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Estimating the distance to uncontrollability: A fast method and a slow one. (English) Zbl 0877.93005

Summary: For a linear control system \((A,B)\), the distance to uncontrollability is characterized by \(\min_{\lambda \in C}\) \(\sigma_{n} ([A-\lambda I,B]),\) where \(\sigma_{n}([A-\lambda I,B])\) is the smallest singular value of the augmented matrix \([A-\lambda I,B]\). Two methods are developed to estimate the distance to uncontrollability, giving both a lower bound and an upper bound. One method is fast, requiring only one spectral decomposition of \(A\) and computations of three smallest singular values and being used for well-conditioned \(A\). The other is slow, requiring computations of a large number of the smallest singular values, but it produces bounds as tight as possible and also a region containing global minimizers. Newton’s method can be used to compute the global minimizers.

MSC:

93B05 Controllability
93B60 Eigenvalue problems
93C05 Linear systems in control theory
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