A time-varying sensor delay is inevitable in a mining cable elevator operating at a maximum distance which is more than 2000-m underground, when the sensor signals are transmitted from a cage at the bottom of the cable to a control center on the ground through a set of wireless devices, including emitters and receivers. This sensor delay may degrade the estimation precision of the state observer or even destroy the stability of the closed-loop system. In this brief, we design an observer which can compensate the sensor delay with a time-varying and arbitrary length to estimate the vibration states of the mining cable elevator, including the time-varying-length mining cable and the cage. The vibration dynamics of the mining cable elevator is described by a wave partial differential equation (PDE)-ordinary differential equation (ODE) coupled system on a time-varying domain, and the sensor delay dynamics is represented by a transport PDE on a time-varying domain. A full-order infinite dimensional observer is constructed to estimate the states of such a wave PDE-ODE-transport PDE coupled system where PDEs are on time-varying domains, only using available boundary values without measurements at the connection points of the PDEs and ODE. The exponential stability of the observer error system is proved via Lyapunov analysis. The effectiveness of the proposed observer is verified via numerical simulation.

中文翻译：

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具有时变传感器延迟的PDE建模矿用电缆电梯的状态观测器设计

当传感器信号从电缆底部的机架通过地面传输到地面的控制中心时，在最大距离地下运行的采矿电缆升降机中不可避免地会发生随时间变化的传感器延迟，该距离超过地下2000 m一组无线设备，包括发射器和接收器。这种传感器延迟可能会降低状态观测器的估计精度，甚至破坏闭环系统的稳定性。在本文中，我们设计了一个观察器，该观察器可以补偿时延和任意长度的传感器延迟，以估算采矿缆索电梯的振动状态，包括时变长度的采矿缆索和保持架。时变域上的波偏微分方程（PDE）-普通微分方程（ODE）耦合系统描述了采矿缆索电梯的振动动力学，而传感器时延动力学则由时变域上的传输PDE表示。可变域。构造了一个全阶无穷维观测器，以估计这种波PDE-ODE-传输PDE耦合系统的状态，其中PDE处于时变域，仅使用可用边界值，而无需在PDE和ODE的连接点进行测量。通过Lyapunov分析证明了观测器误差系统的指数稳定性。通过数值模拟验证了提出的观测器的有效性。构造了一个全阶无穷维观测器，以估计这种波PDE-ODE-传输PDE耦合系统的状态，其中PDE在时变域上，仅使用可用边界值，而无需在PDE和ODE的连接点进行测量。通过Lyapunov分析证明了观测器误差系统的指数稳定性。通过数值模拟验证了提出的观测器的有效性。构造了一个全阶无穷维观测器，以估计这种波PDE-ODE-传输PDE耦合系统的状态，其中PDE处于时变域，仅使用可用边界值，而无需在PDE和ODE的连接点进行测量。通过Lyapunov分析证明了观测器误差系统的指数稳定性。通过数值模拟验证了提出的观测器的有效性。