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

• Published in: IEEE transactions on automatic control Vol. 66; no. 4; pp. 1484 - 1496
• Main Authors: Wang, Min, Wang, Zidong, Dong, Hongli, Han, Qing-Long
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.04.2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Aadaptive control; Backstepping; backstepping-based control; Control stability; Control systems; Control theory; Controllers; Discrete time systems; discrete-time strict-feedback systems; Feedback control; Gaussian distribution; Lyapunov methods; multiplicative noises; Neural networks; neural networks (NNs); Noise; Noise control; Noise intensity; Nonlinear systems; Radial basis function; Stability criteria; Stochastic processes; Weight
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2020.2995576

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 <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula>-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.