Module Theory and Linear System Theory

Abstract

In a series of seminal papers published between 1960 and 1965 [8–13] (see also [14, 15]), R.E. Kalman layed the foundations of what has since become known as Mathematical System Theory. The cornerstones of Kalman’s theory were the celebrated concepts of Controllability, Observability and (Canonical) realization. The first formal introduction of the concepts of controllability and observability as fundamental structural properties of (linear) systems was made by Kalman in [8, 9] and the canonical realization problem and its relation to controllability and observability was first studied extensively in [11]. The relation between the structure of canonical realizations and that of transfer matrices was investigated by Kalman extensively in [12]. The crucial insights in Kalman’s theory derived from the discovery that the concepts of controllability and observability are linked in an essential way to the system’s structure and that many of the system’s structural features are encoded in its controllable and observable behaviors.