Master of Science Thesis in Electrical EngineeringDepartment of Electrical Engineering, Linköping University, 2017

Modeling and Control ofLime Addition in a FlotationProcess

Rasmus Tammia

Master of Science Thesis in Electrical Engineering

Modeling and Control of Lime Addition in a Flotation Process

Rasmus Tammia

LiTH-ISY-EX--17/5071--SE

Supervisor: Gustav Lindmarkisy, Linköping University

Johan BoströmBoliden Mines Technology

Examiner: Martin Enqvistisy, Linköping University

Division of Automatic ControlDepartment of Electrical Engineering

Linköping UniversitySE-581 83 Linköping, Sweden

Copyright © 2017 Rasmus Tammia

Abstract

Flotation is an important and versatile mineral processing technique that is usedto separate hydrophobic materials from hydrophilic. This technique makes it pos-sible to mine complex ores that otherwise would have been regarded as uneco-nomic and non-beneficial. In this case flotation is used to separate copper fromthe unwanted gangue.

The addition of lime is used to control the pH level in the flotation’s pulp, whichgoverns the selectivity of the process, i.e. which minerals are recovered. Currently,fluctuating concentration grades of the produced metals have been observed inBoliden Aitik. Therefore, Boliden proposes a new control strategy which aims tomaintain a constant ratio between the added lime and the incoming ore flow, butat the same time ensuring that the pH level is maintained within allowed limits.

The aim of this thesis is to develop a model that captures the most essential dy-namics of a process stage where lime is added, and then evaluate the suggestedcontrol strategy by studying suitable control structures.

A linear model describing the system dynamics in a specific operating region isobtained by conducting step response experiments on the process. The model isthen used to obtain a model describing the disturbances of the process, therebyyielding a complete model that describes the most important dynamics.

The most promising control structure utilizes the concept of selective control,where a ratio controller is allowed to maintain a constant ratio as long as the pHlevel is within allowed boundaries. The pH level is maintained within the bound-aries with upper and lower bound pH controllers that utilize the concept of anequivalent control objective (known as the strong acid equivalent ) in order toachieve satisfying pH control. The results show that the control structure is ableto maintain a constant ratio, and also ensure that the pH level is kept within theallowed limits.

A cascade inspired pH ratio controller is also studied and evaluated. The resultsshow that this pH ratio controller is only able to maintain a constant ratio as longas the incoming ore flow is constant. However, the outcomes also suggest that theconcentration grades are either sensitive to variations in the ratio between addedreagent and incoming ore flow, or that there is something else that causes themto vary.

iii

Sammanfattning

En viktig och mångsidig metod inom gruvindustrin för separation av hydrofilaoch hydrofoba ämnen är flotation. Denna metod möjliggör utvinning av kom-plexa malmer som annars hade betraktats ekonomiskt ogynnsamma. I detta fallanvänds flotation till att utvinna koppar ifrån malm.

Processens selektivitet, det vill säga vilka mineraler som ska extraheras, bestämsav pH-värdet i processens slurry och denna regleras genom tillsatsen av kalk. Pågrund av att variationer i det utgående kopparkoncentratet i Boliden Aitik harobserverats, föreslår Boliden en ny reglerstrategi. Denna strategi ska eftersträvaatt upprätthålla en konstant kvot mellan den kalk som tillsätts, och det inkom-mande godsflödet, men samtidigt säkerställa att pH-värdet hålls inom tillåtnagränser.

Genom att jämföra lämpliga reglerstrukturer kan den nya reglerstrategin evalue-ras. För att lyckas med detta krävs det att en modell, som beskriver de viktigastedynamiska egenskaperna i ett processteg där kalk tillsätts, tas fram .

En linjär modell för systemdynamiken erhålls via stegsvarsexperiment. Dennamodell används sedan till att ta fram en störningsmodell, och därigenom har enkomplett beskrivning av den huvudsakliga dynamiken erhållits.

Tack vare principen för reglering med väljare kan ett reglersystem beståendes aven kvotregulator och två pH-regulatorer sättas samman. Kvotregulatorn tillåtshålla en konstant kvot så länge pH-värdet är inom de tillåtna gränserna, ochvia en olinjär kompensering av pH-reglermålet ser de två pH-regulatorerna tillatt pH-värdet inte avviker utanför den övre och nedre gränsen. Resultaten visaratt reglersystemet lyckas upprätthålla en konstant kvot, och även se till att pH-värdet bibehålls inom de tillåtna gränserna.

Även en kaskad-inspirerad reglerstruktur utvärderas där resultaten pekar på attreglerstrukturen endast lyckas upprätthålla en konstant kvot så länge det inkom-mande godsflödet är konstant. Resultaten tyder även på att det utgående kop-parkoncentratet antingen är känsligt för variationer i kvoten mellan kalktillsatsoch godsflöde, eller att variationerna förorsakas av någonting annat som ännu ärokänt.

v

Acknowledgments

I would like to thank Martin Enqvist and Gustav Lindmark at Linköping Univer-sity for their help and interesting discussions during this project. I am also verythankful for all the help my supervisor Johan Boström and the staff at Boliden hasgiven me throughout the course of this thesis, and for allowing me to conduct mythesis at Boliden.

Finally, I would like thank my parents for their love and support throughoutmy studies, and my brother Markus for still being my biggest source of inspira-tion.

Skellefteå, May 2017Rasmus Tammia

vii

Contents

Notation xi

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Earlier Studies and Related Work . . . . . . . . . . . . . . . . . . . 3

1.4.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4.2 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Introduction to Flotation 72.1 Principles of Flotation . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Flotation Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Modeling and Controlling the Regulators . . . . . . . . . . . . . . 10

3 Modeling and System Identification 113.1 Linear Parametric Models . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Linear Parametric Model Structures . . . . . . . . . . . . . 113.1.2 Prediction Error Method . . . . . . . . . . . . . . . . . . . . 13

3.2 Modeling Industrial Processes . . . . . . . . . . . . . . . . . . . . . 143.2.1 Process Models Obtained from Step Responses . . . . . . . 14

3.3 Validation of Linear Parametric Models . . . . . . . . . . . . . . . . 153.3.1 Cross Validation . . . . . . . . . . . . . . . . . . . . . . . . . 153.3.2 Residual Analysis . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Control Theory and Strategies 174.1 Control strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1.1 PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.1.2 Ratio Control . . . . . . . . . . . . . . . . . . . . . . . . . . 184.1.3 pH Ratio Control . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 The Strong Acid Equivalent . . . . . . . . . . . . . . . . . . . . . . 204.2.1 An Equivalent Control Objective . . . . . . . . . . . . . . . 20

ix

x Contents

5 Results 235.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.1.1 Step Response Tests . . . . . . . . . . . . . . . . . . . . . . . 235.1.2 Modeling Disturbances . . . . . . . . . . . . . . . . . . . . . 295.1.3 Identification of Closed Loop Systems . . . . . . . . . . . . 34

5.2 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2.1 Ratio Control . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2.2 pH Control - Integral Windup . . . . . . . . . . . . . . . . . 385.2.3 pH Control - Modifying the Derivative . . . . . . . . . . . . 395.2.4 pH Control - Step Responses . . . . . . . . . . . . . . . . . . 395.2.5 pH Control - Disturbances . . . . . . . . . . . . . . . . . . . 405.2.6 Implemented Control System . . . . . . . . . . . . . . . . . 415.2.7 pH Ratio Control . . . . . . . . . . . . . . . . . . . . . . . . 45

6 Conclusions 496.1 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . 496.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

A Appendix 53A.1 OE Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 53A.2 Disturbance Model Parameters . . . . . . . . . . . . . . . . . . . . 53A.3 Control Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 54A.4 Control System Selector . . . . . . . . . . . . . . . . . . . . . . . . . 55

Bibliography 57

Notation

Nomenclature

Notation Interpretation

e(t) Control Errorε(t), ε(t, θ) Prediction Error

η(t) Noise-Free SignalF Controller

G(q, θ) Rational Transfer Function for System DynamicsG SystemG(s) Transfer FunctionH(q, θ) Rational Transfer Function for Disturbance DynamicsKp Static GainL Time Delayn(t) White Gaussian NoisePr , σ VariancepHsp pH Setpoint

ˆΦ PeriodogramΦ Spectral Estimateq Shift Operator

pHcontrol Measured pH with pH ControllerpHratio Measured pH with pH Ratio Controllerr SetpointRε Covariance Function for the Prediction ErrorRu Covariance Function for the Control SignalRε(τ) AutocovarianceRεu(τ) Cross-CovarianceS0 Transfer Function From r to u and/or r to u

xi

xii Notation

Nomenclature

Notation Interpretation

T Time ConstantTs Sample Timeτ Time Lagθ Model Parameter VectorθN Model Parameter Vector that Minimizes VN (θ)u(t) Control Signal/Input Signalu Simulated Output From S0 (pH)u Simulated Output From S0 (Nonlinear Compensation)

VN (θ) Loss Functionω(t) Noise/Disturbance Componenty(t) Outputy(t|θ) Predicted Outputy Output from ARX Model (pH)Y Output from ARX Model (Nonlinear Compensation)yr (t) External Process Variable/Incoming Ore Flow

Y , [OH−] Molar Concentration of Hydroxide IonsYsp Setpoint (Strong Acid Equivalent)Ysp Setpoint (Molar Concentration of Hydroxide Ions)

Abbreviations

Abbreviation Interpretation

ar Autoregressivearx Autoregressive Exogenousarmax Autoregressive Moving Average Exogenousbj Box-Jenkinsoe Output-Errorpi Proportional-Integralpid Proportional-Integral-Derivative

1Introduction

This report presents the methods and results of a master thesis that was car-ried out during the spring of 2017 at Boliden Mines’ Technology department inBoliden, Sweden. The master thesis was carried out by one student enrolled atLinköping University.

This chapter gives a brief introduction to the subject of this master’s thesis. Thechapter describes the problem formulation, the scope, method, and some relatedwork.

1.1 Background

Boliden is a global Swedish mining and smelting company with main focus inproducing copper, zinc, gold, lead and silver. The company conducts exploration,mining and milling in Sweden, Finland and Ireland. Today, Boliden is the indus-try leader in the Nordic region in terms of sustainable metal production fromdeposits by the recycling of used metals.

Sweden’s largest open pit copper mine is located in Aitik, just south of Gällivare.The ore extratced from the open pit undergoes several processing stages in orderto obtain the sought metals. One of these processing stages is called flotation,which is a standard chemical separation technique used in the mining industry,and the purpose of the flotation process is to separate the valuable minerals fromthe unwanted gangue.

A flotation process is a very complex chemical process, and contains a lot of sub-processes and interactions that to this day have not been fully understood. How-

1

2 1 Introduction

ever, the pH levels at several stages in the flotation process play a crucial part(crucial in the sense that the pH level affects the selectiveness in the mineral sep-aration of the process) and a common control method for keeping the pH levelsconstant is the addition of lime. However, the current controller might cause fluc-tuating concentration grades of the produced metals. Boliden has therefore sug-gested an alternative control strategy that controls the ratio between the amountof added lime and the ore flow. The hypothesis is that this will yield a higherexchange in the sought copper, but at the expense of more fluctuating pH levelsin the flotation process. The idea of this thesis is to examine the suggested con-trol strategy and in order to do that, modeling and evaluation of suitable controlstructures have to be performed.

1.2 Problem formulation

Boliden has proposed that instead of focusing on maintaining constant pH levelsat several stages in the flotation process, an alternative would be to control andmaintain a constant ratio between the amount of added lime and the incomingore flow, but at the same time ensure that the pH levels only fluctuate withingiven limits.

The suggested control method will hopefully yield a higher exchange in the soughtcopper and thus generate more profit. By reducing the fluctuating concentrationgrade, a more stable grade of copper concentrate will enable the smelting plant,Rönnskär, to extract the same amount of copper (as well as a higher amount)in their refining process (instead of different amounts and thus not being ableto guarantee customers that the delivered product contains a specific amount ofcopper).

Based on the problem formulation some challenges arise. In order to evaluate thesuggested control strategy and to avoid unnecessary production losses, a modelthat describes the dynamics between added lime, ore flow and pH level is needed.Because of the lack of knowledge on the relationship between lime, ore and pH,black-box modeling techniques are preferred.

1.3 Limitations

Lime is added at several stages in the flotation process, but this thesis only con-cerns one process stage. However, since the main principles of the process stagesare more or less the same (the biggest difference being the dynamics, i.e. slowerand faster dynamics), the methods presented in this thesis are applicable to theremaining process stages where lime is added.

1.4 Earlier Studies and Related Work 3

Reliable models usually require large sets of data. However, because of factorssuch as restrictions on carrying out experiments for data collection (too long ex-periments might cause production losses), and the fact that the process, over thecourse of time, tends to behave differently (e.g. even the seasons have an impacton the process and the acidity of the incoming ore) and has very slow dynamics,it is very hard to obtain a model that accounts for all of these behaviours of theprocess. Therefore, the models in this thesis should merely be viewed as validrepresentations of how the process could behave, since these are based on rathersmall sets of data.

1.4 Earlier Studies and Related Work

This section presents earlier studies and related work. The main goal of pH con-trol strategies (e.g. those presented in [12] and [14]) is to obtain a good pH controlsystem, which is not the case in this thesis. The exact details for why ratio controlwould be a better strategy on a chemical engineering level is beyond the scopeof this thesis. Therefore, more general guidelines and earlier studies of modelingand control of various industrial processes (e.g. as done in [8] and [2]) have servedas a basis for how to approach the problem at hand.

1.4.1 Modeling

One of the most fundamental parts of this project is modeling the relationshipbetween lime addition, ore flow and measured pH level. In [4], the theoreticalframework for system identification is covered (i.e. different methods for identifi-cation, model validation techniques and model construction techniques). Shouldthe modeling require additional signal processing (e.g. data pre-processing toremove high-frequency components, constructing signal models, etc.), then [6]gives a comprehensive treatment of modern signal processing theory.

Earlier studies regarding modeling have been performed at the mining site inAitik. In [8], a case study of modeling the flotation process in Aitik is presented.In this study, conventional system identification methods were applied, and themain idea was to approximate the dynamics of the process with different linearmodels valid in the neighbourhoods of different operating points. These modelswere then intended to be applied for the design of model based control strategies.Validation tests showed a satisfactory agreement between the performance of thevarious models and the measurements. The study served as a basis for future de-sign of model based controllers.

4 1 Introduction

1.4.2 Control

Modeling of industrial systems, and the foundation of discrete-time controllers,are covered in [3]. The book mentions how to take into account and counter un-wanted phenomena (e.g. integral windup). The proposed control strategy for thisthesis was to control the ratio between lime and ore flow, and ratio control strate-gies are also given comprehensive treatments in this book.

Attempts on identifying a state of the art flotation control strategy is somethingthat is done in [2]. However, this proved to be very difficult due to factors suchas the nonlinear behaviour of the process; many interactions between variables(which according to [2] omits control strategies involving a conventional SISOsystem due to non-satisfying results); and lack of important measurable controlvariables. This might very well be the case for the suggested control strategy athand, even though it is not flotation control itself that is considered in this thesis.

In [2] it is suggested that perhaps a model based optimizing control strategy isthe most interesting path to follow. However, this requires a good model and theweakest point proved to be the reliability of the model. Another problem, accord-ing to [2], is the lack of studies and knowledge of automatic control in the fieldof mineral processing.

More advanced control strategies are discussed in [10]. The report argued that amore advanced control strategy might come with some potential drawbacks interms of the limited availability of sensors/actuators as well as possibilities foroperator interactions. This study presented strategies and instrumentation forthe flotation control in Aitik.

The report formulated overall and individual control objectives for the processstages and this would then serve as a basis for development of a more detailedcontrol strategy. It should be noted that [10] only presented ideas and no conclu-sions, so no final statements could be made regarding the suggestions that werepresented in the study.

Again, nonlinearity is a major problem, and [9] clearly states that the advancedcontrol strategies that might be able to handle these problems only have beenstudied in simulation plants. Another aspect to consider, according to [9], is howto successfully implement more advanced control strategies and at the same timemake them manipulable and controllable for process operators that supervise theplant.

In [14], problems associated with pH control are mentioned, namely that pHcontrol is a typical nonlinear and time-delayed system and for this reason it isdifficult to obtain accurate mathematical models for this type of control, and it is

1.4 Earlier Studies and Related Work 5

one of the most complex control objects encountered in the chemical and mineralprocessing industry.

Based on nonlinear pH models, [7] compares a linear and a nonlinear pH controlstrategy. It was concluded that nonlinear pH control is superior if the character-istics of the process are well known. However, since this is mostly not the case, alinear feedback from pH is often as good as a nonlinear feedback, according to [7].

There are numerous ways to account for the nonlinearities of the pH control prob-lem. In [13], an equivalent control objective is proposed which is linear in thestates. The new objective (physically interpreted as the strong acid equivalent ofthe system) thereby replaces the pH control objective. Furthermore, [13] showsthat achieving satisfying results in the equivalent linear control objective alsoyields satisfying pH control.

An overall review of variable transformation approaches, that have been used tocounteract the nonlinearities of the pH control problem, is presented in [1]. Thereview stated that similar concepts to the ones presented in [13] have been suc-cessful in achieving better pH control by compensating for the nonlinearities invarious ways.

Because of the promising results presented in [13], this linear control objectivewas further studied for this thesis.

2Introduction to Flotation

This chapter provides a brief introduction to the concepts of flotation as wellas how these concepts can be applied to mining applications in order to form aflotation process.

2.1 Principles of Flotation

The physio-chemical process known as flotation is a separation process that uti-lizes the difference in surface properties of the valuable minerals and the un-wanted gangue (i.e. the commercially worthless material in the extracted ore).

Before flotation can take place, the ore that is to undergo the flotation process isreduced to finer particles through crushing and grinding. The ore is then mixedwith water to form a slurry (more properly known as pulp) in the flotation cell.By blowing air into a flotation cell, the desired minerals attach to air bubbles thatare transported to the top of the cell to form a froth. The froth is removed fromthe cell, producing a concentrate of the target mineral. However, since most ofthe desired minerals in the pulp rarely are water-repellent in their natural state,it is necessary to add flotation reagents to the pulp. This will then render thesought minerals hydrophobic and thus facilitating air bubble attachment [12].

In order to ensure economical benefits and higher concentrate recovery of a flota-tion process, many cells are put in series to form a bank. The minerals that arenot recovered in a flotation cell move on to the next cell in the bank. These min-erals are referred to as tailings. In Figure 2.1 the principles of flotation are shown.

7

8 2 Introduction to Flotation

Figure 2.1: The main principles of flotation shown in a flotation cell. At thebottom of the cell an agitator is used to achieve mixing and thereby facilitat-ing air bubble attachment. The tailings consist of the minerals that were notrecovered in the previous and current flotation cell (published with permis-sion from Boliden Mineral AB).

The flotation reagents that are used to render the minerals hydrophobic are calledcollectors and these are the most important reagents added to the process sincethe hydrophobic properties govern how well the sought minerals will attach tothe air bubbles. Another type of reagent are the frothers which are used to main-tain a reasonably stable froth. If the froth is unstable, the air bubbles that formthe froth will burst and drop the mineral particles.

Regulators (also known as modifiers) are used to either activate or depress min-eral attachment to the air bubbles in the cell, and they are also used to control thepH level which affects the selectivity of the flotation process. What this means isthat regulators are used to modify the action of the collectors, either by reducingor intensifying their water repellent effect on the minerals’ surface, thus makingthe collectors more selective towards certain minerals. This selectivity is usuallyensured by a high alkalinity in the pulp, and one of the most common regulatorsused to achieve this level of alkalinity is lime [12].

2.2 Flotation Processes 9

2.2 Flotation Processes

The principles of flotation given in Section 2.1 describe what happens in one flota-tion cell. In order to maximize the recovery of the target mineral, the flotation isundertaken at several stages consisting of multiple flotation cells in series at eachstage. Arranging the cells in series form a bank. The stages are called rougher,scavenger and cleaner, and together with the mills that grind the ore they formthe process that is called a flotation process.

In the rougher stage, rougher concentrate is produced. Pulp enters the first cellof the bank giving up a part of its minerals as froth. The minerals that were notrecovered in this cell passes on to the next cell where more mineralised froth isrecovered. This goes on until the last cell in the bank, and the overflow tailingsenter the scavenger stage in the process. Since a finer ore ground would requiremore energy for recovery, the main goal of the roughing stage is to recover asmuch of the sought minerals as possible, but with less emphasis on the producedconcentrate’s quality.

The rougher tailings usually contain minerals that were not recovered in therougher flotation. The objective for the scavenger flotation is to recover any valu-able minerals from these tailings. The froths in the scavenger flotation have lowerconcentrate grade than corresponding rougher froth. In order to be able to re-cover the minerals of interest in the scavenger flotation, the flotation conditionsare more rigorous in this flotation stage than in the rougher flotation.

In order to reject more of the unwanted gangue, the rougher and scavenger con-centrates are usually refloated in cleaner cells. The product yielded from thecleaners are the final concentrate of the flotation process. It is not uncommonto regrind the concentrate from the roughers and the scavengers in regrindingmills in order to achieve more liberation of the valuable minerals in the cleaningstage.

The whole flotation process is shown in Figure 2.2. It should be noted that thisis a very simplified description of a flotation process, but it still shows the mostimportant concepts of the process.

10 2 Introduction to Flotation

Figure 2.2: The figure shows a very simplified description of a flotation pro-cess. At the flotation process in Aitik, there are two parallel flotation lineswhich means that there are two parallell rougher, scavenger and cleanerstages, and also two primary and regrinding mills (published with permis-sion from Boliden Mineral AB).

It is not uncommon to refloat the tailings from the cleaner. This is usually doneby sending the cleaner’s tailings all the way back to the rougher flotation (whichfor example is the case at the flotation process in Aitik).

2.3 Modeling and Controlling the Regulators

In Section 2.1, regulators were described as reagents that are used to either acti-vate or depress mineral attachment to the air bubbles in a flotation cell, and theyare also used to control the pH level. For this thesis, modeling and controlling therelationship between lime and measured pH is considered. In the flotation pro-cess in Aitik, lime is added at several stages in the process. The first stage is in thetwo primary mills. The second stage is in two mixing tanks located between theprimary mill and the rougher flotation (these are the tanks that are used to createthe pulp). And finally, lime is also added before the regrinding mills.

3Modeling and System Identification

The first part of this chapter presents the theory and methods that are appliedwhen conducting system identification. The methods presented in this chapterare all based on what is commonly referred to as black-box modeling. The reasonfor this is because the models are constructed from only measurable signals, andthe models’ parameters have no direct physical interpretation, but are solely usedto describe the properties of the input-output-relationship of the system.

In Section 3.2, a strategy for how to obtain simple models of industrial processesis presented.

3.1 Linear Parametric Models

Linear parametric models are some of the most common model structures to ap-ply when one is unable to describe the dynamics of the system from a physicalpoint of view, or when there is a lack of knowledge of the underlying processes ofthe system. Section 3.1.1 describes commonly occurring linear parametric mod-els and Section 3.1.2 describes how these models are estimated from data.

3.1.1 Linear Parametric Model Structures

Consider a discrete-time model described by the relationship

y(t) = η(t) + ω(t) (3.1)

11

12 3 Modeling and System Identification

In (3.1) ω(t) is used to describe the noise and η(t) is the noise-free signal. Thisexpression can further be expanded to the following

y(t) = G(q, θ)u(t) + H(q, θ)n(t) (3.2)

where G(q, θ) is a rational function described by the the shift operator q and theparameters θ, and u(t) is the input signal. In the same way, H(q, θ) is a rationalfunction used to describe the disturbances that enter the system, and n(t) is whitenoise. G(q, θ) and H(q, θ) in (3.2) can be expressed as

G(q, θ) =B(q)F(q)

=b1q−nk + b2q

−nk−1 + ... + bnbq−nk−nb+1

1 + f1q−1 + ... + fnf q−nf (3.3)

and

H(q, θ) =C(q)D(q)

=1 + c1q

−1 + ... + cncq−nc

1 + d1q−1 + ... + dnd q−nd

(3.4)

which yields the Box-Jenkins model (BJ model), which is the most general linearparametric model of all the commonly occurring model structures (i.e. it candescribe the dynamics for both the noise-free signal u(t), and the white noisen(t)).

In (3.3) and (3.4), nb, nc, nd , nf and nk are the structure parameters (nk is thetime delay), and bi , ci , di and fi are the parameters that are to be estimated fromgiven data (i.e. input and output signals).

An important special case of (3.2) is to ignore modeling the noise that enters thesystem (i.e. setting H(q, θ) = 1 in (3.4)). This will yield the Output-Error model(OE model)

y(t) = G(q, θ)u(t) + n(t) (3.5)

By allowing the denominators for G(q, θ) and H(q, θ) to coincide with each other,i.e.

F(q) = D(q) = A(q) = 1 + a1q−1 + ... + anaa

−na (3.6)

an ARMAX model

A(q)y(t) = B(q)u(t) + C(q)n(t) (3.7)

3.1 Linear Parametric Models 13

is obtained. Another special case is setting C(q) = 1 in (3.7). This will yield anARX model

A(q)y(t) = B(q)u(t) + n(t) (3.8)

The linear models covered in this section are summarized in Table 3.1.

Table 3.1: Linear parametric models.

Model ExpressionBJ model y(t) = B(q)

F(q)u(t) + C(q)D(q)n(t)

OE model y(t) = B(q)F(q)u(t) + n(t)

ARMAX model A(q)y(t) = B(q)u(t) + C(q)n(t)ARX model A(q)y(t) = B(q)u(t) + n(t)

3.1.2 Prediction Error Method

The main goal with the models covered in Section 3.1.1, is to adjust their param-eters so that they will be able to predict a measured signal. A model is deemedgood if it can predict future measurements correctly, given the past measure-ments and inputs. Let θ denote the parameter vector of an obtained model, wherethe model can be used to predict a value of the signal y(t). It is then possible toevaluate how accurate this prediction is by calculating the prediction error

ε(t, θ) = y(t) − y(t|θ) (3.9)

where y(t|θ) in (3.9) is the predicted value of y(t).

When enough data has been collected over an interval t = 1, ..., N , it is possibleto evaluate how well a model, given the parameter value θ, is able to describe thesystem’s dynamics. This can be done by evaluating a loss function

VN (θ) =1N

N∑t=1

(y(t) − y(t|θ))2 =1N

N∑t=1

ε2(t, θ) (3.10)

It is reasonable to choose the value of θ that minimizes (3.10), i.e.

θN = argminθVN (θ) (3.11)

What (3.11) states is that one should always choose the model that has the bestperformance in terms of predicting future values from observed data.

14 3 Modeling and System Identification

3.2 Modeling Industrial Processes

The linear parametric models presented in Section 3.1 often require rich informa-tive data (rich in the sense that the inputs to the system are able to bring forth allinteresting modes and characteristics of the system). This type of data might behard to obtain for numerous reasons. One reason could for example be that the in-puts that are required to acquire accurate models would cause too big productionlosses.

This chapter presents a more simple structured modeling alternative, often ap-plied when modeling industrial plants.

3.2.1 Process Models Obtained from Step Responses

A simple model of an industrial process can be obtained from more cost efficientexperiments. A step response can be studied for obtaining a model of a process. Ifthe input step u(t) is applied on the process, then the measured output y(t) canbe scaled such that it corresponds to a unit step in u(t)

u(t) ={

0, t < 0

1, t ≥ 0(3.12)

A basic model that often is able to capture the most typical characteristics of thestep response is the three parameter model on the form

G(s) =Kp

1 + sTe−sL (3.13)

where L denotes the time delay, T is the time constant and Kp is the static gain.With this model, the step response y(t) from the input in (3.12) becomes

y(t) =

0, t < L

Kp(1 − e−(t−L)/T ), t ≥ L(3.14)

which can be used to obtain the parameters in (3.13) (see [3] for more detailsregarding this topic and for alternatives to the model given in (3.13)).

3.3 Validation of Linear Parametric Models 15

3.3 Validation of Linear Parametric Models

There are several ways to validate linear parametric models. It is possible to an-alyze the collected data (e.g. through spectral or correlation analysis) before theactual modeling. This is a good way to get an indication of the potential presencedisturbances, or guidelines for minimum model order. This pre-analysis can thenbe compared to some of the outcomes from the validation techniques presentedin Sections 3.3.1-3.3.2. This is a good way to ensure whether the model is ableto capture the system’s dynamics or not and if there are any redundancies in themodel. However, the spectral analysis should not be utilized when data is col-lected in the presence of feedback (see [4] or [11] for more details regarding thissituation).

3.3.1 Cross Validation

After data has been collected for system identification, a good guideline for ob-taining more reliable models is to divide the collected data in two parts: estima-tion and validation data (e.g. one third of the collected data for estimation, andtwo thirds for validation). The estimation data can be used to estimate the param-eters θN in a model, whereas the validation data can then be used to evaluate theprediction error in (3.9). This validation technique, called cross-validation, will of-ten yield a more reliable model, but it also requires that enough data is collected[4].

3.3.2 Residual Analysis

Consider again the residuals in (3.9). These should ideally be uncorrelated of theinput signal. If this is not the case, then there are components in ε(t, θ) that arederived from u(t), which means that there are more system dynamics to describethan what the model y(t|θ) has been able to pick up. One way to analyze theresiduals’ correlation with the input signal is to study the cross-covariance

Rεu(τ) =1N

N∑t=1

ε(t + τ)u(t), |τ | ≤ M (3.15)

and check whether the values in (3.15) are close to zero. Furthermore, if the resid-uals really are uncorrelated, then for large values of N , the cross-covariance in(3.15) should approximately be normal distributed with mean value zero andvariance

Pr =1N

∞∑k=−∞

Rε(k)Ru(k) (3.16)

16 3 Modeling and System Identification

where Rε and Ru denote the covariance functions for u and ε. A standard proce-dure to check whether ε(t + τ) and u(t) are correlated for some τ is to plot (3.15)together with the lines ± 3 ·

√Pr and check whether Rεu(τ) has any values outside

of these lines. If that is the case, then ε(t + τ) and u(t) probably are dependent ofeach other for that value of τ .

If the noise signal’s properties are incorporated in the modeling, it is also reason-able to study the autocovariance of the residuals

Rε(τ) =1N

N∑t=1

ε(t + τ)ε(t) (3.17)

where (3.17) can be used to check if the residuals are mutually uncorrelated[4].

4Control Theory and Strategies

This chapter treats the theory behind the control strategies that were investigatedduring this thesis. In order to account for the nonlinearities of the pH controlproblem, an equivalent control objective is presented. The theory behind this con-trol objective, physically interpreted as the strong acid equivalent, is presentedin Section 4.2.

4.1 Control strategies

Several control strategies were proposed for this thesis, all of which are presentedin this section.

4.1.1 PID Control

As previously mentioned, the PID controller (where PID stands for Proportional-Integral-Derivative) is the most common control strategy to come across in themineral processing industry. The reason for this is because of its simplicity yetefficiency in achieving desired outcomes and results from a system or a process.A PID controller can be parameterized according to

u(t) = KP e(t) + KI

t∫0

e(τ)dτ + KDddte(t) (4.1)

17

18 4 Control Theory and Strategies

where the control signal u(t) is the output from the control system, the errore(t) = r(t) − y(t) is the difference between desired and measured output, the con-stants KP and KI determine the gains for the proportional and the integral partof the controller respectively, and the constant KD determines the gain for thederivative of the error e(t). A very simple way to interpret the PID-controller in(4.1) is that KP determines how fast the response will be, KI and the integral parteliminate the steady-state error, and KD and the derivative part serve to reduceoscillations.

A simple block diagram representation of a feedback system with a PID controlleris shown in Figure 4.1.

∑F G

−1

r e u y

Figure 4.1: Block diagram of a feedback control system where the PID con-troller is depicted as the block F and the system is represented by the blockG.

The PID controller often serves as a basis for more sophisticated control strategies(e.g. those covered in Sections 4.1.2-4.1.3) and it is not uncommon to disregardthe derivative part in mineral processing applications (i.e. the controller becomesa PI controller) due to the fact that the less complex PI controller is often able toachieve desired results just as well as the PID controller [12].

4.1.2 Ratio Control

Ratio control is a suitable choice when it is desired to control the ratio betweentwo physical variables [3]. This is a very common situation to come across inmineral processing. A block diagram representation of this strategy is shown inFigure 4.2.

4.1 Control strategies 19

∏ ∑F G

−1

r

yr

ryr e u y

Figure 4.2: Block diagram of a ratio control strategy where the symbol∏

represents multiplication.

The idea in this strategy is to control y(t) so that the ratio between y(t) and yr (t)(which is an external process variable) becomes equal to the reference value r(t).The controller in this case could for example be a PID controller. The error thenbecomes e(t) = ryr (t) − y(t). If the error is equal to zero, the desired ratio is ob-tained [3].

4.1.3 pH Ratio Control

Today, the pH levels in the flotation process are controlled in a PI feedback loopwhere the incoming ore flow is not considered. It is plausible that better controlcan be obtained by manipulating the reactant flow against the incoming ore flow[12]. This can be achieved in a cascade inspired control loop, which is shown inFigure 4.3.

∑F

∏C G

−1

pHsp r u pH

yr

Figure 4.3: Block diagram of the cascade inspired control strategy.

This control strategy translates the error e = pHsp − pH in to a ratio setpoint r.Multiplying the ratio setpoint with the incoming ore flow yr yields the amountof reagent needed to achieve a satisfactory pH level. The reagent flow is scaledthrough the constant C, yielding the control signal for the process.

In Section 4.1.2 a feedback loop was present in the ratio control strategy. How-ever, it turns out that a feedback loop is redundant due to the lack of dynamics

20 4 Control Theory and Strategies

(this will be further explained in Section 5.2), which is why only a scaling factorC is used to translate the reagent flow to the control signal u.

The benefits with this strategy is that the control signal is adjusted with respectto the incoming ore flow before entering the process, instead of adjusting it afterchanges in the pH levels are observed. If the incoming ore flow is more or lessconstant for a long period of time, the ratio would also remain constant if thepH levels are maintained at a constant level. This coincides with the suggestedcontrol strategy that is to be investigated.

A potential risk with this strategy is that should the incoming ore flow yr varymuch, the variable yr will act as a varying gain in the feedback loop which mightcause problems (this is not the case in [12] where the inner ratio controller doesnot lack dynamics and a feedback loop).

4.2 The Strong Acid Equivalent

For this thesis, it is important to obtain a reliable pH controller that ensures thatthe pH levels are maintained within certain boundaries. In the upcoming section,a new control strategy for achieving good pH control is presented.

4.2.1 An Equivalent Control Objective

The control strategies covered in previous sections are preferably applied to lin-ear systems. It is not uncommon to use these strategies to control the pH level ina process. These strategies aim to fulfill the following control objective

pH ≈ pHsp = C (4.2)

where pHsp denotes the setpoint and C is a constant.

However, due to its highly nonlinear nature, the control of pH is recognized asa difficult problem. In [13] a new approach is taken to counter this nonlinearbehaviour. This approach consists of defining an alternative equivalent controlobjective. The physical interpretation of the new control objective is the strongacid equivalent of the system and it takes the following form

Ysp ≈ f (pHsp) = 10−(pHsp) − 10(pHsp−14) = C (4.3)

4.2 The Strong Acid Equivalent 21

where the main advantage using this formulation is that it is linear (i.e. the rela-tionship between the input and output signal is linear) which simplifies controldesign. For a more intuitive interpretation of (4.3) it is reasonable to set

Ysp = −Ysp (4.4)

It is clear that (4.3) is a general expression that can be used in an arbitrary processinvolving pH control, but should the pH level be high with small variations (e.g.pH ∈ [9, 10]), then (4.3) can be approximated as

Ysp ≈ f (pHsp) = 10−(pHsp) − 10(pHsp−14) ≈ −10(pHsp−14) = C (4.5)

and by inserting (4.5) in to (4.4) will yield the expression of the molar concen-tration of hydroxide ions in a process (which is a good measure since it is theaddition of lime that is considered)

Ysp ≈ [OH−]sp = 10(pHsp−14) = C (4.6)

The main principles of the control objectives given in (4.3) and (4.6) are shownin Figure 4.4

f (pHsp)∑

F G f −1(Y )

−1

pHsp Ysp e u Y pH

−Y

Figure 4.4: A linear control structure that can be used to either utilize thestrong acid equivalent, or the molar concentration of hydroxide ions, in orderto obtain an equivalent control objective for the pH control problem.

From Figure 4.4 it can be seen that the controller uses the strong acid equivalent,or the molar concentration of hydroxide ions, to fulfill the control objectives in(4.3) or (4.6), which in turn fulfills the control objective in (4.2) (this is proven in[13]).

It should be noted that the benefits of the approximation that yields (4.6) arequite beneficial since calculating the pH from (4.6) is much easier than from (4.3).Calculating the pH from (4.6) would yield

pH = f −1(Y ) = log10(Y ) + 14 (4.7)

which is easy to implement in a real control system.

5Results

This chapter presents the outcomes and results from this study, along with someshort discussions.

5.1 Modeling

Developing a model that describes the dynamics of the process is important whenstudying how the reagents affect the pH values, and when evaluating controlstrategies.

Since lime is added at several stages in the process, the approach is to developa model only for one part which then will serve as a basis for evaluating controlstrategies, and also as a way to point out on how to obtain models for the remain-ing process stages.

The results presented in this section are valid for the mixing tank BL4101. Thistank is used to generate the pulp for the second line of the flotation process.

5.1.1 Step Response Tests

In Section 3.2 it was shown how a simple model can be obtained through morecost efficient step response tests, which in turn provide useful insight of the timeconstants, time delays and potential nonlinearities of the process.

23

24 5 Results

For the step response tests, the concept of the strong acid equivalent presentedin Section 4.2 is emphasized. The tests are carried out under normal operatingconditions, and both the input signal (added lime) and output signal (measuredpH level) are kept within allowed boundaries. Because of the operating regionsof the process, the approximation of the strong acid equivalent

Y ≈ [OH−] = 10(pH−14) (5.1)

that yields the molar concentration of hydroxide ions is sufficient. The outcomesfrom the step response tests are shown in Figure 5.1.

0 1 2 3 4 5

time [hours]

0

10

20

30

Step Response Tests

0 1 2 3 4 5

time [hours]

0

1

2

3

410

-4

Figure 5.1: Step response experiments. The upper plot shows the input sig-nal u (added lime) and um (u with removed offset), and the lower plot showsthe output signal Y (the molar concentration of hydroxide ions which is arecalculation from the measured pH values). The signals um and Y are usedto determine model parameters.

Due to linear models not being able to capture arbitrary differences between in-put and output signal levels, the offset in the signal u is removed

um = u − 5 (5.2)

where the signal um is used for determining parameters for linear models on theform

G(s) =Kp

1 + sTe−sL (5.3)

5.1 Modeling 25

One model from each step in Figure 5.1 is obtained by determining the parame-ters as described in [3]. The parameters’ values are given in Table 5.1 (e.g. G1(s)denotes the model obtained from the first step in Figure 5.1).

Table 5.1: Parameter values obtained from the step response tests. The unitfor the static gain is molar concentration and for the time delays and con-stants the unit is seconds.

Model ParametersParameters G1(s) G2(s)Static Gain (Kp) 1.18 · 10−5 1.14 · 10−5

Time Delay (L) 60 250Time Constant (T ) 570 450

The results presented in Table 5.1 show that the static gains of the models G1(s)and G2(s) are more or less the same which suggests that a linear model mightsuffice for these operating regions of the process.

The models G1(s) and G2(s) are validated by approximating them in the discrete-time domain and using um as input signal and then comparing the models’ out-puts YG1

and YG2to the value Y . An OE model GOE of the same order as G1 and

G2 is also obtained through system identification, yielding the output YOE (theOE model’s parameters can be found in Appendix A.1). The results are comparedin Figure 5.2.

0 1 2 3 4 5

time [hours]

0

10

20

30

Validation - Serieal Step Responses

0 1 2 3 4 5

time [hours]

0

1

2

3

410

-4

Figure 5.2: Validation of the models G1, G2 and GOE . The dashed lines showthe simulated signals and the black line shows the original signal obtainedfrom the experiments. The initial conditions for all models were set to zero.

26 5 Results

The simulated outputs in Figure 5.2 seem to match the measured output quitewell, but it is not clear whether the nonlinear compensation (5.1) actually hasbeen beneficial. Therefore, the outputs are recalculated to the pH scale. The re-sults are shown in Figure 5.3.

As can be seen from Figure 5.3, all models seem very promising. The only con-cern in their reliability is that when the input signal um becomes equal to zero,the pH drops significantly, and it is very uncertain whether that would occur inthe real process. Setting the lower bound ummin = 2.5% on the input signal yieldsa more reasonable output.

0 1 2 3 4 5

time [hours]

8

9

10

11Validation - Serieal Step Responses (pH)

0 1 2 3 4 5

time [hours]

9

9.5

10

10.5

11

Figure 5.3: Validation of the models in the pH scale. In the upper plot umminis equal to 0%, and in the lower plot ummin is equal to 2.5%. It can be seenthat with the lower bound on the input signal, the models are able to predictthe output more accurately.

Based on the results from the step response tests, and because of the fact that themodels G1 and G2 are more intuitive when tuning simple PID controllers (thiswill become more clear in Section 5.2), GOE is disregarded at this point.

Further validation has been done by studying old process data and also conduct-ing a residual analysis. A dataset was selected so that changes in the output mostprobably were due to changes in the setpoint, and not for disturbances that mayhave entered the process. The results from the validations are shown in Figure5.4.

5.1 Modeling 27

0 50 100 150

time [min]

0

5

10

15Validation - Old Process Data

0 50 100 150

time [min]

9.7

9.8

9.9

10

10.1

Figure 5.4: Validation of the models G1 and G2 from old process data. Theblack line r denotes the set point and the red line the measured output. Thedashed lines represent simulated outputs.

From Figure 5.4 it can be seen that both models are quite accurate at capturingthe main characteristics of the measured pH value. However, because modelingof disturbances is not yet taken in to account, the models will not be able to re-produce an output that coincides with the measured output.

The cross-covariance

Rεum(τ) =1N

N∑t=1

ε(t + τ)um(t), |τ | ≤ M (5.4)

can be normalized to obtain the corresponding cross-correlation (see [6] for moredetails regarding this topic). These correlations have been studied for all models.The results look promising as both cross-correlations suggest that ε(t + τ) andum(t) are uncorrelated. Autocorrelations (i.e. normalized autocovariances) havealso been included to illustrate the need of a disturbance model .

For illustrative purposes, the cross-correlation for GOE is also presented, and it ispossible to argue that the results for models G1 and G2 are slightly better than forGOE . The results from the residual analyses are shown in Figures 5.5 - 5.6.

28 5 Results

-20 -10 0 10 20-1.5

-1

-0.5

0

0.5

1

1.5Autocorrelation

-20 -10 0 10 20

Cross-Correlation

Residual Analysis

Lag

Am

plit

ude

Figure 5.5: Residual analysis for G1 (the outcomes from the residual analysisfor G2 were more or less identical). The light blue shaded areas mark theconfidence intervals.

-20 -10 0 10 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Autocorrelation

-20 -10 0 10 20

Cross-Correlation

Residual Analysis

Lag

Am

plit

ude

Figure 5.6: Residual analysis for GOE . The light blue shaded areas mark theconfidence intervals.

The autocorrelations clearly show that a disturbance model is required to cap-ture the important disturbance dynamics of the process. This shortcoming in themodels will be addressed in the upcoming section. Because of having slightly

5.1 Modeling 29

more matching outputs in Figures 5.2-5.4, G1 is considered as the most promis-ing model.

5.1.2 Modeling Disturbances

Up to this point, the obtained model has the structure of an OE model

y(t) = G(q, θ)u(t) + n(t) (5.5)

which only uses parameters to describe the system dynamics and not the dynam-ics of the disturbances. However, since the process contains a substantial amountof process and measurement disturbances, a good and reliable model that isable to describe these disturbances is important to obtain. A reliable disturbancemodel is also important when evaluating controller performance. The model isderived by first considering a general model structure on the form

y(t) = G(q, θ)u(t) + H(q, θ)n(t) (5.6)

and by rearranging (5.6) to

y(t) − G(q, θ)u(t) = H(q, θ)n(t) (5.7)

it is possible to estimate a disturbance model H(q, θ) since y(t) and u(t) can bemeasured, and G(q, θ) is the model obtained in Section 5.1.1. The disturbancemodel in (5.7) will be assumed to have zero mean white Gaussian noise, withvariance σ , as its input.

The disturbance can be modeled as an AR(n) process (n denotes the model or-der)

A(q)ω(t) = ω(t) + a1ω(t − 1) + ... + anω(t − n) = n(t) (5.8)

which will yield the following noise model

ω(t) = H(q, θ)n(t) =1A(q)

n(t) (5.9)

which can be interpreted as filtered white Gaussian noise. The data used for mod-eling the disturbances is shown in Figure 5.7.

30 5 Results

0 2 4 6 8 10 12 14 16 18

time [hours]

12

13

14

15Data Used for Modeling Noise

0 2 4 6 8 10 12 14 16 18

time [hours]

9.9

9.95

10

10.05

10.1

Figure 5.7: The figure shows the data that is used to model the disturbancesof the process. Before estimating models, the pH scale is recalculated to themolar concentration of hydroxide ions.

By carrying out the calculations according to (5.7) but with um(t) = u(t) − 5 asinput signal to G(q, θ), the disturbance component ω(t) shown in Figure 5.8 isobtained.

0 2 4 6 8 10 12 14 16

time [hours]

-3

-2

-1

0

1

2

310

-5 Disturbance Component

Figure 5.8: The disturbance component ω(t).

5.1 Modeling 31

For an AR model, the minimized loss function

VN (θ) =1N

N∑t=1

(ω(t) − ω(t|θ))2 =1N

N∑t=1

ε2(t, θ) (5.10)

is its prediction error variance. The prediction error variance is compared for ARmodels of different orders in Figure 5.9.

1 2 3 4 5 6 7 8 9 10

model order [n]

2.36

2.38

2.4

2.42

2.44

2.46

2.48

2.510

-13 Loss Function

Figure 5.9: Comparison of the prediction error variance for AR models ofdifferent orders. It can be seen that model orders n = 2 and n = 5 seem likethe most suitable choices.

The autocovariances

Rε(τ) =1N

N∑t=1

ε(t + τ)ε(t) (5.11)

for the AR models of chosen order can be normalized to obtain the autocorrela-tions (see [6] for more details). Thereby, a more convincing statement regardingeach AR model’s reliability can be made. The results from the comparison be-tween the autocorrelations are shown in Figure 5.10 and Figure 5.11.

32 5 Results

-25 -20 -15 -10 -5 0 5 10 15 20 25-0.2

0

0.2

0.4

0.6

0.8

1

Autocorrelation - AR(2)

Lag

Am

plit

ude

Figure 5.10: Autocorrelation analysis for the AR(2) model. The fact that mostof the points are more or less within the confidence interval (the blue areain the figure) suggests that the residuals are mutually uncorrelated of eachother.

-25 -20 -15 -10 -5 0 5 10 15 20 25-0.2

0

0.2

0.4

0.6

0.8

1

Autocorrelation - AR(5)

Lag

Am

plit

ude

Figure 5.11: Autocorrelation analysis for the AR(5) model. Despite a slightimprovement in terms of mutually uncorrelated residuals, a plausible state-ment is that an AR(5) model might be considered as overfitted.

5.1 Modeling 33

From Figures 5.11-5.10 it is clear that a higher model order might not be muchmore beneficial (e.g. it possible to argue that the AR(5) model leads to overfit-ting). Hence, the AR(5) model is disregarded.

Further validation is done by studying the noise component’s estimated signal

spectrum Φωω and periodogram ˆΦωω. These estimates are then compared to the

corresponding spectral estimate and periodogram for the AR(2) model when itsinput is white Gaussian noise with zero mean and variance σ . The outcomes fromthese estimates are shown in Figure 5.12.

Finally, for illustrative purposes, a time domain representation of the disturbancecomponent and simulated output from the AR(2) model is shown Figure 5.13.From Figure 5.13 it can be seen that the simulated disturbance has more or lessthe same characteristics and patterns as the real disturbance component (the dis-turbance model’s parameters, and the chosen value for the variance σ , can befound in Appendix A.2).

10-4

10-3

10-2

frequency [rad/s]

0

0.5

110

-7 Periodogram and Spectral Estimate

10-4

10-3

10-2

frequency [rad/s]

0

0.5

110

-7

Figure 5.12: Comparison of periodograms and spectral estimates. The AR(2)model is able to both attenuate and amplify quite satisfyingly at the samefrequencies as the disturbance component. The variance σ of the input sig-nal to the AR(2) model has to be tuned in order to reproduce a reasonableestimate of the noise component.

34 5 Results

0 2 4 6 8 10 12 14 16

time [hours]

-2

0

2

10-5Disturbance Component and Simulated Disturbance

0 2 4 6 8 10 12 14 16

time [hours]

-2

0

2

10-5

Figure 5.13: Comparison of the disturbance component and the output fromthe AR(2) model.

It should be noted that the reliability of the disturbance model is directly depen-dent on the accuracy and reliability of the model G(q, θ) that was presented inSection 5.1.1. This means that a more accurate disturbance model would requirea more detailed model of G(q, θ).

5.1.3 Identification of Closed Loop Systems

Before the step tests were carried out, some initial attempts on obtaining a modelon the form

y(t) = G(q, θ)u(t) + H(q, θ)n(t) (5.12)

through system identification from old process data were conducted. One majordifficulty to overcome was the presence of feedback due to the implemented con-troller. This causes the control signal u to be dependent of the disturbances thatenter the process which in turn complicates modeling [4]. There are numerousways to account for the presence of a feedback loop in a system, and after eval-uation of various approaches, the most promising results were obtained with atwo-step method [11].

The first step in this method is to identify a model S0 with the setpoint r as inputsignal and the control signal u as output signal. The next step is to generate anew signal u from r and S0 which is independent of the disturbances ω that are

5.1 Modeling 35

entering the system. Finally, the system of interest G, can be estimated using uand the output y. The two-step method is illustrated in Figure 5.14.

S0 G∑r u

ω

y

Figure 5.14: The main principles of the two-step method. The signal u isused to eliminate the impact the disturbances ω have on the control signal uthrough the feedback.

Reliable models obtained from the two-step method usually requires large setsof data as well as a rather varying setpoint r. In the flotation process, the set-points for the pH controllers do not vary much over time, which in turn yieldedunreliable models. An example of the outputs from two ARX models of the sameorder, obtained from the two-step method, is shown in Figure 5.15. The signal yis the output of a linear model in the pH scale, and Y of a model with nonlinearcompensation as presented in Section 4.2.

10 15 20 25 30 35 40 45

time [hours]

9.8

9.9

10

10.1Modeling - Influence of Feedback

10 15 20 25 30 35 40 45

time [hours]

4

6

8

10

12

Figure 5.15: An example of an outcome when performing system identifica-tion on old process data. The blue lines are the original signals, the magentacolored signals originate from modeling in the pH scale, and the red signalsoriginate from the nonlinear compensation.

36 5 Results

It can be seen from Figure 5.15 that both ARX models y and Y follow the samepattern. However, the reliability of the signals u and u can be questioned, whichin turn is caused by the few variations in the setpoint r, resulting in an unreliablemodel of S0.

In spite of treating the data in the same way (e.g. not removing means), and usingthe same type of models and orders throughout the two-step method, there are noindications that the nonlinear compensation has been more beneficial than mod-eling in the pH scale (i.e. when conducting system identification from old processdata). However, due to the promising results presented in [13] as well as the clearrecommendations to not control pH directly in the pH scale, the concepts pre-sented in [13] were emphasized for the rest of the thesis (e.g. these concepts wereemphasized for the models presented Sections 5.1.1-5.1.2).

Despite numerous attempts on using other and larger data sets, as well as apply-ing nonlinear system identification techniques in the pH scale, it was concludedthat dedicated open loop experiments had to be conducted in order to obtainmore reliable models.

5.2 Control 37

5.2 Control

The objective of the new control strategy is to maintain a constant ratio betweenthe added lime and the incoming ore flow, but at the same time to ensure thatthe pH levels are kept within certain boundaries. An equivalent linear control ob-jective is utilized to achieve satisfying pH control. Simple PI and PID controllersare compared against each other and then combined in the final control systemwhich consists of pH and ratio controllers.

In this section, the results from the evaluations of the controllers are presented.It should be noted that there are no specific requirements on the pH controllers(e.g. rise time, overshoot, etc.).

5.2.1 Ratio Control

The ratio control strategy that is presented in Section 4.1.2 is an intuitive way ofcontrolling the ratio between two process variables. This usually requires a trans-fer function G that describes the dynamics between the control signal u and themeasured output y.

In this particular case, there is no need for a transfer function G to describe thedynamics between u and y. The reason for this is that the control signal u [%](i.e. added lime) and the measured lime flow y [l/min] are just related througha scaling factor (i.e. the lime flow is a static linear function of the control signal).This makes the ratio control trivial since replacing the transfer function G andcontroller F with a scaling factor makes the presence of a feedback loop redun-dant (thereby yielding an open loop ratio controller). This significantly simplifiesthe ratio control objective, since the desired ratio will always be achieved as longas the pH is kept within certain limits. The controller is shown in Figure 5.16.

∏C

r

yr

ryr u

Figure 5.16: The absence of dynamics makes the ratio control trivial. Themultiplication with the desired ratio r [l/ton], and the incoming ore flow yr[ton/min], yields the amount of lime ryr [l/min] that is required to maintaina constant ratio. The lime flow is scaled with the constant C to yield thesignal u [%].

38 5 Results

5.2.2 pH Control - Integral Windup

Large changes in setpoints, and a bounded control signal, can cause the integralpart

I = KI

t∫0

e(τ)dτ (5.13)

of a PI/PID controller to accumulate the error during the rise or fall of a step, andthereby overshooting. This usually leads to poor performance of a controller. Inthis case the control signal is limited between 0 − 100%, which in turn is a limiton the lime flow that is added in the process.

Problems with overshooting were observed when evaluating the pH controllers.In Figure 5.17, a comparison between two PI controllers is made, where one ofthe controllers counters the integral windup by adjusting the integral part of thecontroller (see [3] for more details regarding this topic and methods for how toimplement anti-windup in a controller). Despite finer tuning, the overshoot ofthe other controller (blue line in Figure 5.17) is hard to eliminate without anti-windup implementation.

80 90 100 110 120 130 140

time [min]

9.88

9.9

9.92

9.94

Anti-Windup

80 90 100 110 120 130 140

time [min]

5

6

7

8

9

10

Figure 5.17: Comparison between two PI controllers where the blue line isthe simulated pH output when a PI controller is used that does not counter-act the windup behaviour, and the red line is a PI controller that achievesanti-windup by adjusting the integral part of the controller.

5.2 Control 39

5.2.3 pH Control - Modifying the Derivative

Despite challenges with noisy environments, it is possible to implement PID con-trollers so that the measurement noise does not have a significant impact on thecontrol signal. For continuous-time controllers this is usually achieved by low-pass filtering the derivative part of the controller

D(s) =Td s

µTd s + 1(5.14)

that only amplifies high frequency components with a factor of 1/µ, where µ isa design parameter (see [3] for more details on how to approximate (5.14) in thediscrete-time domain).

Setpoint changes can have the negative effect of yielding large values of the PIDcontroller’s control signal. It is possible to introduce design parameters that canbe used to decide how big impact a change in the setpoint should have on thecontrol signal. The design parameter β in

D(t) = Tdd(βr(t) − y(t))

dt(5.15)

can be used to decide how big impact setpoint changes should have on the deriva-tive part (5.15) of the PID controller. Usually, β is set between 0 and 1.

5.2.4 pH Control - Step Responses

Three controllers are evaluated for controlling the pH. In this case a PI and twoPID controllers are compared against each other. Because of the noisy environ-ments in many industrial processes, PI controllers are usually preferred over PIDcontrollers. Nevertheless, two PID controllers are evaluated to see if they can bemore successful in achieving satisfying control. The controllers were tuned ac-cording to tuning rules presented in [3] (the control parameters can be found inAppendix A.3).

Initially, simple step responses for the closed loop system were studied. In Figure5.18 results of the controllers’ step responses are shown.

40 5 Results

40 50 60 70 80 90 100

time [min]

9.9

10

10.1

10.2

10.3

Step Responses

40 50 60 70 80 90 100

time [min]

0

10

20

30

40

50Control Signals

Figure 5.18: Step responses for the three controllers. The magenta coloredline represents a PID controller with design parameter β = 1, and the redcolored line with design parameter β = 0.

It can be seen that the PID controller with β = 1 has a significant peak in thecontrol signal, whereas the PID controller with β = 0 has a smaller peak.

Step responses from a higher to a lower setpoint were also studied. The onlydifference from the responses shown in Figure 5.18 were that the control signalsfor all controllers were bounded from below at 0 %.

5.2.5 pH Control - Disturbances

The disturbance model obtained in Section 5.1.2 is used to simulate disturbancesentering the process. However, the process might be subject to disturbances thatover the course of time very rarely enter the process (e.g. a very acidic incomingore). It is possible that these types of disturbances are unaccounted for in thedisturbance model.

Therefore, such rarely appearing disturbances are simulated along with the dis-turbances that were modeled in Section 5.1.2. In Figure 5.19 the first spike couldfor example represent a very acidic incoming ore, whereas the second spike mightbe caused by a larger amount of lime that was necessary to add at an earlier stagein the process in order to counteract the acidic ore.

5.2 Control 41

0 1 2 3 4 5

time [hours]

9.8

9.9

10

10.1

10.2

10.3

10.4

Step Responses - Disturbances

Figure 5.19: Simulation of disturbances entering the process. In the simu-lations the disturbances enter at the measured output and are then fed backthrough the feedback loop of the controllers.

It can be seen that the controllers are able to handle disturbances entering theprocess. Despite attempts of modifying the PID controller it is possible to arguethat because of its more simple structure, the PI controller is the most preferablecontroller when it comes to controlling the pH levels in the process.

5.2.6 Implemented Control System

After the evaluations of the ratio and pH controllers, a control system consistingof both ratio and pH controllers is designed. By looking at the proposed controlobjective (i.e. maintaining a constant ratio, but at the same time keeping the pHlevel within allowed limits), a reasonable control strategy is to utilize the conceptof selective control. Two pH controllers with setpoints corresponding to the lowerand upper pH limits, and one ratio controller, are connected to a selector that atevery time step determines which control signal to use

umax(t) = max(ur (t), upHu (t)) (5.16)

umin(t) = min(ur (t), upHl (t)) (5.17)

where ur denotes the control signal set by the ratio controller, and upHu andupHl are the control signals set by the upper and lower bound pH controllers. ASimulink model of the implemented control system is shown in Figure 5.20.

42 5 Results

RatioSetpoint uR

RatioController

[Y]

pHSetpoint

uL

pH_sp

LowerBoundpHController

[%]

w

[pH]

[Y]

MixingTank-BL4101

w

Disturbance

LowerLimit

[Y]

pHsetpoint

uU

pH_sp

UpperBoundpHController

UpperLimit

uMin

uR

uMax

u

ControlSignalSelectionScope

10.1

UpperSetpoint[pH]

9.8

LowerSetpoint[pH]

0.22

RatioSetpoint[l/ton]

Figure 5.20: An overview of the control system implemented in Simulink.The pH controllers utilize the equivalent linear control objective presentedin Section 4.2 to control the upper and lower bound pH levels, thereby en-suring that the pH level is maintained within the allowed boundaries. Theratio controller is used to maintain a constant ratio between the added limeand the incoming ore flow.

From Figure 5.20 it can be seen that the signals ur (t), umax(t), and umin(t) aresent to a control signal selector block that determines which control signal touse based on the signals sent from the minimum and maximum blocks (see Ap-pendix A.4 for more details regarding how the control signal selector determineswhich signal to use in order to control the process). For example, should the ratiocontroller yield a pH level that for a short period goes outside the allowed upperlimit, the control signal selector will use the upper bound pH controller to ensurethat the pH level drops so that the level is maintained within the allowed limits.

The control system is first evaluated by investigating if the pH controllers arekeeping the pH within the allowed limits when a relatively large setpoint (0.3l/ton) is set for the ratio controller, when the system is being subjected to distur-bances. This will cause larger variations in the pH values. This is demonstratedin Figure 5.21.

5.2 Control 43

0 1 2 3 4 5 6

time [days]

9.4

9.6

9.8

10

10.2

10.4

Implemented Control System

0 1 2 3 4 5 6

time [days]

ulower

uratio

uupper

Control Signal Selector

Figure 5.21: Simulation of the implemented control system when distur-bances are present. In the upper plot the blue line shows the pH values thatare obtained without the pH controllers. The magenta colored line showswhen the pH controllers are implemented, and the red lines denote the up-per and lower boundaries. The lower plot shows which signal the selectorchooses during simulations to fulfill the control objective.

It should be pointed out that whenever the pH values are within the bound-aries in Figure 5.21, the blue and magenta colored lines coincide with each other,which means that the pH controllers will not intervene since the ratio controllerwill yield a pH that is within the boundaries.

Clearly, the pH controllers are able to maintain the pH within the limits. How-ever, in Figure 5.21 it can be seen that when the pH only slightly surpasses aboundary, it takes some time for the pH controller to take over and control thepH (this can be seen at the beginning of the simulations). It is therefore recom-mended that the boundaries are set to a slightly smaller interval than the actualallowed boundaries.

44 5 Results

The next step is to ensure that the pH controllers will not intervene as long asthe pH is kept within the boundaries, and thereby maintaining a constant ratio.For these simulations, the setpoint for the ratio controller is set to a smaller value(0.22 l/ton). The outcome is shown in Figure 5.22.

Figure 5.22: Simulation of the implemented control system when a smallersetpoint is set for the ratio controller. It should be noted that in these simu-lations, whenever the ratio controller is controlling the process, there seemsto be no disturbances that are causing variations in the control signal. This isdue to the fact that the ratio controller lacks a feedback loop, which in turneliminates the disturbances’ impact on the control signal.

From the simulations demonstrated in Figures 5.21-5.22 it can be concluded thatthe implemented control system is able to maintain a constant ratio as long as thepH is within the allowed limits.

It should be noted that the outcomes from the simulation shown in Figure 5.21is a more extreme scenario, whereas Figure 5.22 demonstrate a more realisticscenario which coincides with the desired outcomes for the new control strat-egy.

5.2 Control 45

5.2.7 pH Ratio Control

The cascade inspired control strategy presented in Section 4.1.3 is often utilizedin various mineral processing applications for achieving better pH control [12].One of the main ideas in a cascade control strategy is to eliminate the distur-bances’ impact on the output in an inner control loop. In this case, the idea is toadjust the control signal with respect to the incoming ore flow, before changes inthe pH levels are observed.

However, it is mentioned that with the absence of dynamics in the inner controlloop, the incoming ore flow can potentially act as a varying gain in the feedbackloop. The main purpose of these simulations is to study how the ratio and the pHlevels vary when the incoming ore flow is accounted for, and also to see if thisstrategy will yield a better control of the pH levels.

During the step response tests presented in Section 5.1.1, the incoming ore flowwas more or less constant. This makes it possible to select parameters for a linearPI ratio controller that is used to set a setpoint value for the ratio r based on theerror pHsp−pH (the control parameters can be found in Appendix A.3). In Figure5.23, the upper plot shows that changes in the ratio are mostly due to changes inadded lime and not incoming ore flow.

0 1 2 3 4 5

time [hours]

0

0.5

1Step Response Tests

0 1 2 3 4 5

time [hours]

0

1

2

3

410

-4

Figure 5.23: During the step response tests the incoming ore flow was moreor less constant. This means that changes in the ratio mostly were due tochanges in the added lime, and not the incoming ore flow, which can be seenin the upper plot. Again, the pH scale is recalculated to the molar concen-tration of hydroxide ions.

46 5 Results

Notice that once again, the nonlinear compensation yielding the molar concentra-tion of hydroxide ions is utilized. By using the same input data (i.e. the same in-coming ore flow) for these simulations as for the simulations of the implementedcontrol system presented in Section 5.2.6, it is possible to study how the cascadeinspired pH ratio control strategy differs from the pH control strategy utilized bythe pH controllers that are controlling the upper and lower bound pH levels inthe implemented control system in Section 5.2.6 (these pH controllers do not ac-count for the incoming ore flow), and how constant of a ratio can be maintained.In this case, the pH setpoint is set to a constant value of 10.2. The results areshown in Figure 5.24.

When comparing the outcomes in Figure 5.24 for the pH ratio control strategy,and the outcomes for the implemented control system in Figure 5.22, it is clearthat the implemented control system is able to maintain a more constant ratiothroughout the simulations (but at the expense of more varying pH levels) com-pared to the pH ratio controller.

Figure 5.24: The blue lines denote how the pH level and the ratio vary whenthe incoming ore flow is unaccounted for by the pH controller (this is thetype of pH controller used in the implemented control system in Section5.2.6), and the magenta colored lines show how corresponding values varywhen utilizing the cascade inspired pH ratio control strategy presented inSection 4.1.3 that accounts for the incoming ore flow. In this case, changesin the ratio are mostly caused by changes in the incoming ore flow and thedisturbances that are entering the process.

5.2 Control 47

However, it is not clear whether an overall better pH control is achieved in thecascade inspired pH ratio control strategy compared to the pH control strategyutilized in the implemented control system in Section 5.2.6. However, it can beseen that sudden changes in the incoming ore flow causes the pH ratio controllerto deviate from the setpoint value 10.2. This confirms that the incoming ore flowacts as a varying gain in the feedback loop due to the absence of dynamics in theinner controller in the cascade strategy.

The simulations show that the pH ratio control strategy is only able to maintaina constant ratio as long as the incoming ore flow is constant. It might be tempt-ing to switch places of the inner and outer loop in the cascade control strategy,and thereby having a ratio controller in the outer loop, and a pH controller inthe inner loop. However, it should be clarified that for cascade control to beworthwhile, the dynamics of the inner loop have to be faster than the outer loop’sdynamics [3].

Based on these simulations, the control system consisting of a ratio controllerand upper and lower bound pH controllers is preferably chosen as the best con-trol strategy for the new control objective considering that it is more successful inmaintaining a constant ratio throughout the simulations. In this particular case,it seems like the pH control system that does not account for the incoming oreflow actually achieves better control performance with regards to controlling thepH, which is another reason for choosing this controller to control the upper andlower pH bounds in the implemented control system.

One important aspect to consider is the reliability of the disturbance model ofthe process. It is possible that the disturbance model is unsuccessful in givingan accurate description of how the incoming ore flow affects the pH levels, andtherefore the results in Figure 5.24 should be interpreted carefully.

However, Figure 5.24 points out something else that is interesting. The new con-trol strategy is to maintain a constant ratio with hopes of yielding a less vary-ing copper concentrate. But based on the simulations it can be seen in Figure5.24 that the ratio does not vary that much over some periods of time. This ei-ther suggests that the copper concentrate is sensitive to variations in the ratio, orthat there is something else in the process that causes the copper concentrate tovary.

6Conclusions

This chapter presents conclusions and a short discussion regarding the resultsobtained in this thesis. Suggestions on future work are also given.

6.1 Discussion and Conclusions

The model developed in this thesis seems to fulfill the requirements of capturingthe main dynamics of the process (in this case the mixing tank, BL4101) aroundcurrent working conditions. Despite its very simple structure, the three parame-ter model has given a better insight of time constants, time delays and nonlineari-ties of the process. However, due to the model’s validity in a very small operatingregion, it is highly recommended to further investigate the dynamics of the pro-cess in a wider operating range. This will enable further studies of whether theequivalent control objective presented in this thesis is more beneficial for control-ling the pH level in the process.

Simulations of the control system, which consists of a ratio controller and two pHcontrollers that utilize an equivalent control objective, show promising resultssince the control system seems to fulfill the desired requirements of keeping aconstant ratio, but at the same time maintaining the pH levels within the allowedboundaries.

Another control strategy that at first glimpse might be reasonable to consider forfulfilling the new control objective, is LQ/MPC control. It would then be possi-ble to have a smaller emphasis on achieving good pH control, and instead focus

49

50 6 Conclusions

on maintaining a more constant ratio. However, due to the absence of dynamicsbetween the added lime, the incoming ore, and the ratio, it would not be possibleto utilize a LQ/MPC strategy to control the ratio. The reason for this is that thesecontrollers require a set of differential equations to control the system (see [5] formore details regarding this topic). Another reason for why a LQ/MPC strategywas not studied was because of the low order of the obtained models (LQ andMPC controllers are preferably applied on higher order systems where for exam-ple PID controllers might fail to give desired results).

6.2 Future Work

This work has served as a starting point for studying the new control strategythat Boliden has presented. However, should this new strategy of maintaininga constant ratio prove to be unsuccessful in fulfilling the requirements of yield-ing a higher exchange in the sought copper, it is highly recommended to furtherstudy the concept of the strong acid equivalent. It is also recommended investi-gate if the pH ratio control strategy that is presented in Section 4.1.3 would yielda higher exchange, or perhaps consider some of the approaches that are reviewedin [1].

As was previously mentioned, models that are valid in a wider operating regionthan the current models would be very interesting to obtain. This would enablestudies of the potential benefits of the strong acid equivalent, but in a widerrange.

Finally, should neither the new control strategy of maintaining a constant ratio be-tween the added lime and the incoming ore flow, or utilizing pH control throughthe nonlinear compensation, yield a better exchange, the source of the problemin a varying copper concentrate is probably caused by something else.

Appendix

AAppendix

A.1 OE Model Parameters

The OE model, GOE

y(t) = G(z)u(t) + n(t) (A.1)

obtained from the step response tests in Section 5.1.1 has the polynomial coeffi-cients

G(z) =4.725 · 10−7

z2 − 0.9592z(A.2)

The model is modeled from data with sample time Ts = 30 sec.

A.2 Disturbance Model Parameters

The AR(2) model

A(z)ω(t) = n(t) (A.3)

that is used to model the disturbances entering the process has the polynomialcoefficients

53

54 A Appendix

A(z) = 1 − 0.9706z−1 − 0.02003z−2 (A.4)

where n(t) in (A.4) is white Gaussian noise with variance σ

n(t) ∼ N (0, σ )

σ = 4.5 · 10−13

The model is modeled from data with sample time Ts = 30 sec.

A.3 Control Parameters

The control parameters for the PI and PID controllers presented in Sections 5.2.3-5.2.6 are found in Tables A.1-A.2.

Table A.1: Parameters for the PI controller.

PI Controller ParametersParameter Value

K 1.6 · 105

Ti 520

Table A.2: Parameters for the PID controller.

PID Controller ParametersParameter Value

K 1.6 · 105

Ti 280Td 28.5µ 0.1

The control parameters for the pH ratio controller are given in Table A.3.

Table A.3: Parameters for the pH ratio controller.

pH Ratio Controller ParametersParameter Value

K 4.32 · 103

Ti 520

A.4 Control System Selector 55

A.4 Control System Selector

In Listing A.1 the code for the selector that is used to determine which controlsignal should be used to control the process is presented.

Listing A.1: Matlab code for the selector that is implemented in the controlsystem.

function [u]=select(uMin,uR,uMax)

%This function is used to determine which of the three%controllers should be given permission to control%the process.

%uR: Control signal from ratio controller.%uMin: Output from the minimum block.%uMax: Output from the maximum block.

if (uMin==uR && uMax~=uR)u=uMax; %The lower bound pH controller is given

%permission to control the process

else if (uMin~=uR && uMax==uR)u=uMin; %The upper bound pH controller is given

%permission to control the processelse

u=uR; %The ratio controller is given%permission to control the process

end

Bibliography

[1] B. Wayne Bequette. Nonlinear Control of Chemical Processes: A Review.Ind. Eng. Chem. Res, Vol. 30(7):1391–1413, 1991. Cited on pages 5 and 50.

[2] A. Berggren, U. Marklund, and A. Sandberg. Flotation control - State of theArt. Unpublished manuscript, Boliden Mineral AB, 2006. Cited on pages 3and 4.

[3] M. Enqvist et al. Industriell Reglerteknik Kurskompendium. LinköpingUniversity, 2014. Cited on pages 4, 14, 18, 19, 25, 38, 39, and 47.

[4] T. Glad and L. Ljung. Modellbygge och simulering. Studentlitteratur AB,Lund, 2:5 edition, 1991, 2004. Cited on pages 3, 15, 16, and 34.

[5] T. Glad and L. Ljung. Reglerteori: Flervariabla och olinjära metoder. Stu-dentlitteratur AB, Lund, 2:7 edition, 1997, 2003. Cited on page 50.

[6] F. Gustafsson, L. Ljung, and M. Millnert. Signal Processing. Studentlitter-atur AB, Lund, 1:3 edition, 2010. Cited on pages 3, 27, and 31.

[7] Tore K. Gustafsson and Kurt V. Waller. Nonlinear and Adaptive Control ofpH. Ind. Eng. Chem. Res, 31(12):2681–2693, 1992. Cited on page 5.

[8] P. Miranda La Hera, B. Ur Rehman, and A. Sandberg. Modelling an indus-trial flotation process: A case study at the mining company Boliden AB. InWorld Automation Congress (WAC), pages 1–5, 2012, Puerto Vallarta, Mex-ico. Cited on page 3.

[9] D. Hoduin, S-L. Jämsä-Jounela, T.C. Carvalho, and L. Bergh. State of the artand challenges in mineral processing control. Control Engineering Practice,(9):995–1005, 2001. Cited on page 4.

[10] H. Jönsson and P. Westerlund. Strategies and instrumentation for flotationcontrol in Aitik. Unpublished manuscript, Boliden Mineral AB, 1999. Citedon page 4.

57

58 Bibliography

[11] L. Ljung. System Identification: Theory for the User. Prentice Hall, 2ndedition, 1999. Cited on pages 15 and 34.

[12] B.A. Wills and T.J. Napier-Munn. Will’s Mineral Processing Technology.Butterworth-Heinemann, 7th edition edition, 2006. Cited on pages 3, 7,8, 18, 19, 20, and 45.

[13] R.A. Wright and C. Kravaris. Nonlinear Control of pH Processes Using theStrong Acid Equivalent. Ind. Eng. Chem. Res, Vol. 30(7):1561–1572, 1991.Cited on pages 5, 20, 21, and 36.

[14] Y.Youlin and WU. Qinghu. A Neural Network PID Control for pH Neutral-ization Process. In 35th Chinese Control Conference (CCC), pages 3480–3483, 2016, Chengdu, China. Cited on pages 3 and 4.