Page 158 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

Matrices and Higher-Order Process Models 133

The State-Space Version
Based on the reader's experience with studying the no-backflow case,
he should be able to rewrite Eq. (5-12) as follows. (Actually, the reader
might better off to try this as an exercise.)

.!!...(~·]=
2
dt
x3
1 1
pA•R12 PA1Rt2
1 -(p~R 12 + p~RJ
p~Rt2
1
0
pA3R23
Note that only the 1, 3 and 3, 1 positions in the A matrix are zero.
The full second row shows that the second tank is coupled with the
other two tanks.

Question 5-3 If, for the three-tank system, the A matrix were diagonal, what
would that mean physically and mathematically?

Answer The absence of off-diagonal terms in the A matrix means that there is no
cross coupling and that each tank acts completely independent of the others. A
diagonal A matrix also means that, instead of a set of three connected differential
equations, there are three separate first-order differential equations that can be
solved separately using techniques already presented in this book.

5-3 Control of Three-Tank System with No Backflow
These two example processes have a potential for control problems
because at high frequencies the phase lag approaches 270°. To make it
even more interesting, let's try integral-only control which we know
adds an immediate 90° of phase lag to whatever is being controlled.
With integral-only control the open-loop transfer function for the
three-tank process with no backflow becomes

R 3 1 1 I I
G(s) = G (s)G (s) = ------- G =-
P c ~s+1~s+l~s+1s c 5
3 2 1