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

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1 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- Conference workshop #4, 24 June 2014, Strasbourg.

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 Satisfiability mod ODE approach n Hybrid reachability approach n Research directions

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

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 Satisfiability mod ODE approach n Hybrid reachability approach 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).

19 17 Outline n Hybrid dynamical systems n Set membership estimation n Satisfiability mod ODE approach n Hybrid reachability approach n Research directions

20 A SAT mod ODE approach n Bounded Model Checking l Can the system reach an unsafe state within k transition steps? l Check satisfiability of a SAT Mod ODE formula 18

21 A SAT mod ODE approach n Bounded Model Checking l Can the system reach an unsafe state within k transition steps? l Check satisfiability of a SAT Mod ODE formula 18

22 A SAT mod ODE approach n Bounded Model Checking l Can the system reach an unsafe state within k transition steps? l Check satisfiability of a SAT Mod ODE formula 18

23 A SAT mod ODE approach n Bounded Model Checking l Can the system reach an unsafe state within k transition steps? l Check satisfiability of a SAT Mod ODE formula 18

24 A SAT mod ODE approach 19

25 19 A SAT mod ODE approach n Bounded Model Checking l Model given: init[0] variables definition trans[i,i+1] dynamic transitions l Target: prop[k]

26 A SAT mod ODE approach n Bounded Model Checking l Model given: init[0] variables definition trans[i,i+1] dynamic transitions l Target: prop[k] l SAT solvers solve the following k-unwound formulas: init[0] prop[0] init[0] trans[0,1] prop[1] init[0] trans[0,1] trans[1,2] prop[2] init[0] trans[0,1] trans[1,2] trans[2,3] prop[3] init[0] trans[0,1] prop[k] l If one formula is satisfiable, the desired property was violated 19

27 A SAT mod ODE approach n isat [Fränzle, et al. 2007] l Interval constraint propagation. Branching... l Conflict-driven learning... n isat-ode [Eggers, et al. 2008, 2014] l ODE allowed in transition dynamics l Enclosure of ODE solutions... l VNODE-LP l (Nedialkov, 2008) l Bounding systems l (Ramdani, et al., 2009) 20

28 A SAT mod ODE approach n isat [Fränzle, et al. 2007] l Interval constraint propagation. Branching... l Conflict-driven learning... n isat-ode [Eggers, et al. 2008, 2014] l ODE allowed in transition dynamics l Enclosure of ODE solutions... l VNODE-LP l (Nedialkov, 2008) l Bounding systems l (Ramdani, et al., 2009) 20

29 21 How to use a SAT mod ODE solver to address set-membership estimation with hybrid systems? l Eggers, Ramdani, Nedialkov, Fraenzle, IFAC SYSID 2012, Bruxelles, pp

30 22 A SAT mod ODE approach n Set inversion. Parameter/State estimation l Branch-&-bound, branch-&-prune, interval contractors (Jaulin, et al. 93) (Raïssi et al., 2004)!!!!! l Unsatisfiability prunes off infeasible parts acts as a contractor characterizes outer approximation!!

31 23 A SAT mod ODE approach l Predicative encoding and set membership estimation. instantiation of variables bounds. a priori knowledge... discrete or continuous transitions

32 23 A SAT mod ODE approach l Predicative encoding and set membership estimation. instantiation of variables bounds. a priori knowledge... discrete or continuous transitions feasible domains for measurements

33 24 A SAT mod ODE approach Outer approximation in Set Membership Estimation SAT mod ODE solver : isat-ode 1. if needed, infere number of transition steps 2. Deduction & no splitting : prune off inconsistant ranges 3. Deduction & splitting : search candidate boxes 4. Evaluate range bounds for solution boxes.

34 25 Set Membership Estimation n Example 1. l parameter estimation in a chemical reaction l a single measurement, uncertain time.

35 25 Set Membership Estimation n Example 1. l parameter estimation in a chemical reaction l a single measurement, uncertain time.

36 25 Set Membership Estimation n Example 1. l parameter estimation in a chemical reaction l a single measurement, uncertain time. init[0] target[1]

37 26 Set Membership Estimation n Example 1. l parameter estimation in a chemical reaction init[0] target[1] CPU time 8h to 56h according tuning. (Intel Core i7 2.6 GHz)

38 27 Set Membership Estimation n Example 2. l Estimation of Hybrid System l (Stursberg, et al. 1997)

39 27 Set Membership Estimation n Example 2. l Estimation of Hybrid System l (Stursberg, et al. 1997) a. Measuring at three discrete time instants. b. Measuring height threshold (Koutsoukos, 2003).

40 Set Membership Estimation n Example 2a. l Estimation of Hybrid System l (Stursberg, et al. 1997) l Measuring at discrete time instants, (3x) added as new transition steps.! l Stage 1 l Infere minimal number of unwinding steps target = one measure. missed or final time not reached. l UNSAT depth 7 7 = max numb. mode switch + 3 measurements reached! 28

41 Set Membership Estimation n Example 2a. l Estimation of Hybrid System l (Stursberg, et al. 1997) l Measuring at discrete time instants,... added as new transition steps.! l Stage 2 l target = all measurements used. l reduce range of initial conditions, l bound range of hybrid state variables. 29

42 Set Membership Estimation n Example 2a. l Measuring at 3 discrete time instants, time x1 x BMC depths BMC depths BMC depths mode_below mode_above mode_dont_care BMC depths 30

43 Set Membership Estimation n Example 2a. l Measuring at 3 discrete time instants, time x BMC depths x mode_below mode_above mode_dont_care BMC depths BMC depths BMC depths 30 CPU time on AMD Opteron GHz 64bits Linux depth 7 87mn bounds 5.7h

44 31 Set Membership Estimation n Example 2b (Koutsoukos, 2003) l Measuring height thresholds l Assume uncertain measurement time.

45 Set Membership Estimation n Example 2b (Koutsoukos, 2003) x1 l Measuring height thresholds l Assume uncertain measurement time time x time mode_below mode_above time 31

46 Set Membership Estimation n Example 2b (Koutsoukos, 2003) x1 l Measuring height thresholds l Assume uncertain measurement time time x mode_below time CPU time on AMD Opteron GHz 64bits Linux bounds 7.3h. mode_above time 31

47 32 Outline n Hybrid dynamical systems n Set membership estimation n Satisfiability mod ODE approach n Hybrid reachability approach n Research directions

48 33 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 )

49 34 Reachability based approach n Predictor-Corrector approach for hybrid systems

50 35 Hybrid Reachability Computation n Guaranteed event detection & localization l An interval constraint propagation approach l (Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011)

51 35 Hybrid Reachability Computation n Guaranteed event detection & localization l An interval constraint propagation approach l (Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011)

52 35 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])

53 35 Hybrid Reachability Computation n Guaranteed event detection & localization l An interval constraint propagation approach l (Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011)

54 36 Hybrid Reachability Computation n Guaranteed event detection & localization l An interval constraint propagation approach l (Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011)

55 Hybrid Reachability Computation n Detecting and localizing events l Improved and enhanced version. A faster version. l (Maïga, et al., IEEE CDC 2013, ECC 2014) 37

56 Hybrid Reachability Computation n Detecting and localizing events l Improved and enhanced version. A faster version. l (Maïga, et al., IEEE CDC 2013, ECC 2014) 37

57 Hybrid Reachability Computation n Detecting and localizing events l Improved and enhanced version l (Maïga, et al., IEEE CDC 2013, ECC 2014) 38

58 Hybrid Reachability Computation n Detecting and localizing events l Improved and enhanced version l (Maïga, et al., IEEE CDC 2013, ECC 2014) 39

59 Hybrid Reachability Computation n Detecting and localizing events l Improved and enhanced version l (Maïga, et al., IEEE CDC 2013, ECC 2014) Bouncing ball in 2D. py x2 px x1 40

60 Hybrid Reachability Computation n Detecting and localizing events l Improved and enhanced version l (Maïga, et al., IEEE CDC 2013, ECC 2014) Bouncing ball in 2D. py px py 1 3 x x1 px 40

61 41 Outline n Hybrid dynamical systems n Set membership estimation n Satisfiability mod ODE approach n Hybrid reachability approach n Research directions

62 42 Research directions n Contractors for a hybrid dynamical systems! l To build upon a hybrid reachability approach n Effective methods 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

63 43 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 A.Eggers, N.Ramdani, N.S.Nedialkov, M.Fränzle. Improving the SAT Modulo ODE Approach to Hybrid Systems Analysis by Combining Different Enclosure Methods, Software & Systems Modeling, doi /s 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, 2014.

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

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,