Survey paper: A survey of some recent results in linear multivariable feedback theory

The theory of controllability and observability has been developed, one might almost say reluctantly, in response to problems generated by technological science, especially in areas related to control, communication, and computers. It seems that the first conscious steps to formalize these matters as a separate area of (system-theoretic or mathematical) research were undertaken only as late as 1959, by KALMAN [1960b-c]. There have been, however, many scattered results before this time (see Section 12 for some historical comments and references), and one might confidently assert today that some of the main results have been discovered, more or less independently, in every country which has reached an advanced stage of “development” and it is certain that these same results will be rediscovered again in still more places as other countries progress on the road to development.

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Lectures on Controllability and Observability

Close relationships are established between convolutional codes and zero-state automata and between cyclic codes and zero-input automata. Furthermore, techniques of automata theory and continuous system theory are used to elaborate on the coding problem; and approaches from coding and automata are used to establish and interpret typical structural conditions in continuous systems. The investigation incorporates basic coding concepts into the currently emerging common basis for automata and continuous systems, and it gives explicit examples of the resulting benefits accruing to each of these areas from the others.

Codes, automata, and continuous systems: Explicit interconnections

Some realizations of (M,R)-systems and their interpretation.

Herein we develop category-theoretic tools for understanding network-style diagrammatic languages. The archetypal network-style diagrammatic language is that of electric circuits; other examples include signal flow graphs, Markov processes, automata, Petri nets, chemical reaction networks, and so on. The key feature is that the language is comprised of a number of components with multiple (input/output) terminals, each possibly labelled with some type, that may then be connected together along these terminals to form a larger network. The components form hyperedges between labelled vertices, and so a diagram in this language forms a hypergraph. We formalise the compositional structure by introducing the notion of a hypergraph category. Network-style diagrammatic languages and their semantics thus form hypergraph categories, and semantic interpretation gives a hypergraph functor. The first part of this thesis develops the theory of hypergraph categories. In particular, we introduce the tools of decorated cospans and corelations. Decorated cospans allow straightforward construction of hypergraph categories from diagrammatic languages: the inputs, outputs, and their composition are modelled by the cospans, while the 'decorations' specify the components themselves. Not all hypergraph categories can be constructed, however, through decorated cospans. Decorated corelations are a more powerful version that permits construction of all hypergraph categories and hypergraph functors. These are often useful for constructing the semantic categories of diagrammatic languages and functors from diagrams to the semantics. To illustrate these principles, the second part of this thesis details applications to linear time-invariant dynamical systems and passive linear networks.

Michael A. Arbib

Papers

A Common Framework for Automata Theory and Control Theory