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Research Papers

Direct Integration Method for Time-Delayed Control of Second-Order Dynamic Systems

[+] Author and Article Information
Zhijie Wen

State Key Laboratory of Mechanical
System and Vibration,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: serowen@sjtu.edu.cn

Ye Ding

State Key Laboratory of Mechanical System
and Vibration,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: y.ding@sjtu.edu.cn

Pinkuan Liu

School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: pkliu@sjtu.edu.cn

Han Ding

School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: hding@sjtu.edu.cn

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received December 30, 2015; final manuscript received November 23, 2016; published online March 22, 2017. Assoc. Editor: Sergey Nersesov.

J. Dyn. Sys., Meas., Control 139(6), 061001 (Mar 22, 2017) (9 pages) Paper No: DS-15-1660; doi: 10.1115/1.4035359 History: Received December 30, 2015; Revised November 23, 2016

A direct integration method (DIM) for time-delayed control (TDC) is proposed in this research. For a second-order dynamic system with time-delayed controllers, a Volterra integral equation of the second kind is used instead of a state derivative equation. With the proposed DIM where matrix exponentials are avoided, semi-analytical representation of the Floquet transition matrix for stability analysis can be derived, the stability region on the parametric space comprising control variables can also be plotted. Within this stability region, optimal control variables are subsequently obtained using a multilevel conjugate gradient optimization method. Further simulation examples demonstrated the superiority of the proposed DIM in terms of computational efficiency and accuracy, as well as the effectiveness of the optimization-based controller design approach.

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Figures

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Fig. 1

The stability region in Example 1 obtained by the DIM (kp = 22.57, number on each contour indicates the corresponding σ value)

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Fig. 5

The two-dimensional (2D) optimization process (kp ≡ 22.57). (a) Full view within the stability region and (b) detailed view of the dashed area showing final results (number on each contour corresponds to |λ|max value).

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Fig. 2

The stability region obtained by the NIM and the DIM (kp = 22.57)

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Fig. 3

Local discretization errors of the NIM and the DIM (kp = 22.57, kr = 20)

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Fig. 4

The stability region in Example 2

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Fig. 6

Responses corresponding to each iteration of the 3D optimization

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