Discrete-time convolution feedback systems †

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Abstract
This paper considers multi-input multi-output discrete-time feedback systems characterized by y = G∗e and e = u — y. Theorem I shows that if the closed-loop impulse response H is stable in the sense that H ∊ ℓ 1 n×n (p), then [Gtilde] (z) =Ñ(z) [ [Dtilde](z)] − where Ñ(z), ;[Dtilde](z) are in ℓ 1 n×n (p) Theorem III gives necessary and sufficient conditions for H ∊ ℓ 1 n×n (p) Finally Theorem III gives necessary and sufficient conditions for stability when [Gtilde] (z) has a finite number of multiple poles in ∣z∣ > p : the case where the leading term of the Laurent expansion at each of these poles is singular is treated in detail.

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