  This paper is concerned with the fuzzy $H_{\infty}$ filter design issue for
nonlinear systems with time-varying delay. To overcome the shortcomings of the
conventional methods with matched preconditions, the fuzzy $H_{\infty}$ filter
to be designed and the T-S fuzzy model are assumed to have different premise
membership functions and number of rules, thus, greater design flexibility and
robustness to uncertainty can be achieved. However, such design will also make
the derived results conservative, to relax the result, a novel integral
inequality which is tighter than the traditional inequalities derived from the
Leibniz-Newton formula is applied, besides, a fuzzy Lypunov function and the
information of the membership functions are also introduced. All the design
methods are presented in LMI-based conditions. Finally, two numerical examples
are given to prove the effectiveness and superiority of the proposed approach.
