Analog implementation of fuzzy controller

Analog Implementation of Fuzzy Controller Alfredo Sanz Department of Electrical Engineering and Computer Science University of Zaragoza, Spain [email protected] Abstract- This paper shows a method based systems. Generally any FIS can be to implement fuzzy controllers with described as a black box with an input analog electronic circuits. A fuzzy vector X=xi,i=1..n, and an output vector controller is defined by a collection of Y=yi,i=1..m. In this system a fuzzyfication fuzzy IF-THEN rules and a set of interface transforms the input in degrees of membership functions characterizing match with linguistic values. Then a the linguistic terms associated with the decision-making unit performs the input and output of the fuzzy controller. inference operations on the rules base. In this paper an electronic circuit with Finally, a defuzzyfication interface Operational Amplifiers is designed from transforms the fuzzy result of the inference a set of membership functions and fuzzy process in a crisp output [4,5]. IF-THEN rules. 1 Introduction Fuzzy rule base Fuzzy logic controllers are proposed[12] to control plants that are mathematically poorly Fuzzyfication Defuzzyfication understood and where experienced human interface interface operators are available for providing X Y qualitative rules of control. Fuzzy controllers make effective use of this information and Decision-making unit provide a linguistic description about the system and control instructions[14]. This method is a model-free approach and Figure 1: Fuzzy inference system provides nonlinear controllers to perform any nonlinear control action. Also from a practical point of view, a fuzzy control is easy to understand, inexpensive to develop Several types of FIS have been proposed and simple to implement. in the literature [2][3]. Depending on the Now there are many methods to type of fuzzy reasoning and fuzzy if-then implement a fuzzy controller. The more rules employed, FIS can be classified in usual implementation of fuzzy controllers is three types: the simulation on a commercial 1. The Output membership function microprocessor. This solution is should be monotonally non- inexpensive but in some real time problems decreasing and the overall output is is not usful. the weighted average of this. Many fuzzy VLSI[13] chips have been 2. The overall fuzzy output is the max. of developed to implement fuzzy controllers, qualified fuzzy output. The final crisp although due to cost reason most of them output may be the center area, have used a serial implementation[1] of bisector area, centroid area, etc. rules and is not possible to use in high [2][3][12]. speed applications with low cost. 3. Takagi and Sugeno's fuzzy if then Fuzzy controllers admit a high degree of rules are used [10,11]. The output of parallel implementation and an analog each rule is a linear combination of solution use this and combine high speed input variables plus a constant term, and low cost. There is possible develop full- and the final crisp output is the custom analog chip[8] but the cost of weighted average of each rule's developping is very high. A discrete output. implementation with analog circuit of fuzzy In the present paper we will focus on the is inexpensive and easy to develop. second type fuzzy systems, but its results can be easily extended to all three fuzzy 2 Fuzzy Logic Controllers systems types. The fuzzy logic system Fuzzy logic controllers are based in Fuzzy performs a mapping from U ⊂ Rn to R. We inference systems (FIS) or Fuzzy-rule- assume that U = U1x....xUn, where U ⊂ Rn, i =1,2,....,n. We now present a description of each of the four block in the fuzzy logic system of Fig.1. where k is the total of the rules in the rule The Fuzzy rule base consists of a base of the controller.. collection of fuzzy IF-THEN rules: 2 Analog implementation R i : IF ( xi is F1i and ... xn is Fni ) (1) We will propose a method to translate a THEN y is G i set of membership functions associated with each input and a set of fuzzy IF-THEN where X = ( x1 ,... xn ) T ∈U and y ∈ R are rules in an electronic circuit of discrete the input and output of the fuzzy logic operational amplifiers and resistors. The system, respectively, F1i and G i are labels basic block will be the Fuzzyfication interface, the Decision making-unit and the of fuzzy set Ui in R, respectively, and Defuzzyfication interface. l=1,2,..k. Each fuzzy rule of (1) define a We will use a circuit of operational fuzzy implication F1i × ... × Fni → G i , which is amplifiers with symmetric power supply Vcc a fuzzy set defined in the space UxR. Many and open-collector output. In the simulation fuzzy implication rules have been proposed we used the LM111, a normal operational in the fuzzy logic literature. We will use amplifier commonly used in comparators Product-operation rule of fuzzy implication: circuits. We will suppose that the input and output µ F i ×...× F i →G i ( x , y ) = µ F i ×...× F i ( x ) µ G i ( y ) are included in the range of [-Vcc,+Vcc] 1 n 1 n (2) Where µ F i ×...× F i ( x ) is defined by 3 Fuzzyfication interface 1 n This block transform an xi input include in the range of [-Vcc,+Vcc] in a vector Ai of µ F i ×...× F i ( x ) = µ F i ( x1 ) ∗ ... ∗ µ F i ( xn ) 1 n 1 n (3) membership functions for the xi .There is necessary to implement different circuit for Here the symbol " ∗ " denote the t-norm, each types of standard membership which corresponds to the conjunction "and" functions: in (1). We will use the most commonly used operation for the t-norm the min. u∗ v = min( u , v ) (4) Z-Type S-Type The Decision making-unit perform a mapping from fuzzy set in U to fuzzy set in R, based upon the fuzzy IF-THEN rules in the fuzzy rule base and the compositional L-Type P-Type rule of inference. We will use a singleton G i then Figure 2: Standard Membership Functions Types y= y µ G ( y) = 1 if Gi i 0 if y≠ y Gi (5) The basic circuit for all this implementation is the differential amplifier of Let Rl be a rule then the value of output y fig 3. by the Rl rule is: R1 R2 y l = µ F l ×...× F l ( x ) yG l X1 1 n (6) Vo R3 The Defuzzyfication interface maps fuzzy X2 set in R to a crisp point in R. The final value of output y is given by R4 k ∑ µ F i ×...× F i ( x ) yG i X3 i =1 1 n y= k ∑ µ F i ×...× F i ( x ) Figure 3: Basic circuit for the Fuzzyfication i =1 1 n (7) interface  Where x1, x2 and x3 are the input and Vo +Vcc is the output. Then: R1 R2 X1 R4 x2 + R3 x3 R R Vo = (1 + 2 ) − 2 x1 R4 + R3 R1 R1 (8) R3 Vo The standard membership functions are defined by their parameters c and s. Then in general, for a range of [0,1] in the R4 membership functions: -Vcc Z (c, s, x ) = Min(1, Max(0, Z0 (c, s, x ))) S (c, s, x ) = Min(1, Max(0, S0 (c, s, x))) Figure 4: Circuit of for implementation of Z- type membership functions P(c1, s1, c2, s2, x) = Min(1, Max(0, Z0 (c1, s1, x )), Max(0, Z0 (c2, s2, x))) (9) Then: a = 2 Vcc s and 1 + 2 Vcc s − 2 s c r= (14) 1 + 2 Vcc s + 2 s c 1 Z 0 ( c , s, x ) = − ( x − c) s 2 The implementation of S-type of 1 membership functions is perform by using S0 ( c , s, x ) = + ( x − c ) s x2 for the input and connecting x1 to 0V. If c 2 (10) =0 connecting x3 to 0 and r = a , else if c< 0 In an electronic implementation with connecting x3 to +Vcc or if c > 0 connecting symmetric power supply Vcc the x3 to -Vcc. Then: membership functions will be: 1 Z (c, s, x) = Min(Vcc, Max( −Vcc, Z0 (c, s, x))) a = 2 Vcc s (1 + ) − 1 r S (c, s, x) = Min(Vcc, Max( −Vcc, S0 (c, s, x))) (15) Vcc P(c1, s1, c2, s2, x) = Min(Vcc, Max( −Vcc, Z0 (c1, s1, x)), r= Max( −Vcc, Z0 (c2, s2, x))) (11) c The implementation of P-type of and membership functions is perform by the min. operation of her S0 and Z0 basic membership functions. This min. operation Z 0 ( c , s , x ) = +2 Vcc ( x − c ) s is implemented by the connection of the (12) open collector output of the operational S 0 ( c , s , x ) = −2 Vcc ( x − c ) s amplifiers. +Vcc Let be the parameters a and r: R1 R2 R2 a= R1 (13) Vo R R3 r= 4 X R3 R4 The implementation of Z-type of membership functions is performed by -Vcc using x1 for the input and connecting x2 to ±Vcc +Vcc and x3 to -Vcc fig.4. Figure 5: Circuit of for implementation of Z- type membership functions  Figure 5 show the DC simulation of output 5 Defuzzyfication interface of fuzzyfication interface for a input from - The basic function of this circuit is to 10V to +10V using PSPICE simulator implement the equation (7). We use two program. The membership functions standard analog adders circuits to defined for this input are: k implement ∑ µ F i ×...× F i ( x ) yG i and i =1 1 n Z1=Z(-5,0.1,x) k P2=P(-0.25,5,0.25,5,x) (16) ∑ µ F i ×...× F i ( x ) . The output of this circuit is S3=S(5,0.1,x) i =1 1 n connected to an analog circuit of multiplier that perform the final value of y. k 10 Z1 The multiplication in ∑ µ F i ×...× F i ( x ) yG i is P2 i =1 1 n 5 S3 performed by the value of resistor that connect the output of the decision making- 0 unit and the addition circuit. This is possible -10 -5 0 5 10 because yG i is a constant. -5 -10 6 Conclusions I have proposed a generalized method to Figure 5: Output of fuzzyfication interface translate the Fuzzy Controller to an analog electronic circuit with operational amplifiers and discrete components. This work is included in my actual proyect to implement 4 Decision making-unit an Integrated Development Envelopment In the decision making-unit we use a (IDEA)[5][6] for developing of Autonomous subcircuit in the premise of each rule to Response Systems (ARS), and her implement the µ F i ×...× F i ( x ) of (3). This compiler from EDA[7] language to 1 n circuit implement a min. operation Figure 6. electronic implementation. X1 X2 References Vo [1] J.L.Huertas, S. Sanchez Solano, A. Barriga, I. Baturone, "Serial Architecture for Fuzzy Controllers: Xn Harware Implementation Using Analog/Digital VLSI Techniques, 2nd. International Conference on Fuzzy Figure 6: Basic circuit for the AND Logic And Neural Networks, Iizuka, operation. 1992 [2] C.-C. Lee. Fuzzy Logic in control systems: fuzzy logic controller-prat 1. IEEE Trans. on Systems, Man, and The Figure 7 shows the simulation of Cybernetics, 20(2):419-435,1990 output circuit of AND operation between the [3] C.-C. Lee. Fuzzy Logic in control membership functions Z1 and P2 defined in Systems: fuzzy logic controller-prat 2. (16) IEEE Trans. on Systems, Man, and 10 Cybernetics, 20(2):419-435,1990 Z1 [4] J.-S. Roger Jang. ANFIS: adactative- P2 5 Networks-Base Fuzzy Inference Z1*P2 systems IEEE Trans. on Systems, 0 Man, and Cybernetics, 1992 -10 -5 0 5 10 (Forthcoming) -5 [5] A. Sanz. Redes Neuronales Artificiales y Sistemas Borrosos: Bases, Relaciones y Aplicaciones. XII -10 Reunión Anual de la Agrupación Figure 7: Output of AND circuit Española de Bioingeniería, 1993 [6] A. Sanz. Sistema para el desarrollo de sistemas de respuesta autónoma (ARS). 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