Event-Triggered Positive Filter Design for Positive Polynomial Fuzzy Systems via Premise Integration and Membership-Function-Dependent Methods

This article focuses on the design of the positive fuzzy filter with an event-triggered mechanism for a positive polynomial fuzzy-model-based system. The main objective is to design a positive and stable fuzzy filter, which makes the
$L_{1}$
performance index of the estimated output signal as good as possible with limited networked bandwidth resources. Since the positive constraints exist in the positive systems, while the disturbance signal of the output signal and the transmission delay in the event-triggered mechanism are taken into account in this research, the augmented system is rebuilt by introducing the weight coefficient of the disturbance signal, and a novel linear copositive Lyapunov function is proposed. In order to adapt to the analysis framework based on the linear Lyapunov function, a novel linear event-triggered condition is proposed. Furthermore, to handle the mismatched premise variables caused by the event-triggered mechanism, the premise variables associated with event-triggered instants are integrated into the interval of continuous premise variable with the help of the novel event-triggered condition, which means that the original multidimensional type-1 membership functions (MFs) are transformed into single-dimensional interval type-2 MFs. Then, an interval type-2 membership-function-dependent method is employed to introduce MF information into the resultant conditions, so that a more optimized
$L_{1}$
performance index is obtained. Finally, an example with simulation results is given to verify the effectiveness of all the fuzzy filter design strategies proposed in this article for optimizing the performance index.