Spectral properties of infinite-dimensional closed-loop systems
Spectral properties of infinite-dimensional closed-loop systems
复制标题
无限维闭环系统的谱特性
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
Chengzhong Xu
中科院分区：
文献类型：
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作者：
G. Weiss;Chengzhong Xu
我们考虑由无限维良定线性系统通过输出反馈得到的反馈系统。因此，我们的框架允许无限的控制和观察操作符。我们的目的是研究开环系统、反馈算子K和闭环系统生成器AK的谱(特别是本征值和本征向量)之间的关系。通过包含开环传递函数的“特征方程”，我们给出了位于开环生成器A的预解集中的部分谱σ(AK)的一个有用的刻画。证明了σ(A)的某些点不是AK的特征值，如果输入和输出都是标量的(使得K是一个数)并且K≠为0。我们特别注意当输出反馈算子K是紧算子时的情形。证明在这种情况下，A的本质谱σe(A)是不变的，也就是说，它等于σe(Ak)。一个相关但困难得多的问题是确定在紧输出反馈下保持不变的σ(A)的最大子集。这个集合，我们称之为A的不动谱，严格包含σe(A)。我们给出了不动谱的一个显式刻画，并研究了我们的结果对于通过互连无穷多个恒等式得到的一类适定系统的结果。
We consider feedback systems obtained from infinite-dimensional well-posed linear systems by output feedback. Thus, our framework allows for unbounded control and observation operators. Our aim is to investigate the relationship between the open-loop system, the feedback operator K and the spectrum (in particular, the eigenvalues and eigenvectors) of the closed-loop generator AK. We give a useful characterization of that part of the spectrum σ(AK) which lies in the resolvent set of A, the open-loop generator, via the “characteristic equation” involving the open-loop transfer function. We show that certain points of σ(A) cannot be eigenvalues of AK if the input and output are scalar (so that K is a number) and K≠0. We devote special attention to the case when the output feedback operator K is compact. It is relatively easy to prove that in this case, σe(A), the essential spectrum of A, is invariant, that is, it is equal to σe(AK). A related but much harder problem is to determine the largest subset of σ(A) which remains invariant under compact output feedback. This set, which we call the immovable spectrum of A, strictly contains σe(A). We give an explicit characterization of the immovable spectrum and we investigate the consequences of our results for a certain class of well-posed systems obtained by interconnecting an infinite chain of identical systems.