This paper presents an accurate and computationally efficient time-domain design method for the proportional–integral–derivative (PID) control of first-order and second-order plants in the presence of discrete time delays. As time delays would generally deteriorate the achievable performance of the PID controllers, their effects should be thoroughly considered in the controller design and parameter tuning process. This paper is thereby motivated to propose a time-domain semi-analytical method for the parameter tuning and stability analysis of PID controllers of the time-delay systems. To facilitate this development, the transfer functions of the investigated plants associated with the PID controllers are first rewritten as linear periodic delayed differential equations (DDEs) in state-space form. Then, the differential quadrature method (DQM) is adopted to estimate the time derivative of the state-space function at each sampling grid point within a duration of the time delay by the weighted linear sum of the function values over the whole sampling grid points. In this way, the DDEs in the time-delay duration are discretized as a series of algebraic equations, and the transition matrix can be obtained by combining these discretized algebraic equations. Thereafter, the stability boundary can be determined and the optimal control gains are obtained by minimizing the largest absolute eigenvalue of the transition matrix. As the minimum problems are commonly solved by the gradient descent approaches, the analytical form of the gradient of the largest absolute eigenvalue of transition matrix with respect to the control gains is explicitly presented. Finally, extensive numeric examples are provided, and the proposed DQM is proven to be an accurate and computationally efficient way to tune the optimal control gains and estimate the stability region in the control gain space.
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October 2015
Research-Article
Optimal Proportional–Integral–Derivative Control of Time-Delay Systems Using the Differential Quadrature Method
Wei Dong,
Wei Dong
State Key Laboratory of
Mechanical System and Vibration,
School of Mechanical Engineering,
e-mail: chengquess@sjtu.edu.cn
Mechanical System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University
,Shanghai 200240
, China
e-mail: chengquess@sjtu.edu.cn
Search for other works by this author on:
Ye Ding,
Ye Ding
1
State Key Laboratory of
Mechanical System and Vibration,
School of Mechanical Engineering,
e-mail: y.ding@sjtu.edu.cn
Mechanical System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University
,Shanghai 200240
, China
e-mail: y.ding@sjtu.edu.cn
1Corresponding author.
Search for other works by this author on:
Xiangyang Zhu,
Xiangyang Zhu
State Key Laboratory of
Mechanical System and Vibration,
School of Mechanical Engineering,
e-mail: mexyzhu@sjtu.edu.cn
Mechanical System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University
,Shanghai 200240
, China
e-mail: mexyzhu@sjtu.edu.cn
Search for other works by this author on:
Han Ding
Han Ding
State Key Laboratory of
Mechanical System and Vibration,
School of Mechanical Engineering,
e-mail: hding@sjtu.edu.cn
Mechanical System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University
,Shanghai 200240
, China
e-mail: hding@sjtu.edu.cn
Search for other works by this author on:
Wei Dong
State Key Laboratory of
Mechanical System and Vibration,
School of Mechanical Engineering,
e-mail: chengquess@sjtu.edu.cn
Mechanical System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University
,Shanghai 200240
, China
e-mail: chengquess@sjtu.edu.cn
Ye Ding
State Key Laboratory of
Mechanical System and Vibration,
School of Mechanical Engineering,
e-mail: y.ding@sjtu.edu.cn
Mechanical System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University
,Shanghai 200240
, China
e-mail: y.ding@sjtu.edu.cn
Xiangyang Zhu
State Key Laboratory of
Mechanical System and Vibration,
School of Mechanical Engineering,
e-mail: mexyzhu@sjtu.edu.cn
Mechanical System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University
,Shanghai 200240
, China
e-mail: mexyzhu@sjtu.edu.cn
Han Ding
State Key Laboratory of
Mechanical System and Vibration,
School of Mechanical Engineering,
e-mail: hding@sjtu.edu.cn
Mechanical System and Vibration,
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 November 24, 2014; final manuscript received June 1, 2015; published online July 14, 2015. Assoc. Editor: M. Porfiri.
J. Dyn. Sys., Meas., Control. Oct 2015, 137(10): 101005 (8 pages)
Published Online: October 1, 2015
Article history
Received:
November 24, 2014
Revision Received:
June 1, 2015
Online:
July 14, 2015
Citation
Dong, W., Ding, Y., Zhu, X., and Ding, H. (October 1, 2015). "Optimal Proportional–Integral–Derivative Control of Time-Delay Systems Using the Differential Quadrature Method." ASME. J. Dyn. Sys., Meas., Control. October 2015; 137(10): 101005. https://doi.org/10.1115/1.4030783
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