Reachability Analysis of Generalized Polygonal Hybrid Systems - PowerPoint PPT Presentation

Reachability Analysis of Generalized Polygonal Hybrid Systems (GSPDIs) Gerardo Schneider Department of Informatics University of Oslo SAC-SV’08 March 16–20, 2008 - Fortaleza university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 1 / 44

Reachability Analysis of GSPDIs Hybrid System: combines discrete and continuous dynamics Examples: thermostat, robot, chemical reaction university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 3 / 44

Reachability Analysis of GSPDIs Hybrid System: combines discrete and continuous dynamics Examples: thermostat, robot, chemical reaction e 3 e 2 R 4 R 2 e 1 x f ? e 4 R 3 R 5 R 1 R 7 e 5 e 8 x 0 R 8 R 6 university-logo e 6 e 7 Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 3 / 44

Outline Polygonal Hybrid Systems (SPDIs) and Motivation 1 Generalized Polygonal Hybrid Systems (GSPDIs) 2 Reachability Analysis of GSPDIs 3 university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 4 / 44

Outline Polygonal Hybrid Systems (SPDIs) and Motivation 1 Generalized Polygonal Hybrid Systems (GSPDIs) 2 Reachability Analysis of GSPDIs 3 university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 5 / 44

Polygonal Hybrid Systems (SPDIs) Preliminaries A constant differential inclusion (angle between vectors a and b ): x ∈ ∠ b ˙ a x’ b a R x university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 7 / 44

Polygonal Hybrid Systems (SPDIs) A finite partition of (a subset of) the plane into convex polygonal sets (regions) x ∈ ∠ b Dynamics given by the angle determined by two vectors: ˙ a e 3 e 2 R 4 R 2 e 1 e 4 R 3 R 5 R 1 R 7 e 5 e 8 R 8 R 6 e 6 e 7 university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 9 / 44

Polygonal Hybrid Systems (SPDIs) A finite partition of (a subset of) the plane into convex polygonal sets (regions) x ∈ ∠ b Dynamics given by the angle determined by two vectors: ˙ a e 3 e 2 R 4 x f R 2 e 1 e 4 R 3 R 5 R 1 R 7 e 5 e 8 x 0 R 8 R 6 e 6 e 7 university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 9 / 44

Polygonal Hybrid Systems (SPDIs) Goodness Goodness Assumption The dynamics of an SPDI only allows trajectories traversing any edge only in one direction university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 11 / 44

Polygonal Hybrid Systems (SPDIs) Goodness Goodness Assumption The dynamics of an SPDI only allows trajectories traversing any edge only in one direction e 4 e 4 e 3 e 3 e 5 e 5 P P b b e 2 e 2 a a e 6 e 6 e 1 e 1 Bad region Good region university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 11 / 44

Polygonal Hybrid Systems (SPDIs) Goodness Goodness Assumption The dynamics of an SPDI only allows trajectories traversing any edge only in one direction e 4 e 4 exit−only exit−only e 3 e 3 inout e 5 e 5 P P b b e 2 e 2 a a e 6 e 6 entry−only e 1 e 1 entry−only Bad region Good region university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 13 / 44

Polygonal Hybrid Systems (SPDIs) Goodness Goodness Assumption The dynamics of an SPDI only allows trajectories traversing any edge only in one direction e 4 e 4 exit−only exit−only e 3 e 3 inout e 5 e 5 P P b b e 2 e 2 a a e 6 e 6 entry−only e 1 e 1 entry−only Bad region Good region Theorem Under the goodness assumption, reachability for SPDIs is decidable university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 13 / 44

Motivation Use of SPDIs for approximating non-linear differential equations Example Pendulum with friction coefficient k , mass M , pendulum length R and y = − ky MR 2 − g sin ( x ) gravitational constant g . Behaviour: ˙ x = y and ˙ R university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 15 / 44

Motivation Use of SPDIs for approximating non-linear differential equations Example Pendulum with friction coefficient k , mass M , pendulum length R and y = − ky MR 2 − g sin ( x ) gravitational constant g . Behaviour: ˙ x = y and ˙ R Triangulation of the plane: Huge number of regions Need to reduce the complexity ... without too much overhead Relax Goodness: GSPDI university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 15 / 44

Motivation Use of SPDIs for approximating non-linear differential equations university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 17 / 44

Outline Polygonal Hybrid Systems (SPDIs) and Motivation 1 Generalized Polygonal Hybrid Systems (GSPDIs) 2 Reachability Analysis of GSPDIs 3 university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 18 / 44

GSPDI: Generalized SPDI Definition An SPDI without the goodness assumption is called a GSPDI e 3 e 2 R 4 R 2 e 1 e 4 R 3 R 5 R 1 R 7 e 5 e 8 R 8 R 6 e 6 e 7 university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 20 / 44

Why Goodness is Good e 3 e 2 e 3 e 2 e 3 e 2 R 4 R 4 R 4 R 2 R 2 R 2 1 1 e 1 e 1 e 1 e 4 e 4 e 4 R 3 R 3 R 3 x f x f x f R 5 R 1 R 5 R 5 R 1 R 1 R 7 R 7 R 7 e 5 e 5 e 5 e 8 e 8 e 8 x 0 x 0 x 0 R 8 R 8 R 8 R 6 R 6 R 6 e 6 e 7 e 6 e 7 e 6 e 7 2 3 e 6 e 7 e 8 e 1 e 2 e 3 e 6 e 7 e 8 e 1 e 2 e 3 e 6 e 7 e 8 ( e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 ) 5 e 9 ( e 6 e 7 e 8 e 1 e 2 e 3 e 4 e 5 ) 5 e 6 e 7 e 8 e 9 4 r e 6 e 7 e 8 e 1 e 2 e 3 e 6 e 7 e 8 ( e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 ) ∗ e 9 r 1 s ∗ university-logo 1 r 2 Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 22 / 44

Why Goodness is Good Theorem An edge-signature σ = e 1 . . . e p can always be abstracted into types of signatures of the form σ A = r 1 s ∗ 1 . . . r n s ∗ n r n + 1 , where r i is a sequence of pairwise different edges and all s i are disjoint simple cycle. There are finitely many type of signatures. university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 24 / 44

Why Goodness is Good Theorem An edge-signature σ = e 1 . . . e p can always be abstracted into types of signatures of the form σ A = r 1 s ∗ 1 . . . r n s ∗ n r n + 1 , where r i is a sequence of pairwise different edges and all s i are disjoint simple cycle. There are finitely many type of signatures. Many proofs (decidability, soundess, completeness) depend on the goodness assumption university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 24 / 44

Problems when Relaxing Goodness Finiteness argument for types of signature is broken for GSPDIs b c Type of signature: ( abcd ) ∗ ( dcba ) ∗ ( abcd ) ∗ d a a d university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 26 / 44

Problems when Relaxing Goodness Finiteness argument for types of signature is broken for GSPDIs b c Type of signature: ( abcd ) ∗ ( dcba ) ∗ ( abcd ) ∗ d a a d Challenge: Reachability analysis of GSPDIs Reduce GSPDI reachability to SPDI reachability; or Provide a completely new decidability proof for GSPDI. university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 26 / 44

Outline Polygonal Hybrid Systems (SPDIs) and Motivation 1 Generalized Polygonal Hybrid Systems (GSPDIs) 2 Reachability Analysis of GSPDIs 3 university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 27 / 44

Getting a Decision Algorithm for GSPDIs Based on that of SPDIs It is enough to consider trajectories without self-crossing 1 It is possible to eliminate all inout edges, preserving reachability 2 It is possible to eliminate all sliding edges, preserving reachability 3 Re-state and prove some results on SPDI reachability useful to 4 GPSDI reachability analysis Prove soundness and termination 5 university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 29 / 44

Getting a Decision Algorithm for GSPDIs Based on that of SPDIs It is enough to consider trajectories without self-crossing 1 It is possible to eliminate all inout edges, preserving reachability 2 It is possible to eliminate all sliding edges, preserving reachability 3 Re-state and prove some results on SPDI reachability useful to 4 GPSDI reachability analysis Prove soundness and termination 5 university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 31 / 44

Getting a Decision Algorithm for GSPDIs Based on that of SPDIs It is enough to consider trajectories without self-crossing 1 It is possible to eliminate all inout edges, preserving reachability 2 It is possible to eliminate all sliding edges, preserving reachability 3 Re-state and prove some results on SPDI reachability useful to 4 GPSDI reachability analysis Prove soundness and termination 5 No decision algorithm for reachability of GSPDIs... We will give a semi-test algorithm! university-logo Gerardo Schneider () Reachability Analysis of GSPDIs SAC-SV’08 – 20.03.2008 31 / 44