PATTERNBASED PROCESS CHARACTERIZATION
AND GAIN SCHEDULING FOR NONLINEAR
CHEMICAL PROCESSES
By
MANISH SINHA
Bachelor of Technology
Indian Institute of Technology
Bombay, India
1993
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
July, 1995.
PATTERNBASED PROCESS CHARACTERIZATION
AND GAIN SCHEDULING FOR NONLINEAR
CHEMICAL PROCESSES
Thesis Approved:
11
PREFACE
Accurate characterization of process dynamics from online sensor data is the key
issue in successful implementation of gain scheduling for controlling chemical processes.
This work presents a development of patternbased gain scheduling for process control.
The approach employs process state maps constructed from windowed slices of multisensor
plant trend data. Process identification is done using principles of similarity based
pattern recognition. This technique provides a straightforward means to associate unique
gain, integral time and/or derivative time controller settings with different states of the
process. Simulation results show that better control performance may be achieved by use
of gain scheduled controller as compared to the conventional fixed feedback systems.
I am sincerely grateful to my thesis advisor, Dr. Rob Whiteley. His guidance
went well beyond that of academic supervision and technical advise, and showed a deep
concern for the professional and personal development of his students. I wish to thank
Dr. Randy S. Lewis and Dr. Gary L. Foutch who served on my committee. A special
thanks also goes to Dr. Eduardo Misawa whose courses gave me a good understanding of
nonlinear control theory.
I also wish to thank all my friends who gave me company during long, late nights
at the lab. A special thanks goes to my roommates for their friendship. They have made
my stay at OSU a most pleasant and enjoyable experience.
111
Chapter
TABLE OF CONTENTS
Page
I INTRODUCTION 1
1.1 Motivation..................................................................................................... 1
1.2 PID Controllers............................................................................................. 2
1.3 Gain Scheduled PID Controllers................................................................... 3
1.4 Contribution of Thesis. 5
1.5 Organization of Thesis.................................................................................. 6
II GAIN SCHEDULING: AN ADAPTIVE CONTROL STRATEGy 7
2.1 Overview 7
2.2 Need for Controller Parameter Adaptation 7
2.3 Gain Scheduling Based on Process Gain 9
2.4 Gain Scheduling Based on Error Diagnostics 11
2.5 Gain Scheduling Using PatternBased Tuning Map 16
2.6 Application of Gain Scheduling 18
2.6.1 Aircraft Control 18
2.6.2 Ship Auto Control 18
2.6.3 Process Control 19
2.7 Need for Future Research 20
III PATTERNBASED PROCESS CHARACTERIZATION 21
3.1 Overview 21
3.2 Formation of PatternBased State Map 22
3.2.1 Selection ofNominal Conditions 22
3.2.2 Selection of Scheduling Variables 23
3.2.3 Representation ofNominal Conditions using MultiSensor
Pattern Trends 26
3.2.4 Arriving at a PatternBased Representation 29
3.3 OnLine Process Characterization 32
IV
3.3.1 Use of Sliding Window for OnLine Process
Characterization 36
3.3.2 Use of Time Smoothened Sensor Patterns 36
3.4 Concluding Remarks 39
IV CONTROLLER PARAMETER CALCULATION 40
4.1 Overview 40
4.2 Scheduling About a Few Nominal Points 41
4.3 Gain Surface Approximation 42
4.4 Choice ofNumber of Closest Neighbors 44
4.5 Interpolation Methods 47
4.5.1 Linear Interpolation 48
4.5.2 Quadratic Interpolation Using Pattern Similarity Measured
by ART2 Neural Network 50
4.5.3 Fuzzy Interpolation 52
4.6 Concluding Remarks 56
V DEMONSTRATION OF PATTERNBASED GAIN SCHEDULING 57
5.1 Overview 57
5.2 Nonisothermal Stirred Tank Reactor 58
5.3 Gain Scheduler Design 60
5.4 Results and Discussion 64
5.4.1 Importance of PatternBased Information 64
5.4.2 Number ofNominal Conditions 71
5.4.3 Interpolation Strategy 77
5.5 Concluding Remarks 85
VI CONCLUSIONS 87
6.1 Future Research 89
REFERENCES 92
APPENDIX: MODEL DEVELOPMENT AND SIMULATION 95
A1.1 Introduction 95
A1.2 Nonisothermal CSTR 95
A1.2.1 Open Loop Behavior 98
A1.3 Estimating Controller Settings at Nominal Operating Points 103
v
Table
LIST OF TABLES
Page
5.1 Tuning Parameters at Various Nominal Operating Conditions 61
5.2 Importance of PatternBased Information for Gain Scheduling 71
5.3 Effect ofNumber of Operating Conditions on Interpolation 76
5.4 Overall Servo Performance of Fixed PI Controllers and Gain Scheduled
Controllers Using Different Interpolation Strategies 85
VI
Figure
LIST OF FIGURES
Page
1.1 Schematic of feedback control loop. 3
2.1 Feedback control loop and adaptation mechanism 10
2.2 Exemplar patterns used for pattern recognition 13
2.3 Schematic of gain scheduler using fuzzy reasoning 14
2.4 Gain scheduling: A programmed adaptive control system 17
3.1 Representation of the transient period as a process evolves to a new state 28
3.2: Pattern representation of an operating condition 30
3.3 Identification of process state using Euclidean norm 35
3.4 Use of sliding a window for establishing the state of the process 37
3.5 Effect of time smoothening to extract trend change 38
4.1 An illustration of a possible gain surface which is to be represented about a
few nominal conditions 43
4.2 Interpolation using a gain map formed from four nominal condition 45
4.3 Effect of number of nominal conditions used for scheduling the controller
parameter 47
VII
4.4 Schematic of linear interpolation 49
4.5 Some typical fuzzy sets to represent controller parameters 52
4.6 Schematic of fuzzy interpolation 55
5.1 Schematic of the demonstration system 59
5.2 Process gain at different operating temperature 60
5.3 Simulink CSTR model with gain scheduled controller 63
5.4 Gain Scheduled control performance using instantaneous raw sensor values 65
5.5 Gain scheduled control performance using time smoothened sensor values 67
5.6 Gain scheduled control performance using raw sensor values and an online
sliding window 69
5.7 Gain scheduled control performance using time smoothened sensor values
and an online sliding window 70
5.8 Controller performance with interpolation using only the 2 closest neighbors 73
5.9 Controller performance with interpolation using only the 4 closest neighbors 74
5.10 Controller performance with interpolation using only 7 closest neighbors 75
5.11 Overall servo performance using Euclidean norm as a similarity measure 78
5.12 Overall servo performance using ART2 based similarity measure for pattern
recognition 79
5.13 Overall servo performance using center of area (COA) fuzzy interpolation 81
5.14 Overall servo performance of a fixed PI controller. Controller settings
corresponding to operating condition number 3 82
VIII
5.15 Overall servo performance of a fixed PI controller. Controller settings
corresponding to operating condition number 4 83
5.16 Overall servo performance of a fixed PI controller. Controller settings
corresponding to operating condition number 7 84
A1.1 Open Loop behavior of the demonstration system 99
A1.2 Existence of multiple steady states 102
A1.3 Typical controller performance at any nominal operating point 105
IX
Abbreviations
ART
COA
CSTR
EXACT
FOPDT
LPV
LTIS
IAE
ISE
ITAE
ODE
OUR
PI
PID
SISO
Roman Letters
e(t)
f
k
pet)
Ps.
x.J1
Ym(t)
Ysp
di Cp
CP1
F
Fe
Ke
Kf
Km
Kp
M
Si
TT
e
X)1
LIST OF NOTATION
Adaptive Resonance Theory
Center of Area
Continuous Stirred Tank Reactor
Expert Adaptive Controller Tuning
First Order Plus Dead Time
Linear Parameter Varying
Linear Time Invariant System
Integral of Absolute Error
Integral of Square Error
Integral of TimeWeighted Absolute Error
Ordinary Differential Equation
Oxygen Uptake Rate
ProportionalIntegral
ProportionalIntegralDerivative
Single Input Single Output
Controller error
sampling frequency (hrI
)
Reaction rate (hrI
)
Value of controlled variable
Controller bias
Normalized sensor data representing jth reading of the iih sensor
Value of measured variable
Value of setpoint
Euclidean norm of pattern vectors
Heat capacity (Btu! IbmOR)
Heat capacity of coolant (Btul IbmoR)
Reactant flowrate (ft3Ihr)
Coolant flowrate (ft3/hr)
Controller gain (ft3/hr oR)
Gain of final control element
Gain of measuring device
Process gain (OR hr/ft3)
Minkowski metric
Similarity measured by ART2 neural network
Temperature (OR)
Coolant temperature (OR)
Sensor data representing 'th 'h J reading of the i1 sensor
x
Greek Letters
~e(t)
~HRXN
1'1
1'D
1'p
81
P
~
0)
Zero value of ith sensor
Rate of change of error
Exothermic heat of reaction (Btu/lb.mol)
Reset time (hr)
Derivative controller parameter (hr/oR)
Process time constant (hr)
Nominal pattern vector
Vigilance parameter
Membership function
Window length (hr)
Xl
CHAPTER I
Introduction
1.1 Motivation
Control of chemical processes has traditionally been performed using linear feedback
controllers. Feedback control adjusts the manipulated variable in order to force the
process to conform to a desired behavior. Such controllers are typically designed on an
assumption that the process dynamics can be approximated by a linear timeinvariant
system (LTIS). With the advances in linear control theory, designing a good feedback
controller for a LTIS plant is a relatively straightforward exercise. Unfortunately, the
dynamics exhibited by chemical processes are typically nonlinear. For such nonlinear
systems the controllers are often detuned to maintain an adequate stability margin but
they perform poorly when the process drifts away from the design condition. Thus there
is an incentive to investigate alternate methods for design of controllers with widely
varying parameter dependent dynamics.
Gain scheduling, or more rigorously controller parameter scheduling, offers a good
solution to compensate for process nonlinearities (Mellichamp et. aI., 1966b, and Pott,
1984). In this approach controller settings are expressed as a function of one or more
measured process outputs and are calculated online to maintain optimum and stable
performance. This control methodology is termed gain scheduling because initially it
was used to accommodate changes in process gain (Kp) only (Astrom, 1983).
1.2 PID Controllers
The most popular feedback controllers used in the chemical industries are PI
(proportionalintegral) and PID (proportionalintegralderivative). As the name suggests,
they adjust the manipulated variable depending on the magnitude of error (proportional),
the cumulative error integrated over time (integral) and the rate of change of error or
derivative of the error (derivative). Derivative action cannot always be used since it is
sensitive to noise in the error signal. This control structure is shown in Figure 1.1. A
controller weights the proportional, integral and derivative action depending on the
tuning parameters. The general equation of a PID controller can be expressed as:
pet) == K c [ e(t) + 1 Je(t*) dt* + ~D de(t) ] + ps
~I dt
(1.1)
Error (e(t)) is defined in equation 1.1 as the difference between the desired value (ysp(t),
the set point) and measured variable YM(t)
2
e(t) == Ysp(t)  Ym(t) (1.2)
where: pet) is the manipulated variable.
Ps is the controller bias.
Kc is the proportional gain of the controller (adjustable).
"CJ is the integral time constant or reset time (adjustable).
"CD is the derivative time constant (adjustable).
The controller bias (Ps) is the value of the manipulated variable at steady state, or
when there is no net error (e(t)), and, hence, no control action. The variable Ps is also
called the controller bias. For a PI controller "CD is zero and thus insensitive to the rate of
change of error.
d(t) Disturbance
y(t)
Measuring
Device
1_+....L.......lo.'.IFinal Controll_~
Element
Controller Mechanism r
y~ e(t)
I t~
l Y.i.m...l..o.\ool'(t+) I
Figure 1.1: Schematic of a feedback control loop.
3
1.3 Gain Scheduled PID Controllers
Gain scheduling adjusts the controller parameters (Ke, 'tr and 'tD) to compensate for
the changes in process dynamics. The key to successful implementation of gain
scheduling lies in accurate characterization of the process dynamics. Mellichamp (1966)
suggested that dynamic information can be inferred from estimation of the process gain
(Kp ). Bristol (1977) describes a pattern recognition method that characterized process
dynamics from response to step input changes. In this work process dynamics are
inferred using a pattern map of the process. All three methods are discussed in the
following chapter.
While theoretically attractive, the practical implementation of gain scheduling
remains difficult. Work by Shamma and Athans (1990) provides guidelines for selection
of scheduling variables and operating conditions. One of the main drawbacks of gain
scheduling is that the controller parameters are adapted in an openloop fashion. There is
no feedback to compensate for an incorrect schedule since the adaptation is performed in
a feedforward mode (Astrom, 1983). Thus the key issue for successful implementation of
gain scheduling is to be able to accurately characterize the process dynamics, i.e.
recognize the current state of the process and the state to which the process is evolving.
Also, the way in which the scheduling is performed becomes critical because smooth
transition in controller parameters is essential if one expects the properties of local linear
controllers to carryover to the gain scheduled control system. This thesis addresses both
Issues.
4
1.4 Contribution of Thesis
This thesis is built on the initial work done by Anderson (1993) which looked at the
use of similarity based pattern recognition to perform gain scheduling. Anderson showed
that a multisensor patternbased gain scheduling system performs better than a
conventional fixed PI controller. Previous work used a neural network based pattern
recognition approach.
The current work focuses more on how the sensor data can be effectively used to
identify the process state rather than the tool to measure similarity between pattern
vectors. Extracting information regarding the state of the process from information
embedded in the multisensor trend patterns plays a key role in successful design of a
gain scheduling control system. Guidelines to select scheduling variables are also
presented. Thus the focus of this thesis has been to develop a novel method to accurately
characterize process dynamics using the tools developed previously by Anderson (1993).
This study has also investigated various interpolation techniques that can be used to
perform gain scheduling. The emphasis has been to develop a strategy that is effective
even during periods of transient operation. Using a nonisothermal continuous stirred tank
reactor (CSTR) as a demonstration system, a formal methodology to design a gain
scheduled controller has been developed.
5
1.5 Organization of Thesis
This thesis is organized into six chapters. In Chapter II different approaches to design
a gain scheduled control system are presented. Our patternbased approach is also
introduced and various successful applications of gain scheduling are discussed.
In Chapter III an online identification technique to characterize different operating
conditions is presented. A pattern based methodology to effectively utilize multisensor
data is developed. Issues, like selection of scheduling variables and use of a sliding
window for process characterization, are also discussed.
Chapter IV presents different strategies that can be used to perform controller
parameter calculations after the state of the process has been established. The chapter
begins with a discussion on the choice of the number of closest neighbors to be used for
performing gain scheduling, followed by the presentation of three different interpolation
strategies.
Chapter V demonstrates our patternbased gain scheduling approach for a simulated
CSTR. First the dynamics of the nonisothermal reactor are described. Next, a patternbased
gain scheduling control system is designed. Finally, the results are presented
which are used to evaluate performance of our scheduling approach.
Concluding remarks, as well as a discussion on issues for future work, are
documented in Chapter VI.
6
CHAPTER II
Gain Scheduling: An Adaptive Control Strategy
2.1 Overview
This chapter presents an overview of different methods that have been used to
perform controller parameter adaptation. This chapter starts with a discussion on a need
for controller parameter adaptation. The key issue in implementation of a gain scheduled
system is identification of the process dynamics. Three methods, as well as the
differences in the methodology used to characterize the process dynamics, are presented.
Finally, some successful commercial applications of gain scheduling are discussed.
2.2 Need for Controller Parameter Adaptation
According to linear state feedback control theory, the best way of controlling any
process is by measuring all the state variables in order to manipulate the process inputs in
some desired fashion (such as the PID algorithm). This is rarely done in practice because
7
all the state variables cannot be measured using online sensors and a rigorous knowledge
of the existing process dynamics is not known. The most common control methodology
utilizes a single input single output (SISO) structure. This is very popular because of the
relative ease of the control system design. The main disadvantage of such a system is
that it disregards the process dynamics of other measured and unmeasured process
variables.
One alternative is to retain SISO methodology, but incorporate the effect of other
process variables on the process dynamics by adaptively changing the controller
parameter settings (e.g. Kc, ~I and ~D). For example, a chemical reactor is often
controlled by measuring the temperature of the reactants and adjusting the coolant flow
rate. It is well known that the dynamics of a reactor will vary with different feed and
product compositions. Thus, the use of more than one variable to measure the process
dynamics in ,a continuous manner becomes imperative. Information extracted from more
than one process variable can then be used to schedule controller gains for the variations
in the process dynamics. Such a control strategy is termed adaptive since the controller
adapts itself to maintain satisfactory control for a nonlinear process.
Different approaches have been investigated to adapt PI controllers to respond to
variations in process dynamics. Different approaches to gain scheduling are discussed in
this Chapter with the aim at implementing this controller adaptation technique in real
time.
8
2.3 Gain Scheduling Based on Process Gain
Mellichamp (1966b) presented an adaptive control system designed to maintain
good control characteristics for processes showing wide variations in process gain. This
method takes advantage of the fact that a good control can be maintained in spite of time
varying process by holding the overall control loop gain constant. The overall openloop
gain for a PID control structure is the product of the gains of all the elements in the
feedback control loop. Openloop gain is expressed in equation 2.1 as:
(2.1)
where: Kp is the process gain.
Km is the gain of the measurement device. (For example thermocouple)
Kc is the controller proportional gain.
Kf is gain of the final control element. (For example flow control valve)
If the objective of controller adaptation is to keep the openloop gain constant, then a
simple gain scheduling can be expressed as shown in equation 2.2. This is further
illustrated in Figure 2.1.
Kc = Constant
KpKmKf
9
(2.2)
The controller gain Kc is adapted to compensate for the variations in the process gain,
Kp . This method can also compensate for the variations in the gains of the other control
elements such as a control valve or other measuring device. The gain associated with the
final control element (Kf) and measuring device (Km) can be identified in a
straightforward manner since they are inherent characteristic of these mechanical devices.
The key step to successfully apply this method is to be able to accurately characterize the
gain of a time varying nonlinear process (Kp). In an earlier paper, Mellichamp (1966a)
presented a method for continuously estimating the gain of a process by applying
sinusoidal perturbations to the process.
Adaptation Mechanisms
~LII.'../S:/\ e t
S~
Ym(t)
Kc=
Controller
(Constant)
Gm=K m
Valve
Kv
Gv= 
1:vS + 1
OQIine Kp
Identifier
Process
Kp
Gp = 
1:pS + 1
y(t)
Auxiliary P cess
Measure ents
Measuring Device
Figure 2.1: Feedback control loop and adaptation mechanism.
Such a control strategy is limited only to those systems whose dynamics can be
approximated by afirst order plus dead time (FOPDT) model. Many processes, such as
10
exothermic reactions, have dynamics which cannot be approximated as FOPDT and are
not amenable to Mellichamp's method. For the demonstration system used in this study
the Kp varies considerably as the process is open loop unstable. This is explained later
when the dynamics of the demonstration system are discussed. Keeping the open loop
gain (Koverau) constant for such a system would lead to unstable controller performance.
Moreover such a scheme neglects variation in other controller settings such as the reset
time, 'tIe
Mellichamp's gain scheduling strategy assumes that the process dynamics can be
characterized by estimating the value of the process gain (Kp). This simple gain
scheduling approach may result in poor control unless the process dynamics are also
considered (Seborg et aI., 1986). For a process which has a long timedelay, this
approach to controller parameter adaptation may result in a performance worse than
conventional PID control unless some kind of time delay compensation is employed
(Wong and Seborg, 1985).
2.4 Gain Scheduling Based on Error Diagnostics
Another way of approaching gain scheduling is as a selftuning algorithm which
adjusts the controller parameters based on the transient error pattern. Bristol (1977) was
first to propose an algorithm to adapt the controller parameters based on a pattern
recognition approach. Typically, with this approach the closed loop is perturbed and the
11
resulting pattern of the response is observed. This pattern is compared with one that is
personally desired. Key characteristics, such as damping and overshoot, are extracted
from the recorded response to characterize the dynamics of the process. Such a selftuning
PID controller automatically adjusts the controller setting to result in a desired
damping and overshoot of the response pattern.
Recently, much work has been done to characterize various process states and
associate them with different pattern characteristics (Cao and McAvoy, 1990; Megan and
Cooper, 1992). In this manner a gain scheduling algorithm can be set up which aims at
driving the error pattern to a desired form. Figure 2.2 shows some typical transient
pattern which can be analyzed to interpret the state of the process. The best response to a
unit step in manipulated variable is represented by the center pattern in Figure 2.2. The
deviation in characteristics of the error pattern recorded from desired pattern can be used
to update the parameters of the controller. A table of pattern characteristics
corresponding to different states of the process (as shown in Figure 2.2) is used. Once the
transient pattern is characterized in reference to the table of different nominal patterns,
the process gain (Kp) and the process time constant ('tp) can be established. A rule along
the lines of ZeiglerNichols (1942) method can then be used to correlate the most
appropriate controller settings.
The aim of a gain scheduler is thus to recognize characteristics of an error pattern as
the process is perturbed by a unit step change in the manipulated variable. Controller
parameter are adapted to drive the error response to the desired form (Cooper and
Lalonde, 1990). This method, like any tuning exercise, is an iterative process. The
12
characteristics of error response are first extracted and then controller adaptation is
performed. This is repeated till the final error characteristic is similar to the desired error
pattern. Highly advanced pattern recognition algorithms such as the ART2A neural
network, have been used for this purpose (Megan and Cooper, 1992, 1994, 1995).
O.SK•p
~~...I r· ~__....
2.0Kp'*
1 .0.,..*p 2.01'p
Figure 2.2: Exemplar patterns used for pattern recognition. (Megan and Cooper, 1992)
Controller tuning based on error diagnostics has been successfully commercialized
(Kraus and Myron, 1984) and offers an excellent way to adapt controller settings for a
time varying system. This commercially available self tuning controller has been dubbed
as "EXACT" which stands for Expert Adaptive Controller Tuning by the Foxboro
13
company. The EXACT controller utilizes expertsystem techniques and artificial
intelligence to tune PID coefficients with no need for process modeling. This has been
successfully used to tune PID coefficients for chemical processes showing large deadtime
and nonlinearities.
Recently, fuzzy reasoning has also been used for this application in an effort to
implement gain scheduling in real time. Zhao et aI. (1993) arrived at a gain scheduling
methodology which utilizes fuzzy rules and reasoning to determine the controller
parameters based on the error signal and its first derivative (rate of change of error). This
approach is schematically presented in Figure 2.3. The strategy of adapting the controller
parameter with reference to error (e(t)) and rate of change of error (ile(t)) is represented in
a form of rulebase. A typical fuzzy rule could be:
"If the error is small and the rate ofchange oferror is large, then reduce the controller
gain (Ke) and increase the reset time (~I)."
Input
+
Fuzzy Rules
and Reasoning
Process Output
Figure 2.3 : Schematic of a gain scheduler using fuzzy reasoning. (Zhao et aI., 1993)
14
For this strategy the gain scheduling is performed only when a step change in setpoint
is made. The scheme is ineffective when the process is in steady state or while the
process is evolving to a new steady state.
Controller parameter adaptation based on error diagnostics cannot be used to schedule
PI controllers in real time. The main reason being that the steady state operation of the
process needs to be perturbed in order to obtain the error diagnostics. Nevertheless, this
technique is an excellent alternative to trial and error tuning (or "hand" tuning) routinely
performed by the process expert to update the controller parameters. In a way this
methodology has automated the "art" of controller tuning by use of artificial intelligence.
2.5 Gain Scheduling Using PatternBased Tuning Map
Perturbations in input variables to record error diagnostics are often not possible since
it disrupts the smooth operation of a plant. Characterizing process dynamics based on the
process gain (Kp) is also not effective for processes which cannot be modeled as an
FOPTD model. Thus there is a big incentive to develop a method to accurately
characterize process dynamics using some other approach. Our approach uses a process
state map constructed from windowed slices of multisensor plant trend data. This
patternbased process state map is used to characterize process dynamics at different
operating conditions. Thus a method to uniquely associate gain, integral and/or
derivative time controller settings with different states of the process is developed.
15
In this approach controller parameter adaptation is performed by treating a nonlinear
plant as a set of localized linear processes (Rugh, 1991). The number of operating
conditions at which the process is linearized to approximate the plant dynamics depends
on the extent of process nonlinearity and are normally chosen to cover the expected range
of plant dynamics. The dynamics of the process are characterized at different operating
conditions and appropriate controller settings are determined for each of these operating
conditions. An automated "tuner" such as Foxboro's EXACT controller may be used to
determine a good set of controller settings at each of the operating conditions. A table of
controller settings at each of the operating conditions is referred to as a "gain map" or
"gain table" in this study. This "gain map" relates the process operating conditions to the
desired controller settings. During plant operation the process state is identified online
and as the process moves from one operating condition to another, the controller
parameters are automatically changed. The adaptation strategy is schematically presented
in Figure 2.4.
Our approach uses a novel patternbased technique for scheduling controller
parameters. More than one process variable is considered over a finite period of time
(window) to accurately characterize the process state. The features extracted from these
multisensor pattern trends are analyzed and the process state is identified by measuring
the similarity of the pattern vector corresponding to the process state with each of the
nominal operating points.
16
Figure 2.4: Gain scheduling: A programmed adaptive control system.
Our patternbased gain scheduling approach involves three basic steps:
• Selection of nominal conditions representing steady operating process conditions.
• Characterization or identification of the process operating state from multisensor
pattern trends.
• Scheduling the controller parameters.
Though this work has looked at implementing gain scheduling for a PI controller, the
strategy is not limited to PID type controllers. Rather, the same strategy can be used for
any linear controller whose parameters can be scheduled to compensate for the
nonlinearities in the process.
17
2.6 Applications of Gain Scheduling
A number of industrial applications of gain scheduling implementations can be
identified. The popularity of such a design technique lies in the fact that it is an
adaptation strategy to the ubiquitous PID algorithm which has time proven utility and
applicability to many of the industrial feedback control problems.
2.6.1 Aircraft control
Early development of gain scheduling included applications in high performance
aircraft for design of an auto pilot (Seborg et aI., 1986). It was found that monitoring the
mach number and dynamic pressure allowed a suitable schedule to be developed
(Astrom, 1987). Astrom's work looked at the effect of state variables other than the
controlled variables (altitude and direction) on the flight dynamics. Since then gain
scheduling has become a primary method to compensate for variations in flight control
problems. It is used extensively in the design of an auto pilot system (Stien, 1980). Gain
scheduling was initially limited to the aircraft industry. With the arrival of computer
control systems, gain scheduling has become easier to implement (Astrom and
Wittenmark, 1989).
2.6.2 Ship Auto Control
This involves use of gain scheduling to compensate for parameter changes due to
environmental changes. These changes include wind velocity, water currents and ship
movements such as sway and yaw. A table look up approach has been used to do gain
scheduling (Kallstrom et aI., 1979). The main goal of this system was to reduce drag in
the ship movement. The tuning values were changed as a continuous function of a ship's
speed. Basically the controller parameters were read directly from a table when the
change in wind velocity and water currents were determined.
18
2.6.3 Process Control
The "EXACT" controller is a commercially available adaptive controller that uses
pattern recognition to achieve desired closed loop characteristics. Although the complete
algorithm is proprietary, its basic features have been published (Bristol and Kraus, 1984).
This self tuning PID controller provides a microprocessor based tool for use at the front
line of process control (Kraus and Myron 1984). The Foxboro company has dubbed this
new controller "EXACT" for Expert Adaptive Controller Tuning.
One notable case where online adaptive control has been widely used is to control
pH of a reacting system. The wide variations in titration curves with changes in buffering
makes pH control ideal for online adaptive control methods. Gain scheduling has been
successfully used in controlling pH of a reacting system (Astrom, 1987). Gain
scheduling has been performed by storing the parameters in a chart form. The feedback
controller can be effectively used in the entire range of operating conditions.
Cardello and San (1988) have looked at gain scheduling for batch bioreactors. Batch
processes are extremely nonlinear and their dynamics show wide variations with time.
They found that Oxygen Uptake Rate (OUR) can be used as a scheduling variable. A
table lookup is used to select the gain depending on the OUR measurement. In
comparison to a fixed PID controller and feedforward  feedback controller, the integral
of the square of the error (ISE) was 20% less for the gain scheduled method. Cardello
and San found that scheduling was an effective method for controlling dissolved oxygen
levels in a batch fermentor. These conclusions are important as the fermentor had large
variations in process load.
Leuba et al. (1992) have extended the concept of gain scheduling and fuzzy logic to
set up an adaptive PI controller. The controller controls the fluid level in Utube steam
generators. The fuzzy logic circuit analyzes the disturbance and based on that
information, decides what the controller effort should be. The gain scheduling aspect of
the system changes controller parameters based on the temperature of the feed water. The
19
gain is adjusted as a linear function of the feed water temperature. This led to a much
smoother and stable performance.
2.7 Need for Future Research
Gain scheduling has emerged as a very powerful and useful technique to reduce the
effect of parameter variations. It is in fact the predominant method to handle parameter
variations in flight control systems. Gain scheduling seems to have gained wide
acceptance for processes whose operating state can be identified by some auxiliary
variables. Auxiliary variables are measurable process variables other than the controlled
variable. From a review of the process control applications where some kind of gain
scheduling has been applied it is clear that controller parameters are scheduled using a
table lookup approach. Once a new operating condition is identified by a process expert,
the controllers are retuned. In applications, such as fluid level control in Utube steam
generators (see subsection 2.6.3), an instantaneous value of a single variable is used to
characterize the state of the process. These techniques have not been widely accepted in
the process industries because it is often difficult to characterize and differentiate one
operating condition from another. There is thus a need to develop a methodology to
apply this technique in an online fashion. Microprocessor based controllers can be easily
programmed to store a table of controller settings and even calculate them online. For
process control systems which have large time constants and time delay there needs to be
a proper method to be able to identify the process condition by looking at the plant
variables or the sensor readings. The aim of this study thus focuses on development of a
technique which uses patternbased information to characterize process dynamics from
online sensor data.
20
CHAPTER III
PatternBased Process Characterization
3.1 Overview
An online identification technique to characterize different operating conditions is
presented in this chapter. The characterization of process dynamics is the key issue for
designing a gain scheduled controller. The first step in setting up a gain scheduled
controller is to approximate the nonlinear plant by a set of linearized models. The
method to represent each of the nominal conditions at which the plant dynamics are
linearized is discussed in this chapter. Once a set of operating conditions are identified
the process dynamics at these conditions are mapped to form a patternbased tuning map.
The key step in this patternbased approach is to map the process dynamics at different
operating conditions. Which scheduling variables are chosen to construct the patternbased
tuning map is also an important issue. The use of such a map to characterize
process dynamics in an online fashion is discussed next.
21
3.2 Formation of PatternBased State Map
A gain scheduling algorithm needs a way to map the controller parameters to
variations in process dynamics. A state map relates the process dynamic characteristics at
different operating conditions to a set of well tuned parameters which will result in
satisfactory and stable control action at those process conditions. A state map of all
possible operating conditions corresponds to a "gain surface". A gain surface can be
described as a plot of controller parameters (for example controller gain, Kc ) at all
possible process operating conditions. The topology of this "surface" is the characteristic
of the process under consideration. We do not propose to associate controller parameters
at all possible operating conditions to arrive at this "gain surface". Rather, the gain
surface is approximated by interpolating the controller parameters when the process
operates between the nominal points.
Formation of state maps typically involves three different steps:
• Selection of nominal conditions.
• Selection of scheduling variables to identify these nominal conditions.
• Represent the nominal conditions by the multisensor pattern trends typically seen
at those conditions
3.2.1 Selection ofNominal Conditions
Nominal conditions are operating conditions at which process dynamics are
represented by a linear model and at which satisfactory controller parameters are
22
available. The choice of nominal operating conditions is often based on the historical
data of the plant. The conditions at which the plant normally operates are the first choice
as nominal operating conditions. Generally, a few more operating conditions are chosen
so that the entire nonlinear operating region can be reasonably approximated.
A process state can often be associated with sensor trends of certain measured
variables. Most chemical processes can be modeled by mass momentum and energy
balances. Mass, energy and momentum in tum can be characterized by variables such as
density, concentration, temperature and flow rate. These characterizing variables are
called state variables and their values define the state of the processing system
(Stephanopoulos, 1990). According to linear control theory, the process dynamics are
directly related to the values of"state variables" rather than the values of measured
variables. For the demonstration system the controller settings are also distributed in the
same manner as the operating conditions in the state space and that there can be one to
one mapping between the two. That is, no two operating conditions have similar
controller parameters associated with them. Thus the nominal conditions are chosen such
that they cover the range of dynamics of the plant and represent the conditions at which
the plant normally operates.
3.2.2 Selection ofScheduling Variables
The next implementation issue is to identify process variables that can then be used to
characterize the process state. Often the state of the process is inferred from the
instantaneous value of the controlled variable. For example the dynamics of a reactor is
23
often inferred from the temperature at which the reaction is occurring. Since the process
dynamics are a function of more than just the controlled variable there is a need to
identify other process variables which can be considered for interpreting the process
dynamics. The most common method for selecting schedulings variables is to look at the
physics of the process and choose variables which can help characterize it. The designer
is often limited to choose the scheduling variables from the measured process variables.
For example often it is not possible to measure the production rate since the product
concentration cannot be analyzed in an online fashion.
Shamma and Athans (1992) in their work on developing a theoretical analysis of gain
scheduled systems present two guidelines for selection of scheduling variables. They are:
1. The scheduling variables should capture plant nonlinearity.
2. The scheduling variables should move slowly.
The first guideline is only a reminder that the plant models are only linearized
approximations to the nonlinear plant. Similarly the second guideline that the scheduling
variables should vary slowly is a reminder that the design model explicitly assumes a
fixed operating condition. Previously these heuristic rules of thumb were verified by
simulations though the work by Shamma (1988) and shows that these guidelines have a
rigorous mathematical justification.
Since the identification of a nominal operating condition is performed by looking at
the patterns of the scheduling variables, the sets of process variables representing a
particular operating condition should be as distinct as possible from "neighboring"
operating conditions. Often the number of process variables which can be measured by
24
online sensors are limited. For example concentration cannot be accurately measured
online. Thus there may be a need to identify certain "virtual variables" which can be
computed online to help establish the current process state. A virtual variable is a
combination of one or more measured variables whose trend provides valuable
information regarding the process dynamics. An example of a virtual variable for a
reactor could be the "concentration" of the reactant inferred from reactor temperature and
a reaction kinetic model. Often it is possible to identify virtual variables that not only
capture information regarding plant dynamics, but also move "slowly" during periods
when the process is under transition from one operating state to another. Since at times it
is not possible to identify slow varying real variables which can be used to uniquely
associate different operating conditions, the use of virtual variables becomes important.
One advantage of using patterns of these scheduling variables is to make process
characterization during periods of transition much more slow and smooth. During
periods of transition some variables fluctuate a lot before they settle down to the value
corresponding to the new steady state. By using patterns of process data the effect of
such fluctuations during period of transition can be reduced. The guidelines to select
scheduling variables to form a pattern map are as follows:
1. Patterns of scheduling variables should capture plants nonlinearites
2. Patterns of scheduling variables should vary slowly as the process moves from one
operating condition to another.
25
3. The pattern representation (in a vector form) of the scheduling variables
characterizing an operating condition should be as distinct from each other as
possible.
3.2.3 Representation ofNominal Conditions Using MultiSensor Pattern Trends.
The final implementation issue in formation of a state map is to arrive at a way to
represent the "nominal conditions". It is generally not possible to accurately characterize
the process dynamics by just measuring the instantaneous value of scheduling variables.
An instantaneous view of the process gives no information regarding the process state
during periods of transition. In another words the process state cannot be accurately
characterized by instantaneous views.
The use of an instantaneous view of the process variables is prone to errors because of
disturbances and transient periods. The fact that the instantaneous view of the process
can result in wrong process characterization will be demonstrated using an illustrative
example. The process state of a reactor is often inferred from the reactant and coolant
temperatures. Gain scheduling is done based on the steady state at which the process is
operating. When the process is in transition and evolving from one steady state to
another an instantaneous view can result in a wrong process characterization. If only the
instantaneous value of a single variable or more than one variable is used to establish the
state of the process, then there is always a danger that the process may "appear" to go
through other steady state operating conditions during transition periods. This is shown
in Figure 3.1. If only an instantaneous value of temperature is used then the process
26
appears to be operating at steady state 3 (883) while the process is actually in transition
(see Figure 3.1a). If two variables (reactant and coolant temperature) are used, even then
the instantaneous view of the process may result in a wrong process characterization (see
Figure 3.1 b). A view of the time history of the process or the pattern trend of temperature
data (see Figure 3.1a) reveals that the process actually went through a period of transition
before evolving to steady state 2 (882). Thus if the process state is viewed from
windowed slices of sensor data then it is possible to accurately characterize the state of
the process. The process dynamics at any operating condition would be very different
depending on whether the process is steady or is in transition at that condition.
Furthermore if the objective of the online gain scheduler is to track the process and
simultaneously schedule parameters depending on the operating conditions, then the idea
of operating condition being represented by a single dimensional vector of process
variables can result in a wrong schedule. There is a need to develop a method that
uniquely identifies each of the operating conditions even during periods of transition.
Traditionally a gain map relating the process operating conditions to fine tuned
controller parameters were made using instantaneous views of single scheduling variable.
We propose to replace the traditional gain map by a more robust, patternbased gain map.
Accurate characterization of a process state requires consideration of more than one
variable for a finite period of time (Anderson and Whiteley, 1993a). In this new approach
the single value scheduling variable is substituted by a multisensor pattern to
characterize the process more accurately and thus significantly improve the gain
scheduling during periods of transient operation. Use of multisensor pattern trends to
27
. .~.. Steady State NO.1. 1 Transition Period 1 Steady State NO.2.
1   1    SS3       I S82
   SS1
I Time
A. Repre senting the transition from one steady state to another using
a single variable.
..... c
co
(5
o
()
~o
Q)
'
..:.::.J. co
'
Q) a.
E
Q)
t
Temperature Of Reactor
B. Representing the transition in reference to two" state variables".
Phase portrait analysis.
Pattern vector representing
steady state SS2
Pattern vector representing
steady state SS1
c. Representing the transition by Ndimensional pattern vectors.
Patternbased representation and analysis.
Figure 3.1 : Representation of the transient period as the process evolves to a new steady
state.
28
identify process dynamics can result in considerable improvement in process
characterization during periods of transition.
3.2.4 Arriving at a PatternBased Representation
The last step in the formation of a patternbased state map is to represent the multisensor
patterns by a vector in a multidimensional pattern space. First the time period
over which the process variable needs to be considered must be specified. This time
period is called the "window length". Window length is thus the time for which the
process needs to be observed before its state can be characterized. The length of the
window is a characteristic of the process for which the controller is being designed.
Some processes show more "sluggish" response to changes in set point than others. How
a window of process data can be transformed to be represented by a point in multidimensional
pattern space is shown in Figure 3.2.
The time constant ('tp) of a process is a measure of the time necessary for the process
to adjust to a change in its input. For a firstorder process, the process evolves to a new
state in about five times the process time constant ('tp) (Stephanopoulos, 1990). The same
relation can be used to arrive at the window length as given in equation 3.1.
Window Length (ro) = 5 x 'tp
Where'tp is the process time constant.
29
(3.1)
I~in~ow_ I
I Len th ~
Temp (T 1)
Coolant
Temp (Te)
Flowrate (F e)
(n1) n
Vector transformation:
P1. {T1 1 ·11 n T 1 n 1 n T e t Fe ~ }
Representation of the pattern in
a multidimensional space.
Figure 3.2: Pattern representation of an operating condition.
In Figure 3.2 it is shown that the a window of process data can be represented as a
vector. To arrive at this vector representation the process data is first normalized.
Normalization is done so that the variable can be expressed as a value between 0 and 1.
This is important for doing any similarity analysis on the pattern vectors since they need
to be expressed on a uniform scale. Equation 3.2 is used for normalization.
30
where
x jix
x 0
spa n
(3.2)
Xi
j is the normalized value corresponding to the ith sample reading of the jth sensor.
Xi
j is the actual sampled value of the ith sample reading of the jth sensor.
Xois the zero of the measurement device.
span is the range of the measurement device.
The normalized pattern vector can be expressed in equation 3.3 as:
I I 1 2 2 8j== (Xl 'X2 , Xd , Xl , .. · · Xd , ..
a a )T) ...., Xl , · . ,Xd
where:
Xi
j is the sample reading ofjth sensor at the ith sensor reading.
(3.3)
d is the number of sampled data points (d = co ) where co is the window length and f
f
is the sampling frequency.)
a: Number of different sensor patterns.
The total number of samples making up a single sensor pattern is d. Since there are a
sensors the dimension of a pattern is "ued". For example, a multisensor pattern
31
composed of 10 discrete samples of three different variables can be represented in a 30
dimension vector space.
In this way all the nominal operating conditions can be represented in pattern space.
Controller parameters are then associated with each of the nominal pattern vectors. This
representation is thus termed as a ''patternbased gain map". Gain map is actually a
misnomer since controller parameters other than the controller gain (Ke) may be
associated with each of the nominal pattern vectors. The following section describes an
online methodology to identify the process state based on the location of the process on
the gain map.
3.3 OnLine Process Characterization
This section describes a method to identify the process state from a window of multisensor
data. This work is based on earlier work done to develop a pattern recognition
methodology to characterize process sensor data (Anderson, 1993).
The state of the process is characterized by measuring the similarity between the
pattern vectors corresponding to the prototypes and the online sensor data. Prototypes
are the pattern vector representation of the nominal operating conditions. During online
operation the sensor readings are first normalized and represented in a pattern vector form
as discussed in the previous section.
32
In this section, the similarity index to compare two pattern vectors will be presented.
The similarity index will then be used to characterize the process state. Various
proximity indices can be used to compare two multidimensional vectors. The most
common index for such patterns is the Minkowski metric:
Minkowski metric (M) is defined by equation 3.5:
where r ~ 1 (3.5)
where y is the vector representing the online sampled data.
8j is the vector representing one of the nominal operating condition in the
patternbased gain map. 81 is also referred as "prototype".
Euclidean Distance (d )is defined in equation 3.6 by:
d (y,8j) = [L I8'/  y/'1 2] 1/2 (3.6)
Euclidean distance or Euclidean norm is a special case of Minkowski metric with n equal
to 2. Euclidean norm, d(y,8j), is a measure of dissimilarity. The smaller the value of
d(y,8j), the closer or more similar the pattern vectors. Euclidean distance is the most
common of the Minkowski metrics. The familiar geometric notions of invariance to
translations and rotations of the pattern space are valid only for Euclidean distance.
Euclidean distance has been widely used in engineering work. Figure 3.3 shows how a
two dimensional vector can be compared with a prototype.
33
The distance between two pattern vectors is measured as a Euclidean norm. As
shown in Figure 3.3, there can be a locus of points having the same Euclidean norm.
Accurate process characterization in those cases requires measurement of the angle also.
Angle is a measure of how the vector points are oriented in the pattern space with respect
to each other. For the purpose of gain scheduling, the process needs to be characterized
in respect to more than one "nominal" operating condition. Euclidean norm as a
similarity measure is adequate for gain scheduling purposes. This can be explained
using Figure 3.3. For a two dimensional vector only two prototypes can be used to infer
the process state in the pattern space. The locus of equidistant points intersect at two
points. Using three prototypes one can uniquely associate the pattern vector in the pattern
space. As shown, there can be only one unique combination of dl :d2:d3 for the pattern
vector shown. For a n dimensional space the number of prototypes needed to uniquely
characterize any pattern vector is n+1. Thus even a distance measure (such as Euclidean
norm) gives a good estimate of "where" the input pattern vector is located. The angle is
not one of the more importance parameter since the similarity is reflected only in
associating how much weightage the controller parameter of the nominal condition will
have on the final controller value.
To determine the location of a pattern with respect to the prototypes, a gravitational
pull analogy can be used. The closer the pattern vector is to any of the prototypes, the
more "pull" or influence it should exert on the controller parameters. The final controller
settings can be determined by combining the "pulls" or proximity indices of two or more.
34
How these "pulls" are analytically combined is a separate issue which will be discussed
in detail in the following chapter.
PatterFrv8"Ctor "
i\; L~cus of patterns having same
I ',d1 Euledian distance form prototype X
\ \ ~ proto~~e
.......
,.,.,  .
I
"
" \
\
\
/
/
t I...~us of patterns having same
// EU~dian distance d2 from
I ~rotoype X2
(
I x~PrototYpe)
\ I dpf _)C~Prototype)
Pattern ~ =  Id3 
Vector ft, ><
I ~~ , tt1  \ Locus of patterns havihg s \ ,,\1(Protojype) EUled~~gt~~S~~n>9,d3fro
Locus of patter~s havin~ same/ /'
Euledian distan~e.d1 from~
protoype X~  ' ' ..... _ _ _ "
......
Figure 3.3: Identification of the process state using Euclidean norm.
35
3.3.1 Use ofSliding Window for OnLine Process Characterization
A sliding window is used to extract the most recent pattern of sensor data from the
process. This is presented schematically in Figure 3.4. The sensor data is normalized and
represented in a column vector form as discussed earlier. If the pattern is identical to one
of the prototypes, then the corresponding values of the controller parameters are used. If
the pattern vector falls between prototypes, then the process is in transition and evolving
to a new steady state. In such cases the controller parameters to be scheduled must be
interpolated.
3.3.2 Use ofTime Smoothened Sensor Patterns
The gain map is formed from a number of nominal operating points. These nominal
points typically correspond to steady state operations. In short, this method uses steady
state process trend information as a basis for interpolation during transient periods. We
propose to use time smoothening to extract the fundamental trend of a process variable
during periods of transition. The value of a sensor is calculated based on an arithmetic
mean of the previous sensor values. This can be considered as a compressed pattern
information. Effect of time smoothening on a wildly fluctuating sensor pattern to extract
its trend is shown in Figure 3.5. This is an illustrative example. The effect of time
smoothening is also shown for the demonstration system in Chapter V.
The strength of our proposed technique is the ability to handle transient conditions.
This method has been developed to match the performance of a skilled operator in
recognizing the process state. An incorrect interpretation of a process trend data can give
36
W
"l
2
1
5:30 6 pm 6:30 7 pm 7:30
Process sensor data
8:30
1
Controller
parameters
corresponding
to nominal cond ition 8
Feedback control loop
Figure 3.4: Use ofa sliding window for establishing the state of the process.
rise to wrong parameters being scheduled and may result in an unstable response which
seriously jeopardizes the safe operation of the plant.
~
I
_ TRANSITION PERIOD .1
r 1\ I
I \ ,.. " T;m.£.er.:!u~    1 
'J '\. / I
Coolant Temperatu e
Raw Sensor Pattern
Temperature
CoolaQt Termperat Ii
Time Averaged Sensor Pattern
*
Figure 3.5: Effect of time smoothening to extract trend change
38
3.4 Concluding Remarks
This chapter describes our patternbased method to characterize the state of a process.
The process dynamics are captured by windowed slices of multisensor trend data. A
sliding window is used to extract sensor data which is represented as a pattern vector.
The current state is compared to any nominal state by measuring the similarity between
pattern vectors. This approach provides an alternative to accurately characterize the state
of the process even during periods of transition. Due to normal fluctuations in process
variables during transition periods, it is not possible to measure process gain (Kp) or look
at error diagnostics to decipher the process state. Thus the methods relating controller
parameters to process gain or characteristics of error pattern (described in Chapter II) fail
during transition periods.
The objective of a gain scheduler is to determine the process state and then schedule
the controller parameters to maintain stable and satisfactory control action. The
information regarding the distance between the current process state and its neighboring
nominal conditions needs to be quantified in the form of a scheduled controller
parameter. An interpolation technique which can perform this task is discussed in the
next chapter.
39
CHAPTER IV
Controller Parameter Calculation
4.1 Overview
The final objective of a gain scheduler is to calculate the controller parameters as the
process moves from one operating state to another. Different strategies to perform
controller parameter calculations after the state of the process has been established is
discussed in this chapter. The best way to calculate controller parameters is to
approximate the gain surface by some continuous function of the process variables or
patterns of process variables. Gain surface, as defined in Chapter 3, is a plot of controller
parameters for all possible operating conditions. Arriving at a gain surface would require
a good analytical model of the process and the controller parameter values at all possible
operating conditions. One reason this is not possible is due to the presence of process
uncertainties. Moreover it is not practical to establish controller parameters at all possible
operating conditions. Thus there is a need to schedule the parameters based on some
approximation of this gain surface.
40
4.2 Scheduling About a Few Nominal Points
We propose to do the gain scheduling about a few nominal operating points. This is
based on the premise that some sort of interpolation between these operating conditions
can be used to approximate the "gain surface." The interpolation strategies discussed in
this chapter use the patternbased state map, discussed in the previous chapter, to
characterize the process dynamics. The process characterization is interpreted from the
similarity measure of the current pattern vectors when compared to the nominal
conditions. The scheduling or interpolation needed to establish the controller parameters
is described in the following sections.
Our gain scheduling approach employs a set of linear stable controllers designed
for different operating conditions. Our gain scheduled controller is thus a linear
parameter varying (LPV) system. Earlier work done by Shamma (1993) shows that for
such LPV systems a smooth transition in parameters is essential if one expects the
properties of the linear controllers to carry over to the gain scheduled control system. No
guarantee on overall stability of the system can be made (Shamma, 1990) while the
process is in transition. Based on Shamma's theoretical analysis, one heuristic to govern
interpolation has emerged. This heuristic states that even when rapid variations in plant
parameters are present the scheduling should be performed slowly. Moreover gain
scheduling is performed in a feedforward manner and there is no feedback for a wrong
schedule. Thus, there is a need to develop a good strategy to interpret information
regarding the state of the process to schedule the controller parameters.
41
4.3 Gain Surface Approximation
Gain surface is a plot of the controller parameters for all possible operating
conditions. Gain scheduling is a relatively straightforward exercise if a function to map
operating conditions onto controller parameters is available. In such a case once the
process state is determined the controller parameters could be directly be read off the gain
surface plot. This is practically impossible since controller parameters at all possible
operating conditions are never known.
We are addressing this problem by approximating the gain surface about a few
nominal conditions. In Figure 4.1 a hypothetical plot of a gain surface is presented. This
plot has been generated using a continuous mathematical function and should not be
confused with the gain surface of the demonstration system used in this study. Moreover
this is a very simplified representation since a gain surface is unlikely to be in a three
dimensional space. Also shown are a few operating conditions at which the controller
parameters are empirically known. The aim of an interpolation strategy is to approximate
the gain surface from the controller values at a few nominal conditions.
Once the process has been characterized at a few nominal operating conditions, a gain
map can be constructed. This gain map relates the patterns of process variables, typically
seen at these operating conditions, to the controller settings that needs to be used at these
conditions. Such a representation of nominal conditions is also shown in Figure 4.1.
42
,._J'  ,,_
J 
~
_J~,
~
.,. ..........
.........
a : An illustration showing a possible gain surface and location of nominal conditions used to
approximate it.
~ .a ~ ..................
Q) a.
E
N
~
ci "E
c: ctI
(5
:Qc) 0u
ctI Q)
.~ £
> Qj eenn "C..
Q) E ...
00
ctI c: w><
Process variable no. 1
Example: the reactor temperature.
b. Contours of conditions having same controller settings.
Figure 4.1: An illustration of a possible gain surface which is to be approximated about a
few nominal conditions.
43
How close the gain surface is approximated depends on the number of nominal
conditions used to approximate it. The curvatures in a gain surface occurs because of
nonlinearities in the process. The extent of the nonlinearity governs the number of
nominal conditions required to approximate the gain surface. For a linear process, the
gain surface can be represented by a flat plane of constant controller parameters for all
operating conditions. The next implementation issue is the choice of the number of
neighbors or the prototypes to be uses for interpolation.
4.4 Choice of Number of Closest Neighbors
Each of the nominal conditions is associated with a set of controller parameters (Ke,
'tJ, 'tn). When the process state is the same as one of the nominal conditions, then the
controller settings corresponding to that operating condition are used. Interpolation is
required when a process state is different from any of the nominal conditions. In such a
case, the number of nominal conditions used to establish the scheduled controller setting
plays is a critical issue.
The effect of different numbers of prototypes used for interpolation is illustrated in
the Figure 4.2. Imagine a gain surface approximated by four nominal operating points. If
only the closest neighbor is used to establish the controller parameter, then the gain
surface can be represented as shown in Figure 4.2b. Each shaded region corresponds to
the settings associated with the nominal condition that is enclosed in that region. Thus
44
the gain surface can be viewed as made of four flat planes. Switching between these
settings will occur whenever the process trajectory goes through more than one region.
This can lead to very rapid variations in controller settings during periods of transition.
1 KC=40 •
A trag
chang
2 conditio • KC=50
3 KC=55 •
Current state
4•KC=45
a Again map showing four nominal
conditions.
b The gain map can be approximated
as 4 different sectors if only one of the
closest neighbor is used interpolate
Figure 4.2: Interpolation using a gain map formed using four nominal conditions.
The effect of different numbers of closest neighbors used for interpolation can be
shown using a similar example. Figure 4.3 illustrates the characteristic of a gain
trajectory with different number of operating points used. Gain trajectory is the record of
changes in controller coefficients with changes in process conditions. When two or more
closest neighbors are used to arrive at the final controller settings, then the final value is a
45
KC=55
3 •
KC=45 • 4
Gain map showing a typical trajectory as a process close to nominal
operating condition 4 changes and evolves as one similar to nominal
condition 1.
57
Using two
Using only one closest
neighbor
CD
nearest neighbor
n:::;J 52 ~
>L..
C..D.
CD
E
n:s 47 L n:s
Q.
L
.!!!
(5 Using all four .L... c 42 nearest neighbor 0u
37
2 3 4 5 6 7 8 9
Sample Number
Figure 4.3: Effect of number of nominal conditions used for scheduling the controller
parameter.
46
weighted average of the controller parameters of the nearest nominal conditions. As
shown in Figure 4.2 the distance (di) is a measure of the distance between vectors in
pattern space. The associated weights are inversely proportional to the distance.
The difference in gain trajectory when different number of closest neighbors are used
is illustrated using the example shown in Figure 4.3. For this example an arbitrarily
chosen gain trajectory is used. When only the closest neighbor is used lot of variations
are seen depending on which sector the process state falls in. Similarly for the case using
two neighbors the gain traj ectory is jagged since pairs of nominal points may compete as
the "closest two" while the process is in transition. For this example, all the nominal
conditions should results in a smooth gain trajectory. This will increase the
computational burden but is necessary if smooth transition in controller parameters is
desired. Later in Chapter V the effect of using a different number of nominal conditions
for interpolation for the test system is demonstrated.
4.5 Interpolation Methods
Different approaches to weight the similarity between pattern vectors can be used.
How the similarity measured is interpreted is a critical issue in performing interpolation.
In this study three different interpolation methods have been investigated. The gain
scheduling is done using patternbased gain map in all the three cases. Different pattern
47
similarity measures to compare online pattern vectors to those representing nominal
conditions can be used. The three different methods are:
1. Linear Interpolation Using Euclidean Distance.
2. Quadratic Interpolation Using Pattern Similarity Measured by ART2 Neural Network.
3. Fuzzy Interpolation.
The first two methods differ in the similarity measure used. The third method can be
used for any similarity measure.
4.5.1 Linear Interpolation
Linear interpolation is based on the premise that the gain surface can be approximated
by a hybrid surface made up of planes of constant slopes. The similarity between two
pattern vectors is measured by the Euclidean norm between the two. Two perfectly
similar patterns will have a Euclidean norm of zero. A schematic of linear interpolation
is shown in Figure 4.5. Linear interpolation uses the lever rule to calculate the final
controller parameter. Equation 4.1 is used to calculate the controller parameter.
n
L(Ki / di)
j
K==
where: K is the final controller parameter
(4.1)
Ki == The controller parameters(e.g. Kc' "CJ or "CD) associated with the ith nominal
condition.
48
di == 1/(8) where 8 is the Euclidean distance used to measure similarity between
pattern vectors.
K =55 Kc=45 Oc
to
K =40" I
c n \ d~ Pattern associated
Nearest ~'d \ r with current state
neighbor ~ ~\~ / of process
....,..,~
.".",.~ * ......... 'S
_ ,..d2f Kc \ 2.. ~" K =50
~  ,.,., c
K =35 ~ \
c \ Net nearest
neighbor
Kc=450
Figure 4.5: Schematic of linear interpolation.
For the case shown in Figure 4.5, the controller gain can be calculated as shown in
Equation 4.2. In this illustrative example only four nearest neighbors are used to arrive at
the final controller gain, Kc:
49
Kc1 Kc2 Kc3 Kc4
+++
Kc = dl d2 d3 d4
111 1
+++dl
d2 d3 d4
(4.2)
In the same manner the other controller parameters can be scheduled. The main
advantage of linear interpolation is the ease of implementation.
4.5.2 Quadratic Interpolation Using Pattern Similarity Measured by ART2 Neural
Network
The ART2 neural network is a modified version of an autonomous learning model
based on Grossberg's adaptive resonance theory (Carpenter and Grossberg, 1987).
Anderson (1993) previously modified the ART2 network to identify regions in pattern
representation space which represent different operating conditions. In this approach the
neural network is trained to recognize patterns previously seen at different nominal
conditions. The nominal operating points are then represented by clusters rather than
points.
Anderson's method to interpret the process state from a window of pattern
information was developed on previous work by Whiteley and Davis (1993a, b). In this
method a sliding window is used to continuously extract the most recent pattern of
operation from the process. This pattern is input to the ART2 network. If the pattern
lies within the cluster of one of the nominal operating points, then the corresponding
values of controller parameters are used. If the pattern lies outside the cluster then some
50
sort of interpolation is applied. When a process pattern is completely identical to one of
the nominal operating points the ART2 similarity value is 1.00.
ART2 based similarity measure is an alternative to Euclidean distance. This
similarity measure depends on the distance as well as the orientation of patterns in the
representation space. Addition of orientation improves the process characterization.
Similarity between a process pattern and a nominal operating point is actually a quadratic
function of the angle between the two (Whiteley et aI., 1993).
The function used to find the quadratic distance between a process pattern and a
prototype is :
(4.3)
where dj is the distance from pattern to the cluster.
p is the radius of the pattern cluster, a user specified variable.
Sj is the ART2 similarity between the pattern and the ith nominal operating point.
The motivation of using the ART2 based quadratic interpolation strategy is to utilize
the orientation as well as the distance between the pattern and the prototype for
calculating the similarity.
51
4.5.3 Fuzzy Interpolation
Fuzzy reasoning is associated with systems involving uncertainties. Fuzzy inference
or fuzzy reasoning, often called approximate reasoning, has become a powerful technique
for practical implementation in control system design (Zimmerman, 1991). We
investigated fuzzy inference because of the uncertainties associated with controller
settings. More often than not these settings are empirically based using a heuristic rule of
thumb or trial and error tuning. There is rarely an exact set of tuning values associated
with any process or operating condition. The use of "excellent," "good" or "bad"
descriptions is often used to describe the efficacy of different controller parameter values.
These linguistic variables (e.g. "excellent") can be represented by fuzzy sets. The
controller parameters used at any nominal condition are also expressed as a fuzzy set.
This is very similar to representing a nominal condition by a cluster in pattern
representation space as described in the previous section. Figure 4.5 shows some typical
fuzzy sets that can be used to represent a controller parameter for example controller gain
coUc
~
LL
C. :.c
~
Q)
.0
E
Q)
~
1.0
Trapezoidal
45 50
Kc
Bell
45 50
Kc
Controller Parameter, Kc
45 50
Kc
Figure 4.5: Some typical fuzzy sets to represent controller parameters.
52
Membership function (y axis) is a measure of the confidence associated with the
controller parameters. In this work membership function is the degree of similarity
between pattern vectors. For the first case shown in Figure 4.5, even when the pattern
vectors are completely similar (membership function = 1.00) there is uncertainty in
choice of gain value (45 50). We have used triangular membership functions to
accommodate the uncertainties in the controller parameters. The spread of which can be
varied depending on how confident a designer is regarding the controller settings.
The controller settings at all operating conditions are thus represented by fuzzy sets.
The degree of overlap is dependent on how different the controller settings are at these
operating conditions and how much uncertainty is associated with them. During online
implementation the process state is characterized using tools presented in Chapter III and
the process pattern is compared with the patterns representing the nominal conditions.
The similarity is normalized and scaled between 0 and 1. The similarity can now be
inferred as a membership function. This is used to clip the fuzzy set of the controller
parameter corresponding to the nominal condition as shown in Figure 4.6. Thus the
process state is compared to each of the nominal operating conditions and its similarity is
represented by the "clipped" fuzzy set. These clipped fuzzy sets can be used together to
calculate the final controller setting. This is called "defuzzification." The number of
clipped fuzzy sets used depends on the number of operating conditions chosen, as
discussed in section 4.4.
53
In this study, the center of area (COA) defuzzification method (Driankov, 1990) has
been used. Center of area looks at the region represented by clipped fuzzy sets and
calculates its center of gravity. The controller parameter corresponding to the center of
gravity of this fuzzy region is the final controller output. An example of this
interpolation method is illustrated using Figure 4.7. In this example only the two closest
neighbors are used to interpolate. It is clear that the current pattern is more similar to the
operating condition "two."
The formula to compute a center of area is:
u* =
where:
k
LUi X f.lu(Ui)
i=l (4.4)
u* = Final interpolated value of controller parameter.
f.li = Normalized similarity value with respect to ith nominal condition.
Ui = The controller setting at the ith nominal condition.
Either the Euclidean norm or the ART2 similarity measures can be used to perform
fuzzy interpolation. In this study we have implemented fuzzy interpolation using
Euclidean norm as a measure of similarity. A sufficient number of nominal conditions
54
t:
~
a.
:2
en L ..ac>
E
a>
E
40 45 50
Gain
Clipped fuzzy set for
operating condition
one.
A combination of clipped fuzzy
set transforming the similarity
with two nearest ne ighbors as
a fuzzy set. The center of area
(COA) gives the interpolated
value.
55
.... COA
;~
150
Interpolated Gain = 48
40
t:
~
a.
.r:.
~
..ac>
E
a>
E
45 50 55
Gain
t:
~
a.
:2
en L ..ac>
E
a>
E
Clipped fuzzy set for
to operating condition
Figure 4.6: Schematic of fuzzy interpolation.
55
should be used so that all the fuzzy sets representing them overlap. This representation
can thus result in a continuous approximation of the gain surface. This also gives a
powerful tool to change the topology of the approximated gain surface by the use of
different kinds of fuzzy sets and different defuzzification methods.
4.6 Concluding Remarks
This chapter presented different interpolation methods using a patternbased tuning
map. The methods presented aim at approximating the gain surface. Linear interpolation
assumes that the gain surface can be represented as planes of constant slopes while the
ART2 and fuzzy methods aim at representing them by piece wise smooth surfaces with
variable topology. Each of the methods can be used to calculate the controller parameters
in an online fashion. The controller parameters scheduled can never be labeled as
"right" or "wrong". The performance of the gain scheduler can be inferred from the error
trajectory of the controlled variable. All these three methods are implemented and
investigated in Chapter V using a nonisothermal reactor as a demonstration system.
56
CHAPTER V
Demonstration of PatternBased Gain Scheduling
5.1 Overview
This chapter demonstrates our patternbased gain scheduling approach for a
simulated stirred tank reactor. The dynamics of the nonisothermal reactor are described
in the next section. A patternbased gain scheduled control methodology is then
developed to control the reactor temperature. The third section presents results which we
use to evaluate the performance of our gain scheduling approach. The importance of
patternbased information to characterize the state of the process is first presented. The
effect of interpolation using a different number of nominal operating points is then
shown. Finally, the performance of this gain scheduled controller at different operating
conditions is demonstrated and compared with the performance of PI controllers having
fixed controller parameters.
57
5.2 Nonisothermal Stirred Tank Reactor
The process used to demonstrate our approach exhibits dramatic changes in the
process gain (Kp). The demonstration system is highly nonlinear and open loop unstable.
A schematic of the nonisothermaljacketed CSTR is shown in Figure 5.1. The system
here has been adapted from Luyben (1990).
The CSTR has one feed and one product stream and the control objective is to
keep the reactor temperature at the desired set point by manipulating the flow rate of the
coolant Fc(t) in the outer jacket. A time delay of 30 seconds is used to compensate for
imperfect mixing. A small amount of white noise is added to the temperature sensor
reading to simulate measurement noise. Reactor parameters and the model equations
appear in the Appendix.
The main source of nonlinearity is the exponential dependence of the reaction rate
on the reactant temperature. This exponential function is popularly known as the
Arrhenius equation. A phase plane analysis for the process dynamics reveals that the
reaction "runs away" at various ranges of temperature. The process gain (Kp) and the
controller gain (Ke) are of opposite signs in the temperature ranges where the reactor is
open loop unstable. What makes open loop unstable systems particularly challenging is
that the controllers can be "detuned" to only a certain extent to account for model
uncertainties but must be tuned "tightly" enough to maintain closedloop stability.
Detuning often leads to an unstable response or results in system entering limit cycles in
58
JACKETED CSTR
Reactor Feed
Stirrer
Cooling
Water
Outlet
Coolin Water
net CSTR
Figure 5.1: Schematic of the demonstration system.
the presence of saturators. Gain scheduling offers an excellent technique to control such
processes.
A look at the process gain at different reactor temperatures is shown in the Figure
5.2. This system shows a wide variation in process gain. As shown in Figure 5.2, there
are two temperatures at which the process gain switches from negative to positive
infinity. The operating region between these two points is open loop unstable. This
system offers proof that the knowledge of process gains alone is not enough to
characterize the process dynamics. Details of a patternbased gain scheduled control
design for this CSTR is presented in the following section.
59
60 r.:=Ir.
4fJ
      ,.           I
2Ll  
60 L..l'l .........,~ I
"?fJ
~u  
c ern
<.9
en
en
Q) u
o~ a..
550 600 650 700
Operating Temperature
Note: For inlet reactant concentration = 1.0Ib.mol A/ft "
Figure 5.2: Process gain at different operating temperatures.
5.3 Gain Scheduler Design
Design of the gain scheduler starts with formation of the patternbased state map.
First the key operating conditions are identified. Since this CSTR is a test system the
conditions at which the reactor normally operates are decided arbitrarily. The nominal
conditions are chosen such that the process dynamics vary from one condition to another.
Reactor temperature and feed concentration are used to label or 'tag' each of these
nominal operating conditions. Eight operating conditions are chosen as nominal
60
operating conditions. These are listed in Table 5.1. The controller settings for each of
these operating conditions have been arrived at by using continuous cycling approach
(Zigler and Nichols, 1942). The parameters were then fine tuned using trial and error
tuning (Jury 1973). Details ofhow the controller settings were established are
documented in the Appendix.
Table 5.1
Tuning Parameters at Various Nominal Operating Conditions.
Inlet Cone. = 0.6 lb.mole A/fe Inlet Cone. = 1.0 lb.mole A/ft'
1 }rsei:==:5:():Q9~:: ::::::..
:r :==; 56t>RR· Kc = 80
Tc::i=5S·SPR::::·· tI = 0.03
3.::: Fe: ::~:24i2ftJhr<
5 :ts.ef¥:560?R
T d560~r{
Tc = S51WR
Fc· ::=;:80.()ft3:/ht
Kc = 120
tI = 0.04
Tset~64dgR: : :::
t :::::6409.tt" .<:: Kc = 50
"Tt:"":: :::i:"61g~R: tI = 0.12
Fe "=15:tf1?Jht
Kc = 40
tI = 0.22
Kc = 80
tI = 0.08
Tset~ 680()f{
T ~680()R
Tc = 655°R
Fe =117.3ft3/hr . . . . . . . . . . . . . . . . . . . . . .. .
Tsef=600?:R.
T: ~:6()O~R
TC:;:::S:S7?:R:"
J*<;:~ 144ft~jlfr:
8
7
6
Kc = 30
tI =0.15
Kc = 35
tI = 0.08
:1'se.f=¥6S:Q?R. "
"T =f:68()8R
Tc ~671oR
Fc =40.()ft3lhi: ,. . .
/r$¢.t:7J54Q~~>/
""T" ":::<k{64d?1t:<::"
tc:"::::::.:~·".6"3:t:?lt::"":
:td//#S3ttft3Ilif<
4
3
2
The next step in the construction of a patternbased gain map is to represent each
of these nominal operating conditions as points in pattern representation space. The
pattern vector was constructed from four scheduling variables. The scheduling variables
61
used are the reactor temperature, coolant temperature and the coolant flowrate. The
reactor temperature set point was used as the fourth scheduling variable. The setpoint is
used to give additional weight to the operating condition where the process is headed. A
six minute sliding window was used. The pattern vector is constructed from ten time
samples for each variable. Thus the sampling frequency is 10/6 min.1 and each
operating state is represented as a 40 dimensional vector in a pattern representation space.
Both Euclidean norm and similarity measure using ART2 neural network can be
used for process characterization. Typically interpolation is done using the four closest
neighbors. The controller can be programmed to use more nominal conditions for
interpolation. For fuzzy interpolation triangular fuzzy sets are used to represent the
controller settings.
In this way a gain scheduling algorithm is designed to control the demonstration
system. The CSTR was simulated on Simulink by MathWorks Inc. as shown in Figure
5.3. Simulink is an extension of Matlab. The CSTR is modeled using four coupled
ordinary differential equations(ODE). The simulation uses a 5th order RungeKutta
method to simultaneously integrate the four differential equations. Details ofCSTR
model development and simulation appears in the Appendix. This simulation can be
changed to test different interpolation strategies. The performance of this gain scheduled
design under different degrees of pattern information and different numbers of operating
condition is investigated and presented in the next section.
62
f(u)
Reaction
Constant
•························..···Fl
Input
Concentration1
Cornponent
Balance
C3
t
Ca
Time
Tset
Kc
Taul
•."w~_"w.~"w.~w ~~w_,,~...•_~,~_._~~'f"::
Ti Energy Balance
Inlet Temperature
Changes
Temp.
Normalized
Data
emu~
Demux
+Euler
+Gear
Gain Scheduler
I I  •.
+RungeKutta 3 +RungeKutta 5 +Adams +Adams/Gear + L1nsim
Start Time: '10_00 '
Stop Time: ,13_00_0 '
Min Step Size: I'0_00_02_5 '
Max Step Size: L10_oO_02_5 '
Tolerance: 1L1_e_7 '
Return Variables:
Imi::::::m:::il::::::::i@::U
1:::jjj;;:liiili;jj;:j:jlil
Figure 5.3: Simulink CSTR model with gain scheduled controller.
63
5.4 Results and Discussion
The performance of the gain scheduled controller is evaluated for different cases
in this section. The aim is to identify the parameters that playa vital role in successful
design of such a system. First the importance of patternbased information is
investigated. The next two sections aim at investigating how the gains should be
scheduled once the process dynamics have been accurately characterized.
5.4.1 Importance ofPatternBased Information
One simple way to perform gain scheduling is based on an instantaneous view of
the process. In other words, no sliding window is used and raw sensor data are used to
extract online process information. On the other extreme process state can be inferred
from time smoothened pattern trends. This section explores the importance of pattern
information for gain scheduling. The simulation system is controlled at various operating
conditions using different degrees of pattern information. The importance of patternbased
information is inferred from controller performance during periods of transition.
The system is initially at the nominal condition of 6400R temperature and
perturbed with changes in set points. The Euclidean norm is used as a similarity measure
and linear interpolation is used to arrive at the controller settings ( Kc and ~I). It is worth
pointing out that the first and third setpoints are the nominal operating points while the
second is not. Figure 5.4 shows performance of the gain scheduled controller using only
the instantaneous view of the process to characterize the process state. In other words
64
II 640
c.
E
Q)
J
600
Sensor Patterns:
Windo\v:
IAE:
Similarity Measure
Interpolation
Controlled Variable, Reactor Temperature
Gain Trajectory
CASE A
Ra\\'
\Vindo\\' length = 0
2.32
Euclidean norn1
Linear
40..r.rr,
80LL~...:....lI.'
Raw Trend Patterns
Fc
0.5 .~'.'~'.''':'''
'.. _..._. 
\"
OL....I...L...L.~.l.~I...I o 0.4 0.8 1.2 1.6 2
Time in Hours
Figure 5.4: Gain scheduled control performance using instantaneous raw sensor values.
The controller gain (Ke) picks up wrong values during periods of transition.
65
the window length is zero. Moreover the sensor readings are not smoothened. At the
first step change, the controller settings rapidly change and the controller settings for a
process state far from the desired plant operating condition is scheduled. As the
fluctuations in process variables die out, the controller settings gradually settle down to
the final value. This behavior is seen at both the set point changes. Such a gain schedule
can be disastrous as it can schedule 'wrong' controller settings that could lead to an
unstable or undesired response.
A time smoothening approach is used to process the sensor signals. Figure 5.5
illustrates how the performance of this controller improves by the use of time smoothened
sensor patterns. The arithmetic mean of the past 10 sensor values is used to identify the
state of the process. Here again, instantaneous values of the smooth sensor signals are
used. Just a simple smoothening leads to large improvement in controller performance.
The improvement in controller performance can be attributed to this smoothening
technique. Time smoothening should not be confused as a signal processing technique to
smooth noisy sensor data. It is actually a way to compress pattern information contained
in a time period for which the signal is smoothened. Such smoothening results in a
sensor trend which is less sensitive to rapid variations in process during periods of
transition. Thus the trend change in the process dynamics can be effectively extracted
from such smoothened pattern trends.
Another way in which use of pattern information can be implemented is by
observing the process for a finite period of time. In this approach a sliding window is
used to extract the features of sensor patterns. The width of this sliding window is
approximately equal to the time it takes for the process under investigation to transform
66
CASE B
Sensor Patterns: l~inle snloothcncd
Sliding Window: Windo\\' length = 0
IAE: 1.85
Sinlilarity Measure: Euclidean nornl
Interpolation: Linear
Controlled Variable, Reactor Temperature
c.
E
Q)
f
600
Gain Trajectory
40..rr~.___,
u
~
c" 60
0(ij
CJ
_80Ll.L...~..J.J
Time Smoothened Trend Patterns
0.5 ",'',',
I
0ll.l....l..ll...LI....L.. J o 0.4 0.8 1.2 1.6 2
Time in Hours
Figure 5.5: Gain scheduled control performance using time smoothened sensor values.
The controller gain (Ke) fluctuates but not as much as shown in case A.
67
from one state to another in response to a step change. Here a six minute sliding window
consisting of 10 time samples is used. As expected the gain trajectory smoothens as a
result of better process characterization. The controller performance for this case is
shown in Figure 5.6.
One result that emerges from these investigations is that use of pattern
information is imperative for design of a gain scheduler. Finally a controller which
extracts features of time averaged sensor patterns over a window length to characterize
the process dynamics, is simulated. Its performance is shown in Figure 5.7. This
controller uses a window length of six minutes and 10 sampled values of time
smoothened sensor outputs.
The results of all the cases discussed so far in this section are tabulated in Table
5.2. The performance of a gain scheduled controller for different cases are compared by
the IAE (integral of absolute error) value of the controller variable. IAE is one of the
most popular indices to measure the performance of a controller. Table 5.2 also shows
controller performance when the process characterization is done using ART2 based
similarity measure. The variation in IAE values as more and more pattern information is
used reemphasizes the importance of pattern information for gain scheduling.
68
CASEC
Sensor Patterns: Ra\\'
Sliding Window: 6 minute sliding \vindo\v
IAE: 1.80
Sinlilarity Measure: Euclidean norm
Interpo lat ion: Linear
Controlled Variable, Reactor Temperature
a: 640
c.
E
Q)
I
600
Gain Trajectory
40,...,.,~..
u
~
C60
·co
CJ
80l.__l.. L__ll.....C======::t::::=========:J
Raw Trend Patterns
0.8 1.2 1.6 2
Time in Hours
0.4
OL__.....IoL..1 L..........L.L..l.. L. J
o
Figure 5.6: Gain scheduled control performance using raw sensor values and an online
sliding window. The controller gain fluctuates but not as much as shown in
case A.
69
(~r\SE I)
Sensor Patterns: linlc sll1oothcncd.
Sliding \Vindo\\': 6 nlinutc sliding \vindo\\'
IAE: 1.80
Sinlilarity Measure: Euclidean nonn
Intcrpolation: Linear
Controlled Variable, Reactor Temperature
a: 640
c.
E
Q) r
600
Gain Trajectory
40 ! ! I I
U
~
C60
Oct;
CJ
80 I I I r
Time Smoothened Trend Patterns
1.6 2
T
OL.................lI................IL..I
o 0.4 0.8 1.2
Time in Hours
0.5 .
Figure 5.7: Gain scheduled control performance using time smoothened sensor values
and a online sliding window. The controller gain settles down to its final
value very fast.
70
Table 5.2
Importance ofPattemBased Information for Gain Scheduling.
CASE PatternBased Information IAE
A Raw sensor patterns. No window information. Linear interpolation: 2.32
B Time smoothened sensor patterns. No window information. Linear interpolation. 1.85
C Raw sensor patterns. Six minute fmite window. Linear interpolation. 1.80
D Time smoothened sensor patterns. Six minute finite window. Linear interpolation. 1.80
Note: Corresponding IAE values when ART2 similarity measure was used are 1.98, 1.81, 1.98 and 1.72 for cases A,
B, C and D respectively.
5.4.2 Number ofNominal Conditions
In the previous section the controllers were scheduled using linear interpolation
based on the four nearest neighbors. Chapter IV illustrated that gain scheduling
performance varies depending on how many nearest neighbors are used for interpolation
purposes. In this section, the effect of a different number of nominal conditions used to
interpolate is shown. The pattern similarity is measured using Euclidean norm and the
final controller settings are calculated using linear interpolation.
The number of nominal conditions used for interpolation is a key issue for
successful implementation of this strategy. Controller performance with interpolation
using 2, 4 and 7 prototypes are shown. In all cases, sensor patterns are time smoothened
and a six minute sliding window is used. Euclidean norm is used as similarity measure.
71
Thus any difference in performance can entirely be attributed to the number of prototypes
used for interpolation.
Figure 5.8 shows the controller performance when only the two nearest neighbors
are used for interpolation. A look at the gain trajectory immediately reveals that during
periods of transition the process state is not correctly identified and results in a wrong
schedule. The process takes almost 25 minutes to settle down to its final value. This is
because when the process is in transition it passes through regions where it could be
distinctly closest to one prototype ('the closest neighbor') but equally closer to two or
more prototypes ('the next closest neighbors'). As the process evolves its interpolation is
constantly done using two or more pairs of prototypes. This results in a jagged gain
trajectory.
Next, the four closest neighbors are used to arrive at the final controller setting.
The resulting controller performance is shown in Figure 5.9. This means that the process
state is observed from four different points in pattern space and all the four of the
similarity measures are used to characterize the state of the process. This gives much
smoother gain trajectory and a drastic improvement in controller performance. Using 7
prototypes further smoothens the gain trajectory (Figure 5.10). Also the IAE value is
reduced by approximately 5 %. The effect of the number of prototypes used for
interpolation is summarized in Table 5.3.
72
CASE E
INTERPOLATION USING 2 PROTOTYPES
Sensor Pattern: Tinle snloothened
Window: 6 nlinute sliding \\'indo\\'
Sinlilarity Measure: Euclidean nornl
Interpolation Linear
IAE: 3.14
Controlled Variable, Reactor Temperature
680r.~r
a: 660
~6401o.....,..,
Q)
~ 620
600L..J...'L.'..J
Controller Gain Trajectory
20r.....,.r~~
(J
~ 40
c~
60
_80L...LL...J..L..J
Controller Reset Time
0.8 1.2 1.6
Time in Hours
0.4
0.2,,.,,,.
Q)
E0.15
J=
Q) 0.1
UJ
~ 0.05
OL......'L'l..J o
Figure 5.8: Controller performance with interpolation using only the 2 closest neighbors.
Gain trajectory is very jagged indicating that transient pattern information
was misinterpreted causing a wrong schedule.
73
CASE F
INTERPOLATION USING 4 PI~OTOTYPES
Sensor Pattern: Tinlc sll100thcncd
Windovt' : 6 nlinutc sliding \\'indo\\'
Sinlilarity Measure Euclidean norm
Interpolation: Linear
IAE: 3.06
Controlled Variable, Reactor Temperature
680
a: 660 f'
~640
l _.
T' ..
Q)
~ 620
600
Controller Gain Trajectory
20
()
~ 40
c~
60
80
Controller Reset Time
0.2
Q)
E0.15 ~ .......... .......
i= .... 0.1 Q) en
~ 0.05 . . . . ., .
0
0.4 0.8 1.2 1.6
0
Time in Hours
Figure 5.9: Controller performance with interpolation using only the 4 closest neighbors.
Gain trajectory is quite smooth.
74
CASE (;
INTERPOLATION USING 7 PROTOTYPES
Sensor Pattern: Tinle snl00thened
Windo\v: 6 nl inute sliding \vindo\v
Sinlilarity Measure Euclidean nornl
Interpolat ion Linear
IAE: '2.97
Controlled Variable, Reactor Temperature
680rrr,r..
a: 660
~640.."'\
Q)
J 620
600'....I.....J........LL.J
Controller Gain Trajectory
20
u
~ 40
c~
60
80
Controller Reset Time
Q)
E
~
+'"
Q)
~ 0.1
a:
0 0.4 0.8 1.2 1.6
Time in Hours
Figure 5.10: Controller performance with interpolation using 7 closest neighbors.
75
CASEG
INTERPOLATION USING 7 PROTOTYPES
Sensor Pattern: Tin1e sI1100thened
Windo\v: 6 n1inute sliding \vindo\v
Sin1ilarity Measure Euclidean norm
Interpolat ion Linear
IAE: 2.97
Controlled Variable, Reactor Temperature
680r..,.,r.
a: 660
E6401
Q)
~ 620
600''......L..I....L.J
Controller Gain Trajectory
20
u
~ 40
c~
60
80
Controller Reset Time
Q)
E
j=
+'
Q)
~ 0.1
a:
0 0.4 0.8 1.2 1.6
Time in Hours
Figure 5.10: Controller performance with interpolation using 7 closest neighbors.
75
Table 5.3
Effect ofNumber of Operating Conditions on Interpolation
Number of Operating
CASE Points Considered for Qualitative Analysis of the Gain IAE
Interpolation Trajectory
Gain trajectory is very jagged indicating that
E 2 transient information is misinterpreted leading to a 3.14
"wrong" schedule.
Gain trajectory is quite smooth and responds
F 4 smoothly during transient periods. 3.06
Gain trajectory is very smooth and very smooth
G 7 changes in controller settings are seen in presence 2.97
of transient condition.
76
5.4.3 Interpolation Strategy
The results shown in the previous two subsections demonstrates the importance of
patternbased information and use of more than two nearest neighbors for gain
scheduling. This subsection investigates performance of a gain scheduled controller
using different interpolation strategies. The aim of an interpolation strategy is to
approximate the gain surface. The trajectory of controller parameters as the process
moves from one operating condition to another is used to evaluate how well the
interpolation strategy approximates the gain surface. First, the system is simulated using
linear interpolation, next a quadratic interpolation based on an ART2 similarity measure
is simulated. Finally, the process is simulated using fuzzy interpolation.
The system is initially at the operating temperature of 6400R and inlet
concentration is 1.0 lb. mole/ft3
. Step changes are made in reactor temperature to
increase the reactor temperature to a maximum possible temperature of 675°R. Then the
temperature is decreased thus covering the entire operating range. For all the simulations
a six minute sliding window and a sampling frequency of 10/6 min.1 is used.
Controller performance using a Euclidean based process state identifier is shown
in Figure 5.11 and with an ART2 based identifier in Figure 5.12. Figures 5.11 and 5.12
show a comparison in controller performance for step changes in set point. It is worth
noting that though the IAE are comparable, the gain trajectory is a bit smoother for the
case in which an ART2 based similarity measure is used. This could be attributed to the
fact that ART2 based similarity not only measures the distance between the pattern
vectors but also takes into consideration its orientation in the pattern space.
77
CASEH
LINEAR INTERllOLATION
Sensor Pattern: Tin1e sn100thened
Window 6 111inute sliding \vindow
Interpolation: Linear using the 4 closest neighbors
Similarity Measure: Euclidean norm
IAE: 5.31
Controlled Variable, Reactor Temperature
a: 680
c.
E640
Q) r
600L.__L_J__L__L__L..__.....__Z;'O'.J
Gain Trajectory
20.r~r,r_.
o ~ 40
c
'«1 60
(!)
801 L. l JL .L.. I
Time Smoothened Trend Patterns
0.5
OL.__J.__.L__l ..L__~___.L... ..I..__'
o 0.4 0.8 1.2 1.6 2 2.4 2.8
Time in Hours
Figure: 5.11: Overall servo performance using Euclidean norm as a similarity measure.
78
CASE I
INTERPOLATION USING ART2 SIMILAI{ITY MEASlJI~E
Sensor Pattern: Time smoothened
Window 6 n1inute sliding \vindow
Interpolation: Quadratic using the 4 closest neighbors
Similarity Measure: ART2
IAE: 5.16
Controlled Variable, Reactor Temperature
a: 680
Q.
E640
Q)
I
600'__....L__I__"" ...a..__'__' """~
Gain Trajectory
20..,rr~.,,
o ~ 40
c:
'«1 60
(!)
80' ~L. • I.'_._._._.__L ' .J
Time Smoothened Trend Patterns
0.5 .. . . Fe .:. ..
QL.__L...__I__L ....I. .l__JL.........J
o 0.4 0.8 1.2 1.6 2 2.4 2.8
Time in Hours
Figure 5.12: Overall servo performance using the ART2 based similarity measure.
79
Moreover, the gain surface approximated using a quadratic interpolation technique can be
inferred to be less discontinuous than the case using linear interpolation. This is based on
an assumption that any discontinuity in gain trajectory is an indication of a discontinuity
in the approximated gain surface.
Fuzzy interpolation is implemented using a triangular fuzzy set to represent
controller settings at each nominal condition. The span of a triangular fuzzy set
representing the controller gain is 20 (ft3hr I
ORI) while the one representing the reset
time ('tl) is 0.02 hr. A center of area defuzzification is used as described in Chapter IV.
The gain trajectory is much more smoother than the previous two methods as shown in
Figure 5.13. This indicates that fuzzy interpolation results in much smoother
approximation of the gain surface.
The performance of fixed gain PI controllers is also shown for comparison
purpose. Figure 5.14 shows controller performance with controller settings for operating
condition number 3 (see Table 5.1). Fixed controllers having controller settings
corresponding to operating conditions 4 and 7 are shown in Figures 5.15 and 5.16,
respectively. It is clear from these simulations that fixed gain controllers perform
effectively only very close to the nominal operating condition. On the other hand, a
simple gain scheduling algorithm superimposed on the PI control structure greatly
enhances the controller performance. The results of the overall servo performances for
the different interpolation strategies are tabulated in Table 5.4.
80
CASEJ
FUZZY INTERPOLATION
Sensor Pattern: Time smoothened
Window: 6 minute sliding window
Interpolation: Fuzzy using the 4 closest
neighbors
Similarity Measure: Euclidean the norm
IAE: . 5.22
Controlled Variable, Reactor Temperature
a: 680
a.
E640
Q)
f
600'__..L..__..a.... '__..........L..__L__~_ ___J
Gain Trajectory
·20~
~o ·40 •••••••••••• ;•..;...•:.•.:..~•••:.....r..~............~•••• : •••••••••••••••••••:•••~_...............~l....:....I.~••••••••••••••
................................................................ .......................................
c
.(ij ·60
C)
80
. . . .
• L....'_'.••_••_•• _•• _•• _•• _•• .•...•1.•._ ••_ ••_ ••_ •• _•• _•• _•• _ _ ••_ ••_.•_• _•• _•• _•• •••_ ••..1....'. '.•• _•• _•• _•• _•••_ ••_ ••...•.1•_.•_• _•• _•• •••_ ••_ ••_.J
Time Smoothened Trend Patterns
0.5
J .
J
~ .
o""__...a.__......a Io L..__...a...I
o 0.4 0.8 1.2 1.6 2 2.4 2.8
Time in Hours
Figure 5.13: Overall servo performance using center of area (COA) fuzzy interpolation.
81
CASEK
FIXED PI AT NOMINAL CONDITION 3
Sensor Pattern: Time snloothened
Fixed Parameters: Kc== 35: 'tJ == 0.08
IAE: 5.31
Comment: Poor perfornlance a\vay frolll the
design conditions
Controlled Variable, Reactor Temperature
(L 680
Q.
E640
Q)
~
600L._.l.l1.'''L.J
Gain Trajectory
20 ! ! ! I
~ 40~
c
'ro 60~
CJ
80 ~" I "( "I" "I"
Time Smoothened Trend Patterns
Ol..__L....L....J.,........&...a.L~.J o 0.4 0.8 1.2 1.6 2 2.4 2.8
Time in Hours
Figure 5.14: Overall servo performance ofa fixed PI controller. Controller settings
corresponding to nominal operating condition number 3.
82
CASEL
FIXED PI AT NOMINAL CONDITION 4
Sensor Pattern: Time smoothened
Fixed Parameters: Kc== 30: tl == 0.15
IAE: 10.55
Comment: Limit cycles.
Controlled Variable, Reactor Temperature
a: 680
a.
E640
Q)
f
600L._l__L__.1...__....l....__1_L.__~___I
Gain Trajectory
20 ! ! ! I
~ 40 1 •• . . .. .
c
'm 60 ..........
C)
80 1 I I I
Time Smoothened Trend Patterns
0.5..
OL__....l...__.L...__L..__~ .L.__.....L...._____L._____I
o 0.4 0.8 1.2 1.6 2 2.4 2.8
Time in Hours
Figure 5.15: Overall servo performance of a fixed PI controller. Controller settings
corresponding to the nominal operating condition 4.
83
CASEM
FIXED PI AT NOMINAL CONDITION 7
Sensor Pattern: Time smoothened
Fixed Parameters: Kc= 50; ~I = 0.12
IAE: 5.26
Comment: Large settling time away from the
design operating condition.
Controlled Variable, Reactor Temperature
a: 680
c.
E640
Q)
~
600
20
Gain Trajectory
I ! r !
(J
~ 40~ 
c
'(ij 60
C!J

80  I I '( I
Time Smoothened Trend Patterns
Figure 5.16: Overall servo performance of a fixed PI controller. Controller settings
corresponding to the operating condition number 7.
Table 5.4
Overall Servo Performance of Fixed PI Controllers and Gain Scheduled
Controllers Using Different Interpolation Strategies.
Case Similarity Measure and Interpolation Strategy Used IAE
Linear Interpolation using the 4 closest nominal conditions.
H 5.31
Using ART2 based pattern similarity measure and quadratic interpolation
I using the highest 4 similarity values. 5.16
Using the Euclidean norm for pattern recognition and a center of area (CGA)
J fuzzy interpolation. 5.22
Fixed PI controller, parameters corresponding to those of nominal operating
K. Condition 3. 5.31
Fixed PI controller, parameters corresponding to those of nominal operating
L condition 4 10.55
(Limit Cycles)
Fixed PI controller, parameters corresponding to those of nominal operating
M condition 7 5.26.
5.5 Concluding Remarks
From the results shown in this chapter it is clear that the successful
implementation gain scheduling is dependent on accurate characterization of the
process dynamics. Use of multisensor trend patterns is critical for accurate
process state characterization. Use of time smoothening is also very important to
monitor the change in process trend towards a different operating condition.
85
Another key issue is use of more than two nominal conditions to schedule
the final controller parameters. The similarity measure itself does not playa very
important role. Thus a simple Euclidean Norm based similarity measure can be
thus for final design. Though the use of quadratic and fuzzy interpolation
strategies does not result in a large improvement in IAE values they definitely
result in a much smoother gain trajectory. This is an indication that the real gain
surface can be better approximated by continuous smooth nonlinear surfaces as
discussed in Chapter IV.
86
CHAPTER VI
CONCLUSIONS
The main contribution of this thesis is that it has led to an improved understanding
that characterization of process dynamics is the key issue in implementation of a gain
scheduled control system design for controlling chemical processes. How the controller
parameters can be scheduled for any possible operating condition based on information
regarding the controller settings at a few discrete operating points is also formalized.
Such an analysis is necessary if gain scheduling is to evolve as a control technique to
compensate for nonlinearites in chemical plants.
Gain scheduling is a standard way of designing flight control systems for aircraft
which operate at wide ranges of altitude and speeds. It is gaining popularity for process
control applications. The main bottleneck in its application in chemical process
industries is that there is no technique to extract information regarding the state of the
process from online sensor data. Thus, the emphasis of this thesis has been to develop a
methodology to predict the state of the process from the online multisensor trend data.
The importance of patternbased information is vital for accurate characterization
of process dynamics. This conclusion can be made based on the results in Chapter V. It
87
is shown that an instantaneous view of the process can lead to a wrong schedule. A
patternbased process characterization is proposed. This method of process
characterization relies heavily on extracting the steady state trends from transient sensor
data. The use of time smoothening as a way to compress pattern information is also
demonstrated. It has been demonstrated that a simple patternbased technique such as
time smoothening can considerably improve the gain scheduling.
The next critical issue is to be able to use the results of this process
characterization scheme to schedule controller settings in an online fashion. How the
interpolation is performed is of vital importance in this adaptive control scheme.
Interpolation plays a very vital role because a process rarely operates at anyone nominal
condition. A formal methodology to perform interpolation based on a few nominal
operating conditions is developed in this study. One main result that emerges from
simulation results shown in Chapter V is that for the given system, at least four nominal
conditions are needed to perform interpolation. The number of nominal conditions
required to be used for interpolation depends on the distribution of the nominal operating
conditions in the patternbased gain map.
The analysis regarding the interpolation strategy in Chapter IV has revealed that
interpolation is a way to approximate the gain surface. The characteristic of the gain
trajectory can indicate how well the gain surface is approximated. Our investigations
show that fuzzy interpolation results in a much smoother gain trajectory. This is
desirable since rapid parameter variations can often lead to instability. Similarity
88
measure based on a simple Euclidean norm between pattern vectors is adequate for gain
scheduling purposes.
The main conclusions are summarized in the list below.
1. Characterization of the process dynamics is the key issue in successful
implementation of a gain scheduled controller.
2. Patterns of multisensor data should be used to accurately characterize the process
dynamics.
3. Time smoothening as a way to compress patternbased information results in
considerable improvement in the performance of gain scheduled systems.
4. More than two nominal conditions should be used for interpolation purposes. The
exact number of nominal conditions to be used for gain scheduling depends on the
distribution of nominal conditions in the pattern representation space.
5. Euclidean Norm as a pattern similarity measure is adequate for gain scheduling
purposes.
6. Fuzzy interpolation results in a much smoother gain trajectory.
6.1 Future Work
The attention is now directed towards future research. The entire thesis is built on
the fact that the relation between process dynamics and controller parameters are known
for a few operating conditions. In this work, the controller parameters for the
demonstration system were determined by use of process reaction curve and continuous
cycling methods. All these methods are based on empirical rules relating the controller
89
parameters to the sensitivity of the manipulated variable to the controlled variable. The
main advantage of an adaptive control strategy is that it can accommodate dynamics of
process variables other than the controlled variable by adjusting the controller parameters.
Hence it is important to investigate the effect of other process variables on the controller
parameters. Such a study can give invaluable insight on the characteristic of the gain
surface relating controller parameters to process dynamics. Such a analysis can also
throw light on the interpolation technique that shows the same characteristic as the real
gain surface.
Another implementation issue that was addressed in this thesis but needs further
investigation is the way to predict the process state that the process is evolving to, during
periods of transition. In this expectationbased gain scheduling, more and more weight
would be given to the controller parameters corresponding to the nominal condition at
which the process would finally settle. We have addressed this by incorporating the
desired set point as a scheduling variable. It is possible to extract information regarding
the trend change in the process by looking at smoothened transient data. With advances
in pattern recognition it is possible to classify different transient trajectories and develop
a method to predict the state towards which the process is headed. The main bottleneck
for such an investigation is the lack of good transient data in process industries.
Analysis of the stability of such control methodology is needed if this method is
to ever evolve as a commercial technique for process control application. This is a
formidable task because there is no function to map the controller parameters to the
process state variables. Absolute stability theory using Popov criteria (Popov, 1962) can
90
be used to establish stability margin and establish boundaries on parameters that will
yield stable controllers for a nonlinear system. Investigations performed by Shamma
(1988) show that main source of instability in such a design scheme is due to rapid
variations in the system parameters. Though this work has developed a method to
minimize rapid variations by use of time smoothened pattern information and a better
interpolation strategy, a theoretical analysis of stability is missing. Such a study would
be based on the linear parameter variation (LPV) analysis ( Desoer and Vidyasagar, 1975,
Shamma, 1993) of control systems.
From a design perspective, this work can be used as a starting point in
development of a gain scheduled controller for chemical processes. This work presents a
scheme to leverage process information from online sensor readings for use in designing
an adaptive controller. Unlike other existing methods proposed to perform gain
scheduling this method does not disrupt the smooth plant operation in order to
characterize its state. One drawback is that the design of a such an adaptive controller is
time consuming. The tuning parameters need to be calculated at the nominal operating
conditions and the scheduling variables need to be identified before the gain scheduled
controller can be implemented. The calculation of controller settings based on the
auxiliary and process variables is often computationally intensive. This is not a major
problem with the advancement of microprocessor technology. It is not difficult to
implement such a methodology on the platform offered by the modern distributed control
systems.
91
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94
APPENDIX
Model Development and Simulation
AI.I Introduction
The appendix can be divided in two parts. In the first section the model
development of the CSTR is presented followed by the discussion of the dynamic
characteristic of the CSTR. The second section documents the tuning techniques used to
establish the controller parameters. Finally, the simulation of the demonstration system
under a gain scheduled controller using a dynamic simulator, Simulink, is presented.
AI.2 Nonisothermal CSTR
Nonisothermal reactors are often the most difficult units to control in a chemical
plant, particularly if the reaction is exothermic. Small deviations in temperature can
significantly change the reaction rate and, consequently, the yield. Furthermore, the
95
increase in reaction rate with increasing temperature tends to make the reactor unstable.
The system discussed here has been adapted from Luyben (1990).
An irreversible exothermic reaction (chemical "A" reacting to form another
chemical "B") is carried out in a perfectly mixed CSTR. The reaction is first order and
exothermic in nature. Negligible heat losses and constant density are assumed. The
reactor has a coolant jacket and the flowrate of coolant is used to control the temperature
of the reactants.
Simple first order reaction is expressed as:
A~ B
where A is the reactant and B is the product.
Arrhenius function governing the reaction rate is expressed as:
k(T) == ko exp (E/RT)
where ko, E and R are constants and T is reactor temperature in oR.
The ODEs describing the system are:
Total continuity:
dV
==FF
dt °
where F° and F are inlet and outlet flowrates in ft3/hr
Reactor Component A continuity :
96
(AI.I)
(AI.2)
(AI.3)
where CAD and CA are the concentration of reactant A in feed and reactor.
Reactor Energy Equation:
d(VT)
p ==p( Fo  F) Cp  L1Hrxn VkCA  UAHx(TTJ)
dt
(AI.4)
(AI.5)
where Cp is the heat capacity of the reacting mixture and L1Hrxn is the exothermic heat of
the reaction. U is the heat transfer coefficient and AHX is the heat transfer surface area.
Jacket energy equation:
CPJ is the heat capacity of the coolant.
(AI.6)
The CSTR model parameters are shown in Table AI.I. These are the CSTR parameters
at a steady condition operating at 600oR.
97
Table AI.I
Nonisothermal Reactor Parameters
Volume VJ = 3.85 ff Rate Constant: ko = 7.08 x101u h{J
Activation Energy: E = 30,000Btu/lb.mol Gas constant R = 1.99 Btu/ lb.mol oR
Heat Transfer Coeff :U= 150 BTU/hr ft'~ oR Heat transfer Area: AHx = 250 ftL
Coolant Inlet Temp. TJO = 530 oR Heat ofRxn:~HRXN=30,000Btu/ Ib.moloR
React. Specific Heat Cp = 0.75 BTU/Ibm oR Coolant Specific HeatCpc=1.0 BTU/IbmoR
Reactant Density: p = 50 lbm/fe Coolant Density PJ = 62.3 lbm/ ftj
Inlet Concentration: 1.0 Ib.mol A/ ftj Set Point Temperature : T~r.l = 600 oR
Ai.2.i Open Loop Dynamic Characteristics
The main source of nonlinearity is the Arrhenius dependence of the reaction rate.
For this example, even a lOR rise in temperature increases the reaction rate by more than
I0 percent, enough to cause significant change in conversion. Furthermore, the increase
in rate with increasing temperature tends to make the reactor unstable. The effect of this
reactor going unstable will be further analyzed based on a steady state analysis later in
this subsection.
Figure AI.I shows that the reactor in absence of a controller runs away at certain
temperature ranges. At 6800R the process is stable in open loop but does not show a
response which can be modeled as a first order plus dead time (FOPDT). The process
runs away at 6000R and settles down to a new steady state. Such a response cannot be
98
Change in coolant flowrate (the manipulated variable)
to see its effect in the reactor tempe rature.
1 2 3 4 5 6
108 rrrr_rrrw
Afters mall increase in flowrate
~107.9 l II
~Nominal values ..............\
IJ """~ 107.8
0
Response at 680 deg R (not FOPDTtype)
Response at 600 deg R (open loop unstable)
Response at 540 Deg R (FOPDT type)
6
Figure A1.1: Open loop behavior of the demonstration system.
99
modeled by a FOPDT since the process is actually opera