SLIDING MODE CONTROL OF FED-BATCH SUGAR … · 2014. 6. 17. · 1.3 Non-linear systems Non-linear system models depict the most faithful representation of natural non-linear systems

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SLIDING MODE CONTROL OF FED-BATCH SUGAR CRYSTALLIZATION PROCESS

By

MERAJ HASAN

Submitted to the Department of Electrical Engineering in partial fulfillment for the Degree of

Master of Science

in Electrical Engineering

Thesis Advisor

Dr. Fahad Mumtaz Malik

College of Electrical and Mechanical Engineering National University of Sciences and Technology, Pakistan

2013

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ABSTRACT

SLIDING MODE CONTROL OF FED-BATCH SUGAR CRYSTALLIZATION PROCESS

By

MERAJ HASAN

ATURAL processes are non-linear in general. To model them is

often a difficult task whereby certain model parameters are not

modeled and an incomplete picture of the process becomes

available to us. Even with such limitations control of such said systems is a

must requirement that cannot be ignored. However design of such a control

is also not a straightforward task especially if the system model is too

complex (as it often is) or too tedious.

Therefore design of control for non-linear system gives us twin problems in

general: the difficult system model and the complex control. While certain

processes can claim to be well-modeled in the sense that their transport

kinetics, fundamental principles and equilibrium relationships can be

expressed by mathematical equations various biological and chemical

systems however do not enjoy this privilege. This is because scientific

knowledge of their fundamental mechanisms is still limited thereby putting

the credibility of their mathematical representations into doubt.

The other problem of tedious control scheme is almost always a result of

greater system model complexity. It is also linked to how accurately a

process can be modeled and what portion of it is poorly modeled or non-

modeled. The choice of control scheme may also depend on how these

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poorly modeled or non-modeled parts of the process have techniques

applied on them to deduce the structure of the model. With the choice of

these techniques is often tied the choice of the control. This is however not

always the case as other factors such as speed, robustness and accurate

control may come into play thereby reducing the importance attached to

deducing the poorly modeled or non-modeled parts of the process.

In this context it becomes important what the requirements from a certain

process are on an industrial or research level as these are often the

motivating factors behind the study, and indeed, control of natural process.

What this thesis aims is to control a natural process highly employed in

industry with a control scheme that has never been used to control it before.

The process is Fed-Batch Sugar Crystallization and the control scheme is

Sliding Mode Control (SMC). It is a non-linear non-stationary process having

presence of non-modeled system dynamics. However the model utilized is

reliable and the purpose of this thesis is to show that, for a particular

simulator under study, the performance of SMC is more reliable and accurate

as compared to the standard control for this process employed in industry:

PID. The robustness of the two schemes is also compared.

An accurate system simulator is developed in this thesis that may aid future

work in this field as the main papers containing extensive study of this

process contain inaccuracies in system equations. While these have been

corrected newer techniques to carry out the process (in addition to the

control) are also suggested which may boost up the speed of the system

considerably. It is expected that this unique combination of process and

control that has not been tried before will open new doors to further increase

the robustness of the control schemes applied hence completely making it

irrelevant the lack of accurate system dynamics in certain parts of the

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process. This could have major economic advantage for those interested in

carrying this research further and implementing it on an industrial scale.

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To

My parents,

Osama, Khalid and Jaffer.

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ACKNOWLEDGEMENTS

I would like to thank Allah The All Mighty for aiding me in the completion of

this thesis through His own perfect ways and for filling me with hope and

comfort in the darkest of times. I thank my mother for her prayers and her

words of encouragement and I thank my father for his prayers and for

helping me in small stuff not related to thesis that would have otherwise

given me much trouble. I thank my thesis supervisor Dr. Fahad Mumtaz

Malik for his vote of confidence and his own unique way of encouragement

and his very pragmatic tips. I thank Dr. Muhammad Salman, Dr. Muhammad

Bilal Malik and Dr. Shahzad Amin Sheikh for obliging to become my GEC

members and for approving my thesis especially Dr. Muhammad Salman for

his small but wonderful tips that alleviated my worries. Finally I thank my

Masters’ fellows Kashif Mehmood, Nouman Masood and Muhammad Ibrahim

for their motivation and help in critical times.

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TABLE OF CONTENTS

Abstract………………………………………………………………………………………………………………i

Acknowledgements……………………………………………………………………………………………v

List of Figures…………………………………………………………………………………………………..ix

List of Tables…………………………………………………………………………………………………….xi

1. Introduction………………………………………………………………………………………………….1

1.1. Feedback control…..………………………………………………………………………..1

1.2. Linear Systems…………..……………………………………………………………………2

1.3. Non-linear systems………………………………………………………………………….3

1.4. Control Problems……..……………………………………………………………………..4

1.5. Sugar Crystallization and robust control………………………………………..6

1.6. Sliding Mode Control and Review of Thesis……………………………………6

2. Review of Sliding Mode Control…………………………………………………………………..8

2.1. Overview………………………………………………………………………………………….8

2.2. Mathematical representation of Sliding Mode Control………………….9

2.3. Sliding Surface………………………………………………………………………………10

3. Review of PID…………………………………………………………………………………………….13

3.1. Overview……………………………………………………………………………………….13

3.2. Implementation…………………………………………………………………………….14

3.3. PID Control Theory……………………………………………………………………….16

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3.3.1. Proportional Term…………………………………………………………….17

3.3.2. Integral Term……………………………………………………………………17

3.3.3. Derivative Term……………………………………………………………….17

3.4. Other forms of PID Controller……………………………………………………….17

3.4.1. Position Form of PID Controller……………………………………….17

3.4.2. Velocity Form of PID Controller……………………………………….18

3.5. Loop Tuning……………………………………………………………………………………19

3.5.1. Manual Tuning (Ziegler-Nichols Method)………………………..19

3.5.2. Software Tuning……………………………………………………………….19

4. Sugar Crystallization Process…………………………………………………………………….20

4.1. Overview……………………………………………………………………………………….20

4.2. Process Description……………………………………………………………………….20

4.2.1. Pan Controlled Operation…………………………………………………22

4.3. Available Measurements……………………………………………………………….22

4.4 Partial Mechanistic and Knowledge-Based Hybrid (KBH) Model….23

4.4.1. Partial Mechanistic Model………………………………………………..23

4.4.1.1. Mass Balance……………………………………………………….24

4.4.1.2. Energy Balance……………………………………………………25

4.4.1.3. Population Balance…..…………………………………………25

4.4.2. KBH Model………………………………………………………………………..26

5. Performance Comparison…………………………………………………………………………..27

5.1. Overview……………………………………………………………………………………….27

5.2. Process Operation………………………………………………………………………….27

5.3. Process Simulator………………………………………………………………………….32

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5.4. Control and Sliding Surface Design………………………………………………34

5.4.1. Sliding Mode Control and Sliding Surface Design……………34

5.4.2. PID Control Design……………………………………………………………36

5.5. Tracking Performance…………………………………………………………………..37

5.5.1. Sliding Surfaces……………………………………………………………….41

5.6. Robust Analysis…………………………………………………………………………….43

6. Conclusions and Future Recommendations……………………………………………..49

6.1. Conclusions……………………………………………………………………………………47

6.2. Future Recommendations…………………………………………………………….49

Appendix…………………………………………………………………………………………………………49

References……………………………………………………………………………………………………..54

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LIST OF FIGURES

1.1. A simplified control system…………………………………………………………………….1

1.2. Output tracking a constant reference of ……………………………………………4

1.3. Stabilization of two system states at the point x=0………………………………5

1.4. Tracking of a sinusoid signal…………………………………………………………………….5

2.1. Discontinuous Relay Control…………………………………………………………………..10

3.1. A feedback loop……………………………………………………………………………………….13

3.2. Analogue Differentiator…………………………………………………………………………..15

3.3. Analogue Integrator……………………………………………………………………………….15

3.4. Relation between digital and analogue electronics in Feedback control

system…………………………………………………………………………………………………………….16

5.1. Schematic of Process Industrial Unit………………………………………………………31

5.2. Main control system body of Process Simulator showing actuator inputs and plant.……………………………………………………………………………………………………….34

5.3. Main body of plant………………………………………………………………………………….35

5.4. Mass Balance equations in Simulink……………………………………………………..35

5.5. Energy Balance equations in Simulink…………………………………………………..36

5.6. Population Balance equations in Simulink……………………………………………..37

5.7. Ms vs time (s)…………………………………………………………………………………………39

5.8. Mi vs time (s).…………………………………………………………………………………………39

5.9. Mw vs time (s).……………………………………………………………………………………….40

5.10. Tm vs time (s).……………………………………………………………………………………..40

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5.11. Volume of Massecuite…………………………………………………………………………..41

5.12. Supersaturation (SMC).………………………………………………………………………..41

5.13: Supersaturation (PID) …………………………………………………………………………42

5.14. Supersaturation…………………………………………………………………………………….42

5.15. Volume fraction of crystals…………………………………………………………………..43

5.16. Sliding Surface for Control loop 1………………………………………………………..43




5.20. SMC response……………………………………………………………………………………….46

5.21. PID control response…………………………………………………………………………….46

5.22. SMC response……………………………………………………………………………………….47


5.24. SMC response……………………………………………………………………………………….48


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LIST OF TABLES

4.1. Characteristics of the crystallizer……………………………………………………………21

4.2. Batch cycle……………………………………………………………………………………………..22

4.3. Typical Operational Data…………………………………………………………………………23

4.4. Measured Input Variables……………………………………………………………………….24

4.5. Measured Output Variables…………………………………………………………………….24

5.1. Summary of Operation…………………………………………………………………….…….31

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CHAPTER 1

INTRODUCTION

1.1 Feedback Control

A generalized control system consists of a plant, actuator, summation

and feedback as depicted in Fig 1.1. The operation can be described as

regulating the output (x) of the plant by varying the input (u) such

that, over a finite time, the output reaches a certain value.

Fig. 1.1: A simplified control system

The achievement of the control scheme depicted above is that it

reduces the error (e) to zero in finite time thereby making sure that

the output becomes equal to the reference (r). It is this simplified

objective of ‘regulation’ that is the crux of this thesis.

The control, u, also called the input to the Plant is the one whose

design is vital to the ‘regulation’ that is aimed to be achieved. Various

techniques exist in order to design the best control for a given system

where the definition of ‘best’ is subjective as it depends on how exactly

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the system that is being controlled (or the plant) should give its

output.

1.2 Linear Systems

System states are defined by state variable x. The nature of these

states informs us whether a system is linear or non-linear. A system

(or plant) may be defined mathematically as follows:

푥̇ = 퐴푥 + 퐵푢

푦 = 퐶푥 + 퐷푢 (1.2.1)

State variable x is n x 1, state output y is p x 1, input u is m x 1. If the

state matrix A can be represented in a linear fashion so that matrix

multiplication with the state variable vector is possible then the system

is linear; otherwise it is not. In Linear systems A, B, C and D (which is

optional) do exist but they do not exist in non-linear system as it is

impossible to represent them as matrices.

The control for linear systems is often straightforward and the

mathematical complexity is reduced. Techniques that work for non-

linear systems may also work well for linear systems. Many times the

system model is a linearized approximation of a non-linear model.

Sometimes it has questionable accuracy whereas at other times the

accuracy of linearization holds only for a very specific range outside of

which the linearized model cannot be said to be a faithful

representation of a non-linear process. However the control system

shown in Fig 1.1 applies to both linear and non-linear system. It is the

system model and the design of the control that is different.

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1.3 Non-linear systems

Non-linear system models depict the most faithful representation of

natural non-linear systems. They are depicted mathematically as

follows:

푥̇ = 푓(푥,푢)

푦 = ℎ(푥, 푢) (1.2.2)

As discussed earlier these systems cannot be modeled via matrices

(state-space form) but have to be written in terms of state variables.

The control of such systems may become very complex if the system

model is complex; as is the case with most natural systems. Another

problem is the lack of complete and accurate model representation.

This is owing to many factors such as lack of scientific evidence as to

the fundamentals of a natural process, measurement uncertainties,

assumptions and lack of empirical data of the process.

Therefore many such systems have unmodeled or partially modeled

system characteristics that pose a difficult problem for control

designers. These aberrations may result in unwanted system

performances and undesirable results. Therefore any control that is to

be designed for such system needs to overcome these problems and

produce optimum results. Research is run in parallel to completely

represent the natural process via its system model. As more of the

process is faithfully represented by the system model it becomes

easier to design an accurate control that would produce the desired

results.

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1.4 Control Problems

Whether the system is linear or non-linear and whatever technique is

used to control it there are certain distinct control problems that are

almost always part of the overall output that is desired from a system.

They are described as follows:

i) Regulation

In this problem the output has to track a constant reference (or

a step) in finite time.

Fig. 1.2: Output tracking a constant reference of

ii) Stabilization

In this problem the system states need to be stabilized at a

certain value in finite time.

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

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Fig. 1.3: Stabilization of two system states at the point x=0

iii) Tracking

While similar to regulation, tracking is actually a more general

form for regulation. While regulation involves tracking a constant

signal or step, tracking as a whole may involve other time-

varying signals such as sinusoids, square and triangular waves.

The objective remains the same however: track the reference,

no matter what it is.

Fig. 1.4: Tracking of a sinusoid signal.

0 2 4 6 8 10 12 14 16 18 20-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 2 4 6 8 10 12 14 16 18 200

0.01

0.02

0.03

0.04

0.05

0.06

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1.5 Sugar Crystallization and Robust Control

Sugar Crystallization process occurs due to nucleation, agglomeration

and growth [1]. It is a non-stationary and non-linear process. Its

available model does count for some unmodeled system dynamics

however the quest for robust control techniques to overcome the poor

understanding of various processes involved is a field for continuous

research.

Sugar Crystallization is an important industrial process which is the

cornerstone for the large-scale production for sugar. Time and product

quality are two important factors in assessing the performance of a

sugar plant. Therefore any control designed for this particular process

should accomplish:

i) Quick tracking of set-points

ii) State stability

iii) High Quality final output

iv) Robustness to unmodeled system dynamics, parameter

fluctuations and actuator irregularities.

Only through achievement of these tasks can a control scheme be

rated as successful.

1.6 Sliding Mode Control and overview of thesis

The technique employed to control the sugar crystallization process in

this thesis is Sliding Mode Control (SMC). Its performance is compared

with a standard PID controller.

While several techniques have been used in the past to control this

process all of them lacked a good performance in one or more of the

objectives stated in topic 1.5. Therefore this thesis will also assess the

performance of SMC in attaining the above stated objectives. The

thesis involved development of a process simulator exclusive for this

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process as well as a corrected bank of equations that describe the

process.

The thesis is the first time SMC has been applied to this process and it

is hoped that this small breakthrough will lead to further research in

this area so that faster, lighter and more robust controls can be

designed that would have a tremendous impact on an industrial scale,

if utilized.

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CHAPTER 2

REVIEW OF SLIDING MODE CONTROL

2.1 Overview

Sliding Mode may arise in systems where the control consists of discontinuous states. These kinds of controls have always had keen research done on them throughout the history of automatic control theory. In particular in the first stage of this research the focus was on “on-off” regulators using relays.

The control switches at high frequency should the so-called sliding modes occur in a system. The study of sliding modes covers different areas ranging from application aspects to mathematical problems. [4]

This popular technique was developed since the 1950s based on existing work done on the “on-off” relays and other methods of discontinuous control and popularized by Utkin’s 1977 paper (see [7] in References). [8]

The main advantages of this technique are

i) It is robust against a large class of model uncertainties and perturbations

ii) Requires less information as compared to other classical control techniques.

iii) A possibility of stabilizing some non-linear systems that may not be stabilized by other feedback-based control schemes.

The reason why the industrial community was slow to catch on to this technique is because of the problem of “chattering”. It will be discussed in detail in Chapter 6, but the phenomenon arose because of high frequency switching employed in the control that the actuators had to cope with. The more the magnitude of the switching gains the higher the chattering. This produced premature wear in actuators and even breaking. [8]

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However solutions were found such as dividing the control into switching and continuous components so as to reduce the switching gain and approximating the signum function with a high-slope saturation function. [3] Other techniques included non-linear gains, higher-order sliding mode control and dynamic extensions. [8] All of these methods work to reduce “chattering” phenomenon. As a result this technique is getting popularized because of ease of implementation, robustness and applicability to complex systems.

Some applications have been developed:

i) Control of Electrical Motors [8] ii) Signal reconstruction and observers [8] iii) Manipulators and mobile robots [4] [8] iv) Power Converters [4] v) Magnetic bearings [8]

Based on these trends several active researchers have combined their efforts to present new trends in sliding mode control. Prominent among them are Utkin, Guldner and Shi (see [4] in References) and Barbot, Perruquetti (see [8] in References).

2.2 Mathematical representation of sliding mode control

We revert back to the system of Figure 1.1. Our aims however this time is to design an input u, also called the control that would achieve the required regulation.

If we employ a discontinuous state of which the input u is a function of then we can enforce sliding mode on the system. If the control switches at high (theoretically infinite) frequency then the sliding modes may arise in the system. We opt for the former (i.e. enforcing sliding mode) by designing control u as:

푢 = 푢0, 푒 > 0−푢0, 푒 < 0 (2.1.1)

or 푢 = 푢0. 푠푖푔푛(푒) [4]

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Fig. 2.1: Discontinuous Relay Control [4]

The aim is to achieve sliding mode. What we mean by that is when the output equates the reference or the error becomes zero then the system is said to be in sliding mode.

If the error is represented by

then the derivative will be

(2.1.2)

as

Hence we can see that the error and its derivatives have opposite signs if

. The tracking error will decay to zero after a finite time interval T. The motion for t>T is known as sliding mode.

2.3 Sliding surface

Sliding surface is the trajectory that system states undertake after system achieves sliding mode. It is where the motion of the system becomes constrained after sliding mode is achieved. The purpose is to bring the trajectory of the system states to a sliding surface s in finite time and keep it there infinitely.

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Let us take the example of a simple system:

푥1̇ = 푥2

푥2̇ = ℎ(푥) + 푔(푥)푢 [3]

We wish to guarantee that the state variable x tends to zero as t tends to infinity. Therefore we design a sliding surface

s = a1x1 + x2

such that

푥1̇ = −푎1푥1

Hence we can be assured that in finite time the system states will stabilize at x = 0.

Additionally we wish to ensure that the system states stay at the sliding manifold s = 0 infinitely. This can be done by ensuring that the sliding manifold s stays at s = 0 infinitely and hence by placing the constraint that

푠̇ = 0

Taking V = (1/2)s2 as a Lyapunov function candidate we have

푉̇ = s푠̇

In order to ensure that the system states, depending now in a reduced form on the sliding manifold, reach and stay on the manifold we must ensure that the input u that we can control includes the sliding surface s and also changes according to the current updated value of s.

This can be done by using the relay control discussed in section 2.1. It is designed as

푢 = −훽(푥)푠푔푛(푠) where

푠푔푛(푠) = 1, 푠 > 0−1, 푠 < 0

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Thus when we do the derivations we can see that 푉̇<0 when we apply this input to the system ensuring that the trajectories stay on the manifold infinitely.

The motion can be summed up via two phases:

i) Reaching phase: here the trajectories start off at their initial conditions and move towards the sliding manifold by virtue of the control u.

ii) Sliding Phase: the trajectories reach the manifold in finite time and slide over it such that their motion is confined to the trajectory s = 0 and the dynamics of the system are given by a reduced-order model.

The control law u depicted above is called the sliding mode control. As soon as the trajectory hits the manifold it should start sliding in ideal sliding mode control. When the system is in sliding mode the error is reduced to zero and ideal tracking is observed (for ideal SMC).

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CHAPTER 3

REVIEW OF PID

3.1 Overview

Proportional-Integral-Derivative or PID is a classical control technique that does not require knowledge of the underlying process that it is controlling. It is a control loop feedback mechanism that is used extensively in industrial control systems. As described in Section 1.1. the objective of feedback control is to calculate an error based on the difference between “setpoint” and process variable (output). This error will be used to adjust the inputs to the plant and generate an output closer to the setpoint.

This error would then be updated and the cycle repeated again via feedback till the output equals the setpoint or the error becomes zero. It will stay zero for the remainder of the operation.

PID algorithm has three parameters namely P, I and K. They manipulate the error to give an output that is fed into an actuator. This is shown in Fig. 3.1.

Fig. 3.1: A feedback loop [9]

The idea of PID controller was conceived in governor designs of 1890s. Some development took place in automatic steering of ships at the start of 20th century. The first research on PID was published in 1922 (Minorsky). [10] He based his research by observing the motion of a Helmsman which not only used present and past errors but predicted future errors based on

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current rate of change. [11] He then represented his observations mathematically. Several other authors published works on PID in the 1930s.

3.2 Implementation

Earlier PID controllers were implemented through mechanical devices. Then the focus shifted to pneumatic controllers and then analogue controllers. They have now been replaced with digital devices such as FPGAs and microcontrollers. Nowadays they are implemented on PLCs or panel-mounted digital controllers. For this thesis, since there is an industrial implementation, both analogue and digital electronics have to be discussed so that one can ascertain which type to model the control scheme on and which type to implement the actual controller in industry. The difference between analogue and digital electronics is discussed below

i) Analogue Electronics: Type of signals utilized is continuous variable signals. They have proportional relation between signal and voltage (or current) hence the meaning of the name in Greek. Analogue circuits are heavier since many devices such as Integrator and Differentiator, that are frequently used, are implemented via bulky hardware consisting of capacitors and resistors.

ii) Digital Electronics: Type of signals utilized take discrete values rather than continuous values in a range. They have to utilize analogue devices in order to interact with real world such as DAC (Digital to Analog Converter) and ADC (Analog to Digital Converter). However integrators and differentiators are implemented in microcontrollers or FPGAs rather than as separate hardware. This reduces the weight, cost and dimensions considerably of the circuit and this is why digital electronics is nowadays preferred.

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Fig. 3.2: Analogue Differentiator

Fig. 3.3: Analogue Integrator

In Figure 3.2 and 3.3 analogue differentiator and integrator are shown. Contrast this with a simple microcontroller where such functions will be implemented digitally.

Figure 3.4 demonstrates how analogue and digital electronics are used together in an industrial control loop. The sensors giving feedback need to have their outputs converted into discrete form just as the output from the plant to the actuators needs to be converted into analogue or continuous form. It should also be noticed that the complex system model is implemented inside the digital controller and does not utilize any analogue electronic devices.

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Fig. 3.4: Relation between digital and analogue electronics in Feedback control system [9]

3.3 PID Control Theory

The input to the plant u also known as the manipulated variable (MV) is a sum of three different components that characterize PID. The variable u is also the output of the digital controller implementing PID as shown in Figure 3.4. It is described by the following equation

(3.3.1)

where

Kp is proportional gain

Ki is integral gain and

Kd is derivative gain.

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3.3.1 Proportional Term

It produces an output proportional to error. The proportional term is given by:

(3.3.2)

3.3.2 Integral Term

The integral term is given by:

(3.3.3)

3.3.3 Derivative Term

The derivative term is given by:

(3.3.4)

3.4 Other forms of PID Controller

Two other forms of PID controller exist, in addition to the classical form discussed above.

3.4.1 Position Form of PID Controller

[12] (3.4.1)

where the parameters Ti and Td replace the conventional derivative and integral parameters in the classical form and are known as Integral time and Derivative time respectively.

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3.4.2 Velocity Form of PID Controller

[12] (3.4.2)

In discrete time the velocity form becomes:

[2] (3.4.3)

It is the continuous time velocity form that is implemented in this thesis in order to compare the performance with the SMC.

3.5 Loop Tuning

Several methods exist to tune the PID control parameters. It can either be done manually or using software.

3.5.1 Manual Tuning (Ziegler-Nichols Method)

One of the most well-known methods to tune PID gains is the Ziegler-Nichols Method. It only requires the PID controller and plant and so can work without a computer.

In this method Ki and Kd are initially set to zero and KP is increased till the output starts to oscillate. That value of KP is known as Ku and the time period of the oscillation is found as Tu. Then the gains are set according to the following formula:

Kp = 0.6 Ku (3.5.1)

Ki = 2Kp/Tu (3.5.2)

Kd = KpKu/Tu (3.5.3)

The value of Kp in eq. 3.5.2 and 3.5.3 is the one calculated in eq. 3.5.1. Similar formulas exist for P, PI and other controllers as well as for controlling overshoot in PID.

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3.5.2 Software Tuning

Simulink has an inbuilt Control tool box that automatically gives optimal gains for a given system. Other software tools are also used.

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CHAPTER 4

SUGAR CRYSTALLIZATION PROCESS

4.1 Overview

Sugar Crystallization is a non-stationary and non-linear process. It occurs through the mechanisms of growth, nucleation and agglomeration. However the underlying process’ relation with operating conditions and process state is poorly known. Lack of scientific knowledge about the crystallization process and hence inadequate knowledge about its fundamental principles has hampered research on the process. [1]

However new methods such as data-based modeling techniques remove the dependency on a priori design techniques. In such techniques the process knowledge is extracted from measurable process data and it attempts to overcome difficulties in knowledge expression.

The sugar crystallization process is Industrial scale fed batch evaporative crystallization type and utilized in cane sugar refining. What is meant by fed batch is that the final crystalline sugar product is produced in batches whereby the base liquor is returned to the process at the end.. Sugar production can be from two sources: beet and cane sugar. The process considered in this thesis utilizes the second type.

Final product is judged by main crystal size distribution (CSD) parameters namely the average mass (AM) and the mass-size distribution function’s coefficient of variation (CV).

4.2 Process Description

Industrial unit considered in this thesis is of the type of fixed calandria with the heavy viscous massecuite circulated via specially designed tubes. The

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unit’s characteristics are as mentioned in Table 4.1. A batch cycle is completed on average in approximately 90 minutes operating time. Its several sequential phases are as listed in Table 4.2.

Table 4.1 [1]

Table 4.2 [1]

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4.2.1 Pan controlled operation

Table 4.3 contains a summary of typical operation data.

Table 4.3 [1]

4.3 Available Measurements

The industrial unit has 15 sensors installed into it. The online measured physical properties are mentioned in in Tables 4.3 and 4.4 consisting of measured input and output quantities, respectively. Table 4.5 contains those constants for each batch. [5]

Supersaturation is measured indirectly from available measured data. The kinetic parameters are known to be affected by the supersaturation (S), temperature (Tm), purity of the solution (Pursol) and the volume fraction of the crystals (vc). Apart from temperature none of these quantities can be measured online. They are function of mass states that are themselves not measured directly. However they can be estimated via certain algorithm. See [6] in Appendix

23

Table 4.4 [1]

Table 4.5 [1]

4.4 Partial Mechanistic and Knowledge-Based Hybrid (KBH) Model

Particle birth or death via aggregation or agglomeration is required to be considered even if the model’s complexity increases. This may be a factor in the consideration of its use in model-oriented control strategies and state estimation.

4.4.1 Partial Mechanistic Model

There are some simplifying assumptions employed in the Process modeling of this thesis.

In the development of model equations it was assumed that (i) the massecuite is homogeneous in the entire volume of the pan; (ii) uniformity of temperature of massecuite; (iii) negligible thermal dynamic delays because of the unit’s thermal capacity; (iv) the volume crystal coefficients

24

and shape remain unchanged during crystallization and (v) the supplied steam is condensed.

Main variables such as crystal contents and supersaturation are calculated from state variables. The evaporation rate has been described by an empirical model.

4.4.1.1 Mass Balance

The following mass balance equations are written for impurities (Mi), water (Mw), crystals (Mc) and dissolved sucrose (Ms):

(4.4.1)

(4.4.2)

(4.4.3)

(4.4.4)

The crystallization rate Jcris consists of the main non-linearity in the process and it is computed via the population balance. The heat input Q and the evaporation rate Jvap are given by the following equations:

(4.4.5)

(4.4.6)

where and kvap are parameters determined experimentally [6]

25

4.4.1.2 Energy Balance

Energy balance is given by Eq. (4.4.7), where Tm denotes the temperature of massecuite. The specific heat capacities and enthalpy terms included are derived from thermodynamical and physical properties given in Appendix.

(4.4.7)

(4.4.8)

(4.4.9)

(4.4.10)

(4.4.11)

4.4.1.3 Population Balance

The basic moments of the number-volume distribution function are

(4.4.12)

(4.4.13)

(4.4.14)

(4.4.15)

26

where and are the kinetic variables linear growth rate, nucleation rate and the agglomeration kernel, respectively. The crystallization rate is given by:

(4.4.16)

Since no equation is provided for growth rate, nucleation and aggregation the model is classified as partial mechanistic. [1]

4.4.2 KBH Model

The overall volume growth rate can be found to be

(4.4.17)

A KBH strategy was proposed (Georgieva, Feyo de Azevedo, et al., 2003) that consisted of combining the a priori knowledge of the process gained by the balances with a neural network approach to extract knowledge in the experimental data available. This is done in place of searching for complete mathematical expressions for these parameters.

The kinetic parameters are described by the following equations:

(4.4.18)

(4.4.19)

(4.4.20)

Kg, Kn and Kag are kinetic constants that were optimized by Georgieva et al., 2003 [1]

27

CHAPTER 5

PERFORMANCE COMPARISON

5.1 Overview

In this chapter performance of the PID control and SMC is compared with

each other when they are both applied to the sugar crystallization process. A

simulator was developed in Simulink that was used to implement the

designed control on the process. The required outputs were then controlled

using this simulator by PID control and SMC. A robust analysis was also

carried out on both control schemes in which certain system parameters

were changed so as to observe the robustness of each scheme to parameter

deviation.

A control procedure was developed by Paz Suarez, Georgieva, et al., 2011

[2] that was modified slightly in this thesis and using that procedure both

PID control and SMC were applied to the plant in question. Details can be

found in Table 5.1.

5.2 Process Operation

Process can be divided into different parts (Simoglou et al., 2005). It was

modified for this thesis and described below:

1. Charging: Fs=0, Ff=max, W=0 and Pvac = 0.5 (hard constraint). The

purpose is to fill up the vessel and bring the initial conditions to the

required values.

2. Concentration: Stirrer is on. Fs=2kg/s. Volume of Massecuite

(suspension of sugar crystals in heavy syrup) is controlled at 12.15 m3

28

using the control of Ff. When supersaturation reaches 1.06, Ff is closed

and Fs reduced to 1.4kg/s.

3. Seeding and setting the grain: The seed crystal is introduced and Fs

continues to be at 1.4kg/s till we reach the required initial conditions

for the next step.

4. Crystallization with liquor (phase 1): Crystallization begins when the

supersaturation is controlled at a fixed point (S=1.15) using Ff for

control. This is done for a fixed time.

5. Crystallization with liquor (phase 2): Supersaturation is kept

constant at 1.15 and the feed valve (Ff) is closed and Fs is used to control

the Supersaturation level of massecuite.

6. Crystallization with Syrup: For economical reasons the liquor is

replaced with a cheaper syrup. Fs is max and using Ff volume fraction of

crystals (vc) is kept constant at 0.43.

There are a few steps after this but they are irrelevant with regards to

this thesis which focuses on control of outputs. They are concerned with

closing the various valves after the required output has been achieved

and these omitted steps involve no control and are undertaken merely to

stop the process using certain conditions. They can be found in Paz

Suarez, Georgieva et al., 2011.

The different parts of this process are moved by distinctive driving forces,

therefore four sequential control loops were designed, shown schematically

in Fig. 5.1. The relation of the loops with the control scheme mentioned

above is described in Table 5.1.

29

Fig. 5.1: Schematic of Process Industrial Unit

Stage Actions Control

Charge W=0 and Pvac = 0.5

(hard constraint)

Initial conditions:

Mw=0.1kg, Mi=0.1kg,

Ms=1kg.

Run for 10 min.

Fs=0, Ff=max, Fw=0

No control

Concentration Stirrer is on.

Fs=2kg/s.

Required Initial

Manipulated Variable

(u) = Ff

Control Loop 1

30

conditions:

Mw=11770kg,

Mi=2.7kg, Ms=257Kg

achieved via the three

input valves being

allowed to run freely till

the required values are

reached.

Volume of Massecuite

(suspension of sugar

crystals in heavy syrup)

is controlled at 12.15

m3

When supersaturation

reaches 1.06, Ff is

closed and Fs reduced to

1.4kg/s.

Seeding and setting the

grain

The seed crystal is

introduced and Fs

continues to be at

1.4kg/s till we reach the

required initial

conditions for the next

step.

No control

Crystallization with

liquor (Phase I)

Crystallization begins

when the

supersaturation is

controlled at a fixed


(u) = Ff

Control Loop 2

31

point (S=1.15) using Ff

for control. This is done

for a fixed time.

Initial Conditions:

Mw=2Kg, Mi=1Kg,

Ms=10Kg


liquor (Phase II)

Supersaturation is kept

constant at 1.15 and

the feed valve (Ff) is

closed and Fs is used to

control the

Supersaturation level of

massecuite.

Initial conditions: same

as in Phase I.


(u) = Fs

Control Loop 3


syrup

For economical reasons

the liquor is replaced

with a cheaper syrup. Fs

is max and using Ff

volume fraction of

crystals (vc) is kept

constant at 0.43.

Initial Conditions:

Mw=0.1kg,

Mi=0.042kg,

Ms=0.01kg.


(u) = Ff

Control Loop 4

Table 5.1: Summary of operation [2]

5.3 Process Simulator

32

The values of certain parameters used in the simulator may have some

discrepancy from values in actual industrial units therefore the control

scheme has to be robust to such discrepancies.

The simulator in question shows how it is possible to control the flow of the

three inputs so as to control certain outputs. When this control scheme is

applied to an actual industrial unit the crystallizer outputs of the unit should

behave as the simulator predicted.

The accuracy of tracking and robustness depends considerably on the initial

conditions given to the simulator. These conditions can be achieved by

letting certain valves open to the fullest at certain times (as described in

Table 5.1) and making sure the purity of the sucrose solution is known.

The following figures show some parts of the simulator.

Fig. 5.2: Main control system body of Process Simulator showing actuator

inputs and plant.

33

Fig. 5.3: Main body of plant

Fig. 5.4: Mass Balance equations in Simulink

34

Fig. 5.5: Energy Balance equations in Simulink

5.4 Control and Sliding Surface Design

The design of PID control and SMC, along with the design of sliding surface s

is discussed below.

5.4.1 Sliding Mode Control and Sliding Surface Design

The control procedure outlined below is general and applicable for all four

control loops. The output y is changed for every loop, except between

second and third. The input u is also changed from Ff to Fs between the four

control loops which results in complete reshaping of the control u.

35

Fig. 5.6: Population Balance equations in Simulink

36

5.4.2 PID Control Design

PID Control was designed via the PID velocity form equation outlined in

Chapter 3. The gains were tuned via Ziegler-Nichols method for tuning. They

came out to be:

Kp = 2

Ki = 400

and

Kd = 250e-6.

A constraint was introduced in this thesis which stated that the gains could

only be tuned for the first control loop after which they would not be altered

in the other three control loops; this could reflect possible limitations in the

hardware implementation of this control where gains could only be set once

and the four loops implemented one after the other based on certain

conditions. Such a constraint however was not employed in Georgieva, Paz

Suarez et al., 2011.

37

5.5 Tracking Performance

The results of charging phase are shown below.

Fig 5.7: Ms vs time (s)

Fig. 5.8: Mi vs time (s)

0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

time

Ms

(Mas

s of

dis

solv

ed s

ucro

se) k

g

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

time

Mi (

Mas

s of

impu

ritie

s) k

g

38

Fig. 5.9: Mw vs time (s)

Fig. 5.10: Tm vs time (s)

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

time

Mw

(Mas

s of

wat

er) k

g

0 1 2 3 4 5 6 7 8 9 1010

20

30

40

50

60

70

80

90

100

time

Tm (T

empe

ratu

re o

f Mas

secu

ite) C

39

The results of the four control loops are as follows that include comparison

with a PID controller:

1. Vm should track set point of 12.15 with control input Ff. (Control loop

1)

Fig. 5.11: Volume of Massecuite

We can see that SMC has slightly better tracking performance than

PID control, for this control loop.

2. S should track set point of 1.15 with control input Ff. (Control loop 2)

Fig. 5.12: Supersaturation (SMC)

40

Fig. 5.13: Supersaturation (PID)

We can see that SMC and PID control have the same performance with the

tracking performance of PID smoother than that of Sliding Mode.

3. S should track set point of 1.15 with control input Fs. (Control loop3)

Fig. 5.14: Supersaturation

As we can see that PID has not tracked the set point at all, while SMC

tracking is close to perfect.

41

4. Vc should track set point of 0.43 with control input Ff. (Control loop4)

Fig. 5.15: Volume fraction of crystals

In this loop the performance of SMC is superior to that of PID as PID has not

settled at the setpoint (or close to it) but rather the response keeps on

increasing.

5.5.1 Sliding Surfaces

The Sliding surfaces for each of the four control loops when SMC is used are

shown below.

1. Sliding Surface for Control loop 1

Fig. 5.16

42


Fig. 5.17


Fig. 5.18

43


Fig. 5.19

5.6 Robust Analysis

Certain parameters in the system were changed in Control loop 2 for both

PID control and SMC. Control loop 2 was chosen because the tracking

performance was same for both PID control and SMC. In all the other loops

the performances were different hence the exact effect on tracking due to

parameter uncertainty may not be visible exactly.

1. In first analysis Feed flow of water, Fw is changed from 1.36 to 1.37 for

both PID control and SMC.

44

Fig. 5.20: SMC response

Fig. 5.21: PID control response

Hence both the control schemes are equally robust to the introduction of

this discrepancy.

0 0.2 0.4 0.6 0.8 1

x 10-3

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

1.2

time (s)

S (S

uper

satu

ratio

n)

Fw=1.37Fw=1.36

45

2. In second analysis Feed flow of water, Fw is changed to 1.38 for both

PID control and SMC.



Hence both the schemes show deterioration in tracking by equal

amount. Therefore they are not robust when Fw>1.37.

0 0.2 0.4 0.6 0.8 1

x 10-3

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

1.2

time (s)

S (S

uper

satu

ratio

n)

Fw=1.38Fw=1.36

0 0.2 0.4 0.6 0.8 1

x 10-3

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

1.2

time (s)

S (S

uper

satu

ratio

n)

Fw=1.38Fw=1.36

46

3. In third analysis Stirrer Power, W, is changed from 10.5A to 50.5 A

(hard constraint) for both PID control and SMC.



We can conclude from above that changing of the parameter W’s values to a

large extent does not affect either of the control schemes. Hence they are

equally robust.

0 0.2 0.4 0.6 0.8 1

x 10-3

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

1.2

time (s)

S (S

uper

satu

ratio

n)

W=50.5AW=10A

0 0.2 0.4 0.6 0.8 1

x 10-3

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

1.2

time (s)

S (S

uper

satu

ratio

n)

W=50.5AW=10A

47

CHAPTER 6

CONCLUSIONS AND FUTURE SUGGESTIONS

6.1 Conclusions

We have observed in section 5.5 that for two of the control loops (3 and 4)

the performance of SMC has been far superior in terms of tracking to PID

control. While in Control loop 1 SMC was only slightly better in Control loop 2

the performance can be said to be equal with PID tracking being termed

smoother.

In terms of robust analysis it was demonstrated that both control schemes

have shown equal level of robustness to parameter discrepancy. The reason

for that is the design of sliding surface: the integral control is essentially a P

controller when relative degree of system is 1, PI when it is 2 and PID when

it is 3. This similarity can be one of the reasons for similar robustness.

Moreover PID and SM also share the qualities of being heuristic towards

tracking problems when the system model is absent or not complete.

SM regularly suffers from chattering whereby several procedures such as

discrete time control are employed to reduce chattering. However in this

thesis we have seen no chattering phenomena as evident from the four

sliding surface responses in Chapter 5. Therefore we can assume that when

this system is implemented on hardware the actuator will not suffer from

wear and tear: the hallmark of chattering.

48

6.2 Future recommendations

It is highly recommended to implement this system on a larger processor

and faster RAM. The processor and RAM used in this thesis were Core 2 Duo

and 2GB respectively. Even with Core i3 and 2 GB RAM the performance was

not significantly improved. Hence the simulations could not complete their

tenure because they ran out of memory. However the accurate tracking in

some control loops may lead to the assumption that the tracking will remain

accurate if the simulation is run for the entire team.

It is also recommended that the system be transferred to discrete time so

that a more realistic analysis could be made on how the control will fair on

hardware.

49

APPENDIX

EQUATIONS FOR PHYSICAL PROPERTIES

50

51

52

53

REFERENCES

[1] Georgieva, P., Meireles, M.J., Feyo de Azevedo, S., 2003. Knowledge

based hybrid modeling of a batch crystallization when accounting for

nucleation, growth and agglomeration phenomena. Chemical Engineering

Science 58, 3699–3707.

[2] Paz Suárez, LP., Georgieva, P., Feyo de Azevedo, S., 2011. Nonlinear

MPC for fed-batch multiple stages sugar crystallization. Chemical engineering

research and design 89, 753–767

[3] Khalil, Hassan K., Non-linear Systems (2002)

[4] Utkin, Guldner, Shi, 1999. Sliding Mode Control of Electromechanical

systems.

[5] Simoglou, A., Georgieva, P., Martin, E.B., Morris, J., Feyo de Azevedo,

S., 2005. On-line monitoring of a sugar crystallization process. Computers &

Chemical Engineering 29, 1411–1422.

[6] Feyo de Azevedo, S., Goncalves, M.J., Chorão, J.M., Bento, L., 1993. On-

line monitoring of white sugar crystallization trough software sensors. Part I.

International Sugar Journal 95, 483–488.

Feyo de Azevedo, S., Goncalves, M.J., Chorão, J.M., Bento, L., 1994. On-line

monitoring of white sugar crystallization trough software sensors. Part II.

International Sugar Journal 96, 18–26.

[7] Utkin, V. “Variable Structure System with Sliding Modes” IEEE TAC, Vol. AC-22, No.2, pp 212-222, 1977

[8] Barbot, Perruquetti. 2002. Sliding Mode Control in Engineering.

[9] Alvi, Atif. Process Industrial control and Instrumentation. PowerPoint

Slides. Skill Development Council Islamabad. Dec 2012 Workshop.

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[10] Minorsky, Nicolas. 1922. "Directional stability of automatically steered

bodies". J. Amer. Soc. Naval Eng. 34 (2): 280–309.

[11] Benett, S. A History of Control Engineering 1930 – 1955.

[12] Ogatta, K., Discrete Time Control Systems.

[13] Ditl, P., Beranek, L., & Rieger, Z. (1990). Simulation of a stirred sugar

boiling pan. Zuckerindustric, 115, 667–676.

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