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)| 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.