Upload
sai-swaroop-mandal
View
222
Download
0
Embed Size (px)
Citation preview
8/19/2019 Files 2 Lectures Lec19
1/14
Interacting vs. Noninteracting Systems
C h a p t e r 6
• 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.
or o A change in one output affects the other outputs.
Otherwise, the process is called noninteracting .
• As an eample, we will consider the two li!uid"level storage
s#stems shown in $igs. %.& and '.1&.
• (n general, transfer functions for interacting processes are more
complicated than those for noninteracting processes.
8/19/2019 Files 2 Lectures Lec19
2/14
C h a p t e r 6
$igure %.&. A noninteracting s#stem:
two surge tan*s in series.
$igure '.1&. Two tan*s in series whose li!uid levels interact.
8/19/2019 Files 2 Lectures Lec19
3/14
C h a p t e r 6
11 1 +%"%-i
dh A q q
dt = −
1 11
1
+%"%-q h R=
/ubstituting +%"%- into +%"%- eliminates q1:
11 1
1
1
+%"0-idh
A q hdt R= −
$igure %.&. A noninteracting s#stem:
two surge tan*s in series.
Mass Balance:
Valve Relation:
8/19/2019 Files 2 Lectures Lec19
4/14
C h a p t e r 6
2utting +%"%- and +%"0- into deviation variable form gives
11 1
1
1+%"01-i
dh A q h
dt R
′′ ′= −
1 11
1+%"0)-q h
R′ ′=
The transfer function relating to is found b#
transforming +%"01- and rearranging to obtain
( )1 H s′ ( )1iQ s′
( )
( )
1 1 1
1 1 1
+%"0&-
13 1i
H s R K
Q s A R s s
′= =
′ + +
where and /imilarl#, the transfer function
relating to is obtained b# transforming +%"0)-.1 1 K R@ 1 1 13 . A R@
( )1Q s′ ( )1 H s′
8/19/2019 Files 2 Lectures Lec19
5/14
C h a p t e r 6
( )
( )1
1 1 1
1 1+%"0%-
Q s
H s R K
′= =
′
The same procedure leads to the corresponding transfer functionsfor Tan* ),
( )
( )) ) )
) ) ) )
+%"00-13 1
H s R K
Q s A R s s
′= =
′ + +
( )
( ))
) ) )
1 1+%"0'-
Q s
H s R K
′= =
′
where and 4ote that the desired transfer
function relating the outflow from Tan* ) to the inflow to Tan* 1
can be derived b# forming the product of +%"0&- through +%"0'-.
) ) K R@ ) ) ).3 A R@
8/19/2019 Files 2 Lectures Lec19
6/14
C h a p t e r 6
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )) ) ) 1 1
) 1 1
+%"05-i i
Q s Q s H s Q s H s
Q s H s Q s H s Q s
′ ′ ′ ′ ′=
′ ′ ′ ′ ′
or
( )
( )) ) 1
) ) 1 1
1 1+%"0-
3 1 3 1i
Q s K K
Q s K s K s
′=
′ + +
which can be simplified to #ield
( )
( ) ( ) ( )
)
1 )
1+%"0-
3 1 3 1i
Q s
Q s s s
′=
′ + +
a second"order transfer function +does unit# gain ma*e sense on
ph#sical grounds6-. $igure %.% is a bloc* diagram showing
information flow for this s#stem.
8/19/2019 Files 2 Lectures Lec19
7/14
7loc* 8iagram for 4oninteracting
/urge Tan* /#stem
$igure %.%. (nput"output model for two li!uid surge tan*s in
series.
8/19/2019 Files 2 Lectures Lec19
8/14
Dynamic Model of An Interacting Process
C h a p t e r 6
( )1 1 )1
1+'"5-q h h
R= −
The transfer functions for the interacting s#stem are:
$igure '.1&. Two tan*s in series whose li!uid levels interact.
8/19/2019 Files 2 Lectures Lec19
9/14
C h a p t e r 6
( )
( )
( )( )
( )
( )
( )
( )
) )) )
)) )
1 1
) )
1 ) ) 11 ) 1 ) ) 1 )
1 )
+'"5%-3 )93 1
13 )93 1
3 1
+'"5)-3 )93 1
where
3 33 3 3 , 9 , and 3 ;
)3 3
i
i
a
i
a
H s R
Q s s s
Q sQ s s s
H s K s
Q s s s
R A R R A R R
′=
′ + +
′ =′ + +
′ ′ +
=′ + +
+ ++@ @
(n Eercise '.10, the reader can show that 9
8/19/2019 Files 2 Lectures Lec19
10/14
Model Comparison
• Noninteracting system
( )
( ) ( ) ( )
1 1 1 ) ) )
)
1 )
where 3 and 3 .
1+%"0-
3 1 3 1i
A R A R
Q s
Q s s s
′=
′ + +
@ @
• Interacting system
( )( )
1 )
)) )
where9 1 and 3 3 3
13 )93 1i
Q sQ s s s
>
′
=′ + +
@
• General Conclusions1. The interacting s#stem has a slower response. +Eample: consider the special case where τ τ1= τ2.)
). ?hich two"tan* s#stem provides the best damping of inlet flow disturbances6
C h a p t e r 6
8/19/2019 Files 2 Lectures Lec19
11/14
C h a p t e r 6
Multiple-Input, Multiple Output
(MIMO !rocesses
• @ost 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 +@(@O- s#stems to distinguish them from the
simpler single"input;single"output +/(/O- s#stems that have
been emphasi=ed so far.
• @odeling @(@O processes is no different conceptuall# than
modeling /(/O processes.
8/19/2019 Files 2 Lectures Lec19
12/14
C h a p t e r 6
• $or eample, consider the s#stem illustrated in $ig. '.1%.
• ere the level h in the stirred tan* and the temperature T are to
be controlled b# adBusting the flow rates of the hot and cold
streams wh and wc, respectivel#.
• The temperatures of the inlet streams T h and T c represent
potential disturbance variables.
• 4ote that the outlet flow rate w is maintained constant and the
li!uid properties are assumed to be constant in the following
derivation.
+'"-
8/19/2019 Files 2 Lectures Lec19
13/14
C h a p t e r 6
$igure '.1%. A multi"input, multi"output thermal miing process.
8/19/2019 Files 2 Lectures Lec19
14/14
C h a p t e r 6