Stability of linear multivariable feedback systems

The stability of linear time-invariant multivariable feedback systems is studied from their open-loop transfer function matrices. It is shown that the stability of a multivariable feedback system depends on the determinant of the loop-difference matrix (det(I + G1G2)) and the characteristic polynomials of its open-loop transfer function matrices. The controllability and observability properties of the feedback system are considered. Hence the stability conditions insure the stability at the output terminals as well as at the state variables of the system. These conditions can be easily checked by using Nyquist plot, the root locus technique, or the Routh-Hurwitz criterion.