VARIABLE STRUCTURE CONTROL APPROACH FOR NON …...reactor is considered whose temperature is to be controlled. The modelling of the reactor is done using mass and energy balance equation

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

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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.

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Grade Obtained in Research

Methodology

(PH001)

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(Advance Control Theory)

(PH002)

BC AB

Supervisor’s Sign

(Dr. Vipul A. Shah)

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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

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of the Copyright Act, written permission from the copyright owners is required, I have

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(a) above for the full term of copyright protection.

viii

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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

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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


4.6.1 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


5.5.1 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


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)


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


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.


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


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].


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


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


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.


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


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.


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]


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.


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]


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.


As discussed in previous section batch reactor dynamic model is defined by the differential

equations as in (4.4)


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


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.


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


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


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


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


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.


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


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


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


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


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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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


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


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

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