Backstepping-Based Fuzzy Adaptive Stabilization of Reaction-Diffusion Equation With State Constraints
针对级联抛物型偏微分方程-常微分方程系统，结合模糊逻辑与反步法设计控制器，确保状态不越界且系统稳定，仿真验证了有效性。
This article proposes a stabilization scheme for a cascaded parabolic partial differential equation (PDE)-ordinary differential equation (ODE) system with state constraints. To begin, by employing the fuzzy-logic system (FLS) technique for the unknown nonlinear functions in the ODE subsystem, the influence of the nonlinearities is successfully eliminated. Then, the infinite and finite-dimensional backstepping methods are skillfully applied to the design of control schemes. Specifically, the infinite-dimensional backstepping transformation and its inverse are utilized to obtain a new PDE subsystem, which is convenient for control design. Next, based on the new target PDE-ODE cascade system, a novel first-order filter is subtly introduced into each step of the finite-dimensional backstepping-based controller design procedure to avoid the issue of "explosion of complexity." Furthermore, a novel barrier Lyapunov function (BLF) is constructed to guarantee that the states of the ODE subsystem do not violate the constraints. The controller designed in this article ensures that all signals in the closed-loop system are bounded and the states can converge to zero. Finally, a simulation example verifies that the proposed scheme can achieve satisfactory performance.