具有$n$状态和$n-1$控制的平坦仿射系统的奇异性
On Singularities of Flat Affine Systems With $n$ States and $n-1$ Controls
摘要：
我们研究了一套平坦的仿射系统的固有奇点$n-1$控制和$n$状态使用的概念，以前介绍的作者Lie-Bäcklund图集。为了这个目的，我们证明了两个容易计算的充分条件，以构建平坦的输出作为一组独立的首次积分的向量场的分布，第一个在一般的情况下，即在一个点的附近，其中的$n-1$控制向量场是独立的，和第二个在退化点，其中$p-1$控制向量场是依赖于$n-p$其他，与$p>1$。我们表明，一组固有的奇点包括一组点的系统不满足强可达秩条件，并包括在一组分布的向量场，介绍了在一般情况下，是奇异的。我们总结了这一分析的三个例子，明显的奇异性的平面系统的通用和非通用退化的情况下。
英文摘要：
We study the set of intrinsic singularities of flat affine systems with $n-1$ controls and $n$ states using the notion of Lie-Bäcklund atlas, previously introduced by the authors. For this purpose, we prove two easily computable sufficient conditions to construct flat outputs as a set of independent first integrals of distributions of vector fields, the first one in a generic case, namely in a neighborhood of a point where the $n-1$ control vector fields are independent, and the second one at a degenerate point where $p-1$ control vector fields are dependent of the $n-p$ others, with $p>1$. We show that the set of intrinsic singularities includes the set of points where the system does not satisfy the strong accessibility rank condition and is included in the set where the distribution of vector fields, introduced in the generic case, is singular. We conclude this analysis by three examples of apparent singularites of flat systems in generic and non generic degenerate cases.
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