Over the past decade a new approach has been introduced to the theory of systems representations. Introduced by Jan C. Willems, it is called the "behavioural" approach. One of its main features is that it is well suited for modelling interconnections of systems. In this book, the author develops representation theory from a behavioural point of view and focuses on various types of ("generalized state space") first-order representations that commonly arise in the process of modelling. It covers minimality, derives transformation groups and offers realization methods that lead directly to minimal realizations. The book further presents generalized notions of controllability indices and observability indices and gives methods to calculate these indices. The book includes a procedure to derive a standard state space description from a general first-order representation. It fortifies the reader's understanding with basic examples from electrical networks and mechanics.
Mathematicians and control engineers doing research on systems that are linear, time-invariant, deterministic, and finite dimensional should find this book a firm basis for understanding both the theory and applications of this behavioural approach.
Part 1 Rational matrices and rational vector spaces: algebraic preliminaries; Euclidean domains of rational functions; pole/zero structure of a rational matrix; Wiener-Hopf structure of a rational matrix; minimal basis of a rational vector space; preliminary results for matrix pencils. Part 2 Representations of linear time-invariant systems: dynamical systems; AR representations; ARMA representations; first-order representations; systems with split external variables. Part 3 Minimality and transformation groups: minimality of a P representation; minimality of a D representation; minimality of a DZ representation; minimality of a DP representation; transformation groups. Part 4 Realization in minimal first-order form: realization in pencil form - the abstract procedure; the pencil realization in terms of a discrete-time behaviour; choosing bases; connections with the Fuhrmann realization. Part 5 Structural invariants: observability indices; controllability indices; the input-output structure.