Singular systems: A new approach in the time domain

A new approach in the time domain is developed for the study of singular linear systems of the form
E\dot{x} = Ax + Bu, y = Cx
with
E
singular. Central to the approach is the fundamental triple
((\alpha E - A)^{-1}E, (\alphaE - A)^{-1}B, C)
with α a real number satisfying det
(\alpha E - A) \neq 0
. Controllability and observability matrices are expressed in terms of the fundamental triple. New tests for impulse controllability and impulse observability are also established. Feedback control problems including pole placement, decoupling, and disturbance localization are studied by use of a modified proportional and derivative feedback of the state in the form of
u = F(\alpha x - \dot{x})+ v
.