A Novel Framework for Backstepping-Based Control of Discrete-Time Strict-Feedback Nonlinear Systems With Multiplicative Noises

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.

Automated summary

This article proposes a novel backstepping-based control framework to address the exponential mean-square stabilization problem for a class of discrete-time strict-feedback nonlinear systems subject to multiplicative noises. By establishing the relationship between system states and controlled errors, and employing radial basis function neural network to handle modeling uncertainties, the framework ensures stability control for the closed-loop system. The control performance is demonstrated through simulation results, showcasing the effectiveness of the proposed scheme.