Scheduled control for robust attenuation of non-stationary sinusoidal disturbances with measurable frequencies

Attenuation of sinusoidal disturbances with uncertain yet online measurable frequencies is considered. The disturbances are modeled as the outputs of an undisturbed parameter-dependent exogenous system with a skew-symmetric system matrix, obtained in response to nonzero initial conditions. The problem is formulated for a parameter-dependent plant as the synthesis of a parameter-dependent controller in a way to ensure internal stability as well as a desired level of steady-state disturbance attenuation in the face of all admissible parameter variations. The solvability of this problem is first related to the existence of bounded solutions to a matrix differential regulator equation subject to an asymptotic norm constraint. Reformulating this as a parameter-dependent state-feedback like synthesis, based on which suitable solutions to the differential regulator equation can be obtained online, tractable solvability conditions are then provided in the form of parameter-dependent matrix inequalities. Controllers that solve the generalized asymptotic regulation problem are also parameterized in terms of the suitable solutions of the differential regulator equation and some free parameter-dependent matrices that are to be designed off-line to ensure stability. A procedure is then developed to design the free parameters in a way to achieve desirable transient behavior. The use of the developed synthesis procedure is illustrated on a simplified version of the course control problem in ship steering.