Modelling of Paper Mill
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Transcript of Modelling of Paper Mill
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Stochastic Averaging Level Control and Its Application to Broke Management in Paper Machines
by SHIRO OGAWA
B. Eng., The University of Tokyo, 1967 M . Eng., The University of Tokyo, 1969
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
in
THE FACULTY OF GRADUATE STUDIES (Department of Electrical and Computer Engineering)
We accept this thesis as conforming to the required standard
THE UNIVERSITY OF BRITISH COLUMBIA December 12, 2003
Shiro Ogawa, 2003
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Abstract
Averaging level control refers to liquid level control of storage tanks, where the objective is to keep
the outlet flow u(t) as smooth as possible against the fluctuating inlet flow, while at the same time
keeping the tank level y{t) within high and low limits. The thesis treats the stochastic averaging
level control problem, where the input disturbance is a stochastic process.
The problem is formulated to minimize a weighted sum of Var[u(r)] and Var[(/)] subject to
the target Var[y(r)]. The state-space linear quadratic optimal control method is used, resulting
in a linear state feedback controller. When the input disturbance is modelled as the output of a
first-order low-pass filter driven by white noise, the optimal controller is a phase-lag network.
Broke storage tank level control is important in stabilizing the paper machine wet end. It is
treated as a special type of averaging level control, where the input disturbance Fb(t) is a two-state
continuous-time Markov process. The spectrum of Fb(t) is obtained and the linear optimal con-
troller is designed with the same methodology as for the general averaging level control problem.
Taking this very specific nature of Fb(t), a new nonlinear control scheme called the minimum
overflow probability controller (MOPC) is designed, and tested against data collected from a paper
machine. The MOPC performs better than the optimal linear controller and manual control.
A new theorem on the state probability distribution of a continuous-time Markov jump system
is presented, which leads to new methods for evaluating the mean and the variance of the state of
a linear jump system, and a new reliable numerical method to calculate the state distributions of
jump systems. These results are utilized to evaluate the overflow probabilities of controllers.
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Contents
Abstract ,,. ii
List of Tables vii
List of Figures ix
Acknowledgements x
1 Introduction and Overview 1
1.1 Background and Motivations 1 1.2 Overview and Structure of the Thesis 2
1.2.1 Format and Notation 4 1.3 Contributions of the Thesis 4
2 Averaging Level Control 7 2.1 Introduction 7 2.2 Mathematical Models 8
2.2.1 Process Model 8 2.2.2 Disturbance Model . . ! 9 2.2.3 Flow Smoothness/Roughness Model 11
2.3 Historical Perspective and Literature Review 13 2.3.1 P/PI Detuning 13 2.3.2 Deterministic Constrained Optimization 16 2.3.3 Constrained Minimum Variance Controller 17
3 Linear Optimal Controllers 19 3.1 Introduction 19 3.2 State-Space Method 20 3.3 Evaluation of Variances 22 3.4 Stationary Input 23
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3.4.1 System Equation and Performance Index 23 3.4.2 State-Space Solution . 24 3.4.3 Noise-Free Observer 25 3.4.4 Controller Parameterization 26 3.4.5 Comparison to PI Controller 29
3.5 Minimum Variance of Outlet Flow 32 3.5.1 Comparison to P Controller 33
3.6 Random Walk Input 35 3.6.1 Effects of Damping Factor 37
3.7 Summary 38 3.8 Mathematical Details 39
3.8.1 Solution of Section 3.4.2 39 3.8.2 Rv(y) and Rv(u) in 3.4.4 39 3.8.3 Rv(y) and Rv(u) in 3.4.5 41
4 Downstream Processes 43 4.1 Introduction 43 4.2 Downstream Process Model 44 4.3 State-Space Solution 47 4.4 Performance Comparison 49
4.4.1 Effects of v(0 weight 50 4.4.2 Var[y2] reduction versus CJP 52
4.5 Summary 54
5 Markov Jump Systems 55 5.1 Introduction 55 5.2 Jump Markov Systems 56 5.3 Holding Time Density 57 5.4 State Distribution 59
5.4.1 Entry Probabilities 63 5.4.2 Examples 63
5.5 State Moments of Linear Jump Systems 71 5.5.1 Example 73
5.6 Linear Quadratic Optimal Control 75 5.7 Summary 77 5.8 Appendix - Mathematical Details 78
5.8.1 Covariance of x(f) 78 5.8.2 Linear Quadratic Control 80
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6 Broke Storage Tank Level Control 84 6.1 Introduction 84
6.1.1 Paper Machine 84 6.1.2 Benefits of Wet End Stabilization 85 6.1.3 Broke Handling 87 6.1.4 Literature Review 87
6.2 Characteristics of Broke Flow 88 6.2.1 Spectrum of Broke Flow ^ 89
6.3 Flow Smoothness Requirements' . 92 6.4 Linear Optimal Controller 92
6.4.1 Jump System Theory 94 6.4.2 Implementation Issues 95
6.5 Nonlinear Controllers . 95 6.5.1 Minimum Overflow Probability Controller 96
6.6 Optimal Change of u(i) 100 6.6.1 Asymptotic Analysis 101 6.6.2 Linear u(i) 103
6.7 Summary 103 6.8 Maximum Principle 104
7 Simulation With Industrial Data 110 7.1 Introduction HO 7.2 Data HO
7.2.1 Process HO 7.2.2 Data Set C 112 7.2.3 Incoming Flow Estimate 113 7.2.4 Statistical Analysis 115 7.2.5 Probability Plots 117 7.2.6 Chi-Square Test 118 7.2.7 Summary 119
7.3 Controller Design 120 7.3.1 QMOPC 120 7.3.2 Linear Controller 122
7.4 Simulation 124 7.4.1 Manual Control 125 7.4.2 Linear Controller 125 7.4.3 QMOPC -127
7.5 Summary 127
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8 Conclusions 130 8.1 Summary 130 8.2 Future Research 132
Bibliography 134
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List of Tables
7.1 Chi-square test of break and normal durations 119
7.2 Overflow probabilities for QMOPC and MOPC 120
7.3 Overflow probabilities for QMOPC 121
7.4 The controller parameter for
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List of Figures
1.1 Averaging level control problem 1
2.1 Storage tank material balance 8
3.1 Parameters of C L vs. K 27
3.2 Gain of C L for K = 0.1,1 and 10 28
3.3 Magnitude of S for C L and C P I 28
3.4 Comparison of C L and PI controller with n - 0.7 30
3.5 Comparison of C L and the PI controller with various n 31
3.6 The ratio of the variance of u(t) between CPD and Cp. x-axis is Rv(y) 34
3.7 Performance change by rj for PI controller 37
4.1 Downstream process and F u 44
4.2 Gain plot of the downstream process transfer function 46
4.3 System for tank level control loop and downstream process 47
4.4 Gains of Co and CL 51
4.5 Magnituedof5 forC D andC/, 51
4.6 Variance reduction of y>i(j) vs variance of v(0 52 4.7 Effects of o)p on variance of y2{t) 52
4.8 Effects of a>p on variance of ^ ( 0 with normalized downstream process cut-off
frequency 53
5.1 Symmetrical binary signal d(t) is filtered by a low-pass filter. 64
5.2 Probability densities of filtered binary signal for B = 0.5,1, and 2 65
5.3 Probability density of filtered binary signal for fi = 10 66
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.1.1 V,
Acknowledgements
During the research of this thesis, I have been fortunate to receive help from many people, and I
would like to express my gratitude to them.
First, to my supervisors, Prof. Guy Dumont, Prof. Michael Davies, and Dr. Bruce Allison.
They warmly encouraged me to take up the challenge, and provided me with steady guidance
throughout my study. Their patience with and critiques for my ever changing ideas, are very much
appreciated. Dr. Allison introduced me to a fruitful area of averaging level control, and kick-
started the research. It was one of highlights of my student life to exchange ideas with him on
many subjects. Also thanks to Dr. Maryam Khanbaghi for introducing Markov jump systems and
sharing her knowledge on broke tank level control with me.
To the Network of Centres of Excellence, Mechanical Wood-Pulps, the Science Council of
British Columbia, and the Pulp and Paper Institute of Canada, I thank for their financial assistance.
I would like to thanks Messrs. Rick Harper and Leo Pelletier of NorskeCanada for supplying
the industrial data. Without their help, this thesis could not have been completed.
To my fellow graduate students, Dr. Junqiang Fan, Stevo Mijanovic, Manny Sidhu, and Dr.
Michael Chong Ping, I would like to express my sincere gratitude for their friendship, and kindness
to share knowledge with me. I appreciate that they treated me as their peer even though I am much
older than them. I felt that I was young again. I thank Dr. Leonardo Kammer of PAPRICAN for
his critique and advice on my work.
The library services in the UBC and the Pulp and Paper Centre was essential to my research,
and I thank the people their. Special thanks goes to Ms. Reta Penco for helping and teaching me in
literature search.
I am fortunate to have many friends who have supported me with their warm friendship. To my
old MacBlo gangs, Glenn Swanlund, Eric Lofkrantz, and Ted Carlson, I thank for their continuing
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friendship. It is comforting to know that my old friends in Japan are always ready to help me. I
would like to thank Dr. Toshio Kojima and Toshinori Watanabe for their encouragement.
I would like to thank two of my friends, Bruce and Dr. Alf Isaksson who have been inspiration
for me to pursue the study of control engineering. Bruce rekindled my enthusiasm for control
theory when he implemented digester level control at Port Alberni more than 20 years ago. Alf's
enthusiasm and dedication to control engineering is contagious.
I would like to thank my supervisor at the University of Tokyo, where I studied 35 years ago as
a master student, Prof. Mitsuru Terao who instilled the spirit of control engineering which I have
kept ever since.
Prof. Karl Astrom, after his retirement, told us that he had the best times in his life during his
graduate study and retirement. No wonder I enjoyed so much because I was a graduate student
and a retiree at the same time! However, one drawback of being an old graduate student is that
I cannot share the joy of completing the research with my parents, whose memory sustained me
during tough times.
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Chapter 1
Introduction and Overview
1.1 Background and Motivations
The research presented in this thesis started with a goal of devising a practical and effective auto-
matic control scheme for broke storage tank level, which was identified as the most influential and
difficult process for stabilizing a paper machine wet end. The importance of stabilizing a paper
machine wet end and the role of the broke storage tank are detailed in Chapter 6. It was soon
realized that the broke storage tank level control problem is a special case of the averaging level
control problem depicted in Figure 1.1. Here, the incoming flow to the storage tank Fd(f) changes
Fd(f) (disturbance)
Downstream Process
Figure 1.1: Averaging level control problem
randomly and the tank level y(t) must be keptbetween the high limit yH and the low limit yi. The
goal of averaging level control is to manipulate the outlet flow Fu(t) to minimize its adverse ef-
fects on the downstream process(es), while at the same time keepingy(t) between^ a n d ^ . The
yH
yL
m r
Storage
Tank
m r
Storage
Tank
I I -A Fu(t)
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1.2. Overview and Structure of the Thesis 2
broke storage tank level control problem has a very specific input disturbance, but the downstream
processes are diffuse and difficult to pin down by a simple mathematical model.
Before embarking on the task of designing a broke storage tank level controller, available tech-
niques for averaging level control, stochastic and deterministic, were reviewed. Averaging level
control can benefit many chemical processes that include surge tanks. Also, many areas in the pa-
per machine wet end can be improved by proper applications of stochastic averaging level control.
Chapters 2, 3 and 4 treat the stochastic averaging level control problem in a unified approach with
optimization methods emphasizing the importance of the input disturbance characteristics and the
downstream processes.
The fact that the input disturbances to the broke storage tank can be modelled by a continuous
time Markov process (Khanbaghi, 1998) motivated the study of Markov jump systems. Although
it was found that jump linear quadratic theory as applied to the broke storage tank level control
problem has no advantages over linear quadratic optimal control theory, a new theorem and nu-
merical methods for evaluating the state variances and the state distributions are devised. The new
methods are utilized in the synthesis and analysis of control schemes in Chapter 7.
With the techniques of stochastic averaging level control and Markov jump system theory,
linear and nonlinear controllers for the broke storage tank level control problem are designed and
analyzed. The new control schemes are tested in simulations with industrial data.
1.2 Overview and Structure of the Thesis
The thesis has three main parts: (1) stochastic averaging level control, (2) Markov jump systems,
and (3) broke storage tank level control. Trie first two parts provide technical tools necessary to
attack the problem treated in the third part.
Averaging Level Control
Three consecutive chapters 2, 3, and 4 cover the stochastic averaging control problem. In Chapter
2, the stochastic averaging level control problem is defined. The importance of the input distur-
bance characteristics and downstream processes is highlighted. Existing techniques for determin-
istic and stochastic averaging level control are reviewed. The popular random-walk assumption on
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1.2. Overview and Structure of the Thesis 3
the input disturbances is scrutinized. A simple first-order low-pass disturbance model is preferred.
Chapter 3 treats the problem with linear quadratic control methods. A state-space approach
with noise-free observer is utilized. The small dimension of the problem makes it possible to obtain
solutions in closed form. Thus, straightforward algorithms and good perspectives are obtained.
In Chapter 4, the downstream process is included explicitly in the problem formulation. The
performance of the resulting controller is compared with ones designed without downstream pro-
cesses. A convenient parameterization is devised to streamline the performance comparison.
Markov Jump Systems
Chapter 5 presents new theorems on state probability distributions and their applications for the variance evaluation and the state distribution calculations. This chapter is independent from the
previous chapters and can be read independently from the rest of the thesis. The new numerical
method for calculating the state distribution is utilized in designing controllers for broke storage
tank level.
Broke Storage Tank Level Control
Chapter 6 provides motivation and background for broke storage tank level control. Difficulties
arise from the input disturbance characteristics, which are quite different from those encountered
in usual averaging level control problems, and defining appropriate outlet flow smoothness re-
quirements. A new nonlinear control scheme called the minimum overflow probability controller
(MOPC) is introduced. Also, the optimal linear controller is designed for comparison.
In Chapter 7, industrial data collected from a paper machine where the broke storage tank level
is controlled manually, is used to estimate the incoming disturbance Fd(f) from a six-month history
of the broke storage tank level y(t) and outlet flow Fu(t). The two controllers designed in Chapter
6 are tuned for Fd(t) and tested by simulation. Comparisons are made among the two controllers
and manual control.
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1.3. Contributions of the Thesis 4
1.2.1 Format and Notation
As the three parts of this thesis deal with subjects more or less independent from each other,
overviews and literature reviews are placed at the beginning of each part, not at the beginning
of the thesis. Detailed mathematical derivations are included at the end of each chapter rather
than at the end of the thesis. This is a compromise between two goals: that the thesis should be
comprehensive and that mathematical details should not impede the smooth logical flow.
Most of the mathematical symbols are standard in the literature. The same symbol is used
with different meanings, depending on the context, if there is no danger of confusion. Economy in
notation is obtained by using the same symbol with different arguments to represent a time signal
in continuous time, discrete time, and in the Laplace transform. For instance, y(t) denotes a process
variable in continuous time, and y(s) the Laplace transform of the same variable. For discrete time,
the standard integer symbols k etc. are used, i.e. y(i) denotes y(t), t = iAt, At being a sample
period. A notation ":=" is used to signify "is defined to equal". Blackboard bold letters: F[event]
means the probability of event and E[X] means the expectation of a random variable X.
All the controllers in the thesis are meant to manipulate the outlet flow Fu(f) to control the
tank level. The formal controller output is denoted with u(t), which may be u(t) = Fu(t) or u{t) =
Fu(t) - E[Fd(t)]. In either case, Var[Fu(r)] = Var[w(r)] and Fu(t) = ii(t). The rate of change in the
outlet flow Fu{t) appears often, and is given a special symbol v(t):
d . d ,
This convention is used throughout the thesis, but the definition of v(r) reappears a few times as
reminder. Sometimes, u{t) is used as the manipulated variable of an abstract problem, such as,
x(t) = Ax(t) + Bu{t). The meaning of u(t) should be clear from the contexts.
1.3 Contributions of the Thesis
Averaging Level Control
The stochastic averaging level control problem is put on a solid theoretical basis by iden-
tifying the importance of input disturbance characteristics and the outlet flow smoothness
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1.3. Contributions of the Thesis 5
definition. Exploiting the continuous time state-space approach, clear perspectives are ob-
tained for synthesis and analysis of the linear optimal controllers.
The random walk input disturbance assumption is given a detailed analysis. Advantages of
stationary disturbances are presented; The optimal linear controller for a first-order low-pass
input disturbance, which is a phase lag network, is synthesized.
A new problem formulation to include the downstream process explicitly is presented. This
helps to clarify the meaning of outlet flow smoothness. With a simplified downstream pro-
cess model, practical controllers are obtained.
A proper state-space formulation is presented when the input disturbance is a random walk.
The key is to make the state vector x(t) stationary, so that Var[x(r)] exists.
Markov Jump System
A new theorem on the state probability distributions is given in Chapter 5, which leads to
new methods for evaluating the mean and the variance of the state of a linear jump system,
and a new reliable numerical method to calculate the state distributions. Unlike approaches
based on simulations, the new method does not have time-discretization approximations
nor randomness due to computer generated random numbers. These results are utilized to
evaluate the overflow probabilities of controllers.
Linear quadratic optimal theory for jump systems, where the state has discontinuity, is ex-
tended to include load-change disturbances in a natural way. This formulation provides a
more flexible means to include Fu(t) in the performance index.
Broke Storage Tank Level Control
The actual incoming flow to a broke storage tank Fb(t) was estimated from the tank level
and outlet flow measurements. This is the first published study of its kind. It was found that
break and normal durations follow the exponential distribution quite well. This is the second
confirmation of the published results of Khanbaghi et al. (1997). However, Fb(t) is not truly
binary as was assumed by the previous studies based on material balance.
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1.3. Contributions of the Thesis 6
A practical nonlinear controller with hard constraints on the outlet flow Fu(t) and its rate of
change Fu(t) is presented. The controller minimizes the overflow probability and guarantees
that the level will stay above the low limit yi. The new method for calculating the state dis-
tribution of Markov jump systems makes the design process straightforward without relying
on simulations. The nonlinear controller is compared to the optimum linear controller and
manual control with the industrial data, and is found to control the tank level better and with
a smaller maximum outlet flow.
The optimal change strategy of Fu(t), where Fu(t) is changed from its initial value UQ to the
final value u\ in a given time T, is investigated. The performance index is the integrated
square error of the downstream process variable. With a simple downstream process model,
the problem is solved with the Maximum Principle's singular control. A closed form solution
is obtained, which makes it possible to compare linear change strategy to the theoretical
optimum.
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Chapter 2
Averaging Level Control
2.1 Introduction
Averaging level control refers to liquid level control of storage tanks, where the objective is to
keep the outlet flow as smooth as possible against the fluctuating inlet flow, while at the same time
keeping the tank level within high and low limits (Figure 1.1). The term "averaging" indicates that
the average of the tank level is significant rather than the level itself. For example, Shinskey (1967)
(page 147) states, "This application is often referred to as 'averaging level control,' because it is
desired that the manipulated flow follow the average level in the tank". The reason that the outlet
flow Fu(t) should be smooth is to minimize its adverse effect on downstream processes. This is
quite different from "normal" regulatory control problems, where the goal is to keep the controlled
process variable y(t) close to its (constant) target yr. Even for regulatory problems, excessive
movements in the manipulated variable u(t) are usually avoided for a number of reasons, including
to avoid saturation in u(t) and to make a controller robust for modelling errors. However, smoothing
of u(t) is never a principal goal of control. An underlying idea for investigating the averaging level
control problem in the thesis is constrained optimization, where the "best" controller is a one that minimizes some performance index J subject to a constraint condition g(x, u) < 0. This idea
provides systematic procedures to design controllers for given situations, which are characterized
by J and g. Also this methodology makes it possible to compare many published control schemes
in a consistent and unified way.
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2.2. Mathematical Models 8
2.2 Mathematical Models
Optimization problems are tackled either analytically or numerically. In either case, mathemat-
ical models for the problem are necessary. Mathematical models for practical problems are al-
ways a compromise between fidelity to the target objects and mathematical tractability. To judge
the fidelity and usefulness of the mathematical models, intimate knowledge of the subject area is
necessary. Thus, construction of practically useful mathematical models is one of the important
activities that distinguish control engineering from applied mathematics. The three main models
needed for averaging level control are:
1. disturbance model
2. process model
3. flow smoothness or roughness model
In the following sections, the mathematical models for averaging level control are discussed in
detail. This is important because historically mathematical models for the input disturbance and the
flow roughness have been treated rather indifferently, although it will be found that they influence
controller design greatly.
2.2.1 Process Model
Figure 2.1 shows a simplified diagram of a material balance around a storage tank, where V(t)
denotes the liquid volume in the tank, sn.dy(t) the tank level. Fj(t) denotes the incoming flow, and Fu(t) the outgoing flow, both expressed in volume!time. Assuming that the liquid is noncompres-
F u ( t )
Figure 2.1: Storage tank material balance. V(t) is the volume of liquid in the tank, and y(f) is the level.
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2.2. Mathematical Models 9
sive, we obtain the following equation on V(t):
V(f)=V(0)+ f[Fd(t')-Fu(t')W (2.1) o
Dividing the both sides of (2.1) by the cross-sectional area of the tank, A, yields,
y(t) = v(0) + Kp f[Fd(t') - Fu(t')]dt', (2.2)
where K p = 1 /A is the process gain. This simple integrator model works well in practice, and is
used throughout the thesis. When the cross-sectional area varies by height, K p is not a constant,
and (2.2) becomes nonlinear. However, linearity can be kept by treating the problem by (2.1).
The level high limit is replaced by the volume high limit. A storage tank for noncompressive
material is a rare example in process control of a simple mathematical model derived from the
first principles that works well. In a physical process, many dynamical subsystems, including
manipulating valves, and measurement sensors with noise and delay, are involved. However, they
can be ignored, at least for liquid storage tanks in a paper machine wet end because the execution
time of level control can be slower than other loops. For instance, a broke storage tank level can be
controlled using control intervals of 30 seconds to 5 minutes, while most pressure or flow loops are
executed at 1 second intervals. Therefore, these dynamics can be ignored by the level controller
that is executing every 30 seconds to a few minutes. This fact is generalized in Marlin (1995), page
The process has no dead time and a phase lag of only 90, indicating that feedback
control would be straightforward for tight level control. This is actually the case in
many systems, since the sensor and valve dynamics are usually negligible.
The system parameter K p can be obtained with a good accuracy from experiments or physical
dimensions of the tank. Thus, robustness problem due to the process uncertainty is not a concern.
2.2.2 Disturbance Model
The role of the disturbance model was rather neglected in the early optimum control literature.
However, its importance, especially in process control, is now well accepted. Still this is not
always the case as stated in Harris and MacGregor (1987):
586:
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2.2. Mathematical Models 10
All optimal controllers (i.e., those optimizing a performance index) must be based
on a specified disturbance (or set-point change) model. Often in the design of such
controllers the disturbance model is not explicitly stated, but rather indirectly implied
in the problem formulation.
In averaging level control, very few papers have paid detailed attention to the input disturbance.
The term "averaging" itself suggests some stationary stochastic process because non-stationary
process cannot be averaged. However, many papers accept the need to bring back the tank level to
a setpoint to maximize the surge capacity. This suggests that the incoming flow is non-stationary
since if Fdif) is stationary there is no need to bring back the tank level to its setpoint as stated in Shinskey (1988). Probably the authors of these papers envisioned the input disturbance as a
combination of stationary rapid fluctuations, which can be averaged out, and less frequent step-type
load changes. This kind of disturbance poses interesting control problems that involve statistical
detection of abrupt changes or hidden Markov processes. However, this is left as one of the future
endeavours.
Most of the performance analyses are based on responses to step disturbances, probably be-
cause of mathematical tractability. If the input disturbance is step-like and stays at a constant
value long enough for the closed loop system to settle down to a quiescent state, then the analyses
based on the step responses would be valid. However many real disturbances do not have this
characteristic and care must be taken in interpreting analyses based on the step assumption.
In the paper machine wet end, disturbances for storage tanks are mostly stochastic. Most of
the studies in a stochastic context assume that the input disturbance is a random walk, which is
non-stationary. The random walk is a popular choice for process control because it introduces
integrating action in the controller. However, in reality, all disturbances are bounded and most
of them can be deemed stationary if observed for a long enough time. In this thesis, stationary
disturbance models are preferred because there is no strong need to eliminate offset for averaging
level control. A simple stationary process with first-order dynamics is used as a disturbance model.
Let d(t) denote the input disturbance deviation variable d(t) := Fj(t)-E[Fd(f)] and w(t) white noise.
It is assumed that d(t) is the output of a first-order low-pass filter with the cutoff frequency
d(s) = -^w(s). (2.3) S + 0)d
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2.2. Mathematical Models 11
This model was used because most disturbances have some kind of roll-off frequency. Also the
input disturbance model for broke tanks, which is a binary-valued random process, can be charac-
terized with respect to the frequency power spectrum with this model.
2.2.3 Flow Smoothness/Roughness Model
All the literature in averaging level control states that one of the control objectives is to keep the
outlet flow smooth. However, the mathematical definition of flow smoothness varies from author
to author. Furthermore, some papers contain no mathematical definition of flow smoothness at
all, so it must be guessed from context. To apply optimization based methods, which minimizes
some performance index it is more convenient to express flow "roughness" or "variability",
denoted with r(u(t)). In deterministic cases, the controller tries to minimize r(u(t)) subject to the level constraints. In stochastic cases, a natural choice for the flow roughness is the mean value of
r(u(t)),E[r(u(t))]. Since the purpose of outlet flow smoothing is to minimize its adverse effect on the downstream
processes, the flow roughness indicator should be selected taking into account the downstream
process. As shown in Section 4.4, for downstream processes that can be approximated by a first-
order system, Var[(r)] = Var[v(r)] works, well if the downstream process time constant is much
shorter than that of the closed loop tank level system. If the downstream process mixes various
material streams, and it is desired to keep mixing ratios constant, then Var[w(r)] rather than Var[v(/)]
might be the better choice. However, if the input disturbance is non-stationary, Var[w(r)] does not
exist, and cannot be used. For broke tank level control, Bonhivers et al. (1999a) and Dabros et al.
(2003) expand the idea of downstream process to a certain paper quality index.
Crisafulli and Peirce (1999) contains an excellent concrete example of the downstream process
and industrial data. The process is a raw cane sugar factory, where crushed cane fibre and water,
called "mixed juice", is heated and fed to a clarifier to extract clear juice. This clear juice is
subsequently crystallized to produce the final product, sugar. The mixed juice goes through two
surge tanks connected in series before it is fed to the clarifier. The paper reports a new feedforward
control scheme to maximize the combined surge capacity of the two tanks resulting in smooth
inlet flow and reduced turbidity of the clear juice. The paper says that, "the main technical control
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2.2. Mathematical Models 12
objective was to minimize the rate of change of the secondary mixed juice flow." Since the inlet
flow to the surge tank fluctuates randomly, E[|v(f)|], or the more mathematically tractable E[v(/)2] =
Var[v(r)], would be an appropriate mathematical measure for flow roughness in this case.
MRCO
The maximum rate of change in the outlet flow (MRCO) is mathematically defined as
MRCO := max |v(0|. 0
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2.3. Historical Perspective and Literature Review 13
2.3 Historical Perspective and Literature Review
It was recognized from the early stage of process control technology development that surge tanks
require different control objectives from normal regulation loops, and many papers and sections of
books have been written for averaging leve^cpntrol since the 1960s. They may be categorized into
the following three groups: (1) P/PI Detuning!;(2) Deterministic Constrained Optimization and (3)
Stochastic Constrained Optimization. Each approach is discussed in detail in the sections below.
2.3.1 P/PI Detuning
Studies published up to the early 1980's constitute this group. The ideas expressed in these papers
reflect control hardware prevalent at that time, pneumatic or electronic single loop controllers.
They are summarized as follows:
1. no mathematical definitions of the input disturbance nor outlet flow smoothness are given.
2. performance analyses are mostly non-mathematical being heavily depending on visual in-
spection of responses for a step disturbance.
3. controller synthesis is mostly ad hoc or heuristic.
This lack of clear definitions both of the input, disturbance and of the meaning of flow smoothness
lead to a number of different schemes claiming good performance.
Proportional Only Control
This is advocated by Shinskey in Shinskey (1988) and Shinskey (1994), although originally he
preferred nonlinear controllers (Shinskey, 1967). Since there is no integrating action, the tank
level tends to be away from its setpoint (off-set). To some authors, this off-set is a shortcoming of
the P controller. However, when the input disturbance is stationary or bounded, the off-set is not
disadvantageous. This point is clearly stated in (Shinskey, 1988): "Not only is there no need to
return the level to some set point such as 50 percent, but this practice actually reduces the effective
capacity of the vessel." Although, the meaning of flow smoothness is not stated clearly, it appears
to be defined as a smaller variation of u{f). It is shown later that the P controller is close to optimal
when the input disturbance is a first-order low-pass stationary process and the flow roughness
is represented by Var[w(r)]. Thus, if me input disturbance is such that the downstream process
-
2.3. Historical Perspective and Literature Review '.;)';[ 14
operates better with a smaller Var[w(/)], then the P controller performs well. However, care must
be taken since for some downstream processes, Var[v(0] is abetter indicator of the flow roughness.
Then a phase-lag network or low-pass filter is a better choice.
PI Tuning Rules
In Cheung and Luyben (19796), tuning rules for PI controllers are given based on analyses of
step responses. The design parameters are the maximum peak height (MPH) and the MRCO. No
justifications for using the MRCO as a flow roughness indicator was given. It seems that the MRCO
was selected for its mathematical tractability in step response analysis. The tuning procedure in
the paper determines the PI controller settings from two design specifications, the MPH and the
MRCO. This procedure potentially leads to a very lightly damped system depending on how the
MPH and the MRCO are selected. A better procedure would be to fix the damping factor n to some
reasonable value between 1 and 0.7, specify the desired MPH, and accept the resulting MRCO as
a price to pay for obtaining the target MPH. This is done in Marlin (1995) with n = 1.
Nonlinear PI Controllers
In Shunta and Fehervari (1976), two kinds of nonlinear PI controllers were proposed, one with
two settings switched by the level error (when the level error is small, slower tuning makes the
outlet flow change more smoothly, and faster tuning kicks in when the level error exceeds a limit
to prevent level limit violation), and the other called "wide range" which changes the proportional
gain and reset time continuously. No tuning guidelines were given in the paper. To estimate the
behaviour of a nonlinear controller against random input disturbance is a difficult problem even
if the input disturbance distribution is available. Tuning of these controllers must be done on a
trial-and-error basis. It seems these controllers did not gain much popularity.
The basic weakness of nonlinear schemes based on the level error is that a large step-like input
disturbance is not detected until the level error exceeds the threshold. It is best to act immediately
against a large input disturbance to minimize any flow roughness indicator based on v(t) - u(t).
As discussed later, the model predictive control (MPC) methods address this problem properly.
-
2.3. Historical Perspective and Literature Review 15
PL Level Controller
A Proportional-Lag (PL) level controller was proposed in Luyben and Buckley (1977) and further
studied in Cheung and Luyben (1979a). It is claimed that their approach maintains most of the
desirable features of proportional-only control but eliminates off-set. It is shown below that the PL
control is a PI controller when measurement noise or delay is negligible. In the PL controller, the
outlet flow u(f) is a sum of a proportional term of the level error and a filtered inlet flow w(t):
u(i) = KLe(t) + w(t), (2.4)
where K L is the proportional gain, e(t) is the level error (y(f) -y r). The filter is a first-order low-pass
filter KF/(TFs + 1) (Cheung and Luyben, 1979a). In the following analysis, KF is set to 1 to avoid an off-set in the level. Also there is no advantages in setting KF 1. Without losing generality,
it is assumed that the level setpoint.yr is zero, so e(t) = y(t). Using the same symbols for Laplace
transforms of time signals, we get
u(s) = KLy(s) + -^-. (2.5) TfS + 1 The process is
y(s) = ^[w(s) - (*)]. (2.6) s
Then, the transfer function from w to y, GpL, is
y ( s ) _ KpTfS ,
v^(sj " (TFs + \)(KLKP + s) K~ls2 + s(KL + (KpTF)-1) + KLT Now, we close the loop with a standard PI controller Kc(l + 1 / T r s ) , where Kc is the proportional gain and Tr the reset time. Then, the transfer function from w to y, Gpj, is
Q .= y W = (2S) P I' w(s) K~ ls 2 + K cs + KcT~l' K )
From (2.7) and (2.8), for a given KL and TF, we can make GPL = GPI by selecting Kc and Tr as
follows: Kc = KL + -r\=-, (2.9) Kpl F
and
Tr = TF^ = TF + -^ (2.10)
Thus, the PL controller has no special advantages over the PI controller. The performance may be
different from the PI controller if dead-time or large measurement noise is involved.
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2.3. Historical Perspective and Literature Review^'V 16
2.3.2 Deterministic Constrained Optimization
Supervisory computers and computer based distributed control systems (DCS) became widely
available in the 1980s, and computation intensive control algorithms such as model predictive
control (MPC) became practical as a result. Papers reviewed in this section are characterized by
their handling of level constraints in a deterministic context (step input disturbance).
DMC
In Cutler (1982), the Dynamic Matrix Control (DMC) algorithm was applied for averaging level
control. The paper does not give mathematical details, but the comments here can be inferred.
In normal DMC, the future trajectory of the controlled variable in discrete time, y([k, k + Np]) (a
positive integer is a prediction horizon) is compared to the desired trajectory yr([k, k + Np]) and the squared sum of the error E^'CKO ~>v(0)2 ^ a function of control increment, Au(k), is used in the performance index to be minimized. For averaging level control, the desired trajectory is set
as a funnel shaped region centred around the level setpoint. The tank level error is deemed to be
zero if it is within the funnel shaped region. Thus, the controller makes no move (Aw(r) = 0) if the
level error is inside the funnelled region, or a smaller move just enough to bring back the level into
the region if the level error is outside. The paper does not mention level constraint handling, which
is one of the strength of DMC. It reported good filtering characteristics of the algorithm when
the input disturbance is small so that the predicted level stays inside the funnel shaped region.
Although a number of simulation results were included, actual applications of this DMC algorithm
to averaging level control problems have not been reported (McDonald and McAvoy, 1986).
Optimal Predictive Controller
In McDonald and McAvoy (1986), the averaging level control problem was studied in continuous
time for a step input disturbance of known amplitude. The control objective is to minimize the
MRCO while keeping the level within constraints. Due to the simplicity of the plant dynamics
and the MRCO criteria, analytical solutions, which are nonlinear, were obtained. The amplitude of
the input disturbance is estimated from the level and the outlet flow, and the implemented control
algorithm takes a form of predictive control, and thus, was named the optimal predictive controller
-
2.3. Historical Perspective and Literature Review 17
(OPC). The OPC does not have the weakness of the level error based nonlinear control schemes.
The authors thought that it was necessary to bring the level to its setpoint, and combined a PI
algorithm with the nonlinear solutions. The matter of how quickly the level should be brought
back to the setpoint can only be discussed with stochastic input disturbances. The MRCO is
difficult to use in a stochastic scenario. However, the paper treated the input disturbance as steps
of undefined length. This makes the tuning procedure ad hoc, heavily depending on simulations.
The contribution of the paper is to put the averaging level control problem in the MPC framework.
MPC
The problem discussed (above) by McDonald and McAvoy (1986) was treated in discrete time in
Campo and Morari (1989). As in continuous time, if u(t) or Au(t) is not constrained, analytical
solutions are available. The need to bring the level back to its setpoint was acknowledged without
discussion. The problem was posed as a finite horizon constrained optimization. To bring the level
to its setpoint, a terminal condition that the predicted level at the end of the control horizon, y(te), is equal to the setpoint yr was included. This makes the control algorithm cleaner than adding
integral action in ad hoc way. The horizon length becomes a tuning parameter. The algorithm is
implemented as a receding horizon controller. With the finite control horizon, the problem can be
solved by linear programming when \u(t)\ and |Aw(r)| are constrained. As was discussed for the case of OPC, the proper selection of the control horizon requires
stochastic arguments, but the paper treats the problem in a purely deterministic context. Neverthe-
less, the paper provides the most complete treatment of the problem for step-like input disturbances
and the MRCO criteria. When \u(t)\ and |Aw(f)| are not constrained, the control algorithm is imple-mentable on any DCS as was reported in Allison and Khanbaghi (2001).
2.3.3 Constrained Minimum Variance Controller
This methodology was introduced by Kelly (1998) for tuning PI controllers, and extended to more
general cases including dead time by Foley et al. (2000). Both papers used a discrete-time for-
mulation to investigate averaging level control as a stochastic optimal control problem. The input
disturbance is assumed to be a random walk. The controller is synthesized to minimize the outlet
-
2.3. Historical Perspective and Literature Review 18
flow roughness (usually expressed as the variance of the outlet flow rate change Var[v(f)] in con-
tinuous time or Var[Aw(&)] in discrete time) keeping the variance of the tank level Var[y(0] to a
target value. The performance index J in discrete time, is set up as follows:
J = Var[y(k)] + pVav[Au(k)]
In discrete time, p can be set to zero to yield the minimum variance controller, where Au(t) is not
constrained. The term "constrained" stems from the fact that p > 0 constrains Var[Aw(r)], and does
not imply that the level is constrained. Constrained optimization of stochastic systems is a very
difficult problem, and no practical algorithm for present day control hardware is available (Batina
et al, 2002). The level is indirectly constrained by specifying its variance. Then, there is some
probability that the level will violate its limits. For most practical applications, this method pro-
vides satisfactory result because input disturbances in the real world are bounded. The gaussian
assumption, which makes the input disturbance unbounded, is a fiction to expedite the mathemati-
cal derivations. However, if the input disturbance happens to be larger than the design specification,
the level will exceed the limit because there is no hard constraint mechanism. This aspect is differ-
ent from OPC and MPC of the previous section, where closed loop hard level constraints are built
into the algorithm.
When there is no deadtime in the process, the optimal controller is a PI controller that makes
the closed loop system second order with the damping factor 77 = V2/2. This fact was made
clear by treating the problem in continuous time (Ogawa et al, 2002). This thesis extends this
methodology to include stationary input disturbance and to include the downstream process in the
performance index.
-
Chapter 3
Linear Optimal Controllers
3.1 Introduction
In this chapter, linear optimal controllers for averaging level control are studied. Although linear
controllers cannot guarantee that the level constraints are met for random walk disturbances, they
are considered for the following reasons:
1. For some problems, it is sufficient to detune existing linear controllers (PI or PID) in order
to make the outlet flow smooth. The study here provides tuning guidelines.
2. The performance of linear controllers can be calculated analytically (and for some simple
problems, in closed form). Usually this is not the case for nonlinear controllers, where
simulation is the only reliable way to assess performance. The linear optimum controllers
provide a reliable benchmark, which any non-linear controller must surpass.
The input disturbance Fd{t) is assumed to be either a non-stationary or stationary stochastic pro-
cess. The performance index is chosen to express the smoothness of the outlet flow Fu{t) in math-
ematically tractable form, usually the variance of Fu(f) or a weighted sum of the variances ofFu(t)
and Fu(t). Note that the latter choice is only available for stationary input disturbances. The output
of a controller is denoted by u(t) and either u(f) = Fu(t), or u(i) = Fu(f) - Fm (Fm = E[Fd(t)]). In
either case, Fu(i) = ii(t) = v(t), and Var[Fu(t)] = Var[w(0] and Var[Fu(r)] = Var[v(0]. Thus in the
sequel, u{t) is used for the variance expression. Letj>(r) denote the tank level andyr its setpoint. A
19
-
3.2. State-Space Method 20
linear controller is sought that minimizes the following performance index J,
J := E[(y(0 - yrf] + p2E[v(t)2 + n(u(t) - E[u(t)])2]
= Var(y(0) + p2[Var(v(f)) + fiVar(u(t)), (3.1)
where // = 0 when the disturbance is non-stationary. The weight p2 is a Lagrange multiplier
(Newton et a l , 1957) so that Var[y(r)] is minimized subject to a constraint
where a constant c is the upper limit of the flow variability. In practice, c is not specified, but
p2 is adjusted to obtain an acceptable Var[y(r)]. Then the resulting optimum controller provides
the minimum flow variability (or maximum smoothness). This approach was pioneered by Kelly
(1998) and Foley et al. (2000) using discrete-time polynomial methods. In the thesis, the prob-
lems are treated in continuous time using state-space methods, so that some characteristics of the
resulting controllers, such as the closed loop damping factor, become evident. One advantage of
the discrete-time approach is its ease of handling dead-time. However, the target applications are
such that dead-time can be safely ignored.
The problem is solved using the state-space linear quadratic (LQ) optimization methods combined
with noise-free observers. Since most of problems in the thesis are single-input-single-output, the
transfer function approach (Wiener-Hopf method) is a viable method, since there is no need for
observer design. However, the state-space approach was adopted for the following reasons:
1. The algebraic Ricatti equations (ARE) can be solved analytically for this size of problem,
and controller parameters, therefore,, can be expressed in closed form. The procedure is
more straightforward than the factorization which is necessary for the Wiener-Hopf method.
When the problem dimension increases, stable computer algorithms and software for solving
the ARE numerically are widely available. Presently, this is not the case for the Wiener-Hopf
approach.
Var[v(0] + //Var[w(r)] < c, (3.2)
3.2 State-Space Method
-
3.2. State-Space Method 21
2. From the nature of the problem, noise-free observers are practical. Thus, the requirement
of an observer is not really a disadvantage for the state-space approach. Rather it may,
in some cases, add flexibility in controller design by providing a framework for selecting
proper (or ad hoc) observers when the measurement noise cannot be ignored (Allison and
Khanbaghi, 2001). Of course, by adding a noise model, the transfer function approach can
produce the optimum controller that includes the observer. However, then the problem size
becomes too large to accept easy factorization by hand and the approach loses the edge.
The basic equations and notation which willibe used throughout this chapter are introduced here.
The plant is expressed in the following state-space form.
where x(t) is the state vector, u(t) is the manipulated variable (not Fu(t) here), w(t) is a Wiener
process, and w(f) white noise. A, B, D are constant matrices of appropriate size. The white noise is
expressed as a formal derivative of the Wiener process w(t). The derivative of the Wiener process
does not exist in a strict mathematical sense, and the ordinary differential equation (ODE) (3.3)
must be written as a stochastic differential equation (SDE). However, the SDE does not provide
any advantage over the (formal) ODE in the development below. Thus the SDE notation is not
used here. Generally, w(f) can be a vector process, but in the thesis w(t) and w(f) are always scalar
processes. The performance index / is
where Q is a positive semi-definite matrix and R a positive definite matrix. It is well known that
in stochastic LQ problems where the white noise enters the system equation additively, the linear
state feedback solutions are identical to those for the deterministic LQ problem (for example see
Fleming and Rishel (1975)). The optimum control u(t)* that minimizes J is a feedback form,
x(i) = Ax(t) + Bu(t) + Dw(t), (3.3)
(3.4)
u(ty = -Kx(t). (3.5)
The feedback gain K is related to a positive-definite solution P of the following algebraic Ricatti
equation (ARE): PA + ATP - PBR-lBTP +Q = 0, (3.6)
-
3.3. Evaluation of Variances 22
and
, K = RrlBTP. (3.7)
A number of stable computer algorithms and programs are available for numerically solving the
ARE. However, the low dimensionality of the problems here makes it possible to obtain closed
form solutions easily. Thus, most of the AREs are solved analytically, so that the equivalence of
the state-space approach and the Wiener-Hopf method can been clearly seen.
3.3 Evaluation of Variances
It is necessary to calculate the variances of process variables to evaluate and compare the con-
trollers. When a variable is the output of a rational transfer function driven by a signal with a
rational spectrum, its variance can be calculated analytically. Ify(0 has a rational power spectrum density
._ N(s)N(-s)
where
Gv(-sr) := v 7 v , (3.8)
N(s) = b-is" 1 + + bis + b0 D(s) = as"+an-is"~l + + a\s + ao,
its variance Var(y(r)) is calculated as
Var[y(01 = ^ - S H ^ . (3-9) N(s)N(-s) _JOO D(s)D(-s)1
Let / denote the above integral when D(s) is an -th order polynomial. Newton et al. (1957)
contains a table that lists / up to n = 10. The first three / are listed below.
h 2
h = 2a0ai
/ 2 = % i l % : ( 3 . 1 0 ) 2aoaia2
bja0ai + (b\ - 2bob2)a0a3 + b\a2a^ 2a0ai(a\a2 - a0a3)
-
3.4. Stationary Input 23
It becomes unwieldy for n > 3 for hand calculations. Astrom (1970) includes a recursive algorithm
and a FORTRAN program for /. In this chapter the following short-hand notation is used.
IA ) - = ^ j J j 7^ 7777^ 7 ~ds- (3.H) D{s)j 2nj J _ 7 0 0 D(s)D(-s)
3.4 Stationary Input
In this section, the input disturbance Fd{i) is assumed to be the output of a first-order low-pass
filter driven by white noise plus a bias. The state-space formulation is straightforward when the
input disturbance is zero-mean. Let F m denote the mean of Fd(i), F m := E[Fd(t)]. The deviation
variable d(i) for the input disturbance is defined.
d ( t ) : = F d ( t ) - F m . (3.12)
Then,
d(s) = -^w(s), (3.13) S + lOd
where o)d is the cut-off frequency and w(s) the Laplace transform of white noise.
3.4.1 System Equation and Performance Index
Since the process is an integrator, the mean of the outlet flow must be equal to Fm to keep the level
stationary. Thus, u(t) is set as a deviation from its mean Fm.
x\(t) = y(f) - y r tank level error
xiit) - d(t) input disturbance
u(t) = Fu(i) - F m outgoing flow deviation (manipulated variable)
As shown below, x2(i) and u(t) appear as the input disturbance and outlet flow for the system
equation. Xi(t) = Kp[Fd(t) - Fu(t)] = Kp[x2(t) + F m - (u(t) + Fm)] = Kp[x2(t) - u(t)]. (3.14)
Since u(t) is stationary, not only the variance of u(t) = v(t) but also u(i) can be included in the
performance index.
/ := E[(y(r) - yr)2] + p2E[v(t)2 + ^ u{t)2} (3.15)
= Var[y(0] + p 2 (Var[v(0] + pVar[M(0]),
-
3.4. Stationary Input 24
where p is a weight for u(t) variation. The control weight is set to p 2 rather than p to simplify the
calculations for the optimal gain below. When p > 0, the resulting controller tries to minimize
the variances of both v(r) and u(t). Those cases are studied later in Chapter 6. Here, cases where
p = 0 are investigated to compare with PI controllers for random-walk disturbances. To include
u{f) = v(f) in the performance index in a straightforward way, v(r) is treated as a formal control
input, and u(f) as an element of the state vector x(t),
x(t) := x2{t) u(t)]T.
Then the performance index J is expressed with the following two matrices Q and R,
J = E{x(tf 1 0 0
0 0 0
0 0 up2 x(t) + v(t)
~R
(3.16)
(3-17)
As X2(f) is the output of a low-pass filter with the cut-off frequency o>d,
x2(s) = S + CJd Then the following system equation is obtained
w{t) => X2(f) = -(OdX2(t) + 0Jdw(t).
d_ Jt
Xl(t)
x2(t) =
u(t)
0 Kp - K p 0 -o>d 0 0 0 0
3.4.2 State-Space Solution
Analytical Solution
The solution of (3.6), P, is expressed with its elements as
P = P\\ Pn Pn
ph P22 Pn
Pl\ P32 P33
Xi(t) 0 0
x2(t) + 0 v(0 + u(t) 1
I T
0
(3.18)
(3.19)
(3.20)
-
3.4. Stationary Input 25
Substitution of A,B,Q, and R of (3.19) and (3.17) into (3.6) yields six second order polynomial
equations on pij, from which a positive definite P can be obtained in closed form (algebraic details
are included in Section 3.8.1). Since
K = BTP/p2 = [pu p 2 i p 3 i ] / p 2 , (3.21)
pn, pn and 7733, which are listed below, are required for K.
P\3 = ~P, P23 = ; ^ P i 3 = p ^ 2 p K p (3.22)
Thus, K = [k\ k2 3] is obtained as follows.
k\ = , k2 = , , k3 = J2KP p (3.23) P coj + ojd^KpT^ + Kp/p' V
3.4.3 Noise-Free Observer
In most applications, x2(t) is not measured, and an observer is necessary. A simple noise-free
observer is constructed by differentiating x\(f). Since = Kp[x2(f)-u(t)] => sx\(s) = Kp[x2(s)-
u(s)],
x2(s) = sKpXx (s) + u(s). (3.24)
Usually, constructing an observer by differentiating the measured variable is not a good practice
due to measurement noise. However, the problems here are formulated with a formal manipulated
variable ii(t) and differentiating once does not introduce real differentiation when the real manip-
ulated variable is u(i). Even if real differentiation is necessary, usually level measurements have
low noise. Also, most applications can be executed with long sampling periods, say 5 seconds to a
few minutes, which make it possible to filter the level measurement adequately if noise is not neg-
ligible. Thus a noise-free observer is practical. See Allison and Khanbaghi (2001) for a Kalman
filtering approach for noisy cases.
The equation u(t) = v(t) = -Kx(t) is expressed in its Laplace transform,
su(s) = -kiXi(s) - k2x2(s) - kiu(s). (3.25)
Substitution of (3.24) into (3.25) yields,
u(s) (s + k2 + k3) = xi (s) (-Jfci - sk2K-1). (3.26)
;5 n n
-
3.4. Stationary Input 26
Let CL(s) denote the transfer function of the controller,
u(s) = CL(sMs)-yr) = CL(s)Xl(s). (3.27)
Then
where
-k\ - sk2K~] s + k\KJk2 s + b CL(s) = = -k2K-1 = Kc , (3.28)
w 5 + ^ + ^ 3 p s + k2 + k3 s + a v '
Kc k2/Kp a = k 2 + h (3.29)
b - k\Kplk2.
The sensitivity function S is
S = 1 = ^ + f3 Kp s + b s2 + s(a + KPKC) + KpKcb' K' }
1 + K - c s s + a
From (3.29) and (3.23),
a + KPKC = k2 + ki+ Kp(-k2/Kp) = h = ^2Kp/p, and
KpKcb = Kp{-k2lKp)kxKplk2 = -^Kp = Kp/p. Thus,
s _ s(s + a) _ s(s + a) s 2 + s y]2Kplp + Kp/p s 2 + y /2co c s + co 2 c '
where wc = ^Kp/p. The closed loop system is a second order system with damping factor rj = V2/2. The second-order maximally flat (Butterworth) filter has the same damping factor. This fact
does not come out clearly in the discrete-time approach (Foley et al, 2000), where the sampling
period introduces additional complexity.
3.4.4 Controller Parameterization
For controller design, the following parameters are given:
K p process gain
o)d input disturbance cut-off frequency
-
3.4. Stationary Input 27
As a tuning parameter, the variance ratio Rv(y) between the input disturbance Fd(t) and the level
y ( t ) is selected.
Rv(y) := Var[y(0] VartxKO]
(3.32) Var[Fd(t)] Var[d(t)]
R v ( y ) is specified to meet the level constraint requirements (the standard deviation of the tank
level is ^Rv(y) times the standard deviation of the incoming flow rate). A convenient way of characterizing the controller is by the bandwidth of the closed loop system relative to o>d,
OJC K := (3.33)
Substitution of y/Kp/p = OJC = KCJd into (3:23.) yields k2 and k-$ in terms of K and CJd- Then a and b are expressed with K and as follows.
K2 + V2/c a - K2 + V2/c + 1 (jjd < did,
, K 2 + V2/C + 1
b = cjd > cod-V2K+ 1
This shows that a < b , thus, the controller C L = K c ( s + b ) / ( s + a) is a phase lag network l . Figure
3.1 shows a and b for K between 0.01 and 100 for o j d = \. Figure 3.2 shows the gain of C L for
Figure 3.1: Plots of a and b of C L = K c ( s + b)/(s + a) for K between 0.01 and 100.
K = 0.1,1, and 10 for Kp = 1. Figure 3.3 shows the magnitude of the sensitivity function S for 'This is why the controller is denoted with Ci, where L stands for "lag".
-
3.4. Stationary Input 28
20 -
10 -
CD
10~2 10"1 10 101 102 to
Figure 3.2: Gains of CL for K = 0.1,1 and 10. (Kp = 1 and cod = 1)
K = 0.1,1 and 10. In addition, for comparison purposes, the magnitude of S for a PI controller,
of which shape is a function of the damping factor n only and does not change by K (see Section
3.4.5), is plotted. The plot marked as PI is for n = 0.7 and K= I.
10~2 10~1 10 101 102 (0
Figure 3.3: Magnitude of the sensitivity function S for CL for K = 0.1,1 and 10. \S \ for CPI with n = 0.7 is
also plotted. (Kp = 1 and o>d = 1)
-
3.4. Stationary Input 29
The controller performance is represented with the variance ratio between d(i) and ii(t), Rv(u), Var[w(0]
Rv(u) := (3.34) Var[d(r)] Thus, for a given Rv(y), smaller Rv(u) indicates a better controller. With the use of the formulae in (3.10), Rv(y) and Rv(u) can be expressed in terms of K (algebraic details are in Section 3.8.2),
K 2 K 4 + 3 V 2 V + 9K 2 + 6V2K + 3 Rviy) = or, V2K(K 2 + V2K+ l ) 3
Rv(ii) = or y/2~K(K 2+ V2/f+ : il). ( K2(\ + V2K)
(K 2 + V2K) + KA (3.35)
[\K2+ yj2K+\)
It is observed that Rv(y) ~ K2/iod and Rv(u) ~ o?d. The variance ratio of the level is proportional to the square of the process gain (this is easy to understand), and is inversely proportional to the square
of the input disturbance cutoff frequency. When cod is high, more energy in the input disturbance is
contained in higher frequencies and it is easier to filter out them with the same closed loop system,
which is a low-pass filter. When o>d is high, the input disturbance is "busy", and increases Rv(u). To subsume the two parameters Kp and o)d in the design parameter, the "standardized" variance
ratio Rv(y) and Rv(u) are defined as follows.
Rv(y) :=
Rv(u) :-
Ry(yWd K 4 + 3 V 2 V + 9K 2 + 6V2~/C + 3 *1
R M =
0J2d V2/c(/c2+ V2K+ 1)
V2/d>2 + V2K + l ) 3
1
U 2 + V2*+ I ;
(3.36)
Rv(y) and ^v(") are convenient in comparing controllers as they are independent of Kp and a>d-
Figure 3.4 shows Rv(y) and ^v(") for CL and a PI controller that makes the closed loop damping
factor n = V2/2.
3.4.5 Compar ison to P I Contro l ler
A phase lag network necessary to implement the optimal controller CL = Kc(s+b)/(s+a) is usually not readily available in present day control system hardware, and it must be base loaded with the
mean incoming flow Fm. On the other hand, PI controllers are a standard feature of every process
control system. Also they don't have to be base-loaded with Fm. In short, the PI controller has a
-
3.4. Stationary Input 30
Figure 3.4: Rv(y) (decreasing graphs) and Rv(ii) (increasing graphs) for CL and a PI controller with r\ = 0.7. x-axis is K = (oc/
-
3.4. Stationary Input 31
If the optimum tuning of the PI controller is sought, it becomes the minimization of Rv(u) subject to
Rv(y) = c\ (a positive constant c\ is the design specification). Rv(y) = c\ means 2T]K(K2 + 2TJK + 1) = 1/cj. Since 2VK(K2 + 2nK + 1) is the denominator of Rv(u), the minimization of Rv(u) becomes minimization of its numerator K3[(1 + 4n2)/e + 8n3]. So the optimum PI controller is characterized
by a pair (K*, if):
(K*,V*) = argmin{/c3[(l + 4n2)K + 8n3]) subject to 2TJK(K2 + 2TJK + 1) = l/c{.
This is a mathematically tractable problem at least numerically. However, since (K*, n*) is a func-
tion of Rv(y), the solution may be impractical. Practically more meaningful cases would result when the PI controller is tuned to a "moderate" (meaning that the closed loop system is not too
oscillatory nor too slow) mode, say n between 0.5 and 2. When the input disturbance is a random
walk, the optimum controller is a PI controller with n = V2/2 as shown in Section 3.6. Also the
critical damping rj = 1 is used often. These two values of n and n = 2 for slow mode, are used
for comparison. Figure 3.5 shows the ratio of ^v(") by the PI controller and Rv(u) by CL- The
performance of the PI controller deteriorates when Rv(y) = Rv(y)a)j/K2 is high. Thus CL is worth consideration for cases of high Rv(y).
2.8 -
2.6 -
10"3 1
-
3.5. Minimum Variance of Outlet Flow 32
3.5 Minimum Variance of Outlet Flow
For some types of downstream processes, the adverse effect of Fu(f) is proportional to the deviation
of Fu(t) from its mean value. Then the natural choice of the performance index is
J=E[Xi(t)2+p2u(t)2]. (3-40)
Now, u(i) is no longer included in J, so the system equation is constructed with u(t) as a control
input.
d_ dt
Xl(t) 0 Kp X\(t) -Kp u(i) + 0 = + u(i) + x2(t) 0 -cod Xlit) 0 0)d
w(t) (3.41)
A B The optimal control w*(r)is a linear state feedback with the feedback gain K =[k\ k 2],
u(s) = -kixi(s) - k2x2(s)
With the noise-free observer as before (x2(s) = KpSX\(s) + u(s)),
(3.42)
u*(s) = x\(s)[-ki - k 2 K p S ] - k 2u(s) u*(s) = - sk2K-1 + k 2
where
c 0 = - -
-x\(s) = (c0 + cis)xi(s), (3.43)
(3.44) l+fe ' " l + k 2
This is a proportional plus derivative (PD) controller. Let P = {pij} denote the solution of the ARE.
Then, the feedback gain K is
K = [kx k2] = BTP/p2 = [-Kppn/p2 - KpPl2/p2]
The ARE is solved analytically with the following result.
Pn = -7T, P\2 = Kp pcod + Kp
(3.45)
(3.46)
Then Kpp +iod 1 co = , Ci = (3.47)
pOJd PUd
The controller CPD(s) = Co + C{S is not proper, and is implemented with a low-pass filter a/(s + a) to make the controller proper as
a s + b CPD{s) - ci(s;+ CQ/CI ) = K r . , (3.48) s + a s + a
where K c = ac\ and b = c 0 /ci . Thus, the PD controller is implemented as a lead-lag controller.
-
3.5. Minimum Variance of Outlet Flow 33
3.5.1 Comparison to P Controller
In Shinskey (1994), the proportional only (P) controller is recommended for averaging level con-
trol. Although flow smoothness was not mathematically defined, it can be understood from the
context that the author sought to reduce the variance of u(t). In the following, the PD controller is
compared to P controllers. Although, the PD controller is implemented as a lead-lag network, its
performance is evaluated as CPD(s) = CQ + c\s to simplify the calculation and to provide the best case scenario for the PD controllers.
PD Controller Performance
The sensitivity function S is
S = 1 = s = 1 . s ( 3 49) 1 + (c0 + cxs)Kp/s 5(1 + Kpci) + KpC0 1 + Kpcx s + \+Kpc\
It is convenient to define a parameter o)c as
From (3.47),
So
coc = -^. (3.50) P
OJc+OJd , KpC0 l + K p c \ = , and = OJC. (3.51)
ojd 1 + Kpc\ S = -J* i _ . (3.52)
COc + OJd S + 0)c
As before, define K as o>c = KOJJ, then
S = . (3.53) 1 + K S + KOJd
After some calculations, Rv(y) is obtained.
The variance ratio between Var[w(r)] and Vax[d(t)], Rv(u) is VarKQ] ir^ + Sic+l)
-
3.5. Minimum Variance of Outlet Flow 34
P Controller
The transfer function of the P controller C p(s) is
Cp Kr (3.56)
Then the sensitivity function with this controller is
1 S = 1 + KcKp/s s + KCKP S + OJC
(3.57)
where coc = KCKP = KOJJ. Then, Rv(y) and Rv(u) for Cp are obtained.
Rv(y) = OJIK(K+ 1) Rv(y) = 1
K(K+ l)
Rv(u) = K+ 1
(3.58)
(3.59)
The reduction of R v (u) by C P D compared to C P is plotted against Rv(y) in Figure 3.6. The maximum
reduction is 0.8638 when Rv(y) = 0.7077. This reduction is 0.93 in terms of standard deviation.
So the PD controller reduces the standard deviation of u{t) by 7 % compared to the P controller
at best. In reality, the reduction is less thanr7 % due to the low-pass filter. This small reduction
probably means that for practical purposes, the PD controller is no better than the P controller.
Figure 3.6: The ratio of the variance of u{f) between Cpo and Cp. x-axis is Rv(y).
-
3.6. Random Walk Input 35
3.6 Random Walk Input
Although stationary processes are preferred as a model of the input disturbance, the random-walk
case is treated here for two reasons: (a) to show that the optimal controller is the same PI controller
for deterministic cases where the input disturbance is a step, and (b) to show a proper way to choose
the state variables for a non-stationary disturbance. Doss et al. (1983) considered the control of
liquid levels in three tanks in series by a state-space LQ approach without explicitly including the
input disturbance in the state , but resorted to other more intuitive approaches that they felt led to
a simpler, more physically acceptable scheme. Harris and MacGregor (1987) reworked the same
problem, and showed that the same satisfactory result can be obtained by the transfer function
method with a disturbance model. If the input disturbance had been included in Doss's work as
shown below, the state-space approach would have produced a satisfactory result.
A continuous-time random walk is a Wiener process w(t). Since w(f) is non-stationary, the
difference between the incoming flow and the outgoing flow, which is stationary, is chosen as
a state variable. With a Wiener process w(t) as the input disturbance, the manipulated variable
u(i) = Fu(t), and the tank level error x\(t) = y(t) -yr, the process is expressed as
= xi(0) + K p f (w(r)-u(T))dT. Jo With X2(t) = w(t) - u(t), w(f) = dw(t)/dt, and v(t) = u(t), the system equation is
(3.60)
d_ dt
X\(t) 0 Kp *i(0 0 v(0 + 0
= + v(0 + x2(i) 0 0 x2(t) -1 1 w(t). (3.61)
B The performance index J is
J := E[xi(t)2 + p2v(02] = E{x(t)T x(t) + v(0 v(0} (3.62)
Note that the outlet flow u(t) itself cannot be included in the performance index because it is non-
stationary and may not have a well-defined variance. The optimum control v(t)* that minimizes /
is a feedback form
v(0* = -Kx(t) (3.63)
-
3.6. Random Walk Input 36
As before, the solution of the ARE, P = {prf is obtained analytically. Substitution of A,B,Q, and R of (3.61) and (3.62) into (3.6) yields the following three equations.
-p22/p2 + \ = 0 PnKp-pnP22/p2 = 0 (3.64)
IpnKp - p\2lp2 = 0
The above three equations and the positive definiteness of P determine ptj as follows.
P n = p , P 2 2 = P y j 2 p K p p n = ^ 2 p / K p
And
(3.65)
(3.66) K = R-lBrP = [-pl2/p2 -P22/P2] = [-l/p - y/2p~Kp/p]
Since u ( t ) = v(t) = - K x ,
su(s) = xi(s)/p + x2(s) yjlpKpIp (3.67)
Usually x2(t) is not available for measurement and an observer is necessary. A simple noise-free
observer is designed below. Since x\(t) = Kpx2(t), in Laplace transform, sx\(s) = Kpx2(s). Thus,
x2(s) = SX\ (s) (3.68)
Substitution of (3.68) into (3.67) yields
su(s) = X\(s)
Thus
fl ^2pKP\ ~ + s-p = xi(s)
u(s) = X\(s) 1
+
(3.69)
(3.70) sp ypKp
This is a PI controller with the proportional gain Kc = yf^, and the reset time Tr = ^2p/Kp. The sensitivity function S = 1/(1 + PC) is
1 S = l + * z ( + / X \ s 2 + Syl2Kp7^ + Kp/p S2 + 2^U>CS + G J I
(3.71)
where OJC = \fKp~/p. So the optimum controller is a PI controller that makes the closed loop system a second order low-pass filter with a damping factor 0.5 VI.
-
3.6. Random Walk Input 37
3.6.1 Effects of Damping Factor
Some PI tuning rules recommend the damping factor n = 1 for integrating processes , for example
Morari and Zafiriou (1989). Effects of n on Var[v(/)] was investigated when n is deviated from the
optimum value rf = 0.5 V2. The PI controller is tuned so that the sensitivity function becomes
s 2 S = -^z = (3.72)
s1 + 2T]0)cS + (x>Lc Then Var[y(r)] and Var[v(f)] become (denoting the variance of w(t) with cr^),
Var|M0] = S . Var[v(0] = ^ I ' c (3.73)
For a given Var[y(J)] target, o)c is adjusted for each n n* to keep Var[y(0] to its target. Then
Var[v()] is compared to the optimum value Var*[v(0],
Var[v(0] (4n2 + l ) 2 V 2 ( 1 x l / 3
Var*[v(0] 4n l V 2 ^ J (3.74)
This is plotted for 77 = [0.5 1.0] in Figure 3.7. Var[v(f)] increases by 5% when 7/ = 1.
1.06H
r\ (damping factor) .1 r
Figure 3.7: Comparison of the optimum controller (n = V2/2) and non-optimum (n = [0.5,1]). X-axis is
for the damping factor n and Y-axis for the variance of u divided by the optimum variance.
-
3.7. Summary 38
3.7 Summary
The averaging level control problem is formulated as a minimization of Var[w] subject to the target
tank level variance Var[y(f)]. The problem is solved by the state-space method combined with
a noise-free observer. When the input disturbance d(t) is modelled as a stationary random pro-
cess (output of a first-order low-pass filter with a cutoff frequency ojd driven by white noise), the
optimum controller CL is a phase lag network,
Ci = Kc . s + a
When the input disturbance is modelled as a random walk, the optimum controller is a PI controller,
Cpi = Kc . s
For both Q, and C/>/, the closed loop system is a second-order system with damping factor V2/2.
A single parameter Rv(y) that characterizes the controller is introduced,
Var[y(Q] (ojd\2
For a stationary input disturbance, Ci and CPI are compared by the ratio of Var[zi(r)] with the two
controllers set up to give the same tank level variance. CL is not much different from CPI when
Rv(y) is small (below 1), but CL performs better for higher Rv(y). For instance, for Rv(y) = 10, CL's Var[w(0] is 1/2.6 of that of CPI.
When the flow smoothness is represented by the variance of u(t) in stead of u(f), the opti-
mum controller, which minimizes Var[u(t)] with a target tank level variance, is a proportional plus derivative (PD) controller, Cpo,
CpD = CQ + C \ S .
However, the performance difference between CPD and a proportional only controller CP is small
(7 % in standard deviation of u(t) at best), and for practical purposes, Cp is recommended.
-
3.8. Mathematical Details 39
3.8 Mathematical Details
3.8.1 Solution of Section 3.4.2
The A R E yields the following seven equations.
() -P2n/P2 + 1 = 0 (b) pnKp - pX2Ab - pnpnlp1 = 0 (c) - pnKp - pnpislp2 = 0 ( d ) 2 p n K p - 2 p 2 2 A b - p \ 3 i p 2 = 0 ( e ) - p l 2 K P + p l 3 K p - p 2 , A b - p 2 3 p 3 3 / p 2
( J ) - 2 p l 3 K p - p 2 3 / p 2 + p p 2 = 0
First from (a), p \ 3 = p , but from (J), p \ 3 must be - p to make p 3 3 real.
(a) - p 2 1 3 /p 2 + 1 = 0 => p l 3 = -p (/) - 2p l 3Kp - p\ 3lp 2 +pp 2 = 0 => p i 3 = p yJ2pK p +pp 2
Then using (c),
(c) -pnKp-pnp 3 3/p 2 = 0 => - /?nA> + ^2pK p + pp2 = 0 => /?n = ^2plK p +p(p/K p) 2
Using (6),
( 6 ) p u K p - p n h - p n P i - i l p 2 = 0 = > p l 2 = ^ l ( p n K p - p u p 2 3 / p 2 ) = A ; \ ^ j 2 p K p + p p 2 + p 2 i / p )
Then, substitute the known to (e),
(e) - p l 2 K p + / ^ .K /? - p n h - P2iPnlpV=> - p u K p -pKp - p 2 i A b - p 2 3 y J 2 p K p + p p 2 / p
So Kp(p + A~bl yj2pKp+pp2) Kp(Abp + y/2pKp + pp2)
p 2 3 = = Ab + ^/2Kp/p+p + KpA-blp-x A2 + Ab ^/2Kp/p+p + Kpp~l
3.8.2 Rv(y) and R v (u) in 3.4.4
As the sensitivity function
-
3.8. Mathematical Details 40
the transfer functions from d(s) to x\(s) GDY and from d(s) to ii(s) G D y are K p ( s + d)
GDY =
GDU =
s2 + y/2o)cs + co2
K p K c ( s 2 + bs) s2 + ^j2cocs + co2
Thus
Var[x!(0] = /v 1 Kp(s + a) = K2pIv s + a
[s + cod s2 + y/2cocs + co2c)
We express coc as Kcod, then a, b, and Kc are expressed as follows: ,,2
U 3 + *2(o>rf+ V2oc) + s( yf2coccod+co2) + 4/o>2,
a = >C + A/2/C
K2 + V2A: + 1 COD = KA(OD < 0)d
K2 + V2/c + 1 b = iOd = KbO)d > 0)d 1 + V2K
Var[x,(0]=^/v s + a
= K:
[s3 + s2(l + K V2)w r f + S(K V2 + /c2)^2 + K 2 ^ ,
2 AC 2 W 3 +^( l - r / cV2)^
2K2co6d ( ( 1 + K V2)(/f V2 + /c2) - K2)
1 +
' P2w 3 V2/rt>2-t- V2/c+ 1) Since Var[d(t)] = Iv(l/(s + cod)) = \/(2cod),
_ Var[x,(Q] _ 2 Rv^ ~ Verity] ~ K
l + J ^ ^ ^ l + K ^ _K2P * 4 + 3 V 2 3 + 9tt2 + 6 V 2 + 3 * co2. V2K(K 2 + V2K+ 1) w2 V2K(K2 + V2K+ l ) 3
So
*v(v) := Rv(yWd K4 + 3 V2K3 + 9K2 + 6 V2K + 3 *2 V2K(K2 + V2/f+ l )
3 (3.75)
Similarly for Var[w(0] and Rv(u), ( 1 KpKc(s2+bs) ) Var[zi(r)] = 7V
+ 5 2 + yJ2coCS + C02) s 2 + bs
U 3 + s2(cod + V2fa>c) + s( ^[2coccod + co2) + codco2
Since K P K C = - k 2 , KpKc
y 2(l + V2/c) V2/c+ 1
3 Here, the variance ratio is concerned, so the input variance can be set to any convenient value.
-
3.8. Mathematical Details 41
So,
Var[zi(0] = ( K\\ + V2K) ^
OJd 2 + Vz* + 1 )
s 2 + bs U 3 + S2(\ + K V2)itd + S(K V2 + K2)CJ2 + K2OJ.
Substituting b = ^j-^^^d, and after some algebra, we get,
Rv(u) = 0>A
V2K(K2 + V2/f + 1) + V5#c + u
*v(u) := /?V(M) 1
w2, ~ V2K(K2 + V2* +1)
V ( l + V2*) ,K2+ V2K + l j
(K2 + V2K) + (3.76)
3.8.3 R v(y) and i?v(w) in 3.4.5 _1 frS
s + T~L . s + b
Then
Thus,
CPI(s) = K c \ \ + ^ - = K c ~ ^ - = K c -
S =
b = T:1
1 + Ikri+b s 2 + sK p K c + K p K c b s 2 + 2rjojcs +
KPKC = 2na>c, and KpKcb = OJ2
Substitution of K P K C - 2T]OJC into K p K c b = o 2 yields
So,
G n v =
GDU =
Kps Kps s 2 + 2no)cs + OJ2 s 2 + 2r)KO)dS + K2to2d
KpKc(s2 + bs) 2 n o j c ( s 2 + sojc/2n) 2nKOJds2 + K2o?ds S2 + 2r\(x>cS + QJ>2 S2 + 2j]KCOdS + K2W2d S2 + 2r}K0)dS + K2OJ2
Var[xi(0] = /v ( , , ^ , , , ) = K2L s + ojd s2 + 2nKiOdS + K2cod j ^ v \5- 3 + ^2(1 +2r)K)a>d + s(2nK+K2)oJd + K20Jd,
2 ((1 +2i7/t)(2^K+r2)w5 - K2ufy 2
-
3.8. Mathematical Details 42
Rv(y) = Var[xi(0]2wrf = cod2r]K(K2 + 2nK + 1)
Rv(y) := Rv(y)co2d 1
2TJK(K2 + 2TJK + 1) (3.77)
Var[w(0] = Iv 1 2nKu>dS2 + K2u)ds
s + cod s 2 + 2r\K0)ds + K2O>2 2rjKO>dS2 + K2co2ds
d , \S3 + s 2(\ +2nK)cod + s(2rjK+K2)a>d + K2cod; {2t]Kcod)2{2T}K + K 2) + K4G>4 ^ 1 + 4n2)/c + St]3)
2cod2r]K(K2 + 2T]K +1) 4TJK(K2 + 2TJK + 1)
R v ( u ) = Yar[u(t)]2tOd = OJ2K\(1+4?]2)K + 8T]3)
2TJK(K2 + 2nK + 1)
-
Chapter 4
Downstream Processes
4.1 Introduction
In the previous chapter, the smoothness of the outlet flow Fu(t) is mathematically captured as the
variance of Fu(t) = v(t) and/or Fu(t), and the problem is solved by optimization with the following performance index (3.15),
J\ = VarLMO] +p2 (Var[v(0] +pVar[U(0]),
wherey\(t) is the tank level and u(t) is the deviation of Fu(t) from its mean value Fm. The reason
for requiring outlet flow smoothness is to minimize adverse effects of tank level control on the
downstream processes. In this chapter, the downstream process variable is explicitly included in
the problem formulation. The averaging level control problem is considered as the minimization
of the downstream variable variance while keeping the tank level between high and low limits. Let
y 2 ( t ) denote the downstream process variable. The minimization of Var[y2(r)] makes the problem
mathematically tractable, and also it makes practical sense for some cases. For example, y 2 ( t ) may
represent a product quality parameter, and y2(t) must be in a specified range for the product to be
acceptable. Minimizing Var[y2(0L then maximizes the product yield.
In this chapter, the following performance index is used.
J := Var[y,(0] +P^ Var[y2(0] +p2Var[v(0] (4.1)
As in Chapter 3, pi and p 2 serve as Lagrange multipliers so that Var[yi(0] is minimized subject to
Pi Var[v(0] + plVar\y2(t)] = c, where c is some constant. Actually, c is not given, and pi and p 2 are
43
-
4.2. Downstream Process Model 44
adjusted to obtain a required Var[yi(r)], which is specified as one of the design parameters. Since
the problems are treated in continuous time, non-zero p\ is necessary to keep the feedback gain K
and Var[v(r)] finite.
4.2 Downstream Process Ivlodel
Figure 4.1 depicts a model of a combined system of the tank level control loop and the downstream
process, where the outlet flow Fu(t) enters at the output of the downstream process P2. The
c2
Fd(i)-
Ci
Fu(t) 6-r*y2(t)
Figure 4.1: The tank level control loop (Ci and P\) and the downstream process P2. The outlet flow Fu(t) enters at the output of the P2.
purpose of the model is to investigate how the downstream process influences tank level controller
design when the statistical relation between Fu(t) and y2(f) is taken into account. The model is not meant for designing a multivariable system, where Fu(t) is determined from both y\(t) and
y2(t). Thus, the interaction between Fu(t) and y2(t) are simplified excluding any deadtime or gain units, which may exist between Fu(t) and y2(t). Therefore, it is assumed that the downstream process variable y2(t) is not available to the tank level control loop. If Fu(f) affects more than one
downstream processes, y2(t) is a hypothetical "representative" process variable that has no real
physical entity.
Downstream processes are represented here by a first-order plus deadtime model.
K2 Pi = -e~TS = K2-
T2S +1 S + 0>2
where K 2 is the steady state gain, T 2 the time constant, co 2 = T 2 the cutoff frequency and T the
-
4.2. Downstream Process Model 45
deadtime. It is assumed that P2 is closed with a PI controller C2,
C2 = Kc ( l + -U = KCS-^-, (4.2) \ Trsj s
where Kc is the proportional gain, Tr the integrating time, and cor = l/Tr. Then the sensitivity
functions'2 is o 1
-
4.2. Downstream Process Model 46
20
10
0
-10 CD T3
-20
-30
-40
-50 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
log(to)
Figure 4.2: The magnitude of Gvi (solid line) and two approximations for io2 - K2 = r = 1 and Kc - 0.3. The dashed line is for OJp = io'p and dotted line is for a>p = l.4a>'p.
Figure 4.2 shows the true Gvi and two approximations with different choice of a>p.
When u enters at the input of the downstream process, Gvi has an additional turn-over fre-
quency a>2 from which \Gvi\ decreases with -40dB/decade. If such a model is used, the resulting
controller, which is more complex, will produce outputs that have more high frequency compo-
nents. Therefore, in the sequel, the downstream process is assumed to be a simple low-pass filter
driven by v(t). Let x3(t) denote the deviation of^ (0 from its mean value:
xi(t):=y2{f)-E\y2(t)l (4.10)
Since x3(t) is the output of a low-pass filter driven by v(r),
x3(i) = -ojpx3(t) + v(t). (4.11)
In reality, there is some "gain" involved between v(t) and x3(t) depending on where and how u(t)
enters the downstream process. Then the downstream model is
x 3 ( s ) = - ^ - v ( s ) , (4.12) S + (Op
where Kj is the gain. The downstream process variable x3(t) is assumed not available for the tank
level control loop, and recovered by a noise-free observer from xi(t) and w(r) using the downstream
process model (4.12). Therefore,
Nc(s)
'The exact value depends on the plant uncertainty and T (Ogawa, 1995), but the IMC rules recommend similar
values (Morari and Zafiriou, 1989) '
-
4.3. State-Space Solution 47
where CD(S) is the controller transfer function. CD is calculated to attain required Var[yi(0] and
Var[v(r)] with the minimum Var[y2] by adjusting pi and p 2 in (4.1). Let CD denote the optimum
controller for Kj = 1, and C'D for Kd \, both with the same Var[yi(r)] and Var[v(f)]. Let cr2,
denote Var[y2(r)] f r Kd = 1 and o\ for Kd \. From (4.12), cr2 = K^a2 if the same controller
is used for both cases. If CD i1 C'D, cr2 Kda2. This contradicts the fact that CD and C'D are the
optimum controllers for each case. Thus, CD must be equal to C'D. This means that Kd does not
influence the optimum controller. However it does influence the value of p 2 to obtain the desired
Var[yi(r)] and Var[v(r)]. Thus, Kd is set to a convenient value for calculation.
4.3 State-Space Solution
The downstream process variable x->,{f) is added to the state variables of Chapter 3.
x \ (0 = y\ (0 ~~ yr tank level error
x2(0 = Fd(i) - Fm input disturbance deviation
*3(0 = yi(f) ~ E[y2(0] downstream deviation variable
u(t) = Fu(t) - Fm outgoing flow deviation (manipulated variable)
In this chapter, the input disturban