ABSTRACT Minimal realisation problems of odd transfer functions for first-degree (multi-linear) nD single-input single-output discrete systems have been studied, but it has not been well solved. This paper provides a new, different method to solve absolutely minimal realisation problems. By methods of limits and algebraic techniques, without using the symbolic approach by Gröbner basis, the requirements of absolutely minimal realisation are transformed into a system of equations represented by the determinants. Since the equations for first-degree 2D systems are solvable by quadratic equations and the conditions for higher-dimensional realisations can be expressed by the results of 2D systems, the absolutely minimal realisations for nD systems can be found by using the realisations of n(n − 1)/2 2D systems. Furthermore, the conditions for existence and construction of the absolutely minimal realisation for the lack of items and not missing two cases are derived from the Pfaffian function of the skew-symmetric matrix. Finally, two numerical examples for 3D and 4D systems are presented to illustrate the basic ideas as well as the effectiveness of the proposed procedure.

中文翻译：

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一阶 nD 系统奇数传递函数的最小实现

摘要 一阶（多重线性）nD 单输入单输出离散系统的奇数传递函数的最小实现问题已经被研究过，但没有得到很好的解决。本文提供了一种新的、不同的方法来解决绝对最小的实现问题。通过极限方法和代数技术，不使用格罗布纳基的符号方法，将绝对极小实现的要求转化为由行列式表示的方程组。由于一阶二维系统的方程可以用二次方程求解，并且高维实现的条件可以用二维系统的结果表示，因此可以通过使用 n(n) 的实现来找到 nD 系统的绝对最小实现− 1)/2 个二维系统。此外，从偏对称矩阵的Pfaffian函数推导出项缺失和不缺失两种情况的绝对最小实现的存在和构造条件。最后，给出了 3D 和 4D 系统的两个数值例子来说明基本思想以及所提出程序的有效性。