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Research Papers

Robust Control and Time-Domain Specifications for Systems of Delay Differential Equations via Eigenvalue Assignment

[+] Author and Article Information
Sun Yi1

Department of Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109-2125syjo@umich.edu

Patrick W. Nelson

Center for Computational Medicine and Biology, University of Michigan, 100 Washtenaw Avenue, Ann Arbor, MI 48109-2218pwn@umich.edu

A. Galip Ulsoy

Department of Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109-2125ulsoy@umich.edu

1

Corresponding author.

J. Dyn. Sys., Meas., Control 132(3), 031003 (Apr 14, 2010) (7 pages) doi:10.1115/1.4001339 History: Received July 01, 2008; Revised February 12, 2010; Published April 14, 2010; Online April 14, 2010

An approach to eigenvalue assignment for systems of linear time-invariant (LTI) delay differential equations (DDEs), based upon the solution in terms of the matrix Lambert W function, is applied to the problem of robust control design for perturbed LTI systems of DDEs, and to the problem of time-domain response specifications. Robust stability of the closed-loop system can be achieved through eigenvalue assignment combined with the real stability radius concept. For a LTI system of DDEs with a single delay, which has an infinite number of eigenvalues, the recently developed Lambert W function-based approach is used to assign a dominant subset of them, which has not been previously feasible. Also, an approach to time-domain specifications for the transient response of systems of DDEs is developed in a way similar to systems of ordinary differential equations using the Lambert W function-based approach.

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Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

Grahic Jump Location
Figure 1

As the eigenvalue moves left, then the stability radius increases consistently, which means improved robustness

Grahic Jump Location
Figure 2

Responses of the system in Eq. 18 with the feedback equation 7 corresponding to the rightmost eigenvalues in Table 3 with different imaginary parts of the rightmost eigenvalues

Grahic Jump Location
Figure 3

Responses of the system in Eq. 18 with the feedback equation 7 corresponding to the rightmost eigenvalues in Table 3 with different real parts of the rightmost eigenvalues

Tables

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