Formal Synthesis of Stabilizing Controllers for Switched Systems - PowerPoint PPT Presentation

Formal Synthesis of Stabilizing Controllers for Switched Systems Pavithra Prabhakar & Miriam García Soto Kansas State University & IMDEA Software Institute HSCC’17 Pittsburgh, PA, USA April, 2017 1

Switching logic synthesis Partition Dynamics f 1 f 2 f 3 f 6 f 5 f 4 Given a set of dynamics and a partition, assign dynamics to each facet such that the resulting switched system is stable. 2

Switched system Family of dynamical systems Partition - finite set of valid facets R = { Ω 1 , Ω 2 , . . . , Ω k } closed convex polyhedra S = ( P , { g p } p ∈ P ) ⇤ R n = [ Ω ∈ R Ω f 1 f 2 x ( t ) ∈ g p ( x ( t )) , ˙ p ∈ P Ω 1 ⇤ ˚ Ω i 6 = ; for every i Ω 6 Ω 2 g p : R n → 2 R n f 3 f 6 ⇤ ˚ Ω i \ ˚ Ω j = ; for every i 6 = j Ω 5 Ω 3 Ω 4 f 5 Switching strategy F = { f 1 , f 2 , . . . , f k } f 4 α : F + → P maximal closed convex subsets of boundary of Ω ’s f i , f j , . . . , f l 7! p Switched system S α = ( P , { g p } p ∈ P , α ) f 1 f 2 f 3 f 4 f 5 Ω 1 Ω 2 Ω 3 Ω 4 Ω 5 3

Stabilization problem Given a system S and a set of valid facets F , find a switching strategy α : F + → P , such that the switched system S α is stable. A system is Lyapunov stable with respect to 0 if for every ε > 0 there exists δ > 0 such that every execution x starting from B δ (0) implies x ∈ B ε (0). ✏ δ x (0) 0 x 4

Overview ⇤ Abstract a game graph G from a family of dynamics S and a set of valid facets F . ⇤ Induce an energy game graph G e from G. ⇤ Compute an energy winning strategy σ from the game graph G e . ⇤ Extract a stabilizing switching strategy α from σ . 5

Abstract Game Graph Construction 6

Quantitative predicate abstraction S = ( { 1 , 2 } , { A 1 , A 2 } ) Quantitative predicate abstraction f 1 f 1 1 5 / 2 f 1 1 , f 4 1 , f 1 1 3 / 10 1 / 2 1 f 2 f 2 2 , f 4 2 , f 1 1 f 2 f 4 1 x = A 1 x x = A 2 x ˙ ˙ 1 2 , f 3 2 , f 2 1 1 / 2 3 / 10 F = { f 1 , f 2 , f 3 , f 4 } 1 , f 2 1 , f 3 1 f 3 f 1 5 / 2 Ω 12 2 f 4 W (( p, f i ) , f j ) = sup {|| x j || f 2 p || x i || : x i ∈ f i , x j ∈ f j , x i Ω ij x j } − − → 1 Ω ij common region of f i and f j f 3 7

Auxiliary cycles divergence f 1 1 Abstraction 1 p, f 1 d 0 d f 2 2 convergence or containment f 1 f 2 1 1 Abstraction g 0 p, f 2 g 1 f 1 f 2 8

Strategy Synthesis 9

Game graph 1 5 / 2 f 1 Game graph is a weighted graph G=(V,E,W) 1 , f 4 1 , f 1 1 3 / 10 ⇤ V = V 0 ∪ V 1 1 / 2 1 1 2 , f 4 2 , f 1 d 2 ⇤ V 0 ∩ V 1 = ∅ 1 1 f 2 f 4 1 1 d 0 ⇤ E ⊆ ( V 0 × V 1 ) ∪ ( V 1 × V 0 ) 1 2 , f 2 2 , f 3 1 ⇤ W : E → Q 1 / 2 3 / 10 1 , f 2 1 , f 3 ⇤ Every node has a succesor 1 f 3 5 / 2 A strategy is a function σ : V ∗ V 0 → V 1 , where V ∗ is the set of finite sequences over V with zero or more elements. 10

Strategy Example S = ( { 1 , 2 } , { A 1 , A 2 } ) 1 5 / 2 f 1 1 , f 4 1 , f 1 1 3 / 10 1 / 2 1 1 2 , f 4 2 , f 1 d 2 1 1 f 2 f 4 1 1 d 0 1 2 , f 2 2 , f 3 1 1 / 2 x = A 1 x x = A 2 x ˙ ˙ 3 / 10 1 , f 2 1 , f 3 1 f 3 5 / 2 Weight of the cycle is 1 / 2 · 5 / 2 · 1 / 2 · 5 / 2 > 1 11

Strategy Example S = ( { 1 , 2 } , { A 1 , A 2 } ) 1 5 / 2 f 1 1 , f 4 1 , f 1 1 3 / 10 1 / 2 1 1 2 , f 4 2 , f 1 d 2 1 1 f 2 f 4 1 1 d 0 1 2 , f 2 2 , f 3 1 1 / 2 x = A 1 x x = A 2 x ˙ ˙ 3 / 10 1 , f 2 1 , f 3 1 f 3 5 / 2 f 1 Weight of the cycle is 1 / 2 · 3 / 10 · 1 / 2 · 3 / 10 < 1 f 4 f 2 f 3 No cycles with weight greater than 1 implies stability. 12

Soundness of abstraction A strategy σ is a winning bounded strategy if there exists M ∈ Z such that for every play τ determined by σ , W( τ ) 6 M. Theorem - stabilizable switching strategy A winning bounded strategy for the game graph G( S , F ), induces a strategy which solves the stabilization problem for the system S and the facets F . 13

Energy game A strategy σ is a winning energy strategy if there exists C ∈ N such that for every play τ = v 1 v 2 . . . determined by σ , C+ P j i =1 W( v i , v i +1 ) > 0. [Brim et al. FMSD’11] Theorem - energy strategy Given a game graph (V,E,W) where W: E → Z , if there exists a winning energy strategy, then there exists a memoryless winning energy strategy. Further, there is an algorithm which returns the memoryless winning energy strategy. 14

Energy game Modification of G( S , F ) to an energy game graph ⇤ Reduce multiplicative game graph to addition game graph. ⇤ Weights are required to be integers. ⇤ Winning energy strategy provides plays lower bounded by a value. Bounded game graph Energy game graph G = (V,E,W) G e = (V,E,W e ) W( e ) = a e b e W e = - lcm G · W lcm G := least common multiple { b e : e ∈ E } Theorem - bounded strategy σ is a winning energy strategy for G e if and only if σ is a winning bounded strategy for G. 15

Reduction to energy game ln(5/2) - lcm G · 0 1 5 / 2 f 1 f 1 1 , f 4 1 , f 4 0 1 , f 1 1 , f 1 1 3 / 10 ln(3/10) - lcm G · 1 / 2 1 0 ln(1/2) - lcm G · 1 0 2 , f 4 2 , f 4 2 , f 1 2 , f 1 d d 2 1 1 - lcm G 0 f 2 0 f 2 f 4 f 4 1 0 d 0 0 1 d 0 1 2 , f 2 2 , f 3 2 , f 2 2 , f 3 0 0 1 1 / 2 ln(1/2) - lcm G · 3 / 10 ln(3/10) - lcm G · 1 , f 2 1 , f 2 1 , f 3 1 , f 3 1 f 3 f 3 ln(5/2) 5 / 2 0 - lcm G · 16

Reduction to energy game ln(5/2) - lcm G · 0 1 5 / 2 f 1 f 1 1 , f 4 1 , f 4 0 1 , f 1 1 , f 1 1 3 / 10 ln(3/10) - lcm G · 1 / 2 1 0 ln(1/2) - lcm G · 1 0 2 , f 4 2 , f 4 2 , f 1 2 , f 1 d d 2 1 1 - lcm G 0 0 f 2 f 2 f 4 f 4 1 0 d 0 0 1 d 0 1 2 , f 2 2 , f 3 2 , f 2 2 , f 3 0 0 1 1 / 2 ln(1/2) - lcm G · 3 / 10 ln(3/10) - lcm G · 1 , f 2 1 , f 2 1 , f 3 1 , f 3 1 f 3 f 3 ln(5/2) 5 / 2 0 - lcm G · Winning energy strategy σ : V 0 → V 1 f 1 7! (1 , f 1 ) f 2 7! (2 , f 2 ) f 3 7! (1 , f 3 ) f 4 7! (2 , f 4 ) 17

Conclusion ⇤ An abstraction technique and game based approach for synthesizing a switching logic for stabilization. ⇤ Our approach can be combined with temporal logic properties to obtain stable controllers which satisfy the temporal logic formulas. 18

Thank you 19