Analysis of singular systems using orthogonal functions

The use of orthogonal functions to analyze singular systems is investigated. It is shown that the differential-algebraic system equation may be converted to an algebraic generalized Lyapunov equation that can be solved for the coefficients of
x(t)
in terms of the orthogonal basis functions. This generalized Lyapunov equation may be considered as a "discrete" equation on the slow subspace of the system, and as a "continuous" equation on its fast subspace. Necessary and sufficient conditions for the existence of a unique solution are given in terms of the relative spectrum of the system. A generalized Bartels/Stewart algorithm based on the
QZ
algorithm is presented for its efficient solution. Relations are drawn with the invariant subspaces of the system.