The optimal root loci of linear multivariable systems

Abstract

Optimally regulated linear systems possess a number of properties attractive to the designer of feedback control loops. On account of these properties both the optimal controller and the compensated system have formed the subject of extensive research in recent years. One aspect of the behaviour of optimal systems which has received little attention, however, concerns the movement of the closed-loop poles of optimal systems when the weight on the control effort in the index of performance is relaxed to arbitrarily low levels. Relevant results on this topic have been derived for scalar systems only while extensions of these to the multivariable case have been presented in a qualitative manner only with the exception of the lowest order of behaviour for which some explicit results exist. Here a new root-locus approach is used in order to study all orders of behaviour. The results derived relate the asymptotic behaviour of the poles of the optimal system to that of the original system and for this reason more light is cast onto the function of the optimal controller. Explicit results are given for all orders up to five although the symmetric nature of the pattern of behaviour established enables by conjecture the extension of these results to all orders. A numerical example is included at the end of this paper for the purposes of illustration.