On the Strong $\mathcal {H}_2$ Norm of Differential Algebraic Systems With Multiple Delays: Finiteness Criteria, Regularization, and Computation

Abstract

The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_2$</tex-math></inline-formula> norm of an exponentially stable system described by delay differential algebraic equations (DDAEs) might be infinite due to the existence of hidden feedthrough terms and, as shown in this article, it might become infinite as a result of infinitesimal changes to the delay parameters. We introduce the notion of strong <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_2$</tex-math></inline-formula> norm of semi-explicit DDAEs, a robustified measure that considers delay perturbations, and we analyze its properties. We derive necessary and sufficient finiteness criteria for the strong <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_2$</tex-math></inline-formula> norm in terms of a finite number of equalities involving multidimensional powers of a finite set of matrices. As the main contribution, we present a strengthened, sufficient, condition for finiteness of the strong <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_2$</tex-math></inline-formula> norm, along with an algorithm for checking it, which has significantly better scalability properties in terms of both the system dimension and the number of delays. We show that the satisfaction of the novel condition is equivalent to the existence of a simultaneous block triangularization of the matrices of the delay difference equation associated with the DDAE. The latter is instrumental to a novel regularization procedure that allows us to transform the DDAE to a neutral-type system with the same transfer matrix, without any need for differentiation of inputs or outputs. This transformation enables computing of the strong <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_2$</tex-math></inline-formula> norm using Lyapunov matrices. Finally, we show by a counterexample that the strengthened condition is, in general, not necessary, inducing open problems, but we also list several classes of DDAEs for which it is necessary and sufficient.