Parameter iden+fica+on with hybrid systems in a - - PowerPoint PPT Presentation
Parameter iden+fica+on with hybrid systems in a bounded-error framework Moussa MAIGA, Nacim RAMDANI, & Louise TRAVE-MASSUYES Universit dOrlans,
Université ¡d’Orléans, ¡Bourges, ¡and ¡LAAS ¡CNRS ¡Toulouse, ¡France. ¡
9-‑11 ¡June ¡2015
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input / control hybrid dynamical system measured
hybrid state and parameter estimation reconstructed variables guaranteed decision making +
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l Non-linear continuous dynamics l Bounded uncertainty
Continuous dynamics Discrete dynamics
l x ∈ Inv(l) l′ x′ ∈ Inv(l′) e : g(x) ≥ 0 ˙ x′ ∈ Flow(l′, x′) x′ = r(e, x) ˙ x ∈ Flow(l, x) x ∈ Init(l)
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initial conditions discrete transition jump
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Discrete transitions Initial conditions freefall Continuous transtions
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l Nonlinear continuous dynamics l Nonlinear guards sets l Nonlinear reset functions
l x ∈ Inv(l) l′ x′ ∈ Inv(l′) e : g(x) ≥ 0 ˙ x′ ∈ Flow(l′, x′) x′ = r(e, x) ˙ x ∈ Flow(l, x) x ∈ Init(l)
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l Nonlinear continuous dynamics l Nonlinear guards sets l Nonlinear reset functions
l x ∈ Inv(l) l′ x′ ∈ Inv(l′) e : g(x) ≥ 0 ˙ x′ ∈ Flow(l′, x′) x′ = r(e, x) ˙ x ∈ Flow(l, x) x ∈ Init(l)
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l Nonlinear … l Bounded uncertainty
lswitching sequence lcontinuous variables
switching sequence for upper bracketting system t 18000 20000 22000 24000 26000 28000 30000 32000 34000 36000 38000 50 100 150 200 250 3009
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f(p)
n ys
Optimisation of J(e(p))
e(p) ys p1 p2
Confidence sets
Yield valid results only if Perturbations, errors and model uncertainties with statistical properties known a priori Model structure is correct, no modeling errors
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f(p)
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Set membersip algorithm Y
p1 p2
Solution set Y
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
Set Membership Algorithm
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l Interval observers
(Raïssi, et al., 2012), (El Thabet, et al. 2014) ….
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l Prediction - Correction / Filtering approaches
(Milanese & Novara, 2011), (Kieffer & Walter, 2011) … 14
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l Branch-&-bound, branch-&-prune, interval contractors … (Jaulin, et al. 93) (Raïssi et al., 2004)
l Interval observers l Prediction-correction / Filtering approaches
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
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l x ∈ Inv(l) l′ x′ ∈ Inv(l′) e : g(x) ≥ 0 ˙ x′ ∈ Flow(l′, x′) x′ = r(e, x) ˙ x ∈ Flow(l, x) x ∈ Init(l)
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l An interval constraint propagation approach
l(Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011)
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l An interval constraint propagation approach
l(Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011)
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˙ x(t) = f (x, p, t), t0 ≤ t ≤ tN, x(t0) ∈ [x0] , p ∈ [p] Time grid → t0 < t1 < t2 < · · · < tN
[xj] [xj+1] actual solution x a priori [˜ xj]
Analytical solution for [x](t), t ∈ [tj, tj+1] [x](t) = [xj] +
k−1
(t − tj)if[i]([xj], [p]) + (t − tj)kf[k]([˜ xj], [p])
l An interval constraint propagation approach
l(Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011)
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l An interval constraint propagation approach
l(Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011)
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l An interval constraint propagation approach
l(Ramdani, et al., Nonlinear Analysis Hybrid Systems 2011)
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l Improved and enhanced version. A faster version.
l(Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014)
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l Improved and enhanced version. A faster version.
l(Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014)
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l Improved and enhanced version
l(Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014)
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l Improved and enhanced version
l(Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014)
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l Improved and enhanced version
l(Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014) Bouncing ball in 2D.
1 2 3 4 5 6 2 4 6 8 10 12 14 x2 x1
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l Improved and enhanced version
l(Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014) Bouncing ball in 2D.
1 2 3 4 5 6 2 4 6 8 10 12 14 x2 x1
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l Branch-&-bound, branch-&-prune, interval contractors … (Eggers, Ramdani et al., 2012), (Maïga, Ramdani et al., 2015)
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l Branch-&-bound, branch-&-prune, interval contractors … (Eggers, Ramdani et al., 2012), (Maïga, Ramdani et al., 2015) Need an inclusion test!
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nHybrid Mass-Spring
l Velocity-dependent damping. Mode switching driven by velocity.
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l case 1 : Parameters acting on continuous dynamics. l CPU time approx. 140 mn!
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l case 2 : parameters acting on discrete transition. l CPU time approx. 40 mn
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l To build upon a hybrid reachability approach
l SM parameter estimation … l SM hybrid state estimation of nonlinear hybrid systems
l Application to actual hybrid systems
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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.149-162, 2011. n A.Eggers, N.Ramdani, N.S.Nedialkov, M.Fränzle, Set-Membership Estimation of Hybrid Systems via SAT Mod ODE. in IFAC SYSID 2012. pp.440-445 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.508-513. n M. Maïga, C. Combastel, N. Ramdani, L. Travé-Massuyès, Nonlinear hybrid reachability using set integration and zonotope enclosures. in ECC 2014, pp.234-239. 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. n M. Maïga, N. Ramdani, L. Travé-Massuyès, Robust fault detection in hybrid systems using set- membership parameter estimation, in IFAC SafeProcess 2015, Paris.