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VARIABLE STRUCTURE CONTROL APPROACH
FOR NON-LINEAR SYSTEMS
A Thesis submitted to Gujarat Technological University
for the Award of
Doctor of Philosophy
in
Instrumentation & Control Engineering
by
Krupa Dhiraj Narwekar
129990917003
under the supervision of
Dr. Vipul A. Shah
GUJARAT TECHNOLOGICAL UNIVERSITY
AHMEDABAD
June-2019
ii
©Krupa Dhiraj Narwekar
iii
DECLARATION
I declare that the thesis entitled Variable Structure Control Approach For Non-Linear
Systems submitted by me for the degree of Doctor of Philosophy is the record of research
work carried out by me during the period from November 2012 to June 2019 under the
supervision of Dr. Vipul A. Shah and this has not formed the basis for the award of any
degree, diploma, associateship, fellowship, titles in this or any other University or other
institution of higher learning.
I further declare that the material obtained from other sources has been duly acknowledged
in the thesis. I shall be solely responsible for any plagiarism or other irregularities, if
noticed in the thesis.
Signature of the Research Scholar: …………………………… Date: 14/06/2019
Name of Research Scholar: Krupa Dhiraj Narwekar
Place: Ahemdabad
iv
CERTIFICATE
I certify that the work incorporated in the thesis Variable Structure Control Approach
for Non-Linear Systems submitted by Mrs .Krupa Dhiraj Narwekar was carried out
by the candidate under my supervision/guidance. To the best of my knowledge: (i) the
candidate has not submitted the same research work to any other institution for any
degree/diploma, Associateship, Fellowship or other similar titles (ii) the thesis submitted is
a record of original research work done by the Research Scholar during the period of study
under my supervision, and (iii) the thesis represents independent research work on the part
of the Research Scholar.
Signature of Supervisor: ……………………………… Date: 14/06/2019
Name of Supervisor: Dr. Vipul A. Shah
Place: Ahmedabad
v
COURSE-WORK COMPLETION CERTIFICATE
This is to certify that Mrs. Krupa Dhiraj Narwekar Enrolment no. 129990917003 is a
PhD scholar enrolled for PhD program in the branch Instrumentation & Control of
Gujarat Technological University, Ahmedabad.
(Please tick the relevant option(s))
He/she has been exempted from the course-work (successfully completed during
M.Phil Course)
He/she has been exempted from Research Methodology Course only (successfully
completed during M.Phil Course)
He/She has successfully completed the PhD course work for the partial requirement
for the award of PhD Degree. His/ Her performance in the course work is as follows-
Grade Obtained in Research
Methodology
(PH001)
Grade Obtained in Self Study Course
(Advance Control Theory)
(PH002)
BC AB
Supervisor’s Sign
(Dr. Vipul A. Shah)
vi
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Linear Systems by Krupa Dhiraj Narwekar has been examined by us. We undertake the
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Name of Research Scholar: Krupa Dhiraj Narwekar
Place : Ahmedabad
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PRIMARY SOURCES
www.me.unlv.eduInternet Source
Krupa Narwekar, V. A. Shah. "Level control ofcoupled tank using higher order sliding modecontrol", 2017 IEEE International Conferenceon Intelligent Techniques in Control,Optimization and Signal Processing (INCOS),2017Publicat ion
Submitted to National University of SingaporeStudent Paper
Submitted to Jawaharlal Nehru TechnologicalUniversityStudent Paper
V. Bandal, B. Bandyopadhyay, A.M. Kulkarni."Design of power system stabilizer using powerrate reaching law based sliding mode controltechnique", 2005 International PowerEngineering Conference, 2005Publicat ion
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PhD THESIS Non-Exclusive License to
GUJARAT TECHNOLOGICAL UNIVERSITY
In consideration of being a PhD Research Scholar at GTU and in the interests of the
facilitation of research at GTU and elsewhere, I, Krupa Dhiraj Narwekar (Full Name of
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including privacy rights, and that I have the right to make the grant conferred by this
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of the Copyright Act, written permission from the copyright owners is required, I have
obtained such permission from the copyright owners to do the acts mentioned in paragraph
(a) above for the full term of copyright protection.
viii
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policy matters related to authorship and plagiarism.
Signature of the Research Scholar:
Name of Research Scholar: Krupa Dhiraj Narwekar
Date: 14/06/2019 Place: Ahmedabad
Signature of Supervisor:
Name of Supervisor: Dr. Vipul A. Shah
Date: 14/06/2019 Place: Ahmedabad
Seal:
ix
THESIS APPROVAL FORM
The viva-voce of the PhD Thesis submitted by Smt. Krupa Dhiraj Narwekar
(Enrollment No.129990917003) entitled Variable Structure Control Approach for
Non-Linear Systems by Krupa Dhiraj Narwekar was conducted on Friday, 14/06/2019
(day and date) at Gujarat Technological University.
(Please tick any one of the following option)
The performance of the candidate was satisfactory. We recommend that he/she be
awarded the PhD degree.
Any further modifications in research work recommended by the panel after 3 months
from the date of first viva-voce upon request of the Supervisor or request of
Independent Research Scholar after which viva-voce can be re-conducted by the same
panel again.
(briefly specify the modifications suggested by the panel)
The performance of the candidate was unsatisfactory. We recommend that he/she
should not be awarded the PhD degree.
(The panel must give justifications for rejecting the research work)
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Name and Signature of Supervisor with Seal 1) (External Examiner 1) Name and Signature
------------------------------------------------------- -------------------------------------------------------
2) (External Examiner 2) Name and Signature 3) (External Examiner 3) Name and Signature
x
ABSTRACT
Due to the advancement in communication technology and simulation software, the
development of model based controllers are becoming more and more popular. The
process industry uses the conventional controllers which are easy to implement and
performs efficiently. This is true when we do not consider the undesirable condition or
uncertainty. Often in the undesirable situation these conventional methods fail to perform.
So the effort has been made to use the model based robust controllers to the process control
application. Variable Structure Systems works on the principle of switching the structure to
reach the equilibrium point. So the controller of the variable structure systems type i.e. the
sliding mode control is used to control the process parameters, level and temperature that
are widely used in any chemical process.
In this work variable structure sliding mode control and higher order sliding mode super
twisting controller is used to control the process parameters usually used in in process
industry which are inherently nonlinear. The two parameters controlled are level and
temperature. A coupled tank system is considered in which the level of tank 2 is controlled
to desired set point. The sliding surface is designed, and the reaching law is chosen so that
the states reach the sliding surface. The control law is designed using equivalent control
method. The results for the conventional first order sliding mode control are plotted in the
presence of sine type of matched disturbance. The chattering is observed in the control law.
The means to reduce the chattering has been adopted by using the modified reaching law
i.e. power rate reaching law. The power rate reaching law gives the faster convergence and
reduced chattering. Further, the level of coupled tank is controlled at desired set point
using higher order sliding mode type super twisting controller. The results of power rate
reaching law and the super twisting controller are plotted and observed for supressed
chattering in the MATLAB Simulink.
For temperature control, a batch reactor model is considered; the concentration of the
chemical in the reactor is dependent on the time dependant temperature trajectory in the
batch reactor. The sliding surface is chosen as an error equation. The sliding mode control
with constant rate reaching law is used to control the temperature of the batch reactor.
Then a power rate reaching law is used for temperature control of reactor. The higher order
sliding mode control is used to reduce the chattering so the same is applied to temperature
xi
control problem. The simulation study is done considering operating constraints using
MATLAB Simulink.
To observe the real time behaviour of the control algorithms, a constant volume stirred
reactor is considered whose temperature is to be controlled. The modelling of the reactor is
done using mass and energy balance equation considering certain assumptions. The sliding
surface is chosen as in the case of batch reactor. The SMC controller is designed for the
reactor system. To reduce the chattering the power rate reaching law based SMC and super
twisting controller is used as previously discussed. The simulation as well as experimental
results is plotted in the presence of matched disturbance. The results are compared on the
basis of performance measures namely; integral square error and integral of absolute error.
xii
Acknowledgement
I thank the Almighty for giving me the strength and capability for pursuing research and
always showing the path by which I am able to complete the research successfully.
I have a deep sense of gratitude towards my research supervisor Dr. Vipul A. Shah, whose
constant guidance throughout the research work has helped me to cross the difficult steps
during the research period. Apart from technical guidance, being a very humble person I
have learnt from him to keep patience and remain calm even in the time of difficult
situation. His support during the DPC meets as well as visits to Dharamsinh Desai
University (DDU), Nadiad for reporting are not to be forgotten by me.
I thank my Doctoral Progress Committee (DPC) members Dr. C. B. Bhatt and
Dr. V. K. Thakar for being my DPC members. They were always cooperative for arranging
the dates for DPC meets and providing their valuable suggestion during the DPC meets. I
especially thank Dr. C.B. Bhatt for allowing the open seminar at his working place and
providing all the support required for conducting the open seminar successfully.
My thanks to the Director, Pandit Dindayal Petroleum University (PDPU), Gandhinagar,
Head (Faculty of Electrical Engineering) (PDPU) and Dr. Anil Marakana, in-charge of
Process Dynamics and Control lab (PDPU), for granting me the permission for using the
experimental setup. My special thanks to Dr. Anil Marakana for technical discussions
before and during the visits at PDPU, Gandhinagar.
I thank my good friends, Mamta Patel, Janki Chotai, Aarti Yadav, Dr. Ankit K. Shah for
motivating me and being there to help me whenever required.
I thank all of them who were readily available for me for technical discussion as and when
required, to name a few are Dr. Dipesh Shah and Dr. Ankit K. Shah.
I dedicate my thesis to my late father Dr. Ramesh Borse who had always dreamt of this
day. Today I am missing him a lot. I thank my mother Mrs. Usha Borse to remain always
present by my side and to help me in whatever way she can. My brother Prasad Borse has
always been a source of motivation for me.
xiii
I thank my husband Mr. Dhiraj Narwekar for giving me strength and support during my
research period. I thank my in laws for their support and especially my mother in law for
taking care of my children during my unavailability for large span of time. Without her I
cannot imagine my research period. I thank my children, Vedant and Ketki for their
innocent love for me and behaving maturely during my absence.
Last but not least I thank my institute Sardar Vallabhbhai Patel Institute of Technology,
Vasad and its management and Principal for motivating me and allowing me to pursue my
research. I am grateful to Dr. Rakesh B. Patel, Head, Instrumentation & Control
Engineering Department for motivating me from time to time, cooperating in departmental
activities while handling the departmental portfolios and academic activities. I thank all
those who have directly and indirectly supported me during my research period in all
respects.
Krupa Dhiraj Narwekar
Date:14/06/2019
Place: Ahmedabad
xiv
Table of Contents
Abstract x
Table of Contents xiv
List of Abbreviation xviii
List of Symbols xix
List of Figures xxi
List of Tables xxiii
List of Appendix xxiv
1 Introduction
1.1 Overview 1
1.2 Problem Definition 2
1.3 Objectives & Scope of the work 3
1.4 Original Contributions from the Thesis 4
1.5 Organisation of the Thesis 5
2 Literature Review
2.1 History of Variable Structure Control (VSC) 6
2.2 Preliminaries of Sliding Mode Control 7
2.2.1 Simulation Results 10
2.3 Chattering Reduction in SMC 12
2.3.1 Saturation function 12
xv
2.3.2 Reaching Law 15
2.3.3 Higher Order Sliding Mode Control 16
2.3.3.1 Relative Degree 17
2.3.3.2 Super Twisting Controller 19
2.4 Summary 20
3 Level Control of Coupled Tank System
3.1 Background 21
3.2 Introduction 21
3.3 Process Description 22
3.4 Controller Design 24
3.4.1 Sliding Mode Controller for Level Control of Coupled Tank 24
3.4.2 Stability Analysis 25
3.4.3 Power Rate Reaching Law for Level Control of Coupled Tank 26
3.4.4 Super Twisting Controller Design for Level Control of Coupled
Tank 26
3.5 Summary 27
4 Temperature Control of Batch Reactor System
4.1 Background 29
4.2 Introduction 29
4.3 Objectives of Control of Batch Reactors 31
4.4 Types of Reactors 33
4.5 Dynamic Model of Batch Reactor 34
xvi
4.6 Controller Design 36
4.6.1 FOSMC Temperature Control of Batch Reactor 38
4.6.2 Stability Analysis 38
4.6.3 Power Rate reaching based SMC for Temperature Control of
Batch Reactor
39
4.6.4 STC for Temperature Control of Batch Reactor 39
4.7 Summary 40
5 Hardware and Interfacing
5.1 Background 41
5.2 Hardware Description 41
5.2.1 Experiment Module 42
5.2.2 Control Module 46
5.3 Interfacing 48
5.4 Dynamic Model of the Reactor 49
5.5 Controller Design 53
5.5.1 FOSMC of the Constant Volume Stirred Reactor 54
5.5.2 Power Rate Reaching Law based SMC 54
5.5.3 STA based Control of Temperature 55
5.6 Summary 56
6 Results and Discussions
6.1 Introduction 57
6.2 Simulation Results of Level Control of Coupled Tank System 57
xvii
6.3 Simulation Results of Temperature Control of Batch Reactor 60
6.4 Experimental Results of Constant Volume Reactor 62
6.5 Discussions 66
6.5.1 Level Control of Coupled Tank System 66
6.5.2 Temperature Control of Batch Reactor 67
6.5.3 Temperature Control of Constant Volume Reactor 67
7 Conclusion & Future Scope
7.1 Conclusion 69
7.2 Contributions 71
7.3 Future Scope 72
List of References 73
List of Publications 79
Appendix A Additional Simulations 80
xviii
List of Abbreviation
CSTR Continuous Stirred Tank Reactor
HOSMC Higher Order Sliding Mode Control
FOSMC First Order Sliding Mode Control
ISE Integral Square Error
IAE Integral of Absolute Error
SMC Sliding Mode Control
SOSMC Second Order Sliding Mode Control
STC Super Twisting Controller
STA Super Twisting Algorithm
VSS Variable Structure Systems
VSC Variable Structure Control
xix
List of Symbols
k Gain of the SMC
s sliding surface
S Sliding manifold
CA Concentration of component A
CB Concentration of component B
R1 Rate Reaction 1
R2 Rate Reaction 2
h1 Height of tank 1(cm)
h2 Height of tank 2(cm)
qL flow rate to tank 1(m3/s)
q1 outflow rate from tank 1(m3/s)
q2 outflow rate from tank 2(m3/s)
g gravitational constant
M the cross sectional area of Tank-1 and Tank 2(cm2)
c12 area of coupling orifice(cm2)
c2 area of the outlet orifice(cm2)
c sliding surface parameter
k1 STC gain 1
xx
k2 STC gain 2
α tuning parameter for Power rate reaching Law
d disturbance
C Heat capacity of the fluid (J/Kg K)
A Area (m2)
V Volume of Fluid in the process vessel (L)
Vh Volume of Fluid in the Heater Tank (L)
T Temperature of the fluid in vessel (°C)
Th Temperature in the heater tank (°C)
T0 Outlet temperature from the heat exchanger (°C)
Tref Reference Temperature (°C)
P Heater input electrical power (Watts)
wi Inlet mass flow rate(kg/s)
w Outlet mass flow rate(kg/s)
Q Heat input to the process vessel from heat exchanger (J/kg)
ρ Fluid Density (kg/m3)
qi Volumetric flow rate into the vessel(L/min)
q Volumetric flow rate out of the vessel (L/min)
qh flow rate through the heat exchanger(L/min)
K A constant for cooler
xxi
List of Figures
FIGURE.
NO. Description
Page
No.
2.1 Concept of sliding mode control 7
2.2 Phase Trajectory with initial condition [x10 x20]= [1 1] 10
2.3 States Convergence from initial condition [x10 x20]= [1 1] 10
2.4 Control signal (u) for system defined by (2.6) 10
2.5 Sliding surface for the systems defined by (2.6) 11
2.6 Phase Portrait ɛ = 0.01 12
2.7 Sates Convergence for ɛ = 0.01 13
2.8 Control Signal (u) for ɛ = 0.01 13
2.9 Phase Portrait for ɛ = =1 13
2.10 States Convergence for ɛ = 1 14
2.11 Control Signal (u) for ɛ = 1 14
2.12 Phases of sliding mode 15
2.13 Phase Portrait: Super Twisting Controller 19
3.1 Coupled Tank Schematic 22
4.1 Batch Reactor 33
4.2 Continuous Reactor 33
4.3 Tubular Reactor 34
4.4 Schematic of Batch Reactor 35
5.1 Experimental Setup for constant volume stirred reactor 42
5.2 Cooler Flow Circuit 43
5.3 Control Module 47
5.4 Safety Switches 48
5.5 NI6009 DAQ module 49
5.6 Process Vessel Schematic 50
6.1 Level of Tank-2 To H=4cm using SMC & Power Rate SMC 58
xxii
6.2 Control Law for Level Control using SMC 58
6.3 Control Law for level control using Power Rate reaching law based
SMC
58
6.4 Level of tank-2 To H=4cm using SMC & STC 59
6.5 Control Law using SMC for Level Control 59
6.6 Control Law using STC for Level Control 59
6.7 Temperature Tracking using SMC for Batch Reactor 60
6.8 Control law for Temperature Tracking using SMC 60
6.9 Temperature Tracking using Power Rate reaching law based SMC 61
6.10 Control Law for temperature tracking using Power Rate reaching
law based SMC
61
6.11 Temperature Tracking using STC for Batch Reactor 61
6.12 Control Law for temperature control using STC 62
6.13 Temperature Tracking using SMC for Constant Volume Reactor 63
6.14 Control Law for Temperature Tracking using SMC 63
6.15 Temperature Profile for SMC 63
6.16 Temperature Tracking using Power Rate Reaching law based SMC 64
6.17 Control law for temperature tracking using Power Rate Reaching
law based SMC
64
6.18 Temperature Profile using Power Rate Reaching law based SMC 64
6.19 Temperature Tracking using STC for Constant Volume Reactor 65
6.20 Control law for temperature tracking using STC 65
6.21 Temperature Profile using STC 65
A.1 Phase Portrait using SMC 81
A.2 Phase Portrait using TSM 82
A.3 Phase Portrait using FTSM 83
xxiii
List of Tables
TABLE
NO.
Description Page
No.
3.1 Coupled Tank Design Constants 27
4.1 Design Parameters for Batch Reactor 40
5.1 Analog Signal Values 45
5.2 Design Specifications for Constant Volume Reactor 53
6.1 Error Analysis using performance measures 68
xxiv
List of Appendix
Appendix A : Additional Simulation 80
Introduction
1
CHAPTER-1
Introduction
1.1 Overview
The process industry has to maintain the various parameters to their desired set point
values as the control of these parameters is important in terms of the product quality, the
manufacturing cost, the energy consumption and several other factors. Therefore the
measurement and control of these parameters in various unit operations is crucial. The
common of these measured and controlled are level, temperature, pressure, flow etc. Also
practically these unit operations are inherently non-linear in nature having dead zones,
friction etc. Therefore, controlling the parameters in these types of systems becomes
challenging task.
Moreover, due to the advancement in communication technology, computer software and
virtual instrumentation, the development of control algorithm using the simulation
softwares has become feasible. These algorithms can be simulated and tested for their
performance on the system and then can be easily implemented on the real time system.
The advantage of this is that the simulation study gives the know-how of the system
behaviour as well as its operating condition. As all of us know that the classical control
techniques have proved to give satisfactory performance but these control techniques
sometimes fail to work in undesired situations. Also due to development of model based
controllers, the critical points in the system can be taken care of while designing the
control algorithms. So model based controllers are becoming more and more popular in the
research community. The popularity of these controllers is due to several features these
controllers possess. Some of the features being the robustness, insensitive to parametric
uncertainty, optimal performance, intelligent behaviour etc. One of these controllers,
Problem Definition
2
Variable Structure Control (VSC) based sliding mode control (SMC) is a robust controller
which is insensitive to parametric uncertainty and matched disturbances [1], [2].
The application of VSC to process industry is due to the above mentioned reasons. The
robustness helps in dealing with the uncertainties which sometimes may disturb the system
under consideration. The sliding surface design consists of the following, one is designing
the sliding surface and second thing is the reaching law [3]. Even though the SMC is
robust, it possesses inherent high frequency oscillations (chattering), which causes fatigue
of mechanical parts in the final control element. So the techniques are devised by many
researchers to reduce chattering like reducing the reaching time, modifying the reaching
law [4],[5],[6],[7],[8]. In recent years, the higher order sliding mode super twisting
controllers is used to control these systems to reduce chattering [9].
1.2 Problem Definition
The control of process parameters is crucial as it affects the overall performance in any
process industry. There is also development of software which has all the functions
required in designing the control algorithm as well as data acquisition systems which helps
in interfacing of the real time systems with these software. So the model based controller
can be easily designed and simulated for their efficacy. In this work, the control of process
parameters is achieved using SMC. The design of SMC includes the selection of sliding
manifold as well as the reaching law. Along with robust performance, the SMC has the
disadvantage of inherent chattering i.e. high frequency oscillations [9]. To reduce this,
researchers have taken several efforts, i.e. to modify the reaching law, design the higher
order sliding mode controller (HOSMC). As, HOSMC super twisting controller (STC)
takes the integration of discontinuous part, the chattering is suppressed to some extent [9],
[10], [11].
As it is well known that the final control element in most of the process industries is
control valve. Most of these control valves are pneumatically operated valves which have
actuator and other mechanical moving parts. Controller output directly affects the final
control element, so considering designing a controller, its effect on final control element
also needs to be studied. Chattering reduction helps in protecting the wear and tear of the
moving parts in the final control element [12], [13].
In this work two case studies are considered namely the level control of coupled tank, the
temperature control in batch reactor. The control algorithms- SMC, power rate reaching
Introduction
3
law based SMC and STC is applied to both the systems on the MATLAB Simulink
environment. To validate the performance of the control strategies, experimental approach
is used in which the laboratory reactor is considered. The temperature of the reactor is
controlled using the control strategies discussed so far.
The problem statement is
“To control the process parameters commonly used in process industry namely level and
temperature, using variable structure control sliding mode control as it is the robust
controller; it induces chattering, so the modified reaching law based SMC and the higher
order sliding mode type STC is used to control these parameters to reduce the chattering
and effectively improves the accuracy”.
1.3 Objectives & Scope of the work
Most of the systems used in industrial environment are inherently non-linear. Control the
parameters related to these systems is mostly done using classical control techniques like
PID. Researchers are continuously trying to develop the control techniques by designing
the adaptive PID, robust PID, Fuzzy PID etc. for these systems to get the optimal and
robust performance [14]. These advanced controllers are mostly designed using model
based control techniques like LQG, State feedback control etc. [15], [16], [17]. Amongst
these control strategies the VSC based SMC and HOSMC are widely applied to the process
control problems because of their features like robustness, insensitive to parametric
uncertainties etc.
Being robust the SMC controller induces chattering which is a drawback of the SMC
controller. To reduce the chattering several techniques are proposed by the researchers,
amongst them is using power rate reaching law and implementing higher order sliding
mode control technique [7],[8],[9]. In this work, the chattering reduction methods are
adopted for level and temperature control in commonly used unit operations. The
objectives can be briefed as:
To design the control algorithm to achieve the desired level of coupled tank using
SMC, Power Rate reaching law based SMC and Super Twisting Controller.
To design the control algorithm for temperature control of batch reactor using the
SMC, Power Rate reaching law based SMC and Super Twisting Controller.
For experimental approach, the constant volume stirred reactor is considered for
temperature control of the reactor. So to develop mathematical model of the reactor
Problem Definition
4
using mass and energy balance equations. Design the SMC controller and higher
order STC
The Scope of the work:
The process parameters which are controlled are namely the level and temperature
in coupled tank and batch reactor respectively on the MATLAB Simulink. For real
time application the temperature control of experimental set up of constant volume
stirred reactor is considered.
The control strategies are applied to process control applications. The chattering
reduction is observed on the Simulink environment for simulation studies and the
performance measures-ISE and IAE are compared for experimental results.
1.4 Original Contributions from the Thesis
The work presented in this thesis consists of two simulation studies-level control of
Coupled tank and Temperature control of batch reactor. The experimental approach is done
in this work for temperature control on experimental set up of constant volume stirred
reactor.
The main contribution of this work can be summarised as
Level control to desired set point using STC for the coupled tank system.
Temperature control of batch reactor to time varying trajectory using STC.
Analysis of chattering reduction using constant rate SMC, Power rate SMC and
higher order sliding mode control for coupled tank system.
Analysis of chattering reduction using constant rate SMC, Power rate reaching law
based SMC and Super Twisting Controller for batch reactor.
The TEQuipment CE117 process trainer is used for experimental approach. The
development of mathematical model of this system using mass and energy balance
equation. Interfacing this kit with the LabVIEW for implementing the control
algorithms. Design of the control algorithm for this system using SMC, power rate
reaching law based SMC and STC to achieve temperature control.
To analyse the chattering reduction by finding the performance measures IAE and
ISE for the control algorithms.
Introduction
5
1.5 Organisation of the Thesis
Chapter 1: Introduction-This chapter includes overview of the work carried throughout
the research work, by including the problem definition, objectives and scope of the work.
Chapter 2: Variable Structure Control and its fundamentals-This chapter covers the
fundamentals of the variable structure control, and some simulations done at the basic
level.
Chapter 3: Level Control of Coupled Tank-Modelling equations of coupled tank, level
control to desired set point using sliding mode control, power rate reaching law based
SMC and SOSMC based STC.
Chapter 4: Temperature Control of Batch Reactor-Basics of reactor, various types of
reactors, modelling of reactor under consideration, temperature control using SMC, SMC
with power rate reaching law, and super twisting controller.
Chapter 5: Hardware and Interfacing-Basics of experimental setup, modelling of the
reactor, interfacing with the computer system for implementing the algorithm on the real
time system. Design the control law for all the three control strategies.
Chapter 6: Result and Discussion-This chapter covers the experimental and the
simulation results. Detail discussion of the experimental and simulated results obtained.
Chapter 7: Conclusion & Future Scope-This chapter includes the conclusion of the work
covered in the whole thesis. The contributions from the work carried out in thesis and the
future scope is further added.
Literature Review
6
CHAPTER-2
Literature Review
2.1 History of Variable Structure Control (VSC)
The history of variable structure control dates back in 1950’s in Soviet Union, where it was
originated. First time it was published outside Soviet Union in English in 1976 by Prof.
Itkis and in 1977 by Prof. Utkin [18],[19]. After its publication a concrete theory was
introduced about the variable structure control to the research community. After which the
researchers in VSC field has no bounds in developing novel algorithms using VSC,
applying VSC in all the aspects of the industry from robotics, to automobile industry, to
biomedical engineering etc.
Variable structure control is a class of systems whereby control law deliberately changes
its structure to reach to the equilibrium point. In other words, Variable structure control
(VSC) is a switching of gains resulting in sliding mode. For example, the gains in each
feedback path switch between two values according to a differential equations or we can
say a rule that depends on the position of the state in the plane at each instant. by
switching the control law drives the non-linear plants states trajectory on the predefined
surface which is designed, and the maintains the states on this surface at each instant of
time. The surface is called a switching surface. When the plant state trajectory is “above”
the surface, a feedback path has one gain and a different gain if the trajectory drops
“below” the surface. This surface defines the rule for proper switching. This surface is also
called a sliding surface (sliding manifold). Ideally, once intercepted, the switched control
maintains the plant’s state trajectory on the surface for all subsequent time and the plant’s
state trajectory slides along this surface [20].
Preliminaries of Sliding Mode Control
7
2.2 Preliminaries of Sliding Mode Control
First the question is what is sliding mode? [3]
Sliding mode: Motion of the system trajectory along a ‘chosen’ line/plane/surface of the
state space
Sliding mode Control: Control designed with the aim to achieve sliding mode
It is usually of VSC type.
The figure shows the brief about the idea behind sliding mode controller design
FIGURE 2.1 Concept of Sliding Mode Control [3].
Although, the feedback-loop response of VSC system mentioned in above section is stable,
is dependent on the parameter. Therefore, the closed-loop response is sensitive to
parameter variation. The VSC that yields the closed-loop response totally insensitive to a
particular class of uncertainties is called Sliding Mode Control (SMC) [21].
Let us understand the concept of sliding mode by considering a double integrator system
( )x u t (2.1)
let us see the results of feedback control system
( ) ( )u t kx t (2.2)
where k >0.
Literature Review
8
In order to analyze the closed loop for this system the phase plot is to be plotted, which is a
plot of velocity with respect to position. Therefore substituting (2.2) in (2.1) and
multiplying by x the equation becomes [1]
xx kxx (2.3)
by integration (2.3) ,
2 2x kx c (2.4)
where c is constant of integration due to initial conditions and positive. If the value of k is
assumed to be k=1, (2.4) is the equation of a circle with center at the origin and radius c .
The graph of x and x , is an ellipse which depends on initial conditions. As seen the states
will not reach the origin rather move in a closed path. This system is stable as the states
remain bounded for all the time but not asymptotically stable.
Consider the control law
1
2 Otherwise
( ) if 0
( )
k x t xxu
k x t
(2.5)
Where 1 2&k k are positive and 1 21k k . The phase plane ( , )x x is partitioned by the
switching rule into four quadrants separated by the axes. The control law given by
2 ( )u k x t will be in effect when in the first quadrant. The states should follow the spiral
motion towards the origin as it traverses through each quadrant. This is the basics of any
variable structure control. To summarize, sliding mode control is special kind of on-off
control. The control signal is applied when the system deviates from the desired point [22].
Consider a system represented by
1 1
2 1
x ax bu d
x x
(2.6)
Mathematically the switching law is defined as
1 1 2 2s c x c x (2.7)
where 1 2 max, , ,and are known and the bounds of d are known with a b c c d d
so rewriting the system in the state space structure
Preliminaries of Sliding Mode Control
9
0 1
1 0 0 0
=
a bx x u d
f gu hd
(2.8)
as (2.7) the switching law is defined as
1 1 2 2
Ts c x c x p x 1 2
Tp c c (2.9)
The convergence is guaranteed if the sliding surface is stable. The stable sliding surface
guarantees the finite time convergence. The stability is proved by Lypunov stability
theorem. The control law guarantees finite time convergence by keeping 0s , the
Lypunov function is given as
21( )
2V x s (2.10)
the input equation (u) is designed such that the Lypunov function is stable, i.e. it satisfies
( )V x ss (2.11)
Substituting (2.7) in (2.10)
( )TV s p x (2.12)
From (2.7) and (2.8)
The control law is defined as
max ( ( ))T T
T T
p f p hd sign s xu
p g p g
(2.13)
by assuming the values of plant parameters, the results are simulated in MATLAB
Simulink environment.
Assuming 1 2 max2, 1, 1, 1, 0.9sin , 0.9, 0.5a b c c d t d
1
1
( ) 3 (0.9 0.5) ( )
3 1.4 ( )
u t x sign s
x sign s
(2.14)
Applying the control law of (2.13) in to system given by (2.6)
Literature Review
10
2.2.1 Simulation Results:
FIGURE 2.2 Phase Trajectory with initial condition [x10 x20]= [1 1]
FIGURE 2.3 States Convergence from initial condition [x10 x20]= [1 1]
FIGURE 2.4 Control signal (u) for system defined by (2.6)
Preliminaries of Sliding Mode Control
11
FIGURE 2.5 Sliding surface for the systems defined by (2.6)
The simulation results are shown to show the practical implementation of basic sliding
mode controller to a system defined by (2.6). As per (2.7) the sliding surface is designed,
the choice of the parameters 1 2 & c c is made so that the sliding surface is stable. The stable
sliding surface guarantees finite time convergence. The values of 1 2 & c c can be found by
pole placement method, LQR method for linear systems. Another approach to design a
sliding surface is that the polynomial formed by the sliding equation must be Hurwitz [23].
As seen from Fig. 2.2 the phase trajectory obtained due the control law designed by sliding
mode control, the states 1 2 & x x converge to the origin from initial condition [1 1]. The
control law as shown in Fig. 2.3 shows the chattering as the states reach the surface. This is
the major drawback of the SMC. This is due to the signum function that performs the
switching action, and introduces discontinuity. So sliding mode control is a discontinuous
control. Fig.2.4 shows the sliding surface, which shows s=0, as the states corresponds to
the convergence of the states.
Therefore with this example several points about the SMC are:
Design Steps of Sliding Mode Control
Switching manifold selection: as per the desirable dynamical characteristics, the
switching manifold is designed.
Discontinuous control design: A discontinuous control strategy is formed to
ensure the finite time reachability of the switching manifolds. The controller
may be either local or global, depending upon specific control requirements [3].
The Sliding surface must be stable with respect to the system
The chattering i.e. high frequency oscillations is an inherent part of SMC
Literature Review
12
2.3 Chattering Reduction in SMC
As discussed in chapter 1 section 1.2, the chattering causes the corrosion or spoiling of the
moving parts of the final control element. In most of the cases final control elements are
the pneumatic or hydraulic valves. These valves have moving parts. The controllers give
the output to the final control element. Therefore the controller output is crucial when
choosing the controller for any application. The SMC controller has many advantages as
discussed earlier, so it is used in many applications, also it has the drawback of chattering
so several chattering reduction techniques are introduced.
2.3.1 Saturation function
The sign function is replaced by saturation function given by sat so that the chattering is
reduced to some extent. The sat function is given by
1
( , ) /
1
s
sat s s s
s
(2.15)
the simulation results are obtained to observe the smooth control law [22].
The control law for the system given by Eq.6, with smooth control law is
13 1.4 ( , )u x sat s (2.16)
With the value of = 0.01, the results obtained are shown in Fig.2.6, Fig.2.7 and Fig.2.8,
namely the phase portrait, the convergence of the states with time and the control signal
respectively. The convergence of the states is same as the sign function but the chattering
has been reduced as compared to sign function.
FIGURE 2.6 Phase Portrait ɛ = 0.01
Preliminaries of Sliding Mode Control
13
FIGURE 2.7 Sates Convergence for ɛ = 0.01
FIGURE 2.8 Control Signal (u) for ɛ = 0.01
With =1, the phase portrait, the states convergence and the control signal is shown in Fig.
2.9, Fig.2.10, and Fig.2.11 respectively. The reaching time is more to reach the sliding
surface than with =0.01, but the chattering is reduced considerably.
FIGURE 2.9 Phase Portrait for ɛ = 1
Literature Review
14
FIGURE 2.10 States Convergence for ɛ = 1
FIGURE 2.11 Control Signal (u) for ɛ = 1
From the results obtained, is can be concluded that replacing the sign function with the sat
function, the chattering drawback is reduced significantly. Though chattering is not
completely eliminated, the amplitude of the chattering can be controlled by the value of .
As the value is reduced the sat function performs similar to the sign function thus
chattering is observed, as the value of =1 increases, the chattering is reduced but the
system performance is affected as the states take more time to converge to the sliding
surface. Thus there is a tradeoff to use the sat function as the smooth function cannot
provide the finite time convergence of the sliding variable (s) to zero in the presence of
external disturbance [24],[25], the states converge to the vicinity of the origin instead of to
the origin itself due to the effect of disturbance.
Preliminaries of Sliding Mode Control
15
2.3.2 Reaching Law
The phases of sliding mode
Reaching- phase: Where the system state is driven from any initial state to reach the
switching surface (the anticipated sliding modes) in finite time.
Sliding-mode phase: Where the system is induced into the sliding motion on the
switching manifolds, i.e., the switching manifolds becomes an attractor [3].
To be more specific the reaching phase is the phase where the describing point initiates and
moves towards the sliding surface. In this phase however tracking error is not controlled
and the system is sensitive to parametric uncertainties and external disturbances and noise.
Thus to minimize the reaching phase or rather eliminate the reaching phase [13].
FIGURE.2.12 Phases of sliding mode [3]
One way is to increase the control input thereby reducing the reaching time, but the system
becomes very sensitive to the un-modelled dynamics, and results in high chattering.
[6][7][8][9].
The reaching law is a differential equation that decides the dynamics of sliding surface.
The differential equation of asymptotically stable sliding surface s(x) is itself a reaching
condition by proper choice of the parameters in the differential equation the quality of the
VSC can be controlled. The general form of reaching condition is [26]
sgn( ) ( )s Q s Kh s (2.17)
where
1[ ...... ], 0m iQ diag q q q
1sgn( ) [sgn( )....... ( )]T
ms s sng s
1[ ,..... ], 0m iK diag k k k
Literature Review
16
1 1( ) [ ( ),.... ( )]T
m mh s h s h s
( ) 0, (0) 0i i i is h s h
From this general form, the three reaching laws are
1. Constant Rate Reaching:
sgn( )s k s (2.18)
This law forces the switching variable ( )s x to reach the sliding manifold S at a constant
rate. The advantage of this law is its simplicity in applying to the system. But it will see
later that the value of k, if taken small, the reaching time increases, and if the value of k is
made large, the chattering is increased severely.
2. Constant plus Proportionate Reaching:
sgn( )s Q s Ks (2.19)
In this the proportional term –Ks are added to the constant term, so the states will move
faster towards the switching manifold at a rate proportional to the value of s .
3. Power Rate Reaching:
sgn( ) 0< <1, i=1 to i i i is k s s m
(2.20)
This reaching law increases the reaching rate when away from the sliding surface, and
reduces the speed, when it reaches nearer to the sliding surface. This result in fast reaching
and chattering is considerably reduced. Thus power rate reaching gives a finite time
reaching. Also, because of the absence of sgn( )Q s the chattering is considerably
reduced.
2.3.3 Higher Order Sliding Mode Control
Sliding mode control methodology is popular control algorithm, for implementation where
robust control is needed amongst the available methods in modern control theory. In spite
of many salient features of sliding mode control such as insensitive to matched
disturbances, finite time convergence, reduced order design, it requires some novel
technique to reduce the high frequency oscillation i.e. chattering [27].
One of the main known drawbacks of the sliding mode control is the high frequency
chattering which is due to the discontinuous right hand side. To avoid chattering several
Preliminaries of Sliding Mode Control
17
approaches are proposed as discussed in previous sections, like using a smooth control law
to modify the reaching law. The idea is to change the dynamics in the vicinity of the
discontinuity surface in order to avoid real discontinuity and also preserve the same
properties of the whole system. The concept is to have new dynamics near the switching
surface that will give sufficiently smooth response. Thus can be used as using an artificial
actuator. The actuator may be functional or have its own dynamics [28]. This actuator will
behave in such a manner that it will indirectly have smooth output when the constraint s=0
is satisfied.
Thus HOSM control approach generates the derivative of the control signal instead of real
control signal itself. [29]
The history of higher order sliding mode dates back when Arie Levant in his PhD
dissertation first proposed the higher order sliding mode to reduce the chattering present in
sliding mode control. [29]
HOSM is a movement on discontinuity set of the dynamic system understood under
Filippov’s sense. The sliding order characterizes the dynamic smoothness degree in the
vicinity of the sliding manifold. The sliding order is a total time derivative of s (including
the zero one) in the vicinity of the sliding mode. Hence the thr order sliding mode is
determined by the equalities
1... 0rs s s s (2.21)
Where r is the relative degree. The ‘rth
sliding mode’ is often commonly called ‘r-sliding’.
2.3.3.1 Relative Degree: Consider a dynamic single input single output system
( ) ( ) , ( )x a x b x u y h x (2.22)
Where a, b, h are sufficiently smooth in a domain nD R . The mapping a:
and :n nD R b D R are called vector fields on D. The derivative y is given by
( ) ( ) ( ) ( )a b
hy a x b x u def L h x L h x u
x
(2.23)
where ( ) ( )a
hL h x a x
x
is called the lie derivative of h with respect to a or along a [30].
The following notations are mentioned below
Literature Review
18
( )
( ) ( )ab a
L hL L h x b x
x
(2.24)
2 ( )( ) ( ) ( )a
a a a
L hL h x L L h x a x
x
(2.25)
1
1 ( )( ) ( ) ( )
kk k aa a a
L hL h x L L h x a x
x
(2.26)
0 ( ) ( )aL h x h x (2.27)
If ( ) 0bL h x , then ( )ay L h x is independent of u. If we continue to calculate the second
order derivative of y, denoted by (2)y ,
(2) 2( )( ) ( ) ( ) ( )a
a b a
L hy a x b x u L h x L L h x u
x
(2.28)
Once again, if ( ) 0,b aL L h x then (2) 2 ( )ay L h x is independent of u, repeating this if h(x)
satisfies 1 ( ) 0i
b aL L h x for i=1,2,3……,r-1 and 1 ( ) 0r
b aL L h x , for some integer r, then u
does not appear in the equations of ( 1), ,...... ry y y and appears in the equation of ( )ry with
a non-zero coefficient:
1( ) ( )r r r
a b ay L h x L L h x u (2.29)
Such integer r has to be less than n or equal to n, and is called the relative degree of the
system.
HOSM can be implemented for any relative degree, but it requires the additional
information of higher derivatives of s shown in (2.29) in real time which is not always
possible, as real time differentiation become basic problem. Therefore SOSMC is
considered in this work. The 2nd
order sliding mode are of two types twisting and super
twisting controller.
The twisting controller is a discontinuous type of second order sliding mode control.
Problem with twisting controller is that it requires additional measurement of x to reduce
the chattering. It is not always possible in real time system that the measurement of x is
available due to the limitation of sensors available for measurement as already discussed in
previous paragraph [31].
Preliminaries of Sliding Mode Control
19
2.3.3.2 Super Twisting Controller: Super twisting controller is a suitable replacement
of the first order sliding mode controller for the system with relative degree one in order to
avoid chattering and to achieve good tracking performance [31].
Consider once again a dynamical system
( ) ( )x a t b t u (2.30)
and suppose that for some constants C, KM, Km, UM, q
,, 0 ( , ) / , 0 1M m M Ma U b C K b t x K a b qU q
The following controller does not need measurement of x . Let
1/2
1
1
sgn( )
sgn( )
M
M
u x x u
u u Uu
x u U
(2.31)
FIGURE 2.13 Phase Portrait: Super Twisting Controller
Lemma 1[31]: With Kmα>C and λ sufficiently large the controller provides for the
appearance of a 2-sliding mode 0x x attracting the trajectories in finite time. The
control u enters in finite time the segment [-UM, Um] and stays there. It never leaves if the
initial value is inside at the beginning.
The controller is called the Super Twisting Controller. The phase portrait of the controller
is as shown in Fig.2.13.
Literature Review
20
2.4 Summary
As discussed in previous section the advantages of SMC based VSC, like the robustness,
insensitive to parameter, disturbance rejection has motivated the researchers to use the
SMC in various applications ranging from automation industry, chemical process industry
to robotic application.
SMC has been used in chemical process as in [32]. FOPDT model is derived and SMC is
used are a controller in chemical process unit [33]. The SMC observers are used in non-
linear process control in [34]. SMC is used is used in temperature control application in
[35]. SMC is also used to control the drum level [36]; SMC is also applied to electro
pneumatic system [37], [38]. SMC is also used in nuclear application [39]. HOSM is
recently used in fed batch processes in [40]. Second order SMC is used in observer-based
fault reconstruction for PEM fuel cell air-feed system in [41]. Adaptive SMC observer is
used in catalytic reduction system for ground vehicles [42]. Terminal sliding mode is used
in CSTR [43]. Second order SMC with fuzzy logic used for optimal energy management in
wind turbine [44]. Fuzzy adaptive SMC in under actuated systems is used in [45]. SMC as
supervisory control is used with adaptive robust PID control in [46]. Sensor less HOSM is
used in induction motor in [47].
In this work the SMC is applied to control the process variables, level and temperature, the
chattering is observed when SMC is applied to the couple tank system. The modified
reaching law, the power rate reaching law is applied as it has fast convergence and
suppresses chattering. The HOSM based algorithms are used widely as it suppresses the
chattering, so in this work the SOSMC based STC algorithm is designed to control the
level and temperature of the coupled tank and the batch reactor respectively. So show the
efficacy of the control algorithms discussed so far, the real time temperature control
problem is considered. The results are analyzed for the IAE and ISE.
Level Control of coupled Tank System
21
CHAPTER-3
Level Control of Coupled Tank System
3.1 Background
In this chapter the level control servo problem is considered. The level of the tank 2 in the
coupled tank system is to be controlled. To show the robust performance the control is
achieved in the presence of input disturbance. The classical SMC tracks the desired level
but the control law shows the chattering. The chattering is undesired so the modification in
the reaching law is done by replacing it with power rate reaching law. Further, the super
twisting controller is used as it is reduces chattering. Thus the results are obtained and the
chattering reduction is observed.
3.2 Introduction
As discussed in detail in chapter 2, the variable structure control approach is widely used in
advance control schemes. In 1977[12], the application of variable structure control based
sliding mode control was discussed elaborately. Sliding mode control consists of two
phases to design the controller, the reaching law equation and the design of the sliding
surface equation. In the reaching phase the difference equation is designed so that the
states reach the sliding surface. Once the states are on the sliding surface they converge to
the desired points [12]. There are many applications in the literature where non-linear
systems are controlled using one order SMC [48], [49], [50], [51], etc. As SMC is
insensitive to the parametric uncertainty, it is applied in critical nonlinear systems and
robotic applications [50], [51], [52] etc. the undesired feature of SMC is chattering. Many
efforts are taken for reducing chattering by designing the adaptive gain controllers [24],
[53], designing a new second order reaching law [8], and cascade sliding mode control
Introduction
22
[54]. The features of SMC are degraded as the efforts to reduce the chattering are taken. As
from the literature it is known that on the sliding surface the controllers gives the robust
performance for disturbances from the input, parametric uncertainties etc., but on the other
hand, on the reaching phase the controller is sensitive to these type of disturbances and
uncertainties, so the efforts to reduce the reaching time can be taken.
When on the sliding surface the states are robust to the disturbances [55]. The switching in
the one SMC is causes the chattering. In [31], the concept of HOSM was introduced to
reduce the chattering. As HOSM preserves the features of SMC and also reduces chattering
[31]. The coupled tank system is part of many in industrial processes. Motivated by the
work done in [56], [57] where the SMC is used to control the level of the coupled tank.
The major contribution in this chapter is to control the height of tank 2 in the presence of
matched disturbance using conventional SMC, power rate reaching law based SMC and
STC.
3.3 Process Description
FIGURE 3.1 Coupled Tank Schematic
The dynamic model of the Coupled Tank is given (3.1) [56].
11
21 2
1
1
L
dhq q
dt M
dhq q
dt M
(3.1)
where
Level Control of coupled Tank System
23
1 12 1 2 1 2
2 2 2 2
2 ( ) for
2 for 0
q c g h h h h
q c gh h
(3.2)
The following constraint qL ≥ 0 is to be satisfied, as the input fluid in the tank can’t be [56].
Also 2 12
1 2
2 2
c g c ga a
M M (3.3)
As in [56] the dynamic equation of the two- tank to design the control law is as follows
1 2
2 21 21 2 22 1 2
2 1 2
1( )
2
x x
z za a ax a a u
M Mz z z
(3.4)
The values of 1 2&z z are the functions of 1 2&x x is given as
1 1
2
1 1 2
2
2
z x
a x xz
a
(3.5)
The objective is to set the height of tank 2 to desired level so, 1 2( ) ( )y t z t h to a
desired value H. The Single input single output problem is considered in which the input
flowrate is the manipulating variable and the controlled variable is the level of tank 2.
So mathematically it can be represented as
1 2
2
1
( ) ( )
x x
x f x b x u
y x
(3.6)
By comparing (3.4) and (3.6) hence
2 21 21 21 2
2 1
2
2
( )
1
2
z za af a a
M z z
ab
M z
(3.7)
The equations thus obtained will be used to design the control law using SMC and STC.
Introduction
24
3.4 Controller Design
The controller design consists of selecting the sliding surface, then the design of reaching
law equation so that the states follow the equation and reach the sliding surface. All the
three control algorithms are discussed in the following sections.
3.4.1 Sliding Mode Controller for Level Control of Coupled Tank
As seen in the previous section the equation suitable for designing the control law has been
derived. Equation (3.6) and (3.7) will be used to design the control law.
Let us first consider the second order system described by (3.6) so as to design the input
equation
Where 2
1 2
Tx x x R , f(x) is a nonlinear function and u is the control input. The y = x1
is the height of tank 2 i.e. the output and the control objective is to stabilize the output to
the output the presence of disturbance from the input channel. The details of the variables
are as follows:
The output variable is height of tank 2
The input to the system is input flow rate qL in to the tank 1.
Therefore the constraints are that for the height to be maintained at desired level in tank 2
qL >0, h1>h2 or h1-h2>0
By assigning the states to the system
Let x1=h2 and x2=h1, qL=u
Therefore our output or the controller variable is x1
the equation for the sliding surface is given by (3.8)
1 2s cx x (3.8)
The value of c should be such that the (3.8) is Hurwitz [58].
For the servo problem the sliding surface is given as
1 2( )s c x H x (3.9)
Where H is the set point for level of tank-2.
Equation (3.10) gives the control law of the form as
equi discontinuousu u u (3.10)
In order to reach the states on the sliding surface, mathematically can be represented as
Level Control of coupled Tank System
25
0s (3.11)
Taking the derivative of the sliding surface, so
0s (3.12)
Also
ss (3.13)
It is called η reachability condition [1]
Taking derivative of (3.8)
1 2s cx x (3.14)
Substituting (3.6), (3.7) in (3.14) so,
2 ( ) ( ) ( )s cx f x b x u ksign s
(3.15)
so, by substituting in (3.15) we get
2 21 21
2 1 2 2
2 1
2 ( ) ( )z za
u M z a a cx ksign sM z z
(3.16)
By proper choice of switching gain k, the controller is designed.
The stability of the sliding surface with the system guarantees finite convergence. the
stability of the sliding surface is proved using Lypunov stability theorem.
3.4.2 Stability Analysis:
For the stable sliding surface the Lypunov stability is proved as
2
1 2 2
1
1 2 2 2
1 2
1
2
0
( ( ) )( ( ) ) 0
( ( ) )( ( ) ( ( ( ) )
( ) ) 0
( ( ) )( ( ) ) 0
v s
v ss
c x H x cx f x bu d
c x H x cx f x b b f x cx
ksign s d
c x H x ksign s d
(3.17)
To make the stable sliding surface the parameter k>0 and also k>>d in (3.17) to reject the
disturbance.
Introduction
26
3.4.3 Power Rate Reaching Law for Level Control of Coupled Tank
As discussed in the literature the one order sliding mode control induces chattering,
therefore looking at the theory discussed in Chapter 2, Section 2.3.2, the power rate
reaching law based SMC has the advantage of fast convergence as well as reduced
chattering.
As per (2.20) the power rate reaching law is given by
sgn( ) 0< <1, i=1 i i i is k s s to m
Substituting (3.6) & (3.15) in (2.20) hence,
2 ( ) ( ) ( )s cx f x b x u k s sign s
(3.18)
Therefore by equivalent control method,
1/22 21 21
2 1 2 2
2 1
2 ( ) ( )pwr
z zau M z a a cx k s sign s
M z z
(3.19)
Thus (3.19) gives the control law for SMC with power rate reaching law. By proper
selection of the gain k, later e simulation results for chattering reduction using power rate
reaching law will be observed.
3.4.4 Super Twisting Controller Design for Level Control of Coupled Tank
As discussed in Chapter 2 Section 2.3.3, several HOSM algorithms are discussed in the
theory of sliding mode control like [58],[59],60],[61], [62].
Amongst them HOSMC, Super Twisting Controller is continuous in nature and hence has
been applied in large number of systems. The Super Twisting Controller is used at the
discontinuous part of the control law [9].
The systems having relative degree one, in those systems we can used STC [9], [63].
The mathematical representation of the STC is given as [9].
1/2
1
2
( )
s ( )
stcu k s sign s v
v k ign s
(3.20)
As discussed earlier, to implement a super twisting controller, the relative degree must be
one. Therefore to prove the relative degree one for the coupled tank system.
Level Control of coupled Tank System
27
So again revisiting (3.18) for the derivative of the sliding surface. From (3.18) it is clear
that input u appears in the first order derivative of the sliding surface, so its relative degree
in one. Hence super twisting controller can be applied to the coupled tank non-linear
model.
The gains k1 and k2 of the STC are set so that the finite time convergence is achieved.
( ) ( ) 0s t s t T (3.21)
By knowing the bounds of the disturbance, d L also 1 1.5 & 2 1.1k L k L , the tuning of
the STC is done for finite time convergence by finding gains k1 and k2 with k1>k2>0. so
that the sliding surface equation s converges to zero in finite time [58].
Therefore, the control law using STC is designed as
1/22 21 21
2 1 2 2 1 2
2 1
2 ( ) ( ) ( )z za
u M z a a cx k s sign s k sign sM z z
(3.22)
By looking at the control law, the integrations of the discontinuous part give a smooth
response which finally reduces chattering.
The design constants are as per the following table [56]
TABLE 3.1: Coupled Tank Design Constants
Sr.
No.
Parameter Value
1 c12(area of coupling Orifice) 0.58 cm2
2 c2(area of outlet Orifice) 0.24 cm2
3 M(cross sectional area of Tank-1 and Tank-2) 208.2 cm2
4 g(gravitational Constant) 981cm2/s
3.5 Summary
The simulations are carried out in the MATLAB Simulink. The simulation results are
shown in the chapter 7 ‘Results and Discussions’. Thus in this chapter FOSMC is
implemented to control the level of tank 2 at desired set point. The chattering was observed
Introduction
28
in the control law. To reduce the chattering, the constant rate reaching law was replaced by
power rate reaching law. The reduced chattering was observed and the convergence was
also faster than the constant rate reaching law. The introduction of HOSMC was due to the
drawback of SMC that it induced chattering, therefore as discussed in the literature, most
suitable replacement of FOSMC i.e. SOSMC Super Twisting controller which has all the
properties in FOSMC , also it reduces chattering. The Simulation results show that the
chattering is reduced using STC.
Background
29
CHAPTER-4
Temperature Control of Batch Reactor System
4.1 Background
In this work temperature control of batch reactor is considered. The temperature gradient in
a batch is a time dependent trajectory which is to be tracked in the presence of disturbance.
The robust tracking is achieved by the means of SMC. Since the FOSMC induces
chattering, the control is achieved through the power rate reaching SMC. The chattering
reduction has been observed. Since the batch reactor is a highly non-linear system, state
dependent coefficient (SDC) factorization method is used to represent the non-linear
system. Also as discussed earlier, STC is a viable replacement of FOSMC; the results are
simulated to achieve the control using super twisting controller in the presence of
disturbance. The chattering is observed in all the three type of controllers namely SMC,
Power Rate Reaching SMC and STC.
4.2 Introduction
The reactor is one of the major unit operations in any process industry. The reactors can be
classified as batch reactor and continuous reactors. Continuous reactors are most widely
used in most of the applications. But the applications such as production of polymers, fine
chemicals and pharmaceuticals, continuous production is not feasible or economical, so
they are operated with batch process. Mixing process with a fixed quantity of chemicals for
predetermined temperature, the chemical processes which are endothermic, as well as
components whose properties change with the temperature, batch process is a choice [64].
In this article a batch reactor model is considered in which two components are mixed to
Temperature Control of Batch Reactor
30
form the third component, whose concentration depends on the temperature maintained in
the batch. In various forms of batch reactor, researchers have applied different strategies of
temperature control. A sequencing batch reactor was considered in which the nitrification
and denitrification was carried out by maintaining the temperature in a predetermined
range [65]. A simple procedure is developed to determine the thermal behavior of a reactor
based on heat balance using PID temperature control [66]. A salt based cooling system is
developed for engineering demonstration reactor i.e. nuclear reactors [67]. The
hydrothermal conversion was studied with temperature control for a new batch reactor in
[68]. In biodiesel reactor temperature control is achieved using split range control [69]. A
cascade-like nonlinear parametric predictive control structure combined with a predictive
functional control concept is presented for bench scaled batch reactor [70]. Dynamic
matrix control is used to control exothermic process, for multi stage batch process to
improve product quality [71].
From the literature it is observed that when dealing with reactor, temperature control is
considered in most of the applications. Since the temperature control is crucial in most of
the processes, the robust control is indeed needed in many situations. There are many
controllers available which maintain the parameter at a predetermined value. Most widely
used and easy to implement is the PID controller, in which researchers have implemented
several modifications to the classical PID control as it has overshoots and oscillations. Like
in [72], have implemented optimize PID controller, LQR and PID are compared for
electric furnace system in [73]. A fractional order fuzzy PID controller is designed for
binary distillation column [74].
As known, classical PID controllers are linear controller applied to linear systems, has
overshoots, and sometimes induces large settling time as well as large oscillations. Also if
these controllers are implemented for nonlinear systems, the systems need to be linearized
in certain operating conditions. Therefore in recent years, attention of researchers towards
designing of model based state dependent controllers which are then applied to control
nonlinear systems, gives good performance with minimum overshoots and reasonable
settling time. These controllers also prove to be robust in the presence of matched as well
as unmatched disturbance as well as modeling uncertainties, by proper selection of the
tuning parameters.
Objectives of Control of Batch Reactors
31
Out of these controllers, variable structure system in which sliding mode control strategy is
most widely implemented since last 5-6 decades. The theory which was very well applied
in Russia several years back was first published in English by Utkin [12]. Sliding mode
controller design is a two-step procedure in which the first being designing the sliding
surface and the design of reaching law. In sliding mode, once the states are on the sliding
surface, the systems become insensitive to the perturbations [1], [12], [75]. There is huge
number of applications of sliding mode controllers in all type of systems.
In the literature many researchers have applied SMC to control process parameters. SMC
based on FOPDT model is used to control two chemical processes [33], Terminal SMC is
used to control CSTR [43], and SMC is applied to temperature control of tempered glass
[57]. SMC is also applied to level control in coupled tank and quadrupole tank [56],
[76],[77]. Temperature control of chemical Batch reactor using SMC is considered in [78]
SMC has several good properties, like insensitive to perturbation, finite time convergence.
Chattering which is the inherent property of SMC which is not acceptable in mechanical
systems so, higher derivatives of the mechanical control variable is included so that later
results are continuous. [59], [78]. This is in general called the higher order sliding mode
control (HOSM). In this, there is twisting controller; it is discontinuous type higher order
SMC, then super twisting controller which is continuous type controller [79].
Motivated by the work done in [80] and [56], in this thesis the control law is designed
using first order SMC, Power rate reaching law based SMC and higher order sliding mode
control based STC technique. The batch reactor dynamic equations are taken from the
literature. The dynamic equations are highly nonlinear. Therefore these equations are
represented using SDC factorization method.
4.3 Objectives of Control of Batch Reactors
The batch reactor control has wide objectives ranging from simple disturbance rejection to
a complete time based cycle to achieve the desired concentration of the final product. Most
important control objective in terms of practical view point is safety, final quality of the
product. Lot of expertise is achieved using continuous process by process engineers by
achieving steady state operating conditions. But there is a short fall of such expertise in
Temperature Control of Batch Reactor
32
batch processes. There is challenge when the control of batch process is considered; the
major reason being some technical and operational considerations as [81]:
Time Variant Characteristics: There is more than one operating point so the control system
cannot be designed. The reason for this is that in batch rector the cycle begins from initial
sate to final state with lot of variation in the two states, as the initial chemicals and the final
product that is formed may be totally different. To elaborate, maintaining the temperature,
the concentration, or the heat produced (exothermic) or the heat absorbed (endothermic),
the reaction rate also change considerably during the batch process. So one operating point
cannot be fixed in this type of more than one varying parameters so control systems design
becomes a difficult problem.
Non-Linear Behaviour: As there are several non-linearities present when the batch process
is considered, like the reaction rates, the temperature dependence, the heat exchanged
between the cooling/heating jackets. The reactor operating range has a large span the
specific operating point does not solve the purpose, or cannot opt approximate linearized
models over the linearized from the control design view point.
Model Inaccuracies: Often it is time consuming to develop the mathematical model for
batch processes. Many a times even critical reactions are unknown. Only few general
equations are known. These problems may lead to modelling inaccuracies.
Few Specific Measurements: The sensors used to measure the temperature, pressure etc. may not be
accurate because of the wide operating range, and also the measurements are carried out by using
non-invasive methods. Sometimes the samples need to be drawn and then analysed for the
measurement purpose. So many a times off line measurements need to be carried out.
Disturbances: Some of the disturbances like addition of wrong solvent may be due to
operator mistake, or the fouling of sensors, presence of impurity in raw materials. Also
when the reaction is taking place the heat produced is an important disturbance which is
varying or unpredictable. In many cases the heat produced is estimated using the
estimation techniques in control system design.
Irreversible Behaviour: In continuous process, there is scope of correction by adding some
solvent or some control action if there is some discrepancy. But in case of batch processes,
it is difficult for corrective action to be taken. Thus from the above points the control of
temperature in the batch reactor is challenging task.
Objectives of Control of Batch Reactors
33
4.4 Types of Reactors
The reactor is defined as the vessel in which two or more compounds are brought together
to obtain the final product. The reaction takes place when it is stimulated by heat supply or
the heat generated because of mixing of two compounds. so the reactor are basically of the
following types [81]
Batch: the process occurs repeatedly, uninterrupted, by uploading the reactants and getting
the finished product.
Semi Batch: In this if two materials are added, one is continuous and other is
discontinuous.
Continuous: In this the reactants are added continuously and can be distinguished between
counter current, co-current and cross current.
The chemical reactors are classified as
Batch Stirred Tank Reactor
Continuous Stirred Tank Reactor
Tubular Reactor
FIGURE 4.1 Batch Reactor [81]
FIGURE 4.2 Continuous Reactor [81]
Temperature Control of Batch Reactor
34
FIGURE 4.3 Tubular Reactor [81]
The continuous reactor and batch reactor work on continuous mixing by which even
distribution of heat as well as concentration is guaranteed.
The batch reactor is used for small portions where the reaction is left for particular period
of time for reaction to take place and to obtain finished product. They have well stirred
tank. The reactors are provided by heater system or cooling system, where a flow of
heating fluid or cooling fluid is used in order to provide heat to the reacting mixture or to
dissipate the excess heat. The various heating/cooling systems are available in industries
are:
a heating fluid or a cooling fluid in a jacket
a provision where cooling fluid is circulated in a coin and a jacket consisting of hot
fluid. So the provision of both the jacket and a coil.
a jacket can be alternatively flushed with heating fluid or a cooling fluid depending
upon the heating of cooling requirement,
4.5 Dynamic Model of Batch Reactor
The dynamic equations of jacketed batch reactor is given by the energy and mass balance
equation. On the basis of mass and energy balance that enters the systems must leave the
systems or accumulate in the system.
IN OUT Acc (4.1)
Where IN denoted energy in, OUT denoted energy out, Acc denotes energy accumulated.
In some cases product term is often added given by
IN PR OUT Acc (4.2)
PR denoted product term.
Objectives of Control of Batch Reactors
35
In the above notation, IN and OUT terms are positive, Acc term can be positive or negative.
PR term can be positive or negative.
The density and the mass heat capacity can be considered constants as we are assuming a
liquid phase
( )rr j
r pr r r pr r
dT Q UST T
dt c V c V (4.3)
The dynamic model of the batch reactor is given by (4.4). The process of batch reactor is
shown in figure (1). The temperature control of the batch reactor is achieved by
manipulating the heating/cooling element which is attached to the input side of the batch
reactor. The reactor consists of the jacket which consists of stem flow thru the jacket. Also
there is coolant coil in the reactor. The stirrer is for even distribution of heat. The coolant
flowing through the coil has the flow rate and heat transfer coefficient. These represent the
mass balance and energy balance equations of the reactor with the assumptions that the
density of the reaction liquid is kept constant and the mixing of the reaction liquid is
perfect [78].
FIGURE 4.4 Schematic of Batch Reactor [78]
Temperature Control of Batch Reactor
36
The reactor takes place in the following sequence with product A and B and the reactor
temperature T.
1 2R RA B C
Therefore CA and CB are the concentrations of product A and B, respectively and T
represents the reactor temperature [78]
2
1
2
1 2
2
1 1 2 2 1 2 1 2
( )
( ) ( )
( ) ( ) ( )
A A
B A B
A B
C R T C
C R T C R T C
T R T C R T C T T u
(4.4)
where
11 10
22 20
( ) exp(273 )
( ) exp(273 )
ER T A
R T
ER T A
R T
(4.5)
As discussed in [78], from the theory of batch reactor, the desired product is B and the
batch cycle is one hour. To get maximum yield of component B the desired temperature
should follow (4.6).
3( ) 54 71exp( 2.5 10 )dT t t (4.6)
Therefore, our objective is to control the temperature in the batch reactor close to (4.6) so
that the maximum yield of component B is achieved and hence to design a control law for
the same. Due to inherent highly nonlinear system it is a challenge to achieve (4.6). As
seen from (4.6), the desired temperature trajectory is non-stationary and time variant. It is a
challenge to control this type of temperature trajectory with the help of conventional
controllers. Also in the presence of matched disturbance as well as with the process
uncertainties, tuning the controller parameters becomes a tedious process. Therefore robust
controller can achieve better tracking than the conventional controllers in the presence of
uncertain conditions.
4.6 Controller Design
As discussed in previous section batch reactor dynamic model is defined by the differential
equations as in (4.4)
Objectives of Control of Batch Reactors
37
For the convenience of designing a controller let us assign state variables and input
variables as x1, x2, x3, u as CA, CB, and T and heating/cooling u element respectively.
Therefore (4.4) changes to
1 2 3 A Bx x x C C T (4.7)
Therefore, with respect to (4.4) and (4.7)
(4.8)
2
2 1 3 1 2 3 2( ) ( )x R x x R x x (4.9)
2
3 1 1 3 1 2 2 3 2 1 2 3 1 2 3( ) ( ) ( )x R x x R x x x x u (4.10)
Rearranging the (4.8), (4.9), (4.10) of batch reactor to represent in the linear structured
quadratic form state dependent coefficients is called SDC factorization [75].
1 1 3 1 1
2 1 1 3 1 2 2 3 2
3 1 1 3 1 1 1 2 2 3 1 2 2 1 3 3 1 2 3
( ) 0 0 0
( ) ( ) 0 0
( ) / ( ) / /
x R x x x
x R x x R k x x u
x R x x x R k x x x x x
(4.11)
Thus the form of equation is given by
(4.12)
By implementing SDC factorization on the batch reactor the obtained equation (4.11), now
the feedback controllers can be used for this nonlinear system, which are used for linear
systems.
The SDC matrices are given as
(4.13)
Therefore equation (4.12) & (4.13) will be used to design the control law using sliding
mode control.
2
1 1 3 1( )x R x x
1 3 1
1 1 3 1 2 2 3
1 1 3 1 1 1 2 2 3 1 2 2 1 3
( ) 0 0
( ) ( ) ( ) 0
( ) / ( ) / /
R x x
f x R x x R x
R x x x R x x x
1 2 3
0
( ) 0g x
x
1 1 1 1
2 2 2 2
3 3 3 3
x f x g
x f x g u
x f x g
Temperature Control of Batch Reactor
38
4.6.1 FOSMC Temperature Control of Batch Reactor
To implement the SMC controller to the batch reactor let us design the control law. First,
comparing (4.11) with the non-linear system equation given by (2.6)
2
1 1 3 1 2 2 3 2 1 2 3
1 2 3
( ) ( ) ( )
( ) ( )
f x R x x R x x x
b x x
(4.14)
As discussed in the literature the first step in designing the SMC controller is the design of
sliding surface.
For the type of temperature control problem considered in this work, an appropriate sliding
surface could be defined to be the difference between the actual and desired Temperature
levels in a reactor.
Ds T T (4.15)
4.6.2 Stability Analysis
As per the theory of sliding surface discussed earlier, the sliding surface must be stable
confined to the system. Let us find the stability of the sliding surface using Lypunov
stability theorem.
Let 21
2V s
V ss
The Lypunov stability theorem 0V
Substituting (4.15) and taking time derivative
3 3( )( ) 0d dT x T x
3( )( ( ( ) ( ) ) 0 d dT x T f x b x u d d is matched type disturbance.
1
3( )( ( ( ) ( )( ( ) ( ( ) ( ) ) 0d d dT x T f x b x b x T f x k s sign s d
3
3
( )( ( ) ( ) ( ) ) 0
( )( ( ) ) 0
d d d
d
T x T f x T f x ksign s d
T x k s sign s d
As seen from the above proof, to maintain the inequality, the value of gain k>0, also to
reject the disturbance the value of k has to be large enough of d.
Objectives of Control of Batch Reactors
39
As discussed in previous section about the theory of sliding surface, the reaching law
equation needs to be chosen, so as to reach the sliding surface. Therefore taking time
derivative of (4.15), hence
ds T T (4.16)
Substituting (4.5) & (4.14) in (4.16)
2
1 1 3 1 2 2 3 2 1 2 3 1 2 3( ( ) ( ) ( ) ) ( )ds T R x x R x x x x u ksign s (4.17)
Therefore by equivalent control method discussed in earlier chapters, the control law for
SMC for batch reactor temperature control is given by
1 2
1 2 3 1 1 3 1 2 2 3 2 1 2 3( ) (( ( ( ) ( ) ) ( ))du x T R x x R x x x ksign s (4.18)
Thus by proper choice of gain the controller will track the time dependant trajectory given
by (4.5).
4.6.3 Power Rate reaching based SMC for Temperature Control of Batch Reactor
As discussed in chapter 2, the power rate reaching law given by (2.20), the control law
needs to be designed. Therefore by substituting (2.20) in the equations of batch reactor
mathematical model given by (4.14), and taking the temperature trajectory given by (4.5).
The control law is given by
1 2
1 2 3 1 1 3 1 2 2 3 2 1 2 3( ) (( ( ( ) ( ) ) ( ))du x T R x x R x x x k s sign s
(4.19)
4.6.4 STC for Temperature Control of Batch Reactor
As discussed earlier, to implement a super twisting controller, the relative degree must be
one. So to find out the relative degree for the temperature control problem of the batch
reactor is the first thing to be done.
The relative degree one is found out by observing (4.17).
The super twisting controller is given by (3.20).
The control law for the batch reactor temperature control problem using STC is derived by
taking the derivative of the sliding surface, then taking the derivative of sliding surface to
guarantee that the states remain at the surface as time tends to infinity. Thus
Temperature Control of Batch Reactor
40
1 2
1 2 3 1 1 3 1 2 2 3 2 1 2 3 1 2( ) (( ( ( ) ( ) ) ( )) ( )du x T R x x R x x x k s sign s k sign s
(4.20)
Equation (4.20) is the control law for STC. By proper selection of gain k1 and k2, the
temperature will be tracked to follow (4.5). The gain value must be so selected that they
supress the matched disturbance and track the temperature trajectory (4.5).
The design parameters are given in the Table 4.1. Where A10, A20=Frequency factor for
reactor AB and BC respectively, α1,α2,β1,β2,γ1,γ2-Coefficients, E1,E2-activation
energy, R-ideal gas constant,Tc-Coolant Temperature.
TABLE 4.1 Design Parameters for Batch Reactors
A10 = 1.1 m3. kmol−1s−1 γ1 = 41.80C. s−1
A20 = 172.2 s−1 γ2 = 83.60C. s−1
E1 = 20900 kJ. kg−1. K−1 α1 = 4.31450C. s−1
E2 = 41800 kJ. kg−1. K−1 α2 = −0.10990C. s−1
R = 8.3143 kJ. kmol−1. K−1 β1 = 1.49620C. s−1
Tc = 250C β2 = 0.05150C. s−1
4.7 Summary
The simulation results are shown in chapter 6, ‘Results and Discussion’. Thus the Batch
reactor temperature is tracked to the desired trajectory. The highlight of this simulation is
that the temperature trajectory is non stationery, i.e. maintaining the temperature profile
during the batch is crucial as the quality of the final product is temperature dependent. The
chattering is observed in SMC, and it is drastically suppressed in SMC with Power rate
reaching law. The application of STC is because it is good replacement for SMC and also
has reduced chattering. This is observed in the simulation results for the STC based control
law. In the next chapter the experimental validation of the three control algorithms with
real time system will be seen.
Hardware and Interfacing
41
CHAPTER-5
Hardware and Interfacing
5.1 Background
In this work the variable structure sliding mode control is used for the temperature control
of the constant volume stirred tank reactor. The SMC law is achieved using the first order
calssical sliding mode control. This result is compared with the modified reaching law in
sliding mode control and the results are observed and compared experimentally for the
chattering analysis. Hence tracking problem is handled using sliding mode control
practically. Also the SOSMC based STC is designed for the same system as it is similar to
the FOSMC except that it reduces chattering.
In the previous chapters the case studies were considered for the same, the level and the
temperature. Since those were simulation results, therefore, real time application of the
algorithms on the system similar to the reactor is carried out in this work. The chattering is
observed in this systems and the analysis is done on the basis of performance measures
namely ISE and IAE. The experimental and simulated results for the system are plotted
and the chattering is observed.
5.2 Hardware Description
This set up is a TEQuipment CE117 process trainer which is used for the experiment
purpose. The apparatus is a fully integrated process control kit which is self-contained [82].
It consists of experimental module and control module. The two are connected with the
multi-way lead.
Experimental Setup
42
5.2.1 Experiment Module
The experimental board is a compact module that includes all the measurement and
process components required for CE117 process trainer. It includes actuator and power
supply required for all the components in its front panel like pumps, heater etc. for the
signal conditioning of the transmitters it incorporates all the circuits. It has two parts a
cooler circuit and a heater circuit which flows cold and hot water respectively.
The cooler flow circuit consist of
FIGURE 5.1 Experimental Setup for Constant Volume Stirred Reactor
a reactor vessel with a drain valve and vent
a storage tank (reservoir)
a D.C. motor with variable speed driven pump
a radiator and a variable speed fan in a cooler unit
a proportional valve with servo control.
The cooler unit is shown in the Fig.5.2
Process Vessel: The Process Vessel is a transparent cylinder. A scale on the front of the
Process Vessel allows the level of water to be accurately measured.
Experimental Module
43
FIGURE 5.2 Cooler Flow Circuit [82]
Pump-2: water is delivered to the cooler and then to proportional valve via a Pump 2 from
the reservoir which then flows to the process vessel. With the gravity water returns to the
reservoir from the process vessel through the drain valve. Through the external input
voltage given to the pump that has input socket mounted on the control module, the flow
rate is controlled.
Bypass valve: In the process flow circuit a needle type Bypass Valve is let the outflow
from the pump to directly return all the water to the reservoir so that the both , the process
vessel and the cooler are bypassed. This helps in providing the way to disturb the system,
or vary the flow rate to the process flow circuit.
Proportional Valve: An electrically controlled Proportional Valve is fitted to remotely
control the flow of water in the Process Flow circuit.
Transmitters: A capacitive level transmitter is mounted in the process vessel on the left
side of it and which is vertically positioned. The transmitter is a simple parallel plate
capacitor. There is a birfurcation at the connecting the top and bottom of the process vessel
through a short tubing that is flexible.
Experimental Setup
44
Signal Conditioning Circuit: the electrical signal from the transmitter measures the change
in the level or temperature as per the transmitter that gives the electric output with respect
to the parameter change in the reactor. The signal conditioning circuit output is calibrated
as 0V for empty process vessel and 10V for the maximum level.
Heat Exchanger Coil: There is a tubular coil at the base of process vessel that is a part of
hot water flow circuit.
TT5: A Platinum Resistance Thermometer (TT5) is fitted in the base plate of the reactor
vessel.
Stirrer: To have uniform temperature throughout the reactor vessel a stirrer is provided at
the base of the process vessel. It has magnetically coupled D.C. motor below the reactor
vessel to drive the stirrer. There is switch that allows the stirrer to be ON or OFF.
Reservoir: The storage tank includes a float type level Switch fitted in the left-hand side of
the storage tank. If the level of the water in the storage tank (Reservoir) goes below a low
level, the level Switch cuts-off the Pump 2 so that it does not dry run. There is an indicator
light on the panel of the Control Module lights when the level Switch senses low water
level. When the level rise above the smallest level, the float type level Switch closes and
the pump supply is activated and the indicator light on the panel goes off.
Cooler unit: Water goes from Pump 2 to the Cooler and afterward enters the reactor vessel.
The Cooler involves conservative arrangement of sections. These entries are associated
through heat with a honeycomb of metal balances that expansion the viable surface region
of the Cooler. A variable speed Fan powers air through the Cooler thus evacuates vitality
(heat) from the water moving through it. With consistent flow of water through the Cooler,
the temperature of the water in the Vessel (and the storage tank) can be decreased.
Thermometers: Platinum Resistance Thermometers are located at the inlet and outlet of the
Cooler. These can be utilized to quantify the temperature difference between the water
streaming all through the Cooler with the goal that the warmth vitality expelled from the
water can be resolved.
Experimental Module
45
TABLE 5.1 Analog Signal Values
Sr.
No.
Item Analog Signal Conversion Details
1
Temperature
Transmitter
Platinum Resistance
Thermometers
0-10V output
Linear
10°C per Volt,
0V=0°C, 10V=100°C
2
Electric Heater
0-10V input
75W per Volt
0V=Heater off,
10V=75W
3 Proportional Valve
0-10Vdc 0V = Closed, 10V =
Open
4 Pump1
Pump2 0-10V
0V=no Flow
10V=Maximum Flow
The Heater flow circuit comprises of
Heater Tank
A heat exchanger coil mounted in the base of the process vessel
a variable speed d.c. motor driven pump (Pump 1)
The heater tank is of stainless steel and has lid to limit the water loss due overflow and
evaporation.
Pump 1 can be driven at different speeds to deliver heated water to the heat exchanger coil
mounted in the base of the process vessel
The water in the heater tank is heated through a heater coil in the heater which has a
variable current to control the heat input to the water.
To measure the temperature of the water in the heater tank a Platinum Resistance
thermometer (TTI) is located in the right hand side of the heater tank. To limit the heater
tank temperature to 60°C, TT1 output is also connected to the circuit that disables the
supply to the heater.
A thermal switch also located in the right-hand side of the heater tank which opens at
70°C. This is provided as a safety back-up to prevent the water in the heater tank from
overheating.
Experimental Setup
46
To measure the temperature of the water when it passed through the heat exchanger in
reactor vessel and before it passes to the heater tank a thermometer (TT2) is located in the
flow pipe. The difference in the TTI-TT2 provides an indication of the effectiveness of the
heat transfer and the energy transferred from the heater tank to the water in the process
vessel.
The heater tank includes a float type level switch mounted in the left hand side of the tank.
If the level of the water in the tank goes below a predetermined low level, the level switche
cuts off the pump-1 to prevent it from dry run. The float switch also disables the heater
supply if the level of the water in the heater tank falls below a minimum to ensure that the
heater element is always covered with water. A marker light on the copy board of the
control module enlightens when the buoy switch has detected low water level. When the
dimension of the water transcended the base dimension, the buoy switch closes and the
siphon supply is empowered afresh and the marker light on the copy board turned off.
5.2.2 The Control Module
The Control Module is a board or the provision to connect all the actuators and the
transmitters in the experimental module to the computer systems. One can measure the
signal from the actuator and can give the signal to the actuator through this control module.
That is connects the CE117 trainer to the computer system where our controlling software
is installed. The control module also has a 4 channel A/D and D/A converter. Fig. 5.3
shows the front panel of control module.
The control module has the detail layout of the experiment module (CE117) that is of
whole process with physical access of the signals from the transmitters and to the
actuators.
Process Vessel Section
The process vessel sections on the control module contain the provision for:
Input signal (S) that can be given to the proportional valve to control its opening
The reading from the pressure transmitter(PT) i.e. output signal
The reading from the level transmitter(LT) i.e. output signal
The reading from the Temperature transmitter(TT5) i.e. output signal
The reading from the Flow transmitter(FT2) i.e. output signal
Experimental Module
47
The ON-OFF switch on this module gives the provision to ON/OFF the stirrer mounted in
the experimental module.
FIGURE 5.3 Control Module [82], [88]
Reservoir Section
The reservoir section in the control module includes:
A 2 mm attachment and a potentiometer to give authority over the conveyance of
Pump 2 in the Process Flow Loop.
A flip change to choose whether the siphon speed is controlled physically
(Manual), or it is controlled utilizing an external source.
Low Water Level Switches and Indicators
The base right-hand corner of the Control Module incorporates two LED pointers. In the
event that the dimension of water in both the vessel as well as hot water tank falls
underneath a base dimension, at that point the Float Level Switches incorporated into the
two tanks will open and cripple the separate siphon (Pump 1 or 2 individually). The
particular activity will be demonstrated by the applicable pointer LED lighting up. The
heating element in the Heater Tank is likewise disabled if the level of water turns out to be
Experimental Setup
48
excessively low. When the dimension in the tank increments over the base, at that point the
Float Level Switch shuts, the siphon supply is re-empowered and the LED marker goes
out.
FIGURE 5.4 SAFETY SWITCHES
5.3 Interfacing
The interfacing of the experimental setup with the computer system is done through
control module. Since the setup comes with the inbuilt CE2000 software. The CE2000 has
all the facilities by which one can control the level, pressure, temperature, flow of the
laboratory reactor using various control laws. The control module has the inbuilt D/A and
A/D Terminals for connecting various transmitters and pump nodes with the computer
system.
In this work, the design of model based controller is incorporated. So there are several
mathematical expressions that need to be incorporated in the equation of the control law.
Therefore, the data acquisition card in used by bypassing the D/A and A/D unit on the
control module.
The LabVIEW software is used to design the control law. The data acquisition card is used
to interface the experimental setup with the LabVIEW software. The DAQ card used is NI
6009. The National Instruments USB-6008/6009 devices provide eight single-ended analog
input (AI) channels, two analog output (AO) channels, 12 DIO channels, and a 32-bit
counter with a full-speed USB interface.
Experimental Module
49
FIGURE.5.5 NI6009 DAQ module
The interfacing thus consists of a LabVIEW and an experimental module interfaced
through the DAQ card. The communication of the data through the DAQ card is carried
out by the use of DAQ Assistant. The analog input is taken from the process vessel
temperature transmitter and given to the DAQ assistant through the DAQ card. Similarly,
the output generated due to the equation of control law is given to the Pump 1 of the
experiment module trough the DAQ Card AO channel. The other analog inputs are taken
from the heater, the inlet temperature to the process vessel, the outlet temperature from the
process vessel.
5.4 Dynamic Model of the Reactor
The hot water from the heater is passed through the coil in the reactor vessel. The tank
water is circulated through the condenser unit into the tank. The interfacing of this kit with
the computer system is done through DAQ NI6009 card of National Instruments as
discussed in previous section. The design of the controller is done in the LabVIEW
environment. The temperature sensor (RTD) and transmitter output is taken through the
DAQ card to the LabVIEW, the control law is designed using sliding mode control law and
the output from the LabVIEW through the DAQ card is given to the pump to manipulate
the flow of hot water through the coil.
The process vessel contains the heat exchanger that supplies an input heat flow rate Q. The
fluid feed to the system is at temperature Ti and mass flow rate wi. The outflow temperature
is T and mass flow rate w. The control variables are the heat input to the heater, fluid input
flow rate, and the outflow rate. The control objective is to control the temperature T and
Experimental Setup
50
the volume of fluid (V) in the reactor. This type of system is the used in many chemical
process systems.
For the reactor vessel of volume V filled with water of density ρ, the mass of water in the
vessel is given by Vρ.
The law of conservation of mass gives the mass balance:
(Rate of Mass Accumulation) = (Rate of mass into the vessel) - (Rate of mass out of the
Vessel)
FIGURE 5.6 Process Vessel Schematic [82]
The mass balance equation is given by
( )
i
d Vw w
dt
(5.1)
The law of conservation of energy is
(Rate of energy accumulation) = (Rate of energy flow into tank) - (Rate of energy flow out
of tank) + (Rate of heat addition to the system)
The energy balance equation is
( )
( ) ( )ref
i i ref ref
d V T TC wC T T wC T T Q
dt
(5.2)
Equation (5.1) and (5.2) can be simplified when assumptions for the fluid in the process
vessel (Stirred Tank) are included. Assume that the density ρ and the specific heat c are
constant.
Experimental Module
51
1
( )i i
dVw w q q
dt (5.3)
Equation (5.3) gives the process dynamics used for the control of the fluid level in the
stirred tank.
Looking at (5.2)
i i
dT dV QV T q T qT
dt dt C (5.4)
This special case which is ‘constant volume stirred tank processes’. This is widely used as
a typical process in chemical engineering. So the process equation can be written as
( )ii
dT W QT T
dt V V C (5.5)
Equation (5.5) gives the relation between the temperature and heat generated by the heat
exchanger. The relation between the flowrate of hot water through the coil in the process
vessel which will vary the temperature is to be needed. So let us consider the experiment
module in more detail.
There are two loops namely the process loop and the heater loop.
Process loop: In the process loop the fluid is pumped from the reservoir at temperature
(Tr) through the cooler where it is cooled to a temperature (Ti), and then into the stirred
tank (Process Vessel). The water returns to the reservoir by means of the drain valve. For
the temperature elements of the cooler system, an energy equation can be written to
balance the energy removed by the cooler with the energy lost by the fluid which runs
through the cooler system.
The law of conservation of the energy is
(Rate of energy removed by the cooler) = (Rate of energy loss by fluid in process loop)
The energy balance equation is
( ) ( )fan r ref i r iKU T T q T T (5.6)
The heater loop: The heater loop consists of a heater tank and a heat exchanger loop. The
heater tank contains an electric heating element and a temperature sensor. The temperature
of the heater tank is controlled by a local control loop. The water is pumped through the
Experimental Setup
52
heat exchanger under the control of the Pump-1. The model of this loop is a simple heat
transfer equation relating the heat energy lost from the heat exchanger to the energy gain in
the process tank. This can be done as a simple heater control experiment or combined with
the multi-loop experiment for the process tank.
The process dynamics for the heater loop are in two parts. One for the heating of the heater
tank, and the other for the heat transfer to the stirred tank via the heat exchanger and the
heater flow loop.
For the heater loop the energy balance is
(Rate of energy transfer to heat exchanger) = (Rate of energy loss by fluid in heater loop)
This is mathematically represented as
0( )h hQ q T T (5.7)
Now since the equation of the temperature (5.5) shows the relation between the
temperature of process vessel (T) and the heat input to the process vessel from the heat
exchanger (Q), the relation between flow rate (qh) to the process vessel through the heat
exchanger is to be derived.
Therefore considering the heater loop equation given by (5.7), getting the relation between
the heat input from the heat exchanger and the flow rate though the process vessel, so
substituting (5.7) in (5.5)
0( )( )i h h
i
dT W q T TT T
dt V VC
(5.8)
Therefore (5.8) will be used to design the control law where output to the system is the
temperature of the stirred tank and the output is the flow rate in to the process vessel.
As seen from the derivation of mass balance equation following assumptions are made.
A1: Fluid Density is constant
A2: Specific heat is constant
A3: Volume of liquid in the tank is kept constant
Considering assumption 3, the volume of the fluid is kept constant at 2 litres by adjusting
the inflow to the tank and outflow to the tank. The other constants are given in table below.
Experimental Module
53
TABLE 5.2: Design Specifications for Constant Volume Reactor
Parameter Values
V (Volume) 2 L
Density (water) ρ 997 Kg/m3
C (heat capacity of water) 4190 J/Kg/K at 25oC
5.5 Controller Design
The controller is to be designed using sliding mode control. As discussed earlier the sliding
mode control design has reaching stage and sliding manifold design step. First is to design
the sliding phase and other is to design a reaching phase. The task of designing the sliding
surface is clear that the sliding surface is the output where the system slides and goes to the
equilibrium position. In tracking problem, the sliding surface is nothing but when the error
i.e. the desired level and the actual level of any parameter say level or temperature reach.
Once the error is zero, the state must remain there up to time t tending to infinity.
Therefore choosing the sliding surface for the laboratory reactor minimises to tracking
problem with error as its sliding surface. So choosing the sliding surface as
ds T T (5.9)
Taking the time derivative of the sliding surface
ds T T (5.10)
Rewriting (5.8)
0( )( )i h
i
W u T Tx T x
V VC
(5.11)
Which takes the form of (2.6)
Substituting (5.11) in (5.10)
0( )( )i h
d i
W u T Ts T T x
V VC
(5.12)
5.5.1 FOSMC of the Constant Volume Stirred Reactor
Experimental Setup
54
The FOSMC is a well understood controller that is used in many applications. The
advantage of FOSMC is that it has finite time convergence. As the states reach the sliding
surface, it os not sensitive to input disturbance and hence is a robust controller. The
FOSMC is applied to the level control problem, then to a temperature control problem
which is simulation based. Working on real time system is a challenge as the real time
system has inherent non linearities, dead zones. So the aim of this work is after proving
simulation results the validation is to be done on the system similar to the reactor which
was simulated in previous chapter. This system is chosen for the purpose of real time
implementation and the chattering is observed due to the FOSMC, then the chattering
reduction techniques have been implemented and the real time results are obtained. The
stability equation for the sliding surface with the system is similar to that done in the batch
reactor case study, which indicates that the value of gain has to be more than zero and large
enough to supress the disturbance.
The control law for FOSMC is given by the equivalent control method and constant rate
reaching law given by (2.18)
0( )( ( ) ) ( )i h
d i
W u T Ts T T x ksign S
V VC
(5.13)
Therefore the control law for FOSMC is given by
5
0
8380[ 2.22 ( ) ( )]d i
h
u T e T x ksign sT T
(5.14)
By substituting the design parameters, the control law is given as (5.14). Replacing with
the flow rate equation, hence
5
0
8380[ 2.22 ( ) ( )]h d i
h
q T e T T ksign sT T
(5.15)
5.5.2 Power Rate Reaching Law based SMC
As discussed in the literature the major disadvantage of the SMC is the fast cycling i.e.
chattering to reduce the chattering several techniques are adopted. The theory of power rate
reaching law also says that the due to taking the power of s, the discontinuous part is
multiplied by the factors which reduces the chattering effect and also improves the
accuracy. Also the finite time convergence is fast as compared to conventional SMC. So in
Experimental Module
55
this work the power rate reaching law is adopted, which is implemented in real time system
and the chattering reduction and accuracy in the temperature tracking has been observed.
For designing the power rate reaching law SMC, let us again consider (5.12)
0( )( ( ) ) ( )i h
d i
W u T Ts T T x k s sign S
V VC
(5.16)
Finding the equation for u , hence
5
0
8380[ 2.22 ( ) ( )]d i
h
u T e T x k s sign sT T
(5.17)
So the equation for flow rate is given by
5
0
8380[ 2.22 ( ) ( )]h d i
h
q T e T T k s sign sT T
(5.18)
By proper choice of gain k and the tuning parameter α output will be obtained.
5.5.3 STA based Control of Temperature
In this work the real time implementation of STC is carried out. The target is to achieve the
desired temperature. The temperature tracking has been carried out with the FOSMC, but it
induced chattering. The literature of STC say that the STC is a replacement of FOSMC,
except it reduces chattering. The effect of chattering reduction is that the accuracy is also
increased. Since the tracking problem is considered, also temperature being crucial
parameter in reactors, so the accurate temperature tracking is required. The STC as
discussed in the literature is SOSMC which takes the second order derivative of the sliding
equation. The advantage of STC over other SOSMC techniques is that it is continuous type
of controller also it does not require additional information of s , which is not always
possible in all the type of applications. So let us design the STC controller for reactor
system, whose dynamics are very well elaborated in previous sections. To implement the
STC, the relative degree is to be one. So first of all verify the relative degree one.
Let us consider (5.12). in (5.12) the first order derivative of sliding surface equation, the u
i.e. the control law appears and hence the relative degree between the sliding surface and
control law is one. The detail of relative degree is given in chapter 2.
Mathematically the representation of STC is given by (3.20)
Experimental Setup
56
So considering (5.12),
5
1 2
0
8380[ 2.22 ( ) ( ) ( )]d i
h
u T e T x k s sign s k sign sT T
(5.19)
So by using the design parameters, the above control law will be used to observe the effect
of STA on real time system. By proper choice of gains k1 and k2, and α, the tracking
problem will be handled.
5.6 Summary
So in this chapter the details about the hardware kit which is used, the mathematical model
of the hardware and the interfacing with the LabVIEW software are seen. The hardware
study is important as the mathematical model is to be derived using its mass and energy
balance equations. The control law is designed with these equations and the constants are
found out for the fluid selected. In the next chapter discussion of the outcomes from the
experimental results will be seen. By proper selection of the tuning parameters the control
law is to be implemented for all the three control algorithms discussed.
Results & Discussions
57
CHAPTER-6
Results and Discussions
6.1 Introduction
In the previous chapter the details of the hardware to be used for experimental validation
are discussed. The simulation results of the level and temperature for the coupled tank and
the batch reactor are discussed in this chapter. Both the simulations are carried out in the
presence of disturbances from the input channel. In this chapter the results of the control
law derived in the previous chapter and implemented on the real time system will be
analysed.
6.2 Simulation Results of Level Control of Coupled Tank System
The control law for all the three controllers namely, FOSMC, Power Rate Reaching SMC,
STC are derived in Chapter 3. The simulation is carried out in the MATLAB (2015a)
Simulink Environment. The simulations are carried out in the presence of input noise given
by d=10sin(t), c=2, k=11.2, L=15 k1=1.5 L =4.74, k2 =1.1*L=16.5. The simulation is
carried out that first the FOSMC based control law in applied and compared with Power
Rate Reaching SMC. The for power rate reaching law is selected as 0.7 as optimum
value for desired output. The control Law for both is observed for chattering. Then the
SMC and STC are compared and again the control law for both the controllers is observed
in terms of chattering. The desired height of tank 2 is set at H=4cm.
Simulation Results
58
FIGURE 6.1 Level of Tank-2 To H=4cm using SMC & Power Rate SMC
FIGURE 6.2 Control Law for level control using SMC
FIGURE 6.3 Control Law for level control using Power Rate reaching law based SMC
Results & Discussions
59
FIGURE 6.4 Level of tank-2 To H=4cm using SMC & STC
FIGURE 6.5 Control Law using SMC for Level Control
FIGURE 6.6 Control Law using STC for Level Control
Simulation Results
60
6.3 Simulation Results of Temperature Control of Batch Reactor
The simulation of the batch reactor mathematical model (4.13) is carried out on MATLAB
(2015a) Simulink. The tuning parameters for all the three controllers namely, SMC, Power
rate reaching SMC and STC are selected such that the temperature tracking is achieved as
well as chattering suppression has been observed. The power rate reaching law k=10 and
α=0.7, for SMC the k=60, for STC as discussed in previous section k1=3.8, k2=11. The
simulation is carried out in the presence of matched disturbance d=5sin (t).
FIGURE 6.7 Temperature Tracking using SMC for Batch Reactor
FIGURE 6.8 Control law for Temperature Tracking using SMC
Results & Discussions
61
FIGURE 6.9 Temperature Tracking using Power Rate reaching law based SMC
FIGURE 6.10 Control Law for temperature tracking using Power Rate reaching law based SMC
FIGURE 6.11 Temperature Tracking using STC for Batch Reactor
Simulation Results
62
FIGURE 6.12 Control Law for temperature control using STC
6.4 Experimental Results of Constant Volume Reactor
The interfacing of the reactor is done with the software for implementation of control law.
The algorithm is developed in the LabVIEW 2015 software and as discussed using DAQ
assistant through the DAQ card, the flow rate is manipulated by varying the pump voltage.
The simulation and the experimental results are plotted in the presence of disturbance from
the input channel for the reactor.
All the results are taken in the presence of d=5sint. The gain is tuned to k=8.5 for getting
optimum output for SMC as well as power rate reaching SMC. α=0.7 is tuned to get the
reduced chattering. The STC gains are set as, L=10, where maxL d . max 5d , so
1 1.5k L , k2= 1.1*L,
k1=4.74, k2=11 for STC Controller.
The mathematical analysis of the tracking error is carried out for all the three control
strategies using Integral of Absolute Error (IAE) and Integral of Square Error (ISE).
The results are taken by taking the sampling frequency of 100 Hz. The figures show the
tracking of temperature, the control law as the chattering reduction needs to be observed.
The figure also shows the temperature profiles of the inlet temperature to the stirred reactor
(Ti), the outlet temperature from the process vessel (T0), the heater temperature (Th). Both
the simulation and experimental results are compared. This validates the model derived
from first principle.
Results & Discussions
63
FIGURE 6.13 Temperature Tracking using SMC for Constant Volume Reactor
FIGURE 6.14 Control Law for Temperature Tracking using SMC
FIGURE 6.15 Temperature Profile for SMC
Simulation Results
64
FIGURE 6.16 Temperature Tracking using Power Rate Reaching law based SMC
FIGURE 6.17 Control law for temperature tracking using Power Rate Reaching law based SMC
FIGURE 6.18 Temperature Profile using Power Rate Reaching law based SMC
Results & Discussions
65
FIGURE 6.19 Temperature Tracking using STC for Constant Volume Reactor
FIGURE 6.20 Control law for temperature tracking using STC
FIGURE 6.21 Temperature Profile using STC
Simulation Results
66
6.5 Discussion
As seen from the figures above and the previous simulation results related to the tracking
of level of coupled tank to the desired set point and the temperature control of batch
reactor, the results are obtained in the presence of matched disturbances. Let us discuss the
results so far obtained.
6.5.1 Level Control of Coupled Tank System
In this work the tracking problem is addressed to achieve the desired level of tank 2 to
H=4cm. The initial height of the tank-1=8cm and tank-2=5cm. The simulation results are
shown in Fig. 6.1, which shows the level of tank-2 reached to the desired level. The
matched disturbance is considered to be d=10*sin(t). The figure shows the level tracking
for SMC as well as Power rate reaching law based SMC. As seen from the Fig.6.1 the tank
level is reached to the desired level by Power rate reaching law based SMC is about t=1.8s
and the convergence using Conventional FOSMC is at about t=5.3s. Fig. 6.2 shows the
control law due to FOSMC, which shows the high frequency oscillations. Let us recollect
from the literature that the once the states reach the sliding surface, the controller induces
discontinuity. This discontinuous movement is referred to as chattering. The large value of
k, increases the switching and as seen from the stability condition k must be large enough
to supress the disturbance, so keeping the gain lower may cost for the robustness of the
controller. So the gain is tuned optimally to k=11.2 that gives the desired performance. The
sliding surface parameter is set at c=2 that makes the polynomial (3.9) Hurwitz, which
makes the sliding surface stable and guarantees the finite time convergence. Fig.6.3 shows
the control law due to power rate reaching law. The value of α is tuned to 0.7. As the α
ranges from 0 to 1, smaller the value of α, the controller behaves much like the
conventional SMC. So the value of α is so chosen that the chattering is reduced and fast
convergence is achieved. Fig. 6.4 shows the tracking of level using FOSMC and STC.
Discussed in the literature, the STC is a viable replacement of FOSMC having all the
advantages of the FOSMC and an added advantage of reduced chattering; this is seen in
Fig. 6.4 the tracking at desired level. The gain for SMC is further tuned to 10.5, as it
showed peak when kept at 11.2, so when compared with STC, the STC converged at
around t=0.68s and SMC reached the desired level at t=3.69s. The control law is seen in
Fig. 6.5 for FOSMC and Fig. 6.6 for STC. The FOSMC shows the chattering as the desired
Results & Discussions
67
level is reached. The chattering is reduced as the control law takes the integration of the
discontinuous part and so the smooth response is obtained. The gains of the STC are L=15,
where maxL d . max 10d , so 1 1.5k L , k2= 1.1*L,
k1=4.26, k2=16.5 for STC Controller. Thus the results show the efficacy of the STC over
the FOSMC.
6.5.2 Temperature Control of Batch Reactor
Since the research work consists of analysing the VSC approach to non-linear system, the
simulation study for the temperature control of batch reactor is considered. The batch
reactor is highly non-linear system. The desired temperature in this case is a non stationery
time dependant trajectory. The simulation is carried out in the presence of input sine type
disturbance d=5sin(t). The temperature in the batch must follow the trajectory to achieve
the maximum yield. Fig. 6.7 and Fig. 6.8 show the trajectory tracking for FOSMC and the
control law for the FOSMC. The tracking is achieved at t=0.98s. High frequency
oscillations are observed in Fig.4.4. as soon as the reactor temperature reaches the desired
temperature profile. The gain is set to k=60 which is high value to reject the disturbance.
Fig. 6.9 and Fig 6.10 show the temperature tracking and the control law for the Power rate
reaching law based SMC. As discussed that the power rate reaching law the properties of
fast convergence, reduced chattering, these can be easily seen from Fig. 6.9 and Fig.6.10.
The gain is set to k=10 and α=0.7. The chattering is considerably reduced by implementing
power rate reaching law SMC. The convergence is also faster at about t=0.3s to achieve
the desired trajectory. Fig. 6.11 and Fig. 6.12 show the temperature tracking using SOSMC
STC. The tracking is achieved in the presence of sine type of disturbance. The gains of the
STC are L=10, where maxL d . max 5d , so 1 1.5k L , k2= 1.1*L, k1=4.26, k2=11 for
STC Controller.
6.5.3 Temperature Control of Constant Volume Reactor
The results of the temperature control of batch reactor are as shown in section 6.2. The
results are found in the presence of sine type disturbance d=5sin (t). The purpose of
choosing this system is to validate the results obtained in the simulation of batch reactor.
The application of VSC approach to the real time system was the objective of this
Simulation Results
68
experiment. Fig. 6.13 shows the temperature tracking to Td=34°C. The control law for the
same is plotted in Fig. 6.14, which shows high chattering. The simulated result also shows
the chattering. The simulated as well as experimental results plotted with reference to the
desired temperature. As seen from the mathematical equation the temperature of the
process vessel is not only dependant on the inlet flow rate but also the heater temperature
(Th), the inlet temperature (Ti), the outlet temperature from the heater tank (T0). The
temperature profile for all the temperatures is plotted in Fig. 6.15. Fig. 6.16 and Fig. 6.17
show the Temperature tracking at Td=45°C and the control law using Power rate reaching
SMC. Fig. 6.18 shows the temperature profiles, when the readings were taken. Fig. 6.17
shows the chattering induced due to Power rate reaching SMC, as seen it is reduced as
compared to the Fig. 6.14. The convergence is faster using power rate reaching law than
the classical SMC. The gain is set to k=8.5 and α=0.7 the controller is implemented.
Fig.6.19 shows the tracking of the temperature using SOSMC based STC. The
implementation of STC is for the reduced chattering. The chattering is reduced as it takes
the integration of the discontinuous part. The control law for STC in Fig. 6.20 shows the
reduced chattering as compared to power rate reaching as well as SMC, experimentally.
Simulation results show the faster convergence with power rate reaching law than the STC
and SMC. The chattering is observed least in STC as compared to the Power Rate SMC
and conventional SMC. The performance measures are calculated to see the quantitative
analysis of the tracking error, for simulated and experimental results for all the three
control algorithms.
TABLE 6.1 Error Analysis using performance measures
Sr.
No.
Performance
Measures
SMC (Constant rate
Reaching law)
SMC(Power Rate
Reaching Law)
STC
Experimental Simulated Experimental Simulated Experimental Simulated
1 IAE 3.42 1.92 3.23 1.85 3.08 1.81 2 ISE 2.32 1.95 2.16 1.89 2.32 1.89
With reference to the Table 6.1, ISE integrates the square of the error over time. ISE is
least in Power rate SMC in experimental results, as it maximises the larger errors and
minimises the smaller errors as it is square of the error. So square of larger number is still
larger and the square of smaller number further becomes small. IAE integrates the absolute
error over time. It nullifies the positive and negative errors. The STC controller has the
least value of the IAE performance measure.
Conclusion & Future Scope
69
CHAPTER-7
Conclusion & Future Scope
7.1 Conclusion
The thesis consists of the application of variable structure controller to the process control
applications which are inherently non-linear. The process parameters considered in this
work are level and temperature. Firstly, the coupled tank is considered for the simulation
study. The level control of the tank-2 is controlled by using VSS based, FOSMC
controller. To design the controller there is a two-step procedure first to design the sliding
surface and to design the equation for the describing points to reach the sliding surface. So
the sliding surface is designed for the coupled tank system. The stable sliding surface is
designed. The constant rate reaching law equation is applied to the dynamic model of
coupled tank system. The limits of the gain values are found out by Lypunov stability
equation for keeping the sliding surface stable with the system and also to supress the
matched disturbance. The simulation is carried out in the Matlab Simulink environment.
The chattering is observed in the control law by applying this controller, so variations of
the SMC are carried out in this work to observe the reduced chattering. Firstly the reaching
law is modified in SMC to power rate reaching law. The tuning parameters are tuned to get
the reduced chattering and fast convergence. The simulation results show the reduced
chattering and fast convergence. The higher order sliding mode controllers are used to
reduce chattering, so the SOSMC based STC is used to control the level of the coupled
tank. Since the STC is used to control the system with relative degree one, the equations
proving the relative degree are also derived. The simulations are carried out using STC and
accordingly gains are tuned for the bounded disturbance. The simulation results show the
reduced chattering as compared to the SMC and Power rate reaching law based SMC.
Conclusion
70
The batch reactor dynamic model was considered for temperature control. Another
important aspect of using batch reactor was the tracking problem included to follow a non
stationary time dependant temperature trajectory so it was a challenging task. As
mentioned earlier the sliding surface is designed, the reaching law equation is implemented
and by equivalent control method the control law is designed to track the desired
temperature trajectory using the conventional FOSMC. The plot of temperature tracking
and the control law is observed. The plot shows the high frequency oscillations. The gain is
tuned to get optimum tracking with disturbance suppression. The power rate reaching law
is substituted in place of classical SMC to reduce the chattering which was observed in the
classical SMC. The gain and the tuning parameter are adjusted to supress disturbance. The
results are simulated in the presence of disturbance from the input side. The results
obtained using power rate reaching based SMC showed reduced chattering and fast
convergence for smaller gain value than the conventional SMC. The STC controller is also
designed for the temperature tracking of the batch reactor. The gains are tuned to supress
the disturbance and get the reduced chattering. The simulation results show the tracking as
well as the control law shows the reduced chattering with respect to the conventional
FOSMC.
The results shown till then were only simulation results therefore the system similar to the
reactor is taken so that the efficacy of the power rate reaching based SMC and the STC are
observed and analysed. For the experimental validation the laboratory reactor is considered
whose temperature is to be controlled at desired value. The mathematical model of the
batch reactor was derived using mass and energy balance equations considering certain
assumptions like constant volume, constant fluid density and constant specific heat. All the
experiments were carried out in the presence of input disturbance. The sliding surface is
chosen as the error equation i.e. the difference between the desired temperature and the
actual temperature. The stability of the sliding surface is proved using Lypunov stability
theorem. The control law based on conventional SMC is derived by equivalent control
method. The simulated results are also plotted along with the experimental result. The plot
of simulated as well as experimental results with respect to the desired temperature is
plotted. The power rate reaching law based SMC is implemented for the reactor model.
The gains are tuned to get the reduced chattering. The results show the faster convergence
with respect to the conventional SMC. The STC is also designed for the reactor model and
the simulation and experimental results are shown. The temperature profiles related to the
Conclusion & Future Scope
71
reactor, like the heater temperature (Th), outlet temperature from the heater (T0), the inlet
temperature to the process vessel (Ti) for all the three control algorithms. The system has
the heater loop and the cooler loop. The cooler loop consists of the condenser fan whose
speed is varied to give the disturbance practically. The inlet temperature to the heater is
decreased by increasing the fan speed; to compensate, the heater temperature rises to
maintain the process vessel temperature to the desired temperature. Finally to get the
quantitative analysis the performance measures ISE and IAE are calculated to see the
performance of the three control algorithms. The quantitative analysis shows that the
power rate reaching law gives the faster convergence in case of ISE. The error analysis is
carried out as the chattering affects the accuracy of the tracking performance. In case of
ISE the STC algorithms performs better with respect to the other two control algorithms.
7.2 Contributions
It is very well known that in process control applications in most of the industries the
conventional PID is the most used controller as it has several advantages such as easy to
implementation, rejection to disturbance, faithful error tracking, no detail information of
the process required etc. sometimes this conventional controller fail to perform in the
presence of undesirable situation. So the model based controllers need to be developed for
these unit operations. Also in the recent years due to advancement in the computer
technology and communication technology, the interfacing of unit operations with the
simulation softwares is becoming easily possible. So the robust controllers are designed by
the researchers form many decades. Amongst the robust controller the VSS based SMC is
most widely applied in many applications from robotic application to biomedical
applications. There are many papers related to application of VSC to chemical process
control. Since in process industry the tracking problems are addressed so implementation
of SMC due to its many advantages of finite time convergence, insensitive to disturbances
its major drawback is that it induces high frequency oscillation i.e. chattering. The
chattering is undesirable as it affects the moving parts in the final control element also it
affects the accuracy and hence the tracking error is induced. So to reduce chattering several
variants of the conventional SMC are proposed which are tried to implement in this work.
The power rate reaching law based SMC and the SOSMC based STC are implemented on
the level control of coupled tank and temperature control of batch reactor. The
experimental validation of the control algorithms is done on the laboratory reactor which is
Conclusion
72
the temperature control problem. To derive the control law for the three control strategies,
the detail mathematical model of the reactor is done in this work. the stable sliding surface
is designed using the Lypunov stability criterion. In the end the detail quantitative analysis
is done for the experimental and simulated results of the reactor.
7.3 Future Scope
The better design of SMC based algorithm can be done by modifying the equation of
sliding surface. Since the applications are considered are of temperature and level control
problems which are inherently slow parameters, some delay compensation based
algorithms can be designed to get faster response. The disturbance considered in this work
is bounded disturbance so the disturbance observers can be implemented that will remove
the constraint on the value of gain of the controller, which makes the system more
vulnerable to external factors. The mathematical model of the reactor is derived using first
principal. The systems identification technique can be adopted and then the controller can
be designed.
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Publications
79
List of Publication
Krupa Narwekar, V. A, Shah , Temperature Control of Reactor using Variable Structure Control,
International Journal of Research and Analytical Reviews, 2018 IJRAR September 2018,
Volume 5, Issue 3,E-ISSN 2348-1269,pp318-322
Krupa Narwekar, V. A, Shah, Level Control of Coupled Tank using Sliding Mode Control, International
Journal of Research, Volume 7, Issue IX, September/2018,ISSN NO: 2236-6124, pp 1025-1031
Krupa Narwekar, V. A. Shah, Temperature Control Using Sliding Mode Control: An Experimental
Approach, ICT4SD 2018 co-located with IRSCNS 2018 Goa, India, Springer conference Proceeding
ASC.
Krupa Narwekar, V. A. Shah, Level control of coupled tank using higher order sliding Mode control,
Intelligent Techniques in Control, Optimization and Signal Processing (INCOS), 2017 IEEE
International Conference, IEEE, Srivilliputhur, India 23-25th
March 2017.
Krupa Narwekar, Dr. V.A.Shah, Robust Temperature Control of Chemical Batch Reactor using Sliding
Mode Control, International Journal of Scientific Research and Management (IJSRM), Issue 07 Pages,
6561-6568, July 2017
Krupa Narwekar, Dr. V.A Shah, Variable Structure Control for Three Tank Mixing Process,
International Conference on multidisciplinary Research Approach for the Accomplishment of
Academic Excellence in Higher & Technical Education through Industrial Process, ISTE Gujarat
Section , 1-2 June 2016, Bangkok Pataya.
Krupa Narwekar, V.A. Shah, Temperature Control Using Higher Order Sliding Mode
Control: An Experimental Approach, Arabian Journal for Science and Engineering, Springer
Publication, communicated
Additional Simulations
80
Appendix A
Additional Simulations
The simulation is carried out to show the fast convergence by using the variations of the
sliding surface. So the design consists of TSM and FTSM which is design of non-linear
sliding surface. So the simulation is carried out on the linearized model of three tank
mixing process.
The Transfer function for the three tank mixing process:
3
0.000312( )
( 0.2)pG s
s
(A.1)
While deriving the mathematical model of the process following assumption are made
[83][84]
All the tanks are well mixed
Dynamics of the valve and sensor are negligible
Dynamics of the valve and sensor are negligible.
No transportation delays (dead time) exist.
A Linear relationship exists between the valve opening and the flow of component
A
Densities of the components are equal
The valve transfer function is taken as unity.
V=volume of each tank=35m3
FB=flow rate of stream B=6.9m3/min
XAi=concentration of A in all tanks and outlet flow=3%A
FA=flow rate of stream A=0.14m3/min
(xA)B=concentration of stream B=1%A
(xA)A=concentration of stream A=100%A
V=valve position==50% open
Terminal Sliding Mode
81
-2 -1.5 -1 -0.5 0 0.5 1 1.5-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
x1
x2
phase potrait
By writing the equations in the state space form
𝑑𝑥(1) = −0.8𝑥(1) − 0.008𝑥(2) − 0.025𝑥(3) (A.2)
Designing the control law
𝑢 = −𝑚𝑥(3) − 𝜌(𝑠𝑖𝑔𝑛(𝑚 ∗ 𝑥(1) + 𝑥(2)) (A.3)
Where ρ and m are tuning parameters.
With initial condition of x0= [-2 -1] to zero as the regulatory problem is considered
FIGURE A.1 Phase Portrait using SMC
The non-linear sliding surface is proposed for fast convergence. This SMC is called
Terminal Sliding Mode.
Terminal Sliding Mode:
The basic principal of TSM can be shown as [85][86] :
Consider the second order systems
𝑥1̇ = 𝑥2 (A.4)
𝑥2 = 𝑓(𝑥1, 𝑥2) + 𝑏(𝑥1, 𝑥2)𝑢(𝑡)̇ (A.5)
First order sliding is given as
𝑠 = 𝑥2 + 𝛽𝑥1𝑞/𝑝 (A.6)
Selecting p>q is the criteria to guarantee the nonlinear sliding surface.
Since ours is the third order system,
𝑠1 = 𝑠0̇ + 𝛽𝑠0𝑞1/𝑝1 (A.7)
𝑠2 = 𝑠1̇ + 𝛽𝑠1𝑞2/𝑝2 (A.8)
And so on for still higher order systems
Additional Simulations
82
Where 𝑠0 = 𝑥1 , 𝛽 > 0, 𝑝1 > 𝑞1
Therefore
𝑠2 = 𝑥(3) + 𝛽1𝑞1/𝑝1𝑥(2)𝑞1𝑝1
−1+ 𝛽2(𝑥(2) + 𝛽1𝑥(1)
𝑞1𝑝1)𝑞2/𝑝2 (A.9)
Substituting (12) in our control law
𝑢 = −𝑚𝑥(1) + 𝜌 ∗ 𝑠𝑖𝑔𝑛(𝑠) (A.10)
Taking
𝑚 = 1, 𝜌 = 1.5, 𝑝1 = 𝑝2 = 2, 𝑞1 = 𝑞2 = 1, 𝛽1 = 𝛽2 = 1
The simulation is done on Matlab environment
The nonlinear sliding surface is observed
FIGURE A.2 Phase Portrait using TSM
It has been observed that the TSM is not giving satisfactory response in terms of
convergence of states in finite time.
Fast Terminal Sliding Mode (FTSM)
The modification of TSM is fast TSM in which the sliding surface equation is modified
[87].
The equation of the sliding surface is given by
𝑠 = 𝑥1̇ + 𝛼𝑥1 + 𝛽𝑥1𝑞/𝑝 (A.11)
Where 𝛼, 𝛽 > 0
The states reach the sliding surface
Terminal Sliding Mode
83
𝑠 = 0
Then
𝑥1̇ = −𝛼𝑥1̇ − 𝛽𝑥1
𝑞/𝑝
The sliding surface equation is
𝑠2 = 𝑥(3) + (𝑥(2) + 𝛽1𝑥(1)𝑞1
𝑝1 + 𝛽2(𝑥(2) + 𝛽1𝑥(1)𝑞1
𝑝1)𝑞2/𝑝2 (A.12)
FIGURE A.3 Phase Portrait using FTSM
Thus in this simulation the variation of SMC was studied in terms of fast convergence. So
from the above simulations it was found that with the design of non-linear sliding surface
results in fast convergence. The drawback of this algorithm is as the order of the system
increase the equation of the sliding surface becomes more complex.
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
x1
x2
Phase potrait