Transcript No Slide Title

Interacting vs. Noninteracting Systems
• Consider a process with several invariables and several output
variables. The process is said to be interacting if:
o Each input affects more than one output.
Chapter 6
or
o A change in one output affects the other outputs.
Otherwise, the process is called noninteracting.
• As an example, we will consider the two liquid-level storage
systems shown in Figs. 4.3 and 6.13.
• In general, transfer functions for interacting processes are more
complicated than those for noninteracting processes.
1
Chapter 6
Figure 4.3. A noninteracting system:
two surge tanks in series.
Figure 6.13. Two tanks in series whose liquid levels interact.
2
Chapter 6
Figure 4.3. A noninteracting system:
two surge tanks in series.
dh1
 qi  q1
dt
Mass Balance:
A1
Valve Relation:
1
q1  h1
R1
(4-48)
(4-49)
Substituting (4-49) into (4-48) eliminates q1:
dh1
1
A1
 qi  h1
dt
R1
(4-50)
3
Chapter 6
Putting (4-49) and (4-50) into deviation variable form gives
dh1
1
A1
 qi  h1
dt
R1
(4-51)
1
q1  h1
R1
(4-52)
The transfer function relating H1  s  to Q1i  s  is found by
transforming (4-51) and rearranging to obtain
H1  s 
R1
K1


Qi  s  A1R1s  1 τ1s  1
(4-53)
where K1 R1 and τ1 A1R1. Similarly, the transfer function
relating Q1  s  to H1  s  is obtained by transforming (4-52).
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Chapter 6
Q1  s  1
1


H1  s  R1 K1
(4-54)
The same procedure leads to the corresponding transfer functions
for Tank 2,
H 2  s 
R2
K2


(4-55)
Q2  s  A2 R2 s  1 τ 2 s  1
Q2  s 
1
1


H 2  s  R2 K 2
(4-56)
where K2 R2 and τ2 A2 R2. Note that the desired transfer
function relating the outflow from Tank 2 to the inflow to Tank 1
can be derived by forming the product of (4-53) through (4-56).
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Q2  s  Q2  s  H 2  s  Q1  s  H1  s 

Qi  s  H 2  s  Q1  s  H1  s  Qi  s 
(4-57)
Q2  s 
1 K 2 1 K1

Qi  s  K 2 τ 2 s  1 K1 τ1s  1
(4-58)
Chapter 6
or
which can be simplified to yield
Q2  s 
1

Qi  s   τ1s  1 τ 2 s  1
(4-59)
a second-order transfer function (does unity gain make sense on
physical grounds?). Figure 4.4 is a block diagram showing
information flow for this system.
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Block Diagram for Noninteracting
Surge Tank System
Figure 4.4. Input-output model for two liquid surge tanks in
series.
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Chapter 6
Dynamic Model of An Interacting Process
Figure 6.13. Two tanks in series whose liquid levels interact.
1
q1   h1  h2 
R1
(6-70)
The transfer functions for the interacting system are:
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H 2  s 
R2
 2 2
Qi  s  τ s  2ζτs  1
(6-74)
Chapter 6
Q2  s 
1
 2 2
Qi  s  τ s  2ζτs  1
H1  s 
K1  τ a s  1
 2 2
Qi  s  τ s  2ζτs  1
(6-72)
where
τ= τ1τ 2 , ζ
τ1  τ 2  R2 A1
, and τ a
2 τ1τ 2
R1R2 A2 /  R1  R2 
In Exercise 6.15, the reader can show that ζ>1 by analyzing the
denominator of (6-71); hence, the transfer function is
overdamped, second order, and has a negative zero.
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Model Comparison
• Noninteracting system
Q2  s 
1

Qi  s   τ1s  1 τ 2 s  1
Chapter 6
where τ1
A1 R1 and τ 2
(4-59)
A2 R2 .
• Interacting system
Q2  s 
1
 2 2
Qi  s  τ s  2ζτs  1
where ζ  1 and τ
τ1τ 2
• General Conclusions
1. The interacting system has a slower response.
(Example: consider the special case where t = t1 t2.
2. Which two-tank system provides the best damping
of inlet flow disturbances?
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Multiple-Input, Multiple Output
(MIMO) Processes
Chapter 6
• Most industrial process control applications involved a number
of input (manipulated) and output (controlled) variables.
• These applications often are referred to as multiple-input/
multiple-output (MIMO) systems to distinguish them from the
simpler single-input/single-output (SISO) systems that have
been emphasized so far.
• Modeling MIMO processes is no different conceptually than
modeling SISO processes.
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• For example, consider the system illustrated in Fig. 6.14.
Chapter 6
• Here the level h in the stirred tank and the temperature T are to
be controlled by adjusting the flow rates of the hot and cold
streams wh and wc, respectively.
• The temperatures of the inlet streams Th and Tc represent
potential disturbance variables.
• Note that the outlet flow rate w is maintained constant and the
liquid properties are assumed to be constant in the following
derivation.
(6-88)
12
Chapter 6
Figure 6.14. A multi-input, multi-output thermal mixing process.
13
14
Chapter 6