Motivated by an engineering application in brake control of cable mining elevators, where the dynamics consist of a brake, a shock absorber, a time-varying-length cable, and a cage, we address a theoretical problem of control of a particular class of coupled hyperbolic PDEs sandwiched between a nonlinear ODE on the actuated side and a linear ODE on the opposite side, with a PDE domain that is time-varying. A state-feedback controller entering a single ODE state is designed to exponentially stabilize the overall system through several backstepping transformations. An observer which only uses the boundary values at the actuated side is constructed to recover all the states of the overall system, based on which a “collocated” type output-feedback control system is proposed. The global exponential stability of the closed-loop system, boundedness, and exponential convergence of the controller, are proved via Lyapunov analysis. The performance is investigated via numerical simulation.

中文翻译：

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扩展类夹心双曲线 PDE-ODE 系统的输出反馈控制

受电缆矿用电梯制动控制工程应用的启发，其中动力学由制动器、减震器、时变长度电缆和笼子组成，我们解决了控制特定类耦合的理论问题双曲 PDE 夹在驱动侧的非线性 ODE 和另一侧的线性 ODE 之间，具有随时间变化的 PDE 域。进入单个 ODE 状态的状态反馈控制器旨在通过多个反步变换以指数方式稳定整个系统。构建了一个仅使用驱动侧边界值的观测器来恢复整个系统的所有状态，在此基础上提出了一种“并置”型输出反馈控制系统。闭环系统的全局指数稳定性，有界性，通过李雅普诺夫分析证明了控制器的指数收敛性。通过数值模拟研究性能。