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

April, 2017

Formal Synthesis of Stabilizing Controllers for Switched Systems

Pavithra Prabhakar & Miriam García Soto

Kansas State University & IMDEA Software Institute

1

HSCC’17 Pittsburgh, PA, USA

Switching logic synthesis

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Given a set of dynamics and a partition, assign dynamics to each facet such that the resulting switched system is stable.

f1 f2

f4

f3 f5

f6

Dynamics Partition

Switched system

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f1 f2 f4 f3 f5 f6 Ω1 Ω2 Ω3 Ω4 Ω5 Ω6

R = {Ω1, Ω2, . . . , Ωk} closed convex polyhedra

⇤ Rn = [Ω∈RΩ ⇤ ˚ Ωi 6= ; for every i ⇤ ˚ Ωi \ ˚ Ωj = ; for every i 6= j

F = {f1, f2, . . . , fk} maximal closed convex subsets of boundary of Ω’s Partition - finite set of valid facets α : F+ → P fi, fj, . . . , fl 7! p Switching strategy ˙ x(t) ∈ gp(x(t)), p ∈ P S = (P, {gp}p∈P) gp : Rn → 2Rn Family of dynamical systems f1 f2 f3 f4 f5 Ω1 Ω2 Ω3 Ω4 Ω5 Switched system Sα = (P, {gp}p∈P, α)

Stabilization problem

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Given a system S and a set of valid facets F, find a switching strategy α : F+ → P, such that the switched system Sα is stable.

δ

✏ x(0) x

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 Bδ(0) implies x ∈ Bε(0).

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Overview

⇤ Abstract a game graph G from a family of dynamics S and a set

• f valid facets F.

⇤ Induce an energy game graph Ge from G. ⇤ Compute an energy winning strategy σ from the game graph Ge. ⇤ Extract a stabilizing switching strategy α from σ.

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Abstract Game Graph Construction

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Quantitative predicate abstraction

f2 f1 1, f4 2, f4 2, f3 1, f3 1, f2 1, f1 2, f1 2, f2 f4 f3

Quantitative predicate abstraction Ωij common region of fi and fj W((p, fi), fj) = sup{||xj|| ||xi|| : xi ∈ fi, xj ∈ fj, xi

p

− − →

Ωij xj}

f1 f2 f4 f3

F = {f1, f2, f3, f4}

˙ x = A2x ˙ x = A1x

S = ({1, 2}, {A1, A2})

1/2 1/2 5/2 5/2 3/10 3/10 1 1 1 1 1 1 1 1

Ω12 1 2

f1 f1 f2 f2

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Auxiliary cycles

f1 f2 f1 f2 f1 f2

d

d0 p, f1

convergence or containment divergence

g g0

p, f2

Abstraction Abstraction 2 1 1 1 1 1

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Strategy Synthesis

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Game graph

Game graph is a weighted graph G=(V,E,W) ⇤ V = V0 ∪ V1 ⇤ V0 ∩ V1 = ∅ ⇤ E ⊆ (V0 × V1) ∪ (V1 × V0) ⇤ W : E → Q ⇤ Every node has a succesor A strategy is a function σ : V ∗V0 → V1, where V ∗ is the set of finite sequences over V with zero or more elements.

f2 f1 1, f4 2, f4 2, f3 1, f3 1, f2 1, f1 2, f1 2, f2 f4 f3

1/2 1/2 5/2 5/2 3/10 3/10 1 1 1 1 1 1 1 1

d d0

1 1 1 2

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Strategy Example

˙ x = A2x ˙ x = A1x

S = ({1, 2}, {A1, A2}) Weight of the cycle is 1/2 · 5/2 · 1/2 · 5/2 > 1

f2 f1 1, f4 2, f4 2, f3 1, f3 1, f2 1, f1 2, f1 2, f2 f4 f3

1/2 1/2 5/2 5/2 3/10 3/10 1 1 1 1 1 1 1 1

d d0

1 1 1 2

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Strategy Example

˙ x = A2x ˙ x = A1x

S = ({1, 2}, {A1, A2})

f2 f1 1, f4 2, f4 2, f3 1, f3 1, f2 1, f1 2, f1 2, f2 f4 f3

1/2 1/2 5/2 5/2 3/10 3/10 1 1 1 1 1 1 1 1

d d0

1 1 1 2

Weight of the cycle is 1/2 · 3/10 · 1/2 · 3/10 < 1

f1 f2 f4 f3

No cycles with weight greater than 1 implies stability.

Soundness of abstraction

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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. 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. Theorem - stabilizable switching strategy

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Energy game

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.

A strategy σ is a winning energy strategy if there exists C∈ N such that for every play τ = v1v2 . . . determined by σ, C+ Pj

i=1W(vi, vi+1) > 0.

[Brim et al. FMSD’11]

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Energy game

Theorem - bounded strategy σ is a winning energy strategy for Ge if and only if σ is a winning bounded strategy for G. W(e) = ae be G = (V,E,W) lcmG := least common multiple {be : e ∈ E} Ge = (V,E,We) We = - lcmG · W Energy game graph Bounded game graph 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.

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Reduction to energy game

f2 f1 1, f4 2, f4 2, f3 1, f3 1, f2 1, f1 2, f1 2, f2 f4 f3

1/2 1/2 5/2 5/2 3/10 3/10 1 1 1 1 1 1 1 1

d d0

1 1 1 2

f2 f1 1, f4 2, f4 2, f3 1, f3 1, f2 1, f1 2, f1 2, f2 f4 f3 d d0

ln(1/2) ln(1/2) ln(5/2) ln(5/2) ln(3/10) ln(3/10)

• lcmG·
• lcmG·
• lcmG·
• lcmG·
• lcmG·
• lcmG·
• lcmG

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Reduction to energy game

f2 f1 1, f4 2, f4 2, f3 1, f3 1, f2 1, f1 2, f1 2, f2 f4 f3

1/2 1/2 5/2 5/2 3/10 3/10 1 1 1 1 1 1 1 1

d d0

1 1 1 2

σ : V0 → V1 f1 7! (1, f1) f2 7! (2, f2) f3 7! (1, f3) f4 7! (2, f4) Winning energy strategy f2 f1 1, f4 2, f4 2, f3 1, f3 1, f2 1, f1 2, f1 2, f2 f4 f3 d d0

ln(1/2) ln(1/2) ln(5/2) ln(5/2) ln(3/10) ln(3/10)

• lcmG·
• lcmG·
• lcmG·
• lcmG·
• lcmG·
• lcmG·
• lcmG

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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

• btain stable controllers which satisfy the temporal logic formulas.

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Thank you