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首页> 外文期刊>IEEE Transactions on Automatic Control >Canonical representation of piecewise-polynomial functions with nondegenerate linear-domain partitions
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Canonical representation of piecewise-polynomial functions with nondegenerate linear-domain partitions

机译：具有非简并线性域分区的分段多项式函数的规范表示

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摘要

Piecewise-linear (PWL) functions are a widely used class of nonlinear approximate functions with applications in both mathematics and engineering. As an extension of this function class, piecewise-polynomial (PWP) functions with a linear-domainpartition represents a more general function class than PWL functions, in that all function pieces in a partitioned domain are (instead of hyperplanes) hypersurfaces described by polynomials. Just like PWL functions, the global expression of PWP functions requires a so-called canonical representation, which is meaningful for practical applications. However, such a canonical representation is still unknown. Our study showed that it is not a straightforward extension of the canonical representation of PWLfunctions; instead, it has a more general form than the latter. In this paper, we discuss the canonical representation of PWP functions with nondegenerate linear-domain partitions. A canonical representation formula is derived and a sufficient conditionfor its existence is given. We show that under some degree constraints, the derived canonical formula reduces to the canonical formula of PWL functions. The consistent variation property, which is a sufficient and necessary condition for the canonicalrepresentation of PWL functions, is found to be less important for PWP functions.
机译：分段线性 （PWL） 函数是一类广泛使用的非线性近似函数，在数学和工程中都有应用。作为该函数类的扩展，具有线性域分区的分段多项式 （PWP） 函数表示比 PWL 函数更通用的函数类，因为分区域中的所有函数段都是由多项式描述的（而不是超平面）超曲面。就像PWL函数一样，PWP函数的全局表达式需要所谓的规范表示，这对实际应用很有意义。然而，这样的规范表示仍然是未知的。我们的研究表明，它不是PWL函数规范表示的直接扩展;相反，它比后者具有更一般的形式。在本文中，我们讨论了具有非简并线性域分区的PWP函数的规范表示。推导了一个规范表示公式，并给出了其存在的充分条件。我们表明，在一定程度的约束下，推导的规范公式简化为PWL函数的规范公式。一致的变分属性是PWL函数规范表示的充分和必要条件，但对于PWP函数来说不那么重要。

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