modelling module

8/10/2019 Modelling Module

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Modeling and Control

Of

an Aero-thermal Chanel

Students:

Danijel Matejevi- 245395

Jrme Clavel - 203156

Marko Petkovi- 205504


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Description of the system:

In this lab we will model a system which is very common industrial situations where

temperature control is required. The presence of transport delays is taken into account.

The apparatus consist of a plastic tube with a fan-controlled airflow through it. Directly after

the fan a heating-element is placed, which heats the air as it passes by. At the end of the tube

we have a temperature sensor that provides an output voltage proportional to the temperature

of the air in the tube. The control objective is to maintain the temperature at a desired level.

Modelling module

Preparations to do for the modelling module

From the figures 1 and 2 we can interpret each of the parameters A1, A2, T1, T2and . In this

case, A1 and A2 represent the maximum amplitudes for the output for some distant time.Concerning T1and T2, for the first transfer function it is the time constant or , as in the fig. ??.

For the second transfer function T2 is 1 + 2, since the pole of the transfer function is of

multiplicity 2. In both cases represents a time delay.

Figure 1: First order response


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Figure 2: Second order response

From the figure 3 we can compare the step responses of each transfer functions. The first

transfer function has a longer rise time while the second one is much steeper in response.

Figure 3: First and second order system step response

For the second function, when the output tries to reach the reference value, it has a small curve

when it start rising. Apart from that, the gain of the second function is higher than that of the

first.

By comparing where the sampling circles are placed on the output, it is impossible to

differentiate two outputs. In order to differentiate them, we would have to have much more

points. Thus, the sampling period chosen is not high enough so we have to increase it. A good

rule of thumb is that the sampling frequency should be 10 times higher than the frequency of

the signal being samples.


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Figure 4: First order transfer function Bode plot

Figure 5: Second order transfer function Bode plot

By comparing the Bode plots in figures 4 and 5 we can see that the second transfer function has

a slope of -40dB/decade after its cut-off frequency which is what we expect from a 2 ndorder

transfer function. If there was a delay, the phase plot would change a lot. We would have a big

shift in phase in this case.

-10

0

10

20

Magnitude

(dB)

10-1

100

101

102

-90

-45

0

Phase(deg)

Bode Diagram

Frequency (rad/s)

-40

-20

0

20

40

Magnitude(dB)

10-1

100

101

102

-180

-135

-90

-45

0

Phase(deg)

Bode Diagram

Frequency (rad/s)

A1(db)

T1

T2

A2(db)


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System calibration

Before we go ahead and start the measurements, we have to calibrate our system in order to

stay on the linear range of the temperature sensor. In order to identify this range we perform a

measurement in order to identify the relation between the input and output of the process. The

fan speed is set to 4 and we increase the input from 0 to 10, each step being 1V. The obtainedresults are shown in table 1 and figure 6

From the previous figure we can identify a linear region is in between the input voltage being

2 and 5 V. During our experiments we have to operate in this region and therefore we have

always to ensure that we operate in that region. The linearized region is also presented in figure

7.

Figure 7: Linear region of operation

From the previous figure we can identify the static gain A being 1.41.

y = 1.41x -2.11

0

1

2

3

4

5

6

2 3 4 5

Linear region

measurement

Linear (measurement)

Proces

input

Proces

output

0 0.1

1 0.27

2 0.4

3 2.12

4 3.64

5 4.946 5.52

7 5.84

8 6.16

9 6.2

10 6.55

Table 1

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10

Calibration

measurement

Figure 6: Calibration of the system


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Step response

In order to observe the step response of the system we remain in or linear region and apply a

step change of the input voltage of 3 V. After the applied step change we could observe the

change of the output signal and this result is shown in figure 8.

Figure 8: Step response in the linear region

The time-domain identification of the step response is given in figure 9. From this figure we

can identify the model of our system as well as its parameters. We can identify that our system

is of second order with two time constants and where = . We can also identify thepresence of the delay d. The parameters of the system are:

= =0.21

= 0 . 9 6 = 0 . 2 3Harmonic response

In order to derive information of the harmonic response of the system we will perform a PRBS

(Pseudo Random Binary Sequence) with 1.5 V of amplitude and 4.5 V offset value in order to

remain in the linear zone. Here we will perform two experiments whereby we will set the

frequency ratio once being 1 and 10. The results obtained from the frequency-domain are show

in the following figures.In figure 10 we observe the frequency response of the system and represent the results with a

Bode diagram. When we set the model being of second order including the delay we can see

that the measured values match very well the theoretical one. Also we can observe that the

parameters slightly differ then the one obtained in the time domain. After a short discussion

among the group members and the teaching assistants we decide to use the time constant values

and the gain from the frequency analysis as more accurate and therefore more valid for later

calculations. The new parameters which will later be used for calculations are:

= =0.27; = 1 . 0 9; = 0 . 2 3;By analysing the figure 11 where the frequency ratio was set to 10, we can again identify the

same parameters for the observed system. By comparing the results for ratio 10 and ratio 1 we


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see that the derived results are the same, but we could also observe that the calculation time

for ratio 10 was much longer due the fact that we have much more samplings (ten times more

than with ratio 1). Therefore we deduce that the most appropriate ratio is 1 in order to have

faster processing time.

Figure 9: Time-domain identification of step response

Figure 10: Frequency-domain identification of PRBS with ratio 1


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Figure 11: Frequency-domain identification of PRBS with ratio 10

Preparation for the control module

Definition of the time constant and the cut-off frequency :

We know that

= 1 and we know that we want a


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The time is the time for which < < , (with = 1) we can easily

compute:

=

ln1

= 0.6

ln10. 9

=0.26[]

Thus we can deduce from the cut-off frequency = = 24.15 []Design of the PID controller (without pure time delay) :

As we have a system with 2 poles = = we will need a PID controller such as :

= (1 1

)

= 1 1

= + = 1.83 ( = 1.09 = =0.52 = =0.13

Sysquake validation:

Figure 12: Sysquake validation


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Figure 13: Sysquake Simulation

We observe in figures 12 and 3 that the time constant in the Sysquake simulation is also around

0.26 and thus validate our PID controller.

Using a Smith predictor

Now, we are requested to apply a Smith predictor in order to see how the system performance

improves or deteriorates with respect to the system parameters. So, first of all, in fig .14 we can

see the result without Smith predictor.

Figure 14: System without Smith predictor


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We designed our control for this behaviour. Now, we have some significant overshoot present

and a way to get rid of these is to apply a Smith predictor. The coefficients of the predictor or

the model are as follows, K = 1.09, T1= T2= 0.27 and the delay was, dh = 0.23. The results

with the predictor can be seen in figure 15.

Figure 15: System with smith predictor

As expected, the predictor corrected the overshoot and we get a step response we want.

In order to see the performance of the predictor we can, for example, change the delay part. In

fig. 16 we set a double delay of the one defined above. We noted that the performance of the

predictor deteriorates significantly. So the Smith predictor is dependent on the system and will

not work for any system you give it. The output will oscillate a lot around reference value and

will never stabilise and converge.

Figure 16: System with double delay time and Smith predictor

We also applied Time-domain identification to confirm our assumptions regarding the predictor

coefficients. In figs. 17 and 18 we see the predictor with the initial delay and increased delay

respectively.


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Figure 17: Time-domain identification of a system with Smith predictor

Figure 18: Time-domain identification of a system with double delay time and Smith predictor


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From the first figure we can confirm that the and K chosen correspond to what we have for

the model. From the second one, the gain is still the same but the time constant is definitely not,

since we increased the delay twice, the time constant increased for the same amount.

In summary, the Smith predictor is good if we have a precise model to work with. Also, a good

trade-off between robustness and performance can be obtained by appropriate tuning of theprimary controller. On the other side, it cannot be used with integrative and unstable processes

and the disturbance rejection response cannot be faster than that of the open loop.

modelling module

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