A new unified input‐to‐state stability criterion for impulsive stochastic delay systems with Markovian switching

Abstract

Summary In this paper, the problems of the input‐to‐state stability (ISS), the integral input‐to‐state stability (iISS), the stochastic input‐to‐state stability (SISS) and the eλt(λ>0)‐weighted input‐to‐state stability (eλt‐ISS) are investigated for nonlinear time‐varying impulsive stochastic delay systems with Markovian switching. We propose one unified criterion for the stabilizing impulse and the destabilizing impulse to guarantee the ISS, iISS, SISS and eλt‐ISS for such systems. We verify that when the upper bound of the average impulsive interval is given, the stabilizing impulsive effect can stabilize the systems without ISS. We also show that the destabilizing impulsive signal with a given lower bound of the average impulsive interval can preserve the ISS of the systems. In addition, one criterion for guaranteeing the ISS of nonlinear time‐varying stochastic hybrid systems under no impulsive effect is derived. Two examples including one coupled dynamic systems model subject to external random perturbation of the continuous input and impulsive input disturbances are provided to illustrate the effectiveness of the theoretic results developed.

Fault‐alarm based hybrid control design for uncertain periodic piecewise time‐varying systems with actuator constraints

“…In the upcoming corollary, the sufficient conditions are established for determining the gain matrices of the controller (18) which ensures the asymptotic stability of the PPTVUS (2) without time-varying delay and actuator bias failure. Precisely, by adopting the strategy analogous to that in Theorem.…”

“…Corollary 2. For the given scalars 𝛼 i > 0 and 𝛾, the states of the resulting system (2) without time-varying delay and actuator bias failure are asymptotically stabilised via the controller (18), if there exist symmetric matrices i,j > 0 and any apt dimensioned matrix  i,j , j ∈ J i , i ∈ L, and positive scalars 𝜅 1i , 𝜅 2i , such that the relations below are met:…”

“…As a consequence, numerous amount of work has been reported by considering time delay in dynamical systems. [15][16][17][18][19][20][21][22][23] Very interestingly, in Reference 16, the authors developed improved stability conditions for linear systems subject to time-varying delays with the aid of Finsler's lemma and an augmented zero equality. The stabilization problem of linear time-varying systems with time delays is addressed in Reference 19 by utilizing non-quadratic Lyapunov functions and functionals.…”

Sufficient conditions on finite‐time input‐to‐state stability of nonlinear stochastic impulsive time‐varying systems

“…As we know, the concept of input‐to‐state stability () was initially presented in References 10,11, which played paramount role because of their broad applications, describing the influences of external inputs on a control system. Quite a few recent results about ‐properties have been published in related fields, such as impulsive systems, 12‐15 networked control systems, 16 Markov systems 17 and hybrid systems 18 . Particularly for impulsive systems, quite a few ‐properties have been studied on account of their wide applications.…”

Delay‐dependent stability analysis and stabilization of stochastic time‐delay systems governed by the Poisson process and Brownian motion

“…where B(t) is a Brownian motion, and h is a delay. Over past years, various kinds of STDSs governed by Brownian motion have got much research attention (see References [22][23][24][25][26][27][28][29][30]. Certainly, when studying STDSs, researchers also want to obtain delay-dependent stability or control criteria.…”