Page 154 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

Matrices and Higher-Order Process Models 129

As is shown in App. G, solving Eq. (5-10) yields the eigenvectors
and eigenvalues of the matrix A. When the eigenvectors have nega-
tive real parts, the process represented by Eq. (5-7) is stable.
Following the development in App. G, the eigenvalues of the A
matrix can be determined by finding the values of A. that satisfy

0 0

0 =0

0

Therefore, the eigenvalues are equal to the poles of the transfer
function G.
It is important that the reader not move on until the material from
this section (and App. G) has been thoroughly digested.

5-2 Third-Order Process with Backflow
Figure 5-5 shows an interconnected three-tank system with forward
and backflow. If we treat each tank separately, the equations of Chap. 3
derived from mass balances can be applied immediately.

dX _ X -X
1
1
pAtdt-U R.2 2
,A dX 2 _ X -X 2 x -X 3
2
1
p .. ~dt- ~2 ~
(5-11)
~ dX 3 _ X -X 3 X 3
2
p dt- ~ R
3
Y=X
3
The variables X , ~, and X represent the levels in each of the
1
3
tanks. The net flow leaving the first tank is