A review of methods for input/output selection <s> Singular vectors <s> This paper presents several systematic procedures for selecting sensor type and locations for multivariable processes. The procedures are based on a singular value analysis of a process gain matrix and provide choices which are a practical compromise between independence and sensitivity. <s> BIB001 </s> A review of methods for input/output selection <s> Singular vectors <s> Abstract This study discusses the choice of appropriate sets of inputs for linear multivariable systems. It is shown, for example, that reduced-order models with better directionality properties can be obtained when fewer inputs are considered. Furthermore, the selection of a good set of inputs for control may become important when both strong and weak inputs are available. The transmission of information from the inputs to the outputs is investigated and appropriate scaling procedures to derive a scaled input matrix are proposed. Singular value decomposition methods facilitate the quantification of the system’s excitation stemming from the various inputs, and thus the selection of an appropriately strong and orthogonal set of inputs. The need for and the implementation and benefits of reducing the number of inputs are illustrated by a large-scale model of a real process. <s> BIB002
The next group of IO controllability measures involves the left (>) or right (;) singular vectors from the SVD in (7). BIB001 suggest three output selection rules. The key idea is to "nd the best compromise between measurements y that are mutually independent and measurements that are sensitive to changes of the inputs u. For this purpose, the SVD of P at steady state is computed. Square IO sets are considered. The xrst rule relies on the SVD of P(0) for the full output set. Those outputs are selected that correspond to the entries of each left singular vector in > with the largest absolute values (note that outputs cannot be distinguished if the same entry occurs multiple times). This is based on the notion that the left singular vectors point into the direction of the "rst (> ), the second (> ), etc., most sensitive combination of outputs. The selected outputs are stated to be sensitive to the inputs and, due to orthogonality of the vectors in >, relatively independent. A similar procedure could be proposed for input selection, involving the largest absolute values of ;. This output selection method is direct, since the SVD is not recomputed for each candidate. For the same reason, however, the method may be ine!ective. The second rule is a modi"ed version of the "rst one. Suppose that the entry in > corresponding to the entry of the largest absolute value of > is large and vice versa. Then there would still be signi"cant interaction between the outputs selected by the "rst rule. To arrive at an output set with reduced interaction, output selection can be based on the di!erences between the absolute-valued entries of the left singular vectors, possibly at the price of reduced sensitivity to u. The third rule resolves the possible ine!ectiveness of the previous ones by recomputing the SVD for all candidate output sets. BIB001 state that a large value of (P(0)) indicates a good sensitivity to inputs and that a small value of (P(0)) indicates a good mutual independence of the outputs. An output set for which the measure: is large should exhibit a good compromise between sensitivity to inputs and mutual independence of outputs. In BIB002 , an input selection method is proposed that aims at selecting the n S inputs with`the strongest and most orthogonal e!ecta on the (controlled) outputs y. This is quanti"ed via the n S largest singular values and the corresponding right singular vectors of the input matrix B in the plant's state-space description in a scaled modal basis. The input selection method is direct if B is generated for the full input set. In a direct approach is suggested to select the n S 5N W inputs from the N S candidates with the largest e!ect on the "xed number of N W outputs y. The SVD of P for the full input set is computed at a relevant frequency. Assuming that P has full row rank N W , the`single-input e!ectivenessa for u H is given by The n S inputs that yield the largest values SH (P) should be selected. It can be expected, but not guaranteed, that this input set gives the largest ratio ""u "" /""u"" , with u the part of u"u #u that is in the null space of P, i.e., Pu "0. In Cao (1995, Section 6.3), the SVD of PRP B is used to de"ne the`single-input disturbance e!ectivenessa, as a similar measure for the disturbance rejection e!ectiveness.