The Language and Fundamental Properties of Behaviors

A technical system as, for example, a machine or an electrical or mechanical network, is typically modeled as a set of equations. The solution set of such a systemof equations is called a behavior,while the individual solutions are the trajectories or signals of the system. In this book, we consider time systems, where the trajectories are functions of a continuous or discrete-time variable which satisfy linear differential or difference equations with constant coefficients. We study behaviors bymeans of their equations or, more precisely, the module of their equations, which allows an algebraic characterization of behavior properties. This approach requires a sufficiently close relationship between the (analytical) signals on the one hand and the (algebraical) equationmodules on the other hand. If the signal space is an injective cogenerator over the ring of operators-a property that is satisfied in all standard cases-such a close relationship is ensured. In fact, an injective cogenerator signal module leads to a perfect duality between the category of systems or behaviors and the category of finitely generated modules with distinguished sets of generators. The language and the mathematical framework introduced here forms the fundament for the rest of the book.