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
Model-based FDI hybrid dynamical system input / measured control outputs + guaranteed - decision making hybrid state reconstructed and parameter variables estimation ‣ Extend to hybrid dynamical systems set-membership approaches for model-based FDI 2
Outline n Hybrid dynamical systems n Set membership estimation n Hybrid reachability approach n Example n Research directions 3
Hybrid Cyber-Physical Systems n Interaction discrete + continuous dynamics n Safety-critical embedded systems n Networked autonomous systems 4
Hybrid Cyber-Physical Systems n Modelling → hybrid automaton (Alur, et al. 1995) l Non-linear continuous dynamics l Bounded uncertainty e : g ( x ) ≥ 0 x ′ = r ( e, x ) l ′ l x ′ ∈ Inv( l ′ ) x ∈ Init( l ) x ∈ Inv( l ) x ′ ∈ Flow( l ′ , x ′ ) x ∈ Flow( l, x ) ˙ ˙ Continuous dynamics Discrete dynamics 5
Hybrid Cyber-Physical Systems n Example : the bouncing ball initial conditions discrete transition jump 6
Hybrid Cyber-Physical Systems n Example : the bouncing ball 7
Hybrid Cyber-Physical Systems n Example : the bouncing ball 6 Initial conditions 5 freefall 4 Continuous transtions 3 X 2 1 0 -1 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 V Discrete transitions 7
Estimation of Hybrid State n Modelling → hybrid automaton l Nonlinear continuous dynamics e : g ( x ) ≥ 0 x ′ = r ( e, x ) l Nonlinear guards sets l ′ l x ′ ∈ Inv( l ′ ) x ∈ Init( l ) x ∈ Inv( l ) x ′ ∈ Flow( l ′ , x ′ ) l Nonlinear reset functions x ∈ Flow( l, x ) ˙ ˙ � l Bounded uncertainty 8
Estimation of Hybrid State n Modelling → hybrid automaton l Nonlinear continuous dynamics e : g ( x ) ≥ 0 x ′ = r ( e, x ) l Nonlinear guards sets l ′ l x ′ ∈ Inv( l ′ ) x ∈ Init( l ) x ∈ Inv( l ) x ′ ∈ Flow( l ′ , x ′ ) l Nonlinear reset functions x ∈ Flow( l, x ) ˙ ˙ � l Bounded uncertainty n Faults as discrete modes !! 8
Estimation of Hybrid State n Modelling → hybrid automaton l Nonlinear … l Bounded uncertainty n Faults as a mode !! � n FDI → State Estimation 38000 36000 → reconstruct system variables switching sequence for upper bracketting system 34000 32000 30000 l switching sequence 28000 26000 l continuous variables 24000 22000 20000 18000 0 50 100 150 200 250 300 t 9
Outline n Hybrid dynamical systems n Set membership estimation n Hybrid reachability approach n Example n Research directions 10
Classical Estimation n Classical estimation is probabilistic C onfi de nce sets y s p 1 y s e(p) Optimisation of J ( e ( p )) f ( p ) p 2 n Yield valid results only if Perturbations, errors and model uncertainties with statistical properties known a priori Model structure is correct, no modeling errors 11
Set Membership Estimation n Unknown but bounded-error framework Solution set p 1 Y Y Set Membership Algorithm Set membersip algorithm f ( p ) p 2 n 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 12
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) … . 13
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) … 14
Set Membership Estimation n Set inversion. Parameter estimation l Branch-&-bound, branch-&-prune, interval contractors … (Jaulin, et al. 93) (Raïssi et al., 2004) 15
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). 16
Outline n Hybrid dynamical systems n Set membership estimation n Hybrid reachability approach n Example n Research directions 17
Reachability based approach n Predictor-Corrector approach for hybrid systems e : g ( x ) ≥ 0 x ′ = r ( e, x ) l ′ l x ′ ∈ Inv( l ′ ) x ∈ Init( l ) x ∈ Inv( l ) x ′ ∈ Flow( l ′ , x ′ ) x ∈ Flow( l, x ) ˙ ˙ 18
Reachability based approach n Predictor-Corrector approach for hybrid systems 19
Hybrid Reachability Computation n Guaranteed event detection & localization l An interval constraint propagation approach l (Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011) 20
Hybrid Reachability Computation n Guaranteed event detection & localization l An interval constraint propagation approach l (Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011) 20
Hybrid Reachability Computation n Guaranteed event detection & localization l An interval constraint propagation approach l (Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011) x ( 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 a priori [˜ x j ] actual solution x � [ x j ] [ x j + 1 ] Analytical solution for [ x ]( t ), t ∈ [ t j , t j +1 ] k − 1 ( t − t j ) i f [ i ] ([ x j ] , [ p ]) + ( t − t j ) k f [ k ] ([ ˜ [ x ]( t ) = [ x j ] + � x j ] , [ p ]) i =1 20
Hybrid Reachability Computation n Guaranteed event detection & localization l An interval constraint propagation approach l (Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011) 20
Hybrid Reachability Computation n Guaranteed event detection & localization l An interval constraint propagation approach l (Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011) 21
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
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
Hybrid Reachability Computation n Detecting and localizing events l Improved and enhanced version l (Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014) 23
Hybrid Reachability Computation n Detecting and localizing events l Improved and enhanced version l (Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014) 24
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. 6 5 4 3 x2 2 1 0 -1 2 4 6 8 10 12 14 x1 25
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. 6 3.5 5 3 4 2.5 2 3 x2 1.5 py 2 1 0.5 1 0 0 -0.5 -1 -1 -6 -3 0 3 6 2 4 6 8 10 12 14 px x1 25
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
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
Inclusion test 27
Inclusion test 27
Inclusion test Frontier of the reachable set = union of zonotopes 28
Inclusion test Frontier of the reachable set = union of zonotopes 28
Inclusion test Frontier of the reachable set = union of zonotopes 28
Outline n Hybrid dynamical systems n Set membership estimation n Hybrid reachability approach n Example n Research directions 29
Parameter identification n Hybrid Mass-Spring l Velocity-dependent damping. Mode switching driven by velocity. 30
Parameter identification n Hybrid Mass-Spring l case 1 : Parameters acting on continuous dynamics. l CPU time approx. 140 mn! 31
Parameter identification n Hybrid Mass-Spring l case 2 : parameters acting on discrete transition. l CPU time approx. 40 mn 32
Outline n Hybrid dynamical systems n Set membership estimation n Hybrid reachability approach n Example n Research directions 33
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