Exponential stability and exponential stabilization of singularly perturbed stochastic systems with time-varying delay.

*(English)*Zbl 1207.93112Summary: The problems of exponential stability and exponential stabilization for linear singularly perturbed stochastic systems with time-varying delay are investigated. First, an appropriate Lyapunov functional is introduced to establish an improved delay-dependent stability criterion. By applying free-weighting matrix technique and by equivalently eliminating time-varying delay through the idea of convex combination, a less conservative sufficient condition for exponential stability in mean square is obtained in terms of \(\varepsilon \)-dependent Linear Matrix Inequalities (LMIs). It is shown that if this set of LMIs for \(\varepsilon =0\) are feasible then the system is exponentially stable in mean square for sufficiently small \(\varepsilon \geqslant 0\). Furthermore, it is shown that if a certain matrix variable in this set of LMIs is chosen to be a special form and the resulting LMIs are feasible for \(\varepsilon =0\), then the system is \(\varepsilon \)-uniformly exponentially stable for all sufficiently small \(\varepsilon \geqslant 0\). Based on the stability criteria, an \(\varepsilon \)-independent state-feedback controller that stabilizes the system for sufficiently small \(\varepsilon \geqslant 0\) is derived. Finally, numerical examples are presented, which show our results are effective and useful.

##### MSC:

93E15 | Stochastic stability in control theory |

93E03 | Stochastic systems in control theory (general) |

93D30 | Lyapunov and storage functions |

93C70 | Time-scale analysis and singular perturbations in control/observation systems |

##### Keywords:

time-delay; singular perturbations; stochastic systems; linear matrix inequality (LMI); exponential stability in mean square; delay-dependent criteria
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\textit{W.-H. Chen} et al., Int. J. Robust Nonlinear Control 20, No. 18, 2021--2044 (2010; Zbl 1207.93112)

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