Linear Quadratic Optimization for Periodic Hybrid Systems Corrado - PowerPoint PPT Presentation

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Linear Quadratic Optimization for Periodic Hybrid Systems Corrado Possieri Dipartimento di Ingegneria Civile e Ingegneria Informatica, Universit` a di Roma Tor Vergata, Roma, Italy Hybrid Dynamical Systems: Optimization, Stability and Applications Trento, 11 January 2017 C. Possieri LQ Optimal Control for a Class of Hybrid Systems 1 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Outline Introduction 1 Finite–Horizon LQ Optimal Control 2 Infinite–Horizon LQ Optimal Control 3 Stabilization 4 Conclusions 5 C. Possieri LQ Optimal Control for a Class of Hybrid Systems 2 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Linear Hybrid Systems with Periodic Jumps Consider the hybrid system governed by the flow dynamics τ ˙ = 1 , x ˙ = Ax + Bu F , whenever ( τ, x ) ∈ [0 , τ M ] × R n , by the jump dynamics τ + = 0 , x + = Ex + Fu J , whenever ( τ, x ) ∈ { τ M } × R n , and τ (0 , 0) = 0 , x (0 , 0) = x 0 . C. Possieri LQ Optimal Control for a Class of Hybrid Systems 3 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Hybrid time domain Each solution is defined on the hybrid time domain T := { ( t, k ) , t ∈ [ kτ M , ( k + 1) τ M ] , k ∈ N } . k 5 4 3 2 1 t 0 τ M 0 2 τ M 3 τ M 4 τ M 5 τ M 6 τ M C. Possieri LQ Optimal Control for a Class of Hybrid Systems 4 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Stabilizability of Periodic Linear Hybrid Systems The tuple ( A, B, E, F, τ M ) is stabilizable , i.e. , for any initial condition x 0 ∈ R n , there exist inputs u F ( · , · ) and u J ( · ) such that t + k →∞ x ( t, k ) = 0 , lim if and only if 1 rank[ Ee Aτ M − sI F R A,B ] = n, ∀ s / ∈ C g , A n − 1 B ] . where R A,B = [ B AB · · · 1 Possieri and Teel. Structural Properties of a Class of Linear Hybrid Systems and Output Feedback Stabilization . IEEE TAC, 2017 C. Possieri LQ Optimal Control for a Class of Hybrid Systems 5 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Detectability of Periodic Linear Hybrid Systems The tuple ( A, E, C F , C J , τ M ) is detectable , i.e. , letting y F ( t, k ) := C F x ( t, k ) , y J ( k ) := C J x ( t k , k − 1) , for any initial state x 0 ∈ R n such that y F = y J = 0 , one has that t + k →∞ x ( t, k ) = 0 , lim with u F = 0 and u J = 0 , if and only if rank[ ( Ee Aτ M ) ′ − sI A,C F ] ′ = n, ( C J e Aτ M ) ′ O ′ ∀ s / ∈ C b , ( C F A n − 1 ) ′ ] ′ . where O A,C F := [ C ′ ( C F A ) ′ · · · F C. Possieri LQ Optimal Control for a Class of Hybrid Systems 6 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Finite–Horizon Linear Quadratic Optimal Control Consider the quadratic cost functional � T J = ( | x ( t, k ) | Q F + | u F ( t, k ) | R F ) dt + 0 K � + ( | x ( t k , k − 1) | Q J + | u J ( k ) | R J ) + | x ( T, K ) | S . k =1 Problem ′ ] ′ that minimizes the functional J Find, if any, u ⋆ = [ u ⋆ ′ u ⋆ F J from the initial condition (0 , x 0 ) , x 0 ∈ R n . C. Possieri LQ Optimal Control for a Class of Hybrid Systems 7 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Cost–to–go function For each ( θ, κ ) ∈ T , 0 � θ � T , x ∈ R n , �� T V ( θ, κ, x ) := inf ( | x ( t, k ) | Q F + | u F ( t, k ) | R F ) dt + u ( · , · ) θ K � � + ( | x ( t k , k − 1) | Q J + | u J ( k ) | R J ) + | x ( T, K ) | S . k = κ +1 The objective is to compute V (0 , 0 , x 0 ) . C. Possieri LQ Optimal Control for a Class of Hybrid Systems 8 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Backward Propagation Through Flow (1/2) For all t ∈ [ Kτ M , T ] , �� T � V ( t, K, x ) = inf ( | x | Q F + | u | R F ) + | x ( T, K ) | S . u ( · ,K ) t k V ( T, K, x ) = | x | S K t Kτ M T C. Possieri LQ Optimal Control for a Class of Hybrid Systems 9 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Backward Propagation Through Flow (2/2) Terminal cost: V ( T, K, x ) = | x | S . Backward Propagation: for all t ∈ [ t K , T ] , V ( t, K, x ) = | x | P ( t,K ) , where − ˙ A ′ P + PA + Q F − PBR − 1 F B ′ P, P = P ( T, K ) = S. Optimal control: for all t ∈ [ t K , T ] , u ⋆ F ( t, K ) = − R − 1 F B ′ P ( t, K ) x, C. Possieri LQ Optimal Control for a Class of Hybrid Systems 10 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Backward Propagation Through Jump At hybrid time ( Kτ M , K − 1) , V ( t K , K − 1 , x ) = inf u {| x | Q J + | u | R J + | Ex + Fu | ¯ S } . k V ( Kτ M , K, x ) = | x | ¯ S K K − 1 t Kτ M C. Possieri LQ Optimal Control for a Class of Hybrid Systems 11 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Backward Propagation Through Jump Terminal cost: V ( T, K, x ) = | x | ¯ S , where ¯ S = S , if K = T/τ M , or ¯ S = P ( t K , K ) , if K � = T/τ M . Backward Propagation: V ( t K , K − 1 , x ) = | x | P ( t K ,K − 1) , where P ( t K , K − 1) = Q J + E ′ ¯ SE − E ′ ¯ SF ( R J + F ′ ¯ SF ) − 1 F ′ ¯ SE. Optimal control: J ( K ) = − ( R J + F ′ ¯ SF ) − 1 F ′ ¯ u ⋆ SEx. C. Possieri LQ Optimal Control for a Class of Hybrid Systems 12 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Solution to the Finite–Horizon LQ Optimal Control There exists a unique solution to the problem, given by − R − 1 F B ′ P ( t, k ) x, u ⋆ F ( t, k ) = u ⋆ − ( R J + F ′ P ( t k , k ) F ) − 1 F ′ P ( t k , k ) Ex, J ( k ) = where P ( t, k ) is the solution to the HRE with flow dynamics − ˙ τ = 1 , − ˙ A ′ P + PA + Q F − PBR − 1 F B ′ P, P = if ( τ, P ) ∈ [0 , τ M ] × R n × n , and jumps dynamics τ + = 0 , Q J + E ′ P + E − E ′ P + F ( R J + F ′ P + F ) − 1 F ′ P + E, P = if ( τ, P ) ∈ { 0 } × R n × n , with τ ( T, K ) = T − Kτ M , P ( T, K ) = S . C. Possieri LQ Optimal Control for a Class of Hybrid Systems 13 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Time scales Let T be a compact time scale , i.e. , a compact subset of R . Let ς ( t ) := inf { s ∈ T : s > t } and µ ( t ) = ς ( t ) − t . Letting f : T → R , define f ( ς ( t )) − f ( s ) f ∆ ( t ) := lim ς ( t ) − s s → t,s ∈ T \{ ς ( t ) } � ˙ f ( t ) , if ς ( t ) � t, = f ( ς ( t )) − f ( t ) , if ς ( t ) > t. ς ( t ) − t C. Possieri LQ Optimal Control for a Class of Hybrid Systems 14 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Linear Quadratic optimal control on time scales Consider the system x ∆ ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) , and the cost functional � b J = ( | x ( t ) | Q ( t ) + | u ( t ) | R ( t ) )∆ t + | x ( b ) | S . a A minimizer for J is given by u = K − 1 Mx where 2 W ∆ − Q + W ς A + A ⊤ W ς + µA ⊤ W ς A + M ⊤ K − 1 M = 0 . with W ( b ) = − S , W ς = W + µW ∆ , and R − µB ⊤ W ς B, K := B ⊤ W ς ( I + µA ) . M := 2 Hilscher and Zeidan. Hamilton–Jacobi theory over time scales and applications to linear–quadratic problems . Nonlinear Analysis, 2012 C. Possieri LQ Optimal Control for a Class of Hybrid Systems 15 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Relation of the hybrid LQR with the LQR on time scales Consider the time scale T = � k ∈ N [ k + kτ M , k + ( k + 1) τ M ] . τ M + 1 2 τ M + 1 3 τ M + 3 4 τ M + 4 τ M 0 2 τ M + 2 3 τ M + 2 t Define the system x ∆ ( t ) = Φ( t, x ( t )) on T where � Ax + Bu F , if ς ( t ) � t, Φ( t, x ) := Ex + Fu J , if ς ( t ) > t, The HRE translates the results for dynamical systems over time scales to the considered hybrid setting. C. Possieri LQ Optimal Control for a Class of Hybrid Systems 16 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Infinite–Horizon Linear Quadratic Optimal Control Consider the cost function � ∞ J ∞ = ( | x ( t, k ) | Q F + | u F ( t, k ) | R F ) dt + 0 ∞ � + ( | x ( t k , k − 1) | Q J + | u J ( k ) | R J ) , k =1 Problem J, ∞ ) ′ ] ′ that ∞ = [ ( u ⋆ F, ∞ ) ′ ( u ⋆ Find, if any, a control input u ⋆ minimizes the cost J ∞ from the initial condition (0 , x 0 ) , x 0 ∈ R n . C. Possieri LQ Optimal Control for a Class of Hybrid Systems 17 / 32

Introduction Finite–Horizon Infinite–Horizon Stabilization Conclusions Limit of the solution to the Hybrid Riccati Equation Assume that the tuple ( A, B, E, F, τ M ) is stabilizable . P T ( t, k ) is the solution to the HRE with P T ( T, K ) = 0 . There exists P ∞ ( σ ) = lim T →∞ P T ( σ, 0) . P ∞ ( σ ) satisfies d dσP ∞ ( σ ) = − A ′ P ∞ ( σ ) − P ∞ ( σ ) A − Q F + P ∞ ( σ ) BR − 1 F B ′ P ∞ ( σ ) , for all σ ∈ [0 , τ M ] , and P ∞ ( τ M ) = Q J + E ′ P ∞ (0) E − E ′ P ∞ (0) F ( R J + F ′ P ∞ (0) F ) − 1 F ′ P ∞ (0) E. C. Possieri LQ Optimal Control for a Class of Hybrid Systems 18 / 32