This work develops a number of new algorithms for the computation of the convex estimate, ¿tco, of the structured singular value, fj. The basis for these algorithms lies in an application of the geometric form of the Hahn-Banach theorem Dual methods are shown to be more useful than their primal counterparts when determining Ha. In particular, a dual method based on a 1-norm optimisation strategy is shown to out perform its 2-norm based counterpart in terms of accuracy, reliability and speed. It is proven that this 1-norm dual method converges to /¿co. The algonthm has been successfully validated using a large selection of random, pseudo random and practically motivated problems. Improvements to the basic algorithm that significantly reduce computation times are outlined and analysed in some detail. This work presents new applications for \i /x provides a novel way of analysing, nonconservatively, the effect of uncertainty in component values on filter performance. It is shown that the problems of computing (1) maximum filter gam, (11) minimum gain and (in) the maximum Euclidean deviation from nominal performance on a polar plot can all be determined using fi theory. The necessary formulations required to deal with these particular problems are developed. In contrast to a grid search or probabilistic approach, using /i provides frequency response information that fully addresses the cross coupling effects of component uncertainty in a rigorous and repeatable fashion. This process has been automated for Butterworth and Chebychev filters of arbitrary order. Third order examples are used to illustrate the approach. L\ methods are used to develop new tuning rules for PID controllers This is achieved using the operator which can reasonably be described as a time domain analogue of /i. Used in conjunction with an application of the robust performance theorem, is used to find the PID settings that best satisfy a given time domain performance objective.