A New Approach of TakagiâSugeno Fuzzy Modeling Using an

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A new approach of T-S fuzzy modeling using improved GA optimization for oxygen content in a coke furnace ridong zhang, jili tao, and Furong Gao Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b01364 • Publication Date (Web): 17 May 2016 Downloaded from http://pubs.acs.org on May 22, 2016

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Industrial & Engineering Chemistry Research

A new approach of T-S fuzzy modeling using improved GA optimization for oxygen content in a coke furnace

Ridong Zhanga,b*, Jili Taob, Furong Gaob a Key Lab for IOT and Information Fusion Technology of Zhejiang, Information and Control Institute, Hangzhou Dianzi University, Hangzhou 310018, P.R. China b Chemical and Biomolecular Engineering Department, The Hong Kong University of Science and Technology, Hong Kong

Abstract: The oxygen content modeling of the coke furnace is important for advanced control design but not an easy job because of various disturbances and nonlinearity. A novel approach is proposed by using an improved genetic algorithm (IGA) combined with the dynamic Auto Regressive with exogenous input (ARX) Takagi-Sugeno (T-S) fuzzy model. The IGA algorithm automatically generates the input variable, the appropriate fuzzy if-then rules and the ARX structure to characterize the dynamic nonlinear feature of the oxygen content by processing the operation data from the industrial coke furnace. And a more comprehensive objective function is constructed considering both the modeling precision and structure simplicity. Hybrid encoding, modified genetic operators, particularly maintain operator, are designed to obtain the satisfactory optimization performance. The modeling accuracy and system structure of T-S fuzzy model are compared with a benchmark Box-Jenkins gas furnace and the complex industrial coke furnace. The results show good modeling accuracy and simple structure of T-S fuzzy model.

Key words：：Oxygen content; Industrial coke furnace; T-S fuzzy ARX model; Improved GA

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1. Introduction Industrial coking is of great importance for supplying various petrochemical products [1]. The oxygen content control of the coke furnace is a complex task among the operation goals in the unit [2]. In such process, modeling is important for advanced controller design but is more difficult because of the nonlinear characteristics, switches of the coke towers and other various disturbances such as inlet oil flow change, feeding temperature variance, etc. [3]. Due to lack of detailed physicochemical knowledge of the process, input/output data based models may be another choice. However, too simple linear model may lead to limited control performance since important dynamics or process information is not incorporated efficiently [4-5]. Takagi-Sugeno (T-S) fuzzy model is a universal approximation tool, which is popular in system modeling [6-10]. And a large number of nonlinear systems have been approximated by fuzzy modeling approach with satisfactory prediction accuracy [11-13]. Generally, the difficulty in fuzzy modeling is the structure identification. How to choose the fuzzy antecedents and consequents have been put forward in [14-18]. The structure evolving learning methods (SELM) for fuzzy systems have been proposed to solve the exponential increase problem of fuzzy rules [14, 15]. A simplified structure evolving method (SSEM) was also presented beginning with only one fuzzy rule [16]. The self-organizing fuzzy neural network (SOFNN) [17] and adaptive neuro-fuzzy inference systems (ANFISs) [18] were able to derive an accurate and compact structure by using recursive learning and adding and pruning techniques. Fuzzy clustering is another technique for fuzzy space partition, which has been widely used to implement the structure identification and parameter estimation. One of the most popular approaches in fuzzy


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modeling is c-means method [19, 20]. However, the inefficient variables cannot be eliminated in above methods, and the learning results are usually locally optimal. As a global optimization method, genetic algorithm (GA) and the other evolutionary computing have received a lot of attentions. Many have been used to design fuzzy rules for system modeling and data classification [21-23]. However, most of them were appropriate to partial structure identification. How to determine the whole structure including the input variables, the linguistic partitioning, the fuzzy rule set and the consequent part, is still largely open, especially for the nonlinear system with more complex dynamics such as the oxygen content of the coke furnace. It is also a challenging problem to develop a simple but efficient fuzzy model that can be efficiently used in system identification and controller design in the industrial field. A good fuzzy model based on input-output data is of both precise modeling accuracy and simple structure [24-27]. It can be regarded as a bi-objective optimization problem, which can be solved by global optimization algorithms, such as weighted sums of objectives (WSO) [28], multi-objective evolution algorithms (MOEAs), etc. [29-32]. Among MOEAs, NSGA-II is superior to several representative algorithms [29]. However, its computing complexity is o(MN), where M is the number of objectives and N is the population size. And its solution is a Pareto set, which is difficult to select by decision maker. The bi-objective optimization problem can be converted into single objective problem by the WSO method, and its weighting coefficient is critical to the optimization results. For T-S fuzzy model, the modeling accuracy can be predicted and its model information can be obtained by expert experiences. Thus, the weighting coefficient is not difficult to be chosen, GA is then applied to solve the optimization problem. Since standard



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GA (SGA) is prone to trap into a local optimal solution and slow to converge, some biological gene operators were effectively adopted in SGA, which greatly improved the global searching capability [33]. In this paper, the hybrid encoding, the improved selection operator, dynamic mutation probability and maintain operator are adopted in SGA to optimize the whole fuzzy structure and parameters. To show the efficiency of the proposed method, firstly, the model of a well-known Box-Jenkins gas furnace [37] is constructed and compared with the typical fuzzy modeling methods in [19, 34-36]. Then the oxygen content modeling in an industrial coke furnace is applied and compared with fuzzy c-means method and NSGA-II. The paper is organized as follows: Section 2 presents the whole work flow of the coke unit, the main channel of oxygen content and its disturbance channel. Section 3 details the T-S fuzzy modeling method and the proposed improved genetic algorithm. Section 4 shows the illustrative example of the well-known Box-Jenkins gas furnace and an oxygen content modeling in an industrial coke furnace. Conclusion is in Section 5. 2 The coke unit [3] The overall process can be seen in Fig.1, where the main job of the coke unit is to coke residual oil. It consists of a fractionating tower (T102), three coke furnaces (F101/1,2,3) and six coke towers (T101/1,2,3,4,5,6). The detailed flow of each part of the unit is further shown in Fig.2. Here furnace (F101/3) is illustrated as an example. Two branches of residual oil (FRC8103, FRC8104) go into the convection room (F101/3) to be preheated to about 330℃, then they combine into one branch and go to the fractionating tower (T102) to exchange heat with gas oil coming from the coke towers (T101/5,6). During the heat exchange, the heavy part of them joins


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together and is divided into two branches (FRC8107, FRC8108) to enter the radiation room of the furnace (F101/3) to be further heated to about 495℃. Finally, they join into one branch and go to the coke towers (T101/5,6) to remove coke. To guarantee production safety and product quality, the oxygen content of the coke furnace should be controlled to a set-point, and the most important issue of advanced controller design is system modeling. In the coke furnace, the oxygen content is adjusted by valve opening for the air inlet blower, which constitutes the main channel. Besides, since the more inlet oil flow need the more oxygen to be consumed, a continuous disturbance of the inlet oil flow changes is fed to the oxygen content. Furthermore, the oxygen content of the furnace often drops and rises drastically during the coke removing due to the fact that part of the inlet gas oil will be removed before heating coke towers. Overall, the inlet oil flow change has made a great influence on oxygen content, which is regarded as the main disturbance. The disturbance channel model needs to be constructed for further controller design or process optimization. Hence, there are two models, i.e., the main channel and the disturbance channel, to be constructed.



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residual oil T101/1

T101/2

T101/3

T101/4

T101/5

T101/6

furnace F101/1

residual oil

furnace F101/2

residual oil

gas oil

circulating oil furnace F101/3 fractionating tower T102

Fig.1 Overall flow of coke unit


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residual oil CAS

FRC8103 37.2 t/h

FRC8104 36.9 t/h

50.8%

AUT

51.4%

TR8155 348.6℃

TR8156 339.6℃

Go to fuel valve

TRC8103 496.6℃

Go to fuel valve

AUT

AUT

AUT

AUT

FRC8107 43.5 t/h

FRC8108 43.6 t/h

79.8%

TRC8105 496℃

85.1%

Furnace 101/3

Fraction tower circulating oil from pumps 102/1,2,3 To T102 TR8129 495.8℃

To T101/5,6

Fig. 2 Detailed flow of coke furnace

The experimental data is gathered from a distributed control system (DCS) CS3000 in an industrial coking unit of a refinery, its sampling period is set 5s in DCS. 1200 input-output measurement samples are selected for system modeling, which is plotted in Figs.3a-3b. At each



sample, 2 inputs and 2 outputs are contained, namely, the outlet oxygen content and the inlet blower valve opening for the main channel, the perturbation of oil flow and its corresponding oxygen content perturbation for the disturbance channel. 6

Oxygen content

Oxygen content

5 4.8 4.6 4.4 4.2

0

200

400

600

800

1000

5.5

5

4.5

1200

0

200

400

Samples

800

1000

1200

800

1000

1200

Oil flow change

850

55

50

45

600

Samples

60

B low er opening

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0

200

400

600

800

1000

1200

800

750

0

Samples

200

400

600

Samples

(a) Blower opening and oxygen content for main channel. (b) Oil flow change and oxygen content for disturbance channel Fig.3 1200 input-output samples for system modeling

3 IGA based T-S fuzzy ARX model It is obvious in Fig.3 that there exist nonlinearity and disturbance in the samples of oxygen content. T-S fuzzy ARX model is then adopted to approximate the dynamics of the nonlinear system. The structure incorporating parameter identification is the most difficult part in fuzzy modeling approach. An IGA is proposed to simultaneously determine the structure and parameters of fuzzy system by using the hybrid encoding, several gene operators and considering the structure complexity into the modeling accuracy.


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y(t )

u (t )

−1

z

M z

−1

z −1

M z −1

e(t )

u (t − 1) u (t − m)

yˆ(t )

y (t − n )

y(t − 1)

Fig. 4 Block diagram of IGA-based T-S fuzzy model identification

The block diagram in Fig. 4 illustrates the whole process of IGA based oxygen content modeling by T-S fuzzy ARX model. In Fig.4, select the main channel as an example, the valve opening of coke unit u (t ), L , u (t − m) and oxygen content y (t − 1), L , y (t − n) are utilized as the input vector of T-S fuzzy ARX model, the model prediction output yˆ(t ) and oxygen content y (t ) is compared, the modeling error e(t ) can then be obtained. The modeling error and the T-S Fuzzy model structure complexity are the two objectives of IGA, which can be changed into single objective by choosing the appropriate weighting coefficient. 3.1 T-S Fuzzy ARX Model

By using the autoregressive with exogenous input (ARX) model structure, T-S fuzzy model possesses the powerful approximating capability with the good predictive feature. A nonlinear mapping between the past input- output data and the predicted output is: yˆ(k ) = f ( X(k ))

(1)

where X(k ) = [ y (k − 1), L , y (k − n), u (k − d ), u (k − d ), L , u (k − d − m)] , n and m are the maximal lags considered for the output and input terms, d is the discrete time delay, and f represents the



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nonlinear relation of the fuzzy model. A T-S fuzzy ARX model often interpolates local linear time-invariant (LTI) ARX submodel, the jth IF-THEN fuzzy rule is shown as follows: Rule j: If

x1 (k )

is

A1 j

and

f j (k ) = BT X(k ) , j = 1,2, L , M , M ≤

is

x 2 (k )

∏

s

i =1

A2 j

and … and

xs (k )

is

Asj

, then

mi .

where B j = [a1j , a2j ,L, anj , b1j , b2j ,L, bmj ]T , the input vector x(k)=[x1(k),...,xs(k)] is usually a subset of

X(k), namely, x(k) ∈ X(k), mi is the number of membership functions of xi (k ) , M is the number of fuzzy rules. The final output of fuzzy model by a weighted mean defuzzification can be expressed as:

∑ α [ x(k )] f (k ) yˆ (k ) = ∑ α [ x(k )] M

j

j =1

j

(2)

M

j

j =1

where α j [ x (k )] delegates the overall value of the premise part of the jth implication for the input input x(k) in the fuzzy inference system (FIS) A j , A j = ∏i =1 Aij , and α j [ x (k )] can be calculated s

as:. α j [ x (k )] = µ1j µ 2j L µ sj

(3)

A Gaussian membership function is adopted and shown as follows:  || xi − cij || 2   σ ij2  

µ i j = exp −

(4)

where c ij and σ ij are the center and width of the Gaussian function in Aij respectively. Define fuzzy basis function (FBF) as: ϕ j [ x (k )] =

α j [ x (k )]

∑

M i =1

α i [ x (k )]

(5)

The output yˆ (k ) can be rewritten as a linear combination of FBFs for the fuzzy consequent


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ARX submodel in the following form: yˆ(k ) = ∑ j =1ϕ j [ x (k )] f j (k ) M

(6)

The input vector, the number of fuzzy rules and its parameters of membership functions determine the fuzzy premise part. The ARX model structure and its parameters comprise the consequent part. Once the complete fuzzy premise part and ARX submodel structure are determined, recursive least square (RLS) method can be utilized to determine the parameters of ARX submodel in terms of input-output data set. Denote θ = [B1T BT2 L BTM ]T Φ (k ) = [φ1[ x (k )] X (k )T φ2 [ x (k )] X (k )T L φM [ x (k )] X (k )T ]T

(7) (8)

Substitute (7) and (8) into (6) yields yˆ(k ) = Φ(k )T θ

(9)

If there are z sampling output data Y=[y(1), y(2),…,y(z)], the value of θ can be obtained by RLS method after z iterations:  θ( k ) = θ( k − 1) + K (k )[ y (k ) − ΦT (k )θ(k − 1)]  T −1  K (k ) = P (k − 1)Φ(k )[Φ ( k ) P ( k − 1)Φ(k ) + 1]  P ( k ) = P ( k − 1) − K ( k )K T (k )[ΦT ( k )P ( k − 1)Φ( k ) + 1] 

(10)

where k=1,2,···,z, K (0) , P (0) are set as relatively small values of (m+n)M-by-1 vector and big values of (m+n)M-by-(m+n)M matrix, respectively. 3.2 IGA for T-S fuzzy model

It is clear that with the increase of input vector dimension, the number of fuzzy rules, ARX submodel structure and the system complexity will increase exponentially. Moreover, the rules



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based on expert knowledge will be complex and usually become impossible without necessary knowledge especially for complex industrial process. Determining the used input variables, the linguistic partitioning, the rule set and ARX submodel structure together involves a complex search space, which is not an easy task to be optimized. To significantly improve the modeling accuracy with the simple system structure, an IGA is designed to solve the optimization problem, and various operators for fuzzy model are combined into the optimization processes. A. Hybrid encoding method In the T-S fuzzy model, considering the similarity of u(k-1),···u(k-m), and y(k-1),···,y(k-n), the input vector x(k) is initially chosen as [y(k-1),u(k-1)], d is set as 1. m and n in X (k ) are generally set in advance according to a priori knowledge, which will directly determine the approximation capability of the fuzzy model. Hence, they are optimized in the limitation of [y(k-1),u(k-1)] to decrease the number of input variables (m1). Moreover, the number of fuzzy rules and their parameters in Eq.(4) are also included in the encoding method. The ith chromosome for encoding the whole fuzzy model is then designed: c11 c12 σ 11σ 12  c21 c22 σ 21σ 22 M M M M  Ci = cr1 cr 2 σ r1σ r 2 M M M M  0 0 0 0 n 0 m 0 

          

(11)

where i = 1, 2, L, N , N is the population size. m1, m and n are the positive integers with 1 ≤ m1 ≤ 2 ,


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1 ≤ m ≤ 4 , 1 ≤ n ≤ 4 . m and n are adopted one bit quaternary coding (0,1,2,3), the decoding is

simply to add one to the quaternary coding. The optimization of m1 is essential to implement the selection of input vector. If m1 is 1, the input vector becomes x(k)=[y(k-1)], ci2 in column 2 and σ i 4 in column 4 are then set to zeros; else the input vector becomes x(k)=[y(k-1),u(k-1)]. Here,

m1 is enumerated in above two cases. r is the number of fuzzy rules generated randomly between [1,9], the rows in [r+1, 9] are also set to zeros. Hence, Ci is a 4-by-10 matrix, however, there are actually at most

r×4+2

parameters to be optimized. The elements cij ,σ ij in Eq.(11) can be

initialized as follows:  ymin + δ ( ymax − ymin ) 1 ≤ i ≤ r , j = 1 cij =  umin + δ (umax − umin ) m1 ≠ 0, 1 ≤ i ≤ r , j = 2 0.1 + δ ( ymax − 0.1) 1 ≤ i ≤ r , j = 3 σi =  0.1 + δ (umax − 0.1) m1 ≠ 0 ,1 ≤ i ≤ r , j = 4

(12)

where δ is a random number produced between 0 and 1, umin and umax are the minimum and maximum of process inputs, ymin and ymax are the minimum and maximum of process outputs. Hence, the input vector, the fuzzy rules and ARX submodel structure are included in the given hybrid encoding method. The parameters θ for ARX submodel can be gained by RLS, if the knowledge bases of fuzzy system are calculated in terms of Eq.(10). N T-S fuzzy model can be gained (C1 , θ1 ),L, (CN , θ N ) .

B. Objectives of T-S fuzzy modeling To improve the modeling accuracy and its generalization capacity, the samples are equally divided into two groups, the former 1/2 data (Y1) is selected to calculate the model parameter θ , and the latter 1/2 data (Y2) is chosen to evaluate the modeling accuracy and its generalization



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capacity at every generation. Moreover, the structure complexity is considered in the objective function, defined as:

Min J (Ci ) =

N1

∧

∑ Y (i) − Y (i) 1

i =1

1

2

N1 +

N2

∧

∑ Y (i) − Y (i) 2

2

N2 + ω(m + n)r

2

(13)

i =1

The objective in Eq.(13) is composed of two part of fuzzy model. The first one is the sum of Root Mean Squared Error (RMSE) for Y1 and Y2, where Y1(i) (i=1,...,N1) are the samples in the ^

dataset Y1, θ can be gained according to the training data Y1 , Y1 (i) (i = 1, L, N1 ) are then derived ^

as the predictions of T-S fuzzy model. Keep θ invariant, Y2 (i) (i = 1, L, N 2 ) can be derived by the same fuzzy model. A testing procedure is incorporated in the objective function, which guarantees the generalization performance. The second part (m+n)r shows the structure complexity of the fuzzy system, ω is a weighting coefficient between (0,1], which reflects the importance degree of structure complexity. Since the order of magnitude of RMSE for fuzzy model can be obtained easily, and the range of structure parameters (m, n, r) is known, ω is selected as ten times less than the order of magnitude of RMSE to guarantee the modeling accuracy.

C. Operators of IGA for T-S fuzzy model In addition to traditional selection, crossover and mutation operators, improved selection operator, the dynamical mutation probability and maintain operator have been designed for the proposed fuzzy model, which may improve the weak local-search capability and premature convergence performance of SGA.


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(1) Selection operator Roulette wheel method is usually used as the selection operator, and the selection probability of an individual Ci , can be calculated as follows. p (Ci ) =

f (Ci ) N

∑

,

f (Ci ) =

f (Ci )

1 J (Ci )

(14)

i =1

In Eq.(14), the individuals with better performance index, i.e., smaller values of objective in Eq.(13), have bigger probability to survive. A set of individuals are selected as parent individuals for reproduction of new individuals. To keep the population diversity, 3N/4 parents are picked according to roulette wheel method, while the left N/4 parents are chosen from the worst N/4 individuals to keep the population diversity. The elitism, namely, the individual with the smallest value of objective function, is directly selected as the parent. (2) Crossover and mutation operators The crossover operation is executed in Eq.(15) with probability pc between current individual ' Ci and the next individual Ci+1, pc is set as 0.9. The offsprings Ci' , Ci+1 are produced after the

crossover operator. C i' = α C i + (1 − α )C i +1 C i'+1 = (1 − α )Ci + α Ci +1

(15)

where α is produced randomly, α ∈ (0,1) , n and m are rounded to the nearest integers. The input vector, the ARX structure and the rules are changed dynamically during the optimization process. However, the operator will be prone to produce more fuzzy rules, and some irrational ones can also be yielded.



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For a better exploration, the individual is mutated with different mutation probability pmi , the individual with better values of objective is assigned smaller mutation probability, shown as follows: pmi = pm 0 −

i ∆pm N

(16)

where pm0 is the initial mutation probability, ∆pm is the maximum change rate, i=1,…, N, the individuals are sorted ascendingly according to the value of objective function. Once the mutation is operated, m, n are mutated in the range of quaternary encoding, r is kept invariant, while the elements of mutated individual are reproduced in terms of Eq.(12). (3) Maintain operator Because GA is essentially a random optimization algorithm, there exist some irrational fuzzy systems during the optimization process. At the same time, crossover operator cannot produce new structure of fuzzy rules, the maintain operator is then designed. 1. Calculate ∆cij = cij - ci, j+1 , if ∆cij < 0.03 , then cij is deleted, and the number of fuzzy rules (r) is decreased. 2. If the number of fuzzy rules is less than 2, a random ∆r is produced satisfying r+ ∆r ≦9, the elements in the new rules are computed according to Eq.(12). 3. If all coefficients in Bj are less than 0.003, the submodel for rule j is regarded inactive, and the corresponding rule is deleted.

d. Optimization process


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The whole optimization process for IGA is described in the following steps: Step 1: Initialize the maximal generation G and the population size N. Generate the chromosomes randomly in the search space for the initial population. Step 2: Decode the chromosome to generate N fuzzy model in terms of Eq.(2) - (10) and compute the performance J for each individual. Step 3: Select the chromosomes to generate 3N/4 parent chromosomes of the next generation according to Roulette wheel selection. And the worst N/4 individuals are directly inherited to keep population diversity and avoid trapping in the local optimization solution early. Step 4: Execute the crossover operator. Repeat this for all the pc×N/2 pairs of parents, and implement the mutation operator with dynamical mutation probability. Step 5: Carry out maintain operator in the new individuals to improve the quality of fuzzy model generated by crossover and mutation operators. Step 6: Repeat steps 2 to 5 until the set maximum evolution generation G is met. Moreover, elitism, the inclusion of the best individual in the next population, is used throughout the optimization process. 4. Case Study In practice, simple linear models are usually used and lead to limited control performance since important dynamics or process information is not incorporated efficiently. The fuzzy arx model is constructed for an industrial coke furnace, and since more information can be incorporated in the model, the subsequent controller design will rely on sufficient information and will beexpected to yield improved control performance.



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To validate the effectiveness of the proposed method, the IGA is run for 10 times at personal computer equipped with Intel Core i5-3470. The parameters of IGA are set as follows: population size N=40, maximal evolution generations G=100, the operator probability pc =0.9,

pm is dynamic changing, pm0 is set to 0.2, ∆pm is set to 0.1. Fuzzy modeling is utilized with a small amount of expert knowledge. That is, the maximal number of fuzzy rules is set as 9, the input vector is chosen from [y(k-1),u(k-1)], the maximal n and m are set as 4 and 4, respectively. At each time, the parameters of IGA and data set are kept invariant. However, the weighting coefficient ω is changed with different cases. The fuzzy model with the smallest RMSE of testing data is regarded as the best result. The Box-Jenkins gas furnace is first simulated and the oxygen content model is established in detail.

4.1 Box-Jenkins gas furnace modeling A well-known real dataset namely the Box-Jenkins gas furnace process [37] is selected as the simulation example, which consists of 296 input-output measurements. At each sampling time k, the input u(k) is the gas flow into the furnace, and the output y(k) is CO2 concentration. The input-output data is divided equally into 2 groups as the training set Y1 and the testing set Y2. Each group includes the dataset with 148 input-output pairs. In this case, ω is set as 0.001, which is an order magnitude less than that of the expected RMSE of fuzzy model. To be comparable with a number of fuzzy models in the literature, mean square error (MSE) is calculated as the performance index of the IGA optimized fuzzy modeling, which is defined as the square of RMSE. After optimization, the input vector is y(k-1), the number of rules r is 6, n is 3 and m is 3. The


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original outputs and the predicted outputs of IGA optimized fuzzy model are compared in Fig. 5, its corresponding training errors and testing errors are exhibited in Fig. 6. Obviously, the training error is less than the testing error. The MSE of the training error (0.015) is much smaller than the MSE of testing error (0.146), as shown in Table 1. The comparison results in the literatures are also listed in Table 1. It can be seen that the number of parameters of IGA is relatively small and the MSEs of IGA for training data and testing data are also good. 62

2

60 1.5 58

Modeling errors

56

Outputs

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


54 52 50

1

0.5

0

48 -0.5 Real Values Fuzzy model outputs

46 44

0

50

100

150

200

250

-1

300

0

50

Samples

100

150

200

250

300

Samples

Fig.5 Comparison of modeling outputs and original outputs

Fig.6 Training error and testing error of fuzzy model

Table 1: Comparison of different fuzzy models for the Box-Jenkins gas furnace Model

Number of Parameters

Number of rules

Training error(MSE)

Testing error(MSE)

Tsekouras(2005)

91

6

0.022

0.236

Li et al (2012)

57

3

0.015

0.147

Pedrycz et al (2014)

56

7

0.017

0.198

Proposed IGA

39

6

0.015

0.146

4.2 Oxygen Content modeling in the coke furnace Oxygen content modeling is critical to realize the model based advanced control. Here, the main channel and disturbance channel are constructed. The sampling input-output data in Fig.3 is first normalized, and then divided equally into 2 groups. Each group includes 600 input output



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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data. The system output of the main channel model and the disturbance channel model is the oxygen content, while the inlet blower valve opening and the perturbation of input fuel are selected as system inputs, respectively. Since there are two objectives in nature in Eq.(13), the proposed methodology is compared with a typical multi-objective genetic algorithm, namely, NSGA-II. The maintain operator is also applied in NSGA-II and the parameters of ARX submodel are derived by using the RLS method. The Pareto front of the main channel and disturbance channel are shown in Fig.7 and Fig.8. Since the range of structure parameter (m,n,r) is relatively small, there are only several solutions in the Pareto front. Furthermore, fuzzy c-means method is also adopted to train the centers of membership functions. The other fuzzy parameters, such as input vector, the number of fuzzy rules and ARX submodel structure, are the same as the best results optimized by IGA. The best fuzzy models obtained by 3 methods are listed in Table 2, where RMSE is Root Mean Squared Error of testing data (Y2). In Table 2, the fuzzy model of IGA can obtain better prediction precision than NSGA-II in terms of the value of RMSE. However, the structure of fuzzy model by using NSGA-II is simplifier than that of IGA. Because fuzzy c-means method utilizes the optimized structure parameters of IGA, the RMSE of main channel and disturbance model using IGA is similar to that of c-means method. The statistical data in 10 runs are listed in Table 3. It can be seen that the mean of RMSE of IGA is superior to that of NSGA-II, partly because the fuzzy model with complex ARX submodel gains better approximation capability. The simple structure of fuzzy model and good modeling precision are obtained after running IGA. Specifically, the run time of IGA is much shorter than NSGA-II.


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To reflect the prediction accuracy of the established model, the comparisons of predicted oxygen content with the measured data of the testing set for the main channel and its disturbance channel are given in Figs.9-12, respectively. Fig.9 shows the comparison of the predicted yield with measured output of the best results by 3 methods for the main channel, while the corresponding prediction error is shown in Fig.10. Fig. 11 shows the fitting curve of prediction values and real value for the disturbance channel by using the same methods, and the estimation errors are depicted in Fig.12. Compared Fig.10 with Fig.12, the maximal modeling error obtained by c-means method and NSGA-II are larger than that obtained by IGA. The same results can be observed comparing with the modeling error of their disturbance models. It can be seen in Table 2 that the proposed method has managed to sustain the error to considerably small values by optimization the whole structure and parameters of T-S fuzzy model. Table 2: Comparison of the best simulation results with 3 methods

Methods IGA NSGA-II c-means

m1

Main channel n m r RMSE

Disturbance channel m1 n m r RMSE

2 1 1

3 1 1 1 3 1

1 1 1

3 3 4

4.5e-3 6.1e-3 4.5e-3

6

12

5.5

11.5

5

11

4.5

10.5

4

10

f2

f2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.5

9.5

3

9

2.5

8.5

2 0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

3 1 3

3 3 2 4 3 3

4.4e-3 7.0e-3 4.4e-3

8 0.012 0.0122 0.0124 0.0126 0.0128 0.013 0.0132 0.0134 0.0136 0.0138

f1

f1

Fig.7 The Pareto front of main channel by NSGA-II

Fig.8 The Pareto front of disturbance channel by NSGA-II



1 Real values IGA NSGA-II c-means

1

0.98

Oxygen content

Oxygen content of disturbances

1.02

0.96

0.94

0.95

0.9

0.92

Real values IGA NSGA-II c-means

0.9

0.88

0

100

200

300

400

500

0.85

600

0

100

200

300

400

500

600

Samples

Samples

Fig.9 Comparisons of disturbance channel outputs for 3 methods

Fig.11 Comparisons of main channel outputs for 3 methods

0.02

0.025 IGA NSGA-II c-means

0.015

IGA NSGA-II c-means

0.02 0.015

0.01

0.01

Modeling errors

Modeling errors

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.005

0

0.005 0 -0.005 -0.01

-0.005

-0.015 -0.01 -0.02 -0.015

0

100

200

300

400

500

-0.025

600

0

100

200

Samples

300

400

500

600

Samples

Fig.10 Modeling errors of disturbance channel for 3 methods

Fig.12 Modeling errors of main channel for 3 methods

Table 3: The performance comparison of 3 methods for 10 runs Methods

m1

n

Main channel m r

(s) 3.5±0.5 217

5.4e-3

1.5±0.5 3.5±0.5 757

6.8e-3

IGA

1.5±0.5 2.5±0.5 2±1

NSGA-II

1.5±0.5 1±0

c-means

1±0

1±0

3±0

1±0

T

m1

RMSE

0.171 4.5e-3

Disturbance channel m r T (s)

n

1.5±0.5 1.5±0.5 2±1 1.5±0.5 2±1 1±0

3±1

214

2.5±0.5 4.5±0.5 851

3±0

3±0

3±0

RMSE

6.1e-3 7.3e-3

0.177 4.6e-3

5. Conclusions In this paper, an improved genetic algorithm was proposed to construct a T-S fuzzy ARX model, which consists of input layer, fuzzy rules layer and output layer. The input variables and fuzzy rules layer are considered to construct premise part of fuzzy model and the output layer is used to form the consequent part. In order to determine both the structure and parameters in the whole fuzzy system, the hybrid encoding is designed to code the structure and parameters in a chromosome. The selection, crossover, mutation operators are used, especially a maintain


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operator including pruning and deleting are proposed to guarantee the validity of the fuzzy system. RLS is applied to obtain the coefficients of ARX submodel. Furthermore, to improve the modeling accuracy and simplify the fuzzy model structure, modeling accuracy, generalization performance and structure complexity of fuzzy system are considered in the objective function. Our proposed method is successfully applied to a benchmark problem and oxygen content modeling of an industrial coke furnace. Simulations show that the proposed method has the ability to obtain a more compact structure with higher or close accuracy comparing with the other works. According to the obtained results, we can conclude that the proposed method is more useful and effective in real complex problem such as oxygen content in coke furnace. The proposed method has a limitation of the input variables and the advanced control is not applied based on the constructed model, so further studies will focus on how to upgrade the proposed method to a dynamic input vector and implement the advanced control.

Acknowledgements The authors acknowledge the Zhejiang Provincial Natural Science Foundation of China (Grant No. LR16F030004 ) and National Natural Science Foundation of China (Grant No. 61273101).

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