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 <inline-formula> <tex-math notation="LaTeX">$\mathcal {L}_{1}$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$\mathcal {L}_{1}$ </tex-math></inline-formula>-gain performance analysis, respectively. Then, a positive <inline-formula> <tex-math notation="LaTeX">$\mathcal {L}_{1}$ </tex-math></inline-formula> observer in a novel standard linear programming condition is designed to guarantee the resulting closed–loop augmented system is positive and stochastically stable with a required <inline-formula> <tex-math notation="LaTeX">$\mathcal {L}_{1}$ </tex-math></inline-formula>-gain performance. Finally, a practical example about the epidemiological model is introduced to show the effectiveness of the main theory.