Robust H2H2 and H∞H∞ memory filter design for linear uncertain discrete-time delay systems

• Filter with arbitrary number of past states and output measures of the system.
• LMI relaxations based on polynomially parameter-dependent variables for filter design.
• Unknown (possibly time-varying) delays in an interval can affect the uncertain system.

This paper is concerned with the problems of robust full-order H2H2 and H∞H∞ filter design for linear uncertain discrete-time systems with multiple state delays. The uncertain parameters affecting the matrices of the system are supposed to be time-invariant and to belong to a polytopic domain. The main novelty is the fact that the filter contains an arbitrary number of past states and past output measures of the system, yielding a filtering system with memory. Linear matrix inequality relaxations based on polynomially parameter-dependent Lyapunov matrices and slack variables are proposed for the H2H2 and H∞H∞ filter design. Due to the extra dynamics introduced through the delayed states, the robust memory filter is able to provide less conservative results in terms of the H∞H∞ and the H2H2 performance when compared to the memoryless case. Throughout the paper, the multiple delays are considered to be fixed and time-invariant, but an extension of the conditions to cope with unknown delays belonging to a given interval is also presented for both time-varying and time-invariant delay cases. Numerical examples are given to demonstrate the improvements of the proposed approach with respect to other methods from the literature.