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Research Papers

An Experimental Study of Robustness of Multi-Objective Optimal Sliding Mode Control

[+] Author and Article Information
Zhi-Chang Qin

Department of Mechanics,
Tianjin University,
Tianjin 300072, China
e-mail: qinzhichang123@126.com

Fu-Rui Xiong

Department of Mechanics,
Tianjin University,
Tianjin 300072, China
e-mail: xfr90311@gmail.com

Jian-Qiao Sun

Professor
Fellow ASME
School of Engineering,
University of California,
Merced, CA 95343;
Tianjin University,
Tianjin 300072, China
e-mail: jqsun@ucmerced.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received December 7, 2015; final manuscript received April 20, 2016; published online June 2, 2016. Assoc. Editor: Nicole Kessissoglou.

J. Vib. Acoust 138(5), 051008 (Jun 02, 2016) (6 pages) Paper No: VIB-15-1510; doi: 10.1115/1.4033494 History: Received December 07, 2015; Revised April 20, 2016

This paper presents an experimental study of robustness of multi-objective optimal sliding mode control, which is designed in a previous study. Inertial and stiffness uncertainties are introduced to a two degrees-of-freedom (DOF) under-actuated rotary flexible joint system. A randomly selected design from the Pareto set of multi-objective optimal sliding mode controls is used in the experiments. Three indices are introduced to evaluate the performance variation of the tracking control in the presence of uncertainties. We have found that the multi-objective optimal sliding mode control is quite robust against the inertial and stiffness uncertainties in terms of maintaining the stability and delivering satisfactory tracking performance as compared to the control of the nominal system, even when the uncertainty is not a small quantity. Furthermore, we have studied the effect of upper bounds of the model estimation error on the stability of the closed-loop system.

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Figures

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Fig. 1

The nominal rotary flexible joint experimental system

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Fig. 2

The experimental results of sliding mode control of the nominal model. This serves as the baseline for robustness studies. Red dashed line in the upper-left figure is the reference signal.

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Fig. 3

The variations of IAEx1, IAEx2, and Ju with the inertial uncertainty Δm of the rotary arm: blue trilateral line: IAEx1; green dot line: IAEx2; red star line: Ju

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Fig. 4

The experimental results of the tracking control with the largest inertial uncertainty Δm=0.256 kg. Red dashed line in the upper-left figure is the reference signal.

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Fig. 5

The variations of the indices with the inertial uncertainty ΔM of the base module: blue trilateral line: IAEx1; green dot line: IAEx2; red star line: Ju

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Fig. 6

The experimental results of the tracking control with the largest inertial uncertainty of the base module ΔM=0.574 kg. Red dashed line in the upper-left figure is the reference signal.

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Fig. 7

The variations of the indices with the number of elastic bands n, representing the stiffness uncertainty: blue trilateral line: IAEx1; green dot line: IAEx2; red star line: Ju

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Fig. 8

The experimental results of the tracking control when the number of elastic bands on each side is n = 1. Red dashed-line in the upper left figure is the reference signal.

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Fig. 9

The variations of the indices with the number N of added elastic bands on the original springs, representing the stiffness uncertainty: blue trilateral line: IAEx1; green dot line: IAEx2; red star line: Ju

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Fig. 10

The experimental results of the tracking control with the maximum number (N = 10) of added elastic bands on the original springs, representing the stiffness uncertainty. Red dashed line in the upper-left figure is the reference signal.

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Fig. 11

The stable regions in the (x2,x3) plane for four different upper bounds of the model estimation error. Legends (a) to (d) correspond to the cases in Eqs. (20)(23). Blue points: stable points which satisfy Js < 0. Lines represent the results of time domain simulations with different initial conditions. Red line: Stable response starting from the initial conditions x1=0, x2=30, x3=45, and x4=0. Green line: Stable response starting from the initial conditions x1=0, x2=−30, x3=−45, and x4=0. Magenta line: Unstable response starting from the initial conditions x1=0, x2=78.26, x3=0, and x4=0.

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