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Local Modeling Approach for Spatially Distributed System Based on Interval Type-2 T-S Fuzzy Sets Mengling Wang, Ning Li, and Shaoyuan Li* Institute of Automation, Shanghai Jiao Tong UniVersity, Shanghai 200240, China
It is important to choose the appropriate type of local model to reduce the model error for a spatially distributed system (SDS) under large uncertainties. In this paper, a new local modeling approach is proposed to estimate the spatial dynamics of SDS based on input output data. The interval type-2 T-S fuzzy model is developed to the local dynamics in consideration of the mutual influence of neighbor regions. The parameters and the proper fuzzy rules of the local models are obtained by using interval type-2 fuzzy satisfactory clustering algorithm. Then the global models can be determined by constructing the local models by smooth interpolation. The accuracies of the modeling methodologies are tested in the shell and tube heat exchanger. 1. Introduction Flexible manipulator, fluid flow process, thermal process, and convection diffusion reaction process are some typical examples of spatially distributed system (SDS).These systems have strong spatial variations that the states, controls and outputs depend on spatial position.1 These systems are generally described by sets of partial differential equations (PDE) with boundary conditions. It cannot be directly used for control design because these equations are infinitedimensional. Methods such as finite differences, Galerkin method, and approximate inertial manifold et al.2-4 have been used to approximate systems to a finite-dimensional ordinary differential equation (ODE). However, these methods require the PDE descriptions of the systems to be known. Actually, many real-world systems have highly complex and nonlinear characteristics that make it difficult to obtain accurate PDE description models. In many cases, the structure and the parameters of the system could be both unknown. The identification of the system is only based on the information from finite sensors. It is necessary to propose a modeling method on the basis of input and output data sets. Today, research about the spatiotemporal modeling, based on input and output data sets, mainly concentrates on local and global approaches.5 For the global method, an infinite number of basis functions can be found to represent the spatial frequencies of the system.6-8 Finite element bases, Fourier series, and Jacobi polynomials are popular approaches to find the basis functions from the data. In ref 9, a nonlinear principal component analysis network (NL-PCA) is designed for the nonlinear dimension reduction; then a nonlinear dimension reduction-based low-order neural model is produced for nonlinear distributed parameter processes. But it is difficult to choose the proper basis functions and the accuracy of modeling is effect directly by the basis functions.6,10-11 For the local approach, it is assumed that the local dynamics is the same and can be obtained by the neighborhood spatiotemporal regions.12-13 Utilizing the measurements at the small spatiotemporal regions, local model can be established.14-15 The global models can be determined by constructing the local models whose dynamics is governed by local action. Applying the approach, the error * To whom correspondence should be addressed. E-mail:
[email protected].
in reconstructing a spatial-temporal model mainly comes from measurement noise and approximation error.16 It makes the modeling circumstance with large uncertainties. Until now, there are no mature local modeling methods to construct an accurate model under large uncertainties for SDSs. To reduce the model error for a SDS under uncertainties, selection of the type of local model and construction of the local models are covered in this paper. To obtain the system model, since the early 1980s, modeling approaches have been employed, including linear regression, autoregressive moving average (ARMA), etc.17,18 Because most industrial processes, especially spatially distributed parameter processes, have highly complex and nonlinear characteristics, the linear modeling method can not well represent the nonlinear characteristics of the processes. Lately, the extraordinary development of many computational intelligence (CI) techniques including fuzzy systems, neural networks, and machine learning has shown that they have great potential application in modeling because of their strong ability to capture the nondeterministic and complex nonlinearity of time series.19-23 In refs 20 and 21, the neural network model based only on process input-output data is developed for the identification of distributed parameter systems, which is used for estimation of the control variable at multiple spatial locations. Fuzzy models is successfully employed for modeling complex nonlinear dynamics compared with linear models.24-26 The priori knowledge about the process can be more easily integrated into the fuzzy model. As we known, the systems in practical industrial processes have nonlinear characteristics under large uncertainty. It makes the modeling problem more difficult. The type-2 fuzzy model was introduced by Zadeh in 1985.27 The membership functions in type-2 fuzzy sets include primary and secondary membership functions. The secondary membership function describes the degrees of fuzzy of primary membership function. The type-2 fuzzy set can be uses when the circumstances are too uncertain to determine exact membership grades such as when the training data is corrupted by unmeasurable noise.27-29 It can model the uncertainties brought from the circumstances and has a good robustness compared with other nonlinear models. Interval type-2 fuzzy set is a simplified type-2 fuzzy set.30 The secondary memberships are equal to one. Compared to
10.1021/ie901278r 2010 American Chemical Society Published on Web 04/05/2010
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Figure 1. Local modeling approach for SDS.
general type-2 fuzzy sets, the interval type-2 fuzzy sets can reduce the computational complexity in type-reduction and have the ability to handle uncertainty. Considering the above aspects, we apply the interval type-2 fuzzy model to build local models instead of linear model and other nonlinear model. In this paper, a new local modeling approach is proposed to describe the spatial dynamics of SDS, in which the hybrid model composed of the finite local interval type-2 fuzzy models. In consideration of the mutual influence of neighbor regions, the local model is constructed which considers the influence of neighbor regions. The interval type-2 fuzzy satisfactory clustering algorithm is proposed to determine a proper number of clusters and define the parameters of the interval fuzzy sets. Bcause a traditional clustering algorithm has to select the number of fuzzy rules only by their experience,33 it leads to a repetitive trial of different number of clusters for a satisfactory result. The identification method in the paper can reduce computational burden on finding the proper fuzzy rules. Then the global models can be determined by constructing the local models by spline interpolation. In the following section, the problem and philosophy is proposed. In the third section, the proposed interval type-2 T-S fuzzy model is described based on input-output data sets. Section 4 will be the part where we conduct a heat exchanger system.31,32,36 In the section, we will analyze the results and discuss the advantages of the proposed method. In the last section, the conclusion is drawn. 2. Problem Formulation and Philosophy Many spatially distributed systems, such as industrial chemical reactor, fluid flow process, and thermal processing, exhibit strong spatial variation. To achieve the spatial control objectives, there are finite spatial sensors and actuators that are used to measure the state and control the processes. And the location and the number of them may be different. Suppose that U(f(x,y,z),t) is the spatiotemporal input and Y(f(x,y,z),t) is the spatiotemporal output, where f(x,y,z) is the spatial variable which represents the location of sensors and actuators, and t is the temporal variable. Applying the local modeling approach, it is assumed that the local dynamics is the same and can be obtained by the neighborhood spatiotemporal regions. Utilizing the measurements at the small local regions, local models can be developed based on the identification theory of lattice dynamical system. The global models can be determined by constructing the local models through information fusion. Figure 1 describes the modeling process by local approach. In this paper, the system is controlled by the finite actuators with temporal signal u(z,t) at certain spatial locations z1, ..., zM.
The output is measured at the N spatial locations z1, ..., zN. Suppose that the structure and the parameters of the process are both unknown. Let x(zm,t) be a spatiotemporal output, where zm (m ) 1, ..., N) represents spatial location and x(zm,t) is the state variable at spatial location zm and time t ) 1, ..., L. The data are discrete in the N spatial locations z1, ..., zN. Using the local modeling approach, the global model can be obtained by fusing the N local models. And the challenging problem for local modeling approach includes how to choose a proper local model to reduce the error in reconstructing a spatial-temporal model and how to define the construction of the local models. The interval type-2 fuzzy set can be used when the circumstances are too uncertain to determine exact membership grades, such as when the training data is corrupted by noise. In this paper, the interval type-2 fuzzy model is used to identify the local model to reduce the computational complexity and have the ability to handle uncertainty from measurement noise and approximation error. Meanwhile, in consideration of the mutual influence of neighbor regions, a local model is proposed that not only considers the variation of local regions but also considers the influence of neighbor regions. Then the local models are combined by smooth interpolation into a complete global model. 3. Interval Type-2 Fuzzy Modeling for SDS Interval type-2 fuzzy set is a kind of type-2 fuzzy sets. The secondary memberships are equal to one. Compared to general type-2 fuzzy sets, it can reduce the computational complexity and have the ability to handle uncertainty. In this paper, the interval type-2 fuzzy model is used to identify the local model based on input and output data sets. The parameters of the interval type-2 fuzzy model are obtained by using interval type-2 fuzzy satisfactory clustering algorithm. Then the global model is proposed for SDS control problems where the PDE description of system is unknown through combining the local models. 3.1. Local Modeling Based on Interval Type-2 Fuzzy Sets. Suppose there are N sensors located at N invariable spatial location and M actuators located at M invariable spatial location and that the location of them may be different. The whole spatial regions are divided into N local regions spatial uniformly corresponding to the number and location of the sensors. The data are collected using an appropriate sampling interval t. The input output data set for each local region can be written as Sm(tj) ) [x(zm, tj), x(zm1, tj), ..., x(zmq, tj), u1(z, tj), ..., uM(z, tj)] (1) where Sm(tj) represents the spatial input and output data set for the m-th local model at discrete time tj, zm is the spatial location of m-th sensor, x(zm,t) is a spatiotemporal state of the m-th local
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model, x(zm1,t), ..., x(zmq,t) are the spatiotemporal state of q neighbor models for the m-th local model, u1(z,t), ..., uM(z,t) are the M spatiotemporal manipulated variables, tj is the discrete time j ) 1, ..., L and m ) 1, ..., N. Then N interval type-2 T-S fuzzy models are used to evaluate the local spatiotemporal dynamics. And the i-th fuzzy rules of the N local models are represented as follows
c
xr(zm, t + 1) )
∑
c
r µrmiam,0,i
i)1
+
c
∑ i)1
µrmi
∑µ
mi r r am,1,i
i)1
Sj(1) + ... +
c
∑
µrmi
i)1 c
∑µ
mi r r am,j,i
i)1
Sj(1 + q + M)(4)
c
F1 f Ri: if x(z1, t) is A1i 1 and x(z11, t) is A1i 2 ... and x(z1q, t) isA1i 1+q and u1(z, t)isA1i 2+q ... and uM(z, t) is A1i 1+q+M, then xi(z1, t + 1) ) a1,0,i + a1,1,ix(z1, t) + a1,2,ix(z11, t) + ... +a1,1+q,ix(z1q, t) + a1,2+q,iu1(z, t) + ... + a1,1+q+M,iuM(z1, t) l l l Fm f Ri: if x(zm, t) is Ami 1 and x(zm1, t) is Ami 2 ... and x(zmq, t) isAmi 1+q and u1(z, t) is Ami 2+q ... and uM(z, t) is Ami 1+q+M, then x (zm, t + 1) ) am,0,i + am,1,ix(zm, t) + am,2,ix(zm1, t) + ... +am,1+q,ix(zmq, t) + am,2+q,iu1(z, t) + ... + am,1+q+M,iuM(z, t) l l l FN f Ri: if x(zN, t) is ANi 1 and x(zN1, t) isANi 2 ... i
and x(zNq, t) is ANi 1+q and u1(z, t) is ANi 2+q ... and uM(z, t) is ANi 1+q+M, then xi(zN, t + 1) ) aN,0,i + aN,1,ix(zN, t) + aN,2,ix(zN1, t) + ... + aN,1+q,ix(zNq, t) + aN,2+q,iu1(z, t) + ... + aN,1+q+M,iuM(z, t) (2) where Fm f Ri is the i-th fuzzy rule of the m-th local model with each rule having 1 + q + M antecedents and 1 + q + M consequents, i ) 1, ..., c and m ) 1, ..., N. xi(zm,t) is output value of the i-th rule for the m-th local model, The antecedent membership function of the i-th rule for the m-th local model is Aimi ) [µlmi,µrmi], and am,j,i ) [am,j,il,am,j,ir] is the consequent parameter in the i-th rule. As we use interval type-2 set in the consequents, then the output of i-th rule for the m-th model is xi(zm,t + 1) ) [xli(zm,t + 1), xri(zm,t + 1), The final output of the m-th interval type-2 fuzzy model is described by c
xl(zm, t + 1) )
∑
c
l µml iam,0,i
i)1
∑ i)1
mi l l am,1,i
+
c
µml i
∑µ i)1
Sj(1) + ... +
c
∑
µml i
i)1 c
∑µ
mi l l am,j,i
i)1
Sj(1 + q + M)(3)
c
∑ i)1
µml i
∑
µrmi
i)1
x(zm, t + 1) ) (xl(zm, t + 1) + xr(zm, t + 1))/2 where µi ) Ami 1 × Ami 2 × ... × Ami 3
(5)
3.2. Parameter Identification. In this paper, the interval Type-2 satisfactory clustering algorithm is used to get the number of rules of this system and helps to determine the parameters of the model. The interval type-2 fuzzy c-means algorithm was proposed to define the parameters of the interval fuzzy sets.34 However, it has to select the number of fuzzy rules c only by their experience. It leads to a repetitive trial of different number of clusters for a satisfactory result, which greatly lowers the computation efficiency on finding the proper fuzzy rules. Therefore a T-S modeling method based on the interval type-2 fuzzy satisfactory clustering algorithm is proposed to determine a proper number of clusters and define the parameters of the interval fuzzy sets. The purpose is to get an interval type-2 T-S model with proper number of rules, lower computational burden. Simply speaking, the modeling method begins with the cluster c ) 2 and the fuzzifer f ) f1. According to the validation criterion, determine whether a new cluster center should be added or not. If the clustering is not satisfied yet, we consider that the division is not proper. Then from the given data set, find a sample that is the most different from the existing cluster centers V1V˜ c as a new center Wc+1. With V1V˜ c+1 as initial cluster centers, compute a new not-random initial partition matrix U0. Then repeat the GK algorithm until a satisfactory result is obtained. As the cluster numbers are determined, set the f ) f2, the antecedent membership functions are determined by this approach. It avoids restarting GK with random conditions by fully using the clustering information. Therefore, the convergence speed is greatly improved. For simulation purposes, in this paper, the required input-output data set is obtained by solving PDEs in the finite differences method. The multiple interval type-2 T-S fuzzy models are structured so that it can approximate the distributed parameter system and estimate the dynamic behavior of the process. Algorithm 1: T-S identification based on satisfactory clustering algorithms. Step 1: Load the input-output data set Sj (j ) 1, ..., L) and set the model number M ) i (i ) 1, ..., N). Step 2: Set initial cluster number c ) 2, and the parameter f ) f1. Step 3: Set initial partition matrix U ) [µi,jf]c×L, divide data set Sj into c parts, and obtain the cluster center Vi ) ∑Lj)1µi,jfSj/∑Lj)1µi,jf, where µi,jf represents the memberships for the data across each cluster. Step 4: Count the cluster covariance matrix Fi ) n n (∑j)1 µi,jm(Sj - Vi)(Sj - Vi)T)/(∑j)1 µi,jm) and matrix (1)/(d+1) -1 Mi ) det (Fi) Fi . The distance between dj
and the cluster center Vi can be obtained by d (dj,Vi) ) (dj - Vi)TMi(dj - Vi), and the memberships can be obtained by 2
d2(dj, Vi)-1/(f-1)
µi,j )
c
∑ d (d , V )
-1/(f-1)
2
j
l
l)1
Step 5: Use Vh ) ∑i)1c[det (Fi)]1/2 to evaluate clustering results. If Vh e VTH, clustering is over, go to step 9, Otherwise, go to Step 6. Step 6: From data set, find a sample Sn, which is most different from all cluster centers. The dissimilarity can be calculated by35
∑
n ) arg min n
(µni - µnj)
(6)
1ei,jec i*j
Step 7: Use new cluster centers V1, ..., Vc, Sn to compute new a initial partition matrix U0, which is not the random partition matrix. Step 8: Let c ) c + 1, U ) U0, and go to Step 2. Step 9: The final cluster number is C. Set f ) f2, then the membership of interval type-2 fuzzy set Aimi ) [µlmi,µrmi] is
{
µlmi ) d2(Sj, Vi)-1/(f1-1)
d2(Sj, Vi)-1/(f1-1)
c
c
∑ d (S , V ) 2
j
-1/(f1-1)
l
if
∑ d (S , V )
-1/(f1-1)
2
j
l)1
l)1
d2(zj, Vi)-1/(f2-1) c
∑ d (S , V ) 2
j
-1/(f2-1)
otherwise
l
l)1
Figure 2. Modeling process.
l
>
d2(Sj, Vi)-1/(f2-1) c
∑ d (S , V )
-1/(f2-1)
2
j
l)1
{
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µr ) mi
d2(Sj, Vi)-1/(f1-1)
d2(Sj, Vi)-1/(f1-1)
c
c
∑ d (S , V )
-1/(f1-1)
2
j
l
if
∑ d (S , V ) 2
j
l)1 2
e
-1/(f1-1)
l
l)1
d2(Sj, Vi)-1/(f2-1) c
∑ d (S , V ) j
d (Sj, Vi)-1/(f2-1)
c
∑ d (S , V )
-1/(f2-1)
2
j
-1/(f2-1)
2
l)1
l
(8)
otherwise
l
l)1
Step 10: The consequent membership function is interval type too. x(zm,t + 1) is the output temperature value at sensor m, and the interval type at sensor m is xi(zm, t + 1) ) [xi(zm, t + 1) - ∆m-1, xi(zm, t + 1) + ∆m] where ∆m-1 ) |x(zm, t + 1) - x(zm-1, t + 1)|/2∆z ) |x(zm+1, t + 1) - x(zm, t + 1)|/2
(9)
Then the consequent parameter amji ) [amjil,amjir] can be obtained through orthogonal least-squared algorithm. The system output is also interval type. The system output for the m-th model is described in eq 5. If the model number is set for 1 to N, the N local models can be obtained by algorithm 1. 3.3. Spatial Reconstruction. For the local approach, it is assumed that the local dynamics is the same and the global models can be determined by constructing the local models. The spatiotemporal data of the whole space can be written as y(z, t) ) f(φ(z), xm(t)), m ) 1, ..., N
(10)
where f(•) represents the smooth interpolation function, φ(z) represents the space basis function of the sensors, and xi(t) refers to the output of the m-th local model. In this paper, we use spline interpolation to reconstruct the global model. 3.4. Generalized Modeling Procedure. The overall flowchart for the modeling algorithm is shown in Figure 2.
l
(7)
4. Case Study: Cross-Flow Heat Exchanger In this section, the local modeling approach based on data driven interval type-2 fuzzy model described in the previous
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Figure 3. Steam heated heat exchanger.
section is applied and evaluated by a simulation study of a crossflow heat exchanger.36 4.1. System Description. Figure 3 presents a steam-heated shell and tube heat exchanger that will be analyzed in the sequel. A fluid enters the heat exchanger at temperature T0 and is heated from the shell side by condensing steam at temperature Tst. The heat transfer coefficient and the fluid properties are considered constant. Applying the energy conservation law in this system while neglecting the diffusive phenomena, the following firstorder hyperbolic partial differential equation36 predicts the variation of the fluid temperature T(z,t) both with time and position ∂T(z, t) ∂T(z, t) + a(Tst(t) - T(z, t)) ) -V ∂t ∂z
Figure 4. Manipulated variable Tst(t).
(11)
The boundary conditions are as follows: T(0, t) ) T0(t), z ) 0 T(z, 0) ) Tss(z), t ) 0 Tss(z) ) Tst + (T0 - Tst) exp(-az/V) a ) hSw /FVCp
(12)
Figure 5. Spatial system output without noise.
We define system input output relation as -1
The values of process parameters are V ) 1 ms , L ) 1 m, a ) 2.92 s-1. In the above equations, the steam temperature Tst(t) is manipulated variable, while the fluid temperature is the controlled variable. The process of this system is shown in Figure 3. 4.2. Development of Interval Type-2 T-S Fuzzy Model. The input-output database was developed by solving eq 11. The method of finite differences is used in this paper. To obtain an initial temperature steady state distribution, eq 11 was solved using ∆x ) 0.01 and ∆t ) 0.001 until t ) 10. Figure 4 shows the manipulated variable Tst(t), and Figure 5 shows the spatial input output data set without adding noise. In actual industrial processes, the measurement data always contain noise and the noise is unmeasureble. It makes the modeling circumstance with large uncertainties. In order to prove the algorithm can reduce the error in reconstructing a spatialtemporal model mainly comes from measurement noise, the real data adding the color noise is used for modeling, while the real data is used for comparison with the modeling output. We add the color noise creating by Markovian chain. The Markovian chain is as follows: V ) [-0.57, 0.38, 0.73] V(k) ) 0.38, p(V(k + 1) ) 0.38) ) 0.3, when p(V(k + 1) ) -0.57) ) 0.3, p(V(k + 1) ) 0.73) ) 0.4 V(k) ) -0.57, p(V(k + 1) ) 0.38) ) 0.5, when p(V(k + 1) ) -0.57) ) 0.3, p(V(k + 1) ) 0.73) ) 0.2 V(k) ) 0.73, p(V(k + 1) ) 0.38) ) 0.6, when p(V(k + 1) ) -0.57) ) 0.1, p(V(k + 1) ) 0.73) ) 0.3
let
Tz(t + 1) ) f(Tz(t), Tz-1(t), Tz+1(t), Ts(t), Tz(t - 1), Tz-1(t - 1), Tz+1(t - 1), Ts(t - 1))(13) where z represents the z-th sensor, z ) 1, ..., N. Color noise is added to the output at each sensor. For the first model, the input output relation is that T1(t + 1) ) f(T1(t), T2(t), Ts(t), T1(t - 1), T2(t - 1), Ts(t - 1)) For the last model, the input output relation is that TN(t + 1) ) f(TN(t), TN-1(t), Ts(t), TN(t - 1), TN-1(t - 1), Ts(t - 1)) So far, there is no mature method to determine the number of sensors. In general, the number of sensor required in the modeling is determined by the system complexity, modeling accuracy, physical consideration, and cost. With more sensors, the model will be more accurate. In this paper, we suppose that the number and the location of sensors are defined previously, and we choose two different system sensor configurations for comparison, that is, there are ten sensors which located at different places. It is also compared with the type-1 fuzzy model and ARX method. The two different cases are as follows: Case 1: Z ) [0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ] Case 2: Z ) [0.15 0.2 0.35 0.4 0.55 0.6 0.75 0.8 0.95 1 ]
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Table 1. Fuzzy Rules for Each Local Model local models N ) 1 N ) 2 N ) 3 N ) 4 N ) 5 N ) 6 N ) 7 N ) 8 N ) 9 N ) 10 case 1 case 2
4 3
2 3
2 3
2 3
2 3
3 2
3 2
2 3
3 2
3 2
If the initial cluster number is set to c ) 2 and the fuzzifier parameter is set to f1 ) 2, a satisfactory cluster number for each model using the method proposed in Agorithm1 from step 1 to step 8 can be obtained. Table 1 describes the fuzzy rules of each local model. If f2 is then changed to 1.8, using eq 7 and eq 8, the antecedent parameter of interval type-2 fuzzy set can be obtained. The consequent parameter of interval type-2 fuzzy set can be obtained through orthogonal least-squared algorithm using eq 9. The input-output relation for ARX model is eq 13 too. Using function ARX in Matlab, the ARX local models can be obtained. After we obtain the ten local models, the global model can be reconstructed by using spline interpolation. In case 1, Figures 6 and 7 represent the color noise and the spatial input output data adding the noise, respectively. Figures 8-10 are the global spatial modeling errors using interval type-2 fuzzy model, type-1 fuzzy model and ARX model respectively. In the second cases, the color noise is the same as case 1. Figure 11 represents the spatial input-output data adding the noise. Figures 12-14 are the errors between the whole spatial output and the global model output of interval type-2 fuzzy model, type-1 fuzzy model, and ARX model, respectively. Table 2 represents the modeling errors of global model for the two cases. Table 3 is the running time for three algorithms in two cases. From Table 2, Table 3, and the above figures, we can find that the local modeling approach based on interval type-2 fuzzy sets proposed in this paper can well describe the whole spatiotemporal dynamics of the system. The performance is better than type-1 fuzzy model and ARX model in these two cases. As the antecedent parameter and consequent parameter
Figure 8. Case 1: Global modeling error of interval type-2 fuzzy.
Figure 9. Case 1: Global modeling error of type-1 fuzzy.
Figure 10. Case 1: Global modeling error of ARX model.
Figure 6. Color noise creation by Markovian chain.
Figure 11. Case 2: Input-output data sets with added noise.
Figure 7. Case 1: Input-output data sets with added noise.
of interval type-2 fuzzy sets are interval type, it needs more running time than other two modeling algorithm. Meanwhile, it is shown that the location of sensors affect the accuracy of modeling a little bit. In case 1, the location of the sensor is distributed uniform. The global modeling errors have small variation using spline interpolation. In case 2, the global modeling errors have large variation than case 1.
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type-2 fuzzy satisfactory clustering algorithm. Then the global models can be determined by constructing the local models by smooth interpolation. The proposed approach achieves a good modeling performance demonstrated by a simulation case study of a shell and tube heat exchanger. Acknowledgment
Figure 12. Case 2: Global modeling error of interval type-2 fuzzy.
This work was supported by the National Nature Science Foundation of China under Grant 60825302, 60774015, the High Technology Research and Development Program of China (Grant: 2007AA041403), and Sponsored by the Program of Shanghai Subject Chief Scientist, and “Shu Guang” project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation. Literature Cited
Figure 13. Case 2: Global modeling error of type-1 fuzzy.
Figure 14. Case 2: Global modeling error of ARX model. Table 2. Comparison of Modeling Performance in RMSE Using Three Algorithms algorithm
case 1
case 2
interval type-2 fuzzy model type-1 fuzzy model ARX model
0.1651 0.2884 0.3654
0.1732 0.2915 0.3676
Table 3. Comparison of the Running Time Using Three Algorithms algorithm
case 1 (s)
case 2 (s)
interval type-2 fuzzy model type-1 fuzzy model ARX model
95 76 36
99 71 37
5. Conclusion A new spatiotemporal modeling approach based on interval type-2 fuzzy sets is proposed in this paper. It can estimate the spatial dynamics of spatially distributed system and reduce the error in reconstructing a spatial-temporal model mainly. The interval type-2 T-S fuzzy model is developed to the local dynamics in consideration of the mutual influence of neighbor regions. The parameters and the proper fuzzy rules of the interval type-2 fuzzy models are obtained by using interval
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ReceiVed for reView August 13, 2009 ReVised manuscript receiVed March 7, 2010 Accepted March 13, 2010 IE901278R
