Global properties of the root-locus map

Abstract

In this paper we use a recently proven “general position lemma” for transfer functions to derive several important qualitative properties, some new, of the root-locus map for multivariable systems. Among the immediate applications which we derive is that it is not in general possible to develop a formula, involving rational operations and the extraction of r-th roots, for an output feedback gain (either complex or real, should a real solution exist) which places a given set of closed-loop poles. This is in sharp contrast to the state feedback situation [2] and is a partial affirmation of a conjecture made in [3]. The technique is to reduce the problem, via global reasoning, to a tractible problem in Galois theory, Having proved this result, the prerequisite global analysis is applied to give the positive result that the pole-placement equations can, however, be solved numerically by the homotopy continuation method. Since this global analysis of the root-locus map also plays a vital role in recent work on generic stabilizability ([3], [10]) and pole placement by output feedback ([6], [9]) and has never yet appeared in its full generality, we thought it would be useful to collect these basic topological and geometric results and derive them in a coherent fashion based on the “general position lemma.”