Our aim will be to pose questions if the techniques and results of geometric control theory developed in the 80-ties and 90-ties of the last century can be helpful in understanding some questions on the signals (curves) studied recently by the use of their signatures, and vice versa.

Signature of a real, parametrized curve $c:[0,T] \to R^n$ is a formal power series of noncommutative variables or a sequence of $n$-tensors $T_n(c)$ over $R^n$ given by iterated integrals of $c$. It defines the curve uniquely up to reparametrizations.

We will show how a canonical problem of nonlinear control theory, called the realization problem, was solved without, and then by using formal noncommutative calculus. The problem consists of deciding if a nonlinear causal input-output map (describing a black box system) can be realized as a nonlinear polydynamical (control) system on a differentiable manifold. The minimal manifold (the state space of the system) is unique up to a diffeomorphism (H. Sussmann 1977), if it exists. Necessary and sufficient conditions for the existence were given by this author (1980), with an earlier partial result of R.W. Brockett who used Volterra series. An approach using formal power series of noncommuting variables was proposed by M. Fliess, modifying the famous K.T. Chen's series proposed in the 50-ties.