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STUDY AND SIMULATION OF THE PREDICTIVE
PROPORTIONAL INTEGRAL CONTROLLER IN COMPARISON
WITH PROPORTIONAL INTEGRAL DERIVATIVE
CONTROLLERS FOR A TWO-ZONE HEATER SYSTEM
by
DEEPA THOMAS MANNATH, B.E.
A THESIS
IN
ELECTRICAL ENGINEERING
Submitted to the Graduate Faculty
of Texas Tech University in Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCE
IN
ELECTRICAL ENGINEERING
Approved
December, 2002
ACKNOWLEDGEMENTS
The compilation of this thesis required many consultations and rigorous,
methodical application. Dr. HenrykTemkin and Dr. Siqing Lu of Applied Materials, to
whom I express my sincerest appreciation for their patient support and invaluable
guidance, graciously accepted these. It was a pleasure to work under them and my
experience holds me in good stead for the future.
Deep appreciation is also extended to Yehuda Demayo, my manager at Applied
Materials, who gave me the opportunity to work on this project and has been a
source of constant support and inspiration since. His insight has been valuable and
has gone a long way into making this project a successful one.
I am grateful to Dr. Mark Holtz and Dr. Tim Dallas for their detailed insights of this
thesis.
Many thanks to Ms. Barbi Dickensheet for reviewing this document when it was
in its incipient stages. Such patience is indeed rare and is much appreciated. Finally,
I would like to thank all of my peers and colleagues in the Department of Electrical
Engineering at Texas Tech University, and in the CPI Division at Applied Materials
for their support and advice.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
ABSTRACT vi
LIST OF TABLES vii
LIST OF FIGURES viii
CHAPTER
1. INTRODUCTION 1
1.1 PI D control 1
1.2 Introduction to Proportional-lntegral-Derivative (PID)
control algorithm 2
1.2.1 Proportional control 2
1.2.2 Overshoot and temperature cycling 2
1.2.3 Eliminating offset 3
1.2.4 Integral action (sometimes called automatic reset) 3
1.2.5 Definition of integral time 4
1.2.6 Integral windup 5
1.2.7 Derivative action (sometimes called rate action) 5
1.2.8 Definition of derivative time 6
1.3 Variations of the PID algorithm 6
1.4 PID control with Feedforward: A special case of PID6 10
1.5 Thesis Outline 10
2. DESCRIPTION OF THE SYSTEM AND THE PROBLEM 12
2.1 Description of the System 12
2.1.1 The SiNgen Centura 300 process chamber 13
2.1.2 Dual Zone Heater Assembly 15
2.1.3 Schematic of the control system 17
2.2 Problems with the existing system 18
2.2.1 Variations in inner and outer zone voltages 18
2.2.2 Disturbances due to change in control algorithm 18
2.3 Goals to be achieved by making necessary changes in the control algorithm 18
3. DESCRIPTION OF THE FUNDAMENTALS AND
THEORY OF PPI CONTROL 20
3.1 Model-based Predictive Control 20
3.2 Predictive PI Controller 22
3.3 Specification of Heater Control 25
3.3.1 Phase I: Overshoot Reduction and Bumpless Transfer 25
3.3.1.1 Current Algorithm 25
3.3.2 Phase II: Total Power Control 26
3.3.2.1 Current Algorithm 26
3.3.2.2 Total Power Control Algorithm 28
3.3.3 Phase III: Autotune of PID parameters 29
3.3.3.1 Current Algorithm 29 3.3.3.2 Relationship of PID Parameters in Gain-Type and
Standard Algorithms 30
3.3.3.3 Autotuning Algorithms 31
3.3.4 Phase IV: Replace PID with PPI 33
3.3.4.1 Current Algorithm 33
3.3.4.2 PPI Algorithms 34
4. MODELING OF THE HEATER 36
4.1 Experimental Data 36
4.1.1 Heater Characteristics 36
4.2 Modeling of the Various Blocks 37
4.3 Simulink For Modeling 38
5. SIMULATIONS AND TUNING 40
5.1 Simulations of The Algorithms 40
5.2 Tuning 44
5.2.1 Tuning of PID 44
5.2.2 Tuning PID: An Example 45
5.2.3 Tuning of PPI 46
5.3 Performance Indices 47
5.4 Simulation Results 48
6. CONCLUSIONS 49
LIST OF REFERENCES 50
ABSTRACT
The PID algorithm is the most popular feedback controller used within the
process industries. This is the algorithm that is currently used on a System in the
DSM (Dielectric Systems and Modules) and the CMD (Contact Metal Deposition)
groups at Applied Materials Inc. The current Algorithm which controls the Ceramic
heater in the LPCVD (Low Pressure Chemical Vapor Deposition) and TTN (Titanium
TiNitride) chambers, had problems, which lead to non-uniform heating. After
examining and listing the problems, it has been concluded that the Predictive
Proportional Integral (PPI) mode of control would provide adequate solutions. In this
project we have studied the heater systems, modeled them and carried out
simulations in Simulink. The results of the simulation were then used to place
Software Requests so that necessary software changes in the Control Algorithm
could be made.
LIST OF TABLES
5.1 Controller settings from Gy and Pu 44
5.2 Example of PID Tuning 45
5.3 Tuning PPI 46
5.4 Results 48
VII
LIST OF FIGURES
1.1 Simplified Block Diagram of a Control System 1
1.2 Response of a PI algorithm to a step in error 4
1.3 Control system with a PID Controller 6
1.4 PI D with Feedforward 10
2.1 SiNgen Chambers on Centura 300 Mainframe 12
2.2 The SiNgen Centura 300 process chamber 13
2.3 SiNgen chamber overview 14
2.4 Dual Zone Heater Assembly 16
2.5 Control System Schematic 17
3.1 Model Predictive Controller (MPC) 21
3.2 Model Predictive Controller (MPC) 22
3.3 Predictive PI control 22
3.4 The Unrealizable Controller 23
3.5 The realizable PPI implementation 24
3.6 Current Algorithm 25
3.7 Cun-ent Algorithm in Current mode 27
3.8 Auto tuning 32
4.1 Power Temperature Curve 36
4.2 Inner Zone Resistance versus Voltage Ratio 37
4.3 Outer Zone Resistance versus Voltage Ratio 37
4.4 Output plots of Various Ordered Systems 38
4.5Simulink Model 39
5.1 Simulink Model for PID1 40
5.2 Simulink Model for PID2 41
5.3 Simulink Model for PID3 41
5.4 Simulink Model for PID4 42
5.5 Simulink Model for PID5 42
VIII
5.6 Simulink Model for PID6 43
5.7 Simulink Model for PPI 43
5.8 Ziegler Nichols Ultimate Cycle method 44
5.9 Pu From Steady Oscillations 45
5.10 Response of a second order system with PID1 46
5.11 Perfonnance Indices for Controllers 47
CHAPTER 1
INTRODUCTION
1.1 PID control
The PID algorithm is the most popular feedback controller used within the
process industries. It has been successfully used for over 50 years. It is a robust and
easily understood algorithm that can provide excellent control perfonnance despite
the varied dynamic characteristics of process plants. Figure 1.1 shows the simplified
block diagram of a control system.
This section attempts to:
• introduce Proportional-lntegral-Derivative (PID) control algorithm,
• discuss the role of the three modes of the algorithm,
• introduce variations in the algorithm,
• discuss the PID in the light of the 300mm LPCVD heater controller.
Load _
u Y e Kp Kc Yr
1 u
>^
r Process Kp
C o n t r o l l e r Kc
manipu la ted v a r i a b l e c o n t r o l l e d v a r i a b l e e r r o r ( Y r - Y ) Process ga in C o n t r o l l e r ga in Set Po in t
efi K
Y
^ Yr
Figure 1.1 Simplified Block Diagram of a Control System
1.2 Introduction to Proportional-lntegral-Derivative (PID) control algorithm
1.2.1 Proportional control
For understanding the operation of the three control modes better, the control
modes when applied to a simple heater are considered.
Consider a controller that can throttle back the heater power well ahead of the
temperature reaching set point. That is, make the power shrink in proportion to
distance from set point. The controller now has a chance to anticipate and head off
temperature overshoot and cycling. This action defines it as a proportional controller.
The proportional mode adjusts the output signal in direct proportion to the
controller input (which is the error signal e). The adjustable parameter to be
specified here is the controller gain Kp. The larger the Kpthe more the controller
output would change for a given error. This is proportional control.
Output u = Kp*e + b
where b=bias, e=error.
The size of error needed to make the Proportional Controller deliver 100% power
is called PROPORTIONAL BAND. It is sometimes expressed as a percentage of
the controller range. Therefore, if this controller has a range of say 0 to 1000°, 40°
represents a 4% proportional band. The gain Kp is defined as 100/%PB (100/4 = 25
in this example).
1.2.2 Overshoot and temperature cycling
If the gain is made large enough, the power will throttle back early enough to
avoid overshoot and temperature cycling. If the temperature could reach set point,
the deviation, which is the controller input, would reach zero, therefore, so would the
output. This does not happen. The temperature settles at some temperature below
set point and some intermediate level of output is delivered. This shortfall of
Controller Output below set point is called offset. From the equation for output of a
Proportional Controller, it is seen that the output equals bias when the error is zero.
The bias is fixed at the normal value of output, say 50%, or it can be adjusted
manually to match the load. This is called manual reset. The Proportional Band
could be further reduced (more controller gain) to get more power and to reduce the
offset at the risk of breaking into temperature cycling again. This is called control
loop instability.
1.2.3 Eliminating offset
Due to the proportional relationship between input and output, the error will
change in correspondence with any change in the output. If the output changes to
adjust to a new load, then the error will also change as follows:
Error e=Kp(u-b)/100.
Say the temperature settles at 180°C, i.e., 20°C below the set point of 200°C with
a corresponding power of 50%. The set point could be reset to about 222°C to get
the controlled temperature to come up close to 200°C. Some simple controllers have
a small knob labeled manual reset, which achieves this without it showing as an
extra 22°C on the set point display. This 22°C deviation is amplified into enough
power to heat the zone to about 200°C.
Problem: There may be times when the process needs say, twice that power to
hold 200°C; for example, when there is higher material throughput. To put out twice
the power, the amplifier would need twice the input (about 44°C offset) othenA/ise;
the temperature would head down again towards 180°C.
1.2.4 Integral action (sometimes called automatic reset)
The integral mode corrects for any offset (error) that may occur between the Set
Point and the process output automatically overtime. The adjustable parameter to
be specified is the integral time T of the controller.
Output u=1/T*ledt.
Reset is often used to describe the integral mode. Reset is the time it takes the
integral action to produce the same change in the manipulated variable (u) as the P
mode's initial static change. Figure 1.2 shows the response of a PI controller to a
step input.
Open loop response of a PI c o n t r o l l e r to a s tep 1 n e
I n i t i a l s tep due to P
T i Time
Figure 1.2 Response of a PI algorithm to a step in error
Now coming to our example of the heater it is not feasible to keep resetting the
keep resetting the set point and waiting around every time the heat demand
changes. What is needed is an automatic and continuous watch on the temperature
and automatic power adjustments, aimed at keeping the deviation at zero. So, let
the controller do this as a second job. Let it watch the deviation and so long as it
persists let the controller put out a gently increasing contribution until there is just
enough power to make the deviation e = 0.
The controller is designed to make the rate of output power growth proportional
to deviation. Therefore, when temperature is close to set point the power is changing
very slowly until at set point the power stops growing and holds at just the level
needed to hold the temperature at set point. For deviations above set point, the
power decays to achieve zero deviation.
This is called integral action. This is a PI (Proportional + Integral) controller.
1.2.5 Definition of integral time
The strength of integral action is expressed in terms of integral time, usually seconds
or minutes. If the deviation = one Proportional Band, the contribution of integral
action will grow to 100% power in one INTEGRAL TIME T. Note that a short integral
time brings a fast growth of power and an eager corrective action.
1.2.6 Integral windup
When a controller that possesses integral action receives an error signal for
significant periods, the integral term of the controller will increase at a rate governed
by the integral time of the controller. This will eventually cause the manipulated
variable to reach 100% (or 0%) of its scale, i.e., its maximum or minimum limits. This
is known as integral windup. A sustained error can occur due to a number of
scenarios, one of the more common being control system override. Override occurs
when another controller takes over control of a particular loop, e.g., because of
safety reasons. The original controller is not switched off, so it still receives an en-or
signal, which through time "SA/inds-up" the integral component unless something is
done to stop this occurring. Many techniques may be used to stop this happening.
One method is known as "external reset feedback." Here the signal of the control
valve is also sent to the controller. The controller possesses logic that enables it to
integrate the error when its signal is going to the control valve, but breaks the loop if
the override controller is manipulating the valve.
1.2.7 Derivative action (sometimes called rate action)
The derivative action anticipates where the process is heading by looking at
the time rate of change of the controlled variable (its derivative). Td is the rate
time and this characterizes the derivative action (with units of minutes). In
theory, derivative action should always improve dynamic response and it does in
many loops. In others, however, the problem of noisy signals makes the use of
derivative action undesirable (differentiating noisy signals translate into
excessive movement of the manipulated variable). Derivative action depends on
the slope of the error, unlike P and I. If the error is constant, derivative action
has no effect.
Now the controller has a third function. It watches for CHANGES of
temperature and puts out a contribution of power that is proportional to RATE
OF CHANGE of temperature, e.g., fast dive, big boost, slow dive gentle boost,
fast rises, big throttle-back, etc. The purpose here is to resist and damp out
unwanted changes and to speed up recovery from temperature disturbances.
This contribution to output power exists only when the temperature is changing.
1.2.8 Definition of derivative time
If the temperature dives at a rate of one proportional band in one DERIVATIVE
TIME Td, the contribution of derivative action is 100% power (and minus 100%
power for temperature climbing).
Output u=Td*de/dt.
This is now a PID (proportional + integral + derivative) controller, also called a 3-
tenn or 3-mode controller.
1.3 Variations of the PID algorithm
In Figure 1.3 a control system with a PID controller is shown.
Yr
.
Y
^ e _
,
u Y e Yr V
PID ^U Driver V Heater
manipulated variable controlled variable error (Yr-Y) Set Point Driver Output
Figure 1.3 Control system with a PID Controller
There can be three representations of the PID control Algorithm
/ Serial: U{s) = K^ 1 \
1 + + V
1
Eis)
Parallel: C/(s) = K Eis) + — E{s) + T,sEis) T.s
Gain : f/(s) = K E{s) + £(5) + /:^5^(5)
where
Kp = proportional gain,
Ti = integral time constant,
Td = derivative time constant,
U(s) = Laplace transformed control signal,
E(s) = Laplace transformed error signal,
K,= 1/T,
K d = 1 / T d .
PID1:
/•
U{s) = K^ 1 \
1 + — + Ts 7T a
Eis)
This is the standard textbook controller. Its value is more academic than
practical as it is not physically realizable. This is due to its relative order, which
is - 1 . It is the most widely used and is known as the non-interacting PID
because the three tuning parameters can be adjusted independently.
PID2:
where
Uis) = K^ / 1 T ^
1 Ts 1 + — + — ^
V - 1 + ^ ^ y Eis)
N = noise filtering constant. It is normally 10, but can be varied depending on
the system.
This is the physically realizable form of PID1 and it has the same
characteristics, but with a noise filtering capability.
PID3:
1 Uis) = K^ 1 + - ^ il + T,s)Eis)
This configuration is usually referred to as the interacting controller. If it is
converted to the PID1 configuration, the effective gain, integral and derivative
actions are functions of the original parameters. This version can be interpreted
as:
Uis) = K^ 1 + 1
V T.s J expr^sEis)
This is a PI controller with a pure predictor term (exp TdS) in which the
prediction horizon is Td. This unrealizable term is approximated as:
expTjs«1-1-TdS.
The proposed controller is actually closely related to this configuration. This
controller is also not realizable because it has a relative order o f - 1 .
PID4:
Uis) = K^ f j X /
1 + -T.s
1 + - "^ 1 + ^ 5
Eis)
PID4 is the same as PID3, but it is physically realizable since it has additional
noise filtering capabilities.
8
PID5:
Uis) = K^ V T.s J \ + -^s
where
Y(s) = Laplace transformed output signal.
This configuration avoids the "derivative kick," which occurs with step
changes in the Set Point (Yr). Consider PID1, a sudden change in the SP
causes the derivative term to become very large and this provides the so-called
"derivative kick" to the final control element. This can be avoided if the derivative
term acts on the measurement and not the error.
PID6:
Uis) = K^ ^Y^is)-Yis) + ^Eis)--^Yis)
where
P = Proportional kick constant,
Yr(s) = Laplace transformed Set Point.
This is similar to PID5, but here the 'proportional kick' is also accounted for.
The Proportional kick constant, p, is used to reduce the overshoot that can be
caused by sudden changes in the Set Point Yr. p is a fractional number and
when there is a sudden change in Yr, p swings into action and helps reduce the
overshoot.
There are two driving types of PID Control.
/ •
Absolute: C/(s) = ^^ \
1 + — + Ts Ts '
Eis)
1 hicremental: At/(s) = K^AEis) + — AEis) + TsAEis)
Ts
9
1.4 PID control with Feedforward: A special case of PID6
PID with Feedforward is a special case of PID6. Its implementation is outlined
in Figure 1.4.
r r
+
J
| j
y
u Y e V r V
P
F l j
i l"-iT^-i^ 1 \\} ^
J r1ve1 H - a - e r ^
V
tnani PL f a t e d v a r i a b l e CO 11 re l i e d v a r ' a b l s e r r o 1 ( ^ ' - Y i '.^^- K D - n t C r " v e O u i p L t
=ccdforv/Q'c f a c t o r
Figure 1.4 PID with FeedfonA/ard
PID + FeedForward: t/(s) = pY^ is) + K^ 1 + — -{-Ts Ts '
Eis)
This is implemented on the Applied Materials Endura SL 300. The advantage is
that the Feed Forward factor automatically sets the balance point. This reduces the
actual range over which temperature needs to be controlled as the temperature
needs to be controlled only around the balance point.
1.5 Thesis Outline
This thesis documents a project undertaken by the DSM (Dielectric Systems
and Modules) and the CMD (Contact Metal Deposition) groups at Applied
Materials Inc. The project examined the problems of the current PID algorithm,
which controls the Ceramic heater in the LPCVD (Low Pressure Chemical Vapor
10
Deposition) and TTN (Titanium TiNitride) chambers. After examining and listing
the problems, it has been concluded that the Predictive Proportional Integral
(PPI) mode of control would provide adequate solutions. The goals of this
project are outlined in the next chapter. In this project we have studied the
heater systems, modeled them and carried out simulations in Simulink. The
results of the simulation were then used to place Software Requests so that
necessary software changes in the Control Algorithm could be made. Finally, it
summarizes the project and discusses the future actions to be taken to
implement these changes.
11
CHAPTER 2 DESCRIPTION OF THE SYTSEM AND THE PROBLEM
2.1 Description of the System
A Centura System shown in Figure 2.1 can hold up to Four Chambers. The
Chamber is attached to the transfer chamber at any of the four locations. Each
chamber has an individual controller, ac box, and gas panel on its frame, which
allows each chamber to run its own process and maintenance independent of
the system. The Factory Interface (Fl) holds cassettes of wafers, which are fed
from the Fl to the Single Wafer Load Locks (SWLLS) and then to the Transfer
Chamber, and finally the wafer is placed into the SiNgen Process Chamber for
nitride deposition.
FACTORY INTERFACE
'CTIAMBEF^ •CONTF^OLLER
Figure 2.1 SiNgen Chambers on Centura 300 Mainframe
12
2.1.1 The SiNgen Centura 300 process chamber
The SiNgen Centura 300 process chamber shown in Figure 2.2, is a single wafer
process chamber.
Figure 2.2 The SiNgen Centura 300 process chamber
It offers a rapid wafer heatup/cooldown cycle and fast deposition rates. It is
designed for thermal chemical vapor deposition of silicon nitride at >600°C
(750°C TO 800°C). Different applications have different requirements for nitride
film deposition. Film Density, film defects, and surface morphology are critical
applications of thin nitride (<100A) in NO stack for advanced gate application
and ONO film stack for capacitor materials. They are less critical for etch stop
and hard mask applications. Good film conformality (step coverage) is important
in spacer and some etch stop applications, but it is not as critical in other nitride
applications where the topography is much less severe.
The SiNgen chamber is equipped with a remote plasma source for in-situ
periodic chamber cleaning. The RPS mounts on the SiNgen chamber lid. NF3
molecules are dissociated in the remote plasma unit. The generated fluorine
13
radicals are delivered to the chamber to remove chamber deposits. The direct
connection between the RPS unit outlet and chamber lid minimizes the
recombination of fluorine radicals, thus maximizing the clean efficiency of the
chamber deposit. Periodic remote NF3 plasma cleans successfully restore
pristine chamber conditions for nitride deposition. A chamber overview is
depicted in Figure 2.3.
CHAMBER LID
CHAMBER BODY
RPS II (REMOTE PUS MA SOURCE)
WATER MANIFOLD
WAFER/HEATER, LIFT ASSEMBLY
Figure 2.3 SiNgen chamber overview.
14
2.1.2 Dual Zone Heater Assemblv
The heater assembly holds the wafer directly below the perforated plate
(showerhead). The heater plate is made of ceramic (AIN), and is heated by the
heating element embedded inside the heater plate. The heater has two separate
heating zones to provide temperature uniformity. They are located on separate
layers inside the heater plate. The heating element of each zone is made of
molybdenum. The heater is heated by resistive heating. The dual zone heating
compensates for the temperature variance therefore providing the desired
temperature uniformity. A ceramic shaft supports the heater plate. During the
process, the interior of the heater shaft is at atmospheric pressure (see Figure
2.4).
1. Heater Plate. The heater assembly holds the wafer during process. The
heater has low separate heating zones. They are located on separate
layers inside the heater plate. The plate is made up of two layers. The top
one is inner zone control and the bottom one is the outer zone control. It
is heated by resistive heating. The wafer is held in place by a 30-mil deep
pocket milled into the surface of the plate. Material: Ceramic
2. Heater Shaft. A shaft that houses the heating element leads and the
Thermocouple lead supports the heater plate. Material: Aluminum.
3. Heater Shaft Clamp. The heater shaft clamp secures the heater shaft to
the aluminum hub. Material: Aluminum.
4. Heating Element Power Lead Insulators. The heating power lead
insulators insulate the heating element power leads from each other and
from the thermocouple lead. Material: Ceramic
5. Aluminum Hub. The heating element power leads and the thermocouple
leads are mounted to the aluminum hub, which acts as a support so that
the leads do not twist or become damaged.
15
THERMOCOUPLE PROBE
HEATER SHAFT CLAMP
HEATING ELEMENT POVifER LEADS
HEATER PLATE
HEATER SHAFT
HEATING ELEMENT LEAD INSULATORS
ALUMItiUMHUB
VITON 0-RING
Figure 2.4 Dual Zone Heater Assembly
6. Heating Element Power Leads. The heating element power leads supply
the power to heat the heating element. Material: Nickel
7. Thermocouple Insulator. The thermocouple insulator insulates the
thermocouple lead from the heating element power leads. Material:
Ceramic.
8. Thermocouple Probe. A thermocouple (TC) assembly monitors the
temperature of the heater plate. The TC only measures the temperature
16
of the center of the heater. The thermocouple probe is a shielded type-K
metallic thermocouple and contains two independent TC junctions:
• Temperature Measurement,
• Over Temperature Protection.
The thermocouple leads pass through the heater shaft to the back of the
heater plate and sends the thermocouple signal from the heater to the
thermocouple amplifier module located on the TC PCB.
^^>{5?)-^ Controller
SP SCR TC Amp
SCR Heater+
TC? Display
Set Point Si l icon Controlled Rect i f ie r Thermocouple Ampli f i e r
Figure 2.5 Control System Schematic
2.1.3 Schematic of the control system
Figure 2.5 shows the schematic of the heater control system. The Set Point is
the required temperature of the heater surface. It can be entered on the screen
at the Factory Interface (Fl). The input to the Controller is the difference
between the Set Point and the amplified Thermocouple measurement. The
Controller output goes to the SCR, which drives the Heater. The Heater
temperature can also be seen at the Display on the Fl screen.
17
2.2 Problems with the existing system
The existing algorithm that controls the heater in the Centura system is a
Proportional Integral Derivative (PID) algorithm. It is unable to provide adequate
control due to the reasons listed below.
2.2.1 Variation in inner and outer zone voltages
The voltage of the inner zone of the heater controls the PID loop, but the
characteristics of the inner zone and the heater temperature vary greatly due to
changes in any of the following factors:
• Operating Temperature,
• Process Conditions,
• Outer/Inner zone voltage ratio.
2.2.2 Disturbances due to change in control algorithm
There are potentially big disturbances when the control strategy changes in the
following cases:
• From Ramp to PID,
• Change in the Feedforward Factor (KRF ),
• Change in the Voltage Ratio (VR).
2.3 Goals to be achieved by making the necessary changes in the control algorithm
The following goals are to be achieved:
• Common code using PPI control to replace PID control for different heaters like
the Ceramic heater in the LPCVD chamber, and in the TTN chamber;
• Multiple set of PPI parameters at different temperature bands;
18
Bumpless transfer between control modes or bands to reduce temperature
overshoot at switching;
Built-in model for reference scheduling;
Tuning guidelines for PPI parameters;
Auto tune of PPI parameters;
Manual Control Mode.
19
CHAPTER 3
DESCRIPTION OF THE FUNDAMENTALS AND THEORY OF PPI CONTROL
3.1 Model-based Predictive Control
A model based predictive controller is a controller that uses a process model
in real time for the computation of the control action to be applied on the
process. This model represents the relationship linking the process input(s) to
the process output.
This model has to be identified: the structure of the model and the
parameters of the model are estimated by an identification algorithm, which
exploits the data collected during specific (open loop) experiments. The model is
used to predict the future process output and to compute the control action in
order to satisfy a given target (SP) for the process variable.
The controller design is only a function of the actual state of the system, x.
There is no predictive action based on the process model. In contrast to the
standard feedback design philosophy, the model predictive controller design is
based upon the feedback information that is the difference between the actual
state of the system x, and a predictive state Xp.
Model predictive controllers as shown in Figure 3.1, basically consist of the
same elements:
• The dynamic model,
• A reference trajectory yr(n) which describes the smooth transition of the
target variable from its current value to the future set point profile C within a
prediction horizon Hp that corresponds to the end of the coincidence horizon He.
This trajectory can be interpreted as the desired behavior of the closed loop
system,
• An objective criterion as a function of the future controller error e(n) between
the reference trajectory and the predicted output over a coincidence horizon
[HI, He].
20
• By minimizing the objective function an optimal profile for the future values of
the manipulated variable is calculated for the coincidence horizon that guides
the predicted target variable as close as possible to the reference trajectory,
• A compensation for modeling errors.
dm clu
X
xp Xsp dm di 'u
u e
process s t a t e p r e d i c t e d s t a t e se t p o i n t s t a t e measured process d is tu rbance unmeasured process d i tu rbance c o n t r o l v a r i a b l e s e r r o r between process s t a t e and p red i c ted s t a t e
Figure 3.1 Model Predictive Controller (MPC)
21
Another example of a Model Predictive Controller would be the one in Figure
3.2. The only modification here is that the Set Point is input to the Model. This is
done to facilitate reference scheduling and this is the algorithm used in this
project.
Set Point
I Model
Yr -
— ^ Clrlr u
1
Driver V
Heater y
Figure 3.2 Model Predictive Controller (MPC)
3.2 Predictive PI Controller
The predictive PI controller is a special case of MPC. Figure 3.3 shows a
schematic of the same.
Set Point
1
Model
Vr
PI u
Driver
Transmitter
V ^ Heater y
Figure 3.3 Predictive PI control
The presence of a time delay in the control loop limits the achievable
perfonnance of a control system. It contributes towards its destabilization by adding
a phase lag, and hence forcing a reduction of the gain of the controller. A pure
predictive term, i.e., e^ '', will cancel the effect of time delay by adding a phase lead
22
and will thus speed up the response. Now, e ^ implies pure prediction and is not
realizable.
The Maclaurin series expansion of e ^ is:
Tps _ _ e = 1 + Tr,S + T n S +
2 2 k k T n S +
2!
Ui(s)= e ^^u(s) ,
3,1
3.2
Using (3.1) in (3.2) and taking the inverse Laplace transform would give
2 2 -pk k U ^ ( t ) = U ( t ) + T ^ d [ u ( t ) ] + T p d [ u ( t ) ] + T p d [ u ( t ) ]
c l ( t ) 2! d ( t ) ^ k! d ( t ) ' < 3. 3
Load
E(s)
' ' - ^ ^ (Kp+1 / TiS ) e sTp
Disturbance
C(s) / A(s)
+
Ul(s
e"^^ B(s) / A(s)
as)
Figure 3.4. The Unrealizable Controller
A realizable approximation would be
2 2 U i * ( t ) = U ( t ) + T p d [ L i | ( t j ] + T p d [ U f ( t ) ]
f ^ d ( t ) 2! d ( t ) ^ 3 . 4
23
where
u^(s) = u^(s) 3,5
N ^ where Tpis the prediction horizon,
N is the noise-filtering constant.
The above equations imply that the current derivatives of a continuous time
signal can be used to predict the future developments of that signal. However taking
derivatives of such a signal in the presence of noise is not feasible because of the
side effect of noise amplification. The simplest solution would be to use filtered
derivatives instead of actual derivatives of the signal. Filtered derivatives are
obtained from the state variable filter.
d ' [ U f < t ) ]
d ( t ) '
s*^ u ( t ) f o r i = 1 , . . . , k
M ( s ) 3 . 6
where, the denominator of 3.5 gives M(s). Figure 3.5 shows the realizable
interpretation of the Predictive PI controller. This controller is actually a PI Controller
enhanced with the predictive term.
Approx. P r e d i c t i o n
U(s)
' ^ - • ^ ^> -» f0<pn /T i5 ) ^
Load Disturbance
C(s) / A(s)
e"^%(s)/A(s)
as)
Figure 3.5. The realizable PPI implementation
24
3.3 Specification of Heater Control
3.3.1 Phase I: Overshoot Reduction and Bumpless Transfer
3.3.1.1 Current Algorithm
The current algorithm for PID on Endura SL 300mm WCVD is shown in
Figure 3.6.
Kff
^ TempRefVr'
Eh®—
Bias
Figure 3.6 Current Algorithm
SCR Heater - i
Filter
The algorithm is as follows:
uik) = Kffy^ik)+K^eik) + u.ik-\) + Keik)-K][yik)-yik-l)] + Bik)
where
x(k) and x(k-1) refers to the current sample and the last sample of variable x;
u is voltage of inner-zone driver;
Uj is the history value of integral part;
Kff is feed forward coefficient;
yr is reference of temperature (this may be different from target temperature);
Kp' is gain of proportional term;
Ki' is gain of integral term;
Kd' is gain of derivative term;
e is difference between yr and y;
B is bias. 25
1. Overshoot reduction algorithm
To reduce overshoot, the following algorithm can be used.
uik) = Kj^y^k)- K'yik) + u.ik -1) + K'eik) -K][yik) - yik -1)] + Bik)
Note the difference of proportional term.
2. Bumpless transfer algorithm
By forcing the initial value to history value of integral part ui (k-1), a bump less
transfer can be achieved. This initialization should be done at one of the
following conditions.
Switch from ramp control to PID control;
Change of any SYSCONs including Kff, Kp", Ki', Kd',B.
uXk-\) = uik-\)-{Kj^y^ik)-K\yik) + Keik)-K][yik)-yik-\)] + Bik)}
where x(k-1) refers to the value of x before switch/change and x(k) refers to the
value of x after switch/change.
3.3.2 Phase II: Total Power Control
3.3.2.1 Current Algorithm
The current algorithm for PID on Endura SL 300mm WCVD is shown in
Figure 3.7. Note that temperature is controlled in current mode.
26
Figure 3.7 Current Algorithm in Current mode
The algorithm is as follows:
uik) = K^y^ik)+ Keik) + u.ik-\) + K'eik) - K][yik) - yik -1)] + Bik)
where
x(k) and x(k-1) refers to the current sample and the last sample of variable x;
u is total power P required for SCR heater driver;
Ui is the history value of integral part;
I is current required for SCR heater driver, and
lik) = •^Pik)/Rik) for resistance R which is a function of temperature T;
Kff is feedfoHA/ard coefficient;
yr is reference of temperature (this may be different from target temperature);
Kp' is gain of proportional term;
Ki' is gain of integral term;
Kd' is gain of derivative term;
e is difference between yr and y;
B is bias.
In the current software, the relationship of resistance R and temperature T is
hard coded for one type of W heater R=0.0128*T+4.75. This should be
27
changed to a generalized function, or at least, implement the two coefficients as
SYSCONs.
3.3.2.2 Total Power Control Algorithm
It is observed that the relationship between total power input to the heater
and temperature output of the heater can be estimated by a linear function.
Therefore, the feedforwad and PID parameters will not change significantly by
using power to control temperature. This is different from using voltage or
current to control temperature, which requires significant change of those
parameters if one of the following factors changes:
• change between voltage mode and current mode SCR driver;
• change between SCR driver and SSR driver;
• change of power ratio of multiple zone heater.
The algorithm for PID control is exactly the same, the difference is the
converter function from power to the input signal to the heater driver.
Current Mode Single-Zone SCR
Iik) = ^Pik)/Rik)
where
Pik) = uik)
and
Rik) = R[Tik)].
Voltage Mode Single-Zone SCR
Vik) = -^Pik)Rik)
where
Pik) = uik)
and
Rik) = R[Tik)].
28
Current Mode Dual-Zone Heater
P,ik) = K^,Pik)
Fik) = i\-K„)Pik)
I,ik)^^P^ik)/R^ik)
F_ik) = ^Pik)/R,ik)
where A,.,, between 0 and 1, is the power fraction of coil 1, and
Pik) = uik)
R,ik) = R^[Tik)]
R,ik) = R,_[Tik)]
Voltage Mode Dual-Zone Heater
P,ik) = K^,Pik)
P,ik) = il-K^,)Pik)
V,ik) = ^P,ik)R,ik)
V,ik) = ^P,ik)R,ik)
where AT ,, between 0 and 1, is the power fraction of coil 1, and
Pik) = uik)
R,ik) = R,[Tik)]
R,ik) = R,[Tik)]
3.3.3 Phase III: Autotune of PID parameters
3.3.3.1 Current Algorithm
The current algorithm for PID on Endura SL 300mm WCVD is shown in
Figure 3.6. Note that temperature is controlled in current mode.
The algorithm is as follows:
uik) = K^y^ ik)-\- Keik) + u,ik -1) + Keik) - K] [yik) - yik -1)] + Bik)
where
• x(k) and x(k-1) refers to the current sample and the last sample of variable x;
• u is total power P required for SCR heater driver;
• Ui is the history value of integral part;
29
I is current required for SCR heater driver, and lik) = Pik)IRik) for
resistance R which is a function of temperature T;
Kff is feedfoHA/ard coefficient;
yr is reference of temperature (this may be different from target temperature);
Kp' is gain of proportional term;
Ki' is gain of integral term;
Kd' is gain of derivative term;
e is difference between yr and y;
B is bias.
In the current software, the relationship of resistance R and temperature T is
hard coded for one type of W heater R=0.0128*T+4.75. This should be
changed to a generalized function, or at least, implement the two coefficients as
SYSCONs.
3.3.3.2 Relationship of PID Parameters in the Gain-Type and Standard Algorithms.
Most auto-tuning algorithms are based on the ISA standard algorithm as
shown in the following equation:
/ r . . . ,. ^
or u = K^ T dt
V
Uis) = K^ ^ 1 ^
V T.s J Eis)
where
• Kp is proportional gain;
• Ti is integal time in second;
• Td is derivative time in second.
We need to give the relationship between the gain-type algorithm and the
standard type.
30
K=K^TJT,
^'<i=^,TJT^
where Ts is sampling time of the control system, 0.100 second for Endura SL
300mm.
3.3.3.3 Autotuning Algorithms
The auto-tuning algorithm will automatically estimate the optimal PID and
feedforward parameters when the user requests to tune the parameters. This
feature can be implemented in the heater calibration screen. The algorithm will
calculate and override the corresponding SYSCONs when the AUTOTUNE
button is clicked.
Autotuning can only be done at a steady-state condition and not in ramp up
or cold down stage. The actual procedure is as follows.
1. Wait until a steady-state condition is reached;
2. Record the power Um/Pm and temperature Ym at the condition;
3. Switch from PID control mode to RELAY control mode;
4. Set the RELAY control parameters, basically (a) reduce power by h watts if
temperature is over 5 degree C and (b) increase power by h watts if
temperature is below to 5 degree C, which is shown in Figure 3.8
5. Record the power and temperature value for about 10 minutes;
6. Calculate the amplitude d of temperature output y and period Tu of
power/temperature as shown in the figure;
31
Figure 3.8 Auto tuning
7. Assume that the heater's model is a first order lag plus a pure time delay as
described in the equation:
Where Km is the gain, Tm is the time lag of first-order model and Dm is the
pure time delay.
8. Calculate the model parameters as follows:
Y
T„=^^JiKjJ^l
T D = - ^
" 2;r
where
Tta
( K-tan
'2KT?^
V T
32
Calculate the standard PID parameters as follows:
Kff
K ^'p
T. =
T,--
Call
K K-K,
= 1/^',,,
2r„,
^K„.iT„,-
••T„.
-IL 4
-DJ
Dulate gain-type
= ^ .
-KJJT,
= KJJT
where Ts is sampling time of the control system, 0.100 second for Endura SL
300mm.
3.3.4 Phase IV: Replace PID with PPI
3.3.4.1 Current Algorithm
The current algorithm for PID on Endura SL 300mm WCVD is shown in
Figure 3.7.
The algorithm is as follows:
uik) = K^y^ ik)+ K\eik) + u. (A: -1) + Keik) - K\ [yik) - yik -1)] + Bik)
where
• x(k) and x(k-1) refers to the current sample and the last sample of variable x;
• u is total power P required for SCR heater driver;
• Ui is the history value of integral part;
• I is current required for SCR heater driver, and lik) = -,JPik)/Rik) for
resistance R which is a function of temperature T;
• Kff is feedforwad coefficient;
• yr is reference of temperature (this may be different from target temperature);
• Kp' is gain of proportional term;
33
• Ki' is gain of integral term;
• Kd' is gain of derivative term;
• e is difference between yr and y;
• B is bias.
In the current software, the relationship of resistance R and temperature T is
hard coded for one type of W heater R=0.0128*T+4.75. This should be
changed to a generalized function, or at least, implement the two coefficients as
SYSCONs.
3.3.4.2 PPI Algorithms
Predictive PI or PPI control has some advantages over standard PID control.
The predictor term in PPI will give a better prediction than the derivative term in
PID. Actually, we can simply use the Predictor to replace the existing 2"' -order
low-pass filter for temperature and then remove the derivative term in the PID.
y,ik) = yik) + yik-\) + ^yik-2)-^y^ik-\)-^y^ik-2) «0 '^0 «0 «0 «0
where
T=TJT^
ao=0.4r'+1.47^+1
a, =-0 .8r '+2 ' r
a, =o.4r'-i.4r+1
&o=4.2r'+3.ir^+i 6i=-8.4r'+2
62=4.2r'-3.ir,+1
• y is temperature;
• yf is the filtered/prective temperature;
• Tp is the predictive time, which is a replacement of derivative time Td;
• Ts is sampling time of the control system, 0.100 second for Endura SL
300mm.
34
The PI control algorithm is then:
uik) = Kj^y^ik)-K^y^ik) + u.ik-\) + K][y^ik)-y^ik)] + Bik)
If auto-tuning algorithm is implemented the PI parameters are still the same
as those in PID and Tp can be selected as 1/3-1/2 of pure delay Dm.
35
CHAPTER 4
MODELING OF THE HEATER
The basic objective of this thesis being to study the behavior of the PPI controller,
a lot of stress has not been laid on modeling. This chapter gives a brief overview of
the data collected for modeling and touches upon the systems used for theoretical
modeling.
The system was modeled from the experimental data collected. The following
sets of data were collected.
4.1 Experimental Data
4.1.1 Heater Characteristics
The power supplied to the heater is varied and the temperature is recorded. The
ratio of temperature and total power does not change greatly. This data was used to
define the SYStem CONStants (SYSCONS) in the system computer. The data from
Figure 4.1 is used to define the SYSCON for gain between power input and
temperature output.
Power vs. Temperature
3.5
S 3
^ 2.5 $ O 2
1.5 650 700 750 800
Tern perature (C)
850
Figure 4.1 Power Temperature Curve
The resistance of the heater coil is only a function of temperature and is
independent of the voltage ratio, Vr. The voltage ratio is the ratio of voltage of
the outer coil with respect to that of the inner coil.
36
R e s i s t a n c e v s . V o l t a g e R a t i o for I n n e r Z o n e
1 1 . 5 0) o c re V) U) 0) IT
1 1
10 .5
10 9.5
• T = 6 7 0 C
• T = 7 5 0 C T = 8 0 0 C
1 2
V o l t a g e Ra t io
Figure 4.2 Inner Zone Resistance versus Voltage Ratio
R e s i s t a n c e v s . V o l t a g e Ra t io for O u t e r Z o n e
11 .5
9> 1 1 u
re 10 .5
I 10 K 9.5
9
• m
T = 6 7 0 C
T = 7 5 0 C
T = 8 0 0 C
1 2
V o l t a g e Ra t io
Figure 4.3 Outer Zone Resistance versus Voltage Ratio
Total power (sum of Inner and Outer Zone powers) is considered for the PID
loop, so that Kff, Kp, Ki and Kd will not change greatly with Vr and Temperature.
4.2 Modeling of the Various Blocks
The heater was modeled as a second order system with the Transfer
Function shown in the block below. This is just a theoretical assumption based
on experience with previous systems. A second order system allowed the
modeling of delay that the resistive coils of the heater gave rise to.
Similariy, the Wafer Disturbance was modeled as a second order system and
Thermocouple Dynamics are modeled as a first-order system.
The delay can be modeled better with higher-order systems:
37
u
t i m e
Step I n p u t O u t p u t o f F i r s t - O r d e r System O u t p u t o f Second-Orde r System O u t p u t o f T h i r d - O r d e r System
Figure 4.4 Output plots of Various Ordered Systems
A second-order system with the following Transfer Function was used.
0.75
30s^ +13S + 1
4.3 Simulink For Modeling
Simulink is a software package built on MATLAB for modeling, simulating,
and analyzing dynamical systems. It supports linear and nonlinear systems,
modeled in continuous time, sampled time, or a hybrid of the two. Systems can also
be multirate, i.e., have different parts that are sampled or updated at different rates.
For modeling, Simulink provides a Graphical User Interface (GUI) for building
models as block diagrams, using click-and-drag mouse operations.
38
step PIP C»ntc,>ll«rl
. •
12bitD/A HeaterDrlverSCR Discrete
Transfer Fon
0 75
30E^+13£-I-1
Heiiter
WdferSeq
-'lOs+O.C
5rjs2+15s+1
Wafe(DI:1urbarice
K* ThttmC'COuple Oifnanao:
- ^ - ^
Temp
Figure 4.5 Simulink Model
This is the Simulink System Model into which the various PID algorithm
models and the PPI model are plugged and simulated. This is the topic of
discussion in the following chapter.
39
CHAPTER 5
SIMULATIONS AND TUNING
There are 6 different types of PID algorithms and 1 PPI algorithm explained in
Chapter 1. This Chapter is devoted to evaluating the performance of the
Predictive PI controller. The PPI is simulated along with the PID structures.
Simulations of these algorithms are carried out in Simulink. The system model
developed in Chapter 4 is used here.
5.1 Simulations of the Algorithms
The Simulink simulation for each of the algorithms is as shown in Figure 5.1
to Figure 5.7.
PID1:
( Uis) = K^
\ ^ + ~ + T,s
T,s Eis)
Yr
€> Y
[ — •
+
Sumi
P
b
1 •
ro poitic
Kp
Ti.s
ntegial
njl1
»-»-
Kp"T?^;^^-^
D C
du/dt
>eriv3tivf
• + ->&
Sum
Figure 5.1 Simulink Model for PID1
40
PID2:
( • ' > 1 Yr
Y
b
1 •
+
Sum1
F
k
k
>I0 pontic
Kp
Ti.s
Integial
nal1
' — r
Kp'Td.s
Td/N.s+1
+ + +
Sum
Transfer Fen
-KD
Figure 5.2 Simulink Model for PID2
PID3:
^(s) = ^ , V T.sj
i\ + T,s)Eis)
Yi
&
h p k
w
1 +
k P • K P ^
V
^
Piopoitionail
F
Sumi 1
Kp
Ti.s
^tegial
!—• k
+
+
Sum
Td
Figure 5.3 Simulink Model for PID3
du/dt
Deiivative • 0
Sum2 -> U
41
PID4:
Uis) = K, '..±" V T.s J
1 + ^^^^ V 1 + ^ s
Eis)
Figure 5.4 Simulink Model for PID4
©1
© Y
1 — •
+
um
k
Pr
1
• Kp
oportio
Kp
Ti.s
Integra
nal1
+
+
Sum
h Kp'Td.s+1
Td/N.s+1
Transfer Fen
•r^ u
PID5:
Uis) = K^ f 1 ^
1 + -T^sj
T,s Eis)--^Y is)
1 + Tf5
©1 Yr
&
Sum1
Pioportionall
Kp_
Ti.s
Integral
Kp'Td.s
Td/N.si-1
Transfer Fen
Figure 5.5 Simulink Model for PID5
-KD Sum
42
PID6:
(7(s) = A- Plis)-Yis) + ^Eis)-^^Yis) TiS l+'^s
O Yr
© •
Sum1
Reference
^ Kp
ProportionaH
Kp_
Ti.s
Integral
Kp'Td.s
Td/N.s<-1
Transfer Fen
Figure 5.6 Simulink Model for PID6
Sum
-K) u
PPI:
Uis) = K^ 1 + -Ts
[(l + TpS + Tp*Tpl 452X1 + 2Tp IN.s + Tp*TplNI N)]Eis)
(ih Yr
&
Proportlonall
Sum1
l<p_
Ti.s
Tp"Tp/4.s2+Tp.s+1
Tp"Tp/N/N.s2+2'Tp/NsH K!) Integral Sum Transfer Fen
Figure 5.7 Simulink Model for PPI
These blocks were plugged into the Controller block of the System Model
developed in Chapter 4 and the PID controllers were tuned using the Ziegler Nichols
method, while a different tuning method, outlined in the next section, was used to
tune the PPI controller.
43
5.2 Tuning
5.2.1 Tuning of PID
The PID controllers are tuned using the Ziegler Nichols Ultimate Cycle
method. Figure 5.8 shows the response of the system at Ultimate Gain. The
classic closed-loop Ziegler Nichols tuning procedure is to advance the gain of
the Proportional only controller until the process is oscillating continuously at
constant amplitude.
10 15 20 25 30 35-nme
Figure 5.8 Ziegler Nichols Ultimate Cycle method
The gain required for achieving this (the ultimate gain, Gu) and the period of
oscillation (the critical period, Pu) provide parameters from which controller
settings are derived as shown in Table 5.1
Table 5.1 Controller settings from Gu and Pu
Controller Type
PI
PID
Kp
Gu/2 _ _
Gu/1.7
Ti
Pu/1.2
Pu/2
Td
Pu/8
44
Note that it is not necessary here to estimate the steady state process gain
as the controller gain, or proportional band, settings are expressed in terms of
that actually set on the controller to make the plant oscillate.
5.2.2 Tuning PID: an example
Increasing only the Proportional gain of PID, ultimate gain Gu, is 10.347 and
substituting in the formula for PID1 in Table 5.1, we have the following values of Kp,
Ti, and Td. The response is as shown in Figure 5.9.
Table 5.2 Example of PID Tuning
Controller Type
PID
Kp
6.086
Ti
10
Td
2.5
3
2
1
CO
^ 0 >-
-1
-J
X Y Plot
Rj
1 \j \i
-
"\ A .
\ 1 V
•
•
50 ICO .< A:<is
15C 200
Figure 5.9 Pu From Steady Oscillations
45
Using these PID values in the PID1 algorithm, the response shown in Figure 5.10 is obtained:
^iXY Graph
2
L5
11-
0.5
< 0 >-
-0.5
-1
-1.5
-2
X Y Plot
A
Unix]
50 100 150 200 250 300 XAxis
Figure 5.10 Response of a second order system with PID1
5.2.3 Tuning of PPI
There are 3 tuning parameters for PPI; therefore we may select a suitable Tp and
tune the PI part by using the Ziegler Nichols Ultimate Cycle method. The following
values of Kp and T , are used, while the best value for Tp is chosen as 5.
Table 5.3 Tuning PPI
Controller Type
PPI
Kp
4.7032
Ti
16.667
Tp
5
46
5.3 Performance Indices
The performance indices shown in the Figure 5.11 are described below.
1,8
1,6
1,4
1 ,2
^ 1.0
° 0 . 8
0 , 6
0 . 4
0 . 2
/ A A r\
/ 1 .
\ 1 \ / ^ B ^
- — 1 1 1 _ J
0 Tr2 4
Time 8 10 12
Figure 5.11 Perfonnance Indices for Controllers
1. Rise time (Yr): The rise time is the time at which the system output first reaches
the desired steady state value.
2. Overshoot (A/B): The overshoot is the ratio of the maximum peak value to the
desired steady state value of the output variable.
3. Integral Square Error (ISE): In order to make it easier to compare all these
designs and comment on the performance of any particular tuning algorithm the
Integral Square Error (ISE) is used as a common performance criterion and a
measure of quality control for all controllers tested.
47
5.4 Simulation Results
The responses of different controllers for unit step input and a wafer disturbance
with an amplitude of 0.5, period of 120s, duty cycle (% of period) of 80 starting at
4000s are examined and tabulated in the following table.
Table 5.4 Results
Kp
T,
Td
P
Tp
Rise time
Max. Over
shoot
ISE
PID1
6.086
10
2.5
108
1.4
10.925
PID2
6.086
10
2.5
108
1.4
8.9921
PID3
6.086
10
2.5
109
1.5
10.525
PID4
6.086
10
2.5
105
1.7
8.1254
PID5
6.086
10
2.5
112
1.5
11.0030
PID6
6.086
10
2.5
0.6
105
1.3
10.6743
PPI
4.7032
16.667
-
-
5
102
1.1
6.3654
48
CHAPTER 6
CONCLUSIONS
In this thesis six PID algorithms and the PPI algorithm were studied and
simulated. The PPI controller is implemented in continuous time and in effect it has
only one tuning parameter (Tp). The PI parameters can be obtained from those of a
first order with time delay approximation of the original high-order system. The
simulations indicate that the controller is robust. It has lesser tuning parameters,
which is an advantage over other PID tuning procedures, since less tuning
parameters imply easier manual or auto tuning. PID controllers are known for their
robustness and it is seen that PPI controllers share the same robustness properties.
The various goals outlined in Chapter 2 have been achieved and implemented
through algorithms for Reference Scheduling, Overshoot Reduction, Bump less
Transfer, Auto tuning. Total Power Control and finally conversion to PPI from PID
control.
It can be concluded that the PPI controller has better performance than the
standard PID tuned by ultimate ZN tuning rules. The results from the simulations are
tabulated in Chapter 5. It can be seen that the critical parameters like Rise Time,
Maximum Overshoot and Integral Square Error are optimum for the PPI case. PPI
gives minimum Rise Time of 102 seconds, least Overshoot of 1.1 and lowest
Integral Square Error of 6.3654.
These results were used to place Software Requests so that necessary software
changes in the Control Algorithm can be made. After these are incorporated into the
System Software, various Test Matrices are proposed to be carried out to test the
practical validity of these results. The most important parameter that will validate the
success of this project would be better thickness uniformity and control of the
deposited films.
49
LIST OF REFERENCES
[I] W. Fred Ramirez, Process Control And Identification, Academic Press, 1994.
U P. P. Kanjilal, Adaptive Prediction and Predictive Control, Peter Peregrinus Ltd., on behalf of IEEE, 1995.
[3] Lennart Ljung, System Identification Theory for the User, Second Edition, PTR Prentice Hall, Upper Saddle River, N.J., 1999.
[4] Wai Lun Lo and A. Besharati Rad, Comparison of Two Auto-Tuning Predictive PI Controllers, International Journal of Modelling and Simulation, Vol. 15, No. 3, 1995.
[5] W. L. Lo, A. B. Rad, K. M. Tsang, Auto-Tuning of Output Predictive PI Controller, ISA Transactions 38 (1999) 25-36.
[6] David Angeli, Alessandro Casavola, Edoardo Mosca, Predictive Pl-control of Linear Plants Under Positional and Incremental Input Saturations, Automatica 36 (2000) 1505-1516.
[7] K. K. Tan, T. H. Lee, F. M. Leu, Predictive PI Versus Smith Control For Dead-Time Compensation, ISA Transactions 40 (2001) 17-29
[8] Frank Allgower, Rolf Findeisen, Zoltan Nagy, Moritz Diehl, Georg Bock, Johannes Schloder, Nonlinear Model Predictive Control for Large Scale Systems, 6' Int. Conf. On Methods and Models in Automation and Robotics, NMAR 2000, pp 43-54, Poland.
[9] A. Besharati Rad and Wai Lun Lo, Predictive PI Controller, Int. J. Control, 1994, Vol. 60, No. 5, 953-975.
[10] John A. Shaw, Process Control Solutions, http://www.iashaw.com/pid/tutorial/
[ I I ] http://www.engin.umich.edu/qroup/ctm/extras/step.html
[12] http://www.che.utexas.edu/~control/sim/pid.htm
50
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