- Email: [email protected]
Fuzzy Modeling of a Column Flotation Process
Copyright © IFAC Automation in Mining, Mineral and Metal Processing, Nancy, France, 2004
ELSEVIER
IFAC PUBLICATIONS w ww _elsev ier. com/localeli fac
FUZZY MODELING OF A COLUMN FLOTATION PROCESS S.M. Vieira,* J.M.C. Sousa,* A. Felicio,t F.O. Dudio t
Technical University of Lisbon, Instituto Superior Tecnico • Dept. ofMechanical Engineering/ GCAR - IDMEC t Dept. Mining Engineering/ CVRM Av. Rovisco Pais, 1049-001 Lisbon, Portugal Phone: +351-21-84/731
Abstract: Real multivariable non linear systems are always difficult to model. Column flotation process is a very complex, nonlinear and multivariable system. When a model of a system like this is needed, it is not always possible to use or identify a common linear model. Fuzzy modeling is a well-known modeling technique, which has been applied to complex and nonlinear processes. This paper integrates fuzzy multivariable modeling with structure identification. One of the hardest tasks when modeling real systems is to define the model structure, specially if it is a multivariable system. To define this structure it is used a common criteria. The results show that is possible to find a better model using this technique, Real data is used for the design and validation of the fuzzy model. Copyright © 2004 IFAC Keywords: Fuzzy modeling, multivariable modeling, structure identification, column flotation,
1. INTRODUCTION
• Finally, the difficulties inherent to the acquisition of accurate measures of many variables of the process.
Column flotation is nowadays an important mineral processing unit. It is a complex multivariable process undergoing several disturbances, such as those originated by changes in feed characteristics and in equipment wearing (Finch and Dobby 1990).
A previous study (M.T. Carvalho and Femandes 1999) carried out with a laboratory column flotation and a two phase air-water system, aiming to develop a linear dynamic model, showed that only some relations between controlled and manipulated variables could be identified, using transient analysis.
Classic and modem approaches of process control rely on the knowledge of dynamic models of the process. However, until now it has not been possible to develop a phenomenological dynamic model of the column flotation, due mainly to the following difficulties:
Trying to overcome these difficulties this paper proposes the identification of a non linear dynamic fuzzy model using input-output data from a laboratory column flotation and a regularity criterion (RC) to find, automatically, the relations between input-output variables (Sugeno and Yasukawa 1993).
• Inaccurate knowledge of the physical and chemicallaws of the sub-processes involved; • Formulation of partial differential nonlinear systems equations of distributed parameters, because the geometry of the column forces the consideration of the axial position; • Need to solve a mathematical problem of a movable interface between the collection and the froth zones;
The paper is organized as follows. The fuzzy modeling technics are described in section 2, The section 3 presents the process of variable selection for each output and describes the procedure of structure identification. The process description is presented in the
435
section 4. The results obtained are shown in section 5 and some conclusions are drawn in section 6.
are fuzzy sets defined in the antecedent space, and Yi is the rule output variable. K denotes the number of rules in the rule base, ai is a parameter vector and bi is a scalar offset. The consequents of the affine TS model are hyperplanes in the product space of the inputs and the output, i.e. lR n x IR. The model output, y, can then be computed by aggregating the individual rules contribution:
2. FUZZY MODELING Fuzzy modeling using measures of the process variables. is a tool that allow an approximation of nonlinear systems when there is no prior knowledge about the system or it is only partially known. Usually, fuzzy modeling follows three steps:
(3)
• Structure identification; • Parameter estimation; • Model validation.
where {3i is the degree of activation of the ith rule: n
(3i =
One of the important advantages of fuzzy models is that they combine numerical accuracy with transparency in the form of linguistic rules. Hence, fuzzy models take an intermediate place between numerical and symbolic models. Traditionally, rule-based fuzzy systems are used (Zadeh 1973). In computational terms, fuzzy models are flexible mathematical structures that are known to be universal function approximators. In fuzzy modeling, the fuzzy "If-Then" rules take the following general form:
and fLAij (Xj) : IR of the fuzzy set A
[0, 1] is the membership function in the antecedent of Ri.
-> ij
3. IDENTIACATION
Fuzzy models use "1[-Then" rules and logical connectives to establish relations between the variables defined to derive the model. The fuzzy sets in the rules serve as an interface amongst qualitative variables in the model, and the input and output numerical variables. The rule-based nature of the model allows a linguistic description of the knowledge. The fuzzy modeling approach has several advantages when compared to other non linear modeling techniques. In general, fuzzy models can provide a more transparent model and can also give a linguistic interpretation in the form of rules. The system to be identified can be represented as a MIMO nonlinear auto-regressive (NARX) model:
The nonlinear identification problem is solved in two steps: Structure identification and parameter estimation. Structure identification: in this step the order of the model and the significant state variables x of the model must be chosen. This is a very important step. In this particular case, once we do not know all the relations between the input-output variables, it will be used a regularity criterion (RC) to find the significant state variable (Sugeno and Yasukawa 1993). To identify the model (2), the regression matrix X and an output vector y are constructed from the available data:
(1)
Here N » n is the number of samples used for identification.
where x is a state vector obtained from several inputs and outputs of the system.
Parameter estimation: The number of rules, K, the antecedent fuzzy sets, A ij , and the consequent parameters, ai, bi are determined in this step, by means of fuzzy clustering in the product space of X x y (Babuska 1998). Hence, the data set Z to be clustered is composed from X and y:
2.1 Takagi-Sugeno fuzzy Models
The Takagi-Sugeno (TS) fuzzy models (Takagi and Sugeno 1985), consists of fuzzy rules where each rule describes a local input-output relation, typically in an affine form. The representation of (1) as a TS model is given by:
~
(4)
j=l
If antecedent proposition then consequent proposition
y = j(x),
IIfLAij(Xj),
Z
= [X,Y1 T
Given the data Z and the number of clusters K, the Gustafson-Kessel fuzzy clustering algorithm (Gustafson and Kessel 1979) is applied to compute the fuzzy partition matrix U. This provides a description of the system in terms of its local characteristic behavior in regions of the data identified by the clustering algorithm, and each cluster defines a rule. Unlike the popular fuzzy c-means algorithm, the Gustafson-Kessel algorithm applies an adaptive distance measure. As
: If Xl is Ail and ... and X n is Ainthen
where i = 1,2, ... , K. Here ~ is the i th rule, x = T [Xl, ... ,xnl is the antecedent vector, Ail, ... ,A in
436
such, it can find hyper-ellipsoid regions in the data that can be efficiently approximated by the hyper-planes described by the consequents in the TS model.
and B, yAB is the model output for the group A input estimated by the model identified using the group B data, yBA is the model output for the group B input estimated by the model identified using the group A data.
The fuzzy sets in the antecedent of the rules are obtained from the partition matrix U, whose ikth element /lik E [0,1] is the membership degree of the data object Zk in cluster i. One-dimensional fuzzy sets A ij are obtained from the multidimensional fuzzy sets defined point-wise in the ith row of the partition matrix by projections onto the space of the input variables x j: .) /lA,j ( Xjk
= proJj.N
n
+1 ( ) /lik ,
Using the two groups of data, we build two fuzzy models, one for each group, A and B, beginning with only one input. At this stage we build a fuzzy model for each input in consideration. Once we are dealing with a MIMO system and this method is only applied to MISO systems, we must consider the outputs not in consideration as inputs for the system. We calculate RC of each model and select the model that minimizes RC. In the next step we consider the input of the selected model fix (belonging to the system's structure) and add another input to our fuzzy model from among the remaining candidates. Now we have two inputs and we select the second input as we do at the first step, according to the minimal value of RC. From now on we proceed as this far until the value of RC increases. When this happens we should have all the relevant input variables for the considered output. Using this method we can reduce the number of iterations, once all the input variables, that produce models with RC bigger than the minimal at the stage, can be excluded from the process. In a generic case the maximum number of iterations is n(n + 1) /2, where n is the number of input variables.
(6)
where proj is the point-wise projection operator. The point-wise defined fuzzy sets A ij are approximated by suitable parametric functions in order to compute /lA,j(Xj) for any value of Xj' The consequent parameters for each rule are obtained as a weighted ordinary least-square estimate. Let BT = [aT; b;J, let X e denote the matrix [X; 1] and let W i denote a diagonal matrix in lR. N x N having the degree of activation, (3i(Xk), as its kth diagonal element. Assuming that the columns of X e are linearly independent and (3i(Xk) > 0 for 1 ~ k ~ N, the weighted leastsquares solution of y = XeB + E: becomes (7)
When using this criterion a fuzzy model is created at each iteration. To do so the number of clusters that best suits the data must be determined. For this purpose it is used the following criterion (Sugeno and Yasukawa 1993):
Rule bases constructed from clusters are often unnecessary redundant due to the fact that the rules defined in the multidimensional premise are overlapping in one or more dimensions. The resulting membership functions will thus be overlapping as well, and more fuzzy sets will describe approximately the same concept. Therefore, the parameters of the model identified must be carefully chosen.
n
S(c) =
c
L L(/lik)ffi(11
Xk -
Vi
11 2
-
1 Vi
-
X
11
2
),
k=li=l
(9) 3.1 RC - Variable Selection where n is the number of data to be clustered, c is the number of clusters (c ~ 2), Xk is the k th data (usually vector), x is the average of data, Vi the center of the i th cluster, /lik is the grade of the k th data belonging to i th cluster and m is a adjustable weight (usually m E [1.5,3]).
When the relevant variables are known as well as the relation between them, this task is not so difficult. The identification of fuzzy model for a column flotation process is a quite complex task. This process has a large number of variables, and the most relevant ones must be chosen. In this paper a certain criterion is minimized to determine which input variables influence each output (Sugeno and Yasukawa 1993). To do so the data is divided into two groups, A and B. As a criterion to this purpose, it is used the regularity criterion, RC, used in "group method of data handling", which is defined as follows:
RC
The number of clusters, c, is determined so that S(c) reaches a minimum as c increases: it is supposed to be a local minimum. In each iteration, S(c) the cluster(s) are determined using the fuzzy C-Means algorithm and the process stops when S(c) increases from one iteration to the next one.
_ JI:~~I(yt-ytB)2/kA+I:~:I(yf'-yf'A)2/kBl -
2
'
The first term of the right-hand side is the variance of the data in a cluster and the second term is that of the clusters themselves. The optimal clustering achieved is the one that minimize the variance in each cluster and maximize the variance between the clusters. The
(8) Where kAand k B are the number of data of the groups A and B, y~and yf are the output data of the groups A
437
parameter m has a great importance in this criterion. The bigger the m is the bigger it will be the optimum number of clusters.
4. PROCESS DESCRIPTION Froth flotation or, shortly, flotation (as froth flotation is the most important of the flotation processes) was introduced in the beginning of the 20 th century and is one of the most versatile separation processes used in mineral processing. Until then, it was not possible to separate several minerals, like most of sulphides, because they exhibit similar density, magnetic susceptibility and conductivity. The flotation process separates fine solid particles based on physic and chemical properties of their surfaces. Industrially, it is a continuous solid-solid separation process performed in a vessel where a threephase system is present: solid particles, air bubbles and water. This pulp is previously conditioned with the controlled addition of small quantities of specific chemical reagents to promote the selective formation of aggregates between solid particles of a given composition and air bubbles. Air is continuously injected in the pulp, giving rise to the formation of air bubbles. The formation of stable froths, capable of transporting particles to the surface of the vessel, is possible if a convenient frother in an adequate concentration is used.
Fig. I. Column flotation scheme (M - Flowmeter, P Pressure sensor). uct hydrophilic particles that were dragged with the aggregates bubbles-hydrophobic particles. The flotation environment is influenced by variables, such as pulps pH, solids concentration, flotation intensity (speed of agitation and air flow rate), particles size and type of water medium.
5. PROCESS IDENTIFICATION
Hydrophobic particles adhere, after collision, to the air bubbles, which move upwards to the top of the vessel where they are recovered as the floated product. Hydrophilic particles settle in the pulp, become the non-floated product or underflow.
This process is difficult to model once it is highly nonlinear and it is not known all the mathematical relations, between variables, that describes the process. There is a few empirical relations for some of the evolved process variables (Carvalho 1998). Usually this is a process controlled by human, but it is an unstable process, except for a very restricted area of operation. The process being unstable makes it more difficult to identify.
Besides mineral processing, it is used in some other fields, such as solvent extraction and recycling. Usually, a vessel which is called flotation cell, with a agitated impeller that keep the pulp in suspension, and the bubbles of air in movement, promoting collision with the particles.
The process in study is highly nonlinear, and the system cannot be totally described by first principles, so is only partly known. In this case it is advantageous to use fuzzy modeling as a way to combine first principles, human knowledge and/or measurements, obtaining an usually called gray-box modeling approach (Babuska 1998).
In the case study, the separation process is performed in a column containing the water medium. The principal difference to the flotation cell is that there is not an impeller to keep the pulp in suspension. Because of that, column flotation is mostly used to separate finer particles than flotation cell equipments. A flow of air is continuously injected in the medium to transport the particles. After collision with air bubbles, which move upwards to the top of the column, hydrophobic particles adhere to them and will be recovered as the floated product. Hydrophilic particles stay in the vessel, becoming the underflow product, recovered in the bottom of the column.
5. I Identification experiments Once the aim of this work was to model the system using fuzzy logic, the experiments to obtain the test data, were design so that all system variables could take all possible values within each variable range. This was accomplished using white noise generation, so we could use a signal with many frequency information and non correlated. The column flotation is a slow
There is also a shower of water in the top of the froth column, used to "wash" from the floated prod-
438
process. When we are dealing with its identification we must be aware that the output variables will take some time to react to the perturbation on the input variables. Knowing this, we constructed the identification signals giving the enough time to the process to respond.
,'~~ o
500
1soo
1000
2000
'11\~~~~1 o
500
, 000
, 500
2000
Tm·ls]
-5~O-----,5OO:::----,'-":OOO:----~,500=-----2000~---......J rm.[sl
Fig. 3. Test results for Fuzzy model using empirical knowledge - V AF = [53.4 85.188.6]% (dashed line: model output; continuous line: system output) H
Fig. 2. System inputs - air flow rate (Q air), rejected flow rate (Qrej), feed flow rate (Qjeed) and washing water flow rate (Qww)
Emp. model:
VAF
53.4 5.3 76.6 4.0
RMS
RC model:
VAF RMS
2)(Yi - YmY)/N,
I Holdup I 88.6 0.6 88.0 0.6
the one using empirical relations. Both indexes show that the performance improve, especially for the level H, which is the most difficult variable to estimate. The validation of the RC model in different data is
'~E~rv:YJ
N
=
Qb 85.1 1.4 92.8 1.0
Table I. Fuzzy models evaluatiOn usmg VAF andRMS
In Fig. 2 it is shown the inputs applied to the system. They are the response of the actuators to the identification signals. The initial signals are steps with the sufficient duration to ensure that the process has enough time to respond. The outputs to be estimated are: the level H, the bias flow rate Qb and the air holdup E, as can be seen in Fig. 3 and 4. The root mean square is used as performance index:
RMS
I
(10)
i=1
o
soo
1 000
1500
2000
,tv~1n¥~}j
where Yi is a system output and Ymi is the correspondent model output. The percentile variance accounted for (VAF) was also used to measure the performance of the model obtained.
o
500
soo
I 000
1
2000
TlrMlsj
V AF = 100%(1 - var(Yi - Ym.)/var(Yi)) (11) where cov is the covariance of the respective vector.
-5i~O---500:::----,~OOO:----~,500=-----2000~---......J nnelsJ
Fig. 4. Test results for Fuzzy model using RC V AF = [76.692.888.0]% (dashed line: model output; continuous line: system output)
5.2 Results The results using the model structure using only the empirical relations between variables is presented in Fig. 3.
presented in Fig. 4, and it is clear that all the outputs show a good performance.
The RC criterion was used to identify the model structure, and it was found that the optimum number of clusters is 4 for each output. The combination of the RC criterion and the knowledge about the process to tune the system delays leads to the best models. The performance indices for both identification techniques is shown in Table I. This table shows that the model obtained with RC has a much better performance than
5.3 Fuzzy model structure This model has 4 inputs and 3 outputs. The sampling period is 4.3 s. The termination tolerance of the clustering algorithm was 0.0 I. In the following, the output-specific information is shown for each output.
439
6. CONCLUSIONS
Output I - Level (H): Rules:
This paper describes a fuzzy model of the flotation column, developed for control. The structure of the model was determined based on an automatic determination of the number of rules; the RC criterion. This simple criterion was very helpful to find the model structure in the column flotation, for which the system dynamics is not completely known. The obtained results shown that the structure of the model is well determined, as the identified fuzzy model is able to describe the system dynamics in an accurate way.
l. If H(k - 1) is All and Qb(k - 1) is Al2 and c(k - 1) is AI3 and Qair(k - 1) as A I 4 and Qrej(k - 2) is A I5 and Q/eed(k - 1) is AI6 and Qww(k - 2) is An then
H(k) = 4.69.10- 1 H(k -1) -1.45·Qb(k-1) +3.67·c(k -1) -1.21· lO- I Qair(k - 1) +4.47· 1O- 2 Qrej(k - 2) +3.75 . 1O- 2 Q/eed(k - I) +6.50· lO- I Qww(k - 2) +2.45·10 2.
if H(k - 1) is A21 and Qb(k - 1) is A 2 2 and c(k - 1) is A 23 and Qair(k - 1) is A24 and Qrej(k - 2) is A25 and Q/eed(k - 1) is A26 and Qww(k - 2) is A27 then H(k) = 4.69· H(k -1) +1.94 ·lOQb(k -1) -1.63 ·10e(k - 1) +6.59.10- 1 Qair(k-l) -1.28·Qrej(k- 2) +1.24·Q /eed(k1) -318· Qww(k - 2) -2.49.10 2 3. If H(k - 1) is A31 and Qb(k - 1) is A32 and e(k - 1) is A33 and Qair(k - 1) is A34 and Qrej(k - 2) is A35 and Q/eed(k - 1) is A36 and Qww(k - 2) is A37 then
Acknowledgments This work is supported by the project POCTIlECMI 3829612001-APTDEC, Funda~ao para a Ciencia e Tecnologia, Ministerio do Ensino Superior e da Ciencia, Portugal, and FEDER.
H(k) = -2.41H(k - 1) -1.67· lOQb(k - 1) +9.50c(k - 1) -4.07.10- 1 Qair(k-l) +1.08Qrej (k-2) -1.13Q /eed(k-l) +2.55Qww(k - 2) +2.10.10 2 Output2 - Bias Flow Rate (Qb): Rules: 1. IfQb(k -1) is All ande(k -1) is A l2 and Qair(k - 1) is AI3 and Qrej(k - 1) is A I4 and Q/eed(k - 1) is AI5 and Qww(k - 1) is A16 then lO- I
7. REFERENCES
l
Qb(k) = -4.52· Qb(k - 1) +7.72 . lO- e(k - 1) -5.32· 1O- 2 Qair(k - 1) +1.44· lO- I Qrej(k - 1) -9.44· 1O- 2 Q/eed(k - 1) +4.12 .1O- 2 Qww(k - 1) +7.40 2.
IfQb(k - 1) is A21 and e(k - 1) is A22 and Qair(k - 1) is A23 and Qrej(k - 1) is A24 and Qjeed(k - 1) is A25 and Qww(k - 1) is A26 then
Qb(k) = 7.05· lO- I Qb(k - 1) -1.84e(k - 1) +8.58 . 1O- 2 Qair(k-l) +3.80.1O- 3 Qrej(k-l) -3.28·1O- 2 Q jeed(k1) -5.56· 1O- 3 Qww(k - 1) -1.41 . 10 1 Output3 - Air Holdup (c): Rules: 1. If H(k - 1) is All and Qb(k - 1) is Al2 and e(k-l) is AI3 and Qair(k-l) is AI4 and Qrej(k-l) is AI5 and Qjeed(k - 1) is AI6 and Qww(k - 1) is An then ECk) = 1.32 .1O- I H(k - 1) -4.25Qb(k -1) -5.66e(k - 1) +1.77· lO- I Qair(k - 1) +4.06· lO- I Qrej(k - 1) -4.73· lO- I Q/eed(k - 1) -1.89· lO- I Qww(k - 1) -1.90.10 1 2. If H(k - 1) is A 21 and Qb(k - 1) is A22 and c(k - 1) is A23 and Qair(k - 1) is A 24 and Qrej(k - 1) is A25 and Q/eed(k - 1) is A26 and Qww(k - 1) is A27 then c(k) = -2.26.10- 1 H(k - 1) +5.01Qb(k - 1) +7.52c(k - 1) -1.61 . 10- 1 Qair(k - 1) -4.93 . 10- 1 Qrej (k - 1) +5.59 . lO- I Q/eed(k - 1) +1.98 .1O- I Qww(k - 1) +1.91.10 1
Babuska, R. (1998). Fuzzy Modeling for Control. Kluwer Academic Publishers. Boston. Carvalho, T. (1998). Fuzzy Control of a Column Flotation Process (in Portuguese). Ph.d. dissertation. Technical University of Lisbon, Instituto Superior Tecnico. Lisbon, Portugal. Finch, J. and J. Dobby (1990). Columnfiotation. Pergamon Press. Gustafson, D.E. and w.c. Kessel (1979). Fuzzy clustering with a fuzzy covariance matrix. In: Proceedings of the 18 th IEEE Conference on Decision and Control. San Diego, CA, USA. pp. 761766. MT. Carvalho, EO. Durao and C. Femandes (1999). Dynamic carachterization of column floatation process: laboratory case study. Minerals Engineering 12(11), 1339-1346. Sugeno, Michio and Takahiro Yasukawa (1993). A fuzzy-logic-based approach to qualitative modeling. IEEE Transactions on fuzzy systems 1(1), 731.
If H(k - 1) is A31 and Qb(k - 1) is A32 and c(k - 1) is A33 and Qair(k - 1) is A34 and Qrej(k - 1) is A35 and Qjeed(k - 1) is A36 and Qww(k - 1) is A37 then
3.
Takagi, Tomohiro and Michio Sugeno (1985). Fuzzy identification of systems and its applications to modelling and control. IEEE Transactions on Systems, Man and Cybernetics 15(1), 116-132. Zadeh, Lotfi A. (1973). Outline of a new approach to the analysis of complex systems and decision processes. IEEE Transactions on Systems, Man and Cybernetics 3, 28-44.
c(k) = -1.48· 1O- 3 H(k - 1) -1.34Qb(k - 1) -4.53· 1O- I c(k-1) +7.57·1O- 2 Qair(k-l) +1.09.10- 1 Qrej(k-1) -1.01.10- 1 Q /eed(k-1) +1.46.10- 1 Qww (k-1) -2.19.10 1 4. If H(k - 1) is A41 and Qb(k - 1) is A42 and e(k - 1) is A43 and Qair(k - 1) is A44 and Qrej (k - 1) is A45 and Q/eed(k - 1) is A 4 6 and Qww(k - 1) is A47 then ECk) = 1.13·1O- I H(k-1) +8.30·1O- I Qb(k-l) +1.00e(k1) -2.52· 1O- 2 Qair(k - 1) -4.18· 1O- 2 Qrej(k - 1) +4.09· 1O- 2 Q/eed(k - 1) -1.58 .1O- I Qww(k - 1) +5.40
440
ELSEVIER
IFAC PUBLICATIONS w ww _elsev ier. com/localeli fac
FUZZY MODELING OF A COLUMN FLOTATION PROCESS S.M. Vieira,* J.M.C. Sousa,* A. Felicio,t F.O. Dudio t
Technical University of Lisbon, Instituto Superior Tecnico • Dept. ofMechanical Engineering/ GCAR - IDMEC t Dept. Mining Engineering/ CVRM Av. Rovisco Pais, 1049-001 Lisbon, Portugal Phone: +351-21-84/731
Abstract: Real multivariable non linear systems are always difficult to model. Column flotation process is a very complex, nonlinear and multivariable system. When a model of a system like this is needed, it is not always possible to use or identify a common linear model. Fuzzy modeling is a well-known modeling technique, which has been applied to complex and nonlinear processes. This paper integrates fuzzy multivariable modeling with structure identification. One of the hardest tasks when modeling real systems is to define the model structure, specially if it is a multivariable system. To define this structure it is used a common criteria. The results show that is possible to find a better model using this technique, Real data is used for the design and validation of the fuzzy model. Copyright © 2004 IFAC Keywords: Fuzzy modeling, multivariable modeling, structure identification, column flotation,
1. INTRODUCTION
• Finally, the difficulties inherent to the acquisition of accurate measures of many variables of the process.
Column flotation is nowadays an important mineral processing unit. It is a complex multivariable process undergoing several disturbances, such as those originated by changes in feed characteristics and in equipment wearing (Finch and Dobby 1990).
A previous study (M.T. Carvalho and Femandes 1999) carried out with a laboratory column flotation and a two phase air-water system, aiming to develop a linear dynamic model, showed that only some relations between controlled and manipulated variables could be identified, using transient analysis.
Classic and modem approaches of process control rely on the knowledge of dynamic models of the process. However, until now it has not been possible to develop a phenomenological dynamic model of the column flotation, due mainly to the following difficulties:
Trying to overcome these difficulties this paper proposes the identification of a non linear dynamic fuzzy model using input-output data from a laboratory column flotation and a regularity criterion (RC) to find, automatically, the relations between input-output variables (Sugeno and Yasukawa 1993).
• Inaccurate knowledge of the physical and chemicallaws of the sub-processes involved; • Formulation of partial differential nonlinear systems equations of distributed parameters, because the geometry of the column forces the consideration of the axial position; • Need to solve a mathematical problem of a movable interface between the collection and the froth zones;
The paper is organized as follows. The fuzzy modeling technics are described in section 2, The section 3 presents the process of variable selection for each output and describes the procedure of structure identification. The process description is presented in the
435
section 4. The results obtained are shown in section 5 and some conclusions are drawn in section 6.
are fuzzy sets defined in the antecedent space, and Yi is the rule output variable. K denotes the number of rules in the rule base, ai is a parameter vector and bi is a scalar offset. The consequents of the affine TS model are hyperplanes in the product space of the inputs and the output, i.e. lR n x IR. The model output, y, can then be computed by aggregating the individual rules contribution:
2. FUZZY MODELING Fuzzy modeling using measures of the process variables. is a tool that allow an approximation of nonlinear systems when there is no prior knowledge about the system or it is only partially known. Usually, fuzzy modeling follows three steps:
(3)
• Structure identification; • Parameter estimation; • Model validation.
where {3i is the degree of activation of the ith rule: n
(3i =
One of the important advantages of fuzzy models is that they combine numerical accuracy with transparency in the form of linguistic rules. Hence, fuzzy models take an intermediate place between numerical and symbolic models. Traditionally, rule-based fuzzy systems are used (Zadeh 1973). In computational terms, fuzzy models are flexible mathematical structures that are known to be universal function approximators. In fuzzy modeling, the fuzzy "If-Then" rules take the following general form:
and fLAij (Xj) : IR of the fuzzy set A
[0, 1] is the membership function in the antecedent of Ri.
-> ij
3. IDENTIACATION
Fuzzy models use "1[-Then" rules and logical connectives to establish relations between the variables defined to derive the model. The fuzzy sets in the rules serve as an interface amongst qualitative variables in the model, and the input and output numerical variables. The rule-based nature of the model allows a linguistic description of the knowledge. The fuzzy modeling approach has several advantages when compared to other non linear modeling techniques. In general, fuzzy models can provide a more transparent model and can also give a linguistic interpretation in the form of rules. The system to be identified can be represented as a MIMO nonlinear auto-regressive (NARX) model:
The nonlinear identification problem is solved in two steps: Structure identification and parameter estimation. Structure identification: in this step the order of the model and the significant state variables x of the model must be chosen. This is a very important step. In this particular case, once we do not know all the relations between the input-output variables, it will be used a regularity criterion (RC) to find the significant state variable (Sugeno and Yasukawa 1993). To identify the model (2), the regression matrix X and an output vector y are constructed from the available data:
(1)
Here N » n is the number of samples used for identification.
where x is a state vector obtained from several inputs and outputs of the system.
Parameter estimation: The number of rules, K, the antecedent fuzzy sets, A ij , and the consequent parameters, ai, bi are determined in this step, by means of fuzzy clustering in the product space of X x y (Babuska 1998). Hence, the data set Z to be clustered is composed from X and y:
2.1 Takagi-Sugeno fuzzy Models
The Takagi-Sugeno (TS) fuzzy models (Takagi and Sugeno 1985), consists of fuzzy rules where each rule describes a local input-output relation, typically in an affine form. The representation of (1) as a TS model is given by:
~
(4)
j=l
If antecedent proposition then consequent proposition
y = j(x),
IIfLAij(Xj),
Z
= [X,Y1 T
Given the data Z and the number of clusters K, the Gustafson-Kessel fuzzy clustering algorithm (Gustafson and Kessel 1979) is applied to compute the fuzzy partition matrix U. This provides a description of the system in terms of its local characteristic behavior in regions of the data identified by the clustering algorithm, and each cluster defines a rule. Unlike the popular fuzzy c-means algorithm, the Gustafson-Kessel algorithm applies an adaptive distance measure. As
: If Xl is Ail and ... and X n is Ainthen
where i = 1,2, ... , K. Here ~ is the i th rule, x = T [Xl, ... ,xnl is the antecedent vector, Ail, ... ,A in
436
such, it can find hyper-ellipsoid regions in the data that can be efficiently approximated by the hyper-planes described by the consequents in the TS model.
and B, yAB is the model output for the group A input estimated by the model identified using the group B data, yBA is the model output for the group B input estimated by the model identified using the group A data.
The fuzzy sets in the antecedent of the rules are obtained from the partition matrix U, whose ikth element /lik E [0,1] is the membership degree of the data object Zk in cluster i. One-dimensional fuzzy sets A ij are obtained from the multidimensional fuzzy sets defined point-wise in the ith row of the partition matrix by projections onto the space of the input variables x j: .) /lA,j ( Xjk
= proJj.N
n
+1 ( ) /lik ,
Using the two groups of data, we build two fuzzy models, one for each group, A and B, beginning with only one input. At this stage we build a fuzzy model for each input in consideration. Once we are dealing with a MIMO system and this method is only applied to MISO systems, we must consider the outputs not in consideration as inputs for the system. We calculate RC of each model and select the model that minimizes RC. In the next step we consider the input of the selected model fix (belonging to the system's structure) and add another input to our fuzzy model from among the remaining candidates. Now we have two inputs and we select the second input as we do at the first step, according to the minimal value of RC. From now on we proceed as this far until the value of RC increases. When this happens we should have all the relevant input variables for the considered output. Using this method we can reduce the number of iterations, once all the input variables, that produce models with RC bigger than the minimal at the stage, can be excluded from the process. In a generic case the maximum number of iterations is n(n + 1) /2, where n is the number of input variables.
(6)
where proj is the point-wise projection operator. The point-wise defined fuzzy sets A ij are approximated by suitable parametric functions in order to compute /lA,j(Xj) for any value of Xj' The consequent parameters for each rule are obtained as a weighted ordinary least-square estimate. Let BT = [aT; b;J, let X e denote the matrix [X; 1] and let W i denote a diagonal matrix in lR. N x N having the degree of activation, (3i(Xk), as its kth diagonal element. Assuming that the columns of X e are linearly independent and (3i(Xk) > 0 for 1 ~ k ~ N, the weighted leastsquares solution of y = XeB + E: becomes (7)
When using this criterion a fuzzy model is created at each iteration. To do so the number of clusters that best suits the data must be determined. For this purpose it is used the following criterion (Sugeno and Yasukawa 1993):
Rule bases constructed from clusters are often unnecessary redundant due to the fact that the rules defined in the multidimensional premise are overlapping in one or more dimensions. The resulting membership functions will thus be overlapping as well, and more fuzzy sets will describe approximately the same concept. Therefore, the parameters of the model identified must be carefully chosen.
n
S(c) =
c
L L(/lik)ffi(11
Xk -
Vi
11 2
-
1 Vi
-
X
11
2
),
k=li=l
(9) 3.1 RC - Variable Selection where n is the number of data to be clustered, c is the number of clusters (c ~ 2), Xk is the k th data (usually vector), x is the average of data, Vi the center of the i th cluster, /lik is the grade of the k th data belonging to i th cluster and m is a adjustable weight (usually m E [1.5,3]).
When the relevant variables are known as well as the relation between them, this task is not so difficult. The identification of fuzzy model for a column flotation process is a quite complex task. This process has a large number of variables, and the most relevant ones must be chosen. In this paper a certain criterion is minimized to determine which input variables influence each output (Sugeno and Yasukawa 1993). To do so the data is divided into two groups, A and B. As a criterion to this purpose, it is used the regularity criterion, RC, used in "group method of data handling", which is defined as follows:
RC
The number of clusters, c, is determined so that S(c) reaches a minimum as c increases: it is supposed to be a local minimum. In each iteration, S(c) the cluster(s) are determined using the fuzzy C-Means algorithm and the process stops when S(c) increases from one iteration to the next one.
_ JI:~~I(yt-ytB)2/kA+I:~:I(yf'-yf'A)2/kBl -
2
'
The first term of the right-hand side is the variance of the data in a cluster and the second term is that of the clusters themselves. The optimal clustering achieved is the one that minimize the variance in each cluster and maximize the variance between the clusters. The
(8) Where kAand k B are the number of data of the groups A and B, y~and yf are the output data of the groups A
437
parameter m has a great importance in this criterion. The bigger the m is the bigger it will be the optimum number of clusters.
4. PROCESS DESCRIPTION Froth flotation or, shortly, flotation (as froth flotation is the most important of the flotation processes) was introduced in the beginning of the 20 th century and is one of the most versatile separation processes used in mineral processing. Until then, it was not possible to separate several minerals, like most of sulphides, because they exhibit similar density, magnetic susceptibility and conductivity. The flotation process separates fine solid particles based on physic and chemical properties of their surfaces. Industrially, it is a continuous solid-solid separation process performed in a vessel where a threephase system is present: solid particles, air bubbles and water. This pulp is previously conditioned with the controlled addition of small quantities of specific chemical reagents to promote the selective formation of aggregates between solid particles of a given composition and air bubbles. Air is continuously injected in the pulp, giving rise to the formation of air bubbles. The formation of stable froths, capable of transporting particles to the surface of the vessel, is possible if a convenient frother in an adequate concentration is used.
Fig. I. Column flotation scheme (M - Flowmeter, P Pressure sensor). uct hydrophilic particles that were dragged with the aggregates bubbles-hydrophobic particles. The flotation environment is influenced by variables, such as pulps pH, solids concentration, flotation intensity (speed of agitation and air flow rate), particles size and type of water medium.
5. PROCESS IDENTIFICATION
Hydrophobic particles adhere, after collision, to the air bubbles, which move upwards to the top of the vessel where they are recovered as the floated product. Hydrophilic particles settle in the pulp, become the non-floated product or underflow.
This process is difficult to model once it is highly nonlinear and it is not known all the mathematical relations, between variables, that describes the process. There is a few empirical relations for some of the evolved process variables (Carvalho 1998). Usually this is a process controlled by human, but it is an unstable process, except for a very restricted area of operation. The process being unstable makes it more difficult to identify.
Besides mineral processing, it is used in some other fields, such as solvent extraction and recycling. Usually, a vessel which is called flotation cell, with a agitated impeller that keep the pulp in suspension, and the bubbles of air in movement, promoting collision with the particles.
The process in study is highly nonlinear, and the system cannot be totally described by first principles, so is only partly known. In this case it is advantageous to use fuzzy modeling as a way to combine first principles, human knowledge and/or measurements, obtaining an usually called gray-box modeling approach (Babuska 1998).
In the case study, the separation process is performed in a column containing the water medium. The principal difference to the flotation cell is that there is not an impeller to keep the pulp in suspension. Because of that, column flotation is mostly used to separate finer particles than flotation cell equipments. A flow of air is continuously injected in the medium to transport the particles. After collision with air bubbles, which move upwards to the top of the column, hydrophobic particles adhere to them and will be recovered as the floated product. Hydrophilic particles stay in the vessel, becoming the underflow product, recovered in the bottom of the column.
5. I Identification experiments Once the aim of this work was to model the system using fuzzy logic, the experiments to obtain the test data, were design so that all system variables could take all possible values within each variable range. This was accomplished using white noise generation, so we could use a signal with many frequency information and non correlated. The column flotation is a slow
There is also a shower of water in the top of the froth column, used to "wash" from the floated prod-
438
process. When we are dealing with its identification we must be aware that the output variables will take some time to react to the perturbation on the input variables. Knowing this, we constructed the identification signals giving the enough time to the process to respond.
,'~~ o
500
1soo
1000
2000
'11\~~~~1 o
500
, 000
, 500
2000
Tm·ls]
-5~O-----,5OO:::----,'-":OOO:----~,500=-----2000~---......J rm.[sl
Fig. 3. Test results for Fuzzy model using empirical knowledge - V AF = [53.4 85.188.6]% (dashed line: model output; continuous line: system output) H
Fig. 2. System inputs - air flow rate (Q air), rejected flow rate (Qrej), feed flow rate (Qjeed) and washing water flow rate (Qww)
Emp. model:
VAF
53.4 5.3 76.6 4.0
RMS
RC model:
VAF RMS
2)(Yi - YmY)/N,
I Holdup I 88.6 0.6 88.0 0.6
the one using empirical relations. Both indexes show that the performance improve, especially for the level H, which is the most difficult variable to estimate. The validation of the RC model in different data is
'~E~rv:YJ
N
=
Qb 85.1 1.4 92.8 1.0
Table I. Fuzzy models evaluatiOn usmg VAF andRMS
In Fig. 2 it is shown the inputs applied to the system. They are the response of the actuators to the identification signals. The initial signals are steps with the sufficient duration to ensure that the process has enough time to respond. The outputs to be estimated are: the level H, the bias flow rate Qb and the air holdup E, as can be seen in Fig. 3 and 4. The root mean square is used as performance index:
RMS
I
(10)
i=1
o
soo
1 000
1500
2000
,tv~1n¥~}j
where Yi is a system output and Ymi is the correspondent model output. The percentile variance accounted for (VAF) was also used to measure the performance of the model obtained.
o
500
soo
I 000
1
2000
TlrMlsj
V AF = 100%(1 - var(Yi - Ym.)/var(Yi)) (11) where cov is the covariance of the respective vector.
-5i~O---500:::----,~OOO:----~,500=-----2000~---......J nnelsJ
Fig. 4. Test results for Fuzzy model using RC V AF = [76.692.888.0]% (dashed line: model output; continuous line: system output)
5.2 Results The results using the model structure using only the empirical relations between variables is presented in Fig. 3.
presented in Fig. 4, and it is clear that all the outputs show a good performance.
The RC criterion was used to identify the model structure, and it was found that the optimum number of clusters is 4 for each output. The combination of the RC criterion and the knowledge about the process to tune the system delays leads to the best models. The performance indices for both identification techniques is shown in Table I. This table shows that the model obtained with RC has a much better performance than
5.3 Fuzzy model structure This model has 4 inputs and 3 outputs. The sampling period is 4.3 s. The termination tolerance of the clustering algorithm was 0.0 I. In the following, the output-specific information is shown for each output.
439
6. CONCLUSIONS
Output I - Level (H): Rules:
This paper describes a fuzzy model of the flotation column, developed for control. The structure of the model was determined based on an automatic determination of the number of rules; the RC criterion. This simple criterion was very helpful to find the model structure in the column flotation, for which the system dynamics is not completely known. The obtained results shown that the structure of the model is well determined, as the identified fuzzy model is able to describe the system dynamics in an accurate way.
l. If H(k - 1) is All and Qb(k - 1) is Al2 and c(k - 1) is AI3 and Qair(k - 1) as A I 4 and Qrej(k - 2) is A I5 and Q/eed(k - 1) is AI6 and Qww(k - 2) is An then
H(k) = 4.69.10- 1 H(k -1) -1.45·Qb(k-1) +3.67·c(k -1) -1.21· lO- I Qair(k - 1) +4.47· 1O- 2 Qrej(k - 2) +3.75 . 1O- 2 Q/eed(k - I) +6.50· lO- I Qww(k - 2) +2.45·10 2.
if H(k - 1) is A21 and Qb(k - 1) is A 2 2 and c(k - 1) is A 23 and Qair(k - 1) is A24 and Qrej(k - 2) is A25 and Q/eed(k - 1) is A26 and Qww(k - 2) is A27 then H(k) = 4.69· H(k -1) +1.94 ·lOQb(k -1) -1.63 ·10e(k - 1) +6.59.10- 1 Qair(k-l) -1.28·Qrej(k- 2) +1.24·Q /eed(k1) -318· Qww(k - 2) -2.49.10 2 3. If H(k - 1) is A31 and Qb(k - 1) is A32 and e(k - 1) is A33 and Qair(k - 1) is A34 and Qrej(k - 2) is A35 and Q/eed(k - 1) is A36 and Qww(k - 2) is A37 then
Acknowledgments This work is supported by the project POCTIlECMI 3829612001-APTDEC, Funda~ao para a Ciencia e Tecnologia, Ministerio do Ensino Superior e da Ciencia, Portugal, and FEDER.
H(k) = -2.41H(k - 1) -1.67· lOQb(k - 1) +9.50c(k - 1) -4.07.10- 1 Qair(k-l) +1.08Qrej (k-2) -1.13Q /eed(k-l) +2.55Qww(k - 2) +2.10.10 2 Output2 - Bias Flow Rate (Qb): Rules: 1. IfQb(k -1) is All ande(k -1) is A l2 and Qair(k - 1) is AI3 and Qrej(k - 1) is A I4 and Q/eed(k - 1) is AI5 and Qww(k - 1) is A16 then lO- I
7. REFERENCES
l
Qb(k) = -4.52· Qb(k - 1) +7.72 . lO- e(k - 1) -5.32· 1O- 2 Qair(k - 1) +1.44· lO- I Qrej(k - 1) -9.44· 1O- 2 Q/eed(k - 1) +4.12 .1O- 2 Qww(k - 1) +7.40 2.
IfQb(k - 1) is A21 and e(k - 1) is A22 and Qair(k - 1) is A23 and Qrej(k - 1) is A24 and Qjeed(k - 1) is A25 and Qww(k - 1) is A26 then
Qb(k) = 7.05· lO- I Qb(k - 1) -1.84e(k - 1) +8.58 . 1O- 2 Qair(k-l) +3.80.1O- 3 Qrej(k-l) -3.28·1O- 2 Q jeed(k1) -5.56· 1O- 3 Qww(k - 1) -1.41 . 10 1 Output3 - Air Holdup (c): Rules: 1. If H(k - 1) is All and Qb(k - 1) is Al2 and e(k-l) is AI3 and Qair(k-l) is AI4 and Qrej(k-l) is AI5 and Qjeed(k - 1) is AI6 and Qww(k - 1) is An then ECk) = 1.32 .1O- I H(k - 1) -4.25Qb(k -1) -5.66e(k - 1) +1.77· lO- I Qair(k - 1) +4.06· lO- I Qrej(k - 1) -4.73· lO- I Q/eed(k - 1) -1.89· lO- I Qww(k - 1) -1.90.10 1 2. If H(k - 1) is A 21 and Qb(k - 1) is A22 and c(k - 1) is A23 and Qair(k - 1) is A 24 and Qrej(k - 1) is A25 and Q/eed(k - 1) is A26 and Qww(k - 1) is A27 then c(k) = -2.26.10- 1 H(k - 1) +5.01Qb(k - 1) +7.52c(k - 1) -1.61 . 10- 1 Qair(k - 1) -4.93 . 10- 1 Qrej (k - 1) +5.59 . lO- I Q/eed(k - 1) +1.98 .1O- I Qww(k - 1) +1.91.10 1
Babuska, R. (1998). Fuzzy Modeling for Control. Kluwer Academic Publishers. Boston. Carvalho, T. (1998). Fuzzy Control of a Column Flotation Process (in Portuguese). Ph.d. dissertation. Technical University of Lisbon, Instituto Superior Tecnico. Lisbon, Portugal. Finch, J. and J. Dobby (1990). Columnfiotation. Pergamon Press. Gustafson, D.E. and w.c. Kessel (1979). Fuzzy clustering with a fuzzy covariance matrix. In: Proceedings of the 18 th IEEE Conference on Decision and Control. San Diego, CA, USA. pp. 761766. MT. Carvalho, EO. Durao and C. Femandes (1999). Dynamic carachterization of column floatation process: laboratory case study. Minerals Engineering 12(11), 1339-1346. Sugeno, Michio and Takahiro Yasukawa (1993). A fuzzy-logic-based approach to qualitative modeling. IEEE Transactions on fuzzy systems 1(1), 731.
If H(k - 1) is A31 and Qb(k - 1) is A32 and c(k - 1) is A33 and Qair(k - 1) is A34 and Qrej(k - 1) is A35 and Qjeed(k - 1) is A36 and Qww(k - 1) is A37 then
3.
Takagi, Tomohiro and Michio Sugeno (1985). Fuzzy identification of systems and its applications to modelling and control. IEEE Transactions on Systems, Man and Cybernetics 15(1), 116-132. Zadeh, Lotfi A. (1973). Outline of a new approach to the analysis of complex systems and decision processes. IEEE Transactions on Systems, Man and Cybernetics 3, 28-44.
c(k) = -1.48· 1O- 3 H(k - 1) -1.34Qb(k - 1) -4.53· 1O- I c(k-1) +7.57·1O- 2 Qair(k-l) +1.09.10- 1 Qrej(k-1) -1.01.10- 1 Q /eed(k-1) +1.46.10- 1 Qww (k-1) -2.19.10 1 4. If H(k - 1) is A41 and Qb(k - 1) is A42 and e(k - 1) is A43 and Qair(k - 1) is A44 and Qrej (k - 1) is A45 and Q/eed(k - 1) is A 4 6 and Qww(k - 1) is A47 then ECk) = 1.13·1O- I H(k-1) +8.30·1O- I Qb(k-l) +1.00e(k1) -2.52· 1O- 2 Qair(k - 1) -4.18· 1O- 2 Qrej(k - 1) +4.09· 1O- 2 Q/eed(k - 1) -1.58 .1O- I Qww(k - 1) +5.40
440