Robust Stability: System-Theoretic Approach

本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多项式）的稳定性（根据严格的Hurwitz性质进行评估）。基础过程不仅证明了Kharitonov最近的结果，而且更重要的是，提供了推广到二元和随后的多元情况的潜力，包括连续时间/空间和离散时间/空间动力系统。这项研究是重要的，因为多维反馈系统已被提出用于各种目的，如迭代图像处理和图像恢复。这种包含反馈回路的图像处理系统有时会在空间和时间上振荡，只有在反馈系统上施加适当的稳定性条件时，才能避免这些不希望的振荡，特别是在实际要求下，特征多项式的系数都是从上到下有界，而不是确定地知道。本研究的第一部分将关注于建立保证多元区间多项式的零集属于共同指定感兴趣的多域的条件。其次，将研究将系统理论方法用于处理区间或“框”型系数不确定性的范围，以适应多项式系数集变化中的依赖关系。最后，将研究第一阶段和可能的第二阶段研究中存在的更一般多域情况下的对应结果，并开始对鲁棒矩阵多项式的稳定性进行研究。总之，本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的稳定性（即那些具有在指定范围内变化的系数不确定性的多项式）。该研究在图像处理和恢复技术方面具有潜在的应用前景。

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.