# 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

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• Introduction Reachability Parameter Synthesis Case Studies Conclusion

Overview

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

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• 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)

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• 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

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• 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

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• 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

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• 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

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

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• Introduction Reachability Parameter Synthesis Case Studies Conclusion

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

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• 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)

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• 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?

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• 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