[Luyben] Process Mod.. - Student subdomain for University of Bath

ANALYSIS OF MULTIVARIABLE SYSTEMS 579
The program given in Table 16.3 uses Equation 16.31 to calculate the elements of
the 3 x 3 RGA matrix. The IMSL subroutine LEQ2C is used to get the inverse of
the matrix. The other subroutines are from Chapter 15. The results are
1.96 -0.66 -0.30
-
RGA = -0.67 1.89 (16.34)
[ -0.29 -0.23
-0.221 1.52
Note that the sums of the elements in all rows and all columns are 1.
B. USES AND LIMITATIONS. The elements in the RGA can be numbers that
vary from very large negative values to very large positive values. The closer the
number is to 1, the less difference closing the other loop makes on the loop being
considered. Therefore there should be less interaction, so the proponents of the
RGA claim that variables should be paired so that they have RGA elements near
1. Numbers around 0.5 indicate interaction. Numbers that are very large indicate
interaction. Numbers that are negative indicate that the sign of the controller
may have to be different when other loops are on automatic.
As pointed out earlier, the problem with pairing on the basis of avoiding
interaction is that interaction is not necessarily a bad thing. Therefore, the use of
the RGA to decide how to pair variables is not an effective tool for process
control applications. Likewise the use of the RGA to decide what control structure
(choice of manipulated and controlled variables) is best is not effective. What
is important is the ability of the control system to keep the process at setpoint in
the face of load disturbances. Thus, load rejection is the most important criterion
on which to make the decision of what variables to pair, and what controller
structure is best.
The RGA is useful for avoiding poor pairings. If the diagonal element in the
RGA is negative, the system may show integral instability: the same situation
that we discussed in the use of the Niederlinski index. Very large values of the
RGA indicate that the system can be quite sensitive to changes in the parameter
values.
163.2 Inverse Nyquist Array (INA)
Rosenbrock (Computer-Aided Control System Design, Academic Press, 1974) was
one of the early workers in the area of multivariable control. He proposed the use
of INA plots to indicate the amount of interaction among the loops.
In a SISO system we normally make a Nyquist plot of the total openloop
transfer function GM B. If the system is closedloop stable, the (- 1, 0) point will
not be encircled positively (clockwise). Alternatively, we could plot (l/G, B). This
inverse plot should encircle the (- 1, 0) point negatively (counterclockwise) if the
system is closedloop stable. See Fig. 16.4.
An INA plot for an Nth-order multivariable system consists of N curves,
one for each of the diagonal elements of the matrix that is the inverse of the