Pole-zero representation of descriptor systems
An algorithm for computing the poles and zeros of a regular descriptor system is given. The system is described by \[ \begin{cases} E\dot x & = Ax+ bu\\ y & = cx+ du,\end{cases} \] where \(\text{det}(\lambda E- A)\neq 0\). It is assumed that the system is irreducible (that is, it has no finite or infinite uncontrollable and/or unobservable modes). The algorithm for computing the poles consists of transforming the matrix pencil \((A- \lambda E)\) into the form \[ \left[\begin{matrix} A_1- \lambda E_1 & *\\ 0 & D(\lambda)\end{matrix}\right] \] by means of orthogonal transformations, where \(E_1\) is nonsingular, and \(D(\lambda)\) is upper triangular with constant diagonal elements. The poles then are the generalized eigenvalues of the pair \((E_1, A_1)\). The same algorithm applied to the matrix pencil \[ \left[\begin{matrix} A- \lambda E & b\\ c & d\end{matrix} \right] \] gives the zeros. Since the algorithm involves only orthogonal transformations, the results are numerically stable as illustrated by examples.
scientific article; zbMATH DE number 797335
scientific article