An improved finiteness test and a systematic procedure to compute the strong [formula omitted] norm of differential algebraic systems with multiple delays

Abstract

We study the strong ℋ2 norm of systems modeled by semi-explicit Delay Differential Algebraic Equations (DDAEs). We recall that the finiteness of the strong ℋ2 norm is linked to an algebraic decision problem that can be solved by checking a finite numbers of equalities. We first improve the verification of the finiteness condition. In particular, the complexity of our new condition removes a dependency on the number of delays. We also show that, without imposing further conditions on the system, the number of checks cannot be further reduced. The methodology relies on interpreting the verification of the finiteness conditions in terms of a Polynomial Identity Testing problem. Second we show, in a constructive way, that if the strong ℋ2 norm is finite, the system can always be transformed into a regular neutral-type system with the same ℋ2 norm, without derivatives in the input or in the output equations. This result closes a gap in the literature as such a transformation was known to exist only under additional assumptions on the system. The transformation enables the computation of the strong ℋ2 norm using delay Lyapunov matrices. Illustrative examples are provided throughout the paper.