Analysis and Computation of the \mathcal 2 Norm of Delay Differential Algebraic Equations

• Published in: IEEE transactions on automatic control Vol. 65; no. 5; pp. 2192 - 2199
• Main Authors: Gomez, Marco A., Michiels, Wim
• Format: Journal Article
• Language: English
• Published: IEEE 01.05.2020
• Subjects: <inline-formula xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> <tex-math notation="LaTeX"> mathcal {H}_2</tex-math> </inline-formula> norm; Controllability; Delay differential algebraic systems; Delays; Differential algebraic equations; Dynamical systems; Lyapunov matrix; Matrix decomposition; Observability
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2019.2938320

We consider a class of dynamical systems described by linear delay differential algebraic equations (DDAEs) called strangeness free, which is broader than the class commonly studied within the control theory field. Two problems arise in the study of the <inline-formula><tex-math notation="LaTeX">\mathcal {H}_2</tex-math></inline-formula> norm of DDAEs: the first one is that it may be infinite even if the system is stable or has no seemingly feedthrough term, and the second one is the computation. In this paper, both problems are addressed. We provide a necessary and sufficient condition for the finiteness of the <inline-formula><tex-math notation="LaTeX">\mathcal {H}_2</tex-math></inline-formula> norm, which is based on controllability and observability properties of the delay-difference part of the system, and we present a formula for computing the <inline-formula><tex-math notation="LaTeX">\mathcal {H}_2</tex-math></inline-formula> norm whenever it is finite, which is obtained by means of a neutral-type system whose transfer matrix is equivalent to the transfer matrix of DDAEs.