Characteristics of LTI descriptor systems

Linear time-invariant (LTI) descriptor systems Eẋ = Ax + Bu with regular matrix pencils λ(E − A) may be separated by an equivalence transformation into a “slow” and a “fast” subsystem. The consistent solution of the fast subsystem can be presented by the input u and its time-derivatives. If these time-derivatives appear explicitly, then the system behaviour is called improper, otherwise it is proper or even strictly proper. This contribution deals with the determination of the related characteristics based on the matrices E, A, B without applying the equivalence transformation: Index k, orders n1, n2 of the subsystems (n1 + n2 = n), (strictly) proper and improper transfer behaviour, degree of improperness of each individual input. These characteristics are calculated by rank conditions of suitable matrices composed of E, A, B. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)