Comparison between Multiobjective GA and PSO for Parameter Optimization of AT2-FLC for a real application in FPGA Yazmin Maldonado Oscar Castillo Division of Graduate Studies Division of Graduate Studies Tijuana Institute of Technology Tijuana Institute of Technology Tijuana, B.C. Mexico Tijuana, B.C. Mexico [email protected] [email protected] Abstract— This paper describes the design of a type-2 average Motor (ReSDCM). The optimization involves taking only fuzzy system on FPGAs and its optimization using multiobjective certain points of the T2-MF in order to give greater efficiency Particle Swarm Optimization (PSO) and a multiobjective Genetic to the algorithm. The PSO and GA have been tested in an Algorithm (GA) for the regulation of speed of a DC motor. Based average type-2 fuzzy logic system (AT2-FLC) for ReSDCM. on the concept of evolution, the PSO algorithm and GA are This paper is organized as follows. In section 2 we present applied to membership functions parameter optimization of type- 2 average fuzzy inference systems. Implementations and the basic concepts of type-2 fuzzy inference systems and the simulations are carried out in FPGA using the Xilinx system basic concepts of FPGA, in section 3 we present the generator. The optimization method was coded in Matlab. The optimization methods such as PSO and GA, in section 4 we results of comparison PSO with GA were analyzed statistically. show the design of AT2-FLC for FPGA. The results and optimization of AT2-FLC for ReSDCM are shown in Section Keywords- AT2-FLC, PSO, GA, T2-MF, FPGA, ReSDCM 5. Finally, Section 6 offers conclusions about this work. I. INTRODUCTION II. TYPE-2 FUZZY INFERENCE SYSTEMS Type-2 fuzzy inference systems (T2-FIS) are used The IT2-FIS consists of four stages: Fuzzification, successfully in many application areas, and there is an Inference, Type Reduction and Defuzzification. We describe increasing interest in the research and implementations of T2- below these stages. FIS because they offer bigger advantages in handling II.1. Fuzzification: The fuzzification maps a numeric ×= × × T ∈ × × ≡ uncertainty with respect to type-1 fuzzy systems. Fuzzy logic value, ( 1... p) X1 X2 ...Xp X into a type-2 fuzzy systems are based on rules, and these rules incorporate ~ ~ μ × = ×=×' set Ax in X. Ax is a singleton fuzzy set if ~ ( ) 1/1 for and linguistic variables, linguistic terms and fuzzy rules. The Ax ' acquisition of these rules is not an easy task for the expert and μ~ (×)=1/0 for all others ×≠× [17][19][27]. Ax is of vital importance in the operation of the controller. The II.2. Inference: Fuzzy reasoning consists of two blocks, the process of adjusting these linguistic terms and rules is usually rules and the inference engine, it works the same way as for done by trial and error, which implies a difficult task, and for type-1 fuzzy systems, except the antecedents fuzzy sets and this reason there have been methods proposed to optimize the consequent are represented by type-2 fuzzy sets. The those elements that over time have taken importance, such as process consists of combining the rules and maps the input to genetic algorithms and particle swarm optimization the output (interval type-2 fuzzy sets), using the Join and Meet [2][4][5][6]. operations [19]. For a FIS-IT2 with p inputs A fuzzy inference system (FIS) consists of three stages: x ∈X ,x ∈X ,...,x ∈X and one output y∈Y, it is assumed that Fuzzification, Inference and Defuzzification [3][10][29][30]. 1 1 2 2 p p Type-1 fuzzy systems (T1-FIS) have exact membership there are M rules, the lth rule in a FIS-IT2 can be written as: functions (MF), while interval type-2 fuzzy systems (IT2-FIS) are described by membership functions with uncertainty 1 ~ l … ~ l ~ l R : If x1 is and F1 and and xp is Fp , Then y is G (1) [9][18]. Most of the fuzzy logic applications with physical systems where l=1,…,M. Once we have the rules it is necessary to require a real-time operation, and higher density calculate the Join(ц) and Meet(п) operations as well as sup- programmable logic devices such as field programmable gate star composition(Ì)[17]. array (FPGA) can be used to integrate large amounts of logic II.3. Type Reductor: The type reductor is used to convert in a single integrated circuit. all type- 2 fuzzy sets to type-1 fuzzy intervals on the output. This paper proposes the optimization of type-2 membership There are several methods to calculate the reduced set, such as functions (T2-MF) using GA and PSO for hardware joint center, center of sums, height, center joint, among others applications such as Regulation Speed of a Direct Current [19]. Equation (2) shows the center of sums type reductor. − = FOU(A1) YCOS (x) − (2) μ (x' ) A1 1 − 1 A (× ' ) ... ... M ∫∫∫∫ i i x' − 111 M M 1 M M 1 l∈∈∈∈[]M []l []M [] f y x1 yyyl ,,,,yr yfyl r fffl r f fl r ∑i=1 FOU (A ) M i − 2 − = f μ ∑i 1 A (x'2 ) where YCOS(x) is a set of intervals determined by two points yl x ∈ X − 1 x' x y y . [y i,y i] corresponds to the centroid type-2 of interval 2 2 r l r y = f (x)∈ X consequent, which can be obtained as Equation (3). − N i i FOU(A1) θ ∈ μ (x' ) x X 1 y 2 − = ∑ i=1 = []i i A1 C i ... 1 N yl , y r A(×' ) G ∫∫ i (3) − θθ∈∈JJ θ 1 yl N y N ∑ i=1 x' 1 x 1 − where yl and yr must be precomputed. Equation (4) shows the FOU (A ) − 2 μ (x' ) calculation. M i i M i i 2 f y f y − A ∑i=1 l l ∑i=1 r r y = y = x l M i r M i x' 2 f f (4) 2 − ∑i=1 l ∑i=1 r There are methods to calculate the centroid of a FIS-IT2 Figure 1. Type-2 average fuzzy system such as [17]. II.4. Defuzzification: Consists in obtaining a numeric value for the output. Using the COS type reductor, the III. OPTIMIZATION METHODS defuzzification is an average value since the range is given by A. Particle Swarm Optimization [yl,yr] [17][19]. The defuzzification is shown in the equation: + The Particle Swarm Optimization (PSO) is a bio-inspired yl yr optimization method. PSO finds the optimal solution by y = (5) ( x) 2 simulating social behavior. PSO is basically developed Apart from the discussed IT2-FIS, there are type-2 through simulation of birds that more in two dimensional generalized fuzzy systems and AT2-FIS, the latter involve space, each particle position and speed. replacing a type-2 fuzzy system using the average of two Since its introduction in 1995, PSO has been improved appropiate T1-FIS, this method [24] is performed as follows: several times and different applications have been achieved. 1. Replace each T2-MF using two T1-MFs with different Most of the modifications to the basic PSO are aimed at degrees of membership in order to obtain the footprint of improving the convergence and the increasing diversity of the uncertainty. 2. To replace the type-2 inference stage, the swarm. inference from each T1-FIS must be obtained. 3. To replace A PSO algorithm maintains a swarm of particles, where the type-reduction system and defuzzification stage of the each particle represents a possible solution. In analogy with type-2 FIS, we obtain the defuzzification of each T1-FIS and the paradigms of evolutionary computation, the particles are the results of the two systems are averaged. transported through a multidimensional search space, where An IT2-FIS and AT2-FIS can be implemented on a general the position of each particle is adjusted according to its own purpose computer, or by a specific use of a microelectronics experience and its neighbors. xi (t) represents the position of realization such as the FPGA [20][21]. particle i in the search space at time t, t denotes the discrete The FPGA is a semiconductor device that contains in its time. The position of the particle is modified by the addition of interior components such as gates, multiplexers, etc. These a velocity vi (t), i.e. the current position [2][4], Equation (6) devices use the VHDL programming language, which is an shows the position of the particle. + = + + (6) acronym that represents the combination of VHSIC (Very xi (t 1) xi (t) vi (t 1) High Speed Integrated Circuit) and HDL (Hardware where xi (0) ~ U (xmin, xmax). The velocity vector reflects both Description Language) [28]. the experimental knowledge of the particle and exchanged the The three basic elements in an FPGA are the logic block social information. The experimental knowledge of a particle (CLB), programmable interconnections and input-output block is often referred to as the cognitive component, which is (IOB). The FPGAs can be used to implement specific proportional to the distance of the particle from its best architectures to accelerate a particular algorithm. Applications position (referred to as the best personal position of the that require a great number of simple operations are suitable particle) found from the beginning. for implementation on FPGAs. A processing element can be PSO can be described as follows; each swarm knows the designed to perform this operation and several instances of it best position of the particle (Plbest) and the best global can be used to perform parallel processing [1][15][23].