Abstract
A model for discrete time stochastic hybrid systems whose evolution can be influenced by some control input is proposed in this paper. With reference to the introduced class of systems, a methodology for probabilistic reachability analysis is developed that is relevant to safety verification. This methodology is based on the interpretation of the safety verification problem as an optimal control problem for a certain controlled Markov process. In particular, this allows to characterize through some optimal cost function the set of initial conditions for the system such that safety is guaranteed with sufficiently high probability. The proposed methodology is applied to the problem of regulating the average temperature in a room by a thermostat controlling a heater.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bujorianu, M.L., Lygeros, J.: Reachability questions in piecewise deterministic Markov processes. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 126–140. Springer, Heidelberg (2003)
Hu, J., Prandini, M., Sastry, S.: Probabilistic safety analysis in three-dimensional aircraft flight. In: Proc. of the IEEE Conf. on Decision and Control (2003)
Prajna, S., Jadbabaie, A., Pappas, G.: Stochastic safety verification using barrier certificates. In: Proc. of the IEEE Conf. on Decision and Control (2004)
Hu, J., Prandini, M., Sastry, S.: Aircraft conflict prediction in the presence of a spatially correlated wind field. IEEE Trans. on Intelligent Transportation Systems 6(3), 326–340 (2005)
Davis, M.H.A.: Markov Models and Optimization. Chapman & Hall, London (1993)
Ghosh, M.K., Araposthasis, A., Marcus, S.I.: Ergodic control of switching diffusions. SIAM Journal of Control and Optimization 35(6), 1952–1988 (1997)
Hu, J., Lygeros, J., Sastry, S.: Towards a theory of stochastic hybrid systems. In: Lynch, N., Krogh, B. (eds.) Hybrid Systems: Computation and Control. LNCS, vol. 1790, pp. 160–173. Springer, Heidelberg (2000)
Lygeros, J., Watkins, O.: Stochastic reachability for discrete time systems: an application to aircraft collision avoidance. In: Proc. of the IEEE Conf. on Decision and Control (2003)
Bujorianu, M., Lygeros, J.: General stochastic hybrid systems: Modelling and optimal control. In: Proc. of the IEEE Conf. on Decision and Control (2004)
Puterman, M.: Markov decision processes. John Wiley & Sons, Inc., Chichester (1994)
Bertsekas, D.P., Shreve, S.E.: Stochastic optimal control: the discrete-time case. Athena Scientific (1996)
Malhame, R., Chong, C.Y.: Electric load model synthesis by diffusion approximation of a high-order hybrid-state stochastic system. IEEE Transactions on Automatic Control AC-30(9), 854–860 (1985)
Milstein, G.: Numerical Integration of Stochastic Differential Equations. Kluver Academic Press, London (1995)
Kumar, P.R., Varaiya, P.P.: Stochastic Systems: Estimation, Identification, and Adaptive Control. Prentice Hall, Inc., New Jersey (1986)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Amin, S., Abate, A., Prandini, M., Lygeros, J., Sastry, S. (2006). Reachability Analysis for Controlled Discrete Time Stochastic Hybrid Systems. In: Hespanha, J.P., Tiwari, A. (eds) Hybrid Systems: Computation and Control. HSCC 2006. Lecture Notes in Computer Science, vol 3927. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11730637_7
Download citation
DOI: https://doi.org/10.1007/11730637_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-33170-4
Online ISBN: 978-3-540-33171-1
eBook Packages: Computer ScienceComputer Science (R0)