db) To: bending Freqenc (rad/sec) Freqenc (rad/sec) 48

49 Example: One-link flexible maniplator torqe Class of admissible es: nknown parameter p ú (m t, x sg ) parameter set: grid of 18 points in [0,.1] [40, 60] M p famil arond a nominal transfer fnction corresponding to parameters p ú (m t, x sg ) For this problem it is not possible to write the coefficients of the nominal transfer fnctions as a fnction of the parameters becase these coefficients are the soltions to transcendental eqations that mst be compted nmericall. Famil of candidate controllers: 18 controllers designed sing LQR/LQE, one for each nominal model Example: One-link flexible maniplator m t x sg t tip position torqe set point t t 49

50 Example: One-link flexible maniplator (open-loop) Example: One-link flexible maniplator (closed-loop with fixed controller) 50

51 Example: One-link flexible maniplator (closed-loop with spervisor control) Example: Uncertain gain C 1 k < 4 The maximm gain margin achievable b a single linear time-invariant controller is 4 Dole, Francis, Tannenbam, Feedback Control Theor,

52 Example: Uncertain gain mlticontroller 1 k < 40 Anderson et. al., Mltiple Model Adaptive Control, Part 1: Finite Controller Coverings, Special George Zames Isse of IJRNC, Example: Uncertain gain otpt reference tre parameter vale monitoring s 52

53 Example: 2-dim SISO linear Class of admissible es: nominal transfer fnction nonlinear parameterized on p 1 1 nstable zero-pole cancellations An re-parameterization that makes the coefficients of the transfer fnction lie in a convex set will introdce an nstable zero-pole cancellation Bt the mlti-estimator is still separable and state-sharing can be sed Example: 2-dim SISO linear otpt reference tre parameter vale (withot noise) 53

54 Example: 2-dim SISO linear otpt reference tre parameter vale (with noise) Example: Distrbance Rejection (nknown freqenc) distrbance freqenc estimate otpt (rejection of one sinsoid) 54

55 Example: Distrbance Rejection (nknown freqenc) distrbance freqenc estimate otpt (rejection of two sinsoids) Example: Distrbance Rejection (nknown freqenc) distrbance freqenc estimate otpt (rejection of a sqare wave) 55

56 Example: Sstem in strict-feedback form Sppose nominal models N p, p P are of the form state accessible Mlti-estimator ( option I ): it is separable so we can do state-sharing When matches the nominal model N p* Matching propert: Assme M = { N p : p P } exponentiall p* P, c 0, λ * >0 : e p* (t) c 0 e -λ* t t 0 Example: Sstem in strict-feedback form Sppose nominal models N p, p P are of the form state accessible To facilitate the controller design, one can first back-step the sstem to simplif its stabilization: after the coordinate transformation the new state is no longer accessible now the control law stabilizes the sstem 56

57 Example: Sstem in strict-feedback form Sppose nominal models N p, p P are of the form Mlti-estimator ( option II ): it is separable so we can do state-sharing When matches the nominal model N p* Candidate controller q = χ(p): makes injected sstem ISS exponentiall Matching propert Detectabilit propert Example: Sstem in strict-feedback form p 2 p 1 α reference α 57

58 Example: Sstem in strict-feedback form Sppose nominal models N p, p P are of the form state accessible In the previos back-stepping procedre: the controller drives γ 0 nonlinearit is cancelled (even when p 1 < 0 and it introdces damping) One cold instead make pointwise min-norm design still leads to exponential decrease of α (withot canceling nonlinearit when p 1 < 0 ) Example: Sstem in strict-feedback form Sppose nominal models N p, p P are of the form state accessible A different recrsive procedre: In this case γ 0 exponentiall pointwise min-norm recrsive design α 0 exponentiall 58

59 Example: Sstem in strict-feedback form Sppose nominal models N p, p P are of the form Mlti-estimator ( option III ): When matches the nominal model N p* Candidate controller q = χ(p): exponentiall Matching propert Detectabilit propert Example: Sstem in strict-feedback form p 2 p 1 α reference α 59

60 Example: Sstem in strict-feedback form p 1 p 2 p 1 p 2 α α pointwise min-norm design feedback linearization design Example: Unstable-zero dnamics nknown parameter estimate otpt (stabilization) 60

61 Example: Unstable-zero dnamics nknown parameter estimate otpt (stabilization with noise) Example: Unstable-zero dnamics nknown parameter reference otpt (set-point with noise) 61

62 Example: Kinematic niccle robot x 2 θ 1 forward velocit 2 anglar velocit x 1 p 1, p 2 nknown parameters determined b the radis of the driving wheel and the distance between them This sstem cannot be stabilized b a continos time-invariant controller. The candidate controllers were themselves hbrid Example: Kinematic niccle robot 62

63 Example: Indction motor in crrent-fed mode ω is the onl measrable otpt λ 2 rotor flx 2 stator crrents ω rotor anglar velocit τ torqe generated Unknown parameters: τ L [τ min, τ max ] load torqe R [R min, R max ] rotor resistance Off-the-shelf field-oriented candidate controllers: 63