Abstraction refinement and plan revision for control synthesis under - PowerPoint PPT Presentation

Abstraction refinement and plan revision for control synthesis under high level specifications Pierre-Jean Meyer Dimos V. Dimarogonas KTH, Royal Institute of Technology July 12 th 2017

Outline Abstraction-based synthesis Global framework Specifications Valid sets Algorithm Cost functions Result Conclusion Abstraction refinement Plan revision IDDFS Sampling period Meyer & Dimarogonas Abstraction refinement and plan revision 2/15

Abstraction-based synthesis Continuous state Discrete state u 1 d e l u 1 l o r x + = f ( x, u, w ) t u 2 n o c u 2 n Abstraction U Synthesis w u 1 Concretization d x + = f ( x, u, w ) u 1 e l l o r x t u n o C Controller Meyer & Dimarogonas Abstraction refinement and plan revision 3/15

Abstraction procedure x + = f ( x , u , w ) ◮ Define the reachable set for any disturbance w ∈ W : RS ( X , U ) = { f ( x , u , w ) | x ∈ X , u ∈ U , w ∈ W } ◮ Partition of the state space ◮ For each partition cell s and control u : compute the reachable set RS ( s , { u } ) ◮ Obtain a non-deterministic transition system: each pair ( s , u ) may have several successors s s u u u u u Meyer & Dimarogonas Abstraction refinement and plan revision 4/15

Global framework Specification conversion Nominal system Initial coarse Main specification x = u ˙ partition P LTL formula θ Abstraction and Controller synthesis Satisfying Nominal plan ψ ∈ P r abstraction S n to follow Controller Abstraction synthesis Disturbed system x = f ( x, w ) + u ˙ yes Success ? Stop Revised no plan ψ ′ Refined partition P ′ Controller Split a Method synthesis partition element choice Abstraction refinement Plan revision Meyer & Dimarogonas Abstraction refinement and plan revision 5/15

Specifications General Linear Temporal Logic ( LTL ) formula: Satisfying plans are lasso-shaped sequences in the state partition ◮ prefix : finite path from the initial cell ◮ suffix : finite path looping on itself Meyer & Dimarogonas Abstraction refinement and plan revision 6/15

Specifications Syntactically co-safe LTL formula: Satisfiable in finite time ◮ prefix : finite path from the initial cell Meyer & Dimarogonas Abstraction refinement and plan revision 6/15

Valid sets Definition (Valid set) Elements of the refined partition X a that can be controlled to follow the desired sequence of cells Finite plan: ψ = ψ 0 ψ 1 . . . ψ r ◮ Final cell: V ( ψ r ) = { ψ r } ◮ for all k ∈ { 0 , . . . , r − 1 } : V ( ψ k ) = { s ∈ X a | s ⊆ ψ k , ∃ u ∈ U a , RS ( s , { u } ) ⊆ V ( ψ k + 1 ) } Meyer & Dimarogonas Abstraction refinement and plan revision 7/15

Algorithm Specification: ψ 2 reach top-left ψ 1 from bottom left Extract initial plan ψ Initial plan: ψ 0 ψ 0 ψ 1 ψ 2 Initialization k := 2 k := k − 1 yes V ( ψ k ) � = ∅ Refine ψ k Revise plan ψ no Update V ( ψ k ) New plan: φ Method Abstraction Plan choice Last step of the plan: ψ 2 refinement revision Initial valid set: V ( ψ 2 ) = { ψ 2 } Meyer & Dimarogonas Abstraction refinement and plan revision 8/15

Algorithm Specification: ψ 2 reach top-left ψ 1 from bottom left Extract initial plan ψ Initial plan: ψ 0 ψ 0 ψ 1 ψ 2 Initialization k := 2 k := k − 1 yes V ( ψ k ) � = ∅ Refine ψ k Revise plan ψ no Update V ( ψ k ) New plan: φ Method Abstraction Plan choice Transition ψ 1 → ψ 2 refinement revision Empty valid set: V ( ψ 1 ) = ∅ Meyer & Dimarogonas Abstraction refinement and plan revision 8/15

Algorithm Specification: ψ 2 reach top-left ψ 1 from bottom left Extract initial plan ψ Initial plan: ψ 0 ψ 0 ψ 1 ψ 2 Initialization k := 2 k := k − 1 yes V ( ψ k ) � = ∅ Refine ψ k Revise plan ψ no Update V ( ψ k ) New plan: φ Method Abstraction Plan choice Transition ψ 1 → ψ 2 refinement revision Refine ψ 1 Valid set still empty: V ( ψ 1 ) = ∅ Meyer & Dimarogonas Abstraction refinement and plan revision 8/15

Algorithm Specification: ψ 2 reach top-left ψ 1 from bottom left Extract initial plan ψ Initial plan: ψ 0 ψ 0 ψ 1 ψ 2 Initialization k := 2 k := k − 1 yes V ( ψ k ) � = ∅ Refine ψ k Revise plan ψ no Update V ( ψ k ) New plan: φ Method Abstraction Plan choice Transition ψ 1 → ψ 2 refinement revision Refine ψ 1 Valid set still empty: V ( ψ 1 ) = ∅ Meyer & Dimarogonas Abstraction refinement and plan revision 8/15

Algorithm Specification: ψ 2 reach top-left ψ 1 from bottom left Extract initial plan ψ Initial plan: ψ 0 ψ 0 ψ 1 ψ 2 Initialization k := 2 k := k − 1 yes V ( ψ k ) � = ∅ Refine ψ k Revise plan ψ no Update V ( ψ k ) New plan: φ Method Abstraction Plan choice Transition ψ 1 → ψ 2 refinement revision Refine ψ 1 New valid set for ψ 1 : V ( ψ 1 ) � = ∅ Meyer & Dimarogonas Abstraction refinement and plan revision 8/15

Algorithm Specification: ψ 2 reach top-left ψ 1 from bottom left Extract initial plan ψ Initial plan: ψ 0 ψ 0 ψ 1 ψ 2 Initialization k := 2 k := k − 1 yes V ( ψ k ) � = ∅ Refine ψ k Revise plan ψ no Update V ( ψ k ) New plan: φ Method Abstraction Plan choice Transition ψ 0 → ψ 1 refinement revision Empty valid set: V ( ψ 0 ) = ∅ Meyer & Dimarogonas Abstraction refinement and plan revision 8/15

Algorithm Specification: ψ 2 reach top-left ψ 1 from bottom left Extract initial plan ψ Initial plan: ψ 0 ψ 0 ψ 1 ψ 2 Initialization k := 2 k := k − 1 yes V ( ψ k ) � = ∅ Refine ψ k Revise plan ψ no Update V ( ψ k ) New plan: φ Method Abstraction Plan choice Transition ψ 0 → ψ 1 refinement revision Refine ψ 0 Valid set still empty: V ( ψ 0 ) = ∅ Meyer & Dimarogonas Abstraction refinement and plan revision 8/15

Algorithm Specification: ψ 2 reach top-left ψ 1 from bottom left Extract initial plan ψ Initial plan: ψ 0 ψ 0 ψ 1 ψ 2 Initialization k := 2 k := k − 1 yes V ( ψ k ) � = ∅ Refine ψ k Revise plan ψ no Update V ( ψ k ) New plan: φ Method Abstraction Plan choice Transition ψ 0 → ψ 1 refinement revision Refine ψ 0 Valid set still empty: V ( ψ 0 ) = ∅ Meyer & Dimarogonas Abstraction refinement and plan revision 8/15

Algorithm Specification: ψ 2 reach top-left ψ 1 from bottom left Extract initial plan ψ Initial plan: ψ 0 ψ 0 ψ 1 ψ 2 Initialization k := 2 k := k − 1 yes V ( ψ k ) � = ∅ Refine ψ k Revise plan ψ no Update V ( ψ k ) New plan: φ Method Abstraction Plan choice Transition ψ 0 → ψ 1 refinement revision Refine ψ 0 Valid set still empty: V ( ψ 0 ) = ∅ Meyer & Dimarogonas Abstraction refinement and plan revision 8/15

Algorithm φ 4 = ψ 2 Specification: reach top-left φ 2 from bottom left Extract φ 3 = ψ 1 initial plan ψ Revised plan: φ 0 φ 1 φ 0 φ 1 φ 2 φ 3 φ 4 Initialization new k := 3 k := k − 1 yes V ( ψ k ) � = ∅ Refine ψ k Revise plan ψ no Update V ( ψ k ) New plan: φ Method Abstraction Plan choice refinement revision Plan revision: keep partial progress: φ 3 φ 4 = ψ 1 ψ 2 Meyer & Dimarogonas Abstraction refinement and plan revision 8/15

Algorithm φ 4 = ψ 2 Specification: reach top-left φ 2 from bottom left Extract φ 3 = ψ 1 initial plan ψ Revised plan: φ 0 φ 1 φ 0 φ 1 φ 2 φ 3 φ 4 Initialization new k := 3 k := k − 1 yes V ( φ k ) � = ∅ Refine φ k Revise plan ψ no Update V ( φ k ) New plan: φ Method Abstraction Plan choice Transition φ 2 → φ 3 refinement revision Empty valid set: V ( φ 2 ) = ∅ Meyer & Dimarogonas Abstraction refinement and plan revision 8/15

Algorithm φ 4 = ψ 2 Specification: reach top-left φ 2 from bottom left Extract φ 3 = ψ 1 initial plan ψ Revised plan: φ 0 φ 1 φ 0 φ 1 φ 2 φ 3 φ 4 Initialization new k := 3 k := k − 1 yes V ( φ k ) � = ∅ Refine φ k Revise plan ψ no Update V ( φ k ) New plan: φ Method Abstraction Plan choice Transition φ 2 → φ 3 refinement revision Refine φ 2 Valid set still empty: V ( φ 2 ) = ∅ Meyer & Dimarogonas Abstraction refinement and plan revision 8/15

Algorithm φ 4 = ψ 2 Specification: reach top-left φ 2 from bottom left Extract φ 3 = ψ 1 initial plan ψ Revised plan: φ 0 φ 1 φ 0 φ 1 φ 2 φ 3 φ 4 Initialization new k := 3 k := k − 1 yes V ( φ k ) � = ∅ Refine φ k Revise plan ψ no Update V ( φ k ) New plan: φ Method Abstraction Plan choice Transition φ 2 → φ 3 refinement revision Refine φ 2 New valid set for φ 2 : V ( φ 2 ) � = ∅ Meyer & Dimarogonas Abstraction refinement and plan revision 8/15

Algorithm φ 4 = ψ 2 Specification: reach top-left φ 2 from bottom left Extract φ 3 = ψ 1 initial plan ψ Revised plan: φ 0 φ 1 φ 0 φ 1 φ 2 φ 3 φ 4 Initialization new k := 3 k := k − 1 yes V ( φ k ) � = ∅ Refine φ k Revise plan ψ no Update V ( φ k ) New plan: φ Method Abstraction Plan choice Transition φ 1 → φ 2 refinement revision Empty valid set: V ( φ 1 ) = ∅ Meyer & Dimarogonas Abstraction refinement and plan revision 8/15