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].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Angeli, D.: A lyapunov approach to incremental stability properties. IEEE Trans. Autom. Control 47(3), 410–421 (2002)
Aubin, J.P., Cellina, A.: Differential Inclusions: Set-valued Maps and Viability Theory. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-69512-4
Baier, C., Katoen, J.P.: Principles of Model Checking. MIT Press, Cambridge (2008)
Belta, C., Yordanov, B., Aydin Gol, E.: Formal Methods for Discrete-Time Dynamical Systems. SSDC, vol. 89. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-50763-7
Clarke, E.M., Grumberg, O., Peled, D.: Model Checking. MIT Press, Cambridge (1999)
Fainekos, G.E., Girard, A., Kress-Gazit, H., Pappas, G.J.: Temporal logic motion planning for dynamic robots. Automatica 45(2), 343–352 (2009)
Fainekos, G.E., Pappas, G.J.: Robustness of temporal logic specifications for continuous-time signals. Theor. Comput. Sci. 410(42), 4262–4291 (2009)
Grädel, E., Thomas, W., Wilke, T. (eds.): Automata Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-36387-4
Hermes, H.: On local and global controllability. SIAM J. Control 12(2), 252–261 (1974)
Kloetzer, M., Belta, C.: Temporal logic planning and control of robotic swarms by hierarchical abstractions. IEEE Trans. Robot. 23(2), 320–330 (2007)
Koiran, P., Cosnard, M., Garzon, M.: Computability with low-dimensional dynamical systems. Theor. Comput. Sci. 132(1–2), 113–128 (1994)
Kress-Gazit, H., Wongpiromsarn, T., Topcu, U.: Correct, reactive, high-level robot control. IEEE Robot. Autom. Mag. 18(3), 65–74 (2011)
Li, Y., Liu, J.: Invariance control synthesis for switched nonlinear systems: an interval analysis approach. IEEE Trans. Autom. Control 63(7), 2206–2211 (2018)
Li, Y., Liu, J.: Robustly complete reach-and-stay control synthesis for switched systems via interval analysis. In: Proceedings of ACC (2018)
Li, Y., Liu, J.: Rocs: A robustly complete control synthesis tool for nonlinear dynamical systems. In: Proceedings of HSCC, pp. 130–135 (2018)
Li, Y., Liu, J.: Robustly complete synthesis of memoryless controllers for nonlinear systems with reach-and-stay specifications. IEEE Trans. Autom. Control 66(3), 1199–1206 (2021)
Li, Y., Sun, Z., Liu, J.: A specification-guided framework for temporal logic control of nonlinear systems. arXiv preprint arXiv:2104.01385 (2021).
Liu, J.: Robust abstractions for control synthesis: completeness via robustness for linear-time properties. In: Proceedings of HSCC, pp. 101–110. ACM (2017)
Liu, J.: Closing the gap between discrete abstractions and continuous control: Completeness via robustness and controllability. In: Proceedings of FORMATS (2021). https://www.math.uwaterloo.ca/~j49liu/papers/2021/liu2021closing.pdf
Liu, J., Ozay, N., Topcu, U., Murray, R.: Synthesis of reactive switching protocols from temporal logic specifications. IEEE Trans. Autom. Control 58(7), 1771–1785 (2013)
Liu, J., Ozay, N.: Abstraction, discretization, and robustness in temporal logic control of dynamical systems. In: Proceedings of HSCC, pp. 293–302 (2014)
Liu, J., Ozay, N.: Finite abstractions with robustness margins for temporal logic-based control synthesis. Nonlinear Anal. Hybrid Syst. 22, 1–15 (2016)
Nam, K., Arapostathis, A.: A sufficient condition for local controllability of nonlinear systems along closed orbits. IEEE Trans. Autom. Control 37(3), 378–380 (1992)
Nilsson, P., Ozay, N., Liu, J.: Augmented finite transition systems as abstractions for control synthesis. Discrete Event Dyn. Syst. 27(2), 301–340 (2017)
Ozay, N., Liu, J., Prabhakar, P., Murray, R.M.: Computing augmented finite transition systems to synthesize switching protocols for polynomial switched systems. In: Proceedings of ACC, pp. 6237–6244 (2013)
Pnueli, A.: The temporal logic of programs. In: Proceedings of FOCS, pp. 46–57. IEEE (1977)
Pola, G., Girard, A., Tabuada, P.: Approximately bisimilar symbolic models for nonlinear control systems. Automatica 44(10), 2508–2516 (2008)
Reissig, G., Weber, A., Rungger, M.: Feedback refinement relations for the synthesis of symbolic controllers. IEEE Trans. Autom. Control 62(4), 1781–1796 (2017)
Royden, H., Fitzpatrick, P.: Real Analysis. Printice-Hall, Boston (2010)
Tabuada, P.: Verification and Control of Hybrid Systems: A Symbolic Approach. Springer, Heidelberg (2009). https://doi.org/10.1007/978-1-4419-0224-5
Tabuada, P., Pappas, G.J.: Linear time logic control of discrete-time linear systems. IEEE Trans. Autom. Control 51(12), 1862–1877 (2006)
Zamani, M., Pola, G., Mazo, M., Tabuada, P.: Symbolic models for nonlinear control systems without stability assumptions. IEEE Trans. Autom. Control 57(7), 1804–1809 (2012)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-030-85037-1_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-85036-4
Online ISBN: 978-3-030-85037-1
eBook Packages: Computer ScienceComputer Science (R0)