Adaptive control of a class of nonlinear systems using neural networks

Layered neural networks are used in nonlinear adaptive control problems. Both the discrete-time case and the continuous-time case are considered. For the discrete-time case, the plant is an unknown SI/SO feedback-linearizable discrete-time system with general relative degree, represented by an input-output model. To derive the linearizing-stabilizing feedback control, a (possibly nonminimal) state space model of the plant is obtained. This model is used to define the zero dynamics, which are assumed to be stable, i.e., the system is assumed to be minimum phase. A layered neural network is used to model the unknown system and generate the feedback control. Based on the error between the plant output and the model output, the weights of the neural network are updated. For the continuous-time case, we work on a SI/SO relative-degree-one system with zero dynamics, and on a MI/MO general relative degree system without zero dynamics. Compared with the discrete-time case, there are two major differences. First, the neural network is used to model nonlinear functions of continuous-time systems, instead of modeling the whole system in the discrete-time case. And second, the control law for the continuous-time case does not involve an explicit system identification process, as appears in the discrete-time case. For both the discrete-time and the continuous-time cases, the convergence results obtained are regional in state space, yet local in parameter space. The results basically say that, for any bounded initial conditions of the plant, if the neural network model contains enough number of nonlinear hidden neurons and if the initial guess of the network weights is sufficiently close to the correct weights, then the tracking error between the plant output and the reference command will converge to a bounded ball. Computer simulations for the continuous-time case are provided.