Aerator Mixer Speed Control System Step Response...
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Aerator Mixer Speed Control System
Step Response Modeling
In this experiment a program that models the aerator mixing system for treatment
waste in Chattanooga was used to produce data used to show the relationship for the input
function and the output function of the system. The ultimate goal of this project was to create a
model for the Aerator system’s output for the ranges of 100 to 600 and 1100 to 1600 RPMs,
given a certain percentage of the motors total power.
This report will describe all relevant background information, the procedure taken and
the results of this experiment. The results will be followed by a discussion of the results and
The waste-water treatment plant uses aerobic microbes to digest waste in
water. It uses an Aerator Mixer to provide the circulation needed for aerobic life. The basic
layout consists of a controls system that receives feedback from the motor and provides output
to it. This allows the motor to maintain a certain operating speed (Henry). The mixer speed
operates on a range of 2-17 RPM. The gearbox between the motor and the aerator is a 1 to 100
ratio, so the motor operates on a range of 200-1700 RPM (Henry). The output signal from the
control system to the motor varies from 0 to 100% of the rated motor power of the motor.
The Schematic of the system can be seen below. The SRC/247, SCZ/247, and ST/247
bubbles are the control system and the cylinder labeled M-247 is the motor.
Figure 1: Aerator System (Henry).
This diagram describes a feedback control loop. In this loop the transducer ST 247 reads the
RPM produced by the motor and transfers this into an electrical signal that is processed by the
analog controller, SRC 247. Once the controller has become aware that a deviation from set
point has occurred, the controller sends a signal, corresponding with the necessary changes in
input to meet the target output of the system, to the final control element SCZ-247. This
element then manipulates the power input to the motor.
The aerator mixer is generalized to a much simpler system as seen below. Where M(t) is
the manipulated variable and the input to our system. It is the percentage of power given to
the motor. The controlled variable, C(t), is the output of our system, or the RPM of the mixer.
Figure 2: Input-output relation (Henry)
This experiment used LabView for data acquisition, and excel for analysis and
presentation. The information path is shown in the schematic below.
Figure 3: Information paths for experiments
As the schematic describes the user gives inputs the LabView program which then tells the
equipment what actions to carry out. In this particular experiment the operator inputs are the
percentage of max motor power, duration of the experiment, step size, and when the step
takes place. The equipment (the aerator mixer system) provides feedback or data
measurements to LabView. LabView then outputs this data in the form of spreadsheets. The
data generated by the equipment consists of sampling signals received by the ST-247
transmitter and the signals sent out by the final control element.
Previous experiments on this system have been done to find the steady state operating
curve. This was done by running the system at a constant input, then graphing the output
generated as a function of time. An example of this can be seen below.
Figure 4 : SSOC for 20% input power
This graph shows the constant input as the green bar and the motor speed as the gold points.
After about 2 seconds the output becomes relatively constant. All the output data after this
point is averaged. Once enough steady state averages were obtained they were graphed
against their respective input power. A table of these averages and the graph are featured
Average output= 300 motor RPMs
Standard Deviation=4.45 RPMs
Figure 5: Average mixer speed at steady state for a constant input
The table shows the input on the left as a percentage of total motor power, and the
output in following column. A confidence interval of 95% is shown in the last column.
Figure 6: steady state operating curve of aerator mixer system
10 1.65 4.09 0.08
15 2.52 3.68 0.07
20 3.4 4.45 0.09
25 4.28 3.75 0.08
30 5.15 3.53 0.07
60 10.4 2.5 0.05
65 11.2 2.4 0.05
70 12.1 2.5 0.05
75 13.0 1.9 0.04
80 13.8 3.9 0.08
85 14.6 4.7 0.09
90 15.5 4.6 0.09
95 16.3 4.3 0.09
100 17.1 4.2 0.08
Input (%)OutPut (RPM of
the mixer)STDev STDev x 2
The operating curve shown in the graph above indicates a linear relation; almost all the data
points lie on the trend line.
In this experiment it is assumed that the relation between the input function, m (t), and
the output function c (t) is a linear first-order differential equation of the form shown below: �������� � ���� � �����
where � is the time constant and � is the steady state gain. The transfer function of this
equation is obtained by taking the Laplace transform, which results in: ��� � � �� � 1�� �
Because the control system for the aerator mixer is a feedback loop there could be a time delay
between the input and output. Once there is a change in the input, this change will travel
through the entire system before it reaches the controller. This time delay will modify the
transfer function to the one shown below ��� � �������� � 1�� �
Here t0 is the time delay or dead time. (Smith & Armando , 1997).
For this experiment the input function is a step function of the form below, where A is the
magnitude of the step, and td is the time at which the step occurs ���� � ���� � ���
The model for the first order plus dead time is derived by taking the inverse Laplace of our
transfer function substituting the previous equation in. ���� � ���� � �� � ���� 1� ����������� ��
The gain, dead time, and the first-order time constant were found by analyzing step
response of the system. As seen on the graph below td is the time when the step response
occurs, and t0 is the time from when the step occurs to when there is an actual response of the
system. A is the size of the input step. Dc is the change in of the steady state output after the
step. The gain is given by � �� . The first-order time constant,�, is the constant that forces the
model response to coincide with the actual response: ��� � �� � ����1� ��� � .632��
This equation is taken from the second edition of Principles and Practice of Automatic Process
Control. Its derivation can be found on pages 310 to 313.
Figure 7: Step response showing model variables
In order to produce an accurate model, multiple parameters for different steps covering each
section of the operating range. These parameters were found for positive step input functions and
negative step input functions.
For the 100-600 motor RPM operating range, six total trials were done. Three of these models
were of a step up situation. Three more were of a step down situation. Each step was of 10% and was of
a power percentage input of 10% to 40%.
For the 1100-1600 motor RPM operating range six total trials were done. Three were step
increases and three were step decreases. The step size was 10% each trial, ranging from 70% to 100%.
Using the labview software, we inputted different power percentages to run the motor in order
to produce outputs for the different runs within our individual operating ranges. The labview then
collected all of our output RPM’s from the motor and organized it into data that could be exported to an
Excel spreadsheet. From this point it was then organized and plotted. The output Rpm’s and input
power percentages were plotted against the time to observe the comparison between a FOPDT model
to actual recovered data.
When observing the constructed plots comparing the FOPDT models to actual collected data for
our ranges. It was obvious that the collected data of the motor driven aerator was very close to the
predicted behavior of the FOPDT model. Below is an example of one of the plots overlayed with the
Figure 8. Represents the plot of actual data with the FOPDT model of this system overlaid. The FOPDT
input and output lines a colored green and purple respectively.
The output plot of the calculate FOPDT model is very close in relation to the actual output of the motor
system (red line). Since the input observed is constantly regulated almost no difference is noticed
between it and the model and can be considered negligible.
Below is a table of some of the sample data that was used to construct the plot in figure 8.
Table 1: The left three columns are experimentally collected data, while the right two columns are
calculated data for the FOPDT model overlaid in figure 8. The box to the far right is values found from
the fit 2 analysis of the experimental data.
0 2 4 6 8 10 12
Value(%) Output(RPM) Input Output
0 20 0 20 340
0.102 20 33.027 20 340
0.206 20 160.527 20 340
0.309 20 250.251 20 340
0.413 20 287.45 20 340
0.516 20 308.812 20 340
0.62 20 322.438 20 340
The FOPDT model calculated data from table 1 uses variable from the box on the far right. These values
are the time step, power step percentage, gain, dead time, and time constant respectively. The inbl and
outbl are the base layer values used to correct position the model to the actual collected data more
accurately for better comparison.
In all output ranges management wanted to be analyzed, the accuracy of the FOPDT model to
the actual recorded data was very accurate (refer to the example for one range in Figure 8).
Table 1 quickly illustrates the values used to construct a FOPDT model for the system’s actual
data. The formulas for the curves that were used to plot the FOPDT model overlaying the actual data
plot was used to represent a perfect operating condition of the motor. In figure 8 it is obvious that the
actual data plot is not exactly matching the FOPDT model, there are several possibilities that would
cause this behavior. The most probable explanation of this non-matching behavior is that the motor in
the aerator assembly is spinning waste water using a fan at a 100:1 ration through a gearbox. The waste
water is most likely not consistent throughout, and would cause an non-constant impedance in the
motor through fan drag.
Additionally, the motor RPMs are reduced through a gear reduction, and depending on the
condition of the gearbox and backlash in gears, there is a chance that there is unseen impedance
through the gearbox. Also, the momentum of turning such a large mixer at this speed is so large that it
would be difficult to slow down as quickly as the FOPDT model requires it too.
If all impendences can be removed from the aerator systems and it allows completely fluid
motion startups and slow downs, only then will it be possible to match the FOPDT model for a unit step
system. However, for the measured system output RPM range that management required us to run; the
aerator system was very close to the FOPDT model estimate for a unit step operating condition.
Considering our analyzed data plot was so accurately close to the FOPDT model, in most running
conditions or operating RPM’s the model could be used to estimate future conditions without having to
run the motor in order to determine data. Such a situation as ours is very helpful when trying to
diagnose a problem with the system; such as if the motor is running at a RPM lower than the FOPDT
model calculated it to be at for a set power percentage. A case like this could be due to a running
problem or a worn out part of the system causing the motor to run lower than normal. Only way to tell
if the system is not performing up to full capacity would be to calculate the projected RPM it should be
running at using the FOPDT model.
Step Up: 10% to 20%
70% to 80%
0 2 4 6 8 10 12
Step Up 10% to 20%
0 2 4 6 8 10 12
Step Response: 70 to 80
Step Down: 20% to 10%
80% to 70%
0 2 4 6 8 10 12
Step Down 20% to 10%
2 4 6 8 10 12
Step Response 80% To 70%