On the discretization of singular systems

Consider the linear time-invariant singular system Eẋ(t) = Ax(t) + Bu(t) (1) where E, A ∈ Rn×n, B ∈ Rm×n ,C ∈ R p×n and D ∈ R p×m . The system is assumed to be regular det [s E − A] = 0. Systems of the above form are usually called singular systems, descriptor systems, generalized state space systems, semistate systems, etc. It is easily seen that when E is non-singular then (1) may be rewritten in a state space form as ẋ(t) = E−1 Ax(t) + E−1 Bu(t) (2)