Stability and performance of hybrid systems

Transcription

1 Stability and performance of hybrid systems Christophe PRIEUR Gipsa-lab CNRS Grenoble 10 juillet 2014 GT SDH ENSAM, Paris 1/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

2 Premiers retours du comité de direction Un grand merci à Pierre pour les 5 années de co-animation. Appel à candidature pour la co-animation proposition au responsable d axe validation par le responsable d axe (avec directeurs du GDR) le changement d animateurs devient effectif, et les sites web du GDR et GT peuvent être mis à jour. 2/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

3 Premiers retours du comité de direction Un grand merci à Pierre pour les 5 années de co-animation. Appel à candidature pour la co-animation proposition au responsable d axe validation par le responsable d axe (avec directeurs du GDR) le changement d animateurs devient effectif, et les sites web du GDR et GT peuvent être mis à jour. 2/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

4 Messages à l intention des GT Principes de base: confiance et libertés. Possibilités de financer une mission par exemple pour un invité étranger ou pour des journées de formations ou pour des réunions exceptionnelles (ouvertures vers des partenaires industriels, et autres d autres GT ou GDR) 1 Bilan après chaque réunion, et garder le site web à jour Evolution des thèmes du GT? Nouveaux GT? Ex: optimisation, bio... Messages à l intention des membres du GDR Macs Refonte du site web du GDR est presque finie Profil personnel et liste de diffusion sont en préparation Rôle des listes de diffusion redéfini 1 comme pour la réunion commune avec MOSAR des 25 et 26 mars 3/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

5 Messages à l intention des GT Principes de base: confiance et libertés. Possibilités de financer une mission par exemple pour un invité étranger ou pour des journées de formations ou pour des réunions exceptionnelles (ouvertures vers des partenaires industriels, et autres d autres GT ou GDR) 1 Bilan après chaque réunion, et garder le site web à jour Evolution des thèmes du GT? Nouveaux GT? Ex: optimisation, bio... Messages à l intention des membres du GDR Macs Refonte du site web du GDR est presque finie Profil personnel et liste de diffusion sont en préparation Rôle des listes de diffusion redéfini 1 comme pour la réunion commune avec MOSAR des 25 et 26 mars 3/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

6 Messages à l intention des GT Principes de base: confiance et libertés. Possibilités de financer une mission par exemple pour un invité étranger ou pour des journées de formations ou pour des réunions exceptionnelles (ouvertures vers des partenaires industriels, et autres d autres GT ou GDR) 1 Bilan après chaque réunion, et garder le site web à jour Evolution des thèmes du GT? Nouveaux GT? Ex: optimisation, bio... Messages à l intention des membres du GDR Macs Refonte du site web du GDR est presque finie Profil personnel et liste de diffusion sont en préparation Rôle des listes de diffusion redéfini 1 comme pour la réunion commune avec MOSAR des 25 et 26 mars 3/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

7 4/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

8 Stability and performance of hybrid systems Christophe PRIEUR Gipsa-lab CNRS Grenoble 10 juillet 2014 GT SDH ENSAM, Paris 5/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

9 Motivations to consider hybrid systems Discontinuous controller may be interesting for the stabilization of nonlinear control systems Use of discontinuous controllers for nonholonomic systems [Astolfi 1996], [Sontag, 1999] but not robust to unstructured noise Patchy feedbacks [Ancona, Bressan, 1999, 2002]: it asks for sufficiently regular perturbations 6/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

10 Two motivations Using jumps between several piecewise controllers may be useful for the asymp. stability, see [Hespanha, Liberzon, and Morse, 04], [van der Schaft, and Schumacher, 2000], [Tavernini,1997], [Michel and Hou, 1999] among others. for the performance, as the speed of convergence or the L 2 -convergence [Beker, Hollot, and Chait, 2004]. Mainly energy-based controllers. Motivation #1 for the study of hybrid systems We get discrete-time controllers with a continuous dynamics between jumps. The closed-loop system has a hybrid dynamics. 7/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

11 Two motivations Using jumps between several piecewise controllers may be useful for the asymp. stability, see [Hespanha, Liberzon, and Morse, 04], [van der Schaft, and Schumacher, 2000], [Tavernini,1997], [Michel and Hou, 1999] among others. for the performance, as the speed of convergence or the L 2 -convergence [Beker, Hollot, and Chait, 2004]. Mainly energy-based controllers. Motivation #1 for the study of hybrid systems We get discrete-time controllers with a continuous dynamics between jumps. The closed-loop system has a hybrid dynamics. 7/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

12 Two motivations Using jumps between several piecewise controllers may be useful for the asymp. stability, see [Hespanha, Liberzon, and Morse, 04], [van der Schaft, and Schumacher, 2000], [Tavernini,1997], [Michel and Hou, 1999] among others. for the performance, as the speed of convergence or the L 2 -convergence [Beker, Hollot, and Chait, 2004]. Mainly energy-based controllers. Motivation #1 for the study of hybrid systems We get discrete-time controllers with a continuous dynamics between jumps. The closed-loop system has a hybrid dynamics. 7/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

13 Motivation #2: quantizer and trig Consider closed-loop system equipped a quantizer in the output and in the input, and subject to sample and hold in the measurements and in the control. P t k? trig. Quantizer q µ? q ν? Quantizer trig. τ j? C 8/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

14 Motivation #2: quantizer and trig Consider closed-loop system equipped a quantizer in the output and in the input, and subject to sample and hold in the measurements and in the control. P t k? trig. Quantizer q µ? q ν? Quantizer trig. τ j? C 8/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

15 Motivation #2: quantizer and trig Consider closed-loop system equipped a quantizer in the output and in the input, and subject to sample and hold in the measurements and in the control. P t k? trig. Quantizer q µ? q ν? Quantizer trig. τ j? C 8/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

16 Large and (more) recent literature on that. [Liberzon, 2003]: linear systems and limited information [Anta, Tabuada, 2010]: sample or not sample? [Nair, Fagnani, Zampieri, Evans. Feedback control under data rate constraints: An overview. Proceedings of the IEEE, 2007] [Donkers, Heemels]: event-triggered in the output with performance, [Girard, 2014]: dynamic event-trig. [Seuret, CP, 2011] and [Postoyan, Anta, Nesic, Tabuada, 2011]: Lyapunov techniques and nonlinear systems for event-trig. Motivation #2 for limited information/control systems The flavor of the dynamics is hybrid. 9/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

17 Motivations Three objectives of this talk: Present a complete frame for hybrid systems Lyapunov theory Robustness for free... Use for the synthesis of stabilizing controllers for nonlinear control systems control systems with isolated nonlinearities (such as saturations) using constructive methods and with a robustness issue. Study for limited information systems here: only linear control systems quantizers + event-trig in the output and in the input 10/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

18 Motivations Three objectives of this talk: Present a complete frame for hybrid systems Lyapunov theory Robustness for free... Use for the synthesis of stabilizing controllers for nonlinear control systems control systems with isolated nonlinearities (such as saturations) using constructive methods and with a robustness issue. Study for limited information systems here: only linear control systems quantizers + event-trig in the output and in the input 10/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

19 Motivations Three objectives of this talk: Present a complete frame for hybrid systems Lyapunov theory Robustness for free... Use for the synthesis of stabilizing controllers for nonlinear control systems control systems with isolated nonlinearities (such as saturations) using constructive methods and with a robustness issue. Study for limited information systems here: only linear control systems quantizers + event-trig in the output and in the input 10/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

20 Outline Part I: Hybrid systems I.1 Basic ideas I.2 Hybrid feedback laws I.3 Robustness for free and other fundamental results Part II: Synthesis of hybrid controllers II.1 For general nonlinear control systems II.2 Reset systems as a particular class of hybrid systems for linear control systems and then with a saturation in the input Optimal reset controllers Part III: Quantizer and event-trig III.1 Linear systems with event-trig only III.2 With an additional quantizer Non-synchronous trig Preliminary results, generalization to NL syst. under progress 11/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

21 Outline Part I: Hybrid systems I.1 Basic ideas I.2 Hybrid feedback laws I.3 Robustness for free and other fundamental results Part II: Synthesis of hybrid controllers II.1 For general nonlinear control systems II.2 Reset systems as a particular class of hybrid systems for linear control systems and then with a saturation in the input Optimal reset controllers Part III: Quantizer and event-trig III.1 Linear systems with event-trig only III.2 With an additional quantizer Non-synchronous trig Preliminary results, generalization to NL syst. under progress 11/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

22 Outline Part I: Hybrid systems I.1 Basic ideas I.2 Hybrid feedback laws I.3 Robustness for free and other fundamental results Part II: Synthesis of hybrid controllers II.1 For general nonlinear control systems II.2 Reset systems as a particular class of hybrid systems for linear control systems and then with a saturation in the input Optimal reset controllers Part III: Quantizer and event-trig III.1 Linear systems with event-trig only III.2 With an additional quantizer Non-synchronous trig Preliminary results, generalization to NL syst. under progress 11/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

23 Outline Part I: Hybrid systems I.1 Basic ideas I.2 Hybrid feedback laws I.3 Robustness for free and other fundamental results Part II: Synthesis of hybrid controllers II.1 For general nonlinear control systems II.2 Reset systems as a particular class of hybrid systems for linear control systems and then with a saturation in the input Optimal reset controllers Part III: Quantizer and event-trig III.1 Linear systems with event-trig only III.2 With an additional quantizer Non-synchronous trig Preliminary results, generalization to NL syst. under progress 11/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

24 I.1 Hybrid systems. Basic ideas (nominal) continuous-time system: ẋ = f (x) 2/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

25 I.1 Hybrid systems. Basic ideas continuous dynamics with uncertainties, noise in the loop, perturbations in the dynamics: ẋ F (x) E.g. F (x) f (x + small error) = ε>0 con ( f (x + B(0, ε)) ) 3/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

26 I.1 Hybrid systems. Basic ideas continuous dynamics with uncertainties and state constraint ẋ F (x), x C C 4/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

27 I.1 Hybrid systems. Basic ideas continuous dynamics ẋ F (x), x C discrete dynamics: discrete time system x + G(x), x D D 5/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

28 I.1 Hybrid systems. Basic ideas continuous dynamics ẋ F (x), x C C discrete dynamics x + G(x), x D mixed dynamics discrete/continuous D ẋ F (x), x C x + G(x), x D 6/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

29 Hybrid systems continuous dynamics with a state constraint ẋ F (x), x C C discrete dynamics x + G(x), x D mixed dynamics discrete/continuous D ẋ F (x), x C x + G(x), x D 7/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

30 I.2 Hybrid feedback laws Given a nonlinear control system: ẋ = f (x, u) ( ) x R n, u U. Definition [CP, Goebel, Teel, 07] A hybrid feedback law consists in a set Q (with a total order, e.g. Q = {0, 1}, Q = {0, 1, 2,...}) for each q Q, sets C q R n and D q R n, a function k q : C q U, a set-valued function G q : D q Q. ( ) closing the loop with u = k q yields a hybrid system 18/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

31 I.2 Hybrid feedback laws Given a nonlinear control system: ẋ = f (x, u) ( ) x R n, u U. Definition [CP, Goebel, Teel, 07] A hybrid feedback law consists in a set Q (with a total order, e.g. Q = {0, 1}, Q = {0, 1, 2,...}) for each q Q, sets C q R n and D q R n, a function k q : C q U, a set-valued function G q : D q Q. ( ) closing the loop with u = k q yields a hybrid system 18/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

32 The closed-loop system may be rewritten as { ẋ Fq (x), x C q, q + G q (x), x D q, (H) where F q (x) contains f (x, k q (x)). Given a hybrid (H), assume the regularity properties (A0) Q (at most) countable. And, q Q, (A q 1) C q and D q are closed (A q 2) F q : R n R n is outer semicontinuous and locally bounded, and F q (x) is nonempty and convex for all x C q. (A q 3) G q : R n Q is outer semicontinuous and locally bounded, and G q (x) is nonempty for all x D q. (A0) (A3) existence of solutions [Goebel, Teel, 06] 19/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

33 The closed-loop system may be rewritten as { ẋ Fq (x), x C q, q + G q (x), x D q, (H) where F q (x) contains f (x, k q (x)). Given a hybrid (H), assume the regularity properties (A0) Q (at most) countable. And, q Q, (A q 1) C q and D q are closed (A q 2) F q : R n R n is outer semicontinuous and locally bounded, and F q (x) is nonempty and convex for all x C q. (A q 3) G q : R n Q is outer semicontinuous and locally bounded, and G q (x) is nonempty for all x D q. (A0) (A3) existence of solutions [Goebel, Teel, 06] 19/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

34 The closed-loop system may be rewritten as { ẋ Fq (x), x C q, q + G q (x), x D q, (H) where F q (x) contains f (x, k q (x)). Given a hybrid (H), assume the regularity properties (A0) Q (at most) countable. And, q Q, (A q 1) C q and D q are closed (A q 2) F q : R n R n is outer semicontinuous and locally bounded, and F q (x) is nonempty and convex for all x C q. (A q 3) G q : R n Q is outer semicontinuous and locally bounded, and G q (x) is nonempty for all x D q. (A0) (A3) existence of solutions [Goebel, Teel, 06] 19/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

35 I.3 Robustness for free of hybrid systems Given a continuous ρ : R n [0, ), vanishing only at x. Let the perturbed system: { ẋ F ρ q (x), x Cq ρ, q + Gq ρ (x), x Dq, ρ (H ρ ) with F ρ q (x) := con F q ( (x + ρ(x)b ) Cq ) + ρ(x)b G ρ q (x) := G q ( (x + ρ(x)b ) Dq ) C ρ q := {x R n ( x + ρ(x)b ) C q } D ρ q := {x R n ( x + ρ(x)b ) D q } 0/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

36 Robustness for free To guarantee the robustness: (A4) The family {C q } q Q is locally finite covering R n ; (A5) The functions G q : R n Q are locally bounded wrt x uniformly with respect to q; (A6) For each q Q, C q D q = R n. Theorem [CP, Goebel, Teel, 07] Assume that the hybrid system (H) satisfy (A0), (A4), (A5), (A6), and (A q 1), (A q 2), (A q 3) for each q in Q. If (H) is asymptotically stable. Then there exists ρ such that (H ρ ) is asymptotically stable. 21/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

37 Theory on hybrid systems Fundamental stability theory results: LaSalle s invariance principle Converse Lyapunov theorems (smooth) For free robustness of stability. In turn, these results give us: new tools for designing hybrid control systems, a better understanding of closed-loop robustness, and motivation to look for Lyapunov proofs of stability. Let us apply this theory for the design of hybrid stabilizers. 22/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

38 Theory on hybrid systems Fundamental stability theory results: LaSalle s invariance principle Converse Lyapunov theorems (smooth) For free robustness of stability. In turn, these results give us: new tools for designing hybrid control systems, a better understanding of closed-loop robustness, and motivation to look for Lyapunov proofs of stability. Let us apply this theory for the design of hybrid stabilizers. 22/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

39 Theory on hybrid systems Fundamental stability theory results: LaSalle s invariance principle Converse Lyapunov theorems (smooth) For free robustness of stability. In turn, these results give us: new tools for designing hybrid control systems, a better understanding of closed-loop robustness, and motivation to look for Lyapunov proofs of stability. Let us apply this theory for the design of hybrid stabilizers. 22/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

40 II Synthesis of hybrid controllers Let the nonlinear control system ẋ = f (x, u) ( ) Assume it is asymp. controllable in x. Based on [Ancona, Bressan, 99] we may build a hybrid feedback law. Robustness is for free. Theorem [CP, Goebel, Teel, 07] Assume the nonlinear system ( ) asymp. controllable in x. Then there exists a hybrid feedback such that the hybrid closed-loop system is robustly asymp. stable. 23/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

41 II Synthesis of hybrid controllers Let the nonlinear control system ẋ = f (x, u) ( ) Assume it is asymp. controllable in x. Based on [Ancona, Bressan, 99] we may build a hybrid feedback law. Robustness is for free. Theorem [CP, Goebel, Teel, 07] Assume the nonlinear system ( ) asymp. controllable in x. Then there exists a hybrid feedback such that the hybrid closed-loop system is robustly asymp. stable. 23/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

42 Connection with Control Lyapunov functions [Sontag, 83] : Asymp. controllability Control Lyapunov function In the previous result, we use the controllability patch by patch Thus we have the notion of Patchy Control Lyapunov Functions This is a ordered family of CLFs From such a hybrid feedback, we may build a PCLF The reciprocal construction is also possible [Goebel, CP, Teel, 09] 24/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

43 Connection with Control Lyapunov functions [Sontag, 83] : Asymp. controllability Control Lyapunov function In the previous result, we use the controllability patch by patch Thus we have the notion of Patchy Control Lyapunov Functions This is a ordered family of CLFs From such a hybrid feedback, we may build a PCLF The reciprocal construction is also possible [Goebel, CP, Teel, 09] 4/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

44 Connection with Control Lyapunov functions [Sontag, 83] : Asymp. controllability Control Lyapunov function In the previous result, we use the controllability patch by patch Thus we have the notion of Patchy Control Lyapunov Functions This is a ordered family of CLFs From such a hybrid feedback, we may build a PCLF The reciprocal construction is also possible [Goebel, CP, Teel, 09] 4/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

45 Illustration #2: adding a hybrid loop for the performance Considering a control system in closed-loop with a (maybe non-stabilizing) controller, a new hybrid loop is suggested to improve the performance guarantee the asymp. stability Aim of this part Find the best controller for a particular class of nonlinear systems Class of reset systems. Control systems with an isolated nonlinearity. Other studies for nonlinear control systems exist (more at the end of this section). 25/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

46 Illustration #2: adding a hybrid loop for the performance Considering a control system in closed-loop with a (maybe non-stabilizing) controller, a new hybrid loop is suggested to improve the performance guarantee the asymp. stability Aim of this part Find the best controller for a particular class of nonlinear systems Class of reset systems. Control systems with an isolated nonlinearity. Other studies for nonlinear control systems exist (more at the end of this section). 25/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

47 Illustration #2: adding a hybrid loop for the performance Considering a control system in closed-loop with a (maybe non-stabilizing) controller, a new hybrid loop is suggested to improve the performance guarantee the asymp. stability Aim of this part Find the best controller for a particular class of nonlinear systems Class of reset systems. Control systems with an isolated nonlinearity. Other studies for nonlinear control systems exist (more at the end of this section). 25/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

48 Illustration #2: adding a hybrid loop for the performance Considering a control system in closed-loop with a (maybe non-stabilizing) controller, a new hybrid loop is suggested to improve the performance guarantee the asymp. stability Aim of this part Find the best controller for a particular class of nonlinear systems Class of reset systems. Control systems with an isolated nonlinearity. Other studies for nonlinear control systems exist (more at the end of this section). 25/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

49 II.2 Class of reset systems First Order Reset Element (FORE) [Ne si`c, Zaccarian, Teel, 2005]: ẋ r = λ r x r + B r e if ey r 0, x r + = 0 if ey r 0. The output is y r = x r. The flow and the jump conditions are defined to improve the performance, in terms of time response or in terms of the overshoot of y 26/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

50 II.2 Class of reset systems First Order Reset Element (FORE) [Ne si`c, Zaccarian, Teel, 2005]: ẋ r = λ r x r + B r e if ey r 0, x r + = 0 if ey r 0. The output is y r = x r. The flow and the jump conditions are defined to improve the performance, in terms of time response or in terms of the overshoot of y 26/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

51 FORE controlling a linear system r e FORE u P(s) y P(s) = 1 s P(s) = s+1 s(s+2) L2 gain in function of λ r [Ne si`c, Zaccarian, Teel, 2005] 7/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

52 Output of a FORE in function of the λ r 10 closed loop output y r = 3 r = 1 =1 r r =5 r = FORE output x r r = 1 r =1 r = Main drawback= Large intermediate values In presence of saturations: may reduce the performance or induce instability 28/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

53 Output of a FORE in function of the λ r 10 closed loop output y r = 3 r = 1 =1 r r =5 r = FORE output x r r = 1 r =1 r = Main drawback= Large intermediate values In presence of saturations: may reduce the performance or induce instability 28/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

54 Reset systems in presence of saturations Control system Closed-loop system ẋ p = A p x p + B p sat(y r ), y = C p x p. ẋ = A f x + BΨ(Kx) if x C, x + = A j x if x D, y = Cx.» Ap B A f = pd r C p B pc r B r C p A r A j =» I 0 0 0, C = ˆ C p 0.» Bp, B = 0, K = ˆ C r D r C p, 29/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

55 Nonlinearity Ψ(Kx) defined by ψ(kx) = sat(kx) Kx. The flow and the jump sets C and D are C = {x R n ; x Q MQx 0} D = {x R n ; x Q MQx 0} [ ] [ ] 0 1 Cp 0 with M = and Q = 1 0 D r C p C r the plant state does not jump (only the control variable). 30/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

56 Piecewise quadratic Lyapunov functions [Loquen, CP, Teel et al, 2010] Define Z = [I n 2 0 (n 2) 2 ] and Θ i = [ 0 1 n 2 sin(θ i ) cos(θ i ) ] T. Class of Lyapunov function: piecewise quadratic functions Definition written in terms of sectors Π i = {x R n ; x S i x 0} from angles θ i. One Lyapunov function by sector V i (x) = x P i x. The continuous time system is unstable but the reset system is stable! 1/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

57 Using quadratic piecewise Lyapunov functions Theorem (see [Loquen, CP, Teel et al, 2010] for more details) If the following linear matrix inequalities in the variables P i = P T i > 0, τ Fi 0, i = 1,..., N, P R = P T R > 0, τ J, τ ɛ1, τ ɛ2 0, γ > 0 are feasible: AT P i + P i A + τ Fi S i P i B d C T γi 0 < 0, i = 1,..., N, γi Z(AT P R + P R A)Z T ZP R B d ZC T γi 0 < 0, γi A T r P 1 A r P R + τ J S R 0 A T r P 1 A r P 1 + τ ɛ1 S ɛ1 0 A T r P 1 A r P N + τ ɛ2 S ɛ2 0 Θ T i (P i P i+1 ) Θ i = 0, i = 0,..., N 1, Θ T N (P N P R )Θ N = 0 then i=0,...,n {x R n ; x P i x 1 if x Π i } is a stability region and the L 2 gain from w to y which is smaller than γ. 2/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

58 Example Let us analysis the stability of r e FORE u P(s) y with (P) given by ẋ p = 0.1x p + sat(y r ) y = x p 3/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

59 Time evolution plant output y(t) d r =5 r =1 r = plant input u(t)=sat(x r) r =5 r =1 r = Top: y(t). Down: u(t) with different λ r 34/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

60 Stability domain 35/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

61 Other results For nonlinear control systems (asymp. stable or not), how to add a hybrid to improve the performance, or to guarantee the asymp. stability. See [CP, Tarbouriech, Zaccarian, 2013] for criterion based on L 2 performance speed of convergence reduction of the output overshoot for synthesis criterion of a hybrid loop. See also ANR project ArHyCo. 36/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

62 III Quantizer and trig Consider a system in closed-loop with a controller, subject to a quantizer in the output and in the input, and subject to sample and hold in the measurements and in the control. P t k? trig. Quantizer q µ? q ν? Quantizer trig. τ j? C A deeper introduction of this problem in the next talk, by Romain? 37/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

63 III Quantizer and trig Consider a system in closed-loop with a controller, subject to a quantizer in the output and in the input, and subject to sample and hold in the measurements and in the control. P t k? trig. Quantizer q µ? q ν? Quantizer trig. τ j? C A deeper introduction of this problem in the next talk, by Romain? 37/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

64 III Quantizer and trig Consider a system in closed-loop with a controller, subject to a quantizer in the output and in the input, and subject to sample and hold in the measurements and in the control. P t k? trig. Quantizer q µ? q ν? Quantizer trig. τ j? C A deeper introduction of this problem in the next talk, by Romain? 37/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

65 Objectives Design Asynchronous event-triggered sampling: The sampling times for outputs and inputs are determined based on certain event-triggering strategies. Dynamic quantization: An update rule is specified for the design parameter (seen as discrete variable) of the quantizers. ( ) ( ) x x = ν k q, and q µk = µ k q q νk ν k µ k Restriction: only linear systems controllers are considered. Approach: we will study each element separately. 38/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

66 Objectives Design Asynchronous event-triggered sampling: The sampling times for outputs and inputs are determined based on certain event-triggering strategies. Dynamic quantization: An update rule is specified for the design parameter (seen as discrete variable) of the quantizers. ( ) ( ) x x = ν k q, and q µk = µ k q q νk ν k µ k Restriction: only linear systems controllers are considered. Approach: we will study each element separately. 38/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

67 System to be controlled P : { ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) Instead of a classic linear controller: { ż(t) = Az(t) + Bu(t) + L(y(t) Cz(t)) (1) u(t) = Kz(t), composed of a Luenberger observer and of a static controller, let us consider the following class of controllers: { ż(t) = Az(t) + Bu(t) + L(qν (y(t k )) Cz(t k )), t [t k, t k+1 ) C : u(t) = q µ (Kz(τ j )), t [τ j, τ j+1 ) (2) Questions: how to design the sequence t k, τ j? How to design the quantizing parameters in q µ and q ν? 39/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

68 To be more precise. Let us assume (A1) (A, B) stabilizable: P c > 0 such that for a given Q c > 0 (A + BK) P c + P c (A + BK) = Q c (A2) (A, C) detectable: P o > 0 such that for a given Q o > 0 (A LC) P o + P o (A LC) = Q o. (A3) All eigenvalues of the matrix A are positive and real. We could avoid (A3). 0/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

69 Sampling times for the the inputs of the controller Intuitive idea Use a separation principle, as for the classical linear controller. For the classical output design method, it is considered the state estimation error x = x z, and let ỹ := y Cz = C x, then we have x(t) = (A LC) x(t) + L(ỹ(t) ỹ(t k )). (3) And derive the state estimation error using only the sampled outputs. t hyp k+1 := sup{t : ỹ(t) ỹ(t k) α x(t), t t k }, as done in [Tabuada], in particular. The trouble with this formula is that we do not know x(t). 1/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

70 Sampling times for the the inputs of the controller Intuitive idea Use a separation principle, as for the classical linear controller. For the classical output design method, it is considered the state estimation error x = x z, and let ỹ := y Cz = C x, then we have x(t) = (A LC) x(t) + L(ỹ(t) ỹ(t k )). (3) And derive the state estimation error using only the sampled outputs. t hyp k+1 := sup{t : ỹ(t) ỹ(t k) α x(t), t t k }, as done in [Tabuada], in particular. The trouble with this formula is that we do not know x(t). 1/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

71 However we could copy the dynamics, to know the current state from the past sampled measures. We could use the variation of constants formula to find a relation between Ỹ k,η := ỹ(t k ) ỹ(t k 1 ). ỹ(t k η+1 ). (4) and x(t), where η is observability index of the pair (A, C). It is done in the next result. 2/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

72 Lemma For any t > t k > t k 1 > > t k η+1, it holds that M k,η (t)ỹk,η = N k,η x(t). (5) where 2 Ce R At k t 3 e As L ds t k Ce R At k 1 t e As L ds 0 0 t M k,η (t) := I ηp ηp k (6) η is observability index of the pair (A, C), 2 N k,η (t) := 6 4 Ce A(t t k ) Ce A(t t k 1). Ce A(t t k η+1) (7) 43/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

73 III.1 Event-trig at the output and at the input Output measurements: Let us define t k recursively defined as follows 0 = t 0 < t 1 <... < t η 1 arbitrary chosen and t k+1 := max{t : e σ(t t k) ỹ(t) ỹ(t k ) α M k,η (t) Ỹk,η, t t k } (8) where α := ε o λ min (Q o ) 2 c P o L, for some ε o (0, 1) and suitable c, σ. Control inputs: Let us define τ j recursively defined as follows τ 0 = 0 and τ j+1 := max{t : z(t) z(τ j ) β z(t), t τ j }. (9) where β := ε c λ min (Q c ) 2 P c B K, for some ε c (0, 1). 44/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

74 III.1 Event-trig at the output and at the input Output measurements: Let us define t k recursively defined as follows 0 = t 0 < t 1 <... < t η 1 arbitrary chosen and t k+1 := max{t : e σ(t t k) ỹ(t) ỹ(t k ) α M k,η (t) Ỹk,η, t t k } (8) where α := ε o λ min (Q o ) 2 c P o L, for some ε o (0, 1) and suitable c, σ. Control inputs: Let us define τ j recursively defined as follows τ 0 = 0 and τ j+1 := max{t : z(t) z(τ j ) β z(t), t τ j }. (9) where β := ε c λ min (Q c ) 2 P c B K, for some ε c (0, 1). 4/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

75 Sampling algorithms It is possible to rewrite the previous dynamics into a hybrid system See e.g. [Seuret, CP, 2011] See also [Abdelrahim, Postoyan, Daafouz and Ne si`c, 2014] for the use of the same framework, without quantizer and with a timer And also the next talk 45/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

76 Sampling algorithms It is possible to rewrite the previous dynamics into a hybrid system See e.g. [Seuret, CP, 2011] See also [Abdelrahim, Postoyan, Daafouz and Ne si`c, 2014] for the use of the same framework, without quantizer and with a timer And also the next talk 45/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

77 Theorem [Tanwani, Fiacchini, CP] Consider the plant (1) and the controller (2) under assumptions (A1)-(A3). If the output measurements are sent to the controller at times t k defined in (8) and the control u( ) is updated at τ j defined in (9), then The closed-loop system (1)-(2) is globally exponentially stable; The minimum inter-sampling time for output measurements separating {t k } k=1 has a positive uniform lower bound on every compact interval [t 0, T ]; The minimum inter-sampling time for control inputs separating {τ j } j=1 has a positive uniform lower bound. In particular k=1 (t k+1 t k ) and j=1 (τ j+1 τ j ) are unbounded. No accumulation point! 6/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

78 Theorem [Tanwani, Fiacchini, CP] Consider the plant (1) and the controller (2) under assumptions (A1)-(A3). If the output measurements are sent to the controller at times t k defined in (8) and the control u( ) is updated at τ j defined in (9), then The closed-loop system (1)-(2) is globally exponentially stable; The minimum inter-sampling time for output measurements separating {t k } k=1 has a positive uniform lower bound on every compact interval [t 0, T ]; The minimum inter-sampling time for control inputs separating {τ j } j=1 has a positive uniform lower bound. In particular k=1 (t k+1 t k ) and j=1 (τ j+1 τ j ) are unbounded. No accumulation point! 6/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

79 Theorem [Tanwani, Fiacchini, CP] Consider the plant (1) and the controller (2) under assumptions (A1)-(A3). If the output measurements are sent to the controller at times t k defined in (8) and the control u( ) is updated at τ j defined in (9), then The closed-loop system (1)-(2) is globally exponentially stable; The minimum inter-sampling time for output measurements separating {t k } k=1 has a positive uniform lower bound on every compact interval [t 0, T ]; The minimum inter-sampling time for control inputs separating {τ j } j=1 has a positive uniform lower bound. In particular k=1 (t k+1 t k ) and j=1 (τ j+1 τ j ) are unbounded. No accumulation point! 6/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

80 Simulation Consider the system with following matrices: [ ] [ ] A := ; B := ; C := [ 1 0 ] Since the matrix A doesn t have any imaginary eigenvalue, it suffices to take η = 2. For this example, we take K = [6, 4.5], L = [2, 1], and Q c = Q o = I 2. We pick ε c = ε o = 0.9, which results in α = 0.03 and β = /53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

81 Figure: Both the states converge to the equilibrium as a result of proposed control strategy. Figure: The sampled output which is transmitted to the controller Figure: Sampled control inputs that are transmitted to the system/ 48/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

82 III.2 Also with quantized output and input Quantized output: ( y ) q ν (y) = νq ν where q( ) denotes the uniform quantizer with sensitivity y and range parameterized by R y, that is: if y R y, then q(y) y y. This way the range of the quantizer q ν ( ) is R y ν and the sensitivity is y ν. Objectives: To design R y and y, and similarly R u and u. 9/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

83 The dynamics of the state estimation error can be written as: ( ( ) y(tk ) x(t) = (A LC) x(t)+l(ỹ(t) ỹ(t k )) ν k L q y(t ) k). ν k ν k and thus with the measurement update rule (8): V o ( x(t)) (1 ε o )λ min (Q o ) x(t) 2 + 2ν k y P o L x(t). Within two measurement updates, the error converges to a ball parameterized by ν k. Trick: select R y and y such that the sequence ν k 0. See [Tanwani, Fiacchini, CP] for more details. 0/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

84 Conclusion Abstract Hybrid systems = systems with a mixed continuous/discrete dynamics We have seen a complete picture for the stability analysis of hybrid systems. In particular: Lyapunov theory For free robustness of stability. Hybrid feedbacks allow to guarantee the stability improve the performance of nonlinear control systems For control systems with a saturation in the input: tractable conditions (LMIs) to compute the optimal reset controller 51/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

85 Conclusion Abstract Hybrid systems = systems with a mixed continuous/discrete dynamics We have seen a complete picture for the stability analysis of hybrid systems. In particular: Lyapunov theory For free robustness of stability. Hybrid feedbacks allow to guarantee the stability improve the performance of nonlinear control systems For control systems with a saturation in the input: tractable conditions (LMIs) to compute the optimal reset controller 51/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

86 Conclusion Abstract Hybrid systems = systems with a mixed continuous/discrete dynamics We have seen a complete picture for the stability analysis of hybrid systems. In particular: Lyapunov theory For free robustness of stability. Hybrid feedbacks allow to guarantee the stability improve the performance of nonlinear control systems For control systems with a saturation in the input: tractable conditions (LMIs) to compute the optimal reset controller 51/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

87 Conclusion Abstract Hybrid systems = systems with a mixed continuous/discrete dynamics We have seen a complete picture for the stability analysis of hybrid systems. In particular: Lyapunov theory For free robustness of stability. Hybrid feedbacks allow to guarantee the stability improve the performance of nonlinear control systems For control systems with a saturation in the input: tractable conditions (LMIs) to compute the optimal reset controller 51/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

88 Conclusion And now? For nonlinear control systems use of observers. Patchy feedbacks depending on the values of the observers [CP, Teel, 2011]: jump rule between 2 output feedback laws For control systems with an isolated nonlinearity anti-windup loop based on a jump rule to improve the performance and/or the stability domain [Tarbouriech, Loquen, CP, 2011]: first result using tractable conditions For control systems with an limited information maximization of the inter-event time, see e.g. [Abdelrahim, Postoyan, Daafouz and Ne si`c, 2014] nonlinear control systems 52/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

89 Conclusion And now? For nonlinear control systems use of observers. Patchy feedbacks depending on the values of the observers [CP, Teel, 2011]: jump rule between 2 output feedback laws For control systems with an isolated nonlinearity anti-windup loop based on a jump rule to improve the performance and/or the stability domain [Tarbouriech, Loquen, CP, 2011]: first result using tractable conditions For control systems with an limited information maximization of the inter-event time, see e.g. [Abdelrahim, Postoyan, Daafouz and Ne si`c, 2014] nonlinear control systems 52/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

90 Conclusion And now? For nonlinear control systems use of observers. Patchy feedbacks depending on the values of the observers [CP, Teel, 2011]: jump rule between 2 output feedback laws For control systems with an isolated nonlinearity anti-windup loop based on a jump rule to improve the performance and/or the stability domain [Tarbouriech, Loquen, CP, 2011]: first result using tractable conditions For control systems with an limited information maximization of the inter-event time, see e.g. [Abdelrahim, Postoyan, Daafouz and Ne si`c, 2014] nonlinear control systems 52/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

91 Thanks to M. Fiacchini, and A. Tanwani, Gipsa-lab, Grenoble T. Loquen, ONERA-Toulouse S. Tarbouriech and L. Zaccarian, LAAS A. Teel, UCSB and you for your attention! 53/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014

92 Thanks to M. Fiacchini, and A. Tanwani, Gipsa-lab, Grenoble T. Loquen, ONERA-Toulouse S. Tarbouriech and L. Zaccarian, LAAS A. Teel, UCSB and you for your attention! 53/53 Christophe PRIEUR Gipsa-lab CNRS Grenoble GT SDH 10/07/2014