If W1/W > 0.5, the correct pairing is W  W1 and x  W2. If W1/W < 0.5, the correct
pairing is W  W2 and x  W1. A value of W1/W = 0.5 yields all mij™s equal to 0.5. This
value for a 2 ¥ 2 system indicates the highest degree of interaction when the inter
action is positive (presented next).
The steps taken to develop the RGM for a 2 ¥ 2 system required only simple
algebra. For a higherorder system the same procedure could be followed; however,
more algebraic steps must be taken to reach a ¬nal solution. For these higherorder
systems matrix algebra can be used to simplify the development of the RGM. The
procedure, as proposed by Bristol [1], is as follows: Calculate the transpose of the
inverse of the steadystate gain matrix and multiply each term of the new matrix by
the corresponding term of the original matrix. The terms thus obtained are the terms
of the relative gain matrix.
This procedure might look out of reach for those unfamiliar with matrix algebra.
But given the utility of this method, it is worthwhile to surpass this dif¬culty. There
are digital computer programs, and even handheld calculators, that can easily crank
out the necessary numbers.
91.2 Positive and Negative Interactions
Positive interaction is the interaction experienced when all the relative gain terms
are positive. It is interesting to see under what conditions this type of interaction
results. Much can be learned from the expression for m11 in a 2 ¥ 2 system
K11 K 22 1
m11 = = (91.12)
K11 K 22  K12 K 21 1  K12 K 21 K11 K 22
190 MULTIVARIABLE PROCESS CONTROL
If there is an odd number of positive Kij™s, the value of m11 will be positive, and fur
thermore, its numerical value will be between 0 and 1.
Positive interaction is the most common type of interaction in multivariable
control systems. In these systems each loop helps the other. To understand what we
mean by this, consider the blending control system shown in Fig. 91.5, and its block
diagram shown in Fig. 91.6. For this process the gains of the control valves are
positive since both valves are failclosed. The gains K11, K21, and K12 are also posi
tive, while the gain K22 is negative. The ¬‚ow controller GC1 is reverse acting, while
the analyzer controller GC2 is direct acting. Assume now that the set point to the
¬‚ow controller decreases; the ¬‚ow controller, in turn, decreases its output to
close the valve to satisfy its new set point. This will cause the outputs from valve
m1
m2
2
1 P
W
W1 2
x1 x2
SP SP
FC AC
FT AT
3
W x
Figure 91.5 Control system for blending process.
 m1
cset +
1
GC1 G11
GV 1
(˜) (˜)
(˜) c1 = W (˜ )
(˜)
G21
(˜)
(˜)
G12
(≠)
(˜)
set
c2 + c2 = x
m2
GV 2
GC 2 G22
(˜ ) (˜)

Figure 91.6 Block diagram showing how signals and variables move.
191
INTERACTION AND STABILITY
GV1 and process G11 to decrease. Because K21 is positive, the output from G21 also
decreases, resulting in lowering the analysis, x. When this happens, the analysis con
troller GC2 also decreases its output. This causes the outputs from GV2 and G12 to
decrease and the output from G22 to increase. Figure 91.6 shows the arrows that
indicate the directions that each output moves. The ¬gure clearly shows that the
outputs from G11 and G12 both decrease. This is what we mean by “both loops help
each other.”
When there are an even number of positive values of Kij™s, or an equal number
of positive and negative values of Kij™s, the value of mij™s will either be less than 0 or
greater than 1. In either case there will be some mij with negative values in the same
row and column. The interaction in this case is said to be a negative interaction. It
is important to realize that for a relative gain term to be negative, the signs of the
open and closedloop gains must be different. This means that the action of the
controller must change when the other loops are closed. For this type of interaction
the loops “¬ght” each other.
92 INTERACTION AND STABILITY
The second question posed at the beginning of the chapter is related to the effect
of the interaction on the stability of multivariable control systems. We ¬rst address
this question to a 2 ¥ 2 system; consider Fig. 91.6.
As explained in Chapter 7, the roots of the characteristic equation de¬ne the sta
bility of control loops. For the system of Fig. 91.6 the characteristic equations for
loop 1 by itself (when loop 2 is in manual) is
1 + GC1GV1G11 = 0 (92.1)
and equally for loop 2,
1 + GC2GV2G22 = 0 (92.2)
The control loops are stable if the roots of the characteristic equation have nega
tive real real parts. To analyze the stability of the complete system shown in Fig.
91.6, the characteristic equation for the complete system must be determined. This
is done using signal ¬‚ow graphs [2], which yields
(1 + GC1GV1G11 ) (1 + GC 2GV 2G22 )  GC1GV1GC 2GV 2G12G21 = 0 (92.3)
The terms in parentheses are the characteristic equations for the individual
loops. Analyzing Eq. (92.3), the following conclusions for a 2 ¥ 2 system can be
reached:
1. The roots of the characteristic equation for each individual loop are not the
roots of the characteristic equation for the complete system. Therefore, it is
possible for the complete system to be unstable even though each loop is
stable. Complete system refers to the condition when both controllers are in
automatic mode.
192 MULTIVARIABLE PROCESS CONTROL
2. For interaction to affect the stability, it must work both ways. That is, both G12
and G21 must exist; otherwise, the last term in Eq. (92.3) disappears, and if