Abstract
We describe a numerical technique for discovering forward invariant sets for discrete-time nonlinear dynamical systems. Given a region of interest in the state- space, our technique uses simulation traces originating at states within this region to construct candidate Lyapunov functions, which are in turn used to obtain candidate forward invariant sets. To vet a candidate invariant set, our technique samples a finite number of states from the set and tests them. We derive sufficient conditions on the sample density that formally guarantee that the candidate invariant set is indeed forward invariant. Finally, we present a numerical example illustrating the efficacy of the technique.
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Notes
- 1.
Runtime measured on an Intel Xeon E5606 2.13Â GHz Dual Processor machine, with 24Â GB RAM, running Windows 7, SP1.
References
Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, vol. 15. SIAM (1994), Philadelphia, PA, USA
Kapinski, J., Deshmukh, J.V., Sankaranarayanan, S., Aréchiga, N.: Simulation-guided lyapunov analysis for hybrid dynamical systems. In: Hybrid Systems: Computation and Control (HSCC). ACM (2014), New York, NY, USA
Khalil, H.: Nonlinear Systems. Prentice Hall (2002), Upper Saddle River, NJ, USA
LaSalle, J.: The Stability and Control of Discrete Processes. Applied Mathematical Sciences. Springer (1986), New York, NY, USA
Parrilo, P.A.: Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. Ph.D. thesis, California Institute of Technology (2000), Pasadena, CA, USA
Topcu, U., Seiler, P., Packard, A.: Local stability analysis using simulations and sum-of-squares programming. Automatica 44, 2669–2675 (2008), Elsevier, Amsterdam, The Netherlands
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Kapinski, J., Deshmukh, J. (2015). Discovering Forward Invariant Sets for Nonlinear Dynamical Systems. In: Cojocaru, M., Kotsireas, I., Makarov, R., Melnik, R., Shodiev, H. (eds) Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science. Springer Proceedings in Mathematics & Statistics, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-12307-3_37
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DOI: https://doi.org/10.1007/978-3-319-12307-3_37
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