Transfer function-based robust filtering: Review and critical evaluation

Promoted by Wilson in his 1989 year work through the convolution and Hankel operator norms, the transfer function approach (TFA) developed by many authors has earlier emerged as a novel trend of sorts in robust estimation of system state to minimize the estimation error bounded norm for the maximized error bounded norm. This paper takes a fresh look at the problem through the bias correction gain $K$ of a recursive filter, reviews and revisits the existing robust $H_2$, energy-to-energy or $H_{\infty }$, energy-to-peak or generalized $H_2$ (G$H_2$), and peak-to-peak or $L_1$ filtering solutions, and critically evaluates their performances. It is shown that the effective $K$ ranges between the larger gain of the optimal Kalman and the smaller gain of the robust unbiased finite impulse response (UFIR) filter. That is, regardless of the robust criterion, the gain produced by the sophisticated TFA turns out to be quite sandwiched by the Kalman and UFIR filters. The filters are tested based on extensive numerical simulations and experimentally in terms of mean square error, robustness, and quality factor.