On the Linearization of Flat Multi-Input Systems via Prolongations

We examine when differentially flat nonlinear control systems with more than two inputs can be rendered static feedback linearizable by a minimal number of prolongations of suitably chosen inputs after applying a static input transformation. We derive sufficient conditions that guarantee such prolongations yield a static feedback linearizable system. For $(x,u)$-flat two-input systems, prior work established precise links between the relative degrees, the highest derivative orders occurring in the flat parameterization, and the minimal dimension of a linearizing dynamic extension, leading to necessary and sufficient criteria for flatness of systems that become static feedback linearizable after at most two prolongations of such suitably chosen inputs. Building on the structure of the time derivatives of a flat output, this work extends this analysis to systems with three inputs.