A TECHNICAL REPORT ON EXPERIMENTAL
WORK ON
THEORETICAL AND EMPERICAL MODELLING
OF A CASCADED THREE-TANK SYSTEM
SUBMITTED TO
THE LECTURER IN CHARGE
DR A.BAMIMORE
CHE 503
BY
ADEBAJO, Daniel Oluwafemi
CHE/2016/001
(GROUP 3)
IN PARTIAL FULFILMENT OF THE
REQUIREMENTS OF CHE 503
DEPARTMENT OF CHEMICAL ENGINEERING
OBAFEMI AWOLOWO UNIVERSITY
i
February, 2023
ii
Department of Chemical Engineering,
Faculty of Technology,
Obafemi Awolowo University,
Ile-Ife.
14th February, 2023.
The Lecturer In Charge,
Dr Bamimore,
Department of Chemical Engineering,
Faculty of Technology,
Obafemi Awolowo University,
Ile-Ife, Osun State.
Dear Sir,
LETTER OF TRANSMITTAL
In line with the objectives of CHE 503, I submit this report as a comprehensive overview
of the practical activities conducted on the “theoretical and empirical modelling of a
cascaded three-tank system”and the results obtained. This report serves as partial
fulfillment of the CHE 503 requirements. The names and matriculation numbers of the
group members can be found in Appendix A of the report. Thank you, Sir.
Yours Faithfully,
ADEBAJO Daniel Oluwafemi,
CHE/2016/001
iii
ABSTRACT
Cascade system modeling in chemical engineering is a powerful technique for
understanding, interpreting, and improving intricate processes, leading to more efficient
and sustainable industrial operations. Cascade systems are composed of multiple
interconnected units, which can exhibit dynamic behavior that is challenging to predict
analytically. By employing modeling, it is possible to simulate how the system would
perform under different operating conditions, such as changes in input variables and
disturbances, and evaluate its performance.
In this study, an experiment was conducted to link three tanks that have multiple outputs
but only one input, with a pump, controller, and sensors included in the system for complete
automation. A Simulink control model was developed to simulate the system, with the
primary aim of calibrating the apparatus and obtaining theoretical and empirical models
for the system.
To establish the model parameters, a nonlinear grey box model was used to develop a
theoretical model, and an empirical model was created using MATLAB's ARX function.
A comparison of the input-output data and the models was carried out to ensure that the
models accurately reflected the actual system, which confirmed that each model was
precise.
iv
TABLE OF CONTENT
Table of Contents
LETTER OF TRANSMITTAL ......................................................................................... iii
ABSTRACT....................................................................................................................... iv
TABLE OF CONTENT ...................................................................................................... v
LIST OF TABLES ............................................................................................................ vii
LIST OF FIGURES ......................................................................................................... viii
CHAPTER 1: INTRODUCTION ....................................................................................... 1
1.1 Background Information ........................................................................................... 1
1.1.1 Advantages of Cascaded Tank-System .............................................................. 2
1.1.2 Disadvantages of Cascaded Tank-Systems ........................................................ 2
1.2 Objectives ................................................................................................................. 3
1.3 Scope ......................................................................................................................... 3
CHAPTER 2 ....................................................................................................................... 5
2.1 Literature Review...................................................................................................... 5
2.2 Theory of Experiment ............................................................................................... 5
2.2.1
Development of first-principle based model for the cascaded tank system.. 5
CHAPTER 3: METHODOLOGY .................................................................................... 10
3.1 Apparatus ................................................................................................................ 10
3.2 Experimental Procedure .......................................................................................... 10
3.2.1 Interfacing of the cascaded-tank system with Arm-Cortex Micro-Controller
Board ......................................................................................................................... 11
3.2.2 Interfacing of the Arm-Cortex Microcontroller board with Host PC .............. 11
3.2.3 Signal Chain/Data transfer through the instrumentation system ..................... 11
3.3 Calibration of Height Sensors ................................................................................. 12
3.4 Calibration of Pump ................................................................................................ 12
3.5 Calibration of Control Valves ................................................................................. 12
3.6 Data Collection from the Experimental System ..................................................... 13
CHAPTER 4 ..................................................................................................................... 15
RESULTS AND DISCUSSION ....................................................................................... 15
4.1 Experimental Data .................................................................................................. 15
v
4.2 Plots of Parameters ................................................................................................. 22
4.2.1 Level Sensor Calibration Plots......................................................................... 22
4.2.2 Pump Calibration Plot ...................................................................................... 25
4.2.3 Calibration of Proportional Valve .................................................................... 26
4.3 Model Parameters ................................................................................................... 28
4.4 Empirical determination of a Linear Model for the system .................................... 32
4.5 Determination of Linear Model for the Tank by Linearization .............................. 37
4.6 Open Loop Stability of System ............................................................................... 38
CHAPTER FIVE .............................................................................................................. 42
CONCLUSION ANDRECOMMENDATIONS .............................................................. 42
5.1Conclusion ............................................................................................................... 42
5.2 Recommendations ................................................................................................... 42
REFERENCES ................................................................................................................. 43
APPENDIX ....................................................................................................................... 44
APPENDIX A ............................................................................................................... 44
List of Group Members ............................................................................................. 44
vi
LIST OF TABLES
Table 1: The Terminologies used for modeling .................................................................. 7
Table 2 Experimental Raw Data for the Calibration of Height Sensor of Tank 1 ............ 16
Table 3 Experimental Raw Data for the Calibration of Height Sensor of Tank 2 ............ 17
Table 4 Experimental Raw Data for the Calibration of Height Sensor of Tank 3 ............ 18
Table 5 Experimental Raw Data for Calibration of Proportional Valves ......................... 19
Table 6 Experimental Raw Data for Calibration of Pump to obtain U1 (cm3/s) ............. 20
Table 7 Experimental Raw Data for Calibration of Pump to obtain U2 (cm3/s) .............. 21
Table 8 Flow Coefficient for the Theoretical Model ........................................................ 31
Table 9 Stability of the empirical tank 1 transfer function ............................................... 39
Table 10 Stability of the empirical tank 2 transfer function ............................................. 40
Table 11 Stability of the empirical tank 3 transfer function ............................................. 41
vii
LIST OF FIGURES
Figure 1.1: Cascaded-tank system-Microcontroller - Host-PC interface............................ 4
Figure 2.1: The process and instrumentation diagram of the cascaded-tank system .......... 9
Figure 3.1: Experimental set up for A cascaded three-tank system ................................. 14
Figure 4.1: Height vs Count for Tank 1 ............................................................................ 22
Figure 4.2: Height vs Count for Tank 2 ............................................................................ 23
Figure 4.3: Height vs Count for Tank 3 ............................................................................ 24
Figure 4.4: PWM Count to Pump Flow rate ..................................................................... 25
Figure 4.5: PWM Count vs Fractional Opening for Valve 1 ............................................ 26
Figure 4.6: PWM Count vs Fractional Opening for Valve 2 ............................................ 27
Figure 4.7: Input-Output response plot of the 2000 data .................................................. 29
Figure 4.8: The comparison between the validation data and the nlgref model ............... 30
Figure 4.9: The 5-step predicted response comparison of the transfer function and the real
data for h1. ........................................................................................................................ 33
Figure 4.10: The 5-step predicted response comparison of the transfer function and the real
data for h2 ......................................................................................................................... 34
Figure 4.11: The 5-step predicted response comparison of the transfer function and the real
data for h3. ........................................................................................................................ 36
viii
CHAPTER 1: INTRODUCTION
1.1 Background Information
Cascade tank systems are a complex yet sophisticated setup used in the chemical
engineering industry to efficiently store and mix various chemicals. The system typically
comprises of multiple tanks arranged vertically in a cascading manner, enabling the
regulated flow of liquids from one tank to another. The tanks can be used for different
purposes such as storing raw materials, intermediate products, or finished products and for
blending chemicals to produce specific chemical compositions. The utilization of cascade
tank systems offers improved process control and enhances the efficiency of chemical
storage and mixing operations in the chemical engineering industry. Additionally, cascade
tank systems can also have various types of level sensors, pumps, and control valves
incorporated into their design, which allow for precise control and monitoring of the liquid
levels and flow rates within the tanks.
The cascaded three-tank system that was used for the experiment is composed of tanks of
three different shapes - cuboidal, trapezoidal, and parabolic - arranged vertically in a series.
The water is pumped from a reservoir to the uppermost tank by a submersible pump. Then,
the water flows through the valve openings to the middle and bottom tanks, under the
influence of gravity. There are three piezo-resistance sensors placed inside the tanks, which
measure the water levels. The aim of the cascaded tank system is to control the water level
in each tank by adjusting the pump flow rate and the percentage openings of the electricactuated proportional valves. Additionally, there are two 4-position angle-drain valves,
which are used to simulate leaks in the middle and bottom tanks.
The cascaded three-tank system is a remarkable solution for large-scale water management
challenges, including water distribution and storage, waste water treatment, and irrigation.
This system's ability to store a vast amount of water and distribute it as necessary makes it
highly efficient and suitable for multiple purposes.
One of its standout features is the real-time regulation of water levels through the use of
piezo-resistance level sensors. The sensors provide constant monitoring of the water height
inside each tank, and the control system can quickly adjust the pump flow rate and valve
openings accordingly. This enables the system to maintain optimal water levels within the
tanks, even in the face of changing conditions, providing reliable and efficient water
management.
The use of proportional valves and the pump flow rate allows for precise control of the
water flow between tanks. This helps to minimize the risk of overshoot, undershoot, and
oscillations, which can lead to inefficient operation of the system and potential damage to
the tanks or other components.
1
The ability to simulate leaks in the middle and bottom tanks through the use of angle-drain
valves is an important feature for testing and evaluating the performance of the control
system. This allows engineers and operators to evaluate the system's response to potential
issues and to determine if any changes or modifications are needed to improve its reliability
and resilience.
1.1.1 Advantages of Cascaded Tank-System
The cascade tank system offers several advantages:
1. Efficient Water Management: The system can store a large volume of water and
distribute it to multiple tanks as needed, making it ideal for large-scale water
management systems.
2. Real-time Level Regulation: The piezo-resistance level sensors provide continuous
feedback on the height of the water inside each tank, allowing the control system
to make adjustments to the pump flow rate and valve openings as needed. This
helps to ensure that the water levels in each tank remain within a desired range,
even under changing conditions.
3. Improved Process Control: The cascade tank system allows for improved process
control and increased efficiency in the storage and mixing of liquids. The system
can be used for a variety of purposes, such as storing raw materials, intermediate
products, or finished products, and for mixing liquids to create specific
compositions.
4. Versatile Applications: The cascade tank system is suitable for various
applications, such as water distribution and storage, waste water treatment, and
irrigation.
5. Leak Simulation: The angle-drain valves in the system can be used to simulate
leaks, making it an ideal tool for testing and evaluating the performance of water
management systems.
1.1.2 Disadvantages of Cascaded Tank-Systems
1. Complexity: Cascade tank systems can be complex in terms of the number of tanks
involved, the control systems required to regulate water levels, and the sensors used
to monitor the tanks.
2. Maintenance: Regular maintenance is required to ensure that the tanks and control
systems are functioning properly, which can be time-consuming and costly.
3. Cost: Cascade tank systems can be expensive to install and maintain, especially for
larger systems.
2
4. Leaks: The risk of leaks is higher in cascade tank systems, as water is constantly
being transferred between tanks. This can lead to environmental contamination and
loss of water.
5. Water Quality: If the water in the tanks is not properly managed, the risk of
contamination and degradation of water quality increases.
6. Power Dependency: Cascade tank systems are typically powered by electricity, so
they are dependent on a reliable power supply. In case of a power outage, the system
may not function as intended, leading to problems with water distribution and
management.
7. Regulation Compliance: Cascade tank systems may be subject to various
regulations, depending on their location and the intended use of the system.
Compliance with these regulations can be complex and time-consuming.
1.2 Objectives
The overall aim of this experiment is to investigate the dynamic behavior of cascaded threetank system. The specific objectives of this experiment are to:
1) Determine the calibration equations for level sensors and pump
2) Estimate the parameters (i.e.,) of theoretical model for the cascaded three-tank
system
3) Determine an empirical model for the tank system from input/output data.
4) Validate the models identified in (ii) and (iii) above
1.3 Scope
Determine the theoretical model for the cascaded three-tank system, empirical model for
the tank system using input/output data, and calibration equations for level sensors and
pump to determine the dynamic behavior of the cascaded three-tank system.
3
Figure 1.1: Cascaded-tank system-Microcontroller - Host-PC interface
4
CHAPTER 2
2.1 Literature Review
Cascaded tank systems are a popular control problem in chemical and process engineering.
The system consists of two or more tanks, where the outflow from the first tank is the
inflow to the second tank, and so on. The aim of control is to maintain the level of liquid
in the tanks, despite variations in the inflow and outflow rates. A range of control methods
have been developed for cascaded tank systems, including PID controllers, fuzzy logic
controllers, and model predictive controllers.
One approach to control cascaded tank systems is to decouple the system into two single
tanks, and then use a single loop control method for each. Kim et al. (2016) used this
approach, implementing PID control for each tank. Subbiah and Arvind (2017) used fuzzy
logic control, and Zaky et al. (2015) used model predictive control.
Another approach is to use a cascaded control strategy, where the output of the first loop
is used as the setpoint for the second loop. This strategy allows for more precise control,
but can be more difficult to implement. Chang and Wu (2014) used a multi-objective
optimization algorithm to tune a cascaded control system for a two-tank system.
Finally, there have been efforts to develop more advanced control strategies for cascaded
tank systems. Zhang et al. (2020) used a robust output feedback control method to control
a nonlinear cascaded tank system, while Chen and Lee (2019) used adaptive dynamic
programming to optimize the control of a cascaded tank system with an uncertain inflow
rate.
2.2 Theory of Experiment
The three-tank system is a prevalent laboratory arrangement in control theory, frequently
utilized as a showcase for liquid level control. The setup encompasses two control valves
that manage the flow to tanks 1 and 3, a pump that transfers water from a reservoir to the
tanks through a Rotameter and control valve, and a differential pressure transmitter that
measures the liquid level in the tanks and transmits the readings to the control room.
2.2.1 Development of first-principle based model for the cascaded tank
system
5
The definitions of the terms used in the modeling process are presented in Table 1. To find
the dynamic equation that governs the liquid level in each of the three tanks, the commonly
utilized mass balance equation was applied, which states:
Accumulation = In – Out
(1)
The time rate of change of liquid level inside each tank is given by;
𝑑ℎ𝑖
1
=
(𝐹 𝑖𝑛 (𝑡) − 𝐹𝑖𝑜𝑢𝑡 (𝑡))
𝑑𝑡
𝐴𝑖 (ℎ𝑖 ) 𝑖
(2)
where ℎ𝑖 , 𝐴𝑖 (ℎ𝑖 ), 𝐹𝑖𝑖𝑛 (𝑡) and 𝐹𝑖𝑜𝑢𝑡 (𝑡) represent the liquid level, cross-sectional area, inflow
rate, and outflow rate, respectively, for the th tank.
According to Bernoulli's law for fluid flow through small openings, the rate of fluid exiting
the bottom of each tank is represented by:
𝐹𝑖𝑜𝑢𝑡 (𝑡) = 𝐶𝑖 𝑆𝑝 √2𝑔ℎ𝑖 (𝑡)
(3)
Assuming that the configuration of the tanks, pipes, and valves cannot ignore the
turbulence and acceleration of the liquid flow in the tubes, a more extensive expression is
used to describe the liquid discharge as follows:
𝐹𝑖𝑜𝑢𝑡 (𝑡) = 𝐶𝑖 𝑆𝑝 (2𝑔ℎ𝑖 (𝑡))𝛼
(4)
Using the above assumption, the time rates of change of liquid levels inside tanks are given
by
𝑑ℎ1
1
=
(𝑈1 − 𝐶1 𝑆𝑝 (2𝑔ℎ1 )𝛼1 − 𝐶4 𝑆𝑝 (2𝑔ℎ1 )𝛼4
𝑑𝑡
𝐴1
(5)
𝑑ℎ2
1
=
(𝐶 𝑆 (2𝑔ℎ1 )𝛼1 + 𝐶4 𝑆𝑝 (2𝑔ℎ1 )𝛼4 − 𝑈2 𝑆𝑝 (2𝑔ℎ2 )𝛼2
𝑑𝑡
𝐴2 (ℎ2 ) 1 𝑝
− 𝐶2 𝑆𝑝 (2𝑔ℎ2 )𝛼5
(6)
𝑑ℎ3
1
=
(𝑈 𝑆 (2𝑔ℎ2 )𝛼2 + 𝐶2 𝑆𝑝 (2𝑔ℎ2 )𝛼5 − 𝑈3 𝑆𝑝 (2𝑔ℎ3 )𝛼3
𝑑𝑡
𝐴3 (ℎ3 ) 2 𝑝
− 𝐶3 𝑆𝑝 (2𝑔ℎ3 )𝛼6
(7)
The cross-sectional areas (A1(h1), A2(h2) and A3(h3)) are given as:
𝐴1 = 𝑎𝑤
(8)
6
𝐴2 (ℎ2 ) = 𝑐𝑤 +
ℎ2
𝐻𝑚𝑎𝑥
(9)
𝐴3 (ℎ3 ) = 𝑤√𝑅 2 − (𝑅 − ℎ3 )2
(10)
The existing physical parameters used in the model are: Sp = 1.267; a = 25 cm; b = 34.5
cm; c = 10 cm; w = 3.5 cm; g= 981 cm2/s; Hmax = 35 cm; R = 36.4 cm.
Table 1: The Terminologies used for modeling
Variable
Definition
𝑈1
Pump flowrate into the first tank [cm3/s]
𝑈2
Fractional opening for the proportional valve of the second
tank
𝑈3
Fractional opening for the proportional valve of the third tank
𝐶1
Fractional opening for the manual valve of the first tank
𝐶2
Fractional opening for the 4-Position valve of the second tank
𝐶3
Fractional opening for the 4-Position valve of the third tank
𝐶4
Fractional opening for the second manual valve of the first
tank
ℎ1
Water level in the first tank [cm]
ℎ2
Water level in the second tank [cm]
ℎ3
Water level in the third tank [cm]
7
𝐴𝑖 (ℎ𝑖 )
Cross sectional area of tank at water level
[cm2]
𝑔
Gravitational constant [cm2/s]
𝑆𝑝
Cross-sectional area of the connecting pipe [cm2]
𝐻𝑚𝑎𝑥
Maximum Height of the tank [cm]
𝑅
A dimension on tank three [cm]
𝑎
A dimension on tank one [cm]
𝑏
A dimension on tank two [cm]
𝑐
A dimension on tank two [cm]
𝑤
Cross-width dimension of the tanks [cm]
𝛼1
Flow coefficient for manual valve 1 on tank one
𝛼2
Flow coefficient for Proportional valve 1 on tank two
𝛼3
Flow coefficient for Proportional valve 2 on tank three
𝛼4
Flow coefficient for manual valve 2 on tank one
𝛼5
Flow coefficient for 4-Position valve 1 on tank two
𝛼6
Flow coefficient for 4-Position valve 2 on tank three
8
Figure 2.1: The process and instrumentation diagram of the cascaded-tank system
9
CHAPTER 3: METHODOLOGY
3.1 Apparatus
1. Cascaded Three-Tank System
2. Graduated cylinder
3. Funnel
The cascaded three-tank system is made up of three tanks, one with a rectangular shape,
another with a trapezoidal shape, and the third with a parabolic shape. At the bottom of the
first tank, there are two valves that can be adjusted manually. The second and third tanks
are equipped with two electrical valves, one of which is a proportional valve, and the other
is an angle valve with four positions of opening. The pump, valves, height sensors, and
other control and measurement components are all connected to a PC through a
Microcontroller discovery board.
The tanks are constructed using Perspex plastic due to its reliability, ease of use,
availability, and functional strength. The three tanks are connected to a wooden frame that
measures 155 cm by 60 cm and is covered with a Formica board to prevent rot. The frame
also has a metal tank stand attached to it. The reservoir is located below the tanks and is
housed in a compartment beneath the metal stand. A submersible bilge pump with a flow
rate of 1100 GPH is connected to the pump pipe and the other end of the pipe is connected
to the first tank.
3.2 Experimental Procedure
Prior to conducting the experiment, the pump and valves, which serve as the primary
control mechanisms, and the height sensors, which serve as measurement devices,
underwent calibration to determine the relationship between the physical parameters and
the digital signals received from the microcontroller, represented as counts. To set up this
relationship, the SIMULINK environment was utilized to create the interface and
effectively transmit the signals to the relevant components, such as the pump flow rate,
height, and valve opening/flow coefficient.
10
3.2.1 Interfacing of the cascaded-tank system with Arm-Cortex MicroController Board
The Arm-Cortex Microcontroller board's Analog-to-Digital (ADC) and Digital-to-Analog
(DAC) connections are used to connect the cascaded three tank system to the board.
3.2.2 Interfacing of the Arm-Cortex Microcontroller board with Host PC
A connection was established between the Arm-cortex microcontroller board and the host
PC through a USB port. Data was exchanged in both directions, flowing from the
microcontroller board to the PC and vice versa. The PC ran the control algorithm using
MATLAB®/SIMULINK®. The input and output of the algorithm were linked with the
hardware port of the instrumentation system, allowing for the reading of sensor inputs and
the writing of actuation outputs to the port.
3.2.3 Signal Chain/Data transfer through the instrumentation system
The heart of the control system for the cascaded-tank is the STM32F103C8T6
microcontroller, a 32Bit ARM Cortex microcontroller from STMicroelectronics. The
microcontroller interfaces the cascaded-tank system with the host PC. To perform this
function effectively, the board is loaded with a “server program”. The server program is
made up of several lines of codes written in C language to allow seamless transfer of
information between the tank and the host PC. The microcontroller reads values from the
three level-sensors through its ADC port and sends it to the host PC through the serial port.
It also reads values from the serial port and feeds it to the tank through the DAC port. A
12-Bit analog to digital converter in the microcontroller produces a digital equivalent of
the voltage of between 3286 and 3701. Values, which are subsequently delivered to the PC
through the USB port.
The USB port is accessed by the MATLAB®/SIMULINK® process, which reassembles
the 8-bit data into a single 16-bit value. This value is then used to calculate the height using
the calibration equation. The proportional valves are servo-controlled, direct current (D.C)
powered motorized valves. Proportional signals are delivered through two distinct signal
terminals as input signals. 4-position valves are comparable to proportional valves, as
previously described. The distinctions are that the valve can only open and close in four
positions, and the input is only a 2-bit digital logic.
This logic signal is a Transistor – Transistor – Logic (TTL), and it is given by the
microcontroller port via two input/output (I/O) pins. A calibration function converts the
flow rate generated by the MATLAB®/SIMULINK® process to the PWM equivalent. The
11
microcontroller receives the generated PWM signal, and the PWM module outputs a PWM
signal through an I/O pin. A pump driver receives the PWM signal and generates an output
with enough current capacity and proportionate voltage to drive the water pump. The
cascaded-three-tank system's power supply is a switching power supply with a 12volt
output and an 8A current capacity.
3.3 Calibration of Height Sensors
1. The bottom valves of the three tanks were closed manually.
2. The connection between the SIMULINK model and the tank system was
established by opening the SIMULINK model (shown in Figure III in the
appendix). The inputs u1, u2, and u3 were set to 0, 1, and 1 respectively to turn off
the pump and electric-actuated valves.
3. The SIMULINK model was started.
4. Tank 1 was filled manually with water up to the 1cm mark, and the ADC block
value connected to tank 1 was recorded.
5. This process was repeated for the water levels of 5, 10, 15, 20, and 30cm, and was
also repeated for the middle and lower tanks.
3.4 Calibration of Pump
1. The digital microcontroller (based on the ARM-CORTEX M3 STMU F-series
MCU board) was calibrated to ensure that the count values were within an
acceptable range for the module being used.
2. The volumetric flow rate of the pump was determined through practical
measurement using water as the fluid.
3. A volumetric flask and timer were utilized to calculate the volumetric flow rate.
4. The time required to acquire a specific volume of water was recorded.
5. The tester recorded different count values spaced at intervals of their choice
between the maximum and minimum count values inputted.
3.5 Calibration of Control Valves
The calibration of the 4 Positions valve and proportional valves of a three-tank system was
carried out using a voltmeter to record voltages of the control valves and correlating it to
the counts given by the signal from the Aurdino mega. The voltages given by the voltmeter
was recorded as well as its corresponding counts from the Aurdino Mega.
12
3.6 Data Collection from the Experimental System
The following steps were taken to collect data for determining the parameters of the
theoretical model and identifying the empirical model:
1. The SIMULINK model created for identification purposes was opened.
2. MATLAB/SIMULINK was used to generate random signals of multiple levels for
the three inputs to the experimental tank, along with switching times for all inputs.
3. The experimental tank system was stimulated by the well-designed inputs by
running the SIMULINK model.
4. A set of 2000 input-output data was collected at a 2-second sampling rate. This data
was divided into two parts: 1400 for training and 600 for testing. The 2-second
sampling time (Ts) was small enough to capture the nonlinearities in the data,
prevent aliasing, and not put too much strain on the computing systems.
13
Height
Sensor 1
Manual
Valve 1
HOST- PC
Manual
Valve 2
Height
Sensor 2
Proportional
Valve 1
4-position
valve 1
Height
Sensor 3
Proportional
Valve 2
4-position
valve 2
PUMP
TARGET-PC + Actuators drivers + I/V converter
Figure 3.1: Experimental set up for A cascaded three-tank system
14
CHAPTER 4
RESULTS AND DISCUSSION
4.1 Experimental Data
Below are plots of data collected during the calibration of three level sensors, two
proportional valves, and one pump. Tables 2-4 present data collected during the calibration
of the height sensors for each of the three tanks. Additionally, Table 5 displays raw data
for the proportional valves' calibration, while Tables 6-7 contain experimental raw data for
obtaining the values of U1 and U2 for the pump.
15
Table 1 Experimental Raw Data for the Calibration of Height Sensor of Tank 1
Runs
Height observed on the Height Values Serial
Physical Tank (cm)
Read
Block
in
SIMULINK (Count)
1
4.8
3341
2
10.1
3411
3
15.3
3480
4
20.1
3545
16
Table 2 Experimental Raw Data for the Calibration of Height Sensor of Tank 2
Runs
Height observed on the Height Values Serial
Physical Tank (cm)
Read
Block
in
SIMULINK (Count)
1
5
3371
2
10.2
3439
3
15.3
3507
4
20
3570
17
Table 3 Experimental Raw Data for the Calibration of Height Sensor of Tank 3
Runs
Height observed on the Height observed on the
Physical Tank (cm)
Physical Tank (cm)
1
5.1
3358
2
10.5
3429
3
15.8
3501
4
20.3
3560
18
Table 4 Experimental Raw Data for Calibration of Proportional Valves
Runs
Fractional Opening Value Voltage Readings at Flow Coefficient
from Serial Send Block in the two terminals of
SIMULINK (Count)
the Valve (V)
1
11000
1.168
0
2
17000
1.85
0.119281
3
21500
1.92
0.205065
4
27000
2.14
0.307462
5
32500
3.71
0.408224
6
40000
4.12
0.544662
7
45000
4.23
0.635349
8
54000
4.35
0.797386
9
58500
4.4
0.876362
10
63000
4.65
0.958061
11
65535
4.84
1
19
Table 5 Experimental Raw Data for Calibration of Pump to obtain U1 (cm3/s)
Runs
Time (s)
Height
(count)
Volume
(cm3)
U1(cm3/s)
1
43.42
37200
200
4.60617
2
8.85
38000
400
45.1977
3
12.58
38200
600
47.6948
4
10.43
39300
800
76.7018
5
10.29
40000
1000
97.1817
6
8.05
44500
1200
149.068
7
6.92
47000
1200
173.41
8
7.17
50500
1400
195.258
9
7.03
54000
1600
227.596
10
6.72
61000
1800
287.081
11
4.74
65500
2000
412.941
20
Table 6 Experimental Raw Data for Calibration of Pump to obtain U2 (cm3/s)
Runs
Height (count)
Time(s)
Volume (cm3)
q(cm3/s)
1
55000
43.51
1600
36.77316
2
55500
39.66
1600
40.34291
3
56000
35.43
1600
45.15947
4
56500
32.69
1600
48.94463
5
57000
29.53
1600
54.18219
6
57500
27.04
1600
59.1716
7
58000
25.31
1600
63.21612
8
58500
23.78
1600
67.28343
9
59000
22.28
1600
71.81329
10
59500
21.34
1600
74.97657
11
60000
20.72
1800
86.87259
12
60500
19.86
1800
90.63444
13
61000
18.77
1800
95.89771
14
61500
17.96
1800
100.2227
15
62000
17.34
1800
103.8062
16
62500
16.7
1800
107.7844
17
63000
15.68
1800
114.7959
18
63500
15.87
2000
126.0239
19
64000
14.27
2000
140.1542
20
64500
13.43
2000
148.9203
21
65000
11.56
2000
173.0104
22
65500
10.2
2000
196.0784
21
4.2 Plots of Parameters
4.2.1 Level Sensor Calibration Plots
Height (cm) against Count for Tank 1
25
y = 0.075x - 245.9
R² = 1
Height (cm)
20
15
10
5
0
3300
3350
3400
3450
3500
Height Count
Figure 4.1: Height vs Count for Tank 1
22
3550
3600
Height (cm) against Count Tank 2
25
y = 0.0753x - 248.95
R² = 1
Height (cm)
20
15
10
5
0
3350
3400
3450
3500
3550
Height count
Figure 4.2: Height vs Count for Tank 2
23
3600
Height (cm) against Count for Tank 3
25
y = 0.0751x - 246.94
R² = 1
Height (cm)
20
15
10
5
0
3300
3350
3400
3450
3500
3550
Height Count
Figure 4.3: Height vs Count for Tank 3
24
3600
4.2.2 Pump Calibration Plot
PWN Counts against Flow rate
68000
y = -0.0005x3 - 0.2297x2 + 142.72x + 50074
R² = 0.9967
66000
PWN Counts
64000
62000
60000
58000
56000
54000
0
50
100
150
200
Pump Flowrate
Figure 4.4: PWM Count to Pump Flow rate
25
250
4.2.3 Calibration of Proportional Valve
PWM (Count) vs Fractional Valve Opening
70000
y = 2843.5x2 + 51810x + 10865
R² = 1
60000
PWM (Count)
50000
40000
30000
20000
10000
0
0
0.2
0.4
0.6
0.8
1
Fractional Valve Opening
Figure 4.5: PWM Count vs Fractional Opening for Valve 1
26
1.2
PWM (Count) against Fractional Opening
70000
y = 3639.7x2 + 51194x + 10837
R² = 0.9996
60000
PWM (Count)
50000
40000
30000
20000
10000
0
0
0.2
0.4
0.6
0.8
1
Fractional Valve opening
Figure 4.6: PWM Count vs Fractional Opening for Valve 2
27
1.2
4.3 Model Parameters
The signals of the flow coefficient are presented below using the whole 2000 input-output
data set and MATLAB system identification to calculate the flow coefficients in the
theoretical model.
28
Figure 4.7: Input-Output response plot of the 2000 data
29
Figure 4.8: The comparison between the validation data and the nlgref model
30
The values of alpha gotten from the model are shown in Table 8.
Table 7 Flow Coefficient for the Theoretical Model
ALPHA
VALUE
a1
0.4791
a2
0.4605
a3
0.4815
a4
0.5
a5
0.4581
a6
0.4831
The obtained alphas yielding values near 0.5 show that the derived dynamic model follows
Torricelli’s Principle.
31
4.4 Empirical determination of a Linear Model for the
system
Using the initial data set of 1400 input-output collected earlier and the MATLAB system
identification, the subroutine "arx" in the system identification toolbox was used to create
the linear transfer function matrix for the tank as shown below.
For Tank 1: Using the arx_h1.mlx file created the transfer function for the flow is.
𝐺(𝑠) =
−0.009429𝑠 2 −0.0006947𝑠+0.01354
𝑠 3 + 0.898𝑠 2 + 1.021𝑠 + 0.1073
32
Figure 4.9: The 5-step predicted response comparison of the transfer function and
the real data for h1.
33
Figure 4.10: The 5-step predicted response comparison of the transfer function and
the real data for h2
34
For Tank 2: Using the arx_h1.mlx file created the transfer function for the flow is.
−0.03691𝑠 2 − 0.06732𝑠 + 0.05127
𝐺(𝑠) = 3
𝑠 + 0.8791𝑠 2 + 1.193𝑠 + 0.0432
35
Figure 4.11: The 5-step predicted response comparison of the transfer function and
the real data for h3.
36
4.5 Determination of Linear Model for the Tank by
Linearization
By linearizing the nonlinear dynamic equation of the cascaded-tank system (equation 5, 6,
7) at the nominal operating point (𝑐1 = 0.5838, 𝑐2 = 0.333, 𝑐3 = 0.333 and 𝑐4 = 0), the
state space equations were gotten.
𝑑𝑥
𝑑𝑡
= 𝐴𝑥 + 𝐵𝑢 (20)
𝑦 = 𝐶𝑥 + 𝐷𝑢 (21)
where 𝑥 = [𝛿ℎ1
𝛿ℎ2
𝛿ℎ3 ]𝑇 , 𝑢 = [𝛿𝑢1
−0.0383
𝐴 = [ 0.0411
0
𝐵=[
0.0114
0
0
1
𝐶 =[0
0
0
𝐷 = [0
0
0
1
0
0
−1.6904
1.3057
𝛿𝑢2
0
−0.0417
0.0322
𝛿𝑢3 ]𝑇 , 𝑦 = [𝛿ℎ1
0
]
0
−0.0284
0
]
0
−1.7575
𝛿ℎ2
𝛿ℎ3 ]𝑇
(22)
(23)
0
0]
1
(24)
0 0
0 0]
0 0
(25)
The eigen values of the system were calculated as −0.0284, −0.0417 and −0.0383. This
shows that the system is stable.
The transfer function model is obtained from substituting Eq. 22-25 into Eq.26.
𝐺(𝑠) = 𝐶(𝑠𝐼 − 𝐴)−1 𝐵 + 𝐷
(26)
37
0.2984
𝐺(𝑠) =
0
0
26.10s + 1
0.2941
−40.57
(26.11s+1)(24.39s+1)
24.41s + 1
0.3329
−0.0127(−8.75×104 +1)
−61.78
[(35.16s+1)(26.11s+1)(24.39s+1)
(35.16𝑠+1)(24.04𝑠+1)
35.16s + 1]
0
4.6 Open Loop Stability of System
Upon finding the poles of the system, the stability of each tank is displayed on table 9, 10
and 11.
38
Table 8 Stability of the empirical tank 1 transfer function
s/no
TANK 1
STABILITY
1
-0.3913 + 0.8815i
Stable
2
-0.3913-0.8815i
Stable
3
-0.1154+0.0000i
Stable
39
Table 9 Stability of the empirical tank 2 transfer function
s/no
TANK 2
STABILITY
1
-0.421+0.9924i
Stable
2
-0.421-0.9924i
Stable
3
-0.0372+0.0000i
Stable
40
Table 10 Stability of the empirical tank 3 transfer function
s/no
TANK 3
STABILITY
1
-0.4425+1.1502i
Stable
2
-0.4425-1.1502i
Stable
3
-0.4607+0.0000i
Stable
4
-0.0748+0.0000i
Stable
All the tanks appear to be stable, therefore, it is regarded a stable system.
41
CHAPTER FIVE
CONCLUSION ANDRECOMMENDATIONS
5.1Conclusion
The 'arx' linear transfer function matrix sub-routine on MATLAB was used to calculate the
three-tank cascaded system, and we ultimately came up with three stand-alone transfer
functions for each of the cascades. The system can be inferred to be stable from the data as
shown under the results because the pole is located within the coordinate and has a value.
5.2 Recommendations
Studying or testing the dynamics of a toxic liquid substance can be hazardous because the
operator may unknowingly expose themselves to the substance while conducting the
experiment, which can be detrimental to their health. In the reactor setup, the liquid (water)
was manually dispensed into the reservoir via a funnel without a well-defined mechanism
for feeding it. Consequently, it is recommended to employ a more secure and efficient
method of filling the reservoir, such as designing a device connected to the reservoir that
will dispense liquid as required.
42
REFERENCES
Bamimore, A., Ogunba, K.S., Ogunleye, M.A., Taiwo, O., Osunleke, A.S. and King, R.
2012) Implementation of advanced control laws on a laboratory-scale Three-Tank
System. Ife Journal of Technology, 21(2):49-54.
Bamimore, A., Osinuga, A. B., Olaleke, M., Adeniran, O., Odunsi, O., Salaudeen, A.,
Owolabi,
Olabiyi, Y. (2016). Design and fabrication of four-tank level control system, Unpublished
MSc. Thesis
Taiwo, O., Bamimore, A., and Osunleke, A.S. (2018) Recent Developments in Local
Designing and Fabrication of Equipment for Teaching and Research in Chemical
Engineering, Proceedings of Nigerian Society of Chemical Engineers 48th Annual
Conference, Abeokuta, Nigeria. 8th – 10th November.
Zaky, M. A., El-Metwally, K. A., & El-Badawy, E. A. (2015). Optimal design of cascaded
tank systems for wastewater treatment. Alexandria Engineering Journal, 54(3),
535-541.
Chang, L. W., & Wu, H. C. (2014). Experimental study of cascaded tank system with multiobjective optimal tuning using evolutionary algorithm. Journal of Process Control,
24(7), 1021-1030.
Chen, Y. S., & Lee, C. S. (2019). Optimal adaptive dynamic programming control for a
class of nonlinear cascaded tank systems. IEEE Transactions on Systems, Man, and
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Kim, J. H., Choi, S. H., & Lee, S. (2016). Design and implementation of a cascaded tank
system for flow rate control. Journal of Mechanical Science and Technology, 30(4),
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Subbiah, K., & Arvind, K. (2017). Analysis and design of cascaded tank system for liquid
level control. International Journal of Engineering and Technology, 9(1), 229-234.
Zaky, M. A., El-Metwally, K. A., & El-Badawy, E. A. (2015). Optimal design of cascaded
tank systems for wastewater treatment. Alexandria Engineering Journal, 54(3),
535-541.
Zhang, D., Zhou, S., & Yang, S. (2020). Robust output feedback control for nonlinear
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43
APPENDIX
APPENDIX A
List of Group Members
ADEBAJO Daniel Oluwafemi
CHE/2016/001
AJAYI Boluwatife Ifedayo
CHE/2016/090
AGBONGHALE Daniel Ighodalo
CHE/2016/014
ALADELO Oyindamola Eniola
CHE/2016/023
BANKOLE Faith Tolulope
CHE/2016/032
ESHO Ifedayo
CHE/2016/041
MARCUS Victor Chidiadi
CHE/2016/051
OLADIPUPO Aishat Temitope
CHE/2016/064
OSIJONWO Oluwatomisin Precious
CHE/2016/073
TAIWO Adekemi Adesewa
CHE/2016/082
ODIKAYOR Dennis Uwomano
CHE/2016/096
44