Space Separation Modeling

ARTICLE pubs.acs.org/IECR

Embedded Interval Type-2 T-S Fuzzy Time/Space Separation Modeling Approach for Nonlinear Distributed Parameter System Mengling Wang,†,‡ Ning Li,‡ Shaoyuan Li,‡ and Hongbo Shi*,† †

Key Laboratory of Advanced Control and Optimization for Chemical Processes of Ministry of Education, East China University of Science and Technology, 130, Meilong Road, Shanghai 200237, China ‡ Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education, Shanghai 200240, China ABSTRACT: Challenging modeling problems include how to obtain a simple and accurate model for a partial differential equation (PDE) unknown nonlinear distributed parameter system (DPS). In this paper, a time/space separation modeling approach based on the interval type-2 T-S fuzzy model (IT2 T-S model) is proposed for nonlinear DPS. First, the spatial-temporal output is divided into a few dominant spatial basis functions and low-dimensional time series by a linear time/space separation method. Second, the interval type-2 T-S fuzzy model is determined from the low-dimensional time series to reconstruct the system dynamics through the spatial basis functions. As the IT2 T-S model employs an interval linear model in the consequent part, the relationship between the input and spatial output can be determined by linear expressions through type reduction and linear time/space reconstruction. Thus, the obtained model is suitable for control design. The simulation presents the accuracies and effectiveness of the proposed modeling methodologies.

1. INTRODUCTION Distributed parameter systems (DPSs) are systems with significant convection, diffusion, and dispersion phenomena. Packed-bed reactors, rapid thermal processing systems, and chemical vapor deposition reactors are some typical examples. These systems have strong spatial variations that the states, controls, and outputs depend on spatial position.1 It is difficult to obtain a model for control design because of the infinitedimensional, spatial-temporal nature and nonlinearities of DPSs.2,3 DPSs are generally described by sets of partial differential equations (PDEs) with boundary conditions, but they cannot be directly used for control design because these equations are infinite-dimensional. Methods such as the Galerkin method, the singular value decomposition (SVD) method, singular perturbations, and the approximate inertial manifold36 have been used to approximate DPSs to the finite-dimensional ordinary differential equations (ODEs) on the basis of finitedimensional systems that accurately describe their dynamic behavior. However, these methods require the PDE descriptions of the systems to be known. Actually, many real-world systems have highly complex and nonlinear characteristics which make it difficult to obtain accurate PDE description models. It is necessary to propose a modeling method for a PDE unknown nonlinear distributed parameter system. If only some parameters of PDEs are unknown, then these parameters can be estimated from the process data.79 In many cases, the structure and the parameters of the system could be both unknown; then the identification approaches are used for modeling DPSs based on input/output data.1012 Time/space separation method and time/space discretization method both can be used for PDE unknown DPS model identification based on input and output data.3,1319 Time/space discretization method usually assumes that the local dynamics is the same at r 2011 American Chemical Society

different spatial locations, and local models can be established based on the identification theory of a lattice dynamic system.13,16,17 They achieve good predictions for many DPSs if the spatial regions are partitioned properly. However, the model dimension may be high because it is determined by the number of local models. Time/space separation method uses spatial basis function expansion to expand the spatiotemporal variable onto an infinite number of spatial basis functions with temporal coefficients. As most DPSs, especially parabolic DPSs, have slow/fast separation properties, it can derive an accurate lowdimensional model by choosing the proper finite number of spatial basis functions.14 That is, the high-dimensional spatiotemporal data are projected into low-dimensional temporal data by finite spatial basis functions. Then a low-dimensional model can be established for dynamic modeling with the help of traditional identification technique. Some time/space separation based modeling methods have been studied before.18,19 In ref 18 the support vector machine (SVM) based time/space separation method is proposed to describe the spatiotemporal dynamic of DPS. In ref 19 neural networks with global spatial basis functions are proposed to describe the spatiotemporal dynamic of DPS. Although the linear time/space separation methods are used in these papers, the nonlinear modeling approaches afterward make these nonlinear models complex, which increases the difficulty in control design. It is crucial to propose an easily used for control design time/space separation modeling approach. Fuzzy models are successfully employed for modeling complex nonlinear dynamics compared with linear models.2022 Received: July 18, 2011 Accepted: November 11, 2011 Revised: October 13, 2011 Published: November 12, 2011 13954

dx.doi.org/10.1021/ie201556u | Ind. Eng. Chem. Res. 2011, 50, 13954–13961

Industrial & Engineering Chemistry Research

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They possess an excellent ability to describe complex systems and to approximate a nonlinear model with a given accuracy. The interval type-2 fuzzy model (IT2) is a simplified type-2 fuzzy set which includes the IT2 Mandani model and the IT2 T-S model.23,24 Compared to a type-1 fuzzy model, it has the ability to approximate nonlinear systems and to handle uncertainty. The IT2 T-S model is a kind of IT2 model which can represent each output space with an interval linear equation.25,26 For the reason that it employs an interval linear model in the consequent part, conventional control theory can be applied for it. Though, nowadays, the interval type-2 models have been used in many fields, most interval type-2 fuzzy models are used to model lumped parameter system (LPS) dynamics.2731 In ref 16 the local modeling approach based on the IT2 T-S model is proposed for a PDE unknown DPS. The local models are established by the IT2 T-S model, which can reduce the modeling error coming from measurement noise and approximation error. Considering the above aspects, in this article, an IT2 T-S based time/space separation modeling approach is proposed to describe the spatial dynamics of nonlinear DPS. Principal component analysis (PCA) is a popular linear time/space separation approach to find the spatial basis functions from the input and output data with a given accuracy. Thus, it is used to obtain the finite dominant spatial basis functions and the low-dimensional time series from spatiotemporal data. Then, the IT2 T-S model is developed to establish the low-dimensional time series. The interval type-2 fuzzy satisfactory clustering algorithm proposed in ref 16 is used to determine the premise parameters of the interval fuzzy sets without preassumption about the fuzzy rules, and the least-squares algorithm is used to determine the consequent parameter on the basis of analyzing the uncertainty of the measurement data. Finally, the spatiotemporal output can be reconstructed back by the spatial basis function. In section 2, the problem and philosophy are proposed. In section 3, the proposed time/space separation modeling method is described based on inputoutput data sets. Section 4 contains a simulation example on a catalytic rod reactor. Also in section 4, we will analyze the results and discuss the advantages of the proposed method. In section 5the conclusion is drawn.

Figure 1. Time/space separation approach for DPS.

modeling approach include how to obtain the spatial basis functions and how to identify the low-dimensional models based on spatiotemporal data. In order to obtain a simple and accurate model, the PCA method is applied to determine the finite spatial basis function ji(z) (i = 1, ..., n), and the interval type-2 fuzzy model is proposed to predict the low-dimensional time series.

3. TIME/SPACE SEPARATION MODELING APPROACH FOR A NONLINEAR DPS In this section, the time/space separation modeling approach is proposed for a nonlinear DPS based on the IT2 T-S model. 3.1. Time/Space Separation and Model Reduction. In practical DPSs, the number of actuators and sensors is finite. The system in this study 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. Thus, eq 1 has to be truncated to a finite dimension description, which is as follows: yn ðz, tÞ ¼

∞

∑

i¼1

ji ðzÞ yi ðtÞ

ð1Þ

where j(z) refers to spatial basis functions and y(t) is the corresponding time series. After time/space separation, the low-dimensional model can be established for dynamic modeling with the help of traditional identification technique. Then the spatiotemporal predictive output can be reconstructed back by the spatial basis function. Figure 1 describes the time/space separation approach. As distributed parameter systems have highly complex and nonlinear characteristics, challenging problems such as the modeling accuracy and efficiency for time/space separation

ð2Þ

The finite basis function spatial basis function among the spatiotemporal output can be obtained by solving the optimal problem min Æ||yðz, tÞ yn ðz, tÞ||2 æ ji ðzÞ s.t.

2. PROBLEM FORMULATION AND PHILOSOPHY Consider a DPS: u(z,t) is the spatiotemporal input and y(z,t) is the spatiotemporal output, where t is the temporal variable, z ∈ Ω is the spatial variable, and Ω is the spatial domain. Using the time/space separation method, the spatiotemporal data y(z,t) can be divided into the spatial basis function j(z) and the low-dimensional temporal data y(t).14 yðz, tÞ ¼

n

∑ ji ðzÞ yi ðtÞ i¼1

yn ðz, tÞ ¼

ðji ðzÞ, jj ðzÞÞ ¼ ( ¼

0, 1,

n

∑ ji ðzÞ yi ðtÞ i¼1 Z

Ω

ji ðzÞ jj ðzÞ dx

i 6¼ j j ∈ L2 ðΩ Þ, i¼j i

i ¼ 1, :::, n

ð3Þ

To solve the above optimal problem, there exists the following eigenvector/eigenvalue problem: Z

Rðz, ζÞ ji ðζÞ dζ ¼ λi ji ðzÞ, Ω i ¼ 1, :::, n

ðji , ji Þ ¼ 1, ð4Þ

where R(z,ζ) = Æy(z,t) y(ξ,t)æ is the spatial two-point correlation function. By using the method of snapshots, the spatial basis function can be expressed as a combination of the snapshots as follows:14 ji ðzÞ ¼ 13955

L

∑ t ¼1

γit yðz, tÞ

ð5Þ

dx.doi.org/10.1021/ie201556u |Ind. Eng. Chem. Res. 2011, 50, 13954–13961


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Substituting eq 5 into eq 4 gives the following equation: Z

L

L

∑

∑

1 yðz, tÞ yðξ, tÞ γik yðξ, kÞ dξ Ω L t¼1 k¼1 ¼ λi

L

∑ γit yðz, tÞ t ¼1

ð6Þ

gives the fraction of the variance retained in the n-dimensional space defined by the spatial basis functions associated with those eigenvalues. A value of more than 0.99 is often taken. When finite spatial basis function ji(z) (i = 1, ..., n) is obtained, the low-dimensional time series can be determined as follows: yi ðtÞ ¼ ðji ðzÞ, yðz, tÞÞ,

R Define Ctk = (1/L) Ω y(ξ,t) y(ξ,k) dξ; then the above eigenvalue problem can be changed into the following problem: C γ i ¼ λi γ i

ð7Þ

The solution of eq 6 yields the eigenvector where γi = [γi1, ..., γiL]T is the ith, which can be used in eq 4 to obtain the spatial basis function ji(z). The maximum number of nonzero eigenvalues is K = min(N,L). Express the eigenvalues λ1 > λ2 > ... > λK and the corresponding spatial basis function j1(z), j2(z), ..., jK(z) in the order of the magnitude of the eigenvalues. The ratio of the sum of the n largest eigenvalues to the total sum η¼

Ri :

n

K

∑ λi = i∑¼ 1 λi i¼1

ð8Þ

i ¼ 1, :::, n

3.2. Low-Dimensional Temporal Modeling. In this section, the interval type-2 T-S fuzzy model is used to evaluate the lowdimensional time series yi(t), i = 1, ..., n. Let u(t) = [u(z1,t), ..., u(zM,t)]; the input data for the model is as follows:

S ¼ ½yðtÞ, yðt 1Þ, ..., yðt dy Þ, uðtÞ, uðt 1Þ, ..., uðt du Þ ð10Þ where p = 2 + dy + du. Then the input/output relationship of the model is represented as follows.

1þd

2þd

p

if yðtÞ is A1i and yðt 1Þ is A2i ... and yðt dy Þ is Ai y and uðtÞ is Ai y ... and uðt du Þ is Ai , then yi ðt þ 1Þ ¼ a0, i þ a1, i yðtÞ þ a2, i yðt 1Þ þ ... þ a1þdy , i yðt dy Þ þ a2þdy , i uðtÞ þ ... þ ap, i uðt du Þ

where Ri is the ith fuzzy rule with each rule having p antecedents and p consequents, i = 1, ..., c. yi(t + 1) is the output value of the ith rule for the model; the antecedent membership function of the ith rule is Aii = [μil,μir]. aj,i = [alj,i,arj,i] is the consequent parameter in the ith rule. As we use interval type-2 set in the consequents, then the output of the ith rule is yi(t + 1) = [yil(t + 1),yir(t + 1)]. The final output of the interval type-2 fuzzy model is described by c

yl ðt þ 1Þ ¼

∑ μil al0i i¼1 c

∑ μil i¼1

þ

∑ μil al1i i¼1 c

∑ μil i¼1

c

∑ μil i¼1

Sj ðpÞ

c

yr ðt þ 1Þ ¼

c

∑

i¼1

μir

ð12Þ

c

þ

μir ar1i ∑ i¼1 c

∑

i¼1

μir

Sj ð1Þ þ ...

c

þ

μir arji ∑ i¼1 c

∑

i¼1

The membership of interval type-2 fuzzy set Aii = [μil,μir] is as follows:

Sj ð1Þ þ ...

c

μir ar0i ∑ i¼1

ð11Þ

c

∑ μil alji i¼1

þ

ð9Þ

μir

Sj ðpÞ

yðt þ 1Þ ¼ ðyl ðt þ 1Þ þ yr ðt þ 1ÞÞ=2

ð13Þ

ð14Þ

3.2.1. Antecedent Parameter Identification for IT2 T-S Model. The interval type-2 fuzzy c-means algorithm was proposed to define the antecedent parameters of the interval type-2 fuzzy sets.32 However, it has to select the number of fuzzy rules c only by their experience. It leads to a repetitive trial of different numbers of clusters for a satisfactory result, which greatly increases the computation efficiency on finding the proper fuzzy rules. In ref 16, the interval type-2 fuzzy satisfactory clustering algorithm is proposed, which uses the cluster validation criterion as termination conditions. It helps to adjust the relationship between the modeling accuracy and the size of the rule base, and to avoid restarting clustering with random conditions. This section presents the outline of the proposed parameter identification approach to obtain the antecedent and consequent parameters of model based on the interval type-2 fuzzy satisfactory clustering algorithm and least-squares identification algorithm. Algorithm 1: Antecedent Parameter Identification Algorithms Step 1. Load the inputoutput data set Sj (j = 1, ..., L). Step 2. Set initial cluster number c = 2 and the parameter f = f1. Step 3. Set initial partition matrix U = [μfi,j]cL, divide data set Sj into c parts, and obtain the cluster center vi = ∑Lj=1 μfi,jSj/∑Lj=1 μfi,j. 13956



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Step 4. Evaluate clustering results. If Vh e VTH, clustering is over; go to step 9. Otherwise, go to step 5. Step 5. From the data set, find a sample Sn which is the most different from all cluster centers. The dissimilarity can be calculated by n ¼ arg min n

∑

ðμni μnj Þ

1 e i, j e c i6¼ j

jΔyc j ¼ ðmax c ðyÞ minc ðyÞÞ=2

ð16Þ

and then the output part of the data cth cluster can be changed as yrj ¼ yj þ Δyc ,

ylj ¼ yj Δyc

ð17Þ

Thus, the whole input/output data Dj ¼ ½ Sj yj can be changed as two groups of input/output data: Dlj ¼ ½ Sj ylj and Drj ¼ ½ Sj yrj . Using the least-squares identification algorithm, the input and output relationships can be described as yil

¼

alr0

þ

alr1 Sj ð1Þ

þ ... þ

alrp Sj ðpÞ

yir ¼ ari0 þ ari1 Sj ð1Þ þ ... þ arip Sj ðpÞ,

ð18Þ i ¼ 1, 2, ..., c ð19Þ

The consequent parameter of the interval type-2 T-S fuzzy model is ½yil , yir ¼ ½ali0 , ari0 þ ½ali1 , ari1 Sj ð1Þ þ ... þ ½alip , arip Sj ðpÞ, i ¼ 1, 2, ..., c

Figure 2. The modeling process.

)

)

Step 6. Use new cluster centers v1, ..., vc and Sn to compute new a initial partition matrix U0, which is not the random partition matrix. Step 7. Let c = c + 1, U = U0, and go to step 2. Step 8. The final cluster number is C, and then the membership of interval type-2 fuzzy set can be counted by eq . 3.2.2. Consequent Parameter Identification for IT2 T-S Model. For the interval type-2 T-S fuzzy model, the interval type-2 set is used in the consequents; the parameters alj,i and arj,i should be identified by the input/output data. Because of the presence of uncertainties of the system, the same or very similar input may produce large differences in the output for the measurement data set. By analyzing the same or very similar input part of the data set, the scope of the output part can be determined. Then the results of the analysis integrated into the interval type-2 T-S fuzzy model consequent parts can completely reflect the actual situation of the linear relation between input and output. According to Algorithm 1, the whole input/output data set was divided into c clusters. For each cluster, set Si Sj e δ to find k group input data, k > 1. Si and Sj are the input data belonging to the cth cluster; δ is identified according to the range of input data. Let maxc(y) and minc(y) refer to the maximum and minimum output parts of the data cth cluster. Let

ð20Þ

Algorithm 2: Consequent Parameter Identification Algorithms Step 1. Obtain the cluster number using Algorithm 1. Step 2. Analyze the input part of the data set, and determine the scope of the output part.

Step 3. Using eqs 16 and 17, the other two groups of input/ output data Dlj ¼ ½ Sj ylj and Drj ¼ ½ Sj yrj can be obtained. Step 4. For data sets Dlj ¼ ½ Sj ylj and Drj ¼ ½ Sj yrj , the parameters of eq 20 can be determined by leastsquares identification algorithm. 3.3. Time/Space Reconstruction. After the low-dimensional model was established by the interval type-2 fuzzy T-S model, the spatiotemporal predictive output can be reconstructed back by the spatial basis function: ^yn ðz, tÞ ¼

n

∑ ji ðzÞ ^yi ðtÞ i¼1

ð21Þ

The overall flowchart for the modeling algorithm is shown in Figure 2.

4. CASE STUDY: A CATALYTIC ROD REACTOR In this section, the modeling approach described in section 3 is applied and evaluated by a simulation study of a catalytic rod reactor.1 4.1. System Description. Figure 3 shows a long thin rod in a reactor which is a typical transport-reaction process in the chemical industry.1 The reactor is fed with pure species A, and a zero-order exothermic catalytic reaction of the form A f B takes place in the rod. A cooling medium that is in contact with the rod is used for cooling. Under the assumptions of constant density and heat capacity of the rod, constant conductivity of the rod, constant temperature at both sides of the rod, and excess of species A in the furnace, the mathematical model, which describes the spatiotemporal evolution of the rod temperature, has the following 13957



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Figure 3. A catalytic rod.

Figure 5. The output database.

Figure 4. The input signal u1(t).

parabolic PDE:1 ∂yðz, tÞ ∂2 yðz, tÞ ¼ þ ∂t ∂z2 þ

βT ðe γ=ð1 þ yÞ e γ Þ T

βu ðbðzÞ uðtÞ yðz, tÞÞ

Figure 6. The spatial basis function.

ð22Þ

subject to the boundary and initial conditions: yð0, tÞ ¼ 0,

yðπ, tÞ ¼ 0,

yðz, 0Þ ¼ sinðzÞ ð23Þ

where y(z,t), u(t), b(z), βT, βu, and γ denote the temperature in the reactor, the manipulated input (temperature of the cooling medium), the actuator distribution, the heat of reaction, the heat transfer coefficient, and the activation energy. The following values are given to the process parameters: βT ¼ 50,

βu ¼ 2,

γ¼4

ð24Þ

Suppose there are four control actuators u(t) = [u1(t), ..., u4(t)]T with the spatial distribution function b(z) = [b1(z), ..., b4(z)]T, bi(z) = H(z (i 1)(π/4)) H(z i(π/4)) (i = 1, ..., 4), and H( 3 ) is the standard Heaviside function. 4.2. Time/Space Separation Based IT2 T-S Modeling. The inputoutput database was developed by solving eq 22 by the finite differences method. The spatial domain is discretized in a number of intervals of equal length Δz, and time is advanced by constant time steps Δt. It was solved using Δt = 0.0004 and Δz = π/120 until t = 2, and the temporal input is as follows: ui ðjÞ ¼ 1:1 þ ð4 þ 2randÞ expð i=5Þ sinðt=ðj=14Þ þ randÞ 0:4 expð i=20Þ sinðt=ðj=2Þ þ 2randÞ i ¼ 1, ..., 4;

j ¼ 1, ..., t=Δt

ð25Þ

where “rand” is a uniform distributed random function on [0, 1]. In this paper, we suppose that there are 20 sensors uniformly distributed in the space domain and choose 200 data uniformly distributed in the time domain. The 200 input signals u1(t) are shown in Figure 4. Figure 5 describes the 20 200 output database for modeling. The dimension of the temporal model is n = 4 using the PCA method. Figure 6 describes the dominant spatial basis functions. Four different models;the type-1 T-S fuzzy model, ARX model, IT2 T-S model (1) (IT2-GK T-S model proposed in this paper), and IT2 T-S model (2) (IT2-heuristic T-S model proposed in ref 33);are trained to predict the system output. The relative L2-norm error (RLNE) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi,sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RLNEðtÞ ¼

N

eðzi , tÞ2 ∑ i¼1

N

yðzi , tÞ2 ∑ i¼1

ð26Þ

is used to compare the four modeling results. Set initial cluster number c = 2 and fuzzifier parameter f1 = 2, f2 = 1.8. According to Algorithm 1 and Algorithm 2, the satisfactory cluster number, antecedent parameters, and consequent parameters can be obtained. For a fair comparison, the same number of rules is used for other the IT2 model and the type-1 fuzzy T-S model. For the IT2-heuristic T-S model, the basic type-1 fuzzy membership function is obtained according to steps 18 of Algorithm 1, the upper membership function is equal to the type-1 membership function, and the lower membership function is obtained by scaling the upper membership function by a factor α = 0.95; the upper consequent parameter 13958



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Figure 10. Global predicted error. Figure 7. Time series y3(t).

Figure 11. The relative L2-norm error. Figure 8. Time series y4(t).

Table 1. Comparison of Modeling Performances in RMSE Using Four Algorithms

Figure 9. System predicted output.

is equal to the type-1 T-S model, and the lower consequent parameter is obtained by scaling the upper membership function by a factor β = 0.98. Using function ARX in Matlab, the time/ space separation based ARX model can be obtained. The root-mean-square error (RMSE) is used to evaluate the accuracy of the model sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N T eðzi , tÞ2 RMSE ¼ NT i ¼ 1 t ¼ 1

∑∑

ð27Þ

where N and T refer to the number of sensors and the length of time, respectively.

algorithm

RMSE

IT2 T-S model (1)

0.018

IT2 T-S model (2)

0.0191

type-1 fuzzy model

0.023

ARX model

0.0312

The reduced dimensional time series y1(t), y2(t), y3 (t), and y4(t) are computed for training the four different models. The measured and predicted temporal coefficients on the training data for y 3(t) and y4(t) are plotted in Figures 7 and 8. The system output and spatial error of the proposed method are shown in Figures 9 and 10, respectively. The relative L2-norm errors for the four algorithms are shown in Figure 11. Table 1 represents the modeling errors of spatiotemporal predictive output for the four different modeling approaches. Table 2 is the running time for the four algorithms. From Table 1, Table 2, and Figures 711, the good prediction capability of the developed time/space separation based interval type-2 T-S fuzzy model is clearly visible. Because the antecedent parameter and consequent parameter of interval type-2 fuzzy sets are interval type, it needs more running time. The performance is better than that of the ARX model and type-1 fuzzy T-S model. Moreover, it shows that, due to the different parameter identification algorithm, the derived interval type-2 T-S fuzzy model performance is better, too. In the sense of accuracy, the proposed method has its practical use in 13959


Industrial & Engineering Chemistry Research Table 2. Comparison of Running Times Using Four Algorithms algorithm

time (s)

IT2 T-S model (1)

63.23

IT2 T-S model (2) type-1 fuzzy model

66.49 52.17

ARX model

37.74

this study, and generally, the proposed method in this paper can produce effectiveness and robustness in DPS prediction.

5. CONCLUSION A new spatiotemporal modeling approach based on interval type-2 fuzzy sets is proposed for nonlinear DPSs based on input/ output data in this article. At first, PCA method is used for time/ space separation and dimension reduction, which derives a small number of dominant spatial basis functions with lowdimensional time series. Then a low-dimensional model can be established based on the interval type-2 T-S fuzzy set, where the parameter identification algorithm is proposed to determine the interval type-2 T-S fuzzy model. Because of the interval linear model in the consequent part, the resulting model is suitable for DPS control. The proposed approach achieves a good modeling performance demonstrated by a simulation case study. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT This work was supported by the National Nature Science Foundation of China (Nos. 60825302, 61074079, 61074061), the High Technology Research and Development Program of China (No.2007AA041403), the Program of Shanghai Subject Chief Scientist, and ‘Shu Guang’ Project of Shanghai Municipal Education Commission and Shanghai Education Development Foundation. ’ REFERENCES (1) Christofides, P. D. Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes; Birkh€auser: Boston, 2001. (2) Armaou, A.; Christofides, P. D. Dynamic optimization of dissipative PDE systems using nonlinear order reduction. Chem. Eng. Sci. 2002, 57 (24), 5083–5114. (3) Baker, J.; Christofides, P. D. Finite-dimensional approximation and control of nonlinear parabolic PDE systems. Int. J. Control 2000, 73 (5), 439–456. (4) Aggelogiannaki, E.; Sarimveis, H. Robust nonlinear H∞ control of hyperbolic distributed parameter systems. Control Eng. Pract. 2009, 17 (6), 723–732. (5) King, E. B.; Hovakimyan, N.; Evans, K A.; Buhl, M. Reduced order controllers for distributed parameter systems: LQG balanced truncation and an adaptive approach. Math. Comput. Modell. 2006, 43 (910), 1136–1149. (6) Christofides, P. D.; Daoutidis, P. Finite-Dimensional Control of Parabolic PDE Systems Using Approximate Inertial Manifolds. J. Math. Anal. Appl. 1997, 216 (2), 398–420.

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