Stability conditions of TS systems based on quadratic and non-quadratic Lyapunov functions

This paper introduces a method for reducing the conservatism in Takagi–Sugeno (TS) fuzzy systems through the use of a non-quadratic Lyapunov function (NQLF), also known as the line integral Lyapunov fuzzy function. By leveraging this function in combination with an efficient methodological approach, the stability analysis of TS systems is significantly improved. The stability conditions for these systems are initially formulated as Bilinear Matrix Inequalities (BMIs), which present a challenge due to their nonlinear nature and computational complexity. To address this difficulty, we propose an iterative algorithm designed to transform the BMI problem into a more tractable form by converting it into a set of Linear Matrix Inequalities (LMIs). LMIs are easier to solve using established optimization techniques, thereby simplifying the stability analysis process without sacrificing its accuracy. This transformation allows for more efficient computation and reduces the conservatism typically associated with BMI-based methods. To validate the effectiveness of our approach, a numerical example is provided, demonstrating how the proposed method outperforms traditional approaches by offering an enhanced stability analysis. The example illustrates the reduction in conservatism, thereby highlighting the practicality and robustness of the approach. Overall, this method offers a promising solution for improving the stability analysis of TS fuzzy systems by reducing the complexity and providing more reliable results. This contribution underscores the potential of using non-quadratic Lyapunov functions to address challenges in system stability with increased computational efficiency.