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Hybrid adaptive compensators for multi-sinusoidal disturbance rejection

The problem of rejecting biased multi-sinusoidal disturbances with frequencies in a given range, acting on an uncertain stable linear system is addressed. Assuming that the system uncertainty allows to design an output feedback linear compensator when the disturbance frequencies are known, an exponentially convergent adaptive observer is proposed to update the compensator parameters when the disturbance frequencies lie in a known given range. The overall proposed compensator is hybrid since its parameters are updated every fixed time-interval, according to a performance index which evaluates the adaptive observer reconstruction error. The main advantage of the proposed hybrid compensator is to drive exponentially the output to zero, from any initial condition which is compatible with the given frequency range, without requiring a sufficiently slow adaptation.