A behavioral approach to classic polynomial matrix descriptions

In this note we aim to show how the well-known denitions and results obtained for the classic polynomial matrix descriptions (PMDs) of a dynamic system (Kailath, Linear Systems, 1980; Rosenbrock, State-Space and Multivariable Theory, 1970; Wolovich, Linear Multivariable Systems, 1974) can be seen as particular instances of the analogous denitions and results obtained within the more general behavioral setting. More precisely, PMDs of dynamic systems correspond to a class of (linear time-invariant) dynamic systems that we will call \extended state-space models". For these models, internal stability as well as the structural properties can be dened and fully characterized, by applying the denitions and criteria introduced within the behavioral approach. As a result, we will obtain the main results of the PMD's approach. Also, the control problem and the observer design problem are here stated, in behavioral terms, and solved. Finally, the introduction of the dual system of an extended state-space model completes the analysis.