Direct Adaptive Control in Unknown Nonlinear Systems that exhibit Brunovski Canonical Form, using Neuro-Fuzzy High Order Neural Networks, with Robustness Analysis

The direct adaptive regulation of nonlinear dynamical systems in Brunovsky form with modeling error effects, is considered in this paper. The method is based on a new Neuro-Fuzzy Dynamical System definition, which uses the concept of Fuzzy Adaptive Systems (FAS) operating in conjunction with High Order Neural Network Functions (HONNFs). Since the plant is considered unknown, we first propose its approximation by a special form of a Brunovsky type fuzzy dynamical system (FDS) assuming also the existence of disturbance expressed as modeling error terms depending on both input and system states. The fuzzy rules are then approximated by appropriate HONNFs. This practically transforms the original unknown system into a neuro-fuzzy model which is of known structure, but contains a number of unknown constant value parameters. The development is combined with a sensitivity analysis of the closed loop in the presence of modeling imperfections and provides a comprehensive and rigorous analysis of the stability properties of the closed loop system. The proposed scheme does not require a-priori information from the expert on the number and type of input variable membership functions making it less vulnerable to initial design assumptions. The existence and boundness of the control signal is always assured by introducing a novel method of parameter hopping and incorporating it in weight updating law. Simulations illustrate the potency of the method and its applicability is tested on the well known benchmarks “Inverted Pendulum” and “Van der pol”, where it is shown that our approach is superior to the case of simple Recurrent High Order Neural Networks (RHONNs).