Formal Abstraction of General Stochastic Systems via Noise Partitioning (2309.10702v1)
Abstract: Verifying the performance of safety-critical, stochastic systems with complex noise distributions is difficult. We introduce a general procedure for the finite abstraction of nonlinear stochastic systems with non-standard (e.g., non-affine, non-symmetric, non-unimodal) noise distributions for verification purposes. The method uses a finite partitioning of the noise domain to construct an interval Markov chain (IMC) abstraction of the system via transition probability intervals. Noise partitioning allows for a general class of distributions and structures, including multiplicative and mixture models, and admits both known and data-driven systems. The partitions required for optimal transition bounds are specified for systems that are monotonic with respect to the noise, and explicit partitions are provided for affine and multiplicative structures. By the soundness of the abstraction procedure, verification on the IMC provides guarantees on the stochastic system against a temporal logic specification. In addition, we present a novel refinement-free algorithm that improves the verification results. Case studies on linear and nonlinear systems with non-Gaussian noise, including a data-driven example, demonstrate the generality and effectiveness of the method without introducing excessive conservatism.
- A. Lavaei, S. Soudjani, A. Abate, and M. Zamani, “Automated verification and synthesis of stochastic hybrid systems: A survey,” Automatica, vol. 146, p. 110617, 2022.
- R. Alur, T. Henzinger, G. Lafferriere, and G. Pappas, “Discrete abstractions of hybrid systems,” Proceedings of the IEEE, vol. 88, no. 7, pp. 971–984, 2000.
- M. Lahijanian, S. B. Andersson, and C. Belta, “Formal verification and synthesis for discrete-time stochastic systems,” IEEE Transactions on Automatic Control, vol. 60, no. 8, pp. 2031–2045, 2015.
- J. Jackson, L. Laurenti, E. Frew, and M. Lahijanian, “Strategy synthesis for partially-known switched stochastic systems,” in Int. Conf. on Hybrid Systems: Computation and Control, 2021.
- M. Dutreix, J. Huh, and S. Coogan, “Abstraction-based synthesis for stochastic systems with omega-regular objectives,” Nonlinear Analysis: Hybrid Systems, vol. 45, p. 101204, 2022.
- T. Badings, L. Romao, A. Abate, D. Parker, H. A. Poonawala, M. Stoelinga, and N. Jansen, “Robust control for dynamical systems with non-gaussian noise via formal abstractions,” Journal of Artificial Intelligence Research, vol. 76, pp. 341–391, 2023.
- R. Givan, S. Leach, and T. Dean, “Bounded-parameter markov decision processes,” Artificial Intell., vol. 122, no. 1-2, pp. 71–109, 2000.
- N. Cauchi, L. Laurenti, M. Lahijanian, A. Abate, M. Kwiatkowska, and L. Cardelli, “Efficiency through uncertainty: Scalable formal synthesis for stochastic hybrid systems,” in ACM Int. Conf. on hybrid systems: computation and control, 2019, pp. 240–251.
- J. Jiang, Y. Zhao, and S. Coogan, “Safe learning for uncertainty-aware planning via interval mdp abstraction,” IEEE Control Systems Letters, vol. 6, pp. 2641–2646, 2022.
- J. Jackson, L. Laurenti, E. Frew, and M. Lahijanian, “Formal verification of unknown dynamical systems via gaussian process regression,” arXiv preprint arXiv:2201.00655, 2021.
- B. C. van Huijgevoort, S. Weiland, and S. Haesaert, “Temporal logic control of nonlinear stochastic systems using a piecewise-affine abstraction,” IEEE Control Sys. Letters, vol. 7, pp. 1039–1044, 2023.
- K. Hashimoto, A. Saoud, M. Kishida, T. Ushio, and D. V. Dimarogonas, “Learning-based symbolic abstractions for nonlinear control systems,” Automatica, vol. 146, p. 110646, 2022.
- S. Adams, M. Lahijanian, and L. Laurenti, “Formal control synthesis for stochastic neural network dynamic models,” IEEE Control Systems Letters, vol. 6, pp. 2858–2863, 2022.
- S. E. Z. Soudjani, C. Gevaerts, and A. Abate, “Faust: F ormal a bstractions of u ncountable-st ate st ochastic processes,” in Tools and Alg. for the Const. and Analys. of Sys. Springer, 2015, pp. 272–286.
- S. Esmaeil Zadeh Soudjani and A. Abate, “Adaptive and sequential gridding procedures for the abstraction and verification of stochastic processes,” SIAM Journal on Applied Dynamical Systems, vol. 12, no. 2, pp. 921–956, 2013.
- Z. Jin, Q. Shen, and S. Z. Yong, “Mesh-based piecewise affine abstraction with polytopic partitions for nonlinear systems,” IEEE Control Systems Letters, vol. 5, no. 5, pp. 1543–1548, 2021.
- F. B. Mathiesen, S. C. Calvert, and L. Laurenti, “Safety certification for stochastic systems via neural barrier functions,” IEEE Control Systems Letters, vol. 7, pp. 973–978, 2022.
- X. Chen, E. Ábrahám, and S. Sankaranarayanan, “Flow*: An analyzer for non-linear hybrid systems,” in Computer Aided Verification. Springer, 2013, pp. 258–263.
- M. Dutreix and S. Coogan, “Efficient verification for stochastic mixed monotone systems,” in 2018 ACM/IEEE 9th International Conference on Cyber-Physical Systems (ICCPS). IEEE, 2018, pp. 150–161.
- S. Bogomolov, M. Forets, G. Frehse, K. Potomkin, and C. Schilling, “Juliareach: a toolbox for set-based reachability,” in ACM Int. Conf. on Hybrid Systems: Computation and Control, 2019, pp. 39–44.