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Research Papers

Eigenvalue Assignment for Control of Time-Delay Systems Via the Generalized Runge–Kutta Method

[+] Author and Article Information
JinBo Niu, LiMin Zhu, Han Ding

State Key Laboratory of Mechanical
System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China

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

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received September 28, 2013; final manuscript received April 18, 2015; published online June 2, 2015. Assoc. Editor: Joseph Beaman.

J. Dyn. Sys., Meas., Control 137(9), 091003 (Sep 01, 2015) (7 pages) Paper No: DS-13-1367; doi: 10.1115/1.4030418 History: Received September 28, 2013; Revised April 18, 2015; Online June 02, 2015

This paper presents an eigenvalue assignment method for the time-delay systems with feedback controllers. A new form of Runge–Kutta algorithm, generalized from the classical fourth-order Runge–Kutta method, is utilized to stabilize the linear delay differential equation (DDE) with a single delay. Pole placement of the DDEs is achieved by assigning the eigenvalue with maximal modulus of the Floquet transition matrix obtained via the generalized Runge–Kutta method (GRKM). The stabilization of the DDEs with feedback controllers is studied from the viewpoint of optimization, i.e., the DDEs are controlled through optimizing the feedback gain matrices with proper optimization techniques. Several numerical cases are provided to illustrate the feasibility of the proposed method for control of linear time-invariant delayed systems as well as periodic-coefficient ones. The proposed method is verified with high computational accuracy and efficiency through comparing with other methods such as the Lambert W function and the semidiscretization method (SDM).

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Figures

Grahic Jump Location
Fig. 1

Resultant rightmost poles corresponding to different computational results of ad in the complex plane with the GRKM

Grahic Jump Location
Fig. 2

Transient responses of DDEs with rightmost poles of different imaginary parts

Grahic Jump Location
Fig. 3

Contour lines of the rightmost poles of Eq. (20) in parameter space (a,ad)

Grahic Jump Location
Fig. 4

Contour lines of desired rightmost characteristic roots of the DDE in Example 3

Grahic Jump Location
Fig. 5

Convergence rate comparisons between the GRKM and the SDM

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