IN THE REAL world, many of physical processes and systems

Transcription

1 470 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 15, NO. 3, JUNE 2007 A Three-Dimensional Fuzzy Control Methodology for a Class of Distributed Parameter Systems Han-Xiong Li, Senior Member, IEEE, Xian-Xia Zhang, and Shao-Yuan Li, Senior Member, IEEE Abstract The traditional fuzzy set is two-dimensional (2-D) with one dimension for the universe of discourse of the variable and the other for its membership degree. This 2-D fuzzy set is not able to handle the spatial information. The traditional fuzzy logic controller (FLC) developed from this 2-D fuzzy set should not be able to control the distributed parameter system that has the tempo-spatial nature. A three-dimensional (3-D) fuzzy set is defined to be made of a traditional fuzzy set and an extra dimension for spatial information. Based on concept of the 3-D fuzzy set, a new fuzzy control methodology is proposed to control the distributed parameter system. Similar to the traditional FLC, it still consists of fuzzification, rule inference, and defuzzification operations. Different to the traditional FLC, it uses multiple sensors to provide 3-D fuzzy inputs and possesses the inference mechanism with 3-D nature that can fuse these inputs into a so called spatial membership function. Thus, a simple 2-D rule base can still be used for two obvious advantages. One is that rules will not increase as sensors increase for the spatial measurement; the other is that computation of this 3-D fuzzy inference can be significantly reduced for real world applications. Using only a few more sensors, the proposed FLC is able to process the distributed parameter system with little complexity increased from the traditional FLC. The 3-D FLC is successfully applied to a catalytic packed-bed reactor and compared with the traditional FLC. The results demonstrate its effectiveness to the nonlinear unknown distributed parameter process and its potential to a wide range of engineering applications. Index Terms Distributed parameter system, fuzzy logic controller, fuzzy set, three-dimensional fuzzy set. I. INTRODUCTION IN THE REAL world, many of physical processes and systems are inherently characterized by the presence of strong spatial variations [1], which are usually called as distributed parameter systems (DPSs). The distinguished feature of distributed parameter systems is that the states, controls, and outputs will depend on spatial position. Many methods were proposed to control DPS over the past three to four decades. The most popular approach, which exists in many engineering applications, is that the distributed nature of the system is ignored and the simplified model is derived based on the assumption of lumped parameters. Therefore, the lumped Manuscript received August 29, 2005; revised March 18, 2006 and May 25, This work was supported in part by Grants from RGC Hong Kong SAR (CityU: 1207/04E and ) and in part by the National Science Foundation of China under Grants and H.-X. Li is with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong ( mehxli@cityu.edu.hk). X.-X. Zhang and S.-Y. Li are with the Institute of Automation, Shanghai Jiao Tong University, Shanghai , China Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TFUZZ parameter system can be controlled by many conventional control methods. The second one, usually called as early lumping [2], is that distributed parameter systems are simply discretized using the finite-difference or the finite-element techniques. These methods will lead to approximate systems of thousands of traditional differential equations without a view to the characteristic of distributed parameter. The last one, called as late lumping [2], is that when designing the controller, all the natural features of DPS are considered and utilized. Therefore, to design the controller, one need more complicated mathematical knowledge and control theory of DPS. The aforementioned approaches are model-based methods, which require accurate mathematical models of the process when designing a controller. Actually, many real world systems have unknown parameters or highly complex and nonlinear characteristics, accurate models are difficult to obtain, or may lead to very complex controllers causing difficulties in application. In the complex environment, model-based control methods may not work well; on the contrary, model-free control methods such as fuzzy logic control (FLC) seem more competitive. Compared to the traditional control paradigm, the fuzzy control paradigm has two practical advantages [3]. First, a mathematical model of the system to be controlled is not required, and second, a satisfactory nonlinear controller can often be developed empirically in practice without complicated mathematics. Therefore, since fuzzy logic control was first introduced in the early 1970s, it has emerged as one of the most active and fruitful areas for research in the application of fuzzy set theory [4]. There have been existed many applications of fuzzy control, such as steam engine control [5], robot control [6], [7], power plant [8], biomedical system control [3]. Until now, most fuzzy controls in the real world are focused on lumped systems. Fewer literatures are found to apply the fuzzy system to the control of distributed parameter systems. In existing literatures, Lin et al. [9] adopted the hierarchical fuzzy logic strategy for a flexible link robot arm control based on the mathematical model; Sagias et al. [10] presented design of a fuzzy adaptive controller for a jacketed plug flow tubular reactor based on a precise PDE model; Sooraksa et al. [11] developed a fuzzy logic based PI+D2 control scheme for both vibration suppression and set-point tracking of a flexible-link robot arm with the exact PDE model; in literature [12], the distributed nature of a flexible robot was considered through a hierarchical fuzzy control structure, where a fuzzy classifier was used at the higher level for space feature extraction and a traditional fuzzy logic controller was used at the lower level. The literature survey shows that the traditional FLC is not inherently designed for the distributed parameter process because /$ IEEE

2 LI et al.: A THREE-DIMENSIONAL FUZZY CONTROL METHODOLOGY FOR A CLASS OF DISTRIBUTED PARAMETER SYSTEMS 471 the fuzzy set used has only two dimensions - one is for the universe of discourse of variable, the other is for the membership degree. The incapability to handle spatial information limits the FLC applications to the lumped system. Therefore, the advantages of fuzzy controller, such as simultaneously handling numerical data and linguistic knowledge [13] and converting the linguistic control strategy based on expert knowledge into an automatic control strategy, cannot be extended to control of the widely existing distributed parameter systems. Obviously, a new fuzzy set is expected to cope with the spatial information. It is noted that some preliminary work has been proposed for using multi-dimensional fuzzy sets [13], [35], [36]. A multivariable antecedent membership function was used for fuzzy modeling of the lumped system [35], [36], where each input variable was regarded as one dimension of the multivariable membership function. A concept of membership grade of membership grade was produced by using a multidimensional fuzzy set (also called as type-2 fuzzy set) [13], [37] to model uncertainties that the traditional fuzzy set was not able to handle directly. A further development using interval type-2 fuzzy set was under investigation for control of the distributed parameter system [38]. However, this approach can not effectively model the spatial information due to the limitation of the interval type-2 fuzzy set. Based on the aforementioned points, in this paper, a three-dimensional fuzzy set (3-D fuzzy set) is proposed based on the spatial nature of the distributed parameter system. The 3-D fuzzy set is composed of a traditional fuzzy set and a third dimension for the spatial information. Based on the concept of the 3-D fuzzy set, a three-dimensional FLC methodology is proposed to control the distributed parameter system. Since there is only one control output in many real industrial environments, the 3-D FLC will behave as the centralized control with distributed inputs from multiple sensors located on the spatial domain. Thus, a centralized control rule base is more appropriate. As there is no precise or quantitative information that can be used for design, the feasible action is to control the overall behavior of the spatial domain instead of accurately manipulating each spatial location. Thus, the 3-D fuzzy inference engine is designed to have two different functions. The first one is to capture the overall behavior feature from distributed inputs, and the second one is to do traditional fuzzy inference with the centralized rule base. The proposed FLC still consists of fuzzification, rule inference, and defuzzification operations. Under the proposed configuration, using a few more sensors, the proposed FLC is able to process the tempo-spatial information with little complexity increased from the traditional FLC, because rules will not increase as sensors increase for the spatial measurement by using one centralized rule base. The 3-D FLC is successfully applied to a catalytic packed-bed reactor and compared with the traditional FLC. The results demonstrate its effectiveness to the nonlinear unknown distributed parameter process and its potential to a wide range of engineering applications. II. PROBLEM FORMULATION A. Distributed Parameter Systems Many processes, such as industrial chemical reactor [14], semiconductor manufacturing [15], [16], solar power plant [17], Fig. 1. Sketch of a DPS with a distributed actuator u and p point measurement sensors. and thermal processing [18], which exhibit highly nonlinear behavior and strong spatial variations, can be represented by the nonlinear partial differential equations [1] as follows: subject to the boundary conditions: and the initial condition where denotes the vector of state variables; is the spatial domain of process; is the spatial coordinate; is the time coordinate; is the vector of manipulated inputs; is the th measured output, ; is a linear diagonal spatial differential operator of the form, which involves first- and second-order spatial derivatives and is dense in Hilbert space H; is a nonlinear vector function; and are constant vectors; and are constant matrices; is a known smooth function of which is determined by the shape (point or distributed) of the th measurement sensor; is a known smooth function of, where describes how the control action is distributed in the spatial interval. As for those processes, some have strong convection characteristics [17], some have strong diffusion phenomena [19], and some have both convection and diffusion characteristics [20]. This highly nonlinear nature usually gives rise to nonlinear control problems that involve the regulation of highly distributed control variables using spatially distributed control actuators and measurement sensors [21]. B. Traditional Fuzzy Control System The nonlinear distributed parameter system discussed above with one actuator is considered as shown in Fig. 1, where point measurement sensors are located at in the one-dimensional space domain respectively and an actuator with some distribution acts on (1) (2) (3)

3 472 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 15, NO. 3, JUNE 2007 Fig D fuzzy set. Fig. 2. Traditional FLCs under different input combinations. (a) Fuzzy twoterm control. (b) Multivariable fuzzy control. (c) Fuzzy two-term control with mean input. the distributed process. Inputs are measurement information from sensors at different spatial locations, i.e., errors and errors in change, where denotes the spatial reference profile, denotes the measurement value from location and denote the th and th sample time. Input and output relationship will be described by fuzzy rules extracted from prior knowledge and expert experience. For controlling such a tempo-spatial process, naturally more sensors will be used for getting more spatial information if the mathematical model of the process is not available. 1) Fuzzy Two-Term Control: If the spatial nature of the distributed parameter system is neglected, then the distributed parameter system is simplified as a lumped system. Consequently, only one sensor is needed for input measurement, and thus a traditional two-term FLC in Fig. 2(a) can be used for PI or PD type control [22], [33]. Since only two inputs are used, the rule base will be two-dimensional with the following structure: If is and is Then is (4) where ; denotes the th rule and denote traditional fuzzy sets; denotes the control action,. This type of FLC often fails to provide satisfactory control performance because it is not inherently designed for the tempospatial process. 2) Multivariable Fuzzy Control: If more sensors are used for spatial measurement, a multivariable FLC could be used in Fig. 2(b) Since sensors are used to provide inputs, the rule base will be -dimensional with the following structure [23]: If is and and is and is and and is Then is (5) where ; denotes the th rule and denote traditional fuzzy sets; denotes the control action,. Under that rule structure, assuming that for each input and labels are designed respectively, there will be at most rules all together. The more sensors are used in space domain, the more spatial information will be processed, which will cause exponential expansion of rules. This is a long existing problem in fuzzy control design without efficient solution so far in the literature survey. Practically, this type of fuzzy system is not feasible to use. C. Fuzzy Two-Term Control With Mean Input A simple way to solve the exponential expansion of rules is to average all spatial measurement as one input. Since eventually two mean inputs are obtained, a two-dimensional rule base can be used as shown in Fig. 2(c) with the rule structure described as follows: If is and is Then is (6) where ;, and denote traditional fuzzy sets of th rule. Though the structure is simpler, the averaged spatial information may not effectively express the actual spatial distribution. In conclusion, using the two-dimensional fuzzy set, all the above traditional fuzzy control systems are not inherently designed to handle the spatial information, which limit their applications to the lumped systems. III. THREE-DIMENSIONAL FUZZY LOGIC CONTROLLER To fully utilize the model-free advantages of fuzzy system, a completely new fuzzy control method should be developed with the inherently designed features for the tempo-spatial process. This kind of fuzzy control system that is able to capture and process the tempo-spatial domain information will be defined as the 3-D FLC in this paper. One of the essential elements of this type of fuzzy system is the 3-D fuzzy set used for modeling the 3-D uncertainty. A. Three-Dimensional Fuzzy Set and Set-Theoretic Operations 1) Three-Dimensional Fuzzy Set: A 3-D fuzzy set is introduced in Fig. 3 by developing a third dimension for spatial in-

4 LI et al.: A THREE-DIMENSIONAL FUZZY CONTROL METHODOLOGY FOR A CLASS OF DISTRIBUTED PARAMETER SYSTEMS 473 Fig. 4. Basic structure of 3-D FLC. formation from the traditional fuzzy set. The definition of 3-D fuzzy set is given as follows. Definition 1: (Definition of 3-D fuzzy set) A 3-D fuzzy set defined on the universe of discourse and on the one-dimensional space is given by When and are continuous, is commonly written as where denotes union over all admissible and. When and are discrete, is commonly written as where denotes union over all admissible and. 2) Set-Theoretic Operations: Based on the definition of 3-D fuzzy set, the set-theoretic operations on 3-D fuzzy sets are given here. Let and be two 3-D fuzzy sets. Definition 2: (Definition of Union on 3-D fuzzy sets) the union and, i.e.,, is given by (7) (8) (9) (10) where, and is the t-conorm. Definition 3: (Definition of Intersection on 3-D fuzzy sets) the intersection and, i.e.,, is given by (11) where, and is the t-norm. Definition 4: (Definition of Complement on 3-D fuzzy set) The complement of, i.e.,, is given by where. (12) Using this 3-D fuzzy set, a 3-D fuzzy membership function (MF) can be developed to describe a relationship between input and the spatial variable with the fuzzy grade. Thus, a new FLC can be further developed in the next section for processing this fuzzy spatial distribution. B. Configuration of 3-D FLC and Its Control Strategy Since the 3-D fuzzy set is inherently defined to express the spatial information, it would be useful to develop fuzzy control for the distributed parameter process that has tempo-spatial nature. Theoretically, the 3-D fuzzy set or 3-D global fuzzy MF is the assembly of 2-D traditional fuzzy sets at every spatial location. However, the complexity of this global 3-D nature may cause difficulty in developing the FLC. Practically, this 3-D fuzzy MF can be approximately constructed by 2-D fuzzy MF at each sensing location. The more sensors are used in the space domain, the better 3-D global fuzzy MF could be obtained. As there is only one control output and no precise or quantitative information that can be used for design, the feasible action is to control the overall behavior of the spatial domain instead of accurately manipulating each spatial location. Thus, a centralized rule base might be more appropriate, which can also avoid the exponential explosion of rules as in (5) when sensors increase. The new FLC will have the same basic structure as the traditional one, which is composed of fuzzification, rule inference and defuzzification as shown in Fig. 4. Due to its unique 3-D nature, some detailed operations of this new FLC will be different from the traditional one as shown in Fig. 5 for spatial information processing and compressing. Crisp inputs from the space domain are first transformed into one 3-D fuzzy input via the 3-D global fuzzy MF. Then this 3-D fuzzy input will go through the spatial information fusion and dimension reduction to become a traditional 2-D fuzzy input, in which the spatial information is embedded. After that, a traditional fuzzy inference is carried out with a crisp output produced from the traditional defuzzification operation. 1) Fuzzification: Similar to the traditional FLC, there are two different fuzzifications: singleton fuzzifier and non-singleton fuzzifier [24]. Because of involving the spatial dimension, their definitions are given as follows. Definition 5: (Definition of singleton fuzzifier) Let be a 3-D fuzzy set, be a crisp input,, and be a point

5 474 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 15, NO. 3, JUNE 2007 Fig. 5. Operation details of 3-D FLC. in one-dimensional space. Singleton fuzzifier maps into in at location, then is a fuzzy singleton with support if for, and for all other with. Definition 6: (Definition of Non-Singleton Fuzzifier) Let be a 3-D fuzzy set, be a crisp input,, and be a point in one-dimensional space. Non-singleton fuzzifier maps into in at location, then is a fuzzy singleton with support if for decreases from 1 as moves away from with, and for all other with. If finite sensors are used, this 3-D fuzzification can be considered as the assembly of the traditional 2-D fuzzification at each sensing location. Therefore, for discrete measurement sensors located at shown in Fig. 1, is defined as crisp spatial input variables in space domain, where denotes the crisp input at the measurement location for the spatial input variable denotes the domain of. The variable is marked by to distinguish from the ordinary input variable, indicating that it is a spatial input variable. The fuzzification for each crisp spatial input variable is uniformly expressed as one 3-D fuzzy input in the discrete form as follows:... (13) Then, the fuzzification result of crisp inputs can be represented by (14) where denotes the t-norm operation; it has been assumed that the membership function is separable [13]. 2) Rule Inference: a) Rule Base: Using the 3-D fuzzy set, the th rule in the rule base can be expressed as follows: If is and and is Then is (15) where denotes the th rule denotes spatial input variable, and denotes 3-D fuzzy set; denotes the control action,, and denotes a traditional fuzzy set; is the number of fuzzy rules. b) Inference: The inference engine of the 3-D FLC is expected to transform a 3-D fuzzy input into a traditional fuzzy output. Thus, the inference engine must have the ability to cope with spatial information. In this paper, it is designed to have three operations: spatial information fusion, dimension reduction, and traditional inference operation as shown in Fig. 5. The inference process is about the operation of 3-D fuzzy set including union, intersection and complement operation, which are defined in the Section III-A. Considering the fuzzy rule expressed as the (15), the rule presents a fuzzy relation Thus, a traditional fuzzy set is generated via combining the 3-D fuzzy input and the fuzzy relation represented by rules.

6 LI et al.: A THREE-DIMENSIONAL FUZZY CONTROL METHODOLOGY FOR A CLASS OF DISTRIBUTED PARAMETER SYSTEMS 475 Fig. 6. Spatial information fusion at each crisp input x =[x (z); x (z)]. 3) Spatial Information Fusion: This first operation in the inference is to transform the 3-D fuzzy input into a 3-D set, appearing as a 2-D fuzzy spatial distribution at each input in Fig. 6. The set is defined by an extended sup-star composition on the input set and antecedent set. Fig. 6 gives a demonstration of spatial information fusion in the case of two crisp inputs from the space domain, i.e.,. As shown in Fig. 6, this spatial MF, approximately describing a functional distribution in space domain, is produced through the extended sup-star operation on two input sets from singleton fuzzification and two antecedent sets in a discrete space at each input value. The more sensors are used in the space domain, the better spatial distribution can be obtained. An extended sup-star composition employed on the input set and antecedent sets of the th rule, is denoted by with the grade of the MF derived as (16) where ; multiple antecedents are connected by and (t-norms); denotes the t-norm operation. Remark 1: The sup-star composition [23] is used in the traditional FLC as the compositional operator of the input and implication, where star denotes an operator, such as min and product. To make it difference, the sup-star composition used in (17) for the composition of the input set and antecedent sets will be called the extended sup-star composition in this paper. The sup-min [25] and sup-product [26] composition, which are commonly used in the traditional FLC, will be used here. 4) Dimension Reduction: The set shows an approximate fuzzy spatial distribution for each input, in which physical information such as the sensor locations and their relative distance between each other are approximately embedded. The dimension reduction operation is to compress the spatial distribution information into 2-D information. As shown in Fig. 6, the 3-D set could be simply regarded as a 2-D spatial MF on the plane for each input. Thus, the centroid operation in (18) might be a good option to compress this 3-D set into a 2-D set that can approximately describe the overall impact of the spatial distribution with respect to the input. (18) (17) where denotes a continuous plane curve and denotes the arc length of. Since finite sensors are employed in practice, the set is discrete. A polygonal path in right side of Fig. 6 is formed by connecting adjacent points with straight-line segment, and is

7 476 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 15, NO. 3, JUNE 2007 regarded as an approximation of the spatial distribution. The centroid operation in discrete form can be expressed in (19). (19) Fig. 7. Structure of 3-D Fuzzy two-term FLC. where segments are formed through sensors with as the MF grade at the location ; is the length of the straight-line segment; is the distance from axis to the centroid of straight-line segment; The physical meaning of (19) can be easily seen from its expansion in (19.a) where (19.a) where denotes the number of fired rules and denotes the composite output fuzzy set. 6) Defuzzification: After the inference, a traditional fuzzy set is produced. Then, the same defuzzification used in the traditional FLC will be taken to produce a crisp output. For engineering applications, literature [13] provides numerous candidates for defuzzifier, including maximum, mean-of-maxima, centroid, center-of-sums, height, modified height, and center-of-sets. However, there is no standard defuzzifier, thus, the center-of-sets [27] is chosen as the defuzzifier in this paper due to its simple computation. In center-of-sets defuzzification, the consequent set of each fired rule is replaced by a singleton situated at its centroid, whose amplitude equals the firing level, then the result of defuzzification is the centroid of these singletons. The output expression is given as follows: It is actually a weighted operation for capturing the overall behavior feature of the spatial domain with, and as the time-variant weights of and, respectively. These weights are nonlinear function of sensor locations and their MF grades. Shape of spatial MF will affect these weights due to fixed sensor location. Remark 2: The overall behavior of the spatial domain is defined in the mechanism (18), implemented in (19) through multiple-sensor measurement, and finely tuned via input scaling gains. Using one sensor will be difficult to do that. Thus, even though each controller formed at each sensing location looks like the same due to using the same control rule base, it may still behave differently by using different input scaling gains. After the overall behavior processing, the fuzzy input becomes the traditional 2-D input, then only the traditional fuzzy inference is required. 5) Traditional Inference Operation: This is the last operation in the inference, where implication and rules combination similar to those in the traditional inference engine are carried out. After the dimension reduction, the distance from -axis to the centroid of the reduced-dimensional fuzzy set is regarded as the firing level [13] (degree of firing [34]) of the th rule. The implication operation is executed through the following equation (20) where Mamdani implication is used; stands for a t-norm; is the membership grade of the consequent set of the fired rule ; is the output fuzzy set of the fired rule. Finally, the inference engine combines all the fired rules (21) (22) where is the centroid of the consequent set of the fired rule, which represents the consequent set in (15), and is the number of fire rules. C. Design of 3-D FLC for Distributed Parameter Process If only error and error in change are used as the crisp spatial inputs for 3-D FLC, then a 3-D two-term FLC as shown in Fig. 7 will be designed for a distributed parameter system. It is assumed that point measurement sensors are located at respectively. and are the scaled spatial inputs for 3-D two-term FLC, where denotes the scaled spatial error, denotes the scaled spatial error in change, and denote the scaling factors of actual error and error in change at the sensing location respectively. Then, a simple 2-D rule base is generated with in (15). The 3-D two-term FLC properly utilizes the spatial information only with a simple rule base, taking full advantages of the traditional fuzzy control described in Subsection II-B. Fig. 8 gives an evolution sketch of 3-D two-term FLC from the traditional two-term FLC, the multivariable FLC and the two-term FLC with mean input. For convenience, the 3-D FLC represents the 3-D two-term FLC. Currently, there is no mature method that could guide design of 3-D FLC for the distributed parameter system. Except the same problem encountered in design of the traditional FLC, the new challenges are summarized here. Design a proper 3-D global fuzzy MF for input signals. This 3-D fuzzy MF should be able to describe the functional distribution over the entire space. Fortunately, this 3-D fuzzy MF can be considered as the assembly of 2-D fuzzy MF at each spatial location. Thus, design of this 3-D

8 LI et al.: A THREE-DIMENSIONAL FUZZY CONTROL METHODOLOGY FOR A CLASS OF DISTRIBUTED PARAMETER SYSTEMS 477 Fig. 9. Membership function for e ;1e, and 1u. TABLE I 2-D LINEAR RULE BASE Fig. 8. Evolution of 3-D Fuzzy two-term FLC. fuzzy MF can be simplified as design of the 2-D fuzzy MF of each sensing input, and construction of these 2-D fuzzy MFs will directly lead to a 3-D global fuzzy MF for the FLC. The overall behavior of the spatial domain is defined in the mechanism (18), implemented in (19) through multiple-sensor measurement, and finely tuned via input scaling gains. Design of mechanism (18)/(19) will be closely related with how you measure and control the spatial performance. Properly design sensor locations for spatial measurement, which will affect capturing the overall behavior feature of the spatial domain and the subsequent control performance. Input scaling gains are useful means for finely tuning weights of spatial membership grades, which are not easy to design. Design of scaling gains may involve two important roles, one is definitely with performance, and the other might be related with stability. For simplicity, scaling gains for each spatial input in our simulation are chosen to be the same. Without the well-developed approach, all these have to be considered using the practical knowledge of the process when designing for the distributed parameter process [28]. In general, design procedure of 3-D FLC can be summarized as follows. 1) Properly design the overall behavior measurement of the spatial domain (19) according to control requirement. 2) Properly allocate measurement sensors in spatial domain according to the process nature and overall behavior measurement. 3) Properly design 2-D fuzzy MFs for each input from the spatial domain. These 2-D fuzzy MFs will form a 3-D global fuzzy MF on the entire spatial domain. 4) Design and tune scaling gains for all inputs and output for the overall behavior measurement, and the control performance. 5) Design control rule base according to the control requirement. This design is similar to those in the traditional FLC [31] [33]. In general, all the previously developed design method for the traditional FLC [31] [33] could be applied here to help design of rules, MF and even scaling gains. IV. APPLICATION IN DISTRIBUTED PARAMETER SYSTEM CONTROL A. Catalytic Packed-Bed Reactor We consider the controls of the temperature distribution in a long and thin nonisothermal catalytic packed-bed reactor [2], [29], [19]. A reaction of the form takes place on the catalyst. The reaction is endothermic and a jacket is used to heat the reactor. A dimensionless model that describes this nonlinear tubular chemical reactor is provided as follows: Subject to the boundary conditions (23) (24) where, and denote the dimensionless temperature of the gas, the catalyst, and jacket, respectively. The values of the process parameters are given as follows:

9 478 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 15, NO. 3, JUNE 2007 Fig D FLC controlled catalyst temperature and manipulated input profiles. A sin distribution heat source is used with. Suppose that the spatial reference profile is given as, then the entire spatial temperature should follow this reference. It is assumed that only catalyst temperature measurement is available. The process model (23) (24) will be simulated according to the method of lines [30] for evaluating the proposed control scheme. B. Design of 3-D FLC For simplicity and convenience, five point sensors are uniformly located along the length of the reactor with for collecting the spatial distribution of the temperature. Therefore, at each sampling time, five spatial temperatures are measured as. Two spatial inputs for 3-D FLC are error and error in change, i.e., and, where and are the th and th sample time. Initially, the scaling factor for each is set to be 1.5, the scaling factor for each is set to be 0.5, and the scaling factor for is set to be 1.0. All the scaled input variables and output variable are normalized into. Due to finite sensors used, the MFs for error and change in error are discrete. As described previously, the 3-D global fuzzy MF is the assembly of 2-D fuzzy MFs at each sensing location. Each and as well as the incremental control output are 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). For convenience, the MF is chosen as triangular shape as indicated in Fig. 9. Then, the actual output is expressed as (25) where and are the th and th sample time. A linear control rule base will be used as in Table I with the following format [31], [32] If is and is Then is O (26) where and are scaled spatial input variables, representing error and error in change, respectively; and denote 3-D fuzzy sets, which are assembled by 2-D fuzzy sets PS and NS at each sensing location; is the incremental control action; O is a 2-D fuzzy set. The rule weight is defaulted as unity. Singleton fuzzifier is used for fuzzification, the minimum is used for the t-norm in the intersection operation, the maximum is used for the -conorm in the union operation, and the center-of-sets is used for defuzzification. The 3-D FLC controlled temperature performance and manipulated input of the catalytic process are presented in Fig. 10, where the catalyst temperature tracks the reference profile well along the reactor length from zero initial condition. C. Simulation Comparisons The proposed 3-D FLC will be compared with the traditional fuzzy logic controller described in Section II-B, i.e., fuzzy twoterm control with one sensing input at and fuzzy twoterm control with mean input. All the rules, MFs, and scaling gains in the traditional FLC and the 3-D FLC are set to be the same. The quantitative performance criteria should be able to consider the performance on the entire space domain. The commonly used criteria in control of non-pde process, such as steady-state error (SSE), integral of the absolute error (IAE), and integral of time multiplied by absolute error (ITAE), are modified and defined in (27) (29), respectively, for the distributed parameter process (27) (28) (29) where is the time when the system is at the steady state; the space domain region is set as, and ; and the simulation duration is set as 8 s in the paper. Simulation performance is compared in Fig. 11 when there is no parameter perturbation and in Fig. 12 when there is a

10 LI et al.: A THREE-DIMENSIONAL FUZZY CONTROL METHODOLOGY FOR A CLASS OF DISTRIBUTED PARAMETER SYSTEMS 479 TABLE II PERFORMANCE COMPARISONS Fig. 11. Catalyst temperature profiles under ideal condition. (a) 3-D FLC. (b) Fuzzy two-term control with mean input. (c) Fuzzy two-term control with one sensing input. (Dotted line: Reference profile. Solid line: Actual temperature profile at steady state) Fig. 12. Catalyst temperature profiles under disturbed condition. (a) 3-D FLC. (b) Fuzzy two-term control with mean input. (c) Fuzzy two-term control with one sensing input (Dotted line: Reference profile. Solid line: Actual temperature profile at steady state).

11 480 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 15, NO. 3, JUNE 2007 parameter perturbation (50% increase of that is equivalent to 50% increase of velocity of gas). The quantitative comparisons for both ideal and perturbed conditions are given in Table II. The simulation results show that the 3-D FLC performs better than the traditional FLCs. The fuzzy two-term control with one sensing input has a worse performance no matter in ideal or disturbed conditions, since it only uses one sensing input. Although both the 3-D FLC and the fuzzy two-term control with mean input utilize more sensing information, the proposed 3-D FLC can still achieve better performance in all conditions. This is mainly attributed to the inherent 3-D fuzzy set that can explicitly model the spatial information and the 3-D inference engine that can process the spatial information. V. CONCLUSION In this paper, a completely new fuzzy control methodology is proposed for a class of tempo-spatial processes. Different to the traditional FLC, this novel FLC has the 3-D fuzzy set and 3-D inference engine that are able to process the spatial information. Similar to the traditional FLC, this 3-D FLC also consists of fuzzification, rule inference, and defuzzification. With the help of the novel structure design, a simple 2-D rule base can still be used for 3-D fuzzy MFs. Thus, rules will not increase as sensors increase for measuring spatial information, which minimizes the computation for real world applications. Finally, the 3-D FLC is successfully applied to a catalytic packed-bed reactor and compared with the traditional FLC. The results demonstrate its effectiveness to the nonlinear unknown distributed parameter process and its potential to a wide range of engineering applications. REFERENCES [1] P. D. Christofides, Control of nonlinear distributed process systems: Recent developments and challenges, AIChE J., vol. 47, no. 3, pp , [2] W. H. Ray, Advanced process control. 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12 LI et al.: A THREE-DIMENSIONAL FUZZY CONTROL METHODOLOGY FOR A CLASS OF DISTRIBUTED PARAMETER SYSTEMS 481 Han-Xiong Li (S 94 M 97 SM 00) received the the B.E. from the National University of Defence Technology, China, in 1982, the M.E. degree in electrical engineering from Delft University of Technology, Delft, The Netherlands, in 1991, and the Ph.D. degree in electrical engineering from the University of Auckland, Auckland, New Zealand, in Currently, he is an Associate Professor in the Department of Manufacturing Engineering and Engineering Management, the City University of Hong Kong. He is a Chang Jiang Scholar an Honorary Professorship awarded by the Ministry of Education, China. Over the last twenty years, he has had opportunities to work in different fields, including military service, industry, and academia. His research interests include intelligent control and learning, intelligent modelling, and control of distributed parameter systems. Dr. Li serves as an Associate Editor for the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, PART B. He was awarded the Distinguished Young Scholar fund by the China National Science Foundation in Xian-Xia Zhang received the B.S. degree in automatic control from the University of Science and Technology Beijing, Beijing, China, in 1998, and the M.E. degree in measurement techniques and instrumentation from Shanghai University, Shanghai, China, in She is currently working toward the Ph.D. degree at Shanghai Jiao Tong University, Shanghai, China. Her research interests include fuzzy system, intelligent control, and distributed parameter system. Shao-Yuan Li (SM 05) was born in He received the B.S. and M.S. degrees from Hebei University of Technology, Hebei, China, in 1987 and 1992, respectively, and the Ph.D. degree from the Department of Computer and System Science of Nankai University, China, in He is currently a Professor with the Institute of Automation, Shanghai Jiao Tong University, Shangai, China. His research interests include fuzzy systems and nonlinear system control.