Differential Flatness of Two One-Forms in Arbitrary Number of Variables

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Muruhan Rathinam and Richard M. Murray
Systems and Control Letters, 36:317-326, 1999.

Given a differentially flat system of ODEs, flat outputs that depend only on original variables but not on their derivatives are called zero-flat outputs and systems possessing such outputs are called zero-flat. In this paper we present a theory of zero-flatness for a system of two one-forms in arbitrary number of variables $(t,x^1,\dots,x^N)$. Our approach splits the task of finding zero-flat outputs into two parts. First part involves solving for distributions that satisfy a set of algebraic conditions. If the first part has no solution then the system is not zero-flat. The second part involves finding an integrable distribution from the solution set of the first part. Typically this part involves solving PDEs. Our results are also applicable in determining if a control affine system in $n$ states and $n-2$ controls has flat outputs that depend only on states. We illustrate our method by examples.