Kronecker Summation Method and Multiple Delay Systems

A new procedure is presented for determining the kernel and the offspring hyperplanes for general LTI dynamics with multiple delays. These hyperplanes, as they are very recently introduced in a concept paper (Sipahi and Olgae, 2005), form the basis of the overriding paradigm which is called the "Cluster Treatment of Characteristic Roots (CTCR)". In fact, these two sets of hyperplanes exhaustively represent the locations in the domain of the delays where the system possesses at least one pair of imaginary characteristic roots. To determine these kernel and offspring we use the extraordinary features of "Extended Kronecker Summation" operation in this paper. The end result is that the infinite dimensional problem reduces to a finite dimensional one (and preferably into an eigenvalue problem). Following the procedure described in this paper we are able to shorten the computational time considerably in determining these hyperplanes. We demonstrate these concepts via an example case study which treats a 3-delay system. For this case another perspective, called the "building block", is utilized to display the kernel in 3D space of the "spectral delays"