Graph-theoretic approach to symbolic analysis of linear descriptor systems

AbstractContinuous descriptor systems EẋAx+Bu, yCx, where E is a possibly singular matrix, are symbolically analyzed by means of digraphs. Starting with four different digraph characterizations of square matrices and determinants, the author favors the Cauchy-Coates interpretation. Then, an appropriate digraph representation of the matrix pencil (sE−A) is given, which is followed by a digraph interpretation of det(sE−A) and the transfer-function matrix C(sE−A)−1B. Next, a graph-theoretic procedure is derived to reveal a possibly hidden factorizability of the determinant det(sE−A). This is very important for large-scale systems. Finally, as an application of the derived results, an electrical network is analyzed symbolically

Digraph based determination of Jordan block size structure of singular matrix pencils

AbstractThe generic Jordan block sizes corresponding to multiple characteristic roots at zero and at infinity of a singular matrix pencil will be determined graph-theoretically. An application of this technique to detect certain controllability properties of linear time-invariant differential algebraic equations is discussed

On generalized inverses of singular matrix pencils

Linear time-invariant networks are modelled by linear differential-algebraic equations with constant coefficients. These equations can be represented by a matrix pencil. Many publications on this subject are restricted to regular matrix pencils. In particular, the influence of the Weierstrass structure of a regular pencil on the poles of its inverse is well known. In this paper we investigate singular matrix pencils. The relations between the Kronecker structure of a singular matrix pencil and the multiplicity of poles at zero of the Moore–Penrose inverse and the Drazin inverse of the rational matrix are investigated. We present example networks whose circuit equations yield singular matrix pencils