### D.11.1 Control theory background

Control systems are usually described by differential (or difference) equations, but their properties of interest are most naturally expressed in terms of the system trajectories (the set of all solutions to the equations). This is formalized by the notion of the system behavior. On the other hand, the manipulation of linear system equations can be formalized using algebra, more precisely module theory. The relationship between modules and behaviors is very rich and leads to deep results on system structure.

The key to the module-behavior correspondence is a property of some signal spaces that are modules over the ring of differential (or difference) operators, namely, the injective cogenerator property. This property makes it possible to translate any statement on the solution spaces that can be expressed in terms of images and kernels, to an equivalent statement on the modules. Thus analytic properties can be identified with algebraic properties, and conversely, the results of manipulating the modules using computer algebra can be re-translated and interpreted using the language of systems theory. This duality (algebraic analysis) is widely used in behavioral systems and control theory today.

For instance, a system is controllable (a fundamental property for any control system) if and only if the associated module is torsion-free. This concept can be refined by the so-called controllability degrees. The strongest form of controllability (flatness) corresponds to a projective (or even free) module.

Controllability means that one can switch from one system trajectory to another without violating the system law (concatenation of trajectories). For one-dimensional systems (ODE) that evolve in time, this is usually interpreted as switching from a given past trajectory to a desired future trajectory. Thus the system can be forced to behave in an arbitrarily prescribed way.

The extreme case opposed to controllability is autonomy: autonomous systems evolve independently according to their law, without being influenceable from the outside. Again, the property can be refined in terms of autonomy degrees.

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