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 modulebehavior 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 retranslated 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 torsionfree. This concept can be refined by the socalled
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 onedimensional 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.
