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On the computation of the gap metric. (English) Zbl 0669.93039
The paper presents an expression for the gap metric that is computable using well-known techniques. It also establishes the equivalence between the gap and the graph metrics.
Let \(P_ i,i=1,2\), be linear, time-invariant systems, considered as operators acting on \({\mathbb{H}}_ 2\). For \(i=1,2\), the graph of \(P_ i\) is \(graph(P_ i)=G_ i{\mathbb{H}}_ 2\) where \(G_ i\) is an inner matrix function. The gap between \(P_ 1\) and \(P_ 2\) is \[ \delta (P_ 1,P_ 2):=\| {\mathbb{P}}_{G_ 1{\mathbb{H}}_ 2}-{\mathbb{P}}_{G_ 2{\mathbb{H}}_ 2}\| \] where \({\mathbb{P}}_{G_ i{\mathbb{H}}_ 2}\) denotes the orthogonal projection with range \(G_ i{\mathbb{H}}_ 2\). The above expression defines a metric on the space of (possibly unstable) systems. The gap metric has been introduced by Krein and Krasnosel’skij, and Sz.-Nagy [cf. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Vol. I. (New York 1961; for a review of the 1981 English edition see Zbl 0467.47001). In control theory, it has been introduced by G. Zames and A. K. El-Sakkary [Unstable Systems and feedback: The gap metric. Proc. Allerton Conf., Oct. 1980, pp. 380-385.] [see also A. K. El-Sakkary, IEEE Trans. Autom. Control AC-30, 240-247 (1985; Zbl 0561.93047)] as being appropriate for the study of uncertainty in feedback systems. The gap metric induces the weakest topology in which feedback stability is a robust property. Using the commutant lifting theorem it is shown that \[ \delta (P_ 1,P_ 2)=\max \{\inf_{Q\in H_{\infty}}\| G_ 1-G_ 2Q\|_{\infty},\inf_{Q\in {\mathbb{H}}_{\infty}}\| G_ 2-G_ 1Q\|_{\infty}\}. \] The computation of the infima above has been studied extensively over the last few years and can be carried out using well-known techniques [cf. B. A. Francis, A course in \({\mathbb{H}}_{\infty}\) control theory, Lect. Notes Control Inf. Sci. 88 (1987; Zbl 0624.93003), Ch. 8].
An alternate metric with similar properties has been introduced by M. Vidyasagar [IEEE Trans. Autom. Control AC-29, 403-418 (1984; Zbl 0536.93042)]. This, known as the graph metric, is defined by: \[ d(P_ 1,P_ 2)=\max \{\inf_{Q\in {\mathbb{H}}_{\infty},\| Q\|_{\infty}\leq 1}\| G_ 1-G_ 2Q\|_{\infty},\inf_{Q\in {\mathbb{H}}_{\infty},\| Q\|_{\infty}\leq 1}\| G_ 2-G_ 1Q\|_{\infty}\}. \] It is shown that \(2\delta (P_ 1,P_ 2)\geq d(P_ 1,P_ 2)\geq \delta (P_ 1,P_ s)\).
Reviewer: T.T.Georgiou

MSC:
93C05 Linear systems in control theory
93D15 Stabilization of systems by feedback
47B38 Linear operators on function spaces (general)
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