On the asymptotic root-loci of linear multivariable systems

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Abstract
A new derivation for the asymptotes of root-loci of linear multivariable systems is presented. The analysis utilizes the matrix return difference relationship to produce scalar limits delineating the angle and pivots of the root-loci asymptotes, and details the nature of the related limiting input vector directions. A general matrix series reduction procedure is developed to yield the different orders of root-loci patterns and the necessary and sufficient conditions for the termination of the reduction process presented. The three parts comprising the operations necessary to extract the root-loci asymptotes are drawn together to give a complete algorithm. Finally some results pertaining to the asymptotic input vector directions are given, along with observations on the relationship between this analysis and that for asymptotic optimal root-loci.

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