New results on delay-dependent stability analysis and stabilization for stochastic time-delay systems

Abstract

SUMMARYThis paper investigates the problems of stability analysis and stabilization for stochastic time‐delay systems. Firstly, this paper uses the martingale theory to investigate expectations of stochastic cross terms containing the Itô integral. On the basis of this, an improved delay‐dependent stability criterion is derived for stochastic delay systems. In the derivation process, the mathematical development avoids bounding stochastic cross terms, and neither model transformation method nor free‐weighting‐matrix method is used. Thus, the method leads to a simple criterion and shows less conservatism. Secondly, on the basis of this stability result, this paper further proposes a state‐feedback controller that exponentially stabilizes the stochastic delay system by a strict LMI. Therefore, unlike previous results, it is not necessary to transform the nonlinear matrix inequalities into LMIs by the cone complementarity linearization method or parameter‐tuning method, which always yield a suboptimal solution. Finally, examples are provided to demonstrate the reduced conservatism of the proposed conditions.Copyright © 2013 John Wiley & Sons, Ltd.

New Stabilization of Continuous‐Time Delayed Systems Based on Partially Delay‐Dependent Controllers

“…Thus, during the past decades, much attention has been paid to the derivation of less conservative delay-dependent conditions. For example, some results [1,5,11,17,22,23] were established by exploiting the slack variable technique, while the other results [2][3][4]14,24,26,28] were obtained based on the Jensen inequality. Particularly, in order to further reduce the Manuscript received September 2, 2015; revised December 13, 2015; accepted February 16, 2016.…”

“…The presence of time delay often degrades system performance, even leads to instability. Motivated by these facts, various research topics of time-delayed systems have emerged, for example, stability [1][2][3][4][5][6][7][8][9], stabilization [10][11][12][13][14][15][16], output control [17,18], H ∞ control and filtering [19][20][21][22][23][24][25][26], synchronization [27,28], and so on, in which both delay-independent and delay-dependent results are involved. Because of making use of the length of delay, delay-dependent results are less conservative, especially when time delay is small.…”

“…Thus, during the past decades, much attention has been paid to the derivation of less conservative delay-dependent conditions. For example, some results [1,5,11,17,22,23] were established by exploiting the slack variable technique, while the other results [2][3][4]14,24,26,28] were obtained based on the Jensen inequality. Particularly, in order to further reduce the Manuscript conservatism, some improved results [29,30] were got by considering the probability distribution of delay as the Bernoulli distribution, while the probability distribution of time delay in references [31][32][33][34][35][36] were assumed to satisfy a Markov chain.…”

Necessary and sufficient conditions of risk‐sensitive optimal control and differential games for stochastic differential delayed equations

“…We also note that, as in the nondelay case, the “completion of squares” method can be used for a class of linear‐quadratic (LQ) problems with the linear SDDE and the quadratic cost . Furthermore, the stabilization of the SDDE was considered by Song et al, and the LMI conditions for H ∞ control of the linear SDDE was identified by Yang et al The H ∞ control problem for the linear SDDE with perturbation was studied by Li and Yang …”

“…14 We also note that, as in the nondelay case, the "completion of squares" method can be used for a class of linear-quadratic (LQ) problems with the linear SDDE and the quadratic cost. 15 Furthermore, the stabilization of the SDDE was considered by Song et al, 16 and the LMI conditions for H ∞ control of the linear SDDE was identified by Yang et al 17 The H ∞ control problem for the linear SDDE with perturbation was studied by Li and Yang. 18 We note that the stochastic optimal control and differential games for SDDEs studied in the references mentioned above can be regarded as risk-neutral since the corresponding objective functional involves only the first moment of the integral cost.…”