On the relationships between the unbounded asymptote behaviour of multivariable root loci, impulse response and infinite zeros

Using singular-value decomposition techniques, and making systematic use of the Schur complement for a partitioned matrix, an investigation is carried out of how the input and output spaces associated with a square transfer matrix can be decomposed in terms of the way in which a system responds to vector impulses of various orders. The results so obtained are then used to characterize the forms of behaviour of the unbounded asymptotes of the multivariable root locus. A discussion is given of the asymptotes and infinite zeros.