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Iterative Learning Fuzzy Inference System Iterative Learning Fuzzy Inference System S. Ashraf*, E. Muhammad**, F. Rashid and M. Shahzad, * NUST, Rawalpndi, Pakistan, ** NUST, Rawalpindi, Pakistan, *** PAEC, Pakistan, **** NUST, Pakistan Abstract—This paper presents a learning fuzzy controller were published in another seminal paper [5]. Since then which can adapt with changing performance requirements. fuzzy controllers have found many successful industrial During the past decade we have witnessed a rapid growth in applications, have shown significant improvements in the number and variety of applications of fuzzy logic system performance and have proved to be more robust ranging from consumer electronics and industrial process than conventional controllers [6,7,8]. control to decision support system and financial systems. The challenging tasks associated with fuzzy control The fuzzy controller designer faces the challenge of choosing design has always been, to choose appropriate the appropriate membership functions, minimum rule base membership functions, minimum rule base and the most and the most suitable fuzzifier and defuzzifier. Having made suitable fuzzifier and defuzzifiers. Having made these these choices, the fuzzy controller has to be tuned to deliver choices, the fuzzy controller has to be tuned to deliver the the desired response. Multiple simultaneous adjustments desired response. Multiple simultaneous adjustments (rules, membership functions and gains) make the optimum (rules, membership functions and gains) make the tuning even more difficult. It is now realized that complex optimum tuning even more difficult. Many techniques real world problems require intelligent systems that combine knowledge, techniques and methodologies from have been used to overcome this difficulty including various sources. These intelligent systems are supposed to neural networks [9,10], a phase plane technique for FPI possess humanlike expertise within a specific domain, adapt rule base design [11], and gain phase margin analysis themselves and learn to do better in changing environment. technique [12]. In this paper we combine fuzzy techniques with iterative Fuzzy controller is a rule based control system [13]. learning to formulate a scheme that can automatically find Before any rules are made, member ship functions are the appropriate fuzzy controller to meet our design defined. Membership functions have uncertainties requirements. The scheme is adaptive and can handle the associated with them. These uncertainties arise because uncertainties arisen by difference in perception about a different people have different perceptions about a concept. Extensive literature survey shows that designing concept [14]. Manipulation of perception plays a key role fuzzy controllers with desired performance specifications is in human recognition, design and execution process [14]. not a trivial task. Even the specification of linguistic An uncertainty may be either due to a fuzzy linguistic variables, key concept in fuzzy system design, can be variable or due to randomness in the occurrences of some different from different experts. This scheme tries to fill this parameters. The former is associated with words, and the gap. The results show that the scheme is robust, cost fact that words can mean different things to different effective and relatively simple to implement. It makes use of people, and the latter is associated with unpredictability. the non linearity inherent in the fuzzy systems. This scheme Probability theory is used to handle random uncertainty, has the potential to make consumer electronics, decision and Fuzzy Systems are used to handle linguistic support systems and all other countless number of areas uncertainty, and sometimes fuzzy systems can be used to where fuzzy has made in roads, perform better. handle both kinds of uncertainties [15]. To handle linguistic uncertainties, Type-2 Fuzzy sets (T2 FS) and their related logic was developed [16,17]. In such I. INTRODUCTION uncertainties it is difficult to determine the exact Modern day controllers are becoming more and more Membership functions (MF) for a fuzzy system (FS). All complex as the performance requirements, complexity of of these uncertainties translate into uncertainties about tasks and processes increases. On the other hand, human fuzzy set membership function [6]. As an example, operators take care of complex control tasks with good suppose the variable of interest is motor speed, denoted by results and apparently without difficulty and without x where x ∈[0,100] and this gives a speed of 0 to 100 knowing the mathematical model of the system. In an era rev/sec. One of the terms that might characterize the where systems and their models are getting more and amount of perceived speed is ‘fast’. Now if one asks 10 more complex, increasing the difficulties of the controller experts to locate the ends of an interval for fast speed on designers, simpler controller with similar performance are the scale of 0-100, different experts will give different always welcomed. The technology that can convert human ranges for a particular application in mind. These ranges thinking or knowledge into controller design is fuzzy are usually averaged before membership functions are logic[1,2,3]. This concept gave a continuum of grades of defined. Because of these differences fuzzy controller membership to classes of objects like “the class of tall designer has to experiment or look for other techniques to men”, “the class of real numbers greater than ten” and find the membership functions, rule bases or both to meet “the class of high speed cars”. In 1973 Zadeh published a the required performance specifications. With respect to seminal paper [4] which established the foundation for membership function definitions, there is always a region fuzzy control. In 1975 Mamdani and Assilian established of uncertainty where their end points can lie. Within this the basic framework for fuzzy controller and applied the fuzzy controller to control a steam engine. Their results Proceedings of International Bhurban Conference on Applied Sciences & Technology Islamabad, Pakistan, January 7 – 10, 2008 region of uncertainty there can be infinite number of endpoints. This situation is shown in figure 1. Figure 2. Block diagram of a fuzzy control system Here rkj () represents the reference signal for kN= 1... and j = 1...∞ . Variable j represents the iteration number and k represents the samples. The error is represented by ekj (), the input to the plant is uk()and the next plant output is yk(1)+ . Factors j j Figure 1. Triangular MFs when base end points have uncertainty g and g are the input and output scaling factors. The associated with them e u Fuzzy Logic Controller (FLC) adjusts the input to the plant. The output of the FLC is dependent on the choice of This region is called the footprint of uncertainty (FOU) all four blocks of a fuzzy controller. These four blocks of in Type 2 Fuzzy System (T2 FS) theory. Somewhere in a fuzzy controller are presented in figure 3. this FOU are located the lower and upper extremities of our desired MF. There can be N membership functions in this region of uncertainty. Our aim is to find MFMFxorMFxorMFxdesired = 12() ()...N () In T2 FS each potential MF is assigned a weight, extending the concept into third dimension. This makes representation and computation extremely difficult. For a fuzzy controller we need to define both input and output membership functions. This paper proposes to adjust input MFs to achieve design specifications. To adjust these MFs an iterative learning process is proposed. It is linked with steady state error and overshoot, which are used to specify design requirements. Figure 3. A typical structure of a fuzzy controller Iterative learning control (ILC) has shown great success in tasks where the process is repetitive [18,19]. The basic idea behind ILC is that the information learnt from the previous trial is used to improve the control input for the In figure 3, U and V are the universes of discourse for next trial. The control input is adjusted to decrease the input and output membership functions. difference between the desired and the actual output. In Fuzzifier is defined as a mapping from a real valued the proposed approach control input is adjusted indirectly * ' by iteratively learning fuzzy controller parameters, point x ∈U to fuzzy set A in U [3]. The input to the according to our design specifications. fuzzifier is crisp. Typically a fuzzifier could be a Although fuzzy has been mixed with other Singleton fuzzifier, a Gaussian fuzzifier or a Triangular technologies, researchers have rarely mixed iterative fuzzifier. Singleton fuzzifier is represented as learning with fuzzy. Researchers who have done so have mainly focused it for learning the model of the system * [20,3] or for learning the weights of the Artificial Neural ⎧1 if x= x ⎫ μA'()x = ⎨ ⎬ Network (ANN) [9]. ⎩⎭0 otherwise This paper uses 2-D theory [21,22] to represent variables. This 2-D representation is very convenient to In a fuzzy inference engine, fuzzy logic principles are present and handle iterative processes. used to combine the fuzzy IF-THEN rules in the fuzzy ' II. BASICS OF FUZZY CONTROL rule base into a mapping from a fuzzy set A in U to a fuzzy set B ' in V . There are many choices of inference Although the founding father of fuzzy logic, Zadeh [1] engines. Some of the most popular are product, minimum, initially expected fuzzy logic’s main applications in Lukasiewicz, Zadeh and Dienes-Rescher inference economics, medicines, psychology, biology and engines. linguistics, most of the real applications have been developed in engineering system control. A typical block Fuzzy rule base is the heart and soul of the fuzzy diagram of a fuzzy control system is shown in figure 2. system. It contains rules of the form Proceedings of International Bhurban Conference on Applied Sciences & Technology Islamabad, Pakistan, January 7 – 10, 2008 IF x is A and…and x is A THEN y is B Here NB stands for Negative Big, Z stands for Zero and 1 1 n n PB for Positive Big.
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