Constrained real-time optimization of a grinding circuit using steady-state linear programming supervisory control

Powder Technology 124 (2002) 254 – 263 www.elsevier.com/locate/powtec

Constrained real-time optimization of a grinding circuit using steady-state linear programming supervisory control Richard Lestage a,*, Andre´ Pomerleau b,1, Daniel Hodouin c,2 a

Centre de Recherche sur la De´fense Valcartier (CRDV), 2459 blvd Pie XI, Val-Be´lair, Quebec, Canada G3J 1X5 Groupe de Recherche sur les Applications de l’Informatique a` l’Industrie Mine´rale (GRAIIM), De´partement de ge´nie e´lectrique et de ge´nie informatique, Universite´ Laval, Sainte-Foy, Quebec, Canada G1K 7P4 c Groupe de Recherche sur les Applications de l’Informatique a` l’Industrie Mine´rale (GRAIIM), De´partement de ge´nie des mines, de la me´tallurgie, el des mate´riaux, Universite´ Laval, Sainte-Foy, Quebec, Canada G1K 7P4

b

Abstract This paper presents an application of real-time optimization (RTO) to a simulated ore grinding plant. The control and optimization methods are based on a dynamic linear model of the process. A linear programming (LP) method is used on-line to find the optimum controller set-point as a function of the process-operating constraints. The optimizer selects set-point values that maximize circuit throughput subject to constraints on circulating load, pump box level and hydrocyclone overflow and underflow densities. At the regulatory control level, performances of unconstrained and constrained multivariable predictive controllers are compared and discussed. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Grinding mill; Real-time optimization; Linear programming; Dynamic simulation; Grinding circuit control

1. Introduction Ore comminution processes are huge consumers of energy, which determine the performance of the subsequent ore beneficiation processes. Their optimization is then an important issue in the mineral processing industry [21,28]. As the concept of optimization is rather vague, optimization studies require a clear definition of the goals to be aimed at and of the level of optimization that is to be applied. Let us first define the four different optimization levels that may be considered for a grinding circuit. At a first level, optimization can be performed at the design stage of the process for the selection of the flowsheet configuration and the equipment size [19]. The objective is to achieve the metallurgical specifications while minimizing the investments and operating costs, and maximizing the reliability, flexibility, and expandability of the

*

Corresponding author. Tel.: +1-418-844-4000x4713; fax: +1-418844-4502. E-mail addresses: [email protected] (R. Lestage), [email protected] (A. Pomerleau), [email protected] (D. Hodouin). 1 Tel.: +1-418-656-2131x2987; fax: +1-418-656-3159. 2 Tel.: +1-418-656-5003; fax: +1-418-656-5343.

system. A second optimization level is the optimal tuning of the operating conditions, such as the selection of the mill RPM, the grinding media load, the classifier settings (number of hydrocyclones, vortex finder and apex diameters. . .). The objective is to generate a particle size distribution, a slurry density, and a production capacity, which are optimal for the economic performance of the subsequent separation process, usually assessed by the net smelter return (NSR). At a third level, optimization is performed on the operating set-points: feed-rate, slurry densities, particle size distribution of the product, circulating load. Most of the time the objective is to maximize the throughput while keeping the product fineness at a target value. Finally, the lower level in the hierarchy of process optimization is optimal control [9,23,31,32]. The objective of optimal control is to keep the process at its set-point, while minimizing a performance criterion containing the squared deviations to the set-points and the squared amplitudes of the control actions. The present study considers mainly the third optimization level, i.e. the selection of the optimal set-points of the controller. However, to illustrate the method on a simulated grinding circuit, the fourth level, optimal control, will be also simulated, although this is not the primary objective of the study. Fig. 1 shows the organization of the third level optimizer and the fourth level optimal controller. There are

0032-5910/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 ( 0 2 ) 0 0 0 2 8 - 1

R. Lestage et al. / Powder Technology 124 (2002) 254–263

Fig. 1. Hierarchical optimization scheme.

various approaches to real-time optimization (RTO) at this third level. The qualities of the process model used as well as the evaluation of the economic benefits associated to RTO are important issues of these approaches [8,16,17,29]. The RTO scheme proposed in this study of grinding circuit optimization is based on a linear programming (LP) technique. The method has already been successfully applied to various processes [20,22,30,35,36]. The steady-state part of the grinding process is linearized around nominal operating conditions, thus allowing to apply an LP method to maximize a linear efficiency criterion subject to equality and inequality constraints. The set-points selected by the LP – RTO are then distributed to the control loops at the lower level. The latter may use conventional controllers or model-based controllers. In the case of LP optimization, the process is to be operated on constraints. The controllers have to manage these constraints, which should also be enforced during transient regimes. An optimal method is proposed in this study to solve this problem. It is based on a constrained predictive control algorithm. The paper is organized into three parts. In the first one, a closed-loop grinding circuit is described. In the second part, an LP – RTO method is presented in a sufficiently general form to allow its application to any other process. The paper goes through the model expression, the constant factor updating, the formulation of the objective function and the constraints, the calculation of the set-points, a discussion on the feasibility and, finally, the presentation of the optimal constrained controller for the lower level of the hierarchical optimization strategy. In the third part, the application of the above steps to the closed-loop grinding circuit is detailed. Simulated results are then discussed, showing the efficiency of this hierarchical method based on an LP steady-state optimal supervision of the set-points of an optimal multivariable controller.

semi-autogenous mill or any other adequate comminution device can replace the rod mill. Normally, a grinding operation must be tuned in such a way that the subsequent separation process is optimal and thus gives a metal recovery and concentrate grade which maximize the NSR (the revenue obtained by selling the concentrates to a smelter) [33,34]. At constant production capacity and for a given ore composition, there is such an optimal distribution of the particle size and composition distributions delivered by the grinding plant. Unfortunately, it is almost impossible to operate a grinding circuit with such targets because the particle composition distribution is unknown [2,38]. Furthermore, when the production capacity is allowed to vary, there is a compromise to be found between the concentrate quality and quantity. Finally, the optimal operating point may strongly vary with the metal market prices and the ore grade [3]. Most of the time, these complex relationships between the revenue at the separation stage and the grinding circuit tuning (particle size and composition distributions and production capacity) are impossible to track. For that reason, mineral processing plant optimization is broken down into separate optimization stages of the grinding and separation processes, thus leading to a sub-optimal operation. Because of the competition between the ground product fineness, which tends to improve concentrate grades, and the production capacity, which increases the quantity of sold metal, the optimization of a grinding circuit is usually formulated as one of these three objectives:


maximize production at constant fineness maximize fineness at constant production
optimize a weighted function of both above objectives


The sale contract between the mineral processing plant and the smelter is usually written in such a way that it is more profitable to increase production than metal recovery and/ or concentrate grade. As a consequence, the objective of

2. Process description A typical rod-mill ball-mill grinding circuit is depicted in Fig. 2. The ore is fed directly to the rod-mill, which discharges into a pump box. The pump feeds a group of hydrocyclone classifiers. The hydrocyclone overflow is the finished product directed to the flotation stage. The hydrocyclone underflow product is recycled to the pump box after being reground in a ball mill. In some plants, a ball mill, a

255

Fig. 2. Ore grinding circuit.

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maximizing the tonnage at constant fineness is the suboptimal goal that is adopted by most mineral processing plants. In practice, the tonnage is increased up to a point where constraints are encountered. As the objective is directly a linear function of a manipulated variable (the ore feedrate), the LP approach is particularly well suited to real-time optimization of grinding operations. Two manipulated variables are available for the ore feedrate maximization subject to constraints: the water addition to the pump and the rod-mill feedrate. The optimum is limited by process constraints, which may depend on design parameters or operating conditions of the circuit [4,7,11,12,24,25,37]. In many cases, the feedrate is limited by the ball-mill capacity expressed by a maximum circulating load. The pumping capacity to the hydrocyclones can also be a limiting factor, due either to maximum pump speed or to pump box level. Mill overload constraints can be detected by measurement of the sound level, bearing oil pressure, or mill power. Additional constraints can be put on the hydrocyclone overflow density for flotation requirements, on the hydrocyclone feed density and pressure for hydrocyclone operation requirements. A grinding control study also reported the measurement of the angle of the hydrocyclone underflow discharge to prevent roping (an inefficient operating regime where the underflow pattern is no more a spray) caused by an excessive hydrocyclone flow rate. For a properly designed circuit, only operation constraints such as the limited mill capacity must be allowed to prevail. Design constraints such as limited pumping capacity, fully open valves and insufficient hydrocyclones capacity should not be permanent limitations in order to take full advantage of the mill capacity.

3. Linear programming optimizer for constrained processes A real process is necessarily restricted by constraints. Depending upon the structure of the optimization criterion and the location of the constraints, the optimum can be located on constraints or can be unconstrained. An optimization criterion linear with respect to the manipulated variables leads necessarily to an optimum on constraints. Finding an optimum subject to constraints is a task very different from finding an unconstrained optimum. When the optimal process operation is located on constraints, it is important to identify the active constraints and the values of the constrained variables, in order to maintain the process on these constraints. Only constrained optimization is addressed in this paper. The method is valid for linear or linearized objective function involving linear constraints. 3.1. Process model Both the optimizer and the optimal controller are based on a process model. Various compromises must be consid-

ered when selecting the model. On one hand, a simple model offers poor prediction capability; however, fast model updating can be used to minimize model-process discrepancy, since the model contains a low number of parameters to be updated. On the other hand, a complex model has a better overall prediction ability but, since it is complex, parameter updating is more difficult to achieve. The method proposed here relies on a simple model approach with fast parameter update. The optimizer calculations are based on the steady-state values of the dynamic linear model of the process used in the optimal controller. Linear dynamic transfer functions can be built easily by using identification strategies based on process excitation. Inputs are grouped into two classes: manipulated inputs u and measured disturbances p. Their -related transfer functions are gathered into the transfer matrices G(s) and Gp(s), leading to the dynamic equation: YðsÞ ¼ GðsÞU ðsÞ þ Gp ðsÞPðsÞ þ EðsÞ

ð1Þ

where Y(s) is the vector of variables to be controlled or constrained (including internal states as well as output variables) and where E(s) accounts for modeling errors and unmeasured disturbances. The steady-state linear multivariable model can then be written in the following form: y ¼ Ku þ Kp p þ e

ð2Þ

where K and Kp are the gain matrices given by the steadystate part of the dynamic model matrices G(s) and Gp(s). The vector e, called the model constant or bias, is updated to account for deviations between the model and the true process. These deviations result from model inaccuracies due to the linearization of the process, the time evolution of the process gains and the unmeasured persistent disturbances. It is assumed here that the uncertainties on K and Kp in the vicinity of the constrained optimum create only small errors on the location of the optimum. Otherwise, adaptation of K and Kp is necessary. However, adaptive methods for the estimation of matrices G(s) and Gp(s) to deduce K and Kp are complex and sensitive to process noise and will not be studied here. Conditions that ensure convergence to the optimum, when errors are present in the gain matrix, have been given by Forbes and Marlin [15]. 3.2. Model bias update The model constant is selected in order to eliminate the difference between the process and the model outputs at the actual operating point. For this purpose, the time-varying model constant, which is the difference between the process and the dynamic model, is continuously measured. This value is then filtered and used as the steady-state constant e for steady-state optimization (Fig. 3). The filter time constant is a tuning factor that allows the designer to select the controller set-point update dynamics. It is important to notice that the optimizer is an additional feedback path in

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Assuming that the gradient of the linearized J function is constant in the vicinity of the optimum, Dj is the direction of the fastest increase, when J is to be maximized. The gradient is the optimization direction because it points in the direction of the steepest ascent of the objective function. 3.4. Constraints Constraints can be any linear function of the variables u, y and state variables having physical meanings. The most obvious constraints are actuator limits. Equality constraints on manipulated variables can also be used. Constraints on outputs may be used either to fix a given production quality or to prevent unwanted process operation. All the constraints can finally be gathered in the following matrix and vector linear equations:

Fig. 3. Model bias estimator.

the control loop (see Fig. 1). In the presence of large modeling errors, a too aggressive set-point update may destabilize the closed-loop system. 3.3. The optimizer objective function The objective function is a scalar value J that represents the performance. The objective function can be any function J of the measured and manipulated variables: J ¼ f ðu, yÞ

ð3Þ

In order to use linear programming techniques, this objective function must be a linear function of u and y. When the non-linearity is weak and the optimum close to the operating point (u0,y0), the objective function can be linearized using a first-order Taylor’s approximation:

  @f ðu,yÞ

@f ðu,yÞ

Jc ¼
ðu  u0 Þ þ y  y0

@uT @yT
u0 ,y u0 ,y 0 0   ð4Þ þ f u0 ,y0 This linear function in u and y can now be transformed into a single function of u with the use of the steady-state model of Eq. (2): J ¼ DTJ u þ gðp,eÞ

ð5Þ

where 2


@f ðu,yÞ

DJ ¼ 4
@uT


u 0 ,y


@f ðu,yÞ

gðp,eÞ ¼
@uT


3T


@f ðu,yÞ

þ
@yT


u 0 ,y

0

u0 ,y

 K5

ð6Þ

0


@f ðu,yÞ

ðu0 Þ þ
@yT
0

 ðKp  p þ e  y0 Þ þ f ðu0 ,y0 Þ

Ae u ¼ be ðp,eÞ

ð8Þ

Ai uVbi ðp,eÞ where each row of matrices A and vectors b represents a constraint. 3.5. Optimization Assuming that e and p will hold their value in the future, -the optimization problem can be formulated. The objective function term g( p,e) of Eq. (5) does affect the optimum -value but not the optimum location defined as the value u*. So, it can be removed from the objective function without altering the solution u* of the following optimization problem: max DTJ u u

subject to Ae u ¼ be ðp,eÞ

ð9Þ

Ai uVbi ðp,eÞ Since this problem contains a linear objective function with linear constraints, the optimal solution u* can be found by using linear programming (LP) techniques. The Simplex or the Modified Simplex is the most usual methods for solving this kind of problem [5,6]. As the model constant e of the steady-state equation is continuously updated, the optimization can be performed at each sampling instant even if the process is not strictly in steady-state operation. The final LP optimizer structure is shown in Fig. 4. 3.6. Set-point calculations

u0 ,y

0

ð7Þ

The optimal set-point values z* of the output variables, which correspond to the manipulated variable value u*,

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4. Optimal control The primary objective of the control strategy is to bring the process to its optimal condition u* and y*. The usual control objective of a predictive controller is to minimize squared deviations to set-point while minimizing the quadratic sum of the variations of the actions. This calculation is made on the predicted trajectory of the process model over the prediction horizon N: min

uðkÞ,KuðkþN 1Þ

¼

N h X

IðkÞ

ðuðk þ i  1Þ  uðk þ i  2ÞÞT Rðuðk þ i  1Þ

i¼1

uðk þ i  2ÞÞ þ ðyðk þ iÞ

Fig. 4. Real-time linear programming optimization scheme.

y*ðkÞÞT Qðyðk þ iÞ  y*ðkÞÞ can be calculated using the steady-state model of the process: z* ¼ Cy* ¼ CðKu* þ Kp p þ eÞ

ð10Þ

where C is the matrix which extracts, from the set of output variables, the variables that are under set-point control. 3.7. Behavior of the LP –RTO method The LP strategy pushes the process in the direction where the objective function improves until no further improvement is possible without violating one of the constraints. A fundamental property of LP optimization is that, except for some degenerated cases, the optimum is located at constraint intersections. The bias update influences the location of these constraints. As a consequence, the optimum may not be always on the same constraints. Real-time optimization is needed in order to keep the process at its optimum since these constraints continuously move due to measured and unmeasured disturbances. Degenerated cases happen when either no solution, or multiple solutions, or unbounded solutions to the LP problem are found. When no feasible solution exists, constraints are too tight and must be relaxed. Any solution of the multiple solutions case can be used with equal success. The presence of an unbounded solution means that not enough constraints have been defined. This case is impossible if lower and upper bounds constraints are imposed to all manipulated variables. There are limitations to the LP optimizer performance since set-points only are respecting constraints. It is possible that, when applying these set-points, the controllers use a transient path which violates the constraints. This may occur since only some outputs variables are usually controlled. To prevent constraint violation during transients, a control strategy that takes into account constraints on auxiliary variables must be used [18,26,27].

i

ð11Þ

where the diagonal matrices Q and R weight the relative cost of output and input deviations to the optimal values. The main advantage of predictive controllers is their ability to handle hard constraints. While linear programming optimization selects set-points that respect constraints for steady-state operation, constraints at the control level can be used to ensure that the transient response also respect these constraints. Hard constraints on the input variables can be applied very easily. Hard equality constraints should not be applied to output variables. Solving a control criterion that requires an exact output value to be specified is equivalent to a deadbeat controller. To prevent robustness problems associated to deadbeat controllers, only inequality output constraints are applied. The control criterion of Eq. (11) is then subject to the constraints: Ny yðk þ iÞVny Mu uðk þ I  1Þ ¼ mu

for bio½0,N

ð12Þ

Nu uðk þ i  1ÞVnu

5. Application of the LP – RTO method to a simulated grinding process For the purpose of demonstrating the LP – RTO method, the process of Fig. 2 is simulated using the software DYNAFRAG [10,13]. The dynamic simulator is based on a kinetic model of the breakage process involving a fragment distribution function and a rate function [1]. The transport and mixing properties in the mill are represented by a series of interactive perfectly mixed tanks. The hydrocyclone model is a semi-empirical model describing the classifier efficiency as a function of particle size and operating conditions [14]. The simulator has been calibrated and validated with industrial data obtained from a concentrator processing a Pb– Zn – Cu complex sulfide ore. The water added to the rod mill is

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Table 1 Transfer functions Process variables

u1, rod mill feedrate (t/h)

u2, water to pump box (m3/h)

% t=h G11 ðsÞ ¼ ð1 þ 5300sÞð1 þ 750sÞ % 0:2ð1  900sÞ t=h G21 ðsÞ ¼ ð1 þ 5200sÞð1 þ 750sÞ

% m3 =h G12 ðsÞ ¼ ð1 þ 3200sÞð1 þ 60sÞ % 0:012ð1 þ 39500sÞ 3 m =h G22 ðsÞ ¼ ð1 þ 4400sÞð1 þ 50sÞ

0:255ð1  5600sÞe600s

y1, overflow % solids

y2, % – 47 Am

0:14ð1 þ 4050sÞ

t=h t=h G31 ðsÞ ¼ ð1 þ 5700sÞð1 þ 400sÞ % 5:749 t=h G41 ðsÞ ¼ ð1 þ 5500sÞð1 þ 210sÞ 13:8

y3, mill throughput (t/h)

y4, pump box level (%)

automatically adjusted in such a way that the solid composition in the slurry passing through the rod mill is constant. The proposed LP –RTO strategy is applied to this simulated grinding process. There are two manipulated variables available in the grinding circuit:

% m3 =h G42 ðsÞ ¼ ð1 þ 4700sÞ 1:962

The numerical values of the transfer functions are obtained from the responses of the simulator around a nominal operating regime. The results are shown in Table 1, from which the following gain matrix K can be extracted: 2

0:255 6 :02 6 K ¼ Gð0Þ ¼ 4 13:8 5:749




u1: the rod-mill feed (t/h)
u2: the pump box water addition (m3/h) and four output variables:


y1: the hydrocyclone overflow density (% solids) y2: the fraction of particles smaller than 325 mesh (47 Am) in the circuit product (%)
y3: the tonnage through the ball mill (t/h)
y4: the pump box level (%)


In the present example, the ground product fineness ( y2) target is 48% of particles smaller than 47 Am. This is obtained through metallurgical tests showing that this fineness gives adequate average recoveries and grades. Constraints on auxiliary variables limit the possible production. The tonnage through the ball mill ( y3) must not exceed 820 t/h, otherwise the mill is overloaded. The hydrocyclone overflow density ( y1) must not be over 52% to prevent sedimentation problems. It must not also be below 48% to fit flotation feed requirements. Finally, the pump box level ( y4) must stay between 15% and 85% of the overflowing level.

t m3 ð1 þ 5000sÞð1 þ 5sÞ 4:2ð1  700sÞ

G32 ðsÞ ¼

3 0:14 0:012 7 7 4:2 5 1:962

ð14Þ

in the units defined in Table 1. Although the simulator is non-linear, the variation of K, close to the optimum location, is small and does not lead to a wrong optimum location. However, this point has not been verified by the application of a non-linear constrained programming technique to the simulator. 5.2. Optimizer for the grinding circuit The LP optimizer is designed to keep the process at its operating constraints while maximizing the tonnage u1 at a given value for y2 = 48% (the percentage of particles smaller than 47 Am). The optimization criterion is:

u J ¼ ½1 0 1 u2



¼ DTJ u

ð15Þ

The equality constraint can be written:

5.1. The linear dynamic model of the grinding circuit For this two-input – four-output grinding circuit, the transfer matrix is defined as:

48% ¼ ½k21 k22 u þ e2

ð16Þ

which gives for Eq. (8): 2

G11 ðsÞ 6 G21 ðsÞ GðsÞ ¼ 6 4 G31 ðsÞ G41 ðsÞ

3 G12 ðsÞ G22 ðsÞ 7 7 G32 ðsÞ 5 G42 ðsÞ

Ae ¼ ½k21  k22

ð17Þ

be ðeÞ ¼ 48% þ e2

ð18Þ

ð13Þ

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R. Lestage et al. / Powder Technology 124 (2002) 254–263

The inequalities are written: 2 3 2 3 820 t=h y3 6 y1 7 6 52% 7 6 7 6 7 6 y1 7V6 48% 7 6 7 6 7 4 y4 5 4 85% 5 15% y4

ð19Þ

Using the model of Eq. (2), the coefficients of Eq. (8) are: 3 2 k32 k31 6 k11 k12 7 7 6 7 ð20Þ A1 ¼ 6 k k 11 12 7 6 4 k41 k42 5 k41 k42 where the gain values are taken from Eq. (14), and with the following values of bi: 2 3 820 t=h  e3 6 52%  e1 7 6 7 7 ð21Þ bi ðeÞ ¼ 6 6 48% þ e1 7 4 85%  e4 5 15% þ e4 The vector of constraint b(e) must be updated when the model constant e changes. Even though a filter can be used for the evaluation of e, it was not necessary in this specific application to use a filter to slow down the model updating loop dynamics. The optimal set-points z2 and z3 are now calculated from the relationships:



  z2 0 1 0 0 0 1 0 0 y* ¼ ðKu* þ eÞ ¼ 0 0 1 0 z3 0 0 1 0 ð22Þ and used as set-points for the controllers.

Fig. 6. Optimum location for disturbed operating conditions.

The optimizer selects the set-point value z2 = 48% and the z3 value that maximizes u1, while keeping the process inside its inequality constraint domain. Fig. 5 shows how the constraints limit the production capacity of the circuit for the nominal operating conditions. The line ‘‘y2 = set-point’’ represents the combination of u1 and u2 values which gives the appropriate 48% value. Fig. 5 shows that the maximum value of the circulating load y3 (820 t/h) limits the feedrate u1. In this case, the optimizer selects the set-point so that y2 is 48% and y3 is at its maximum value. This is for the normal operating conditions. In the presence of disturbances, the constraint may move as shown in Fig. 6 for the following variations of the bias: 2

3 1% 6 0% 7 7 De ¼ 6 4 þ40 t=h 5 10%

ð23Þ

The maximum u1 is now set by the minimum y1. The y3 setpoint must then be set at 792 t/h by the optimizer, since the minimum y1 is reached instead of the maximum of y3. In another situation, y4 could be the limiting factor. 5.3. Optimal controller

Fig. 5. Optimum location for normal operating conditions.

A proper control strategy must be designed to track the set-points selected by the LP optimizer. The controller is designed to manipulate u1 and u2 in order to get the outputs y2 and y3 at their set-points z2 and z3. The controller is a moving horizon predictive controller, which minimizes the criterion given in Eq. (8). It is tuned with a control horizon

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These specifications ask for a decrease of the response time of the controlled process in comparison to the open loop dynamics (see Table 1). 5.4. Simulation results for the grinding circuit

Fig. 7. Grinding simulation results: unconstrained controller.

N = 6, a sampling period of 300 s, and the following weighting parameters: 2

0 0 0 6 0 25 0 Q¼6 4 0 0 0:1 0 0 0

1 R¼ 0

0 0:1

3 0 07 7 05 0

ð24Þ

 ð25Þ

The simulation sequences (Figs. 7 and 8) begin at time zero with normal operating conditions. The high circulating load y3 limits the u1 rod-mill feed allowable. The set-points are then set to the target particle size ( y2 = 48%) and to the highest circulating load admissible ( y3 = 820 t/h). At time 5 h, an ore hardness increase occurs. It causes an increase in the ball-mill tonnage y3 and a coarser particle size y2. To handle this situation, the optimizer lowers the y3 set-point in order to keep the process at the lowest y1 constraint. At time 10 h, the operator changes the number of hydrocyclones in operation, the pump box upper level y4 becomes the limiting factor. At time 15 h, the specification on the particle size fineness is set to 47% minus 47 Am, and the upper pump box level constraint is still at its limit. Figs. 7 and 8 show simulation results for two different control strategies. In Fig. 7, the predictive controller does not take into account the constraints on the overflow density y1 and the pump level y4, while tracking the set-points. This leads to a transient violation of the lower y1 and upper y4 constraints, when the optimizer selects a new y3 set-point. In Fig. 8, the controller is designed to respect constraints in priority to set-points. The constraint is satisfied on the setpoint change, but at the cost of a slight deviation of y2 to the z2 set-point. The final selection of the best control strategy depends on the importance of a transient deviation to the particle size set-point in comparison to a transient constraint violation.

Only y2 and y3 are controlled with this Q matrix, because they are the two main variables and the system possesses only two degrees of freedom (input variables). The performances of the controller are not sensitive to the selection of the weighting factors since the closed loop time responses are forced through the following reference models for set point tracking: y1 ðsÞ 1 ¼ * y1 ðsÞ 1 þ 500s

ð26Þ

y2 ðsÞ 1 ¼ y* 1 þ 1000s ðsÞ 2

ð27Þ

y3 ðsÞ 1 ¼ y3*ðsÞ 1 þ 1000s

ð28Þ

y4 ðsÞ 1 ¼ * y4 ðsÞ 1 þ 1000s

ð29Þ Fig. 8. Grinding simulation results: constrained controller.

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6. Conclusion Real-time optimization is used to find the operating conditions that give the best performances achievable as disturbances occur in a process. Various optimization structures exist in order to achieve this goal. LP optimization is a simple method that can be used when the optimum is located on process constraints and when the objective function, as well as the process model, is linear or linearized. For small non-linearities, bias update is a simple method that gives a continuous correction to the process model. The LP optimizer can efficiently handle steady-state inputs, states and outputs, equality and inequality constraints. However, for transient constraints handling, a control system that enforces constraints must be used. An application to a simulated ore grinding circuit shows that this optimization scheme is efficient for throughput maximization while maintaining the product fineness at a constant set-point, without violating overflow density, circulating load and sump level constraints, except during transient regimes. Because the simulator has been validated on various plant data and is able to mimic realistic disturbances such as ore grindability variations, it is believed that the strategy is applicable to real plants. However, as in any other strategy, the transfer matrix should be properly identified and some safety features to back off to manual setpoint control for very abnormal operating conditions should be implemented.

Acknowledgements This study is part of a generic project on the application of automatic control to the mineral processing and extractive metallurgy industries. It is supported by the Natural Resources Departments of the Quebec and Canada Governments as well as by a consortium of 14 Canadian mining companies.

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