System Theory: a Hilbert Space Approach

Intuitively, a system is a black box whose inputs and outputs are functions of time (or vectors of such functions). As such, a natural model for a system is an operator defined on a function space. This observation and its corollary to the effect that system theory is a subset of operator theory, unfortunately, proved to be the downfall of early researchers in the field. The projection theorem was used to construct optimal controllers that proved to be unrealizable, operator factorizations were used to construct filters that were not causal, and operator invertibility criteria were used to construct feedback systems that were unstable. The difficulty lies in the fact that the operators encountered in system theory are defined on spaces of time functions and, as such, must satisfy a physical realizability (or causality) condition to the effect that the operator cannot predict the future. Although this realizability condition usually takes care of itself in the analysis problems of classical applied mathematics, it must be externally imposed on the synthesis problems that are central to system theory.