Study of Control Theory Construction Based on Computer Algebra

本研究的目的是在计算机代数的基础上建立新的控制系统理论。1990年，主要工作是研究二维系统理论，取得了以下成果：1）二维传递函数的互质性与二维系统的稳定性：二维传递函数的互质性与一维传递函数的互质性有很大的不同。在此基础上，提出了零互质性、因子互质性等概念。此外，在二维传递函数矩阵的情况下，小互质性是必要的。本文定义了Ω互质性的概念，从而统一讨论了二维系统的互质性和稳定性条件。2）二维系统可镇定补偿器的综合算法：该算法归结为求解双边矩阵方程AX+BY=C。本文在REDUCE的基础上开发了求解矩阵方程的软件包，阐明了二维系统与计算机代数之间存在着深刻的联系。1991年，在上述结果的基础上，将二维系统理论推广到一般的n维系统理论，得到了如下结果。3）n维系统的稳定性：近年来，n-D系统的新稳定性概念-实用稳定性-被引入。建立了实用可镇定补偿器的综合算法。4）n-D系统的动力学仿真：n-D系统的动力学响应非常复杂。然后利用计算机图形学和REDUCE软件开发了用户友好的n-D系统仿真器。

The aim of this study is to construct the new control system theory based on the computer algebra. In 1990, the main work was to research the 2-D system theory and the following results were obtained.1) Coprimeness of 2-D transferfunction and stability of 2-D system : The coprimeness is quite difference from the one of 1-D transferfunction. Then the concepts such as zero-coprimeness and factor coprimeness should be considered. In addition, minor coprimeness was necessary in the case of 2-d transferfunction matrix. Here the new concept - OMEGA -coprimeness was defined and consequently both coprimeness and stability condition for 2-D system could be discussed uniformly.2) Synthesis algorithms of stabilizable compensator for 2-D system : The algorithm is reduced to solving the bilateral matrix equation AX+BY=C. In this work the software package for the matrix equation has been developed based on REDUCE and it was clarified that there existed the deep relationship between the 2-D system and the computer algebra.In 1991, based on the above results, 2-D system theories were extended to the general n-D system theories and following results were obtained.3) Stability of n-D system : Recently the new stability concept for n-D system, -practical stability- was introduced. Here the synthesis algorithms of the practical stabilizable compensator were established. And the Computer Aided Design system was also developed.4) Dynamical simulation of n-D system : The dynamical response of n-D ststem is very complex. Then the user friendly simulator for n-D system was developed by using computer graphics and REDUCE.