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This article is concerned with the exponential mean-square stabilization problem for a class of discrete-time strict-feedback nonlinear systems subject to multiplicative noises. The state-dependent multiplicative noise is assumed to occur randomly based on a stochastic variable obeying the Gaussian white distribution. To tackle the difficulties caused by the multiplicative noise, a novel backstepping-based control framework is developed to design both the virtual control laws and the actual control law for the original nonlinear system, and such a framework is fundamentally different from the traditional n-step predictor strategy. The proposed design scheme provides an effective way in establishing the relationship between the system states and the controlled errors, by which a noise-intensity-dependant stability condition is derived to ensure that the closed-loop system is exponentially mean-square stable for exactly known systems. To further cope with nonlinear modeling uncertainties, the radial basis function neural network (NN) is employed as a function approximator. In virtue of the proposed backstepping-based control framework, the ideal controller is characterized as a function of all system states, which is independent of the virtual control laws. Therefore, only one NN is employed in the final step of the backstepping procedure and, subsequently, a novel adaptive neural controller (with modified weight updating laws) is presented to ensure that both the neural weight estimates and the system states are uniformly bounded in the mean-square sense under certain stability conditions. The control performance of the proposed scheme is illustrated through simulation results.
Wang et al. (Mon,) studied this question.