Pole-zero representation and transfer function of descriptor systems

Concerns the problem of pole-zero representation of linear time-invariant generalized state space or descriptor systems described by Edx(t)/dt=Ax(t)+bu(t), y(t)=cx(t)+du(t) where x(t)∈Rⁿ, u(t), y(t)∈R and det(λE-A)≠0, i.e., the pencil (λE-A) is regular. The transfer function of this system is G(X)=c(λE-A)_-1b+d. If the descriptor matrix E has full rank, the system is nonsingular, otherwise it is singular. If the system is nonsingular, theoretically, we can obtain an equivalent state space realization by premultiplying the state equation by E-1. Once we have this 4-tuple, we can easily obtain its pole-zero representation. However, obtaining this by first computing the transfer function can be numerically quite sensitive. A small perturbation in coefficient of the transfer function can lead to significant loss of accuracy in numerical computation of poles and/or zeros. A pole-zero representation algorithm was proposed by Varga (1989). The present approach has several features that make it more efficient and reliable