Bounds for closed-loop transfer functions of multivariable systems

A multivariable feedback system y(s)=G(s)x(s), x(s) = u(s)- F(s)y(s) is treated where G(s) ≜ (g
kl
(s)) is the transfer function matrix of a plant and F(s) ≜ diag(f
1
(s),...,fn(s)) is that of a controller. A new bound for the transfer function h
j
(s) that relates y
j
(s) to u
j
(s) when f
j
(s)≡ 0 is given. The main result reads |h
j
(s)- g
jj
(s)| < aj(s) if |f
k
(s)
^-1
+ g
kk
(s)| > a
k
(s) for k = 1,... ,n; k≠j. Here, A ≜ diag(a
1
(s),...,a
n
(s)) is a diagonal matrix which makes A-B a semi-M-matrix where B ≜ (b
kl
) is given by b
kk
=0, b
kl
= |g
kl
(s)| (k≠l). A similar result is also obtained for the inverse transfer function.