A new parameter-dependent approach to discrete-time robust $H-{2}$ filtering

This paper revisits the problem of robust $H-{2}$ filtering for discrete-time systems with parameter uncertainties. Given a stable system with parameter uncertainties residing in a polytope with $s$ vertices, the focus is on designing a robust filter such that the filtering error system is robustly asymptotically stable and has a guaranteed estimation error variance for the entire uncertainty domain. A new polynomial parameter-dependent idea is introduced to solve the robust $H-{2}$ filtering problem, which is different from the quadratic framework that entails fixed matrices for the entire uncertainty domain, or the linearly parameter-dependent framework that uses linear convex combinations of $s$ matrices. This idea is realized by carefully selecting the structure of the matrices involved in the products with system matrices. A linear matrix inequality (LMI) condition is obtained for the existence of admissible filters, and based on this, the filter design is cast into a convex optimization problem, which can be readily solved via standard numerical software. The merit of the proposed method lies in its less conservativeness than the existing robust filter design methods, as illustrated via a numerical example.