Closed-form minimax time-delay filters for underdamped systems

This paper derives closed-form solutions for the parameters of a time-delay filter designed to be robust to uncertainties in frequencies to be cancelled. It is shown that the slope of the magnitude plot of the two time-delay filter is zero at the nominal frequency indicating that it is a local maximum. This information is used for deriving the solution of the parameters of the time-delay filter in closed form. Three time-delay filters are also designed which force a zero of the filter to be located at the nominal frequency of the system. Uniform and non-uniform distributions of the penalty over the uncertain regions are permitted in this formulation. The applicability of the proposed technique for the control of multi-mode systems is also illustrated.