Plant directionality, coupling and multivariable loop-shaping

We study the multivariable loop-shaping problem when the plant is ill-conditioned and feedback properties at both plant input and output are of interest. Directionality and coupling are described, using the singular value decomposition and principal angles between high- and low-gain subspaces. Our first results are new first-order approximations to feedback properties at one loop-breaking point in terms of the open loop transfer function at that point. Secondly, we study the relation between feedback properties at different loop-breaking points in the event that the plant is ill-conditioned. The nominal relation is studied first. Then robustness is studied, using a first-order approximation to feedback properties at one point with respect to uncertainty at the other. This analysis yields conditions that must be satisfied by feedback properties at one point to prevent poor feedback properties at the other. Finally, these two sets of results are combined to show how feedback properties at both points may be...