Multidimensional Systems Design: Novel Directions

当一个一维（1-D）系统的模型被精确地知道时，许多感兴趣的性质，例如稳定性，可以通过分析一个自变量的一组多项式来研究。这种分析的一个困难是多项式的集合可能是无限的，问题是如何把对这个无限集合的研究简化为对一个有限的低阶集合的研究。此外，多项式集中出现的系数通常是不精确已知的，只能给出系数的界。这就提出了有关属性对参数变化的鲁棒性的问题。对于一维系统，这两个问题已经相当好地解决了。本研究的目的是将上述关于一维系统的某些结果推广到二维或三维系统。这样的系统在光学图像处理和机器人等许多工程应用中都很重要。特别是，两个一般的研究任务是系数扰动下n维多项式的性质不变性，多项式的不可约性和一类谱分解。

When a model of a one-dimensional (1-D) system is known precisely, many properties of interest, e.g. stability, can be investigated through the analysis of a set of polynomials in one independent variable. A difficulty in such an analysis is that the set of polynomials may be infinite and the question arises of how to reduce the study of this infinite set to the study of a finite set of low order. Furthermore, the coefficients appearing in the set of polynomials are generally not known precisely and only bounds on the coefficients can be given. This brings up the question of the robustness of the properties of interest to parameter variations. For 1-D systems, these two questions have been reasonably well resolved. The goal of this research is to generalize certain results regarding the above questions for 1-D systems to two- or three-dimensional systems. Such systems are important in a number of engineering applications such as optical image processing and robotics. In particular, the two general research tasks are property invariance of n-dimensional polynomials under coefficient perturbations, and polynomial irreducibility and a type of spectral factorization.