A review of methods for input/output selection <s> The condition number <s> Abstract With increased process integration it has become very important to evaluate and compare the dynamic operability characteristics (dynamic resilence) of alternate designs. Based on recent results in multivariable frequency response theory a new framework is developed for this purpose. It is shown that dynamic resilence is determined by characteristics inherent in the system and that it is independent of the imposed controller structure and type. This gives the new method considerable intuitive appeal and allows it to avoid the lengthy optimization procedures which are typical for the previously published techniques. <s> BIB001 </s> A review of methods for input/output selection <s> The condition number <s> The control structure selected has a strong effect on the performance of a closed loop system with respect to its disturbance rejection capability. But the methods used for its selection usually result in a combinatorial problem. To overcome this difficulty, an input pre-screening criterion, the input-disturbance alignment (IDA) measure, is proposed. Some important features of the new indicator and its relationship with other well-known controllability measures are presented. The case study included shows that the IDA can be used for efficient selection of manipulated variables from a large number of candidate inputs. These then form the best control structure for disturbance rejection. <s> BIB002
The frequency-dependent condition number is another common IO controllability measure. In general, an IO set should be chosen which results in a small condition number. The condition number of a column or row vector (which have only one singular value) equals one and so it is not selective if n S "1 or n W "1. BIB001 shows that a small condition number (P) corresponds to good robustness against full-block (unstructured) multiplicative uncertainty. This would justify selecting IO sets giving rise to a small (P). Skogestad and Postlethwaite (1996, Section 6.10) state that the performance of plants (IO sets) with a small (P) is robust to both diagonal and unstructured multiplicative uncertainty at the input of the plant. Section 3.6 sophisticates the use of (P) to address robustness. Reeves (1991, Section 5 .2) proposes a method to reduce the number of candidate inputs and outputs prior to applying the computationally more involved method in Section 3.6. Starting with the full N S ;N W IO set, that single input or single output is eliminated which produces the reduced (N S !1);N W or N S ;(N W !1) IO set with the smallest (P), computed at a relevant frequency. In this way, the size of the IO set is reduced gradually, until it is manageable for other techniques. The "nal IO set is not guaranteed to be the IO set of that dimension for which (P) is smallest. For that, (P) should have been computed for all IO sets of that dimension.  consider the disturbance condition number. For the kth disturbance d I in Fig. 4 , it is with 14 BI (P, P B )4 (P). Here, P BI is the kth column of P B corresponding to d I and "" ) "" denotes the Euclidean vector norm (the length) at a given frequency. For nonsquare P, P\ should be replaced by PR. The disturbance condition number is a measure of the input magnitude which is needed to reject a disturbance in the direction P BI , relative to rejecting a disturbance with the same magnitude, but in the direction requiring the least control e!ort. Input sets yielding a small BI (P, P B ) are most e!ective for disturbance rejection. The essential di!erence between input selection with (P) and with BI (P, P B ) is that (P) aims at rejecting disturbances with large magnitudes by using a given input magnitude, whereas BI (P, P B ) aims at small input magnitudes for a given disturbance direction, irrespective of the disturbance's magnitude. The de"nition of the`input disturbance alignmenta in BIB002 shows close resemblance with (15): Input sets with BI (P, P B ) close to one are preferred. The projection norm of P BI on the column space of P is then large, which serves good disturbance rejection.