Dual Event-Triggered Polynomial Dynamic Output Control for Positive Fuzzy Systems via an IT2 Membership Function Relaxation Method

The co-design problem of dual event-triggered (DET) mechanism and polynomial dynamic output-feedback (PDOF) controller is investigated for positive polynomial fuzzy systems (PPFSs) with uncertainty and disturbance constraints. Specifically, a 1-norm DET mechanism compatible with the positivity of PPFSs is proposed to asynchronously update measurement outputs and PDOF control signals. However, synthesizing this DET-PDOF controller proves challenging due to the coupling of multiple unknown PDOF controller gain matrices within the positivity and stability conditions, which results in complex nonconvex terms. By introducing auxiliary variables and constraints, sufficient conditions for DET-PDOF controller solution are given to ensure both the $L_{1}$ -gain performance and strict positivity of PPFSs with uncertainty and disturbance. Moreover, existing stability analysis results that ignore membership functions (MFs) tend to be conservative, implying that the obtained DET-PDOF controller is effective only within a limited triggered threshold range, leading to worse transmission performance. Therefore, a multivariate optimization method based on an improved genetic algorithm (IGA), which accounts for the system states and PDOF controller variables, is developed to substantially expand the admissible DET threshold range while effectively suppressing dual-triggering frequencies. Finally, a numerical example and a two-linked tank system with parameter uncertainty are provided to validate the feasibility of the proposed scheme.