- Email: [email protected]
A Combined MPC for Milling and Flotation – A Simulation Study
18th IFAC Symposium on Control, Optimization and Automation in 18th IFAC Symposium on Control, Optimization and Automation in Mining, Mineral and Metal Processing 18th IFAC Symposium on Optimization and in 18th IFAC Symposium on Control, Control, Optimization and Automation Automation in Available online at www.sciencedirect.com Mining, Mineral and Metal Processing Stellenbosch, South Africa, AugustOptimization 28-30, 2019 and Automation in 18th IFAC Symposium on Control, Mining, Mineral and Metal Processing Mining, Mineral and Metal Processing 18th IFAC Symposium on Control, Optimization and Automation in Stellenbosch, South Africa, August 28-30, 2019 Mining, MineralSouth and Metal Processing Stellenbosch, Africa, August Stellenbosch, Africa, August 28-30, 28-30, 2019 2019 Mining, MineralSouth and Metal Processing Stellenbosch, South Africa, August 28-30, 2019 Stellenbosch, South Africa, August 28-30, 2019
ScienceDirect
IFAC PapersOnLine 52-14 (2019) 24–29 A Combined MPC for Milling and Flotation – A Simulation Study A Combined MPC for Milling and Flotation –– A Simulation Study A Combined MPC for Milling and Flotation A Simulation Study A Combined MPC for Milling and Flotation – A Simulation Study A Combined MPC for Milling and Flotation – A Simulation Study A CombinedKevin MPC for Milling and Flotation – A Simulation Study Brooks*, Merinda Westcott* and Margret Bauer**
Kevin Brooks*, Merinda Westcott* and Margret Bauer** Kevin Kevin Brooks*, Brooks*, Merinda Merinda Westcott* Westcott* and and Margret Margret Bauer** Bauer** Kevin Brooks*, Merinda Westcott* and Margret Bauer** Kevin Brooks*, Merinda Westcott* and Margret Bauer** *BluESP, 53 Platina St, Randburg, South Africa *BluESP, 53 Platina Platina St, Randburg, South Africa Africa (E-mail: [email protected], [email protected] ) *BluESP, 53 Randburg, South *BluESP, 53 Platina St, St, Randburg, South Africa (E-mail: [email protected], [email protected] ) *BluESP, 53 Platina St, Randburg, South Africa **Department of Electrical, Electronic and Computer Engineering, (E-mail: [email protected], [email protected] ) (E-mail: [email protected], [email protected] ) *BluESP, 53 Platina St, Randburg, South Africa **Department of Electrical, Electronic and Computer Engineering, (E-mail: [email protected], [email protected] ) University ofof Pretoria, Lynwood Road, Pretoria, South Africa **Department Electrical, Electronic and Computer Engineering, **Department of Electrical, Electronic and Computer Engineering, (E-mail: [email protected], [email protected] ) University of Pretoria, Lynwood Road, Pretoria, South Africa **Department of Electrical, Electronic and Computer Engineering, (E-mail: [email protected] ) Africa University of ofofPretoria, Pretoria, Lynwood Road, Pretoria, South Africa University Lynwood Road, Pretoria, South **Department Electrical, Electronic and Computer Engineering, (E-mail: [email protected] )) Africa University of Pretoria, Lynwood Road, Pretoria, South (E-mail: [email protected] (E-mail: [email protected] ) Africa University of Pretoria, Lynwood Road, Pretoria, South (E-mail: [email protected] ) (E-mail: [email protected] )
Abstract: In the metals and mining industry, an ore-milling step followed by froth flotation is a very common Abstract: In the metals industry, an ore-milling step flotation is a very common processing It is and alsomining common that advanced controls offollowed one formby orfroth another are installed on these Abstract: In the and mining industry, an step followed by froth flotation is Abstract: Inroute. the metals metals and mining industry, an ore-milling ore-milling step followed byor froth flotation is aa very very common common processing route. It is also common that advanced controls of one form another are installed on these Abstract: In thecommon metals and mining industry, an ore-milling step followed byor froth flotation is apaper very common units. It is less for the advanced controllers to be combined in any formal way. This explores processing route. It is also common that advanced controls of one form another are installed on these processing route. It is and also common thatcontrollers advanced controls offollowed one in form another areThis installed on these Abstract: In thecommon metals mining industry, an ore-milling step byor froth flotation is apaper very common units. It is less for the advanced to be combined any formal way. explores processing route. It ismodel also common that advanced controls of be onecombined, form or another areThis installed on these the way two typical predictive control structures could and what are the benefits of units. It is less common for the advanced controllers to be combined in any formal way. paper explores units. It is less common for the advanced controllers to be combined in any formal way. This paper explores processing route. It is also common that advanced controls of one form or another are installed on these the way two typical model predictive control structures could be combined, and what are the benefits of units. It is less common for the advanced controllers to be combined in any formal way. This paper explores doing so. Use is made of existing linear dynamic models of plants the authors have implemented model the way two typical model predictive control structures could be combined, and what are the benefits of the way model predictive control structures be combined, andhave whatimplemented are the benefits of units. Itso.istwo lesstypical common forexisting the advanced controllers to be could combined inthe any formal way. This paper explores doing Use is made of linear dynamic of plants authors model the way two typical model predictive control could be combined, andhave whatimplemented are the benefits of predictive control (MPC) on. The results showstructures thatmodels the coordinated system can behave differently to the doing so. Use is made of existing linear dynamic models of plants the authors model doing so. Use is made of existing linear dynamic models of plants the authors have implemented model the way two typical model predictive control structures could be combined, and what are the benefits of predictive control (MPC) on. The results show that the coordinated system can behave differently to the doing so. Use is made of existing linear dynamic models of plants the authors have implemented model separate controllers, depending on the steady-state optimisation coefficients applied. predictive control (MPC) on. The results show that the coordinated system can behave differently to the predictive control (MPC) on. The results show that the coordinated system can behave differently to the doing so. Use is made of existing linear dynamic models of plants the authors have implemented model separate controllers, depending on the steady-state optimisation coefficients predictive control (MPC) on. The results show that the coordinated systemapplied. can behave differently to the separate controllers, depending on steady-state optimisation coefficients applied. separate controllers, depending on the the steady-state optimisation coefficients applied. predictive control (MPC) on. The results show that the coordinated can behave differently to the © 2019, IFAC (International Federation of Automatic Control) Hosting bysystem Elsevier Ltd. All rights reserved. Keywords: Identification and Modelling, Advanced Process Control, Model predictive and optimizationseparate controllers, depending on the steady-state optimisation coefficients applied. separate controllers, depending on the steady-state optimisation coefficients applied. Keywords: Identification and Modelling, Advanced Process Control, Model predictive and optimizationbased control, Milling, Flotation, Plantwide Control Process Keywords: Identification and Modelling, Modelling, Advanced Process Control, Control, Model Model predictive predictive and and optimizationoptimizationKeywords: Identification and Advanced based control, Milling, Flotation, Plantwide Control Keywords: Identification and Modelling, Advanced Control, Model predictive and optimizationbased control, Milling, Flotation, Plantwide Control Process based control, Milling, Flotation, Plantwide Control Keywords: Identification and Modelling, Advanced Process Control, Model predictive and optimizationbased control, Milling, Flotation, Plantwide Control based control, Milling, Flotation, Plantwide Control through a combination of pulp level manipulation, air injection 1. INTRODUCTION through combination of pulp level manipulation, air injection rates andaaa reagent dosage changes. through combination of pulp 1. INTRODUCTION through combination ofchanges. pulp level level manipulation, manipulation, air air injection injection 1. INTRODUCTION rates and reagent dosage 1. INTRODUCTION through a combination of pulp level manipulation, air injection rates and reagent dosage changes. The recovery of metals mined ores frequently involves the through rates and reagent dosage changes. a combination of pulp level manipulation, air injection 1. from INTRODUCTION Thereand is an extensive reagent dosageliterature changes.on the industrial application 1. from INTRODUCTION The recovery recovery of metals metals from mined ores oresbyfrequently frequently involvesBoth the rates processing steps of milling following froth flotation. The of mined involves the rates and reagent dosage changes. There is an extensive literature on the industrial application The recovery of metals from mined ores frequently involves the advanced control on milling circuits. Some examples include There extensive literature on industrial application processing steps of milling milling following byfrequently froth flotation. Both There is is an an extensive literature on the the industrial application The recovery of metals from mined oresby involves the advanced milling andsteps flotation are two adjacent several consecutive processing steps of following froth flotation. Both control on milling circuits. Some examples include processing of milling following froth flotation. Both is an extensive literature onal.the industrial application The recovery of metals from mined oresbyof frequently involves the There Silva and Tapia (2009), Steyn et (2010), Karelovic et al. advanced control on milling circuits. Some examples include milling and flotation are two adjacent of several consecutive advanced control on milling circuits. Some examples include There is an extensive literature onal.the industrial application processing of milling following byoffroth flotation. steps in and oresteps beneficiation. The remaining processing stepsBoth are Silva milling flotation are adjacent several consecutive and Tapia (2009), Steyn et (2010), Karelovic et al. milling and flotation are two two adjacent several consecutive advanced control on milling circuits. Some examples include processing steps of milling following byoffroth flotation. Both (2013) and Steyn and Sandrock (2013). While there is a large Silva and Tapia (2009), Steyn et al. (2010), Karelovic et al. steps in ore beneficiation. The remaining processing steps are Silva and Tapia (2009), Steyn et al. (2010), Karelovic et al. advanced control on milling circuits. Some examples include milling and flotation two adjacent of processing several consecutive not considered in this are work. steps in ore beneficiation. The remaining steps are (2013) and Steyn and Sandrock (2013). While there is a large steps in ore beneficiation. The remaining processing steps are Silva and Tapia (2009), Steyn et al. (2010), Karelovic et al. milling and flotation are two adjacent of several consecutive amount of literature on flotation control of simulated systems, (2013) and Steyn and Sandrock (2013). While there is a large not considered in this work. (2013) and Steyn and Sandrock (2013). While there is a large Silva and Tapia (2009), Steyn etcontrol al. (2010), Karelovic et al. steps in ore beneficiation. The remaining processing steps are amount not considered in this work. of literature on flotation of simulated systems, not considered in this work. (2013) and Steyn and Sandrock (2013). While there is a large steps in ore beneficiation. The remaining processing steps are there is less published on implementations on industrial scale amount of flotation control simulated There are different types of mills, classified according to the (2013) amount of literature literature onSandrock flotation control of of simulated systems, and Steyn andon (2013). While there issystems, a large not considered in this work. there is less published implementations on scale amount of literature on on flotation control systems, not considered in this work. There aremedium different types of mills, classified according to the the plants. Some examples include Cortesof etsimulated al.industrial (2008), who there is less published on implementations on industrial scale grinding used. Two types commonly utilised are There are different types of mills, classified according to there is less published on implementations on industrial scale amount of literature on flotation control of simulated systems, There are different types of mills, classified according to the plants. Some examples include Cortes et al. (2008), who there is less published on implementations on industrial scale grinding medium used. Two types commonly utilised are describe the application of MPC for control of froth velocities plants. Some examples include Cortes et al. (2008), who There are different types of mills, classified according to the autogenous (AG) mills in which theclassified orecommonly grindsaccording itself, and to semigrinding used. Two types utilised are plants.is Some examplesonofinclude Cortes eton al.industrial (2008), who there less published implementations scale grinding medium used. Two types commonly utilised are describe There aremedium different types of mills, the the application MPC for control of froth velocities plants. Some examples include Cortes et al. (2008), who autogenous (AG) mills in which the ore grinds itself, and semimeasured by a camera, a similar approach described by describe the application of MPC for control of froth velocities grinding medium used. Two types commonly utilised are (SAG) mills, where size reduction is achieved autogenous (AG) mills in which the ore grinds itself, and semidescribe the application of MPC for control of froth velocities plants. Some examples include Cortes et al. (2008), who autogenous (AG) mills in which thesize orecommonly grinds itself, and semigrinding medium used. Two types utilised are measured by aa camera, aaMPC similar approach described by describe the application of for control of froth velocities autogenous (SAG) mills, where reduction is achieved Dawson and Koorts (2014) using fuzzy logic, and Muller et al. measured by camera, similar approach described by autogenous (AG) mills in which the ore grinds itself, and semithrough the (AG) addition of in steel ballsthe tosize the grinds mil. itself, autogenous (SAG) mills, where size reduction is and achieved measuredand a camera, similar approach described by describe thebyapplication of aMPC for control ofand froth velocities autogenous (SAG) mills, where reduction is achieved mills which ore semi- Dawson Koorts (2014) using fuzzy logic, Muller et al. measured by a camera, a similar approach described by through the addition of steel balls to the mil. (2011), who discuss the use of fuzzy logic for mass pull control, Dawson and Koorts (2014) using fuzzy logic, and Muller et al. autogenous (SAG) mills, where size reduction is achieved through the the addition addition of steel steelwhere balls to tosize the reduction mil. Dawsonwho and (2014) using fuzzy logic, anddescribed Muller et by al. byKoorts a camera, aof similar approach through of balls mil. autogenous (SAG) mills, is achieved measured the use fuzzy logic for mass pull control, Dawson and discuss Koorts (2014) using fuzzy logic, and Muller etand al. Flotation inaddition the mining industry is the a process with a higher level MPC for grade and recovery. Brooks (2011), who discuss the use of fuzzy logic for mass pull control, through the of steel balls to the mil. for selectively (2011), (2011), who discuss the use of fuzzy logic for mass pull control, Dawson and Koorts (2014) using fuzzy logic, and Muller et al. through the addition of steel balls to the mil. Flotation in the mining industry is a process for selectively with a higher level MPC for grade and recovery. Brooks and (2011), who discuss the use of fuzzy logic for mass pull control, concentrating and separating valuable minerals from gangue Flotation in the mining industry is a process for selectively Koorts (2017) describe anfor MPC flotation scheme using x-ray with aa higher level MPC grade and recovery. Brooks and Flotation in the mining industry is a process forfrom selectively with higher level MPC for grade and recovery. Brooks and (2011), who discuss the use of fuzzy logic for mass pull control, concentrating and separating valuable minerals gangue Koorts (2017) describe an flotation scheme using x-ray Flotation ininathe mining industry is with a process forfrom selectively a higher level MPC forMPC grade and Munalula recovery. Brooks anda minerals process slurry. Along mechanised mining, concentrating and separating valuable minerals gangue fluorescence analysers and Brooks and (2017) use Koorts (2017) describe an flotation scheme using concentrating andmining separating valuable minerals gangue with Flotation inathe industry is with a process forfrom selectively Koorts (2017)analysers describe anforMPC MPC flotation scheme(2017) using x-ray x-ray with a higher level MPC grade and Munalula recovery. Brooks anda minerals in in process slurry. Along mechanised mining, fluorescence and Brooks and use concentrating and separating valuable minerals from gangue Koorts (2017) describe an MPC flotation scheme using x-ray flotation is widely considered to have been one of the great minerals a process slurry. Along with mechanised mining, cascaded MPC scheme involving cameras and online grade fluorescence analysers and Brooks and Munalula (2017) use aa minerals in a process slurry. Along with mechanised mining, concentrating and separating valuable minerals from gangue fluorescence analysers and Brooks and Munalula (2017) use Koorts (2017) describe an MPC flotation scheme using x-ray flotation in is widely widely considered to have have been one of of the the great cascaded MPC scheme involving cameras and online grade minerals a process slurry. industry. Along with mechanised mining, fluorescence analysers and Brooks and Munalula (2017) use a breakthroughs in the mining The process involves flotation is considered to been one great measurements. cascaded MPC scheme involving cameras and online grade flotation is widely considered to have been one of the great fluorescence minerals in a process slurry. industry. Along with mechanised mining, cascaded MPC schemeand involving cameras and online analysers Brooks and Munalula (2017) grade use a breakthroughs in the the mining The process involves flotation is widely considered to have been one of the great measurements. cascaded MPC scheme involving cameras and online grade using a collector to render the valuable mineral surface breakthroughs in mining industry. The process involves measurements. breakthroughs in the The process involves flotation widely considered to have been one of the great cascaded measurements. MPC of scheme involving cameras and considered online grade using aa is collector to mining render industry. the valuable mineral surface interaction flotation and milling has been in breakthroughs in theto mining industry. The process involves measurements. hydrophobic, thus promoting its attachment to injected air The using collector render the valuable mineral surface using a collector to render the valuable mineral surface breakthroughs in the mining industry. The process involves measurements. The interaction of flotation and milling has been considered in hydrophobic, thus promoting its attachment to injected air improving overall circuit performance through improved The interaction of flotation and milling has been considered using a collector to render the valuable mineral surface bubbles subsequent recovery the froth overflow from an The interaction of flotation and milling has been considered in hydrophobic, thus to promoting itsinattachment attachment to injected injected air in hydrophobic, thus promoting its to air using a and collector render the valuable mineral surface overall circuit performance through improved The interaction of flotation and milling been(2009) considered bubbles and subsequent recovery in the froth overflow from an improving control of the milling circuit. Wei andhas Craig show in improving overall circuit performance through improved hydrophobic, thus promoting its attachment to injected air agitated tank. The two most important aspects when bubbles and subsequent recovery in the froth overflow from an improving overall circuit performance through improved The interaction of flotation and milling has been considered inaa bubbles and subsequent recovery in the froth overflow from an hydrophobic, thus promoting its attachment to injected air of the milling circuit. Wei and Craig (2009) show improving overall circuit performance through improved agitated and tank. Thevaluable two most important aspects when curve relating flotation recovery to the particle size distribution control of the milling circuit. Wei and Craig (2009) show bubbles subsequent recovery in important the froth overflow from an control concentrating the minerals through milling and agitated tank. The two most aspects when control of the milling circuit. Wei and Craig (2009) show improving overall circuit performance through improvedaa agitated tank. Thevaluable two most aspects when bubbles and subsequent recovery in important the froth overflow from an curve recovery to the particle size distribution control of the flotation milling Wei and Craig(2013) (2009) show concentrating the minerals through milling and of therelating flotation feed. circuit. Steyn and Sandrock use thisaa curve relating flotation recovery to the particle size distribution agitated tank. Thevaluable two most important aspects when flotation are the grade produced and the recovery achieved. concentrating the minerals through milling and curve relating flotation recovery to the particle size distribution control of the milling circuit. Wei and Craig (2009) show concentrating the minerals throughaspects millingwhen and of agitated tank. Thevaluable two most important the flotation feed. Steyn and Sandrock (2013) use this curve relating flotation recovery to the particle size distribution flotation are the grade produced and the recovery achieved. method but note that “a representative model of the primary of flotation feed. Steyn and Sandrock (2013) use this concentrating the valuable minerals through achieved. milling and curve flotation are the grade produced and the of the therelating flotation feed. Steyn and Sandrock (2013) use this flotation recovery to the particle size distribution flotation areobjectives thethe grade produced and circuit the recovery recovery achieved. concentrating valuable minerals through millingstated and method but note that “a representative model of the primary of the flotation Steyn andperformance Sandrock (2013) use thisa The control the milling are generally mill product sizefeed. to overall plant alone is not method but note that “a representative model of the primary flotation are the grade of produced and the recovery achieved. method but note that “a representative model of the primary of the flotation feed. Steyn and Sandrock (2013) use thisa flotation are the grade produced and the recovery achieved. Thetocontrol control objectives of the milling milling circuit are generally stated mill product size to overall plant performance alone is not method but note that “a representative model of the primary as be providing a product of consistent flow, density and The objectives of the circuit are generally stated trivial task”. mill product size to overall plant performance alone is not aa The control objectives of the milling circuit are generally stated mill product size to overall plant performance alone is not method but note that “a representative model of the primary as to tocontrol be size providing a product product ofaim consistent flow, density and trivial task”. The objectives of theThe milling circuit are generally stated mill product size to overall plant performance alone is not a particle distribution. of a flotation circuit is to as be providing a of consistent flow, density and trivial task”. as tocontrol be size providing a product ofaim consistent flow, density The objectives of theThe milling circuit are generally trivial task”. size mill product to overall plant performance alone is not a particle distribution. of maximum a flotation flotation circuitstated isand to The presence of several MPC implementations in connected as to be providing a product ofaim consistent flow, recovery density and trivial task”. strike a balance between achieving at particle size distribution. The of a circuit is to particle size distribution. The of maximum a flotation circuit isand to as to be providing a product ofaim consistent flow, recovery density trivial task”. presence of several MPC implementations in connected strike balance between achieving at The processes is common in the chemical industry. In distributed The of MPC implementations in connected particle distribution. The aim grades. of maximum a flotation circuitis isthe to economically viable concentrate Recovery strike aaa size balance between achieving recovery at The presence presence of several several MPC implementations in distributed connected strike balance between achieving maximum recovery at particle size distribution. The aim of a flotation circuit is to processes is common in the chemical industry. In The presence of several MPC implementations in connected economically viable concentrate grades. Recovery is the MPC, a series optimisation problems are decomposed into a processes is common in the chemical industry. In distributed strike a balance between achieving maximum recovery at fraction of valuable metal in the feed reporting to the economically viable concentrate grades. Recovery is processes is common in the chemical industry. In distributed The presence of several MPC implementations in connected economically viable concentrate grades. Recovery is the strike a balance between achieving maximum recovery at MPC, a series of optimisation problems are decomposed into a processes is common in the chemical industry. In distributed fraction of valuable metal in the feed reporting to the set of subproblems (Camponogara et al., 2002). The individual MPC, a series of optimisation problems are decomposed into economically viable concentrate grades. Recovery is the concentrate, while the grade is the mass fraction of the metal of fraction of valuable metal in the feed reporting to MPC, a series of optimisation problems are decomposed into aa processes is common in the chemical industry. In distributed fraction of while valuable metalis in the feed reporting to economically viable concentrate grades. Recovery is the set of subproblems (Camponogara et al., 2002). The individual MPC, a series of optimisation problems are decomposed into a concentrate, the grade the mass fraction of the metal of MPCs are referred to as ‘agents’ andet can exchange information. set of (Camponogara al., 2002). The individual fraction valuable metal the feed reporting to the interest inof particular stream. These variables arethe controlled concentrate, the is the mass fraction of metal of set of subproblems subproblems (Camponogara etcan al.,are 2002). The individual MPC, aare series of optimisation problems decomposed into a concentrate, while the grade grade is in the massvariables fraction of the metal of MPCs fraction ofaa while valuable metal in the feed reporting to the referred to as ‘agents’ and exchange information. set of subproblems (Camponogara et al., 2002). The individual interest in particular stream. These are controlled MPCs are as information. concentrate, while the grade is the massvariables fraction of the metal of set interest particular stream. These are controlled are referred referred to to as ‘agents’ ‘agents’ and andetcan can exchange information. of subproblems (Camponogara al.,exchange 2002). The individual interest in in aa while particular stream. These arethe controlled concentrate, the grade is the massvariables fraction of metal of MPCs are referred to as ‘agents’ and can exchange information. interest in a particular stream. These variables are controlled MPCs MPCs are referred to as ‘agents’ and can exchange information. 2405-8963in© IFAC (International Federation of Automatic Control) interest particular stream. These variables are controlled Copyright ©a2019, 2019 IFAC 24 Hosting by Elsevier Ltd. All rights reserved. 24 Peer review©under of International Federation of Automatic Copyright 2019 responsibility IFAC 24 Control. Copyright © 24 Copyright © 2019 2019 IFAC IFAC 24 10.1016/j.ifacol.2019.09.158 Copyright © 2019 IFAC 24 Copyright © 2019 IFAC 24
24 24 24 24 24
2019 IFAC MMM Stellenbosch, South Africa, August 28-30, 2019 Kevin Brooks et al. / IFAC PapersOnLine 52-14 (2019) 24–29
25
Distributed MPC has been app lied to power generation and to the chemical industry (Venkat et al., 2008), however – to the authors’ knowledge – not in the minerals processing industry. This article investigates the performance of two individual controllers for milling and flotation versus the performance of a combined, large-scale MPC controller for both processes combined. The paper is structured as follows. Section 2 briefly describes the milling and flotation processes studied here. Section 3 gives the linear process model derived from step responses. Section 4 describes the MPC control strategy both for two individual controllers and for one combined controller, comparing the results of both control strategies. Conclusions are drawn in Section 6. 2. PROCESS DESCRIPTION Fig. 2. Rougher Bank Showing Control Loops and Analysers
The process studied consists of a SAG mill feeding a flotation plant consisting of two rougher cells. Although industrial plants have many more cells, this configuration was chosen for simplicity. The mill is shown in Fig. 1. Run of Mine ore is fed to the mill together with the recycle flow of screen oversize and water. The mill reduces the particle size and the slurry exits the mill into the discharge sump. The screen undersize flows to the surge tank from where it is pumped to the first rougher flotation cell.
by plant testing. The models required are represented as unit step responses. The dynamic response of a controlled variable (CV or output) to a unit step in an MV (manipulated variable or input) or FF (feedforward or disturbance variable). The model prediction of an output is then given by (Garcia et al., 1989): 𝑦𝑦(𝑘𝑘) = ∑𝑛𝑛−1 𝑖𝑖=1 𝐻𝐻𝑖𝑖 ∆𝑢𝑢(𝑘𝑘 − 1) + 𝐻𝐻𝑛𝑛 𝑢𝑢(𝑘𝑘 − 𝑛𝑛)
(1)
where y(k) is an output at discrete time k, n is the number of truncated coefficients, u is the input and Hi as well as Hn are the step response coefficients. The difference equation for the input is abbreviated as Δu(k) = u(k) – u(k–1). The extension of Eq. 1 to the case where there are multiple inputs is straightforward since the assumption of linearity allows for superposition. The step response coefficients, Hi, are obtained by performing a planned experiment. The data of the experiment is used in the fitting routine. In the software used here, either finite impulse response or a subspace technique is used for the fitting (Verhaegen and Dewilde, 1992, Darby et. al., 2009). The linear dynamic models for the two plants are shown in the form of unit step responses in Fig. 3 and Fig. 4. The manipulated variables (MVs) and feedforwards (FFs) are down the rows, while the controlled variables (CVs) are across the columns. The SAG mill MVs are the solids feed rate, the inlet water, discharge sump and surge tank water addition, and flows out of the discharge sump and surge tank. The CVs are the load, power, discharge sump and surge tank level, the discharge sump and product densities and the product PSD. The manipulated variables for flotation are the pulp level, the two air flows and reagent addition flowrate. The controlled variables are concentrate and tailings grade, the reagent dosage (amount of reagent per tonne of metal in feed), and the calculated steady-state recovery. The feedforward variables are the feed flow, grade, density and PSD.
Fig. 1. SAG Mill Showing Base Level Controls It is assumed that online measurement of the densities and the particle size distribution (PSD) of feed and product are available. The rougher flotation circuit is shown in Fig. 2. The slurry flows by gravity from Cell 1 to Cell 2. The froth to pulp interface level is controlled in Cell 2, while the air addition rate is controlled to a setpoint in both Flotation cells. A reagent, known as a collector, is added to Cell 1. Composition measurements are available for the feed, concentrate and tails streams. 3. LINEAR MODELS
The inclusion of calculated dosages as CVs with reagent flows as MVs is unusual but allows the controller to smoothly change reagent flows on changes of total metal feed. This functionality can be included by use of PID control; this method, however, neglects the dynamics associated with rejecting the disturbance.
AspenTech’s DMCPlus was used as the MPC engine for this study. This algorithm employs a description of the plant in the form of finite step responses, which are generally obtained
25
2019 IFAC MMM 26 Kevin Brooks et al. / IFAC PapersOnLine 52-14 (2019) 24–29 Stellenbosch, South Africa, August 28-30, 2019
Fig. 3. Unit Step Response Models for the Mill with time scale in minutes
Fig. 4. Unit Step Response Models for Flotation with time scale in minutes. The time scales have the units of minutes; the indicated time to steady-state is ten minutes. The responses have been sped up by a factor of 6 for ease of simulation. The actual times to steady-state of the systems are in the region of sixty minutes
The DMC Plus algorithm has been described by Garcia et al., (1989). The equations will not be repeated here. Suffice it to say that a quadratic program formulation is used to solve for an optimal move plan subject to MV and CV constraints.
𝑦𝑦̂𝑖𝑖,𝑠𝑠𝑠𝑠 ≥ 𝑦𝑦𝑖𝑖,𝑚𝑚𝑚𝑚𝑚𝑚 − 𝜀𝜀𝑖𝑖
(4)
In order to facilitate the handling of multiple constraints, a rank can be assigned to each dependent variable. This allows the controller to relax a dependent variable’s constraints as described in Eq. 3 and Eq. 4 in order to honour another higherranked variable’s constraints.
Before solving the dynamic optimisation problem, the algorithm checks for steady-state feasibility by solving the optimisation problem: 𝑛𝑛
(3)
where 𝜙𝜙 is the objective function value, nc is the number of controlled variables, 𝜀𝜀𝑖𝑖 are slack variables, 𝑊𝑊𝑖𝑖 are weights, 𝑦𝑦̂𝑖𝑖,𝑠𝑠𝑠𝑠 is the steady-state value of CV i and 𝑦𝑦𝑖𝑖,𝑚𝑚𝑚𝑚𝑚𝑚 and 𝑦𝑦𝑖𝑖,𝑚𝑚𝑚𝑚𝑚𝑚 are the high and low limits on CV i.
4. OPTIMISATION AND THE DMC ALGORITHM
𝑐𝑐 min 𝜙𝜙 = ∑𝑖𝑖=1 𝜖𝜖𝑖𝑖2 𝑊𝑊𝑖𝑖
𝑦𝑦̂𝑖𝑖,𝑠𝑠𝑠𝑠 ≤ 𝑦𝑦𝑖𝑖,𝑚𝑚𝑚𝑚𝑚𝑚 + 𝜀𝜀𝑖𝑖
The objective function in Eq. 2 is minimised subject to the current constraints on the MVs and CVs. If there are slack
(2)
subject to:
26
2019 IFAC MMM Stellenbosch, South Africa, August 28-30, 2019
Kevin Brooks et al. / IFAC PapersOnLine 52-14 (2019) 24–29
27
Figure 5. A subset of Mill MPC dynamics in response to a limit change at Product PSD at time 05:04:00 (bottom right subplot). variables that are non-zero, this implies that the steady-state solution is not feasible. In this case, the CV limits are relaxed until the solution is feasible.
The mill controller is configured to maximise the feed subject to the current constraints on densities and the particle size distribution (PSD) of the product. The controller includes three CVs that have integrating models; the load, sump level and rougher feed level. The first of these is configured to control to a value since tight control of load improves grinding performance. The latter two are configured to use as much as possible of the surge capacity. This assists in smoothing the flows out; this is advantageous in stabilising the flotation feed.
Should the minimisation of Eq. 2 prove that all the slack variable values are zero, then there exist one or more feasible steady-state solutions. In this case, the following economic optimisation is solved: 𝑁𝑁
𝑁𝑁𝑗𝑗
𝑖𝑖 min 𝐽𝐽 = ∑𝑖𝑖=1 𝑐𝑐𝑖𝑖 Δ𝑢𝑢𝑖𝑖,𝑠𝑠𝑠𝑠 + ∑𝑗𝑗=1 𝑐𝑐𝑗𝑗 |Δ𝑢𝑢𝑗𝑗,𝑠𝑠𝑠𝑠 |
(5)
The flotation controller has the control objectives to control the overall reagent dosage in a range while maintaining concentrate and tail grade between limits. The optimisation objective of the controller is to maximise recovery.
where J is the objective function value, Ni is the number of MVs that have economic directions, Nj is the number of MVs whose movement is to be minimised and ci and cj are cost factors.
The two individual controllers give good results in controlling the plants in the specified ranges and reacting to disturbances. The results of increasing the PSD limit percentage is shown in Fig. 5. This can be seen in the red lower curve of PROD_PSD, the PSD after milling, which is changed from 80% to 85%. The MPC reacts smoothly and in a coordinated fashion. Despite the objective to maximise feed, the MPC reduces feed as this is the only method to ensure the finer grind. The vertical dashed line represents current time; the plots extend two times to steadystate in the past, and one in the future.
The Δ𝑢𝑢𝑖𝑖,𝑠𝑠𝑠𝑠 and Δ𝑢𝑢𝑗𝑗,𝑠𝑠𝑠𝑠 are the changes in the values of the MVs at the present time to those at steady-state. These are the variables chosen to minimise Eq. 5 subject to the current MV and CV constraints. The steady-state values of the MVs are imposed on the dynamic solution at the end of the control horizon. 4.1 TWO MODEL PREDICTIVE CONTROLLERS The model matrices as derived through the step responses in Fig. 3 and Fig. 4 were used to implement model predictive control on each unit separately. For this study, the plant and the controller have identical models i.e. no plant model mismatch is included.
The flotation controller manages the competing objective of grade and recovery well. This can be seen in Fig. 6 where the MPC responds to an increase in the recovery low limit of one percent. Control is smooth; the increased recovery is achieved 27
2019 IFAC MMM 28 Kevin Brooks et al. / IFAC PapersOnLine 52-14 (2019) 24–29 Stellenbosch, South Africa, August 28-30, 2019
at the cost of the use of extra reagent. In this formulation, the effect of the mill parameters on the flotation bank is modelled by including variables feed flow, density, grade and PSD as feed-forward variables in the flotation controller. This approach does not leverage the information on future predictions of CVs and MVs that are available from the mill controller.
(CV) PROD_PSD (FF) Milling
(CV) RGH_DENS (FF)
Flotation
(MV) RGH_FD (FF) Fig. 7. Linking of milling and flotation through CVs (outputs) and MVs to feed forwards.
The two separate controllers are sub-optimal in the sense that the flotation controller cannot affect the variables that are fed to it. For instance, if the economics are such that increased recovery requires a finer feed, this change would have to be made by a human rather than the system. For these reasons combining of the two MPCs is investigated.
To combine the controllers some thought must be given to the resulting models and how they should be formulated. If an MV in an upstream controller is repeated as a feed-forward in a downstream controller then this is straightforward. The replacement of the feed-forward by a manipulated variable and its associated move-plan yields an immediate predictive advantage. The case is more complicated if a CV in an upstream controller is used as a feed-forward in a downstream controller. One method would be to re-visit the data originally used for model development and attempt to derive the models of the effect of mill inputs on flotation outputs. This approach has some difficulties, since
Fig. 8. Convoluted Models for Grades, Dosage and Recovery versus Mill MVs identifying the effects in the face of other disturbances as well as the relatively long settling times can be challenging. It is possible to derive the required models by mathematical manipulation. In this case, the models of the downstream controller are found by convolution of the upstream CV model with the downstream CV models that depend on the feedforward: 𝐶𝐶(𝑗𝑗, 𝑘𝑘) = ∑𝑝𝑝 ∑𝑞𝑞 𝐴𝐴𝑈𝑈 (𝑝𝑝, 𝑞𝑞)𝐴𝐴𝐷𝐷 (𝑗𝑗 − 𝑝𝑝 + 1, 𝑘𝑘 − 𝑞𝑞 + 1)
(6)
where AU is the matrix of the upstream (milling) and AD the matrix of the downstream system (flotation). As an example, the rougher feed density and product PSD are CVs in the mill controller and a feed-forward in the flotation controller, while the rougher feed flow is an MV in the mill controller. On combining the matrices, models are formed for the grades, reagent dosage and recovery versus these three mill MVs.
Fig. 6. Flotation MPC: Change in Recovery Low Limit at 04:38 4.2 THE COMBINED CONTROLLER The aim of combining the two MPCs is to achieve tighter control, particularly in the reaction of the flotation circuit to upstream disturbances. In addition, it is hoped that a superior operating point will be achieved. Fig. 7. outlines the signals that link the behaviour of the two systems.
The time to steady state of the convoluted model is double that of the individual models. This reflects the fact that two systems that operate in series have been connected from a modelling point of view. This longer time to steady state can have implications for the control of the system. In principle, the convolution of Eq. 6 removes the intermediate variables from 28
2019 IFAC MMM Stellenbosch, South Africa, August 28-30, 2019 Kevin Brooks et al. / IFAC PapersOnLine 52-14 (2019) 24–29
the matrix altogether. The variable can be retained in the matrix if desired, but care must then be taken not to over specify the problem.
29
The approach is by no means limited to the unit operations considered. As the use of MPC becomes more commonplace is the mineral processing industry, larger controllers will evolve with improved optimisation scope.
The results of a change in the feed PSD for the combined controller are shown in Fig. 9. In this case, the controller recognises that there will be a drop in recovery and increases the air to the flotation cells.
REFERENCES Bergh, L. G., & Yianatos, J. B. (2011). The long way toward multivariate predictive control of flotation processes. Journal of Process Control, 21(2), 226-234. Brooks, K.S. and Koorts, R., 2017. Model Predictive Control of a Zinc Flotation Bank Using Online X-ray Fluorescence Analysers. IFAC-PapersOnLine, 50(1), pp.10214-10219. Brooks, K. and Munalula, W., 2017. Flotation Velocity and Grade Control Using Cascaded Model Predictive Controllers. IFAC-PapersOnLine, 50(2), pp.25-30. Camponogara, E., Jia, D., Krogh, B. H., & Talukdar, S. (2002). Distributed model predictive control. IEEE control systems magazine, 22(1), 44-52. Cortés, G., Verdugo, M., Fuenzalida, R., Cerda, J., & Cubillos, E. (2008). Rougher flotation multivariable predictive control: concentrator A-1 division Codelco Norte. Proceedings of the V International Mineral Processing Seminar 1(6) 316-325. Darby, M.L., Harmse, M. and Nikolaou, M., 2009. MPC: Current practice and challenges. IFAC Proceedings Volumes, 42(11), pp.86-98. Dawson, P. and Koorts, R., 2014. Flotation Control Incorporating Fuzzy Logic and Image Analysis. IFAC Proceedings Volumes, 47(3), pp.352-357. Garcia, C.E., Prett, D.M. and Morari, M (1989). Model Predictive Control: Theory and Practice – A Survey. Automatica 25(3), 335-348. Karelovic, P., Razzetto, R. and Cipriano, A., 2013. Evaluation of MPC strategies for mineral grinding. IFAC Proceedings Volumes, 46(16), pp.230-235. Silva, D.A. and Tapia, L.A., 2009. Experiences and lessons with advanced control systems for the SAG mill control in Minera Los Pelambres. IFAC Proceedings Volumes, 42(23), pp.25-30. Steyn, C.W., Brooks, K.S., De Villiers, P.G.R., Muller, D. and Humphries, G., 2010. A Holistic Approach to Control and Optimization of an Industrial R Ball Milling Circuit. IFAC Proceedings Volumes, 43(9), pp.137-141. Steyn, C.W. and Sandrock, C., 2013. Benefits of optimisation and model predictive control on a fully autogenous mill with variable speed. Minerals Engineering, 53, pp.113-123. Venkat, A. N., Hiskens, I. A., Rawlings, J. B., & Wright, S. J. (2008). Distributed MPC strategies with application to power system automatic generation control. IEEE transactions on control systems technology, 16(6), 11921206. Verhaegen, M. and Dewilde, P., 1992. Subspace model identification part 2. Analysis of the elementary outputerror state-space model identification algorithm. International journal of control, 56(5), pp.1211-1241. Wei, D. and Craig, I.K., 2009. Economic performance assessment of two ROM ore milling circuit controllers. Minerals Engineering, 22(9-10), pp.826-839.
Fig. 9. Combined MPC: Increase in feed PSD at 10.22 5. CONCLUSION MPC on the commonly found mineral processing operations of milling and flotation is now an industrial reality. This work demonstrates that by careful consideration of common variables, these MPCs may be combined in a mathematically rigorous manner. The resulting larger controller displays more co-ordinated behaviour and can adjust milling circuit parameters to meet flotation circuit requirements. The combined controller presented here has the potential to improve overall circuit performance beyond that achieved by two local MPCs. The size of the application is not unusually large when measured against applications in the petrochemical industry.
29
ScienceDirect
IFAC PapersOnLine 52-14 (2019) 24–29 A Combined MPC for Milling and Flotation – A Simulation Study A Combined MPC for Milling and Flotation –– A Simulation Study A Combined MPC for Milling and Flotation A Simulation Study A Combined MPC for Milling and Flotation – A Simulation Study A Combined MPC for Milling and Flotation – A Simulation Study A CombinedKevin MPC for Milling and Flotation – A Simulation Study Brooks*, Merinda Westcott* and Margret Bauer**
Kevin Brooks*, Merinda Westcott* and Margret Bauer** Kevin Kevin Brooks*, Brooks*, Merinda Merinda Westcott* Westcott* and and Margret Margret Bauer** Bauer** Kevin Brooks*, Merinda Westcott* and Margret Bauer** Kevin Brooks*, Merinda Westcott* and Margret Bauer** *BluESP, 53 Platina St, Randburg, South Africa *BluESP, 53 Platina Platina St, Randburg, South Africa Africa (E-mail: [email protected], [email protected] ) *BluESP, 53 Randburg, South *BluESP, 53 Platina St, St, Randburg, South Africa (E-mail: [email protected], [email protected] ) *BluESP, 53 Platina St, Randburg, South Africa **Department of Electrical, Electronic and Computer Engineering, (E-mail: [email protected], [email protected] ) (E-mail: [email protected], [email protected] ) *BluESP, 53 Platina St, Randburg, South Africa **Department of Electrical, Electronic and Computer Engineering, (E-mail: [email protected], [email protected] ) University ofof Pretoria, Lynwood Road, Pretoria, South Africa **Department Electrical, Electronic and Computer Engineering, **Department of Electrical, Electronic and Computer Engineering, (E-mail: [email protected], [email protected] ) University of Pretoria, Lynwood Road, Pretoria, South Africa **Department of Electrical, Electronic and Computer Engineering, (E-mail: [email protected] ) Africa University of ofofPretoria, Pretoria, Lynwood Road, Pretoria, South Africa University Lynwood Road, Pretoria, South **Department Electrical, Electronic and Computer Engineering, (E-mail: [email protected] )) Africa University of Pretoria, Lynwood Road, Pretoria, South (E-mail: [email protected] (E-mail: [email protected] ) Africa University of Pretoria, Lynwood Road, Pretoria, South (E-mail: [email protected] ) (E-mail: [email protected] )
Abstract: In the metals and mining industry, an ore-milling step followed by froth flotation is a very common Abstract: In the metals industry, an ore-milling step flotation is a very common processing It is and alsomining common that advanced controls offollowed one formby orfroth another are installed on these Abstract: In the and mining industry, an step followed by froth flotation is Abstract: Inroute. the metals metals and mining industry, an ore-milling ore-milling step followed byor froth flotation is aa very very common common processing route. It is also common that advanced controls of one form another are installed on these Abstract: In thecommon metals and mining industry, an ore-milling step followed byor froth flotation is apaper very common units. It is less for the advanced controllers to be combined in any formal way. This explores processing route. It is also common that advanced controls of one form another are installed on these processing route. It is and also common thatcontrollers advanced controls offollowed one in form another areThis installed on these Abstract: In thecommon metals mining industry, an ore-milling step byor froth flotation is apaper very common units. It is less for the advanced to be combined any formal way. explores processing route. It ismodel also common that advanced controls of be onecombined, form or another areThis installed on these the way two typical predictive control structures could and what are the benefits of units. It is less common for the advanced controllers to be combined in any formal way. paper explores units. It is less common for the advanced controllers to be combined in any formal way. This paper explores processing route. It is also common that advanced controls of one form or another are installed on these the way two typical model predictive control structures could be combined, and what are the benefits of units. It is less common for the advanced controllers to be combined in any formal way. This paper explores doing so. Use is made of existing linear dynamic models of plants the authors have implemented model the way two typical model predictive control structures could be combined, and what are the benefits of the way model predictive control structures be combined, andhave whatimplemented are the benefits of units. Itso.istwo lesstypical common forexisting the advanced controllers to be could combined inthe any formal way. This paper explores doing Use is made of linear dynamic of plants authors model the way two typical model predictive control could be combined, andhave whatimplemented are the benefits of predictive control (MPC) on. The results showstructures thatmodels the coordinated system can behave differently to the doing so. Use is made of existing linear dynamic models of plants the authors model doing so. Use is made of existing linear dynamic models of plants the authors have implemented model the way two typical model predictive control structures could be combined, and what are the benefits of predictive control (MPC) on. The results show that the coordinated system can behave differently to the doing so. Use is made of existing linear dynamic models of plants the authors have implemented model separate controllers, depending on the steady-state optimisation coefficients applied. predictive control (MPC) on. The results show that the coordinated system can behave differently to the predictive control (MPC) on. The results show that the coordinated system can behave differently to the doing so. Use is made of existing linear dynamic models of plants the authors have implemented model separate controllers, depending on the steady-state optimisation coefficients predictive control (MPC) on. The results show that the coordinated systemapplied. can behave differently to the separate controllers, depending on steady-state optimisation coefficients applied. separate controllers, depending on the the steady-state optimisation coefficients applied. predictive control (MPC) on. The results show that the coordinated can behave differently to the © 2019, IFAC (International Federation of Automatic Control) Hosting bysystem Elsevier Ltd. All rights reserved. Keywords: Identification and Modelling, Advanced Process Control, Model predictive and optimizationseparate controllers, depending on the steady-state optimisation coefficients applied. separate controllers, depending on the steady-state optimisation coefficients applied. Keywords: Identification and Modelling, Advanced Process Control, Model predictive and optimizationbased control, Milling, Flotation, Plantwide Control Process Keywords: Identification and Modelling, Modelling, Advanced Process Control, Control, Model Model predictive predictive and and optimizationoptimizationKeywords: Identification and Advanced based control, Milling, Flotation, Plantwide Control Keywords: Identification and Modelling, Advanced Control, Model predictive and optimizationbased control, Milling, Flotation, Plantwide Control Process based control, Milling, Flotation, Plantwide Control Keywords: Identification and Modelling, Advanced Process Control, Model predictive and optimizationbased control, Milling, Flotation, Plantwide Control based control, Milling, Flotation, Plantwide Control through a combination of pulp level manipulation, air injection 1. INTRODUCTION through combination of pulp level manipulation, air injection rates andaaa reagent dosage changes. through combination of pulp 1. INTRODUCTION through combination ofchanges. pulp level level manipulation, manipulation, air air injection injection 1. INTRODUCTION rates and reagent dosage 1. INTRODUCTION through a combination of pulp level manipulation, air injection rates and reagent dosage changes. The recovery of metals mined ores frequently involves the through rates and reagent dosage changes. a combination of pulp level manipulation, air injection 1. from INTRODUCTION Thereand is an extensive reagent dosageliterature changes.on the industrial application 1. from INTRODUCTION The recovery recovery of metals metals from mined ores oresbyfrequently frequently involvesBoth the rates processing steps of milling following froth flotation. The of mined involves the rates and reagent dosage changes. There is an extensive literature on the industrial application The recovery of metals from mined ores frequently involves the advanced control on milling circuits. Some examples include There extensive literature on industrial application processing steps of milling milling following byfrequently froth flotation. Both There is is an an extensive literature on the the industrial application The recovery of metals from mined oresby involves the advanced milling andsteps flotation are two adjacent several consecutive processing steps of following froth flotation. Both control on milling circuits. Some examples include processing of milling following froth flotation. Both is an extensive literature onal.the industrial application The recovery of metals from mined oresbyof frequently involves the There Silva and Tapia (2009), Steyn et (2010), Karelovic et al. advanced control on milling circuits. Some examples include milling and flotation are two adjacent of several consecutive advanced control on milling circuits. Some examples include There is an extensive literature onal.the industrial application processing of milling following byoffroth flotation. steps in and oresteps beneficiation. The remaining processing stepsBoth are Silva milling flotation are adjacent several consecutive and Tapia (2009), Steyn et (2010), Karelovic et al. milling and flotation are two two adjacent several consecutive advanced control on milling circuits. Some examples include processing steps of milling following byoffroth flotation. Both (2013) and Steyn and Sandrock (2013). While there is a large Silva and Tapia (2009), Steyn et al. (2010), Karelovic et al. steps in ore beneficiation. The remaining processing steps are Silva and Tapia (2009), Steyn et al. (2010), Karelovic et al. advanced control on milling circuits. Some examples include milling and flotation two adjacent of processing several consecutive not considered in this are work. steps in ore beneficiation. The remaining steps are (2013) and Steyn and Sandrock (2013). While there is a large steps in ore beneficiation. The remaining processing steps are Silva and Tapia (2009), Steyn et al. (2010), Karelovic et al. milling and flotation are two adjacent of several consecutive amount of literature on flotation control of simulated systems, (2013) and Steyn and Sandrock (2013). While there is a large not considered in this work. (2013) and Steyn and Sandrock (2013). While there is a large Silva and Tapia (2009), Steyn etcontrol al. (2010), Karelovic et al. steps in ore beneficiation. The remaining processing steps are amount not considered in this work. of literature on flotation of simulated systems, not considered in this work. (2013) and Steyn and Sandrock (2013). While there is a large steps in ore beneficiation. The remaining processing steps are there is less published on implementations on industrial scale amount of flotation control simulated There are different types of mills, classified according to the (2013) amount of literature literature onSandrock flotation control of of simulated systems, and Steyn andon (2013). While there issystems, a large not considered in this work. there is less published implementations on scale amount of literature on on flotation control systems, not considered in this work. There aremedium different types of mills, classified according to the the plants. Some examples include Cortesof etsimulated al.industrial (2008), who there is less published on implementations on industrial scale grinding used. Two types commonly utilised are There are different types of mills, classified according to there is less published on implementations on industrial scale amount of literature on flotation control of simulated systems, There are different types of mills, classified according to the plants. Some examples include Cortes et al. (2008), who there is less published on implementations on industrial scale grinding medium used. Two types commonly utilised are describe the application of MPC for control of froth velocities plants. Some examples include Cortes et al. (2008), who There are different types of mills, classified according to the autogenous (AG) mills in which theclassified orecommonly grindsaccording itself, and to semigrinding used. Two types utilised are plants.is Some examplesonofinclude Cortes eton al.industrial (2008), who there less published implementations scale grinding medium used. Two types commonly utilised are describe There aremedium different types of mills, the the application MPC for control of froth velocities plants. Some examples include Cortes et al. (2008), who autogenous (AG) mills in which the ore grinds itself, and semimeasured by a camera, a similar approach described by describe the application of MPC for control of froth velocities grinding medium used. Two types commonly utilised are (SAG) mills, where size reduction is achieved autogenous (AG) mills in which the ore grinds itself, and semidescribe the application of MPC for control of froth velocities plants. Some examples include Cortes et al. (2008), who autogenous (AG) mills in which thesize orecommonly grinds itself, and semigrinding medium used. Two types utilised are measured by aa camera, aaMPC similar approach described by describe the application of for control of froth velocities autogenous (SAG) mills, where reduction is achieved Dawson and Koorts (2014) using fuzzy logic, and Muller et al. measured by camera, similar approach described by autogenous (AG) mills in which the ore grinds itself, and semithrough the (AG) addition of in steel ballsthe tosize the grinds mil. itself, autogenous (SAG) mills, where size reduction is and achieved measuredand a camera, similar approach described by describe thebyapplication of aMPC for control ofand froth velocities autogenous (SAG) mills, where reduction is achieved mills which ore semi- Dawson Koorts (2014) using fuzzy logic, Muller et al. measured by a camera, a similar approach described by through the addition of steel balls to the mil. (2011), who discuss the use of fuzzy logic for mass pull control, Dawson and Koorts (2014) using fuzzy logic, and Muller et al. autogenous (SAG) mills, where size reduction is achieved through the the addition addition of steel steelwhere balls to tosize the reduction mil. Dawsonwho and (2014) using fuzzy logic, anddescribed Muller et by al. byKoorts a camera, aof similar approach through of balls mil. autogenous (SAG) mills, is achieved measured the use fuzzy logic for mass pull control, Dawson and discuss Koorts (2014) using fuzzy logic, and Muller etand al. Flotation inaddition the mining industry is the a process with a higher level MPC for grade and recovery. Brooks (2011), who discuss the use of fuzzy logic for mass pull control, through the of steel balls to the mil. for selectively (2011), (2011), who discuss the use of fuzzy logic for mass pull control, Dawson and Koorts (2014) using fuzzy logic, and Muller et al. through the addition of steel balls to the mil. Flotation in the mining industry is a process for selectively with a higher level MPC for grade and recovery. Brooks and (2011), who discuss the use of fuzzy logic for mass pull control, concentrating and separating valuable minerals from gangue Flotation in the mining industry is a process for selectively Koorts (2017) describe anfor MPC flotation scheme using x-ray with aa higher level MPC grade and recovery. Brooks and Flotation in the mining industry is a process forfrom selectively with higher level MPC for grade and recovery. Brooks and (2011), who discuss the use of fuzzy logic for mass pull control, concentrating and separating valuable minerals gangue Koorts (2017) describe an flotation scheme using x-ray Flotation ininathe mining industry is with a process forfrom selectively a higher level MPC forMPC grade and Munalula recovery. Brooks anda minerals process slurry. Along mechanised mining, concentrating and separating valuable minerals gangue fluorescence analysers and Brooks and (2017) use Koorts (2017) describe an flotation scheme using concentrating andmining separating valuable minerals gangue with Flotation inathe industry is with a process forfrom selectively Koorts (2017)analysers describe anforMPC MPC flotation scheme(2017) using x-ray x-ray with a higher level MPC grade and Munalula recovery. Brooks anda minerals in in process slurry. Along mechanised mining, fluorescence and Brooks and use concentrating and separating valuable minerals from gangue Koorts (2017) describe an MPC flotation scheme using x-ray flotation is widely considered to have been one of the great minerals a process slurry. Along with mechanised mining, cascaded MPC scheme involving cameras and online grade fluorescence analysers and Brooks and Munalula (2017) use aa minerals in a process slurry. Along with mechanised mining, concentrating and separating valuable minerals from gangue fluorescence analysers and Brooks and Munalula (2017) use Koorts (2017) describe an MPC flotation scheme using x-ray flotation in is widely widely considered to have have been one of of the the great cascaded MPC scheme involving cameras and online grade minerals a process slurry. industry. Along with mechanised mining, fluorescence analysers and Brooks and Munalula (2017) use a breakthroughs in the mining The process involves flotation is considered to been one great measurements. cascaded MPC scheme involving cameras and online grade flotation is widely considered to have been one of the great fluorescence minerals in a process slurry. industry. Along with mechanised mining, cascaded MPC schemeand involving cameras and online analysers Brooks and Munalula (2017) grade use a breakthroughs in the the mining The process involves flotation is widely considered to have been one of the great measurements. cascaded MPC scheme involving cameras and online grade using a collector to render the valuable mineral surface breakthroughs in mining industry. The process involves measurements. breakthroughs in the The process involves flotation widely considered to have been one of the great cascaded measurements. MPC of scheme involving cameras and considered online grade using aa is collector to mining render industry. the valuable mineral surface interaction flotation and milling has been in breakthroughs in theto mining industry. The process involves measurements. hydrophobic, thus promoting its attachment to injected air The using collector render the valuable mineral surface using a collector to render the valuable mineral surface breakthroughs in the mining industry. The process involves measurements. The interaction of flotation and milling has been considered in hydrophobic, thus promoting its attachment to injected air improving overall circuit performance through improved The interaction of flotation and milling has been considered using a collector to render the valuable mineral surface bubbles subsequent recovery the froth overflow from an The interaction of flotation and milling has been considered in hydrophobic, thus to promoting itsinattachment attachment to injected injected air in hydrophobic, thus promoting its to air using a and collector render the valuable mineral surface overall circuit performance through improved The interaction of flotation and milling been(2009) considered bubbles and subsequent recovery in the froth overflow from an improving control of the milling circuit. Wei andhas Craig show in improving overall circuit performance through improved hydrophobic, thus promoting its attachment to injected air agitated tank. The two most important aspects when bubbles and subsequent recovery in the froth overflow from an improving overall circuit performance through improved The interaction of flotation and milling has been considered inaa bubbles and subsequent recovery in the froth overflow from an hydrophobic, thus promoting its attachment to injected air of the milling circuit. Wei and Craig (2009) show improving overall circuit performance through improved agitated and tank. Thevaluable two most important aspects when curve relating flotation recovery to the particle size distribution control of the milling circuit. Wei and Craig (2009) show bubbles subsequent recovery in important the froth overflow from an control concentrating the minerals through milling and agitated tank. The two most aspects when control of the milling circuit. Wei and Craig (2009) show improving overall circuit performance through improvedaa agitated tank. Thevaluable two most aspects when bubbles and subsequent recovery in important the froth overflow from an curve recovery to the particle size distribution control of the flotation milling Wei and Craig(2013) (2009) show concentrating the minerals through milling and of therelating flotation feed. circuit. Steyn and Sandrock use thisaa curve relating flotation recovery to the particle size distribution agitated tank. Thevaluable two most important aspects when flotation are the grade produced and the recovery achieved. concentrating the minerals through milling and curve relating flotation recovery to the particle size distribution control of the milling circuit. Wei and Craig (2009) show concentrating the minerals throughaspects millingwhen and of agitated tank. Thevaluable two most important the flotation feed. Steyn and Sandrock (2013) use this curve relating flotation recovery to the particle size distribution flotation are the grade produced and the recovery achieved. method but note that “a representative model of the primary of flotation feed. Steyn and Sandrock (2013) use this concentrating the valuable minerals through achieved. milling and curve flotation are the grade produced and the of the therelating flotation feed. Steyn and Sandrock (2013) use this flotation recovery to the particle size distribution flotation areobjectives thethe grade produced and circuit the recovery recovery achieved. concentrating valuable minerals through millingstated and method but note that “a representative model of the primary of the flotation Steyn andperformance Sandrock (2013) use thisa The control the milling are generally mill product sizefeed. to overall plant alone is not method but note that “a representative model of the primary flotation are the grade of produced and the recovery achieved. method but note that “a representative model of the primary of the flotation feed. Steyn and Sandrock (2013) use thisa flotation are the grade produced and the recovery achieved. Thetocontrol control objectives of the milling milling circuit are generally stated mill product size to overall plant performance alone is not method but note that “a representative model of the primary as be providing a product of consistent flow, density and The objectives of the circuit are generally stated trivial task”. mill product size to overall plant performance alone is not aa The control objectives of the milling circuit are generally stated mill product size to overall plant performance alone is not method but note that “a representative model of the primary as to tocontrol be size providing a product product ofaim consistent flow, density and trivial task”. The objectives of theThe milling circuit are generally stated mill product size to overall plant performance alone is not a particle distribution. of a flotation circuit is to as be providing a of consistent flow, density and trivial task”. as tocontrol be size providing a product ofaim consistent flow, density The objectives of theThe milling circuit are generally trivial task”. size mill product to overall plant performance alone is not a particle distribution. of maximum a flotation flotation circuitstated isand to The presence of several MPC implementations in connected as to be providing a product ofaim consistent flow, recovery density and trivial task”. strike a balance between achieving at particle size distribution. The of a circuit is to particle size distribution. The of maximum a flotation circuit isand to as to be providing a product ofaim consistent flow, recovery density trivial task”. presence of several MPC implementations in connected strike balance between achieving at The processes is common in the chemical industry. In distributed The of MPC implementations in connected particle distribution. The aim grades. of maximum a flotation circuitis isthe to economically viable concentrate Recovery strike aaa size balance between achieving recovery at The presence presence of several several MPC implementations in distributed connected strike balance between achieving maximum recovery at particle size distribution. The aim of a flotation circuit is to processes is common in the chemical industry. In The presence of several MPC implementations in connected economically viable concentrate grades. Recovery is the MPC, a series optimisation problems are decomposed into a processes is common in the chemical industry. In distributed strike a balance between achieving maximum recovery at fraction of valuable metal in the feed reporting to the economically viable concentrate grades. Recovery is processes is common in the chemical industry. In distributed The presence of several MPC implementations in connected economically viable concentrate grades. Recovery is the strike a balance between achieving maximum recovery at MPC, a series of optimisation problems are decomposed into a processes is common in the chemical industry. In distributed fraction of valuable metal in the feed reporting to the set of subproblems (Camponogara et al., 2002). The individual MPC, a series of optimisation problems are decomposed into economically viable concentrate grades. Recovery is the concentrate, while the grade is the mass fraction of the metal of fraction of valuable metal in the feed reporting to MPC, a series of optimisation problems are decomposed into aa processes is common in the chemical industry. In distributed fraction of while valuable metalis in the feed reporting to economically viable concentrate grades. Recovery is the set of subproblems (Camponogara et al., 2002). The individual MPC, a series of optimisation problems are decomposed into a concentrate, the grade the mass fraction of the metal of MPCs are referred to as ‘agents’ andet can exchange information. set of (Camponogara al., 2002). The individual fraction valuable metal the feed reporting to the interest inof particular stream. These variables arethe controlled concentrate, the is the mass fraction of metal of set of subproblems subproblems (Camponogara etcan al.,are 2002). The individual MPC, aare series of optimisation problems decomposed into a concentrate, while the grade grade is in the massvariables fraction of the metal of MPCs fraction ofaa while valuable metal in the feed reporting to the referred to as ‘agents’ and exchange information. set of subproblems (Camponogara et al., 2002). The individual interest in particular stream. These are controlled MPCs are as information. concentrate, while the grade is the massvariables fraction of the metal of set interest particular stream. These are controlled are referred referred to to as ‘agents’ ‘agents’ and andetcan can exchange information. of subproblems (Camponogara al.,exchange 2002). The individual interest in in aa while particular stream. These arethe controlled concentrate, the grade is the massvariables fraction of metal of MPCs are referred to as ‘agents’ and can exchange information. interest in a particular stream. These variables are controlled MPCs MPCs are referred to as ‘agents’ and can exchange information. 2405-8963in© IFAC (International Federation of Automatic Control) interest particular stream. These variables are controlled Copyright ©a2019, 2019 IFAC 24 Hosting by Elsevier Ltd. All rights reserved. 24 Peer review©under of International Federation of Automatic Copyright 2019 responsibility IFAC 24 Control. Copyright © 24 Copyright © 2019 2019 IFAC IFAC 24 10.1016/j.ifacol.2019.09.158 Copyright © 2019 IFAC 24 Copyright © 2019 IFAC 24
24 24 24 24 24
2019 IFAC MMM Stellenbosch, South Africa, August 28-30, 2019 Kevin Brooks et al. / IFAC PapersOnLine 52-14 (2019) 24–29
25
Distributed MPC has been app lied to power generation and to the chemical industry (Venkat et al., 2008), however – to the authors’ knowledge – not in the minerals processing industry. This article investigates the performance of two individual controllers for milling and flotation versus the performance of a combined, large-scale MPC controller for both processes combined. The paper is structured as follows. Section 2 briefly describes the milling and flotation processes studied here. Section 3 gives the linear process model derived from step responses. Section 4 describes the MPC control strategy both for two individual controllers and for one combined controller, comparing the results of both control strategies. Conclusions are drawn in Section 6. 2. PROCESS DESCRIPTION Fig. 2. Rougher Bank Showing Control Loops and Analysers
The process studied consists of a SAG mill feeding a flotation plant consisting of two rougher cells. Although industrial plants have many more cells, this configuration was chosen for simplicity. The mill is shown in Fig. 1. Run of Mine ore is fed to the mill together with the recycle flow of screen oversize and water. The mill reduces the particle size and the slurry exits the mill into the discharge sump. The screen undersize flows to the surge tank from where it is pumped to the first rougher flotation cell.
by plant testing. The models required are represented as unit step responses. The dynamic response of a controlled variable (CV or output) to a unit step in an MV (manipulated variable or input) or FF (feedforward or disturbance variable). The model prediction of an output is then given by (Garcia et al., 1989): 𝑦𝑦(𝑘𝑘) = ∑𝑛𝑛−1 𝑖𝑖=1 𝐻𝐻𝑖𝑖 ∆𝑢𝑢(𝑘𝑘 − 1) + 𝐻𝐻𝑛𝑛 𝑢𝑢(𝑘𝑘 − 𝑛𝑛)
(1)
where y(k) is an output at discrete time k, n is the number of truncated coefficients, u is the input and Hi as well as Hn are the step response coefficients. The difference equation for the input is abbreviated as Δu(k) = u(k) – u(k–1). The extension of Eq. 1 to the case where there are multiple inputs is straightforward since the assumption of linearity allows for superposition. The step response coefficients, Hi, are obtained by performing a planned experiment. The data of the experiment is used in the fitting routine. In the software used here, either finite impulse response or a subspace technique is used for the fitting (Verhaegen and Dewilde, 1992, Darby et. al., 2009). The linear dynamic models for the two plants are shown in the form of unit step responses in Fig. 3 and Fig. 4. The manipulated variables (MVs) and feedforwards (FFs) are down the rows, while the controlled variables (CVs) are across the columns. The SAG mill MVs are the solids feed rate, the inlet water, discharge sump and surge tank water addition, and flows out of the discharge sump and surge tank. The CVs are the load, power, discharge sump and surge tank level, the discharge sump and product densities and the product PSD. The manipulated variables for flotation are the pulp level, the two air flows and reagent addition flowrate. The controlled variables are concentrate and tailings grade, the reagent dosage (amount of reagent per tonne of metal in feed), and the calculated steady-state recovery. The feedforward variables are the feed flow, grade, density and PSD.
Fig. 1. SAG Mill Showing Base Level Controls It is assumed that online measurement of the densities and the particle size distribution (PSD) of feed and product are available. The rougher flotation circuit is shown in Fig. 2. The slurry flows by gravity from Cell 1 to Cell 2. The froth to pulp interface level is controlled in Cell 2, while the air addition rate is controlled to a setpoint in both Flotation cells. A reagent, known as a collector, is added to Cell 1. Composition measurements are available for the feed, concentrate and tails streams. 3. LINEAR MODELS
The inclusion of calculated dosages as CVs with reagent flows as MVs is unusual but allows the controller to smoothly change reagent flows on changes of total metal feed. This functionality can be included by use of PID control; this method, however, neglects the dynamics associated with rejecting the disturbance.
AspenTech’s DMCPlus was used as the MPC engine for this study. This algorithm employs a description of the plant in the form of finite step responses, which are generally obtained
25
2019 IFAC MMM 26 Kevin Brooks et al. / IFAC PapersOnLine 52-14 (2019) 24–29 Stellenbosch, South Africa, August 28-30, 2019
Fig. 3. Unit Step Response Models for the Mill with time scale in minutes
Fig. 4. Unit Step Response Models for Flotation with time scale in minutes. The time scales have the units of minutes; the indicated time to steady-state is ten minutes. The responses have been sped up by a factor of 6 for ease of simulation. The actual times to steady-state of the systems are in the region of sixty minutes
The DMC Plus algorithm has been described by Garcia et al., (1989). The equations will not be repeated here. Suffice it to say that a quadratic program formulation is used to solve for an optimal move plan subject to MV and CV constraints.
𝑦𝑦̂𝑖𝑖,𝑠𝑠𝑠𝑠 ≥ 𝑦𝑦𝑖𝑖,𝑚𝑚𝑚𝑚𝑚𝑚 − 𝜀𝜀𝑖𝑖
(4)
In order to facilitate the handling of multiple constraints, a rank can be assigned to each dependent variable. This allows the controller to relax a dependent variable’s constraints as described in Eq. 3 and Eq. 4 in order to honour another higherranked variable’s constraints.
Before solving the dynamic optimisation problem, the algorithm checks for steady-state feasibility by solving the optimisation problem: 𝑛𝑛
(3)
where 𝜙𝜙 is the objective function value, nc is the number of controlled variables, 𝜀𝜀𝑖𝑖 are slack variables, 𝑊𝑊𝑖𝑖 are weights, 𝑦𝑦̂𝑖𝑖,𝑠𝑠𝑠𝑠 is the steady-state value of CV i and 𝑦𝑦𝑖𝑖,𝑚𝑚𝑚𝑚𝑚𝑚 and 𝑦𝑦𝑖𝑖,𝑚𝑚𝑚𝑚𝑚𝑚 are the high and low limits on CV i.
4. OPTIMISATION AND THE DMC ALGORITHM
𝑐𝑐 min 𝜙𝜙 = ∑𝑖𝑖=1 𝜖𝜖𝑖𝑖2 𝑊𝑊𝑖𝑖
𝑦𝑦̂𝑖𝑖,𝑠𝑠𝑠𝑠 ≤ 𝑦𝑦𝑖𝑖,𝑚𝑚𝑚𝑚𝑚𝑚 + 𝜀𝜀𝑖𝑖
The objective function in Eq. 2 is minimised subject to the current constraints on the MVs and CVs. If there are slack
(2)
subject to:
26
2019 IFAC MMM Stellenbosch, South Africa, August 28-30, 2019
Kevin Brooks et al. / IFAC PapersOnLine 52-14 (2019) 24–29
27
Figure 5. A subset of Mill MPC dynamics in response to a limit change at Product PSD at time 05:04:00 (bottom right subplot). variables that are non-zero, this implies that the steady-state solution is not feasible. In this case, the CV limits are relaxed until the solution is feasible.
The mill controller is configured to maximise the feed subject to the current constraints on densities and the particle size distribution (PSD) of the product. The controller includes three CVs that have integrating models; the load, sump level and rougher feed level. The first of these is configured to control to a value since tight control of load improves grinding performance. The latter two are configured to use as much as possible of the surge capacity. This assists in smoothing the flows out; this is advantageous in stabilising the flotation feed.
Should the minimisation of Eq. 2 prove that all the slack variable values are zero, then there exist one or more feasible steady-state solutions. In this case, the following economic optimisation is solved: 𝑁𝑁
𝑁𝑁𝑗𝑗
𝑖𝑖 min 𝐽𝐽 = ∑𝑖𝑖=1 𝑐𝑐𝑖𝑖 Δ𝑢𝑢𝑖𝑖,𝑠𝑠𝑠𝑠 + ∑𝑗𝑗=1 𝑐𝑐𝑗𝑗 |Δ𝑢𝑢𝑗𝑗,𝑠𝑠𝑠𝑠 |
(5)
The flotation controller has the control objectives to control the overall reagent dosage in a range while maintaining concentrate and tail grade between limits. The optimisation objective of the controller is to maximise recovery.
where J is the objective function value, Ni is the number of MVs that have economic directions, Nj is the number of MVs whose movement is to be minimised and ci and cj are cost factors.
The two individual controllers give good results in controlling the plants in the specified ranges and reacting to disturbances. The results of increasing the PSD limit percentage is shown in Fig. 5. This can be seen in the red lower curve of PROD_PSD, the PSD after milling, which is changed from 80% to 85%. The MPC reacts smoothly and in a coordinated fashion. Despite the objective to maximise feed, the MPC reduces feed as this is the only method to ensure the finer grind. The vertical dashed line represents current time; the plots extend two times to steadystate in the past, and one in the future.
The Δ𝑢𝑢𝑖𝑖,𝑠𝑠𝑠𝑠 and Δ𝑢𝑢𝑗𝑗,𝑠𝑠𝑠𝑠 are the changes in the values of the MVs at the present time to those at steady-state. These are the variables chosen to minimise Eq. 5 subject to the current MV and CV constraints. The steady-state values of the MVs are imposed on the dynamic solution at the end of the control horizon. 4.1 TWO MODEL PREDICTIVE CONTROLLERS The model matrices as derived through the step responses in Fig. 3 and Fig. 4 were used to implement model predictive control on each unit separately. For this study, the plant and the controller have identical models i.e. no plant model mismatch is included.
The flotation controller manages the competing objective of grade and recovery well. This can be seen in Fig. 6 where the MPC responds to an increase in the recovery low limit of one percent. Control is smooth; the increased recovery is achieved 27
2019 IFAC MMM 28 Kevin Brooks et al. / IFAC PapersOnLine 52-14 (2019) 24–29 Stellenbosch, South Africa, August 28-30, 2019
at the cost of the use of extra reagent. In this formulation, the effect of the mill parameters on the flotation bank is modelled by including variables feed flow, density, grade and PSD as feed-forward variables in the flotation controller. This approach does not leverage the information on future predictions of CVs and MVs that are available from the mill controller.
(CV) PROD_PSD (FF) Milling
(CV) RGH_DENS (FF)
Flotation
(MV) RGH_FD (FF) Fig. 7. Linking of milling and flotation through CVs (outputs) and MVs to feed forwards.
The two separate controllers are sub-optimal in the sense that the flotation controller cannot affect the variables that are fed to it. For instance, if the economics are such that increased recovery requires a finer feed, this change would have to be made by a human rather than the system. For these reasons combining of the two MPCs is investigated.
To combine the controllers some thought must be given to the resulting models and how they should be formulated. If an MV in an upstream controller is repeated as a feed-forward in a downstream controller then this is straightforward. The replacement of the feed-forward by a manipulated variable and its associated move-plan yields an immediate predictive advantage. The case is more complicated if a CV in an upstream controller is used as a feed-forward in a downstream controller. One method would be to re-visit the data originally used for model development and attempt to derive the models of the effect of mill inputs on flotation outputs. This approach has some difficulties, since
Fig. 8. Convoluted Models for Grades, Dosage and Recovery versus Mill MVs identifying the effects in the face of other disturbances as well as the relatively long settling times can be challenging. It is possible to derive the required models by mathematical manipulation. In this case, the models of the downstream controller are found by convolution of the upstream CV model with the downstream CV models that depend on the feedforward: 𝐶𝐶(𝑗𝑗, 𝑘𝑘) = ∑𝑝𝑝 ∑𝑞𝑞 𝐴𝐴𝑈𝑈 (𝑝𝑝, 𝑞𝑞)𝐴𝐴𝐷𝐷 (𝑗𝑗 − 𝑝𝑝 + 1, 𝑘𝑘 − 𝑞𝑞 + 1)
(6)
where AU is the matrix of the upstream (milling) and AD the matrix of the downstream system (flotation). As an example, the rougher feed density and product PSD are CVs in the mill controller and a feed-forward in the flotation controller, while the rougher feed flow is an MV in the mill controller. On combining the matrices, models are formed for the grades, reagent dosage and recovery versus these three mill MVs.
Fig. 6. Flotation MPC: Change in Recovery Low Limit at 04:38 4.2 THE COMBINED CONTROLLER The aim of combining the two MPCs is to achieve tighter control, particularly in the reaction of the flotation circuit to upstream disturbances. In addition, it is hoped that a superior operating point will be achieved. Fig. 7. outlines the signals that link the behaviour of the two systems.
The time to steady state of the convoluted model is double that of the individual models. This reflects the fact that two systems that operate in series have been connected from a modelling point of view. This longer time to steady state can have implications for the control of the system. In principle, the convolution of Eq. 6 removes the intermediate variables from 28
2019 IFAC MMM Stellenbosch, South Africa, August 28-30, 2019 Kevin Brooks et al. / IFAC PapersOnLine 52-14 (2019) 24–29
the matrix altogether. The variable can be retained in the matrix if desired, but care must then be taken not to over specify the problem.
29
The approach is by no means limited to the unit operations considered. As the use of MPC becomes more commonplace is the mineral processing industry, larger controllers will evolve with improved optimisation scope.
The results of a change in the feed PSD for the combined controller are shown in Fig. 9. In this case, the controller recognises that there will be a drop in recovery and increases the air to the flotation cells.
REFERENCES Bergh, L. G., & Yianatos, J. B. (2011). The long way toward multivariate predictive control of flotation processes. Journal of Process Control, 21(2), 226-234. Brooks, K.S. and Koorts, R., 2017. Model Predictive Control of a Zinc Flotation Bank Using Online X-ray Fluorescence Analysers. IFAC-PapersOnLine, 50(1), pp.10214-10219. Brooks, K. and Munalula, W., 2017. Flotation Velocity and Grade Control Using Cascaded Model Predictive Controllers. IFAC-PapersOnLine, 50(2), pp.25-30. Camponogara, E., Jia, D., Krogh, B. H., & Talukdar, S. (2002). Distributed model predictive control. IEEE control systems magazine, 22(1), 44-52. Cortés, G., Verdugo, M., Fuenzalida, R., Cerda, J., & Cubillos, E. (2008). Rougher flotation multivariable predictive control: concentrator A-1 division Codelco Norte. Proceedings of the V International Mineral Processing Seminar 1(6) 316-325. Darby, M.L., Harmse, M. and Nikolaou, M., 2009. MPC: Current practice and challenges. IFAC Proceedings Volumes, 42(11), pp.86-98. Dawson, P. and Koorts, R., 2014. Flotation Control Incorporating Fuzzy Logic and Image Analysis. IFAC Proceedings Volumes, 47(3), pp.352-357. Garcia, C.E., Prett, D.M. and Morari, M (1989). Model Predictive Control: Theory and Practice – A Survey. Automatica 25(3), 335-348. Karelovic, P., Razzetto, R. and Cipriano, A., 2013. Evaluation of MPC strategies for mineral grinding. IFAC Proceedings Volumes, 46(16), pp.230-235. Silva, D.A. and Tapia, L.A., 2009. Experiences and lessons with advanced control systems for the SAG mill control in Minera Los Pelambres. IFAC Proceedings Volumes, 42(23), pp.25-30. Steyn, C.W., Brooks, K.S., De Villiers, P.G.R., Muller, D. and Humphries, G., 2010. A Holistic Approach to Control and Optimization of an Industrial R Ball Milling Circuit. IFAC Proceedings Volumes, 43(9), pp.137-141. Steyn, C.W. and Sandrock, C., 2013. Benefits of optimisation and model predictive control on a fully autogenous mill with variable speed. Minerals Engineering, 53, pp.113-123. Venkat, A. N., Hiskens, I. A., Rawlings, J. B., & Wright, S. J. (2008). Distributed MPC strategies with application to power system automatic generation control. IEEE transactions on control systems technology, 16(6), 11921206. Verhaegen, M. and Dewilde, P., 1992. Subspace model identification part 2. Analysis of the elementary outputerror state-space model identification algorithm. International journal of control, 56(5), pp.1211-1241. Wei, D. and Craig, I.K., 2009. Economic performance assessment of two ROM ore milling circuit controllers. Minerals Engineering, 22(9-10), pp.826-839.
Fig. 9. Combined MPC: Increase in feed PSD at 10.22 5. CONCLUSION MPC on the commonly found mineral processing operations of milling and flotation is now an industrial reality. This work demonstrates that by careful consideration of common variables, these MPCs may be combined in a mathematically rigorous manner. The resulting larger controller displays more co-ordinated behaviour and can adjust milling circuit parameters to meet flotation circuit requirements. The combined controller presented here has the potential to improve overall circuit performance beyond that achieved by two local MPCs. The size of the application is not unusually large when measured against applications in the petrochemical industry.
29