Robust Linear $H_{\infty}$ Filter Design for a Class of Uncertain Nonlinear Systems: An LMI Approach

This paper addresses the design of robust $H_{\infty}$ filters for a class of nonlinear systems subject to uncertain, possibly time-varying, parameters. The nonlinear system is described by a differential-algebraic representation, which can model the whole class of systems with rational functions of the state and uncertain parameters, as well as some trigonometric nonlinearities. The admissible values of the uncertain parameters and their rate of variation are assumed to belong to a given polytope. A linear matrix inequality methodology is proposed for designing full-order robust linear filters with a minimized upper-bound on the $\mathcal{L}_2$-gain of the noise-to-estimation error operator for all admissible uncertainty. In order to improve the performance of the latter filter, the methodology is then extended to the design of high-order linear filters. Numerical examples are presented to demonstrate the effectiveness of the proposed filter designs.