Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state

Abstract

Although the mean square stabilization of hybrid systems by feedback control based on discretetime observations of state and mode has been studied by several authors since 2013, the corresponding almost sure stabilization problem has received little attention. Recently, Mao was the first to study the almost sure stabilization of a given unstable system (t) = f(x(t)) by a linear discrete-time stochastic feedback control Ax([t/τ]τ)dB/(t) (namely the stochastically controlled system has the form dx(t) = f(x(t))dt + Ax([t/τ]τ)dB/(t), where B(t) is a scalar Brownian, τ > 0, and [t/τ] is the integer part of t/τ. In this paper, we consider a much more general problem. That is, we study the almost sure stabilization of a given unstable hybrid system (t) = f(x(t), r(t)) by nonlinear discrete-time stochastic feedback control u(x([t/τ]τ)dB(t) (so the stochastically controlled system is a hybrid stochastic system of the form dx(t) = f(x(t), r(t))dt + u(x([t/τ]τ))dB(t), where B(t) is a multi-dimensional Brownian motion and r(t) is a Markov chain.

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Acknowledgements

The authors would like to thank Leverhulme Trust (Grant No. RF-2015-385), Royal Society (Grant No. WM160014, Royal Society Wolfson Research Merit Award), Royal Society and Newton Fund (Grant No. NA160317, Royal Society–Newton Advanced Fellowship), Engineering and Physics Sciences Research Council (Grant No. EP/K503174/1), National Natural Science Foundation of China (Grant Nos. 61503190, 61473334, 61403207), Natural Science Foundation of Jiangsu Province (Grant Nos. BK20150927, BK20131000), and Ministry of Education (MOE) of China (Grant No. MS2014DHDX020) for their financial support. The first author would like to thank Chinese Scholarship Council for awarding him the scholarship to visit the University of Strathclyde for one year.

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Correspondence to Xuerong Mao.

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Song, G., Lu, Z., Zheng, BC. et al. Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state. Sci. China Inf. Sci. 61, 70213 (2018). https://doi.org/10.1007/s11432-017-9297-1

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Keywords

  • Brownian motion
  • Markov chain
  • generalized Itô formula
  • almost sure exponential stability
  • stochastic feedback control