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

580 MULTIVARIABLE PROCESSES
G,+fB plane
w=o
FIGURE 16.4
SISO Nyquist and inverse Nyquist plots.
product of the GMCsj and &) matrices. If this product is defined as. Qeiw) =
GMfi,) &,,), then the INA plots show the diagonal elements of Q&. It is convenient
to use the nomenclature
and let the 0th element of the Q matrix be ~ij.
(16.35)
(16.36)
The three plots for a 3 x 3 system of 9r1, Qz2, and G33 are sketched in Fig. 16.5.
Then the sum of the magnitudes of the off-diagonal elements in a given row
of the $ matrix is calculated at one value of frequency and a circle is drawn with
this ra&us. This is done for several frequency values and for each diagonal
element.
I1 = I612 I + I613 I + . . . + I GIN I
f-2 = I421 I + I423 I + . . . + I d2N I
(16.37)
The resulting circles are sketched in Fig. 16.5 for the Gil plot. The circles are
called Gershgorin rings. The bands that the circles sweep out are Gershgorin
bands. If all the off-diagonal elements were zero, the circles would have zero
radius and no interaction would be present. Therefore the bigger the circles, the
more interaction is present in the system.
If all of the Gershgorin bands encircle the (- 1, 0) point, the system is
closedloop stable as shown in Fig. 16.6~. If some of the bands do not encircle the