Extended Kronecker summation for determining the kernel and offspring of LTI systems with multiple delays

A new concept is presented for determining the kernel and the offspring hypersurfaces for general LTI dynamics with multiple delays. These hypersurfaces, as they are very recently introduced in a concept paper (Sipahi and Olgac 2005), form the basis of the overriding paradigm which is called the "Cluster Treatment of Characteristic Roots (CTCR)". In fact, these two sets of hypersurfaces 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 hypersurfaces. We demonstrate these concepts via some example case studies.