A review of methods for input/output selection <s> The relative gain array <s> Large elements in the RGA imply a plant which is fundamentally difficult to control. (1) The plant is very sensitive to uncorrelated uncertainty in the transfer matrix elements. (2) The closed-loop system with an inverse-based controller is very sensitive to diagonal input uncertainty. With a diagonal controller, the system is not sensitive to diagonal input uncertainty, but the controller does not correct for the strong directionality of the plant and may therefore give poor performance even without uncertainty. <s> BIB001 </s> A review of methods for input/output selection <s> The relative gain array <s> Abstract The paper presents results on frequency-dependent tools for analysis, structure selection and design of control systems. This includes relationships between the relative gain array (RGA) and right half plane zeros, and the use of the RGA as a sensitivity measure with respect to individual element uncertainty and diagonal input uncertainty. It is also shown how frequency-dependent plots of the closely related performance relative gains (PRGA) and a new proposed disturbance measure, the closed-loop disturbance gains (CLDG), can be used to evaluate the achievable performance (controllability) of a plant under decentralized control. These controller-independent measures give constraints on the design of the individual loops, which when satisfied guarantee that the overall system satisfies performance objectives with respect to setpoint tracking and disturbance rejection. <s> BIB002 </s> A review of methods for input/output selection <s> The relative gain array <s> This paper studies deviations of open-loop properties in the presence of modeling uncertainties. Our aim is to gain insights into how open-loop properties and thus potentially closed-loop properties may vary in the face of a diagonally structured uncertainty. We give several estimates for the worst case deviations of the open-loop transfer function in terms of certain structured singular values and their bounds, and also in terms of certain scaled plant condition numbers, the relative gain array, and the block relative gains. Our analysis shows that the estimates in terms of the structured singular values and bounds are tight in general, so are those in terms of the condition numbers for certain cases studied previously in the literature. We show that the worst case deviation will be large when the estimates stated in terms of the structured singular values, or under certain circumstances in terms of the condition numbers, are large. On the other hand, an example is constructed to show that the relative gain array and block relative gains may be optimistic measures in assessing these deviations. The developments here support and reinforce previous conjec- tures and results which assert that plants with large condition numbers and/or relative gains are potentially difficult to control. <s> BIB003 </s> A review of methods for input/output selection <s> The relative gain array <s> This paper explores the application of recent techniques to the problem of structure selection and robust multivariable controller design for a high performance gas turbine engine. The engine under consideration is a Rolls Royce: 2-spool reheated turbofan, used to power modern military aircraft. Efficient and effective tools for control structure design are described and applied to the problem at hand. It is stressed that a good structure selection is very important, it results in simpler controllers, and in general, a simpler design exercise. A 2-DOF H/sub infinity / controller design is presented. Work is currently in progress on gain scheduling of such controllers. <s> BIB004
The "nal IO controllability measure that is often employed for IO selection is the frequency-dependent relative gain array (RGA) for a nonsingular, square matrix: with`. * a denoting element-by-element multiplication (Hadamard or Schur product). For a nonsquare P, the inverse in (19) is again replaced by the pseudo-inverse  ). The RGA is independent of the scaling of the plant's inputs and outputs if n S "n W , while it is independent of output scaling if n S 'n W and independent of input scaling if n S (n W . The RGA was introduced by  as a measure for interactions in decentralized control systems. As a result, there is a bulk of literature proposing RGA-related conditions for CC selection, but the RGA is also useful for IO selection. A frequently encountered rule is that IO sets causing large RGA elements should be avoided, since the corresponding plants would be di$cult to control, see, e.g., BIB003 . BIB002 and BIB001 show that plants with large absolutevalued RGA elements (1) are very sensitive (especially around the bandwidth) to diagonal multiplicative input uncertainty if an inverse-based controller`K"P\a is used and (2) are very sensitive to element-by-element uncertainty in P. In practice, it is often plausible to assume the "rst type of uncertainty, e.g., in case of neglected actuator dynamics. However, it is disputable to assume the second type, because the individual elements of P(s) are usually coupled in some way. Motivated by (1), one step of the IO selection procedure in BIB004 is based on the magnitude of RGA elements. Reeves (1991, Section 5.2) proposes two RGA-based heuristics to reduce the N;N full IO set (N"N S "N W ) to a smaller, n;n IO set (n"n S "n W ). These heuristics may not produce the IO set that is optimal with respect to the intended objective. The xrst heuristic employs the following lower bound to the condition number at a given frequency  : "2 max("" (P)"" , "" (P)"" )!14 (P) with "" (P)"" : "max H G " GH " the maximum absolutevalued row sum and with "" (P)"" : "max G H " GH " the maximum absolute-valued column sum (the subscript`ia denotes the induced matrix norm). Actually, (P) is a lower bound to the minimum condition number (P) under the best-possible IO scalings: D and D are diagonal matrices with real, positive entries. Computation of (P) involves a convex optimization and so is rather easy, but computation of (P) is easier. Based on the fact that (P)4 (P) and the notion that IO sets yielding a small plant condition number are preferred,  suggests to discard that single input and output from the N;N IO set for which the (N!1);(N!1) reduced IO set exhibits the smallest (P). The second heuristic relies on the aforementioned notion that a large RGA element GH only permits a small uncertainty in the corresponding plant element P GH .  proposes to identify the row and column indices of the largest RGA element and to discard the corresponding input and output from the N;N IO set. For the (N!1);(N!1) IO sets obtained with either of these heuristics, the RGA is recomputed and additional inputs and outputs may be eliminated. This procedure is repeated until the IO set with the intended dimension is reached. As shown in ,  , and Skogestad and Postlethwaite (1996, Section 10.5), to some extent the RGA may be used for a direct approach to IO selection. Consider P corresponding to the full IO set and suppose N S ON W . The RGA for the nonsquare P could then be used for squaring down the plant to have dimension min H GH 41 that is much smaller than one. In this way, the squared IO set consists of the inputs with the largest e!ect on the outputs and of the outputs which can best be a!ected by the inputs. In Cao (1995, Section 6.4), the RGA-like matrix B : is used in a similar way to reject inputs which are less e!ective for disturbance rejection.