Stochastic Adaptive Control for Systems with Nonlinear Parameterization: Almost Sure Stability and Tracking
This addresses adaptive control for nonlinear stochastic systems in applications like neural networks and social dynamics, offering a novel approach to a relatively unexplored problem with practical implications.
The paper tackled adaptive control for nonlinear stochastic systems with nonconvex parameter dependencies by proposing an online nonlinear weighted least-squares estimator and an adaptive control algorithm, achieving global stability and long-run average tracking of reference trajectories without restrictive assumptions like persistent excitation.
This paper concerns the adaptive control problem for a class of nonlinear stochastic systems in which the state update is given by a nonlinear function of linear dynamics plus additive stochastic noise. Such systems arise in a wide range of applications, including recurrent neural networks, social dynamics, and signal processing. Despite their importance, adaptive control for these systems remains relatively unexplored in the literature. This gap is primarily due to the inherently nonconvex dependence of the system dynamics on unknown parameters, which significantly complicates both controller design and analysis. To address these challenges, we propose an online nonlinear weighted least-squares (WLS)-based parameter estimation algorithm and establish the global strong consistency of the resulting parameter estimates. In contrast to most existing results, our consistency analysis does not rely on restrictive assumptions such as persistent excitation conditions of the trajectory data, making it applicable to stochastic adaptive control settings. Building on the proposed estimator, we further develop an adaptive control algorithm with an attenuating excitation signal that can effectively combine adaptive estimation and feedback control. Finally, we are able to show that the resulting closed-loop system is globally stable and that the system trajectory can track, in a long-run average sense, the reference trajectory generated with the true system parameters. The proposed methods and theoretical results are finally validated through simulations in two nonlinear interaction network applications.