Equivalence between adaptive cancellation algorithms and linear time-varying compensators

The internal model principle of control theory specifies that, in order to completely reject a sinusoidal disturbance, a compensator should have poles on the imaginary axis at the frequency of the disturbance. It was found earlier that several adaptive algorithms for the cancellation of disturbances of fixed frequency were equivalent to linear time-invariant compensators satisfying the internal model principle. The paper investigates issues associated with disturbances of varying frequency. The main result is that the equivalence can be extended to a linear time-varying system equivalence. However, one must be careful in the choice of the state-space model of the compensator. The linear time-varying system equivalence is shown to hold true for a large class of algorithms for adaptive disturbance cancellation. Keywords: sinusoidal disturbances, periodic disturbances, internal model principle, adaptive control, time-varying systems, linear parameter varying systems.