Positive $$\mathscr {L}_1$$ Observer Design

Abstract

AbstractThis chapter concerns the positive \(\mathscr {L}_1\) observer design for positive nonlinear semi-Markov jump systems (S-MJSs) via the expansion of Taylor formula and the fuzzy Lyapunov function approach, where semi-Markov jump parameters, positivity, T-S fuzzy, and external disturbance are all considered. A fuzzy Lyapunov function approach is introduced into the study of positive systems with less conservativeness. Through the fuzzy linearization technique, positive nonlinear S-MJSs can be described as positive T-S fuzzy S-MJSs. Then, some sufficient conditions for stochastic stability and \(\mathscr {L}_1\)-gain performance analysis are proposed with the fuzzy Lyapunov function. Ulteriorly, a parameter solving algorithm for positive \(\mathscr {L}_1\) observer is designed in a novel standard linear programming condition, to guarantee the closed-loop system positive and stochastically stable with a required \(\mathscr {L}_1\)-gain performance. In the end, the proposed method is applied into the epidemiological model to show its effectiveness.