Invariance properties in the root sensitivity of time-delay systems with double imaginary roots

✍ Scribed by Elias Jarlebring; Wim Michiels

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 340 KB
• Volume: 46
• Category: Article
• ISSN: 0005-1098
• DOI: 10.1016/j.automatica.2010.03.014

✦ Synopsis

If iω ∈ iR is an eigenvalue of a time-delay system for the delay τ 0 then iω is also an eigenvalue for the delays τ k := τ 0 + k2π /ω, for any k ∈ Z. We investigate the sensitivity, periodicity and invariance properties of the root iω for the case that iω is a double eigenvalue for some τ k . It turns out that under natural conditions (the condition that the root exhibits the completely regular splitting property if the delay is perturbed), the presence of a double imaginary root iω for some delay τ 0 implies that iω is a simple root for the other delays τ k , k = 0. Moreover, we show how to characterize the root locus around iω. The entire local root locus picture can be completely determined from the square root splitting of the double root. We separate the general picture into two cases depending on the sign of a single scalar constant; the imaginary part of the first coefficient in the square root expansion of the double eigenvalue.