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Estimating infinitesimal generators of stochastic systems with formal error bounds: a data-driven approach

Authors:

Abolfazl Lavaei,

Ameneh Nejati,

Sadegh Soudjani,

Majid ZamaniAuthors Info & Claims

HSCC '21: Proceedings of the 24th International Conference on Hybrid Systems: Computation and Control

Article No.: 28, Pages 1 - 2

https://doi.org/10.1145/3447928.3456660

Published: 19 May 2021 Publication History

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Abstract

In this work, we propose a data-driven technique for a formal estimation of infinitesimal generators of continuous-time stochastic systems with unknown dynamics. In the proposed framework, we first approximate the infinitesimal generator of the solution process via a set of data collected from solution processes of unknown systems. We then put some proper assumptions on dynamics of systems and quantify the closeness between the infinitesimal generator and its approximation while providing a priori guaranteed confidence bound. We show that both the time discretization and the number of data play significant roles in providing a reasonable closeness precision.

Motivations and Related Works. Infinitesimal generator of a stochastic process (i.e., a partial differential operator that encodes a great deal of information about the process) plays a principal role in many analysis frameworks including: (i) stability analysis of continuous-time stochastic systems via Lyapunov functions [1], (ii) establishing similarity relations between two continuous-time stochastic systems via stochastic simulation functions [2], (iii) constructing finite abstractions of continuous-time stochastic control systems [3], and (iv) safety analysis of continuous-time stochastic systems via barrier certificates [4], to name a few. In the relevant literature, only [5] studies the estimation of infinitesimal generators, whose approach is based on the assumption of knowing the precise model and discretizing both time and state. To the best of our knowledge, our work is the first to propose a method for estimating the infinitesimal generator of stochastic systems with unknown dynamics while providing formal guarantees. The complete version of this work can be found in [6].

References

[1]

A. R. Teel, A. Subbaraman, and A. Sferlazza, "Stability analysis for stochastic hybrid systems: A survey," Automatica, vol. 50, no. 10, pp. 2435--2456, 2014.

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[2]

A. A. Julius and G. J. Pappas, "Approximations of stochastic hybrid systems," IEEE Transactions on Automatic Control, vol. 54, no. 6, pp. 1193--1203, 2009.

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[3]

M. Zamani, P. Mohajerin Esfahani, R. Majumdar, A. Abate, and J. Lygeros, "Symbolic control of stochastic systems via approximately bisimilar finite abstractions," IEEE Transactions on Automatic Control, vol. 59, no. 12, pp. 3135--3150, 2014.

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[4]

S. Prajna, A. Jadbabaie, and G. J. Pappas, "A framework for worst-case and stochastic safety verification using barrier certificates," IEEE Transactions on Automatic Control, vol. 52, no. 8, pp. 1415--1428, 2007.

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[5]

L. Donati, M. Heida, B. G. Keller, and M. Weber, "Estimation of the infinitesimal generator by square-root approximation," Journal of Physics: Condensed Matter, vol. 30, no. 42, p. 425201, 2018.

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[6]

A. Nejati, A. Lavaei, S. Soudjani, and M. Zamani, "Data-driven estimation of infinitesimal generators of stochastic systems," in Proceedings of the 7th IFAC Conference on Analysis and Design of Hybrid Systems (ADHS), 2021.

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Estimating infinitesimal generators of stochastic systems with formal error bounds: a data-driven approach

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Published In

HSCC '21: Proceedings of the 24th International Conference on Hybrid Systems: Computation and Control

May 2021

300 pages

ISBN:9781450383394

DOI:10.1145/3447928

Program Chairs:
Sergiy Bogomolov
Newcastle University, UK
,
Raphaël Jungers
UCLouvain, Belgium

Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the Owner/Author.

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Association for Computing Machinery

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Published: 19 May 2021

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HSCC '21: 24th ACM International Conference on Hybrid Systems: Computation and Control

May 19 - 21, 2021

Tennessee, Nashville

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HSCC '21 Paper Acceptance Rate 27 of 77 submissions, 35%;

Overall Acceptance Rate 153 of 373 submissions, 41%

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Dual Bounds for Periodical Stochastic Programs

Limiting Discounted-Cost Control of Partially Observable Stochastic Systems

A Global Stochastic Maximum Principle for Forward-Backward Stochastic Control Systems with Quadratic Generators

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