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

e-BaCCuSS Alexandre Vieira - On the optimal control of linear complementarity systems Bernard Brogliato - Christophe Prieur Alexandre Vieira 1 - Bernard Brogliato 1 - Christophe Prieur 2 Introduction Direct Method 1 Univ. Grenoble Alpes, INRIA Grenoble - 2 Univ. Grenoble Alpes, GIPSA-Lab Necessary conditions 26th September 2017 Numerics : the indirect method Conclusion alexandre.vieira@inria.fr, bernard.brogliato@inria.fr, christophe.prieur@gipsa-lab.fr. 1 / 26

Introduction e-BaCCuSS Problem: Alexandre � T Vieira - C ( u ) = ( x ( t ) ⊺ Qx ( t ) + u ( t ) ⊺ Uu ( t )) dt → min Bernard Brogliato - 0 Christophe Prieur such that: x ( t ) = Ax ( t ) + Bv ( t ) + Fu ( t ) ˙ Introduction 0 ≤ v ( t ) ⊥ Cx ( t ) + Dv ( t ) + Eu ( t ) ≥ 0 Direct Method x ( 0 ) = x 0 , x ( T ) free Necessary conditions where A ∈ R n × n , B , F ∈ R n × m , C ∈ R m × n , D , E ∈ R m × m , T > 0, x : [ 0 , T ] → R n Numerics : the indirect and u , v : [ 0 , T ] → R m , Q and U matrices of according dimensions, supposed method symmetric positive definite. Conclusion Hypothesis : D is a P-Matrix. Motivation: Mechanics, Electronic Circuits, Chemical reactions 2 / 26

A difficult problem e-BaCCuSS Alexandre Vieira - Bernard Brogliato - Existence of optimal solution not proved (classical Fillipov theory does not Christophe apply here due to lack of convexity). Cesari (2012), Theorem 9.2i and onwards Prieur Special cases arise when E = 0 : switching modes are activated when the state Introduction reaches some threshold defined by the complementarity conditions. Georgescu et Direct Method al. (2012), Passenberg et al. (2013) Necessary conditions Since u is also involved = ⇒ mixed constraints; makes use of non-smooth Numerics : analysis. Clarke and De Pinho (2010) the indirect method Conclusion 3 / 26

Direct method e-BaCCuSS Alexandre Vieira - First way to compute numerical approximation: direct method. Bernard Brogliato - Christophe N − 1 Prieur � � x ⊺ k Qx k + u ⊺ � min k Uu k Introduction k = 0 Direct s.t. x k + 1 − x k Method = Ax k + 1 + Bv k + Fu k , ∀ k ∈ 0 , ..., N − 1 h Necessary conditions 0 ≤ v k ⊥ Cx k + Dv k + Eu k ≥ 0 Numerics : the indirect method = ⇒ Mathematical Program with Equilibrium Constraints (MPEC) Conclusion 4 / 26

Direct method e-BaCCuSS Alexandre Such optimality problems are hard to tackle. Complementarity constraints: Vieira - Bernard Brogliato - v ≥ 0 Christophe Prieur Cx + Dv + Eu ≥ 0 Introduction v ⊺ ( Cx + Dv + Eu ) = 0 Direct Method violate usual constraint qualifications. Necessary conditions Numerics : Need to redefine usual qualification for this problem, the indirect method and associated stationarity properties. Conclusion 5 / 26

Direct method e-BaCCuSS Alexandre Denote λ H , λ G multipliers associated to 0 ≤ v ⊥ Cx + Dv + Eu ≥ 0. Vieira - Bernard Brogliato - Weak stationarity: λ G i = 0 if v i > 0 = ( Cx + Dv + Eu ) i and λ H i = 0 if Christophe Prieur v i = 0 < ( Cx + Dv + Eu ) i Strong stationarity: Weak stationarity + λ G i , λ H i ≥ 0 if Introduction Direct v i = 0 = ( Cx + Dv + Eu ) i Method Necessary Property conditions Numerics : If ( x ∗ , u ∗ , v ∗ ) is a minimum for MPEC, it is weak stationary. the indirect method Here, if we suppose E invertible, the optimal solution is strong stationary. Conclusion 6 / 26

Direct method e-BaCCuSS Alexandre Suppose E invertible = ⇒ algorithm converging to a strong stationary point. Vieira - Bernard [1] : Algorithm relaxing smartly the complementarity constraint, adding a Brogliato - Christophe parameter that continuously converge to 0. Prieur [2] : Complementarity added in the cost, creating a barrier problem solved Introduction with interior point method. Direct Method Under some conditions, both converge to strong stationary points. Necessary conditions Numerics : the indirect method [1] C. Kanzow and A. Schwartz. A new regularization method for mathematical programs with Conclusion complementarity constraints with strong convergence properties. SIAM Journal on Optimization, 23(2):770–798, 2013. [2] S. Leyffer, G. López-Calva, and J. Nocedal. Interior methods for mathematical programs with complementarity constraints. SIAM Journal on Optimization, 17(1):52–77, 2006. 7 / 26

Why do we bother ? e-BaCCuSS This method works well... For small precision. Possible pseudominima? Alexandre Vieira - Bernard Brogliato - = ⇒ Indirect method. Christophe Prieur Suppose an optimal solution exists = ⇒ Search for necessary conditions. Introduction Two reasons for that: Direct Method Useful for analyzing the solution (continuity, sensitivity...) Necessary Indirect method needs a good initial guess: direct method used for that. conditions Numerics : Really general necessary conditions were obtained in [1]. But as such, they are not the indirect method really practical (complicated hypothesis, really general equations...). Conclusion [1] L. Guo and J. J. Ye. Necessary optimality conditions for optimal control problems with equilibrium constraints (2016). 8 / 26

Weak stationarity e-BaCCuSS Define S = { ( x , u , v ) | 0 ≤ v ⊥ Cx + Dv + Eu ≥ 0 } and the partition of { 0 , ..., m } : Alexandre Vieira - I 0 + ( x , u , v ) = { i | v i ( t ) = 0 < ( Cx ( t ) + Dv ( t ) + Eu ( t )) i } Bernard t Brogliato - Christophe I + 0 ( x , u , v ) = { i | v i ( t ) > 0 = ( Cx ( t ) + Dv ( t ) + Eu ( t )) i } Prieur t I 00 Introduction t ( x , u , v ) = { i | v i ( t ) = 0 = ( Cx ( t ) + Dv ( t ) + Eu ( t )) i } Direct Method Necessary conditions Theorem Numerics : Let ( x ∗ , u ∗ , v ∗ ) be a local minimizer of radius R ( · ) . Suppose Im ( C ) ⊆ Im ( E ) . Then the indirect method there exist an arc p and measurable functions λ G : R → R m , λ H : R → R m such Conclusion that the following conditions hold: 1 the transversality condition: p ( T ) = 0 9 / 26

Weak stationarity e-BaCCuSS Alexandre Vieira - Theorem Bernard Brogliato - 2 the Weierstrass condition for radius R : for almost every t ∈ [ t 0 , t 1 ] , Christophe Prieur � � u � � u ∗ ( t ) �� Introduction � � ( x ∗ ( t ) , u , v ) ∈ S , − � < R ( t ) Direct � � v v ∗ ( t ) Method � ⇒ � p ( t ) , Ax ∗ ( t ) + Bv + Fu ) � − 1 Necessary = 2 ( x ∗ ( t ) ⊺ Qx ∗ ( t ) + u ⊺ Uu ) conditions Numerics : ≤ � p ( t ) , Ax ∗ ( t ) + Bv ∗ ( t ) + Fu ∗ ( t )) � − 1 the indirect 2 ( x ∗ ( t ) ⊺ Qx ∗ ( t ) + u ∗ ( t ) ⊺ Uu ∗ ( t )) method Conclusion 10 / 26

Weak stationarity e-BaCCuSS Alexandre Vieira - Theorem Bernard Brogliato - 3 the Euler adjoint equation: for almost every t ∈ [ 0 , T ] , Christophe Prieur p ( t ) = − A ⊺ p ( t ) + Qx ∗ ( t ) − C ⊺ λ H ( t ) ˙ Introduction Direct 0 = F ⊺ p ( t ) − Uu ∗ ( t ) + E ⊺ λ H ( t ) Method 0 = B ⊺ p ( t ) + λ G + D ⊺ λ H ( t ) Necessary conditions i ( t ) , ∀ i ∈ I + 0 0 = λ G ( x ∗ ( t ) , u ∗ ( t ) , v ∗ ( t )) Numerics : t the indirect method i ( t ) , ∀ i ∈ I 0 + 0 = λ H ( x ∗ ( t ) , u ∗ ( t ) , v ∗ ( t )) t Conclusion 11 / 26

Euler equation e-BaCCuSS How can we solve the following BVP? Alexandre Vieira - Bernard x = Ax + Bv + Fu ˙ Brogliato - Christophe p = − A ⊺ p + Qx − C ⊺ λ H ˙ Prieur 0 = F ⊺ p − Uu + E ⊺ λ H Introduction 0 = B ⊺ p + λ G + D ⊺ λ H Direct Method 0 = λ G i ( t ) , ∀ i ∈ I + 0 Necessary ( x ( t ) , u ( t ) , v ( t )) t conditions 0 = λ H i ( t ) , ∀ i ∈ I 0 + ( x ( t ) , u ( t ) , v ( t )) Numerics : t the indirect method x 0 = x ( 0 ) , Conclusion 0 = p ( T ) 12 / 26

Euler equation How can we solve the following BVP? e-BaCCuSS Alexandre Vieira - x = Ax + Bv + Fu ˙ Bernard Brogliato - p = − A ⊺ p + Qx − C ⊺ λ H ˙ Christophe Prieur 0 = F ⊺ p − Uu + E ⊺ λ H → isolate u Introduction 0 = B ⊺ p + λ G + D ⊺ λ H → isolate λ G Direct Method Necessary i ( t ) , ∀ i ∈ I + 0 0 = λ G ( x ( t ) , u ( t ) , v ( t )) conditions t Numerics : i ( t ) , ∀ i ∈ I 0 + 0 = λ H ( x ( t ) , u ( t ) , v ( t )) the indirect t method Conclusion x 0 = x ( 0 ) , 0 = p ( T ) 13 / 26

Strong stationarity e-BaCCuSS Alexandre Vieira - 0 = λ G i ( t ) , ∀ i ∈ I + 0 Bernard ( x ( t ) , u ( t ) , v ( t )) t Brogliato - Christophe 0 = λ H i ( t ) , ∀ i ∈ I 0 + ( x ( t ) , u ( t ) , v ( t )) Prieur t We miss a piece of information: what happens on I 00 Introduction ? t Direct Method Proposition Necessary conditions Let ( x ∗ , u ∗ , v ∗ ) be a local minimizer and suppose E invertible. Then ( x ∗ , u ∗ , v ∗ ) is Numerics : strongly stationary, meaning: the indirect method i ( t ) ≥ 0 , ∀ i ∈ I 00 λ G i ( t ) ≥ 0 , λ H t ( x ( t ) , u ( t ) , v ( t )) Conclusion 14 / 26

Strong stationarity e-BaCCuSS Alexandre Vieira - Bernard Brogliato - Christophe Prieur 0 = λ G ∀ i ∈ I + 0 i ( t ) , ( x ( t ) , u ( t ) , v ( t )) t Introduction 0 = λ H ∀ i ∈ I 0 + i ( t ) , ( x ( t ) , u ( t ) , v ( t )) t Direct Method λ G i ( t ) ≥ 0 , λ H ∀ i ∈ I 00 i ( t ) ≥ 0 , t ( x ( t ) , u ( t ) , v ( t )) Necessary conditions Almost like a linear complementarity problem! Numerics : the indirect method Conclusion 15 / 26