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Design of Experiments as a Prerequisite for Development of Fuzzy Models?
Intelligent Control and Automation Sciences 5th IFAC International Conference on Belfast, Ireland, August 21-23, 5th IFACNorthern International Conference on 2019 Intelligent Control and Automation Sciences Available online at www.sciencedirect.com Intelligent Control andConference Automationon Sciences 5th IFAC International Belfast, Ireland, August 2019 Belfast, Northern Northern August 21-23, 21-23, 2019 Intelligent ControlIreland, and Automation Sciences Belfast, Northern Ireland, August 21-23, 2019
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52-11 (2019)for 140–145 Design of ExperimentsIFAC as PapersOnLine a Prerequisite Development of Fuzzy Models? Design Design of of Experiments Experiments as as a a Prerequisite Prerequisite for for Development Development of of Fuzzy Fuzzy Models? Models? Design of Experiments as a Prerequisite for Development of Fuzzy Models?
Sašo Blažič *, Igor Škrjanc *
Sašo Sašo Blažič Blažič *, *, Igor Igor Škrjanc Škrjanc ** Sašo Blažič *, Igor Škrjanc * * University of Ljubljana, Faculty of Electrical Engineering, Slovenia (e-mail:Faculty [email protected]) University of of Ljubljana, Ljubljana, of Electrical Electrical Engineering, Engineering, Slovenia Slovenia ** University Faculty of (e-mail: [email protected]) * University of Ljubljana, of Electrical Engineering, Slovenia (e-mail:Faculty [email protected]) [email protected]) Abstract: Design of experiments is an(e-mail: approach where the experiments for measurement acquisition should be planned very carefully in order to obtain high-informative data. This paper tackles a problem of
Abstract: Design of is approach where experiments for acquisition should Abstract: of experiments experiments is an an where the the problems experiments for measurement measurement acquisition constructingDesign mathematical models of approach complex nonlinear based on measurement data ofshould poor be planned very carefully in order to obtain high-informative data. This paper tackles a problem of be plannedDesign very carefully order obtain high-informative This paper a problem of Abstract: of experiments is anto approach where the experiments measurement acquisition should information content. Two in practical examples are dealt with. data. The for first one is tackles a textile wastewater
constructing mathematical of nonlinear problems based on measurement data of constructing mathematical models of tocomplex complex nonlinear problems based measurement of poor decolourisation process, andmodels theorder second one an atmospheric corrosion of structural metal materials. In poor both be planned very carefully in obtain high-informative data. Thisonpaper tackles adata problem of information content. Two practical examples are dealt with. The first one is a textile wastewater information content. Two practical examples areconstructing dealt problems with.a The first is a textile wastewater cases the measurement datamodels were not for very accurate model. In such cases it is constructing mathematical of adequate complex nonlinear based onone measurement data of poor decolourisation process, and the second one atmospheric corrosion of structural metal materials. In decolourisation process, and practical the second one an an atmospheric corrosion structural materials. In both both necessary to make a trade-off between model complexity and model information content. Two examples are dealt with. accuracy. Theof first one ismetal a textile wastewater cases the measurement data were not adequate for constructing a very accurate model. In such cases it cases the measurement were not adequate for constructing a very such cases it is is decolourisation process, data and the second one an atmospheric corrosion of accurate structuralmodel. metal In materials. In both necessary todesign make aoftrade-off trade-off between model complexity and model model accuracy. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: experiments, fuzzy model, radial basis network, decolourisation process, atmospheric necessary to make a between model complexity and accuracy. cases the measurement data were not adequate for constructing a very accurate model. In such cases it is corrosion. necessary to make aoftrade-off between model complexity basis and model accuracy. Keywords: Keywords: design design of experiments, experiments, fuzzy fuzzy model, model, radial radial basis network, network, decolourisation decolourisation process, process, atmospheric atmospheric corrosion. corrosion. Keywords: design of experiments, fuzzy model, radial basis network, decolourisation process, atmospheric corrosion.
data of the process are collected without the supervision of an expert for experimental modelling. In such cases the data are data process are without of data of of the theless process are collected collected without the the supervision supervision of an an 1. informative, and consequently the experimental This paper discusses a very important issue when dealing typically 1. INTRODUCTION INTRODUCTION expert modelling. In cases the data of for theexperimental process are collected without thebut supervision ofare an expert for experimental modelling. In such such cases the data data are modelling becomes much more difficult it still can be with model discusses building on measurement data. Having 1. INTRODUCTION typically less informative, and consequently consequently the experimental experimental This paper aabased very important issue when dealing expert forless experimental modelling. In such cases the data and are typically informative, and the This paper discusses very important issue when dealing achieved by reducing the expectation about the model high-informative data based is theon prerequisite when building modelling becomes much more difficult still with model building measurement data. Having typically less informative, and consequently theit modelling becomes much more difficult but but itexperimental still can can be be This abased very important issue when dealing with paper model discusses building on measurement data.should Having adapting the modelling techniques. mathematical models. To achieve this, experiments be achieved by reducing the expectation about the model and high-informative data is the prerequisite when building modelling becomes much more difficult but it still can be achieved by reducing the expectation about the model and with model building based measurement data. Having high-informative data isdealing theon prerequisite when building carefully planned. When with nonlinear models such adapting the modelling techniques. mathematical models. To achieve this, experiments should be In this paper we will show two practical problems we dealt achieved by reducing the expectation about the model and adapting the modelling techniques. high-informative dataartificial the neural prerequisite when building mathematical models. Toisachieve this, networks experiments be as fuzzy planned. models or theshould problem carefully When with nonlinear models such with in paper the Both cases are similar: they both deal with a thepast. modelling techniques. mathematical models. To dealing achieve this, experiments should be adapting carefully even planned. When dealing with nonlinear models such In this will two problems dealt becomes more critical due to the high number of model In thiscomplex paper we we will show show two practical practical problems we we dealt as fuzzy fuzzy planned. models or or artificial neural networks the problem problem quite nonlinear process, the measurement database carefully When dealing with nonlinear models such as models artificial neural networks the with in the past. are they both deal parameters thatmore need critical to be estimated/trained. In this weBoth will cases show twosimilar: practical problems wewith dealtaa with in paper the Both cases are similar: they bothuncertainty, deal with becomes due to high model small in past. both cases and includes quite some as fuzzy even models or critical artificial the of problem becomes even more dueneural to the the networks high number number of model is quite complex nonlinear process, the measurement database with incomplex thedata past. Bothnocases are be similar: they both deal with a quitenew nonlinear process, the measurement database parameters that need to be estimated/trained. could longer measured due uncertainty, to practical Design ofeven experiments approach that aims to deal with and becomes due to the high number of model parameters thatmore need critical to is be an estimated/trained. is small in both cases and includes quite some quite complex nonlinear process, themodel measurement database is small in both cases and includes quite some uncertainty, reasons. Our task is to build the that extracts the this problem the influence (input) parameters that by needfirst to is beidentifying estimated/trained. andsmall new in data could noand longer be measured measured due uncertainty, to practical practical Design of experiments an approach that aims to deal with is both cases includes quite some and new data could no longer be due to Design of experiments is an approach that aims to deal with available information while knowing that the measurement variables and by the first consequence (output)influence variables,(input) then reasons. Our task is to build the model that extracts the this identifying and new Our data couldissmaller notolonger be measured due to practical reasons. task build thenormally model that extracts Design of experiments is an variation, approach the that then aimsspreading to deal with this problem problem by first identifying the influence (input) is much than expected forthea determining the area of their and the database available information while knowing that the measurement variables and the consequence (output) variables, then reasons. Our task is to build the model that extracts the available information while knowing that the measurement this problem identifying theto influence variables and by the first consequence (output) variables, then problem of this complexity. influence variables of the experiment cover the (input) whole database is much smaller than normally expected for aa determining the area of their variation, and then spreading the available information while knowing that the measurement database is much smaller than normally expected for variables and the consequence (output) variables, then determining the area of their variation, and then spreading the space of these variables that the model will be used for. problem of this complexity. influence variables of the to cover the This paper ismuch organised as follows. Sectionexpected 2 presents smaller than normally forthea problem ofisthis complexity. determining the area ofexperiments their variation, and spreading the database influence variables the experiment experiment to then cover thethewhole whole Adopting design of of only makes low space of these these variables that the not model will be the used for. problem process of decolourisation. Section 3 presents atmospheric this complexity. influence variables of the experiment to cover whole space of variables that the model will be used for. is as follows. Section 22 presents number of design experiments (the least possible?), but alsothe results This This paper paperprocess. is organised organised follows. Section presents the the Adopting of only makes Sectionas4 brings some conclusions. space of these variables that the not model will be used for. corrosion Adopting design of experiments experiments not only makes the low low process of decolourisation. Section 3 presents atmospheric in a reliable model on a broad range of input variables This paper is organised as Section follows. 3Section 2 presents the process of decolourisation. presents atmospheric number of experiments (the least possible?), but also results Adoptingof design ofof experiments not only makes low corrosion process. Section 4 brings some conclusions. number experiments (the least possible?), but not alsothe variation. Design experiments is certainly aresults new process decolourisation. Section 3 presents atmospheric corrosionofprocess. Section 4 brings some conclusions. in a reliable model on a broad range of input variables 2. PROCESS OF DECOLOURISATION number ofsince experiments least possible?), but also results in a reliable model on(the a broad range of book input variables approach it goes back to the famous of corrosion process. Section 4 brings some conclusions. variation. Design of experiments is certainly not aaFisher new in a reliable model on a broad range of input variables variation. Design of experiments is certainly not new 2. PROCESS PROCESS OF DECOLOURISATION (1935). The approach can be used infamous differentbook areasofsuch as This section deals withOF theDECOLOURISATION process of decolourisation of a 2. approach it goes back to Fisher variation. since Design of experiments isfamous certainly notof a2011), new approach since itcircuit goes back to the the book Fisher computer-aided design (Rémond et al., textile wastewater that is the result of an industrial dyeing 2. PROCESS OF DECOLOURISATION (1935). approach can be used different areas as deals with the process decolourisation of approach since it goes back to thein book ofsuch Fisher (1935). The The approach can be(Chaabane used infamous different areas such as This This section section deals with the techniques process of of of decolourisation of aa production system design et al., 2009), the process. There exist many decolouring (dos computer-aided circuit design (Rémond et al., 2011), textile wastewater that is the result of an industrial dyeing (1935). Theofapproach be(Tůmová used(Rémond in different areas as textile computer-aided circuitcandesign et2018), al., such 2011), This section deals with decolourisation ofwea wastewater that isthe theprocess result of the an industrial dyeing evaluation experiments et et al., al., control Santos, 2007). Inexist our case we deal with process where production system design (Chaabane 2009), the process. There many techniques of decolouring (dos computer-aided circuit design (Rémond et al., 2011), production system design (Chaabane et 2017), al., 2009), the textile wastewater that is the result of an industrial dyeing process. There exist many techniques of decolouring (dos design (Aengchuan, and Phruksaphanrat, tableting measure absorbance of we thedeal wastewater A thatwhere defines evaluation of experiments (Tůmová et al., 2018), control Santos, 2007). In our case the process we production system design (Chaabane al., 2009), the evaluation of experiments (Tůmová et et al., 2018), control There many of decolouring (dos Santos, 2007). Inexist our that case wetechniques deal with with the process where we process optimisation (Belič et al., 2009), control design of process. “dirtiness” (0 means the water does not absorb any light design (Aengchuan, and Phruksaphanrat, 2017), tableting measure absorbance of the wastewater A that defines evaluation of experiments (Tůmová et al., 2018), control design (Aengchuan, and Phruksaphanrat, 2017), tableting Santos, 2007). In our case we deal with the process where we measure absorbance of the wastewater A that defines servo systems (Precup et al., 2008), etc. The design of – it is completely transparent, whiledoes a larger number means it process optimisation (Belič et al., 2009), of (0 means that the water not absorb any light design and Phruksaphanrat, 2017),adesign tableting process (Aengchuan, optimisation (Belič et al., difficult 2009), control control design of “dirtiness” measure absorbance of the wastewater A that defines “dirtiness” (0 means that the water does not absorb any light experiments becomes much more when dynamic is “dirty”). The decolourisation process is done by adding the servo systems (Precup et al., 2008), etc. The design of it transparent, while aa larger number means it processsystems et for al., 2009), design of of –“dirtiness” servo (Precup et al., 2008), etc.control The content design (0 that the the water not absorb any light –peroxide it is is completely completely transparent, while larger means it model ofoptimisation the becomes process is(Belič sought since frequency O2means , and applying UVdoes lamp fornumber a by certain time. H2The experiments much more difficult when aadesign dynamic is “dirty”). decolourisation process is done adding the servo systems (Precup et al., 2008), etc. The of experiments becomes much more difficult when dynamic – it is completely transparent, while a larger number means it is “dirty”). The decolourisation process is done by adding the the excitation becomes extremely important. model process sought for frequency of O applying the UV lamp for a certain time. peroxide experiments more difficult when acontent dynamic 2The 2,, and model of of the the becomes process is ismuch sought for since since frequency content of is We“dirty”). built H the model of this process that is intended for control decolourisation process is done by adding the O and applying the UV lamp for a certain time. peroxide H 2 2 the becomes Unfortunately, very often it happens it is necessary to peroxide model of the process isextremely sought forimportant. sincethat frequency content of the excitation excitation becomes extremely important. design. Modelling and control ofthat this process by using the UV lamp for a certain time. a H 2O2, and applying We built the of process is intended control We built the model model of this this process thatby is Kumar intendedetfor for control construct a mathematical model of a certain process, and the the excitation becomes extremely important. fuzzy-model approach has been done al. (2015) Unfortunately, very very often often it it happens happens that that it it is is necessary necessary to to design. Modelling and control of this process by using a Unfortunately, We builtModelling the model of processofthat intendedby forusing controla design. andthis control thisis process construct of and approach been by Kumar (2015) Unfortunately, very oftenmodel it happens that it process, is necessary to fuzzy-model construct aa mathematical mathematical model of aa certain certain process, and the the design. Modelling andhas control of this fuzzy-model approach has been done done by process Kumar et etbyal. al.using (2015)a Copyright IFAC construct ©a 2019 mathematical model of a certain process, and the 140 fuzzy-model approach has been done by Kumar et al. (2015) 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. 1. INTRODUCTION
Copyright 2019 IFAC 140 Peer review© of International Federation of Automatic Copyright ©under 2019 responsibility IFAC 140Control. 10.1016/j.ifacol.2019.09.131 Copyright © 2019 IFAC 140
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but this paper deals with model development based on a smaller database.
causes that much more data are necessary for building the model. This is why 4-dimensional input space is avoided. In this case, the model will have the following form:
Before proceeding with control design a solid model of the process to be controlled has to be available. This is especially the case when model-based control methods are utilised. The main problem in this case lies in the data that the model was derived from. It is very well-known that several model forms (e.g. fuzzy models, artificial neural networks, spline models, piece-wise linear models etc.) exist that are capable of describing the nonlinear processes with arbitrarily small modelling errors. On the other hand, none of them is able to extrapolate the information to the parts of the space where no measured data or small amount of measured data was available.
D = f (C p , PUV , T )
Af = (1 − D) Ai = (1 − f (C p , PUV , T )) Ai
2.2 Radial basis network model of the decolourisation process The radial basis network (RBN) will be used for approximating the mapping given by Eq. (2). The model of the process will be obtained based on the measurement data for the Irgalan Gelb 3R KWL dye: PUV [W], Cp [mg/l], T [min], and the absorbance A (measured only at a few different times T). The data collected include the measurements of three repeated groups of experiments, each consisting of 15 experiments. The number of samples is rather low because samples of wastewater are taken manually and being analysed. The measurement error seems to be quite high. Some obvious outliers were manually removed. Since there are three inputs to the network and one output, the results are given in the form of three-dimensional planes where x and y axes correspond to the two inputs, the third input is fixed, while the decolourisation factor is shown on the z-axis.
Decolourisation factor given by the formula Ai − Af Ai
(1)
where Ai is initial absorbance (before the decolourisation process), and Af is final absorbance (after the decolourisation process). Consequently, D = 0 means that the process was completely unsuccessful while D = 1 means that the decolourisation is perfect. •
(3)
A remark is also needed here regarding the presence of the time of the experiment (T) in model given by Eq. (2). Since the measurements are taken at quite arbitrary times, the model is given in a form where time is used as explicit model input.
The first thing we have to determine when considering model structure is the selection of model inputs and outputs. The first open issue is the selection of the model output. At least two choices are possible:
D=
(2)
Note that decolourisation factor is given as the output, since we do not model the dependence upon the initial absorbance. The actual output of the model Af is then obtained from Eq. (1):
2.1 Model structure selection
•
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Final absorbance Af.
Since the model will be used for control design purposes, the decision has to be carried out focusing on the control aspect of the problem. The goal of the control is to decolourise the waste water. We have to define a certain reference. In our opinion some target final absorbance has to be defined as a control goal. If this were not the case and the decolourisation factor were selected as system output (and consequently control reference), all waste waters would be treated the same irrespective of the actual “dirtiness”. That would result in almost the same control actions in all circumstances which, clearly, denies the reason for building the complex nonlinear model. To sum up, the best solution for the system output is final absorbance Af. The selection of model inputs seems to be easier, but still there are some open questions regarding the topic. The candidates for the system inputs are: the concentration of the peroxide H2O2 (Cp), the power of the UV lamp (PUV), and the time of the experiment (T). But since the final absorbance probably depends nonlinearly upon the initial absorbance, the latter would have to be one of the inputs to the system. But having 4 inputs instead of 3
Fig 1. RBN approximation for PUV = 1200 W
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Figures 1, 2, and 3 show the decolourisation factor as a function of the peroxide concentration and the time while the power of the UV lamp is fixed (at 1200, 1400, and 1600 W). The measurements are shown with circles, and the network output at the same inputs with crosses. Figures 4 to 6 shown similar graphs for the fixed values of the peroxide concentration. Although it can be seen that the data set is quite poor (important data are missing in some operating conditions and the variability of the data is very high in repeating experiments), some qualitative properties of the process are clearly seen.
Fig. 4. RBN approximation for Cp = 0.7 mg/l
Fig. 2. RBN approximation for PUV = 1400 W
Fig. 5. RBN approximation for Cp = 4.5 mg/l
Fig. 3. RBN approximation for PUV = 1600 W 2.3 Control design Control problem in this case is to determine the control actions that will result in a purified water. By control actions we mean Cp, PUV and T applied to the process. Of course the cost of control rises with rising control actions.
Fig. 6. RBN approximation for Cp = 8.3 mg/l
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3. ATMOSPHERIC CORROSION PROCESS
Optimal control is proposed to solve the control problem. The most important thing when applying optimal control is to define a suitable cost function. The cost has to include all important aspects of the control problem. In our case, we decided to choose the following cost function: J = kd g ( Af ) + k p
Cp Cmax
+ ke
PUV T PmaxTmax
The process of atmospheric corrosion is modelled in this section. It is modelled by means of Takagi-Sugeno fuzzy model. The obtained model could serve for prediction of atmospheric corrosion in near future under simulated climatic changes (acidifying of atmosphere, lower concentrations of SO2 in Europe, higher concentrations of O3, NOX, global warming etc.).
(4)
where g(Af) is the function that defines the cost due to unsatisfactory final absorbance: 0 g ( Af ) = Af − Asatis
Af < Asatis Af ≥ Asatis
Atmospheric corrosion of structural metal materials is very complex, non-linear process depending on various meteorochemical and material parameters. Till nowadays the impact of these parameters on the process of degradation of materials is not fully understood. Thanks to existing longterm climate programmes in Europe (ECE/EMEP and ICP) and in Asia/Africa (RAPIDC) meteorochemical data such as temperature, relative humidity, amount of precipitation, concentration of main pollutants are daily measured; and corrosion mass losses of selected metals (carbon steel, copper, zinc and aluminium) are annually measured. There also exist measurement of corrosion of various metals in many locations in the world. The modelling of these phenomena has been dealt in research (Simillion et al., 2014; Halama et al. 2011). Unfortunately, the results made from the existing huge databases are not allowed to be published. This paper uses a very small free database, but significant information can be collected from it. The analysis of this database also shows major problems that are usually encountered when dealing with the problem of atmospheric corrosion.
(5)
If the final absorbance is less than a certain threshold Asatis (the acceptable level of the final absorbance), the first term of the cost function, the punishment of the unsatisfactory decolourisation, will be 0. If, however, the final absorbance is greater, the first term will be positive. The second and the third term in Eq. (4) represent the cost due to H2O2 consumption and the cost due to energy consumption, respectively, with constants Cmax, Pmax and Tmax representing maximal values of Cp, PUV and T. One of the most subtle tasks is choosing the values of the constants kd, kp and ke that represent the weights of the decolourisation cost, the peroxide cost, and the energy cost, respectively. This problem is not treated in this paper. Inserting (3) into (4), we obtain the control cost function: J (C p , PUV , T , Ai ) = kd g ((1 − f (C p , PUV , T )) Ai ) + k p
Cp Cmax
+ ke
PUV T Pmax Tmax
(6)
3.1 Linear static model
The cost function depends on four variables assuming we know the mapping f (in our case it is approximated by the RBN model).
As already said, this paper is based on a relatively small atmospheric corrosion database. Actually, there are only 32 measurements. Each measurement consists of exposition time t (in years), temperature T (in 0C), relative humidity HR (in %), SO2 concentration CSO2 (in µg.m-3), precipitation p (in mm), pH pH, and corrosion mass C (in g.m-2). We want to obtain the model of this system with the latter variable being the output, and the other six variables being the inputs. Since there are only 32 points available in the six-dimensional space and the measurements could be of a poor quality, we started the modelling by finding a static linear system:
The optimal control is given by minimising the cost function (6) with respect to three control variables (Cp, PUV and T). Since the minimisation is done in four dimensional space (Cp, PUV, T, and Ai) each optimal control variable depends upon the fixed value of initial absorbance (that is of course known prior to the decolourisation process). Initial absorbance is the only measurement and the only input to the control. Minimisation of the cost function can be done off-line to come up with the control functions: PUV = h1 ( Ai ) T = h2 ( Ai )
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ti y= C= i i
(7)
C p = h3 ( Ai )
Ti
H R i
C SO 2i
p i
θ1 T ψ= pHi = 1 32 (2) i θ, i θ 6
where tilde denotes the normalised variable that is obtained by subtracting the mean and dividing this difference by the standard deviation of this variable. By normalising the variables all variables have zero mean and the variance of 1 and thus their contribution to the final model is made equal.
The three control functions h1, h2, and h3 can be simply implemented on a dedicated hardware using e.g. tabular form. It does not have to be stressed that the shape of these three control functions depends heavily on the choice of design parameters kd, kp and ke. The experts from the textile engineering validated the proper behaviour of the control.
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The solution in this case can be obtained by using simple least squares method: θT = [ 0.8162 0.0564 0.0913 0.4277 −0.0123 −0.1084] (3)
Note that the set of measurements is very small and we cannot afford splitting it into identification data and validation data. The other reason for not splitting it is that the data lie in a quite broad region of the input space and thus extrapolation to other regions is not so straight-forward. The most important thing is that identification of the linear model could be performed on a smaller database, but identification of the Takagi-Sugeno model requires much more data.
Table 1: Mean-square errors of the T-S model (3 and 7 MFs) and scheduling variable xk
Because of the lack of proper validation, a sort of verification will be done in the following. The model error is defined as: ~ ei = Ci − ψ Ti θˆ , i = 1,2,32
1 32 2 ∑ ei 32 i =1
(5)
M = 0.1096
(6)
3.2 Takagi-Sugeno static model The first step in constructing the model is structural identification, i.e. we need to select the appropriate structure of the system. We cannot afford a very complex structure because the number of measurements is very low. We decided to test each of the six variables as scheduling variables. Six different fuzzy systems were obtained where three Gaussian membership functions were utilized. Since all signals are normalised and roughly span the interval [-3,3], the membership functions had the centres at -2, 0 and 2, respectively, while σ-parameters were 1 in all cases. Fuzzy model was obtained for all six cases where all input variables served as scheduling variables. In the case of the input variable xk (k = 1, 2,3, 4,5, 6) the regressor vector was set up in the following way: k
− c1 )2
ψ Tk = x1 xk −1
2
, µ 2 = e − ( xk − c2 ) , µ3 = e − ( xk − c3 ) xk
µ1 µ1 + µ2 + µ3
xk
µ3 µ1 + µ2 + µ3
1
2
3
4
5
6
M3k
0.0790 0.0928 0.0975 0.1095 0.0907 0.1092
M7k
0.0642 0.0669 0.0708 0.0945 0.0789 0.0959
If we compare the second and the third row of Table 1, we can observe that the biggest drop in variance is achieved if we use x1 as a scheduling variable in both cases. In the first case x1 is followed by x5, x2, and x3. In the second case x1 is followed by x2, x3, and x5. As expected these results are similar, but they are not the same since they depend upon a specific position of the membership functions. So, if both results are taken into account, the proposed scheduling variables would be x1 in the case when there is only one, and x1 and x2 if two scheduling variables are selected. The last TS model based on original data will be obtained by taking x1 and x2 as the premise variables (each fuzzified by using 3 membership functions – the same as in the one-premisevariable case), used only for fuzzification of variable x2 (the tests have shown that this choice gives the best results) while the other signals (x1 and x3 to x6) enter the regressor vector in the original form. In this case the MSE is 0.0582 but the number of estimated parameters is 3x3+5=14.
For the linear model (3) the mean-square error is:
µ1 = e − ( x
k
(4)
The mean-square error is defined as: M =
over the identification data set. This model is calculated analogously to the linear case. The results are shown in Table 1 (second row) for different values of k, i.e. for different scheduling variables. Another experiment was done with 7 membership functions with centres located at -3, -2, -1, 0, 1, 2, 3, while σ were still 1. Mean-square errors are shown in Table 1 (third row).
xk
Fig. 7 shows the comparison of the measured outputs and the outputs of the three models: the linear model (6 estimated parameters), the T-S model with the premise variable x1 and 7 membership functions (12 estimated parameters) and the T-S model with the premise variables x1 and x2 (each fuzzified with 3 membership functions) – 14 estimated parameters. It would be much better if the graph in original space could be plotted, but this is impossible since the input space is 6dimensional. This is why the sample number i is plotted on the abscise axis of Fig. 7.
2
µ2 (7) µ1 + µ2 + µ3
xk +1 x6
Similarly as in the case of the linear model, the signals used for identification were normalised. Parameter estimate θˆ k was then obtained by using least squares. It consists of eight parameters (3 “fuzzy” and 5 “linear”). We are particularly interested in the mean-square error of the model calculated 144
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Kumar, K., Deep, S., Suthar, S., Dastidar, M. G., and Sreekrishnan, T. R. (2016). Application of fuzzy inference system (FIS) coupled with Mamdani’s method in modelling and optimization of process parameters for biotreatment of real textile wastewater, Desalination and Water Treatment, 57 (21), pp. 9690-9697. Precup, R., Preitl, S., Rudas, I. J., Tomescu, M. L., and Tar, J. K. (2008). Design and Experiments for a Class of Fuzzy Controlled Servo Systems, IEEE/ASME Transactions on Mechatronics, vol. 13, no. 1, pp. 22-35. Rémond, E., Nercessian, E., Bernicot, C., and Mina, R. (2011). Mathematical approach based on a “Design of Experiment” to simulate process variations, 2011 Design, Automation & Test in Europe, Grenoble. dos Santos, A. B., Cervantes, F. J., and van Lier, J. B.(2007). Review paper on current technologies for decolourisation of textile wastewaters: Perspectives for anaerobic biotechnology, Bioresource Technology, 98 (12), pp. 2369-2385.
Fig. 7. The measured output, the output of the linear model, and two outputs of T-S models plotted against the sample number 4. CONCLUSIONS
Simillion, H., Dolgikh, O., Terryn, H., and Deconinck, J., (2014). Atmospheric corrosion modelling, Corrosion Reviews, 32 (10), pp. 73-100.
This paper brings up a problem that we are all very much aware of: How to find a model of a nonlinear process based on data of poor information content? The paper does not provide a definitive answer but it does open some discussion. Surely, it is necessary to make a trade-off between model complexity and model accuracy in such cases. If it is possible, some new experiments could be designed and the measurements taken. But in the two presented cases no new measurements were possible.
Tůmová, O., Kupka, L., and Netolický, P. (2018). Design of Experiments approach and its application in the evaluation of experiments, 2018 International Conf. on Diagnostics in Electrical Engineering, Pilsen.
REFERENCES Aengchuan, P., and Phruksaphanrat, B. (2017). A comparative study of design of experiments and fuzzy inference system for plaster process control, Proceedings of the World Congress on Engineering 2017 Vol I, WCE 2017, London. Belič, A., Škrjanc, I., Zupančič-Božič, D., Vrečer, F. (2009). Tableting process optimisation with the application of fuzzy models, International Journal of Pharmaceutics, vol. 389, no. 1/2. Chaabane, C., Chaari, T., and Trentesaux, D. (2009). Integrated sizing support system using simulation and design of experiments, 2009 International Conference on Computers & Industrial Engineering, Troyes, pp. 735741. Fisher, R. A. (1935). The design of experiments. Oliver & Boyd, Oxford. Halama, M., Kreislova, K., and Van Lysebettens, J. (2011). Prediction of Atmospheric Corrosion of Carbon Steel Using Artificial Neural Network Model in Local Geographical Regions. Corrosion, 67(6):065004.
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52-11 (2019)for 140–145 Design of ExperimentsIFAC as PapersOnLine a Prerequisite Development of Fuzzy Models? Design Design of of Experiments Experiments as as a a Prerequisite Prerequisite for for Development Development of of Fuzzy Fuzzy Models? Models? Design of Experiments as a Prerequisite for Development of Fuzzy Models?
Sašo Blažič *, Igor Škrjanc *
Sašo Sašo Blažič Blažič *, *, Igor Igor Škrjanc Škrjanc ** Sašo Blažič *, Igor Škrjanc * * University of Ljubljana, Faculty of Electrical Engineering, Slovenia (e-mail:Faculty [email protected]) University of of Ljubljana, Ljubljana, of Electrical Electrical Engineering, Engineering, Slovenia Slovenia ** University Faculty of (e-mail: [email protected]) * University of Ljubljana, of Electrical Engineering, Slovenia (e-mail:Faculty [email protected]) [email protected]) Abstract: Design of experiments is an(e-mail: approach where the experiments for measurement acquisition should be planned very carefully in order to obtain high-informative data. This paper tackles a problem of
Abstract: Design of is approach where experiments for acquisition should Abstract: of experiments experiments is an an where the the problems experiments for measurement measurement acquisition constructingDesign mathematical models of approach complex nonlinear based on measurement data ofshould poor be planned very carefully in order to obtain high-informative data. This paper tackles a problem of be plannedDesign very carefully order obtain high-informative This paper a problem of Abstract: of experiments is anto approach where the experiments measurement acquisition should information content. Two in practical examples are dealt with. data. The for first one is tackles a textile wastewater
constructing mathematical of nonlinear problems based on measurement data of constructing mathematical models of tocomplex complex nonlinear problems based measurement of poor decolourisation process, andmodels theorder second one an atmospheric corrosion of structural metal materials. In poor both be planned very carefully in obtain high-informative data. Thisonpaper tackles adata problem of information content. Two practical examples are dealt with. The first one is a textile wastewater information content. Two practical examples areconstructing dealt problems with.a The first is a textile wastewater cases the measurement datamodels were not for very accurate model. In such cases it is constructing mathematical of adequate complex nonlinear based onone measurement data of poor decolourisation process, and the second one atmospheric corrosion of structural metal materials. In decolourisation process, and practical the second one an an atmospheric corrosion structural materials. In both both necessary to make a trade-off between model complexity and model information content. Two examples are dealt with. accuracy. Theof first one ismetal a textile wastewater cases the measurement data were not adequate for constructing a very accurate model. In such cases it cases the measurement were not adequate for constructing a very such cases it is is decolourisation process, data and the second one an atmospheric corrosion of accurate structuralmodel. metal In materials. In both necessary todesign make aoftrade-off trade-off between model complexity and model model accuracy. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: experiments, fuzzy model, radial basis network, decolourisation process, atmospheric necessary to make a between model complexity and accuracy. cases the measurement data were not adequate for constructing a very accurate model. In such cases it is corrosion. necessary to make aoftrade-off between model complexity basis and model accuracy. Keywords: Keywords: design design of experiments, experiments, fuzzy fuzzy model, model, radial radial basis network, network, decolourisation decolourisation process, process, atmospheric atmospheric corrosion. corrosion. Keywords: design of experiments, fuzzy model, radial basis network, decolourisation process, atmospheric corrosion.
data of the process are collected without the supervision of an expert for experimental modelling. In such cases the data are data process are without of data of of the theless process are collected collected without the the supervision supervision of an an 1. informative, and consequently the experimental This paper discusses a very important issue when dealing typically 1. INTRODUCTION INTRODUCTION expert modelling. In cases the data of for theexperimental process are collected without thebut supervision ofare an expert for experimental modelling. In such such cases the data data are modelling becomes much more difficult it still can be with model discusses building on measurement data. Having 1. INTRODUCTION typically less informative, and consequently consequently the experimental experimental This paper aabased very important issue when dealing expert forless experimental modelling. In such cases the data and are typically informative, and the This paper discusses very important issue when dealing achieved by reducing the expectation about the model high-informative data based is theon prerequisite when building modelling becomes much more difficult still with model building measurement data. Having typically less informative, and consequently theit modelling becomes much more difficult but but itexperimental still can can be be This abased very important issue when dealing with paper model discusses building on measurement data.should Having adapting the modelling techniques. mathematical models. To achieve this, experiments be achieved by reducing the expectation about the model and high-informative data is the prerequisite when building modelling becomes much more difficult but it still can be achieved by reducing the expectation about the model and with model building based measurement data. Having high-informative data isdealing theon prerequisite when building carefully planned. When with nonlinear models such adapting the modelling techniques. mathematical models. To achieve this, experiments should be In this paper we will show two practical problems we dealt achieved by reducing the expectation about the model and adapting the modelling techniques. high-informative dataartificial the neural prerequisite when building mathematical models. Toisachieve this, networks experiments be as fuzzy planned. models or theshould problem carefully When with nonlinear models such with in paper the Both cases are similar: they both deal with a thepast. modelling techniques. mathematical models. To dealing achieve this, experiments should be adapting carefully even planned. When dealing with nonlinear models such In this will two problems dealt becomes more critical due to the high number of model In thiscomplex paper we we will show show two practical practical problems we we dealt as fuzzy fuzzy planned. models or or artificial neural networks the problem problem quite nonlinear process, the measurement database carefully When dealing with nonlinear models such as models artificial neural networks the with in the past. are they both deal parameters thatmore need critical to be estimated/trained. In this weBoth will cases show twosimilar: practical problems wewith dealtaa with in paper the Both cases are similar: they bothuncertainty, deal with becomes due to high model small in past. both cases and includes quite some as fuzzy even models or critical artificial the of problem becomes even more dueneural to the the networks high number number of model is quite complex nonlinear process, the measurement database with incomplex thedata past. Bothnocases are be similar: they both deal with a quitenew nonlinear process, the measurement database parameters that need to be estimated/trained. could longer measured due uncertainty, to practical Design ofeven experiments approach that aims to deal with and becomes due to the high number of model parameters thatmore need critical to is be an estimated/trained. is small in both cases and includes quite some quite complex nonlinear process, themodel measurement database is small in both cases and includes quite some uncertainty, reasons. Our task is to build the that extracts the this problem the influence (input) parameters that by needfirst to is beidentifying estimated/trained. andsmall new in data could noand longer be measured measured due uncertainty, to practical practical Design of experiments an approach that aims to deal with is both cases includes quite some and new data could no longer be due to Design of experiments is an approach that aims to deal with available information while knowing that the measurement variables and by the first consequence (output)influence variables,(input) then reasons. Our task is to build the model that extracts the this identifying and new Our data couldissmaller notolonger be measured due to practical reasons. task build thenormally model that extracts Design of experiments is an variation, approach the that then aimsspreading to deal with this problem problem by first identifying the influence (input) is much than expected forthea determining the area of their and the database available information while knowing that the measurement variables and the consequence (output) variables, then reasons. Our task is to build the model that extracts the available information while knowing that the measurement this problem identifying theto influence variables and by the first consequence (output) variables, then problem of this complexity. influence variables of the experiment cover the (input) whole database is much smaller than normally expected for aa determining the area of their variation, and then spreading the available information while knowing that the measurement database is much smaller than normally expected for variables and the consequence (output) variables, then determining the area of their variation, and then spreading the space of these variables that the model will be used for. problem of this complexity. influence variables of the to cover the This paper ismuch organised as follows. Sectionexpected 2 presents smaller than normally forthea problem ofisthis complexity. determining the area ofexperiments their variation, and spreading the database influence variables the experiment experiment to then cover thethewhole whole Adopting design of of only makes low space of these these variables that the not model will be the used for. problem process of decolourisation. Section 3 presents atmospheric this complexity. influence variables of the experiment to cover whole space of variables that the model will be used for. is as follows. Section 22 presents number of design experiments (the least possible?), but alsothe results This This paper paperprocess. is organised organised follows. Section presents the the Adopting of only makes Sectionas4 brings some conclusions. space of these variables that the not model will be used for. corrosion Adopting design of experiments experiments not only makes the low low process of decolourisation. Section 3 presents atmospheric in a reliable model on a broad range of input variables This paper is organised as Section follows. 3Section 2 presents the process of decolourisation. presents atmospheric number of experiments (the least possible?), but also results Adoptingof design ofof experiments not only makes low corrosion process. Section 4 brings some conclusions. number experiments (the least possible?), but not alsothe variation. Design experiments is certainly aresults new process decolourisation. Section 3 presents atmospheric corrosionofprocess. Section 4 brings some conclusions. in a reliable model on a broad range of input variables 2. PROCESS OF DECOLOURISATION number ofsince experiments least possible?), but also results in a reliable model on(the a broad range of book input variables approach it goes back to the famous of corrosion process. Section 4 brings some conclusions. variation. Design of experiments is certainly not aaFisher new in a reliable model on a broad range of input variables variation. Design of experiments is certainly not new 2. PROCESS PROCESS OF DECOLOURISATION (1935). The approach can be used infamous differentbook areasofsuch as This section deals withOF theDECOLOURISATION process of decolourisation of a 2. approach it goes back to Fisher variation. since Design of experiments isfamous certainly notof a2011), new approach since itcircuit goes back to the the book Fisher computer-aided design (Rémond et al., textile wastewater that is the result of an industrial dyeing 2. PROCESS OF DECOLOURISATION (1935). approach can be used different areas as deals with the process decolourisation of approach since it goes back to thein book ofsuch Fisher (1935). The The approach can be(Chaabane used infamous different areas such as This This section section deals with the techniques process of of of decolourisation of aa production system design et al., 2009), the process. There exist many decolouring (dos computer-aided circuit design (Rémond et al., 2011), textile wastewater that is the result of an industrial dyeing (1935). Theofapproach be(Tůmová used(Rémond in different areas as textile computer-aided circuitcandesign et2018), al., such 2011), This section deals with decolourisation ofwea wastewater that isthe theprocess result of the an industrial dyeing evaluation experiments et et al., al., control Santos, 2007). Inexist our case we deal with process where production system design (Chaabane 2009), the process. There many techniques of decolouring (dos computer-aided circuit design (Rémond et al., 2011), production system design (Chaabane et 2017), al., 2009), the textile wastewater that is the result of an industrial dyeing process. There exist many techniques of decolouring (dos design (Aengchuan, and Phruksaphanrat, tableting measure absorbance of we thedeal wastewater A thatwhere defines evaluation of experiments (Tůmová et al., 2018), control Santos, 2007). In our case the process we production system design (Chaabane al., 2009), the evaluation of experiments (Tůmová et et al., 2018), control There many of decolouring (dos Santos, 2007). Inexist our that case wetechniques deal with with the process where we process optimisation (Belič et al., 2009), control design of process. “dirtiness” (0 means the water does not absorb any light design (Aengchuan, and Phruksaphanrat, 2017), tableting measure absorbance of the wastewater A that defines evaluation of experiments (Tůmová et al., 2018), control design (Aengchuan, and Phruksaphanrat, 2017), tableting Santos, 2007). In our case we deal with the process where we measure absorbance of the wastewater A that defines servo systems (Precup et al., 2008), etc. The design of – it is completely transparent, whiledoes a larger number means it process optimisation (Belič et al., 2009), of (0 means that the water not absorb any light design and Phruksaphanrat, 2017),adesign tableting process (Aengchuan, optimisation (Belič et al., difficult 2009), control control design of “dirtiness” measure absorbance of the wastewater A that defines “dirtiness” (0 means that the water does not absorb any light experiments becomes much more when dynamic is “dirty”). The decolourisation process is done by adding the servo systems (Precup et al., 2008), etc. The design of it transparent, while aa larger number means it processsystems et for al., 2009), design of of –“dirtiness” servo (Precup et al., 2008), etc.control The content design (0 that the the water not absorb any light –peroxide it is is completely completely transparent, while larger means it model ofoptimisation the becomes process is(Belič sought since frequency O2means , and applying UVdoes lamp fornumber a by certain time. H2The experiments much more difficult when aadesign dynamic is “dirty”). decolourisation process is done adding the servo systems (Precup et al., 2008), etc. The of experiments becomes much more difficult when dynamic – it is completely transparent, while a larger number means it is “dirty”). The decolourisation process is done by adding the the excitation becomes extremely important. model process sought for frequency of O applying the UV lamp for a certain time. peroxide experiments more difficult when acontent dynamic 2The 2,, and model of of the the becomes process is ismuch sought for since since frequency content of is We“dirty”). built H the model of this process that is intended for control decolourisation process is done by adding the O and applying the UV lamp for a certain time. peroxide H 2 2 the becomes Unfortunately, very often it happens it is necessary to peroxide model of the process isextremely sought forimportant. sincethat frequency content of the excitation excitation becomes extremely important. design. Modelling and control ofthat this process by using the UV lamp for a certain time. a H 2O2, and applying We built the of process is intended control We built the model model of this this process thatby is Kumar intendedetfor for control construct a mathematical model of a certain process, and the the excitation becomes extremely important. fuzzy-model approach has been done al. (2015) Unfortunately, very very often often it it happens happens that that it it is is necessary necessary to to design. Modelling and control of this process by using a Unfortunately, We builtModelling the model of processofthat intendedby forusing controla design. andthis control thisis process construct of and approach been by Kumar (2015) Unfortunately, very oftenmodel it happens that it process, is necessary to fuzzy-model construct aa mathematical mathematical model of aa certain certain process, and the the design. Modelling andhas control of this fuzzy-model approach has been done done by process Kumar et etbyal. al.using (2015)a Copyright IFAC construct ©a 2019 mathematical model of a certain process, and the 140 fuzzy-model approach has been done by Kumar et al. (2015) 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. 1. INTRODUCTION
Copyright 2019 IFAC 140 Peer review© of International Federation of Automatic Copyright ©under 2019 responsibility IFAC 140Control. 10.1016/j.ifacol.2019.09.131 Copyright © 2019 IFAC 140
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but this paper deals with model development based on a smaller database.
causes that much more data are necessary for building the model. This is why 4-dimensional input space is avoided. In this case, the model will have the following form:
Before proceeding with control design a solid model of the process to be controlled has to be available. This is especially the case when model-based control methods are utilised. The main problem in this case lies in the data that the model was derived from. It is very well-known that several model forms (e.g. fuzzy models, artificial neural networks, spline models, piece-wise linear models etc.) exist that are capable of describing the nonlinear processes with arbitrarily small modelling errors. On the other hand, none of them is able to extrapolate the information to the parts of the space where no measured data or small amount of measured data was available.
D = f (C p , PUV , T )
Af = (1 − D) Ai = (1 − f (C p , PUV , T )) Ai
2.2 Radial basis network model of the decolourisation process The radial basis network (RBN) will be used for approximating the mapping given by Eq. (2). The model of the process will be obtained based on the measurement data for the Irgalan Gelb 3R KWL dye: PUV [W], Cp [mg/l], T [min], and the absorbance A (measured only at a few different times T). The data collected include the measurements of three repeated groups of experiments, each consisting of 15 experiments. The number of samples is rather low because samples of wastewater are taken manually and being analysed. The measurement error seems to be quite high. Some obvious outliers were manually removed. Since there are three inputs to the network and one output, the results are given in the form of three-dimensional planes where x and y axes correspond to the two inputs, the third input is fixed, while the decolourisation factor is shown on the z-axis.
Decolourisation factor given by the formula Ai − Af Ai
(1)
where Ai is initial absorbance (before the decolourisation process), and Af is final absorbance (after the decolourisation process). Consequently, D = 0 means that the process was completely unsuccessful while D = 1 means that the decolourisation is perfect. •
(3)
A remark is also needed here regarding the presence of the time of the experiment (T) in model given by Eq. (2). Since the measurements are taken at quite arbitrary times, the model is given in a form where time is used as explicit model input.
The first thing we have to determine when considering model structure is the selection of model inputs and outputs. The first open issue is the selection of the model output. At least two choices are possible:
D=
(2)
Note that decolourisation factor is given as the output, since we do not model the dependence upon the initial absorbance. The actual output of the model Af is then obtained from Eq. (1):
2.1 Model structure selection
•
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Final absorbance Af.
Since the model will be used for control design purposes, the decision has to be carried out focusing on the control aspect of the problem. The goal of the control is to decolourise the waste water. We have to define a certain reference. In our opinion some target final absorbance has to be defined as a control goal. If this were not the case and the decolourisation factor were selected as system output (and consequently control reference), all waste waters would be treated the same irrespective of the actual “dirtiness”. That would result in almost the same control actions in all circumstances which, clearly, denies the reason for building the complex nonlinear model. To sum up, the best solution for the system output is final absorbance Af. The selection of model inputs seems to be easier, but still there are some open questions regarding the topic. The candidates for the system inputs are: the concentration of the peroxide H2O2 (Cp), the power of the UV lamp (PUV), and the time of the experiment (T). But since the final absorbance probably depends nonlinearly upon the initial absorbance, the latter would have to be one of the inputs to the system. But having 4 inputs instead of 3
Fig 1. RBN approximation for PUV = 1200 W
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Figures 1, 2, and 3 show the decolourisation factor as a function of the peroxide concentration and the time while the power of the UV lamp is fixed (at 1200, 1400, and 1600 W). The measurements are shown with circles, and the network output at the same inputs with crosses. Figures 4 to 6 shown similar graphs for the fixed values of the peroxide concentration. Although it can be seen that the data set is quite poor (important data are missing in some operating conditions and the variability of the data is very high in repeating experiments), some qualitative properties of the process are clearly seen.
Fig. 4. RBN approximation for Cp = 0.7 mg/l
Fig. 2. RBN approximation for PUV = 1400 W
Fig. 5. RBN approximation for Cp = 4.5 mg/l
Fig. 3. RBN approximation for PUV = 1600 W 2.3 Control design Control problem in this case is to determine the control actions that will result in a purified water. By control actions we mean Cp, PUV and T applied to the process. Of course the cost of control rises with rising control actions.
Fig. 6. RBN approximation for Cp = 8.3 mg/l
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3. ATMOSPHERIC CORROSION PROCESS
Optimal control is proposed to solve the control problem. The most important thing when applying optimal control is to define a suitable cost function. The cost has to include all important aspects of the control problem. In our case, we decided to choose the following cost function: J = kd g ( Af ) + k p
Cp Cmax
+ ke
PUV T PmaxTmax
The process of atmospheric corrosion is modelled in this section. It is modelled by means of Takagi-Sugeno fuzzy model. The obtained model could serve for prediction of atmospheric corrosion in near future under simulated climatic changes (acidifying of atmosphere, lower concentrations of SO2 in Europe, higher concentrations of O3, NOX, global warming etc.).
(4)
where g(Af) is the function that defines the cost due to unsatisfactory final absorbance: 0 g ( Af ) = Af − Asatis
Af < Asatis Af ≥ Asatis
Atmospheric corrosion of structural metal materials is very complex, non-linear process depending on various meteorochemical and material parameters. Till nowadays the impact of these parameters on the process of degradation of materials is not fully understood. Thanks to existing longterm climate programmes in Europe (ECE/EMEP and ICP) and in Asia/Africa (RAPIDC) meteorochemical data such as temperature, relative humidity, amount of precipitation, concentration of main pollutants are daily measured; and corrosion mass losses of selected metals (carbon steel, copper, zinc and aluminium) are annually measured. There also exist measurement of corrosion of various metals in many locations in the world. The modelling of these phenomena has been dealt in research (Simillion et al., 2014; Halama et al. 2011). Unfortunately, the results made from the existing huge databases are not allowed to be published. This paper uses a very small free database, but significant information can be collected from it. The analysis of this database also shows major problems that are usually encountered when dealing with the problem of atmospheric corrosion.
(5)
If the final absorbance is less than a certain threshold Asatis (the acceptable level of the final absorbance), the first term of the cost function, the punishment of the unsatisfactory decolourisation, will be 0. If, however, the final absorbance is greater, the first term will be positive. The second and the third term in Eq. (4) represent the cost due to H2O2 consumption and the cost due to energy consumption, respectively, with constants Cmax, Pmax and Tmax representing maximal values of Cp, PUV and T. One of the most subtle tasks is choosing the values of the constants kd, kp and ke that represent the weights of the decolourisation cost, the peroxide cost, and the energy cost, respectively. This problem is not treated in this paper. Inserting (3) into (4), we obtain the control cost function: J (C p , PUV , T , Ai ) = kd g ((1 − f (C p , PUV , T )) Ai ) + k p
Cp Cmax
+ ke
PUV T Pmax Tmax
(6)
3.1 Linear static model
The cost function depends on four variables assuming we know the mapping f (in our case it is approximated by the RBN model).
As already said, this paper is based on a relatively small atmospheric corrosion database. Actually, there are only 32 measurements. Each measurement consists of exposition time t (in years), temperature T (in 0C), relative humidity HR (in %), SO2 concentration CSO2 (in µg.m-3), precipitation p (in mm), pH pH, and corrosion mass C (in g.m-2). We want to obtain the model of this system with the latter variable being the output, and the other six variables being the inputs. Since there are only 32 points available in the six-dimensional space and the measurements could be of a poor quality, we started the modelling by finding a static linear system:
The optimal control is given by minimising the cost function (6) with respect to three control variables (Cp, PUV and T). Since the minimisation is done in four dimensional space (Cp, PUV, T, and Ai) each optimal control variable depends upon the fixed value of initial absorbance (that is of course known prior to the decolourisation process). Initial absorbance is the only measurement and the only input to the control. Minimisation of the cost function can be done off-line to come up with the control functions: PUV = h1 ( Ai ) T = h2 ( Ai )
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ti y= C= i i
(7)
C p = h3 ( Ai )
Ti
H R i
C SO 2i
p i
θ1 T ψ= pHi = 1 32 (2) i θ, i θ 6
where tilde denotes the normalised variable that is obtained by subtracting the mean and dividing this difference by the standard deviation of this variable. By normalising the variables all variables have zero mean and the variance of 1 and thus their contribution to the final model is made equal.
The three control functions h1, h2, and h3 can be simply implemented on a dedicated hardware using e.g. tabular form. It does not have to be stressed that the shape of these three control functions depends heavily on the choice of design parameters kd, kp and ke. The experts from the textile engineering validated the proper behaviour of the control.
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The solution in this case can be obtained by using simple least squares method: θT = [ 0.8162 0.0564 0.0913 0.4277 −0.0123 −0.1084] (3)
Note that the set of measurements is very small and we cannot afford splitting it into identification data and validation data. The other reason for not splitting it is that the data lie in a quite broad region of the input space and thus extrapolation to other regions is not so straight-forward. The most important thing is that identification of the linear model could be performed on a smaller database, but identification of the Takagi-Sugeno model requires much more data.
Table 1: Mean-square errors of the T-S model (3 and 7 MFs) and scheduling variable xk
Because of the lack of proper validation, a sort of verification will be done in the following. The model error is defined as: ~ ei = Ci − ψ Ti θˆ , i = 1,2,32
1 32 2 ∑ ei 32 i =1
(5)
M = 0.1096
(6)
3.2 Takagi-Sugeno static model The first step in constructing the model is structural identification, i.e. we need to select the appropriate structure of the system. We cannot afford a very complex structure because the number of measurements is very low. We decided to test each of the six variables as scheduling variables. Six different fuzzy systems were obtained where three Gaussian membership functions were utilized. Since all signals are normalised and roughly span the interval [-3,3], the membership functions had the centres at -2, 0 and 2, respectively, while σ-parameters were 1 in all cases. Fuzzy model was obtained for all six cases where all input variables served as scheduling variables. In the case of the input variable xk (k = 1, 2,3, 4,5, 6) the regressor vector was set up in the following way: k
− c1 )2
ψ Tk = x1 xk −1
2
, µ 2 = e − ( xk − c2 ) , µ3 = e − ( xk − c3 ) xk
µ1 µ1 + µ2 + µ3
xk
µ3 µ1 + µ2 + µ3
1
2
3
4
5
6
M3k
0.0790 0.0928 0.0975 0.1095 0.0907 0.1092
M7k
0.0642 0.0669 0.0708 0.0945 0.0789 0.0959
If we compare the second and the third row of Table 1, we can observe that the biggest drop in variance is achieved if we use x1 as a scheduling variable in both cases. In the first case x1 is followed by x5, x2, and x3. In the second case x1 is followed by x2, x3, and x5. As expected these results are similar, but they are not the same since they depend upon a specific position of the membership functions. So, if both results are taken into account, the proposed scheduling variables would be x1 in the case when there is only one, and x1 and x2 if two scheduling variables are selected. The last TS model based on original data will be obtained by taking x1 and x2 as the premise variables (each fuzzified by using 3 membership functions – the same as in the one-premisevariable case), used only for fuzzification of variable x2 (the tests have shown that this choice gives the best results) while the other signals (x1 and x3 to x6) enter the regressor vector in the original form. In this case the MSE is 0.0582 but the number of estimated parameters is 3x3+5=14.
For the linear model (3) the mean-square error is:
µ1 = e − ( x
k
(4)
The mean-square error is defined as: M =
over the identification data set. This model is calculated analogously to the linear case. The results are shown in Table 1 (second row) for different values of k, i.e. for different scheduling variables. Another experiment was done with 7 membership functions with centres located at -3, -2, -1, 0, 1, 2, 3, while σ were still 1. Mean-square errors are shown in Table 1 (third row).
xk
Fig. 7 shows the comparison of the measured outputs and the outputs of the three models: the linear model (6 estimated parameters), the T-S model with the premise variable x1 and 7 membership functions (12 estimated parameters) and the T-S model with the premise variables x1 and x2 (each fuzzified with 3 membership functions) – 14 estimated parameters. It would be much better if the graph in original space could be plotted, but this is impossible since the input space is 6dimensional. This is why the sample number i is plotted on the abscise axis of Fig. 7.
2
µ2 (7) µ1 + µ2 + µ3
xk +1 x6
Similarly as in the case of the linear model, the signals used for identification were normalised. Parameter estimate θˆ k was then obtained by using least squares. It consists of eight parameters (3 “fuzzy” and 5 “linear”). We are particularly interested in the mean-square error of the model calculated 144
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Kumar, K., Deep, S., Suthar, S., Dastidar, M. G., and Sreekrishnan, T. R. (2016). Application of fuzzy inference system (FIS) coupled with Mamdani’s method in modelling and optimization of process parameters for biotreatment of real textile wastewater, Desalination and Water Treatment, 57 (21), pp. 9690-9697. Precup, R., Preitl, S., Rudas, I. J., Tomescu, M. L., and Tar, J. K. (2008). Design and Experiments for a Class of Fuzzy Controlled Servo Systems, IEEE/ASME Transactions on Mechatronics, vol. 13, no. 1, pp. 22-35. Rémond, E., Nercessian, E., Bernicot, C., and Mina, R. (2011). Mathematical approach based on a “Design of Experiment” to simulate process variations, 2011 Design, Automation & Test in Europe, Grenoble. dos Santos, A. B., Cervantes, F. J., and van Lier, J. B.(2007). Review paper on current technologies for decolourisation of textile wastewaters: Perspectives for anaerobic biotechnology, Bioresource Technology, 98 (12), pp. 2369-2385.
Fig. 7. The measured output, the output of the linear model, and two outputs of T-S models plotted against the sample number 4. CONCLUSIONS
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This paper brings up a problem that we are all very much aware of: How to find a model of a nonlinear process based on data of poor information content? The paper does not provide a definitive answer but it does open some discussion. Surely, it is necessary to make a trade-off between model complexity and model accuracy in such cases. If it is possible, some new experiments could be designed and the measurements taken. But in the two presented cases no new measurements were possible.
Tůmová, O., Kupka, L., and Netolický, P. (2018). Design of Experiments approach and its application in the evaluation of experiments, 2018 International Conf. on Diagnostics in Electrical Engineering, Pilsen.
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