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Multivariable Predictive Control of a Pilot Flotation Column

Danny Calisaya*, ric Poulin*, Andr Desbiens*, Ren del Villar, Alberto Riquelme**Department of Electrical and Computer Engineering

Department of Mining, Materials and Metallurgical Engineering

LOOP (Laboratoire dobservation et doptimisation des procds), Universit Laval, Qubec, QC, Canada

AbstractThe aim of this work is the control of hydrodynamic

variables of a pilot flotation column working with a three-phase

system (air-water-ore) in an industrial environment. Since

hydrodynamic variables are closely related to metallurgical and

economical performances of the unit, the implementation of

such a control strategy is crucial to optimize its operation. The

hydrodynamic variables here considered are the gas hold-up in

the collection zone and the fraction of wash-water underneath

the interface. They are controlled by manipulating gas flow rate

and wash-water flow rate respectively. Hydrodynamic variables

are controlled through a constrained model predictive control

(MPC) strategy. This choice is dictated by the interdependencyof these controlled variables and the capability of predictive

controllers to handle process constraints. Identification tests

lead to a representative model of the system under nominal

operating conditions but indicate variations of process behavior

with changing operating regimes. Results show good control

performances, confirming the potential use of MPC to control

hydrodynamic variables for real-time optimization of full-scale

flotation columns.

I.INTRODUCTION

N the effort to enrich the ore from the mine, mineralseparation by flotation columns is an important area of

research for the metallurgical industry. Although the firstcommercial use of this process goes back almost 30 years,the research on the control of hydrodynamic variables of the

process as well as sensors to measure them are not yet fullydeveloped. This step is necessary for the development andthe implementation of an indirect (or hierarchical) real-timeoptimization strategy [1]. For the first time, simultaneousconstrained control of gas hold-up in the collection zone andthe fraction of wash-water underneath the interface in anindustrial environment are presented in this paper.

Different advanced control techniques, such as adaptiveself-tuning control [2], fuzzy predictive control [3],decentralized control [4], multivariable nonlinear predictivecontrol [5], or predictive control based on a neural network

model [6] have been tested for optimizing and controlling thecolumn flotation process. From all these techniques, themodel predictive control (MPC) is the interest of severalresearchers given its capability of handling interactions andconstraints which is necessary for real-time optimization. Tomention a few, constrained nonlinear predictive control

based on IMC-optimization [7], multivariable predictivecontrol of laboratory flotation columns, [8]-[10] could becited as examples. Extensive and critical reviews regarding

instrumentation, control, and optimization of flotationcolumns can be found in [11] and [1].

This article presents the development, the implementation,and the evaluation of MPC to control strategic hydrodynamicvariables of a three phase (air-water-ore) pilot flotationcolumn in an industrial environment (Agnico-Eagle, LarondeDivision). The hydrodynamic variables are the volumetricwash-water fraction underneath the interface wand the gashold-up g in the collection zone. This work extends a

previous study achieved on a two-phase (water-air) system

[12], where bias rate, i.e. net downward flow rate of waterthrough the interface, is the variable used to assess thecleanliness of the froth [13]. In the present context, theestimation of bias is done using the volumetric wash-waterfraction underneath the interface instead of bias directlysince these variables are linearly related [14] andmeasurement of w is simpler. The method could be easilyimplemented on a full scale column using a commercial gsensor and the proposed sensor for w. Such an approachwould not require modification of existing local controlloops.

Section II gives a brief description of the flotation columnprocess and introduces manipulated and controlled variablesas well as the related instrumentation. Section III discussesthe identification of the system and the validation ofresulting dynamic linear models. The MPC algorithm andtuning is presented in section IV. Section V presentsindustrial results and analyses controller performances,whereas conclusions are given in Section VI.

II.PROCESS DESCRIPTION

Fig. 1 is a schematic of a flotation column showing itsthree input streams: pulp feed, air injection, and wash-wateraddition, as well as its two output streams: concentrate andtailings. In normal operating conditions, the column contentshows two distinct regions in terms of the amount of air

content: the collection zone in the lower part, with less than30% air, and the cleaning zone (upper part) with more than70% air.

To perform the separation between valuable mineral andgangue, some chemical reagents (collectors, frothers,activators or depressants, and pH modifiers) are usuallyadded, either at an earlier stage (conditioning) or at thecolumn itself. The produced froth carries the valuablemineral particles, due to its hydrophobic properties (naturalor induced) and overflows into the column concentrate

I

2012 American Control Conference

Fairmont Queen Elizabeth, Montral, Canada

June 27-June 29, 2012

978-1-4577-1096-4/12/$26.00 2012 AACC 4022


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launder, whereas the hydrophilic particles (gangue) arewithdrawn through the bottom port as tailings. The addedwash-water helps in improving the concentrate quality,releasing the entrained undesirable particles from the froth [1].

Fig. 1. Diagram of a flotation column

The pilot flotation column used in this work consists in a7.32 m height and 151 mm diameter Plexiglas tube, providedwith the necessary feed and tailing ports, froth overflowlaunder, a perforated ring for wash-water addition over thefroth, and a porous sparger at the bottom for air injection.The column is fully automated, with local control loops toregulate all flow rates (tailings, wash-water, and air) andfroth-depth, as shown in Fig. 2. A controller forhydrodynamic variables (w and g) and, obviously, thenecessary sensors for measuring all relevant variables, aredetailed here after. The pulp processed during testscorresponds to the feed of the 3rd cleaner column of thecopper circuit, containing 15% solids and having 99% of theore particle-size smaller than 100 m.

The froth depth (interface) is measured using theconductivity profile based sensor and determined by analgorithm relying on the largest slope method developed by[15], a weighted average interpolation of the two possiblefroth depth values [16]. The conductivities are measured by aField-Programmable Gate Array (FPGA) [17].

The measure of gas hold-up (volumetric fraction of gas) inthe collection zone is made using a sensor installed near theinterface and the method described by [18]-[19] andevaluated by Maxwell's equation for electrolyte mixtures[20], proposed by [19], and used by [12]. The final relation

for measuring the gas hold-up is:

sglsl

sglslg

kk

kk

5.0100

(1)

where ksl is the conductivity of pulp only (solid and liquid)and ksgl is the conductivity of pulp-gas mixture. These aremeasured by the siphon and open cells respectively (Fig. 2).

The conventional bias rate variable is replaced in this workby the concept of fraction of wash-water underneath the

interface proposed by [14] and adapted by [21]. Consideringthat the gas hold-up is measured sufficiently close to theinterface and assuming no solid conductivity, then theadditivity rule in a three-phase system leads to the followingexpression:

)1(

1

15.0'

100swf

g

gsglf

w kk

kk

(2)

where kf is the conductivity of feed pulp, ksglis the

conductivity of pulp-gas mixture underneath the interface, kw

is the conductivity of wash-water ands

is the percentage

of solid into pulp. More details about this relation are foundin [21].

Fig. 2. Process and instrumentation diagram of the pilot flotation

column

The first manipulated variable is the gas superficialvelocity. It is calculated from the gas mass flow meter

reading refg at reference conditions (21.1oC and 101.3

kPa) and the gas specific gravity refg :

crefg

refg

c

refgref

gAA

QJ

(3)

Tailings

Air (Jg)

Concentrate

Wash water

(Jw)

Ew

g

Collection

zone

Cleaningzone

Feed

Interface

w

PT

FPGA

Wash

Water

Feed

Concentrate

Tailings

Input

Opto 22

TT2

Speed

Reference

Level

Set-Point

Air

Set-Point

FT

FIC

Mass Flow

Controller

Wash Water

Set-Point

LC

Air

Syphon

Cell

Open

Cell

TT1

FC FT

Conductivities

Level

Estimation

4023


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where refgQ is the gas volumetric flow rate and Acis the cross

sectional area of column. The superficial gas velocityJgmustbe compensated for temperature and pressure at testconditions as shown below:

16.294

15.273

23.1033

23.1033 T

PJJ refgg (4)

where the absolute pressure P is measured in cm H2O andthe temperature Tis in Celsius degrees. The pressure PT andtemperature sensors TT1 are shown in Fig. 2

The second manipulated variable is the wash-watersuperficial velocity Jw, obtained by dividing the volumetricflow rate by the column cross sectional area:

c

ww

A

QJ (5)

The froth depth is controlled using a PI controller

manipulating the speed of tailings pump. The wash-waterflow rate is controlled by a PI controller implemented with aMoore Mycro 353 controller. Graphical interfaces and dataacquisition are performed by a HMI/SCADA softwareiFIX working under a Windows XP operating system.An Opto 22 I/O system is used to centralize sensor andactuator signals as shown in Figs. 2 and 3. Algorithms for thefroth depth control and multivariable predictive control of gand ware implemented in MatLab. All signals are sampledat a two seconds interval.

Fig. 3. HMI/SACAD system of the pilot flotation column

III. IDENTIFICATION OF THE PROCESS TRANSFER FUNCTIONS

The identification of the process transfer functions hasbeen made using the iterative prediction error minimizationmethod provided by MatLab System IdentificationToolbox. Based on prior knowledge about the system, low-order continuous-time models were selected [12].

The identified transfer functions, and the standard-deviation of each parameter, are as follows:

1)2.62(2.358

)3.38(2.272

)(

)()(

)2.2(74

,

s

e

sJ

ssG

s

w

wJww

(6)

1)2.10(5.2

)5.1(04.0

)(

)()(

)7.2(2.56

,

s

e

sJ

ssG

s

w

gJwg

(7)

1)2.77(195

)3.12(2.45

)(

)()(

)2.4(8.30

,

s

e

sJ

ssG

s

g

wJgw

(8)

1)4.27(8.89

)7.4(0.25

)(

)()(

)1.0(8.74

,

s

e

sJ

ssG

s

g

gJgg

(9)

The standard deviation of the gain, time constant, anddelay of ww JG , and gg JG , is less than 25%, which

allows to consider the model found as a correctapproximation of the real process. Equation 7 shows astandard deviation larger than nominal values for the gainand time constant suggesting that the relation betweenJwandg is not significant, which is reasonable from a physical

point of view. The addition of wash-water has few impacts

on the gas hold-up in the collection zone. The transferfunction wg JG , will thus be considered as zero for control

purposes. The standard deviation of the parameters of

gw JG

, are in the range of 35%. This slightly high standard

deviation can be explained by disturbances within theprocess (variations in the feed composition) during tests.

Fig. 4. Identification and validation data forww J

G, and

wg JG , : input/output data (-), model (- -)

Speed

reference

(Feed)RS232/

Modbus

FPGAMoore(Mycro

353)

Windows XP

MatLab

IFix RS 485

I/O Opto22

Speed

reference

(Wash Water)

Air Set-Point

Electrodes

Temperatures Pressure Air

flow rate

Wash Water

flow rate

Speed

reference

(Tailing)

g

(%)

w

(%)

Jw

(cm/s)

g

(%)

w

(%)

Jw

(cm/s)

Time s

Time (s)

Identification

Validation

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Fig. 4 shows the fit of models for identification andvalidation data for

ww JG

, and wg JG , where the operating

point has been removed. In case of ww JG , , both

identification and validation tests shows a good fit. Asmentioned above for wg JG , , no strong relation betweenJw

and g can be observed in Fig. 4. The identification andvalidation tests for gw JG , and gg JG , are presented in

Fig. 5. For both transfer functions, the model fitting isreasonable.

Finally, the percentage of variance explained by thedifferent models (best fit) are gathered in Table I foridentification and validation data. In case of ww JG , and

gg JG

, transfer functions, the percentage for validation

data is higher than the one obtained for identification data.This is explained by the relatively higher disturbance levelduring identification test. Regarding wg JG , , the best fit is

nearly zero as expected.

Fig. 5. Identification and validation data forgw J

G, and gg J

G , :

input/output data (-), model (- -)TABLE I

EXPLAINED VARIANCE FOR IDENTIFICATION AND VALIDATION DATA

Transfer Function Identification Validation

ww JG

, 50 % 63 %

wg JG

, 1 % 0.5 %

gw JG , 41 % 34 %

gg JG

,

68 % 75 %

In summary, the transfer functions of the system used fordesigning the controller in the next section are:

)(

)(

190

250

1195

45

1358

272

)(

)(

75

3074

sJ

sJ

s

e

s

e

s

e

s

s

g

w

s

ss

g

w

(10)

IV.DESIGN OF MPC

The control strategy used for the pilot flotation column wasdeveloped using the MatLab MPC toolbox. The MPC

controller output, Tn kukuku u )()()( , where nu is thenumber of manipulated variables, is defined by:

)|()1()( kkukuku (11)

It is the result of the minimization of the following

optimization problem with respect to the sequence of inputincrements and to the slack variables:

u

p y

c

n

j

ju

jj

H

i

n

j

jyj

ekkHukku

ekikuwikr

kikyw

1

222

1

0 1)),|1(),...|((

)|()1(

)|1(min

(12)

The optimization problem is subject to followingconstraints:

)()/()( iukikuiu jmaxjjmin (13)

eiykikyeiy jmaxjjmin )()/()( (14)

0)/( khku , 1,, pc HHh (15)

0e (16)

where )|( kku is the first element of the optimal sequence,

Hc and Hp are the control and prediction horizon

respectively, ny is the number of system outputs,yj

w is the

weight for the output variablej, ujw is the weight for change

in each manipulated variable and ju is the change in

manipulated variable j.The predicted output is jy and jr is

the reference value over the whole prediction horizon.The slack variable e is used to relax the constraints

imposed to controlled variables. The weight on the slackvariable penalizes the violation of constraints and is given

by:u

jyj ww

,max10

5 (17)

The larger is with respect to input and output weights, themore the constraint violation is penalized as explained in [22].

g

(%)

w

(%)

Jg

(cm/s

)

g

(%)

w

(%)

Jg

(cm/s)

Time (s)

Time (s)

Identification

Validation

4025


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The global specification for MPC tuning was to obtain aclosed-loop response with dynamics similar to the open-loopsystem with a limited coupling between g and w. Theselected control period is 10 seconds. This value takes intoaccount the overall dynamics of the system whereas allowingthe controller to properly reject disturbances. The predictionhorizon Hp was set to 150 control periods and Hc to one.Constraints imposed to the manipulated and controlled

variables as well as operating points and are listed inTable II.

TABLE IICONSTRAINTS ON MANIPULATED AND CONTROLLED VARIABLES

Variable Unit Minimum Maximum

wJ cm/s 0.05 0.18

gJ cm/s 0.1 0.5

w

% 10 40

g % 5 20

The most critical constraints are the maximum of Jw, theminimum of w, and the minimum of Jg. The first preventsbreaking the froth, the second ensures its cleanliness, and thelast is necessary to sustain froth. Other constraints ensureworking within the preselected and feasible operating range.Weight values for manipulated and controlled variables aregiven in Table III.

Fig. 6 presents a simulation of the closed-loop system. Itshows a set-point change onw at 100 seconds and two set-

point changes on gat 2 000 and 4 000 seconds. For each set-point change, the controller smoothly adjusts the manipulatedvariables to reach the reference within an acceptable timeresponse whereas the coupling effect on wis limited whichis coherent with the previously stated specification.

Fig. 6. Simulation of control performance: manipulated/controlledvariable (-), set-point (- -), constraint (- .)

TABLE IIIWEIGHTS ON MANIPULATED AND CONTROLLED VARIABLES

Variable Weight (wu) Variable Weight (wy)

wJ 210

w

0.26

gJ 210 g

0.26

V.INDUSTRIAL RESULTS

The controller designed in Section IV was implemented onthe pilot flotation column with the same parameter settings.The froth depth set-point was kept constant at 36 cm duringtests.

Fig. 7 presents control performances obtained for the firsttest (Test A). The set-point of the fraction of wash-waterunderneath the interface was changed at 100, 2 000, 3 600,5 400 and 7 600 seconds, whereas the set-point of gas hold-up was kept constant. For the first three steps on w, it isworth noticing that the manipulated variable Jw ismomentarily saturated. The time response for w isacceptable but the control action Jw is slightly more

aggressive than the one observed in simulation. As expected,set-point changes on whas few impacts on g.

Fig. 7. Test A: manipulated/controlled variable (-), set-point (- -),

constraint (- .)

Fig. 8 presents the results of the second test (Test B) wherea disturbance on wwas introduced at 450 seconds and thegas hold-up set-point changed at 1 600, 3 600 and 6 400seconds. In this case, control actions are rather smoothindicating that the time response could be reduced by a

proper readjustment of MPC parameters. However, both testresults generally comply with performance obtained insimulation.

Jg

(cm/s)

g

(%)

Jw

(cm/s)

w

(%)

Time (s)

Jg

(cm/s)

g

(%)

Jw

(cm/s)

w

(%)

Time (s)

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Fig. 8. Test B: manipulated/controlled variable (-), set-point (- -)

VI.CONCLUSIONS

The paper described the development, implementation andevaluation of a controller for hydrodynamic variables of a

pilot flotation column operating in an industrial environment.The considered variables are the gas hold-up and the wash-water fraction underneath the interface. The selected controlalgorithm is constrained multivariable MPC. Results haveshown that MPC is able to control the process whilerespecting constraints and has delivered interesting

performances in the context of an industrial campaign for

both set-point tracking and disturbance rejection, undervarious operating conditions. It is thus reasonable to considerthat such a controller could be successfully integrated in areal-time optimization strategy. Further works will notablyfocus on finding relationships between froth depth, gas hold-up and wash water fraction underneath the interface and themetallurgical and economic performance of the unit. Thefinal objective is the real-time optimization of a full-scalecolumn.

ACKNOWLEDGEMENT

The authors would like to acknowledge Agnico-Eagle

Laronde Division, XStrata-Nickel, COREM and NSERC forproviding the necessary funding for this project, and toAgnico-Eagle personnel for on-site support.

REFERENCES

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[3] S. M. Vieira, J. M. C. Sousa, and F. O. Durao, "Real-time fuzzypredictive control of a column flotation Process," Fuzzy SystemsConf., 2007. FUZZ-IEEE, London, pp. 1-6, 2007.

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[8] O. D. Chuk, V. Mut, E. Nez, and L. Gutierrez, "Multivariablepredictive control of froth depth and gas holdup in column flotation,"Tokyo, Japan, pp. 87-91, 2001.

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pp. 19-24, 2010.

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[12] M. Maldonado, A. Desbiens, and R. del Villar, "Potential use of modelpredictive control for optimizing the column flotation process.," Int.Journal of Mineral Proc., 2009.

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[16] M. Maldonado, A. Desbiens, and R. del Villar, "An update on theestimation of the froth depth using conductivity measurements,"Minerals Engineering, vol. 21, no. 12-14, pp. 856860, 2008.

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[19] F. Tavera and R. Escudero, "Gas hold-up and solids hold-up inflotation columns: on-line measurement based on electricalconductivity," Transactions of the Institutions in MMM Proc. andExtractive Metallurgy, vol. 11, 2002.

[20] J. C. Maxwell, A Treatise of electricity and magnetism OxfordUniversity press, Oxford University Press, London UK, 1892.

[21] R.-M. Estban, "Validation industrielle de la mesure du diffrentieldeau dans une colonne de flottation." MS thesis, Department ofMining, Materials and Metallurgical Engineering, Laval University,Qubec, Canada, 2011.

[22] H. Mukai and E. Polak, "A second-order method for the generalnonlinear programming problem," Journal of Optimization Theory andApplications, vol. 26, no. 4, pp. 515-532, 1978.

Jg

(cm/s)

g

(%)

Jw

(cm/s)

w

(%)

Time (s)

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