State-Observer Design of a PDE-Modeled Mining Cable Elevator With Time-Varying Sensor Delays

摘要

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.