On the strong H2 norm of differential algebraic systems with multiple delays: finiteness criteria, regularization and computation

The H2 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 paper, it might become infinite as a result of infinitesimal changes to the delay parameters. We first introduce the notion of strong H2 norm of semi-explicit DDAEs, a robustified measure that takes into account delay perturbations, and we analyze its properties. Next, we derive necessary and sufficient finiteness criteria for the strong H2 norm, in terms of a frequency sweeping test over a hypercube, and in terms of a finite number of equalities involving multi-dimensional powers of a finite set of matrices. As the main contribution, we present a strengthened, sufficient, condition for finiteness of the strong H2 norm, along with an algorithm for checking it, which has significantly better scalability properties in terms of both the dimension of the system 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 to the DDAE. The latter is instrumental to a novel regularization procedure that allows to transform the DDAE to a neutral type system with the same transfer matrix, without any need for differentiation of inputs or outputs. As we illustrate, this transformation enables for instance to compute the strong H2 norm using an established approach grounded in Lyapunov matrices. Finally, we investigate the conservatism of the sufficient finiteness condition.We show by a counterexample that the condition is in general not necessary, inducing open problems, but we also list several classes of DDAEs for which it is necessary and sufficient