On Adaptive Control For Stochastic Nonlinear Systems With Unmodeled Dynamics

Because of its wide practical background, the investigation on stability analysis and adaptive controller design of stochastic nonlinear systems has been the focus in automatic control community during the past decades. Combined with backstepping technique, dynamic surface control and adaptive control, this paper discusses the stability problem of stochastic nonlinear systems with unmodeled dynamics, time-varying delays and the actuator dead-zone. Utilizing the approximation capability of neural networks and Young’s inequality, several kinds of adaptive controllers are given. The main contributions are highlighted as follows:Firstly, utilizing the approximation capability of neural networks and Young’s inequality, an adaptive NN dynamic surface control scheme is proposed for a class of stochastic pure-feedback nonlinear systems with unmodeled dynamics. The design extends the approach of dynamic surface control (DSC) to the stochastic systems, the explosion of complexity in traditional backstepping design and circular arguments in the process of approximation are both avoided. This scheme also reduces the calculation on a large scale by introducing the quartic Lyapunov function instead of the integral type. The control signal and adaptation law are independent of the nodes of neural network. By theoretical analysis, it is proved that all signals in the closed-loop systems are bounded in probability. A simulation example further demonstrates the effectiveness of the control scheme.Secondly, a novel adaptive control scheme is investigated based on the backstepping design for a class of stochastic nonlinear systems with unmodeled dynamics and time-varying state delays. The radial basis function neural networks are used to approximate the unknown nonlinear functions obtained by using Ito differential formula and Young’s inequality. The unknown time-varying delays and the unmodeled dynamics are dealt with by constructing appropriate Lyapunov-Krasovskii functions with exponential factor and introducing available dynamic signal. It is proved that all signals in the closed-loop system are bounded in probability and the error signals are semi-globally uniformly ultimately bounded (SGUUB) in mean square or the sense of four-moment. Simulation results illustrate the effectiveness of the proposed design.Thirdly, a novel adaptive neural control scheme is presented for a class of stochastic strict-feedback nonlinear systems with unmodeled dynamics and dead-zone model using stochastic small-gain theorem. Different from the existing work, the unmodeled dynamics contains the stochastic diffusion terms, and the prerequisite of dynamic disturbances is relaxed. Also, by introducing the appropriate coordinate transformation and Young’s inequality, the proposed scheme removes the matching condition of the system functions. The controller and adaptation law are independent of the nodes of neural network. At each step, only one parameter needs to be adjusted online, as a result, the computational burden is reduced greatly. The stability analysis is given to show that all the signals in the closed-loop system are ISpS in probability. The effectiveness of the proposed design is illustrated by simulation results.