Filter Design for Positive T–S Fuzzy Continuous-Time Systems With Time Delay Using Piecewise-Linear Membership Functions

Abstract

This article focuses on the filtering problem and stability analysis for positive Takagi-Sugeno (T-S) fuzzy systems with time delay under L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -induced performance. Due to the importance of estimation of system states but the few filter design results on positive nonlinear systems, it is an attractive and meaningful topic well worth studying. In order to fully exploit and take advantage of the positivity of positive T-S fuzzy systems, many commonly used methods, for instance free-weighting matrix approach and similarity transformation are probably not suitable for positive systems. To address the hard-nut-to-crack problem, an auxiliary variable is introduced so that the augmentation approach can be employed to carry out the positivity and stability analysis of filtering error systems. In addition, another obstacle that cannot be ignored is the existence of nonconvex terms in the stability and positivity conditions. For getting around this barrier, some iterative linear matrix inequality algorithms have been proposed in the literature. However, considering the weakness that these methods cannot guarantee the convergence to a numerical solution and the iterative process is exhaustive, we present an effective matrix decoupling method to convert the nonconvex conditions into convex ones in this article. Furthermore, a linear copositive Lyapunov function, which incorporates the positivity of system states and time delay at the same time is chosen so that the positivity characteristic of filtering error systems can be captured further. However, because of plenty of valuable information of membership functions being ignored, hence, the obtained results are conservative. For the sake of relaxing the conservativeness, the advanced piecewise-linear membership functions approximate method is utilized to facilitate the stability and positivity analysis. Therefore, the relaxed stability and positivity conditions, which are cast as sum of squares (SOS) are obtained and can be solved numerically. Finally, the effectiveness of the designed fuzzy filtering strategy with satisfying L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -induced performance are demonstrated by a simulation example.