Files 2 Lectures Lec19

8/19/2019 Files 2 Lectures Lec19

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Interacting vs. Noninteracting Systems

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• 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.


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$igure %.&. A noninteracting s#stem:

two surge tan*s in series.

$igure '.1&. Two tan*s in series whose li!uid levels interact.


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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:


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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′


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( )

( )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@


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( )

( )

( )

( )

( )

( )

( )

( )

( )

( )) ) ) 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.


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7loc* 8iagram for 4oninteracting

/urge Tan* /#stem

$igure %.%. (nput"output model for two li!uid surge tan*s in

series.


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Dynamic Model of An Interacting Process

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( )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.


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( )

( )

( )( )

( )

( )

( )

( )

) )) )

)) )

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


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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

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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.


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• $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.

+'"-


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$igure '.1%. A multi"input, multi"output thermal miing process.


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