Asymptotic behaviour of root-loci of linear multivariable systems

The theory of the asymptotic behaviour of the root-loci of linear, time-invariant, multivariable, feedback systems is developed. It is shown that each of the system zeros attracts, and is a terminating point of, one of the root-loci, as the feedback gain tends to infinity. The root-loci, that are not attracted by the zeros, tend to infinity in a special pattern that is dictated by the eigen-properties of the elementary matrices of the system. To complete the geometric description of the asymptotic behaviour of the root-loci, the concept of infinite zeros and their order is introduced. Each infinite zero of order r attracts one root-locus and, together with r−1 other infinite zeros of the same order, the corresponding asymptotes form a Butterworth configuration of order r around a special point defined as a ‘ pivot ’. A detailed algorithm for the calculation of the finite and infinite zeros is given and is illustrated by examples. A synthesis technique is then proposed by which a constant feedback controller may be found such that all the root-loci asymptotes lie in the complex left-half plane and thus ensure system stability for high gains.