Explicit Symmetries of Strict Feedforward Control Systems

We show that any symmetry of a smooth strict feedforward system is conjugated to a scaling translation and any 1-parameter family of symmetries to a family of scaling translations along the first variable. We compute explicitly those symmetries by finding the conjugating diffeomorphism. We deduce, in accordance with our previous work, that a smooth system is feedback equivalent to a strict feedforward form if and only if it gives rise to a sequence of systems, such that each element of the sequence, firstly, possesses an infinitesimal symmetry whose flow is conjugated to a 1-parameter families of scaling translations and, secondly, it is the factor system of the preceding one, that is, is reduced from the preceding one by its symmetry. We illustrate our results by computing the symmetries of the Cart-Pole system