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Research Papers

Axial Vibration Suppression in a Partial Differential Equation Model of Ascending Mining Cable Elevator

[+] Author and Article Information
Ji Wang

State Key Laboratory of
Mechanical Transmission, and
College of Automotive, Engineering,
Chongqing University,
Chongqing 400044, China
e-mail: wangji@cqu.edu.cn

Shumon Koga

Department of Mechanical and
Aerospace Engineering,
University of California, San Diego,
La Jolla, CA 92093-0411
e-mail: skoga@ucsd.edu

Yangjun Pi

State Key Laboratory of
Mechanical Transmission, and
College of Automotive, Engineering,
Chongqing University,
Chongqing 400044, China
e-mail: cqpp@cqu.edu.cn

Miroslav Krstic

Fellow ASME
Department of Mechanical and
Aerospace Engineering,
University of California, San Diego,
La Jolla, CA 92093-0411
e-mail: krstic@ucsd.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received October 10, 2017; final manuscript received May 2, 2018; published online June 4, 2018. Assoc. Editor: Davide Spinello.

J. Dyn. Sys., Meas., Control 140(11), 111003 (Jun 04, 2018) (13 pages) Paper No: DS-17-1515; doi: 10.1115/1.4040217 History: Received October 10, 2017; Revised May 02, 2018

Lifting up a cage with miners via a mining cable causes axial vibrations of the cable. These vibration dynamics can be described by a coupled wave partial differential equation-ordinary differential equation (PDE-ODE) system with a Neumann interconnection on a time-varying spatial domain. Such a system is actuated not at the moving cage boundary, but at a separate fixed boundary where a hydraulic actuator acts on a floating sheave. In this paper, an observer-based output-feedback control law for the suppression of the axial vibration in the varying-length mining cable is designed by the backstepping method. The control law is obtained through the estimated distributed vibration displacements constructed via available boundary measurements. The exponential stability of the closed-loop system with the output-feedback control law is shown by Lyapunov analysis. The performance of the proposed controller is investigated via numerical simulation, which illustrates the effective vibration suppression with the fast convergence of the observer error.

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Figures

Grahic Jump Location
Fig. 1

The mining cable elevator: (a) original model and (b) simplified model

Grahic Jump Location
Fig. 2

The hoisting velocity z˙*(t)

Grahic Jump Location
Fig. 3

The open-loop responses of the plant Eqs. (27)(30). The large vibration is caused both at the moving cage and at the midpoint of the cable: (a) the axial vibration at the moving cage and (b) the axial vibration at the midpoint.

Grahic Jump Location
Fig. 4

The closed-loop responses of the plant Eqs. (27)(30) with the PD controller (157) (dashed line) and the proposed state-feedback controller (61). While both controllers achieve the convergence to zero, the proposed controller achieves faster convergence with less overshoot: (a) the axial vibration at the moving cage and (b) the axial vibration at the midpoint of the cable.

Grahic Jump Location
Fig. 5

The responses of the closed-loop system Eqs. (27)(30) and the observer design Eqs. (75)(78) with the output-feedback control law (118). The observer achieves convergence to the actual distributed state, and the associated output-feedback controller retains similar performance to the state-feedback: (a) the observer error of the axial vibration at the midpoint and (b) the axial vibration at the midpoint.

Grahic Jump Location
Fig. 6

The hoisting acceleration

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Fig. 7

The closed-loop responses of the accurate plant Eqs. (11)(13) with Eq. (18) under the PD controller (157) (dashed line) and the proposed output-feedback controller (118): (a) the axial vibration at the moving cage and (b) the axial vibration at the midpoint of the cable

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