research-article

Probabilistic Safety Verification of Stochastic Hybrid Systems Using Barrier Certificates

Authors:

Zhengfeng Yang,

Xuandong LiAuthors Info & Claims

ACM Transactions on Embedded Computing Systems (TECS), Volume 16, Issue 5s

Article No.: 186, Pages 1 - 19

https://doi.org/10.1145/3126508

Published: 27 September 2017 Publication History

Abstract

The problem of probabilistic safety verification of stochastic hybrid systems is to check whether the probability that a given system will reach an unsafe region from certain initial states can be bounded by some given probability threshold. The paper considers stochastic hybrid systems where the behavior is governed by polynomial equalities and inequalities, as for usual hybrid systems, but the initial states follow some stochastic distributions. It proposes a new barrier certificate based method for probabilistic safety verification which guarantees the absolute safety in a infinite time horizon that is beyond the reach of existing techniques using either statistical model checking or probabilistic reachable set computation. It also gives a novel computational approach, by building and solving a constrained optimization problem coming from verification conditions of barrier certificates, to compute the lower bound on safety probabilities which can be compared with the given threshold. Experimental evidence is provided demonstrating the applicability of our approach on several benchmarks.

References

[1]

Alessandro Abate, Joost-Pieter Katoen, John Lygeros, and Maria Prandini. 2010. Approximate Model Checking of Stochastic Hybrid Systems. European Journal of Control 16, 6 (2010), 624--641.

[2]

Matthias Althoff, Olaf Stursberg, and Martin Buss. 2008. Stochastic reachable sets of interacting traffic participants. In Proc. of the IEEE Intelligent Vehicles Symposium. IEEE, 1086--1092.

[3]

Rajeev Alur, Costas Courcoubetis, Nicolas Halbwachs, Thomas A. Henzinger, P.-H. Ho, Xavier Nicollin, Alfredo Olivero, Joseph Sifakis, and Sergio Yovine. 1995. The algorithmic analysis of hybrid systems. Theoretical computer science 138, 1 (1995), 3--34.

Digital Library

[4]

Michaël Bensimhoun. 2009. N-Dimensional Cumulative Function, And Other Useful Facts About Gaussians and Normal Densities. Jerusalem, Israel, Tech. Rep (2009).

[5]

Olivier Bouissou, Alexandre Chapoutot, Adel Djaballah, and Michel Kieffer. 2014. Computation of parametric barrier functions for dynamical systems using interval analysis. In Decision and Control (CDC), 2014 IEEE 53rd Annual Conference on. IEEE, 753--758.

[6]

Manuela L. Bujorianu. 2004. Extended stochastic hybrid systems and their reachability problem. In International Workshop on Hybrid Systems: Computation and Control. Springer, 234--249.

[7]

Manuela L. Bujorianu and John Lygeros. 2003. Reachability questions in piecewise deterministic Markov processes. In International Workshop on Hybrid Systems: Computation and Control. Springer, 126--140.

Digital Library

[8]

Liyun Dai, Ting Gan, Bican Xia, and Naijun Zhan. 2017. Barrier Certificates Revisited. Journal of Symbolic Computation 80 (2017), 62--86.

Digital Library

[9]

Thao Dang and Thomas Martin Gawlitza. 2011. Discretizing Affine Hybrid Automata with Uncertainty. Springer Berlin Heidelberg, 473--481.

Digital Library

[10]

Thao Dang and Romain Testylier. 2012. Reachability Analysis for Polynomial Dynamical Systems Using the Bernstein Expansion. Reliable Computing 17, 2 (2012), 128--152.

[11]

Christian Ellen, Sebastian Gerwinn, and Martin Fränzle. 2015. Statistical model checking for stochastic hybrid systems involving nondeterminism over continuous domains. International Journal on Software Tools for Technology Transfer 17, 4 (2015), 485--504.

Digital Library

[12]

Martin Fränzle, Ernst Moritz Hahn, Holger Hermanns, Nicolás Wolovick, and Lijun Zhang. 2011. Measurability and Safety Verification for Stochastic Hybrid Systems. In Proceedings of the 14th International Conference on Hybrid Systems: Computation and Control (HSCC’11). ACM, 43--52.

Digital Library

[13]

William Glover and John Lygeros. 2004. A stochastic hybrid model for air traffic control simulation. Proc. of the International Workshop on Hybrid Systems: Computation and Control (2004), 372--386.

[14]

Ernst Moritz Hahn, Arnd Hartmanns, Holger Hermanns, and Joost-Pieter Katoen. 2013. A compositional modelling and analysis framework for stochastic hybrid systems. Formal Methods in System Design 43, 2 (2013), 191--232.

Digital Library

[15]

João Hespanha. 2004. Stochastic Hybrid Systems: Application to Communication Networks. Proc. of the International Workshop on Hybrid Systems: Computation and Control (2004), 47--56.

[16]

Joao P. Hespanha. 2014. Modeling and analysis of networked control systems using stochastic hybrid systems. Annual Reviews in Control 38, 2 (2014), 155--170.

[17]

Jianghai Hu, John Lygeros, and Shankar Sastry. 2000. Towards a theory of stochastic hybrid systems. In International Workshop on Hybrid Systems: Computation and Control. Springer, 160--173.

Digital Library

[18]

A. Agung Julius. 2006. Approximate Abstraction of Stochastic Hybrid Automata. In Proceedings of the International Workshop on Hybrid Systems: Computation and Control. 318--332.

Digital Library

[19]

James Kapinski, Jyotirmoy V. Deshmukh, Sriram Sankaranarayanan, and Nikos Aréchiga. 2014. Simulation-guided lyapunov analysis for hybrid dynamical systems. In Proc. of the Hybrid Systems: Computation and Control (HSCC). ACM, 133--142.

Digital Library

[20]

Hui Kong, Fei He, Xiaoyu Song, William N. N. Hung, and Ming Gu. 2013. Exponential-condition-based barrier certificate generation for safety verification of hybrid systems. In Computer Aided Verification. Springer, 242--257.

[21]

Michal Kočvara and Michael Stingl. 2005. PENBMI User’s guide (Version 2.0). (2005). Available at http://www.penopt.com.

[22]

Marta Kwiatkowska, Gethin Norman, Jeremy Sproston, and Fuzhi Wang. 2004. Symbolic model checking for probabilistic timed automata. Lecture notes in computer science 3253 (2004), 293--308.

[23]

Kim G. Larsen and Axel Legay. 2016. Statistical Model Checking: Past, Present, and Future. In International Symposium on Leveraging Applications of Formal Methods. Springer, 3--15.

[24]

Axel LegayEmail and Mahesh Viswanathan. 2015. Statistical model checking: challenges and perspectives. International Journal on Software Tools for Technology Transfer 4 (2015), 369--376.

Digital Library

[25]

André Platzer. 2011. Stochastic differential dynamic logic for stochastic hybrid programs. In International Conference on Automated Deduction. Springer, 446--460.

Digital Library

[26]

Stephen Prajna and Ali Jadbabaie. 2004. Safety verification of hybrid systems using barrier certificates. In International Workshop on Hybrid Systems: Computation and Control. Springer, 477--492.

[27]

Stephen Prajna, Ali Jadbabaie, and George J. Pappas. 2004. Stochastic safety verification using barrier certificates. In Decision and Control, 2004. CDC. 43rd IEEE Conference on, Vol. 1. IEEE, 929--934.

[28]

Stephen Prajna, Ali Jadbabaie, and George J. Pappas. 2007. A framework for worst-case and stochastic safety verification using barrier certificates. IEEE Trans. Automat. Control 52, 8 (2007), 1415--1429.

[29]

Stefan Ratschan and Zhikun She. 2007. Safety verification of hybrid systems by constraint propagation-based abstraction refinement. ACM Transactions on Embedded Computing Systems 6, 1 (2007), 573--589.

Digital Library

[30]

Sriram Sankaranarayanan, Xin Chen, and Erika Abrahám. 2013. Lyapunov function synthesis using Handelman representations. In The 9th IFAC Symposium on Nonlinear Control Systems. 576--581.

[31]

Mohamed Amin Ben Sassi, Romain Testylier, Thao Dang, and Antoine Girard. 2012. Reachability analysis of polynomial systems using linear programming relaxations. In Automated Technology for Verification and Analysis. Springer, 137--151.

Digital Library

[32]

Fedor Shmarov and Paolo Zuliani. 2015. Probreach: verified probabilistic delta-reachability for stochastic hybrid systems. In Proceedings of the 18th International Conference on Hybrid Systems: Computation and Control. ACM, 134--139.

Digital Library

[33]

Fedor Shmarov and Paolo Zuliani. 2016. Probabilistic Hybrid Systems Verification via SMT and Monte Carlo Techniques. In Haifa Verification Conference. Springer, 152--168.

[34]

Andrew Sogokon, Khalil Ghorbal, Paul B. Jackson, and André Platzer. 2016. A Method for Invariant Generation for Polynomial Continuous Systems. In Verification, Model Checking, and Abstract Interpretation. Springer, 268--288.

Digital Library

[35]

Jeremy Sproston. 2000. Decidable model checking of probabilistic hybrid automata. In International Symposium on Formal Techniques in Real-Time and Fault-Tolerant Systems. Springer, 31--45.

Digital Library

[36]

Qinsi Wang, Paolo Zuliani, Soonho Kong, Sicun Gao, and Edmund M. Clarke. 2015. SReach: A Probabilistic Bounded Delta-Reachability Analyzer for Stochastic Hybrid Systems. In Proceedings of the 13th International Conference on Computational Methods in Systems Biology CMSB 2015. Springer, 15--27.

[37]

Xia Zeng, Wang Lin, Zhengfeng Yang, Xin Chen, and Lilei Wang. 2016. Darboux-type barrier certificates for safety verification of nonlinear hybrid systems. In Proceedings of the International Conference on Embedded Software (EMSOFT). IEEE, 1--10.

Digital Library

[38]

Lijun Zhang, Zhikun She, Stefan Ratschan, Holger Hermanns, and Ernst Moritz Hahn. 2010. Safety verification for probabilistic hybrid systems. In International Conference on Computer Aided Verification. Springer, 196--211.

Digital Library

Cited By

Nejati APrakash Nayak SSchmuck A(2024)Context-triggered Games for Reactive Synthesis over Stochastic Systems via Control Barrier CertificatesProceedings of the 27th ACM International Conference on Hybrid Systems: Computation and Control10.1145/3641513.3650136(1-12)Online publication date: 14-May-2024
https://dl.acm.org/doi/10.1145/3641513.3650136
Wang YZhou WFan JWang ZLi JChen XHuang CLi WZhu Q(2024)POLAR-Express: Efficient and Precise Formal Reachability Analysis of Neural-Network Controlled SystemsIEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems10.1109/TCAD.2023.333121543:3(994-1007)Online publication date: Mar-2024
https://doi.org/10.1109/TCAD.2023.3331215
Anand MLavaei AZamani M(2024) Compositional synthesis of control barrier certificates for networks of stochastic systems against -regular specifications Nonlinear Analysis: Hybrid Systems10.1016/j.nahs.2023.10142751(101427)Online publication date: Feb-2024
https://doi.org/10.1016/j.nahs.2023.101427
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Probabilistic Safety Verification of Stochastic Hybrid Systems Using Barrier Certificates

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cover image ACM Transactions on Embedded Computing Systems

ACM Transactions on Embedded Computing Systems Volume 16, Issue 5s

Special Issue ESWEEK 2017, CASES 2017, CODES + ISSS 2017 and EMSOFT 2017

October 2017

1448 pages

ISSN:1539-9087

EISSN:1558-3465

DOI:10.1145/3145508

Editor:
Sandeep K. Shukla
Indian Institute of Technology, India

Issue’s Table of Contents

Copyright © 2017 ACM.

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ACM Journals for the Design of Smart and Connected Systems

Publication History

Published: 27 September 2017

Accepted: 01 June 2017

Revised: 01 May 2017

Received: 01 April 2017

Published in TECS Volume 16, Issue 5s

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Shanghai Natural Science Foundation
National Natural Science Foundation of China
China Scholarship Council

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Cited By

Nejati APrakash Nayak SSchmuck A(2024)Context-triggered Games for Reactive Synthesis over Stochastic Systems via Control Barrier CertificatesProceedings of the 27th ACM International Conference on Hybrid Systems: Computation and Control10.1145/3641513.3650136(1-12)Online publication date: 14-May-2024
https://dl.acm.org/doi/10.1145/3641513.3650136
Wang YZhou WFan JWang ZLi JChen XHuang CLi WZhu Q(2024)POLAR-Express: Efficient and Precise Formal Reachability Analysis of Neural-Network Controlled SystemsIEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems10.1109/TCAD.2023.333121543:3(994-1007)Online publication date: Mar-2024
https://doi.org/10.1109/TCAD.2023.3331215
Anand MLavaei AZamani M(2024) Compositional synthesis of control barrier certificates for networks of stochastic systems against -regular specifications Nonlinear Analysis: Hybrid Systems10.1016/j.nahs.2023.10142751(101427)Online publication date: Feb-2024
https://doi.org/10.1016/j.nahs.2023.101427
Gray AForets MSchilling CFerson SBenet L(2024)Verified propagation of imprecise probabilities in non-linear ODEsInternational Journal of Approximate Reasoning10.1016/j.ijar.2023.109044164:COnline publication date: 1-Jan-2024
https://dl.acm.org/doi/10.1016/j.ijar.2023.109044
Nadali ATrivedi AZamani M(2024)Transfer of Safety Controllers Through Learning Deep Inverse Dynamics ModelIFAC-PapersOnLine10.1016/j.ifacol.2024.07.43658:11(129-134)Online publication date: 2024
https://doi.org/10.1016/j.ifacol.2024.07.436
Wu HFeng SGan TWang JXia BZhan N(2024)On Completeness of SDP-Based Barrier Certificate Synthesis over Unbounded DomainsFormal Methods10.1007/978-3-031-71177-0_16(248-266)Online publication date: 13-Sep-2024
https://doi.org/10.1007/978-3-031-71177-0_16
Wetzlinger MKochdumper NBak SAlthoff M(2023)Fully-Automated Verification of Linear Systems Using Reachability Analysis with Support FunctionsProceedings of the 26th ACM International Conference on Hybrid Systems: Computation and Control10.1145/3575870.3587121(1-12)Online publication date: 9-May-2023
https://dl.acm.org/doi/10.1145/3575870.3587121
Jahanshahi NLavaei AZamani M(2023)Compositional Construction of Safety Controllers for Networks of Continuous-Space POMDPsIEEE Transactions on Control of Network Systems10.1109/TCNS.2022.318664910:1(87-99)Online publication date: Mar-2023
https://doi.org/10.1109/TCNS.2022.3186649
Nejati ALavaei AJagtap PSoudjani SZamani M(2023)Formal Verification of Unknown Discrete- and Continuous-Time Systems: A Data-Driven ApproachIEEE Transactions on Automatic Control10.1109/TAC.2023.325514168:5(3011-3024)Online publication date: May-2023
https://doi.org/10.1109/TAC.2023.3255141
Zhu QLi WHuang CChen XZhou WWang YLi JFu F(2023)Verification and Design of Robust and Safe Neural Network-enabled Autonomous Systems2023 59th Annual Allerton Conference on Communication, Control, and Computing (Allerton)10.1109/Allerton58177.2023.10313451(1-8)Online publication date: 26-Sep-2023
https://doi.org/10.1109/Allerton58177.2023.10313451
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