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Bounds for closed-loop transfer functions of multivariable systems

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2 Author(s)
Araki, M. ; Kyoto University, Yoshida, Kyoto, Japan ; Nwokah, O.

A multivariable feedback system y(s)=G(s)x(s), x(s) = u(s)- F(s)y(s) is treated where G(s) ≜ (gkl(s)) is the transfer function matrix of a plant and F(s) ≜ diag(f1(s),...,fn(s)) is that of a controller. A new bound for the transfer function hj(s) that relates yj(s) to uj(s) when fj(s)≡ 0 is given. The main result reads |hj(s)- gjj(s)| < aj(s) if |fk(s)-1+ gkk(s)| > ak(s) for k = 1,... ,n; k≠j. Here, A ≜ diag(a1(s),...,an(s)) is a diagonal matrix which makes A-B a semi-M-matrix where B ≜ (bkl) is given by bkk=0, bkl= |gkl(s)| (k≠l). A similar result is also obtained for the inverse transfer function.

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Automatic Control, IEEE Transactions on  (Volume:20 ,  Issue: 5 )