In this work, the fuzzy observer-based control problem is investigated for a class of nonlinear coupled systems, which consists of a hyperbolic partial differential equation (PDE) containing nonlinearities and a nonlinear ordinary differential equation (ODE). The nonlinear coupled system is represented as a Takagi–Sugeno (T–S) fuzzy coupled hyperbolic PDE-ODE model. Based on the T–S fuzzy model, a novel Lyapunov functional approach is proposed to design a fuzzy observer based control strategy. More specifically, a fuzzy observer is presented to estimate the state variables of the fuzzy coupled PDE-ODE system with the measurements of the PDE, and the exponential convergence of the observer error is proved. Then, a fuzzy controller is given utilizing the estimated states as feedback variables, and it is proved that the evolution profiles of the PDE and the trajectory of the ODE in the closed-loop fuzzy system converge exponentially to the desired values, respectively. The sufficient existence conditions of the fuzzy observer based controller are formulated in terms of a set of space differential linear matrix inequalities (SDLMIs). A recursive algorithm based on the finite-difference approximation and the linear matrix inequality techniques are provided to solve the SDLMIs. Finally, the results are applied to case study of a predator–prey system, and the simulations are performed to illustrate the effectiveness of the proposed observer based control law.

中文翻译：

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非线性耦合双曲 PDE-ODE 系统的基于模糊观测器的控制


在这项工作中，研究了一类非线性耦合系统的基于模糊观测器的控制问题，该系统由包含非线性的双曲偏微分方程（PDE）和非线性常微分方程（ODE）组成。非线性耦合系统表示为 Takagi–Sugeno (T–S) 模糊耦合双曲 PDE-ODE 模型。基于T-S模糊模型，提出了一种新的Lyapunov泛函方法来设计基于模糊观测器的控制策略。更具体地说，提出了一种模糊观测器，用偏微分方程的测量来估计模糊耦合PDE-ODE系统的状态变量，并证明了观测器误差的指数收敛性。然后，利用估计状态作为反馈变量给出模糊控制器，并证明闭环模糊系统中偏微分方程的演化曲线和常微分方程的轨迹分别指数收敛于期望值。基于模糊观测器的控制器的充分存在条件用一组空间微分线性矩阵不等式（SDLMI）来表述。提供了一种基于有限差分近似和线性矩阵不等式技术的递归算法来求解SDLMI。最后，将结果应用于捕食者-被捕食者系统的案例研究，并进行模拟以说明所提出的基于观察者的控制律的有效性。