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δ-Freeness of a class of linear systems

Starting from a simple example of linear delayed system (with 2 inputs and 2 outputs) commonly used in process control, we show that, as for flat systems (see [1]). an explicit parametrization of all the trajectories can be found. Once more this leads to an easy motion planning. More generally speaking, we prove that this property, called δ-freeness (see [2. 4]) is general among higher dimensions linear delayed systems. More theoretically speaking, we use the module framework and consider a linear delayed system as a finitely generated module over the ring R[d/dt,δ], where δ is one or a set of delay operators. We show that this system is δ-free. That is we can find a basis of its corresponding module over the localized ring R[d/dt, δ, δ-1]. An applicable way to exhibit such a basis is explicitly described.