Matlab Lab Report: Filter Design, System Identification and Control

This Matlab lab report investigates the effects of filters on signal processing and their application in control systems. The report details the modeling of first and second-order control systems using experimental data and the System Identification Toolbox, respectively. It differentiates between the step responses of first and second-order systems. The study includes the design of proportional and PI controllers to meet specific design specifications, analyzing their impact on the control system's performance. The report also analyzes the impact of filters and their time constants, and how to determine proportional and integral gains for controller design. Experimental results are presented to evaluate the performance of proportional and PI controllers in achieving the design targets. The report uses Harvard referencing style to support arguments with books, peer-reviewed journal articles, and conference proceedings.

Matlab Lab Report
Name of the student
Name of the Institution

AIM AND OBJECTIVE
 Investigate the effects of filters
 To model first order control system using experimental data
 To model second order system using System Identification Toolbox
 To differentiate between first and second order systems utilizing step response
 Design of proportional and PI controllers to meet design specifications
 To investigate effects of proportional and PI controller on a control system
RESULTS AND ANALYSIS
Impact of using a filter
A filter in signal processing, is a device that eliminates or suppresses unwanted components from
a certain signal. A filter is used widely in control systems, electronics, communication devices,
image processing among other applications. Filters are normally used together with PI, PD or
PID controllers to achieve robust controlling system. In signal processing, filters reduce the level
of noise or distortion from a signal, tunes or smoothen a signal and can also recover signals. The
behavior of a filter can be analyzed or described by the transfer function in frequency domain of
either first or second order.
The transfer function of a filter is expressed by the equation, H(s) = C(s) / R(s) where C(s)
represent the output and R(s) the input of a filter. Transfer function of a filter can be expressed in
form of their time constant, τ. Transfer function of a filter differs depending on the type, as there
are several of them. The filter used in the experiment was a first order system, with the transfer
function given as
G(s)filter = 1
τs+1 , with time constant, τ set to default, that is 0.15. Time constant, τ characterizes
how fast the filter responds to the step input of first order systems, a higher time constant has
less steep slope and lower time constant has more steep slope of the step response.
The impacts of using filters was demonstrated in the experiment by analyzing the output signal
of both unfiltered and filtered model of the DC motor. MATLAB’S Signal Processing Toolbox
offers filter designer app that enabled us to design a digital filter for this purpose. The signal of
both filtered and unfiltered output were plotted and the results displayed in figure 1. The blue
curve represents the filtered output. It has been smoothened by the filter, no noise or distortion is
experienced.

Figure 1: un-filtered and filtered output.
How to model first order systems
First order systems are defined as those whose output and input relationship are a first order
differential equation, an equation with one order derivative. First order systems have only one
element of storage, corresponding to the single order of its differential equation. That is, the
order of the differential equation is equal to the number of storage elements.
In general, the transfer function of a first order system is given by the equation below
G(s) = Output
Input = K
τs+1 ………………………..equation 1
Where K is the dc gain of the system and τ is the time constant. DC gain of a particular system is
defined as the ratio between the input signal and a steady or constant value of the output.
Equation 1 can be rewritten as
Output, c(t) = K
τs+1 * input, u(t)
The transfer function can be transformed to time domain by computing the inverse Laplace
transform of the equation. This gives

c(t) = K – exp(-t/τ), since u(t) = 1
For a dc gain of 10 and time constant of 1.5s the response of the system is as shown below. Time
constant, τ is the point at which the output is 63% of its steady state value, for this example the
steady state value is 10.
Figure 2: response of a system. Available from http://engineering.ju.edu.jo/Laboratories/04%20-
%20First%20Order%20System.pdf Accessed on February 9, 2020.
We can apply the same principle to derive a transfer function of the model in our experiment.
The steady state value 115.
DC gain, K = steady state value / 1
= 115 RPM
63% of steady state value = 0.63 * 115 = 72.5 RPM
Time value from figure 3 that corresponds to speed of 72.5 is approximately 0.35s
Thus, the transfer function of the first order system is given by
G(s) = Output
Input = K
τs+1 = 115
0.35 s+ 1

Figure 3: step response from experimental data
Second order model identified by System Identification Toolbox
System Identification Toolbox provides functions, blocks and an app that can be used to
construct wide range of models using measured input and output values. System Identification
Toolbox provides a platform for the user to create and use dynamic system’s models that are not
easily modelled using specification or first principal. Time domain input and output data were
imported to identify continuous time transfer function of the model. The System Identification
Toolbox then performs system identification to estimate the parameters of a user model.
Estimated model from the System Identification Toolbox is as shown below

Thus, the transfer function of the second order system derived from System Identification
Toolbox is
G(s) = 1.501∗105
s2+ 205.3 s+1313 …………….equation 2
The difference between simulated results and experimental results of a model identified using
System Identification is investigated as shown below. The coefficients of transfer function of
equation 2 (transfer function derived from System Identification Toolbox) are used to modify the
original block diagram so as give us simulated results. Step response of the two systems were
then plotted as shown in the figure 4. The experimental output signal is subjected to noise since
the model was estimated using un-filtered data.
Figure 4: experimental versus simulated output of DC motor.

Step response of both first and second order models are compared, as displayed in figure 5
below. From the figure, the first order model has a slower response as compared to second order
system.
Figure
5: step
response of first and second order systems
How to determine proportional gain of a controller
The transfer function of first order model obtained in step 10 of section 1 is given by:
Gp(s) = 115
0.35 s+ 1
And since the general form of the equation is K
τs+1 , time constant, τ = 0.35
With the proportional controller acting alone, the closed loop transfer of the system is given by
G(s)cl = Kp∗G (s)
1+ Kp∗G( s)

=
Kp∗115
0.35 s+1
1+ Kp∗115
0.35 s+1
= 115∗Kp
0.35 s+(1+115 kp)
Thus, time constant of the closed loop system is given by
τ = 0.35
1+ 115∗Kp
The equation above should give a time constant of 80% of the initial constant (0.35)
τ = 0.8 * 0.35 = 0.28
0.28 = 0.35
1+115∗Kp
0.28 + 32.2 * Kp = 0.35
Kp = 0.011
How to determine integral gain, Ki
Design specification is, for the system to achieve damping ratio of 0.5
A transfer function of a second order system is represented by the equation
G(s) = ω n2
s2+ 2∗Zeta∗ῳn∗s +ω n2 , where zeta is the damping ratio and ῳn is the natural frequency
The system consisting of integral controller and the plant is given represented by the transfer
function below
G(s) = Gp(s) * Gcontroller(s) = 115
0.35 s+ 1 * ki
s
= 115∗ki
0.35 s2+ s
= 328.6∗ki
s2+ 0.35 s
Closed loop transfer function is given by

G(s)cl = = 328.6∗ki
s2+0.35 s+328.6 Ki , …………………equation 3
G(s) = ω n2
s2+ 2∗Zeta∗ῳn∗s +ω n2 ……………….equation 4
Equating equation 3 and 4 we have
2 * Zeta * ῳn = 0.35
ῳn^2 = 328.6 Ki
damping ratio, zeta is given as 0.5
thus
2 * 0.5 * ῳn = 0.35
Therefore, ῳn = 0.35
But, ῳn^2 = 328.6 * Ki
Substituting for wn, we have
0.35^2 = 328.6 * Ki
Ki = 0.136
Transfer function of the PI controller can now be expressed as
G(s) = kp + ki
s
= 0.011 + 0.136
s
Impacts of proportional controller
The impact of a proportional controller is to reduce the steady state error of the system making it
more stable. The proportional controller can also be used to make slow system respond faster to
the input step.

REFERENCES
Kherfane, S., Hadjeri, S. and Zidi, S.A., 2018. THE IMPACT OF A FUZZY SELF-TUNING PI
CONTROLLER SUPPORTED BY STATCOM ON LCC-HVDC SYSTEM CONNECTED TO
WEAK AC GRID IN BOTH SIDE. Journal of Engineering Science and Technology, 13(5),
pp.1237-1252.
Messner, W.C., 2018, June. The Last Jedi: Step Response Identification of Overdamped Second
Order Systems. In 2018 Annual American Control Conference (ACC) (pp. 368-373). IEEE.
Taylor, G.S., Kostic, M.N., Bowling, D. and Starr, G.P., Stryker Corp, 2019. Thermal system
with step response. U.S. Patent Application 16/291,358.
Wang, J., Zhou, X. and Ma, Y., 2019, August. Mechanism Analysis and Simulation Study of
Static Difference Generated by Proportional Controller. In 2019 IEEE International Conference
on Mechatronics and Automation (ICMA) (pp. 51-57). IEEE.
Yakimenko OA. Engineering Computations and Modeling in MATLAB®/Simulink®. American
Institute of Aeronautics and Astronautics, Inc.; 2019 Dec 15.