Relaxed event-triggered tracking control of positive T–S fuzzy systems via a membership function method

In this paper, the sampled-data-based event-triggered tracking control for the positive nonlinear system is discussed. The event-triggered mechanism naturally causes a mismatch between the membership functions of the system model and the fuzzy controller. Meanwhile, the positive constraint and tracking behavior increase the complexity of system analysis and bring conservative analysis results. How to achieve positive and event-triggered tracking control while ensuring good tracking performance has become challenging. To address the challenge, firstly, the positive T–S fuzzy model is established to characterize the positive tracking nonlinearsystem. Subsequently, a novel Lyapunov–Krasovskii functional is constructed, which takes into account the tracking errors and transmission delays induced by the event-triggered mechanism. Furthermore, the piecewise linear membership functions (PLMFs) are used to relax the conservativeness of analysis results. Then, to ensure the system positivity and obtain the approximation errors of PLMFs, an outer constraint is proposed to handle the mismatched membership functions caused by the sampled-data-based event-triggered mechanism. Finally, the proposed approaches are demonstrated to achieve good tracking performance while reducing communication resources through a numerical example and a two-linked tank practical example.