Cancellations in multivariable continuous-time and discrete-time feedback systems treated by greatest common divisor extraction

This correspondence considers a multivariable system with proper rational matrix transfer functions G
0
and G
f
in the forward and feedback branches, respectively. It develops a strictly algebraic procedure to obtain polynomials whose zeros are the poles of the matrix transfer functions from input to output (H
y
), and from input to error (H
e
). G
0
and G
f
are given in the polynomial matrix factored form
N_{0}D_{0}^{-1}
and
D_{f}^{-1}N_{f}
. The role of the assumption det [
I + G_{f}(\infty)G_{0}(\infty)] \neq 0
and the relation between the zeros of det [
I + G_{f}G_{0}
] and the poles of H
y
and H
e
are indicated. The implications for stability analysis of continuous-time as well as discrete-time systems are stressed.