On the optimal control of linear complementarity systems Vieira - - PowerPoint PPT Presentation

IPL Modeliscale Alexandre On the optimal control of linear complementarity systems Vieira - Bernard Brogliato - Christophe Prieur Alexandre Vieira 1 - Bernard Brogliato 1 - Christophe Prieur 2 Introduction Why is it so hard ? 1 Univ. Grenoble Alpes, INRIA Grenoble - 2 Univ. Grenoble Alpes, GIPSA-Lab Direct approach Necessary conditions 22nd May 2018 and indirect method Minimal time Conclusion alexandre.vieira@inria.fr, bernard.brogliato@inria.fr, christophe.prieur@gipsa-lab.fr. 1 / 38

Introduction IPL Modeliscale c(u1) min c ( x u , u ) Alexandre c(u2) Vieira - Bernard s.t. ˙ x u ( t ) = f ( x u ( t ) , u ( t )) , Brogliato - Christophe ( x u ( t ) , u ( t )) ∈ S ( t ) , Prieur x u ( 0 ) = x 0 , Introduction c(u3) Why is it so x u ( T ) free . hard ? Direct approach Necessary c(u4) conditions and indirect method c(u5) Minimal time x(t) Conclusion 0 T 2 / 38

Introduction IPL Modeliscale Problem: Alexandre � T Vieira - Bernard ( x ( t ) ⊺ Qx ( t ) + u ( t ) ⊺ Uu ( t ) + v ( t ) ⊺ Vv ( t )) dt → min C ( u ) = Brogliato - Christophe 0 Prieur such that:  x ( t ) = Ax ( t ) + Bv ( t ) + Fu ( t ) ˙ Introduction  Why is it so  hard ? 0 ≤ v ( t ) ⊥ w ( t ) = Cx ( t ) + Dv ( t ) + Eu ( t ) ≥ 0 Direct approach  x ( 0 ) = x 0 , x ( T ) free  Necessary conditions and indirect where T > 0, x : [ 0 , T ] → R n absolutely continuous, v : [ 0 , T ] → R m , method u : [ 0 , T ] → R m u , A , B , C , D , E , F , Q , V and U matrices of according dimensions, Minimal time Conclusion U supposed symmetric positive definite, Q and V positive semi-definite. Motivation: Mechanics, Electronic Circuits, Chemical reactions 3 / 38

A difficult problem IPL Modeliscale Alexandre Vieira - 0 ≤ v ( t ) ⊥ Cx ( t ) + Dv ( t ) + Eu ( t ) ≥ 0 Bernard Brogliato - Christophe Prieur Existence of optimal solution not proved (classical Fillipov theory does not Introduction apply here due to lack of convexity). Cesari (2012), Theorem 9.2i and onwards Why is it so hard ? Direct Special cases arise when E = 0 and D P-matrix : switching modes are approach activated when the state reaches some threshold defined by the Necessary conditions complementarity conditions. Georgescu et al. (2012), Passenberg et al. (2013) and indirect method Since u is also involved = ⇒ mixed constraints; makes use of non-smooth Minimal time analysis. Clarke and De Pinho (2010) Conclusion 4 / 38

Direct Method IPL Modeliscale A first way to compute numerically an approximate solution: direct method. Alexandre Vieira - Bernard N Brogliato - Christophe � min x i Qx i + u i Uu i Prieur i = 0 Introduction  x i + 1 = x i + h ( Ax i + Bv i + Eu i ) Why is it so hard ?   Direct s.t. 0 ≤ v i ⊥ Cx i + Dv i + Eu i ≥ 0 approach Necessary  x 0 fixed  conditions and indirect method ⇒ Mathematical Program with Equilibrium Constraints (MPEC). = Minimal time Since it is a convex cost with linear constraints: there exist a solution for any fixed h . Conclusion 5 / 38

Direct Method IPL Modeliscale 2 ways to solve numerically this problem: Alexandre Vieira - 1 Cost penalization on the complementarity constraint v ⊺ ( Cx + Dv + Eu ) = 0: Bernard Brogliato - didn’t work well on examples. Christophe Prieur 2 Relaxation of the complementarity: choose a sequence ε k ≥ 0 converging to 0 and create a sequence of optimization problems where the constraint Introduction Why is it so v ⊺ ( Cx + Dv + Eu ) = 0 is replaced by v ⊺ ( Cx + Dv + Eu ) ≤ ε k : works well. hard ? Direct approach [1] S. Leyffer, G. López-Calva, and J. Nocedal. Interior methods for mathematical programs with Necessary conditions complementarity constraints. SIAM Journal on Optimization, 17(1):52–77, 2006. and indirect method [2] C. Kanzow and A. Schwartz. A new regularization method for mathematical programs with Minimal time complementarity constraints with strong convergence properties. SIAM Journal on Optimization, Conclusion 23(2):770–798, 2013. 6 / 38

Direct Method � t Denote x 1 ( t ) = 0 i 1 ( s ) ds + x 1 ( 0 ) (the charge of C IPL i 1 the capacitor, in coulomb) and x 2 ( t ) = i 2 ( t ) (the Modeliscale electric current, in ampere). Then the evolution of Alexandre Vieira - this system is described as: Bernard u 2 Brogliato - D λ Christophe x 1 ( t ) = − 1 RC x 1 ( t ) + x 2 ( t ) − 1 R λ ( t ) + 1 Prieur  ˙ R u 2 ( t ) ,   R  Introduction   x 2 ( t ) = − 1 LC x 1 ( t ) − 1 L λ ( t ) + 1  Why is it so  ˙ L ( u 2 ( t ) − u 1 ( t )) , hard ?   Direct  u 1 approach 1 Necessary 0 ≤ λ ( t ) ⊥ RC x 1 ( t ) − x 2 ( t )  L conditions    and indirect   method + 1 R λ ( t ) − 1  i 2  R u 2 ( t ) ≥ 0 ,   Minimal time Conclusion The constants are chosen as R = 10 Ω , C = 80 000 µ F, and L = 2H. 7 / 38

Direct Method IPL C � 1 Modeliscale x 1 ˙ [ Rx 2 ( t ) 2 + u 1 ( t ) 2 + u 2 ( t ) 2 ] dt min Alexandre Vieira - 0 Bernard x 1 ( t ) = − 1 RC x 1 ( t ) + x 2 ( t ) − 1 R λ ( t ) + 1 Brogliato -  ˙ R u 2 ( t ) , u 2 Christophe D λ  Prieur     x 2 ( t ) = − 1 LC x 1 ( t ) − 1 L λ ( t ) + 1   ˙ L ( u 2 ( t ) − u 1 ( t )) , Introduction   R  Why is it so  hard ?   1 Direct   approach 0 ≤ λ ( t ) ⊥ RC x 1 ( t ) − x 2 ( t ) s.t. u 1 Necessary conditions  + 1 R λ ( t ) − 1  L and indirect  R u 2 ( t ) ≥ 0 ,   method    x 2 Minimal time  x ( 0 ) = ( 200 C , 50 A )    Conclusion    x ( 1 ) = ( 50 C , 0 A ) 8 / 38

Direct Method C 200 IPL x1 x 1 ˙ Modeliscale 150 x2 100 Alexandre Vieira - 50 Bernard 0 Brogliato - u 2 D λ Christophe −50 Prieur −100 −150 R Introduction 0.0 0.2 0.4 0.6 0.8 1.0 λ always naught! t Why is it so hard ? Direct 1000 approach u 1 u1 800 Necessary u2 600 L conditions and indirect 400 method 200 x 2 0 Minimal time −200 Conclusion −400 −600 0.0 0.2 0.4 0.6 0.8 1.0 t 9 / 38

Direct Method 5 � 10 IPL x1 4 Modeliscale � x ( t ) � 2 2 + u ( t ) 2 � � min dt , x2 3 Alexandre x3 0 2 Vieira - 1   0 1 0   0  Bernard 0 Brogliato -   x ( t ) +  ,  x ( t ) = ˙ 0 0 1 0 −1 Christophe     Prieur  −2  0 0 0 v ( t ) + u ( t )   −3   Introduction −4  � �  0 ≤ v ( t ) ⊥ 1 0 0 x ( t ) + u ( t ) ≥ 0 , a.e. on [ 0 , 10 ] 0 2 4 6 8 10 Why is it so  t hard ? s.t. Direct  − 2  1800 approach   v1  , 1600  x ( 0 ) = 1 Necessary    1400 conditions   − 1 and indirect  1200  method  1000   x ( T ) free ,  800 Minimal time 600 Conclusion 400 Resolution with Direct Method and relaxation of 200 0 the complementarity. Library used: IPOPT and 0 2 4 6 8 10 t CasADI. h = 10 − 3 . 10 / 38

Why do we bother ? IPL Modeliscale Suppose an optimal solution exists = ⇒ Search for necessary conditions. Alexandre Two reasons for that: Vieira - Bernard Useful for analyzing the solution (continuity, sensitivity...) Brogliato - Christophe Prieur The direct method mostly works fine! But very slow for high precision or big systems. Possible pseudominima? Introduction Why is it so hard ? Really general necessary conditions were obtained in [1]. But as such, they are not Direct approach really practical (complicated hypothesis, really general equations...). Necessary conditions and indirect Can it be enhanced in the case of LCS? method Minimal time [1] L. Guo and J. J. Ye. Necessary optimality conditions for optimal control problems with equilibrium Conclusion constraints (2016). 11 / 38

Weak stationarity IPL Define S = { ( x , u , v ) | 0 ≤ v ⊥ Cx + Dv + Eu ≥ 0 } and the partition of { 1 , ..., m } : Modeliscale Alexandre I 0 + ( x , u , v ) = { i | v i ( t ) = 0 < ( Cx ( t ) + Dv ( t ) + Eu ( t )) i } Vieira - t Bernard Brogliato - I + 0 Christophe ( x , u , v ) = { i | v i ( t ) > 0 = ( Cx ( t ) + Dv ( t ) + Eu ( t )) i } Prieur t I 00 t ( x , u , v ) = { i | v i ( t ) = 0 = ( Cx ( t ) + Dv ( t ) + Eu ( t )) i } Introduction Why is it so hard ? Direct approach Theorem Necessary conditions Let ( x ∗ , u ∗ , v ∗ ) be a local minimizer of radius R ( · ) . Suppose Im ( C ) ⊆ Im ( E ) . Then and indirect method there exist an absolutely continuous function p : [ 0 , T ] → R n and measurable Minimal time functions λ G : R → R m , λ H : R → R m such that the following conditions hold: Conclusion 1 the transversality condition: p ( T ) = 0 12 / 38

Weak stationarity IPL Modeliscale Theorem Alexandre Vieira - Bernard 2 the Weierstrass condition for radius R : for almost every t ∈ [ t 0 , t 1 ] , Brogliato - Christophe Prieur � u ∗ ( t ) � � u � �� ( x ∗ ( t ) , u , v ) ∈ S , � � − � < R ( t ) Introduction � � v ∗ ( t ) v Why is it so � hard ? ⇒ � p ( t ) , Ax ∗ ( t ) + Bv + Fu ) � − 1 Direct 2 ( x ∗ ( t ) ⊺ Qx ∗ ( t ) + u ⊺ Uu + v ⊺ Vv ) approach = Necessary conditions ≤ � p ( t ) , Ax ∗ ( t ) + Bv ∗ ( t ) + Fu ∗ ( t )) � and indirect method − 1 2 ( x ∗ ( t ) ⊺ Qx ∗ ( t ) + u ∗ ( t ) ⊺ Uu ∗ ( t ) + v ∗ ( t ) ⊺ Vv ∗ ( t )) Minimal time Conclusion 13 / 38