New definition of Membership function and its Hardware implementation with a new strategy

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1 New definition of Membership function and its Hardware implementation with a new strategy A. Ebrahimi*, S. Aminifar**, M. Daneshwar***, Gh. Yosefi*** * Department of Mathematics ** & *** Department of Electrical Engineering Islamic Azad University,Mahabad Branch Mahabad Iran **Corresponding author s.aminifar@iau-mahabad.ac.ir Abstract: This paper introduces a new mathematical method for introducing Membership function in fuzzy systems. This new mathematical definition relies on concept of continuousvalued logic (CVL) which can be applied to control the pulse width of generated Pulse Width Modulation. The introduced Membership function is in the form of triangle with curved brim in order of two. For the first time, degree of membership becomes correspondent to the frequency of a signal. MATLAB simulations show more soft and reliable controlling. Using this membership function provides more simplicity in fuzzy logic circuitry. New circuits have designed for implementing this Idea and these circuits have used for implementing a twoinput one-output fuzzy logic controller. The circuit simulation has done with HSPICE. Implemented controller has used practically for controlling the temperature of a room. Keywords: Membership function modeling, CVL-based controller, frequency, FLC 1. Introduction The overall performance of a fuzzy logic system (FLS) depends on proper choosing of membership functions (MFs), the rules of inference engine, and the scale factors of the inputs and the outputs. Of course choosing these criteria needs enough theoretical and practical knowledge, and without considering the details, an FLS may not give better results than what can be achieved using conventional controllers. Usually the desired performance is achieved by manually tuning the parameters based on experience and trial and error; but in critical applications, after designing a basic FLS, an optimization algorithm is employed to optimize the performance. Fuzzy system optimization problem is addressed in several literature. Some articles have focused on choosing proper rules in inference engine [1,]. Also in some papers, tuning of the input and output scale factors (which is known as context adaptation) are proposed [3,4,5]. But because of higher sensitivity of fuzzy systems to their MFs [6], many researches are performed on the MF optimization. Various methods are used to address this issue, which are either derivative-based or derivative-free methods. The derivative-free approaches are desirable because of robustness. Slow convergence is their main drawback. Some of these methods are genetic algorithm [7], neural network [1,8], evolutionary programming [10], etc. Convergence of derivative-based methods is fast. But they are limited to specific types of inference and MFs. The most common approaches are least squares [11], back propagation [1]. In this paper we introduce the triangle membership function with curved rims. The power of curve is two which is correspondent to linguistic hedge (LH) "very" in terms of fuzzy sets.. Modified Triangle Membership Function Modified triangle membership function MTMF is extended form of triangular ISSN: ISBN:

2 membership function; of course they are in turn a special form of LR functions introduced by Dubois and Prade [5]. Fig.1 shows typical shapes of MTMFs. \ appropriate power; hence MTMFs can be interpreted as the extended form of those functions which are generated by applying LHs. For example, applying hedge very to a triangular MF results in a function which has the same effect of a MTMF with equal upper and lower half-width powers of power of x =. 3. Continuous-valued logic Fig.1 various MTMFs Note that the formulas for a trapezoidal MF can be obtained easily from a triangular MF, and the derivative formulas of the fuzzy output with respect to MF parameters are zero when the input is in the core of the MF (the area which the membership values are 1); therefore, to avoid further complexity, we eliminate the discussion of trapezoidal MFs. Triangular MF has two straight lines, but MTMF is composed of two curves. The ith MTMF is uniquely determined by its function: MD=Membership degree(x) = A(1-X ) The degree of membership of the jth crisp input in its ith fuzzy set is obtained by the following equation: MD(x) = A (1-X ) if x is bigger than middle point MD(x) = A (-1+X ) if x is smaller than middle point Special case is when power = 1; in this case the curves are reduced to straight lines and the MFs are triangular. Therefore three parameters remain for triangular functions which are modal point, and upper and lower half-widths. Furthermore, if we see the fuzzy system as a black box, and only consider its input to output characteristic, we can say that modifying conventional MFs by applying linguistic hedges (LHs) results in the same effect of defining a MTMF with The overall performance in normal logic we have just two states 0 and 1. In this paper we use the concept of continuousvalued logic in which there is all values between 0 and 1 and all these numbers are considered as different logics. Defining continues-valued logic provides extreme logic states. Using extreme logic states helps to control a system with high nonlinearity, easier and softer. Continues-valued logic was initially produced and developed on Microchips known as PIC Microcontrollers. The success of the resulting designs led to an introduction of continues-valued logic in Switch Mode Power Supply (SMPS) where the appropriate output voltage and current are picked up by the Microcontroller that drives a closed-loop circuit [3]. By using continues-valued logic concept, Microcontroller can conveniently become substituted for an Operational Amplifier that is employed in the application such as voltage regulators. Initializing the device for high impedance input configuration on non-schmitt ports provides a stress free condition in threshold voltage Area and gives free detection choice of the values 0 and 1. When Microcontroller starts oscillating, due to the duty cycle of the generated oscillation, the importance of values closer to 1 or 0 can be determined. The resulting 50% duty cycle provides another state in Threshold voltage area which is the position of system HALT or Doing Nothing state. In comparison with Fuzzy designs, using continues-valued logic creates opportunities for faster, softer, cheaper and simpler design concepts. Using continues-valued logic can minimize the design blocks not only by ignoring ISSN: ISBN:

3 conventional ADA converters, but also being used as an average reference voltage of its narrow domain that is typically 1. to 1.5 volts. There is always an input definition boundary for any logic gate specifying true or false, yes or no and (1) or (0) means. There are two inspection methods to specify (0) or (1) in a logic input: # Schmitt trigger buffer # TTL level buffer The Schmidt Trigger concept makes use of "positive feedback" in order to toggle sudden changes from one level to another. While a TTL buffer input follows the ordinary V/I Laws to transfer data and only this concept is incorporated in threshold voltage Area, where input is driven by a voltage at the threshold stage and causes instability of the output. The bouncing phenomena of the output generate a PWM that is appropriate to the level of the threshold area [3, 4]. As in the circuit of figure 1, the voltage boundary in a TTL buffer input adheres the treated theory in section [5, 7]. The driving program is as simple as in below equation: Port X () = Port X (1) While the voltage sets to mid threshold voltage area, the output I /O shows an unpredictable value that changes within 0 and 1 and its frequency is dependent to the clock frequency of the device. Duty cycle of the generated wave also varies due to the lower and higher threshold voltage boundary definitions. Fig. : Input boundary voltage definition for TM PIC mid-range Microcontrollers While the MCU is initialized to generate an internally stabilized PWM instead of instable mode which is discussed in previous section, duty cycle is controlled using the Continues-valued Logic concept. By looking through the specifications given by the manufacturers of the MCUs, the hardware architecture involved with PWM generator is designed by an incremental n-bit register counter [5]. Adding or deducting to the n register value will increase or decrease the mark (high level) of duty cycle due to the output comparator as figured in. Fig.3: PWM register counter Comparator PWM The resolution of incremented and decremented duty cycle depends on the bits of the register. For instance PIC14T686 from Microchip employs a 1- bit (4096step) counter/register. 4. Hardware implementation fuzzifier circuit of 4.1. Fuzzifier Circuit The main Idea for implementing membership function generator has shown in figure 4. First of all, we apply crisp data to a system which produces a ramped triangular as Fig.4(b) which its frequency is vary as illustrated in Fig.(c). Finally generated wave which its pick to pick voltage is located in threshold voltage area as described in Fig.1, applied to an MCU. A CVL Pulse has generated which the duty cycle varies correspondent to input crisp data by the function in order two as shown in Fig.4(c). In figure 5 the proposed circuit has shown. Of course in this circuit by choosing R1, R and R3 the output pick to pick voltage is set to threshold voltage area. Because of this, there is no need to voltage shifter. ISSN: ISBN:

4 Input Crisp Data I=f(\t) v Input Crisp Data Schmitt Trigger Integrator (b) t Shift Pick to Pick of generated ramped voltage To the Threshold Voltage Area Microcontroller oscillate proper to input voltage f v Membership degree = Frequency of generated triangle waveform (c) Input crisp data FuzzifiedCVL PWM Output (a) Fig.4: The main Idea of fuzzifier circuit The proposed circuit contains a Schmitt trigger and an integrator and a Microcontroller. The frequency in output of the Op-Amp is calculated by following equation: f = (R1/(4*R*R4*C))[1-(Vcrisp/Vq) ] This equation shows that the output frequency varies respect to (Vcrisp). This causes to provide the triangle membership function which the rims are the function of second order. (d) 4.. Inference engine And defuzzification part In our controller in inference engine, we have used Max detector for making rules. As we know, our outputs from previous stage (fuzzifier stage) are CVL PWM pulses. Because of this, the Max circuit is simplified to a simple comparator which is realizable by an Op-Amp. For the defuzzification, considering the properties of CVL pulse, the only thing we need is a weighted adder, which is realizable with resistors and Op-Amp. All these circuits have shown in figure 6. The final signal is in the form of CVL pulse which is directly applicable to the process under control. Some of simulation results have shown in figure 7. Pick to pick not greater than Threshold Voltage Area in TTL Microcontroller chips C R6 t R5 1 R3 R1 Vq R4 vz MCU R vz Generated Ramp Input Crisp Data Output generated PWM CVL FuzzifiedPulse Fig.5: The proposed Circuit for Fuzzifier ISSN: ISBN:

5 CVL PWM Fuzzifiedpulse Crisp Input A fuzzifier Comparators R1 R R3 R4 Output CVL R5 R6 Rf Crisp Input B fuzzifier Fig.6: The structure of proposed controller 5. Simulation results In order to show the efficiency of the MTMfs in CVL-based Fuzzy Controller and compare the results with triangular MFs in conventional fuzzy controllers, we consider the problem of approximating a two-dimensional function given by f (x1, x) = cos(x1) + cos(x) where 0<x1< and 0<x<. With steps of 0.1 for x1 and x, i.e. 1 points for each, the ideal output has a total of N = 1 1 = 441 samples. And have simulated twoinput one-output fuzzy system in figure 6 and used to approximate this function. The domain of inputs are partitioned into three MFs labeled as: Small (S),, Medium (M), and Large (L). The output MFs are six singletons named as Negative Large (NL), Negative Small (NS), Normal (N), Zero (Z), Positive Small (PS) and Positive Large (PL). The rule base of the inference engine is normal distributed rule base. In MATLAB simulations one time we have used TMF and one time MTMFs. we see improvement in the performance as a result of using the MTMF in this process. Fig.7: MATLAB simulation results Fig.8: simulation result of proposed controller Figure 7 shows a comparison between simple triangle membership fuzzy controller and modified triangle membership CVL-based fuzzy controller. The figure 7(a) is the control surface which is desirable. And figures 7(b) and 7(c) show the simulation results of TMF and MTMF CVL-based control surfaces, respectively. As in the figure 7 is clear the surface of our proposed controller is so similar to desired one than conventional method. Figure 8 shows generated continuous-valued logic pulse which is produced in output point of one of fuzzifier blocks of controller Fig.6 which is applied to control the temperature of the room which controlled by a fan. Red diagram in Fig.8 shows the RPM of fan in started conditions in the control process. And black diagram of Fig.8 shows the output of defuzzifier. The results are more reliable than controller with simple triangular membership functions. 6. Conclusion A new mathematical method for introducing Membership function in fuzzy systems named MTMF was introduced. The concept of CVL injected to membership function. The new membership function output was the frequency of a PWM. MTMFs are extended form of triangular/trapezoidal types, hence can offer better performance simply because they have more compatible with fuzzy sets. Simulation results verified that the response of the system improves significantly when using the MTMFs instead of triangular types. This concept applied to an implemented controller practically for controlling the temperature ISSN: ISBN:

6 of a room. The achievements are simplicity in inference engine and defuzzification circuits and good results of control process. MATLAB simulations show more soft and reliable controlling. New circuits have designed for implementing this Idea and these circuits have used for implementing a two-input one-output fuzzy logic controller. The circuit simulation has done with HSPICE. The most noticeable point for maximum achievement of these advantages is that CVL effect has a great convergance in closed-loop control circuitary. Acknowledgements: The authors would like to thank Islamic Azad University; Mahabad branch for funding this research. References [1] Huertas JL, Sanchez-Solano S, Barriga A, "hardware implementation using AID VLSI techniques" In: Proceedings of the 8th Int. Confer. On [1] W. Barada, H. Singh, Generating optimal adaptive fuzzy-neural models of dynamical systems with applications to control, IEEE Trans. Systems Man Cybernet. Part C 8 (1998) [] I. Baturone, S. Sanchez-Solano, A. Barriga, J.L. Huertas, Implementation of CMOS fuzzy controllers as mixed-signal integrated circuits, IEEE Trans. Fuzzy Systems 5 (1) (1997) 1 19 (special issue on Hardware Implementations). [3] O. Cordon, F. Herrera, L.Magdalena, P. Villar,Agenetic learning process for the scaling factors, granularity and contexts of the fuzzy rule-based system data base, Inform. Sci. 136 (001) [4] O. Cordon, F. Herrera, P. Villar, Analysis and guidelines to obtain a good uniform fuzzy partition granularity for fuzzy rule-based systems using simulated annealing, Internat. J. Approx. Reasoning 5 (000) [5] D. Dubois, H. Prade, Fundamentals of Fuzzy Sets, first ed., Kluwer Academic Publishers, Boston, 000. [6] M. Figueiredo, F. Gomide, Design of fuzzy systems using neurofuzzy networks, IEEE Trans. Neural Networks 10 (1999) [7] T. Gwin, F. Goide, W. Perycz, Context adaptation in fuzzy processing and genetic algorithms, Internat. J. Intell. Systems 13 (003) [8] R.E. Kalman, A new approach to linear filtering and prediction problems, Trans. ASME J. Basic Eng. 8 (series D) (1960) [9] L. Magdalena, F. Monasterio-Huelin, Fuzzy logic controller with learning through the evolution of its knowledge base, Internat. J. Approx. Reasoning 16 (1997) [10] P.S. Maybeck, Stochastic Models, Estimation, and Control, Vol. 1, Academic Press, New York, [11] K.M. Passino, S. Yurkovich, Fuzzy Control, Addison-Wesley, Longman, Reading, MA, New York, 1998Fuzzy Logic and Neural Networks, Iizuka, Japan, July 17-, 004. p [1] Walter G.Jung, IC Op-Amp Cookbook, third edition, SAMS publications, pp. 7 17, (1999). [13] Microchip Technology Incorporation reference books, PICmicro Mid-Range MCU Family, Reference Manual, pp. (9-1)-(9-14). (007) [14] Microchip Technology Incorporation reference books, PIC1F683 Data Sheet with Nano-Watt Technology), (004) [15] Christophe P. Basso, Switch Mode Power Supply Spice Cookbook, Mc Graw Hill, pp (001) [6] Bart Kosko, Fuzzy Thinking: The new Science of Fuzzy Logic, University of Southern California, (1996) [17] Alereza Nazimi, Flts For mixed Analog Digital closed loop circuitry, master thesis, (008) [18] Temaritre L, Patyra MJ, Mlynek D. Analysis and design of CMOS fuzzy logic controller in current-mode. IEEE J Solid State Circuits 1999;9(3):317-. [19] Yamakawa T. A fuzzy inference engine in nonlinear analog model and its application to a fuzzy logic control. IEEE Trans Neural Networks 003;4(3): ISSN: ISBN: