Page 87 - Distributed model predictive control for plant-wide systems

Structure Model and System Decomposition 61

Table 4.1 The meaning of the value for RGA elements
Value Interpretation

= 1 Open-loop gain and closed-loop gain are identical
ij
Interaction does not affect the pairing of (y , u )
j
i
= 0 Open-loop gain is zero
ij
The jth input does not affect the jth output
0 < < 1 Closed-loop interaction increases gain
ij
Interaction is most severe when = 0.5
ij
> 1 Closed-loop interaction reduces gain
ij
Higher values indicate more severe interaction
< 0 Closed-loop gain is in the opposite direction from the
ij
open-loop gain
In order to better comprehend the meaning of the RGA, the meaning of the values of the
elements of the RGA is shown in Table 4.1.
Therefore, considering all of the above shown in Table 4.1, the pairing strategy should match
the variables where is nearest to 1 while avoiding the variables where is zero or negative.
ij ij

4.4.3 Pairing Rules

Before introducing the paring rules, let us first introduce the Niederlinksi index (NI) which is
used to analyze the stability of the control loop pairings using the result of the RGA [86]:
det(G)
(4.28)
NI = ∏
n
g
i=1 ii
The goal of the RGA and NI analysis is to quantitatively determine the optimal variable pair-
ing for a given process. A negative NI value indicates instability in the control loop. Intuitively,
we prefer to pair variables u and y so that is close to 1 at all frequencies for decentralized
ij
j
j
control; the reason is that the gain from u to y is not affected by closing the other loops.
j
j
However, one should avoid pairings where the sign of the steady-state gain from u to y may
j
j
change depending on the control of the other outputs, because this will yield instability with
integral action in the loop. More precisely, the basic rules that should be remembered when
someone attempts to obtain an optimal pairing of control loops in for decentralized control are
as follows:
• Rule 1: Select pairing with positive RGA coefficients close to unity in the frequency range
where performance is sought in the frequency range of interest; the control loops will be
effectively decoupled.
• Rule 2: If the NI value is negative, the loop pairing for that control system configuration is
unacceptable.

In addition, it should be noted that NI should not be used for systems that have time delays
(dead time), as the stability predictions assume immediate feedback to the controller.