Laboratory Experiment 2: Implementation of a Speed Controller
MTRN3020
Modelling and Control of a Mechatronics
I verify that the contents of this report are my own work.
Arthur Ching
z5162176
12/04/20
1. Introduction
This laboratory experiment was designed to create a speed controller implemented in a motorgenerator system. To design the controller, the so-called Ragazzini’s method and the direct analytical
design method is required. The controller design involves only a small amount of calculations leading
to a difference equation.
The first part of the experiment connects the system to a DC motor for initial rotational power. The
speed is shifted from 1000 rpm to 2000 rpm to simulate a plant generating power. Thus, using an
encoder, the shaft speed is measured at a constant sampling rate.
The second part of the experiment connects the system to a power grid. A load is connected to the
generator, thus simulating the changes in the motor speed as a result of a change in the number of
resistors connected. The number of resistors connected is determined by our unique student z-ID.
The report illustrates and documents the experimental process and analysis used to obtain both the
simulation and experimental results. As a result, this experiment can draw a conclusion between the
differences of each procedures.
2. Aims of the Experiment
The aims of the experiment are to design and model a controller using the direct analytical design
method and Ragazzini’s method to control the speed of a motor. The main goal of the controller is to
maintain that speed at the specified value regardless of load changes. Simulation and implementation
of the experiment designed through MATLAB’s Simulink should determine the difference between the
experimental and theoretical methods.
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3. Experimental Procedure
The experiment requires a motor that is connected to a generator, where the generators are driven
by an electric motor. A pulse width modulation (PWM) amplifier will generate an applied voltage
which becomes the control input to the system, thus the controller output becomes the input to the
PWM amplifier. The controller decides based on the error of the speed. The electric motor drives a DC
generator with identical parameters as the motor. A bank of 15 resistors is connected in parallel, each
with 150Ω of resistance, signalling the output of the generator. If there are zero resistors connected,
then the generator system runs on “no load”. If there are a certain number of resistors connected,
then there will be a current in the generator circuit, thus resulting in a load torque on the motor,
driving the generator. The speed will be affected as the load changes.
Pre-Lab Calculations
1. Download SpeedController.mexw64.
2. Run MATLAB
3. Change Folder to where you put SpeedController.mexw64
4. Find
your
desired
time
constant
and
sampling
time
from
the
spreadsheet
MTRN3020SpeedControllerLabTimesAndData_2020.xlsx.
5. Execute
the
following
command
in
MATLAB
>> y = SpeedController(YourDesiredTimeConstantInSeconds, YourSamplingTimeInSeconds)
Example
for
54
ms
desired
time
constant
and
8
ms
sampling
time
is
>> y = SpeedController(0.054, 0.008)
6. This will give you five values. We will use these numbers to generate your data files.
The five generated values y,
Y=
𝟎. 𝟎𝟎𝟎𝟏𝟏𝟕𝟕𝟏𝟕𝟓𝟐
− 𝟎. 𝟎𝟎𝟎𝟎𝟗𝟒𝟖𝟐𝟖𝟔𝟓
2
𝟎. 𝟗𝟑𝟑𝟔𝟑𝟎𝟕𝟔
𝟎. 𝟎𝟔𝟔𝟑𝟔𝟗𝟐𝟑
𝟏
Table 1: System Parameters
System Parameter
Desired Time Constant (s)
Sampling Time (s)
Value
0.054
0.008
Method
Part A
1. Enter calculated system values into the computer from SpeedController.
2. Observe the change in the motor speed of the motor-generator system.
3. Record experimental results.
4. Compare results with simulated results.
5. Superimpose the results on a graph together to draw conclusions.
Part B
1. Connect the Power Grid to the system.
2. Insert z-ID and system values to determine the number of resistors.
3. Observe the change in the motor speed of the motor-generator system.
4. Record experimental results.
5. Compare results with simulated results.
6. Superimpose the results on a graph together to draw conclusions.
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4. Controller Design Calculation
Figure 1 is the block diagram to be used for this experiment.
Figure 1: Speed Control Block Diagram
Step 1
Download the noload.m file from Moodle and load it into MATLAB. This file contains the response of
the motor when a voltage of 24 Volts is applied. The controller 𝐺𝑐 (𝑧) relates the error in
counts/second to PWM Units, thus the motor speed won’t need to be converted to any other units.
To obtain the transfer function that relates the applied voltage to the speed, it is required to use first
order response fitting to the motor response data. This will become,
𝐴
1+ 𝜏s
𝐺𝑝1(𝑠) =
(1)
Since the speed cannot be analysed in this experiment, the encoder that is used will sample the counts.
Thus, the speed is calculated, and a transfer function is formed relating voltage to the counts, by
adding an integrator. This results in,
𝐺𝑝 (𝑠) =
𝐴
𝑠(1 + 𝜏 𝑠)
(2)
To find the values of the system gain A and the time constant 𝜏. By extrapolating the set of data from
the noload.m file and plotting a graph of time (Column 1) as x-axis and the motor speed (Column 3) as
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y-axis. This will produce an experimental plot of the response of the motor. The values for A and 𝜏
can be determined using MATLAB as shown in Figure 2.
Figure 2: MATLAB Value Calculation
Since the system gain A, is simply the steady state of the graph, using the theoretical response of w(t)
allows us to find the time constant value. This is shown in Equation 3.
𝑤(𝑡) = 24𝐴 (1 − 𝑒 −
𝑡
𝜏)
(3)
To find the time constant value, at t = 𝜏 ,
𝑤(𝑡) = 24𝐴(1 − 𝑒 −1)
(4)
From MATLAB,
𝑨 = 𝟑𝟏𝟔𝟖𝟕. 𝟓 𝑎𝑛𝑑 𝒘(𝒕)𝝉 = 𝟒𝟖𝟎𝟕𝟐𝟕. 𝟕
By plotting a line of 𝒘(𝒕)𝝉 = 𝟒𝟖𝟎𝟕𝟐𝟕. 𝟕 across the theoretical graph, this can
approximately determine the time constant, as seen in Figure 3.
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Figure 3: Finding the time constant 𝜏
This results in the time constant as 𝝉 = 𝟎. 𝟎𝟑𝟕𝒔. Using these values, allows a comparison between
the theoretical and experimental, as seen in Figure 4.
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Figure 4: Theoretical vs. Experimental MATLAB Plot
Step 2
By combining all blocks in the block diagram, except 𝐺𝑐 (𝑧), the plant transfer function can be obtained
as follows.
𝐺𝑝 (𝑧) = 𝑍 [
𝐴 (𝑧 − 1)
24 (1 − 𝑒 −𝑠𝑇 )
]
𝑠
126
𝑠(1 + 𝜏𝑠)
𝑧𝑇
(5)
(𝑧 − 1)
𝑧𝑇
(6)
Break this into two parts,
𝐺𝑝 (𝑧) = 𝐴(𝑧)
Using the following commands in MATLAB, found from Moodle, this allows the calculation of A(z). The
system gains A, and the 𝜏 values are found from the previous part, whilst the T (sampling time) value
is found from the excel spreadsheet found on Moodle.
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>> num = [24*A];
>> den = 126*[tau 1 0];
>> [numd, dend] = c2dm(num,den,T,’zoh’);
>> roots(numd)
>> roots(dend)
This results in,
𝑟𝑜𝑜𝑡𝑠(𝑛𝑢𝑚𝑑) = −0.930487042825578
1
0.805561107667721
𝑟𝑜𝑜𝑡𝑠(𝑑𝑒𝑛𝑑) =
The c2dm function will automatically take into account of the zero-order hold (ZOH). 𝐴(𝑧) can now be
written in the form of,
𝐴(𝑧) =
𝐶(𝑧 − 𝑧1 )
(𝑧 − 1)(𝑧 − 𝑝1 )
(7 )
Where the constant C is calculated from the MATLAB commands.
By substituting (7) into (6), 𝐺𝑝 (𝑧) is now,
𝐺𝑝 (𝑧) =
𝐶(𝑧 − 𝑧1 )
𝑧𝑇(𝑧 − 𝑝1 )
(8 )
The following values can now be substituted into (8), where:
•
𝐶 = 4.863343080223641
•
𝑇 = 0.008 (𝑆𝑎𝑚𝑝𝑙𝑖𝑛𝑔 𝑇𝑖𝑚𝑒)
•
𝑧1 = −0.930487042825578
•
𝑝1 = 0.805561107667721
𝐺𝑝 (𝑧) =
𝐶(𝑧 − 𝑧1 )
4.8633(𝑧 + 0.9305)
=
0.008𝑧(𝑧 − 0.8056)
𝑧𝑇(𝑧 − 𝑝1 )
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(9)
Step 3
To obtain the 𝐹(𝑧), the given design specifications will need to be used. However, the controller must
also have a zero steady state error which can be achieved by eliminating the ringing zero introduced
by the negative zero in the number of 𝐺𝑝 (𝑧).
The location of the closed loop pole in the s-plane can be found through a first order response with a
desired time constant of 0.054 seconds,
𝑠= −
1
1
= −
= −18.5151852
𝜏
0.054
(10)
Thus, an approximation for 𝑧 can be obtained,
𝑧 = 𝑒 𝑠𝑇 = 𝑒 −18.5151852 ×0.008 = 0.8623033568
(11)
However, this system has a zero at 𝑧 = −0.930487, which is within the unit circle, but is on the
negative real axis. This would mean a ringing in the stable system and would be considered
undesirable. To eliminate the ringing, applying the stability constraint to make this zero absorbed by
the numerator of 𝐹(𝑧) will help this problem. Thus, the desired 𝐹(𝑧) becomes,
𝐹(𝑧) = 𝑏0
(𝑧 + 0.9305)
𝑧(𝑧 − 0.8623)
(12)
This also satisfy the causality constraint, thus substituting 𝐹(1) = 1, will result in 𝑏0 as,
𝑏0 =
(1 − 0.8623)
≈ 0.07132741124
(1 + 0.9305)
(13)
Hence, 𝐹(𝑧) is,
𝐹(𝑧) = 0.07133
(𝑧 + 0.9305)
𝑧(𝑧 − 0.8623)
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(14)
Step 4
To obtain the controller transfer function, knowing 𝐺𝑝 (𝑧) and 𝐹(𝑧) it is possible to use,
𝐹(𝑧)
1
𝐺𝑝 (𝑧) (1 − 𝐹(𝑧))
(15)
𝑧(𝑧 − 0.8623) − 0.07133(𝑧 + 0.9305)
𝑧(𝑧 − 0.8623)
(16)
𝑧 2 − 0.93363𝑧 − 0.066369
𝑧(𝑧 − 0.8623)
(17)
𝐺𝑐 (𝑧) =
Firstly, we need to find 1 − 𝐹(𝑧),
∴ 1 − 𝐹(𝑧) =
=
By substituting 𝐺𝑝 (𝑧), 𝐹(𝑧) and Eqn. 17, the controller transfer function is,
𝐺𝑐 (𝑧) =
=
(𝑧 + 0.9305)
0.07133
0.008𝑧(𝑧 − 0.8056)
𝑧(𝑧 − 0.8623)
∙
𝑧 2 − 0.9336𝑧 − 0.066369
4.8633(𝑧 + 0.9305)
𝑧(𝑧 − 0.8623)
(18)
0.07133(𝑧 + 0.9305)
0.008𝑧(𝑧 − 0.8056)
∙
4.8633(𝑧 + 0.9305) 𝑧 2 − 0.9336𝑧 − 0.066369
(19)
𝑀(𝑧) 0.000117336𝑧 2 − 0.00009453𝑧
≈
𝑧 2 − 0.9336𝑧 − 0.066369
𝐸(𝑧)
(20)
0.000117336 − 0.00009453𝑧−1
1 − 0.9336𝑧 −1 − 0.066369𝑧 −2
(21)
=
The coefficients were calculated through simplification, thus by cross-multiplying the coefficients, we
can simplify it to the implementation,
𝒎(𝒌) = 𝟎. 𝟗𝟑𝟑𝟔 𝒎(𝒌 − 𝟏) + 𝟎. 𝟎𝟔𝟔𝟑𝟕 𝒎(𝒌 − 𝟐) + 𝟎. 𝟎𝟎𝟎𝟏𝟏𝟕𝟑 𝒆(𝒌) − 𝟎. 𝟎𝟎𝟎𝟎𝟗𝟒𝟓𝟑 𝒆(𝒌 − 𝟏)
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5. Simulink Block Diagram
Part A – Design Verification
The Simulink block diagram for this part is designed correctly, if the actual response shows
zero steady state error and a first order response that shows a time constant of 𝜏𝑑 . The block
diagram was adjusted with no load to generate a plot displaying the motor speed changing
from 1000 rpm to 2000 rpm.
Figure 5: Simulink Block Diagram for Part A
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Part B – Disturbance Rejection
For the controller to be successfully designed, the resistors that are connected through the
LoadSim will momentarily drop the speed, however the controller will make the system get
back to its specified speed. The only difference between Part A and Part B, is the repeating
sequence (stair) that’s been added to the number of resistors. The values of the repeating
sequence are based on the z-ID 5162176 which has been converted into hexadecimal.
Figure 6: Simulink Block Diagram for Part B
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6. Part A – Discussion
The results produced by Part A’s Simulink block diagram was superimposed with the experimental
data as show in Figure 7.
Figure 7: Simulation vs Experimental Data for Part A
The time constant used for the design of the speed controller has proven to be very effective, thus
reflecting a successful result. It can be seen that the simulation performed a lot more smoothly
compared to the experimental results, however both responses are consistent. The prominent noise
that happens during the steady state, could be a result of various physical limitations. For example,
systematics errors that may have affected the results, could be the two motors connected to the
central shaft that drives the system. The motion of the motors may have produced small vibrations,
which could have affected the experimental data. To reduce the effect of this error, dampening
measures such as lubrication between the motors could help reduce the vibration together.
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7. Part B – Discussion
Figure 8: Simulation vs Experimental Data for Part B
In Figure 8, it is evident that the simulation and experimental data are similar and consistent. The
speed controller that was designed proved to be successful in rejecting the disturbances effectively.
Despite a slight initial overshoot in the motor speed for the experimental data, both results are in
accordant with each other. The spikes on the plot are a result of the changes to the resistor loads. The
simulation data drops faster and lower in motor speed than the experimental data but recovers faster.
The ambient noise within the experimental data could be a result of the resistors and wires. Since this
experiment, is set up as an electrical power-grid, applying a varying load pattern. In reality, these
resistors and wires are imperfect and will have slight deviations from their labelled resistance within
a tolerance. The load draws current through the changing resistors, thus allowing a built up in error
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through resistive inconsistencies. To minimise this effect, it may be possible to create a more sensitive
simulation which could result in a more accurate graph as it considers the varying changes of the load.
8. Conclusion
In conclusion, the method for designing a zero steady state error controller was successful, as a result
of proper planning and choice of the desired transfer functions and the plant. Using the direct
analytical design method, a speed controller was able to be developed and implemented, thus
controlling the rotation of a motor. Despite the simulation results being much smoother, the
experimental results proved that there are a large number of factors that may affect the system. The
prominent noises from the experimental data is inevitable, as a result of physical limitations, however
it is possible to improve on the experiment method, and use higher quality equipment, that can
imitate the simulated result.
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