Abstract This paper employs observer-based feedback control technique to discuss the design problem of output feedback fuzzy controllers for a class of nonlinear coupled systems of a parabolic partial differential equation (PDE) and an ordinary differential equation (ODE), where both ODE output and pointwise PDE observation output (i.e., only PDE state information at some specified positions of the spatial domain) are available. Initially, a Takagi-Sugeno (T-S) fuzzy PDE-ODE model is constructed to exactly represent the nonlinear coupled system via the local sector nonlinearity approach. Then, based on the T-S fuzzy PDE-ODE model, a Lunberger-type fuzzy observer is constructed by the ODE output and the pointwise PDE observation output to estimate the nonlinear coupled system state. A fuzzy controller is proposed by the estimated state. Sufficient conditions for existence of the proposed observer-based fuzzy controller are developed such that the exponential stability of the augmented closed-loop system is ensured, and presented in terms of bilinear matrix inequalities (BMIs). A local algorithm via the convex optimization technique is proposed to solve these BMIs. Finally, numerical simulation results are provided to support the proposed design method.

中文翻译：

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非线性抛物线 PDE-ODE 耦合系统的基于观测器的输出反馈模糊控制

摘要 本文采用基于观测器的反馈控制技术，讨论了一类抛物偏微分方程（PDE）和常微分方程（ODE）非线性耦合系统的输出反馈模糊控制器的设计问题，其中 ODE 输出和逐点 PDE 观测输出（即，只有空间域某些指定位置的 PDE 状态信息）可用。最初，通过局部扇区非线性方法构建了一个 Takagi-Sugeno (TS) 模糊 PDE-ODE 模型来精确表示非线性耦合系统。然后，基于TS模糊PDE-ODE模型，通过ODE输出和逐点PDE观测输出构造Lunberger型模糊观测器来估计非线性耦合系统状态。通过估计状态提出模糊控制器。提出的基于观测器的模糊控制器存在的充分条件被开发，以确保增强闭环系统的指数稳定性，并以双线性矩阵不等式 (BMI) 表示。提出了一种通过凸优化技术的局部算法来解决这些 BMI。最后，提供了数值模拟结果来支持所提出的设计方法。