On the stability of multivariable feedback systems

Abstract

In this paper, we consider the stability of a multivariable unity feedback system with an r × r rational proper transfer matrix G(s) in the forward path. The analysis is facilitated by introducing an “equivalent” scalar transfer function g(s) which consists of the sum of the determinants of all leading principal minors of G(s). It is shown that the characteristic roots of the multivariable closed loop system consists of two sets, one consisting of the closed loop characteristic roots of the scalar unity feedback system with g(s) in the forward path, and a second set consisting of the subset of open loop characteristic roots which do not appear as poles of g(s). The root locations (RHP or LHP) of the first set of roots can be determined by applying the Nyquist criterion to g(s). This test can be a measurement based test constructed from the individual frequency responses gij(jω) of the entries of G(s). Examples are included for illustration.