### Abstract

Original language | English |
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Number of pages | 16 |

Journal | Science in China Series F - Information Sciences |

Early online date | 13 Jun 2018 |

DOIs | |

Publication status | E-pub ahead of print - 13 Jun 2018 |

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### Keywords

- brownian motion
- Markov chain
- generalised Itô’s formula
- almost sure exponential stability
- stochastic feedback control

### Cite this

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**Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state.** / Song, Gongfei; Lu, Zhenyu; Zheng, Bo-Chao; Mao, Xuerong.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state

AU - Song, Gongfei

AU - Lu, Zhenyu

AU - Zheng, Bo-Chao

AU - Mao, Xuerong

PY - 2018/6/13

Y1 - 2018/6/13

N2 - Although the mean square stabilisation of hybrid systems by feedback controls based on discretetime observations of state and mode has been studied by several authors since 2013 (see, e.g., [17,19,27,31]), the corresponding almost sure stabilisation problem has little been investigated. Recent Mao [18] is the first to study the almost sure stabilisation of a given unstable system x(t) = f(x(t)) by a linear discretetime 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 will consider a much more general problem. That is, we will to study the almost sure stabilisation of a given unstable hybrid system x(t) = f(x(t), r(t)) by nonlinear discrete-time stochastic feedback control u(x([t/τ]τ), r([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), r([t/τ]τ))dB(t)), where B(t) is a multi-dimensional Brownian motion and r(t) is a Markov chain.

AB - Although the mean square stabilisation of hybrid systems by feedback controls based on discretetime observations of state and mode has been studied by several authors since 2013 (see, e.g., [17,19,27,31]), the corresponding almost sure stabilisation problem has little been investigated. Recent Mao [18] is the first to study the almost sure stabilisation of a given unstable system x(t) = f(x(t)) by a linear discretetime 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 will consider a much more general problem. That is, we will to study the almost sure stabilisation of a given unstable hybrid system x(t) = f(x(t), r(t)) by nonlinear discrete-time stochastic feedback control u(x([t/τ]τ), r([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), r([t/τ]τ))dB(t)), where B(t) is a multi-dimensional Brownian motion and r(t) is a Markov chain.

KW - brownian motion

KW - Markov chain

KW - generalised Itô’s formula

KW - almost sure exponential stability

KW - stochastic feedback control

UR - http://engine.scichina.com/publisher/scp/journal/Sci%20China%20Info%20Sci%20F?slug=Overview

U2 - 10.1007/s11432-017-9297-1

DO - 10.1007/s11432-017-9297-1

M3 - Article

JO - Science in China Series F - Information Sciences

JF - Science in China Series F - Information Sciences

SN - 1009-2757

ER -