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Closing the Gap Between Discrete Abstractions and Continuous Control: Completeness via Robustness and Controllability

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Book cover Formal Modeling and Analysis of Timed Systems (FORMATS 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12860))

Abstract

A central theoretical question surrounding abstraction-based control of continuous nonlinear systems is whether one can decide through algorithmic procedures the existence of a controller to render the system to satisfy a given specification (e.g., safety, reachability, or more generally a temporal logic formula). Known algorithms are mostly sound but not complete in the sense that they return a correct controller upon termination, but do not offer guarantees of finding a controller if one exists. Completeness of abstraction-based nonlinear control in the general setting, therefore, remains an open question. This paper investigates this theoretical question and presents two sets of main results. First, we prove that sampled-data control of nonlinear systems with temporal logic specifications is robustly decidable in the sense that, given a continuous-time nonlinear control system and a temporal logic formula, one can algorithmically decide whether there exists a robust sampled-data control strategy to realize this specification when the right-hand side of the system is slightly perturbed by a small disturbance. Second, we show that under the assumption of local nonlinear controllability of the nominal system around an arbitrary trajectory that realizes a given specification, we can always construct a (robust) sampled-data control strategy via a sufficiently fine discrete abstraction. In a sense, this shows that temporal logic control for controllable nonlinear systems is decidable.

Supported by the Natural Sciences and Engineering Research Council of Canada, the Canada Research Chairs Program, and the Ontario Early Researcher Award Program. The paper has an extended version with Appendix available at [19].

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Authors and Affiliations

  1. University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Jun Liu

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  1. Jun Liu
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Correspondence to Jun Liu .

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Editors and Affiliations

  1. Université Paris-Est Créteil, Créteil, France

    Catalin Dima

  2. CNRS, Université de Paris, Paris, France

    Mahsa Shirmohammadi

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Liu, J. (2021). Closing the Gap Between Discrete Abstractions and Continuous Control: Completeness via Robustness and Controllability. In: Dima, C., Shirmohammadi, M. (eds) Formal Modeling and Analysis of Timed Systems. FORMATS 2021. Lecture Notes in Computer Science(), vol 12860. Springer, Cham. https://doi.org/10.1007/978-3-030-85037-1_5

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  • DOI: https://doi.org/10.1007/978-3-030-85037-1_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-85036-4

  • Online ISBN: 978-3-030-85037-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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