Parameter iden+fica+on with hybrid systems in a bounded- error framework

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1 Parameter iden+fica+on with hybrid systems in a bounded- error framework Moussa MAIGA, Nacim RAMDANI, & Louise TRAVE- MASSUYES Université d Orléans, Bourges, and LAAS CNRS Toulouse, France.!! SWIM 2015, Praha 9-11 June 2015

2 2 Model-based FDI input / control hybrid dynamical system measured outputs + - guaranteed decision making hybrid state and parameter estimation reconstructed variables Extend to hybrid dynamical systems set-membership approaches for model-based FDI

3 3 Outline n Hybrid dynamical systems n Set membership estimation n Hybrid reachability approach n Example n Research directions

4 Hybrid Cyber-Physical Systems n Interaction discrete + continuous dynamics n Safety-critical embedded systems n Networked 4 autonomous systems

5 5 Hybrid Cyber-Physical Systems n Modelling hybrid automaton (Alur, et al. 1995) l Non-linear continuous dynamics l Bounded uncertainty Continuous dynamics e : g(x) 0 x = r(e, x) x Init(l) l x Inv(l) l x Inv(l ) ẋ Flow(l, x) ẋ Flow(l,x ) Discrete dynamics

6 6 Hybrid Cyber-Physical Systems n Example : the bouncing ball initial conditions discrete transition jump

7 7 Hybrid Cyber-Physical Systems n Example : the bouncing ball

8 Hybrid Cyber-Physical Systems n Example : the bouncing ball 6 5 Initial conditions 4 freefall Continuous transtions 3 X V Discrete transitions 7

9 8 Estimation of Hybrid State n Modelling hybrid automaton l Nonlinear continuous dynamics l Nonlinear guards sets l Nonlinear reset functions x Init(l) e : g(x) 0 x = r(e, x) l l x Inv(l) x Inv(l ) ẋ Flow(l, x) ẋ Flow(l,x )! l Bounded uncertainty

10 8 Estimation of Hybrid State n Modelling hybrid automaton l Nonlinear continuous dynamics l Nonlinear guards sets l Nonlinear reset functions x Init(l) e : g(x) 0 x = r(e, x) l l x Inv(l) x Inv(l ) ẋ Flow(l, x) ẋ Flow(l,x )! l Bounded uncertainty n Faults as discrete modes!!

11 Estimation of Hybrid State n Modelling hybrid automaton l Nonlinear l Bounded uncertainty n Faults as a mode!!! n FDI State Estimation reconstruct system variables l switching sequence l continuous variables switching sequence for upper bracketting system = t 9

12 10 Outline n Hybrid dynamical systems n Set membership estimation n Hybrid reachability approach n Example n Research directions

13 Nacim Ramdani et al. Set-membership identification Classical Estimation n Classical estimation is probabilistic y s p 1 Confidence sets y s f(p) e(p) Optimisation of J(e(p)) n p 2 Yield valid results only if Y p 1 Perturbations, errors and model uncertainties Méthode ensembliste with statistical f(p) properties known apriori n Y Model structure is correct, no modeling errors Ensemble solution p 2

14 y s p 1 s Set e(p) Membership Estimation Optimisation de J(e(p)) n Unknown but bounded-error framework Y p 1 Y Set Set Membership membersip algorithm Algorithm f(p) Solution set p 2 n p 2 Hypothesis Uncertainties and errors are bounded with known prior bounds A set of feasible solutions S = {p P f(p) Y} = f 1 (Y) P Nacim Ramdani et al. Set-membership identification... 12

15 13 Set Membership Estimation n State estimation with continuous systems l Interval observers (Moisan, et al. 2009), (Meslem & Ramdani, 2011), (Raïssi, et al., 2012), (El Thabet, et al. 2014).

16 14 Set Membership Estimation n State estimation with continuous systems l Prediction - Correction / Filtering approaches (Raïssi et al., 2005), (Meslem, et al, 2010), (Milanese & Novara, 2011), (Kieffer & Walter, 2011)

17 Set Membership Estimation n Set inversion. Parameter estimation l Branch-&-bound, branch-&-prune, interval contractors (Jaulin, et al. 93) (Raïssi et al., 2004) 15

18 16 Set Membership Estimation n State estimation with Continuous systems l Interval observers l Prediction-correction / Filtering approaches Reachability + Set inversion n State estimation with Hybrid systems l Piecewise affine systems (Bemporad, et al. 2005) l ODE + CSP (Goldsztejn, et al., 2010) l Nonlinear case (Benazera & Travé-Massuyès, 2009) l SAT mod ODE (Eggers, et al., 2012) (Maïga, et al. 2015).

19 17 Outline n Hybrid dynamical systems n Set membership estimation n Hybrid reachability approach n Example n Research directions

20 18 Reachability based approach n Predictor-Corrector approach for hybrid systems x Init(l) e : g(x) 0 x = r(e, x) l l x Inv(l) x Inv(l ) ẋ Flow(l, x) ẋ Flow(l,x )

21 19 Reachability based approach n Predictor-Corrector approach for hybrid systems

22 20 Hybrid Reachability Computation n Guaranteed event detection & localization l An interval constraint propagation approach l (Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011)

23 20 Hybrid Reachability Computation n Guaranteed event detection & localization l An interval constraint propagation approach l (Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011)

24 20 Hybrid Reachability Computation n Guaranteed event detection & localization l An interval constraint propagation approach l (Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011) ẋ(t) =f (x, p, t), t 0 t t N, x(t 0 ) [x 0 ], p [p] Time grid t 0 < t 1 < t 2 < < t N apriori[ x j ] actual solution x [x j ] [x j+1 ] Analytical solution for [x](t), t [t j, t j+1 ] [x](t) =[x j ] + k 1 i=1 (t t j ) i f [i] ([x j ], [p]) + (t t j ) k f [k] ([ x j ], [p])

25 20 Hybrid Reachability Computation n Guaranteed event detection & localization l An interval constraint propagation approach l (Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011)

26 21 Hybrid Reachability Computation n Guaranteed event detection & localization l An interval constraint propagation approach l (Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011)

27 Hybrid Reachability Computation n Detecting and localizing events l Improved and enhanced version. A faster version. l (Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014) 22

28 Hybrid Reachability Computation n Detecting and localizing events l Improved and enhanced version. A faster version. l (Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014) 22

29 Hybrid Reachability Computation n Detecting and localizing events l Improved and enhanced version l (Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014) 23

30 Hybrid Reachability Computation n Detecting and localizing events l Improved and enhanced version l (Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014) 24

31 Hybrid Reachability Computation n Detecting and localizing events l Improved and enhanced version l (Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014) Bouncing ball in 2D x x1 25

32 Hybrid Reachability Computation n Detecting and localizing events l Improved and enhanced version l (Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014) Bouncing ball in 2D x2 py px x1 25

33 Set Membership Estimation n Parameter estimation with hybrid systems l Branch-&-bound, branch-&-prune, interval contractors (Eggers, Ramdani et al., 2012), (Maïga, Ramdani et al., 2015) 26

34 Set Membership Estimation n Parameter estimation with hybrid systems l Branch-&-bound, branch-&-prune, interval contractors (Eggers, Ramdani et al., 2012), (Maïga, Ramdani et al., 2015) Need an inclusion test! 26

35 27 Inclusion test

36 27 Inclusion test

37 28 Inclusion test Frontier of the reachable set = union of zonotopes

38 28 Inclusion test Frontier of the reachable set = union of zonotopes

39 28 Inclusion test Frontier of the reachable set = union of zonotopes

40 29 Outline n Hybrid dynamical systems n Set membership estimation n Hybrid reachability approach n Example n Research directions

41 30 Parameter identification n Hybrid Mass-Spring l Velocity-dependent damping. Mode switching driven by velocity.

42 Parameter identification n Hybrid Mass-Spring l case 1 : Parameters acting on continuous dynamics. l CPU time approx. 140 mn! 31

43 Parameter identification n Hybrid Mass-Spring l case 2 : parameters acting on discrete transition. l CPU time approx. 40 mn 32

44 33 Outline n Hybrid dynamical systems n Set membership estimation n Hybrid reachability approach n Example n Research directions

45 34 Research directions n Contractors for hybrid dynamical systems! l To build upon a hybrid reachability approach n Effective methods for set membership estimation l SM parameter estimation l SM hybrid state estimation of nonlinear hybrid systems! n Combine with decision making for FDI l Application to actual hybrid systems

46 35 Focused References n N. Ramdani and N. S.Nedialkov, Computing reachable sets for uncertain nonlinear hybrid systems using interval constraint propagation techniques, Nonlinear Analysis: Hybrid Systems, 5(2), pp , n A.Eggers, N.Ramdani, N.S.Nedialkov, M.Fränzle, Set-Membership Estimation of Hybrid Systems via SAT Mod ODE. in IFAC SYSID pp n M. Maïga, N. Ramdani, L. Travé-Massuyès, A fast method for solving guard set intersection in nonlinear hybrid reachability, in IEEE CDC 2013, pp n M. Maïga, C. Combastel, N. Ramdani, L. Travé-Massuyès, Nonlinear hybrid reachability using set integration and zonotope enclosures. in ECC 2014, pp n M. Maïga, N. Ramdani, L. Travé-Massuyès, C. Combastel, A CSP versus a zonotope-based method for solving guard set intersection in nonlinear hybrid reachability, Mathematics in Computer Science, n M. Maïga, N. Ramdani, L. Travé-Massuyès, Robust fault detection in hybrid systems using setmembership parameter estimation, in IFAC SafeProcess 2015, Paris.

Set- membership es-ma-on of hybrid dynamical systems.

Set- membership es-ma-on of hybrid dynamical systems. Towards model- based FDI for hybrid systems Prof. Nacim RAMDANI Université d Orléans, Bourges. France. nacim.ramdani@univ- orleans.fr!! ECC14 Pre-