Robust Stability: System-Theoretic Approach

这项研究涉及一个系统理论的方法来确定 的稳定性（根据严格的Hurwitz性质评估） 区间多项式（即具有系数不确定性的那些 其在特定范围内独立变化）具有真实的或复数 系数其背后的程序不仅证明了 Kharitonov的结果，但更重要的是，提供了潜力， 推广到二元，随后，多元 连续时间/空间和离散时间/空间动力学的情况 系统. 这项研究很重要，因为多维反馈 已经提出了用于各种目的的系统 图像处理和图像恢复。 这样的图像处理系统 包含反馈回路的系统有时会在空间中振荡 只有在以下情况下才能避免这些不希望的振荡 对反馈系统施加适当的稳定性条件， 特别是在实际要求下， 的特征多项式分别从上到下有界 而不是被确定地知道。 第一部分 研究将涉及创造条件， 多元区间多项式的零集属于 通常指定的感兴趣的多域。 第二，范围 采用系统理论方法来处理 区间或“盒装”类型的系数不确定性的情况下， 多项式系数集的变化的依赖性是 如果允许，将被调查。 最后， 第一阶段和可能的第二阶段取得的成果， 在更一般的多域情况下存在的研究将是 鲁棒矩阵稳定性的研究 将启动多项式。总之，本研究的目的是 开发一种系统理论方法， 区间多项式（即具有系数 在指定范围内变化的不确定性）。 本研究 图像处理和恢复技术的潜在应用。

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.