Robust generalized asymptotic regulation against non-stationary sinusoidal disturbances with uncertain frequencies

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 and neutrally stable exogenous system that evolves from nonzero initial conditions. The problem is considered for a plant that depends partially on the uncertain parameters. Moreover, both the plant and the exogenous system are allowed to have dependence on another parameter vector that is measurable during online operation. The problem is then formulated as the synthesis of a controller that is scheduled on the measurable parameter in a way to guarantee robust internal stability and attenuate the disturbance according to a desired profile in steady state. 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. It is also clarified how the transient behavior of the controller can be improved by some additional constraints. The order of the synthesized controller is equal to the order of the plant plus the order of the exogenous system.