modelling of paper mill

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

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.

ii

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

iii

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

i v

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

v

8 Conclusions 130 8.1 Summary 130 8.2 Future Research 132

Bibliography 134

vi

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

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

viii

5.4 0i, 02 and

.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

x

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.

xi

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)

1

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

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.

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


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


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.


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.

Chapter 2


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.

7

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.


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:


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


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


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

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.


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.

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


(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


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]),


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)


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


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


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".


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)


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


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/


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.


3.5.1 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)


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)


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.


3.6.1 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


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.


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


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


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


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

modelling of paper mill

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