research-article

Safety Verification of Nonlinear Hybrid Systems Based on Invariant Clusters

Authors:

Sergiy Bogomolov,

Christian Schilling,

Thomas A. HenzingerAuthors Info & Claims

HSCC '17: Proceedings of the 20th International Conference on Hybrid Systems: Computation and Control

Pages 163 - 172

https://doi.org/10.1145/3049797.3049814

Published: 13 April 2017 Publication History

Abstract

In this paper, we propose an approach to automatically compute invariant clusters for nonlinear semialgebraic hybrid systems. An invariant cluster for an ordinary differential equation (ODE) is a multivariate polynomial invariant g(u, x)=0, parametric in u, which can yield an infinite number of concrete invariants by assigning different values to u so that every trajectory of the system can be overapproximated precisely by the intersection of a group of concrete invariants. For semialgebraic systems, which involve ODEs with multivariate polynomial right-hand sides, given a template multivariate polynomial g(u, x), an invariant cluster can be obtained by first computing the remainder of the Lie derivative of g(u,x) divided by g(u, x) and then solving the system of polynomial equations obtained from the coefficients of the remainder. Based on invariant clusters and sum-of-squares (SOS) programming, we present a new method for the safety verification of hybrid systems. Experiments on nonlinear benchmark systems from biology and control theory show that our approach is efficient.

References

[1]

R. Alur, T. Dang, and F. Ivancić. Progress on reachability analysis of hybrid systems using predicate abstraction. In HSCC. 2003.

[2]

E. Asarin, T. Dang, and A. Girard. Reachability analysis of nonlinear systems using conservative approximation. In HSCC. 2003.

[3]

A. Ayad. A survey on the complexity of solving algebraic systems. In International Mathematical Forum, volume 5, 2010.

[4]

S. Bogomolov, A. Donzé, G. Frehse, R. Grosu, T. T. Johnson, H. Ladan, A. Podelski, and M. Wehrle. Guided search for hybrid systems based on coarse-grained space abstractions. STTT, 2015.

[5]

S. Bogomolov, G. Frehse, M. Greitschus, R. Grosu, C. S. Pasareanu, A. Podelski, and T. Strump. Assume-guarantee abstraction refinement meets hybrid systems. In HVC, 2014.

[6]

S. Bogomolov, G. Frehse, R. Grosu, H. Ladan, A. Podelski, and M. Wehrle. A box-based distance between regions for guiding the reachability analysis of SpaceEx. In CAV, 2012.

Digital Library

[7]

S. Bogomolov, C. Herrera, M. Muniz, B. Westphal, and A. Podelski. Quasi-dependent variables in hybrid automata. In HSCC, 2014.

Digital Library

[8]

S. Bogomolov, C. Schilling, E. Bartocci, G. Batt, H. Kong, and R. Grosu. Abstraction-based parameter synthesis for multiaffine systems. In HVC, 2015.

[9]

X. Chen, E. Ábrahám, and S. Sankaranarayanan. Taylor model flowpipe construction for non-linear hybrid systems. In RTSS, 2012.

Digital Library

[10]

D. A. Cox, J. Little, and D. O'Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, 2007.

[11]

T. Dang, O. Maler, and R. Testylier. Accurate hybridization of nonlinear systems. In HSCC, 2010.

Digital Library

[12]

K. Ghorbal and A. Platzer. Characterizing algebraic invariants by differential radical invariants. In TACAS. 2014.

[13]

A. Girard and S. Martin. Synthesis for constrained nonlinear systems using hybridization and robust controllers on simplices. IEEE Trans. Automat. Contr., 57(4), 2012.

[14]

A. Goriely. Integrability and nonintegrability of dynamical systems, volume 19. World Scientific, 2001.

[15]

E. Goubault, J. Jourdan, S. Putot, and S. Sankaranarayanan. Finding non-polynomial positive invariants and Lyapunov functions for polynomial systems through Darboux polynomials. In ACC, 2014.

[16]

S. Gulwani and A. Tiwari. Constraint-based approach for analysis of hybrid systems. In CAV, 2008.

Digital Library

[17]

T. A. Henzinger. The theory of hybrid automata. In LICS, 1996.

[18]

Y. Jiang, H. Liu, H. Kong, R. Wang, M. Hosseini, J. Sun, and L. Sha. Use runtime verification to improve the quality of medical care practice. In ICSE, 2016.

Digital Library

[19]

T. T. Johnson and S. Mitra. Invariant synthesis for verification of parameterized cyber-physical systems with applications to aerospace systems. In AIAA Infotech at Aerospace Conference, 2013.

[20]

H. Kong, F. He, X. Song, W. N. Hung, and M. Gu. Exponential-condition-based barrier certificate generation for safety verification of hybrid systems. In CAV, 2013.

Digital Library

[21]

H. Kong, X. Song, D. Han, M. Gu, and J. Sun. A new barrier certificate for safety verification of hybrid systems. The Computer Journal, 57(7):1033--1045, 2014.

[22]

J. Liu, N. Zhan, and H. Zhao. Computing semi-algebraic invariants for polynomial dynamical systems. In EMSOFT, 2011.

Digital Library

[23]

A. Platzer and E. Clarke. Computing differential invariants of hybrid systems as fixedpoints. In CAV, 2008.

Digital Library

[24]

S. Prajna and A. Jadbabaie. Safety verification of hybrid systems using barrier certificates. HSCC, 2004.

[25]

S. Prajna, A. Papachristodoulou, P. Seiler, and P. A. Parrilo. SOSTOOLS and its control applications. 2005.

[26]

R. Rebiha, A. V. Moura, and N. Matringe. Generating invariants for non-linear hybrid systems. TCS, 594, 2015.

[27]

E. Rodrıguez-Carbonell and A. Tiwari. Generating polynomial invariants for hybrid systems. HSCC, 2005.

Digital Library

[28]

S. Sankaranarayanan. Automatic invariant generation for hybrid systems using ideal fixed points. In HSCC, 2010.

Digital Library

[29]

S. Sankaranarayanan, H. Sipma, and Z. Manna. Constructing invariants for hybrid systems. HSCC, 2004.

[30]

B. Sassi, M. Amin, A. Girard, and S. Sankaranarayanan. Iterative computation of polyhedral invariants sets for polynomial dynamical systems. In CDC, 2014.

[31]

R. F. Stengel. Flight dynamics. Fluid Dynamics, 1, 2004.

[32]

G. Stengle. A Nullstellensatz and a Positivstellensatz in semialgebraic geometry. Mathematische Annalen, 207(2), 1974.

[33]

A. Tiwari. Abstractions for hybrid systems. Formal Methods in System Design, 32(1), 2008.

Digital Library

[34]

A. Tiwari and G. Khanna. Nonlinear systems: Approximating reach sets. HSCC, 2004.

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Safety Verification of Nonlinear Hybrid Systems Based on Invariant Clusters

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cover image ACM Conferences

HSCC '17: Proceedings of the 20th International Conference on Hybrid Systems: Computation and Control

April 2017

288 pages

ISBN:9781450345903

DOI:10.1145/3049797

Program Chairs:
Goran Frehse
University Grenoble Alpes
,
Sayan Mitra
University of Illinois at Urbana Champaign

Copyright © 2017 ACM.

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Published: 13 April 2017

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HSCC '17: 20th International Conference on Hybrid Systems: Computation and Control

April 18 - 20, 2017

Pennsylvania, Pittsburgh, USA

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HSCC '17 Paper Acceptance Rate 29 of 76 submissions, 38%;

Overall Acceptance Rate 153 of 373 submissions, 41%

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https://doi.org/10.1007/978-3-031-19759-8_13
Boreale M(2021)Automatic pre- and postconditions for partial differential equationsInformation and Computation10.1016/j.ic.2021.104860(104860)Online publication date: Dec-2021
https://doi.org/10.1016/j.ic.2021.104860
Sogokon AMitsch STan YCordwell KPlatzer A(2021)Pegasus: sound continuous invariant generationFormal Methods in System Design10.1007/s10703-020-00355-z58:1-2(5-41)Online publication date: 20-Jan-2021
https://dl.acm.org/doi/10.1007/s10703-020-00355-z
He HWu J(2020)A New Approach to Nonlinear Invariants for Hybrid Systems Based on the Citing Instances MethodInformation10.3390/info1105024611:5(246)Online publication date: 2-May-2020
https://doi.org/10.3390/info11050246
Kochdumper NAlthoff M(2020)Computing Non-Convex Inner-Approximations of Reachable Sets for Nonlinear Continuous Systems2020 59th IEEE Conference on Decision and Control (CDC)10.1109/CDC42340.2020.9304022(2130-2137)Online publication date: 14-Dec-2020
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Boreale M(2020)Complete algorithms for algebraic strongest postconditions and weakest preconditions in polynomial odesScience of Computer Programming10.1016/j.scico.2020.102441(102441)Online publication date: Mar-2020
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Boreale M(2020)Automatic Pre- and Postconditions for Partial Differential EquationsQuantitative Evaluation of Systems10.1007/978-3-030-59854-9_15(193-210)Online publication date: 3-Nov-2020
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Sogokon AMitsch STan YCordwell KPlatzer A(2019)Pegasus: A Framework for Sound Continuous Invariant GenerationFormal Methods – The Next 30 Years10.1007/978-3-030-30942-8_10(138-157)Online publication date: 23-Sep-2019
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