In this article, we investigate the parabolic partial differential equations (PDEs) systems with Neumann boundary conditions via the Takagi–Sugeno (T–S) fuzzy model. On the basis of the obtained T–S fuzzy PDE model, a novel fuzzy state controller which is associated with the boundary state of position and the mean value coefficient matrix derived through the mean value theorem of integral is designed to analyze the asymptotic stability of the parabolic PDE system. Without sampling the nonlinear parameter of the system, new stability conditions are deduced in the form of linear matrix inequalities (LMIs). Moreover, compared with the novel fuzzy state controller, more conservative conditions based on another fuzzy state controller are also provided. Finally, we explore the state-feedback controller into the Fisher equation as an application. Simulation results show that the proposed method is effective.

The output regulation of nonlinear distributed parameter systems (DPSs) with time-varying delay is studied in this paper. By employing the Takagi-Sugeno (T-S) fuzzy model, the nonlinear time-delay DPSs are described as a fuzzy partial differential equation (PDE). To deal with the limited capability of communication channels, the state information and output regulation error are quantized by a kind of dynamic quantizer. The actuator failures which occur frequently in practice are considered in two cases: to be unknown and to follow the Bernoulli stochastic distribution. Then, the boundary-control-based output regulation controllers are designed based on the derived T-S fuzzy model. And fuzzy controller design strategies are proposed to ensure the system state and the output regulation error are finitetime bounded (FTB). By constructing appropriate Lyapunov functionals, sufficient conditions for the control gains and the quantizer's adjusting parameters are provided in forms of linear matrix inequalities (LMIs). Finally, an application of the output regulation strategies into the cooling process in a hot strip mill is explored. The feasibility and effectiveness of the design strategies are indicated through the simulation results.

This article focused on the investigation of boundary dissipative control for a class of nonlinear parabolic partial differential equation (PPDE). By employing the Takagi–Sugeno (T–S) fuzzy modeling framework, the nonlinear system is characterized as a category of linear PPDE. An output-based nonparallel distributed fuzzy state observer is developed to estimate the unmeasured system state. Dynamic errors with undetermined parameters for the PPDE subject to appropriate boundary conditions are presented. To overcome performance limitations of traditional controllers in managing systems with spatially distributed characteristics, a boundary-based nonparallel distributed dynamic control scheme incorporating time delays and actuator failures is developed, building upon the established observer framework. Meanwhile, to enhance the efficiency of network resources, signals of the measurement output, observer state, and controller state are quantized through a specific dynamic quantizer scheme. Then, the finite-time dissipative performance of the fuzzy system is rigorously analyzed through the constructed Lyapunov functionals. Conditions and parameters design algorithm for the fuzzy closed-loop systems are established to ensure the specified control performance. In addition, a method involving arbitrary given matrices is employed to decouple the established nonlinear conditions. Finally, the efficacy of the proposed dissipative control strategy is validated through simulation results of the hot strip mill cooling process.