This document presents a new approach to analyzing the robustness of the relative gain array (RGA) for uncertain systems. It derives bounds on the RGA elements for a 2x2 uncertain system and provides sufficient conditions to determine if the plant remains non-singular over the uncertainty set. An example is provided to illustrate the bounds on the magnitude and phase of the RGA in the frequency domain for an uncertain system. The analysis of the RGA's robustness to uncertainties can help assess decisions made based on the nominal plant model.

1. A New Approach to the Dynamic
RGA Analysis of Uncertain Systems
Miguel Casta˜o (miguel.castano@ltu.se)
n
Wolfgang Birk (wolfgang.birk@ltu.se)
Lule˚ University of Technology, Sweden
a
At a each given frequency ω all the possible values of λp11(jω)
Abstract can be approximated using the following parametrization of the
curve enclosing the possible values of Wc
Uncertainties in a process model can be translated into uncer-
C |Wc|max + (c1 · r2 + c2 · r1) · (cos(θ) − 1)
tainties in the DRGA which might invalidate the decision on
the variable paring in decentralized control based on the nomi- +r1 · r2 · (cos(2θ) − 1),
nal plant. (c1 · r2 + c2 · r1) · sin(θ) + r1 · r2 · sin(2θ) ,
The bounds of the DRGA of a 2x2 uncertain system of the form
θ ∈ [0, 2π]
Gpij (s) = Gij (s)(1 + Wij (s) · ∆ij (s)), ∀i, j = {1, 2} will be
where c1 = (1 − |W11 |2)−1 ; c2 = (1 − |W22 |2)−1
analyzed.
r1 = (|Wc|max − |Wc|min − 2 · c1 · r2)/2 · c2
Theory
r2 = max({|Wij |}) · c2
i,j
 
λp11 1 − λp11 
RGA(Gp(s)) =  0.15 0.95
1 − λp11 λp11
 
Limits of |λp |
11
0.1 0.9 Nominal |λ11|
Random Values of |λ |
p
1 0.05
0.85 11
λp11 = 0.8
G12·G21 · (1+W12∆12)(1+W21∆21)
Magnitude
0
1/(1−Gc ⋅ Wc)
1 − G11·G22 (1+W ∆ )(1+W ∆ )
Im
0.75
11 11 22 22 −0.05
0.7
denoting Gc = G12G21 and W = (1+W12∆12)(1+W21∆21) , if
−0.1
0.65
G11G22 c (1+W11∆11)(1+W22∆22) −0.15 0.6
|Wij | ≤ 1; ∀i, j = {1, 2} then, at a given frequency −0.2
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
0.55
10
−2 −1
10 10
0
10
1 2
10
Re Frequency
|Gc · Wc| ∈ |Gc| · |Wc|min , |Gc| · |Wc|max
2 2 2 2 0rad/sec
ΦGcWc ∈ − asin(|Wij |), asin(|Wij |) Possible values of |λp11(jω)| at (left) . Bounds of 10
i=1 j=1 i=1 j=1 |λp11| in frequency domain (right) .
with Based on sampled computations at different frequencies, the
(1−|W12|)(1−|W21|) (1+|W12|)(1+|W21|) bounds of the RGA can be depicted in frequency domain.
|Wc|min = (1+|W |)(1+|W |) , |Wc|max = (1−|W |)(1−|W |)
11 22 11 22
An analysis of the RGA to uncertainties could then be done to
If Gc ·Wc can equal 1 in the uncertainty set at a given frequency, asses or reject decisions taken from the nominal plant.
the plant can become singular and the RGA will be unbounded.
Conclusions
Example
• Uncertainties in a process model can affect the validity of de-

5 2.75
 
0.25 s+0.06667
 cisions based on the nominal RGA.
 s+1
G(s) =

0.4s+1 0.25s+1  , W (s) = 3s2+4s+1 
• Bounds of the RGA of a 2x2 uncertain system have been de-
 −2 3  0.24 0.3832s+0.01533 
   

0.4s+1 0.3s+1 2s+1 0.02s2+2.01s+1 rived.
The possible values of |Gc · Wc| and ΦGcWc are represented in • Sufficient conditions to asses the non-singularity of the plant
frequency domain. 1 is never a possible value of (Gc · Wc), thus in the uncertainty set have been derived, and thus sufficient
the plant remain non-singular and the RGA is bounded in all conditions for the RGA to be bounded.
the uncertainty set at the represented frequencies. • IMs help to understand a complex process (open loop or closed
0.9
0.8
Bounds of |G ⋅ W |
Nominal |Gc|
c C
−130
−140
Bounds of Φ(g ⋅ w )
Nominal Φ(g )
c
c c loop) and give useful information about how it should be con-
0.7
−150
−160
trolled. Analysis of robustness of an IM can be done to asses
or reject the decisions based on the nominal plant.
Phase (Degrees)
0.6
−170
Magnitude
0.5 −180
−190
0.4
−200
0.3
−210
0.2
−220
0.1 −230
−2 −1 0 1 2 −2 −1 0 1 2
10 10 10 10 10 10 10 10 10 10
Frequency (rad/sec) Frequency (rad/sec)
Bounds of |Gc · Wc| (left) and ΦGcWc (right) represented in
frequency domain. Possible analysis of interactions in a stock preparation plant.