A Fuzzy Lyapunov Function Approach to Positive Ll Observer Design for Positive Fuzzy Semi-Markovian Switching Systems With Its Application

This paper concerns a positive G1 observer for positive nonlinear semi-Markovian switching systems (MSSs) via the expansion of Taylor formula and the fuzzy Lyapunov function approach, in which semi-Markovian switching parameters, positivity, Takagi-Sugeno (T-S) fuzzy, and external disturbance are all considered in a unified framework. A fuzzy Lyapunov function approach with less conservativeness is introduced into the research of positive systems. In the system under consideration, positive S-MSSs with the semi-Markovian process can describe more complex systems in a practical control process. The main motivation of this paper is that the practical system subject to positivity and abrupt changes can be described by positive nonlinear S-MSSs, which always needs to consider the external disturbance. First, by using the normalized membership function approach, positive nonlinear S-MSSs can be represented by local positive T-S fuzzy S-MSSs. Second, by constructing the fuzzy Lyapunov function, some sufficient conditions are proposed for stochastic stability and G1-gain performance analysis, respectively. Then, a positive G1 observer in a novel standard linearprogramming condition is designed to guarantee the resulting closed-loop augmented system is positive and stochastically stable with a required G1-gain performance. Finally, a practical example about the epidemiological model is introduced to show the effectiveness of the main theory.