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Answered: - Problem Statement After some experimental tests with step inputs,


Please can I get help with this process control homework? Document is attached. Thanks.


Problem Statement

 

After some experimental tests with step inputs, a (very slow) industrial-scale polymerization reactor is modeled

 

as

 


 

[

 


 

23 e?0.2 s

 

Y (s)

 

4.6 s+1

 

=

 

?0.2 s

 

Z (s)

 

4.7 e

 

2.2 s +1

 


 

[ ]

 

[ ]

 


 

?12 e?0.4s

 

?4.2 e?0.4s

 

1.8 s+ 1 U ( s)

 

3.4 s+1

 

+

 

?0.4 s

 

?0.4s D (s)

 

5.8 e

 

V (s ) ?0.60 e

 

1.8 s+ 1

 

2.0 s+1

 


 

[ ]

 


 

11\* MERGEFORMAT ()

 


 

where uppercase letters refer to Laplace transforms of corresponding lowercase letters; the outputs y , z

 

refer to reactor conditions, measured in real time; the manipulated inputs u , v refer to feed flow rates; the

 

disturbance d refers to the flow rate of a purge stream through the reactor; and time is measured in hours.

 

(All variables are in deviation form around a steady state.) Through some multivariable analysis it was decided to

 

regulate (i.e. to control by feedback) the output y by manipulating the input u , using part of the model in

 

eqn. 1, namely

 

?0.2s

 


 

?0.4s

 


 

23 e

 

?4.2 e

 

Y ( s )=

 

U ( s )+

 

D ( s ) ? G ( s ) U ( s ) +Gd ( s ) D(s )

 

4.6 s +1

 

3.4 s+1

 


 

22\* MERGEFORMAT ()

 


 

Various aspects of controller design will be examined below.

 

1. Starting with the Laplace-domain equation Y ( s )=G(s)U (s) from eqn. 2, write the ordinary

 

differential equation (ODE) connecting y (t) and its derivatives with u ( t ) , as well as the general

 

solution y (t) of that (ODE).

 

2. Write and plot the response of

 


 

y (t) when u ( t ) =? (t) , i.e. a unit impulse. Assume

 


 

y ( 0 )=0 .

 


 

3. Design a feedback controller for this process using the IMC method. Write both the transfer function

 

and ODE corresponding to that controller.

 

4. Tune the time constant ? of the above controller for robust stability, and show closed-loop

 

simulations indicating the closed-loop block diagram and the responses of y (t) and u(t)

 

unit step change in the disturbance d (t) for the following cases:

 

(a) Perfect model/process agreement; and

 

(b) Errors in the gain, time constant, and time delay of G ( s ) equal to ( +30 ,?30 ,+30 ) ,

 

( +30 ,+30 ,?30 ) . (For each parameter, p , assume preal =p model + error .)

 


 

to a

 


 

and

 


 

5. Approximate the controller of the above part (4.) by a PI controller, and repeat the closed-loop

 

simulations of part (4.). How do the results of this part compare to the results of part (4.)?

 


 

y is noisy, a first-order low-pass filter with transfer function

 

1

 

L ( s )=

 

is added, to smoothen out measurements. The resulting closed loop (in the traditional

 

?s+ 1

 


 

6. Because the measurement of

 


 

feedback and IMC forms) is as shown in the following figures, with corresponding closed-loop transfer

 

functions

 


 

Y ( s )=G y

 


 

SP

 


 

?y

 


 

( s ) Y SP ( s )+G r ? y ( s ) Gd ( s)D ( s ) +Gn ? y ( s ) N ( s )

 

?

 


 

33\* MERGEFORMAT ()

 


 

R (s)

 


 

U ( s )=G y

 


 

SP

 


 

?u

 


 

( s ) Y SP ( s )+G r ?u ( s ) G d ( s)D ( s ) +Gn ? u ( s ) N ( s )

 

?

 

R (s)

 


 

44\* MERGEFORMAT ()

 


 

Express the transfer function Q ( s ) in terms of G ( s ) , C ( s ) , L ( s ) , and write the closed-loop

 

transfer functions G y ? y ,G d ? y , G n ? y , G y ? u ,G d ?u , G n ?u in eqns. 3 and 4 in terms of each of the

 

following two sets of transfer functions:

 

(a) G ,C ,G d , L ; and

 

(b) G ,Q , G d , L.

 

SP

 


 

SP

 


 

7. Using the IMC feedback configuration of the above part (6.) select a Q ( s ) similar to that of parts (3.)

 

and (4.), and show the shape (asymptotes) of magnitude plots for the frequency response of the closedloop transfer functions G y ? y ,G d ? y , G n ? y , G y ? u ,G d ?u , G n ?u parametrized in terms of ?

 

and ? . Based on these plots, explain whether you would prefer ? > ? or ? < ? . (Assume that

 

process and model match perfectly. For G d ? y plot only the very low and very high frequency

 

asymptotes.)

 

SP

 


 

SP

 


 

8. Using band-limited white noise with power equal to 0.0001 and sample time equal to 0.001, show

 

closed-loop simulations indicating the closed-loop block diagram and the responses of y (t) and

 

u(t) to a unit step change in the disturbance d (t) for the following cases:

 

(a) Perfect model/process agreement, ?=0 ;

 

(b) Perfect model/process agreement, ?=? ; and

 

2

 


 

(c) Errors in the gain, time constant, and time delay of G ( s ) equal to ( +30 ,?30 ,+30 ) ,

 

?=? . (For each parameter, p , assume preal =p model + error .)

 


 

9. Because the disturbance d (t) can be measured, that measurement can be used in a feedforward

 

controller to augment the feedback controllers of parts (4.) and (5.). Design that feedforward controller,

 

i.e. derive its transfer function C FF ( s) and corresponding differential equation.

 


 

10. Using the controller C FF (s) of part (9.) and the IMC controller of part (6.), show closed-loop

 

simulations indicating the closed-loop block diagram and the responses of y (t) and u(t) to a

 

unit step change in the disturbance d (t) for the following cases:

 

(a) Perfect model/process agreement, ?=? ; and

 

(b) Errors in the gain, time constant, and time delay of G ( s ) equal to ( +30 ,?30 ,+30 ) , ? = ? .

 

(For each parameter, p , assume preal =p model + error .)

 


 

2

 


 

 


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