Finite-Frequency $\mathcal {H}_{-}/\mathcal {H}_{\infty }$ Memory Fault Detection Filtering Design for Uncertain Takagi–Sugeno Fuzzy Affine Systems

This article is concerned with the finite-frequency
$\mathcal {H}_{-}/\mathcal {H}_{\infty }$
memory fault detection filtering problem for discrete-time Takagi–Sugeno fuzzy affine systems with norm-bounded uncertainties. The objective is to design a piecewise affine memory filter by using system historical information such that the resulting closed-loop filtering error system is asymptotically stable with the prescribed finite-frequency
$\mathcal {H}_{-}/\mathcal {H}_{\infty }$
performance. Based on the generalized Kalman–Yakubovič–Popov lemma combined with the celebrated
$\mathcal {S}$
-procedure, new sufficient conditions for the fuzzy affine filtering error system to have the finite-frequency
$\mathcal {H}_{-}/\mathcal {H}_{\infty }$
performance are given at first. By further using piecewise fuzzy quadratic Lyapunov functions and Projection lemma, the filtering analysis results for the filtering error system to be asymptotically stable with the prescribed finite-frequency
$\mathcal {H}_{-}/\mathcal {H}_{\infty }$
performance are obtained. Then, the filtering synthesis is carried out with the aid of matrix inequality convexification techniques, and the synthesis results are described in terms of linear matrix inequalities. It is further shown that a better filtering performance can be achieved by using more system historical information. Finally, simulation is provided to verify the effectiveness of the proposed approach.