关于微分平坦性的充要条件
On Necessary and Sufficient Conditions for Differential Flatness
摘要：
本文在无穷阶喷流流形的微分几何框架下，在消去输入变量后，用隐式表示法研究微分平坦非线性系统的特征。我们将Fliess et al.（1999）中引入的Lie-Bäcklund等价的概念扩展到这个隐式的上下文，并将注意力集中在与平坦系统相关的Lie-Bäcklund同构上，称为平凡化。它们可以用不定$\ddt$的多项式矩阵局部刻画，其值域等于隐变分系统多项式矩阵的核.这样的多项式矩阵对于计算由所有可能平凡化的微分生成的微分形式的理想是有用的。引入微分形式的强闭理想的概念，证明了平坦性等价于后一理想的强闭性，进而等价于广义活动标架结构方程解的存在性.两个顺序的程序，以有效地计算平坦的输出推导出各种例子和后果。
英文摘要：
This paper is devoted to the characterization of differentially flat nonlinear systems in implicit representation, after elimination of the input variables, in the differential geometric framework of manifolds of jets of infinite order. We extend the notion of Lie-Bäcklund equivalence, introduced in Fliess et al. (1999), to this implicit context and focus attention on Lie-Bäcklund isomorphisms associated to flat systems, called trivializations. They can be locally characterized in terms of polynomial matrices of the indeterminate $\ddt$, whose range is equal to the kernel of the polynomial matrix associated to the implicit variational system. Such polynomial matrices are useful to compute the ideal of differential forms generated by the differentials of all possible trivializations. We introduce the notion of a strongly closed ideal of differential forms, and prove that flatness is equivalent to the strong closedness of the latter ideal, which, in turn, is equivalent to the existence of solutions of the so-called generalized moving frame structure equations. Two sequential procedures to effectively compute flat outputs are deduced and various examples and consequences are presented.
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