Robust Stability: System-Theoretic Approach

这项研究涉及一种系统理论方法来确定具有实系数或复系数的区间多项式(即具有在指定范围内独立变化的系数不确定性的区间多项式)的稳定性(根据严格的Hurwitz性质进行评估)。基本过程不仅证明了Kharitonov的最新结果，更重要的是，为推广到连续时间/空间和离散时间/空间动力系统的双变量以及随后的多变量情况提供了可能性。这项研究很重要，因为多维反馈系统已经被提出用于各种目的，如迭代图像处理和图像恢复。这种包含反馈环路的图像处理系统有时已知在空间和时间上振荡，只有在对反馈系统施加适当的稳定性条件的情况下才能避免这些不希望的振荡，特别是在满足特征多项式的系数都是上下有界的实际要求的情况下，而不是确定地知道。这项研究的第一部分将致力于建立条件，以保证多元区间多项式的零集属于共同指定的感兴趣的多域。其次，将研究为处理区间或“盒式”系数不确定性而开发的系统理论方法的适用范围，以适应允许多项式系数集的变化中的相关性的情况。最后，将研究在更一般的多域情况下存在的第一和可能第二阶段的研究结果的对应结果，并将开始研究鲁棒矩阵多项式的稳定性。总之，这项研究的目的是发展一种系统理论方法，允许确定区间多项式(即具有在指定范围内变化的系数不确定性的那些)的稳定性。该研究对图像处理和复原技术具有潜在的应用价值。

This research concerns a system-theoretic approach to determine the stability (assessed in terms of the strict Hurwitz property) of interval polynomials (i.e. those having coefficient uncertainties which vary independently over specified ranges) having real or complex coefficients. The underlying procedure not only proves the recent results of Kharitonov but, more importantly, offer potentialities for generalization to the bivariate and, subsequently, the multivariate cases for both continuous-time/space and discrete-time/space dynamical systems. This research is important because multidimensional feedback systems have been proposed for various purposes such as iterative image processing and image restoration. Such image processing systems that contain feedback loops are sometimes known to oscillate in space and time and these undesirable oscillations can only be avoided if proper stability conditions are imposed on the feedback systems, especially subject to the practical requirement that the coefficients of the characteristic polynomial are each bounded from above and below instead of being known with certainty. The first part of this research will be concerned with establishing conditions to guarantee that the zero-set of multivariate interval polynomials belong to commonly specified polydomains of interest. Second, the scopes for adapting the system-theoretic approach developed to handle the interval or "boxed" type of coefficient uncertainties to the case when dependencies in the variation of the polynomial coefficient set are allowed, will be investigated. Finally, the counterpart of the results derived in the first and, possibly, the second phase of research which exist in the case of more general polydomains will be investigated and research into the stability of robust matrix polynomials will be initiated. In summary, the aim of this research is to develop a system-theoretic approach allowing the determination of the stability of interval polynomials (i.e. those having coefficient uncertainties which vary over specified ranges). This research has potential applications to image processing and restoration techniques.