Robust H∞ filtering for polytopic uncertain systems via convex optimisation

A robust H∞ filtering technique is proposed for convex polytopic uncertain systems. This class of uncertainty can describe the parametric uncertainty more precisely, without conservatism, than the norm-bounded uncertainty. By applying a bounded real lemma to the error dynamics and using the Schur complement with the appropriate change of variables, a nonlinear matrix inequality is obtained. It is then shown that the congruence transformation, with some newly defined variables, converts this nonlinear matrix inequality into the convex optimisation problem for the design of robust H∞ filters, which is expressed by linear matrix inequality and can be solved very efficiently by so called interior point algorithms. The optimal tolerance level can be directly computed without the aid of the conventional bisection method, and the proposed algorithm does not require the additional search procedures needed for dealing with the norm-bounded uncertainty. Numerical examples are given to show that the proposed filter is more robust than the robust H₂ filter against the parameter variation, as well as the noise in the worst-case frequency range and to illustrate the advantage of describing the uncertainty as polytopic rather than norm bounded.