Adaptive Algorithms for the Rejection of Sinusoidal Disturbances Acting on Unknown Plants

The rejection of periodic disturbances is a problem frequently encountered in control engineering, and in active noise and vibration control in particular. The paper presents a new adaptive algorithm for situations where the plant is unknown and may be time-varying. The approach consists in obtaining online estimates of the plant frequency response and of the disturbance parameters. The estimates are used to continuously update control parameters and cancel or minimize the effect of the disturbance. The dynamic behavior of the algorithm is analyzed using averaging theory. Averaging theory is used to approximate the nonlinear time-varying closed-loop system by a nonlinear time-invariant system. 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 canceled even if the true parameters of the system are not exactly determined. Simulations, as well as extensive experiments on an active noise control testbed, illustrate the results of the analysis and demonstrate the ability of the algorithm to recover from abrupt system changes or track slowly-varying parameters. Extensions of the algorithm to systems with multiple inputs/outputs and disturbances consisting of multiple frequency components are provided.