Averaging analysis of a sinusoidal disturbance rejection algorithm for unknown plants

The paper discusses the dynamic behavior of an adaptive algorithm for the rejection of a sinusoidal disturbance acting on an unknown system. Averaging theory is used to approximate the nonlinear time-varying closed-loop system by a nonlinear time-invariant system. Then, it is shown that the four-dimensional averaged system has a two-dimensional equilibrium surface, which can be divided into stable and unstable subsets. Trajectories generally converge to a stable point of the equilibrium surface, implying that the disturbance is asymptotically cancelled even if the true parameters of the system may not be exactly determined. Simulations demonstrate the results of the analysis.