Robust generalized asymptotic regulation against non-stationary sinusoidal disturbances

Attenuation of sinusoidal disturbances with uncertain and arbitrarily time-varying frequencies is considered in the form of a generalized asymptotic regulation problem. The disturbances are modeled as the outputs of a parameter-dependent unexcited exogenous system that evolves from nonzero initial conditions. The parameter dependence is assumed to be in such a form that the state of the exogenous system has constant norm at all times. Considering a partially parameter-dependent system, the problem is then formulated as the synthesis of a linear time-invariant controller with which the closed-loop respects a desired level of attenuation profile in steady-state and exhibits sufficiently fast transient response for all admissible parameter variations. The main result of the paper is a synthesis procedure based on a convex optimization problem, which is identified by a set of parameter-dependent linear matrix inequalities and can be rendered tractable through standard relaxation schemes. The order of the synthesized controller is equal to the order of the plant plus the order of the exogenous system.