Lyapunov-Based Stability Analysis of Adaptive Neural-Network Controllers for Nonlinear Perturbed Systems

A Lyapunov-based framework for stability analysis and synthesis of adaptive neural-network (NN) controllers for a class of uncertain second-order nonlinear systems (SNS) with bounded external perturbations and unmodelled dynamics is presented. Online learning is employed for the reconstruction of the plant nonlinearity with the use of a radial-basis-function (RBF) network whose weights are adapted using a direct adaptation law deduced from a single composite Lyapunov function. The proposed controller couples the weight update to a persistent robustifying action, while the closed-loop stability is guaranteed throughout the learning transient, in contrast to schemes that guarantee stability after learning has converged. Using a composite Lyapunov function in the filtered tracking error and the weight-estimation error, we prove that all closed-loop signals are uniformly ultimately bounded (UUB) and that the tracking error converges to an explicitly characterized residual set whose radius is governed by the network reconstruction accuracy, the disturbance bound and the design gains. A σ-modification ensures parameter boundedness without persistency of excitation, and a robustness theorem shows that bounded parametric perturbations of the plant preserve stability and enlarge the ultimate bound only gradually (a graceful degradation, rather than a loss of the guarantee). The open-loop plant (a forced double-well Duffing oscillator) is characterized by means of equilibrium and Jacobian analyses. A bifurcation diagram and the largest Lyapunov exponent are presented, which show a chaotic regime (with λ1≈0.17). Numerical experiments indicate that the proposed controller is able to suppress the chaotic motion with a small value of the ultimate bound, and maintain a smooth reference motion with a small and constant RMS error of order 10−3, which is approximately 26 times less than the RMS error obtained with a tuned fixed-gain baseline, and the theoretical dependence of the ultimate bound on the disturbance and the design gains is confirmed by sensitivity sweeps.