EXPERIMENT 4

FIRST-ORDER AND SECOND-ORDER CONTROL SYSTEMS

OBJECTIVES
This experiment and lab-work give the student a basic understanding of PID Controller design (P,
PI, PD, and PID), the step response and impulse response, also the first order and second-order
system characteristics.

REFERENCES
1. Modern Control Engineering 5th Edition, K. Ogata

2. P. Kalita, and J.K. Barman, “Performance Analysis of Non-linear Jacketed CSTR based on
Different Control Strategies.”, Journal of Electrical and Electronics Engineering (AJEEE),
vol. 1, Sep 2017.

3. http://ctms.engin.umich.edu/CTMS/index.php?example=CruiseControl&section=System
Modelingm, Cruise Control System, CTMS, University of Michigan
4. https://en.wikipedia.org/wiki/Continuous_stirred-tank_reactor, CSTR, Wikipedia.org

EQUIPMENT REQUIRED
- MATLAB Software R2014 and up version (make sure you already download and install
it in your computer)
- The Simulation Script Package (intip.in/simulasikontrol)

PRE-EXPERIMENT TASK
1. What do you know about the PID controller? Explain it in a concise way!
2. What is the application of the PID controller in the real world? Give 3 examples!
3. What is step response and impulse response?
4. Why is the PID controller frequently employed to control various kinds of systems?
5. How to tune the PID controller?
6. What is an underdamped, critically damped, and overdamped system?

1
INTRODUCTION
First-order and second-order control systems play a critical role in the modern regime of control
engineering. Those two types of systems can be used to encompass the model of many systems in
the industrial plant until the cutting-edge systems. The way of controlling the system is also a
straightforward one because those two have their own unique characteristics, so we can specify
the design requirements and specifications that we would like to apply. The basic and conventional
controller widely used in the first order and second-order systems is The PID Controller.

Proportional-Integral-and-Derivative Controller or simply abbreviated as PID Controller is a

simple and straightforward control algorithm that is widely used in various real-world applications.
The PID controller is oftentimes utilized in every type of plant because it is indeed uncomplicated-
to-tune and cheap yet powerful for controlling and regulating the plant. The PID Controller mainly
consists of three main components, namely the Proportional Controller (P), Integral Controller (I),
and Derivative Controller (D). Figure 1 shows the block diagram of the PID Controller

Figure 1. The PID Controller Block Diagram

Furthermore, because this controller is vastly used in many applications, thus it is indeed
compulsory to learn how this controller works and how to tune this controller. For an illustration,
the actual use of the PID Controller can be seen in a Continuous Stirred Tank Reactor (CSTR).
The PID Controller is utilized to control the temperature of the chemical agent inside the tank.
Hence the temperature of the tank can reach the desired and safe level reference value. Figure 2
illustrates the CSTR plant.

2
 Figure 2. CSTR Plant Illustration

SIMULATION ENVIRONMENT PREPARATION

Follow the following steps for prepping the simulation environment script package on the
MATLAB software,

1. First, download the software package from this link. (intip.in/simulasikontrol)

2. Extract the downloaded software anywhere you want
3. After that, there will be two files. Namely simulasip4.fig and simulasip4.m

4. Open the simulasip4.m with your MATLAB by either clicking two times the file or
directly open the file from MATLAB.

3
5. Next, after the file opened, then click the run button on MATLAB and wait for the
loading process.

6. Finally, when the application is appearing, your simulation environment is ready to use!

4
➢ Further explanation about the simulation environment, as we look at the picture above there
are several numbers and signs that define each featured menu of the simulation
environment.
A. The menu marked by number 1 is the feature that allows you to choose the order of
the system either first-order or second-order.

B. The menu marked by number 2 is the feature that allows you to rather activate PI,
PD, P, or PID Controller.

C. The menu marked by number 3 is the feature that allows you to enter the value of
second-order parameters.

D. The menu marked by number 4 is the feature for saving, activate or not the grid on
the plot, and also zoom in-out the plotted response. The save button automatically
saves the plotted figure inside the folder used to open the script with “.fig”
extension file.
E. The menu marked by number 5 is for entering the value of the used controller.
F. The menu marked by number 6 is the feature that allows you to enter the value of
first-order parameters.

5
 G. The menu marked by number 7 is the feature for testing the system with step signal
and impulse signal. In addition, you can also enter your desired step amplitude
(reference signal) for the step signal.
H. The menu marked by number 8 is the menu for displaying the system response plot.

➢ All of the systems will be automatically changed to the closed-loop form with the unity
feedback and the block diagram as can be seen below,

Where,
● C(s) is the controller transfer function
● G(s) is the plant transfer function

6
EXPERIMENT
Experiment 1. The First-Order System
Cruise Control System
Automatic cruise control is an excellent example of a feedback control system found in many
modern vehicles. The purpose of the cruise control system is to maintain a constant vehicle speed
despite external disturbances, such as changes in wind or road grade. This is accomplished by
measuring the vehicle speed, comparing it to the desired or reference speed, and automatically
adjusting the throttle according to control law.

Assuming the mass of the vehicle (m) is equal to 2000 kg and the damping coefficient equal
to 60 N.s/m. By taking the Laplace transform of the governing differential equation and assuming
zero initial conditions, we find the transfer function of the cruise control system to be [3],

1 1
𝐺(𝑠) = =
𝑚𝑠 + 𝑏 2000𝑠 + 60
Note: The further modeling process can be seen in [3].

A. Operational Procedure
1. Select the first-order on the system order’s menu on the simulation application
window.

2. Later on, set the system parameter as the aforementioned transfer function of the
cruise control system.

3. First, test the system with an impulse signal to obtain the impulse response, then
save and record the response!.

4. Second, look at Table 1, from the transfer function of the cruise control system
above determine the time constant of the system.

7
 5. Jump to Table 2, do the experiment by applying the specified parameters from the
P controller gain table to simulation software, later fill the blank! (Reference point
is equal to step amplitude). We can get the information about the time domain
characteristics (settling time, etc) from the command window of MATLAB after
you run the simulation.

6. Don’t forget to save the figure after simulation and enclose it in your final lab-
work report!
7. Repeat the step for Table 3-4.

B. Experimental Data

Table 1. Cruise Control First Order System Characteristics

Time Constant (𝜏)

8
 Table 2. P Controller on First Order System

Reference Proportional Steady-State Steady-State Rise-time

Point Gain (Kp) Value Error (%) (s)

45 1

45 20

45 30

45 50

45 100

Table 3. PI Controller on First Order System

Reference Proportional Integral Steady-State Steady-State Rise-

Point Gain (Kp) Gain (Ki) Value Error (%)
time (s)

60 20 0.5

60 20 5

60 20 20

60 20 200

60 20 500

9
 Table 4. PD Controller on First Order System

Reference Proportional Derivative Steady-State Steady-State Rise-time

Point Gain (Kp) Gain (Kd) Value Error (%) (s)

30 20 1

30 20 5

30 20 20

30 20 40

30 20 100

C. Experiment and Additional Task 1

Answer the following questions concisely and correctly!

➔ Experiment Task 1

1. What conclusions can you get from the impulse response test on the system? How does the
system response converge to zero? Explain your reason for that!
2. From the data which was got from the experiment, answer the following questions:
a. What is the effect of increasing the proportional gain (Kp) on the system response?
b. What is the characteristic of a Proportional Controller (P)?

c. From Table 3, what is the effect of increasing the integral gain (Ki) on the system
response?

d. From Table 4, what is the effect of increasing the derivative gain (Kd) on the system
response?

e. What is the effect of using a PI Controller and PD Controller instead of merely a P

Controller?

10
➔ Additional Task 1

Suppose we have a first-order system as follows,

𝐾
𝜏𝑠 + 𝑁
Where,
K = Your Last Digit NRP / 100

Note: Last NRP Digit is your last three-digit or two-digit number depend on your NRP
number, for instance (0711xxxxxxx088). Thus the last digit is 88

1. Test the impulse response for the system and save the figure!

2. Simulate the system by inputting step signal and varying the time constant (τ) as well N
as follows, then observe and note the effect!
a. τ = 0.001s and N = 20
b. τ = 0.1s and N = 20
c. τ = 5s and N = 20
d. τ = 5s and N = -20
What is the effect of varying the time constant (τ) on the system response?

3. From the data obtained from question number 1, is the system stable for all time constant
and N-values combination? Why could it be that?
4. Do the following experiment on the system above and fill the table below,

11
 Reference Kp Kd Ki Steady-State Steady-State Settling-Time (s)
Point Value Error (%)

10 100 0 0

10 100 10 0

10 100 0 5

5. Design a PI controller for that system, so the system response follows the desired response
characteristics and giving a zero-offset error!

6. Draw the pole-zero map before and after the system is getting compensated by the PI
controller!

Experiment 2. The Second-Order System

The Continuous Stirred Reaction Tank (CSTR)

The continuous stirred-tank reactor (CSTR), also known as a vat- or back mix reactor, mixed
flow reactor (MFR), or a continuous-flow stirred-tank reactor (CFSTR), is a common model for a
chemical reactor in chemical engineering and environmental engineering. A CSTR often refers to
a model used to estimate the key unit operation variables when using a continuous agitated-tank
reactor to reach a specified output. The mathematical model works for all fluids: liquids, gases,
and slurries.

12
 Figure 3 The Real-Life Illustration of CSTR Plant

The behavior of a CSTR is often approximated or modeled by that of an ideal CSTR, which
assumes perfect mixing. In a perfectly mixed reactor, the reagent is instantaneously and uniformly
mixed throughout the reactor upon entry. Consequently, the output composition is identical to the
composition of the material inside the reactor, which is a function of residence time and reaction
rate. The CSTR is the ideal limit of complete mixing in reactor design, which is the complete
opposite of a plug flow reactor (PFR). In practice, no reactors behave ideally but instead fall
somewhere in between the mixing limits of an ideal CSTR and PFR [4].

In addition, the control objective of the CSTR plant, in this case, is to control the temperature
of the chemical inside it. Because the modeling process is the way too long to explain, Kalita et al.
[2] have modeled the system’s transfer function with the form of the second-order model as
follows,

0.991𝑠 + 16.62
𝐺(𝑠) = 2
𝑠 + 28.39𝑠 + 24.06

A. Operational Procedure
1. With the same simulation environment, now change the system type to a second-
order system by clicking the menu on the simulation application window.

13
 2. Later on, set the system parameter as the aforementioned transfer function of the
CSTR system.

3. First, test the system with an impulse signal to obtain the impulse response, then
save and record the response!.

4. Second, look at Table 5, from the transfer function of the CSTR system above find
the damping ratio and natural frequency of the system.
5. Moving on to Table 6, do the experiment by applying the specified parameters from
the table to simulation software later fill the blank!

8. Latter, don’t forget to save the figure after simulation and enclose it in your final
lab-work report!

Table 5. CSTR Second Order System Characteristics

System’s Damping Ratio (ζ) System’s Natural Frequency (𝜔𝑛 )

Table 6. PID Controller on Second Order System

Ref Kp Kd Ki Steady-State Steady-State Settling- Rise-Time Overshoot

Value Error (%) Time (s) (%)
(s)

10 10 0.5 5

10 10 5 10

10 10 20 20

10 50 0.5 5

10 50 5 10

10 50 20 20

14
 10 80 0.5 5

10 80 5 10

10 80 20 20

C. Experiment and Additional Task 2

Answer the following questions concisely and correctly!

➔ Experiment Task 2
1. What information about the nature of the system can we get from Table 5?
2. Why does the impulse response drive to zero? Explain this!
3. What is the effect of zero, which occurs on the CSTR system?
4. What do you know about the time domain specifications? Why is it important?

5. From Table 6, What is the effect of changing Proportional Gain (Kp), Integral Gain
(Ki), and Derivative Gain (Kd) on the PID Controller?

6. Design a PID Controller for the CSTR system, so the response has the following
characteristics! (Settling Time 2% with ts = 2s and Maximum Overshoot
Percentage 5%)

7. Implement the designed PID Controller from question number 6 to the Operational
Amplifier form!

➔ Additional Task 2
1. Find the transfer function and determine the nature (overdamped, critically damped,
or underdamped) of the RLC circuit below! (suppose the R = (Your Last Digit
NRP) Ohm, L = 300 mH, and C = 2 µF)

15
2. Why do we tend to learn about the first-order and second-order systems in detail
but not the higher-order one such as third-order or above?

3. Suppose you have a MIMO (Multi-Input and Multi-Output) system. Can we design
only one PID controller to control the system entirely? Explain your thoughts about
that!

4. Find and read a paper or journal article about the application of PID to the real-
world plant, then write the summary (min 4 lines) and the article title for the answer
to this question!

16