Adaptive control of nonlinear systems using Hopfield-based dynamic neural network

Pin-Cheng Chen, Tsu-Tian Lee, Chi-Hsu Wang, Ping-Zong Lin

National Taipei University of Technology

Taiwan

In this paper, we propose a direct adaptive control scheme using Hopfield-based dynamic neural network for SISO nonlinear systems with external disturbances. A Hopfield-based dynamic neural network is used to approximate the ideal controller, and a compensation controller is used to suppress the effect of approximation error and disturbance. The weights of Hopfield-based dynamic neural network are on-line tuned by the adaptive laws derived in the sense of Lyapunov, so that the stability of the closed-loop system can be guaranteed. In addition, the tracking error can be attenuated to a desired level by selecting some parameters adequately. Simulation results illustrate the applicability of the proposed control scheme. The designed parsimonious structure of the Hopfield-based dynamic neural network makes the practical implementation of the work in this paper much easier. The case of Hopfield-based neural network without the self-feedback loop is also studied and shown to have inferior results than that of Hopfield neural network with the self-feedback loop.

In recent years, owing to their massive parallelism, fast adaptability, and inherent approximation capabilities, neural network (NN) has been used for controlling a wide class of complex nonlinear systems under the restriction that complete model information is not available. One of the important classes of NNs, the static neural networks (SNNs), has achieved much success in nonlinear control as a function approximator or a system identifier. However, the complex structure of the NNs' make the practical implementation of the control schemes infeasible, and the numbers of the hidden neurons in the NNs' hidden layers are hard to be determined. Another well-known disadvantage is that SNNs are quite sensitive to the major change which has never been learned in the training phase. Despite the immense popularity of SNNs, some researchers adopt dynamic neural networks (DNNs) to solve the control problem of nonlinear systems. An important motivation is that a smaller DNN is possible to provide the same functionality of a much larger SNN. In addition, SNNs are unable to represent dynamic system mapping without the aid of tapped delay, which results in long computation time, high sensitivity to external noise, and a large number of neurons when high dimensional systems are considered. This drawback severely affects the applicability of SNNs to system identification, which is the central part in some control techniques for nonlinear systems. Owing to their dynamic memory, DNNs have good performance on identification, state estimation, trajectory tracking, etc., even with the unmodeled dynamics. Take these advantages, some researchers first identify the nonlinear system according to the measured input and output, and then calculate the control low based on the DNN model.

Hopfield model was first proposed by Hopfield J.J. in 1982 and 1984. Because a Hopfiled circuit is quite easy to be realized and has the property of decreasing in energy by finite number of node-updating steps, it has many applications in different fields. In this paper, a so-called direct adaptive control scheme using Hopfield-based dynamic neural network (DACHDNN) for SISO affine nonlinear systems is proposed. The control object is to force the system output to track a given reference signal. The ideal controller is approximated by the internal state of a Hopfield-based DNN, and a compensation controller is used to dispel the effect of the approximation error and bounded external disturbance. The synaptic weights of the Hopfield-based DNN are on-line tuned by adaptive laws derived in the Lyapunov sense. The control law and adaptive laws provide stability for the closed-loop system with external disturbance. Furthermore, the tracking error can be attenuated to a desired level provided that the parameters of the control law are chosen adequately. Finally, a three-order dynamic system is used to demonstrate the effectiveness of the proposed control scheme.