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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
Transcript

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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