In this study, the authors first develop a direct method used to solve the linear nonhomogeneous time-invariant difference equation with the same number for inputs and outputs. Economic cybernetics is the crystallization for the integration of economics and cybernetics. It analyzes the stability, controllability, and observability of the economic system by establishing a system model and enables people to better understand the characteristics of the economic system and solve economic optimization problems. The economic model generally applies the discrete recurrence difference equation. The significant analytic approach for the difference equation is the z-transformation technique. The z-transformation state of the economic cybernetics state-space difference equation generally is a rational function with the same power for the numerator and the denominator. The proposed approach will take the place of the traditional methods without all annoying procedures involving the long division of some complicated polynomials, the expanded multiplication of many polynomial factors, the differentiation of some complicated polynomials, and the complex derivations of all partial fraction parameters. To highlight the novelty of this research, this study especially applies the proposed theorems originally belonging to engineering to the field of economic applications.

中文翻译：

---

解决经济复杂系统的新方法

在这项研究中，作者首先开发了一种直接方法，用于求解输入和输出具有相同数量的线性非齐次时不变差分方程。经济控制论是经济学和控制论整合的结晶。它通过建立系统模型来分析经济系统的稳定性，可控制性和可观察性，使人们能够更好地理解经济系统的特征并解决经济优化问题。经济模型通常应用离散递归差分方程。差分方程的重要解析方法是z变换技术。该ž控制论状态空间差分方程的-变换状态通常是一个有理的函数，对分子和分母具有相同的幂。所提出的方法将取代传统方法，而无需进行任何烦人的过程，包括复杂的多项式的长除法，许多多项式因数的扩展乘法，某些复杂的多项式的微分以及所有部分分数参数的复杂导数。为了强调这项研究的新颖性，本研究特别将最初属于工程学的拟议定理应用于经济应用领域。