On the Causality of Linear Time Invariant Descriptor Systems

Abstract

In this paper questions of the following type will be investigated: ‘which general linear time invariant (l.t.i.) time continuous causal or noncausal system can be approximated or represented by (l.t.i.) time discrete descriptor systems?”, and ‘what happens with the causality or acausality of these systems in the approximation process?” Various answers to these questions will be presented as statements of two theorems. These answers strongly depend on the dimension of the state space of the systems considered. Among these the following statements are of primary interest: Finite dimensional (l.t.i.) time continuous descriptor systems are always causal, whereas finite dimensional (l.t.i.) time discrete descriptor systems may be causal or noncausal. On the other hand, infinite dimensional (l.t.i.) time discrete and time continuous descriptor systems may be causal or noncausal. Each general (l.t.i.) system can be approximated by finite dimensional (l.t.i.) time discrete descriptor systems, and represented (exactly) by an infinite dimensional (l.t.i.) time continuous descriptor system.