Stability and Stabilization of Positive Fuzzy Systems via Polynomial Lyapunov Functions With Application to Tank Level Control

In this paper, new stability conditions are obtained by designing a Lyapunov function that contains polynomials of the system states and membership functions. An iterative algorithm is designed to solve the stabilization problem. Since more free variables are introduced by using the square root of the system states and square matrix representation, the proposed conditions are less conservative than existing ones. Furthermore, the time derivative of the membership functions is dealt with by applying polynomial switching mechanism, while the average dwell time approach is employed to address the switching signal. The effectiveness of the proposed method is demonstrated through three numerical examples, including a practical tank level control system, confirming its applicability to real-world scenarios. This paper shows that for positive fuzzy systems, better results can be achieved by utilizing the positivity of the system states and the information of the membership functions. Note to Practitioners—Positive fuzzy systems have found wide applications in engineering domains such as chemical process control, biological systems, and water resource management, where system variables are inherently non-negative. However, traditional control methods often fail to fully exploit the structural characteristics of such systems, resulting in conservative designs and limited performance. Motivated by these challenges, this paper proposes a new stabilization approach that utilizes the positivity of the system states and the information of the membership functions. A Lyapunov function is constructed by incorporating the square roots of state polynomials and polynomial terms of the membership functions, and an iterative algorithm is developed to solve the associated stabilization conditions. Furthermore, the relationship between the stability region and the degree of the polynomials is investigated, which has important implications for enhancing overall system performance.