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Pseudospectral components and the distance to uncontrollability. (English) Zbl 1078.93008
The authors consider the control system defined by \(x' = Ax + Bu\) for \(A \in \mathbb{R}^{p \times p}\), \(B \in \mathbb{R}^{p \times q}\), \(x(t) \in \mathbb{R}^{p}\), \(u(t) \in \mathbb{R}^{q}\). This system is called controllable if, given initial and final states \(x(0)\) and \(x(T)\), there exists a control function \(u(t)\) giving a trajectory \(x(t)\) with the given endpoints. The main results are the following:
Theorem: Given any real \(\varepsilon\) and matrices \(P, Q \in \mathbb{C}^{m \times n}\), \(m \leq n\), suppose there exists no complex \(z\) for which the singular value \(\sigma_{\min}(P + zQ)\) is both multiple and equals \(\varepsilon\). Then the set \(\Lambda = \{ z \in \mathbb{C} : \sigma_{\min}(P + zQ) \leq \varepsilon \}\) has no more than \(2m(4m-1)\) components.
Theorem: For any matrices \(P, Q \in \mathbb{C}^{m \times n}\), \(m \leq n\), and any real \(\varepsilon\), the set \(\{ z \in \mathbb{C} : \sigma_{\min}(P + zQ) \leq \varepsilon \}\) has no more than \(2m(4m-1)\) components.
In sections 5 and 6 the authors study algorithms for computing the distance to uncontrollability.

MSC:
93B05 Controllability
15A18 Eigenvalues, singular values, and eigenvectors
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
Software:
ARPACK; eigs; Eigtool
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