Control-Affine Systems Compatible with the Multi-Chained Form and Their X-Maximal Flatness

In this paper, we introduce the concept of x-maximal flatness. A control system is x-maximally flat if the number of new states gained by each successive derivation of the flat output is the largest possible. Firstly, we show that the only control-linear systems that are xmaximally flat are those that are static feedback equivalent to the m-chained form. Secondly, we generalize that result from control-linear systems to control-affine systems whose control-linear subsystem is static feedback equivalent to the m-chained form. We prove that they are x-maximally flat if and only if the drift exhibits a triangular form compatible with the m-chained form (and recently characterized in [1] and [2]). We also show that if we skip the assumption of the x-maximal flatness, the latter condition is not necessary for x-flatness of control-affine system whose associated controllinear subsystem is static feedback equivalent to the mchained form.