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

DESIGN OF CONTROLLERS FOR MULTIVARIABLE PROCESSES 603
more sluggish will be the setpoint and load responses. The method yields settings
that give a reasonable compromise between stability and performance in
multivariable systems.
3. Using the guessed value of F and the resulting controller settings, a multivariable
Nyquist plot of the scalar function IV&,,) = - 1 + Det [L + GYCioI &,,J
is made. See Sec. 16.1.2, Eq. (16.5). The closer this contour is to the (- 1, 0)
point, the closer the system is to instability. Therefore the quantity W/(1 + W)
will be similar to the closedloop servo transfer function for a SISO loop
G, B/(1 + G, B). Therefore, based on intuition and empirical grounds, we
define a multivariable closedloop log modulus L,,
I I
W
L,, = 20 log,, -
1+w
(17.10)
The peak in the plot of L,, over the entire frequency range is the biggest log
modulus L,“,““.
4. The F factor is varied until Lz”,“”1s equal to 2N, where N is the order of the
system. For N = 1, the SISO case, we get the familiar +2 dB maximum
closedloop log modulus criterion. For a 2 x 2 system, a t4 dB value of LE’,“”
is used; for a 3 x 3, +6 dB; and so forth. This empirically determined criterion
has been tested on a large number of cases and gives reasonable performance,
which is a little on the conservative side.
This tuning method should be viewed as giving preliminary controller settings
which can be used as a benchmark for comparative studies. Note that the
procedure guarantees that the system is stable with all controllers on automatic
and also that each individual loop is stable if all others are on manual (the F
factor is limited to values greater than one so the settings are always more conservative
than the Ziegler-Nichols values). Thus a portion of the integrity question
is automatically answered. However, further checks of stability would have
to be made for other combinations of manual/automatic operation.
The method weighs each loop equally, i.e., each loop is equally detuned. If it
is important to keep tighter control of some variables than others, the method
can be easily modified by using different weighting factors for different controlled
variables. The less-important loop could be detuned more than the moreimportant
loop.
Table 17.1 gives a FORTRAN program that calculates the BLT tuning for
the Wood and Berry column. The subroutine BLT is called from the main
program where the process parameters are given. The resulting controller settings
are compared with the empirical setting in Table 17.2. The BLT settings usually
have larger gains and larger reset times than the empirical. Time responses using
the two sets of controller tuning parameters are compared in Fig. 17.1.
Results for the 3 x 3 Ogunnaike and Ray column are given in Table 17.2
and in Fig. 17.2. The “ +4” and “ + 6” refer to the value of L,“,“” used. Both of
these cases illustrate that the BLT procedure gives reasonable controller settings.