Parameter-dependent H2H2 and H∞H∞ filter design for linear systems with arbitrarily time-varying parameters in polytopic domains

In this paper, the problem of filter design for linear continuous-time systems with arbitrarily fast time-varying parameters is investigated. The time-varying parameters belong to a polytope with known vertices, affect all the system matrices and are assumed to be available online for implementation of the filters. Necessary and sufficient parameter-dependent linear matrix inequality (LMI) conditions for the existence of a parameter-dependent filter assuring that the estimation error dynamics is quadratically stable and satisfies bounds to the H2H2 or to the H∞H∞ norms are given. A sequence of standard LMI conditions assuring the existence of homogeneous polynomially parameter-dependent (HPPD) solutions to the parameter-dependent LMIs for filter design is provided in terms of the vertices of the polytope (no gridding is required), yielding parameter-dependent filters of arbitrary degree assuring quadratic stability of the error dynamics for the H2H2 or the H∞H∞ cases. As the degree of the HPPD solutions increases, less and less conservative LMI conditions are obtained, tending to the necessary conditions that assure optimal values for the H2H2 or the H∞H∞ performance of the estimation error dynamics under quadratic stability. Numerical examples illustrate the results, showing that parameter-dependent filters can provide better attenuation levels than the ones obtained with robust filters, at the price of a more complex filtering strategy.