Robust ${\mathscr {H}}_{\infty }$ Filtering for a Class of Two-Dimensional Uncertain Fuzzy Systems With Randomly Occurring Mixed Delays

This paper is concerned with the robust <inline-formula><tex-math notation="LaTeX">${\mathscr {H}}_{\infty }$ </tex-math></inline-formula> filtering problem for a class of 2-D uncertain fuzzy systems with randomly occurring mixed delays (ROMDs). The underlying 2-D systems are described by the Fornasini–Marchesini model, and the uncertainty is expressed in a linear fraction form. An improved Takagi–Sugeno (T–S) fuzzy model corresponding to the spatial promise variables is adopted to represent the complicated 2-D nonlinear system. The mixed delays consisting of both discrete and distributed delays are allowed to appear in a random manner governed by two sets of Bernoulli distributed white sequences with known probability. A full-order fuzzy filter is constructed to estimate the output signal such that, in the presence of parameter uncertainties and ROMDs, the dynamics of the estimation errors is asymptotically stable with a prescribed <inline-formula><tex-math notation="LaTeX">${\mathscr {H}}_{\infty }$ </tex-math></inline-formula> disturbance attenuation level. Based on the stochastic analysis technique and the Lyapunov-like functional, sufficient conditions are established to ensure the existence of the desired filters, and the explicit expressions of such filters are derived by means of the solution to a class of convex optimization problems that can be solved via standard software packages. A numerical example is provided to demonstrate the effectiveness of the developed filter design algorithms, and the filter performances with and without fuzzy rules are also compared.