Fuzzy Modeling with Emphasis on Analog Hardware Implementation Part I: Design

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1 Fuzzy Modeling with Emphasis on Analog Hardware Implementation Part I: Design SHAKTI KUMAR Haryana Engineering College Jagadhari, Yamunanagar INDIA K.K. AGGARWAL GGS Indraprastha University Delhi INDIA ARUN KHOSLA Department of Electronics and Communication Engineering National Institute of Technology Jalandhar INDIA Abstract: - In this paper, a new approach of fuzzy modeling with emphasis on analog hardware implementation is suggested. In the suggested approach, the input space of each input variable is partitioned independently through the modified form of fuzzy c-means (FCM) clustering algorithm to generate the membership functions. By considering all the possible combinations between the membership functions of the input variables and the cluster centers, the complete rule-base is generated through reference to the available input-output data. The output membership functions are taken to be singletons for easy and straightforward implementation in analog domain. The created rule-base is then optimized through the exhaustive search techniques suggested by the authors. The suggested approach has been applied on fuzzy controller data for rapid Nickel-Cadmium batteries charger developed [1]. The data for the batteries charger has been obtained through experimentation with an objective to charge the batteries as fast as possible. Special purpose analog modules, designed by the authors and realized with operational amplifiers (op-amps), diodes, resistors are considered for hardware implementation. The proposed fuzzy modeling design approach is described in Part I. Part II describes the synthesis and analog implementation. Key Words: - fuzzy modeling, FCM, membership functions, analog modules 1 Introduction Fuzzy models are generally identified through following two approaches: 1) Knowledge-driven design, in which the fuzzy system is identified using the human knowledge 2) Data-driven design, in which the fuzzy system is derived from the input-output data Design and implementation of a fuzzy system depends on the application towards which it is addressed. Non-real time applications like data-analysis or 1

2 decision-making are software implemented on a personal computer and provide a high flexibility since they support fuzzy systems with an arbitrary number of rules, without any limitations concerning number and types of membership functions and wide range of inferencing mechanisms. Fuzzy logic toolbox for Matlab is such an application package. On the other hand real-time applications are generally implemented using general-purpose processors. Although the fuzzy systems are inherently parallel systems, but generalpurpose processors are exclusively sequential and the three processing stages of a fuzzy system viz. fuzzification, inference and defuzzification are performed serially thus resulting in a high response time. Fuzzy chips employing parallel architectures [2] have been proposed in order to reduce the processing time and both analog and digital techniques are used for implementation. These chips are costly and don t provide any flexibility for accommodating more number of input & output variables and fuzzy rules than the specified numbers. Analog circuits have been found to very attractive for implementing arithmetic functions used in fuzzy controllers. Parallelism can be employed through analog circuits and hence they offer the advantage of high speed. Analog fuzzy modules are general modules for fuzzification such as membership function generating circuits, modules related to inference engine, such as fuzzy operator implementation circuits and defuzzification circuits. The modules can be organized by designer to implement fuzzy controller specific to his particular application. The implementation through analog fuzzy modules is very flexible, since it is capable of accommodating any number of input and output variables in contrast to fuzzy chips, where number of inputs and outputs are fixed. This paper presents a new fuzzy modeling approach with emphasis on analog hardware implementation. General-purpose analog fuzzy modules are considered for implementation. This paper is organized as follows. In Section 2, the algorithm for building fuzzy model from the available inputoutput data is explained. The created fuzzy model may contain certain redundant rules, which can be identified and removed through the exhaustive search technique suggested by the authors. The exhaustive search technique is discussed in Section 3, which is applied on the rule-base generated. Section 4 compares the hardware requirements of the reduced system with the system considering all the rules. Finally, conclusions are drawn in Section 5. 2 The Proposed Algorithm STEP I: Generation of the Membership functions and cluster centers through the modified FCM algorithm Fuzzy clustering algorithms [3, 4, 5] are unsupervised algorithms used for partitioning the data into a pre-defined number of clusters with fuzzy boundaries and are used extensively for classification, approximation problems and recognition of geometrical shapes in image processing. Consider the data matrix Z consists of vectors z k, k=1, 2,.,N, contained in its column and each vector is n-tuple. These vectors are partitioned into c clusters, 2

3 and clusters are represented by prototype vectors, v i = [v i1, v i2,.,v in ] T R n, i = 1, c. Prototype matrix is represented by V and has v i in its i th -column. The fuzzy partition is represented by the matrix U R c x N, whose element, µ i,k [0,1] and represents the membership degree of the data vector z k in i th -cluster. FCM clustering algorithm partitions the data Z into c overlapping clusters so as to minimize the objective function defined in equation (1). c N m 2 J ( Z; V, U ) = ( µ ) d ( z, v ) (1) i= 1 k = 1 i, k k i The exponent m (1, ) determines the fuzziness of the clusters and for most of the applications, m=2 [4]. d( z k, vi ) defines the distance of data vector zk from the cluster prototype v i. The minimization of equation (1) is carried out subject to the constraints defined in equations (2) & (3). c i= 1 µ 1, k=1,.,n (2) i, k = N 0 < µ i, k < N, i=1,.,c (3) k = 1 Equation (2) implies that the membership coefficients for each data point must add to unity and equation (3) ensures that the clusters are neither empty nor contain all the points to degree 1. The application of FCM algorithm implies the minimization of equation (1) subject to the constraints defined in equations (2) & (3) and leads to the finding fuzzy partition matrix U, and prototype matrix V. Some important observations about FCM [6] are: i) FCM always converges for m>1 ii) iii) FCM finds local minima of the objective function, because it is derived from the gradient of the objective function The results of FCM not only depends on the parameters m & c, but also on the choice of initial prototypes Following parameters are to be defined by the user for applying FCM algorithm: 1. Number of clusters (c) 2. m (>1) 3. termination criteria, ε (say)>0 In the modified clustering algorithm, the partitioning of each input variable is done independently i.e data matrix Z represents one-dimensional data. The modified FCM algorithms is listed as below: Step 1: Initialize partition matrix U (0) (cxn) U = [µ ik ] Step 2: At step r calculate cluster center V (r) =[ v ij ], using v n k = 1 ij = n µ x k = 1 m ik µ m ik kj, j= 1, 2, 3.. m Step 3: Update partition matrix for r th step by 1 1 ( ) ( ) 2 /( 1) r+ = c r r m µ ik ( d ik / d jk ) for I k = φ j= 1 µ r+1 ik = 0, for all classes I, where I Є I k where I k ={i 2 c n; d ik (r ) =0} and I k ={1,2.,c} - I k Step 4: If U (r+1) - U (r) ε,stop; else set r=r+1 and return to step 2. 3

4 Step 5: Find the cluster centers V 1 and V 2 where, V 1 is the cluster center with minimum distance from the first data point V 2 is the cluster center with maximum distance from the first data point Step 6: For V 1 : If µ i1 = 0.0 then z=c+1 µ zk = 1- µ ik for x k < V 1 else µ zk = 1 for x k < V 1 where, k= 1,2..n Figure 1. Membership functions for the input variable Temperature (T) For V 2 : If µ in = 0.0 then z=c+1 µ zk = 1- µ ik for x k > V 2 else µ zk = 1 for x k > V 2 where, k= 1,2..n The purpose of this step is to place membership functions with appropriate shape at the two extremes. The modified FCM algorithm is applied on the battery charger data. The membership functions generated are shown in figures 1 and 2 for the two input variables viz. temperature and temperature gradient respectively. For the input variable temperature, 12 clusters were defined and for the other input variable temperature gradient, 4 clusters were taken. The cluster centers for the two input variables are shown in Table 1. In an air-conditioning environment, the ambient temperature is close to 25 o C, the universe of discourse for the variable temperature has been taken from o C, in contrast to the training data, where the values are between 0-50 o C. Figure 2. Membership functions for the input variable Temperature gradient (dt) STEP II: Generating Rule-base from the cluster centers generated in the Step I and training data From the cluster centers generated in the Step I and the training data available, rules can be generated. The number of rules generated for the given case are 12x4=48. The rules generated are represented as fuzzy associated memory and are shown in Table. 2. The rules can be read as under: Rule 1 If T is T1 and dt is dt1 then Charging current is 8 Rule 2 If T is T2 and dt is dt1 then Charging current is 8 4

5 Table 1: Cluster Centres Temperature Cluster number Cluster Centre Temprature Gradient Cluster number Cluster Centre This is in contrast to the approach suggested by Mendel et. al [8, 9], where a rule is generated corresponding to every data point. A fuzzy rule base, be it generated from experts or by some learning or identification schemes may contain redundant, weakly contributing or outright inconsistent rules. It is therefore, highly desirable to extract the more pertinent elements of a given rule set. Researchers have formulated various techniques for this objective, yet there is no uniformly accepted approach for designing a fuzzy rule set efficiently and effectively. Various orthogonal transformation methods [10, 11, 12, 13, 14] have been proposed for selecting important fuzzy rules from a given rule base. K. Nozaki et.al [15] proposed a method for automatically generating fuzzy if-then rules from numerical data. Genetic algorithms have also been used [16, 17, 18, 19, 20, 21, 22, 23, 24] for optimizing fuzzy membership functions and fuzzy rule base. In this paper, the redundant rules are eliminated using the exhaustive search technique proposed by the authors [7]. The technique is explained in the following section. 3 Exhaustive Search Algorithm [7] Exhaustive Search Technique is based on the identity + Y =, which means that the term Y is contained in and is not affecting the output and hence can be dropped. The proposed algorithm involves searching for such rule combinations where variables can be dropped and the rules can be merged to get a reduced rule-base. The exhaustive search algorithm is easy to apply if all the membership functions corresponding to input variables are listed in sequence. The algorithm searches such rule groupings, in which only one of the input variables is changing and all the consequents are similar. If all the rules are listed in sequence, then the rules forming the group based on the above criteria shall be adjacent. Whenever such a grouping is found, one of the variables in the grouping shall be dropped and all the rules in the grouping can be merged to get a single rule. The basic step of the algorithm can be understood from the Table 3. The suggested algorithm searches for such combinations as found for the above three rules and is being represented by (1,2,3), thereby 5

6 Table 2: Fuzzy Rules Generated T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 dt dt dt dt suggesting that two rules can be dropped and the variable at third position can be removed. Table 3: Rules 1, 2 & 3 in indexed form Rule No. I 1 I 2 I 3 O (1,2,3) Thus the three rules can be replaced by single rule represented in indexed form as under: Rule I 1 I 2 I 3 O No Exhaustive search algorithm is applied on the rule-base generated and is shown in the figure 4. Through this approach, the numbers of rules have been reduced from 48 to 9. The reduced rules are listed as below: Rule No. Rule 1. If T is T1-T7 (merging of membership functions from T1-T7) then output is 8 2. If T is T8 then output is 6 3. If T is T9 and dt is dt1 then output is 4 4. If T is T9 and dt is dt2 then output is 4 5. If T is T9 and dt is dt3 then output is 4 6. If T is T9 and dt is dt4 then output is 4 7. If T is T10 then output is 2 8. If T is T11 then output is 1 9. If T is T12 then output is Hardware Savings The analog fuzzy modules built around op-amp and other components can be organized to realize the fuzzy system designed in Section 2. The organization of these modules to implement the system is shown in figure 5. The numbers of different modules required are listed in Table 4. If the redundant rules are dropped, the hardware requirements shall decrease. Number of different modules required for analog hardware implementation after removing the redundant rules by applying exhaustive search techniques as described in Section 3 are shown in table 5. Table 4 Number of different modules required for analog hardware implementation (without applying exhaustive search algorithm) Membership 16 Function generators Circuits 48 Implication Circuits 48 MA circuit for With 48 inputs defuzzification 6

7 Rule 4 Rule 3 Rule 7 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 dt dt dt dt Rule 1 Rule 2 Rule 6 Rule 9 Rule 8 Figure 3 Reduced Rule-Base Rule 5 Fuzzification Composition Implication T MF generator MF generator dt MF generator MA (Defuzzification) Crisp Output Figure 4 Organization of analog modules for fuzzy system implementation 7

8 Table 5 Number of different modules required for analog hardware implementation (after applying exhaustive search algorithm) Membership 10 Function generators Circuits 9 Implication 9 Circuits MA circuit for With 9 inputs defuzzification 5 Conclusions In Part I of the paper, we have used modified FCM clustering algorithm to design the fuzzy system from the available input-output data. The designed system is optimized through exhaustive search technique to remove the redundant rules. The approach is especially suitable if the system is to be implemented through analog hardware modules. The results are specific to system under consideration and shall vary from system to system. If the fuzzy system is implemented on a generalpurpose microprocessor, the reduced rule-base shall reduce the processing time. In part II, the synthesis and implementation of the design is presented. References [1]. Arun Khosla, Shakti Kumar and K.K. Aggarwal, Design and Development of RFC-10: A Fuzzy Logic Based Rapid Battery Charger for Nickel-Cadmium Batteries, Proceedings of HiPC 2002 Workshop on Soft Computing (WoSCo'02), Bangalore, India, pp [2]. I. Baturone, Á. Barriga, S. Sánchez- Solano, C.J. Jiménez-Fernández, D.R. Lopez., Microelectronic Design of Fuzzy-Logic Based Systems, CRC Press, [3]. Frank Hoppner, Frank Klawonn, Rudolf Kruse and Thomas Runkler, Fuzzy Cluster Analysis: Methods for Classification, Data Analysis and Image Processing, John Wiley, [4]. H.Hellendoorn, D. Driankov (Eds.), Fuzzy Model Identification: Selected Approaches, Springer, [5]. J.C. Bezdek, Pattern Recognition with Fuzzy Objective Fuzzy Algorithms, Plenum Press, [6]. John Yen and Reza Langari, Fuzzy Logic: Intelligence, Information and Control, Pearson Education, 2003 (First Indian Reprint). [7]. Arun Khosla, Shakti Kumar and K.K. Aggarwal, Hardware Reduction for Fuzzy based Systems Via Exhaustive Search Technique, National Seminar on Emerging Convergent Technologies and Systems (SECTAS-2002), Dayalbagh Educational Institute, Agra, India, March 1-2, [8]. Mendel J.M., Fuzzy Logic Systems for Engineering: A Tutorial, Proceedings of the IEEE, Vol. 83, March pp [9]. Li-in Wang and J.M. Mendel, Generating fuzzy rules by learning from examples, IEEE Transactions on Systems, Man and Cybernetics, Vol. 22, December pp [10]. G.C. Mouzouris and J.M. Mendel, "Designing fuzzy logic systems for uncertain environments using a singular value-qr decomposition method," Proceedings of the Fifth IEEE International Conference on Fuzzy Systems, New Orleans, LA, pp , [11]. J.Yen and L.Wang, "An SVD-based fuzzy model reduction strategy," Proceedings of the Fifth IEEE International conference on Fuzzy Systems, New Orleans, LA, pp ,

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