First-Order Representations of Linear Systems

1 Introduction.- 2 Rational matrices and vector spaces.- 3 Representations of linear time-invariant systems.- 4 Minimality and transformation groups.- 5 Realization in minimal first-order form.- 6 Structural invariants.- 7 Conclusions.

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.