Theory and implementation of adaptive algorithms for rejecting disturbances having multiple sinusoidal components

This dissertation is concerned about adaptive algorithms for rejecting disturbances having multiple sinusoidal components. A discrete-time adaptive algorithm is proposed to reject such disturbances in the case where the frequencies are unknown and possibly time-varying. A reference sensor is not assumed to be available. The stability of the algorithm is analyzed using averaging theory, and the design of the parameters is based on the linearized averaged system. While the algorithm is first designed for rejecting periodic disturbances with one sinusoidal component, it is also extended to deal with cases where the disturbance has multiple sinusoidal components. A frequency separation method is proposed to prevent the frequency estimates from converging to the same value.; The dissertation also considers the problem of rejecting disturbances with two sinusoidal components in the case where the frequencies are unknown and closely spaced. A natural approach consists in canceling the components using two separate adaptive algorithms combined in a single scheme. However, experiments in active noise control applications have shown that convergence using such an approach could be very slow. The alternative approach of this dissertation consists in representing the disturbance signal as a single sinusoid with time-varying magnitude and phase. The theoretical basis and the limitations of such a representation are first discussed. Then, an adaptive disturbance rejection algorithm is proposed and the resulting nonlinear system is analyzed using some approximations. Active noise control experiments demonstrate that the proposed algorithm has better convergence properties than an algorithm designed to cancel the two frequency components separately. In some cases, however, the cost is a small residual error on the output signal.; The last problem considered in this dissertation is the rejection of sinusoidal disturbances with known but rapidly varying frequencies. The dissertation shows that a large class of adaptive algorithms for disturbance cancellation yields control systems that are equivalent to linear compensators implementing the internal model principle. The fact had been known to be true for periodic disturbances with fixed frequency. However, the dissertation shows that the result can be extended to disturbances of time-varying frequency (i.e., frequency-modulated signals), regardless of the rate of variation of the frequency. In particular, several adaptive controllers are shown to be equivalent to linear time-varying compensators, with a pseudo-gradient algorithm being equivalent to a polytopic linear parameter varying compensator. The equivalence provides an opportunity to apply knowledge gained either in adaptive control or in robust linear control to the other field.