On the Computation of $\pi$-Flat Outputs for Linear Time-Delay Systems

This paper deals with linear time-varying, delay systems. Extensions of the concept of differential flatness \cite{Fliess_95} to this context have been first proposed in \cite{Mounier_95,Fliess_96} (see also \cite{Rudolph_03,Chyzak_05}), by the introduction of $\pi$-flat output. Roughly speaking, it means that every system variable may be expressed as a function of a particular output $y$, a finite number of its time derivatives, time delays, and predictions, the latter resulting from the prediction operator $\pi^{-1}$. We propose a simple and constructive algorithm for the computation of $\pi$-flat outputs based on concepts of polynomial algebra, in particular Smith-Jacobson decomposition of polynomial matrices. Some examples are provided to illustrate the proposed methodology.