Reachability Computation and Parameter Synthesis for Polynomial Dynamical Synthesis for Polynomial Dynamical Systems Tommaso Dreossi April 4, 2016 ... Overview What’s this thesis about: Formal analysisofdynamical systems Dynamical system: mathematical model used to describe the

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  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    Reachability Computation andParameter Synthesis for Polynomial

    Dynamical Systems

    Tommaso Dreossi

    April 4, 2016

    1 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    Overview

    Whats this thesis about:

    Formal analysis of dynamical systems Dynamical system: mathematical model used to describe the

    evolution of a system

    Why dynamical systems?

    Help to model, understand, make predictions Dynamical systems are ubiquitous

    2 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    OverviewFormal Verification

    Dynamical systems are fundamental in the design of complexsystems (e.g., cyber-physical systems)

    Find application in safety-critical situations

    It is important to formally verify a system Important questions are:

    Does the system reach an unsafe state? (Reachability) Can we correctly tune the system? (Parameter synthesis)

    3 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    Overview

    Two important problems:

    Reachability: Compute all the reachable states from a set ofinitial conditions

    Parameter Synthesis: Find a set of parameters such that thesystem satisfies a given a property

    t

    x

    X0,P X1X2 X3 X4

    Reachability

    Compute X0,X1,X2,X3,X4, . . .

    t

    x

    X0,P

    Parameter Synthesis

    Find P P

    Xi and P can be infinite bad for nonlinear dynamics

    4 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    Overview

    Two important problems:

    Reachability: Compute all the reachable states from a set ofinitial conditions

    Parameter Synthesis: Find a set of parameters such that thesystem satisfies a given a property

    t

    x

    X0,P X1X2 X3 X4

    Reachability

    Compute X0,X1,X2,X3,X4, . . .

    t

    x

    X0,P

    Parameter Synthesis

    Find P P

    Xi and P can be infinite bad for nonlinear dynamics

    4 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    Overview

    Two important problems:

    Reachability: Compute all the reachable states from a set ofinitial conditions

    Parameter Synthesis: Find a set of parameters such that thesystem satisfies a given a property

    t

    x

    X0,P X1X2 X3 X4

    Reachability

    Compute X0,X1,X2,X3,X4, . . .

    t

    x

    X0,P

    Parameter Synthesis

    Find P P

    Xi and P can be infinite bad for nonlinear dynamics

    4 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    OverviewState of the Art

    Reachability computation:

    Lot results on linear systems Hundreds of variables [FLGD+11, KV00, Fre05] No efficient solutions for nonlinear systems Low dimensions ( 10) [CAS13, KGCC15]

    Parameter synthesis:

    Analytic/optimization techniques (scalability issues) Simulation based approaches (not formal/exhaustive)

    [Don10, MMB03, HWT96]

    No formal approaches dealing with infinite sets

    5 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    OverviewOur contributions

    We consider discrete-time polynomial dynamical systems (nonlinear) andinfinite compact sets (for both states and parameters)

    Reachability analysis:

    Image computation based on boxes, parallelotopes, andparallelotope bundles

    Bernstein coefficients (new efficient algorithm, symbolicapproach)[DD14, DDP14, DDP16]

    Parameter synthesis:

    Formalization using Signal Temporal Logic (STL) Definition of synthesis semantics for STL Algorithm to synthesize parameter sets using STL[DDP15]

    Implementation:

    Sapo1: C++ tool that gathers all the developed methods1https://github.com/tommasodreossi/Sapo

    6 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    OverviewOur contributions

    We consider discrete-time polynomial dynamical systems (nonlinear) andinfinite compact sets (for both states and parameters)

    Reachability analysis:

    Image computation based on boxes, parallelotopes, andparallelotope bundles

    Bernstein coefficients (new efficient algorithm, symbolicapproach)[DD14, DDP14, DDP16]

    Parameter synthesis:

    Formalization using Signal Temporal Logic (STL) Definition of synthesis semantics for STL Algorithm to synthesize parameter sets using STL[DDP15]

    Implementation:

    Sapo1: C++ tool that gathers all the developed methods1https://github.com/tommasodreossi/Sapo

    6 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    OverviewOur contributions

    We consider discrete-time polynomial dynamical systems (nonlinear) andinfinite compact sets (for both states and parameters)

    Reachability analysis:

    Image computation based on boxes, parallelotopes, andparallelotope bundles

    Bernstein coefficients (new efficient algorithm, symbolicapproach)[DD14, DDP14, DDP16]

    Parameter synthesis:

    Formalization using Signal Temporal Logic (STL) Definition of synthesis semantics for STL Algorithm to synthesize parameter sets using STL[DDP15]

    Implementation:

    Sapo1: C++ tool that gathers all the developed methods1https://github.com/tommasodreossi/Sapo

    6 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    OverviewOur contributions

    We consider discrete-time polynomial dynamical systems (nonlinear) andinfinite compact sets (for both states and parameters)

    Reachability analysis:

    Image computation based on boxes, parallelotopes, andparallelotope bundles

    Bernstein coefficients (new efficient algorithm, symbolicapproach)[DD14, DDP14, DDP16]

    Parameter synthesis:

    Formalization using Signal Temporal Logic (STL) Definition of synthesis semantics for STL Algorithm to synthesize parameter sets using STL[DDP15]

    Implementation:

    Sapo1: C++ tool that gathers all the developed methods1https://github.com/tommasodreossi/Sapo

    6 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    OverviewRoadmap

    1 Reachability Analysis

    1 How to transform/approximate a set?2 Bernstein coefficients for polynomials3 Box-based reachability4 Parallelotope-based reachability5 Parallelotope bundle-based reachability

    2 Parameter Synthesis

    1 Problem formalization via STL2 Synthesis semantics3 Synthesis algorithm

    3 Application

    1 Tool overview2 Case studies

    4 Conclusion

    7 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    OverviewRoadmap

    1 Reachability Analysis

    1 How to transform/approximate a set?2 Bernstein coefficients for polynomials3 Box-based reachability4 Parallelotope-based reachability5 Parallelotope bundle-based reachability

    2 Parameter Synthesis

    1 Problem formalization via STL2 Synthesis semantics3 Synthesis algorithm

    3 Application

    1 Tool overview2 Case studies

    4 Conclusion

    7 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    ReachabilityComputation

    Problem

    Given a dynamical system f : Rn Rn and a set of initialconditions X0 Rn, compute the reachable sets up to time T N

    How to compute/represent nonlinear set transformations?(nonconvexity)

    Idea: Over-approximate sets with simpler objects (polytopes)

    8 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    ReachabilityComputation

    Problem

    Given a dynamical system f : Rn Rn and a set of initialconditions X0 Rn, compute the reachable sets up to time T N

    How to compute/represent nonlinear set transformations?(nonconvexity)

    Idea: Over-approximate sets with simpler objects (polytopes)

    8 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    ReachabilityComputation

    Problem

    Given a dynamical system f : Rn Rn and a set of initialconditions X0 Rn, compute the reachable sets up to time T N

    How to compute/represent nonlinear set transformations?(nonconvexity)

    Idea: Over-approximate sets with simpler objects (polytopes)

    8 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    ReachabilitySingle Step

    Polytopes as solution of linear systems

    X0 Dx c (D, c : template and offset)

    How to find c j?

    c j maxxXi

    Dj f (x)

    Nonlinear optimization problemHow to bound a polynomial?

    9 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    ReachabilitySingle Step

    Polytopes as solution of linear systems

    X0 Dx c (D, c : template and offset)

    How to find c j?

    c j maxxXi

    Dj f (x)

    Nonlinear optimization problemHow to bound a polynomial?

    9 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    ReachabilitySingle Step

    Polytopes as solution of linear systems

    X0 Dx c (D, c : template and offset)

    How to find c j?

    c j maxxXi

    Dj f (x)

    Nonlinear optimization problemHow to bound a polynomial?

    9 / 32

  • Introduction Reachability Parameter Synthesis Case Studies Conclusion

    ReachabilitySingle Step

    Polytopes as solution of linear systems

    X0 Dx c (D, c : template and offset)

    How to find c j?

    c j maxxXi

    Dj f (x)

    Nonlinear optimization problemHow to bound a polynomial?

    9 / 32