Implementation of stable adaptive neural networks for feedback linearization

For a class of single-input single-output continuous-time nonlinear systems, a multilayer neural network-based controller that feedback linearizes the system is presented. Control action is used to achieve tracking performance for a state feedback linearizable but unknown nonlinear system. We show that indirect adaptive schemes will learn how to control the plant, result in bounded internal signals, and achieve stable tracking for a reference input asymptotically. The multilayer neural network (NN) is used to approximate given plant to any desired degree of accuracy and generate the feedback control. Based on the error between the plant output and the desired output, the weight-update rule of NN is derived to satisfy Lyapunov stability. A projection method is employed so that NN weights are bounded. It is shown that all the signals in the closed-loop system are uniformly bounded under mild assumptions. The initialization of NN weights is straightforward. The performance of an indirect adaptive scheme is demonstrated through the control of an inverted pendulum system and a system with internal dynamics.