Small-Gain Approach for Adaptive Optimal Control of Switched Nonlinear Systems With Unstable Dynamics

This article addresses the adaptive optimal tracking control problem of nonlinear switched systems with unstable unmodeled dynamics. One challenge is how to find the optimal control strategy for a modeled x-dynamic system containing the unmodeled dynamics. Another challenge is how to achieve both optimality and stability of the closed-loop system with unstable dynamics. To this end, a novel solution combines adaptive dynamic programming and the small-gain approach, integrating integral penalties on control, states, and unstable dynamics via actor–critic learning and nonzero game mechanisms. The small-gain approach links closed-loop optimality and stability. For unstable dynamics, stability analysis employs a combined fast-slow switching strategy with gain allocation satisfying the small-gain theorem conditions; optimality analysis compensates unstable dynamics via adaptive parameter estimation and derives the optimal controller via a two-player game. Under small-gain conditions, the designed optimal controller ensures closed-loop output tracks reference signals with ultimately bounded errors. Finally, simulation results validate the proposed approach.