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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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##### References:
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