Minerals Engineering 16 (2003) 10611068 This article is also available online at: www.elsevier.

com/locate/mineng

Level control strategies for otation cells

Laboratory of Process Control and Automation, Helsinki University of Technology, Kemistintie 1, 02150 Espoo, Finland

Abstract Flotation is a dicult process to run eciently. One way to make otation performance better is to improve cell level control. However, controlling pulp levels in otation cells is a complex control task because of strong interactions between the levels in otation cells. Therefore advanced controllers are needed to give good level control. This paper deals with a model of six otation cells in series. Simulations are performed to compare dierent control strategies. Four control strategies are considered: one SISO controller and three dierent MIMO controllers including a new multivariable controller. It is shown that level control performances of the MIMO controllers are signicantly better than that of the classical SISO controller. 2003 Elsevier Ltd. All rights reserved.
Keywords: Flotation machines; Froth otation; Process control; Mineral processing

1. Introduction Level control of otation cells is a very complex task due to high interactions between the process variables. A control action implemented at any point in the otation circuit tends to be transmitted to both upstream and downstream units, and sometimes with amplication. Large variations in the ow rate to the rst cell and varying composition of the raw ore also cause problems. Flotation cells are conventionally controlled by isolated PI-controllers. PI control works well when the cell being controlled is isolated. However, in a otation circuit where interactions are strong, PI control does not meet the requirements of high control performance. Hence a considerable amount of research has been carried out over the last few years to develop better control techniques for otation circuits (J ams a-Jounela et al., 2001). Niemi et al. (1974), Koivo and Cojocariu (1977) used a single cell model when developing an optimal control algorithm via applications of the maximum principle. Andersen et al. (1979) and Zargiza and Herbst (1987) reported an application of state feedback control and a Kalman lter for rougher otation control. Hammoude and Smith (1981) used a linear model to develop a minimum-variance controller for recleaning. New advanced
*

control method has been also recommended for the control of otation processes by (Stenlund and Medvedev, 2000). The aim of this research is to study and compare dierent control strategies from the point of view of cell level control. In addition, a new control strategy is presented and implemented. Its performance is compared to three dierent strategies: one traditional SISO control strategy, and two MIMO control strategies. The strategies are compared by the means of special performance indices. In the following the mathematical model of a otation cell is rst developed and a ve cells in series cell conguration is constructed. The simulations are then performed in order to determine suitable control parameters for controllers of the cell levels. Simulation was performed with Matlab 5.2.0 and its Simulink library.

2. Mathematical modeling of otation cells in series In the otation process the pulp is fed into the rst cell and the froth is collected in the launders. The feed can be measured by a ow measurement, and the remaining pulp ows into the next cell. The magnitude of the ow depends on the pressure dierence between two adjacent cells, the position of control valves, and the viscosity and density of the pulp. The magnitude of the pressure dierence can be determined from the physical

Corresponding author. E-mail address: sirkka-l@hut. (S.-L. J ams a-Jounela).

0892-6875/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.mineng.2003.06.004

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height dierence of the cells. The pulp level in a cell is measured and controlled by adjusting the control valve. In the following a otation cell is considered as a tank of perfectly mixed pulp. Since the pulp is perfectly mixed, the density is the same throughout the cell, i.e. there are no spatial density gradients in the cell. As the froth ow is small compared with the pulp ow, it is ignored in the outgoing ows. The impact of the air feed on the pulp level is also ignored. The cells under study do not have the properties of ideal tanks because the cross-sectional area of the cell is not constant. A mathematical model for the physical properties will be developed and discussed next. 2.1. Single cells in series In a otation process several single cells are connected in series as shown in Fig. 1. For the rst cell in the series: p oV 1 q F1 q KCv u1 y1 y2 h1 1 ot where q feed rate to the rst cell, y1 pulp level in the rst cell, y2 pulp level in the second cell, hn physical dierence in height between the cells, u1 control signal, K constant coecient, F1 outow from the rst cell, and Cv valve coecient. The equations for cells 2, 3, and 4 are respectively (i 2, 3, and 4) p oV i Fi1 Fi KCv ui1 yi1 yi hi1 ot p 2 KCv ui yi yi1 hi The equation for the last cell (n 5) p oV n Fn1 Fn KCv un1 yn1 yn hn1 ot p KCv un yn hn

p oyi Ki1 Cv ui1 yi1 yi hi1 Ai ot p KCv ui yi yi1 hi Ai where i 2, 3, and 4. p oyn KCv un1 yn1 yn hn1 An ot p KCv un yn hn An where n 5. 2.2. Double cells in series

Mathematical models for the double cells can be derived in a similar manner (J ams a-Jounela et al., 2003). The principle dierences compared with the mathematic models of single cells in series are that, in a double cell, both pulp levels are controlled by manipulating a control valve in the second cell outow, as can be seen from Fig. 2. The cells are physically on the same level. The pulp level in the second cell is measured and compared with the set point. This signal is used as input to a PI-controller. The output signal of the controller is the desired valve position as denoted by u1 . The cells in a double cell are connected via a ange. The ange limits the maximum ow between the cells. The pressure difference in the cells is the only driving force for the ow, and depends on the density of the pulp. The velocity of outow from the rst cell can be derived using Bernoulls equation, resulting in p v 2gy1 y2 7

3
q

y1 F1

y2 F2 u1

In the case of an ideal tank, the cross-section of a cell is assumed to be constant. The pulp levels in the cells can therefore be written as p oy1 q F1 q KCv u1 y1 y2 h1 4 A1 A1 ot

y3 F3

Fig. 2. Double otation cell series.

Fig. 1. Schematic diagram of otation cells.

P. K ampj arvi, S.-L. J ams a-Jounela / Minerals Engineering 16 (2003) 10611068

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where g gravity, y1 pulp level in the rst cell, and y2 pulp level in the second cell. The volume ow across the ange can be calculated as p 8 F1 vAflange 2gy1 y2 ll hl where Aflange cross-sectional area of the ange, ll length of the ange, and hl height of the ange. The pressure drop due to owing resistances is assumed to be negligible and can be ignored. The change of the pulp volume in the rst cell with respect to time can be written as p oV 1 q F q 2gy1 y2 ll hl 9 ot The outow of the second double cell is similar to that of a single cell, and the ow into the second cell is the outow of the rst. Therefore the mathematical model is p oV 2 F1 F 2gy1 y2 ll hl ot p 10 K2 Cv u1 y2 y3 h1 2.3. Modelling the Outokumpu otation cells Fig. 3 shows the cross-section of the otation cells under study. The pulp level changes as a function of the cell volume. The change is linear from the bottom of the cell to the starting level of the launders and boosters, denoted by Hlowerpart in Fig. 3. The cross-section of the pulp subsequently decreases on moving upwards because the launders and boosters reduce the volume of the cell. Calculation of the pulp volume can be divided into two sections: the pulp level below and above the level of

Hlowerpart . When the volume is smaller than the volume of the cylinders Hlowerpart , the pulp level is determined as htot V pR2 tot 11

where Rtot is the radius of the cell. If the pulp volume is equal or greater than Hlowerpart , the volume is written as V Vlowerpart Vupperpart h VB h VR h 12 where Vlowerpart volume of the cell below the level of Hlowerpart , Vlowerpart volume of the cylinder above Hlowerpart , VB volume of the boosters, and VR volume of the launders. A level variable, the zero point of which is equal to Hlowerpart is denoted by h. The volumes of the launder and booster are obtained by geometric relations, and substituting them in Eq. 12 results in  p  2 h3 K B h2 pKB Rtotal hpKR R2 total 3 V Vlowerpart 0 13 where KB constant coecient of the booster dynamics and KR constant coecient of the launder dynamics. The third order equation has a solution in Hlowerpart Htotal , where Htotal is the total height of the cell. KB , KR and Vlowerpart are constants and specic for each cell size. The total level of pulp can be determined by summarizing h and Hlowerpart . The eect of pulp level non-linearity can be seen in Fig. 4, in which the pulp level is presented in ideal tank conditions as a function of the pulp volume. The cell type is TC-50, the maximum volume of which is 50 m3 . 2.4. Valve sizing and characteristic curve of the valves Valve sizing is based on the Cv value, which is calculated according to the ISA standard as follows:

R r,out LR rC

LC r2

LB
rB

HC HB

4.6 4.4 4.2

TC50, linear and nonlinear pulp level

4 3.8 level 3.6 3.4

H lowerpart

3.2 3 2.8 2.6 40 45 50 55 60

R total
Fig. 3. Cross-section of a otation cell of Outokumpu Mintec.

volume

Fig. 4. Pulp level as a function of pulp volume.

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r q Cv 1:17Q Dp

14

Measurement yset + e

Fin m .

Feedforward + + u F Flotation y cell dynamics

where Q is the ow rate (m3 /h), Cv the valve capacity coecient, q the pulp density (kg/m3 ), and Dp the pressure dierence over the valve. The ow rate (m3 /h) through the cell is calculated as Vcell Q 1: 2 s=60
3

PI controller

Valve

ym

Measurement

15
Fig. 5. Control diagram of feed-forward controller.

where Vcell is the cell volume (m ) and s the pulp retention time in the cell. The valves in the models are sized for a ow which has a retention time of 1.5 min in one cell. The ow rate to the rst cell is also calculated using this retention time value. Characteristic curves of the control valves are produced by Larox Flowsys and were used in the Simulink models in order to cause realistic and non-linear behavior for the valves.

m1 ( s )

H11 (s ) H 21 (s ) H12 (s )

+ +

y1 (s )

m2 (s )

+ H 22 ( s) +

y 2 (s )

Fig. 6. Control diagram of a basic decoupling controller.

3. Control strategies The dierent control strategies are discussed and described in the following sections. These strategies are selected because they can be used with basic PI-controllers and without any additional instrumentation. Traditionally in otation cell series there is only one ow measurement in the beginning of the series and level PIcontrollers in every cell. 3.1. Feed-forward controller A ow feed-forward controller monitors disturbances in the inow to the rst cell and uses proportional action to close or open the valves of the cell in order to compensate for disturbances. Compensation is linearly dependent on the dierence between the current inow and the normal inow. The measurement signal is ltered in order to prevent the feed-forward control from reacting to random variation in the ow. However, this kind of controller does not provide any extra performance improvement in the event of disturbances occurring somewhere else in the cell series. The model of the feedforward controller is shown in Fig. 5. 3.2. Decoupling controller A decoupling controller is based on dierential equations (1)(3). The purpose of the decoupling controller is to eliminate the crosswise eects of control loops, and hence the stability of a single control circuit depends only on its own stability features. The basic model of the decoupling controller is shown in Fig. 6. The mathematical criterion to be fullled for decoupling a tank i will be DFi in DFi out 0 16 where DFi in is a change of inow to tank i. Using the valve functions from Eqs. (1)(3) the equation can be written as follows p 17 Fi Ki Cvi ui Dhi where Dhi is the level dierence over the valve. Substituting in Eq. (16), it becomes p Ki1 Ci1 ui1 Dui1 Dhi1 DDhi1 p 18 Ki Ci ui Dui Dhi DDhi 0 Solving this equation for the change in the control signal gives q Dui Ki1 Ci1 =Ki Ci u0i1 Dh0i1 =Dh0i ui 19 f u0i1 ; Dh0i1 ; Dh0i Eventually, the control signal for a tank i becomes ui uPI f ui1 ; Dhi1 ; Dhi 20 where uPI is the control signal from a PI-controller. In order to handle the variations from the inow, the feedforward is attached to the rst tank. 3.3. Multivariable controller similar to FloatstarTM A multivariable controller (Schubert et al., 1995) controls the total inventory of material in the upstream tanks. In this control strategy, controlling a valve is

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inuenced not only by the dierence between a set point and the measured level in the tank, but also the dier-

e2 e1 P1 + P2 + + yset + + e u Valve F Flotation cell dynamics y P3 e3

PI controller

ences between set points and the measured levels in all the tanks in upstream. These variables are summed and fed to the PI-controller of the cell. Furthermore, the variables are scaled by a suitable factor depending on the valve size, position and process. In this strategy each control valve can be regarded as a sluice gate of a dam. When a damned inventory is too high in upstream, the valves are opened more than usual, even when there seems to be no need to take such an action on the basis of the levels in the neighboring vessels. The control diagram is shown in Fig. 7. 3.4. New multivariable controller When some disturbance occurs in tank, it has an effect to level in the previous tank in upstream. Previously

ym

Measurement

Fig. 7. Control diagram of a multivariable controller similar to FloatstarTM (tank 4).

e2 e3 e1 P1 P2 P3

+ + +
Filter

P5

e5

Fin
Feedforward

Fin,m F

Measurement

yset

e
PI controller

u +
Valve

y
Flotation cell dynamics

+ ym

Measurement

Fig. 8. The control diagram of MV controller (tank 4).

Fig. 9. Simulink model of the ow dynamics in a otation cell.

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described multivariable control does not take disturbances of this kind into account. Also strategy does not take into account disturbances arising from pulp feed. Therefore a ow feed-forward has been added to the system and the dierence between set point and measured level in next tank is also added to the controller of previous tank. The control diagram is shown in Fig. 8.

4. Simulations In the simulations a conguration of six TC-50 cells in series was studied in accordance with the ideal tank assumption. Therefore the eects of boosters and launders were not considered. The valves were 100% oversized according to the ISA standard, and the retention time in each cell was 1.5 min. Control strategies included conventional PI-controllers with feed-forward control, decoupling controller, a multivariable controller similar to FloatstarTM and a feed-forward multivariable controller. The simulation results of a +3 cm change in the set points of the cell levels at times 100, 150, 200, 250, 300 and 350 s are presented in the following. Making 20% change in the feed to the rst otation cell was also simulated with dierent strategies. The set point of the cell level is lowest in the rst cell, and the set point values increase on moving towards the last cell in the series, where the operating range of the level controller is

smaller. The simulation schemes were constructed with Matlab 6.0.0 Simulink software. The Simulink model of the ow dynamics in the otation cell is presented in Fig. 9. The controllers were tuned and compared using the following indices. The IAE index (integral of the absolute value of the error) integrates the absolute value of errors, and even-handedly weights all the deviations. ISE (integral of the square error) gives more weight to big deviations from the set point. ISE IAE Z Z
t2

y t ysp t dt
t2

21 22

tt1

jy t ysp tj dt

tt1

5. Simulation results The simulations of the congurations of six TC-50 cells in series resulted in parameters for the PI-controllers. Integration times in the traditional system with a feed-forward controller were between 15 and 50 s and proportional gains between 0.8 and 1.2. Because MIMO control strategies respond better to disturbances, the PI parameters were set faster. Integration times in all the PI-controllers were set to 15 s and gain to 1. In the de-

4.25 4.2 4.15 4.1 4.05 4 (a) 0 100 200 300 400 500

4.25 4.2 4.15 4.1 4.05 4 (b) 0 100 200 300 400 500

Fig. 10. Feed-forward controller. On the left the response to a )20% change in pulp feed, and on the right the response to set point changes.

4.25 4.2 4.15 4.1 4.05 4 0 100 200 300 400 500

4.25 4.2 4.15 4.1 4.05 4 0 100 200 300 400 500

Fig. 11. Decoupling controller. On the left the response to a )20% change in pulp feed, and on the right the response to set point changes.

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coupling controller the PI-parameters were between 15 50 s and 11.4 s, correspondingly. The responses of the feed-forward controller to disturbances in pulp feed and to set point changes are presented in Fig. 10. As can be seen, the )20% change in pulp feed is aecting all the cell levels in the series. The set point changes in a cell also have undesirable eects on the adjacent cells. There is always a considerably large perturbation in the level of the next cell every time a set point change is made in the system. The responses of the decoupling controller are illustrated in Fig. 11. The decoupling controller is a MIMO controller, and it also takes into account the interactions between cells. As can been seen from the graphs,

the decoupling controller eectively eliminates disturbances arising from changes in the pulp feed. Furthermore, set point changes in the cells do not aect to the other cells. The responses of congurations in which a controller similar to FloatstarTM and the new multivariable controller are used are shown in Figs. 12 and 13. The new multivariable controller seems to be more robust than the other controller, especially during changes in pulp feed. The IAE and ISE indices, which depict the performance of controllers, are shown in Tables 14. As was to be expected, the traditional SISO control with ow feedforward had the poorest gures in all cases.

4.25 4.2 4.15 4.1 4.05 4 0 100 200 300 400 500

4.25 4.2 4.15 4.1 4.05 4 0 100 200 300 400 500

Fig. 12. Multivariable controller similar to FloatstarTM . On the left the response to a )20% change in pulp feed, and on the right the response to set point changes.

4.25 4.2 4.15 4.1 4.05 4 0 100 200 300 400 500

4.25 4.2 4.15 4.1 4.05 4 0 100 200 300 400 500

Fig. 13. New multivariable controller. On the left the response to a )20% change in pulp feed, and on the right the response to set point changes.

Table 1 The performance indices for the feed-forward controller Feed-forward controller 1 2 3 4 5 6 ISE(+20%) 3.1 2.9 2.7 2.2 2.0 0.5 IAE(+20%) 348 361 371 341 322 159 ISE()20%) 3.7 3.1 3.3 2.7 2.4 0.4 IAE()20%) 325 360 391 371 361 147 ISE(s.p.c.) 2.7 4.0 5.0 5.33 6.5 3.7 IAE(s.p.c.) 298 520 712 818 970 604

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Table 2 The performance indices for the decoupling controller Decoupling controller 1 2 3 4 5 6 ISE(+20%) 1.1 0.0008 0.0000008 0.00000005 0.00003 0.04 IAE(+20%) 327 10.2 0.3 0.05 1.2 29.5 ISE()20%) 0.7 0.0003 0.00000004 0.000000001 0.000004 0.019 IAE()20%) 263 8.0 0.1 0.01 0.4 19.5 ISE(s.p.c.) 2.0 2.2 2.2 2.2 2.1 2.1 IAE(s.p.c.) 220 242 241 241 228 179

Table 3 The performance indices for the multivariable controller similar to FloatstarTM MV controller 1 2 3 4 5 6 ISE(+20%) 7.81 0.1 0.03 0.0001 0.0001 0.01 IAE(+20%) 499 65.0 25.2 2.2 2.7 18.5 ISE(20%) 9.02 0.07 0.03 0.00006 0.00008 0.002 IAE(20%) 527 58.6 26.5 1.4 1.5 7.8 ISE(s.p.c.) 2.2 2.3 2.3 2.3 2.3 1.7 IAE(s.p.c.) 202 243 248 249 245 151

Table 4 The performance indices for the new multivariable controller New MV controller 1 2 3 4 5 6 ISE(+20%) 1.79 0.03 0.007 0.0001 0.0009 0.005 IAE(+20%) 216 30.7 12.3 1.1 4.3 8.1 ISE()20%) 2.1 0.02 0.007 0.00009 0.0005 0.003 IAE()20%) 231 27.4 13.2 1.7 3.5 6.9 ISE(s.p.c.) 2.2 2.2 2.2 2.2 2.2 1.9 IAE(s.p.c.) 204 202 211 211 212 149

6. Conclusions All the simulated congurations were successfully tuned. It is noticeable that the classical SISO strategy with feed-forward controller cannot even approach the performances of the MIMO controllers. This is due to high interactions between the control loops, which SISO systems cannot take into account. The dierences between dierent MIMO systems are somewhat smaller. All the controllers performed robustly to disturbances in pulp feed and to set point changes. The decoupling controller had the best IAE and IDE indices. However, the decoupling controller is sensitive to model uncertainties (Skogestad and Postelwaite, 1996). This also means that process changes can strongly degrade the control performance.

References
Andersen, R., Gronli, B., Olsen, T., Kaggernd, I., Ramslo, K., Sandvik, K., 1979. An optimal control system of the rougher otation at the Folldal Verk concentrator, Norway. In: Proceedings of the 13th International Mineral Processing Congress. New York, USA, pp. 15171540.

Hammoude, A., Smith, H., 1981. Experiments with self-tuning control of otation. In: Proceedings of the 3rd IFAC Symposium on Automation in Mining, Mineral and Metal Processing. Oxford, UK, pp. 213218. J ams a-Jounela, S.-L., Laurila, H., Karesvuori, J., Timperi, J., 2001. Evaluation of the future automation trends in control and fault diagnosticsa case study in otation plant. In: 10th IFAC Symposium on Automation in Mining, Mineral and Metal Processing. J ams a-Jounela, S.-L., Dietrich, M., Halmevaara, K., Tiili, O., 2003. Control of pulp levels in otation cells. Control Engineering Practise, 7381. Koivo, H., Cojocariu, R., 1977. An optimal control for a otation circuit. Automatica 13 (1), 3745. Niemi, A., Maijanen, J., Nihtil a, M., 1974. Singular optimal feed forward control of otation. In: IFAC/IFORS Symposium on Optimization MethodsApplied Aspects. Varna, Bulgaria, pp. 277283. Schubert, J.H., Henning, R.G.D., Hulbert, D., Craig, I.K., 1995. Flotation controla multivariable stabilizer. In: XIXth IMPC, San Fransisco, vol. 3, pp. 237241. Skogestad, S., Postelwaite, I., 1996. Multivariable Feedback Control: Analysis and Design. John Wiley & Sons. Stenlund, B., Medvedev, A., 2000. Level control of cascade coupled otation tanks. Future trends in automation in mineral and metal processing. In: J ams a-Jounela, S.-L., Vapaavuori, E. (Eds.), IFAC Workshop 2000, Helsinki, Finland, pp. 194199. Zargiza, R., Herbst, J.A., 1987. A model based feed forward control scheme for otation plants. In: 116th AIME annual meeting. Denver, CO, USA, pp. 2327.