Relative gain array

The relative gain array (RGA) is a classical widely-used^{[citation needed]} method for determining the best input-output pairings for multivariable process control systems.^[1] It has many practical open-loop and closed-loop control applications and is relevant to analyzing many fundamental steady-state closed-loop system properties such as stability and robustness.^[2]

Definition

Given a linear time-invariant (LTI) system represented by a nonsingular matrix $\mathrm {G}$ , the relative gain array (RGA) is defined as

\mathrm {R} =\Phi (\mathrm {G} )=\mathrm {G} \circ {(\mathrm {G} ^{-1})}^{T}.

where $\circ$ is the elementwise Hadamard product of the two matrices, and the transpose operator (no conjugate) is necessary even for complex $\mathrm {G}$ . Each ${i,j}$ element $\mathrm {R} _{i,j}$ gives a scale invariant (unit-invariant) measure of the dependence of output $j$ on input $i$ .

Properties

The following are some of the linear-algebra properties of the RGA:^[3]

Each row and column of $\Phi (\mathrm {G} )$ sums to 1.
For nonsingular diagonal matrices $\mathrm {D}$ and $\mathrm {E}$ , $\Phi (\mathrm {G} )=\Phi (\mathrm {D} \mathrm {G} \mathrm {E} )$ .
For permutation matrices $\mathrm {P}$ and $\mathrm {Q}$ , $\mathrm {P} \Phi (\mathrm {G} )\mathrm {Q} =\Phi (\mathrm {P} \mathrm {G} \mathrm {Q} )$ .
Lastly, $\Phi (\mathrm {G} ^{-1})=\Phi (\mathrm {G} )^{T}=\Phi {(\mathrm {G} ^{T})}$ .

The second property says that the RGA is invariant with respect to nonzero scalings of the rows and columns of $\mathrm {G}$ , which is why the RGA is invariant with respect to the choice of units on different input and output variables. The third property says that the RGA is consistent with respect to permutations of the rows or columns of $\mathrm {G}$ .

Generalizations

The RGA is often generalized in practice to be used when $\mathrm {G}$ is singular, e.g., non-square, by replacing the inverse of $\mathrm {G}$ with its Moore–Penrose inverse (pseudoinverse).^[4] However, it has been shown that the Moore–Penrose pseudoinverse fails to preserve the critical scale-invariance property of the RGA (#2 above) and that the unit-consistent (UC) generalized inverse must therefore be used.^[5] ^[6]

References

^ Bristol, E.H. (1966), "On a new measure of interaction for multivariable process control", IEEE Transactions on Automatic Control, 1: 133–134, doi:10.1109/TAC.1966.1098266
^ Chen, Dan; Seborg, D.E. (2002), "Relative Gain Array Analysis for Uncertain Process Models", AIChE Journal, 48 (2): 302–310, doi:10.1002/aic.690480214
^ Johnson, C.R.; Shapiro, H.M. (1986), "Mathematical aspects of the relative gain array (A◦A^−T)", SIAM Journal on Algebraic and Discrete Methods, 7 (4): 627–644, doi:10.1137/0607069
^ van de Wal, M.; de Jager, B. (2001), "A review of methods for input/output selection", Automatica, 37 (4): 487–510, doi:10.1016/S0005-1098(00)00181-3
^ Uhlmann, Jeffrey (2019), "On the Relative Gain Array (RGA) with Singular and Rectangular Matrices", Applied Mathematics Letters, 93: 52–57, arXiv:1805.10312, Bibcode:2018arXiv180510312U, doi:10.1016/j.aml.2019.01.031, S2CID 44092817
^ Qasim Al Yousuf, Rafal; Uhlmann, Jeffrey (2021), "On Use of the Moore-Penrose Pseudoinverse for Evaluating the RGA of Non-Square Systems", Iraqi Journal of Computers, Communications, Control & Systems Engineering, 3 (21): 89–97, arXiv:2106.09766, doi:10.33103/uot.ijccce.21.3.8, S2CID 235485155

[1] Bristol, E.H. (1966), "On a new measure of interaction for multivariable process control", IEEE Transactions on Automatic Control, 1: 133–134, doi:10.1109/TAC.1966.1098266

[2] Chen, Dan; Seborg, D.E. (2002), "Relative Gain Array Analysis for Uncertain Process Models", AIChE Journal, 48 (2): 302–310, doi:10.1002/aic.690480214

[3] Johnson, C.R.; Shapiro, H.M. (1986), "Mathematical aspects of the relative gain array (A◦A^−T)", SIAM Journal on Algebraic and Discrete Methods, 7 (4): 627–644, doi:10.1137/0607069

[4] van de Wal, M.; de Jager, B. (2001), "A review of methods for input/output selection", Automatica, 37 (4): 487–510, doi:10.1016/S0005-1098(00)00181-3

[5] Uhlmann, Jeffrey (2019), "On the Relative Gain Array (RGA) with Singular and Rectangular Matrices", Applied Mathematics Letters, 93: 52–57, arXiv:1805.10312, Bibcode:2018arXiv180510312U, doi:10.1016/j.aml.2019.01.031, S2CID 44092817

[6] Qasim Al Yousuf, Rafal; Uhlmann, Jeffrey (2021), "On Use of the Moore-Penrose Pseudoinverse for Evaluating the RGA of Non-Square Systems", Iraqi Journal of Computers, Communications, Control & Systems Engineering, 3 (21): 89–97, arXiv:2106.09766, doi:10.33103/uot.ijccce.21.3.8, S2CID 235485155

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