Spatially Constrained Fuzzy-Clustering-Based Sensor Placement for ...

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 18, NO. 5, OCTOBER 2010

Spatially Constrained Fuzzy-Clustering-Based Sensor Placement for Spatiotemporal Fuzzy-Control System Xian-Xia Zhang, Member, IEEE, Han-Xiong Li, Senior Member, IEEE, and Chen-Kun Qi, Member, IEEE

Abstract—Many industrial processes are spatiotemporal dynamic systems. A three-dimensional fuzzy-logic controller (3-D FLC) has been recently developed to process the inherent capability of spatiotemporal dynamic systems. Sensor placement, which is always crucial to the control of spatiotemporal dynamic systems, is also critical to the design of the 3-D FLC. In this paper, a new sensor-placement strategy is developed. Its main feature is to position the sensor by utilizing the main characteristics of spatial distribution. The key technique is to use a spatial-constrained fuzzy c-means algorithm to extract the characteristics of spatial distribution. For an easy implementation, a systematic sensor-placement design scheme in four steps (i.e., data collection, dimension reduction, data clustering, and sensor locating) is developed. Finally, control of a catalytic packed-bed reactor is taken as an application to demonstrate the effectiveness of the proposed sensor-placement scheme. Index Terms—Fuzzy-clustering algorithm, fuzzy-logic control (FLC), sensor placement, three-dimensional (3-D) fuzzy set, type-2 fuzzy system.

I. INTRODUCTION N THE real world, many industrial processes and systems have the characteristics of spatial distribution. The states, controls, and outputs of the processes depend on the space position as well as on the time [1]. These processes are spatiotemporal dynamic systems, which are also called distributed parameter systems (DPSs) [2]. Traditionally, a spatiotemporal dynamic system is controlled by a model-based method [2], where the spatially distributed feature is either ignored completely or considered with the complex mathematical theory. A good mathematical model is definitely required.

I

Manuscript received March 8, 2010; accepted May 21, 2010. Date of publication July 15, 2010; date of current version September 29, 2010. This work was supported in part by the General Research Fund project from the Research Grants Council of Hong Kong under CityU: 117208, in part by the projects from the National Science Foundation of China under Grant 50775224 and Grant 60804033, in part by the project from the Shanghai Science and Technology Commission Foundation under Grant 09dz2201200, in part by the project from the Shanghai Municipal Natural Science Foundation under Grant 09ZR1409600, and in part by the project from the National Basic Research Program of China under Grant 2009CB724301. X.-X. Zhang is with the Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronics and Automation, Shanghai University, Shanghai 200072, China (e-mail: [email protected]). H.-X. Li is with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong, and also with the School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China. C.-K. Qi is with the School of Mechanical Engineering, Shanghai Jiao Tong University, State Key Laboratory of Mechanical Systems and Vibration, Shanghai 200240, China ([email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TFUZZ.2010.2058810

However, the process model may not be easily obtained in many complex situations, and then, a model-free control method has to be used. This leads to the recent development of the novel three-dimensional fuzzy-logic control (3-D FLC) [3], which has the inherent capability to process spatiotemporal dynamic systems and can be considered as the special application of a type-2 fuzzy system [4], [5]. The 3-D FLC uses one kind of type-2 fuzzy set, which is called 3-D fuzzy set, in this paper, and executes a 3-D inference engine. The 3-D fuzzy set is composed of the traditional fuzzy set and a third dimension for the spatial information. It is actually a kind of spatiotemporal fuzzy-control system with the traditional model-free advantage. Since it is just in the beginning phase, there are still many things need to be studied and improved. As described in the previous work [3], sensor placement will affect the capture of the spatial distribution and the subsequent control performance. Therefore, the placement of the sensors in the space domain for better performance will be the key issue in this paper. Sensor placement has been studied over the past several decades, and various methods have been brought forward from different point of views. Most of the methods aimed at achieving better state estimation or state observer. The possible criteria for sensor placement are the trace of the optimal filtering error covariance function [6], observability measures [7], [8], cost function of the estimation error in the closed-loop infinitedimensional system [9], convergence properties of the observer [10], cost functions related to Gramian observability matrix [11], etc. Besides, some methods combined the optimal sensor location and the control parameters [11]. For instance, in [12], a simple H∞ controller was designed, and optimal sensor location was obtained by minimizing H2 -norm of the closed-loop plant. These methods are usually called model-based methods because accurate mathematical models are required. However, many real-world systems are unknown or too complex to obtain models, for which the model-based method may not work well. Therefore, some other methods for sensor placement were proposed. In [13], the determinant of Fisher information matrix formed by sensitivity functions with respect to the unknown parameters was taken as the criterion for parameter estimation, and the Gram determinant formed by sensor responses was taken as the criterion for observer estimation. In [14], the optimal sensor location was to select the most-suitable secondary process variable as soft-sensor inputs for batch distillation by exploiting the properties of principal-component analysis (PCA) on the sensitivity matrix. Tongpadungrod et al. [15] identified the position of an applied load via the optimal sensor location that was achieved by genetic algorithm (GA) under the criterion related to PCA. Sadegh and Spall [16] minimized the redundant information

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ZHANG et al.: SPATIALLY CONSTRAINED FUZZY-CLUSTERING-BASED SENSOR PLACEMENT

provided by multiple sensors by maximizing the overall sensor response while minimizing the correlation among the sensor inputs for signal detection in complex structure. These methods brought forward useful solutions to the problems in terms of their features, respectively. Since only data information is used, they are usually called data-driven methods. Because of no mathematical models used, the data-driven methods are very promising in practical engineering applications. In this study, we aim to develop a new data-driven sensorplacement strategy for the spatiotemporal fuzzy-control system, namely, the main characteristics of spatial distribution is first extracted by a fuzzy-clustering method along the space dimension of the spatiotemporal dynamic data, and then, it is employed for the sensor placement. The main feature of the strategy is to place the sensor position by utilizing the main characteristics of spatial distribution, which means that the sensors will be properly placed to catch the crucial spatial information. The key technique is to apply spatial-constrained fuzzy c-means algorithm (SCFCM) to extract the characteristics of spatial distribution. SCFCM, which has been modified from the previous robust FCM algorithm (RFCM) [17], is a new fuzzy-clustering method that can cope with spatiotemporal data. For an easy implementation, a systematic sensor-placement design scheme in four steps (i.e., data collection, dimension reduction, data clustering, and sensor locating) is developed. The paper is divided into six sections. In Section II, preliminaries on spatiotemporal dynamic systems and 3-D FLC are described. In Section III, the spatial-constrained fuzzy-clustering algorithm is presented in detail. Section IV describes the systematic sensor-placement design scheme for the spatiotemporal 3-D fuzzy-control system. Application to the catalytic packedbed reactor and experimental results are given in Section V. Finally, conclusions and future works are given in Section VI.

II. PRELIMINARIES A. Spatiotemporal Dynamic System The distinguished feature of a spatiotemporal dynamic system is that the system behavior varies with the space and the time. Usually, the system can be described mathematically by nonlinear partial differential equations, and its practical examples include an industrial chemical reactor [1], a solar power plant [18], thermal processing [19], etc. In this study, we take a catalytic packed-bed reactor [2], [3], [20], [21] as an example. As shown in Fig. 1, the catalytic packed-bed reactor is long and thin. It is fed with gaseous reactant C from the right side, and the zero-order gas-phase reaction C → D is carried out on the catalyst. Since the reaction is endothermic, a jacket is used to heat the catalyst. The control problem considered here is to control the catalyst temperature throughout the reactor in order to maintain a desired degree of reaction rate by manipulating the jacked temperature. The example reveals its inherent spatiotemporal nature and gives rise to control problems that involve the regulation of highly distributed control variables using spatially distributed control actuators and measurement sensors.

Fig. 1.

Catalytic packed-bed reactor.

Fig. 2.

Basic structure of 3-D FLC.

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B. Three-Dimensional Fuzzy-Logic Controller The 3-D FLC was first proposed in [3] for spatiotemporal dynamic systems. The configuration of the 3-D FLC is designed to have the inherent ability to deal with spatial information, as shown in Fig. 2. At the conceptual level, the 3-D FLC has a similar functional structure to the traditional FLC that has the three basic operations of fuzzification, rule inference, and defuzzification. However, it will differ in the technical details because of the 3-D processing requirement [22]. Generally, the 3-D FLC will be involved with the following basic designs: 3-D membership function (MF), 3-D fuzzification, 3-D rule base, 3-D rule inference, defuzzification, and spatial scaling factors. For the detailed description and features, see [3]. In addition, the 3-D FLC will relate to an important design problem, i.e., how to place multiple sensors in the space domain. With proper sensor placement, the sensors can capture the main behavior feature of the space domain. As a result, control performance of the spatiotemporal 3-D fuzzy-control system can be enhanced eventually. III. SPATIAL-CONSTRAINED FUZZY-CLUSTERING ALGORITHM Fuzzy-clustering method is one of the data-driven learning tools for unlabeled data [23]. It can mine underlying knowledge (or data structure) from a dataset that is difficult for humans to manually identify. FCM [30] is one of the most-popular fuzzyclustering algorithms and has been utilized in a wide variety of applications. Mathematically, FCM attempts to find a partition for a set of input patterns while minimizing an objective function. Their mathematical expressions are given in Appendix A. FCM has many advantages, including straightforward implementation, fairly robust behavior, applicability to multichannel data, and the ability to model uncertainty within the data [17]; however, it cannot deal with spatial-data information effectively. For image data, especially in the image segmentation problem, many efforts were made to investigate the FCM incorporated with spatial context, for instance, spatially smoothing the

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MFs [24], [25] and directly modifying the objective function by adding spatial constraint to neighboring pixels [17], [26]. In this study, we will develop the earlier work [17] and add two forms of spatial constraints into FCM for spatiotemporal dynamic data. The two forms of spatial constraints are spatial-pattern constraint and spatial-cluster constraint. The spatial-pattern constraint is realized by adding a penalty term into the objective function [17], which aims at restricting the MFs in FCM to be spatially smooth. On the other hand, the spatial-cluster constraint is realized by adding spatial adjacent constraint into the probabilistic constraint, which aims at restricting the cluster evolution in FCM to be spatially smooth. Both the spatial constraints enhance the capability of the proposed algorithm to keep the space continuity. Thus, the proposed algorithm is called SCFCM. Let X = {x1 , x2 , . . . , xN } be the spatiotemporal dynamic data from a spatiotemporal dynamic system, where xj = (xj 1 , . . . , xj d )T ∈ d . Input patterns, i.e., x1 , x2 , . . . , xN , have inherent space order (or spatial interaction) [26]. The features (attributes, dimensions, or variables) of the jth pattern, i.e., xj 1 , . . . , xj d , have inherent time order. Similar to FCM, SCFCM attempts to find a partition (c-fuzzy clusters) for the input patterns X while minimizing the following objective function:

λ1 , . . . , λN to enforce the constraint in (2), we have   c
c
N N



β  2   = µm µm µm JSFCM k j Dk j + kj ql 2 j =1 j =1 k =1

−

N
j =1

c
N


2 µm k j Dk j

k =1 j =1

  c
N


β   + µm µm kj ql 2 j =1 k =1

(1)

q ∈Q k l∈N j

λj 

q ∈Q k l∈N j

k =1






µtj − 1 .

(4)

t∈φ s

Taking the partial derivative of (4) with respect to µk j , it yields   

∂JSFCM −1  2  − λj . (5) Dk j + β = mµm µm ql kj ∂µk j q ∈Q k l∈N j

Setting (5) to 0, we have  µk j = 

 m Dk2 j + β

λj  q ∈Q k

1/(m −1)  l∈N j

µm ql



.

(6)

Using (2), we have  1/(m −1)

λ j    µtj =   2 +β m m D µ t∈φ s t∈φ s tj q ∈Q t l∈N j q l =1

JSFCM (U, V ) =



(7)

which leads to 1/(m −1) λj m 

1/(m −1) −1
1  . (8)   = 2 +β m Dtj q ∈Q t l∈N j µq l t∈φ s

where U = [µk j ]c×N is the fuzzy-partition matrix, µk j ∈ [0, 1] is the membership coefficient of the jth pattern in the kth cluster; V = [v1 , . . . , vc ] is the cluster-center (i.e., prototype or mean) matrix, m ∈ [1, ∞) is the fuzzy exponent, β is a parameter, Dk2 j is the Euclidean distance between the jth pattern and the kth cluster, Qk = C\{k}, and C = {1, 2, . . . , c} is the set of cluster serial number, Nj = {j − rN , . . . , j + rN }\{j}, and rN is the radii of set of neighbors of the jth pattern. The objective function (1) is subject to the following constraints:


µtj = 1

(2)

µk j = 0

(3)

t∈φ s

where j ∈ ps (s = 1, 2, . . . , c), and ps is the set of patterns belonging to the sth cluster (vs ), φs is the set of neighbors of the cluster vs (when 2 ≤ s ≤ c − 1, φs = {vs−1 , vs , vs+1 }; when s = 1, φs = {v2 }; when s = c, φs = {vc−1 }), and k ∈ {C\φs }. Similar to FCM, an alternating optimization (AO) algorithm is derived by minimizing (1). In detail, using Lagrange multipliers

Substituting (8) into (6), we have the necessary condition for µk j to be at a local minimum of JSFCM as follows: −1/(m −1)    Dk2 j + β q ∈Q k l∈N j µm ql µk j =  −1/(m −1) . (9)    2 +β m D µ tj t∈φ s q ∈Q t l∈N j q l Taking the partial derivative of (4) with respect to vk , we get N m j =1 µk j xj vk = N . (10) m j =1 µk j Finally, the steps of our SCFCM algorithm can be described as follows. 1) Provide initial values for the center vk (k = 1, . . . , c), and set initial µk j (k = 1, . . . , c; j = 1, . . . , N ). 2) Compute memberships using (3) and (9). 3) Compute centers using (10). 4) If the algorithm has converged, then quit. Otherwise, go to step 2). We define convergence to be when the maximum change in the objective function is less than a threshold value. In theory, based on the results given in [27], we can use the theory of Zangwill [28] to prove that arbitrary sequences generated by

ZHANG et al.: SPATIALLY CONSTRAINED FUZZY-CLUSTERING-BASED SENSOR PLACEMENT

the above iteration procedures (1 <m < ∞) always terminates at a local minimum of JSFCM or, at worst, always contains a subsequence that converges to a local minimum of JSFCM . In SCFCM, m, β, and rN are adjustable parameters, and they will influence SCFCM in different aspects. Similar to that in FCM, m (m > 1) controls the degree of fuzziness in the resultant MFs [29]. As m approaches unity, the MFs become crisper and approach binary functions. For spatiotemporal data, it will lead to a better clustering result in space. As m increases, the MFs become increasingly fuzzy. For spatiotemporal data, it will lead to overmuch cluster result in the space; in other words, there exists some clusters with null pattern. The following simulation example in Section V will validate this phenomenon. Term β (β ≥ 0) controls the tradeoff between minimizing FCM objective function and obtaining smooth MFs [17]. When β = 0, (1) is equivalent to (A.1), and SCFCM loses the spatial constraint to the MFs. When β > 0, the dependency on the jth pattern neighbors causes µk j to be large, while the neighboring membership values of other cluster is small. Conversely, if the neighboring membership values are small, the membership value at µk j is increased. The result is a smoothing effect that causes neighboring membership values in the same cluster to be similar to one another. Term rN controls the spatial smoothness among patterns [17]. The smaller rN is used, the better spatial smoothness among patterns can be obtained. Usually, it can be set as 1 or 2. Remark: SCFCM is developed from the RFCM given in [17], but they have different principles and objectives, which are as follows. 1) SCFCM is used for spatiotemporal dynamic data, while RFCM is used for static data. 2) SCFCM is used to mine the main characteristics of spatial distribution, while RFCM is used for pixel classification. 3) SCFCM introduces two forms of spatial constraints (i.e., pattern constraint and cluster constraint) into FCM, while RFCM only contains one form of spatial constraint (i.e., pattern constraint). Due to these differences, SCFCM has the advantage of analyzing spatiotemporal dynamic data. This can be validated in Section V. IV. SPATIAL-CONSTRAINED FUZZY-CLUSTERING-BASED SENSOR-PLACEMENT DESIGN The systematic sensor-placement design scheme for spatiotemporal 3-D fuzzy-control systems consists of four steps: data collection, dimension reduction, data clustering, and sensor locating, as shown in Fig. 3. First, a set of data is collected as the fundamental data for sensor placement. To acquire sufficiently rich information from the system, a high-dimensional dataset is usually produced. However, the high-dimensional dataset will lead to a result that the subsequent processing system works inefficiently or even ineffectively. Therefore, it is useful to first reduce the dimension of the data to a manageable size, keeping as much of the original information as possible, and then feed the reduced-dimension data into the processing system. The third step realizes the data-clustering function using SCFCM, which aims at extracting the main spatial-distribution nature

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Fig. 3. Systematic sensor-placement design for spatiotemporal fuzzy-control systems.

of spatiotemporal dynamic systems. In terms of the result of data clustering, the fourth step is finally used to determinate the sensor location. Step 1—Data collection: The ideal test signal for nonlinear systems is white noise with a Gaussian-amplitude distribution (i.e., the Gaussian white noise) [31]; however, the white-noise test signal has some disadvantages in practical applications. For instance, it is not easily realized in the engineering application, because the industrial equipment, such as the control valve, cannot act as frequently as the white noise. Due to this physical limitation, the pseudorandom multilevel signals (PRMS-s) are used as a compromise. As artificially generated signals, they have features similar to those of the Gaussian white noise; hence, they become good substitutes of the white-noise signals in practical applications and then turn to be the most-popular choice for the persistently exciting perturbation signals for nonlinear system identification [31]. Remark: There is some resemblance between the system identification and the data clustering, i.e., for sensor placement studied here, because both of them are doing the same kind of work—modeling and, thus, require the data to be sufficiently rich. System identification can be considered as a “fine” modeling (in control field) with (linear) known structure and unknown parameters need to identify, while the data clustering can be imagined as a “coarse” modeling (in machine learning field) that mines underlying knowledge (or data structure) from data. 1) For system-identification problem, the input signal must satisfy persistent excitation condition to guarantee the unique solution of parameter estimation, namely, the identifiability. 2) For the data-clustering problem, the dynamic behavior of the system should be excited sufficiently so that the data from the system are rich enough. In this study, we choose PRMS-s with maximal length [31] as the input test signals. By selecting a proper PRMS with maximal length as the

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input test signal, the nonlinear system could be excited sufficiently. When choosing a PRMS with maximal length as the test signal, parameters including number of the levels, lowest and uppermost values of input signal, length of the period, sampling time, and minimum switching time (i.e., clock time), should be selected. For its detailed features and parameter selection, see [31]. With a proper PRMS with maximal length as the test signal, we can collect output measurements from the systems in the time interval [t0 , t1 ]. Let Y = {y1 , y2 , · · · , yN } be the collected dataset, where yj = [yj 1 , · · · , yj P ] is the collected sampling data from the jth (1 ≤ j ≤ N ) sensing location in the time interval [t0 , t1 ], and yj s is the sampling measurement at the sth sampling period 1 ≤ s ≤ P . In practice, to acquire dynamics of the system adequately, a large P is better so that more measurements from the system are collected [32]. To be consistent with the data expression in Section III, the original high-dimensional dataset Y can be explained as the following: Y = {y1 , y2 , . . . , yN }, with yj = (yj 1 , . . . , yj P )T ∈ P where yj is the jth pattern (1 ≤ j ≤ N ), yj s is the sth feature of the jth pattern (1 ≤ s ≤ P ), N is the number of patterns, and P  1 is the number of features. Step 2—Dimension reduction: After the first step, a highdimensional dataset is produced. If the dimensionality of the data space is high enough, the distance between the nearest points is no different from that of other points [33]. Accordingly, clustering algorithms based on the distance measure may no longer be effective in a high-dimensional space [34]. Therefore, the data dimension must be reduced before being fed into the clustering processing system. Popular techniques for dimension reduction without labels are PCA, factor analysis, and projection pursuit [35]. PCA and factor analysis aim to reduce the dimension such that the representation is as faithful as possible to the original data. Both techniques realize the function of reducing dimensionality by removing redundancy. On the other hand, projection pursuit aims at finding “interesting” projections by addressing relevance problem. In essence, these techniques apply transformation on the original feature space [35]. Moreover, the transformation would still require collecting all the features to obtain the reduced set, which is sometimes not desired. In this study, we are interested in subsets of the original features and employ a method of dimension reduction via principal variables (i.e., features in this paper) for feature selection [36]. The method is to select p (p < P ) principal features based on the spectral decomposition of the sample correlation matrices and the partial correlation matrices. The selected principal features can best (in some sense) represent the original features. A stepwise selection procedure and its stopping rule are given in Appendix B. Step 3—Data clustering: After the second step, a dataset with reduced dimensionality is produced. Therefore, SCFCM can be used for the data analysis, which aims at mining the underlying

Fig. 4.

Initialization of fuzzy-partition matrix.

Fig. 5.

Test signal generated by PRQS.

data structure to uncover the inherent spatial-distribution nature of the spatiotemporal dynamic systems. After the second step, it is assumed that the original highdimensional dataset Y is reduced into a low-dimensional dataset X = {x1 , x2 , . . . , xN }, with xj = (xj 1 , . . . , xj p )T ∈ p where xj is the jth pattern (1 ≤ j ≤ N ), xj s is the sth feature of the jth pattern (1 ≤ s ≤ p), N is the number of patterns, p (p P ) is the number of features with reduced dimensionality, and P is the number of features with original dimensionality. SCFCM given in Section III will be directly executed on X. Using the aforementioned four steps of SCFCM, we obtain a fuzzy-partition matrix U = [µk j ]c×N and cluster-center matrix V = [v1 , . . . , vc ] and, eventually, find a data grouping. It should be noted that the initialization of U and V will influence the clustering result. Similar to FCM, the AO algorithm of SCFCM is sensitive to the starting points, which leads to different solutions. Some common cluster-center initialization methods for FCM can be used for the initialization of SCFCM, including using c points randomly drawn from a hyperbox containing data X, choosing the first c distinct data points of X, and using c points uniformly distributed along the diagonal of the hyper box containing data X [30]. In this study, the evenly distributed center initialization method is employed, which is coincident with human intuition, since one usually chooses the evenly distributed sensor placement when he lacks domain knowledge. On the other hand, to keep the space continuity, the fuzzy-partition matrix is initialized using local-space-influence characteristics [37]. For instance, in this paper, the MFs are chosen as the standard triangular shape with the center vi (i = 1, . . . , c) shown in Fig. 4. Step 4—Sensor locating: After the third step, we can obtain data groups that represent the space structure in a body of

ZHANG et al.: SPATIALLY CONSTRAINED FUZZY-CLUSTERING-BASED SENSOR PLACEMENT

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Fig. 6. Three-dimensional graph representation with the best two principal-component scores and space coordinate based on different features. (a) 624 features. (b) Four selected features.

Fig. 7.

Data-clustering result analyzed by SCFCM with graphical representation through the best two principal components. (a) m = 1.5. (b) m = 4.

unlabeled data. In this step, we aim to determine sensor location for the spatiotemporal dynamic systems in terms of the data groups and their centers. A natural way might be to take the group centers as the direct criterion for sensor location. This is reasonable since group centers are the most-representative points in the data groups. Then, a simple way to achieve the sensor location is to use the interpolation method [38] in terms of the known data groups and their centers.

tubular chemical reactor is given as follows: εp

∂Tg (z, t) ∂Tg (z, t) =− + αc (Ts (z, t) − Tg (z, t)) ∂t ∂z − αg (Tg (z, t) − u(t))

γTs (z, t) ∂Ts (z, t) ∂ 2 Ts (z, t) = + B exp 0 ∂t ∂z 2 1 + Ts (z, t) − βc (Ts (z, t) − Tg (z, t)) − βp (Ts (z, t) − b(z)u(t))

(17)

subject to the boundary conditions V. APPLICATION TO CATALYTIC PACKED-BED REACTOR The catalytic packed-bed reactor shown in Fig. 1, as described in Section II, is a typical spatiotemporal dynamic system. The dimensionless mathematical model that describes this nonlinear

z = 0,

Tg (z, t) = 0,

z = 1,

∂Ts (z, t) =0 ∂z

∂Ts (z, t) =0 ∂z (18)

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Fig. 8. Data-clustering result analyzed by RFCM with graphical representation through the best two principal components

where Tg (z, t) and Ts (z, t) denote the dimensionless temperature of gas and catalyst, respectively, which are spatially dependent with z ∈ [0, 1], b(z)u(t) denotes the spatiotemporal heating source with the distribution b(z) and the manipulated input u(t), and u(t) denotes the dimensionless temperature of jacket. The values of the process parameters [20] are given as follows: εp = 0.01, B0 = −0.003,

γ = 21.14, αc = 0.5,

βc = 1.0, and

Fig. 9. Three-dimensional graphical representation of data-clustering result with the best two principal-component scores and space coordinate TABLE I SENSOR LOCATIONS WITH EVENLY DISTRIBUTED INITIALIZATION

βp = 15.62

αg = 0.5.

In this application, we aim to control the catalyst temperature Ts (z, t) throughout the reactor in order to maintain a desired degree of reaction rate using the measurements of catalyst temperature from q sensing locations z  = [z1 z2 . . . zq ], and manipulating one spatially distributed heating source [i.e., b(z) = 1 − cos(πz)]. The spatial reference profile is given as Tsd (z) = 0.42 − 0.2 cos(πz), 0 ≤ z ≤ 1, and then, the entire spatial catalyst temperature should follow this reference. The mathematical model [see (17) and (18)] is only for the process simulation for evaluation of the sensor-placement scheme and not for control design. The method of lines [39] is used to simulate the model. A. Sensor-Placement Design Step 1—Data collection: The pseudorandom quinary signal (PRQS) with maximum length of 624 is used as the test signal. According to some a priori knowledge of the system, we choose the parameters of PRQS as the following: The number of the levels is 5, the lowest and uppermost values of input are 0.1474 and 0.0632, respectively, the length of the period is 624, the sampling time is 0.1s, and the minimum switching time (i.e., clock period) is 0.1s. The test signal generated by PRQS is shown in Fig. 5. With the above parameters, we can reproduce the signal and the whole test experiment any time. The space domain of the system is discreted into 81 points uniformly. In the time interval [0, 62.3], we collect the catalytic temperature measurements from the system with the sampling period set as 0.1s over the space domain. The collected temperature measurements are denoted by Y = {y1 , y2 , . . . , yN }, with yj = (yj 1 , . . . , yj P ) ∈ P

TABLE II PERFORMANCE INDEX COMPARISONS (2–9 SENSORS)

where yj is the catalytic temperature measurement from the jth (1 ≤ j ≤ N ) sensing location in the time interval [0, 62.3], yj s is the catalytic temperature measurement at the sth (1 ≤ s ≤ P ) sampling period, N is the number of discrete space points, i.e., N = 81, and P is the number of sampling measuring, i.e., P = 624. In other words, we have obtained a high-dimensional dataset Y that contains 81 patterns, each of which holds 624 dimensions. In the next step, the dimension-reduction operation will be carried out to reduce the dimensionality of Y . Step 2—Dimension reduction: Using the dimensionreduction method based on principal variable, as described in Section IV and Appendix B, four principal features, i.e., the 24nd, 157th, 471th, and 469nd features, are selected, respectively. The stopping rule α is 0.99, which indicates that the selected features capture 99% variation in the total variation of all features. The effectiveness of dimension reduction can be shown in Fig. 6, where the plot in (a) is based on all 624 features, and the plot in (b) is based on four selected features. It should be noted that the plot in Fig. 6 is the graphical representation of high-dimensional data based on PCA, which is a commonly used

ZHANG et al.: SPATIALLY CONSTRAINED FUZZY-CLUSTERING-BASED SENSOR PLACEMENT

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Fig. 10. Control-performance comparison with different sensor-placement schemes. (a) With evenly distributed sensor placement. (b) With proposed sensor placement.

technique for the visualization of high-dimensional unlabeled data [40]. For more attention about the technique, see [40]. There are not so many differences between the two configurations of PCs. This illustrates the meaningfulness of feature selection in PCA since selected features can provide almost the same result as the original features. In the following step, the four principal features will be used to represent the original 624 features. In other words, we have a low-dimensional dataset given by X = {x1 , x2 , . . . , xN }, with xj = (xj 1 , . . . , xj p )T ∈ p where xj is the jth pattern or the catalytic temperature measurement from the jth sensing location, 1 ≤ j ≤ N , xj s is the sth feature of the jth pattern, or the catalytic temperature measurement at the sth selected sampling instant (1 ≤ s ≤ p), N is the number of patterns, or the number of discrete space points, i.e., N = 81, and p is the number of features with reduced dimensionality, or the number of selected measuring instants, i.e., p = 4. Step 3—Data clustering: For the low-dimensional dataset X, SCFCM is employed to analyze data and find a data grouping. As a result, data groups and their centers are extracted.

The five-sensor-placement problem is taken as an example. First, we initialize the cluster-center and fuzzy-partition matrix. The initial cluster centers, i.e., initial sensor locations, are chosen as [0, 0.25, 0.5, 0.75, 1], which are uniformly distributed over the space. Then, based on the initial cluster centers, the fuzzy-partition matrix is initialized with the same shape, as shown in Fig. 4, where only five centers are used. The parameters m, β, and rN are set as m = 1.5, β = 0.5, and rN = 2. In addition, the stop rule of AO algorithm is given as follows: The minimal improved value for iteration is 10−6 , and the maximal iteration number is 500. Afterward, using the steps of SCFCM, as described in Section III, the data groups and their centers are shown in Fig. 7(a). The influence of the parameter m to SCFCM is also investigated. Fig. 7(b) presents the data-grouping result, when m increases to 4, where one overmuch cluster center (i.e., larger marker: square) is produced. The same result takes place in other case. This is because the MFs become increasingly fuzzy as m increase; furthermore, it will lead to overmuch cluster as m increases to certain value. In this application, we find that a better scope for m would be [1.2, 1.7] through numerous of simulation experiments. Additionally, simulation result of RFCM in [17] for the fivesensor-placement scheme is shown in Fig. 8. In RFCM, the same

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parameters are employed. From Fig. 8, we can find that RFCM leads to space discontinuity—an incorrect grouping result. Compared with RFCM, SCFCM keeps a better space continuity. Step 4—Sensor locating: Based on the data-grouping result, we employ interpolation method to determine the sensor location for the catalytic packed-bed reactor. Herein, we use piecewise cubic Hermite interpolation method. If the parameter m is 1.5, the final sensor locations for the five-sensor-placement problems are [0.1075, 0.2775, 0.4703, 0.7390, 0.9054], as shown in Fig. 9. Other results are listed in Table I. B. Control-Performance Validation The effectiveness of the proposed sensor-placement method is validated on the 3-D fuzzy-controlled catalytic packed-bed reactor. The design of 3-D FLC is given in Appendix C. We carried out two different sensor-placement schemes on the 3-D fuzzy-controlled catalytic packed-bed reactor. One is that two to nine sensors are evenly distributed in the space domain, respectively. The other is that the proposed sensor-placement scheme in Table I is used. The control-performance comparison of the two schemes is given in Table II, where SSE, IAE, and ITAE [3] stand for steady-state error, integral of the absolute error, and integral of time multiplied by absolute error for spatiotemporal dynamic systems, respectively. We also give graph comparison. Two-sensor-placement problems are taken as examples. The 3-D FLC-controlled catalyst temperature varying with time and space, manipulated input, and the catalyst temperature at steady state are presented in Fig. 10. Sensors in Fig. 10(a) are evenly placed, while sensors in Fig. 10(b) are placed with the proposed scheme in Table I. From Fig. 10, we can find that the proposed sensor-placement scheme has improved the control performance. The simulation results indicate that all of the proposed eight sensor-placement schemes (from two to nine sensors) outweigh its corresponding evenly distributed sensor-placement scheme, in particular, using two sensors, the control performance is improved greatly. The results validate the effectiveness of the proposed sensor-placement method for the spatiotemporal 3-D fuzzy-control system. In addition, from Table II, we can find that three sensors are enough for the measurement in this application. VI. CONCLUSION AND FUTURE WORKS A systematic sensor-placement methodology based on SCFCM is proposed for spatiotemporal 3-D fuzzy-control systems. SCFCM is realized by introducing two spatial constraints to FCM. One is for the spatial constraint among patterns, and the other is for the spatial constraint among clusters. Using SCFCM, the main characteristics of spatial distribution can be extracted, and then, sensors can be located accordingly to capture crucial spatial information of the system. The developed sensorplacement scheme is composed of four steps: data collection, dimension reduction, data clustering, and sensor locating. Finally, a catalytic parked-bed reactor is taken as an example to demonstrate the effectiveness of the proposed sensor-placement scheme for the 3-D fuzzy-control system.

In this study, SCFCM is proposed as a key technique to solve the sensor-placement problem for the spatiotemporal fuzzycontrol systems. There are many challenge works for the future research. The first one is to find a good cluster initialization method. Since the data from spatiotemporal dynamic systems contain the feature of spatial ordering, a proper initialization method should be investigated to help get a global optimal solution of sensor location. Another one is concerned with the cluster validation. For spatiotemporal dynamic systems, the cluster validation can be used to determine the optimal sensor numbers. Cluster validation techniques in FCM might not be appropriate to SCFCM because of the ordering of spatial data; therefore, a new cluster-validation technique should be developed in the future. APPENDIX A MATHEMATICAL DESCRIPTION OF FCM Mathematically, FCM attempts to find a partition (c-fuzzy clusters) for a set of input patterns X = {x1 , x2 , . . . , xN }, where xj = (xj 1 , . . . , xj d )T ∈ d is the jth pattern, and each measure xj s is said to be a feature (i.e., attribute, dimension, or variable), while minimizing the following objective function:

JFCM (U, V ) =

c
N


2 µm k j Dk j

(A.1)

∀j = 1, . . . , N

(A.2)

k =1 j =1

under the probabilistic constraint c


µk j = 1

k =1

where U = [µk j ]c×N is the fuzzy-partition matrix, µk j ∈ [0, 1] is the membership coefficient of the jth pattern in the kth cluster, V = [v1 , . . . , vc ] is the cluster-center (i.e., prototype or mean) matrix, m ∈ [1, ∞) is the fuzzy exponent, and Dk2 j is the distance measure between the jth pattern and the kth cluster. The function optimization is done by an AO procedure [23]. For AO solutions of the FCM clustering model, the equations to update U and V are the necessary conditions for local extrema of the objective function. After initializing U or V, the following two operations are successively carried out: 1

µk j = c s=1

(Dk j /Dsj )2/(m −1)

j =1

µm k j xj

N

vk =  N

j =1

µm kj

.

(A.3)

(A.4)

These operations are reiterated until convergence when cluster prototypes, or the fuzzy-partition matrix, or the objective function are stable with respect to a given tolerance.

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APPENDIX B METHOD OF DIMENSION REDUCTION VIA PRINCIPAL VARIABLES (SEE [36]) Let us assume that  the sample covariance and correlation matrices of Y are and R, respectively. The aim is to select some subset of p principal features from P original features. Let us suppose that the set of features F is  partitioned into subsets F (1) and F (2). The covariance matrix can be written as  
Σ11 Σ12 = . Σ21 Σ22 Then, the partial covariance matrix for F (2) given F (1) is Σ22.1 = Σ22 − Σ21 Σ−1 11 Σ12 and the partial correlation matrix, i.e., R22.1 , is obtained by scaling Σ22.1 so that diagonal elements are unity. If we begin with a correlation matrix R, then we further define the unscaled partial correlation matrix −1 S˜22.1 = R22 − R21 R11 R12 .

(B.1)

Let us consider the P × P correlation matrix R, which is assumed full rank for convenience. It can be expressed in terms of its spectral decomposition as R=

P


where η1 ≥ η2 ≥ · · · ≥ ηP > 0 are the ordered eigenvalues of R, and α1 , . . . , αP are the associated eigenvectors. Then, we have

R =

P
i=1

where hj =

P  i=1

ηi2

=

P P

j =1

i=1

 2 rij

Membership function for ei , ∆ei , and ∆u.

4) Repeat the process: Calculate h values, identify candidate variables, select the best variable, and compute a partial correlation matrix for the remaining candidates. The stopping rule for the selection process can be defined as [41] α=

(l) 1 − tr(S˜22.1 ) tr(R)

(B.3)

(l) where R is the initial correlation matrix, S˜22.1 is the matrix (B.3) given selection of features f1 ,. . ., fl , and α is the proportion that the selected feature capture variation in the total variation of all features. Therefore, the selection procedure could stop once we have identified that the selected subset represents at least 90% of the original variation of the data.

APPENDIX C

ηi αi αiT

i=1

2

Fig. 11.

=

P


hj

(B.2)

j =1

2 rij .

The decomposition of (B.2) suggests that we may examine the values h1 , · · · , hP , which are the sum of the squared correlations between feature fj and other features. Large values of hj are obtained when feature fj has, on average, high loadings on important PCs. As the values {hj }Pj=1 combine information from both the eigenvalues and the loadings, we would expect choices based on them to be more robust to the sensitivity issues raised by selection via single loadings on important PCs. A stepwise feature-selection scheme is given as follows. 1) In order to remove initial scale effects, typically begin with the correlation matrix R, rather than the raw covariance matrix. 2) Determine the hj values for each feature fj (j = 1, . . . , P ), identify the variable with the largest hj value, and select it. 3) Form the unscaled partial correlation matrix S˜22.1 using (B.1) for the remaining variables given the feature(s) that have already selected.

DESIGN OF THREE-DIMENSIONAL FUZZY-LOGIC CONTROLLER FOR THE CATALYTIC PACKED-BED REACTOR The error of spatial catalyst temperature and its error change are taken as two spatial inputs for 3-D FLC, i.e., e∗z = {e∗1 , · · · , e∗p } and ∆e∗z = {∆e∗1 , . . . , ∆e∗p }, where e∗i = Tsd (zi ) − Ts (zi , k), and ∆e∗i = e∗i (k) − e∗i (k − 1). Let kez = {ke1 , . . . , kep }, krz = {kr1 , . . . , krp }, and k∆ u be the spatial scaling factors of e∗z , ∆e∗z , and the incremental control action ∆u, respectively. Then, the scaled error ez and the change error ∆ez are the direct spatial inputs of 3-D FLC, where ez = {e1 , . . . , ep }, with ei = kei e∗i , and ∆ez = {∆e1 , . . . , ∆ep }, with ∆ei = kri ∆e∗i . 1) 3-D membership function: Since 3-D MF can be regarded as the assembly of traditional 2-D MF from each sensing input, the design of 3-D MF is transformed to the design of 2-D MF. The 2-D MF, for each sensing input, can be chosen as triangular shape, as shown in Fig. 11, where each input is classified into seven linguistic labels as positive large (PL), positive middle (PM), positive small (PS), zero (O), negative small (NS), negative middle (NM), and negative large (NL). The reasons to use triangular-shaped MFs include the following. a) They have been one of the most-commonly used MFs [4]. b) They have a simple parametric representation [42], [43]. c) Determining the membership degrees can be achieved with low computational effort [42].

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2) 3-D fuzzification: Since 3-D fuzzification can be regarded as the assembly of the traditional 2-D fuzzification at each sensing location, the design of 3-D fuzzification is transformed to the design of traditional 2-D fuzzification. In this study, singleton fuzzification is used. 3) 3-D rule base: The linear-control rule base is used; for instance, one of the rules is expressed as “if ez is PM, and ∆ez is NB then, ∆u is NS,” where PM and NB are 3-D fuzzy sets, which are assembled by 2-D fuzzy sets PM and NB at each sensing location, ∆u is the incremental control action, whose MFs are triangular shape and classified into seven linguistic labels, as shown in Fig. 11, NS is the 2-D fuzzy set, and the rule weight is defaulted as unity. 4) 3-D rule inference: The spatial t-norm in spatialinformation-fusion operation is chosen as “minimum.” The centroid approach [3] is used for the dimensionreduction operation. In traditional inference operation, “minimum” and “maximum” are used for the t-norm in the intersection operation and for the t-conorm in the union operation, respectively. 5) Defuzzification: The center-of-sets-type defuzzifier is used. 6) Spatial scaling factors: The scaling factor for each e∗i is set to be 1.5, the scaling factor for each ∆e∗i is set to be 0.5, and the scaling factor for ∆u is 1.0. ACKNOWLEDGMENT The authors would like to thank anonymous referees for their helpful and precious comments. REFERENCES [1] P. D. Christofides, Nonlinear and Robust Control of Partial Differential Equation Systems: Methods and Applications to Transport-Reaction Processes. Boston, MA: Birkh¨auser, 2001. [2] W. H. Ray, Advanced Process Control. New York: McGraw-Hill, 1981. [3] H. X. Li, X. X. Zhang, and S. Y. Li, “A three-dimensional fuzzy control methodology for a class of distributed parameter system,” IEEE Trans. Fuzzy Syst, vol. 15, no. 3, pp. 470–481, Jun. 2007. [4] J. M. Mendel, Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions. Upper Saddle River, NJ: Prentice-Hall, 2001. [5] H. Hagras, “A hierarchical type-2 fuzzy logic control architecture for autonomous mobile robots,” IEEE Trans. Fuzzy Syst., vol. 12, no. 4, pp. 524–539, Aug. 2004. [6] S. Kumar and J. H. Seinfeld, “Optimal location of measurements for distributed parameter estimation,” IEEE Trans. Autom. Control, vol. AC23, no. 4, pp. 1198–1210, Aug. 1978. [7] W. Waldraff, D. Dochain, S. Bourrel, and A. Magnus, “On the use of observability measures for sensor location in tubular reactor,” J. Process Control, vol. 8, no. 5, pp. 497–505, 1998. [8] A. Armaoua and M. A. Demetriou, “Optimal actuator/sensor placement for linear parabolic PDEs using spatial H2 norm,” Chem. Eng. Sci., vol. 61, no. 22, pp. 7351–7367, 2006. [9] C. Antoniades and P. D. Christofides, “Integrating optimal actuator/sensor placement and robust control of uncertain transport-reaction processes,” Comput. Chem. Eng., vol. 26, no. 2, pp. 187–203, 2002. [10] A. A. Alonso, I. G. Kevrekidis, J. R. Banga, and C. E. Frouzakis, “Optimal sensor location and reduced order observer design for distributed process systems,” Comput. Chem. Eng., vol. 28, no. 1, pp. 27–35, 2004. [11] I. Bruant, “Optimal piezoelectric actuator and sensor location for active vibration control, using genetic algorithm,” J. Sound Vib., 2010. [12] M. Guney and E. Eskinat, “Optimal actuator and sensor placement in flexible structures using closed-loop criteria,” J. Sound Vib., vol. 312, pp. 210–233, 2008.

[13] A. Vande Wouwer, N. Point, S. Porteman, and M. Remy, “An approach to the selection of optimal sensor locations in distributed parameter systems,” J. Process Control, vol. 10, no. 4, pp. 291–300, 2000. [14] E. Zamprogna, M. Barolo, and D. E. Seborg, “Optimal selection of soft sensor inputs for batch distillation columns using principal component analysis,” J. Process Control, vol. 15, no. 1, pp. 39–52, 2005. [15] P. Tongpadungrod, T. D. L. Rhys, and P. N. Brett, “An approach to optimize the critical sensor locations in one-dimensional novel distributed tactile surface to maximize performance,” Sens. Actuators A, vol. 105, pp. 47–54, 2003. [16] P. Sadegh and J. C. Spall, “Optimal sensor configuration for complex systems,” in presented at the Proc. Amer. Control Conf., Philadelphia, PA, 1998. [17] D. L. Pham, “Spatial models for fuzzy clustering,” Comput. Vis. Image Understanding, vol. 84, no. 2, pp. 285–297, 2001. [18] R. N. Silva, J. M. Lemos, and L. M. Rato, “Variable sampling adaptive control of a distributed collector solar field,” IEEE Trans. Control Syst. Technol., vol. 11, no. 5, pp. 765–772, Sep. 2003. [19] C. C. Doumanidis and N. Fourligkas, “Temperature distribution control in scanned thermal processing of thin circular parts,” IEEE Trans. Control Syst. Technol., vol. 9, no. 5, pp. 708–717, Sep. 2001. [20] P. D. Christofides, “Robust control of parabolic PDE systems,” Chem. Eng. Sci., vol. 53, no. 16, pp. 2949–2965, 1998. [21] K. A. Hoo and D. Zheng, “Low-order control-relevant models for a class of distributed parameter systems,” Chem. Eng. Sci., vol. 56, no. 23, pp. 6683–6710, 2001. [22] H. X. Li, “Three-dimensional fuzzy logic system for modeling & control— Special applications of type-2 fuzzy system,” IEEE eNewslett., System, Man, Cybern. Soc., no. 27, Jun. 2009. [23] S. Nascimento, Fuzzy Clustering Via Proportional Membership Model. Amsterdam, The Netherlands: IOS, 2005. [24] Y. A. Tolias and S. M. Panas, “On applying spatial constraints in fuzzy image clustering using a fuzzy rule-based system,” IEEE Signal Process. Lett., vol. 5, no. 10, pp. 245–247, Oct. 1998. [25] Y. A. Tolias and S. M. Panas, “Image segmentation by a fuzzy clustering algorithm using adaptive spatially constrained membership functions,” IEEE Trans. Syst., Man, Cybern., A, Syst., Humans, vol. 28, no. 3, pp. 359– 369, May 1998. [26] A. W. C. Liew, S. H. Leung, and W. H. Lau, “Fuzzy image clustering incorporating spatial continuity,” Proc. Inst. Electr. Eng. Vision, Image Signal Process., vol. 147, no. 2, pp. 185–192, Apr. 2000. [27] J. C. Bezdek, “A convergence theorem for the fuzzy isodata clustering algorithms,” IEEE Trans. Pattern Anal. Mach. Intell., vol. PAMI-2, no. 1, pp. 1–8, Jan. 1980. [28] W. Zangwill, Nonlinear Programming: A Unified Approach. Englewood Cliffs, NJ: Prentice-Hall, 1969. [29] W. Pedrycz, Knowledge-Based Clustering: From Data to Information Granules. Hoboken, NJ: Wiley, 2005. [30] J. C. Bezdek, J. Keller, R. Krisnapuram, and N. R. Pal, Fuzzy Models and Algorithms for Pattern Recognition and Image Processing. Norwell, MA: Kluwer, 1999. [31] R. Haber and L. Keviczky, Nonlinear System Identification— Input–Output Modeling Approach, Volume 1: Nonlinear System Parameter Identification. Dordrecht, The Netherlands: Kluwer, 1999. [32] T. S¨oderstr¨om and P. Stoica, System Identification. Hemel Hempstead, U.K.: Prentice-Hall Int., 1989. [33] K. Beyer, J. Goldstein, R. Ramakrishnan, and U. Shaft, “When is nearest neighbor meaningful,” in Proc. 7th Int. Conf. Database Theory, 1999, pp. 217–235. [34] R. Xu and D. Wunsch II, “Survey of clustering algorithms,” IEEE Trans. Neural Netw., vol. 16, no. 3, pp. 645–678, May 2005. [35] H. Liu and H. Motoda, Computational Methods of Feature Selection. Boca Raton, FL: Chapman & Hall/CRC, 2008. [36] J. A. Cumming and D. A. Wooff, “Dimension reduction via principal variables,” Comput. Stat. Data Anal., vol. 52, no. 1, pp. 550–565, 2007. [37] C. Bailey-Kellogg and F. Zhao, “Influence-based model decomposition for reasoning about spatially distributed physical systems,” Artif. Intell., vol. 130, no. 2, pp. 125–166, 2001. [38] D. Kincaid and W. Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3rd ed. Pacific Grove, CA: Brooks/Cole, 2002. [39] W. E. Schiesser, The Numerical Methods of Lines Integration of Partial Differential Equations. San Diego: Academic, 1991. [40] I. T. Jolliffe, Principal Component Analysis. New York: SpringerVerlag, 1986.

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[41] J. A. Cumming “Clinical decision support,” Ph.D. dissertation, Durham Univ, Durham, U.K., 2006. [42] K. Michels, F. Klawonn, R. Kruse, and A. N¨urnberger, Fuzzy ControlFundamentals, Stability and Design of Fuzzy Controllers. Berlin, Germany: Springer-Verlag, 2006. [43] W. Pedrycz, “Why triangular membership functions?,” Fuzzy Sets Syst., vol. 64, no. 1, pp. 21–30, 1994.

Xian-Xia Zhang (M’10) received the B.E. degree in automatic control from the University of Science and Technology Beijing, Beijing, China, in 1998, the M.E. degree in measurement techniques and instrumentation from Shanghai University, Shanghai, China, in 2003, and the Ph.D. degree in control theory and control engineering from Shanghai Jiao Tong University, in 2008. She is currently an Assistant Professor with the School of Mechatronics and Automation, Shanghai University. Her research interests include type-2 fuzzy systems, intelligent control, distributed parameter systems, learning algorithms, and system identification.

Han-Xiong Li (S’94–M’97–SM’00) received the B.E. degree in aerospace engineering from the National University of Defense Technology, Changsha, China, the M.E. degree in electrical engineering from the Delft University of Technology, Delft, The Netherlands, and the Ph.D. degree in electrical engineering from the University of Auckland, Auckland, New Zealand. He is currently a Full Professor with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong. Over the past 20 years, he has had been engaged in different fields, including military service, industry, and academia. His research experience and recent accomplishments include the design and control of thermal cure processes and fluid dispensing processes for integrated circuit packaging, intelligent modeling and control of spatiotemporal dynamic systems, spatial–temporal fuzzy systems for process control, and probabilistic fuzzy systems for process modeling. He has authored or coauthored more than 100 Science Citation Index journal papers (half of them in IEEE TRANSACTIONS and American Society of Mechanical Engineers Transactions) with h-index 20. His current research interests include intelligent modeling and control, process design and control, and distributed parameter systems. Dr. Li is an Associate Editor of the IEEE TRANSACTIONS ON SYSTEMS, MAN AND CYBERNETICS—PART B: CYBERNETICS and the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS. He received awarded the Distinguished Young Scholar (overseas) Award from the China National Science Foundation in 2004 and a Chang Jiang Chair Professorship from the Ministry of Education, China, in 2006.

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Chen-Kun Qi (S’09–M’10) received the Ph.D. degree from the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong, in 2009 and the B.E. degree in mechanical engineering and the M.E. degree in automatic control, both from Shanghai Jiao Tong University, Shanghai, China, in 2001 and 2004, respectively. He is currently an Assistant Professor with the School of Mechanical Engineering, Shanghai Jiao Tong University. His research interests include industrial-process modeling and control, system identification, distributed parameter systems, robotics, and intelligent learning.