Asymptotic Stabilization of Chain Integrator Systems via Adaptive Neural Control

This work proposes an Adaptive Neural Control for the asymptotic stabilization of a chain of integrators at the origin. The proposed approach addresses the stabilization of the integrator chain by means of a control law whose applied signal is structurally bounded to $(-1,1)$ by the hyperbolic tangent architecture, i.e., $u\left(t\right)=tanh\left(z\right)$, where z represents a weighted linear combination of the system states and a bias term. Furthermore, an adaptation law for the weights is proposed, based on the classical backpropagation algorithm for neural networks. The stability analysis is conducted using singular perturbation theory, demonstrating that, under a sufficiently high learning rate, the closed-loop system exhibits a Standard Singular Perturbation Form. This formulation allows for the analysis of the system across two distinct time scales: the adaptation dynamics (fast subsystem) and the state dynamics (slow subsystem). Based on this formulation, explicit conditions on the learning rate and the initial conditions are derived to guarantee local asymptotic stability using Tikhonov’s theorem. These conditions characterize the region of attraction and ensure that the adaptive neural controller stabilizes the system. Numerical simulations were carried out to evaluate the controller’s performance under three different scenarios: ideal conditions, initialization outside the region of attraction, and a low learning rate. These scenarios illustrate the closed-loop system behavior and validate the theoretical conditions required for asymptotic stability. Furthermore, comparative numerical simulations were conducted on an Inverted Pendulum on a Cart system to benchmark the proposed Adaptive Neural Control against Linear Quadratic Regulator, Sliding Mode Control, and Nested Saturation Function controllers. Based on the Integral of Time-weighted Squared Error performance index, the Adaptive Neural Control demonstrated a significant reduction in control effort, achieving performance improvements of up to 95.02% compared to the aforementioned strategies.