Process Control, 3P4

Assignment 7Kevin Dunn, kevin.dunn@mcmaster.ca Due date: 07 April 2014 (at class)

This assignment gives you practice with multiloop control.

Question 1 [5]

From a previous final exam

A 2x2 subsystem, derived from step inputs to a heavy oil fractionator, lead to the following transfer function model:

y1 =1.2e−27s

45s+ 1m1 +

4.5e−27s

50s+ 1m2

y2 =1.4e−8s

19s+ 1m1 +

4.0

13s+ 1m2

where

• y1 = top end point

• y2 = intermediate reflux temperature

• m1 = top draw rate

• m2 = intermediate reflux duty

Use a relative gain array analysis to calculate the appropriate loop pairing for this system.

Solution

The RGA analysis requires the steady state gain matrix, K = G(s = 0):

(y1y2

)=

1.2e−27s

45s+ 1

4.5e−27s

50s+ 11.4e−8s

19s+ 1

4.0

13s+ 1

( m1

m2

)y(s) = G(s)m(s)

Now set it to steady state, G(s = 0)

(y1y2

)=

(1.2 4.51.4 4.0

)(m1

m2

)y(s) = Km(s)

The relative gain array is Λ = K ⊗(K−1

)TΛ =

(−3.2 4.24.2 −3.2

)which indicates that the following pairing should be made (pick the pairing with λij values closest to 1, and positive)

• the top end point should be controlled by manipulating the intermediate reflux duty

• the intermediate reflux temperature should be controlled by changing the top draw rate

Question 2 [27 = 5 + 6 + 8 + 8]

From a previous final exam

Step tests on a distillation column with a methanol-water feed resulted in the following transfer function model:

y1 =10e−7s

15s+ 1u1(s)−

17e−2s

21s+ 1u2(s) +

4e−8s

15s+ 1d(s)

y2 =6e−7s

10s+ 1u1(s)−

17e−3s

12s+ 1u2(s) +

5e−3s

13s+ 1d(s)

where:

• y1 = overhead methanol mole fraction

• y2 = bottoms methanol mole fraction

• u1 = reflux flow rate

• u2 = reboiler steam flow rate

• d = feed flow rate

1. Draw a diagram of the equipment, showing the sensor and manipulated variable locations.

2. Determine the appropriate loop pairing for this system.

3. Draw a complete block diagram of the controlled system showing clearly all the transfer function elements.

4. Since the feed flow rate can be readily measured, design a feedforward control system for this process, andshow on your block diagram how it would be implemented along with the feedback control system. You mayexpress your design in terms of the transfer function element identifiers g11(s), gd(s) etc. without substitutingthe analytical expressions for the transfer functions.

Solution

1. An example of a generic distillation column is shown here; reboiler and reflux entry points vary from column tocolumn. The key features of the manipulated variables, controlled variables and disturbances are shown though.

2

2. Appropriate loop pairing can be selected using the relative gain array:

(y1y2

)=

10e−7s

15s+ 1− 17e−2s

21s+ 16e−7s

10s+ 1− 17e−3s

12s+ 1

( u1u2

)

K =

(10 −176 −17

)Λ =

(2.5 −1.5−1.5 2.5

)which indicates that the following pairing should be made (pick the pairing with λij values closest to 1, andpositive)

• the overhead methanol mol fraction should be controlled by manipulating the reflux flow rate

• the bottoms methanol mole fraction should be controlled by changing the reboiler steam flow rate

Note that the feed flow to the column is a disturbance. In most real systems we cannot adjust that feed flow rate,since it is set by operational constraints (we have to process N tons of material per day), so we cannot vary it.

3. The block diagram, with pairing and feedback and disturbances is shown below:

4. The feed flow is a valid disturbance to apply feedforward to, because it causes unacceptable deviation in the twocontrolled variables, it is measured, changes in either the reflex flow, (u1), or the reboiler steam flow, (u2), donot have a causal relationship to the flow, and the dynamics from flow to the two outputs is comparable (if a bitlonger) than from the two manipulated variables to the two controlled variables.

5. This question combines knowledge from two areas of the course: feedforward control and multiloop control.

The parts in red have been added to the block diagram to show how feedforward can be used in this context.The key insight is to recognize the same feedforward controller has to be used for both controlled variables, y1and y2.

3

Deriving the feedforward controller is no different to the single loop case: start from the controlled variable andwork backwards:

y1 = gd1d+ g12Gffd+ g11Gffd

Since we would like y1(s) = 0, we can solve for the feedforward controller:

0 = d (gd1 + g12Gff + g11Gff )

0 = gd1 + g12Gff + g11Gff

Gff (s) = − gd1g12 + g11

Similarly, we can derive a feedforward controller for the second output:

0 = d (gd2 + g21Gff + g22Gff )

Gff (s) = − gd2g21 + g22

Which one of these two to pick would depend on which output we would like to be least affected by thedisturbance.

A compromise is possible. Ignoring the dynamics of the system and dealing purely with the steady state gainswe would get (a suboptimal, but still perfectly functional controller):

Gff,1(s) = − gd1g12 + g11

= − 4

−17 + 10= 0.57

Gff,2(s) = − gd2g21 + g22

= − 5

6− 17= 0.45

Picking a value of Gff ≈ 0.5 will work well for both outputs in this scenario then.

END

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