Hybrid fuzzy modeling of chemical processes

Hybrid fuzzy modeling of chemical processes

Fuzzy Sets and Systems 130 (2002) 265 – 275 www.elsevier.com/locate/fss Hybrid fuzzy modeling of chemical processes Yin Wang ∗ , Gang Rong, Shuqing ...

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Fuzzy Sets and Systems 130 (2002) 265 – 275

www.elsevier.com/locate/fss

Hybrid fuzzy modeling of chemical processes Yin Wang ∗ , Gang Rong, Shuqing Wang National Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou 310027, People’s Republic of China Received 7 January 2000; received in revised form 22 June 2001; accepted 6 November 2001

Abstract Fuzzy models have been proved to have the ability of modeling all plants without any priori information. However, the performance of conventional fuzzy models can be very poor in the case of insu/cient training data due to their poor extrapolation capacity. In order to overcome this problem, a hybrid grey-box fuzzy modeling approach is proposed in this paper to combine expert experience, local linear models and historical data into a uniform framework. It consists of two layers. The expert fuzzy model constructed from linguistic information, the local linear model and the T-S type fuzzy model constructed from data are all put in the 5rst layer. Layer 2 is a fuzzy decision module that is used to decide which model in the 5rst layer should be employed to make the 5nal prediction. The output of the second layer is the output of the hybrid fuzzy model. With the help of the linguistic information, the poor extrapolation capacity problem caused by sparse training data for conventional fuzzy models can be overcome. Simulation result for pH neutralization process demonstrates its modeling c 2001 Elsevier Science B.V. All ability over the linear models, the expert fuzzy model and the conventional fuzzy model.  rights reserved. Keywords: Process modeling; Hybrid fuzzy modeling; Linear model; pH neutralization process

1. Introduction Process modeling is a very important step in the control, diagnosis and optimization of the process system. For most cases, the model is constructed from the data because 5rst principal models are very di/cult to be obtained. Arti5cial neural networks (ANN), fuzzy systems, etc., can all be used to model chemical processes using historical data. Unfortunately, insuf5cient data hamper the accuracy of these models because they rely completely on the data when inducing process behavior. Many methods have been proposed to overcome this problem. Obviously, incorporating the prior



Corresponding author. Fax: +86-571-795-1125. E-mail address: [email protected] (Y. Wang).

knowledge into the models will make the identi5cation better conditioned. The regulation method was 5rst introduced into the process model to satisfy the smoothness assumption [6]. In 1994, Thompson and Kramer [7] proposed a hybrid method which utilizes static 5rst principle equations to control the extrapolation of the hybrid model and ANN to compensate the inaccuracy in the 5rst principle equations. Johansen [2] demonstrated that many types of prior knowledge including smoothness can be included as a term in the criterion or as constraints. Recently, Johansen proposed a hybrid model composed of a number of local models valid in diJerent operating regimes [3]. The local models are combined by smooth interpolation into a complete global model. It is compatible with the engineer’s understanding of the system. However, this approach requires the union of all the local operating regimes covering the whole operating region, i.e.,

c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter  PII: S 0 1 6 5 - 0 1 1 4 ( 0 1 ) 0 0 2 4 2 - 1

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we need a local model at every operating regime. This assumption is very di/cult to be satis5ed in real-time application. For the above work, little has been done to incorporate the linguistic information from experts or operators into the modeling approach. In this paper, a hybrid fuzzy modeling approach is proposed to combine linguistic information, local linear models and historical data into a uniform framework. It consists of two layers. The local linear model and the T-S type fuzzy model constructed from data are all put in the 5rst layer. Using fuzzy set theory, the linguistic information is used to form an expert fuzzy model of the process and is also put in the 5rst layer. Layer 2 is a fuzzy decision module, which is employed to decide which model in the 5rst layer should be employed to make the 5nal prediction. The output of the second layer is the output of the hybrid fuzzy model. With the help of the expert fuzzy model, the extrapolation capacity of the hybrid fuzzy model is greatly improved. Simulation result for pH neutralization process demonstrates its modeling ability over the local linear model, the expert fuzzy model and the conventional fuzzy model. This paper is organized as follows. The basic structure of the conventional T-S type fuzzy model and its identi5cation algorithm are discussed in Section 2. Section 3 describes the general structure of the proposed hybrid fuzzy model. The simulation result is presented in Section 4 and, 5nally, concluding remarks are given in Section 5.

are the orders of the process while d1 ; : : : ; dp are time delay, respectively. A T-S type fuzzy model can be constructed to model process (1) with its fuzzy IF–THEN rules expressed as following: Rule j: If x1 (k) is A1j and x2 (k) is A2j and : : : and xm (k) is Amj Then yF (k + 1) = BjT ∗ U (k);

j = 1; 2; : : : ; M;

(2)

where X (k) = [x1 (k); : : : ; xm (k)]T is a scaled subset of U (k); Bj = [bj1; bj2; : : : ; bj(ny + nu1 + · · · + nup )]T are parameters in the conclusion part, yF (k + 1) is the predicted output of the fuzzy model, M is the number of rules. The output of fuzzy model can be expressed as following [8]: M T j=1 j [ X (k)]Bj U (k) yF (k + 1) = ; (3) M j=1 j [ X (k)] where j [X (k)] is the gaussion membership function m of the inferred fuzzy set Aj (Aj = i=1 Aij ), de5ned by   X (k) − aj 2 j [ X (k)] = exp − ; (4) j2 where aj = [aj1 ; aj2 ; : : : ; aj; m ]T is the center of the gaussion membership function of the fuzzy set Aj while j is the width. Other forms of membership functions can also be used here. Denition 1. De5ne fuzzy basis function (FBF) [8] as j [ X (k)] : (5) j [ X (k)] = M l=1 l [ X (k)]

2. Conventional fuzzy modeling approach 2.1. Basic structure of the T-S type fuzzy model Since an multi-input–multi-output (MIMO) system can always be separated into a group of multi-input– single-output (MISO) systems, we will only consider an MISO system here. Assume the process to be modeled is described as following:

The output y F (k +1) can then be expressed as a linear combination of FBFs in the following form:

y(k + 1) = f[ U (k)];

2.2. Conventional T-S type fuzzy model identi9cation

(1)

where U (k) = [y(k); y(k − 1); : : : ; y(k − ny + 1); u1 (k − d1 ); : : : ; u1 (k − d1 − nu1 + 1); : : : ; up (k − dp ); : : : ; up (k −dp −nup +1)]T , ui is the ith input, ny ; nu1 ; : : : ; nup

yF (k + 1) =

M 

j [ X (k)]BjT U (k):

(6)

j=1

For most existing T-S type fuzzy model identi5cation approach, the membership function is decided a

Y. Wang et al. / Fuzzy Sets and Systems 130 (2002) 265 – 275

priori, i.e., those FBFs j [ X (k)] have already been decided. The parameters in the conclusion part (Bj ) can then be easily estimated using least-squares algorithm. Denote  = [B1T

B2T

T T BM ] ;

···

’(k) = [1 [X (k)]U (k)T ···

(7)

(8)

Substitute (7) and (8) into (6) yields yF (k + 1) = ’(k)T ;

(9)

Also denote  = [’(1) Yd = [yd (2)

’(2) yd (3)

··· ···

’(N )]T ; yd (N + 1)]T :

(10) (11)

Rearrange (9) for k = 1; 2; : : : ; N yields Yd = ;

(12)

(12) is in the form of a linear regression model. If the reverse of matrix (T ) exists, the parameters matrix  can be uniquely determined by the conventional least-squares algorithm as following:  = [T ]−1 T Yd :

predictions. That is to say conventional fuzzy models are poor in their extrapolation capacity. To solve this problem, we should try to use any useful information available. Here, we are going to combine the linguistic information, local linear models with the fuzzy model to construct a hybrid fuzzy model. 3. Hybrid fuzzy modeling approach

2 [X (k)]U (k)T

M [X (k)]U (k)T ]T ;

267

(13)

If  is poorly conditioned (i.e., some singular values of  are near or even equal to zero), however, the parameter estimates given by (13) will change dramatically if elements of Yd are modi5ed slightly and result in poor predictions of y F when used with new  data [5]. This is always the case because su/cient training data are very di/cult to be obtained for most processes. To solve this problem, PLS algorithm [10] or singular value decomposition (SVD) approach [11] should be used instead of conventional least-squares algorithm to give a robust estimation on . An alternative is to partition the input space according to the training data using clustering algorithm so that  is better conditioned [1,9]. In such circumstance, a sparse fuzzy rule base will be created. When new input variables fall out of any existing fuzzy set Aj , i.e., MAX{j [X (k)]} is very small, the fuzzy model (6) will fail to give any reliable

In this section, a two-layered hybrid fuzzy modeling approach is presented. The general structure of the hybrid fuzzy model is shown in Fig. 1. The 5rst layer is a basic models layer, the expert fuzzy model constructed from linguistic information, the T-S type fuzzy model constructed from data and the local linear model are all put here. The second layer is a fuzzy decision layer, which employs a fuzzy decision module to choose which model in the 5rst layer should be used to make the 5nal prediction. The output of the second layer is the output of the hybrid fuzzy model. 3.1. Basic models layer 3.1.1. The expert fuzzy model (EFM) For almost every process, we can get some linguistic information from domain experts or operators. Some of them can be described by a set of fuzzy rules as following: Rule j: If xE1 (k) is AE1j and xE2 (k) is AE2j and : : : and xEn (k) is AEnj Then yE (k + 1) = fEj [ U (k)]; j = 1; 2; : : : ME ;

(14)

where X E (k) = [xE1 (k) ; : : : ; xEn (k)]T ⊂ U (k); fEj [U (k)] is a scalar output function, yE (k + 1) is the predicted output of the EFM while ME is the number of rules for the EFM. The output of the EFM can be expressed as following [8]: ME j=1 j [ X E (k)] ∗ fEj [ U (k)] ; (15) yE (k + 1) = ME j=1 j [ X E (k)] function of the where j [X E (k)] is the membership n inferred fuzzy set AEj (AEj = i=1 AEij ). For conventional fuzzy models, they always require that for any

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Y. Wang et al. / Fuzzy Sets and Systems 130 (2002) 265 – 275

Process

the expert fuzzy model U(k)

the local linear model the T-S type fuzzy model

y(k+1)

yE(k+1) yL(k+1) yF(k+1)

Fuzzy decision module

yH(k+1) -

+

e(k+1) Fuzzy decision layer

Basic models layer

the hybrid fuzzy model Fig. 1. General structure of the hybrid fuzzy model.

input X E (k), there always exists a fuzzy set AEj such that j [X E (k)]¿0. This is not required for the EFM because there are still two other models which can be used for compensation. Denition 2. For any given input X E (k), the reliability of 5ring the EFM is de5ned by ’E =

ME 

j [ X E (k)]:

(16)

j=1

3.1.2. Local linear models (LLM) The process is always working at some operating points. In the neighborhood of any operating point, a linear model can be easily obtained using classical identi5cation methods. It can be described by an ARMAX model in the following form: At lth operating point: yL (k + 1) = DlT U (k); l = 1; 2; : : : ; ML ;

(17)

where yL (k + 1) is predicted output of the local linear model while Dl = [dl1; dl2; : : : ; dl(ny + nu1 + · · · + nup )]T are the parameters of the LLM. Although these local linear models are only valid in local area, they provide a rough description of the process and are widely used in the design of model predictive controller (MPC) for the process systems. 3.1.3. The T-S type fuzzy model (FM) Since the general structure of the T-S type fuzzy model and its parameters identi5cation have already

been discussed in Section 2, we will only present its structure learning algorithm here. Assume there are N samples given as {U (i); yd (i + 1)}; i = 1; 2; : : : ; N , where yd is the desired output. We should 5rst partition the input space into several fuzzy sets using the nearest neighborhood clustering algorithm [8] as following: 1. Starting from the 5rst data pair {U (1); yd (2)}, establish a cluster center a1 = X (1), set M = 1. Select a radius  and a positive parameter ¿1. 2. Consider the kth input=output data pair {U (k); yd (k + 1)} (k = 2; 3; : : : ; N ), and there are M clusters with their centers located at a1 ; a2 ; : : : ; aM . We need to 5nd the nearest cluster to X (k), i.e. |X (k) − ajk | = min |X (k) − ah |: 16h6M

(18)

Then, if |X (k) − ajk |¿, which means the kth data points is outside of all the existing M clusters, establish a new cluster with its center located at X (k), set M = M + 1; aM = X (k). 3. If k is greater than N , 5nish the clustering algorithm. Otherwise, let k = k + 1 go to step 2. Using the clustering algorithm, the centers for all the M membership functions of the FM are found. Set j = , then the premise part of the FM has been decided. The parameters in the conclusion part can then be easily estimated by the conventional leastsquares algorithm as described in Section 2.2. The local linear models in (17) can be directly introduced into the FM. Assume the operating point of

Y. Wang et al. / Fuzzy Sets and Systems 130 (2002) 265 – 275

the lth local linear model is characterized by X l . For l = 1; 2; : : : ; ML , 5nd the nearest cluster to X l , i.e., |X l − ajk | = min |X l − ah |:

(19)

16h6M

If |X l − ajk |¿, which means the lth operating point is outside of all the existing M clusters, establish a new cluster with its center located at X l , set M = M + 1; aM = X l (k); BM = Dl . Denition 3. For any given input X (k), the reliability of 5ring the FM is de5ned by ’F =

M 

j [ X (k)]:

(20)

j=1

Although the EFM and the FM share the same function form, there are large diJerences between them.

269

If ’F is LOW and ’E is LOW then yH (k + 1) = ’E yE (k + 1) + ’F yF (k + 1) + (1 − ’E − ’F )yL (k + 1);

(21)

where is yL (k + 1) is the predicted output using one of the local linear models in (17). It is compatible with engineers’ understanding of the process. If the FM is reliable, we can use it only because the FM can provide the most accurate prediction if there are enough training data. If the FM is not so reliable, but the EFM can provide some reliable compensation, use these two models to form the hybrid fuzzy model output. If neither the EFM nor the FM gives reliable predictions, a local linear model from (17) should be used as a default model to give a reliable prediction. The output of the hybrid fuzzy model can then be expressed as following:

yH (k + 1) "F; H (’F )yF (k + 1) + "F; L (’F )"E; H (’E ) =

’E yE (k+1)+’F yF (k+1) ’E +’F

+ "F; L (’F )"E; L (’E )[’E yE (k + 1) + ’F yF (k + 1) + (1 − ’E − ’F )yL (k + 1)]

"F; H (’F ) + "F; L (’F )"E; H (’E ) + "F; L (’F )"E; L (’E )

; (22)

The EFM is concentrated at global mapping and its fuzzy sets are partitioned is a coarse way. It is a rough description of the process which provides directional information about the process. On the contrary, the FM is good at local mapping. It is used to compensate the modeling error between the hybrid fuzzy model and the process. Its fuzzy sets are partitioned in a 5ne way. 3.2. Fuzzy decision layer For the above-mentioned three models, which one should be used to give the 5nal prediction? If this section, a fuzzy decision module is introduced to make a selection among these three models. Its fuzzy selection rules can be expressed as follows: If ’F is HIGH then yH (k + 1) = yF (k + 1): If ’F is LOW and ’E is HIGH then yH (k + 1) = (’E yE (k + 1) + ’F yF (k + 1))=(’E + ’F )

where "F denotes the membership function of fuzzy set for ’F , while "E denotes the membership function of fuzzy set for ’E , L and H represent for LOW and HIGH, respectively. The membership function for ’F and ’E are shown in Fig. 2. For the three sub-models in the hybrid fuzzy model, the EFM and the LLM are obtained before the hybrid fuzzy model is constructed and are kept unchanged since then. The identi5cation of the hybrid fuzzy model is performed through the adaptation of the FM.

4. Simulation result The proposed hybrid fuzzy modeling approach will now be applied to a simulated pH neutralization process [4]. The stirred tank system shown in Fig. 3 has three inlet streams—an acid stream (HNO3 ), a base stream (NaOH) and a buJer stream (NaHCO3 ). The liquid level in the tank is allowed to vary freely. Nominal operating conditions are given in Table 1. Two

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Y. Wang et al. / Fuzzy Sets and Systems 130 (2002) 265 – 275 Table 1 Nominal conditions for the pH neutralization process

η(ϕ F) Low

High

A = 207 cm2 Cv = 8:75 ml=cm=s pK1 = 6:35 pK2 = 10:25 Wa1 = 3 × 10−3 mol=l Wa2 = −3 × 10−2 mol=l Wa3 = −3:05 × 10−3 mol=l Wb1 = 0

ϕF η(ϕ η ϕ E) Low

High

Wb2 = 3 × 10−2 mol=l Wb3 = 5 × 10−5 mol=l ' = 0:5 min q1 = 16:6 ml=s q2 = 0:55 ml=s q3 = 15:6 ml=s h = 14:0 cm pH4 = 7:0

1 W˙ a4 = [(Wa1 − Wa4 )q1 + (Wa2 − Wa4 )q2 Ah + (Wa3 − Wa4 )q3 ];

ϕE Fig. 2. Membership functions for ’F and ’E .

1 W˙ b4 = [(Wb1 − Wb4 )q1 + (Wb2 − Wb4 )q2 Ah + (Wb3 − Wb4 )q3 ];

q3 Wa3 Wb3

q2 Wa2 Wb2 q1 Wa1 Wb1

Wa4 + 10pH4 −14 + Wb4 − 10−pH4 = 0;

pH4 pH q4 Wa4 Wb4 Fig. 3. pH neutralization system.

reaction invariants for any stream are de5ned as 2− Wa = [H+ ] − [OH− ] − [HCO− 3 ] − 2[CO3 ];

(23)

2− Wb = [H2 CO3 ] + [HCO− 3 ] + [CO3 ]:

(24)

The 5rst invariant represents a charge balance while the second represents a balance on the carbonate ion. The process model consists of three nonlinear ordinary diJerential equations and a nonlinear output equation for the pH neutralization system: 1 h˙ = (q1 + q2 + q3 − Cv h0:5 ); A

(25)

(26)

(27)

1 + 2 × 10pH4 −pK2 1 + 10pK1 −pH4 + 10pH4 −pK2 (28)

where h is the liquid level, Wa4 and Wb4 are the invariants of the eSuent stream, q1 , q2 and q3 are the acid, buJer and base Towrate, respectively. The sampling period is chosen as 0:25 min. It is desired to control pH4 by the base Towrate q3 . The interesting operation range is de5ned by 06pH614 and 06q3 640 ml=s. In order to examine the extrapolation ability of the hybrid fuzzy model, two data sequences are generated which are shown in Fig. 4. The data set used for identi5cation is shown in Fig. 4(a), which covers a small part of the operating range. While the validation data sequence is shown in Fig. 4(b), which covers a large part of the operating range. Since the pH measurement has time delay and is always not so reliable, it is not a good choice to choose its past values directly as model inputs. The past model predictions are then used as model inputs, i.e., we want to construct a dynamic model of the process. The general structure of dynamic process model is shown in Fig. 5, where Z −1 means unit time delay.

Y. Wang et al. / Fuzzy Sets and Systems 130 (2002) 265 – 275

271

4.1. Prediction result of the EFM For the above pH neutralization process, we can easily obtain a local linear model near pH = 7:0 using a step response as following: y(k + 1) = 7:0 + 0:86[y(k) − 7:0] + 0:08[u(k) − 15:6]:

(29)

From our experience of the pH process, the gain is very high near pH = 7:0 and fair low when pH is far from 7.0. And we also know that If pH is larger than 7:0 Then the gain is 9% of the gain near 7:0: If pH is lower than 7:0 Then the gain is 12% of the gain near 7:0:

(30)

Using the linguistic information (30) and the local linear model (29), an EFM can be constructed as following: If pH is larger than 7:0 Then yE (k + 1) = 7:0 + 0:86[yE (k) − 7:0] + 0:007[u(k) − 15:6]: If pH is near 7:0 Then yE (k + 1) = 7:0 + 0:86[yE (k) − 7:0] + 0:08[u(k) − 15:6]: If pH is lower than 7:0 Then yE (k + 1) = 7:0 + 0:86[yE (k) − 7:0] + 0:01[u(k) − 15:6]: (31)

Fig. 4. Data sequences used for identi5cation and validation.

Process

y(k+1)

… U(k)

Z-1 Process dynamic model Z-1 Z-2

… Fig. 5. General structure of the dynamic process model.

yH(k+1)

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Y. Wang et al. / Fuzzy Sets and Systems 130 (2002) 265 – 275

µ 1.0

0.0 3.0

7.0

10.0

pH(k)

Fig. 6. Membership function of the EFM in simulation.

The membership function of the EFM is shown in Fig. 6. The prediction result of the EFM on the validation data sequence is shown in Fig. 9(a) while modeling error of the EFM is shown in Fig. 9(b).

Fig. 7. Membership functions of the T-S type fuzzy model in simulation.

4.2. Prediction result of the LLM We choose the local linear model (29) for comparison. The prediction result of the LLM on validation data set is shown in Fig. 9(a) while modeling error of the LLM is shown in Fig. 9(b). 4.3. Prediction result of the FM Here a T-S type fuzzy model is trained on the identi5cation data.  = 0:5,  = 1:5. 4 rules are generated after structure learning. The membership functions of the FM are shown in Fig. 7. And the trained T-S type fuzzy model is shown as following: IF pH(k) is A1 THEN yF (k + 1) = 1:5458yF (k) − 0:7331yF (k − 1) + 0:1652yF (k − 2) + 0:0103u(k) +0:0087u(k − 1) − 0:0086u(k − 2); IF pH(k) is A2 THEN yF (k + 1) = 0:4670yF (k) − 0:0806yF (k − 1) + 0:1735yF (k − 2) + 0:1501u(k) +0:0453u(k − 1) + 0:0046u(k − 2); IF pH(k) is A3 THEN yF (k + 1) = −0:6453yF (k) + 1:9018yF (k − 1)

− 1:1110yF (k − 2) + 0:6728u(k) −0:2379u(k − 1) − 0:0106u(k − 2); IF pH(k) is A4 THEN yF (k + 1) = 1:2618yF (k) − 0:5675yF (k − 1) + 0:1443yF (k − 2) + 0:1213u(k) − 0:0554u(k − 1) + 0:0191u(k − 2): After parameters identi5cation, the FM is employed to give predictions on validation data set. The prediction result of the FM on the validation data sequence is shown in Fig. 9(a) while modeling error of it is shown in Fig. 9(b). 4.4. Prediction result of the hybrid fuzzy model Now the hybrid fuzzy modeling approach will be applied to the pH neutralization process. Since the T-S type fuzzy model has already been obtained in Section 4.3, it can be used to give predictions on the validation data set directly. The membership functions of the fuzzy decision module are shown in Fig. 8. The prediction result of the hybrid fuzzy model on the validation data sequence is shown in Fig. 9(a) while modeling error of it is shown in Fig. 9(b).

Y. Wang et al. / Fuzzy Sets and Systems 130 (2002) 265 – 275 Table 2 MSE of diJerent models on validation data set

η(ϕ η ϕ F) 1

High

Low

0 0

Model

MSE

Hybrid model EFM LLM FM

0.34 1.61 3.37 5.37

ϕF

0.5

4.5. Comparison of the steady-state response of four models with pH neutralization process

η(ϕ ϕ E) 1

273

High Low

0 0

1

ϕE

Fig. 8. Membership functions of the fuzzy decision module in simulation.

The mean square error (MSE) of these four models on validation data set are also compared in Table 2. From Fig. 9 and Table 2, we can see that the hybrid fuzzy model achieves the best result.

For the dynamic process model, it is a dynamic system. If the input keeps unchanged, the output of dynamic process model will arrive at certain point if it is stable. This means there exists a static relation between the input and the steady-state value of the output. This relation is well known as steady-state response in process industry. Here we are going to examine the steady-state responses of these four models and the real process. The simulation results are compared in Fig. 10. From the above comparison, we can see that (i) In the region where 126q3 616, the hybrid fuzzy model and the FM 5t the real process

Fig. 9. (a) Comparison between the prediction result of the hybrid fuzzy model, the EFM, the LLM, the FM and the validation data set. (b) Modeling error of the hybrid fuzzy model, the EFM, the LLM and the FM on the validation data set.

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Y. Wang et al. / Fuzzy Sets and Systems 130 (2002) 265 – 275

Fig. 10. Comparison of the steady-state response of the hybrid fuzzy model, the EFM, the LLM, the FM and the real process.

quite well. The reason is that there are su/cient training data in this region. Both of them can be trained to approximate the real process perfectly. But for the EFM, some diJerences exist compared to the real process because the EFM is a rough description of the real process. (ii) In the region where q3 ¡12, the FM shows very poor performance. This is caused by sparse training data. It demonstrates the poor extrapolation ability of the FM. The EFM is better than the FM. The hybrid fuzzy model shows the same result as the EFM due to lack of training data here. (iii) In the region where q3 ¿16, the hybrid fuzzy model shows much better performance than the other two. The reason is that there are some training data in the region 166q3 618. With the help of training data and linguistic information, the extrapolation capacity of the hybrid fuzzy model is greatly improved. (iv) The LLM is quite diJerent from the real process except in the point near pH = 7:0. The reason is that the LLM is obtained at that point. However, since pH neutralization is a strong nonlinear process, as we can see from the steady-state response of it, the LLM does not have the ability to 5t

the real process in other operation points, which demonstrates the local property of the LLM. 5. Conclusion In this paper, a hybrid fuzzy model is proposed. Using fuzzy set theory, it combines the expert fuzzy model, local linear models and the compensation fuzzy model into a whole framework. With the help of the linguistic information and the default local linear model, the poor extrapolation capacity problem caused by sparse training data for conventional fuzzy models can be overcome. Future work is to design model-based controller based on the hybrid fuzzy model. References [1] S.G. Cao, N.W. Rees, G. Feng, Analysis and design for a class of complex control systems, Part I: Fuzzy modeling and identi5cation, Automatica 33 (1997) 1017–1028. [2] T.A. Johansen, Identi5cation of non-linear systems using empirical data and prior knowledge—and optimization approach, Automatica 32 (1996) 337–356.

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