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CORRESPONDENCE Further Theoretical Results on “Relative Gain Array for Norm-Bounded Uncertain Systems” Bijan Moaveni* and Ali Khaki Sedigh Faculty of Electrical Engineering, K. N. Toosi UniVersity of Technology, Tehran Seied-Khandan 16315, Iran Sir: The issue of input-output pairing for uncertain multivariable systems has only very recently been addressed.1-3 In the recent work by Kariwala et al.,1 an approach to input-output pairing analysis for uncertain multivariable system was presented. They presented a “method for obtaining a bound on the magnitude of the worst-case relative gain, calculated at steady state and also at higher frequencies”. This involved the presentation of a method for calculating a tight bound on the worst-case relative gain, and it derived the necessary and sufficient conditions for the sign change of the relative gain over the uncertainty set. The proposed method uses signal representation for uncertain relative gain. Thus, the key theoretical points in their work1 can be summarized as follows: (1) Present a method for obtaining a bound on the magnitude of the worst-case relative gain, calculated at steady state and at higher frequencies. (2) Present the necessary and sufficient conditions for the sign change of the relative gain of a norm-bounded uncertain system. (3) Present a signal-based representation of the relative gain for uncertain systems. However, three main drawbacks are associated with the method. First, its implementation in large-scale or complex systems is computationally costly. Second, it cannot solve the sensitivity problem, because sufficiently large relative gains result in closed-loop sensitivity issues.4 Finally, it cannot detect the appropriate input-output pair when there is no sign change in the relative gains and most of them are positive. In this correspondence paper, we present a theorem and propose a method to compute the variation bounds in the relative gains. This computes the upper bound of each variation in the relative gain. Here, computed bounds will be loose, but their computations are faster for large-scale systems. In addition, the variation bounds of the relative gains can show the magnitude of large-value relative gains and can solve the sensitivity problem.
Γu )
) (G + ∆) X (G + ∆)
Therefore, if σ j (∆) < σ(G), then
||δij|| e
2σ j (eTi G)σ j (EjG-1)σ j (∆) σ(G) - σ j (∆)
) δup ij
(2)
and the variation bound of an uncertain RGA is u up λij - δup ij e λij e λij + δij
(3)
Proof. According to Figure 1, each uncertain relatiVe gain can be represented as
λuij ) eTi (G + ∆)ejeTj (G + ∆)-1ei ) eTi (G + ∆)Ej(G + ∆)-1ei (4) and the corresponding certain relatiVe gain is
λij ) eTi GejeTj G-1ei ) eTi GEjG-1ei
(5)
Therefore, δij ) λuij - λij ) eTi (G + ∆)Ej(G + ∆)-1ej - eTi GEjG-1ei
δij ) eTi (G + ∆)Ej(G + ∆)-1ei - eTi GEjG-1ei
As given in Kariwala et al.,1 the uncertain relative gain array (RGA) for multivariable systems with a nominal model G and additive uncertainty ∆ can be computed as -T
Γu ) [λuij] ) [λij + δij] ) (G + ∆) X (G + ∆)-T
(6)
where
1. A Theorem on Relative Gain Array (RGA) for Uncertain Systems
[λuij]
and each uncertain relative gain (λuij) can be presented as shown in Figure 1.1 Now, to compute the variation bounds of the relative gains, we propose the structure shown in Figure 2 to represent λuij: Therefore, the following theorem is presented to compute the upper bound of ||δij|| and the variation bounds of uncertain relative gains. Theorem. Consider the linear multivariable system G and the corresponding RGA, Γ ) [λij], where, in the presence of additive uncertainty,
(1)
* To whom correspondence should be addressed. Tel.: +98 21 88462066. Fax: +98 21 88469084. E-mail address: b_moaveni@ dena.kntu.ac.ir.
(7)
and
δij ) eTi G{(In + G-1∆)Ej(In + G-1∆)-1 - Ej}G-1ei
(8)
Thus,
δij ) eTi G{G-1∆Ej - EjG-1∆}(G + ∆)-1ei Now, using 2-norm, one obtains
10.1021/ie071104m CCC: $37.00 © 2007 American Chemical Society Published on Web 10/31/2007
(9)
Ind. Eng. Chem. Res., Vol. 46, No. 24, 2007 8289
Figure 1. Signal-based presentation of the uncertain relative gain. (Adapted from Kariwala et al.1)
same column in uncertain RGA, the nominal input-output pairing remains valid for all parameter variations. Otherwise, if there is an overlap between variation bounds of the same row or the same column in uncertain RGA, the nominal inputoutput pairing may change, because of parameter variations. Figure 2. Representation of uncertain relative gain.
3. Conclusion
||δij|| ) ||eTi G{G-1∆Ej - EjG-1∆}(G + ∆)-1ei|| ) σ j (eTi G{G-1∆Ej - EjG-1∆}(G + ∆)-1ei) and using norm properties,2 one obtains ||δij|| e
σ j (eTi G)σ j (G-1∆Ej - EjG-1∆)
(10)
(11)
σ(G + ∆)
where, if σ j (∆) < σ(G), then eq 11 can be rewritten as
σ j (eTi G){σ j (G-1∆Ej) + σ j (EjG-1∆)}
||δij|| e
e σ(G) - σ j (∆) 2σ j (eTi G)σ j (EjG-1)σ j (∆) σ(G) - σ j (∆)
Literature Cited
) δup ij (12)
Therefore, the Variation bound of the uncertain relatiVe gain is u up λij - δup ij e λij e λij + δij
In this correspondence paper, a theorem is given based on the main results of Kariwala et al.1 for input-output pairing analysis for uncertain multivariable systems. A method to compute the relative gains’ variation bound of RGA to inputoutput pairing analysis is provided. The results can decrease the computational load in large-scale uncertain systems, solve the sensitivity analysis problem, and propose the appropriate pair, when there is no sign change for relative gains.
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(1) Kariwala, V.; Skogestad, S.; Forbes, J. F. Relative gain array for norm-bounded uncertain systems. Ind. Eng. Chem. Res. 2006, 45, 1751. (2) Chen, D.; Seborg, D. E. Relative gain array analysis for uncertain process models. AIChE J. 2001, 48, 302. (3) Khaki-Sedigh, A.; Moaveni, B. Relative gain array analysis of uncertain multivariable plants. In Proceedings of 7th European Control Conference (ECC03), Cambridge, U.K., 2003. (4) Skogestad, S.; Morari, M. Implications of large RGA elements on control performance. Ind. Eng. Chem. Res. 1987, 26, 2323.
2. Comments The above theorem shows the simplicity of δup ij computation and the variation bounds of the relative gains. If there is no overlap between the variation bounds of the same row and the
ReceiVed for reView August 13, 2007 Accepted October 1, 2007 IE071104M
