Admissibility and Admissibilization of Singular Polynomial Fuzzy Systems with Time-Varying Delay
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Admissibility and Admissibilization of Singular Polynomial Fuzzy Systems with Time-Varying Delay Zhiguang Feng1,2 • Yang Yang1 • Zhengyi Jiang2 • Yuxin Zhao1 • Xin Yuan1
Received: 9 July 2020 / Revised: 2 September 2020 / Accepted: 12 September 2020 Ó Taiwan Fuzzy Systems Association 2020
Abstract This paper addresses the admissibility analysis and the polynomial fuzzy controller design for a class of singular polynomial fuzzy systems with time-varying delay. Firstly, according to Lyapunov stability theory, a sufficient condition is obtained by means of sum-of-squares (SOS) to ensure that the closed-loop singular polynomial fuzzy system is admissible. Secondly, by the help of the two equivalent sets, SOS-based admissibility conditions are proposed to avoid solving the equality conditions and the polynomial fuzzy controller is designed. Finally, three examples validate the usefulness and superiority of the proposed design procedure compared with existing results. Keywords Polynomial fuzzy controller Singular polynomial fuzzy systems Sum-of-squares (SOS) Timevarying delay
& Yang Yang [email protected] Zhiguang Feng [email protected] Zhengyi Jiang [email protected] Yuxin Zhao [email protected] Xin Yuan [email protected] 1
College of Automation, Harbin Engineering University, Harbin 150001, China
2
School of Mechanical, Materials, Mechatronic and Biomedical Engineering, University of Wollongong, Wollongong 2522, NSW, Australia
1 Introduction The study of singular systems is generally carried out at the two levels described by differential equations and algebraic equations, which enables singular systems to describe and express more widely the dynamic systems, and has been applied to practical systems in the past few decades, such as biological systems [1], mechanical engineering systems [2], bio-economic systems [3] and so on. In addition, timedelay is the main cause of system stability deterioration and performance degradation, and it is difficult to avoid the existence of time-delay in many physical systems. Therefore, discussions regarding singular time-delay systems have been extensively reported, and the research methods of standard systems with time-delay have been popularized to singular systems with time-delay in recent years. For example, the reachable set estimation [4], admissibility analysis [5], sliding mode control [6] and a-dissipativity analysis [7] of singular systems with time-delay have been studied using the novel integral inequality, the reciprocally convex method and the time-delay partitioning technique of normal systems with time-delay, respectively. In addition, in the real world, most of the time-delays are timevarying rather than constant, and it is more practical to study singular systems with time-varying delay [8, 9]. Moreover, as we know that the study of the state-feedback control of singular systems with time-varying delay is not fully investigated, which motivates our current research. On the other hand, the Takagi–Sugeno (T-S) fuzzy model is a very powerful tool
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