A Software Tool: Type-2 Fuzzy Logic Toolbox

Transcription

1 A Software Tool: Type-2 Fuzzy Logic Toolbox MUZEYYEN BULUT OZEK, ZUHTU HAKAN AKPOLAT Firat University, Technical Education Faculty, Department of Electronics and Computer Science, Elazig, Turkey Received 18 January 2006; accepted 15 January 2007 ABSTRACT: The concept of type-2 fuzzy set was initially proposed as an extension of classical (type-1) fuzzy sets. Type-2 fuzzy sets are very useful in circumstances where it is difficult to determine an exact membership function for a fuzzy set; hence they are very effective for dealing with uncertainties. However, type-2 fuzzy sets are more difficult to use and understand than type-1 fuzzy sets. Even in the face of these difficulties, type-2 fuzzy logic has found applications in many fields. In this article, a new Type-2 Fuzzy Logic Toolbox written in MATLAB programming language is introduced. The main aim is to help the user to understand and implement type-2 fuzzy logic systems easily. Type-2 Fuzzy Logic Controller Block is also prepared for use in SIMULINK. Since general type-2 fuzzy logic systems are very complicated, they are not preferred in applications. Thus, only interval type-2 fuzzy logic systems are considered in the proposed Type-2 Fuzzy Logic Toolbox. Since MATLAB Fuzzy Logic Toolbox users are familiar with its windows, all the menus of the developed software are prepared in the same format of the MATLAB Fuzzy Logic Toolbox. ß 2008 Wiley Periodicals, Inc. Comput Appl Eng Educ; Published online in Wiley InterScience ( DOI / cae Keywords: type-2 fuzzy logic; type-2 fuzzy sets; interval sets; type-2 fuzzy logic toolbox INTRODUCTION Recently, a new class of fuzzy logic systems type-2 fuzzy logic systems which are useful for incorporating uncertainties is introduced [1 4]. Type-2 fuzzy logic is able to handle uncertainties because it can model them and minimize their effects. Unfortunately, type-2 fuzzy sets are more difficult to use and understand than traditional type-1 fuzzy sets. Therefore, their use is not widespread yet. Even in the face of these difficulties, type-2 fuzzy logic has found applications in the classification of coded video Correspondence to Z. H. Akpolat (zh_akpolat@yahoo.com). ß 2008 Wiley Periodicals Inc. streams, co-channel interference elimination from nonlinear time-varying communication channels, connection admission control, extracting knowledge from questionnaire surveys, forecasting of timeseries, function approximation, pre-processing radiographic images and transport scheduling [3]. Type-2 fuzzy logic has been also used in some control applications [5 9]. A short summary of the existing literature on type-2 fuzzy sets can be found in [1 3]. Currently, there are some software programs like Fuzzy Logic Toolbox of MATLAB 1 /SIMULINK (MATLAB is a registered trademark of The Math- Works, Inc., Natick, MA) by which the simulation of type-1 fuzzy logic systems can be done easily but, there is no such software for the simulation of type-2 1

2 2 OZEK AND AKPOLAT fuzzy logic systems. In this article, a new Type-2 Fuzzy Logic Toolbox is introduced for direct implementation of type-2 fuzzy logic systems in MATLAB and SIMULINK. Type-2 Fuzzy Logic Controller Block is also prepared for the use of SIMULINK applications. General type-2 fuzzy logic systems are not preferred in applications due to their computational complexity. Thus, only interval type-2 fuzzy logic systems are considered in the proposed Type-2 Fuzzy Logic Toolbox. Since MATLAB Fuzzy Logic Toolbox users are familiar with its windows, all the menus of the developed software are prepared in the same format of the MATLAB Fuzzy Logic Toolbox. TYPE-2 FUZZY LOGIC SYSTEMS Fuzzy set theory, introduced by Zadeh [10], has found wide applications in many fields as well as in control systems [2]. Fuzzy logic control has emerged as a practical alternative to the conventional control techniques since it provides a decision making mechanism which allows the designer to put expert knowledge into the controller. However, the classical or traditional fuzzy logic systems (type-1 fuzzy logic systems) cannot fully handle the linguistic, measurement and parameter uncertainties [1]. In order to reduce the effects of uncertainties, a new class of fuzzy logic systems type-2 fuzzy logic systems are introduced recently [2]. The concept of type-2 fuzzy set was initially proposed as an extension of ordinary (or type-1) fuzzy sets by Zadeh [11]. Mendel and his students [1 4] have recently introduced a complete theory of type-2 Fuzzy Logic Systems (FLSs) which are again expressed by IF-THEN rules but, their consequent and/or antecedent sets are type-2 fuzzy sets. These sets are fuzzy sets whose membership grades are not crisp values; instead, they are type- 1 fuzzy sets. Type-2 fuzzy sets are very useful in circumstances where it is difficult to determine an exact membership function for a fuzzy set; hence they are useful for dealing with uncertainties. Consider the transition from ordinary sets to fuzzy sets: when the membership of an element in a set cannot be determined as 0 or 1, type-1 fuzzy sets are used. Similarly, when we have difficulties in the determination of membership grade even as a crisp number in [0,1], type-2 fuzzy sets are then used. type-2 fuzzy set. Let us consider blurring the type-1 membership function shown in Figure 1a by shifting the points on the membership function to the left or to the right but, not necessarily by the same amounts, as shown in Figure 1b. Then, at a certain value of x, say x 1, the membership function (or the membership degree) is not a crisp value any more; instead, it takes on values wherever the vertical line intersects the blur. Those values need not all be weighted the same; hence, an amplitude distribution can be assigned to all of those points. Assume that this is done for all x 2 X (X is the universe of discourse), then, a three dimensional membership function is created, which is called type- 2 membership function that characterizes a type-2 fuzzy set [2]. For the clarity of explanations through the article, some definitions are given below: Definition-1: A type-2 fuzzy set, denoted ~A, is characterized by a type-2 membership function m A ~ ðx; uþ, where x 2 X and u 2 J x [0,1], that is, ~A ¼ fððx; uþ; m A ~ ðx; uþþj8x 2 X; 8u 2 J x ½0; 1Šg ð1þ in which 0 m A ~ ðx; uþ 1. ~A can also be expressed as Z Z m A ~ ðx; uþ ~A ¼ J x ½0; 1Š ð2þ ðx; uþ x2x u2j x where RR denotes union over all admissible x and u. Definition-2: At each value of x, say x ¼ x 1, the 2D plane whose axes are u and m A ~ ðx 1; uþ is called a vertical slice of m A ~ ðx; uþ. It is m ~A ðx ¼ x 1; uþ for x 1 2 X and 8u 2 J x1 ½0; 1Š, that is, m ~ A ðx ¼ x 1; uþ m ~ A ðx 1Þ¼ Z u2jx 1 f x1 ðuþ u J x1 ½0; 1Š ð3þ TYPE-2 FUZZY SETS AND MEMBERSHIP FUNCTIONS A type-2 membership function is actually a three dimensional membership function that characterizes a Figure 1 (a) A type-1 membership function. (b) Blurred type-1 membership function.

3 A SOFTWARE TOOL 3 in which 0 f x1 ðuþ 1. Because x 1 2 X, 1 is droped on m A ~ ðx 1Þ and it is referred to m A ~ ðxþ as a secondary membership function; it is a type-1 fuzzy set, which is also referred to as a secondary set. Definition-3: A type-2 fuzzy set can be expressed as the union of all secondary sets, that is, using (3), ~A, in a vertical-slice manner, can be re-expressed as ~A ¼fðx; m A ~ ðxþþj8x 2 Xg ð4þ or ~A ¼ Z x2x m ~ A ðxþ=x ¼ Z x2x R f x ðuþ=u u2j x x J x ½0; 1Š ð5þ Definition-4: The domain of a secondary membership function is called the primary membership of x. In (5), J x is the primary membership of x, where J x ½0; 1Š for $\forall x \in X: Definition-5: The amplitude of a secondary membership function is called a secondary grade. In (1), m ~ A ðx 1; u 1 Þðx 1 2 X; u 1 2 J x1 Þ is a secondary grade; in (5), f x (u) is a secondary grade. Definition-6: When f x ðuþ ¼1; 8u 2 J x ½0; 1Š, then the secondary membership functions are interval sets, and if this is true for 8x 2 X, an interval type-2 membership function is obtained. Interval secondary membership functions reflect a uniform uncertainty at the primary memberships of x. Definition-7: Uncertainty in the primary memberships of a type-2 fuzzy set, ~A, consists of a bounded region that we call the footprint of uncertainty (FOU). It is the union of all primary memberships, that is, Example-1: The shaded region in Figure 2a is the FOU for a type-2 fuzzy set. The primary memberships, J x1 and J x2, and their associated secondary membership functions m A ~ ðx 1Þ and m A ~ ðx 2Þ are shown at the points x 1 and x 2. The upper and lower membership functions, m A ~ ðxþ and m ~A ðxþ, are also shown in Figure 2a. The secondary membership functions, which are interval sets, are shown in Figure 2b. Details of the definitions above can be found in Ref. [1]. STRUCTURE OF A TYPE-2 FUZZY LOGIC SYSTEM The structure of a type-2 Fuzzy Logic System (FLS) is shown in Figure 3. It is actually very similar to the structure of an ordinary type-1 FLS. It is assumed in this article that the reader is familiar with type-1 FLSs and thus, in this section, only the similarities and differences between type-2 and type-1 FLSs are underlined. The fuzzifier shown in Figure 3, as in a type-1 FLS, maps the crisp input into a fuzzy set. This fuzzy set can be a type-1, type-2 or a singleton fuzzy set. In singleton fuzzification, the input set has only a single point of nonzero membership. The singleton fuzzifier is the most widely used fuzzifier due to its simplicity and lower computational requirements. However, this kind of fuzzifier may not always be adequate especially in the cases of uncertainties [1]. Therefore, the non-singleton fuzzification, which is more FOUð~AÞ ¼[ x2x J x ð6þ Definition-8: An upper membership functions and a lower membership function are two type-1 membership functions that are bounds for the FOU of a type-2 fuzzy set ~A. The upper membership function is associated with the upper bound of FOUð~AÞ, and is denoted m A ~ ðxþ, 8x 2 X. The lower membership function is associated with the lower bound of FOUð~AÞ, and is denoted m A ~ ðxþ, 8x 2 X, that is, and m ~ A ðxþ ¼FOUð ~AÞ 8x 2 X ð7þ m ~ A ðxþ ¼FOUð ~AÞ 8x 2 X ð8þ Because the domain of a secondary membership function has been constrained in (1) to be contained in [0,1], lower and upper membership functions always exist. Figure 2 (a) The FOU for a type-2 fuzzy set. (b) The secondary membership functions.

4 4 OZEK AND AKPOLAT Figure 3 The structure of a type-2 Fuzzy Logic System. effective as far as the uncertainties are concerned, is used in most studies. In type-1 non-singleton fuzzification, measurement x i ¼ x 0 i is mapped into a fuzzy number; that is, the inputs are modeled as type- 1 fuzzy numbers and membership functions are associated with them. In other words, a type-1 nonsingleton fuzzifier is one for which m Xi ðx 0 i Þ¼1 (i ¼ 1,...,p) and m Xi ðx i Þ decreases from unity as x i moves away from x 0 x. A type-2 FLS whose inputs are modeled as type-1 fuzzy numbers is referred to as a type-1 non-singleton type-2 FLS. Similarly, a type-2 FLS whose inputs are modeled as type-2 fuzzy numbers is referred to as a type-2 non-singleton type-2 FLS [1]. In type-1 FLSs, IF-THEN rules are generally used, which has the form of (the lth rule) R l : IF x 1 is F1 l and...and x p is Fp l ; ð9þ THEN y is G l l ¼ 1;...; M where x i s are inputs (i ¼ 1,...,p), Fi l s are antecedent sets, G l s are consequent sets and y is the output. The difference between type-2 and type-1 FLS is associated with the nature of the membership function but, this is not important while constructing the rule base. Hence, the structure of the rules does not change in the type-2 case, the only difference is that some or all of the fuzzy sets involved are type-2. It should be noted that we will have a type-2 FLS as long as at least one of the antecedent or consequent sets is a type-2 fuzzy set. The lth rule in a type-2 FLS has the form of R l : IF x 1 is ~F 1 l and...and x p is ~F p l ; ð10þ THEN y is ~G l l ¼ 1;...; M where implies that the fuzzy set is a type-2 fuzzy set. The inference engine of a type-1 FLS provides a mapping from input type-1 fuzzy sets to output type-1 fuzzy sets by using all rules. The antecedents in a rule are connected by t-norm which corresponds to intersection of the fuzzy sets. By using the sup-star composition, the membership grades in the input fuzzy sets are combined with those in the output fuzzy sets and then, all the rules may be combined by t-conorm operation (union of fuzzy sets) or by defuzzfication using the weighted summation. The inference process in a type-2 FLS is very similar to that in a type-1 FLS. The rules are combined by the inference engine that provides a mapping from input type-2 fuzzy sets to output type-2 fuzzy sets. In order to do this, intersections, unions and compositions of type-2 fuzzy sets are required. In a type-1 FLS, a crisp output is produced by the defuzzifier from the output of the inference engine, which is actually a fuzzy set. On the other hand, in a type-2 FLS, the output of the inference engine is normally a type-2 fuzzy set. Using the Zadeh s extension principle [1], a type-reduced set, that is a type-1 fuzzy set, is obtained from the type-2 output sets of the FLS. This operation is called type reduction. The type reduction is an important calculation for Type-2 FLSs. It is a new and complicated concept, and details of type reduction methods can be found in Refs. [1 3]. Hence, the type-reduced set can be defuzzified using well known techniques (e.g., centroid, bisector, mean of maximum, smallest of maximum and largest of maximum) to obtain a crisp (type-0) output from a type-2 FLS. A general type-2 FLS is very complicated because of type reduction. Interval type-2 fuzzy sets given by Definition-6 are the most widely used type-2 fuzzy sets because they are simple to use and calculations simplify a lot when the secondary membership functions are interval sets in which case the type-2 FLS is called an interval type-2 FLS. In the existing literature on FLSs, two most popular FLSs are the Mamdani and Takagi-Sugeno-Kang (TSK)

5 A SOFTWARE TOOL 5 systems. Up to this point, even though it is not referred to them as such, all the FLSs were Mamdani FLSs. The Mamdani and TSK FLSs are both characterized by IF-THEN rules and have the same antecedent structures, however, they differ in the structure of the consequent parts. The consequent of a TSK rule is a linear or nonlinear function of input variables, whereas the consequent of a Mamdani rule is a fuzzy set. In a type-1 Mamdani FLS, the output of the inference engine is a type-1 fuzzy set and defuzzification is used to obtain a crisp output (type-0 set). On the other hand, the output of a type-1 TSK FLS is a crisp value and defuzzification is not required. Similarly, although a type reduction procedure exists in type-2 Mamdani FLSs, there is no type reduction needed for type-2 TSK FLSs [1]. THE DEVELOPED SOFTWARE: TYPE-2 FUZZY LOGIC TOOLBOX The developed software Type-2 Fuzzy Logic Toolbox consists of MATLAB 1 -based functions, that is, M-files (MATLAB is a registered trademark of The MathWorks, Inc.) and it is designed to ensure a user friendly tool for the simulation of interval type-2 fuzzy logic systems. Some of the functions made freely available by Mendel [1] are also used in The Type-2 Fuzzy Logic Toolbox. The developed software can be run by typing fuzzy2 on the command line of MATLAB. The Type-2 Fuzzy Logic Toolbox provides a simple point-and-click interface that guides the user effortlessly through the steps of FLS design. It extends the MATLAB technical computing environment with tools for the design of systems based on type-2 fuzzy logic. Graphical User Interfaces (GUIs) guide the user through the steps of type-2 fuzzy inference system design. The toolbox lets the user implement complex type-2 FLSs using simple logic rules. It can be used as a stand-alone type-2 fuzzy inference engine. Alternatively, type-2 fuzzy inference blocks can be used in Simulink and the type-2 fuzzy systems can be simulated within a comprehensive model of the entire dynamic system by using the Type-2 Fuzzy Logic Controller Block that is prepared and added to the Simulink Library. Like all MATLAB toolboxes, the Type-2 Fuzzy Logic Toolbox can be customized. The user can inspect algorithms, modify source code, and add membership functions, defuzzification techniques, implication, aggregation AND, OR and type reduction methods. Windows, Editors, and Viewers of the Type-2 Fuzzy Logic Toolbox Fuzzy inference is a method that interprets the values in the input vector and, based on user-defined rules, assigns values to the output vector. Using the GUI editors and viewers in the Fuzzy Logic Toolbox, the user can build the rules set, define the membership functions, and analyze the behavior of a fuzzy inference system (FIS). The provided editors and viewers are FIS editor, membership function editor, rule editor, rule viewer and surface viewer.fis Editor Figure 4 shows the FIS Editor that displays general information about a type-2 fuzzy inference system. It displays actually a menu bar that allows the user to open related GUI tools, open and save systems, and so on. The File menu let the user to open a new interval type-2 fuzzy system which can be a singleton type-2 Mamdani FIS, type-1 nonsingleton type-2 Mamdani FIS, type-2 nonsingleton type-2 Mamdani FIS or type-2 Sugeno FIS with no variables and no rules called Untitled. Under file menu, the other options that the user can select are Open from disk... to load a system from a specified.fis file on disk. Save to disk...to save the current system to a.fis file on disk. Save to disk as... to save the current system to disk with the option to rename or relocate the file. Open from workspace...to load a system from a specified FIS structure variable in the workspace. Save to workspace... to save the system to the currently named FIS structure variable in the workspace. Save to workspace as... to save the system to a specified FIS structure variable in the workspace. Close window to close the GUI. The Edit menu let the user add another input or output to the current system. The user can also delete a selected variable or undo the most recent change under edit menu. Under View menu, the options are Edit MFs...to invoke the Membership Function Editor. Edit rules... to invoke the Rule Editor. View rules... to invoke the Rule Viewer. View surface... to invoke the Surface Viewer. As shown in Figure 4, there are six pop-up menus provided to change the functionality of the six basic

6 6 OZEK AND AKPOLAT Figure 4 FIS Editor of Type-2 Fuzzy Logic Toolbox. steps in the fuzzy implication process on the FIS Editor: And method: Choose min, prod, or Custom, for a custom operation. Or method: Choose max or Custom, for a custom operation. Implication method: Choose min, prod, or Custom, for a custom operation. This selection is not available for Sugeno-style fuzzy inference. Aggregation method: Choose max, sum, or Custom, for a custom operation. This selection is not available for Sugeno-style fuzzy inference. Type reduction method: Choose center of sets, center of sums, centroid, height, modified height or Custom, for a custom operation. This selection is not available for Sugeno-style fuzzy inference. Defuzzification method: For Mamdani-style inference, choose centroid or Custom, for a custom operation. For Sugeno-style inference, choose wtaver (weighted average) or wtsum (weighted sum). Membership Function Editor The membership function editor shown in Figure 5 allows the user to display and edit the membership functions associated with the input and output variables of the FIS. On the Membership Function Editor, there is a menu bar that allows the user to open related GUI tools, open and save systems, and so on. The File menu for the Membership Function Editor is the same as the one found on the FIS Editor. Under Edit menu, the user can select the options of Add MF, Add custom MF, Remove current MF, Remove all MFs, and Undo. On the other hand, under View menu, the user can select the options of Edit FIS properties, Edit rules, View rules, View surface. Rule Editor Figure 6 shows the rule editor that allows the user to view and edit fuzzy rules. On the Rule Editor, there is a menu bar that lets the user to open related GUI tools, and to open and save systems, change the format of the rules and so on. The menus of the Rule Editor are similar to the menus of the other editors. Rule Viewer The Rule Viewer shown in Figure 7 lets the user to view detailed behavior of a FIS to help, diagnose the behavior of specific rules or study the effect of changing input variables. The menu bar on the Rule Viewer allows the user to open related GUI tools, to open and save systems, and so on. The menus of the Rule Viewer are similar to the menus of the other editors. Surface Viewer The Surface Viewer shown in Figure 8 generates a 3-D surface from two input variables and the output of

7 A SOFTWARE TOOL 7 Figure 5 Membership Function Editor of Type-2 Fuzzy Logic Toolbox. an FIS. The menu bar on the Surface Viewer allows the user to open related GUI tools, open and save systems, and so on. The menus of the Surface Viewer are similar to the menus of the other editors. Type-2 Fuzzy Logic Controller Block The type-2 fuzzy logic controller block shown in Figure 9 is prepared and added into the library of Figure 6 Rule Editor of Type-2 Fuzzy Logic Toolbox.

8 8 OZEK AND AKPOLAT Figure 7 Rule Viewer of Type-2 Fuzzy Logic Toolbox. Figure 8 Surface Viewer of Type-2 Fuzzy Logic Toolbox.

9 A SOFTWARE TOOL 9 ACKNOWLEDGMENTS We would like to thank Prof. J. M. Mendel and his former PhD students N. Karnik and Q. Liang, who made free M-file functions available online for type-2 FLSs [1]. REFERENCES Figure 9 fuzzy logic toolbox in Simulink. So, the controller representing a type-2 fuzzy logic system can be easily used in Simulink files. CONCLUSIONS Type-2 Fuzzy Logic Controller Block. Type-2 logic systems have been an attractive research area in recent years. However, they are more difficult to understand and implement than conventional type- 1 fuzzy logic systems. In this study, a new software tool developed for helping users to understand, design and analyze interval type-2 fuzzy logic system is presented. The developed software called Type-2 Fuzzy Logic Toolbox is actually a collection of MATLAB1 based M-files (MATLAB is a registered trademark of The MathWorks, Inc.). The format and menus of the developed software are designed similar to the original Fuzzy Logic Toolbox of MATLAB since the users of MATLAB are familiar with them. [1] J. M. Mendel, Uncertain rule-based fuzzy logic systems: Introduction and new directions. Prentice Hall PTR, Upper Saddle River, NJ, [2] N. N. Karnik, J. M. Mendel, and Q. Liang, Type-2 fuzzy logic systems, IEEE Trans Fuzzy Syst 7 (1999), [3] J. M. Mendel and R. I. B. John, Type-2 fuzzy sets made simple, IEEE Trans Fuzzy Syst 10 (2002), [4] Q. Liang and J. M. Mendel, Interval type-2 fuzzy logic systems: Theory and design, IEEE Trans Fuzzy Syst 8 (2000), [5] K. C. Wu, Fuzzy interval control of mobile robots, Comput Elect Eng 22 (1996), [6] C.-H. Lee and Y.-C. Lin, Control of nonlinear uncertain systems using type-2 fuzzy neural network and adaptive filter, Proceedings of the 2004 IEEE International Conference on Networking, Sensing and Control, March , [7] P. Melin and O. Castillo, A new method for adaptive model-based control of non-linear plants using type-2 fuzzy logic and neural networks, Proceedings of the 12th IEEE International Conference on Fuzzy Systems, FUZZ 03, May 25 28, [8] A. Homaifar, Y. Shen, and B. V. Stack, Vibration control of plate structures using PZT actuators and type II fuzzy logic, Proceedings of the 2001 American Control Conference, June 25 27, 2001, [9] Q. Liang, N. N. Karnik, and J. M. Mendel, Connection admission control in ATM networks using surveybased type-2 fuzzy logic systems, IEEE Trans Syst Man Cybern Part C 30 (2000), [10] L. A. Zadeh, Fuzzy sets, Informat Contr 8 (1965), [11] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-1, Inf Sci 8 (1975),

10 10 OZEK AND AKPOLAT BIOGRAPHIES Muzeyyen Bulut Ozek was born in Elazig, Turkey, in She received the BSc degree in 2002 from Firat University, Elazig, Turkey, and the MSc degree in 2004 from the Department of Electronics and Computer Science, Firat University, where she is currently a PhD student. She is also a teacher of computers at Elazig Primary School, Elazig. Her research interests include fuzzy logic, expert systems, and artificial intelligence techniques. Zuhtu Hakan Akpolat was born in Elazig, Turkey, in He received the BSc degree in 1989 from Hacettepe University, Ankara, Turkey, and the MSc degree in 1992 from Firat University, Elazig, Turkey, both in electrical and electronics engineering. He received the PhD degree in electrical engineering from the University of Nottingham, United Kingdom, in He is currently a professor in the control division of the Department of Electronics and Computer Science, Technical Education Faculty, Firat University. His research interests include control of electrical drives, fuzzy logic, sliding mode control, intelligent control techniques, and mechanical load emulation methods.