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

Abstract

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.