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Modelling and Computer Control of a Ball Mill Unit
Copyright © [FAC Mining. Mineral and Metal Processing. Montreal. Canada 1980
MODELLING AND COMPUTER CONTROL OF A BALL MILL UNIT P. Heitzmann, C. Humbert and
J. Ragot
Laboratoire d'Automatique et de Recherche Appliquee, BP 452, 54001 Nancy Cedex, France Abstract. This article describes a method of on-line control of the size of output particles in a ball mill and classifier unit. This is not achieved through direct measuring of particle-size but through action on pulp flow and density. Kewwords. Ball mill, computer control, identification, least squares approximations, models, predictive control, state estimation. INTRODUCTION Experiments and theoretical studies on fragmentation phenomena in ball mills have shown that for constant physical characteristics of the ore at the input, particle size at the output depends essentially on the flow of solid that goes through (Ragot and co-workers, 1977 ; Lynch, 1977). The operation process of classifiers - devices which work both as a sensor, sensitive to size, and as control unit capable of separation - has also been throughly examined by researchers (Lynch, Rao, 1975 ; Hodouin, Birubi, Everell, 1977). Models have been defined that express separation taking into account the physical parameters of the ore, pulp flow and density at the input, and the geometrical structure of the classifier. The control engineer only uses effective control variables, the other variables are considered as constant (e.g. Apex diameter) or of constant mean value (e.g. the physical characteristics of the ore). Thus, the relation between the cutting point, dSO, and the control variables can be expressed as :
Z
percentage of solid
e
relative to water
m
relative to ore
the dimensions of the ball mill used were - length
=
- diameter
meter 0 . 8 meter.
The variables we can control are : - ore flow, QI' and water flow, Q2' at mill input, - water flow, Qe , added to the sump to lower 6 pulp density since dowstream circuits (classifiers, flotation cells) operate at a lower density than that in the mill, - pump flow, Q9' at sump output. For obvious practical reasons weight values inside the mill are not available. These are determined indirectly through measurements taken at the sump stage, the sump playing the part of an observer. Ball mill equations
where K1 , K2 , K3 are constant parameters, Z, the percentage of solid and Q, the flow of pulp at the classifier input.
The variations of volume, V , and weight, w , 4 4 inside the mill are related to the volumetric and massic flow by the following relations :
This relation immediately shows how important it is to have a good knowledge and control of the various loads, flows and densities of pulp in the circuit, to ensure control over particle size. MATHEMATICAL MODEL OF THE BALL MILL AND
o
W4
S~P
volumetric flow (l/h)
d
density
V
volume (1)
w
weight (kg)
Qt d l + Qe l de - QS
dS
o
where X means dX/dt.
Ref erences (see fig. I) Q
=
Consider an open loop circuit: Q 11
=
O.
The other equations necessary for a complete definition of the model are based on two hypotheses :
349
P. Heitzmann, C. Humbert and J. Ragot
3S0
I) The ball mill works as an ideal mixer the pulp inside the mill and at its output have the same density.
2) The ball mill works through an overflowing
process : outflow varies in proportion with volume variations :
~vhere
X
(Qs' d S ' VS' d S)! is the state vector.
U
(QI' Qe 2 , Qe , Qg )! is the control 6 vector.
=
The state equation is not linear. The only parameters introduced are T4 and V4 min' equation linearizing The means working point (MWP) is defined by : QIO' dlO' Qe 20 , QSO' dSO' VSO' dSO' Qe 60 , QgO and it verifies the mass balanced equa-
hence after integration
where V min is the minimum volume of pulp 4 retained. Combining the equations gives
o
@I
d4 =
(d I - d 5) + Qe 2 (d e - d 5)] / V4
tions QSO QIO QSO d SO = QgO = Q SO QgO d = SO
+ Qe 20
QIO d + Qe
Qe
de
20
60
A Taylor expansion limited to the first order gives I
o
2
+
Q d + Qe de 60 SO SO
and with hypothesis 2 :
Qe
lO
- T4
o
o
o
o
-g
o
o
o
o
o
c
o
-d
o
o
o
o
(de - d )] S 1
+ V4 minJ
a
Sump equations The equations are determined in a similar way except that the outflow is a controlled value the overflow process hypothesis must be ruled out. The mass balanced equations are
e
-f
o
o
o
o
+
o
Ws
QSd S + Qe 6 de - Qgd g
=
To avoid sedimentation an agitator is added to the sump which now behaves as an ideal mixer :
d
=
VSO d
0
S
= [Q s
(d - dS) + Qe 6 (de S
The model can be expressed as 0
X
f
(~,
!!.)
d8~
SO VSO Q + Qe 60 SO VSO d - I SO
d b
a =
e
d
o
-b
with:
c
After combining the equations
-I
g
=
T
- d
lO
Q 4
SO
SO
+ V
f 4
min
mln
QSO
/ Vs Coefficients a, b, c, d introduced during linearizing only depend on the mean working point and are therefore known if the MWP is
351
Modelling and Computer Control of a Ball Mill Unit
known, too. Coefficients e, f, g introduced in the ball mill equations are related to the MWP as well as to T4 and V4 min. Real time identification presented later will allow knowledge of the values of these parameters. SIMULATION
measurements in the sump are perturbed by - the variations of the mill outflow, QS' the variations of the water added, Qe , 6 - outflow pumping, Q9. The T4 and V4 min coefficients in turn may vary for any number of reasons
Sensors and actuators For direct measuring of the necessary variables we chose the following components currently available on the market - a dosometer-type mass flow control unit
variable pulp viscosity, - variable ore characteristics, ball wearing out,
(QI) ,
- major shifts of the working point. - lieliflux-tyrewater flow sensors (Qe , Qe ), 2 6 - bubble sensors to measure pulp level, hence volume (VS), - bubble sensors or a gamma-ray densimeter to measure pulp density (d ) in the sump, S
- an electromagnetic pulp flow meter at sump output (Q9)' - a controllable outflow pump. The computer The computer used is an a real time monitor.
SE~S
This slow drifting of the coefficients will be considered as equal to zero within a time interval called the "horizon". The generalized least square method (G.L.S.M.) If the model of a linear system is defined by the equation Y= X e, reducing the deviation between the system output and the model output E = Y - Y= Y - X 9 by minimizing a quadratic criterio~ leads ~ after linearizing the criterion with respect to the parameters - to the determination of a parameter estimator :
Solar 16/40 with
In real time operation the "Task" is a basic programm unit which performs specific functions such as measurements, filtering, etc
To remove the bias of this estimator we use the residue whiting technique suggested by Clarke (1967) ; Steiglitz and ~c Bride (1965). Flow response of the ball mill
A number of so-called "service" tasks have been programmed, whose purpose is to : - communicate with the user, - test the presence and validity of the signals coming from the sensors,
The linear model of the flow response is taken in its discrete form. Its transfer function using the z transform is : -T/T4 K(I-e
_I
)z
- e- T / T 4 z-I
- choose which of the components will be used in the ball mill and classifier unit through the means of electrical relay units,
where T is the sampling period.
- send commands to the process,
Theoretically the gain K = b /(] + a]) is
register all the states of the circuit, - give reports,
l
equal to one and we merely have to determine q] or b] with G.L.S.M. Practically we shall derive K from the two values obtained with G.L.S.M..
- perform the various start/stop sequences. To these service tasks we must add signal processing and control value computing tasks. IDENTIFICATION The ball mill parameters to be determined are the overflow volume, V . , and the flow4 ml.n dependent time constant, T . 4 The method of identification will face very noisy signals since pulp flow and density
A gain, K, other than unity on a certain horizon will show the inadequacy of the method due to a lack of information. Density response of the circuit The density response of the circuit is used to identify the value of V4 min via coefficient g defined previously. The general principle is shown in fig. 2 a.
352
P. Heitzmann, C. Humbert and J. Ragot
Parameter, identification through non linear programming
ble matrix in the
G.L.S.~.
(Bouatouch, 1977).
Pseudo-random binary sequences (P.R.B.S.)
A non-linear programming approach is used here since it avoids linearizing density equations. We min1m1se the following quadratic criterion by adjusting coefficient V4 min' I k r-C(k) = "2 . L Lds (i) -dS (i~ 2 T 1=k-N+I
where N . T is the time horizon.
A gradient algorithm is used to find the value of the coefficient P+I
vP
V4 min
4 min
-
k
~C
(k) V4 min
L
i = k - N+ I
rd
LS
(i) - d
S
(i)]
adS =---T
av 4 min To find out the adS / av 4 min term of this equation, the following equations are added to the initial set of the model equations ~ = f (~~) and solved by Kutta ~unge's method: a aV 4 min a SV 4 min
d d
s (----crt) d ds
(
d
For first order systems, I/:;. T is usually chosen greater than 5 times the time constant and AT slower than the time constant divided by 5. To the logical value of "I" corresponds a "a" amplitude, and to "0" a "-a" amplitude. "a" is chosen to generate a minimum but sufficient variation when the slowest step occurs.
P indicates the iteration number, with
Consider an M stage shift register driven by a clock with a T period and having a feedback made up of the addition modulo 2 (exclusive OR) of same stages. The P.R.B.S. is the signal given by any of the registers. It is periodic: its period is 1 /:;. T and its length is 1. They are same possible combination of signals to be added modulo 2 to have a maximum length I = 2 lo\~ 1.
ads
.) dt (av 4 m1n d (ads -dt V min) 4
These algorithms have been programmed and integrated into the real time system as a task (Hasdorff~.
Input signals We must be able to achieve any required variations of pulp inflow and density around the mean working point without interfering with particle size at the output . The simplest way of doing this is to act on water inflow.
Note - If we take a step signal as input there is still the problem of choosing the right starting point and identification horizon, whereas the horizon alone matters when P.R.B. S. are used. In that case, the horizon must be equal to a sequence length, which is a considerable advantage in real time identification. Since permanent superposal of P.R.B.S. allows the start at any time of an identification task which will retrieve the last N values in the mass memory (with N x T = I x/:;. T, T the sampling period, I the sequence lenght, /:;.T he length of the shortest step in the sequence (Foulard, Gentil, Sandraz, 1977). Results from simulated values The circuit was simulated with the non-linear model. Noiseless responses were used to test the identification methods. The main conclusions about GLSM are that: - there is immediate convergence of T4 towards the simulated value,
Step input
- the convergence of v 4 min is obtained after about twenty passages through the residue whiting filter. This is due to the adaptation error : the model for simulation used non linear equations and the P.R . B.S., linearized ones,
The most commonly used input signal is the step since it allows a visual estimation of phenomena and the use of numerical identification methods. It is essential, however, to take a few precautions
- using an increasing or decreasing step alone leads to significant errors in the parameter values because in that case the relations defined between the mean values are not verified. For example:
- its duration must be long enough to allow the output to reach a steady state,
d SO = (QIO d l +Qe 20 +Qe 60 ) / (QIO+ Qe 20+ Qe 60)
- the identification horizon must take into account both the stable and the transient states, but the former's period must not be too long compared with that of the transient states as this would lead to a non inverti-
(QI d I + Qe 2 + Qe ) / (QI + Qe 2 + Qe 6 ) 6
=r
dS
where X is the mean value of x (t).
Modelling and Computer Control of a Ball Mill Unit
3S3
The G.L.S.M. works well with symetrical steps (increasing and decreasing) or with P.R.B.S.
Filter
With non linear programming, convergence towards the right value is fast and the optimal criterion is in the 10-S range.
A particular simple and efficient one is given in the recurrent expression where f(kT) is the measured value at time kT, and fX(kT) the filtered value :
Results from actually measured values
fX(kT) = a [f(kT) + qf
With the G.L.S.M. the value of V4 min after the first passage is totally wrong: 32 1. While the space between the balls alone is 60 1 (this value was obtained by filling in the space between balls with water, the engine not running). After about 20 iterations we obtain a value of 140 liters.
+ q2f [(k -
2)T~
(k-I)T] ...
or
Hence a + q = I to have uni ty gain. -I
With the non linear programming method the value of V min quickly reached 120 1. Near 4 the optimum value the criterion stayed high (2.SS) compa~ed with that obtained in simulation. This is due essentially to measurement noises and errors. OF THE STATE VARIABLE VALUES
ESTI~ATION
Using the z operator the transfer function of the filter can be expressed as : - q
- qz
+
V min) d S 4
5 Q S
+
(QI
Qe 2 - QS) / T4
5 d
S
[Q I (d I - d S) [T4
QS +
+
Qe
2
(I
-~sU
/
V 4 min]
Actually we shall use the discrete form instead and Kutta Runge's method of integration. Ore and water inflow measurements are little disturbed by noise and, what is more, the transfer functions act as low pass filters thus giving estimated state values that are little disturbed by noise. This approach, however, is not based on feedback principles and therefore does not take into account possible disturbances causes by erratic ore feeding, chemical additions or unmeasured classifier backflow. From sump values
0
Q9 - Qe 6
d
d
S
S
+
+
(de Larminat, Thomas, 1977) Volume control in the sump The diagram of fig. 3a outlines the general principle of regulation. If we want to use the overall value of the signals we must take into account the non linearities that exist in the circuit (e.g. pump non-revertibility and saturation). In regulation, the signals to be considered are variations round a determined value and pump operations can be expressed by a first-order system with a unity gain and a T time constant. The outflow of pulp from th1J1 ball mill, QS' and the flow of water added, Qe , act as an overall disturbance, 6 AQ 7 = ~QS + ~Qe6· With CI(s), as the controller transfer function, the system equation is derived from the diagram and the linearized model equations
~VS(s)
=
AVSC(S) Cl (s) + ~Q7(s) (I + Tm(s) CI(s) + s (I + Tm s)
Proportional feedback controller. The proportional controller coefficient, K , is deterI mined so that the positioning error due to a ~Q7 perturbation should be equal to a given value
The sump equations give d and Q in terms of S S the VS' d S ' Qe 6 , Q9 measured values QS
-T/TF - e _ z-I e- T / TF
CIRCUIT CONTROL
Applying the values of ore and water inflows to the model allows estimation of the weight in the mill, pulp outflow and density using the following equations : (T 4 Q S
or
which immediately points out the similarity with a first-order filter with a, TF, time constant.
From input values
W 4
-I
Vs
(d S - I) Qe / Q 6 S
0 +
d S "s / Q S
in which we can see derived functions of noisy values that will amplify noise unless a filter ~s added.
Cancelling the positioning error through the means of a proportional integral controller CI(s) = KI (I + I / Tis) was abandoned for stability reasons as it may indeed not have been possible to comply with the stability condition, Ti > Tm' satisfactorily.
P. Heitzmann, C. Humbert and J. Ragot
3S4
Feedforward controller. Rath~r than let any perturbation occur and, only then, move in with a feedback "corrector" we shall measure the source of the disturbance, 6Q7' and modify the control variable, ~Q9. Only disturbance 6Qe
6
- water added - is di-
rectly measurable as the data on pulp flow at mill output, 6Q , are not available ; an observer, achieve~ by a computer using the model of the mill, will give an estimation of 6Q S called 6Q . The signal, an image of the disS turbance, is applied to pump control with due respect to scale. Feedforward control must take place only if the volume has reached its preset value, hence the feedforward signal will be algebraically added to that from the feedback. Pulp density control in the sump Water is added when a measured density is higher than a preset value. The control diagram using linearized equations is given in fig. 3b. The equation of the closed-loop system is a~QS(s)
+
b C (s) 6d 8C (s) + c AdS(s) 2 s + d + b C (s)
2
P.I. controller. The use of a proportional controller was rejected as it led to a positioning error that was too high with a value fo the proportional coefficient physically achievable. We used a P.I. controller yith a transfer function of C2 (s) = K2 (I + T s) i2 Ti2 is chosen so that the integration constant should be equal to the time constant of the open loop system; T. = I/d. In which case 2 (6d ) is 1 8 M
1S
the
Feedforward controller. As in the preceding case the variations at the output variable, d 8 , induced by the disturbances, Q and d ' S S are cancelled out by action on the control variable Q . e6 o d8
0 =
~S
(d S - d 8) + Qe 6 (I - d 8)] / V8
This is true only if d So Qe
8
= d
8C
Note - The feedforward control signals are derived from ball mill input signals which, themselves, had been previously filtered by the mill transfer functions acting as low-pass filters. The commands are therefore less noisy than those sent by the feedback controllers. With both regulators working together the gain of the feedback controller will be reduced, and there will be less high frequency fluctuations of the control variables sump water inflow and pump inflow. Pulp volume and density control in the sump combined with control of ore and water coming into the ball mill are the first stage in automatizing the system. If we now consider a grinding circuit with a classifier we may assume that the grindability of the ore may become lower (e.g. more big particles than usual at the input or variations of the physical caracteristics of the ore). Number of big particles at the mill output will increase, and as there are sent back to the mill by the classifier, they will cause a flow increase, which in turn, will accelerate this process of insufficient grinding. This simple example shows how difficult it is for the above circuit to cope with any changes in ore characteristics successfully. With that in mind we suggest a different approach which consists in controlling pulp throughflow. Pulp flow control in the mill Theoretically the simplest solution would be to keep the amount of ore inflow at a constant level by adapting new inflow to the measurements given by a flowmeter at the classifier output. We could also add a gamma-ray densimeter to act on ore and water separately. But this would hardly be feasible in practice since pulp flowing out of the classifier is full of air bubbles which prevent correct functioning of the sensors.
8C (S)
In a steady state lid 8 (t) = t'.d 8C (t). 6d 8C set point.
Since flow, QS' and density, d ' are not meaS surable, the values used are those given by the simulation model. The operation principle is shoVTI in the functional diagram using the linearized equations (fig. 3b).
.
, the feedforward control variable, is 6FF equal to Qe = Q (d - d ) / (d - ). 6FF 8C 8C S S Practically the feedback and feed forward signals are algebraically added.
Another solution, more easily feasible with the available technology, is to regulate mill outflow through acting on new product input while considering the back flow from the classifier as a disturbance. The diagram (fig. 4a) clearly shows this principle. C (s) is the 6 transfer function of the controller and P is the water/ore opportioning unit. C6 (s) QSC(s) + Q)) (s) tl + sT 4 ) + C6 (s) \,e can take a P.T. controller with Ti =T . 4
355
Modelling and Computer Control of a Ball Mill Unit
Hence T4 QII(s) s K 6 + T4 T4 I + s (I+T s) ( I + - s) 4 K6 K6
input - the classifier backflow and sump water additions being considered as disturbances (see fig. Sa),
QsC(s)
Q (s)
s
The general transfer function using a proportional controller CS(s) = K ' is : S
[Ks
(J / T4)
Pulp density control in the mill
- Q (s~ 9
The water/ore apportioning unit splits the overall new product flow, Q3' in two, according to ore density, d , ana the preset densil ty, d ' 3C
If flow control alone is used, d ' is kept 3C constant but if densit y perturbations causes by the classifier are taken into account, action on d to cancel them will have to be 3C taken. Note that recycled pulp is specified by its flow, QI I' and density, d ll · The balanced equations are :
A Taylor expansion around the mean working point defined by Q30' d 30 , QIIO' QsO' d so ' = Q and verifying the relations Q 30 + QIIO sO and d d gives llO 30 0
g lidS + v lid 3 + w MII
t;d 5 with g
QsO/(T4 QsO + V4 min)
v
Q30 / (T 4 QsO + V4 min)
w
QIIO / (T 4 QsO + V4 min)
Note that g = v + w. The use of the Laplace transform leads to the definition of a control diagram (see fig. 4b).
s
"With a P.r. controller C (s) = K7 (J .. I /Ti 7s) 7 where the integration constant, Ti , is equal 7 to the time constant of the open-loop system Ti7 = I /g .
+ QI I (s) + (Qe 6 ( s)
+ S T ) 4
+ (I / T 4) s + KS / T 4
For an aperiodic response the damping factor, ~, must be greater than or equal to, one ~ = 1/21T4 Kg >,.1. Hence KS(I /(4 T4 ). The positioning error is (QII + Qe 6 - Q9) / KS' An important error, but one that does not disturb the operation process if one has been careful to take an adequate volume in the sump. To this system may be added a density control loop (see fig. 6) along with a feedforward controller to cancel Qe induced perturbations. In this type of control 6 the s"mp off~ct:vely acts as a buffer. Total mass flow control The principles of this control were set out (Lynch, 1977), The amount of water added to the sump is constant and pulp volume is controlled by action on pump intake after level measurements (as described above), The total sump mass outflow, given by the product of the signals from an electromagnetic flowmeter into the signals from a gamma-ray densimeter, is controlled by action on new ore input (see fig, sb). Generally the response time for volume control is lower than that of the other time constants in the circuit lIQ = lIQ9' To s simplify the equations, and only to that purpose the pulp backflow from the classifier will be considered as having the same density as that of new product input, lid I I = O. This simplification will in no way contradict the general principle of the control suggested here. If the mean working point is the reference point (in regulation systems only and not in servo systems) IIQ = 0 the relation 9C between mass outflow and classifier backflow variations is given by : d
(I + s/d) + (a/d) Q 90 90 (I + s 7d) (I + s T4) + C9 (s) (Cl 90 (I + s / d)
The response is given by : vC (s) 8d sC (s) + wild I I (s) 7 s + g + vC (s) 7
2
(I
VSC ( s)
+ (a/d) Q9~
or
d sO + (d
/d)s 90 (I +s!d)(1 +s T ) + 4 C9 (s) [d so + (d 90 /d)
sJ
QS9 is the mass flow, the others being voluFlow control using sump volume variations The use of an ordinary pump, instead of one with controllable intake, will be possible if we control sump volume by acting on new ore
metric flows. We shall use a P.I.D. controller, whose time constants can be easily adjusted through a compensation method Cg(s) = Kg + I / (T;-9 s ) + Td sJ
[I
356
P. Heitzmann, C. Humbert and J. Ragot
with Ti
grinders). This method is therefore directly competitive with indirect measurements such as noise level, grinder weight and oil pressure in the bearings.
Hence :
(l+
~
s
d
cr)(I+ST 4 )[I+(K 9 d 50 )+~)Sl 50 J
From which we can see that the smaller the sump (d high) is, the faster the response. Implementation The control algorithms have been programmed and entered as real time tasks to achieve : - Control flexibility. Most of the controller time coefficients are dependent on the M.W.P. or on the values of the coefficients in the model. The use of a computer allows automatic adjustment of these controller coefficients. Online action by the user enables modification of the various coefficients at any time for example, for proportional coefficients the operator will look for a trade-off between a fast time response and low noise in the command signals. At all times, the computer will watch out for possible physically inadmissible amplitudes and will, if necessary, activate the corresponding alarm systems. - Strategy flexibility. Changing over from one type of control to another is easy, one merely has to switch over from one task to another. - Expansion flexibility. The modularity of the basic programs allows the user to add special purpose functions. CONCLUSION Achieving the model of the overall grinding system by translating all the phenomena of such a complex process into equations was out of the question as it involu~d probabilistic notions of heterogeneous partirle grinding (Andrews, ~ika, 1975 ; Saint-Etienne, 1979 ; A:I.M.E., 1980). We relied instead on very slmple hypotheses and adopted a macroscopic approach of the problem. This is a compromise between the drawing up of a "black box" like model and a microscopic model. Thus we chose either a linear or a non linear form according to the mathematical implements available. Two methods of identification have been compared: non-linear programming and G.L.S.M. both of which produced quite satisfactory results in spite of noise and non-linearities. Thus, certain variables, normally unavailable for measurements such as flow and density in regulation loaps and pulp load in the ballmill, could be estimated. Pulp load in the mill in particular is very important for mineral engineers because it is most likely related to nominal flow, grind quality and ball wear out (which very high in autogenous
The classifier is characterized by its selection curve, which depends essentially on pulp flow and density at its input. This is why we used regulators that allow adjustment of these two values. The process can thereby react to changes in the characteristics of the incoming ore to keep output particle size constant. All the tasks that perform the operations we have described above (service, identification and control tasks) allow easy choice of required points, high circuit stability and storing of all action taken. This control over the circuit now allows setting out series of measurements to look into the correlations that exist between controlled variables and ore characteristics. In the longer term this study will be part of place in a larger study : the output of the grinding circuit will constitute a flotation process command. REFERENCES AI~
(1980). Fine particles processing. Proceedings of the International Symposium and Fine Particles Processing, Las Vegas, 24 28 February.
Andrews, J. R. G. and Mika, T. S. (1975). Comminution ot a heterogeneous material development of a model for liberation phenomena. 11 th International Mineral Processing Congress, Cagliari, 20-26 April. Bouatouch, K. (1977). Recherche des caracteristiques optimales des signaux d'entree pour des methodes statistiques d'identifi~atio~._These de specialite Automatique, Unlverslte de Nancy, 25 Avril. Clarke, D. W. (1967). Generalized leastsquares estimation of the parameters of a dynamic model. IFAC Symposium, Prague. de Larminat, P. and Thomas, Y. (1977). Automatique des systemes lineaires. Flammarion Sciences. Foulard. C., Gentil, S., Sandraz, J. P. (1977). Commande et regulation par calculateur numerique. Editions Eyrolles. Hasdorff, L. Gradient ootimization and non linear control. Wil~y Ed. Hodouin, D., Berube, M. A., Everell, M. D. (1977). ~eport on the sampling campaign performed on the grinding circuit of heath steel mines. LTD. (N.B.). Universite Laval. Canada. Lynch, A. J. (1977). Mineral crushings and grinding circuits. Elsevier Scientific Puhlishing Company, vol. I.
Modelling and Computer Control of a Ball Mill Unit Lynch, A. J., Rao, T. C. (1975). Modelling and scale up of hydrocyclone classifiers. 11th International Minera1 Processing Congress, Cagliari. 20-26 April.
Steiglitz, K., Mc Bride, 1. E. (1965). A technique for the identification of linear systems. IEEE Transactions on Automatic Control. AC-IO, 0ctober, pp. 461-463.
Ragot, J., Degoul, P., Roesch, M., Heitzmann, P. (1977). Axiomatique formalisee des processus de fragmentation. Industrie Minerale - Mineralurgie. Avri 1.
P. Heitzman (1979) - Conception et realisation de la commande d'un circuit de broyageclassification. These Specialite Automatique. Universite Nancy 1.
Saint-Etienne, v. (1979). Modelisation d'un processus mineralurgique. Approche experimentale de la dynamique du broyage discontinu et simulation. These de 3eme Cycle, INPL-ENSG, Nancy.
ACKNOHLEDGMENTS This work was supported in part by
D.G.~.S.T.
nO 75.07.1498 and 77.07.1564.
CLASSIFIER
WATER
Qe
2
\/
357
BALL MILL
(
WATER
Qe
figure I GRINDING CIRCUIT SUMP
6
P. Heitzmann, C. Humbert and J. Ragot
358
llQ
llQ 1
process
Mill llQe
2
s
Ma Sump
MS llQe6-
+ 1
computer
l+s/d
+ l+s/g
d
figure 2 a DENSITY IDENTIFICATION
______________________________________ ______________________
~,
k.
~
~see
r
Q
ore flow at mill input
l'~
.~--------------------,~------~----------------~------~ Qe 2 water flow at mill input
:!===================================~. Qe
6
water flow to the
:~========================================~ d
S
densit y in the sump
--2&011-
.~--------------------------------------------------------~ figure 2 b STEP RESPONSE OF THE SYSTE~ with pulp level and density controllers in the sump
359
Modelling and Computer Control of a Ball Mill Unit
Mill
s
set point
figure 3 a b
PULP VOLUME CONTROL IN THE SUMP PULP DENSITY CONTROL IN THE SUMP
Mill
t-..;,-- - ---I I Model : ~--..,.
feedforword
L-_
I
I
I I _J water contra
+
360
P. Heitzmann, C. Humbert and J. Ragot
+
-
llQ
Sc
llQ
C (.) 6
p
1+8T llQe
S
I
4
I
I
2
a
I I llQ d
S
+
30
+ lid 3
+ lld
S
C (5) 7
lld
S
1+"&
I I
b
J
---1
lld
figure 4
PULP FLOW AND DENSITY CONTROL IN THE MILL
s
•
Modelling and Computer Control of a Ball Mill Unit
361
P
s
+
figure 5 a
b
PULP VOLUME CONTROL IN THE SUMP
MASS FLOW CONTROL
P
II~-!.:::--I.----\
t.Q =t.Q
S
t.Q s9 mass flow
9
901----,-
362
P. He i tzmann, C. Humbert and J. Ragot
water flow SP
\/
water flow 4
density SP
SP
/
/1 /
/,
FF
,;
density SP
/
....- - 1 1
4
I
---_/ pulp SP
SP
flow SP
~,--------------------------<9C
WF WC WA P
water flow measurement PC water flow controller OA water actionner PD proportionner DC FF 'pulp flow feedforword controller PL PF pulp flow measurement or estimationLC t d d PA O ISO 2n 0 3r control strategy
)--4__----~--------~
pulp flow controller ore control pulp density measurement or estimation density controller pulp level measurement level controller pump control figure 6
MODELLING AND COMPUTER CONTROL OF A BALL MILL UNIT P. Heitzmann, C. Humbert and
J. Ragot
Laboratoire d'Automatique et de Recherche Appliquee, BP 452, 54001 Nancy Cedex, France Abstract. This article describes a method of on-line control of the size of output particles in a ball mill and classifier unit. This is not achieved through direct measuring of particle-size but through action on pulp flow and density. Kewwords. Ball mill, computer control, identification, least squares approximations, models, predictive control, state estimation. INTRODUCTION Experiments and theoretical studies on fragmentation phenomena in ball mills have shown that for constant physical characteristics of the ore at the input, particle size at the output depends essentially on the flow of solid that goes through (Ragot and co-workers, 1977 ; Lynch, 1977). The operation process of classifiers - devices which work both as a sensor, sensitive to size, and as control unit capable of separation - has also been throughly examined by researchers (Lynch, Rao, 1975 ; Hodouin, Birubi, Everell, 1977). Models have been defined that express separation taking into account the physical parameters of the ore, pulp flow and density at the input, and the geometrical structure of the classifier. The control engineer only uses effective control variables, the other variables are considered as constant (e.g. Apex diameter) or of constant mean value (e.g. the physical characteristics of the ore). Thus, the relation between the cutting point, dSO, and the control variables can be expressed as :
Z
percentage of solid
e
relative to water
m
relative to ore
the dimensions of the ball mill used were - length
=
- diameter
meter 0 . 8 meter.
The variables we can control are : - ore flow, QI' and water flow, Q2' at mill input, - water flow, Qe , added to the sump to lower 6 pulp density since dowstream circuits (classifiers, flotation cells) operate at a lower density than that in the mill, - pump flow, Q9' at sump output. For obvious practical reasons weight values inside the mill are not available. These are determined indirectly through measurements taken at the sump stage, the sump playing the part of an observer. Ball mill equations
where K1 , K2 , K3 are constant parameters, Z, the percentage of solid and Q, the flow of pulp at the classifier input.
The variations of volume, V , and weight, w , 4 4 inside the mill are related to the volumetric and massic flow by the following relations :
This relation immediately shows how important it is to have a good knowledge and control of the various loads, flows and densities of pulp in the circuit, to ensure control over particle size. MATHEMATICAL MODEL OF THE BALL MILL AND
o
W4
S~P
volumetric flow (l/h)
d
density
V
volume (1)
w
weight (kg)
Qt d l + Qe l de - QS
dS
o
where X means dX/dt.
Ref erences (see fig. I) Q
=
Consider an open loop circuit: Q 11
=
O.
The other equations necessary for a complete definition of the model are based on two hypotheses :
349
P. Heitzmann, C. Humbert and J. Ragot
3S0
I) The ball mill works as an ideal mixer the pulp inside the mill and at its output have the same density.
2) The ball mill works through an overflowing
process : outflow varies in proportion with volume variations :
~vhere
X
(Qs' d S ' VS' d S)! is the state vector.
U
(QI' Qe 2 , Qe , Qg )! is the control 6 vector.
=
The state equation is not linear. The only parameters introduced are T4 and V4 min' equation linearizing The means working point (MWP) is defined by : QIO' dlO' Qe 20 , QSO' dSO' VSO' dSO' Qe 60 , QgO and it verifies the mass balanced equa-
hence after integration
where V min is the minimum volume of pulp 4 retained. Combining the equations gives
o
@I
d4 =
(d I - d 5) + Qe 2 (d e - d 5)] / V4
tions QSO QIO QSO d SO = QgO = Q SO QgO d = SO
+ Qe 20
QIO d + Qe
Qe
de
20
60
A Taylor expansion limited to the first order gives I
o
2
+
Q d + Qe de 60 SO SO
and with hypothesis 2 :
Qe
lO
- T4
o
o
o
o
-g
o
o
o
o
o
c
o
-d
o
o
o
o
(de - d )] S 1
+ V4 minJ
a
Sump equations The equations are determined in a similar way except that the outflow is a controlled value the overflow process hypothesis must be ruled out. The mass balanced equations are
e
-f
o
o
o
o
+
o
Ws
QSd S + Qe 6 de - Qgd g
=
To avoid sedimentation an agitator is added to the sump which now behaves as an ideal mixer :
d
=
VSO d
0
S
= [Q s
(d - dS) + Qe 6 (de S
The model can be expressed as 0
X
f
(~,
!!.)
d8~
SO VSO Q + Qe 60 SO VSO d - I SO
d b
a =
e
d
o
-b
with:
c
After combining the equations
-I
g
=
T
- d
lO
Q 4
SO
SO
+ V
f 4
min
mln
QSO
/ Vs Coefficients a, b, c, d introduced during linearizing only depend on the mean working point and are therefore known if the MWP is
351
Modelling and Computer Control of a Ball Mill Unit
known, too. Coefficients e, f, g introduced in the ball mill equations are related to the MWP as well as to T4 and V4 min. Real time identification presented later will allow knowledge of the values of these parameters. SIMULATION
measurements in the sump are perturbed by - the variations of the mill outflow, QS' the variations of the water added, Qe , 6 - outflow pumping, Q9. The T4 and V4 min coefficients in turn may vary for any number of reasons
Sensors and actuators For direct measuring of the necessary variables we chose the following components currently available on the market - a dosometer-type mass flow control unit
variable pulp viscosity, - variable ore characteristics, ball wearing out,
(QI) ,
- major shifts of the working point. - lieliflux-tyrewater flow sensors (Qe , Qe ), 2 6 - bubble sensors to measure pulp level, hence volume (VS), - bubble sensors or a gamma-ray densimeter to measure pulp density (d ) in the sump, S
- an electromagnetic pulp flow meter at sump output (Q9)' - a controllable outflow pump. The computer The computer used is an a real time monitor.
SE~S
This slow drifting of the coefficients will be considered as equal to zero within a time interval called the "horizon". The generalized least square method (G.L.S.M.) If the model of a linear system is defined by the equation Y= X e, reducing the deviation between the system output and the model output E = Y - Y= Y - X 9 by minimizing a quadratic criterio~ leads ~ after linearizing the criterion with respect to the parameters - to the determination of a parameter estimator :
Solar 16/40 with
In real time operation the "Task" is a basic programm unit which performs specific functions such as measurements, filtering, etc
To remove the bias of this estimator we use the residue whiting technique suggested by Clarke (1967) ; Steiglitz and ~c Bride (1965). Flow response of the ball mill
A number of so-called "service" tasks have been programmed, whose purpose is to : - communicate with the user, - test the presence and validity of the signals coming from the sensors,
The linear model of the flow response is taken in its discrete form. Its transfer function using the z transform is : -T/T4 K(I-e
_I
)z
- e- T / T 4 z-I
- choose which of the components will be used in the ball mill and classifier unit through the means of electrical relay units,
where T is the sampling period.
- send commands to the process,
Theoretically the gain K = b /(] + a]) is
register all the states of the circuit, - give reports,
l
equal to one and we merely have to determine q] or b] with G.L.S.M. Practically we shall derive K from the two values obtained with G.L.S.M..
- perform the various start/stop sequences. To these service tasks we must add signal processing and control value computing tasks. IDENTIFICATION The ball mill parameters to be determined are the overflow volume, V . , and the flow4 ml.n dependent time constant, T . 4 The method of identification will face very noisy signals since pulp flow and density
A gain, K, other than unity on a certain horizon will show the inadequacy of the method due to a lack of information. Density response of the circuit The density response of the circuit is used to identify the value of V4 min via coefficient g defined previously. The general principle is shown in fig. 2 a.
352
P. Heitzmann, C. Humbert and J. Ragot
Parameter, identification through non linear programming
ble matrix in the
G.L.S.~.
(Bouatouch, 1977).
Pseudo-random binary sequences (P.R.B.S.)
A non-linear programming approach is used here since it avoids linearizing density equations. We min1m1se the following quadratic criterion by adjusting coefficient V4 min' I k r-C(k) = "2 . L Lds (i) -dS (i~ 2 T 1=k-N+I
where N . T is the time horizon.
A gradient algorithm is used to find the value of the coefficient P+I
vP
V4 min
4 min
-
k
~C
(k) V4 min
L
i = k - N+ I
rd
LS
(i) - d
S
(i)]
adS =---T
av 4 min To find out the adS / av 4 min term of this equation, the following equations are added to the initial set of the model equations ~ = f (~~) and solved by Kutta ~unge's method: a aV 4 min a SV 4 min
d d
s (----crt) d ds
(
d
For first order systems, I/:;. T is usually chosen greater than 5 times the time constant and AT slower than the time constant divided by 5. To the logical value of "I" corresponds a "a" amplitude, and to "0" a "-a" amplitude. "a" is chosen to generate a minimum but sufficient variation when the slowest step occurs.
P indicates the iteration number, with
Consider an M stage shift register driven by a clock with a T period and having a feedback made up of the addition modulo 2 (exclusive OR) of same stages. The P.R.B.S. is the signal given by any of the registers. It is periodic: its period is 1 /:;. T and its length is 1. They are same possible combination of signals to be added modulo 2 to have a maximum length I = 2 lo\~ 1.
ads
.) dt (av 4 m1n d (ads -dt V min) 4
These algorithms have been programmed and integrated into the real time system as a task (Hasdorff~.
Input signals We must be able to achieve any required variations of pulp inflow and density around the mean working point without interfering with particle size at the output . The simplest way of doing this is to act on water inflow.
Note - If we take a step signal as input there is still the problem of choosing the right starting point and identification horizon, whereas the horizon alone matters when P.R.B. S. are used. In that case, the horizon must be equal to a sequence length, which is a considerable advantage in real time identification. Since permanent superposal of P.R.B.S. allows the start at any time of an identification task which will retrieve the last N values in the mass memory (with N x T = I x/:;. T, T the sampling period, I the sequence lenght, /:;.T he length of the shortest step in the sequence (Foulard, Gentil, Sandraz, 1977). Results from simulated values The circuit was simulated with the non-linear model. Noiseless responses were used to test the identification methods. The main conclusions about GLSM are that: - there is immediate convergence of T4 towards the simulated value,
Step input
- the convergence of v 4 min is obtained after about twenty passages through the residue whiting filter. This is due to the adaptation error : the model for simulation used non linear equations and the P.R . B.S., linearized ones,
The most commonly used input signal is the step since it allows a visual estimation of phenomena and the use of numerical identification methods. It is essential, however, to take a few precautions
- using an increasing or decreasing step alone leads to significant errors in the parameter values because in that case the relations defined between the mean values are not verified. For example:
- its duration must be long enough to allow the output to reach a steady state,
d SO = (QIO d l +Qe 20 +Qe 60 ) / (QIO+ Qe 20+ Qe 60)
- the identification horizon must take into account both the stable and the transient states, but the former's period must not be too long compared with that of the transient states as this would lead to a non inverti-
(QI d I + Qe 2 + Qe ) / (QI + Qe 2 + Qe 6 ) 6
=r
dS
where X is the mean value of x (t).
Modelling and Computer Control of a Ball Mill Unit
3S3
The G.L.S.M. works well with symetrical steps (increasing and decreasing) or with P.R.B.S.
Filter
With non linear programming, convergence towards the right value is fast and the optimal criterion is in the 10-S range.
A particular simple and efficient one is given in the recurrent expression where f(kT) is the measured value at time kT, and fX(kT) the filtered value :
Results from actually measured values
fX(kT) = a [f(kT) + qf
With the G.L.S.M. the value of V4 min after the first passage is totally wrong: 32 1. While the space between the balls alone is 60 1 (this value was obtained by filling in the space between balls with water, the engine not running). After about 20 iterations we obtain a value of 140 liters.
+ q2f [(k -
2)T~
(k-I)T] ...
or
Hence a + q = I to have uni ty gain. -I
With the non linear programming method the value of V min quickly reached 120 1. Near 4 the optimum value the criterion stayed high (2.SS) compa~ed with that obtained in simulation. This is due essentially to measurement noises and errors. OF THE STATE VARIABLE VALUES
ESTI~ATION
Using the z operator the transfer function of the filter can be expressed as : - q
- qz
+
V min) d S 4
5 Q S
+
(QI
Qe 2 - QS) / T4
5 d
S
[Q I (d I - d S) [T4
QS +
+
Qe
2
(I
-~sU
/
V 4 min]
Actually we shall use the discrete form instead and Kutta Runge's method of integration. Ore and water inflow measurements are little disturbed by noise and, what is more, the transfer functions act as low pass filters thus giving estimated state values that are little disturbed by noise. This approach, however, is not based on feedback principles and therefore does not take into account possible disturbances causes by erratic ore feeding, chemical additions or unmeasured classifier backflow. From sump values
0
Q9 - Qe 6
d
d
S
S
+
+
(de Larminat, Thomas, 1977) Volume control in the sump The diagram of fig. 3a outlines the general principle of regulation. If we want to use the overall value of the signals we must take into account the non linearities that exist in the circuit (e.g. pump non-revertibility and saturation). In regulation, the signals to be considered are variations round a determined value and pump operations can be expressed by a first-order system with a unity gain and a T time constant. The outflow of pulp from th1J1 ball mill, QS' and the flow of water added, Qe , act as an overall disturbance, 6 AQ 7 = ~QS + ~Qe6· With CI(s), as the controller transfer function, the system equation is derived from the diagram and the linearized model equations
~VS(s)
=
AVSC(S) Cl (s) + ~Q7(s) (I + Tm(s) CI(s) + s (I + Tm s)
Proportional feedback controller. The proportional controller coefficient, K , is deterI mined so that the positioning error due to a ~Q7 perturbation should be equal to a given value
The sump equations give d and Q in terms of S S the VS' d S ' Qe 6 , Q9 measured values QS
-T/TF - e _ z-I e- T / TF
CIRCUIT CONTROL
Applying the values of ore and water inflows to the model allows estimation of the weight in the mill, pulp outflow and density using the following equations : (T 4 Q S
or
which immediately points out the similarity with a first-order filter with a, TF, time constant.
From input values
W 4
-I
Vs
(d S - I) Qe / Q 6 S
0 +
d S "s / Q S
in which we can see derived functions of noisy values that will amplify noise unless a filter ~s added.
Cancelling the positioning error through the means of a proportional integral controller CI(s) = KI (I + I / Tis) was abandoned for stability reasons as it may indeed not have been possible to comply with the stability condition, Ti > Tm' satisfactorily.
P. Heitzmann, C. Humbert and J. Ragot
3S4
Feedforward controller. Rath~r than let any perturbation occur and, only then, move in with a feedback "corrector" we shall measure the source of the disturbance, 6Q7' and modify the control variable, ~Q9. Only disturbance 6Qe
6
- water added - is di-
rectly measurable as the data on pulp flow at mill output, 6Q , are not available ; an observer, achieve~ by a computer using the model of the mill, will give an estimation of 6Q S called 6Q . The signal, an image of the disS turbance, is applied to pump control with due respect to scale. Feedforward control must take place only if the volume has reached its preset value, hence the feedforward signal will be algebraically added to that from the feedback. Pulp density control in the sump Water is added when a measured density is higher than a preset value. The control diagram using linearized equations is given in fig. 3b. The equation of the closed-loop system is a~QS(s)
+
b C (s) 6d 8C (s) + c AdS(s) 2 s + d + b C (s)
2
P.I. controller. The use of a proportional controller was rejected as it led to a positioning error that was too high with a value fo the proportional coefficient physically achievable. We used a P.I. controller yith a transfer function of C2 (s) = K2 (I + T s) i2 Ti2 is chosen so that the integration constant should be equal to the time constant of the open loop system; T. = I/d. In which case 2 (6d ) is 1 8 M
1S
the
Feedforward controller. As in the preceding case the variations at the output variable, d 8 , induced by the disturbances, Q and d ' S S are cancelled out by action on the control variable Q . e6 o d8
0 =
~S
(d S - d 8) + Qe 6 (I - d 8)] / V8
This is true only if d So Qe
8
= d
8C
Note - The feedforward control signals are derived from ball mill input signals which, themselves, had been previously filtered by the mill transfer functions acting as low-pass filters. The commands are therefore less noisy than those sent by the feedback controllers. With both regulators working together the gain of the feedback controller will be reduced, and there will be less high frequency fluctuations of the control variables sump water inflow and pump inflow. Pulp volume and density control in the sump combined with control of ore and water coming into the ball mill are the first stage in automatizing the system. If we now consider a grinding circuit with a classifier we may assume that the grindability of the ore may become lower (e.g. more big particles than usual at the input or variations of the physical caracteristics of the ore). Number of big particles at the mill output will increase, and as there are sent back to the mill by the classifier, they will cause a flow increase, which in turn, will accelerate this process of insufficient grinding. This simple example shows how difficult it is for the above circuit to cope with any changes in ore characteristics successfully. With that in mind we suggest a different approach which consists in controlling pulp throughflow. Pulp flow control in the mill Theoretically the simplest solution would be to keep the amount of ore inflow at a constant level by adapting new inflow to the measurements given by a flowmeter at the classifier output. We could also add a gamma-ray densimeter to act on ore and water separately. But this would hardly be feasible in practice since pulp flowing out of the classifier is full of air bubbles which prevent correct functioning of the sensors.
8C (S)
In a steady state lid 8 (t) = t'.d 8C (t). 6d 8C set point.
Since flow, QS' and density, d ' are not meaS surable, the values used are those given by the simulation model. The operation principle is shoVTI in the functional diagram using the linearized equations (fig. 3b).
.
, the feedforward control variable, is 6FF equal to Qe = Q (d - d ) / (d - ). 6FF 8C 8C S S Practically the feedback and feed forward signals are algebraically added.
Another solution, more easily feasible with the available technology, is to regulate mill outflow through acting on new product input while considering the back flow from the classifier as a disturbance. The diagram (fig. 4a) clearly shows this principle. C (s) is the 6 transfer function of the controller and P is the water/ore opportioning unit. C6 (s) QSC(s) + Q)) (s) tl + sT 4 ) + C6 (s) \,e can take a P.T. controller with Ti =T . 4
355
Modelling and Computer Control of a Ball Mill Unit
Hence T4 QII(s) s K 6 + T4 T4 I + s (I+T s) ( I + - s) 4 K6 K6
input - the classifier backflow and sump water additions being considered as disturbances (see fig. Sa),
QsC(s)
Q (s)
s
The general transfer function using a proportional controller CS(s) = K ' is : S
[Ks
(J / T4)
Pulp density control in the mill
- Q (s~ 9
The water/ore apportioning unit splits the overall new product flow, Q3' in two, according to ore density, d , ana the preset densil ty, d ' 3C
If flow control alone is used, d ' is kept 3C constant but if densit y perturbations causes by the classifier are taken into account, action on d to cancel them will have to be 3C taken. Note that recycled pulp is specified by its flow, QI I' and density, d ll · The balanced equations are :
A Taylor expansion around the mean working point defined by Q30' d 30 , QIIO' QsO' d so ' = Q and verifying the relations Q 30 + QIIO sO and d d gives llO 30 0
g lidS + v lid 3 + w MII
t;d 5 with g
QsO/(T4 QsO + V4 min)
v
Q30 / (T 4 QsO + V4 min)
w
QIIO / (T 4 QsO + V4 min)
Note that g = v + w. The use of the Laplace transform leads to the definition of a control diagram (see fig. 4b).
s
"With a P.r. controller C (s) = K7 (J .. I /Ti 7s) 7 where the integration constant, Ti , is equal 7 to the time constant of the open-loop system Ti7 = I /g .
+ QI I (s) + (Qe 6 ( s)
+ S T ) 4
+ (I / T 4) s + KS / T 4
For an aperiodic response the damping factor, ~, must be greater than or equal to, one ~ = 1/21T4 Kg >,.1. Hence KS(I /(4 T4 ). The positioning error is (QII + Qe 6 - Q9) / KS' An important error, but one that does not disturb the operation process if one has been careful to take an adequate volume in the sump. To this system may be added a density control loop (see fig. 6) along with a feedforward controller to cancel Qe induced perturbations. In this type of control 6 the s"mp off~ct:vely acts as a buffer. Total mass flow control The principles of this control were set out (Lynch, 1977), The amount of water added to the sump is constant and pulp volume is controlled by action on pump intake after level measurements (as described above), The total sump mass outflow, given by the product of the signals from an electromagnetic flowmeter into the signals from a gamma-ray densimeter, is controlled by action on new ore input (see fig, sb). Generally the response time for volume control is lower than that of the other time constants in the circuit lIQ = lIQ9' To s simplify the equations, and only to that purpose the pulp backflow from the classifier will be considered as having the same density as that of new product input, lid I I = O. This simplification will in no way contradict the general principle of the control suggested here. If the mean working point is the reference point (in regulation systems only and not in servo systems) IIQ = 0 the relation 9C between mass outflow and classifier backflow variations is given by : d
(I + s/d) + (a/d) Q 90 90 (I + s 7d) (I + s T4) + C9 (s) (Cl 90 (I + s / d)
The response is given by : vC (s) 8d sC (s) + wild I I (s) 7 s + g + vC (s) 7
2
(I
VSC ( s)
+ (a/d) Q9~
or
d sO + (d
/d)s 90 (I +s!d)(1 +s T ) + 4 C9 (s) [d so + (d 90 /d)
sJ
QS9 is the mass flow, the others being voluFlow control using sump volume variations The use of an ordinary pump, instead of one with controllable intake, will be possible if we control sump volume by acting on new ore
metric flows. We shall use a P.I.D. controller, whose time constants can be easily adjusted through a compensation method Cg(s) = Kg + I / (T;-9 s ) + Td sJ
[I
356
P. Heitzmann, C. Humbert and J. Ragot
with Ti
grinders). This method is therefore directly competitive with indirect measurements such as noise level, grinder weight and oil pressure in the bearings.
Hence :
(l+
~
s
d
cr)(I+ST 4 )[I+(K 9 d 50 )+~)Sl 50 J
From which we can see that the smaller the sump (d high) is, the faster the response. Implementation The control algorithms have been programmed and entered as real time tasks to achieve : - Control flexibility. Most of the controller time coefficients are dependent on the M.W.P. or on the values of the coefficients in the model. The use of a computer allows automatic adjustment of these controller coefficients. Online action by the user enables modification of the various coefficients at any time for example, for proportional coefficients the operator will look for a trade-off between a fast time response and low noise in the command signals. At all times, the computer will watch out for possible physically inadmissible amplitudes and will, if necessary, activate the corresponding alarm systems. - Strategy flexibility. Changing over from one type of control to another is easy, one merely has to switch over from one task to another. - Expansion flexibility. The modularity of the basic programs allows the user to add special purpose functions. CONCLUSION Achieving the model of the overall grinding system by translating all the phenomena of such a complex process into equations was out of the question as it involu~d probabilistic notions of heterogeneous partirle grinding (Andrews, ~ika, 1975 ; Saint-Etienne, 1979 ; A:I.M.E., 1980). We relied instead on very slmple hypotheses and adopted a macroscopic approach of the problem. This is a compromise between the drawing up of a "black box" like model and a microscopic model. Thus we chose either a linear or a non linear form according to the mathematical implements available. Two methods of identification have been compared: non-linear programming and G.L.S.M. both of which produced quite satisfactory results in spite of noise and non-linearities. Thus, certain variables, normally unavailable for measurements such as flow and density in regulation loaps and pulp load in the ballmill, could be estimated. Pulp load in the mill in particular is very important for mineral engineers because it is most likely related to nominal flow, grind quality and ball wear out (which very high in autogenous
The classifier is characterized by its selection curve, which depends essentially on pulp flow and density at its input. This is why we used regulators that allow adjustment of these two values. The process can thereby react to changes in the characteristics of the incoming ore to keep output particle size constant. All the tasks that perform the operations we have described above (service, identification and control tasks) allow easy choice of required points, high circuit stability and storing of all action taken. This control over the circuit now allows setting out series of measurements to look into the correlations that exist between controlled variables and ore characteristics. In the longer term this study will be part of place in a larger study : the output of the grinding circuit will constitute a flotation process command. REFERENCES AI~
(1980). Fine particles processing. Proceedings of the International Symposium and Fine Particles Processing, Las Vegas, 24 28 February.
Andrews, J. R. G. and Mika, T. S. (1975). Comminution ot a heterogeneous material development of a model for liberation phenomena. 11 th International Mineral Processing Congress, Cagliari, 20-26 April. Bouatouch, K. (1977). Recherche des caracteristiques optimales des signaux d'entree pour des methodes statistiques d'identifi~atio~._These de specialite Automatique, Unlverslte de Nancy, 25 Avril. Clarke, D. W. (1967). Generalized leastsquares estimation of the parameters of a dynamic model. IFAC Symposium, Prague. de Larminat, P. and Thomas, Y. (1977). Automatique des systemes lineaires. Flammarion Sciences. Foulard. C., Gentil, S., Sandraz, J. P. (1977). Commande et regulation par calculateur numerique. Editions Eyrolles. Hasdorff, L. Gradient ootimization and non linear control. Wil~y Ed. Hodouin, D., Berube, M. A., Everell, M. D. (1977). ~eport on the sampling campaign performed on the grinding circuit of heath steel mines. LTD. (N.B.). Universite Laval. Canada. Lynch, A. J. (1977). Mineral crushings and grinding circuits. Elsevier Scientific Puhlishing Company, vol. I.
Modelling and Computer Control of a Ball Mill Unit Lynch, A. J., Rao, T. C. (1975). Modelling and scale up of hydrocyclone classifiers. 11th International Minera1 Processing Congress, Cagliari. 20-26 April.
Steiglitz, K., Mc Bride, 1. E. (1965). A technique for the identification of linear systems. IEEE Transactions on Automatic Control. AC-IO, 0ctober, pp. 461-463.
Ragot, J., Degoul, P., Roesch, M., Heitzmann, P. (1977). Axiomatique formalisee des processus de fragmentation. Industrie Minerale - Mineralurgie. Avri 1.
P. Heitzman (1979) - Conception et realisation de la commande d'un circuit de broyageclassification. These Specialite Automatique. Universite Nancy 1.
Saint-Etienne, v. (1979). Modelisation d'un processus mineralurgique. Approche experimentale de la dynamique du broyage discontinu et simulation. These de 3eme Cycle, INPL-ENSG, Nancy.
ACKNOHLEDGMENTS This work was supported in part by
D.G.~.S.T.
nO 75.07.1498 and 77.07.1564.
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