Extended kronecker summation for cluster treatment of LTI systems with multiple delays

A new procedure is presented for determining the kernel and the offspring hypersurfaces for general linear time invariant (LTI) dynamics with multiple delays. These hypersurfaces, as they have very recently been introduced in a concept paper [R. Sipahi and N. Olgac, Automatica, 41 (2005), pp. 1413-1422], 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 the kernel and offspring we use the extraordinary features of the "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. One of the examples treats a 3-delay system. For this case another interesting perspective, called the "building block," is also utilized to display the kernel in three-dimensional space in the domain of "spectral delays."