Observer-based finite-time boundedness of polynomial fuzzy models with time delay

This paper investigates the observer-based finite-time boundedness (FTB) problem for a class of time delay polynomial fuzzy (TDPF) models. A polynomial observer estimates unknown system states using measured outputs. It employs the same membership functions and delay as the TDPF model. Based on these estimated states, the controller implements to ensure the FTB of the TDPF model by using the parallel distributed compensation (PDC) conception. An appropriate Lyapunov–Krasovskii functional yields sufficient conditions with coupled variables. These ensure the FTB of the augmented model combining state vector estimates and estimation errors. Next, these conditions are separated into two sets. A decoupling technique method is adopted to transform the first set into sum of squares (SOS) conditions which can be numerically and partially symbolically solved by using MATLAB toolbox SOSTOOLS. Based on the obtained feasible solution, the polynomial observer-based controller gains are computed. Furthermore, the FTB is verified by checking the second set, which consists of a single condition for parameters related to FTB. An algorithm is proposed to elucidate the design procedure. In the end, a numerical example and an application to the inverted pendulum system are presented to demonstrate the developed design.