On the relationships between 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 characterise the forms of the behaviour of the unbounded asymptotes of the multivariable root locus. A discussion is given of the asymptotes and infinite zeros.