стр. 73 
Dm1 m2

set
m1
c1 +
GC1 G11
GV 1
c1
G21
G12
set
c2 + m2 c2
GC 2 G22
GV 2

Controller Process
Block diagram for a general 2 ВҐ 2 system.
Figure 94.1
195
DECOUPLING
cset +  m1
1
GC1 G11
GV 1
c1
D21
G21
D12 G12
cset + m2 c2
2
GC 2 G22
GV 2

Controller Process
Decoupler
Block diagram for a general 2 ВҐ 2 system with decoupler.
Figure 94.2
and D12 so that
Dc1
=0
Dm2 m1
From block diagram algebra,
Dc1 = D12GV1G11Dm2 + GV 2G12 Dm2 (94.1)
Dc 2 = D21GV 2G22 Dm1 + GV1G21Dm1 (94.2)
Setting Dc1 = 0 in Eq. (94.1),
GV 2G12
D12 =  (94.3)
GV1G11
and setting Dc2 = 0 in Eq. (94.2),
GV1G21
D21 =  (94.4)
GV 2G22
Usually, we lump the valve transfer functions with the process unit itself; therefore,
GP12
D12 =  (94.5)
GP11
GP 21
D21 =  (94.6)
GP 22
where GPij = GVjGij.
196 MULTIVARIABLE PROCESS CONTROL
There are several things that should be pointed out. If one looks at the method
to design the decoupler, and at its objective, one is reminded of the feedforward
controllers. The disturbance to a loop is the manipulated variable of the other loop.
Remembering that each process transfer function contains a Kij, a tij, and a toij, decou
pler D21 looks as follows:
K 21 t 22 s + 1 ( to 21 to 22 )s
GP 21
D21 =  = e
K 22 t 21 s + 1
GP 22
Thus, similar to feedforward controllers, the decoupler is composed of steadystate
and dynamic compensations. The difference is that, unlike feedforward controllers,
decouplers form part of the feedback loops and therefore they affect the stability.
Because of this, the decouplers must be selected and designed with great care.
For more extensive discussion on decoupling, such as partial or steadystate
decoupling and decoupling for n ВҐ n systems, the reader is referred to Smith and
Corripio [2].
94.2 Decoupler Design from Basic Principles
In Section 94.1 we showed how to design decouplers using block diagram algebra;
thus the decouplers obtained are linear decouplers. In this section we present the
development of a steadystate decoupler from basic engineering principles. The
resulting algorithm is a nonlinear decoupler. The procedure is similar to the one
presented in Chapter 7 for designing feedforward controllers.
Consider the blending tank shown in Fig. 91.5. In this process there are two com
ponents, salt and water; thus two independent mass balances are possible. We start
with a total mass balance:
W = W1 + W2 (94.7)
A mass balance on salt is used next:
W1x1 + W2x2  Wx = 0 (94.8)
From Eq. (94.7)
W1 = W  W2 (94.9)
From Eq. (94.8) and using Eq. (94.9) yields
x  x1
W2 = W1 (94.10)
x2  x
Realize that Eqs. (94.9) and (94.10) provides the manipulated variables W1 and
W2. However, we have two equations, Eqs. (94.9) and (94.10), and four unknowns,
W1, W2, W, and (x  x1)/(x2  x). Thus there are two degrees of freedom. Well, we
have two controllers, and we can let the controllers provide two of the unknowns.
197
REFERENCES
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