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Research Papers

# Stability Analysis of LTI Systems With Three Independent Delays—A Computationally Efficient Procedure

[+] Author and Article Information
Rifat Sipahi1

Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115rifat@coe.neu.edu

Hassan Fazelinia, Nejat Olgac

Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269-3139

A case where stability transition takes place at $s=0$ may also occur and this can be easily detected. This degenerate case is disregarded in the rest of the paper.

On a desktop computer with 3 GHz Pentium 4 processor and 2 Gbyte of memory.

1

Corresponding author.

J. Dyn. Sys., Meas., Control 131(5), 051013 (Aug 19, 2009) (9 pages) doi:10.1115/1.3117220 History: Received August 25, 2008; Revised January 08, 2009; Published August 19, 2009

## Abstract

A practical numerical procedure is introduced for determining the stability robustness map of a general class of higher order linear time invariant systems with three independent delays, against uncertainties in the delays. The procedure is based on an efficient and exhaustive frequency-sweeping technique within a single loop. This operation results in determination of the complete description of the kernel and the offspring hypersurfaces, which constitute exhaustively the potential stability switching loci in the space of the delays. The new numerical procedure corresponds to the first step in the overarching framework, called the cluster treatment of characteristic roots. The results of this treatment can also be represented in another domain (called the spectral delay space) within a finite dimensional cube called the building block, which is much simpler to view and analyze. The paper also offers several case studies to demonstrate the practicality of the new numerical methodology.

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## Figures

Figure 1

(a) Stability picture, τ3=0.2 s. Stability regions shaded: kernel (thick red) and offspring (thin blue). (b) Stability picture, τ3=0.4 s. Stability regions shaded: kernel (thick red) and offspring (thin blue).

Figure 2

Parametric variation of stability posture for varying τ3: τ3=0.2 (red), τ3=0.25 (blue), τ3=0.3 (black), τ3=0.35 (magenta)

Figure 3

Building block (BB) representation of the system in example 1

Figure 4

Cross-sectional outlook of the BB in example 1: kernel (red, thick) and the offspring (blue, thin) for various values of τ3ω

Figure 5

Stability pictures for τ3=0.05 s, 0.07 s, and 0.09 s. Stable regions shaded: kernel (thick red) and offspring (thin blue)

Figure 6

Parametric variation of stability posture for varying τ3: τ3=0.05 (red), τ3=0.055 (blue), τ3=0.060 (black), τ3=0.065 (magenta)

Figure 7

Building block (BB) representation of the system in example 2

Figure 8

Cross-sectional outlook of the BB in example 2: kernel (red, thick) and the offspring (blue, thin) for various values of τ3ω

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