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Technical Brief

# Differential Quadrature Method for Stability and Sensitivity Analysis of Neutral Delay Differential Systems

[+] Author and Article Information
Wei Dong

State Key Laboratory of 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,
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,
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,
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 25, 2015; final manuscript received October 26, 2016; published online February 13, 2017. Assoc. Editor: Yang Shi.

J. Dyn. Sys., Meas., Control 139(4), 044504 (Feb 13, 2017) (7 pages) Paper No: DS-15-1589; doi: 10.1115/1.4035167 History: Received November 25, 2015; Revised October 26, 2016

## Abstract

This work develops a computationally efficient stability analysis method for the neutral delay differential systems. This method can be also conveniently applied for the optimal parameter tuning of related control systems. To facilitate this development, at each sampling grid point, the time derivative of the concerned differential system is first estimated by the differential quadrature method (DQM). The neutral delay differential system is then discretized as numbers of algebraic equations in the concerned duration. By combining the obtained discretized algebraic equations, the transition matrix of the two adjacent delay time durations can be explicitly established. Subsequently, the stability boundary is estimated, and the optimal parameters for the controller design are evaluated by searching the global minimum of the spectral radius of the transition matrix. In order to solve such optimization problems with the gradient descent algorithms, this work also analytically formulates the gradient of spectral radius of transition matrix with respect to the concerned parameters. In addition, a strong stability criterion is introduced to ensure better robustness. Finally, the proposed method is extensively verified by numeric examples, and the proposed differential quadrature method demonstrates good accuracy in both parameter tuning and stability region estimation for the neutral delay differential systems.

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## References

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## Figures

Fig. 3

The contour of the real part of the rightmost poles corresponding to varying α and h

Fig. 4

Stability region of the system (34)

Fig. 5

The stability region of the system (34) with varying ka

Fig. 1

The contour of the spectral radius corresponding to varying α and h

Fig. 6

The optimal control gains in the parametric space

Fig. 7

The comparative experiments with the optimal control gains

Fig. 2

The response of system (32) when varying α and h

Fig. 8

The stability region determined by the semidiscretization method with different number of grid points (a) n = 5, (b) n = 20, and (c) n = 40

Fig. 9

The stability region determined by the DQM with different number of grid points (a) n = 5, (b) n = 10, and (c) n = 20

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