Abstract
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.
Highlights
Linear nonhomogeneous time-invariant difference equations are ubiquitous in many engineering and mathematical fields [1]
We develop a direct method used to solve the linear nonhomogeneous time-invariant difference equation with the same number for inputs and outputs. e z-transformation state of the economic cybernetics statespace difference equation generally is a rational function with the same power for the numerator and the denominator, and the boresome processes involving the long division of both complicated polynomials, the expanded multiplication of all polynomial factors, the differentiation of both complicated polynomials, and the complex derivations of all partial fraction parameters are inevitable for existing traditional methods
We propose a direct method used to solve the linear nonhomogeneous time-invariant difference equation with the same number for inputs and outputs in this study and explicitly express the general form of the solution sequence with shorter time than that of the traditional approaches
Summary
Linear nonhomogeneous time-invariant difference equations are ubiquitous in many engineering and mathematical fields [1] They appear in the engineering theory of discrete-time systems and control theory of discrete-time systems as fundamental models of the discrete-time systems [2,3,4], and discrete-time signal processing as fundamental recurrence equations of sampled signals [5]. In algebraic combinatorics, they are one of the most significant topics, tying different special sequences with their generating functions [6, 7]. Economic cybernetics provides basic theory and more advanced methods for us to study complex economic systems
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