Identification and control of wet grinding processes ...

Identification and control of wet grinding processes - Application to the Kolwezi

concentrator

December 2013

Ecole Polytechnique de Bruxelles

Thèse présentée par Moïse MUKEPE KAHILU en vue de l’obtention du grade de Docteur en Sciences de l’Ingénieur

Promoteur : Prof. Michel KINNAERT Co-promoteurs : Prof. Pierre KALENGA NGOY M. Prof. Jean-Marie MOANDA NDEKO

Abstract

Enhancing mineral processing techniques is a permanent challenge in the mineral and metal industry. Indeed to satisfy the requirements on the final product (metal) set by the consuming market, control is often applied on the mineral processing whose product, the ore concentrate, constitutes the input material of the extractive metallurgy. Therefore much attention is paid on mineral processing units and especially on concentration plants. As the ore size reduction procedure is the critical step of a concentrator, it turns out that controlling a grinding circuit is crucial since this stage accounts for almost 50 % of the total expenditure of the concentrator plant. Moreover, the product particle size from grinding stage influences the recovery rate of the valuable minerals as well as the volume of tailing discharge in the subsequent process. The present thesis focuses on an industrial application, namely the Kolwezi concentrator (KZC) double closed-loop wet grinding circuit. As any industrial wet grinding process, this process offers complex and challenging control problems due to its configuration and to the requirements on the product characteristics. In particular, we are interested in the modelling of the process and in proposing a control strategy to maximize the product flow rate while meeting requirements on the product fineness and density. A mathematical model of each component of the circuit is derived. Globally, the KZC grinding process is described by a dynamic nonlinear distributed parameter model. Within this model, we propose a mathematical description to exhibit the increase of the breakage efficiency in wet operating condition. In addition, a relationship is proposed to link the convection velocity to the feed ore rate for material transport within the mills. All the individual models are identified from measurements taken under normal process operation or from data obtained through new specific experiments, notably using the G41 foaming as a tracer to determine material transport dynamics within the mills. This technique provides satisfactory results compared to previous studies. Based on the modelling and the circuit configuration, both steady-state and dynamic simulators are developed. The simulation results are found to be in agreement with the experimental data. These simulation tools should allow operator training and they are used to analyse the system and to design the suitable control strategy. As the KZC wet grinding process is a Multi-Input Multi-Output (MIMO) system, we propose a decentralized control scheme for its simplicity of implementation. To overcome all the control issues, a Double Internal Model Control (DIMC) scheme is proposed. This strategy is a feedforward-feedback structure based on the use of both a modified Disturbance Observer (DOB) and a Proportional-Integral Smith-Predictor (PI-SP). A duality between the DOB and PI-SP is demonstrated in design method. The latter is exploited to significantly simplify the design procedure. The designed decentralized controllers are validated in simulation on the process linearized model. A progressive implementation of the control strategy is proposed in the context of the KZC grinding circuit where instrumentation might not be obvious to acquire.

ACKNOWLEDGEMENTS 2013

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ACKNOWLEDGEMENTS

This thesis is the result of a four-year work performed partly at the Université Libre de Bruxelles (ULB) in Belgium and partly at the Université de Lubumbashi (UNILU) in the Democratic Republic of Congo. The project budget and the Belgian stay were financially supported by the Belgian Technical Cooperation (BTC) which is therefore gratefully acknowledged. As this thesis cannot be the result of a single person, I would like to thank all the people who helped me to achieve this work. First of all, my greatest thanks are due to my promoter, Professor Michel Kinnaert, head of the Control Engineering and System Analysis Department of ULB, who initiated me in research. He devoted a considerable part of his limited time on this thesis and guided me with his relevant ideas and advices. Thanks for encouraging me to systematically think in a critic and in an analytic way. Many thanks are due to my co-promoters, Professor Pierre Kalenga Ngoy Mwana and Professor Jean-Marie Moanda Ndeko for their interest in my work and for their precious advices. I also owe my best thanks to Professor Alain Delchambre, president of ULB, who met me in my country during his supporting teaching at the Electromechanical Department of UNILU. He was always open to me and put me in touch with Professor Michel Kinnaert. Many thanks for supporting and encouraging me. A friendly personal contact resulted from these interactions. My great thanks are also for Professor Raymond Hanus for having received me in the Control Engineering and System Analysis Department and for his precious ideas and encouragements. I would like to express my respectful acknowledgement to Professor Philippe Bogaerts, president of this thesis committee, for his pertinent contributions during the board for the thesis follow-up. I gratefully acknowledge Professor Emmanuele Garone, for his relevant ideas on optimization and Model Predictive Control (MPC) issues. Moreover, I would like to thank the personnel of Kolwezi Concentrator (KZC) for their help and collaboration during the experimental work and the data acquisition. I would like to express my gratitude to my colleagues from the Control Engineering and Systems Analysis Department for their encouragements and their contribution to the good atmosphere of the lab, especially during the famous coffee breaks. Thanks to all people who indirectly contributed to this work by their comments and suggestions.

ACKNOWLEDGEMENTS 2013

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Special thanks to my parents, Leonard Mukepe and Marie-Claire Tshisuwa, for their permanent and immeasurable support. Many thanks to my sisters and brothers, especially to Léon Zeka, and to my friend Daniel Kasongo. Thanks are also due to all my family in-law and to all the MUSKA Engineering’s people. Finally, I wish to express all my admiration and all my gratefulness to my beloved wife Nicole Ihemba who has accompanied me with her strong love during these four years. She has suffered uncounted days of separation, accepting this difficult situation with an incredible patience. Without her encouraging support, it would have been much harder for me to finalise this work. To Monica and Jovic, my beloved children, thank you for their patience and unconditional love. As my offspring, they are my heart.

CONTENTS 2013

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CONTENTS

Chapter I: INTRODUCTION .................................................................................... 1

I.1 Motivations ......................................................................................................... 1 I.2 State of the art .................................................................................................... 2 I.3 Contributions of this study ............................................................................... 3 I.4 Outline ................................................................................................................ 5

Part I: PROCESS MODELLING AND SIMULATION Chapter II: WET GRINDING CIRCUIT OF THE KOLWEZI CONCENTRATOR ................................................................................. 9

II.1 General concepts of mineral processing ........................................................ 9

II.1.1 Objectives and processes of mineral processing ....................................... 9 II.1.2 Grinding process ..................................................................................... 10

II.2 Grinding process of the Kolwezi concentrator ............................................. 11 II.2.1 Preliminary description ............................................................................ 11 II.2.2 Objective of KZC ..................................................................................... 12 II.2.3 Dry crushing ............................................................................................ 12 II.2.4 Wet grinding ............................................................................................ 12 II.2.5 Flotation ................................................................................................... 15 II.2.6 Thickening and filtration ........................................................................... 15 II.2.7 Current control mode ............................................................................... 15

Chapter III: STATE OF THE ART FOR GRINDING PROCESSE S MODELLING ....................................................................................... 17

III.1 Fragmentation or breakage modelling ......................................................... 17

III.1.1 Introduction ............................................................................................. 17 III.1.2 Energy-based models ............................................................................. 18 III.1.3 Phenomenological models ...................................................................... 21

III.2 Material transport modelling ......................................................................... 26 III.2.1 Notion of material transport within a mill ................................................. 26 III.2.2 Residence time distribution ..................................................................... 26 III.2.3 Distributed parameter model ................................................................... 27

III.3 Complete model of a grinding mill process ................................................. 28 III.4 Modelling of hydrocyclone classification .................................................... 29

III.4.1 Selectivity function (Tromp curve) of a hydrocyclone classifier ............... 29 III.4.2 Recycled and circulating loads ............................................................... 30 III.4.3 Experimental determination of a tromp curve .......................................... 31 III.4.4 Mathematical models of a hydrocyclone classifier................................... 31 Chapter IV: MODELLING OF THE KZC GRINDING PROCESS ....................... 33

IV.1 Modelling purpose ......................................................................................... 33 IV.2 Rod mill model ............................................................................................... 34

IV.2.1 Mathematical model ............................................................................... 34 IV.2.2 Model parameters .................................................................................. 35

IV.3 Hydrocyclone classifiers model ................................................................... 36 IV.3.1 Mathematical model ............................................................................... 36

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IV.3.2 Model parameters .................................................................................. 36 IV.4 Ball mill model ............................................................................................... 36

IV.4.1 Mathematical model ............................................................................... 36 IV.4.2 Model parameters .................................................................................. 38

IV.5 Model of the other components .................................................................... 38 IV.6 Overall representation ................................................................................... 38

Chapter V: PARAMETER IDENTIFICATION AND SYSTEM SIMU LATION .... 41

V.1 Identification procedure ................................................................................. 41

V.1.1 Modelling objectives ................................................................................ 41 V.1.2 Experimental field (Measurements) ......................................................... 41 V.1.3 Model structure ....................................................................................... 42 V.1.4 Optimization procedure ........................................................................... 42

V.2 Mill transport parameters ............................................................................... 43 V.3 Mill grinding parameters ................................................................................ 46 V.4 Hydrocyclone classifier parameters ............................................................. 52 V.5 Steady-state simulator ................................................................................... 53

V.5.1 Simulator implementation ........................................................................ 53 V.5.2 Simulation results .................................................................................... 56

V.6 Dynamic simulator ......................................................................................... 58

V.6.1 Simulator implementation ........................................................................ 58 V.6.2 Simulation results .................................................................................... 59

Part II: CONTROLLER DESIGN AND VALIDATION Chapter VI: STATE OF THE ART FOR WET GRINDING PROCE SSES CONTROL ........................................................................................... 67

VI.1 Typical wet grinding process ....................................................................... 67

VI.1.1 Description ............................................................................................. 67 VI.1.2 Dynamic and steady-state features ........................................................ 69 VI.1.3 Typical variable pairing ........................................................................... 70

VI.2 Instrumentation ............................................................................................. 71 VI.2.1 General concepts ................................................................................... 71 VI.2.2 Sensors .................................................................................................. 71 VI.2.3 Actuators ................................................................................................ 72 VI.2.4 Instrumentation for the typical wet grinding circuit .................................. 72

VI.3 Decentralized control strategies .................................................................. 73 VI.3.1 Classical control ..................................................................................... 74 VI.3.2 Advanced control ................................................................................... 76

VI.4 Centralized control strategies ...................................................................... 78 VI.4.1 Decoupling control.................................................................................. 78 VI.4.2 Model-based Predictive Control.............................................................. 80 VI.4.3 Adaptive and nonlinear control ............................................................... 82

VI.5 Discussion ..................................................................................................... 82 Chapter VII: ANALYSIS OF THE SYSTEM MODEL ........................................... 85

VII.1 Model linearization ....................................................................................... 85

VII.1.1 Selection of the initial steady-state operating point ................................ 85

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VII.1.2 Continuous-time linearized model ......................................................... 86 VII.1.3 Discrete-time linearized model .............................................................. 90

VII.2 Operating point optimization ....................................................................... 94 VII.3 Pairing of controlled and manipulated variabl es ....................................... 95

VII.3.1 Problem statement ................................................................................ 95 VII.3.2 Steady-state criteria .............................................................................. 96 VII.3.3 Dynamic criteria .................................................................................... 98

VII.4 Interaction effects ....................................................................................... 101 VII.5 Uncertainty characterization ...................................................................... 101

VII.5.1 General concept of uncertainty ............................................................ 101 VII.5.2 System model uncertainties ................................................................ 104

VII.6 Discussion .................................................................................................. 108 Appendix VII.1 .................................................................................................... 110 Appendix VII.2 .................................................................................................... 111

Chapter VIII: DECENTRALIZED CONTROL FOR THE KZC GRI NDING PROCESS ....................................................................................... 113

VIII.1 DIMC Structure .......................................................................................... 113

VIII.1.1 Control issues .................................................................................... 113 VIII.1.2 Control objectives ............................................................................... 115 VIII.1.3 Presentation of the DIMC ................................................................... 116

VIII.2 Robust DIMC design .................................................................................. 119 VIII.2.1 Robustness problem .......................................................................... 119 VIII.2.2 Duality between PI-SP and DOB design ............................................. 122

VIII.2.3 Robust design of DOB and PI-SP ....................................................... 127 VIII.3 Application to KZC grinding process ....................................................... 132

VIII.3.1 Instrumentation of the KZC grinding circuit ......................................... 132 VIII.3.2 Simulation on the linearized model ..................................................... 133 VIII.3.3 Progressive implementation ............................................................... 137 Chapter IX: CONCLUSIONS AND PERSPECTIVES ........................................ 143

IX.1 Conclusions ................................................................................................. 143

IX.1.1 Modelling and simulation ....................................................................... 143 IX.1.2 Control .................................................................................................. 144

IX.2 Future research directions .......................................................................... 145 IX.2.1 Modelling and simulation ....................................................................... 145 IX.2.2 Control .................................................................................................. 146 Bibliography .......................................................................................................... 147

LIST OF ACRONYMS AND NOTATIONS 2013

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LIST OF ACRONYMS AND NOTATIONS

Acronyms AG: Autogeneous ANN: Artificial Neural Networks ��1 : Ball mill 1

��2 : Ball mill 2 D: Decoupler DC: Decoupling Control DI: Density Indicator DIMC: Double Internal Model Control DNA: Direct Nyquist Array DOB: Disturbance Observer DRGA: Dynamic Relative Gain Array Ds: Distributor FI: Flow Indicator FOTD: First Order with Time Delay FV: Flow control Valve

��1 : Hydrocyclone 1

��2 : Hydrocyclone 2 IAE: Integral of Absolute Error IMC: Internal Model Control ISA: Instrumentation Systems and Automation Society KZC: Kolwezi Concentrator LI: Level Indicator MC: Multivariable Control MPC: Model-based Predictive Control MIMO: Multiple-Input Multiple-Output MPT: Multi-Parametric Toolbox MOL: Method Of Lines NMPC: Nonlinear Model Predictive Control ODE: Ordinary Differential Equation OP: Operating Point PC: Pre-Compensator PI: Proportional-Integral, Pressure Indicator PID: Proportional-Integral-Derivative PI-SP: Proportional-Integral Smith Predictor PDE: Partial Differential Equations PSA: Particle Size Analyser PSD: Particles Size Distribution RGA: Relative Gain Array

� : Rod mill RMPC: Robust Model Predictive Control RMSR: Root Mean Square of Residuals RTD: Residence Time Distribution


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SAG: Semi-autogeneous SISO: Single-Input Single-Output SP: Smith Predictor SVD: Singular Value Decomposition VSD: Variable Speed Driver WI: Weight Indicator

Notations

Arabic alphabet

: Dynamic matrix � : Multivariable MPC dynamic matrix � : Ore grindability

: Input matrix

�,� : Breakage, repartition or distribution function

∗ : Triangular fragmentation matrix

� : Transfer function of controller, output matrix

�� : Circulating load of a hydrocyclone classifier

�� : Mass of constituent �� : Tracer concentration in the material at the outlet of the mill

�� : Recycled load of a hydrocyclone classifier

� : Uniform diffusion coefficient, Laplace transform of disturbance signal on output ��: Ore grindability (inverse of ore hardness) �� : Feed particle size variations

�� : Size for which the hydrocyclone selectivity is 25 %

�� : Size for which the hydrocyclone selectivity is 75 %

��: Size expressing the product fineness

��,�� : Hydrocyclone classifier cut-point

�� : Diffusion coefficient of constituent �� !: Laplace transform of estimated lumped disturbance brought back to the manipulated variable �" : Total lumped disturbance

� " : Estimated total lumped disturbance

�#" : Error between �" and � " �$,%&,'!" : Rod mill output slurry density

�( : Mean value of the solid ore density in Kolwezi copper belt

�) : Water density

⋀ : Matrix defining the relative gain array

∆, : Additive uncertainty

∆-, : Radius of disk . ∆& : Multiplicative uncertainty


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∆-& : Bound on multiplicative uncertainty

/ : Laplace transform of error signal

/0 : Specific breakage energy (according to Bond)

E2 : Specific breakage energy (according to Charles)

E3 : Specific breakage energy (according to Hukki)

/4 : Specific breakage energy (according to Kick) /5 : Specific breakage energy (according to Rittinger)

/67 : Specific breakage energy (according to Svensson and Murkes)

8 : Matrix transfer function between the controlled variables and the manipulated variables 9 : Probability of fragmentation of grains according to their sizes

8�� : (;, <)"> entry of 8

? : Matrix transfer function between the controlled variables and the external disturbances

@A : Corrected coefficient matrix

?�� : (;, <)"> entry of ?

� : Material hold-up in the mill per unit of length �� : Initial spatial profile of the hold-up B : Identity matrix, Cutoff imperfection coefficient

C: Cost function

CD6 : Least squares cost function

K2 : Charles’s constant

F : Sampling instant FG : Additional parameter in the selection function K3 : Hukki’s constant

H4 : Kick constant

HIJ,K : Proportional gain of nominal PI controller

H5 : Rittinger constant

H67 : Constant of Svensson and Murkes

FL,MN : Supplementary parameter in the convective velocity model

O : Length of the mill

P : Column vector expressing the masses per particles size

�� : Mass of particles in the ;"> size interval Q : Laplace transform of measurement noise signal RS : number of spatial grid points

RT : number of spatial intervals

U : Transfer function of the plant

V : Laplace complex variable

U� : Mass fraction of material crossing through the sieve of mesh ; UW : Cumulated particles size distribution of material after fragmentation

UG : Peclet number

UK: Nominal transfer function of the plant


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UBK : Nominal transfer function of PI primary controller

X: Error weighting

Y: Transfer function of DOB low-pass filter

YG : Water flow rate YZ : Feed mass flow rate

YZ,%GW : Feed mass flow rate at a reference state Y7 : Feed mass flow rate of a hydrocyclone classifier

Y&∗ : Mass flow rate coming out of the rod mill

YI : Product mass flow rate of a hydrocyclone classifier

Y5 : Recirculating mass flow rate of a hydrocyclone classifier

Y%&,'!" : Rod mill output slurry flow rate

[�: Mass flow rate of constituent ��

\ : Input weighting matrix

�� : Mass fraction of material retained on the sieve of mesh ; ] : Diagonal selection matrix

]� : Specific rate of breakage or selection function of interval size ; ^ : Time

^, : Mean of the residence time distribution

^G : Median of the residence time distribution

_( : Sampling period

`� : Fresh ore feed rate

`� : Rod mill feed water rate, dilution water flow rate for the typical wet grinding circuit `a : Dilution water flow rate

`a,& : Dilution water mass flow rate b : Uniform convective velocity b�: Convection velocity of constituent �� b%GW : Uniform convective velocity at a reference state c : Total mass of particles

dZ,� : Feed mass fraction of sizee�

c�: Bond’s index, characteristic of the material

d� : Mass fraction of particles of size ; d�

� : Initial spatial the mass fraction of size e� f : Vector of state variables

g: Spatial coordinate along the mill axis

h : Laplace transform of output signal

i� : Product particle size or fineness

i� : Product flow rate, circulating load for the typical wet grinding circuit

ia : Product density

h& : Vector of measured variables

i$ : Predicted output

h%: Reference or setpoint

e: Size of particles, Z-transform complex variable


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e� : Size of particles in the ;"> size interval

e� : Size of particles in the <"> size interval

‖… ‖l : �l − Rno� Greek alphabet

p : Parameter depending on material in the selection function q�,� : Cumulative breakage function

r : Vector of deviations between h& and h

r� : Probability of particles of size e� to cross through the discharge grid

s : Sensitivity function

t: Parameter depending on material in the cumulative breakage function

u� : Dimensionless selection function of interval size ; �� : Constituent of a tubular chemical reactor

v�� : Hydrocyclone selectivity function or tromp curve w : Delay of the tracer impulse

w�� : Second hydrocyclone model parameter w�� : (;, <)"> Relative gain

wM: Time constant of DOB low-pass filter

wM,K : Time constant of nominal DOB low-pass filter

x : Complementary sensitivity function

. : Family of plants

y : Standard deviation of the residence time distribution

z : Time constant

z{ : Time delay

| : Vector of model parameters |} : Vector of model parameter estimates ~ : Sensitivity weighting function

y� : ;"> Singular value

LIST OF FIGURES 2013

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LIST OF FIGURES

II.1: Simplified chain of processes in mineral processing and metallurgical plants for a metallic ore .................................................................................................. 10 II.2: Typical wet grinding circuit................................................................................... 11 II.3: Flowsheet of the KZC grinding circuit .................................................................. 13 II.4: Rod Mill ............................................................................................................... 14 II.5: Hydrocyclone Classifiers ..................................................................................... 14 II.6: Ball Mills .............................................................................................................. 14 III.1: Classification into size intervals .......................................................................... 21 III.2: Mass flow rates at the hydrocyclone ends .......................................................... 30 IV.1: Representation of the KZC grinding process as a MIMO system ....................... 39 V.1: Laboratory scale slurry flotation .......................................................................... 44 V.2.a): RTD fitting of the rod mill ................................................................................. 45 b): RTD fitting of the ball mill................................................................................. 45 V.3.a): Rod mill PSD fitting-simple validation .............................................................. 49 b): Evolution of the PSDs for the three intervals of size classes within the rod mill 49 V.4.a): Rod mill PSD fitting-cross validation................................................................ 50 b): Evolution of the PSDs for the three intervals of size classes within the rod mill 50 V.5.a): Ball mill PSD fitting-cross validation ................................................................ 51 b): Evolution of the PSDs for the three intervals of size classes within the ball mill 51 V.6.a): Hydrocyclone 1 fitting curve – simple and cross validations ............................ 52 b): Hydrocyclone 2 fitting curve – simple and cross validations ........................... 53 V.7: Evolution of the PSDs for the three intervals of particle sizes within the rod mill .. 57 V.8: Steady-state characteristics of the system .......................................................... 58 V.9: Time-mill axis evolution of the rod mill content .................................................... 60 V.10: Time evolution of the product PSD .................................................................... 61 V.11: Time evolution of the product flow rate .............................................................. 61 V.12: Time evolution of the product density ................................................................ 61 V.13: Time evolution of the product fineness variations around nine operating points 62 V.14: Time evolution of the product flow rate variations around nine operating points . 62 V.15: Time evolution of the product density variations around nine operating points .. 63 VI.1: Typical wet grinding circuit: single-stage closed-loop ball mill grinding ............... 68 VI.2: Representation of the typical wet grinding process as a MIMO system .............. 69 VI.3: Typical wet grinding circuit with its instrumentation-ISA norm ............................ 72 VI.4: PID control ......................................................................................................... 76 VI.5: IMC diagram ...................................................................................................... 77 VI.6: DOB based control ............................................................................................. 78 VI.7: Decoupling control ............................................................................................. 79 VI.8: Philosophy of the MPC scheme ......................................................................... 80

VII.1: Principle of the initial steady-state operating point selection .............................. 86 VII.2: Step variations of manipulated variables ........................................................... 88 VII.3: Step variations of disturbances ......................................................................... 89


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VII.4: Time evolution of the product fineness after step variations of input variables ... 89 VII.5: Time evolution of the product flow rate after step variations of input variables .. 90 VII.6: Time evolution of the product density after step variations of input variables ..... 90 VII.7: Comparisons between continuous-time and discrete-time models .................... 92 VII.8: Frequency evolution of relative gains ................................................................ 99 VII.9: DNA with Gershgorin’s bands for the first choice of variable pairing ................ 100 VII.10: DNA with Gershgorin’s bands for the second choice of variable pairing ........ 100 VII.11: Interaction effects between manipulated and controlled variables ................. 102 VII.12.a): Additive uncertainty representation ........................................................... 104 b): Multiplicative uncertainty representation ................................................... 104 VII.13: Neighbourhood space around the nominal operating point ............................ 104 VII.14: Step variations of manipulated variables ...................................................... 105 VII.15: Variations of output variables after step variations of manipulated variables around the nominal and the eight boundary operating points ........................ 106 VII.16: Family of uncertain plants ............................................................................. 108 A-VII.1: Output variations after step change of feed particle size distribution ............ 110 VIII.1: Standard form of PI-SP .................................................................................. 116 VIII.2: IMC form of PI-SP .......................................................................................... 117 VIII.3: Standard form of DOB .................................................................................... 118 VIII.4: IMC form of DOB ........................................................................................... 118 VIII.5: DIMC scheme ................................................................................................ 119 VIII.6: Block diagram for definition of sensitivity ......................................................... 120 VIII.7: Block diagram for DOB design ....................................................................... 123 VIII.8: Block diagram for PI-SP design...................................................................... 124 VIII.9: Design of DOB and PI-SP .............................................................................. 128 VIII.10: Reconstruction of lumped disturbance on the product fineness for both nominal and uncertain models .................................................................................. 129 VIII.11: Reconstruction of lumped disturbance on the product flow rate for both nominal and uncertain models .................................................................................. 129 VIII.12: Reconstruction of lumped disturbance on the product density for both nominal and uncertain models .................................................................................. 129 VIII.13: Bode’s diagram of transfer function between the total lumped disturbance and its estimate for the channel 1 ....................................................................... 130 VIII.14: Bode’s diagram of transfer function between the total lumped disturbance and its estimate for the channel 2 ....................................................................... 130 VIII.15: Bode’s diagram of transfer function between the total lumped disturbance and its estimate for the channel 3 ....................................................................... 130 VIII.16: Step response of product fineness ............................................................... 131 VIII.17: Step response of product flow rate ............................................................... 131 VIII.18: Step response of product density ................................................................. 131 VIII.19: KZC wet grinding circuit with its instrumentation-ISA norm ........................... 132 VIII.20: Time evolutions of fresh ore feed rate and product fineness for both nominal and mismatch cases – linear simulation ...................................................... 134 VIII.21: Time evolutions of rod mill feed water flow rate and product flow rate for both nominal and mismatch cases – linear simulation ......................................... 135 VIII.22: Time evolutions of dilution water flow rate and product density for both nominal


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and mismatch cases – linear simulation ...................................................... 135 VIII.23: Estimation of lumped disturbance on the product fineness for both nominal and mismatch cases – linear simulation ............................................................. 135 VIII.24: Estimation of lumped disturbance on the product flow rate for both nominal and mismatch cases – linear simulation ............................................................. 136 VIII.25: Estimation of lumped disturbance on the product density for both nominal and mismatch cases – linear simulation ............................................................. 136 VIII.26: Time evolutions of fresh ore feed rate and product fineness for both nominal and mismatch cases in one closed-loop configuration – linear simulation .... 138 VIII.27: Time evolutions of product flow rate and product density for both nominal and mismatch cases in one closed-loop configuration – linear simulation........... 139 VIII.28: Estimation of lumped disturbance on the product fineness for both nominal and mismatch cases in one closed-loop configuration – linear simulation........... 139 VIII.29: Time evolutions of fresh ore feed rate and product fineness for both nominal and mismatch cases in two closed-loop configuration – linear simulation .... 140 VIII.30: Time evolutions of dilution water flow rate and product density for both nominal and mismatch cases in two closed-loop configuration – linear simulation .... 141 VIII.31: Estimation of lumped disturbance on the product fineness and on the product Density for both nominal and mismatch cases in two closed-loop configuration –linear simulation......................................................................................... 141 VIII.32: Time evolution of product flow rate for both nominal and mismatch cases in two closed-loop configuration – linear simulation ............................................... 141

LIST OF TABLES 2013

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LIST OF TABLES

IV.1.a): Rod mill transport parameters ........................................................................ 35 b): Rod mill grinding parameters .......................................................................... 36 IV.2: Hydrocyclone � parameters (� = 1,2) ................................................................ 36 IV.3.a): Ball mill transport parameters ........................................................................ 38 b): Ball mill grinding parameters .......................................................................... 38 V.1: Internal transport parameters estimation for rod/ball mill ..................................... 44 V.2: Transport parameters estimation for rod/ball mill ................................................. 44 V.3: Grinding parameters estimation for rod/ball mill .................................................. 48 V.4: Parameter estimation for hydrocyclone classifiers ............................................... 52 V.5: Steady-state simulation results ............................................................................ 56 V.6: Signs of gains between manipulated and output variables .................................. 59 VI.1: Dynamic and steady-state features of the typical grinding circuit........................ 70 VI.2: Typical variable pairing ...................................................................................... 70 VII.1: Initial steady-state operating point ..................................................................... 86 VII.2: Optimal operating point ..................................................................................... 95 A-VII.2.1: Neighbourhood space around the nominal operating condition after +/- 10 % variations of manipulated variables .......................................................... 111

A-VII.2.2: Identification of boundary parameters of the linearized system model ...... 111 VIII.1: Step changes of setpoints .............................................................................. 115 VIII.2: Step-sinusoidal variations of external disturbances ........................................ 115 VIII.3: Duality between PI-SP and DOB design ........................................................ 126 VIII.4: Tuning parameters of the controllers .............................................................. 127 VIII.5: Performance indices in step setpoint changes ............................................... 127 VIII.6: Performance indices in step setpoint changes for all the control system – linear simulation ........................................................................................................ 134 VIII.7: Gains of frequency response between external disturbances and controlled variables at frequency equal to 10[rad/h] in the nominal case .................... 137 VIII.8: Gains of frequency response between external disturbances and controlled variables at frequency equal to 10[rad/h] in the mismatch case ................. 137

VIII.9: Reduction ratio of gains of frequency response at frequency of 10[rad/h] obtained thanks to the DOB and the PI-SP in the nominal case .................... 137 VIII.10: Reduction ratio of gains of frequency response at frequency of 10[rad/h] obtained thanks to the DOB and the PI-SP in the mismatch case ................ 137 VIII.11: Reduction ratio of gains of frequency response at frequency of 10[rad/h] obtained thanks to the DOB and the PI-SP in the nominal case for one closed- loop configuration ........................................................................................ 140 VIII.12: Reduction ratio of gains of frequency response at frequency of 10[rad/h] obtained thanks to the DOB and the PI-SP in the mismatch case for one closed-loop configuration ................................................................................................ 140 VIII.13: Reduction ratio of gains of frequency response at frequency of 10[rad/h]

LIST OF TABLES 2013

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obtained thanks to the DOB and the PI-SP in the nominal case for two closed- loop configuration ........................................................................................ 142 VIII.14: Reduction ratio of gains of frequency response at frequency of 10[rad/h] obtained thanks to the DOB and the PI-SP in the mismatch case for two closed-loop configuration ............................................................................. 142

Chapter I: INTRODUCTION 2013

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Chapter I: INTRODUCTION

I.1 Motivations

Mineral processing refers to the set of techniques and methods for extraction of the valuable minerals contained in the blocks of ore coming from the mine. The raw material is thus valorised. This can be performed only after reducing the coarse blocks by means of mineral preparation methods into a size such that one can separate efficiently each valuable mineral from useless minerals. This separation resorts to physical, chemical or physicochemical methods and allows obtaining a concentrate of maximal value and the lowest possible tailing.

Using mineral processing methods constitutes a permanent challenge in the mineral and metal industry. In a first stage, the process has to be designed. The second step is concerned with the starting up and the commissioning of the installation combined with adjustments of some parameters. Once the running-in is completed, the plant operator should keep the process under control and, in order to maintain or increase the profit margin, he has to optimize it. This requires a deep knowledge of the process behaviour.

To satisfy the requirements on the final product (metal) set by the consuming market, one can either make changes and/or modifications on the mineral processing stage or on the extractive metallurgy step. Changing some values of parameters and variables of the extractive metallurgy may induce a high cost (Bouchard, 2001). Thus, control is often applied on the mineral processing whose product, the ore concentrate, constitutes the input material of the extractive metallurgy. This is why much pressure and attention are paid on mineral processing units and especially on concentration plants.

As the ore size reduction procedure is the critical step of a concentrator, it turns out that controlling a grinding circuit will be of much benefit and crucial for all the concentrator. Indeed, the grinding process accounts for almost 50 % of the total expenditure of the concentrator plant (Chen et al., 2008). Moreover, the product particle size from grinding stage influences the recovery rate of the valuable minerals as well as the volume of tailing discharge in the subsequent process. This is the first justification for studying an efficient way to optimize and control a grinding process such as that of the Kolwezi concentrator.

An industrial wet grinding process offers complex and challenging control problems, among which:

- the necessity of satisfying rapidly to requirement variations of the consuming market with regard to the final product, i.e. the metal. This requirement is typically expressed in terms of ensuring maximum production with the best possible quality;


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- the necessity of adapting to the random variations of the raw material properties and characteristics;

- the difficulty to highlight the influence of the water on the process efficiency; - the presence of a physical closed-loop, i.e. the material rejected by a classifier

is recycled back to a ball mill inlet; - the existence of nonlinear relationships between state variables such as the

material hold-up and the particle size distribution within a mill.

The above five stated issues constitute the second motivation of this study.

I.2 State of the art

Several studies and investigations have been carried out on the mineral grinding processes, from their modelling to their optimization and control. Thus, much scientific literature has been devoted to this framework in the last thirty years. In the reference books by Lynch (Lynch, 1977) and Austin (Austin et al., 1984) followed by King (King, 2001), many studies and implementations have been performed and regularly surveyed. Other very interesting and precious works have been achieved by academic and/or industrial specialists. We can quote, with references therein, (Hulbert, 1989; Hodouin & Del Villar, 1994; Duarte et al., 1999; Pomerleau et al., 2000; Boulvin, 2001; Hodouin et al., 2001; Liu & Spencer, 2004; Lepore, 2006; Chen et al., 2008; Chen et al., 2009; Ozkan et al., 2009; Weig & Craig, 2009; Yang et al., 2010; Hodouin, 2011). Most of these studies are based on a typical grinding process. This typical configuration is a single closed-loop grinding circuit consisting mainly of one ball-mill, one pump sump and one hydrocyclone classifier.

Modelling

Deriving a model for a process is drastically linked to the expected use of the model system. In the case of grinding circuits, one can distinguish two main classes of models; on the one hand, steady-state or static models used for dimensioning or optimizing the circuit, and on the other hand, dynamic models allowing to proceed with dynamic simulation and the design of a control strategy.

Initially, most of the models found in the literature were static and/or empirical. For breakage operations, in these empirical models (Rittinger, 1857; Kick, 1883; Bond, 1952), the energy consumed is the main variable to be minimized. Hence, such models are useful for the design of circuits.

Nowadays, the literature proposes more and more dynamic models for grinding processes and circuits (Hodouin, 1994; Boulvin, 2001; Liu & Spencer, 2004; Lepore, 2006; Ozkan et al., 2009) but most of them are either simple lumped parameter dynamic models or dynamic models in which the transport dynamics is included as time delay, i.e. mean residence time, in the state variables. Yet two simultaneous phenomena must be combined to derive a meaningful model of a grinding process: the breakage or fragmentation phenomenon and the transport of material within a mill. To the author’s knowledge, the latter approach has only been applied to cement grinding circuits (Boulvin, 2001; Lepore, 2006).


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Simulation

Many simulation techniques and packages already exist for flowsheet simulation in the mineral processing industry (Liu et al., 2004; Reyes-Bahena, 2001; Napier-Munn & Lynch, 1992) but most of them are either based on a steady-state analysis, for instance METSIM, USIM PAC, MODSIM, Limn and JKSimMet, JKMRC Simulator, SIMBAL, GSIM, CAM and Bruno, or on simple lumped parameter dynamic models, e.g. Aspen Dynamics, SysCAD and MinOOcad (Herbst & Blust, 2000).

Control

In the literature, one can find several studies devoted to the challenge of grinding circuits control. The investigations in this framework cover many aspects from theoretical up to practical considerations. The related results include classical control as well as advanced control techniques from decentralized Proportional-Integral-Derivative (PID) control to centralized Model-based Predictive Control (MPC) schemes (Hodouin, 1994; Duarte et al., 1999; Pomerleau et al., 2000; Chen et al., 2008; Chen et al., 2009; Weig & Craig, 2009; Yang et al., 2010; Hodouin, 2011). Focusing on the typical wet grinding process, most of these control strategies are based on a two input-two output scheme. The controlled variables are typically the product particle size and the circulating load while the manipulated variables are usually the fresh ore feed rate and the dilution water flow rate.

I.3 Contributions of this study

The current dissertation is focusing on an industrial grinding plant, namely the grinding process of the “KolweZi Concentrator”, KZC in short, from the Democratic Republic of Congo. This grinding circuit consists mainly of one rod-mill in an open-loop and two ball mills, one pump sump, one distributor and two hydrocyclone classifiers in a double closed-loop. Hence, the KZC grinding process has a more complex topology than the typical one and offers more challenging issues in modelling and control.

Therefore in this thesis, we have developed and extended to wet operating/working condition and to mineral processing the complex model used in dry cement grinding. Thus, a phenomenological approach combining population mass balance and pulp dynamics transport is used for rod mill and ball mills modelling. This leads to nonlinear partial differential equations (PDE) containing six parameters for each mill. While the repartition function is almost the same for both dry grinding and wet grinding, a mathematical description is proposed for the selection function which should be higher in wet grinding than in dry grinding (Ozkan et al., 2009). Another mathematical relationship which describes the link between the convective velocity and the feed ore rate has been suggested. A steady-state model with two parameters is sufficient to describe the hydrocyclones classifiers (King, 2001) while the pump sump is modelled by a perfect mixer. The other components of the KZC flowsheet are simply considered as time delay elements.

For control purposes, product particle size, product flow rate and product density are regarded as controlled variables while fresh ore feed rate, rod mill feed


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water and dilution water flow rate are the three manipulated variables. Therefore, the KZC circuit can be considered as a Multi-Input Multi-Output (MIMO) dynamical system with couplings, time delays, strong external disturbances such as ore grindability (inverse of ore hardness) and feed ore particle size distribution, and internal disturbances caused by model mismatches. This control study includes explicitly the control of the product density which has usually been neglected in the control of the typical grinding process. Yet, the density of the pulp from the grinding stage influences significantly the performance of the subsequent stage. The process under study is thus considered as a 3x3 MIMO system.

All model parameters are determined from experimental data by using the nonlinear least squares algorithm. It should be noted that, due to the unavailability of sensors on the process, we could not perform on-line measurements of the variables in real-time. Hence we could not use a colorant neither a radioactive tracer to evaluate the transport dynamics of material within the mills. An alternative approach has been developed on the basis of the G41 foaming as tracer. This new experimental procedure provided satisfactory results in comparison with previous studies (Rogovin et al., 1988). Similarly due to a lack of instrumentation, Tyler series sieves have been employed for measurements of particle size distributions.

For global simulation, all the individual models are connected according to the circuit configuration provided by the flowsheet of the installation. Both a steady-state simulator and dynamical simulator based on three classes of particle sizes have been developed within the MATLAB/Simulink software. The resulting steady-state simulator is in good agreement with the recorded data, and the dynamic simulator exhibits the expected qualitative behaviour. Both tools could allow operator training. Finally, the dynamical simulator has been analysed in order to highlight the steady-state and dynamic features of the process. These features characterize the main control issues and led us to choose a suitable control structure and design the corresponding controller.

The process model has been linearized around a steady-state operating point. A two-step approach has been used to determine the operating point around which linearization has been performed. First, an initial estimate of the optimal operating point is determined on the basis of a simulation study. Thus, a linearized model of the process is determined around this initial operating point. Next, a correction to the initial operating point has been computed by solving a convex optimization problem. The optimization problem consisted in maximizing the product flow rate while meeting specific constraints on product quality and accounting for actuator limitations. To do so, it has been assumed that the linearized model remains valid for the considered correction range.

A Double Internal Model Control (DIMC) has been developed as the most suitable decentralized control scheme of the KZC wet grinding process. This linear control strategy has been selected in order to cope with all the control issues and is based on the linearized system model. The main control issues have been identified as follows: strong coupling effects, model mismatches regarded as internal disturbances, strong external disturbances, large time delays and model parameter uncertainties. These uncertainties are essentially due to operating point changes. The control


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objective consists in holding the process on the optimal operating point despite the internal and external disturbances. The proposed DIMC structure is composed of a Proportional-Integral (PI) Smith Predictor (SP) and a Disturbance Observer (DOB) on each channel of the paired variables. A duality between the PI-SP and the DOB has been established from a design point of view. A robust design has been performed to allow the controller to work properly on the actual plant whose model is uncertain. The validation of the control scheme has been made on the basis of simulation results from the linearized model. A progressive implementation of the control structure is also explained in the framework of the KZC installation. This is helpful to show the benefit of the control if the plant does not possess all the required instrumentation.

In summary, our contributions in the framework of wet grinding processes can be listed as follows:

- the generalisation of the grinding model to a complex topology containing a double closed-loop;

- the mathematical description of the selection function which takes into account the positive effect of water on the efficiency of the grinding;

- the mathematical description of the link between the convective velocity and the feed ore rate;

- the use of the G41 foaming as tracer yielding a new experimental method to determine the dynamics of material within a mill;

- the derivation of an analytic solution of the steady-state model in both open-loop and double closed-loop configurations, and its use to estimate the grinding model parameters;

- the control of the product density as the third controlled variable; - the proposition of a DIMC as a suitable decentralized control scheme; - the use of the PI-SP and the exploitation of the duality between the PI-SP and

the DOB for controller design; - the use of a robust performance criterion to design the controller and the DOB; - the possibility of a progressive implementation of the control scheme.

I.4 Outline

This dissertation is divided into two parts: process modelling and simulation (chapters II, III, IV and V), and controller design and validation (chapters VI, VII and VIII).

First part

Chapter II provides the complete description of the Kolwezi concentrator and especially its grinding stage which is a double closed-loop wet grinding circuit. The flowsheet is presented and the main components are depicted. In Chapter III, we present an overview of the state of the art in the modelling of grinding processes. This state of the art is focused on the grinding phenomenon and the classification by hydrocyclone. Chapter IV is concerned with the modelling of the KZC grinding circuit. The models of the main components of the installation are derived. The concatenation of all individual models according to the circuit configuration allows setting the global model.


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In Chapter V, we first describe the methodology used to identify the parameters of the different models. Experiments and measurements performed on site are explained. The obtained data and the least squares algorithm are employed to estimate the optimal parameters. Finally, we present both the steady-state simulator as well as the dynamic simulator and we discuss the simulation results. Second part In Chapter VI, a survey of the various control schemes used in the framework of grinding circuits is presented. Based on the typical wet grinding circuit, this survey covers the state of the art of the main control methods from decentralized controllers to multivariable controllers. Chapter VII presents a systematic analysis of the process under study. Aspects as system model linearization, operating point optimization, variables pairing, interaction effects and uncertainty characterization are studied. This chapter ends by highlighting the main features of the process. Finally, in Chapter VIII, the DIMC scheme is described. The robust design method is systematically explained. The duality between the PI-SP and the DOB designs is revealed. The controller is validated by means of various simulations conducted on the linearized model. The possibility of a progressive implementation of the control structure in the KZC context is demonstrated.

Part I :

PROCESS MODELLING AND SIMULATION

Chapter II: WET GRINDING CIRCUIT OF THE KOLWEZI CONCENTRATOR

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Chapter II: WET GRINDING CIRCUIT OF THE

KOLWEZI CONCENTRATOR

Introduction The present chapter aims at describing the industrial plant which is the object of this study, namely the grinding process of the Kolwezi concentrator. First, the framework of the current dissertation is defined. To this end, the general concepts of mineral processing are presented. The goal and the chain of processes in mineral processing are briefly given. After, the emphasis is made on the critical aspect of grinding stage. The typical grinding flowsheet, most encountered in the literature, is particularly highlighted. Finally, the wet grinding process of the Kolwezi concentrator is presented. While the other stages of the plant are briefly explained, the grinding one is described and specified. This description will later be our reference in the current study, on modelling and control, especially in the comparison with the typical circuit. .

II.1 General concepts of mineral processing

II.1.1 Objectives and processes of mineral processi ng

A raw ore cannot be used as such as a final product for industrial or commercial uses. It needs to be treated for preparing usable materials that can be either specific minerals released from the ore, or more usually metals, alloys, or compounds such as oxides. The aim of mineral processing for most of ores is to concentrate the valuable minerals contained in raw ores for the subsequent stages such as metal extraction, fabrication of consumption products. Usually, minerals are first liberated from the ore matrix by comminution and particle size separation processes, and then separated from the tailings using processes capable of selecting particles based on their physical or chemical properties, such as surface hydrophobicity, specific gravity, magnetic susceptibility, chemical reactivity, and colour.

The processes in mineral processing can thus be classified into the following distinct categories (Hodouin, 2011):

- minerals liberation processes (crushing, grinding and size classification); - minerals separation processes (flotation, magnetic or gravimetric separation,

sorting, leaching, etc.); - concentrate pre-treatment for subsequent metal extraction (drying,

agglomeration, sintering, etc.); - peripheral processes (feeders, pumps, conveyors, thickeners, systems for

concentrate shipping, tailing disposal, mine backfilling, effluent treatment, reagent dosage, etc.).


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But in general, the transformation chain in mineral processing is a technically coherent sequence of processes. This chain is shown in Figure II.1 for a metallic ore.

Figure II.1: Simplified chain of processes in mineral processing and metallurgical plants for a metallic ore The current dissertation is just focusing on one operation among the above processes chain. Therefore, the following subsection is concerned with the grinding step since it is the critical stage of the mineral processing chain.

II.1.2 Grinding process

The grinding process is a size reduction operation often used in the mineral industry to liberate the valuable minerals from the tailings (Pomerleau et al., 2000). It is a fundamental operation process, and in many respects the most important unit operation in a mineral processing plant (Chen et al, 2008). Indeed, grinding process represents almost half of the total operating costs associated with the mining operation, and the product particle size greatly influences the recovery rate of the valuable minerals and the volume of tailing discharge in the subsequent processes. Low qualified rate of product particle size can cause unacceptable economic loss and could be harmful for pollution control. This is why the grinding process is the critical stage of a mineral concentrator plant.

The true objective of grinding is to obtain a proper liberation of the minerals coupled with a proper particle size distribution in order to maximize recovery and concentrate value without overgrinding, while keeping specific comminution energy and reagent consumption as low as possible, taking into account the prevailing economic conditions. These objectives being highly complex, simpler ones must be formulated. For effective concentration, the grinding process has simply to maintain the following output variables stable, mainly including the product particle size distribution, circulating load and mill solid concentration (Chen et al., 2008).

Unlike the cement processing and other processing units, the grinding mills used in mineral processing units runs basically in wet condition, i.e. the input materials of the process are the fresh ore and the water. The latter may be fed into different places of the circuit. There are several configurations of grinding circuits ranging from single-stage to multiple-stage and from open-loop to closed-loop grinding circuits. Figure II.2 shows the typical flowsheet most encountered in the literature specialised on modelling and control of mineral grinding processes. It is a single-stage and simple closed- loop circuit consisting of one ball-mill, one pump sump and one hydrocyclone classifier.


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Figure II.2: Typical wet grinding circuit

The next section presents the wet grinding circuit which is the object of the current study. This circuit is compared to the typical one in order to underline the differences leading to the additional challenges characterized in subsequent chapters.

II.2 Grinding process of the Kolwezi concentrator

II.2.1 Preliminary description

The industrial plant, object of this study, is the “KolweZi Concentrator”, KZC in short. This plant belongs to Gécamines which is a Democratic Republic of Congo state company. Gécamines is the greatest mining and mineral processing company in the DRC and is acting in the Katanga province. Its head quarter is located in the Lubumbashi town. The Kolwezi town, where the KZC is situated, constitutes the so-called “west group” of Gécamines. KZC is a mineral processing installation dealing especially with mineral concentration. This concentrator is fed by raw material coming from mines and produces a copper-cobalt concentrate after rejecting the tailings. Four steps summarize the KZC process: dry crushing and screening, wet grinding and classification, flotation, thickening and filtration. In the next subsections, the overall objective of the KZC plant is given. Then the four stages are presented. The emphasis is made on the grinding process which is the heart of KZC and the centre of our study.


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II.2.2 Objective of KZC KZC receives dolomitic or siliceous blocks of ore from three different deposits:

- Deposit K1: the ore of this deposit is rich in copper and cobalt with a high grade in manganese.

- Deposit K2: this ore is poor in copper and cobalt but rich in manganese. - Deposit K3: the ore contained in this deposit is rich in copper but poor in cobalt

and manganese. The production capacity of KZC is actually of 50��/ℎ/�� (fifty tons-dry per hour and per rod mill). The feed mean grades are of 2.5% in copper and 0.2% in cobalt.

The required grades of the produced concentrate are of 15 − 18% in copper and 0.98% in cobalt. The target of KZC is to produce the best copper-cobalt concentrate in terms of quantity and quality.

II.2.3 Dry crushing The dry crushing section is the first step of the KZC process and it is also the first stage of the ore mechanical preparation or size reduction. This section is fed by coarse blocks of 1500to 1800 as size and should feed into

the wet grinding section the ore with 19as maximum size. The KZC dry section is a three sub-steps size reducing procedure: primary crushing by the 1/10 Arbed jaw crusher, secondary crushing by the 7� standard Symons cone

crusher and tertiary crushing by the 7� short head Symons cone crusher. II.2.4 Wet grinding The wet grinding section is the critical step of KZC. Its flowsheet is shown in Figure II.3. This circuit is composed mainly of one rod mill, two ball mills, two hydrocyclones classifiers, one sump pump, one slurry distributor and several pipes and channels for material transport. With reference to the mineral processing language, this kind of flowsheet can be called “double-closed loop modified traditional circuit” (Bouchard, 2001) or “double-closed loop preclassification circuit” (Hodouin et al., 1994). After the dry crushing and screening stage, the copper-cobalt ore is sent to a stock pile. Then, a set of conveyor belts transport the raw material to the wet grinding circuit. The ore is fed into the rod mill together with the feed water.

In an open circuit, the tumbling action of the rods within the revolving mill crushes the feed ore to finer sizes. The slurry is discharged from the rod mill to a sump pump where it is diluted by adding water, and then pumped to a distributor which separates the flow into two streams. Each separated slurry flow is fed into a


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Ds

hydrocyclone classifier whose underflow stream containing the larger particles is recycled back to a ball mill for regrinding while the overflow stream containing the finer particles is considered as one part of the product. The set made by the sump pump, the pulp distributor, the two hydrocyclones classifiers and the two ball mills constitutes the double closed-loop circuit. The process product is thus the sum of the two overflow streams produced by the classifiers.

Figure II.3: Flowsheet of the KZC grinding circuit

Caption

- HC 1: Hydrocyclone 1 - HC 2: Hydrocyclone 2 - Ds: Distributor

This second step of the KZC mechanical preparation process is simply called “milling” and its aim is just that of providing the subsequent flotation section with a maximum flow rate of pulp characterized by:

- 80%of particles with size smaller than the reference size of 74� ;

- a density ranging between 1.3 and 1.4.

The Particles Size Distribution (PSD) is measured by means of 48 and 200

��sieves of Tyler sieves series having respectively 0.3 and 0.074 as mesh dimension. One litre of pulp is poured on the two sieves and the method consists in checking if nothing is retained on the 48�� sieve and around 20% should be

retained on the 200��sieve. The pulp density is deduced from the weighing of one litre of pulp by means of a Marcy balance. This milling circuit is composed of the following unitary operations: fine milling by rod mill, size classification by hydrocyclones and ultra-fine milling by ball mills. Figures II.4, II.5 and II.6 show respectively the photographs of the rod mill, the hydrocyclones classifiers and the ball mills in use at KZC, with characteristics therein.

HC 2

Crushing and screening

Ball Mill 2

Ball Mill 1

Sump Pump

Stock Pile

Conveyor Belt

Fresh Ore

Feed

Water Feed

DilutionWater

HC 1

Product

Rod Mill


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Figure II.4: Rod Mill

Dimensions: 7’x12’ Brand: Marcy (Metso Minerals)

a) b)

Figure II.5: Hydrocyclone Classifiers Kind: Static classifier

a) Tricône b) Simple

Figure II.6: Ball Mills Kind: Ball Mill 8’x84’’ or Tricône 9’x3’x6’x8’ Brand: Hardinge (Metso Minerals)


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II.2.5 Flotation Flotation is the mineral concentration method employed at KZC. The role here is to concentrate the mineral or grow up the grade of useful mineral with an acceptable efficiency. The input of this section is the hydrocyclones overflow and the outputs are the concentrate sent to the last step of thickening and filtration, and the tailing which is taken out. The characteristics of the flotation cells are the following:

• Brand: Auto Kumpo • Composition: - 20 primary cells.

- 20 secondary cells. - 16 tertiary cells. - Capacity per cell: 8�.

- Motor: Asynchronous, 25��, < 1000��/��, 550 .

II.2.6 Thickening and filtration This last step of the KZC process is concerned with removing out the water contained in the concentrated pulp coming from the flotation. For this purpose, a solid-liquid separation method combining thickening and filtration is used. One usually completes the process by drying the mineral mud in order to obtain a dry concentrate. II.2.7 Current control mode Currently, there is no control system installed on the KZC plant and furthermore, there is no instrumentation on the plant except two valves for the rod mill feed water and the dilution water. The process is both manually controlled and not optimised. Hence, the KZC target is never reached. Therefore in this work, we are going to propose an automatic control structure in order to maintain the KZC wet grinding process on the optimal operating point maximizing the production while meeting constraints on the product fineness and density.

Conclusion This chapter has first presented the introductory notions of mineral processing. The chain of processes involved in this framework has been shown. Then, as our work is applied on an industrial concentrator plant, the process under study has been described. On the one hand the typical grinding circuit proposed by the literature has been provided and on the other hand the KZC grinding process has been described. Its main components have also been given. It can be seen that the KZC grinding circuit is a wet grinding operation and exhibits a more complex topology with two stages of milling and a double physical closed-loop. These characteristics will be used in the next chapters in order to determine suitable approaches for modelling and control of the KZC wet grinding process since it is manually controlled.

Chapter III: STATE OF THE ART FOR GRINDING PROCESSE S MODELLING

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Chapter III: STATE OF THE ART FOR GRINDING

PROCESSES MODELLING

Introduction The goal of this chapter is to give an overview of the different modelling approaches used in the scope of grinding circuits. By observing the typical and the KZC grinding circuits, we realise that two main operations are involved, namely the mill grinding and the hydrocyclone size classification. The grinding operation within a mill may be split into two elementary operations, i.e. the fragmentation and the material transport. Therefore, the fragmentation or breakage models are presented first. Next, the chapter addresses the issue of modelling the transport dynamics of material within a mill. After, the two models are combined to form the final model of a grinding mill process. The last section is concerned with the modelling of the size classification of particles by the hydrocyclone. The models described in this chapter will be employed as reference from which the appropriate model of the KZC wet grinding plant will be derived.

III.1 Fragmentation or breakage modelling

III.1.1 Introduction

Nowadays, several kinds of breakage machines are used in industry. The energy consumed by those machines constitutes a very high cost (Boulvin, 2001). Moreover, only 2 to 20 % (according to the grindability of the material and the kind of installation) of the energy supplied is efficiently dedicated to the breakage (Labahn & Kohlhaas, 1983) while the rest is dissipated trough heat. Deriving a dynamical model of the process should allow more understanding of its behaviour and eventually improving its performance.

The attempts to quantify the breakage phenomena have been the object of several theoretical studies including energy-based approaches, morphological aspects and the study of transformations occurring in the breakage devices (Jdid et al., 2006). Most of the mathematical models found in the literature are steady-state and empirical (Rittinger, 1857; Kick, 1883 and Bond, 1952). These models have been derived after many years of experiments and observation works. In these models, the energy consumed for the fragmentation constitutes the main variable to be minimised. The steady-state models proposed in the literature should allow the circuit design and they are also called “energy-based models”.

In 1941, R.L Brown proposed an original approach of fragmentation modelling (Boulvin, 2001). This approach is based on the concept of “population model”. The particles of the material are set in intervals of size. The fragmentation phenomenon is


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explained by the “disappearance or death” and “appearance or birth” of particles within each particle size interval or class. This is the phenomenological modelling.

In the following subsections, the two approaches introduced above are detailed as a state of the art for the modelling of the particles fragmentation or breakage. III.1.2 Energy-based models Three main theories have been reported in order to describe the relationship between the energy consumed by the material and the reduction of its size during the fragmentation process. Nowadays, the assumptions proposed to determine this relationship are not rigorously proved because one does not yet know how to measure the quantity of energy really absorbed by the particles during their fragmentation. One can only measure the total energy consumed by the fragmentation device or equipment. In 1857, Von Rittinger postulated that the specific energy for the breakage of a given material is directly proportional to the amount of specific surface newly created. The specific energy is defined as the quantity of energy to be supplied in order to grind a unit mass of the material. This results in the following relationship:

�� = ��(�� − �) (3.1)

where �� : Specific breakage energy (according to Rittinger).

�� : Rittinger constant.

��: Specific surface of the material after fragmentation.

�: Specific surface of the material before fragmentation.

The Rittinger constant �� represents in fact the specific energy of breakage per unit of specific surface produced. The Rittinger relationship can also be expressed with regard to the size ‘z’ of particles:

�� = ��

�� (�) − � ��

�� (�)� (3.2)

where �(�): Cumulated particles size distribution of material before fragmentation. ��(�) : Cumulated particles size distribution of material after fragmentation.

As in practice it is difficult to perform the integration from �� = 0; one commonly uses the following pseudo-Rittinger form:

�� = �� , ! − �

��, !" (3.3)

where �,#$ : Size corresponding to 80 % of passing in the initial particle size

distribution.


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��,#$ : Size corresponding to 80 % of passing in the final particle size

distribution.

The drawback of this law lies in the fact that Rittinger did not take into consideration the material deformation before its fragmentation. The amount of surface produced might be proportional to the required work only if it is proportional to the applied stress multiplied by the deformation length.

Kick proposed in 1883 a very simple model for breakage of solid particles. This model expresses the specific breakage energy of a homogeneous material with regard to initial and final particles sizes:

�% = �% log ��, !��, !" (3.4)

where�%: Specific breakage energy (according to Kick).

�%: Kick constant. The required work to reduce a given mass of material is therefore the same for a given reduction ratio regardless of the initial size of the material. This is obviously not compatible with practice. In addition, the material is not really homogeneous and the fragmentation depends on its imperfections (cracks, dislocations, etc.). Since none of the two previous laws was in agreement with the overall results observed during industrial fragmentation operations, Bond proposed in 1952 a law whose form is similar to the pseudo-Rittinger’s, by analysing several experimental results. The general form of this law is given by:

�) = * � �$+��, ! − �$

+��, !" (3.5)

where�): Specific breakage energy (according to Bond).

*: Bond’s index, characteristic of the material.

Originally based on the Griffith’s fractures theory, Bond’s study has been proved physically unfounded after analysis. Nevertheless, this law is still regularly applied in practice for the sizing of breakage/grinding facilities and it is a compromise between those of Rittinger and Kick. With no theoretical value, it has become an empirical reference in the world of grinding: Bond’s index is used universally. This index is determined by a grindability test (Bond testing) performed according to a standardized procedure in a laboratory scale mill (Austin and al., 1984).

The inadequacy of Rittinger’s, Kick’s and Bond’s laws and their contradictory nature led to search for other relationships between size reduction and energy consumed. Thus, Svensson and Murkes considered that, in the expression of Bond,

the exponent of 100/� is not a constant et refused the choice of �#$. They then proposed a more general form:


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�./ = �./ �0�$$�1

� − 0�$$�2

� " (3.6)

where�./: Specific breakage energy (according to Svensson and Murkes).

�./: Constant of Svensson and Murkes.

�3: Final mean size (in µm) of particles which can be calculated approximately

as the geometric mean of final sizes��,�$, ��,4$, ��,5$, ��,6$ and ��,7$.

��: Initial mean size (in µm) of particles which can be calculated as the

geometric mean of initial sizes�,�$, �,4$, �,5$, �,6$ et �,7$.

8: Parameter to be determined for each material and for a given fragmentation mode. Charles proposes a general law as follows:

��9 = −�9 :�� (3.7)

where E< : Specific breakage energy (according to Charles).

K< : Charles’s constant. After integration of this relationship, one derives:

- for 8 = 1, the Kick’s law;

- for 8 = 1.5,the Bond’s law;

- and for 8 = 2, the Rittinger’s law.

For Hukki, the relationship between the energy required for particles size reduction is a composite relationship of those of Rittinger, Kick and Bond. This can be expressed by:

��A = −�A :��(B) (3.8)

where EC : Specific breakage energy (according to Hukki).

KC : Hukki’s constant.

D(�) : Probability of fragmentation of grains according to their sizes.

In addition to the assumptions made by the authors of the first three energy-based laws, it takes into account the probability of fragmentation D(�) of grains according to their original dimensions. This probability is equal to unity for large particles and tends to zero for ultrafine particles.

In conclusion, all works carried out to determine the energy required to reduce the sizes of materials have been so far aiming at a single mathematical formulation to link closely the energy consumed to the particles size in all areas of fragmentation,


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from crushing to the ultrafine grinding. However, many fragmentation tests performed on different materials show that Rittinger's law is consistent with fine to ultrafine fragmentation, that the Kick’s law applies well in the case of a coarse fragmentation and the Bond’s law covers areas of coarse to fine grinding. The latter law is also used for mill sizing (Jdid et al., 2006). The other laws are generally difficult to handle and face particularly complex calculations and/or tests to determine some parameters (proportionality constant, value of the exponent 8, etc.). For fine to ultrafine grinding, whatever the law in question, the energy consumption increases monotonically but not always linearly with the fineness of the particles.

III.1.3 Phenomenological models

In contrast to energy-based models, phenomenological models are not aiming at determining the energy consumed by the operation of fragmentation. They allow a more physical description of this process (Boulvin, 2001). To characterize the evolution of particle size distribution resulting from a fragmentation process, mathematical models have been developed based on the concept of RL Brown (1941) that does not consider the material to be broken as a whole but as a population of fragments varying in size. This population is divided into disjoint size classes. Each size class or interval includes particles of similar size. Grinding a given size class is characterized by the disappearance or death of some of the fragments belonging to the broken class and by the creation or birth of smaller fragments going to be added to other classes. Figure III.1 shows an example of a classification into size intervals. The idea of Brown sparked of course other works.

Figure III.1: Classification into size intervals

First came the stochastic formulation of grinding by Epstein in 1947 (Jdid et al., 2006). This author considers that a fragmentation operation breaks up into a series of unit steps, each consisting of two principal operations which are the selection of a fraction of the material and the fragmentation of that fraction. He introduced the concept of fragmentation probability �(�), defined as the probability of a particle of

size � to be fragmented at the EFG step of the fragmentation process and the notion of

particle size distribution H,I(�, �I) which is the cumulative mass distribution of

particles of size � < �I coming from the fragmentation of a unit mass of size �I .

. . .

Size interval 1 Size interval 2 Size interval j


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Then, other researchers among whom Broadbent and Callcott, Sedlatschek and Bass, Austin Klimpel, and Luckie and their co-workers Reid, Herbst and Fuerstenau (Jdid et al., 2006), used the concept of Epstein to obtain results covering modelling, simulation and prediction of particle size distribution of the breakage or fragmentation process.

Studying the fragmentation process is therefore separated in two fundamental functions, i.e. the selection function and the breakage function. Both functions are regarded as time-continuous and particles size-discrete. They are presented in the next two paragraphs.

The fragmentation model described below was presented for the first time by Sedlatschek and Bass (Boulvin, 2001). To describe the kinetics of fragmentation, each size class is characterized by a coefficient �(�) called specific rate of breakage or selection function. In developing this model, a key assumption is introduced: it is assumed that the selection function of a given size class is not influenced by other classes. This assumption leads to linear equations. The selection function provides the proportion of particles, from a given size interval, which is broken per unit time. It expresses thus the speed of mass evolution within this size interval during the fragmentation process.

If we denote by K the mass of particles in the EFG size interval at time L, the mass variation of this size interval per unit time yields :

:��(F):F = −�K(L) (3.9)

with 0 ≤ � ≤ 1

The proportionality coefficient �(�) gives the specific rate of breakage of interval size

E and its dimension is the inverse of time.

If * is the total mass of particles and N(L) the mass fraction of particles of size E, (3.9) can be written:

:[P�(F)Q]:F = −�N(L)* (3.10)

If * is constant, we simply obtain:

:P�(F):F = −�N(L) (3.11)

Each size interval has its own value of selection function. The set made by the values � allows one to define the selection matrix which is in fact a diagonal matrix :


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� =STTTTU�� 00 �3

⋯ 0⋯ 0 ⋯ 0⋯ 0⋮ ⋮0 0 ⋮⋯ �⋮⋯ 0⋮ ⋮0 0 ⋮⋯ 0⋮⋯ ��XYYYYZ (3.12)

That is � = �E[\� (3.13)

E = 1, 8

The value of ��from the finest size class is equal to zero because no finer size than

the 8FG can be theoretically generated. The selection function is typically given by the simple power law equation in the literature on dry grinding (Boulvin, 2001):

� = [(�)] (3.14)

where [[ℎ_�] denotes the grindability effect (inverse of the ore hardness) and `[a. b] is a parameter depending on material. The selection function is thus expressed in inverse of time since it can be seen as exhibiting the effect of the fragmentation velocity.

To complete the fragmentation model, it is necessary to highlight the features of the particle sizes of the product resulting from breakage. This introduces directly the concept of breakage function c,I(�_�, �, �d�, �I).This function describes how

particles ranging in the size class e generate after fragmentation particles of smaller

size E. As a distribution function, c,I has been found depending not only on � and �I but also on upstream and downstream interval sizes with respect to �, i.e. �_� and

�d�. Referring to (Austin et al., 1984), this function is independent of grinding conditions but it is a feature of the broken material. Therefore it can be determined from laboratory tests.

The fragmentation function can be written in the form of a triangular matrix as follows:

c∗ =STTTTU0 0c3,� 0 0 00 0 ⋯ 0⋯ 0c4,� c4,3cg,� cg,3

0 0cg,4 0 ⋯ 0⋯ 0⋮ ⋮c�,� c�,3⋮ ⋮c�,4 c�,g ⋮⋯ 1

XYYYYZ (3.15)

where c,I = 0Dhie ≥ E[8�c�,� = 1 (3.16)

and each column should verify:


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∑ c,I�lId� = 1 (3.17)

Let us introduce now the cumulative breakage function for which the most used mathematical description will be presented in the sequel. This function m,I(�_�, � , �I)describes the cumulative percentage of particles belonging to the

interval e being broken and crossing the sieve E whose aperture mesh is smaller than

that of sieve e. m,I = ∑ cn,e8n=E (3.18)

The c,I coefficients can be obtained by:

c,I = m,I − md�,I (for 8 > E > e ≥ 1) and c�,I = m�,I (3.19)

Hence for the 8 size intervals, the corresponding matrix can be expressed:

m =

STTTTTUm�,� 1m3,� m3,3

1 11 1 ⋯ 1⋯ 1m4,� m4,3mg,� mg,3m4,4 1mg,4 mg,g

⋯ 1⋯ 1⋮ ⋮m�,� m�,3⋮ ⋮m�,4 m�,g

⋮⋯ m�,�

XYYYYYZ (3.20)

Observing the breakage process leads one to realize that the size distribution of the broken product has always a particular and characteristic shape. Wherefore, many laws of particles size distribution have been proposed (Lynch, 1977; Austin et al., 1984; Boulvin, 2001 and King, 2001). A simplified mathematical description of the cumulative breakage distribution is given by:

m,I = ��p2�q "r (3.21)

where s[a. b] is a parameter depending on material. In the following and last paragraph of the current subsection, the equation of the population mass balance is provided. This equation combines the two above functions to set the final phenomenological model of a fragmentation operation. The population mass balance of the material expresses that the variation in time of the particle mass in size interval E is due to two simultaneous phenomena:

- The disappearance of particles of initial size � by breakage into smaller particles:

−�K(L) (3.22)


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- The appearance of particles of size � from breakage of particles of larger size �I (e < E) :

∑ c,I_�Il� �IKI(L) (3.23)

Thus, this time variation can be written as:

:��(F):F = −�K(L) + ∑ c,I_�Il� �IKI(L) (3.24)

The above Equation (3.24) is the phenomenological model of the fragmentation process. In matrix form, we obtain:

:/(F):F = (c∗ − u)�v(L) (3.25)

where v(L): Column vector expressing the masses per particles size.

c∗ : Triangular fragmentation matrix defined by (3.15).

u : Identity matrix.

� : Diagonal selection matrix defined by (3.12). Equation (3.24) takes the following form if * is constant:

:P�(F):F = −�N(L) + ∑ c,I_�Il� �INI(L) (3.26)

This equation can be reformulated in terms of “crossing trough sieves” (Equation 3.27) or in terms of “retained on sieve” (Equation 3.28) by using the cumulative breakage function m,I.

:w�(F):F = ∑ m,I_�Il� �INI(L) (3.27)

where � = ∑ Nx(L)�xl denotes the mass fraction of material crossing through the

sieve of mesh E. :��(F):F = −∑ mE,eE−1e=1 �eNe(L) (3.28)

where y = ∑ Nx(L)_�xl� expresses the mass fraction of material retained on the sieve

of mesh E. In a mill, the ground material needs to be transported and discharged out. This is why the fragmentation model alone is not sufficient to describe the dynamic phenomenon involved in a grinding mill. The breakage modelling should be supplemented by the material transport modelling. The latter is the goal of the next section.


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III.2 Material transport modelling

III.2.1 Notion of material transport within a mill

Previous studies (Rogovin et al., 1988) have shown that the material transport dynamics within a mill is evaluated by Residence Time Distribution (RTD) experiments using either a colorant or a radioactive tracer. A pulse of tracer is introduced at the inlet of the mill. The response measured at the outlet provides the mean residence time of the material within the mill and the axial dispersion coefficient of the tracer. Currently, image-based techniques are also used and allow online measurements of the product particle size distribution (Herbst & Blust, 2000; Maerz & Palangio, 2000). This approach of evaluating the RTD also allows to highlight the influence of operating variables (rotation speed, grinding media filling, material filling, etc.) on the transport dynamics (Boulvin, 2001; Hennart et al., 2010). Mathematical models for the dynamics of material transport are classified into two categories, i.e. lumped parameter dynamic models and distributed parameter dynamic models. For the first category, the mill is divided into several grinding reactors characterized each by a transfer function. The series connection of these transfer functions reproduces the pulse response as measured on the process through dynamic tests performed by means of a tracer. Regarding the second category, the mill is considered as a tubular reactor in which reagents move continuously along the axis. A set of partial differential equations describes the movement of all size classes. Transport of each class results in both convection and diffusion terms. In the two following sections, mathematical developments are given on the RTD and the distributed parameter model. III.2.2 Residence time distribution

As previously stated, the transport dynamics is usually assessed in practice by measuring the RTD of the material within the mill. The experiment consists of the following steps: - mark a sample of material using a colorant (e.g. fluorescein) or a radioactive tracer; - put that sample in the feed of the circuit; - measure at regular time intervals the quantity of tracer contained in the material at

the output of the circuit.

Some mathematical models are proposed in the literature to represent the response of the tracer. The logarithmic normal distribution given by Y. Mori (Mori et al., 1967) is expressed by: z{(L) = | }~�2! �

�F√3� ��a − (}~�2! F_}~�2! F�)13�1 � (3.29)


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where z�(L): Tracer concentration in the material at the outlet of the mill.

� : Delay of the tracer impulse, computed as the ratio between the amount of tracer and the material feed rate in the mill.

L� : Median of the RTD.

� : Standard deviation of the distribution. The shape of the RTD depends on the plant operating conditions such as the rotation speed of the mill, the grinding media load, the material load, the amount of plasticizer added to the material, the ventilation, etc. III.2.3 Distributed parameter model

In this approach, the mill is seen as a tubular chemical reactor in which the reaction is in fact the fragmentation of the material and the mass balance of the various

constituents � results in a system of partial differential equations. Similarly to the

concept of population of particles, a constituent � is a class of particles of size �. For each size interval, the mass flow rate is expressed by the sum of a convection term and a diffusion term. Thus, the set of partial differential equations related only to the transport phenomenon can be written:

��(�,F)�F = − ��(�,F)

�� (3.30)

where �(�, L): Mass flow rate of constituent � at location � and at time L [n\/�]. z(�, L): Mass of constituent � per unit of length [n\/K]. �: Spatial coordinate along the mill axis. The above set of equations can be developed as follows:

��(�,F)�F = − �� E(�)zE(�, L) − HE(�) �zE(�,L)�� (3.31)

(for E = 1. . .8, 8 being the number of constituents)

where H(�) : Diffusion coefficient of constituent � [K2/�]. �(�): Convection velocity of constituent � [K/�]. The ‘8’ initial and the ‘28’ boundary conditions supplementing the partial differential equations (3.31) are given by:

- Initial conditions (for E = 1Lh8 ) : z(�, 0) = z$(�); 0 ≤ � ≤ � (3.32)


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where z$(�) is the mass of constituent � per unit of length at the initial time.

- Input boundary conditions (for E = 1Lh8 ):

�(0)z(0, L) − H(0) ��(�,F)�� l$ =��,(L);∀L > 0 (3.33)

where ��,(L) is the feed mass flow rate of constituent �. These input boundary conditions express the continuity of the mass flow rate at the inlet side of the mill for the ‘8’ constituents.

- Output boundary conditions (for E = 1Lh8 ) :

�(�)z(�, L) − H(�) ��(�,F)�� l� = ��(�)z(�, L); ∀L > 0 (3.34)

where � : Probability of particles of size � to cross through the discharge grid.

�: Length of the mill.

The output flow rate of constituent � is equal to the ratio � multiplied by the

convection term at the outlet side of the mill.

We present in the following section how the combination of the fragmentation phenomenological model and the distributed parameter model of the material transport gives rise to the complete model of a grinding mill process.

III.3 Complete model of a grinding mill process

The population mass balance (3.26) combined with the slurry dynamics transport (3.31) leads to the following basic nonlinear partial differential equations expressed for each of the 8 size intervals (1 ≤ E ≤ 8):

�(�N)�L = − �� N − H �� (�N)� − ��N

+∑ c,I�I�NI_�Il� (3.35)

where: - �(�, L) denotes the material hold-up in the mill per unit of length at location �

and at time L. - For particles of size �, N(�, L) is the mass fraction at location � and at time

L, �(�) is the convection velocity at location �, H(�) is the diffusion

coefficient at location �, and � is the selection function or specific rate of breakage. - c,I is the breakage or repartition function characterizing breakage of particles

of size e into particles of smaller size E.


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The partial differential equations (3.35) are supplemented by the initial and boundary conditions given as follows:

�(�, 0)N(�, 0) = �$(�)N$(�);∀� ∈ [0, �] (3.36)

�(0)�(0, L)N(0, L) − H(0) �� [�(�, L)N(�, L)]��l$ =��(L)N�,(L);∀L > 0 (3.37)

�(�)�(�, L)N(�, L) − H(�) �� [�(�, L)N(�, L)]��l� = ��(�)�(�, L)N(�, L); ∀L > 0 (3.38)

Where �$(�) and N$(�) are the initial spatial profiles of the hold-up and of the mass

fraction of size �, respectively, ��(L) and N�,(L) are the feed mass flow rate and the

feed mass fraction of size�, respectively. In mineral processing units, the step of particles size reduction by means of grinding mill is usually combined with the size classification to make a closed-loop. The typical model used for the hydrocyclone size classification is given in the last section hereafter.

III.4 Modelling of hydrocyclone classification

III.4.1 Selectivity function (Tromp curve) of a hyd rocyclone classifier In mineral processing, classification components allow separating coarse from finer particles and are also used to build closed-loop circuits. The hydrocyclone is ranged into the steady-state classifiers category. Coarse particles are separated from finer by means of cyclonic effect.

The selectivity function A9(�) (also called Tromp curve by the practitioners) expresses for each particle size the rejected mass fraction (or percentage) of the material at the inlet.

A9(�) = Mass flow rate of particles of size 'zi' rejected by the classifier

Mass flow rate of particles of size 'zi' fed into the classifier (3.39)

The features of such a tromp curve are as follows:

- the overall withdrawing (also called by-pass) : lowest value of the selectivity function;

- the maximum withdrawing: separation efficiency of the ‘super-fines’ ; - the cutoff mesh or tromp size �5$ : particle size for which the selectivity function

is 0.5; - the cutoff imperfection coefficient:

u = :¡¢_:1¢3:¢! (3.40)


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�/

��

�w

where �75 : Size for which the selectivity is 75 %.

�35: Size for which the selectivity is 25 %. The selectivity function is linked to the inlet and outlet flow rates of material. This is why the notions of recycled and circulating loads are defined in the next section in the context of a closed-loop. III.4.2 Recycled and circulating loads

To characterize an operating point for an installation working in closed-loop, recycled load and circulating load concepts must be considered. Let us consider Figure III.2 showing the three mass flow rates at the ends of the hydrocyclone classifier.

Figure III.2: Mass flow rates at the hydrocyclone ends

The recycled load is defined as the ratio between recirculating mass flow rate and the product mass flow rate:

¤y = ¥¦¥§ (3.41)

where �� and �w are respectively the recirculating mass flow rate and the product mass flow rate. The circulating load is the ratio between the classifier feed mass flow rate and the product mass flow rate:

¤¤ = ¥¨¥§ = ¥§d¥¦

¥§ = 1 t ¤y (3.42)

where �/ is the feed mass flow rate. Let us explain how to determine experimentally the selectivity function in the following subsection before giving the typical mathematical models of the hydrocyclone classifier in the last subsection.


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III.4.3 Experimental determination of a tromp curve

The selectivity function can be obtained from measurements of the particles size distribution at the three classifier ends (inlet, overflow outlet, and underflow outlet) (Boulvin, 2001). Let K, a, and i be respectively the mass fraction of particles belonging to size interval E at the inlet, overflow outlet, and underflow outlet of the classifier. The mass conservation principle applied to each particle size leads to the following equations:

i�� = A9(�)K�/ (3.43)

a�w = [1 − A9(�)]K�/ (3.44)

Combining Equations (3.39), (3.41), (3.42), (3.43) and (3.44) provides the expression of selectivity in terms of measurable quantities:

A9(�) = ©��

(��_ª�)(©�_ª�) (3.45)

In practice, a and i are directly measured at the two outlet sides of the classifier at the

same time. The particles size distribution K at the inlet side of the classifier is simply

reconstructed from a and i.

III.4.4 Mathematical models of a hydrocyclone class ifier

The hydrocyclone classifier has very fast dynamics compared to the mill so that the quasi-steady-state assumption holds and a static model can be adopted.

Typical hydrocyclone models most used are given below (King, 2001; Austin et al., 1984) for 1 ≤ E ≤ 8:

- Rosin-Rammler model

A9(�) = 1 − ��a «− � ��!,¬"

|¬® (3.46)

where �5$,A9 = �$,A9(0.693)2

²¬ and �25,�¤�75,�¤ = ��a(− �.5635��¤ )

- Log-normal model

A9(�) = �√3�� a ³− ´1

3 µ�¶�(��)_·�(¸¢!,¬)²¬ �_¹ �b (3.47)

where :1¢,¬:¡¢,¬ = ��a(−1.349��¤)


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- Exponentiel model

A9(�) = ��ª�|¬� B�¸¢!,¬"�_��ª�|¬� B�¸¢!,¬"�d��ª(|¬)_3

(3.48)

where :1¢,¬:¡¢,¬ =

¶�»¼��½¾²¬¿À1ÁÂ Ã¶�[4��ª(|¬)_3]

- Logistic model

A9(�) = ��d� B�¸¢!,¬"

p²¬ (3.49)

where �25,�¤�75,�¤ = ��a³_3.�763|¬ µ

The first parameter �5$,A9 is the classifier corrected cut-point and �A9 is the second

model parameter.

The ratio :1¢,¬:¡¢,¬ is commonly called the Sharpness Index (SI).

There is no specific indication in the use of each of the above four laws and their practical results are all satisfactory (Austin et al., 1984).

Conclusion The models of a grinding mill and a hydrocyclone classifier have been reviewed in this chapter. The fragmentation process is described either by energy-based models or by phenomenological models. The main concept used in the description of the breakage phenomenon is the division of the material into size intervals of particles. The model of the transport dynamics of material within a mill is characterized by two phenomena, namely convection and diffusion of particles. The complete model of a grinding mill is a set of nonlinear partial differential equations while a set of steady-state models is sufficient to describe the hydrocyclone classifier. In the following chapter, these models will be adapted in the context of the KZC grinding circuit. The process configuration will be taken into account in deriving the overall model of the circuit under study.

Chapter IV: MODELLING OF THE KZC GRINDING PROCESS 2013

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Chapter IV: MODELLING OF THE KZC

GRINDING PROCESS

Introduction Throughout the current chapter, the KZC grinding circuit is thoroughly studied. The goal of this part of our study is to derive the dynamical model of this process. From models given in the previous chapter, the modelling of the KZC double closed-loop wet grinding process is determined on the basis of realistic assumptions and the circuit flowsheet. The first section explains the purpose of the expected model. The model of the rod mill operating in an open loop is presented in the second section. The next section deals with the modelling of the two hydrocyclone classifiers. After, the models of the two ball mills are described in a double closed-loop configuration. Then, the models used for the other circuit components are briefly explained while the last section shows the overall representation of the KZC grinding circuit. All model parameters are indicated and will later be estimated from experimental data.

IV.1 Modelling purpose

In a mineral concentration plant, the particle size of the product coming from the crushing and the grinding stages is very important because of its great influence on the downstream stage. Follow-up of the product fineness is required in any breakage installation. Whatever product that does not comply with the particle size criteria is taken out and considered as a loss for the plant. This leads one naturally to control the values of this variable. As per discussion with the KZC management and practitioners, we will focus our study only on the KZC wet grinding circuit which is considered as the heart of the plant and requires an automatic control. For this circuit, three output variables should be controlled, i.e. the product fineness, the pulp density and the production flow rate. Thus, in this thesis we will propose a suitable control strategy in order to help KZC to produce the best possible pulp in both quality (particles size and density) and quantity (production flow rate) in its grinding stage. To reach this target, it is necessary to understand and study the mechanisms on which the rod/ball mill grinding are based as well as the hydrocyclone classification phenomena. Therefore, in this chapter we shall derive dynamical models able to exhibit the dynamic behaviour of the process. In the previous chapter, we mentioned that energy-based models are static and do not aim at describing the fragmentation operation. This is why phenomenological models will be used to express the dynamical behaviour of the rod mill and the two ball mills. But the hydroclyclone classifiers have very fast dynamics compared to the rod/ball mills so that the quasi-steady-state assumption holds and static models will be adopted. The pump sump will be modelled by a perfect mixer, the


34

slurry distributor by its separation coefficient and the rest of the components by simple time delay models. The other fast dynamics will also be neglected, notably those of sensors and actuators. Therefore, according to the circuit flowsheet, we shall successively derive mathematical models for the rod mill, the hydrocyclone classifiers and the ball mills, respectively in the following three sections.

IV.2 Rod mill model

IV.2.1 Mathematical model The model from the previous chapter will be adapted here to the conditions of the KZC installation. Some extra assumptions and practical considerations are required. Firstly, let us consider the efficiency of the breakage in wet operating condition. During our measurement tasks carried out on the installation and according to the recorded data, it was noticed that the specific rate of breakage is higher in wet grinding than in dry grinding. Therefore, from Equation (3.14), the following expression is proposed for the rate of breakage:

�� = �1 + ��( �)� (4.1)

where �� is an additional parameter and � is the water flow rate.

For the cumulative breakage function, the relationship (3.21) is valid. Indeed, from a practical point of view, the repartition function is almost the same in both dry and wet conditions. But it was observed that � = � (Boulvin, 2001).

��,� = ��

��

(4.2)

Secondly, let us make an assumption on the two material transport parameters. The convective velocity and diffusion coefficient will be supposed to be both size and spatial coordinate-independent. In the rest of the work, we will denote their uniform values �and � respectively. The logarithmic normal distribution (3.29) describing the RTD is therefore given by the following simplified expression (Boulvin, 2001):

� (!) = " # $ √& ' () *+, -− (#/0 ()1

2 ' ( 3 (4.3)

The resulting uniform convective velocity for a fixed feed mass flow rate and the diffusion coefficient are respectively given by (4.4) and (4.5).

� = 4!� (4.4)

where !5 is the mean of the RTD.


35

� = #1

$(6 78

9:;�< �=$

(4.5)

In practice, the mean residence time of material within a mill is inversely proportional to the feed flow rate. In other terms, the convective velocity should change with the feed ore flow rate. In order to highlight this dependence, a simple linear model is proposed:

��>� = �?�@ A1 + �0,BC�BC/BC,D6E�

BC,D6E F (4.6)

where the subscript G*H alludes to a reference state, and �0,BC is a supplementary

model parameter. The reference state will later be characterized by fixed values of fresh ore feed rate, rod mill water flow rate and dilution water flow rate.

In addition, from Equation (3.38), we shall set I� = 1 since the KZC rod mill has a non-selective discharge grid. By taking into account the above considerations, the final model of the rod mill is therefore written as:

J�KL��J! = −� J�KL��J+ + � J

$�KL��J+$ − ��KL�

+∑ N�,��KL��/O�PO (4.7)

The initial and boundary conditions supplementing the partial differential equations (4.7) are expressed as follows:

K�+, 0�L��+, 0� = KR�+�L�R�+�;∀+ ∈ [0, 4] (4.8)

�K�0, !�L��0, !� − � VVW [K�+, !�L��+, !�]XWPR =>�!�L>,��!�;∀! > 0 (4.9)

VVW [K�+, !�L��+, !�]XWP# = 0;∀! > 0 (4.10)

IV.2.2 Model parameters Table IV.1 gives the inventory of the rod mill model parameters. The subscript GZ refers to rod mill.

Table IV.1.a): Rod mill transport parameters

b): Rod mill grinding parameters

a) Transport

parameters Designat ion

�?�@,?[[Zℎ/O] Convection velocity of reference (Equation 4.6)

�?[[Z$ℎ/O] Diffusion coefficient (Equation 4.7)

�0,BC,?[[,. ^] Velocity-feed ore rate dependence parameter (Equation 4.6)


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b)

Grinding parameters

Designation

��,?[[ℎZ/_] Water effect parameter (Equation 4.1)

�?[[ℎ/O] Grindability (inverse of hardness) (Equation 4.1)

�?[[,. ^] Material depending parameter (Equations 4.1 and 4.2)

IV.3 Hydrocyclone classifiers model

IV.3.1 Mathematical model As stated in Section IV.1, the quasi-steady-state assumption holds for the two hydrocyclone classifiers since their dynamics are insignificant with respect to those of the mills. The retained modelling is given by the logistic function (3.49) thanks to its simplicity (Austin et al., 1984). IV.3.2 Model parameters The parameter inventory of the two hydrocyclone classifiers is provided in Table IV.2.

Table IV.2: Hydrocyclone ` parameters �` = 1,2� Hydrocyclone b

parameters Designation

cde�[,. ^] Second model parameter (Equation 3.49)

fgR,de�[ZZ] Cut-point (Equation 3.49)

IV.4 Ball mill model

IV.4.1 Mathematical model The KZC flowsheet (Figure II.3) shows that the two identical ball mills are working in a double closed-loop by means of the two hydrocyclone classifiers. The first closed-loop consists of: ball mill 1 - pump sump - distributor - hydrocyclone 1; and the second closed-loop consists of: ball mill 2- pump sump – distributor - hydrocyclone 2. Each ball mill will be described by the same model as the rod mill except the input boundary conditions because of the presence of the double closed-loop. Hence, the set of equations (4.7), the initial conditions (4.8) and the output boundary conditions (4.10) apply. But the input boundary conditions (4.9) should change and include the double closed-loop effect stated and given by Equations (4.11) below for ball mill 1 and (4.12) for ball mill 2:

h�K�0, !�L��0, !� − � JJ+ [K�+, !�L��+, !�]iWPRjk[O

= l'm,OndeO� ��op[∗ �!�L?[,��!�r +[�K�4, !�L��4, !�]k[O + [�K�4, !�L��4, !�]k[$s;∀! > 0 (4.11)


37

h�K�0, !�L��0, !� − � JJ+ �K�+, !�L��+, !�iWPRjk[$

= l'm,$nde$� ��op[∗ �!�L?[,��!�r +��K�4, !�L��4, !�k[O + ��K�4, !�L��4, !�k[$s;∀! > 0 (4.12) with [∗ �!� = ?[,tu(�!� + ^_,[�!� (4.13) where: - l'm,O�,. ^ and l'm,$�,. ^ are the distributor coefficients related to respectively the closed-loop 1 and closed-loop 2; - ndeO� ��,. ^ and nde$� ��,. ^ are the selectivity functions for respectively the hydroclone 1 and hydrocyclone 2; - ?[,tu(�!��!/ℎ is the rod mill output slurry flow rate; - ^_,[�!��!/ℎ is the dilution water mass flow rate; - the term ol'm,OndeO� ��p[∗ �!�L?[,��!�rw�!/ℎ denotes the mass flow rate of particles of size � recycled by hydrocyclone 1. Flow [∗ �!� corresponds to the mass flow rate coming out of the rod mill to which the dilution water is added. - the term ol'm,$nde$� ��p[∗ �!�L?[,��!�rw�!/ℎ denotes the mass flow rate of particles of size � recycled by hydrocyclone 2. - the term ol'm,OndeO� ��K�4, !�L��4, !�k[Ow�!/ℎ stands for the mass flow rate of particles of size � expressing the return effect of particles rejected by the hydroclone 1 throughout the closed-loop 1. - the term ol'm,OndeO� ��K�4, !�L��4, !�k[$w�!/ℎ denotes the cross effect of closed-loop 2 on closed-loop 1. It expresses the mass flow rate of particles of size � recycled by hydrocyclone 1 but coming out of ball mill 2. - the term ol'm,$nde$� ��K�4, !�L��4, !�k[$w�!/ℎ stands for the mass flow rate of particles of size � expressing the return effect of particles rejected by the hydroclone 2 throughout the closed-loop 2. - the term ol'm,$nde$� ��K�4, !�L��4, !�k[Ow�!/ℎ denotes the cross effect of closed-loop 1 on closed-loop 2. It expresses the mass flow rate of particles of size � recycled by hydrocyclone 2 but coming out of ball mill 1. The subscripts xZ1, xZ2 and �y refer respectively to ball mill 1, ball mall 2 and distributor.

As the distributor splits the flow rate into two sub-streams, its coefficients are linked by:

l'm,O = 1 − l'm,$ (4.14)

The set of equations (4.11), (4.12) and (4.13) show how in the double closed-loop, the two ball mills with their respective hydrocyclones classifiers are influencing each other, i.e. the two closed-loops are interdependent.


38

IV.4.2 Model parameters

In the sequel, the two identical ball mills are supposed to be described by the same model. Table IV.3 depicts the inventory of the corresponding model parameters. The subscript xZ refers to ball mill.

Table IV.3.a): Ball mill transport parameters

b): Ball mill grinding parameters

a) Transport

parameters Designation

�?�@,k[�Zℎ/O Convection velocity of reference (Equation 4.6)

�k[�Z$ℎ/O Diffusion coefficient (Equation 4.7)

�0,BC,k[�,. ^ Velocity-feed ore rate dependence parameter (Equation 4.6)

b)

Grinding parameters

Designation

��,k[�ℎZ/_ Water effect parameter (Equation 4.1)

�k[�ℎ/O Grindability (inverse of hardness) (Equation 4.1)

�k[�,. ^ Material depending parameter (Equations 4.1 and 4.2)

The models of the main components of the KZC grinding circuit have been derived in the three above sections. The next section just lists the types of models used for the other components.

IV.5 Model of the other components The pump sump is adequately modelled by a perfect mixer (mass conservation principle), the slurry distributor by its separation coefficient and the rest of the components, i.e. pipes and channels, by simple time delay models. All the above models of different components require to be put together according to the circuit flowsheet (Figure II.3) to form the global model of the KZC grinding circuit. The following section exhibits the overall representation of the process with emphasis on the input variables and the output variables from a control viewpoint.

IV.6 Overall representation All the individual models are connected as per the circuit configuration (Figure III.3) to set the global model of the circuit under study.


39

z{�|�

KZC GRINDING PROCESS:

DOUBLE CLOSED-LOOP WET GRINDING CIRCUIT

}~�|� }{�|� }��|� ��|�

�{�|� �~�|�

z~�|�

For control purposes, product particle size �O�!��,. ^, product flow rate �$�!��!/ℎ and product density �_�!��,. ^ are the three controlled variables while fresh ore feed

rate ^O�!��!/ℎ, rod mill feed water ^$�!��Z_/ℎ and dilution water flow rate

^_�!��Z_/ℎ are the three manipulated variables. Therefore, the KZC circuit can be considered as a “Multiple-Input Multiple-Output” (MIMO) nonlinear dynamical system with couplings, time delays, strong external disturbances such as ore hardness or grindability fO�!��1/ℎ and feed particle size

variations f$�!��,. ^, and internal disturbances caused by model mismatches (Yang et al., 2010). The first external disturbance fO�!� = ��!� is involved in Equation (4.1) giving the expression of the selection function and it is a physical characteristic of the feed ore. The second one f$�!� = L>,��!� is involved in Equation (4.9) expressing the continuity

of the flow rate at the inlet side of the rod mill. It is a dimensional characteristic of the feed ore. In practice, these two external disturbances vary randomly. Figure IV.1 below shows the overall map of the KZC grinding circuit.

Figure IV.1: Representation of the KZC grinding process as a MIMO system

Conclusion The modelling of the KZC grinding process has been performed in this chapter. Unlike the models found in the literature, two mathematical descriptions have been proposed inside the model. The first description expresses the positive effect of water on the efficiency of the fragmentation via the selection function. The second one takes into consideration the inverse dependence of the mean residence time of material within a mill with the feed flow rate. This has been expressed via the convective velocity. Another remark is that the equations of the physical double closed-loop have been established. At any operating condition, the two closed-loops influence each other. The global representation showed that the KZC grinding process is a 3x3 MIMO system with two external disturbances. All parameters involved in the model have been given in different tables. The following chapter is thus concerned with the estimation of these parameters and the process simulation so that we can better understand the behaviour of the KZC double closed-loop wet grinding circuit.

Chapter V: PARAMETER IDENTIFICATION AND SYSTEM SIMU LATION 2013

41

Chapter V: PARAMETER IDENTIFICATION

AND SYSTEM SIMULATION

Introduction The parameters of the KZC grinding circuit model have been estimated by using the classical approach commonly used for this purpose. The main steps of such a procedure are notably as follows: (1) define the modelling objectives; (2) build up the experimental field (experimental data and measurements); (3) provide the model structure (4) chose an adequate identification criterion (cost objective function); (5) use an appropriate optimization algorithm; (6) present the obtained results (parameter estimates); (7) perform the model validation (simple and cross validations). More details on these seven steps are provided in the first section. The three following sections are devoted respectively to the estimation of mill transport parameters, mill grinding parameters and hydrocyclone classifier parameters. In the last two sections, simulation results are presented with regard to the steady-state simulator and the dynamic simulator respectively. These simulation results constitute an important indication of the process behaviour.

V.1 Identification procedure

V.1.1 Modelling objectives A complete mathematical model of the KZC grinding circuit has been given in the previous chapter. This global modelling integrates all the mathematical models of the circuit components and all model parameters have also been clearly introduced. As already stated, the goal of our modelling is first to allow to analyse and understand the behaviour of the installation by simulations. In this way, we should be able to highlight its main features. Secondly, we will use the model for control purpose by proposing a suitable control strategy in the context of the process under study. V.1.2 Experimental field (Measurements) Experiments and measurements were conducted on the real installation at steady-state operating points. In fact, three sets of experimental data have been recorded for each circuit component. It must be pointed out that all the individual models have been identified from measurements taken under normal process operations or from data obtained through specific experiments. Indeed, due to the unavailability of sensors on the process, we could not perform on-line measurements of the variables in real-time. Hence we could not use a colorant neither a radioactive tracer to evaluate the transport dynamics of material within the mills. An alternative approach has been developed on the basis of


42

the G41 foaming as tracer. This new experimental procedure provided satisfactory results in comparison with previous studies (Rogovin et al., 1988). Similarly due to a lack of instrumentation, Tyler series sieves have been employed for measurements of particle size distributions. Measurements of the fresh ore rate have been performed by using the weighbridge. The Marcy balance was used to determine the slurry density. The water flow rates have been simply deduced from the knowledge of the solid flow rates and the densities. It must also be mentioned that due the lack of instrumentation, it was not possible for us to evaluate the accuracy as well as the error of each measurement. V.1.3 Model structure If we consider our modelling objectives and due to the small amount of the collected data (3 sets), it is meaningful to simplify our model structure with regard to the number of particle size intervals. This is why a reduced-order model, based on only three classes of size intervals, i.e.,1.18 < �� ≤ 19, 0.074 < � ≤ 1.18, and �� ≤0.074[��], has been adopted (Lepore, 2006). The third class is called the “fineness”.

V.1.4 Optimization procedure The parameter estimation problem can be stated as follows: given a set of experimental data or measurements for different operating points, find the values of the parameters which minimize a cost function expressing the deviations between model

predictions and the experimental data. Mathematically this requires finding estimates ��

of the parameters � which yield the smallest possible value of the objective function. Given the difficulty to quantify measurements errors, we decided to resort to the simplest approach compatible with the considered model class, namely the nonlinear least squares method. Let �� be the vector of measured variables and �� the vector of corresponding variables estimated by the nonlinear model. The parameter identification criterion by means of nonlinear least squares is given by: �� = �� !"�� (5.1)

where !"�� denotes the least squares cost function expressed by: !"�� = #$��#�� (5.2) with #��: the vector of deviations between �� and �� given by:

#�� = �� − �� (5.3) The “lsqcurvefit algorithm” from the Matlab 7.10.0.499 (R2010a) software has been employed to solve the nonlinear curve-fitting (data-fitting) problem referring to


43

Equation (5.1) in least-squares sense. The “lsqcurvefit” function uses the same algorithm as “lsqnonlin”. But “lsqcurvefit” simply provides a convenient interface for data-fitting problems. Here it is the “trust-region-reflective algorithm” which has been used. The results will be illustrated by superimposing the model prediction and the experimental data. The estimated parameters will be provided together with the Root Mean Square of Residuals (RMSR), as an indication of the accuracy of the estimates. The simple validation aims at reproducing the experimental data used to estimate the parameters while the cross validation aims at demonstrating the predictive capability of the model. As a rule of thumb, two-thirds of measurements sets are used for simple validation. The last third is reserved for cross validation. In the next three sections, the parameter estimation procedure explained above is applied to the circuit components. V.2 Mill transport parameters As already stated, previous studies (Rogovin et al., 1988) have shown that the material transport dynamics within a mill is evaluated by RTD experiments using either a colorant or a radioactive tracer. As the KZC installation has limited instrumentation, it was not possible for us to use such a tracer. Thus, a new experimental method for the determination of the RTD has been developed. This method is based on the use of the G41 foaming as tracer. The quantification of the foaming effect at regular time intervals allows one to determine the material RTD. Here are the main stages of the methodology:

- inject a quantity of G41 foaming (0.5 litre) to the inlet side of the mill and initialize the stop watch at this moment;

- take at regular time intervals a quantity of pulp (2.5 litres) on the outlet side of the mill;

- by means of a laboratory scale flotation machine, perform the flotation of the various samples of slurry under the same conditions. Figure V.1 depicts this procedure;

- measure the volume of the quantity of pulp poured by foaming effect. The RTD is given by the curve of the foam volume/concentration evolution versus time.

First, the transport parameters have been estimated according to Equations (3.29), (4.4) and (4.5) for the following values of the manipulated variables: &'� = 50[)ℎ+�], &' = 10.6[��ℎ+�] and &'� = 43.7[��ℎ+�]. The internal1 transport parameters �.�, 01, )̂3� have been estimated from Equation (3.29). The model used can be stated as follows:

1 Hidden parameters which do not explicitly appear in the model.


44

4$,� � 5�.1, 01, )̂3� (5.4)

If we know the concentration 4$,� from measurements by the G41 foaming method

explained above, the only unknowns will be the three parameters �.�, 01, )̂3� to be estimated. As a result by using the nonlinear least squares algorithm, Figure V.2 a) and Figure V.2 b) depict the RTD fitting curves while the estimated internal parameters are provided in Table V.1. The RMSR is equal to 2.699110+7�8. &� for the rod mill and to

9.806410+7�8. &� for each ball mill.

The values of )̂9 have been deduced from the RTD and are equal to 179.9786�:�for the rod mill and 150.1432�:� for each ball mill.

Figure V.1: Laboratory scale slurry flotation

Table V.1: Internal transport parameters estimation for rod/ball mill

Parameter estimate

Rod mill Ball mill

.;�:� 74.0558 30.1497 01�:� 0.4082 0.1705

)̂<�:� 411.3615 165.4341

Equations (4.4) and (4.5) have been used to calculate the corresponding values of the

final transport parameters =>�?<@, ABC. As a result, Table V.2 gives the estimated values

of the final transport parameters.

Table V.2: Transport parameters estimation for rod/ball mill

Parameter estimate

Rod mill Ball mill

>�?<@��*+�� 0.730810 0.586810

AB�� *+�� 0.518410 0.100810

Flotation machine

Flotation cell

Stop watch

Pulp/Slurry

Recovery pan Water cleaner


45

a)

b)

Figure V.2.a): RTD fitting of the rod mill

b): RTD fitting of the ball mill

Secondly, transport parameters estimated under &'� = 45[)ℎ+�] while &' and &'� do not change revealed that according to (4.6), the values of the supplementary

estimated parameters D�E,FG are equal to 1.3300[8. &] and 1.1595[8. &] for the rod

mill and the two identical ball mills, respectively.

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Time [s]

Foa

min

g co

ncen

trat

ion

[p.u

]

Measurements

Model

0 50 100 150 200 250 300 350 400 4500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Time [s]

Foa

min

g co

ncen

trat

ion

[p.u

]

Measurements

Model


46

V.3 Mill grinding parameters Measurements are supposed to be taken in a steady-state process operation. In this condition, an analytic solution of Equation (4.7) can be derived (Boulvin, 2001).

Let H � I!; JK = !"LE ; M< = !EN ;OK = EPQLFG be dimensionless variables where M< is the

Peclet number. The steady-state model is then represented by the following spatial ordinary differential equations (1 ≤ � ≤ �):

0 = − RSL�T�RT + �VWRXSL�T�RTX − J�O��H� + ∑ Z�,[J[O[�H��−1[=1 (5.5)

with the boundary conditions:

OK�0�− 1M3 \O�=HC\H ]H=0 = ^_,� (5.6)

RSL�T�RT `Ta� = 0 (5.7)

Equations (5.5), (5.6) and (5.7) form a so-called “two-point boundary-value problem”. The analytical solution can be found in two steps:

1) Homogeneous solution

The equations without cross terms or second members are:

0 = − RSL�T�RT + �VWRXSL�T�RTX − J�O��H� (5.8)

The homogeneous solution is therefore given by: Ob,K�H� = cK,K 3d8�eKH� + fK,K 3d8�gKH� (5.9)

with:eK = M</2 + i�M</2� + ΓKM< (5.10)

gK = M</2 − i�M</2� + ΓKM< (5.11)

Coefficients cK,K and fK,K are obtained by using the boundary conditions.

2) Particular solution

Due to the triangular structure of the problem, the particular solution can have the form of the homogeneous one but must include all the contributions from the upstream size intervals (1 ≤ [ ≤ � − 1):

Ok,K�H� = ∑ lcK,m 3d8=emHC + fK,m 3d8=gmHCnK+�ma� (5.12)

with:em = M</2 + i�M</2� + ΓmM< (5.13)

gm = M</2 − i�M</2� + ΓmM< (5.14)

The complete analytical solution, sum of the homogeneous and the particular solutions, is expressed by (5.15):


47

OK�H� = ∑ lcK,m 3d8=emHC + fK,m 3d8=gmHCnKma� (5.15)

Coefficients cK,m and fK,m are obtained by substituting the expression (5.15) of the

solution in the equations (5.5), (5.6) and (5.7). Since: RSL�T�RT = ∑ =cK,mem3opT + fK,mgm3qpTCKma� (5.16)

RXSL�T�RTX = ∑ =cK,mem 3opT + fK,mgm 3qpTCKra� (5.17)

Then:

0 = RXSL�T�RTX − M< RSL�T�RT − M<JKOK�H� + M< ∑ ZK,mJmOm�H�K+�ma�

= ∑ =cK,mem 3opT + fK,mgm 3qpTCKra� − M< ∑ =cK,mem3opT + fK,mgm3qpTCKma�

−M<JK ∑ =cK,m3opT + fK,m3qpTCKma� + M< ∑ ZK,mJml∑ =cm,s3otT + fm,s3qtTCKsa� nK+�ma� (5.18)

By rearranging the summation subscripts, we can write:

∑ u3eDT vc�,D=eD2 − M<eD − M3J�C + M< ∑ Z�,[J[K+�mas c[,Dw + 3gDT vf�,D=gD2 −K+�sa�M<gD − M3J�C + M< ∑ Z�,[J[K+�mas f[,Dwx + l3eLTcK,K=eK − M<eK − M3J�C +3gLTfK,K=gK − M<gK − M3J�Cn = 0 (5.19)

From the characteristic equations of (5.5), we deduce that:

yeK − M<eK − M<JK = 0gK − M<gK − M<JK = 0 (5.20)

Hence, from (5.19) and (5.20), the values of cK,s and fK,s (for D = 1to� − 1) are

given by:

|cK,s = − VWotX+VWot+VW}L ∑ ZK,mJmK+�mas cm,sfK,s = − VWqtX+VWqt+VW}L ∑ ZK,mJmK+�mas fm,s (5.21)

To determine the coefficients cK,K and fK,K, we can use the boundary conditions:

In H = 0: M< ∑ vc�,[ + f�,[w − ∑ vc�,[e[ + f�,[g[wKma�Kma� = M<^~,K (5.22)

or

cK,K�M3 − eK� + fK,K�M3 − gK� + ∑ lcK,m=M3 − emC + fK,m=M3 − gmCn�−1[=1 = M3^_,� (5.23)


48

In H � 1:

∑ =cK,mem3op + fK,mgm3qpCKma� = 0 (5.24)

or

cK,KeK3oL + fK,KgK3qL + ∑ =cK,mem3op + fK,mgm3qpCK+�ma� = 0 (5.25)

Thus, the values of the coefficients cK,K and fK,K will be determined by solving the set of

algebraic equations made up of (5.23) and (5.25).

The model used in the grinding parameter estimation problem can then be stated as follows: OK�1� = ��^~,K, JK, ZK,m� (5.26)

Or simply by OK�1� = ��^~,K, D< , �, �� (5.27)

where the latter function is obtained by substituting expressions deduced from (4.1), (4.2) and (3.19) for respectively �K, �K,m and ZK,m.

We recall that only three classes of size intervals, i.e.,1.18 < �� ≤ 19, 0.074 < � ≤1.18, and �� ≤ 0.074[��] are considered. If for each size interval � we know the inlet Particle Size Distribution (PSD) ^~,K and the

outlet PSD OK�1�, the only unknowns will be the three grinding parameters �D< , �, �� to be determined. As a result, with a RMSR equal to 1.828010+7[8. &] for the rod mill and 2.499410+7[8. &] for the ball mill, Table V.3 provides the values of the estimated parameters. As illustrations for the rod mill, Figure V.3.a) shows the mill PSD fitting (simple validation result) and Figure V.3.b) displays the computed evolution of the three PSDs associated to the three classes of particle sizes within the mill while Figure V.4 illustrates the same PSD features for the cross validation.

Table V.3: Grinding parameters estimation for rod/ball mill

Parameter estimate

Rod mill Ball mill

DB3[ℎ�−3] 0.0031 0.0004 �1[ℎ+�] 5.4000 5.4200 �1[8. &] 1.0971 1.0114


49

a)

b)

Figure V.3.a): Rod mill PSD fitting-simple validation b): Evolution of the PSDs for the three intervals of size classes within the rod mill

10-2

10-1

100

101

102

-0.2

0

0.2

0.4

0.6

0.8

1

Particle size [Logarithmic scale]

Cum

ulat

ive

unde

rsiz

e [p

.u]

Measurements

Model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mill axis [pu]

PS

D [

pu]

Class 1

Class 2Class 3


50

a)

b)

Figure V.4.a): Rod mill PSD fitting-cross validation b): Evolution of the PSDs for the three intervals of size classes within the rod mill

Similarly to the rod mill, Figures V.5.a) and b) show the fitting curve related to the ball mill for the cross validation.

10-2

10-1

100

101

102

-0.2

0

0.2

0.4

0.6

0.8

1


Cum

ulat

ive

unde

rsiz

e [p

.u]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mill axis [pu]

Cum

ulat

ive

unde

rsiz

e [p

u]

Class 1

Class 2Class 3


51

a)

b)

Figure V.5.a): Ball mill PSD fitting-cross validation b): Evolution of the PSDs for the three intervals of size classes within the ball mill

Observing the Figures V.3.b) and V.4.b) we realize that, by the grinding effect in the rod mill, the mass fraction of the third class (the smallest particles) is increasing with the mill axis while the mass fraction of the first class (coarse particles) is decreasing and the mass fraction of the second class (medium size particles) is ranging between the two extreme evolutions. Figure V.5.b) shows that for the ball mills which are operating in a second grinding stage, the particles of the first class almost completely disappear. Only two size intervals are present. As expected, the mass fraction of the second class is decreasing

10-2

10-1

100

101

102

-0.2

0

0.2

0.4

0.6

0.8

1


Cum

ulat

ive

unde

rsiz

e [p

.u]

Measurements

Model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mill axis [pu]

PS

D [

pu]

Class 1

Class 2Class 3


52

along the mill by grinding effect and hence the mass fraction of the third class is increasing with the mill axis.

V.4 Hydrocyclone classifier parameters

We know that the selectivity function can be obtained from measurements of the PSD at the classifier inlet, overflow outlet and underflow outlet. The model used in the hydrocyclone parameter estimation problem is stated by matching the equations (3.45) and (3.46):

�=\��,P� , .P�C = ��K, 8K, �K� (5.28)

where the functions � and �� correspond to the right hand sides of (3.46) and (3.45), respectively. If for each size interval � we know the PSD at each classifier end, the only unknowns

will be the two parameters �\��,P� , .P�� to be determined.

Results, with a RMSR equal to 0.0077[8. &] for the first and 0.0029[8. &] for the second hydrocyclone classifier, are given in Table V.4. Figures V.6.a) and b) illustrate the corresponding fitting curves where measurements 1 and measurements 3 have been used for simple validation whereas measurements 2 have been reserved for cross validation.

Table V.4: Parameter estimation for hydrocyclone classifiers

Parameter estimate

Hydrocyclone 1

Hydrocyclone 2

.;P�[8. &] 4.9150 4.0686 \;��,P�[��] 2.2791 2.2164

a)

10-2

10-1

100

101

102

-0.2

0

0.2

0.4

0.6

0.8

1


Sel

ectiv

ity f

unct

ion

[p.u

]

Model

Measurements 1Measurements 2

Measurements 3


53

b)

Figure V.6.a): Hydrocyclone 1 fitting curve – simple and cross validations b): Hydrocyclone 2 fitting curve – simple and cross validations

Observing Figures V.6 a) and b), we realize that for the two classifiers the coarse particles are totally recycled while the entire finest particles are produced together with almost 90 % of the medium size particles.

From the recorded data and referring to Equations (4.11), (4.12), (4.13) and (4.14), the distributor coefficients �N�,� and �N�, have been found equal to 0.4845 and 0.5155,

respectively. The time delays for material transport in pipes and channels are ranging between 0.03 and 0.91[ℎ]. At this stage of our study, the modelling of the KZC grinding process is completely achieved. The model structure is known and all model parameters have been identified. Let us now implement simulators based on the model and according to the circuit flowsheet. Hence, the two following sections present respectively steady-state and dynamic simulations of the process under study. Simulation results are discussed and compared to the experimental data.

V.5 Steady-state simulator V.5.1 Simulator implementation The steady-state simulator allows determining the values of process variables for a steady-state operation. It can then be exploited either for the optimization of the system by computing the optimal operating conditions or in order to indicate a reference state for control. Our steady-state simulator has been developed within the MATLAB/Simulink software. For global simulation, all the individual models are

10-2

10-1

100

101

102

-0.2

0

0.2

0.4

0.6

0.8

1


Sel

ectiv

ity f

unct

ion

[p.u

]

Model

Measurements 1Measurements 2

Measurements 3


54

connected according to the circuit configuration provided by the flowsheet of the installation. Hereafter are details on the implementation of the steady-state simulator of the KZC wet grinding circuit:

- Open circuit: Rod mill

As the rod mill is operating in an open circuit, the algebraic equations standing for its steady-state behaviour are given by Equations (5.5), (5.6) and (5.7). The simulator is thus based on the analytical solution given by the set of expressions (5.15), (5.23) and (5.25). The main steps of the computation are (for � = 1, 2, 3):

1) - Calculation of cK,K and fK,K from (5.23) and (5.25).

- Calculation of cK,m and fK,m for 1 ≤ [ ≤ � − 1, from (5.21).

2) Calculation of the PSD evolution along the mill axis O?�,K�H� and the output

PSD O?�,K�1� from (5.15).

3) Calculation of the output mass flow rate by the principle of flow rates continuity

�'?�,�� = &'� + &' ,� (5.29)

where: - �'?�,��[)/ℎ] is the static value of the rod mill output slurry flowrate;

- &'� = �'~[)/ℎ] is the static value of the fresh ore feed rate; - &' ,�[)/ℎ] is the static value of the rod mill water mass flow rate. The extra :&�:4��8) � alludes to the mass flow rate.

4) Calculation of the output slurry density

\̅k,?�,�� = \:\^(&�1+&�2,�)\:&�2,�+\^&�1

(5.30)

where: - \̅k,?�,��[8. &] is the static value of the rod mill output slurry density;

- \� = 2.7 is the mean value of the solid ore density in Kolwezi copper belt as provided by the practitioners; - \Q = 1 is the water density.

- Double closed-loop: Two ball mills, pump sump, distributor, two hydrocyclones, two ball mills The steady-state behaviour of the physical double closed-loop is expressed by

the following algebraic equations where the two loops are interdependent. The :&�:4��8) 1 alludes to the first closed-loop and the :&�:4��8) 2 to the second closed-loop. For the first closed-loop:


55

0 = − RSL,�(T�)RT�

+ �VW,�

RXSL,�(T�)RT�X − JK,�OK,�(H�) + ∑ ZK,m,�Jm,�Om,�(H�)K+�ma�

(5.31)


OK,�(0) − 1M3

\O�,1=H1C\H1 ]H1=0 = e��,�,1�N�,� uF'�∗

��O?�,K(1) + F'��,��

��K,�(1) +

F'��X,��

�K, (1)x (5.32)

RSL,�(T�)RT�

`T�a�

= 0 (5.33)

where: - �'�∗ = �'?�,�� + &'�,� is the mass flow rate including the rod mill output mass

flow rate and the dilution water mass flow rate;

- �'��,�� and �'�� ,�� are the mass flow rates at the outlets of the ball mills

on the first closed-loop and the second closed-loop, respectively. For the second closed-loop:

0 = − RSL,X(TX)RTX

+ �VW,X

RXSL,X(TX)RTXX − JK, OK, (H ) + ∑ ZK,m, Jm, Om, (H )K+�ma�

(5.34)


OK, (0) − 1M3

\O�,2=H2C\H2 ]H2=0 = e��,�,2�N�, uF'�∗

��O?�,K(1) + F'��X,��

��K, (1) +

F'��,��

�K,�(1)x (5.35)

RSL,X(TX)RTX

`TXa�

= 0 (5.36)

The solutions of the set of algebraic equations (5.31), (5.32), (5.33) and (5.34), (5.35), (5.36) are respectively given by:

OK,�(H�) = ∑ lcK,m,� 3d8=em,�H�C + fK,m,� 3d8=gm,�H�CnKma� (5.37)

and OK, (H ) = ∑ lcK,m, 3d8=em, H C + fK,m, 3d8=gm, H CnKma� (5.38)

where coefficients coefficients cK,m,�, em,�, fK,m,�, gm,�, cK,m, , em, and gm, are

determined as explained in section V.3 by expressions (5.16) to (5.25).


56

The main steps of the computation are (for � = 1, 2, 3):

1) Calculation of eP�,K,� and eP�,K, from (5.28) and (3.46).

2) - Calculation of cK,K,�, fK,K,�,cK,K, and fK,K, .

- Calculation of of cK,m,�, fK,m,�,cK,m, and fK,m, for 1 ≤ [ ≤ � − 1.

3) - Calculation of the PSD evolution along the mill axis OK,�(H�) and OK, (H ) from (5.37) and (5.38). - Calculation of the output PSD OK,�(1) and OK, (1).

4) Calculation of the product PSD �̂k?,K

�̂k?,K = Q��,L,��Q��,L,X∑ =Q��,L,��Q��,L,XC�L �

(5.39)

with:

- �̂ 8�,�,1 = (1 − e��,�,1)�A:,1 ¡��∗

&�1O��,�(1) + ��1,¢&)

&�1��,1(1) + ��2,¢&)

&�1��,2(1)£

(5.40)

- �̂ 8�,�,2 = (1 − e��,�,2)�A:,2 ¡��∗

&�1O��,�(1) + ��2,¢&)

&�1��,2(1) + ��1,¢&)

&�1��,1(1)£

(5.41)

It should be noted that the product fineness ¤'�is equal to �̂k?,�. 5) Calculation of the product flow rate

¤' = &'� + &' ,� + &'�,� (5.42)

6) Calculation of the product density

¤'� = R¥R¦(��X,��,�)R¥(��X,��,�)�R¦��

(5.43)

V.5.2 Simulation results Simulation results demonstrate an excellent agreement with the experimental data. Table V.5 shows the simulation results obtained for the operating condition given

by: &'� = 54[)ℎ+�],&' = 28.88[��ℎ+�],&'� = 9.33[��ℎ+�],\̅� = �1?� = 5.4[ℎ+�] and \̅ = ^~ = [0.48880.35580.1554].

Table V.5: Steady-state simulation results

Controlled variables

Residuals

¤� = 0.3973[8. &] +0.0001 ¤ = 92.2100[)ℎ+�] −0.0006 ¤� = 1.5840[8. &] +0.0003


57

The residuals correspond to the relative discrepancy between the measured and simulated outputs. Figure V.7 depicts the computed evolution of the three PSDs associated to three intervals of particle sizes within the rod mill. This map corresponds to the expected evolution and confirms the grinding effect.

Class 1 Class 2 Class 3 Figure V.7: Evolution of the PSDs for the three intervals of particle sizes within the rod mill Let us now give the steady-state characteristics of the process by matching workspace values of manipulated variables to the corresponding values of the output variables provided by the simulator. This information is more meaningful in terms of overall steady-state behaviour of the KZC circuit. To do so, we first state the boundary conditions (5.44) expressing the workspace in which the manipulated variables should be varying. This workspace also expresses the physical constraints on the manipulated variables due to the limitations of actuators as reported by the practitioners.

| 30 ≤ &� ≤ 80[)ℎ+�]10 ≤ & ≤ 90[��ℎ+�]0 ≤ &� ≤ 50[��ℎ+�]

(5.44)

As a result covering all the various steady-state operating conditions included in the system workspace, Figure V.8 displays a map of the three inputs-three outputs evolutions simulated from &'� = 54[)ℎ+�],&' = 28.88[��ℎ+�],&'� = 9.33[��ℎ+�], \̅� = 5.4[ℎ+�] and \̅ = [0.48880.35580.1554] as reference state. Each column shows the variations of the three output variables depending on the variations of the corresponding manipulated variable while the other input variables are kept unchanged. Observing this map, we realize that the system is globally nonlinear with respect to the gains despite the fact that we can therein highlight some affine evolutions. Indeed the production flow rate versus each manipulated variable corresponds to an affine function. It must be pointed out that the product fineness is not changing with the dilution water flow rate as could be expected.

After the steady-state simulator, we present in the coming section the dynamic simulator.

-1

0

1

2

0

0.2

0.4

-1

0

1

2

0

0.5

1

-1

0

1

2

0

0.2

0.4


58

Figure V.8: Steady-state characteristics of the system

V.6 Dynamic simulator V.6.1 Simulator implementation The dynamic simulation is useful in investigating and solving control issues. The dynamic simulator has also been developed within the MATLAB/Simulink software. Except for the three mills, all other components of the circuit have been modelled as in the steady-state simulator. For the mills, the Method Of Lines (MOL) (Schiesser, 1991; Vande Wouwer et al., 2004) is used to implement their dynamic behaviour. The PDE system ((4.7), (4.8), (4.9) and (4.10)) is transformed by spatial discretisation into an equivalent system of Ordinary Differential Equation (ODE), which is solved by means of the ODE15s solver from MATLAB. Given the tubular configuration of the mills, the simple method of the finite differences is appropriate for the spatial discretization. In this way, the mill axis is discretized in a small number �: of spatial intervals, typically

�: = 10. The set of equations (4.7) simply become the following set of ODEs:

\§K(D, ))\) = −>§I,K(D, )) + A§II,K(D, )) − �K§K(D, ))

+ ∑ ZK,mK+�ma� �m§m(D, )) (5.45)

(for � = 1to�, � being the number of size intervals and D = 1to��, �� being the number of the spatial grid points).

where: - §K(D, )) = �(ds, ))^K(ds, ))is the mass of material per unit length in the size

interval � and at the point D of the spatial grid.

- §I,K(D, ))is the approximation of the first order derivative of §K(D, )) by the

finite differences formula.

30 40 50 60 70 800.2

0.4

0.6

0.8

1

Fresh ore feed rate [t/h]

Pro

duct

fin

enes

s [p

.u]

30 40 50 60 70 8060

80

100

120

140


Pro

duct

ion

flow

rat

e [t

/h]

30 40 50 60 70 80

1.3

1.4

1.5

1.6


Pro

duct

den

sity

[p.

u]

10 20 30 40 50 60 70 80 900.74

0.76

0.78

0.8

0.82

0.84

Rod mill water flow rate [m3/h]

Pro

duct

fin

enes

s [p

.u]

10 20 30 40 50 60 70 80 9040

60

80

100

120


Pro

duct

ion

flow

rat

e [t

/h]

10 20 30 40 50 60 70 80 901

1.2

1.4

1.6

1.8


Pro

duct

den

sity

[p.

u]

0 10 20 30 40 50 60-0.5

0

0.5

1

1.5

2

Dilution water flow rate [m3/h]

Pro

duct

fin

enes

s [p

.u]

0 10 20 30 40 50 6070

80

90

100

110

120


Pro

duct

ion

flow

rat

e [t

/h]

0 10 20 30 40 50 601.1

1.15

1.2

1.25

1.3

1.35


Pro

duct

den

sity

[p.

u]


59

- §II,K(D, )) is the approximation of the second order derivative of §K(D, )) by

the finite differences formula.

The calculation of the approximations §I,K(D, )) and §II,K(D, )) is performed by finite

differences respectively on three points for §I,K(D, )) and on five points for §II,K(D, )). Four our case � = 3 and �� = �: + 1 = 11. Thus, this spatial discretization method leads naturally to a system of 33 ODEs for each mill. V.6.2 Simulation results

Figure V.9 shows a set of 3-D evolutions of respectively class 1, class 2 and class 3 (fineness) mass within the rod mill while Figure V.10, Figure V.11, and Figure V.12 depict the time evolutions of the three output variables respectively ¤�()) [8. &], ¤ ()) [)ℎ+�] , and ¤�()) [8. &] after step variations of &�()) [)ℎ+�] from 35 to 42 at

) = 3600 :, & ()) [��ℎ+�] from 50 to 60 at ) = 7200 :, &�()) [��ℎ+�] from 6 to 7.2

at ) = 9000 :, \�()) [ℎ+�] from 5.4 to 6.1 at ) = 10800 : and the vector \ ()) [8. &] from [0.6 0.3 0.1] to [0.4 0.4 0.2] at ) = 14400 : where the first entry decreases while the other two increase. The various steady-state operating conditions obtained after the input changes are equal to those provided by the steady-state simulator, as could be expected. The lack of instrumentation of the process does not allow one to validate the global transient behaviour. Yet, the qualitative effect of each manipulated variable agrees with the expected interaction between the different variables reported by the plant operators (see Table V.6). For disturbance input effects, each output variable increases with the decrease of the ore hardness and with the increase of finer particles mass within the feed ore particle size distribution.

Table V.6: Signs of gains between manipulated and output variables

y1 y2 y3

u1 - + +

u2 + + -

u3 0 + -

Output variables

Man

ipul

ated

variab

les

+: Positive gain -: Negative gain

0: Gain equal to zero

In addition, Figure V.13, Figure V.14 and Figure V.15 display how the output variables are varying with step variations of the manipulated variables on nine different operating points (OP), i.e. eight OPs corresponding to the eight boundary points of the workspace defined by (5.44), and the last one chosen so that the product flow rate is almost maximum. The step variations, with the same amounts as described above for Figures V.9, V.10, V.11 and V.12, are performed so that we remain in the workspace of


60

the manipulated variables. These curves show that large differences of gains and time delays can be observed between the responses around the nine operating points. We will come back to this issue when we will be dealing with controller design for this process.

a)

b)

c)

Figure V.9: Time-mill axis evolution of the rod mill content a) Class 1; b) Class 2; c) Class 3

01

23

4

00.5

11.5

2

x 104

0

2000

4000

6000

8000

Mill axis (m)Time (s)

Mas

s of

cla

ss 1

(kg

/m)

1000

2000

3000

4000

5000

6000

7000

01

23

4

00.5

11.5

2

x 104

1

1.5

2x 10

4


Mas

s of

cla

ss 2

(kg

/m)

1.2

1.3

1.4

1.5

1.6

1.7

1.8

x 104

01

23

4

00.5

11.5

2

x 104

0

5000

10000

15000


Mas

s of

cla

ss 3

(kg

/m)

5000

6000

7000

8000

9000

10000

11000

12000

Mas

s of

cla

ss 1

[kg/

m]

Mas

s of

cla

ss 2

[kg/

m]

Mas

s of

cla

ss 3

[kg/

m]

Time [s]

Time [s]

Time [s]

Mill axis [m]

Mill axis [m]

Mill axis [m]


61

Figure V.10: Time evolution of the product PSD

Figure V.11: Time evolution of the product flow rate

Figure V.12: Time evolution of the product density

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

0.2

0.4

0.6

Time (s)

Prod

uct P

SD (

p.u)

Class 1Class 2Class 3

0 2000 4000 6000 8000 10000 12000 14000 16000 1800090

95

100

105

Time (s)

Prod

uct f

low

rat

e (t

/h)

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

1.58

1.6

1.62

Time (s)

Prod

uct d

ensi

ty (

p.u)

Time [s]

Time [s]

Time [s]

Pro

duct

den

sity

[p.u

] P

rodu

ct fl

ow r

ate

[t/h]

P

rodu

ct P

SD

[p.u

]


62

Figure V.13: Time evolution of the product fineness variations

around nine operating points

Figure V.14: Time evolution of the product flow rate variations around nine operating points

0 5 10 15 20 25-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

Pro

duct

fin

enes

s va

riatio

n [p

.u]

Time [h]

OP 1

OP 2OP 3

OP 4

OP 5

OP 6

OP 7OP 8

OP 9

0 5 10 15 20 25-25

-20

-15

-10

-5

0

5

10

Time [h]

Pro

duct

flo

w r

ate

varia

tion

[t/h

]

OP 1

OP 2OP 3

OP 4

OP 5

OP 6

OP 7OP 8

OP 9


63

Figure V.15: Time evolution of the product density variations around nine operating points

Conclusion

In this chapter, the model parameters have been estimated and the process simulators have been presented. All model parameters have been identified from measurements taken under normal process operation or from data obtained through specific experiments, notably using the G41 foaming. The nonlinear least squares algorithm has been used for this purpose. For global simulation, all the individual models are connected according to the circuit configuration provided by the flowsheet of the installation. Both a steady-state simulator and dynamical simulator based on three classes of particle sizes have been developed within the MATLAB/Simulink software. The steady-state simulator is in good agreement with the recorded data and the dynamic simulator exhibits the expected qualitative behaviour. Both tools should allow operators and practitioners training. They will also be used in the remaining of this work to deal with fine tuning of the optimal operating conditions and validation of the control laws.

It must be pointed out that this step our study constitutes the end of the first part. In this part, a complete mathematical model and global simulators of the KZC grinding circuit have been given. In the second part, we will be focusing on the selection, design and validation of the suitable decentralized control strategy for the KZC grinding process.

0 5 10 15 20 25

-0.1

-0.05

0

0.05

0.1

Time [h]

Pro

duct

den

sity

var

iatio

n [p

.u]

O P 1

O P 2

O P 3

O P 4

O P 5

O P 6

O P 7

O P 8

O P 9

Part II :

CONTROLLER DESIGN AND VALIDATION

Chapter VI: STATE OF THE ART FOR WET GRINDING PROCE SSES CONTROL

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67

Chapter VI: STATE OF THE ART FOR WET GRINDING

PROCESSES CONTROL

Introduction The objective of this chapter is to present a survey of the various control schemes used in the framework of grinding circuits. From a control point of view, grinding processes are essentially MIMO systems. Therefore two main approaches namely decentralized and centralized control can naturally be used for their control. First, a typical wet grinding circuit encountered in the literature is described with emphasis on its flowsheet configuration, variables and instrumentation. After, a state of the art of the main control methods from decentralized Proportional-Integral-Derivative (PID) controllers to multivariable predictive controllers is presented. The last section is devoted to a critical discussion of the control strategies in order to highlight their respective advantages as well as drawbacks. This comparative study will be useful later to choose the most suitable control scheme for the KZC grinding circuit. .

VI.1 Typical wet grinding process VI.1.1 Description

As said in Chapter II, the most common circuit configuration encountered in the literature is the single-stage and closed-loop circuit shown in Figure VI.1 (Wei & Craig, 2009). This typical wet grinding flowsheet uses most often a ball mill in its single-stage grinding and this is why it is usually referred to as ball mill grinding circuit (Chen et al., 2009; Yang et al., 2010). Besides the ball mill, this circuit consists of a pump sump and a hydrocyclone. During the production process, the coarse ore is fed into the mill together with the mill feed water. The tumbling action of the balls within the revolving mill crushes the feed ore to finer sizes. The slurry with finer particle is discharged from the mill to a pump sump. Such slurry is diluted in the pump sump by adding dilution water, and then pumped to a hydrocyclone for classification. After the classification process, the slurry is separated into two streams: an overflow stream containing the finer particles which is regarded as the desired product and an underflow stream containing the larger particles regarded as the circulating load which is recycled back to the ball mill for regrinding.

Generally, the production objective is to maintain at a setpoint value the percentage of the particles smaller than the reference size or fineness (for instance 80% smaller than 74 ��). Therefore the product particle size ��[%�. �] is the most important controlled variable in a grinding circuit. Mine ore has to be ground to a specified size called the liberation size so that the imbedded mineral particles are exposed for effective recovery in the downstream processing.


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Figure VI.1: Typical wet grinding circuit: single-stage closed-loop ball mill grinding

It should be pointed out that grinding to a size much smaller than the liberation size may also bring about some drawbacks. Besides the higher power consumed in grinding, too fine particles or slimes are difficult to recover, which will also produce a larger volume of tailing discharge in the subsequent concentration and filtration process.

In parallel, the circulating load ��[�/ℎ] defined as the mass per hour of solids entering the ball mill from the hydrocyclone is to be maximized. High recirculation at a fixed product size means lower energy consumption. Excessive circulating load can result in classifier, mill or pump overload conditions. Note that mill solid concentration (% solids) and sump pump level are other important variables that need to be controlled. In practice, two local controllers are usually designed for those two controlled variables: the sump level is maintained within a limited range by regulating the slurry pumping rate [��/ℎ], while the mill solid concentration is kept around a

desired setpoint by regulating the mill feed water [��/ℎ] in (constant or variable) proportion to the fresh ore feed rate (Pomerleau et al., 2000; Chen et al., 2009;Yang et al., 2010). The selected controlled variables are therefore usually the product size distribution ��[%�. �]and the circulating load ��[�/ℎ]. Strong external disturbances in grinding circuits, such as variations of ore grindability (inverse of ore hardness [ℎ]) ��[1/ℎ] and feed ore particle size ��[%�. �], may cause continuous fluctuations of the product particle size. An analysis of the process behaviour reveals that ore hardness and feed particle size affect the product particle size and circulating load through the dynamics including time delays and inertial elements. It should be pointed out that the second disturbance is not


2013

69

considered as a variation of feed particle size distribution but rather and simply as a variation of feed particle fineness (Pomerleau et al., 2000; Chen et al., 2009; Yang et al., 2010). This leads to conclude that output variables are not sensitive to variations of particle size out of the ore fineness class (particles smaller than the reference size). The fresh ore feed rate ��[�/ℎ] and the dilution water flow rate ��[��/ℎ] to the sump box are the available manipulated variables.

The typical wet grinding process is then a MIMO system and the relationships among the output variables, manipulated variables and external disturbances is globally illustrated in Figure VI.2.

Figure IV.2. Overall typical grinding circuit

Figure VI.2: Representation of the typical wet grinding process as a MIMO system

VI.1.2 Dynamic and steady-state features

To gain insight the dynamics as well as the steady-state behaviour of the described typical grinding circuit, one should necessarily implement simulation tools. Indeed, the application of the dynamic simulation approach can help understanding the complex, nonlinear behaviour and the dynamic interactions in such a grinding process. It is a cheap and effective means of investigating circuit optimization and it is also extremely useful in developing and testing new ideas for process software-sensors and control (Liu & Spencer, 2004). Several studies and investigations have been carried out on the dynamic behaviour of grinding mill processes (Lynch, 1977; Austin et al., 1984; Hodouin & Del Villar, 1994; Boulvin, 2001 and Lepore, 2006). As already said in Chapter III, many simulation techniques and packages already exist for flowsheet simulation in the mineral processing industry but most of them are based either on a steady-state analysis or simple lumped parameter dynamic models using particularly first-order with time delays transfer functions to describe the relationship between any input variable and any output variable. Table VI.1 below summarizes the dynamic and steady-state features with the ranges of parametric values for the typical grinding circuit.

��(�) ��(�)

��(�)

��(�)

��(�)

��(�)


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70

Table VI.1: Dynamic and steady-state features of the typical grinding circuit (Hodouin & Del Villar, 1994; Pomerleau et al., 2000; Chen et al., 2009; Yang et al., 2010).

Variable

Relationship Sign of

gain Gain (absolute

value) 1 Time constant [�] Time delay

[�] �� – 0.4 – 2.5 0.02 – 0.06 0.005 – 0.020 �� + 0.2 – 5.0 0.05 – 0.20 0.001 – 0.006 �� + 0.5 – 3.0 0.05 – 0.20 0.005 – 0.020 �� + 0.3 – 4.0 0.01 – 0.10 0.001 – 0.010 �� – 0.5 – 8.0 0.01 – 0.20 0.005 – 0.017 �� – 0.8 – 9.0 0.01 – 0.20 0.005 – 0.020 �� + 1.0 – 35.0 0.01 – 0.20 0.005 – 0.015 �� + 1.0 – 42.0 0.05 – 0.20 0.005 – 0.020

VI.1.3 Typical variable pairing

Variable pairing is an issue only if one should design a decentralized control scheme and there is no concern in the multivariable control case. This is why the most efficient pairing between manipulated and output variables must be chosen prior to tune decentralized controllers. Indeed, when variable pairing is not properly selected, interactions between controlled and control variables can result in undesirable control loop interactions, leading to poor control performance.

The well-known Bristol’s Relative Gain Array (RGA) and its extension in the dynamic framework, namely the Dynamic Relative Gain Array (DRGA), are the two criteria frequently applied to grinding processes (Pomerleau et al., 2000; Yang et al., 2010).The typical variable pairing is given in Table VI.2 below, i.e. the fresh ore feed rate ��(�) is usually paired with the product fineness ��(�) and the dilution water flow

rate ��(�) with the circulating load ��(�). Table VI.2: Typical variable pairing

Manipulated

variable Output variable ��

��

The flowsheet and the variables of the typical wet grinding circuit have been described in this section. However, it is worth noting that a critical requirement for achieving improved grinding control is on-line measurement of output variables and particularly one-line particle size analysis (Coghill et al., 2002). Indeed, optimisation and control cannot be performed without a minimum amount of information on the process variables and the efficiency of the operating strategy is totally dependent upon the

1 Units of gains are :(�� ) [ℎ/�], (�� ) [ℎ/��], (�� ) [. �], (�� ) [�/��],

(�� ) [ℎ], (�� ) [. �], ], (�� ) [�], (�� ) [�/ℎ]


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quality of the information used. Nowadays, adequate instrumentation is commercially available (Hodouin, 2011; Wei & Craig, 2009). Thus, the following section will briefly describe the most used sensors and actuators in grinding circuits as well as a usual instrumentation required for the typical wet grinding process control.

VI.2 Instrumentation

VI.2.1 General concepts

According to their role, sensors used in grinding processes can be classified into two categories: sensors used for equipment protection and sensors providing measurements for process control (Hodouin & Del Villar, 1994). For the protection of equipment, we can quote for instance: temperature sensors for motors and mill gears, pressure sensors for gears oil, thickness sensors for oil film in some lubricating systems, detecting systems for ore level within feed hoppers and detectors for metal within ore material. Among sensors for variable measurements, we can distinguish specific sensors used in mineral processing notably particle size analysers and more common sensors such as flow meters, densimeters, weighbridges, etc. Generally speaking, information acquisition on grinding process variables provided by the literature covers various aspects of this specifically critical stage in a control strategy: measurement instrumentation, data reconciliation, patter recognition, fault detection and isolation, soft sensors, image processing, process and controller performance monitoring (Hodouin et al., 2001; Sbarbaro et al., 2003). In addition to the sensors, two other kinds of instruments are important for grinding circuits control:

- actuators, such as motorized valves, variable speed drivers, conveyor belt feeders, etc.;

- information processing units such as programmable controllers, interfaces process-information processing system, information transmission networks, computers and operator consoles.

VI.2.2 Sensors

For many years, Tyler series sieves have been used for discontinuous measurement of particle size distributions after prior sampling of the slurry (Bouchard, 2001) and they can be seen as a laboratory analysis method. But nowadays, reliable devices for online measurement of particle size are commercially available. There are various kinds of devices using different physical phenomena or principles for particle size measurement: ultra-sound absorption, laser diffraction, displacement of probes, pulp sedimentation, video image processing, optical detection, ultrasonic and velocity spectrometry combined with gamma-ray transmission (Wei & Craig, 2009; Coghill et al., 2002). Some of them provide only one point of the particle size distribution curve, for instance the cumulative percentage of undersize from a given mesh, while others provide complete information (Salter, 1991).


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Other measurement devices commonly and successfully used in various engineering fields are also employed in grinding circuits. Among them, we can list slurry and water flow meters, weighbridge for ore, densimeters, pressure gauges, watt meters, level meters, weight meters, etc.

VI.2.3 Actuators

Besides sensors, a control system also requires actuators to transmit control actions provided by decision devices such as controllers. Most important in a grinding process are variable aperture feeders, variable speed drivers or conveyors, variable speed pumps and valves.

The control of fresh ore rate is done by varying the speed of feed conveyor belts or the aperture of feeders if possible. The use of variable speed pumps to feed pulp into hydrocyclones makes the control system more flexible. In some installations, one can vary the diameter of hydrocyclone apex in order to control the classification. Valves are used to control water flow rates in a wet grinding process.

VI.2.4 Instrumentation for the typical wet grinding circuit

By using the standard norm from the Instrumentation Systems and Automation Society (ISA), Figure VI.3 below gives an overview without details of a typical instrumentation which can be installed on the typical wet grinding circuit prior dealing with the stage of control.

Figure VI.3: Typical wet grinding circuit with its instrumentation-ISA norm

Caption: * Actuators - VSD: Variable Speed Driver


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- FV: Flow control Valve - Green colour: useful actuators for control * Sensors - FI: Flow Indicator - PSA: Particle Size Analyser - WI: Weight Indicator - LI: Level Indicator - DI: Density Indicator - PI: Pressure Indicator - Red colour: useful sensors for control It should be pointed out that for the purpose of controlling the overall process, only two sensors and two actuators are needed. They are respectively associated to the controlled variables and the manipulated variables:

- Sensors

PSA 15: Product particle size or fineness [p.u] FI 14: Circulating load flow rate [kg/h]

- Actuators

VSD 1: Fresh ore feed rate [kg/h] FV 2: Dilution water flow rate [m3/h] In the above sections, the process description and its required instrumentation have been provided with the final aim of control. In the following two sections, the main control schemes for the typical wet grinding process are surveyed. For each control structure, the emphasis is made on the type of model used and the method employed to design the controller.

VI.3 Decentralized control strategies Single-Input Single-Output (SISO) elementary loops constitute one of the most encountered control configurations in grinding circuits. Each loop links one manipulated variable to one controlled variable. This is the decentralized approach which lies on the multi-loop strategy (Bakule, 2008; Vu & Lee, 2011). As already indicated, one of the main issues to be solved with this method is the choice of the most efficient variables pairing prior to focus on the study of control strategies. In the sequel, we will refer to the pairing given in Table VI.2 for the typical wet grinding circuit. Let �() be the 2x2 matrix transfer function between the controlled variables

and the manipulated variables, and �() the 2x2 matrix transfer function between the controlled variables and the external disturbances as per the relationships provided in Figure VI.2. These matrices are defined as follows where denotes the Laplace complex variable:


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�() = ��() ��()��() ��()� (6.1)

�() = ��() ��()��() ��()� (6.2)

The output variables are expressed by:

� �() �()� = ��() ��()��() ��()��!�()!�()� + ��() ��()��() ��()��#�()#�()� (6.3)

In the following subsections, classical and advanced decentralized control schemes are surveyed for this grinding circuit. VI.3.1 Classical control The Feedback control structure with the Proportional-Integral-Derivative (PID) controllers $�() and $�() (Figure VI.4) is the most used closed-loop classical algorithm in grinding processes as stated in (Hodouin & Del Villar, 1994; Edwards et al., 2002; Ivezic & Petrovic, 2003; Wei & Craig, 2009, Hodouin, 2011). It is usually the reference to compare different algorithms (Pomerleau et al., 2000). There are several reasons for the domination of PID controllers. Firstly they are easy to understand and implement and there is no need for high levels of skill, which is often in short supply, as required for advanced controllers. A further hindrance to the implementation of advanced control is that plants are often reluctant to allow advanced vendors to monitor their processes remotely, a step which could alleviate the shortage of skilled manpower. PID control, as applied to grinding circuits, is discussed in (Desbiens et al., 1997; Flament et al., 1997; Pomerleau et al., 2000; Edwards et al., 2002). In order to tune the controllers better, one can use heuristic methods where each controller is designed on the basis of the transfer function according to the variables pairing. The controllers $�() and $�() are respectively adjusted with

respect to ��() and ��(). The disadvantage of this design method is that, once loops are closed, these controllers act within a complex combination of four transfer functions �() of the system and the controllers themselves. Hence, it usually yields poor performance. A more suitable design method, referred to as extended PID control, is described hereafter. To best tune the controllers, a frequency-based method, in which the open-loop characteristics are deduced from the closed-loop dynamics is used (Desbiens et al., 1996). First, for tuning a controller, it is imperative to know the dynamics of the process to be controlled. With SISO systems, the method consists in opening the loop and evaluating the transfer function seen by the controller. For MIMO systems, the loop with the controller under consideration is opened while the other loops are kept closed. For the typical wet grinding process, the transfer function seen by the controller of the first loop, $�(), is:


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��() = ��() − &'(())&('())*(())�+*(())&((()) (6.4)

The transfer function seen by the second controller $�() is: ��() = ��() − &'(())&('())*'())�+*'())&((()) (6.5)

It may be seen that there exists an interdependence between $�() and $�(). Indeed, their individual tuning depends on the choice of the other controller. Consequently, to evaluate the unknowns $�() and $�(), further information is necessary. This supplementary information is given by the closed-loop specifications which can be translated into open-loop characteristics. For each closed-loop, the control objectives are the absence of permanent error with respect to step like reference changes and disturbances, and a second-order setpoint tracking response with specified dynamics. Thus, the corresponding closed-loop transfer functions can be selected as follows: �*,�() = �+-.')(�+-''))(�+-(')) (6.6)

�*,�() = �+-.()(�+-'())(�+-(()) (6.7)

where /01, /�1 and /�1 are time constants of �*,1(); 2 = 1,2. The subscript $5 refers to closed-loop.

The open-loop characteristics deduced from the tracking closed-loop specifications are: �6,�() = $�()��() = �+-.'))(-''+-('+-.'+-''-(')) (6.8)

�6,�() = $�()��() = �+-.())(-'(+-((+-.(+-'(-(()) (6.9)

with �6,1() = &789())�:&789()) ; 2 = 1,2.

The subscript <5 refers to open-loop. The transfer functions $�() and $�() of the controllers are obtained by solving the previous equations. Knowing the required sign of the controller gain and the necessary presence of an integrator, the appropriate solution can be selected without any difficulty. The last step in tuning the controllers consists in defining the transfer functions $=�() and $=�() that best approximate the frequency response $�(>?) and $�(>?), while ensuring a stable closed-loop system. The structure of each controller is imposed and


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the determination of the parameters is performed using a constrained objective function. The following structure is used: $=1() = @7(�+-9))(�+-A')+-A()()-9)(�+-B')+-B()() ;2 = 1,2. (6.10)

Figure VI.4: PID control

Notations: C,�(): Product fineness setpoint C,�(): Circulating load setpoint !�(): Fresh ore feed rate !�(): Dilution water flow rate #�(): Ore hardness #�(): Feed ore fineness �(): Product fineness �(): Circulating load VI.3.2 Advanced control The first advanced or modern control strategy developed for grinding circuits is the Internal Model Control (IMC) (Hodouin & Del Villar, 1994) whose diagram is represented in Figure VI.5. It can be seen that the control system includes two blocks labelled controllers and models. For each closed-loop, the effect of the parallel path with the model is to subtract the effect of the manipulated variable from the process output. As the model does not mimic the dynamic behaviour of the process perfectly, the feedback signal expresses the influence of (unmeasured) disturbances, the effect of coupling and the effect of model error. The model error gives rise to a feedback in the true sense of the word. Generally speaking, the grinding processes are usually described by First Order with Time Delay (FOTD) transfer functions. Then, Proportional-Integral (PI) controller is used (Chen et al., 2009; Yang et al., 2010). For controller tuning, the time constant is set so that the zero of the controller compensates the pole of the system. The remaining parameter, i.e. the controller gain, is tuned in order to get the required dynamics in setpoint tracking by means of heuristic methods.

�() C,�() +

#�()

+

�() �()

+

+ +

$�() $�()

#�()

+

−

− C,�() �()

!�() !�()

Typical Grinding Process Controllers


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.

Figure VI.5: IMC diagram

To improve the system performance in the presence of strong disturbances, an alternative form of IMC has been applied to the typical grinding process. This strategy is called “Disturbance Observer” (DOB) based control (Chen et al., 2009; Yang et al., 2010). This approach aims at dealing with disturbances directly by controller design and an effective solution is to develop disturbance estimation techniques. It is reported that DOB has fine abilities in dealing with disturbances including model mismatches, strong external disturbances and coupling effects. It does not rely on precise disturbance models. A DOB is usually a two-input-one-output system which can be used for the feedforward compensation design. The inputs are the manipulated variable and the controlled variable, while the output is an estimate of the disturbances.

A DOB estimates a lumped disturbance #DE,1�) consisting of both external and internal

disturbances. The estimate of the disturbance is employed in the feedforward compensation design to attenuate the effects of the disturbance. The control strategy is depicted in Figure VI. 6. It consists of two compound controllers, one per loop. Each controller includes a PI feedback part and a feedforward compensation part for the disturbances by using a DOB. The standard DOB is not appropriate since the considered systems have time delays. So a modified DOB is used accounting for time

delays. It should be noted that within the DOB, FG1,1�) is related to �H1,1�) by:

�H1,1�) = FG1,1�)I:-J99) (6.11)

Compared with other control strategies, this scheme demonstrates the following remarkable features: capability of handling model mismatches and strong external disturbances, simplicity of parameter tuning, low computation burden and complexity, and easy industrial implementation.

�H��() +

+

�H��()

Typical Grinding Process

�() C,�() +

#�()

+

�() �()

+

+ +

$�() $�()

#�()

+ −

− C,�() �()

!�() !�()

Controllers

Models

−

−


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The PI feedback controllers are tuned as for the IMC standard form. The design of the DOB mainly depends on the design of the filters K1(), 2 = 1,2. The time constants of the filter are the most meaningful parameters which determine the ability to resist disturbances. Indeed, each K1() is designed to be a low-pass filter with steady-state gain of 1 so that:

- in the domain of low-frequency, K1() approaches to 1, guaranteeing that the estimate of the lumped disturbance is approximately equal to the lumped disturbance. This means that the effects of the disturbance can be attenuated by a feedforward compensation design based on the DOB;

- in the domain of high-frequency, K1() approximately equals to zero, which ensures that the system possesses the features of the open-loop system, thus the high-frequency measurement noise can be filtered out.

Figure VI.6: DOB based control,2 = 1,2

#DE,1�), 2 = 1,2, denote the estimate of the output lumped disturbance brought back to

input. Often, K1(), 2 = 1,2, are selected as first-order-low-pass filters which can be expressed as: K1() = �LM,9)+� (6.12)

The time constants NO,1 of the filters are selected to satisfy the performance and

stability requirements of the closed-loop control systems.

VI.4 Centralized control strategies VI.4.1 Decoupling control

I:-J99)

K1()

FG11:�()


1() 1∗() +

#1()

+

�() �()

+ $1() + − −

!1() Controllers

− +

Disturbance Observers

#DE,1�)


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The design of a MIMO model-based control strategy can be done directly from the model of the multivariable grinding process. This is Multivariable Control (MC). But we can also carry on this design in two stages, i.e. firstly we decouple the grinding process with a pre-compensation or decoupling transfer matrix positioned upstream of the system. The set made by the Pre-Compensator (PC) or Decoupler (D) and the process is a decoupled system consisting of simple SISO systems. Secondly, decentralized controllers are separately designed on the basis of the decoupled system. This technique (Figure VI.7) is called Decoupling Control (DC) (Hodouin & Del Villar, 1994). As grinding circuits possess multiple and large time delays, the design of an accurate decoupler requires complex calculations.

Figure VI.7: Decoupling control The design method presented in the literature (Hodouin & Del Villar, 1994; Pomerleau et al., 2000) allows getting a pseudo-decoupler #() where some filtering poles are naturally added in order to make it realizable.

#() = Q 1 :&H('())&H''())&H'(())&H((()) 1R (6.13)

where �H1S() stands for the (2, >) entry of the estimated matrix transfer function �H() of

the system (2 = 1,2; > = 1,2�.

�T() = �() ∗ #() = U�T,��() 00 �T,��()W (6.14)

SISO PID controllers $�() and $�() can then respectively be tuned with respect to �T,��() and �T,��() by heuristic methods.

#()


�() �∗() +

#�()

+

�() �()

+

+

+ $�() $�()

#�()

+

−

− �∗() �() !�() !�()

SISO Controllers Decoupler

Centralized Controller


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VI.4.2 Model-based Predictive Control

Even though Model-based Predictive Control (MPC) is available since the end of the seventies, the first investigations of its applicability to grinding plants appeared a decade later (Hodouin, 2011). Nowadays this powerful technique is becoming more and more popular in the mineral industries for its outstanding features such as: capability of handling MIMO processes despite interaction between variables, ability to control difficult processes such as processes with a time delay part and a non-minimum-phase part, capability of handling constraints on both manipulated and controlled variables, simplicity of modelling (generally depending on step/impulse response), etc. (Niemi et al., 1997; Ramasamy et al., 2005; Chen et al., 2007; Chen et al., 2008; Wei & Craig, 2009; Remes et al., 2010 ; Yang et al., 2010). Unlike other multivariable control strategies, MPC was conceived primarily by industry. The underlying philosophy of the MPC scheme consists of a predictive model, a reference trajectory, a feedback correction and rolling optimization. MPC can be stated as follows: at any sampling instant, given a reasonably accurate predictive model and the desired future closed-loop behavior (or a reference trajectory), plan the set of future control moves in such a way that the predicted output is as close to the reference trajectory as possible without any violations in the operating constraints. Figure VI.8 depicts the philosophy of the MPC scheme.

Figure VI.8: Philosophy of the MPC scheme

The MPC strategy is implemented in a receding horizon framework. At any sampling instant, the optimization problem is formulated over a prediction horizon and a future control variable trajectory is calculated that minimizes the objective function while satisfying all the constraints. Only the first move is applied to the plant and this step is repeated for the next sampling instant. A multivariable MPC algorithm was originally proposed by Cutler et al. (Yang et al., 2010). Considering the typical grinding process with 2 manipulated variables and 2 controlled variables, the quadratic performance objective function of a constrained MPC algorithm possesses the following form:


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min∆E \(]) = [�C(]) − �̂)(])]-_`�C(]) − �̂)(])a + [∆�(])]-b[∆�(])] (6.15)

subject to the following constraints:

- manipulated variable constraints: �c1d ≤ �(]) ≤ �cfg (6.16)

- manipulated variable rate constraints: h�c1d ≤ �(]) − �(] − 1) ≤ h�cfg (6.17)

- controlled variable constraints: �c1d ≤ �(]) ≤ �cfg (6.18) where �C(]) denotes the setpoints of the 2 controlled variables in the future i sampling

instants, and �̂)(]) denotes the predicted outputs of the 2 controlled variables in the

future i sampling instants. ∆�(]) stands for the changes of the 2 manipulated

variables in the future j sampling instants. i and j represent the prediction horizon

and control horizon, respectively. _(2i × 2i) and b(2j × 2j) represent the error weighting and input weighting matrix, respectively. The model predictions of the outputs are stated as: lm(] + 1) = lm0(] + 1) + n∆�(]) (6.19)

where lm(] + 1) and lm0(] + 1) are the model prediction vector and the initial

predicted output vector without control action, respectively. n(2i × 2j) is a multivariable MPC dynamic matrix formed from the unit step response coefficients. Due to the effects of measurement noise, external disturbances and model mismatches, the predicted outputs are generally not accurate and should be corrected by the real outputs �(]) = [��(]), ��(])]- to realize closed-loop control. Construct I(]) as the vector of predicted errors for the 2 controlled variables: I(]) = [I�(]), I�(])]- = [�1(])− �m1(]), �2(])− �m2(])]/ (6.20)

where �̂1(]), 2 = 1,2, is the model prediction of 2 controlled variables at the (])op sampling instant. Thus the predicted outputs after correction is expressed by: lm=(] + 1) = lm(] + 1) + qrI(]) (6.21)

where lm=(] + 1) is the predictive output vector after correction, qr(2i × 2) is the corrected coefficient matrix usually defined by unitary coefficients2.

2 This assumption does not ensure the stability of the convergence of the method.


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The desired closed-loop behaviour and MPC tuning parameters have in principle a straightforward relationship. If tighter control of some specific controlled variables is desired, it can be achieved by choosing a corresponding higher weight in matrix _. Also excess variations in manipulated variable moves can be suppressed by an appropriate choice of b. VI.4.3 Adaptive and nonlinear control

All the above control algorithms can be made adaptive/self-tuning if the controller is equipped with a recursive estimator for process model parameters. The adaptive methods can yield serious sensitivity issues when the model has got a large number of parameters and should thus be properly supervised to guarantee a safe running of the control system (Desbiens et al., 1994; Najim et al., 1995; Poulin et al., 1996). All the above control algorithms are based on a linearized model of the typical wet grinding process around a steady-state operating point. Nonlinear models have also been used for the control of grinding processes. A well-studied control scheme consists in substituting nonlinear models for the linear models in the IMC scheme of Figure VI.5. More particularly Artificial Neural Networks (ANN) have been used to describe the nonlinear dynamics of grinding circuits (Flament et al., 1993; Stange, 1993). A neural network is a black-box model and allows expressing relationship between process input and output variables by means of a parallel structure containing elements which perform nonlinear transformations. The feasibility of applying neural networks have been studied on simulators (Hodouin & Del Villar, 1994; Conradie & Aldrich, 2001; Duarte et al., 2001).

VI.5 Discussion The overview from the previous sections shows that there are numerous publications in the field of wet grinding process control. The choice of a suitable control structure and the design method mainly depend on the process features and the control objectives. The typical grinding process is a 2x2 MIMO system with low time delays ranging between 0.001 up to 0.020ℎ (Table VI.1). It possesses strong coupling effects and strong external disturbances. We will now summarize the advantages and drawbacks of decentralized control schemes as well as centralized control schemes applied to this typical grinding circuit. The most beneficial decentralized control structure is the DOB based control. Indeed, it is the only one which solves efficiently the issues of coupling effects and unmeasured external disturbances. The DOB reconstructs a lumped disturbance on each manipulated variable-controlled variable channel. The lumped disturbance includes the coupling effects, the external disturbances and the model errors. Although this control strategy yields acceptable performance on the typical grinding process, still there are certain topics that have not received the required attention:


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- How to deal with grinding processes exhibiting large time delays (outside of the range considered in Table VI.1) ?

- How to cope efficiently with dynamic model uncertainties due to operating point changes both in the DOB and in the controller ?

- How to exploit a possible duality between controller and DOB design ?

The advantage of the decoupling control over the classical one is the compensation of the couplings. This compensation is not very accurate since one uses a pseudo-decoupler. It is not obvious to cope with the other control issues, i.e. time delay effects, strong disturbances and model uncertainties, by designing the SISO controllers on the basis of the decoupled system models. MPC is the most appropriate centralized control scheme for the typical grinding process and is potentially able to cope with all control issues except one. Indeed, MPC demonstrates limited ability in the presence of strong disturbances. The reason is that it does not deal with disturbances directly by controller design. Usually the controller cannot react directly and fast to reject these disturbances, although this controller can finally suppress the disturbances through feedback regulation in a relatively slow way. Thus the controller generally becomes sluggish when meeting strong disturbances (Chen et al., 2009; Yang et al, 2010). Hence, here also there are certain topics which require attention such as:

- How to cope efficiently with strong disturbances ? - How to deal with dynamic model uncertainties due to operating point changes ?

The above questions will help us in the orientation of our research in order to propose the most suitable control strategy for the KZC grinding circuit.

Conclusion In this chapter, an overview of control schemes applied to the typical wet grinding circuit have been presented and discussed. This state of the art shows that there are various methods from decentralized control to centralized control in the framework of grinding processes. Most of these interesting methods use linear control laws and are based on a linearized model of the grinding process around a steady-state operating point. The critical discussion of control schemes found in the literature reveals that there are still some topics which have not yet received the required attention. These topics will be taken into consideration in the study of control schemes for the KZC wet grinding process. Prior dealing with its control, the KZC circuit must be analysed in order to highlight its dynamic as well as steady-state features. This is the goal of the next chapter.

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85

Chapter VII: ANALYSIS OF THE SYSTEM MODEL

Introduction

The current chapter deals with the analysis of the system model. To use linear control tools later on, the KZC nonlinear model should first be linearized. The determination of the operating point around which linearization should be performed is the object of the first part of this chapter. A two-step approach is used. First, an initial estimate of the optimal operating point is determined on the basis of a simulation study. Next, a linearized model of the process is determined around this initial operating point, and a correction to the initial operating point is computed by solving a convex optimization problem, assuming the linearized model remains valid for the considered correction range. The optimization problem consists in maximizing the product flow rate while meeting specific constraints on product quality and accounting for actuator limitations. For decentralized control purpose, the most efficient pairing between manipulated variables and output variables should be evaluated. Both steady-state and dynamic criteria are applied to this end. Next, dynamic model uncertainties are described. These uncertainties are due to operating point changes and are expressed in terms of bounds on variations of linearized model parameters. Finally, a discussion of both nominal model with the selected variables pairing and uncertain plant is done so that the main features of the system are highlighted.

VII.1 Model linearization

VII.1.1 Selection of the initial steady-state opera ting point

The initial steady-state operating point, around which the system model will be linearized, has been found by trials and errors guided by the knowledge of the process behaviour. We know that our control goal is to maximize the production of the KZC grinding circuit by guarantying in the same time the product quality, i.e. maximize the product flow rate �� under constraints on the product particle size ��, �� ≅ 0.8, and the product density ��, 1.3 ≤ �� ≤ 1.4. We have also to keep in mind that the ability of each actuator is limited by saturations. This implies that the manipulated variable workspace should be taken into account for any simulation study so that we comply with the practical conditions. These actuator limitations are summarized as follows as per the data provided by the practitioners:

30[�ℎ��] ≤ �� ≤ 80[�ℎ��]10[��ℎ��] ≤ �� ≤ 90[��ℎ��]0[��ℎ��] ≤ �� ≤ 50[��ℎ��] � (7.1)

The first disturbance, i.e. ore grindability �� is kept at its estimated value of 5.4[ℎ��] whereas the second one, i.e. feed ore particle size distribution �� is


86

maintained at its typical value of [0.60.30.1]� according to the recorded data1. Various scenarios have been tested in simulation by changing the three manipulated variables, i.e. fresh ore feed rate, rod mill water flow rate and dilution water flow rate, within the bounds defined by the saturations of actuators. The operating point over which the product flow rate is the highest possible within the bounds defined by the constraints on the product particle size and product density is retained. This approach is summarized in Figure VII.1 below.

Figure VII.1: Principle of the initial steady-state operating point selection

As a result, Table VII.1 gives the selected operating point found by this approach.

Table VII.1: Initial steady-state operating point

Manipulated variables Output variables �� = 35[�ℎ��] �� = 0.7898[ . �] �� = 50[��ℎ��] �� = 81.2410[�ℎ��] �� = 6[��ℎ��] �� = 1.3053[ . �]

For twenty-four hours by twenty-four hours operating organization, the daily production capacity will be of 1,949.784 tons of pulp. By proceeding so, this operating point should be intuitively as close as possible to the expected optimal point and can be considered as an approximate optimal operating point. The accurate optimal operating point will be derived later by correcting this approximate one via the solution of a duly stated convex optimization problem. Hence, the target of the control will be of holding the system on the optimal operating condition. Prior to the optimization, both continuous-time and discrete-time linearized models are determined in the following sub-sections. VII.1.2 Continuous-time linearized model

The phenomenological model derived for a grinding mill is nonlinear. Hence, the KZC grinding circuit consisting of three mills is overall nonlinear too. Moreover, the physical double-closed loop makes the system more complex notably due to the recycling effect that induces supplementary nonlinearities. However, it is always possible to derive a linear model which will be valid only around the selected operating

1 Mean values of feed ore mass fractions within the three size classes provided by practitioners.

30[�ℎ��] ≤ �� ≤ 80[�ℎ��] 10[��ℎ��] ≤ �� ≤ 90[��ℎ��]

�� ≅ 0.8

�̅� = 5.4[ℎ��] = 5.58

KZC Process: Double Closed-Loop Wet Grinding Circuit

�̅� = [0.60.30.1]#

1.3 ≤ �� ≤ 1.4

��[�ℎ��] ↗

0[��ℎ��] ≤ �� ≤ 50[��ℎ��]


87

point. To do so, we resorted to use of the dynamic simulator to generate step responses associated to the three input variables. In accordance with most references on the topic, we have chosen the “First Order transfer function with Time Delay” (FOTD) model to describe the dynamic relationship between each input variable and each output variable. This leads, for the three manipulated variables and the three output variables, to the linear lumped parameter

model given by the following matrix transfer function where denotes the complex Laplace variable:

%� � =&''''(

�).)�*+).),,, -.� /�).+)�0 - 0.* �)12

).),,, -.� /�).03 - 0

�.��,�).�3 -.� /�).��, - ).+3�*

).))�* -.� /�).�)+0 - ).+,0�).)),� -.�

/�).)�,, -

).))*)).�3 -.�

/�).��, - �,0., �)12

).))�* -.� /�).�� - �).))3�).)),� -.�

/�).)�,, -

455556

(7.2)

The time constants and time delays are expressed in hours. We can realize that the third manipulated variable, i.e. the dilution water flow rate ��, has no effect on the

product fineness �� as said previously. Similarly, the matrix transfer function of the two measurable external disturbance channels is expressed as follows:

7� � =&''''(

).)�+3).),8� -.� /��.))�� - ).��*)).)33+ -.�

/��.)))0 -

�.8�,� 91:.2;<2 =��.+�+, 91>.::?@ =

).)33+ -.�

��.�))) 91:.22AB =��).��0 91>.:;?> =

).)33+ -.�

).)��0 91:.2;A2 =�).)�� 91>.:>2C =

).)+0, -.�

).)88) 91:.222B =�).)8�) 91>.:CC@ =

).),�8 -.� 455556

(7.3)

We considered just one entry of the second external disturbance, i.e. the feed ore particle size distribution ��. This entry corresponds to the third class since the mass variations of particles of the first and the second classes have been found not affecting the output variables for a fixed mass of the third class called ore fineness. This result is confirmed in Appendix VII.1. Globally, this can also be explained by the fact that particles belonging to the first and second classes are ground by the two stages of grinding, namely the rod mill and the two ball mills, so that the coarse particles almost disappear and only a few part of medium size particles remains in the product. Therefore, the output variables are not sensitive to any particular mass variation within the two first classes only. The mass variation within the third class is consequently dominant and the most significant as disturbance. As a validation, Figures VII.4, VII.5 and VII.6 show how the linearized model fits the nonlinear model after step variations of the input variables depicted in Figure VII.2 for the three manipulated variables and in Figure VII.3 for the two disturbances.


88

a)

b)

c)

Figure VII.2: Step variations of manipulated variables a) Fresh ore feed flow rate b) Rod mill feed water flow rate c) Dilution water flow rate

0 2 4 6 8 10 12 14 16 18 20 2234

35

36

37

38

39

40

41

42

43

Time [h]

Fre

sh o

re fee

d ra

te [t/h]

2 4 6 8 10 12 14 16 18 20 2248

50

52

54

56

58

60

62

Time [h]

Rod

mill fee

d w

ater

[m

3 /h]

0 2 4 6 8 10 12 14 16 18 20 22

6

6.2

6.4

6.6

6.8

7

7.2

Time [h]

Dilu

tion

wat

er flo

w rat

e [m

3 /h]

Rod

mill

feed

wat

er fl

ow r

ate

[m3 /h

] D

ilutio

n w

ater

flow

rat

e [m

3 /h]

Fre

sh o

re fe

ed r

ate

[t/h]


89

a)

b)

Figure VII.3: Step variations of disturbances a) Ore grindability b) Feed ore fineness

Figure VII.4: Time evolution of the product fineness after step variations of input variables

0 2 4 6 8 10 12 14 16 18 20 225.3

5.4

5.5

5.6

5.7

5.8

5.9

6

6.1

Time [h]

Ore

grin

dabi

lity

[h- 1

]

0 2 4 6 8 10 12 14 16 18 20 22

0.1

0.12

0.14

0.16

0.18

0.2

Time [h]

Fee

d or

e fin

enes

s [p

.u]

2 4 6 8 10 12 14 16 18 20 220.6

0.65

0.7

0.75

0.8

Time [h]

Pro

duct

fin

enes

s [p

.u]

Nonlinear model

Linearized model

Ore

grin

dabi

lity

[h-1]

Fed

d or

e fin

enes

s [p

.u]

Pro

duct

fine

ness

[p.u

]


90

Figure VII.5: Time evolution of the product flow rate after step variations of input variables

Figure VII.6: Time evolution of the product density after step variations of input

variables We realize that high order dynamics are appearing from the nonlinear system but they are neglected within the linearized model since we have chosen FOTD form for its description for the sake of simplicity. The effects of these discrepancies will be later taken into account as one part of internal disturbances in the stage of control design. Moreover, for decentralized control purpose, we will see that the dynamics corresponding to the unpaired variables will be considered as coupling effects.

VII.1.3 Discrete-time linearized model

To ease the statement of the convex optimization problem, a discrete-time setting is used. The sampling period #D has been taken equal to 0.1[ℎ]. This value is such that the order of the discrete-time linearized model between the three manipulated variables and the three controlled variables is limited to 11 maximum. By using the “first-order hold” discretization method, the discrete-time matrix transfer function corresponding to %� �, where E stands for the z-transform complex variable, is given by :

2 4 6 8 10 12 14 16 18 20 22

80

82

84

86

88

90

Time [h]

Pro

duct

flo

wra

te [t/h]

Nonlinear model

Linearized model

2 4 6 8 10 12 14 16 18 20 22

1.26

1.28

1.3

1.32

1.34

1.36

Time [h]

Pro

duct

den

sity

[p.

u]

Nonlinear model

Linearized model


91

%F�E� =&''''(�).)��G�).))3,G�).�)3� E�+ ).)))+G.).)))�G�).�)3� E��) 0).��3�G.).�**,G�).+8)� E�� ).+��3G.).)�*�3,G E�� ).+��0G.).)�8�8G E��).))�,G.).))��G�).+8)� E�� ).)),*G�).)))�G E�� ).))3�G�).)))�G E��45

5556 (7.4)

where the dynamics corresponding to the poles equal to 3.087010��+ and 4.567010�� are neglected and the time delays are approximated as follows in terms of sampling period: HF,�� = 6.039 ≅ 6[#D]; HF,�� = 9.5 ≅ 10[#D];HF,�� = HF,�� = 3.114 ≅ 3[#D]; HF,�� = 2.069 ≅ 2[#D]; HF,�� = HF,�� = 0.344 > 0 ≅ 1[#D]; HF,�� = 2.222 ≅ 2[#D]. As an illustration, Figure VII.7 displays how the discrete-time model fits the continuous-time one after step variations of the manipulated variables. The recursive equations in small variations around the steady-state operation point and relative to the model (7.4) are given by:

KLLLMLLLN �O��P� = 0.1052�O��P − 1� − 0.0112�O��P − 6� − 0.0054�O��P − 7�+0.0006�O��P − 10� + 0.0003�O��P − 11��O��P� = 0.6703�O��P − 1� + 0.2152�O��P − 3� + 0.1884�O��P − 4�+0.6335�O��P − 2� − 0.4063�O��P − 3� − 0.0122�O��P − 4�+0.6219�O��P − 1� − 0.3896�O��P − 2� − 0.0183�O��P − 3��O��P� = 0.6703�O��P − 1� + 0.0014�O��P − 3� + 0.0012�O��P − 4�−0.0048�O��P − 2� + 0.0031�O��P − 3� + 6.710�3�O��P − 4�−0.0051�O��P − 1� + 0.0032�O��P − 2� + 1.3410�,�O��P − 3�

(7.5)

Finally, a discrete-time state space model containing thirty-three state variables can be derived as follows:

STU�P + 1� = VTU�P� + WXY�P�ZU�P� = [TU�P� (7.6)

Where the state variables are:

TU�P� = [�O�P�� , �O�P − 1�� , … , �O�P − 10��]� (7.7)

The state space equations can thus be written as:


92

a1) b1)

a2) b2)

a3) b3)

Figure VII.7: Comparisons between continuous-time and discrete-time models a1), a2) and a3) Step variations of feed ore flow rate, rod mill water flow rate and dilution water flow rate, respectively. b1), b2) and b3) Step responses of product fineness, product flow rate and product density, respectively.

0 0.5 1 1.5 2 2.5 3 3.535

36

37

38

39

40

41

42

Time (h)

Fee

d or

e flo

w rat

e (t/h

)

Continuous-time model

Discrete-time model

0 0.5 1 1.5 2 2.5 3 3.50.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8


Discrete-time model

0 0.5 1 1.5 2 2.5 3 3.550

51

52

53

54

55

56

57

58

59

60

Rod

mill fee

d wat

er flo

w rat

e (m

3/h)


Discrete-time model

0 0.5 1 1.5 2 2.5 3 3.580

82

84

86

88

90

92

94

96

98


Discrete-time model

0 0.5 1 1.5 2 2.5 3 3.56

6.5

7

7.5

Dilu

tion

wat

er flo

w rat

e (m

3/h)


Discrete-time model

0 0.5 1 1.5 2 2.5 3 3.51.24

1.25

1.26

1.27

1.28

1.29

1.3

1.31

1.32

Time (h)


Discrete-time model

Dilu

tion

wat

er fl

ow r

ate

[m3 /h

] R

od m

ill fe

ed w

ater

flow

rat

e [m

3 /h]

Fee

d or

e flo

w r

ate

[t/h]

Pro

duct

fine

ness

[p.u

]

Time [h] Time [h]

Pro

duct

flow

rat

e [t/

h]

Time [h] Time [h]

Pro

duct

den

sity

[p.u

]

Time [h] Time [h]


93

&''''''''''(�O�P + 1��O�P��O�P − 1��O�P − 2��O�P − 3��O�P − 4��O�P − 5��O�P − 6��O�P − 7��O�P − 8��O�P − 9�45

5555555556

=

&''''''''''( Φ�0�^�0�^�0�^�0�^�0�^�0�^�0�^�0�^�0�^�0�^�

Φ�0�^�_�^�0�^�0�^�. . .. . .. . .. . .. . .. . .

Φ�0�^�0�^�_�^�0�^�. . .. . .. . .. . .. . .. . .

Φ,. . .. . .0�^�_�^�0�^�. . .. . .. . .. . .. . .

0�^�. . .. . .. . .0�^�_�^�0�^�. . .. . .. . .. . .

Φ+. . .. . .. . .. . .0�^�_�^�0�^�. . .. . .. . .

Φ8. . .. . .. . .. . .. . .0�^�_�^�0�^�. . .. . .

0�^�. . .. . .. . .. . .. . .. . .0�^�_�^�0�^�. . .

0�^�. . .. . .. . .. . .. . .. . .. . .0�^�_�^�0�^�

Φ�). . .. . .. . .. . .. . .. . .. . .. . .0�^�_�^�

Φ��0�^�. . .. . .. . .. . .. . .. . .. . .0�^�0�^�455555555556

&''''''''''( �O�P��O�P − 1��O�P − 2��O�P − 3��O�P − 4��O�P − 5��O�P − 6��O�P − 7��O�P − 8��O�P − 9��O�P − 10�45

5555555556

+ W�O�P� (7.8)

with: �O = [�O1�O2�O3]#; �O = [�O��O��O�]� (7.9)

Φ� = `0.1052 0 00 0.6703 00 0 0.6703a ;Φ� = `0 0 00 0.6335 −0.38960 −0.0048 0.0032 a ; Φ� = ` 0 0 00.2152 −0.4063 −0.01830.0014 0.0031 1.3410�,a ; Φ, = ` 0 0 00.1884 −0.0122 00.0012 6.710�3 0a ; Φ+ = `−0.0110 0 00 0 00 0 0a ;Φ8 = `−0.0054 0 00 0 00 0 0a ;

Φ�) = `0 0.0006 00 0 00 0 0a ;Φ�� = `0 0.0003 00 0 00 0 0a ; (7.10)

The dynamic matrix is given by:

V =

&''''''''''( Φ�0�^�0�^�0�^�0�^�0�^�0�^�0�^�0�^�0�^�0�^�

Φ�0�^�_�^�0�^�0�^�. . .. . .. . .. . .. . .. . .

Φ�0�^�0�^�_�^�0�^�. . .. . .. . .. . .. . .. . .

Φ,. . .. . .0�^�_�^�0�^�. . .. . .. . .. . .. . .

0�^�. . .. . .. . .0�^�_�^�0�^�. . .. . .. . .. . .

Φ+. . .. . .. . .. . .0�^�_�^�0�^�. . .. . .. . .

Φ8. . .. . .. . .. . .. . .0�^�_�^�0�^�. . .. . .

0�^�. . .. . .. . .. . .. . .. . .0�^�_�^�0�^�. . .

0�^�. . .. . .. . .. . .. . .. . .. . .0�^�_�^�0�^�

Φ�). . .. . .. . .. . .. . .. . .. . .. . .0�^�_�^�

Φ��0�^�. . .. . .. . .. . .. . .. . .. . .0�^�0�^�455555555556

(7.11)

The input matrix is expressed by:

W = ` W�_�^�0�8^�a; B� = `0 0 00 0 0.62190 0 −0.0051a (7.12)

The output matrix is given by: [ = [_�^�0�^�)] (7.13)

The above state space model is used in the following section to compute the optimal operating point on the basis of a convex optimization problem.


94

VII.2 Operating point optimization

We recall that to linearize the system model, we have chosen an operating point which was close to our expectations, i.e. close to the required value of the product fineness, compatible with the density product boundary values and providing a highest possible production flow rate. Now we will correct this operating point by computing explicitly the optimal one over which the plant should be maintained despite the presence of disturbances. The model used for the system optimization is the discrete-time state space model given by Equations from (7.6) to (7.13). The control goal of the KZC wet grinding process is to “maximize the production flow rate �� under required values of the product fineness �� and

density ��. While �� should be kept on a specified value, �� has to be maintained between two extreme values. More precisely:

- the product fineness must reach the so-called �*), i.e. 80 % of particles

undersize of 74d��200e/fℎ�; in other words, the mass fraction of particles

of the third class within the product stream must be of 0.8[ . �]; - the product density must range between 1.3 and 1.4, i.e. in a sample of pulp

taken out on the product stream the mass of solid particles must be 30 to 40% more than that of the water.

Mathematically this problem can be stated as: Determine the control sequence �O��0�, … , �O��h − 1�that minimizes the following cost function: i = ∑ −�O��P�klm� (7.14)

subject to:

nTU�P + 1� = VTU�P� + WXY�P�TU�0� = 0ZU�P� = [TU�P� (7.15)

KLLMLLN0.8 − o�1 − �p1 ≤ �q1�P� ≤ 0.8 + o�1 − �p1[ .�]1.3 − �p3 ≤ �q3�P�≤ 1.4 − �p3[ .�]30 − �p1 ≤ �q1�P� ≤ 80− �p1 r�ℎ−1s10 − �p2 ≤ �q2�P� ≤ 90− �p2 r�3ℎ−1s0 − �p3 ≤ �q3�P� ≤ 50− �p3[�3ℎ−1]

(7.16)

where ot� = 10�, is an infinitesimal positive number which makes the computation

feasible. The horizon h is chosen large enough so that the transient takes a tiny part of the horizon and the achieved equilibrium can be considered to correspond to the optimal steady-state, in the sense that it maximizes the steady-state productivity.


95

The solution of the open-loop convex optimization problem (7.14), (7.15) and (7.16) yields the optimal operating point given in Table VII.2 which is close to the initial steady-state operating point shown in Table VII.1 and over which the model has been linearized.

Table VII.2: Optimal operating point

Manipulated variables Output variables ��,uvw = 34.3200[�ℎ��] ��,uvw = 0.8000[ . �] ��,uvw = 45.0000[��ℎ��] ��,uvw = 77.4735[�ℎ��] ��,uvw = 6.5000[��ℎ��] ��,uvw = 1.3257[ . �]

This optimization has been performed by using the revised simplex method which solved the problem as a linear programming problem. The MATLAB Multi-Parametric Toolbox (MPT), containing a code of the mentioned method, has been employed to this end. It must be pointed out that by comparing the two operating points, we realize that the optimal operating point reaches the strong requirement on the product fineness while the density is maintained within its boundary values. But improving the product quality results in decreasing the production capacity. Indeed, a reduction of 3.7675 tons of pulp per hour is observed with respect to the initial operating point. The daily optimal production is 1,859.364 tons of pulp. This decrease of pulp production does not imply the decrease of production for the whole plant. Indeed, the product quality is the critical requirement for high recovery within the subsequent flotation process. If the pulp fineness is guaranteed, the flotation will yield good recovery. Otherwise, despite a high flow rate of pulp, the subsequent process will produce much tailings and less concentrate. Therefore, the optimal output variables will be used in the next chapter as setpoints or references in the control study. However, the manipulated variables should efficiently be paired with the controlled variables before focusing on the design of a decentralized controller. The next section solves this issue by means of both steady-state and dynamic criteria.

VII.3 Pairing of controlled and manipulated variabl es

VII.3.1 Problem statement

Most of the concepts used in the current subsection and in the two followings are derived from (Kinnaert, 1995). When one is faced with the control of a MIMO system, the selection of the most effective control configuration may not be obvious. Indeed, due to process interaction, each manipulated variable is affecting several controlled variables, and it might be difficult to choose the best pairing between those variables. Besides, once such a choice has been made, one may wonder whether conventional single loop feedback controllers can yield satisfactory performance, despite the interaction between the loops. Such a control scheme is called a multi-loop or decentralized strategy. It offers


96

several advantages such as ease of implementation, simplified design and decentralized tuning, and ease of making the system failure tolerant. Hence the interest of using multi-loop control when possible. To be able to decide when this approach is indeed suitable, several measures of interaction have been developed and proposed in the literature. In this work, we will first discuss two steady-state criteria, i.e. the Bristol’s Relative Gain Array (RGA) and the Singular Value Decomposition (SVD). We should not be fully confident in the results provided by the steady-state interaction measures since they can lead to misleading information because of the system dynamics. Therefore dynamic considerations have to be taken into account in order to cover all the frequency range of interests of the system. Two dynamic criteria will then be applied, namely the Dynamic generalization of RGA (DRGA) and the Direct Nyquist Array (DNA) with Gershgorin’s bands. Thus, the best pairing between manipulated variables and controlled variables will be chosen on the basis of these steady-state and dynamic criteria.

VII.3.2 Steady-state criteria

VII.3.2.1 Bristol’s Relative Gain Array

The RGA is the most widely used criterion and also one of the first systematic interaction measures which was developed (Bristol, 1966; Seborg et al., 1989). It provides both a measure of the process interaction and a recommendation concerning an adequate pairing of output and manipulated variables. It only requires steady-state information on the process model.

Let us consider the linearized system model given by Equation (7.2). The relationship between manipulated variables and output variables can be written:

ZU� � = %� �XY� � (7.17)

where ZU� � and XY� � are the deviations of the output and the manipulated variables

with respect to their steady-state value. Both ZU� � and XY� � are vectors of dimension

3 and %� � is the 3x3 process transfer matrix. The RGA is based on the notion of relative gain defined as follows: xyz = l{|}{| ; (with ~ = 1,2,3 and � = 1,2,3 ) (7.18)

where: Pyz = �U{�Y| when XYl = 0∀P ≠ � �yz = �U{�Y| when ZUl = 0∀P ≠ � (7.19)

Pyz is the steady-state gain between the �� manipulated variable and the ~�� output

variable in open-loop; thus Pyz = %yz�0�.�~� is evaluated by keeping all the output

variables constant, except the ~�� one. This situation could be achieved in practice by


97

controlling the 2 manipulated variables XYl, P ≠ �, in such a way that ZUl = 0, P ≠ ~,

using controllers with an integrating action. The ratio xyz is a dimensionless number

which is independent of the scaling of the variables.

The RGA is defined as the matrix ⋀ with xyz as �~, �� entry. The computation of ⋀ is straightforward. Indeed, it turns out that the matrix with �~, �� elements 1/�yz, ~ = 1,2,3; � = 1,2,3, is [%�0��]�. Thus the RGA is obtained through element by

element multiplication of %�0� and [%�0��]� (Kinnaert, 1995):

⋀ = %�0��[%�0��]� (7.20)

where � denotes the so-called schur product and not the conventional matrix product. It must be pointed out that the RGA is normalized, namely the sum of the elements of each row or each column of ⋀ is one.

Hence, based on the RGA, the Bristol’s criterion is stated as follows (Bristol, 1966): “the measure corresponding to the paired variables should be positive and as close to one as possible”.

For our linearized system, the RGA is given by:

�7V = `0.2868 0.7132 00.3961 4.0031 −3.39920.3171 −3.7163 4.3992 a (7.21)

According to the Bristol’s statement, the RGA proposes the following pairing (in green colour): ��, ��, ��, �� and ��, ��: let us call it “first choice”.

VII.3.2.2 Singular Value Decomposition

The main difference between a scalar SISO system and a MIMO system is the presence of different input and output directions in the latter. The Singular Value Decomposition (SVD) provides a useful way of quantifying multivariable directionality. Briefly, any complex matrix 7 may be decomposed into a SVD as follows: 7 = Z�X� (7.22) where:

- Z and X are unitary matrices, and X� is the hermitian transpose of X, i.e. the complex conjugate transpose of X. By definition a square matrix e with complex entries is unitary if ee� = e� e = _.

- � = ��̿ 00 0�, with �̿ = �~��, . . . , ��, � = ��P�7� and �� ≥. . . ≥ �� are

the singular values of 7. The numbers �y� can be shown to make up the eigenvalues of the matrix 77� (possibly with some zeros added).

The ratio ��7� = �� ⁄ is called the condition number. If 7 is the matrix transfer

function of a process, a large ��7� (typically >10) indicates that the process is ill-conditioned. This kind of process is particularly difficult to control essentially due to uncertainty which often affects the process model and which prevents from knowing


98

precisely the input directions corresponding to the different gains.

Based on the SVD of the steady-state matrix transfer function %�0� of our linearized model, we should pair the input corresponding to the row containing the

element of largest magnitude in the ~�� right singular vector of %�0� with the output

corresponding to the row containing the largest magnitude element in the ~�� left

singular vector (~ = 1,2,3� (Kinnaert, 1995).

The SVD of %�0� leads to:

X = `0.7995 0.4256 0.42390.6006 −0.5812 −0.54910.0127 0.6936 −0.7203a , Z = `−0.0094 −0.7427 0.66961.0000 −0.0079 0.00530.0013 0.6696 0.7427a (7.23)

� = `1.5313 0 00 0.0158 00 0 6.630410�,a (7.24)

The condition number is � = 2.309510� > 10, and hence the process is ill-conditioned. By observing matrices X and Z, the SVD proposes the pairing (in green colour) ��, ��, ��, �� and ��, �� which is the same as the choice suggested by the RGA, i.e. the “first choice”. VII.3.3 Dynamic criteria

VII.3.3.1 Dynamic Relative Gain Array

As the RGA only relies on the plant steady-state gain, it can lead to misleading information. The DRGA is simply an extension of the RGA in the dynamic framework. It will be calculated over the frequency range of interest. By analysing the matrix transfer function of the KZC wet grinding process, we realise that the fastest dynamics is H�� = 0.0028[ℎ] and the slowest one is HF,�� = 0.95[ℎ]. The corresponding highest

and lowest cross-over frequencies are respectively given by �� = 357.1429[��/ℎ] and �� = 1.0526[��/ℎ]. Thus, a frequency range � ∈ �10��, 10��[��/ℎ] covers the bandwidth of the KZC wet grinding circuit. The DRGA is simply defined as follows: ⋀�� = %��[%��]� (7.25)

The evolution of the relative gains of the system over the frequency range � ∈�10��, 510��[��/ℎ] is depicted in Figure VII.8.


99

Figure VII.8: Frequency evolution of relative gains

The stabilised values of the relative gains, typically for 410� ≤ � ≤ 10�[��/ℎ], yield the following DRGA:

��7V = ` 1.0031 −0.0031 0−0.0155 14.0014 −12.98590.0124 −12.9983 13.9859 a (7.26)

The selection criterion is the same as that of the RGA and the proposed pairing is clearly: ��, ��, ��, �� and ��, ��. This is the “second choice”.

VII.3.3.2 Direct Nyquist Array with Gershgorin’s ba nds

The DNA is an array of graphs, the �~, �� graph being the Nyquist plot of �yz� �. By looking at those curves, one can compare the gain of each diagonal element �yy� � with the magnitude of the off-diagonal elements of the corresponding row or the corresponding column for different pairings of the manipulated and controlled variables. The best pairing will correspond to the situation where the largest elements of the DNA display lie on the main diagonal. To visualize the degree of interaction in the system, one can add the so-called Gershgorin’s bands on the DNA display (Jensen et al., 1986). For each diagonal graph of the DNA display, the Gershgorin’s band is a casing-shaped curve generated by the frequency evolution of the Gershgorin’s circle. The latter being the closed disc centered at %yy�� with radius �y��; where �y�� is the

sum of the gains of the non-diagonal entries of %yz�� in the ~�� row; ~ = 1,2,3 and � = 1,2,3. The variable pairing is then based on the principle of main diagonal dominance, i.e. the best pairing is selected such that the Gershgorin’s bands do not enclose the origins of the corresponding main diagonal Nyquist graphs. Figures VII.9 and VII.10 depict maps for the DNA with Gershgorin’s bands respectively

0 100 200 300 400 500

1

1.2

1.4

Frequency [rad/h]

Rel

ativ

e ga

in (

1,1)

[p.

u]

0 100 200 300 400 500-0.4

-0.2

0

Frequency [rad/h]

Rel

ativ

e ga

in (

1,2)

[p.

u]

0 100 200 300 400 500-1

0

1

Frequency [rad/h]

Rel

ativ

e ga

in (

1,3)

[p.

u]

0 100 200 300 400 500-2

-1

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for the first choice and the second choice over the frequency range � ∈ �10��, 10��. If we apply the principle of dominance of main diagonal elements, we realize that the second choice is more effective than the first. Indeed, in Figure VII.10a) the Gershgorin’s band does not enclose its origin while all those of Figure VII.9 enclose their origins. Although the other two Gershgorin’s bands of Figure VII.10 also encircle their origins, the second choice is better than the first one thanks to Figure VII.10a). Thus we would be tempted to select: ��, ��, ��, �� and ��, ��.

a) b) c)

Figure VII.9: DNA with Gershgorin’s bands for the first choice of variable pairing a) ��, �� ; b) ��, �� and c) ��, ��

Blue colour: Nyquist graph Red colour: Gershgorin’s band

a) b) c)

Figure VII.10: DNA with Gershgorin’s bands for the second choice of variable pairing a) ��, �� ; b) ��, �� and c) ��, �� Blue colour: Nyquist graph

Red colour: Gershgorin’s band

VII.3.3.3 Final choice

Which pairing should we accept ? There is really a dilemma because the two steady-state criteria advice the first choice while the two dynamic criteria yield the second one. An important indication is the fact that dynamic criteria cover the process bandwidth and control design should account for process dynamics. Consequently, the second choice is more suitable and will be definitively used in the sequel, i.e. ��, ��, ��, �� and ��, ��. Though the selected pairing is the most efficient, the couplings between variables are not eliminated. To highlight the amount of interaction effects between variables, the retained configuration has been analysed by simulations. This is explained in the following section.

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VII.4 Interaction effects

To exhibit the interaction effects between manipulated variables and output variables, we can observe Figure VII.11 which depicts variations of output variables after step changes of manipulated variables. These simulation results reveal strong couplings between variables. This is one of the main issues to deal with in the controller design. These simulations consist in small variations around the selected operating point by using the linearized model. Yet in reality, the KZC grinding process is nonlinear. Hence, any change of the operating point will result in another dynamic behaviour of the process, i.e. a change of the linearized model. If the same model structure is used, this change will simply result in variations of the model parameters. In the following section, we express these variations as unstructured modelling uncertainties. This information will be used later on for robust controller design.

VII.5 Uncertainty characterization

VII.5.1 General concept of uncertainty

“The term uncertainty refers to the differences or errors between models and reality, and whatever mechanism is used to express these errors will be called a representation of uncertainty. Indeed, the quality of a model depends on how closely its responses match those of the true plant. Since no single fixed model can respond exactly like the true plant, we need, at least a set of models. However, the modelling problem is much deeper because the universe of mathematical models from which a model set is chosen is distinct from the universe of physical systems. Therefore, a model set which includes the true physical plant can never be constructed. A good model should be simple enough to facilitate design, yet complex enough to give confidence that designs based on the model will work on the true plant” (Zhou et al., 1996).

Linear time invariant models of the type used in this part of our work describe actual plant dynamics only approximately. This is supported by Figures VII.4, VII.5 and VII.6 where the curves of the linearized model, FOTD transfer functions, do not fit accurately the dynamics of the nonlinear simulator. The latter being itself an approximation. The model uncertainty can have several sources. Most important, real processes are nonlinear. If the process model is obtained via linearization, then it is accurate only in the neighbourhood of the reference state chosen for the linearization. In other cases the process might be represented quite accurately by a linear model. However, different operating conditions could lead to changes in the parameters in the linear model. For example, increased throughput and flowrates usually result in smaller deadtimes and time constants. In the two cases above, the sources and the structure of the uncertainty are known quite accurately. However, there is always some “true” uncertainty even when the underlying process is essentially linear: the physical parameters are never known exactly and fast dynamic phenomena (e.g., valve dynamics) are usually neglected in the model. Therefore, at high frequencies, even the model order is unknown (Morari & Zafiriou, 1989).


102

Figure VII.11: Interaction effects between manipulated and controlled variables

For the process under study, all three sources are involved. In the sequel, we will put emphasis on modelling uncertainties due to operating point changes. Indeed, given the absence of on-line data from the KZC plant, we cannot properly characterize the modelling uncertainty associated to parameter estimation errors. Besides, as will appear in the design of the decentralized controller later on, uncertainties due to the choice of a low complexity model (FOTD transfer function) does not play a dominant role. Indeed, for the SISO models associated to the input-output pairs that are considered for controller design, a good fit of the transients observed with the nonlinear simulator is achieved.

0 2 4 6 8 10 12 14 16 18 2034

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Uncertainty can be described in many different ways: bounds on the parameters of a linear model, bounds on nonlinearities, frequency domain bounds, etc. (Morari & Zafiriou, 1989). In this dissertation, we will start from a description based on uncertainty bounds on the parameters of the linearized model. We will also consider the SISO case according to the variable pairing. To account for model uncertainty we will assume that the dynamic behaviour of the KZC grinding plant is described not by a single linearized time invariant model but by a family� of linear time invariant models. The family� of plants will be defined in the frequency domain. We will also assume that the transfer function magnitude and phase at a particular frequency � is not

confined to a point but can lie in a disk �� with radius ∆�� on the Nyquist plane.

The union of all disks �� constitutes the family of possible plants� and can be viewed as a Nyquist band. Algebraically, the family� of plants is defined by:

�� = ¡: |¡�~�� − ¡¤�~��| ≤ ∆��¥ (7.27)

Here ¡¤�~�� is the nominal plant or the model defining the centre of all disks. Any member of the family� satisfies:

¡�~�� = ¡¤�~�� + ∆��~�� (7.28)

with: |∆��~��| ≤ ∆�� (7.29)

Equation (7.28) is referred to as an additive uncertainty description and (7.29) states a bound on the allowed additive uncertainty.

By defining: ∆¦�~�� = ∆§�y¨�©ª�y¨� (7.30)

and ∆�¦�� = ∆�§�¨�|©ª�y¨�| (7.31)

the family� (7.27) can be represented as: �� = «¡: |©�y¨��©ª�y¨�||©ª�y¨�| ≤ ∆�¦��¬ (7.32)

Thus, any member of the family� satisfies:

¡�~�� = ¡¤[1 + ∆¦�~��] (7.33)

with: |∆¦�~��| ≤ ∆�¦�� (7.34)

Equation (7.33) is referred to as a multiplicative uncertainty description and (7.34) states a bound on the allowed multiplicative uncertainty. The uncertainty usually increases with frequency. The reason is that our models tend to describe well the steady-state and low-frequency behaviour of processes but become inaccurate for high-frequency inputs. Figure VII.12 below shows a block diagram representation of the two types of uncertainties.


104

a) b)

Figure VII.12.a): Additive uncertainty representation b): Multiplicative uncertainty representation

It should be noted that only the multiplicative uncertainty will be used in this work for its convenience in robust control study.

VII.5.2 System model uncertainties

To describe the uncertainties for the KZC grinding process, we will first assume that the nominal operating point corresponds to the initial operating point around which the system model has been linearized (Table VII.1). Since the retained pairing of variables is ��, ��, ��, �� and ��, ��, the three nominal models are set to:

¡¤,y� � = %yy� �; ~ = 1,2,3 (7.35)

where %� � is given by (7.2).

A simple way to define uncertainty bounds on the model parameters is to study how operating point changes affect these parameters. Hence, a neighbourhood space around the nominal operating point may be defined. Let us consider upper and lower bounds on each manipulated variable. This results in eight boundary points for both manipulated and controlled variables. Figure VII.13 provides an example of such a map.

Figure VII.13: Neighbourhood space around the nominal operating point - 0: Nominal operating point - 1,2, …, 8: Eight boundary operating points To determine the range of change of the manipulated variables to be used for characterizing the uncertainty, we have compared the nominal operating condition and the optimal one. It turns out that considering a 10 % change on all nominal values of the manipulated variables defines an operating region that contains the optimal operating point. Table A-VII.2.1 in Appendix VII.2 gives values of the eight boundary

¡¤

∆�

+ + ¡¤

∆¦

+ +

��

��

��

��

1 2

3

6 5

7 8

4 1 2 3

4

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8

0 0


105

operating points.

Step response simulations have been carried out in order to estimate the twenty-four FOTD transfer functions for the 3x3 KZC grinding circuit. These simulations are based on the selected variables pairing and step variations of the manipulated variables are made at the different boundary operating points (Figure VII.14).

a)

b)

c)

Figure VII.14: Step variations of manipulated variables a) Fresh ore feed flow rate b) Rod mill feed water flow rate c) Dilution water flow rate

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Simulation results are depicted in Figure VII.15 hereafter.

a)

b)

c)

Figure VII.15: Variations of output variables after step variations of manipulated variables around the nominal and the eight boundary operating points a) Variation of product fineness b) Variation of product flow rate c) Variation of product density

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Nominal operating pointOperating point no.6Operating point no.5Operating point no.4Operating point no.1Operating point no.7Operating point no.8Operating point no.3Operating point no.4

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Twenty-four FOTD have been identified in a straightforward way to fit these curves obtained with the nonlinear simulator. The corresponding result is given in Table A-VII.2.2 of Appendix VII.2. The bounding FOTD transfer functions can be deduced from Figure VII.15 by looking for the farthest curve from the nominal one. The corresponding model parameters are green-marked in the table. In the sequel, these three transfer functions will be referred as the mismatch model. We may conclude here that:

- two model parameters, i.e. steady-state gain and time delay, are affected by uncertainties in the channel from �� to ��;

- only time delay is affected by uncertainty in the channel from �� to ��; - only steady-steady gain is affected by uncertainty in the channel from �� to ��.

Generally speaking, the bound on multiplicative uncertainty for FOTD transfer functions is expressed by:

∆�¦�� = ®p®ª �.¯ªy¨�.�̄y¨ /�y¨��̄°�¯°ª� − 1 (7.36)

where ±¤,H� and HF¤ stand respectively for steady-state gain, time constant and

time delay of the nominal model while ±,p Hp and HF̅ stand for those of the bounding

model. The application of the above relationship on our model configurations, given by Equation (7.2) and Table A-VII.2.2, results in the bounds on multiplicative uncertainties given by:

²∆��,11�� = ³1.2151/−0.25y¨ − 1³∆��,22�� = ³/−0.1222y¨ − 1³∆��,33�� = 0.2453

(7.37)

where ∆�¦,�� covers uncertainties on the steady-state gain and time delay of ¡¤,� whereas ∆�¦,�� only covers uncertainty on time delay of ¡¤,� and ∆�¦,�� only

that on the steady-state gain of ¡¤,�. We realize that time constant of each channel is

not affected by operating point changes. Figure VII.16 shows Nyquist diagrams of the three uncertain models ¡�, ¡� and ¡� with

uncertainty disks over the frequency range � ∈ �10��, 10��. The radius of each disk represents, at a given frequency, the module of the related bound on multiplicative uncertainty multiplied by the nominal model gain. Before concluding this chapter, the next section analyses the behaviour of the KZC grinding circuit. This critical discussion leads to a summary of the main characteristics of the system.


108

a) b)

c)

Figure VII.16: Family of uncertain plants a) Uncertain model between �� b) Uncertain model between �� c) Uncertain model between ��

VII.6 Discussion

The KZC grinding circuit has been analysed in different ways and through different aspects in the previous sections. Let us now summarize this analysis on a control point of view. By observing Figures VII.2, VII.3, VII.4, VII.5 and VII.6, we realize that a good fit of the transients observed with the nonlinear simulator is achieved with respect to the SISO models related to the retained input-output pairs. The small model mismatches observed in this case will be later on regarded as internal disturbances for controller design purpose. The other transients, including their respective model mismatches, associated to the unpaired variables will be considered as coupling effects. We also realize that the process is strongly affected by the two external disturbances. Moreover, time delays between input and output variables are observed. These time delays are due to material transport and are ranging from 0.0344 to 0.6039[ℎ] around the nominal operating point. These time delays are larger than that of the typical grinding circuit (Table VII.1). From Figures VII.14 and VII.15, we conclude that the KZC grinding process is sensitive to operating point changes. Uncertainty bounds, on the parameters of the linearized model, corresponding to operating point changes have been described. The resulting unstructured multiplicative uncertainty will be used later on for robust controller design.

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In summary, the main features of the KZC grinding process are highlighted as follows: strong coupling effects, model mismatches regarded as internal disturbances, strong external disturbances, large time delays, model parameter uncertainties due to operating point changes.

Conclusion

The KZC grinding process has been analysed throughout this chapter. This analysis led us to retain four main features of the system, namely model mismatches, strong external disturbances, large time delays and model parameter uncertainties. These features reveal that this process is not easy to control. They will then be considered as the main control issues. The control objective will therefore be of holding the process under the optimal operating point despite the effects of these issues. Thus, the following chapter will focus on this control problem by proposing a suitable decentralized scheme.


110

Appendix VII.1: Validation of a one-entry feed part icle size distribution as second disturbance Figure A-VII.1.1 below confirms the independence of output variables with mass variations only in classes 1 and 2 of feed particle size distribution as shown by the disturbances linearized model given by Equation (7.3). Indeed, keeping the same variation of the feed ore fineness in both scenarios and whatever changes in the other two size classes, we observe the same variations of the output variables.

a1) a2)

b1) b2)

c1) c2)

Figure A-VII.1: Output variations after step change of feed particle size distribution a1), b1) and c1): from [0.60.30.1]� to [0.50.30.2]�

a2), b2) and c2): from [0.40.50.1]� to [0.20.60.2]�

18 19 20 21 220.785

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Appendix VII.2: Eight boundary operating points

Table A-VII.2.1: Neighbourhood space around the nominal operating condition after +/- 10 % variations of manipulated variables

Variable

��[�/ℎ] ��[��/ℎ] ��[��/ℎ] ��[ . �] ��[�/ℎ] ��[ . �] B

ound

ary

oper

atin

g po

int

1 38.5000 45.0000 5.4000 0.7134 81.8728 1.3686

2 38.5000 55.0000 5.4000 0.7253 88.3999 1.3103

3 31.5000 55.0000 5.4000 0.8761 79.8132 1.2524

4 31.5000 45.0000 5.4000 0.8690 73.3043 1.3068

5 38.5000 45.0000 6.6000 0.7134 82.6518 1.3608

6 38.5000 55.0000 6.6000 0.7253 89.1789 1.3042

7 31.5000 55.0000 6.6000 0.8761 80.5922 1.2467

8 31.5000 45.0000 6.6000 0.8690 74.0833 1.2995

Table A-VII.2.2: Identification of boundary parameters of the linearized system model

Transfer function

%�� %�� %�� ±��[ℎ/�] H��[ℎ] HF,��[ℎ] ±��[ℎ/��] H��[ℎ] HF,��[ℎ] ±��[ℎ/��] H��[ℎ] HF,��[ℎ]

Bou

ndar

y op

erat

ing

poin

t

1* -0.0155 0.0444 0.4439 0.6518 0.0028 0.1609 -0.0066 0.0042 0.0344

2 -0.0154 0.0444 0.4599 0.6518 0.0028 0.1569 -0.0051 0.0042 0.0344

3 -0.0219 0.0444 0.8761 0.6518 0.0028 0.3180 -0.0047 0.0042 0.0344

4* -0.0226 0.0444 0.8539 0.6518 0.0028 0.3291 -0.0061 0.0042 0.0344

5 -0.0155 0.0444 0.4489 0.6518 0.0028 0.1609 -0.0063 0.0042 0.0344

6 -0.0154 0.0444 0.4589 0.6518 0.0028 0.1569 -0.0050 0.0042 0.0344

7 -0.0219 0.0444 0.8579 0.6518 0.0028 0.3169 -0.0046 0.0042 0.0344

8* -0.0226 0.0444 0.8539 0.6518 0.0028 0.3272 -0.0059 0.0042 0.0344

Caption: - %yy: FOTD transfer function between the ~�� manipulated variable and the ~�� controlled variable, ~ = 1,2,3; - ±yy: steady-state gain of %yy, ~ = 1,2,3;

- Hyy: time constant of %yy, ~ = 1,2,3;

- HF,yy: time delay of %yy, ~ = 1,2,3;

- green colour : Operating point corresponding to bounding parameters (mismatch model).

Chapter VIII: DECENTRALIZED CONTROL FOR THE KZC GRI NDING PROCESS

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Chapter VIII: DECENTRALIZED CONTROL FOR THE

KZC GRINDING PROCESS

Introduction This chapter aims at designing controller for the KZC grinding process. This is the ultimate objective of the current thesis. The control system proposed here is a decentralized structure based on the previously specified paired variables. The control goal is to hold the process on the optimal operating point despite disturbances. This goal can be split into two sub objectives as follows: reach the reference state characterized by the optimal values of controlled variables and reject the effects of disturbances on the controlled variables. That said, we propose a combined feedback-feedforward controller to reach this double control goal. The suggested controller is implemented in an IMC structure. This is why the control scheme developed in this work is called “Double Internal Model Control” (DIMC). The two sub objectives are handled by two sub controllers chosen in order to deal with the main control issues presented in the previous chapter. Thus in each channel, a “Proportional Integral-Smith Predictor” (PI-SP) controller is used for the setpoint reaching while a “Disturbance Observer” (DOB) is employed for the disturbance rejection. The duality between these two controller components is demonstrated with respect to the design method. Robust design is used to cope with model uncertainty so that the controller may work properly on the actual plant whose model is uncertain. Results and conclusions are then provided. Validation of the controller is performed on both the nominal model and the mismatch model. Hence, this chapter is organised as follows. The first section shows the DIMC structure while the second one presents the robust design procedure. In the following section, the designed control scheme is applied to the KZC grinding process. To take into account the context of the KZC circuit where instrumentation might not be obvious to acquire, a possible progressive implementation of the control structure is studied. The scenarios are discussed for a possible application on the real plant. .

VIII.1 DIMC Structure VIII.1.1 Control issues

Couplings of variables

It has been shown in Chapters V and VII that the KZC process really possesses interaction between manipulated and output variables. Each manipulated variable is affecting several output variables except the dilution water which has quasi no effect on the product fineness. To determine a decentralized control, the variable pairing selected in Chapter VII is used.


2013

114

As argued Chapter VI, to compensate for coupling effects, one can use either a PC positioned upstream of the system or a DOB acting in a feedforward scheme and in which coupling effects on each channel are regarded as disturbances. The PC runs in a centralized control scheme while each DOB works in a decentralized configuration. Therefore in this dissertation, only DOB structures will be used. Time delays

Time delay is a fundamental characteristic of many multivariable processes in practice such as grinding circuits. The KZC grinding process exhibits larger time delays than the typical grinding circuit. Therefore specific attention is required on this issue here. In general, time delay or dead-time within a dynamic system is due to transport of fluids or materials in pipes or channels and to measurement delay or analysis time. Consequences of time delay are a delay of disturbance detection in the case of fluids or materials transport, and a delay of the control action compared to the measurement, i.e. control action and measurement are not synchronous. Hence, the control system performance is decreased, as compared to a delay free system.

To cope with time delay systems, the “Smith Predictor” (SP) (Smith, 1957) is the most used technique. It is based on introducing a compensation loop inside the controller with the purpose of cancelling the effect of the time delay on the dynamics of the closed-loop. For our case study, a primary “Proportional-Integral” (PI) controller will be incorporated in the SP structure. This configuration will be called “PI-SP” controller.

Strong external disturbances

The KZC grinding circuit is characterized by two strong disturbances, i.e. ore grindability and feed ore fineness. The disturbance effects must be rejected out of the controlled variables and this requirement can be handled by a feedback structure. When the specifications of the disturbance rejection, imposed on a controlled system, are not met by a feedback controller, the control configuration can be extended with a DOB (Coelingh et al., 2000). Hence to reject the effects of the two strong external disturbances on each control channel of the KZC circuit, a DOB will be implemented in a feedforward scheme.

Variations of dynamics with operating point changes

The KZC grinding process model is nonlinear. Thus, the model has been linearized around a nominal steady-state operating point. Bounds on model parameters uncertainty have been described in Chapter VII. With this consideration, the controller must be robust with respect to these variations of model dynamics. The robust control problem involves the following items (Morari & Zafiriou, 1989):

- process model; - model uncertainty bounds; - type of inputs (i.e., setpoints and disturbances); - performance objectives.

The first two have been described in the previous chapter. The following sub section is hence devoted to the last two ones.


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VIII.1.2 Control objectives The ultimate objective of control system design is that the controller should work “well” when implemented on the actual plant. This goal may be best understood if it is split into a series of sub objectives. Since the model is an approximate description of the true plant, it is reasonable to require stability when the controller is applied to the plant model. Thus, the minimal requirement on the closed-loop system is nominal stability. Beyond this, all design techniques aim to make the error resulting from external inputs small. This defines nominal performance. For the KZC grinding process, it has been shown that there are two types of external inputs, namely setpoints and external disturbances. Small step setpoint changes that lie in the range defined in Table A-VII.2.1 with respect to output variables will be considered. As the aim is to show how the controlled variables reach the optimal values, typical changes of 1% provided in Table VIII.1 will be adopted.

Table VIII.1: Step changes of setpoints

Setpoint Small change ( �%) ��,�� . �� From 0.7920 to 0.8000 ��,��ℎ�� From 76.6988 to 77.4735 ��,��ℎ�� From 1.3390 to 1.3257

Generally, the ore hardness and feed particle size fluctuate continuously during the whole process of production (J.Yang et al., 2010). Thus, a sinusoidal external disturbance is likely to match the features of the real practice better than a common step disturbance does. But for control design purpose, these two kinds of disturbances will be taken into consideration. Therefore, mixt step-sinusoidal external disturbances will be considered in this work. From the viewpoint of practitioners, Table VIII.2 provides realistic variations of the two external disturbances. In this table, step variations only consist in amplitude variations while both amplitude variations and frequency are involved in sinusoidal variations. These variations will be injected around the nominal values.

Table VIII.2: Step-sinusoidal variations of external disturbances

Disturbance Amplitude 1 Frequency ��ℎ�� 10�%� 10� !�/ℎ� �� . �� 50�%� 10� !�/ℎ�

It is necessary that our knowledge about the model uncertainty, presented in Section VII.5, be incorporated into the controller design procedure. Otherwise the controllers are bound to fail on the actual plant. We will require that the controller be designed such that the closed-loop system is stable and meets the performance specifications for all members of the family # of possible plants. Any closed-loop

1 Amplitude of each step variation and each sinusoid is expressed in % of the nominal value of

the corresponding disturbance given by its typical value (Figure VII.1).


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property which holds for a family of plants will be referred to it as robust. Thus, it is important to require stability when the controller is applied to the real plan. This is robust stability. Therefore, our controller design objective is robust performance which is achieved simply by satisfying both nominal performance and robust stability. In addition, due to actuator saturations combined with integrating actions in the controllers, it is imperative to deal with the windup problem. An anti-windup scheme is thus required. More clearly, the control goal to be achieved in this work is to “hold the process on the optimal operating point despite disturbances”. This goal can be split into two sub objectives as follows:

(1) reach the reference state characterized by the optimal values of controlled variables after step setpoints change;

(2) reject the effects of disturbances on the controlled variables.

The relevant controller should incorporate two dynamics: one for setpoint change and another for disturbance rejection. In the next subsection, we demonstrate that such a controller may be based on the model of the plant and may produce two control actions. This is why this control structure is called “Double Internal Model Control” (DIMC). In the sub section coming below, we propose a combined structure of PI-SP and DOB as the feedback-feedforward controller suitable for the DIMC. VIII.1.3 Presentation of the DIMC

PI-SP scheme

The classical time delay compensation was conceived in 1957 and is known after its originator as the Smith Predictor method. The standard form of this control structure including a PI primary controller is shown in Figure VIII.1 below. It requires a model of the plant excluding dead time and a separate series function block that estimates the cumulative time delay of the plant. The red arrow indicates the block which contains the tuning parameters.

Figure VIII.1: Standard form of PI-SP

Control variable

Controlled variable

External d isturbance s

Uncertain Plant with time delay

PI

$

System Model

without time delay

System time delay

%

% $

$ $ Output error

Predicted feedback

Setpoint

Predicted output

Estimated output

$ $

Smith Predictor


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The model without time delay is in fact the predictor part of the model. The output error is added to the estimated undelayed model output to form the feedback signal. There are many variations of the SP scheme and it is well known that the SP technique is extremely sensitive to model errors (Santacesaria & Scattolini, 1993). A simple rearrangement of the block diagram in Figure VIII.1 which leaves all input-output relationships unaffected makes the similiarity with the IMC structure apparent. Figure VIII.2 shows the IMC form of PI-SP.

Figure VIII.2: IMC form of PI-SP

The PI-SP is used to improve the closed-loop performance for systems with time delay. However, here are some properties of PI-SP (Morari & Zafiriou, 1986):

(1) with the predictor in place, the PI controller can be tuned just like for the system model without time delay;

(2) PI-SP controllers work well for setpoint changes but not for disturbances;

(3) PI-SP controllers are sensitive to model errors.

For the above reasons, the PI-SP will be robustly designed.

DOB scheme

The DOB structure is exactly that presented in subsection VI.3.2 and shown in Figure VI.6. It is recalled here for the convenience of the reader. Figure VIII.3 depicts the standard form of the DOB structure while Figure VIII.4 shows its IMC form. We would also like to recall that this DOB estimates a lumped disturbance consisting of both external and internal disturbances. It will then be used to deal with external disturbances, coupling effects and model mismatch issues. Robust design will also be used to cope with the model uncertainty issues.

Control variable

Controlled variable

External d isturbance s

Uncertain Plant with time delay

PI

$

System time delay

%

% $

%

$

Output error

Setpoint $ $

System Model

without time delay

System Model

without time delay

IMC Controller Predictor

Model Predict


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Figure VIII.3: Standard form of DOB

Figure VIII.4: IMC form of DOB

DIMC scheme

To overcome all the control issues for the KZC grinding circuit, one can use simultaneously the PI-SP and the DOB control schemes. This combination results in the DIMC scheme whose structure is given in Figure VIII.5.

Disturbance observer

External disturbances

%

$

Estimated disturbance

Controlled variable

Control variable

Inverse System Model without

time delay

System time delay

Low-pass Filter

Other control variables Uncertain

Plant with

time delay $

$

Low-pass Filter

$ %

IMC Controller


% $

Estimated disturbance

Controlled variable

Control variable


time delay

System time delay

Low-pass Filter

Other control variables Uncertain

Plant with time delay $

$ $ %

System Model without

time delay

Model


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Figure VIII.5: DIMC scheme

It should be noted that the above DIMC structure constitutes the retained decentralized control strategy for the KZC grinding process. By its structure, this control strategy will overcome the following already stated control issues: coupling effects, time delays and strong external disturbances. The last control issue, namely the model uncertainty due to operating point changes, will be handled by a robust design of both PI-SP and DOB.

We can realize that PI-SP and DOB complement each other to deal with all the control issues for the KZC grinding circuit control. Moreover, their implementation in an IMC scheme contains a very important feature, namely a duality between their robust designs. The next section is concerned with the robust DIMC design and exploits this duality to simplify the controller design procedure.

VIII.2 Robust DIMC design VIII.2.1 Robustness problem

The general concepts presented here are taken from the book of (Morari & Zafiriou, 1989).

DOB Controller


% $

Controlled variable

Control variable


time delay

System time delay

Low-pass Filter

Other control variables

Uncertain Plant with time delay $

$ $ %

System Model without

time delay

Model

$ System Model without

time delay System

time delay

Model

%

PI

System Model

without time

delay

$

%

Setpoint $

%

PI-SP

DOB

PI-SP Controller

Controller


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Sensitivity and complementary sensitivity functions

Let us consider the following closed-loop system whose setpoint signal is denoted by & �, output signal by ' �, disturbance signal on output by ( �, error

signal by ) � and measurement noise by * �. + � and , � represent the transfer functions of the controller and of the plant, respectively.

Figure VIII.6: Block diagram for definition of sensitivity The sensitivity function - � relates the reference & � and the noise * � to the error ) �. It also expresses the effect of the disturbance ( � on the output ' �.

- � ≜ /0�10� = % /0�

30� = 40�50� = �

�670�80� (8.1)

The sensitivity function - � is important in judging the performance of a feedback controller. It is desirable to make it as “small” as possible. A function usually associated to the sensitivity function is the complementary sensivity function 9 � which derives its name from the equality:

- � $ 9 � = 1 (8.2)

The complementary sensitivity 9 � is given by:

9 � ≜ 40�10� = % 40�

30� = 70�80��670�80� (8.3)

The following requirements should be noted for 9 �:

(1) we can say that 9 � relates the reference to the output. From this viewpoint, it should be made as close to unity as possible;

(2) we can also say that 9 � relates the effect of measurement noise on the output. From this viewpoint, it should be made small.

These two requirements are contradictory which illustrates one of the basic trade-offs in feedback design: good setpoint tracking and disturbance rejection - ≈ 0, 9 ≈ 1� must be traded off against suppression of measurement noise - ≈ 1, 9 ≈ 0�. This trade-off is usually irrelevant in process control since measurement noise is effective in high frequency range. Moreover, model uncertainty also imposes an upper bound on the magnitude of the complementary sensitivity function. With the poor models

, � ) � $ * �

$

( � $ $ + � %

& � ' � ; �

$


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generally available for process control, this bound is much more restrictive than the measurement noise constraint. Therefore, measurement noise will be neglected for the controller design in this work. Robust performance It has been indicated that robust stability is the minimum requirement that a control system has to satisfy to be useful in a practical environment where model uncertainty is an important issue. The robust stability theorem is stated as follows by means of the <= − >? @ :

Assume that all plants , in the family #

# = A,: |7DE��7FDE�||7FDE�| ≤ ∆IJK�L (8.4)

have the same number of right-half-plane poles and that a particular controller +

stabilizes the nominal plant ,M. The system is robustly stable with the controller + if and

only if the complementary sensitivity function 9M � for the nominal plant ,M satisfies the following bound: ‖9M∆IJ‖= ≜ O� E|9M∆IJ| < 1 (8.5)

However, robust stability alone is not enough. If the bound (8.5) is satisfied for

a family #, then there exist a particular plant , ∈ # for which the closed-loop system

is on the verge of instability and for which the performance is arbitrarily poor. Thus we also have to make sure that some performance specifications are met for all plants in

the family #. This requires to derive conditions for robust performance.

The robust performance theorem is stated as follows:

Assume that all plants , in the family #

# = A,: |7DE��7FDE�||7FDE�| ≤ ∆IJK�L (8.6)

have the same number of right-half-plane poles. Then the closed-loop system will meet the performance specification

‖-R‖= ≜ O� E|-R| < 1∀, ∈ # (8.7)

If and only if the nominal system is closed-loop stable and the sensitivity function -M � and the complementary sensitivity function 9M � satisfy:


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|9M∆IJ| $ |-MR| < 1∀K (8.8)

where RK� stands for a weighting function. Trivially, robust performance (8.8) implies robust stability (8.5) and nominal

performance (8.7). The interdependence of -M and 9M makes it a challenge to meet (8.8). The latter is also called mixed sensitivity criterion.

The frequency dependent performance weight RK� can be simplified and set as a constant. This is usually expressed in terms of the maximum allowed peak height

(R�� of the sensitivity function -M. Typically 0.3 < R < 0.9. We will assume here the

maximum peak of -M equal to 2 as suggested by (Morari & Zafiriou, 1989). Thus,

R = 0.5 and the robust performance criterion is reduced to:

∆IJ|9M| $ 0.5|-M| < 1∀K (8.9)

The <= % >? @ robust performance criterion (8.9) expresses the control optimization problem to be solved with respect to the controller parameters. It is our controller design criterion. Prior to apply this criterion to our case study, we define in the next subsection, the duality between DOB and PI-SP with regard to sensitivity functions. This feature will be helpful in the controller design. For our case, thanks to the duality between DOB and PI-SP, the <= % >? @ robust performance criterion (8.9) will only be applied to the DOB. The PI-SP controller will then be designed from the DOB by simply setting the time constant ratio between the two closed-loops. VIII.2.2 Duality between PI-SP and DOB design

DOB design

Let us begin with the DOB design. As already indicated, we are going to use the modified DOB structure presented in Subsection VIII.1.3. This scheme is characterized by adding the plant time delay on the control variable channel entering the DOB. Hence, the DOB compares two synchronous signals to generate a convenient estimated lumped disturbance. It is thanks to this particularity that the duality between PI-SP and DOB design is demonstrated at the end of this subsection. To better describe the DOB design, let us consider for a channel T= 1,2,3� of paired

variables �D % �D� the DOB scheme restructured in Figure VIII.7 below and where:

- ;D � and 'D � are respectively the control and the controlled variables;

- (D � is the cumulative external disturbance on the controlled variable; - (UV,D � is the estimate lumped disturbance brought back to the manipulated

variable; - ;W � is the manipulated variables from the other channels and expresses the

coupling between manipulated and controlled variables X= 1,2,3� ≠ T; - ,D � is the transfer function of the uncertain plant model and will be later on

referred to as the mismatch model;


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- ,Z,M,D � = [>,T1$\>,T is the transfer function of the undelayed part of the nominal

process model with ,M,D � = ,Z,M,D �]�^_,F,`0; ,M,D � is therefore the FOTD

transfer function of the nominal plant; - aM,D � is the nominal transfer function of DOB filter.

Figure VIII.7: Block diagram for DOB design

In the nominal case, where the plant model is represented by its nominal transfer function ,M,D �, the sensitivity and the complementary sensitivity functions are given

by (8.10) and (8.11), respectively.

-M,D � = 4`0�5`0� = 1 % aM,D �]�^_,F,`0 (8.10)

9M,D � = 1 % -M,D � = aM,D �]�^_,F,`0 (8.11)

By adopting the first-order-low-pass filter given by Equation (6.12) which is rewritten into Equation (8.12), Equations (8.10) and (8.11) become respectively (8.13) and (8.14).

aM,D � = 11$ba,>,T (8.12)

The filter time constant bc,M,D is the unique parameter to be designed for each channel

T. -M,D � = 1 %

def_,F,`g�6hi,F,`0 (8.13)

9M,D � = ]%\�,>,T 1$ba,>,T (8.14)

Thus, bc,M,D will be designed by solving the following optimization problem derived from

(8.9):

∆IJ,D j ]%\�,>,TXK�1$ba,>,TXK�j $ 0.5 j1 % ]%\�,>,TXK�

1$ba,>,TXK�j < 1∀K (8.15)

;W �

aM,D �]�^_,F,`0

$ ,Z,M,D�� aM,D �

'D � (D � $ ,D � $ $

% ;D �

%

(UV,D �


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From Figure VIII.7, (UV,D � is given by:

(UV,D � = aM,D �,Z,M,D�� (k,D � (8.16)

where (k,D � is the total lumped disturbance at the plant output and includes the

cumulative external disturbance (D �, the coupling effects due to ;W �and the model

mismatches.

The estimated total lumped disturbance (Uk,D �is therefore expressed by: (Uk,D � = aM,D �]�^_,F,`0(k,D � (8.17)

It turns out that the total lumped disturbance is filtered and delayed to obtain its estimate. Hence for a sinusoidal disturbance, as the KZC grinding process possesses large time delays, this may lead to an estimated disturbance which is in phase opposition with respect to the real disturbance. We can also note that the DOB will not fully suppress the dynamic couplings since their estimates are also delayed. However, in steady-state the disturbance will be rejected as demonstrated hereafter.

Define (lk,D �as the error between (k,D �and (Uk,D �: (lk,D � = (k,D � % (Uk,D � = �1 % aM,D �]�^_,F,`0�(k,D � (8.18)

According to the final-value theorem, one obtains: ��k,D∞� = limk→= ��k,D� = lim0→Z (lk,D � = 0 (8.19)

It can be found from (8.19) that the lumped disturbance can be asymptotically suppressed. Hence, this modified DOB is a practical approach to overcome processes with large time delays. PI-SP design

Let us now explain our proposed approach to design the PI-SP. Consider Figure VIII.8 which depicts the block diagram for PI-SP that we use for design purpose.

Figure VIII.8: Block diagram for PI-SP design

$

;W �

,Z,M,D � %

,M,D �

'D � (D � $ ,D � $ $

% '�,D �

$

)D � ,rM,D �

$


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In this above figure:

- '�,D � and 'D � are respectively the setpoint (or reference) and the controlled

variables; - )D � is the error signal; - ,rM,D � is the nominal transfer function of the PI primary controller.

The nominal (for ,D � = ,M,D �) sensitivity and complementary sensitivity functions are expressed in this case by:

-M,D � = /`0�4s,`0� = 1 %

7tF,`0�7F,`0��67tF,`0�7u,F,`0� (8.20)

9M,D � = 1 % -M,D � = 7tF,`0�7F,`0�

�67tF,`0�7u,F,`0� (8.21)

We are now going to recall the three basic properties for the PI-SP controller stated in Subsection VIII.1.3:

(1) with the predictor in place, the PI controller can be tuned just like for the system model without time delay;

(2) PI-SP controllers work well for setpoint changes but not for disturbances;

(3) PI-SP controllers are sensitive to model errors.

Let us also recall that the nominal process model is given by the FOTD transfer

function ,M,D � = ,Z,M,D �]�^_,F,`0 where ,Z,M,D � = [>,T1$\>,T is its undelayed part.

Thus, a convenient way to control the undelayed part ,Z,M,D � of ,M,D � is to use a PI

feedback controller whose zero compensates the stable pole of the system model. Therefore ,rM,D � is selected to be expressed by:

,rM,D � = [7t,M,D v1 $ 1\>,T �w (8.22)

The proportional gain [7t,M,D of the PI controller is the unique parameter to be designed

for each channel T. Hence Relationships (8.20) and (8.21) simply become:

-M,D � = 1 % def_,F,`g�6hxy,F,`0 (8.23)

9M,D � = ]%\�,>,T 1$b,r,>,T (8.24)


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where: b7t,M,D = \>,T �[,r,>,T[>,T (8.25)

Thus, b7t,M,D will be designed by solving the following optimization problem derived

from (8.9):

∆IJ,D j ]%\�,>,TXK�1$b,r,>,TXK�j $ 0.5 j1 % ]%\�,>,TXK�

1$b,r,>,TXK�j < 1∀K (8.26)

By comparing the set of equations (8.13), (8.14) and (8.15) to the set of (8.23), (8.24) and (8.26), we realize that there is an equivalency between DOB design and PI-SP design. This is stated as the design duality between DOB and PI-SP as per Table VIII.3 below.

Table VIII.3: Duality between PI-SP and DOB design

Function DOB PI-SP Sensitivity function

-M,D � = 1 % def_,F,`g�6z{,|,}0

-M,D � = 1 % def_,F,`g�6z~�,|,}0

Complementary sensitivity function

9M,D � = def_,F,`g�6hi,F,`0

9M,D � = def_,F,`g�6hxy,F,`0

We perceive that both DOB and PI-SP possess the same sensitivity and complementary sensitivity functions except the tuning parameter. Moreover, the robust performance criterion is also the same (8.15 and 8.26). Thus, thanks to this duality feature, we can just design one of the two sub controllers and deduce the other one in a straightforward way. We can therefore move from DOB to PI-SP and vice versa. As in each channel T the DOB is acting in the inner closed-loop while the PI-SP is involved in the outer closed-loop, the inner closed-loop should be faster than the outer closed-loop. If we take a typical ratio of two, the DOB closed-loop will be set to be two times faster than the PI-SP closed-loop. This is expressed by the following relationship in terms of time constants: b7t,M,D = 2bc,M,D (8.27)

In conclusion, by exploiting the duality between DOB and PI-SP the design of the controllers is much simplified. The time constant bc,M,D of the DOB filter will be first

designed on the basis of Equations (8.13), (8.14) and (8.15). Next, the proportional gain [7t,M,D of the PI-SP is deduced on the basis of (8.27) and (8.25). This procedure is

applied in the following subsection.


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VIII.2.3 Robust design of DOB and PI-SP

By applying the design procedure described in the previous subsection, we find the tuning parameters given in Table VIII.4. The corresponding design curves are depicted in Figure VIII.9.

Table VIII.4: Tuning parameter of the controllers

Parameter First closed -loop Second closed-loop

Third closed -loop

bc,M,D 0.0450�ℎ�

0.0500�ℎ� 0.0038�ℎ� [7t,M,D %2.652310��/ℎ� 4.3000�@�/� %8.607010��@�/ℎ�

As a validation, the DOB capability to reconstruct the disturbance is shown in Figures VIII.10, VIII.11 and VIII.12 for the first, second and third closed-loop, respectively. The external disturbances are injected according to variations provided in Table VIII.2. We realize that in the nominal case, the time evolution of the estimated total disturbance is close to that of the total disturbance despite the presence of a time delay. Indeed, as the system time delay is intentionally introduced within the DOB structure, the estimated disturbance should be delayed with respect to the disturbance itself. As already stated and justified by Equation (8.17), this yields an estimated disturbance which is in phase opposition with regard to the real disturbance on product flow rate (Figure VIII.11). To gain insight the comparison between the disturbance and its estimate, and moreover to highlight the phase effect, Bode’s diagrams of the

transfer functions aM,D �]�^_,F,`0T = 1,2,3� which link (Uk,D �to (k,D � are depicted

in Figures VIII.13, VIII.14 and VIII.15. The phase shift does clearly appear in each Bode’s plot and particularly at frequency of sinusoidal disturbance, i.e. 10� !�/ℎ�. In the mismatch case (Figures VIII.10, VIII.11 and VIII.12), the recorded discrepancy between the disturbance and its estimate is due to uncertainty bounds considered on model parameters. However, when the DOB will be associated with the PI-SP to form the DIMC scheme, these errors between the disturbance and its estimate will be handled by the PI-SP in the outer closed-loop. The PI-SP capability to reach the optimal controlled variable is attested by Figures VIII.16, VIII.17 and VIII.18 for the three channels. Step setpoint changes occurred at time = 1�ℎ�correspond to those provided in Table VIII.1. Two performance indices, namely the overshoot and the Integral of Absolute Error (IAE) are employed to evaluate the control performance. While there is no overshoot, IAE values are given in Table VIII.5 for both nominal and uncertain cases. We recall here that the uncertain or mismatch model corresponds to the three bounding transfer functions (Table A-VII.2.2).

Table VIII.5: Performance indices in step setpoint changes

Performance indices

Product fineness Product flow rate

Product density

IAE (nominal case) 0.0205�ℎ� 2.8934�� 0.0020�ℎ� IAE (mismatch case) 0.0800�ℎ� 2.8938�� 0.0014�ℎ�


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a)

b)

c)

Figure VIII.9: Design of DOB and PI-SP a) First channel; b) Second channel; c) Third channel

10-3

10-2

10-1

100

101

102

103

0.2

0.4

0.6

0.8

1

1.2

Pulsation [rad/h]

Rob

ust pe

rfor

man

ce c

riter

ion

[p.u

]

Limit of the robust performence criterion

DOB for 1st closed-loop

PI-SP for 1st closed-loop

10-3

10-2

10-1

100

101

102

103

0

0.2

0.4

0.6

0.8

1

1.2

Pulsation [rad/h]

Rob

ust pe

rfor

man

ce c

riter

ion

[p.u

]


DOB for 2nd closed-loop

PI-SP for 2nd closed-loop

10-3

10-2

10-1

100

101

102

103

0

0.2

0.4

0.6

0.8

1

1.2

Pulsation [rad/h]

Rob

ust pe

rfor

man

ce c

riter

ion

[p.u

]


DOB for 3rd closed-loop

PI-SP for 3rd closed-loop


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Figure VIII.10: Reconstruction of lumped disturbance on the product

fineness for both nominal and uncertain models


flow rate for both nominal and uncertain models


density for both nominal and uncertain models

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Time [h]

Dis

turb

ance

on

prod

uct fin

enes

s [p

.u]

Disturbance

Estimate disturbance for nominal case

Estimate disturbance for mismatch case

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-0.5

0

0.5

1

1.5

2

2.5

3

Time [h]

Dis

turb

ance

on

prod

uct flo

w rat

e [t/h

]

Disturbance



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-5

0

5

10

15

20x 10

-3

Time [h]

Dis

turb

ance

on

prod

uct de

nsity

[p.

u]

Disturbance




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Figure VIII.13: Bode’s diagram of transfer function between the total lumped disturbance and its estimate for the channel 1



-15

-10

-5

0

Mag

nitu

de (dB

)

100

101

102

-3600

-2880

-2160

-1440

-720

0

-15

-10

-5

0

Mag

nitu

de (dB

)

100

101

102

-1440

-1080

-720

-360

0

-0.8

-0.6

-0.4

-0.2

0

100

101

102

-225

-180

-135

-90

-45

0

Frequency [rad/h]

Frequency [rad/h]

Frequency [rad/h]

Mag

nitu

de [d

B]

Pha

se [°

] M

agni

tude

[dB

] P

hase

[°]

Mag

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B]

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]


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a) b)

Figure VIII.16: Step response of product fineness a) Fresh ore feed rate

b) Product fineness

a) b)

Figure VIII.17: Step response of product flow rate a) Rod mill feed water flow rate

b) Product flow rate

a) b)

Figure VIII.18: Step response of product density a) Dilution water flow rate;

b) Product density

0 2 4 6 8 10 12 14 16 1834.3

34.35

34.4

34.45

34.5

34.55

34.6

34.65

34.7

34.75

Time [h]

Fre

sh o

re fee

d ra

te [t/h]

Fresh ore feed rate for nominal case

Fresh ore feed rate for mismatch case

0 2 4 6 8 10 12 14 16 180.791

0.792

0.793

0.794

0.795

0.796

0.797

0.798

0.799

0.8

0.801

Time [h]

Pro

duct

fin

enes

s [p

.u]

Setpoint

Response for nominal case

Response for mismatch case

0 5 10 15 20 25 30 3543.6

43.8

44

44.2

44.4

44.6

44.8

45

45.2

Time [h]

Rod

mill fee

d wat

er flo

w rat

e [m

3 /h]

Rod mill feed water flow rate for nominal case

Rod mill feed water flow rate for mismatch case

0 5 10 15 20 25 30 3576.6

76.7

76.8

76.9

77

77.1

77.2

77.3

77.4

77.5

Time [h]

Pro

duct

flo

w rat

e [t/h

]

Setpoint



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

4

4.5

5

5.5

6

6.5

Time [h]

Dilu

tion

wat

er flo

w rat

e [m

3 /h]

Dilution water flow rate for nominal case

Dilution water flow rate for mismatch case

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51.325

1.33

1.335

1.34

Time [h]

Pro

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den

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[p.

u]

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In the mismatch case, each controller tries to increase its action in order to overcome the effects of model mismatch on the controlled variable. This can result in a small IAE exhibited with respect to the nominal case for the product density whose uncertainty only affects the steady-state gain. We should point out that these linear simulations for validation purpose of controller design have been performed separately on each closed-loop. In the next section, our design is applied to the whole KZC grinding process. First, the required instrumentation is presented prior to demonstrate the ability of the designed DIMC on both the nominal and the mismatch process models.

VIII.3 Application to KZC grinding process VIII.3.1 Instrumentation of the KZC grinding circui t

Figure VIII.19: KZC wet grinding circuit with its instrumentation-ISA norm Caption: * Actuators - VSD: Variable Speed Driver - FV: Flow control Valve * Sensors - FI: Flow Indicator - PSA: Particle Size Analyser - WI: Weight Indicator - LI: Level Indicator - DI: Density Indicator - PI: Pressure Indicator

FI 01

WI 04

PSA 02

FI 03

PSA 05

PSA 06

PSA 22

FI 08

DI 09

FI 10

PI 11

PSA 12

DI 23

FI 24 PSA

25

DI 15

PSA 13

FI 14

LI 07

PSA 20

FI 19

DI 18

PSA 17

WI 16

WI 21

VSD 1

FV 2

FV 3


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The necessity of instrumentation for grinding processes has been stated in Section VI.2. By means of the standard norm from the ISA, Figure VIII.19 above shows without details a typical instrumentation which can be installed on the KZC wet grinding circuit before dealing with its control. It should be noted that for the control of the overall KZC grinding process, only three sensors and three actuators are required. They are respectively associated to the controlled variables and the manipulated variables as follows:

- Sensors

PSA 25: Product particle size or fineness � . �� FI 24: Product flow rate �/ℎ� DI 23: Product density � . ��

- Actuators

VSD 1: Fresh ore feed rate �/ℎ� FV 2: Rod mill feed water flow rate �@�/ℎ� FV 3: Dilution water flow rate �@�/ℎ� It is assumed that the required instrumentation is installed on the process when the design procedure is applied to the KZC grinding circuit in the next two subsections. VIII.3.2 Simulation on the linearized model

The scenario of the simulation is described as follows. We start in manual control with �� = 34.7501�/ℎ�, �� = 43.8116�@�/ℎ� and �� = 3.9906�@�/ℎ� yielding �� = 0.7920� . ��, �� = 76.6988�/ℎ� and �� = 1.3390�@�/ℎ�. At time

= 2�ℎ�, we switch from manual to automatic control such that we remain at the same

state. At time = 4�ℎ�, we perform step changes of setpoints in order to reach the

optimal values of controlled variables. External disturbances ��1/ℎ� and �� . �� are

injected according to Table VIII.2 at times = 48�ℎ� and = 72�ℎ�, respectively. Figures VIII.20, VIII.21 and VIII.22 show the time evolutions of manipulated and controlled variables for channels 1, 2 and 3, respectively. The oscillations and overshoots observed in transient behaviour of controlled variables are due to coupling effects which are not completely suppressed by the DOB as already indicated. Indeed, the amount of these coupling effects on each controlled variable has been illustrated by DNA with Gershgorin’s bands in Figure VII.10 and time evolutions of Figures VII.4, VII.5 and VII.6. Time evolutions of lumped disturbance and its estimates are similarly depicted in Figures VIII.23, VIII.24 and VIII.25. This simulation has been simultaneously performed on both nominal and uncertain models. The time evolution of each estimated disturbance is close to that of the disturbance itself except for the presence of the corresponding time delay and the possible reduction of the magnitude of the estimated sinusoidal disturbance due to the DOB low-pass filter.


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We realize that the designed controllers perform well for both setpoint reaching and disturbance rejection and in both nominal and mismatch cases. To gain insight the performance of the designed control scheme, we can analyse separately setpoint reaching and disturbance suppression. For setpoint reaching, Table VIII.6 provides values of overshoot and IAE as performance indices. The maximum overshoot is 75.30% occurring on the product

flow rate and the maximum IAE is 6.4591�/ℎ� also occurring on the product flow rate.

Observing the simulated curves from time = 0 to 48�ℎ�, we perceive that the robust DIMC performs well in both nominal model and mismatch model cases. The three robust PI-SP allow the control system to reach the optimal controlled variables while the three robust DOB exhibit good estimation of lumped disturbance which consists in this case of coupling effects and model mismatches. But the three DOB do not suppress completely dynamic coupling effects since the estimate disturbance is dynamically delayed with respect to the disturbance. The dynamic couplings between variables are attenuated before being totally suppressed in the steady-state. Table VIII.6: Performance indices in step setpoint changes for all the control system -

linear simulation

Performance indices

Product fineness Product flow rate

Product density

Overshoot (nominal case)

24.79�%� 75.30�%� 1.00�%� Overshoot

(mismatch case) 14.19�%� 54.35�%� 1.40�%�

IAE (nominal case) 0.0337� . �� 6.4591�/ℎ� 0.0047� . �� IAE (mismatch case) 0.0241� . �� 3.6666�/ℎ� 0.0042� . ��

a) b)

Figure VIII.20: Time evolutions of fresh ore feed rate (a) and product fineness (b) for both nominal and mismatch cases – linear simulation

0 10 20 30 40 50 60 70 80 90 10033.5

34

34.5

35

35.5

36

36.5

37

Time [h]

Fre

sh o

re fee

d ra

te [t/h]

Nominal case

Mismatch case

0 10 20 30 40 50 60 70 80 90 100

0.76

0.78

0.8

0.82

0.84

0.86

Time [h]

Pro

duct

fin

enes

s [p

.u]

Setpoint




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a) b) Figure VIII.21: Time evolutions of rod mill feed water flow rate (a) and product flow rate

(b) for both nominal and mismatch cases – linear simulation

a) b) Figure VIII.22: Time evolutions of dilution water flow rate (a) and product density (b) for

both nominal and mismatch cases – linear simulation

Figure VIII.23: Estimation of lumped disturbance on the product fineness for both

nominal and mismatch cases – linear simulation

0 10 20 30 40 50 60 70 80 90 10010

15

20

25

30

35

40

45

50

Time [h]

Rod

mill fee

d wat

er flo

w rat

e [m

3/h]

Nominal case

Mismatch case

0 10 20 30 40 50 60 70 80 90 100

75

76

77

78

79

80

81

82

83

Time [h]

Pro

duct

flo

w rat

e [t/h

]

Setpoint



0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

35

Time [h]

Dilu

tion

wat

er flo

w rat

e [m

3/h]

Nominal case

Mismatch case

0 10 20 30 40 50 60 70 80 90 1001.322

1.324

1.326

1.328

1.33

1.332

1.334

1.336

1.338

1.34

Time [h]

Pro

duct

den

sity

[p.

u]

Setpoint



0 10 20 30 40 50 60 70 80 90 100-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Time [h]

Dis

turb

ance

on

the

prod

uct fin

enes

s [p

.u]

Disturbance for nominal case

Estimate disturbance for nominal caseDisturbance for mismatch case



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Figure VIII.24: Estimation of lumped disturbance on the product flow rate for both

nominal and mismatch cases – linear simulation

Figure VIII.25: Estimation of lumped disturbance on the product density for both

nominal and mismatch cases – linear simulation The magnitude of sinusoidal disturbance may not be accurately estimated, as depicted particularly in Figure VIII.23, because of the DOB low-pass filter.

For disturbance suppression, we can observe the simulated curves from time = 48 to 96�ℎ�. We analyse here the effects of external disturbances on the controlled variables. As for coupling effects in steady-state, we note that step variations of external disturbances are totally suppressed by the DOB from the controlled variables. For the continuous sinusoidal disturbances, the controlled variables exhibit also sinusoidal fluctuating around the optimal values since the simulations are performed with the linearized model. In the nominal case, Table VIII.7 provides gains of frequency response between each external disturbance and each controlled variable at the frequency of 10� !�/ℎ� corresponding to that of the sinusoidal external disturbances. Similarly, Table VIII.8 provides those of the mismatch case.

0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

20

25

30

Time [h]

Dis

turb

ance

on

the

prod

uctio

n flo

w rat

e [t/h

]



Disturbance for mismatch case


0 10 20 30 40 50 60 70 80 90 100-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Time [h]

Dis

turb

ance

on

the

prod

uct de

nsity

[p.

u]



Disturbance for mismatch case



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Table VIII.7: Gains of frequency response between external disturbances and controlled variables at frequency equal to 10� !�/ℎ� in the nominal case

�� . �� /ℎ� �� . �� 1/ℎ� 0.0296�ℎ� 0.7485�� 0.0005�ℎ� �� . �� 0.1989� . �� 4.8462�/ℎ� 0.0047� . ��

Table VIII.8: Gains of frequency response between external disturbances and controlled variables at frequency equal to 10� !�/ℎ� in the mismatch case

�� . �� /ℎ� �� . �� 1/ℎ� 0.0424�ℎ� 0.7185�� 0.0006�ℎ� �� . �� 0.2814� . �� 4.1010�/ℎ� 0.0046� . ��

Thanks to the DOB and the PI-SP, the notable reduction ratio of frequency response gains obtained with respect to the frequency response gains in open-loop without DOB and PI-SP at the frequency of 10� !�/ℎ� is given in Tables VIII.9 and VIII.10, respectively in the nominal case and the mismatch case.

Table VIII.9: Reduction ratio of gains of frequency response at frequency of 10� !�/ℎ� obtained thanks to the DOB and the PI-SP in the nominal case

�� . �� /ℎ� �� . �� 1/ℎ� 55.53�%� 78.12�%� 13.32�%� �� . �� 77.57�%� 97.67�%� 18.80�%�

Table VIII.10: Reduction ratio of gains of frequency response at frequency of 10� !�/ℎ� obtained thanks to the DOB and the PI-SP in the mismatch case

�� . �� /ℎ� �� . �� 1/ℎ� 79.55�%� 74.99�%� 15.99�%� �� . �� 94.15�%� 82.65�%� 18.40�%�

VIII.3.3 Progressive implementation

During our experiments on the field we perceived that the KZC grinding circuit has no instrumentation, especially sensor, installed. For control purpose, the instrumentation described in Subsection VIII.3.1 is required. We propose here a progressive implementation when all the instrumentation is not available. If the plant


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management can only acquire sensor and actuator for only one closed-loop, we would suggest getting that of the closed-loop or channel 1 since the product fineness is the most important controlled variable. If instrumentation of two closed-loops might be acquired, one will add the closed-loop or channel 3 since the product density is the second product quality characteristic. It is obvious to regard it as the next important controlled variable. Hence, the product flow rate will be considered in the last position. Indeed, the product flow rate is supposed to be maximized after complying with the product quality requirements. We have therefore established a prioritization of the closed-loops in this order:

(1) first priority: first closed-loop ��, ��; (2) second priority: third closed-loop ��, ��; (3) last priority: second closed-loop ��, ��.

In the following lines, we present results for respectively one closed-loop and two closed-loops. The emphasis is made on the benefit we can gain in these configurations where instrumentation is not complete.

One closed-loop configuration

We repeat the same scenario explained in Subsection VIII.3.2 but we close only the first loop while the other loops are kept in manual control mode by means of optimal manipulated variables ��,�0k and ��,�0k (Table VII.2). The simulation results

are shown in Figures VIII.26, VIII.27 and VIII.28.

a) b)

Figure VIII.26: Time evolutions of fresh ore feed rate (a) and product fineness (b) for both nominal and mismatch cases in one closed-loop configuration – linear simulation

0 10 20 30 40 50 60 70 80 90 10034

34.5

35

35.5

36

36.5

37

Time [h]

Fre

sh o

re fee

d ra

te [t/h]

Nominal case

Mismatch case

0 10 20 30 40 50 60 70 80 90 100

0.76

0.78

0.8

0.82

0.84

0.86

Time [h]

Pro

duct

fin

enes

s [p

.u]

Setpoint




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a) b) Figure VIII.27: Time evolutions of product flow rate (a) and product density (b) for both

nominal and mismatch cases in one closed-loop configuration – linear simulation

Figure VIII.28: Estimation of lumped disturbance on the product fineness for both nominal and mismatch cases in one closed-loop configuration – linear simulation We remark that the product fineness reaches its optimal value and the product density does not reach its optimal value but lies in its prescribed range set between 1.3 and

1.4. The product flow rate also does not reach its optimal value. Hence, we can conclude that in this configuration only the product quality is guaranteed.

The reduction of frequency response gains with regard to the open-loop without DOB and PI-SP at the frequency of 10� !�/ℎ� is given in Tables VIII.11 and VIII.12, respectively in the nominal and the mismatch cases.

0 10 20 30 40 50 60 70 80 90 10075.5

76

76.5

77

77.5

78

78.5

79

79.5

80

Time [h]

Pro

duct

flo

w rat

e [t/h

]

Setpoint



0 10 20 30 40 50 60 70 80 90 1001.325

1.33

1.335

1.34

1.345

1.35

1.355

1.36

Time [h]

Pro

duct

den

sity

[p.

u]

Setpoint



0 10 20 30 40 50 60 70 80 90 100-0.02

0

0.02

0.04

0.06

0.08

0.1

Time [h]

Dis

turb

ance

on

the

prod

uct fin

enes

s [p

.u]



Disturbance for mismatch caseEstimate disturbance for mismatch case


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Table VIII.11: Reduction ratio of gains of frequency response at frequency of 10� !�/ℎ� obtained thanks to the DOB and the PI-SP in the nominal case for one closed-loop configuration

�� . �� /ℎ� �� . �� 1/ℎ� 56.29�%� 46.11�%� 58.62�%� �� . �� 66.38�%� 49.07�%� 68.00�%�

Table VIII.12: Reduction ratio of gains of frequency response at frequency of 10� !�/ℎ� obtained thanks to the DOB and the PI-SP in the mismatch case for one closed-loop configuration

�� . �� /ℎ� �� . �� 1/ℎ� 79.36�%� 53.56�%� 66.61�%� �� . �� 94.15�%� 53.00�%� 76.40�%�

Two closed-loop configuration

We apply the same simulation scenario but only the second loop is kept in manual control mode. The corresponding results are depicted in Figures VIII.29, VIII.30, VIII.31 and VIII.32. We note that both product fineness and product density reach their optimal values while the product flow rate fails to reach its. As in the previous configuration, we conclude that only the product quality is guaranteed.

a) b)

Figure VIII.29: Time evolutions of fresh ore feed rate (a) and product fineness (b) for both nominal and mismatch cases in two closed-loop configuration – linear simulation

0 10 20 30 40 50 60 70 80 90 10034

34.5

35

35.5

36

36.5

37

Time [h]

Fre

sh o

re fee

d ra

te [t/h]

Nominal case

Mismatch case

0 10 20 30 40 50 60 70 80 90 100

0.76

0.78

0.8

0.82

0.84

0.86

Time [h]

Pro

duct

fin

enes

s [p

.u]

Setpoint

Response for nominal caseResponse for mismatch case


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a) b) Figure VIII.30: Time evolutions of dilution water flow rate (a) and product density (b) for both nominal and mismatch cases in two closed-loop configuration – linear simulation

a) b)

Figure VIII.31: Estimation of lumped disturbance on the product fineness (b) and on the product density (b) for both nominal and mismatch cases in two closed-loop configuration –linear simulation

Figure VIII.32: Time evolution of product flow rate for both nominal and mismatch

cases in two closed-loop configuration – linear simulation

0 10 20 30 40 50 60 70 80 90 100

4

5

6

7

8

9

10

Time [h]

Dilu

tion

wat

er flo

w rat

e [m

3/h]

Nominal case

Mismatch case

0 10 20 30 40 50 60 70 80 90 1001.32

1.322

1.324

1.326

1.328

1.33

1.332

1.334

1.336

1.338

1.34

Time [h]

Pro

duct

den

sity

[p.

u]

Setpoint



0 10 20 30 40 50 60 70 80 90 100-0.02

0

0.02

0.04

0.06

0.08

0.1

Time [h]

Dis

turb

ance

on

the

prod

uct fin

enes

s [p

.u]




0 10 20 30 40 50 60 70 80 90 100-0.01

-0.005

0

0.005

0.01

0.015

0.02

Time [h]

Dis

turb

ance

on

the

prod

uct de

nsity

[p.

u]




0 10 20 30 40 50 60 70 80 90 100

76

77

78

79

80

81

82

83

84

Time [h]

Pro

duct

flo

w rat

e [t/h

]

Setpoint




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The reduction ratio of gains of frequency response with respect to the open-loop without DOB and PI-SP at the frequency of 10� !�/ℎ� is given in Tables VIII.13 and VIII.14, respectively in the nominal and the mismatch cases.

Table VIII.13: Reduction ratio of gains of frequency response at frequency of 10� !�/ℎ� obtained thanks to the DOB and the PI-SP in the nominal case for two closed-loop configuration

�� . �� /ℎ� �� . �� 1/ℎ� 56.47�%� 77.54�%� 23.98�%� �� . �� 66.26�%� 96.99�%� 27.20�%�

Table VIII.14: Reduction ratio of gains of frequency response at frequency of 10� !�/ℎ� obtained thanks to the DOB and the PI-SP in the mismatch case for two closed-loop configuration

�� . �� /ℎ� �� . �� 1/ℎ� 79.92�%� 87.30�%� 21.32�%� �� . �� 94.27�%� 94.19�%� 24.00�%�

Conclusion In this chapter, the control scheme has been selected and the related controller has been designed such that the controlled variables are kept at their optimal values despite coupling effects, large time delays, strong external disturbances and model parameter uncertainties. To deal with all these issues, a DIMC structure has been proposed in a feedback-feedforward scheme with a PI-SP and a DOB. By comparing these two components of the controller, we perceived a duality relationship between them. This duality has been exploited to drastically simplify the robust controller design. DOB has been designed first on the basis of a robust performance criterion and PI-SP has been directly deduced thanks to the duality feature. The designed controller has been validated by simulation on both the nominal model and the mismatch model. If all the required instrumentation is not available, we have suggested a progressive implementation of the control scheme based on a prioritization strategy of closed-loops. The priority has been established as follows: product fineness (closed-loop 1), product density (closed-loop 3) and product flow rate (closed-loop 2). It has been demonstrated that we can reach both product particle size and product density requirements by means of either one closed-loop configuration or two-closed loop configuration. Thus, the minimum instrumentation required to satisfy the product quality requirements is just one PSA and one VSD. Obviously, if one wants to reach all the optimal values of controlled variables, all the three loops must be closed and one FI, one DI and two FV must supplement the instrumentation for the second and the third closed-loop. In conclusion, the designed robust PI-SP and DOB perform well on both nominal and mismatch models.

Chapter I X: CONCLUSIONS 2013

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Chapter IX: CONCLUSIONS AND PERSPECTIVES

The size reduction process is really the critical step of a concentrator plant. In fact, grinding process accounts for almost 50 % of the total expenditure of the concentrator plant and the product particle size from this stage influences the recovery rate of the valuable minerals as well as the volume of tailing discharge in the subsequent process. Furthermore, due to the requirements on the plant production and the product features, an industrial grinding circuit usually offers complex and challenging control problems. In the current thesis, we have treated the problem of identification and control of the Kolwezi concentrator (KZC) wet grinding process. To this end, our systematic methodology has first consisted in modelling and simulation of the process under study. Next, a suitable control scheme has been proposed and the related controller designed. In the following section, we present our conclusions.

IX.1 Conclusions

IX.1.1 Modelling and simulation

In order to understand the behaviour of the KZC grinding process, we needed to model it. First, the circuit flowsheet has been described in Chapter II. Compared to the typical grinding process which just possesses a ball mill in closed-loop with a hydrocyclone classifier, the KZC wet grinding circuit has a more complex topology. Indeed, this circuit consists mainly of one rod-mill in an open-loop and two ball mills, one pump sump, one distributor and two hydrocyclone classifiers in a double closed-loop. From a state of the art for grinding processes modelling presented in Chapter III, we have developed and extended to wet operating condition and to mineral processing the complex model used in dry cement grinding. We used a phenomenological approach combining population mass balance and pulp dynamics transport for rod mill and ball mill modelling. This led us to nonlinear partial differential equations (PDE) containing six parameters for each mill.

The phenomenological approach does not consider the material to be broken as a whole but as a population of fragments varying in size. This introduces the concept of particle size classes or intervals. Grinding a given size class is characterized by the disappearance or death of some of the fragments belonging to the broken class and by the creation or birth of smaller fragments going to be added to other classes. Studying the fragmentation process is therefore subject to two fundamental functions, i.e. the selection function and the breakage or repartition function. While the latter is almost the same for both dry grinding and wet grinding, we have proposed a mathematical description for the selection function which is higher in wet grinding than in dry grinding. In addition, a mathematical relationship which describes the dependence of the convective velocity with the feed ore rate has been suggested. A two-parameter steady-state model has been found sufficient to describe


144

the hydrocyclone classifiers whereas the pump sump has been modelled by a perfect mixer and the other components by time delay elements.

The KZC wet grinding process has been considered as a 3x3 Multi-Input Multi-Output (MIMO) dynamical system whose product particle size, product flow rate and product density are regarded as controlled variables while fresh ore feed rate, rod mill feed water and dilution water flow rate are the three manipulated variables. In Chapter V, we have identified the model parameters and built simulators for our process. All model parameters have been determined from experimental data by using the nonlinear least squares algorithm. An analytic solution of the steady-state model has been derived in both open-loop and double closed-loop configurations and exploited to estimate the grinding model parameters. Due to the lack of sensors on the process, we could not perform on-line measurements of the variables in real-time. Hence we could not use a colorant neither a radioactive tracer to evaluate the transport dynamics of material within the mills. Therefore we have developed an alternative approach on the basis of the G41 foaming as tracer. This new experimental procedure has provided satisfactory results in comparison with previous studies. Similarly, Tyler series sieves have been employed for measurements of particle size distributions while pulp density has been deduced from weight measurements by means of the Marcy balance.

For global simulation, all the individual models have been connected according to the circuit configuration provided by the flowsheet of the installation. Both a steady-state simulator and dynamical simulator based on three classes of particle sizes have been developed within the MATLAB/Simulink software. The resulting steady-state simulator is in good agreement with the recorded data, and the dynamic simulator exhibits the expected qualitative behaviour. Both tools have been used to train some operators. Finally, the dynamical simulator has been analysed in order to highlight the steady-state and dynamic features of the process.

IX.1.2 Control

The state of the art for wet grinding processes control has been presented in Chapter VI. The main control schemes applied to the typical grinding process have been described. In Chapter VII, the system model has been analysed from a control point of view. The main features of the KZC grinding plant seen as the main control issues are as follows: strong coupling effects, model mismatches regarded as internal disturbances, strong external disturbances such as ore grindability (inverse of ore hardness) and feed ore particle size distribution, large time delays and model parameter uncertainties. These uncertainties are essentially due to operating point changes since the system model has been linearized around the optimal operating point. Indeed, to apply linear control tools, a two-step approach has been used to determine the operating point around which the system model has been linearized. On the basis of a simulation study, an initial estimate of the optimal operating point has been first determined. Hence, a linearized model of the process has been derived around this initial operating point. Next, the initial operating point has been corrected by solution of a convex optimization problem. Assuming that the linearized model remains valid for the considered correction range, the optimization problem consisted in maximizing the product flow rate under specific constraints on product quality and


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actuator limitations. The optimal values of controlled variables have then been used as references or setpoints in our control strategy.

We have proposed to explicitly include in our control study the control of the product density which is usually neglected in the literature. Indeed, the density of the pulp from the grinding stage influences significantly the performance of the subsequent stage. To deal with all the control issues, a Double Internal Model Control (DIMC) has been developed in Chapter VIII as the most suitable decentralized control scheme for the KZC wet grinding process. This linear control strategy is based on the variable pairing selected in Chapter VII by means of both steady-state and dynamic criteria. The control goal consists in holding the process on the optimal operating point despite the internal and external disturbances. The proposed DIMC structure is composed of two parts, namely a Proportional-Integral (PI) Smith Predictor (SP) and a Disturbance Observer (DOB) on each channel of the paired variables. A duality between the PI-SP and the DOB has been demonstrated from a design point of view. The �� robust performance criterion has been used for robust design to allow the controller to perform properly on the actual plant whose model is uncertain. The validation of both control scheme and controller design has been made on the basis of simulation results from the linearized model. To show the benefit of the control if the plant does not get the required instrumentation, we have finally studied a possible progressive implementation of the control structure in the framework of the KZC installation.

Next, we suggest some perspectives as future work directions.

IX.2 Future research directions

IX.2.1 Modelling and simulation

Nowadays, the trend is the replacement of rod mills by autogeneous (AG) or semi-autogeneous (SAG) mills in wet grinding processes. An interesting challenge should be to study if the model developed in this work can be extended to AG/SAG grinding mills which are usually greater in size and are known to yield a higher grinding efficiency than rod mills. Does the modelling approach developed in our dissertation suit for this kind of grinding mills ? This question will probably lead to a kind of generalization of wet grinding mill modelling.

In addition, deriving a model for a process on which all the required instrumentation is installed should be another challenging problem. Here the sub objectives can be oriented but not limited to:

- develop a full model with more than three particle size classes; - estimate the parameters on the basis of on-line measurements to confirm the

model structure and the prediction capability of the estimated parameters; - validate the transient behaviour of the dynamic simulator by comparing

simulation results to on-line measurements; - confirm the experimental methodology based on the use of the G41 foaming as

a tracer by comparing results from this method with those from the use of a conventional tracer (colorant or radioactive) in the same operating conditions.


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IX.2.2 Control

For control of grinding processes, future research works could be carried out in the research directions proposed in the following lines. First, the proposed decentralized control could be extended to a MIMO DIMC in a full multivariable control. This should lead to a MIMO DOB and a MIMO PI-SP. Is the design duality feature demonstrated in this thesis valid in the case of a such MIMO controller ? Results and conclusions of this future research will probably open other interesting ways to better understand the duality between the modified DOB and the PI-SP in robust design. Secondly, we started but did not complete the application of Model Predictive Control (MPC). In the linear case we would like to suggest that this powerful control strategy be applied to grinding processes with considerations of uncertainty on time delays. If these uncertainties are properly characterized, Robust MPC (RMPC) design might be performed and applied to the grinding plants. Finally, we expect that general control study will be carried out on the real nonlinear grinding processes. A Nonlinear MPC (NMPC) strategy might be able to overcome all control issues within the complete range of operating points or workspace of a whole nonlinear wet grinding circuit such as the KZC one. As a final conclusion, we realize that challenging and exciting problems still remain in the framework of identification and control of wet grinding processes. The contributions and results from this thesis are not absolute but may be criticised and used later on for further research. We therefore pass on the baton to other researchers with the hope of promoting research activities in this field.

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