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Variable Structure Control Approach for
Non-linear Systems
Ph.D. Synopsis
Submitted to
Gujarat Technological University
For the degree
Of
Doctor of Philosophy In
Instrumentation & Control Engineering
By
Mrs. Krupa D. Narwekar
Enrolment No: 129990917003
Supervisor:
Dr. Vipul A. Shah
HoD, IC Engineering Department
Dharamsinh Desai University
Nadiad, Gujarat
Table of Contents
1. Title of the thesis and Abstract…………………………………………………... 1
2. Brief description on the state of the art of the research topic……………………. 1
3. Definition of the Problem………………………………………………………... 2
4. Objective and Scope of work……………………………………………………. 2
5. Original contribution by the thesis………………………………………………. 3
6. Methodology of Research, Results / Comparisons……………………………….
4
6.1 Level Control of Coupled Tank……………………………………………… 6
6.2 Temperature Control of Batch Reactor………………………………………. 9
6.3 Temperature Control of Laboratory Reactor………………………………… 11
7. Achievements with respect to objectives………………………………………… 15
8. Conclusion……………………………………………………………………….. 15
9. Copies of papers published and a list of all publications arising from the thesis... 16
References 17
1 | P a g e
1. Title of the Thesis & Abstract
Title: Variable Structure Control Approach for Non-linear Systems
Abstract: Due to the advancement in communication technology and simulation software,
the development of model based controllers are becoming more and more popular. In this
work variable structure sliding mode controller and higher order sliding mode super twisting
controller is used to control the process parameters usually used in many unit operations in
process industry which are inherently nonlinear. The two parameters controlled are level and
temperature control. For level, a coupled tank system is considered in which the level of tank
2 is controlled to desired set point. For Temperature, a batch reactor model is considered; the
concentration of the chemical in the reactor is dependent on the temperature trajectory in the
batch. So temperature control of the batch reactor to the time dependant trajectory is done.
The sliding mode control with constant rate reaching law is used to control both the
parameters. The variation in the reaching law is done by using power rate reaching law. Since
the higher order sliding mode control is implemented to reduce the chattering so the same is
applied to both the systems. The simulation study is done considering, both the systems
operating constraints using Matlab Simulink. To observe the real time behaviour of the
control algorithms, a laboratory batch reactor is considered whose temperature is to be
controlled. The chattering reduction is observed in sliding mode control with power rate
reaching law and super twisting controller. The results are compared on the basis of
performance measures namely; integral square error and integral of absolute error.
2. Brief Description on the State of the Art of the Research Topic
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 [1], [2]. Therefore the
measurement and control of these parameters in various unit operations is crucial. The
commonly measured and controlled of these parameters are level, temperature, pressure, flow
etc. Also practically these unit operations are inherently non-linear in nature having dead
zones, friction etc. Therefore when controlling the parameters in these type 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 softwares has become
2 | P a g e
feasible. The advantage of this is that the simulation study gives the know-how of the system
behaviour as well as its operating condition. Also development of advanced controllers is due
to several features model based controllers possess. Some of the features being the
robustness, insensitive to parametric uncertainty, optimal performance, intelligent behaviour
etc. One of these controllers, Variable Structure Control (VSC) sliding mode control (SMC)
is a robust controller which is insensitive to parametric uncertainty and matched disturbances
[3],[4]. The sliding surface design consists of two steps one is designing the sliding surface
and second thing is the reaching law [5]. Even though the SMC is robust, it possesses
inherent high frequency oscillations (chattering), which causes wear and tear 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
[6][7][8][9][10]. In recent years, the higher order super twisting controllers is implemented in
these systems to reduce chattering [11].
3. Definition of the Problem
In this work two case studies are considered to control the process parameters using VSC
approach, namely the level control of coupled tank, the temperature control in batch reactor.
The control algorithms-SMC, power rate reaching law SMC and STC is applied to both the
systems on the Matlab Simulink environment. To validate the performance of the control
algorithm, 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 and
observe the effect of controller to reduce chattering.
4. Objective & Scope of 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 [16]. These advanced controller are mostly designed using model based control
techniques like LQG, State feedback control etc. [17][18][19]. 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,
3 | P a g e
amongst them is using power rate reaching law, using higher order sliding mode control
technique[9][10][11].
The objective of the work:
Designing the control algorithm to achieve the desired level of coupled tank using
SMC. To observe the reduced chattering using Power Rate SMC and Super Twisting
Controller.
Design the control algorithm for temperature control of batch reactor using the SMC.
To observe the reduced chattering using Power Rate SMC and Super Twisting
Controller.
For experimental approach, the laboratory reactor is considered for temperature
control of the reactor. So to develop mathematical model of the reactor using mass
and energy balance equations. Using the mathematical model, design the SMC
controller and 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 laboratory 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.
5. Original Contribution by the Thesis.
The work presented in this thesis consists of two case studies-level control of Coupled tank
and Temperature control of batch reactor. The experimental approach is done in this work for
temperature control of laboratory reactor.
The main contribution of this work can be summarised as
Level control of coupled tank using STC
Temperature control of batch reactor 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 SMC and
higher order sliding mode control for coupled tank system
4 | P a g e
The TEQuipment CE117 laboratory reactor 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
SMC and STC to achieve temperature control. Finding the performance measure-ISE
and IAE.
6. Methodology of Research, Results / Comparisons
The tasks carried out in this work are
Level control of coupled tank -A simulation approach
Temperature control of Batch reactor -A simulation approach
Temperature control of laboratory reactor-An experimental Approach
The history of VSS up until the early 70’s has been described in [14]. The two-step procedure
for sliding mode control design was clearly stated:
1. Sliding surface design;
2. Discontinuous (relay or unit) controllers ensuring the sliding modes.
As mentioned above the sliding mode controller in its basic form consist of designing a sliding
surface and the reaching law to ensure the sliding mode [20][21][22]. The sliding surface
design is of reduced order to that the system. The states slide on the sliding surface and reach
the equilibrium only if the sliding surface is stable. So the important part in SMC design is
choosing the suitable the stable sliding surface [3] [4][5]
Let us consider a second order system given by
(1)
where is the state vector, f(x), b(x) are the nonlinear function in x, u is the
input, d is the matched disturbance.
The sliding surface is designed as
(2)
The reaching law should be such that the states reach the Sliding surface [15]. Reaching the
sliding surface mathematically represents equation (3)
(3)
1 2s cx x
5 | P a g e
So one of the equations to take the state from initial condition to the sliding surface can be
represented by equation (4)
(4)
This is referred to as constant rate reaching law.
The constant rate reaching law gives the output directly proportional to the gain k which
causes the system to become over sensitive.
The disadvantage of constant rate reaching law is up to some extent suppressed the power
rate reaching law
(5)
This reaching law takes the power of the sliding surface with the product of gain. so the
chattering is suppressed to some extent.
The higher order sliding mode controller is implemented in recent years because of their
property of reduced chattering [23]. In higher order sliding mode control, the finite time
convergence is guaranteed not only on s=0 but the higher derivatives of the s [5]. In this
work we have considered the second order sliding mode control so the convergence is
guaranteed as in equation (6).
0s s (6)
The second order super twisting controller can be mathematically represented by
(7)
As seen from equation (7), the integration of discontinuous part is evaluated in the control
law thus giving the smooth response. The super twisting controller is implemented if the
relative degree r=1. Relative degree one means control u explicitly occurs in the first
derivative of the sliding surface [25][26].
From the theory of SMC and HOSMC it is clear that for all the case studies considered and
experiments following methodology is to be followed
Mathematical model of the system under consideration
Designing the sliding surface
Stability of the sliding surface with respect to the system under consideration
Design the control law for sliding mode control
Design the control law for power rate sliding mode control
Design the control law using super twisting controller ensuring relative degree one
1 ( )s c sign s
1/2
11
22
( )
s ( )
u c s sign s v
v c ign s
1 ( )s c s sign s
6 | P a g e
Observing the chattering for all three controllers and discussion of the result
6.1 Level Control of Coupled Tank:
The coupled tank system is widely used in process control applications as well as
many laboratories for experimentation [27], [28],[29]. The schematic of the couple tank is as
shown in Fig.1.
Fig.1. Schematic of the Coupled Tank System
The single input single output model is considered in this case [27]. The two tanks are
connected to each other. Input flow q is through the pump. The constraint q≥0 as the pump
will always the pump the water in tank 1
The controlled variable is height of tank 2
The input to the system is input flow rate q in to the tank 1.
Therefore the constraints are that for the height to be maintained at desired level in tank 2
q>0, h1>h2 or h1-h2>0
By assigning the states to the system
Let x1=h2 & x2=h1, q=u
Therefore our output or the controller variable is x1
The system equations can be written as
(8)
1 2
221 2 1 2
2 2
2 1 2
221 2 12
2 1
2
2
1* 2 1
2 2
1* 2
2
1
2
x x
z za a a ax a u
C Cz z z
z za a afx a
C z z
ab
C z
7 | P a g e
The sliding surface is to be designed for the coupled tank problem. Let us consider equation
(2)
By modifying the equation (2) for tracking problem
(9)
where H is the desired level of tank 2
c will be selected such that the equation (2) is Hurwitz.
The control law is designed as
(10)
The control law is given as
(11)
By using the power rate reaching law the control law becomes
(12)
For implementing STC the control law is given as
(13)
It is to be noted that the controller is continuous one as the discontinuous part is integrated
and then incorporated in the control law. Thus is behaves in a continuous fashion and there-
by reduces the chattering. To apply a super twisting controller to the coupled tank system, it
is necessary that it has the relative degree one with the control law [24]. Therefore first we
will prove the relative degree. Taking the derivative of Equation (9) and substituting (8), we
observe that the control law first explicitly occurs in first derivative of s, so the relative
degree is one. So we can apply super twisting controller to the coupled tank dynamic model.
The gains k1 and k2 of the STC, are tuned so that it guarantees finite time convergence of the
sliding sets[25][26].
The results are as follows.
Fig 2. Level of tank 2 from initial condition 5cm to desired height =4 cm
1
2 1( ( ))u b fx cx c s sign s
1
2 1( ( ))u b fx cx c sign s
8 | P a g e
Fig 3. Control law constant rate reaching law
Fig.4 Control Law Power Rate SMC
Fig.5 Height of Tank 2=4cm
Fig. 6 Control law for SMC
Fig. 7 Control law for STC
The simulation is done in the presence of input noise of d=10sin(t). The results show the
comparison of sliding mode control with power rate reaching SMC, which is used to reduce
the chattering in the input signal. Two sets of comparisons are done; one with SMC with
power rate reaching law and SMC with STC. The finite time convergence is seen in both the
9 | P a g e
cases fig. 2 and fig. 5. The Control law in both cases shows the suppressed chattering in case
of power rate reaching SMC fig. 3 a.d fig 4 as well as in STC fig 7 and fig.7. The gain value
c1=10 and in STC case the gain are set as max(d)=10 so L>dmax. So assuming the value of
L=15
c11=1.5*sqrt(L);=5.8
c22=1.1*L=16.5
6.2 Temperature Control of Batch Reactor
In many application temperature control of the batch reactor is considered [30][31][32].
The batch reactor is a chemical reactor which mixes two chemicals chemical A and chemical
B. The two chemicals mix together; the concentration of B is dependent on the time
dependant temperature trajectory in the batch [33]. In short, the concentration is maintained if
the temperatures in the batch are maintained to desired temperature profile. Therefore the
batch reactor problem reduces to temperature control of batch reactor.
The mass and energy balance equations are given by [33].
(14)
where
For designing the control law, the equation (14), is modified as
1 2 3 A Bx x x C C T
(15)
(16)
(17)
As discussed in [33], 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 the following equation.
2
1
2
1 2
2
1 1 2 2 1 2 1 2
( )
( ) ( )
( ) ( ) ( )
A A
B A B
A B
C k T C
C k T C k T C
T k T C k T C T T u
11 10
22 20
( ) exp(273 )
( ) exp(273 )
Ek T A
R T
Ek T A
R T
2
2 1 3 1 2 3 2( ) ( )x k x x k x x
2
1 1 3 1( )x k x x
2
3 1 1 3 1 2 2 3 2 1
2 3 1 2 3
( ) ( )
( )
x k x x k x x
x x u
10 | P a g e
1( ( ) )d stcu b T f x u
(18)
The first step in SMC is designing the sliding surface. The designing of sliding surface,
implies that s=0. Since it is a tracking problem, the error should tend to zero, as time tends to
infinity, which also satisfies the condition.
Therefore, the sliding surface is chosen as
(19)
The control law designed for SMC, Power rate SMC and SMC are as follows
c1>0 (20)
1
1( ) ( ( ) s ( ))du b x T f x c s ign s c1>0, 0<α<1 (21)
(22)
Fig. 8 Temperature tracking for batch Reactor using SMC
Fig. 9 Temperature tracking for batch Reactor using Power rate SMC
Fig. 10 Control Law for SMC
Fig.11 Control Law for Power Rate SMC
3( ) 54 71exp( 2.5 10 )dT t t
Ds T T
1
1( ) ( ( ) ( ))du b x T f x c sign s
11 | P a g e
Fig. 12 Temperature tracking for batch Reactor using STC
Fig.13 Control Law for Power Rate SMC
The simulation is done in the presence of d=5sinωt, accordingly the gain values are tuned for
each of the controllers. As seen from the tracking is seen in each of the controllers, but
chattering suppression is observed in the Power rate reaching law and super twisting
controller. The power rate reaching law c1=10 and α=0.7, for SMC the c1=60, for STC as
discussed in previous section c11=6.70,c22=22.
6.3 Temperature Control of Laboratory Reactor.
For experimental approach, the system selected is the laboratory reactor. The laboratory
reactor is as shown in Fig. (14)
Fig.14 Laboratory reactor
This set up is a TEQuipment CE117 process trainer which consists of cylindrical transparent
vessel, which can be filled with water whose temperature is to be controlled at desired level.
The CE117 kit is interfaced with LabVIEW by using DAQ card NI6009. The input from the
transmitter is given to analog input terminal of the DAQ card which inputs the data to the VI
of HOSM and the control signal generated from the LabVIEW is given to the analog output
terminal of the DAQ card which is connected to the pump of the process kit. The pump
12 | P a g e
voltage is manipulated and hence the flow of hot water from the heat exchanger is varied to
maintain the desired temperature in the tank.
In this work the control objective is to maintain the Temperature T at the desired level by
varying the input flow rate from the heat exchanger qh.
The mass balance equation is given as
(23)
The energy balance equation is
(24)
The following assumptions are made
A1: Fluid Density is constant
A2: Specific heat is constant
A3: Volume of liquid in the tank is kept constant
The equation can be rewritten as
(25)
In this application the relation between Q (heat input to the process vessel from the heat
exchanger) and qh (flow rate through the heat exchanger) is to be obtained, as the
manipulating variable is qh.
So considering the heater loop, the energy balance equation is
(26)
Therefore substituting (21) in (20)
(27)
The sliding surface is designed as
(28)
By equivalent control method substituting the values of dT/dt, and finding the equation for qh,
which is the inlet flow rate, the control law is given by
5
1
0
8380[ 2.22 ( ) ( )]h d i
h
q T e T T c sign sT T
(29)
for power rate reaching law
13 | P a g e
5
1
0
8380[ 2.22 ( ) | | ( )]h d i
h
q T e T T c s sign sT T
(30)
for STC
(31)
Fig.15 Temperature Tracking using SMC
Fig.16 Control Law for SMC
Fig. 17 Temperature Tracking using Power Rate SMC
14 | P a g e
Fig.18 Control Law for Power Rate SMC
Fig. 19 Temperature Tracking using STC
Fig.20 Control Law for STC
From the figure it is observed that the tracking of the desired temperature has been achieved
by using sliding mode control technique. The gain is tuned to c1=8.5 for getting optimum
output. By observing the control law, it is seen that in sliding mode control the switching of
the controller is seen and also the tracking is achieved but oscillations are seen in the
temperature above and below the set point. By using the power rate reaching SMC, the
oscillations are reduced and hence the temperature fluctuations are reduced above and below
the set point. The super twisting controller as discussed in the previous case studies is also
observed in this experiment. the gains are set as , L=10 is considered so c11=6.7082,c22=22.
The control law gives smooth response with respect to the sliding mode control and power
15 | P a g e
Rate reaching SMC. The mathematical analysis of the tracking error is carried out for all the
three controllers using Integral of Absolute Error (IAE) and Integral of Square Error (ISE).
Table: 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
7. Achievements with respect to objectives
In the beginning, the objectives are mentioned, which were to design the controllers for the
two case studies, and to experimentally validate the control algorithm which is simulated.
After going through the methodology and results, it is clear that all the objectives are attained
successfully. The mathematical analysis of the experimental work is also carried out.
8. Conclusion
The VSC based controller is designed for the level and temperature control and hence the
servo problem is addressed with these controllers. As we know 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 when we consider designing a
controller its effect on final control element also needs to be studied. In this work when we
observe chattering in SMC, the modified SMC i.e. power rate SMC is used to supress
chattering. The second order STC is also implemented to reduce the chattering effect. The
simulation study considered in case studies of coupled tank and Batch reactor problem, the
suppressed chattering is observed. It is also very important to observe the effect in real time.
As real time systems have the inherent friction, dead zones etc. so the challenge of designing
the controller for real time is achieved in this work and reduced chattering is also observed.
16 | P a g e
9. Copies of papers published and a list of all publications arising from the thesis
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(the papers
in this conference are sent to Scopus)
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
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 and technical education through
industrial process, ISTE Gujarat Section
17 | P a g e
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