Adaptive control of nonlinear time-varying systems with unknown parameters and model uncertainties

This paper investigates the adaptive control problem for nonlinear time-varying systems with unknown parameters and model uncertainties. A novel class of switching functions is designed, and its construction method is detailed, along with a proof of the continuity of its $n-1$ order derivatives. Two simple examples are provided to illustrate how the proposed congelation of variables method handles unknown high-frequency time-varying parameters in both the feedback and input paths. A new neural network control scheme is then developed, integrating an adaptive neural network controller with a robust controller. The smooth transition between these two controllers is ensured by the novel switching function, which guarantees global system stability. Furthermore, by combining the congelation of variables method with adaptive backstepping, a new adaptive tracking control scheme is proposed. This scheme effectively handles unknown high-frequency time-varying parameters while achieving asymptotic tracking of arbitrary reference signals. Simulation results show that the proposed novel adaptive control method delivers superior control accuracy while reducing energy consumption: it achieves an order of magnitude improvement over the traditional adaptive robust control method and two orders of magnitude improvement over the conventional sliding mode control method.