Jun Hu, Zidong Wang, Huijun Gao; A delay fractioning approach to robust sliding mode control for discrete-time stochastic systems with randomly occurring non-linearities. IMA J Math Control Info 2011; 28 (3): 345-363. doi: 10.1093/imamci/dnr008
In this paper, the robust sliding mode control (SMC) problem is investigated for a class of uncertain discrete-time stochastic systems with randomly occurring non-linearities and time delays. The randomly occurring non-linearity, which describes the phenomena of a class of non-linear disturbances occurring in a random way, is modelled according to a Bernoulli distributed white sequence with a known conditional probability. By constructing a novel Lyapunov–Krasovskii functional, the idea of delay fractioning is applied to cope with the robust SMC problem with time delays. Sufficient conditions are derived to ensure the stability of the systems dynamics in the specified sliding surface. Such conditions are characterized in terms of a set of linear matrix inequalities with an equality constraint. A discrete-time SMC law is synthesized to guarantee the reaching condition of the discrete-time sliding mode surface. Finally, a simulation example is presented to illustrate the effectiveness of the proposed results.