## 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 ? &gt; ? or ? &lt; ? . (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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