System theory - A Hilbert space approach

System theory: A Hilbert space approach
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Avraham Feintuch, Richard Sacks | 1982 | ISBN: 0122517504 | English | 300 pages | PDF | 3 MB

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.

In an effort to alleviate these and similar problems encountered in the design of regulators, passive filters, and stochastic systems, the theory of operators defined on a Hilbert resolution space was developed in the mid-1960s. In essence, a Hilbert resolution space is simply a Hilbert space to which a time structure has been axiomatically adjoined, thereby allowing one to define such concepts as causality, stability, memory, and passivity in an operator theoretic setting. The present text is therefore devoted to an exposition of the theory of operators defined on a Hilbert resolution space and the formulation of a theory of system based thereon.