Multiple-valued Logic Approach to Fuzzy Controllers Implementation

WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 Multiple-valued Logic Approach to Fuzzy Controllers Implementation V. VARSHAVSKY, V. MARAKHOVSKY1, I. LEVIN2, H. SAITO3 1,3 The University of Aizu Aizu-Wakamatsu City, Fukushima Prefecture, 965-8580 JAPAN [email protected], [email protected] 2 Tel Aviv University Ramat Aviv, Tel Aviv 69978 ISRAEL [email protected] Abstract: - This paper offers a new technique for designing fuzzy controllers as analog hardware devices on bases of CMOS implementation of multi-valued logical functions. This approach is based on using a summing amplifier with saturation as a building block that can be considered as a multi-threshold logical element. The functional completeness in an arbitrary-valued logic of a summing amplifier with saturation is proven. This fact gives a theoretical background for an analog implementation of fuzzy devices. In contrast with the traditional software approach to fuzzy controller implementations based on explicit fuzzification, fuzzy inference, and defuzzification procedures, hardware implementations of fuzzy controllers as analog devices have advantages of higher speed, lower power consumption, smaller die area and more. Universal and proper design methods for such type of hardware are offered. The paper illustrates design examples for real industrial fuzzy controllers and provides SPICE simulation results of their functioning. Key-Words: - Fuzzy Logic, Fuzzy Controller, Fuzzy rules, Fuzzy inference, Fuzzification and Defuzzification, Multi-Valued Logic, Multi-threshold Element, Functional Completeness, Summing Amplifier. 1 Introduction Fuzzy logic control is a methodology bridging 1. The fuzzification is a transformation of analog artificial intelligence and traditional control theory. (continuous) input variables to linguistic ones, e.g., This methodology is usually applied in the only transformation of temperature into the terms cool, cases when accuracy is not of high necessity or warm, hot or transformation of speed into the terms importance. On the other hand, as it is stated in [1], negative big (NB), negative small (NS), zero (Z)”, “Fuzzy Logic can address complex control positive small (PS), positive big (PB). Such problems, such as robotic arm movement, chemical transformation is realized by introduction of so- or manufacturing process control, antiskids braking called membership functions, which define both a systems or automobile transmission control with range of value and a degree of membership. For more precision and accuracy, in many cases, than linguistic variables it is important not only which traditional control techniques … . Fuzzy Logic is a membership function a variable belongs to, but also methodology for expressing operational laws of a a relative degree (weight) to which it is a member. system in linguistic terms instead of mathematical A variable can have a weighted membership in equations.” several membership functions at the same time. Wide spread of the fuzzy control and high 2. The fuzzy inference maps input linguistic effectiveness of its applications in a great extend is variables onto output linguistic variables on the base determined by formalization opportunities of a system of fuzzy rules of the type “IF A THEN B” necessary behavior of a controller as a “fuzzy” For instance: “IF the temperature is worm THEN the (flexible) representation. This representation usually speed is Positive Small (PS)” or “IF the speed is is formulated in the form of logical (fuzzy) rules Negative Big (NB) THEN force is ZERO”. Since under linguistic variables of a type “If A then B”. input linguistic variables are weighted, the output The Fuzzy Logic methodology [2, 3] comprises linguistic variables can be obtained weighted as three phases: well. Traditional fuzzy logic approach comprises Mamdani-type and Sugeno-type inference methods. ISSN: 1109-9445 Issue 6, Volume 4, June 2007 109  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 The Mamdani-type method is more intuitive and This work shows that for a sufficient wide set of assumes the output variables as a fuzzy set. Fuzzy applications, fuzzy controllers can be implemented rules in it contain a fuzzy precondition part (after IF) as rather simple CMOS devices, which can be used and a fuzzy consequence part (after THEN). The in embedded systems or as an IP core. What is the Sugeno-type method expects the output variables to basic idea of the proposal? be singletons or dealing with consequents that are A fuzzy controller is a deterministic device, for equations. So it is better suited for mathematical which one and only one value of the output analog analysis, nonlinear system modeling and variable corresponds to each value combination of interpolation. the input analog variables. It means that the fuzzy 3. In the defuzzification phase, the weighted values controller should realize an analog of output linguistic variables obtained as a result of 1 function Y = f ( x1 , x2 ,..., x n ) . fuzzy inference have to be transformed to analogue There are two important questions: (continuous) variables. This procedure is also based 1. How to transit from a standard specification of a on membership functions. Two major methods are fuzzy logic function to the specification of used for defuzzification: corresponding analog function? – The maximum defuzzification method, wherein an 2. How to transit from an analog function output value is determined by the linguistic specification and/or from a standard specification variable with the maximum weight; of a fuzzy logic function to corresponding CMOS – The centroid calculation defuzzification method, implementation? wherein an output value is determined by the First of all, let us address to membership weighted influence of all the active output functions. In most cases [2 – 4], membership membership functions. functions have a triangle or trapeze form (see Fig.1). As a rule, or at least in a great part of applications, a fuzzy controller is a transformer of α input analog signals into an analog output signal. A A B C D E F G 1 linguistic variable is a subjective characteristic of an input analog variable, values of which are transformed on bases of given membership functions into a set of weighted values of T corresponding linguistic variables. This procedure is Fig.1 Types of membership functions. called a fuzzification and it contains as its composite part the analog-digital transformation. In Fig.1 linguistic points (variables) A and B are A set of combinations of weighted linguistic cold, C is fresh, D and E are worm, F and G are hot. variables corresponds to each value combination of These points determine the connection of the input analog variables. On bases of a system of linguistic variables with values of the analog fuzzy inference rules it is possible to receive the set variable T (T is temperature). Relatively to these of weighted output linguistic variables. Using these points and similar points for other analog input variables and their membership functions, with help variables we can compose a table of fuzzy rules of one of well known defuzzification methods it is connecting combinations of input linguistic possible to form values of the analog output variables with output linguistic variables. variable. The defuzzification procedure also On bases of membership functions we can put includes digital-analog transformation. into accordance to the input and output linguistic At present the most wide-spread way of fuzzy variables a set of integer numbers splitting by logic control implementation is using the appropriate way all diapason of changing of programmable fuzzy controllers, which are available corresponding analog variables. Then the table of on the market together with the means of computer fuzzy rules will to determine by obvious way the aided programming (e.g. Motorola’s 8-bit 68HC11 function of multi-valued logic, values of which and 16-bit 68HC12 microcontrollers or specialized define the digit representation of the output fuzzy processors of Siemens 80C517/80C535 families). However, in spite of the implementation evidence and fuzzy controllers’ accessibility this approach to controller implementation possesses 1 It should be noticed that in suppressing majority of some disadvantages, e.g. such as high cost and low publications on fuzzy controllers, this function is given as throughput (that is especially important when fuzzy a response surface and practically without exception this control in the control contour is used) etc. surface has a piecewise linear form. ISSN: 1109-9445 Issue 6, Volume 4, June 2007 110  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 linguistic variable on chosen value combinations of multi-valued input variables. In other words, according to our concept, for a broad class of fuzzy controller specifications it is possible to construct corresponding tables connecting input and output membership functions. Frequently membership functions evenly divide the ranges of output variables’ variations. If it is not so, the membership functions can be brought to even scale by increasing the number of gradations or, as it will be shown later, by introducing a certain equalization procedure for logical levels. Therefore, specification tables represent nothing but tables Fig.2 Summing amplifier: a) general designation, determining a specific multi-valued logical function. b) CMOS implementation using symmetrical And what is more, for a number of implementations invertors. it is possible to neglect weighting and determining input linguistic variables and simply to use Vout a) continuous-valued variables. Vdd The above idea was in the focus of our research. We dealt with searching for simple basic multi- valued functions, which, from the one hand, would present a complete functional basis in the multi- Vdd / 2 valued logic, and from the other hand, could be ∑ n − Vdd / 2 R0 V efficiently implemented by CMOS technology. Rj (V j − dd ) 2 j =1 b) y k 2 Hardware Implementation of Fuzzy ∑ω x n j j Controllers −k k j =1 −k 2.1 Summing Amplifier as a Multi-Valued Logical Element Fig.3 Summing amplifier’s behavior: a) within Summing amplifier’s behavior, accurate to the voltage coordinates; b) within multi-valued members of the infinitesimal order that is variable coordinates. determined by the amplifier’s gain factor in disconnected condition (Fig.2), is described as Let us split the source voltage Vdd on m = 2k + 1 follows: voltage levels. Then replacing the input voltages V j − Vdd / 2 by m-valued logical variables ⎧ n R V V ⎪Vdd if ∑ 0 j=1 R j (V j − dd ) ≤ − dd 2 2 x j = ( 2 ⋅V j − Vdd )k /Vdd and the output voltage Vout ⎪ ⎪ by m-valued variable y and designating R0 / R j = ω j ⎪V n R V Vout = ⎨ dd − ∑ 0 (V j − dd ) in other cases (1) the system (1) can be represented as (2). ⎪ 2 j =1 R j 2 n ⎪ V n R V y ( X ) = S ( ∑ω j ⋅ x j ) = ⎪0 ⎪ if dd ≤ 2 ∑ 0 j=1 R j (V j − dd ) 2 j =1 ⎩ ⎧ n ⎪+ k if ∑ ω j ⋅ x j ≤ −k where Vdd is the supply voltage, V j is the voltage ⎪ j =1 (2) ⎪ n n on jth input, R j is the resistance of jth input, R0 is ∑ ∑ = ⎨− ω j ⋅ x j if k > ω j ⋅ x j > -k the feedback resistance, and Vdd / 2 is the midpoint ⎪ j =1 j =1 ⎪ n of the supply voltage. Dependence of Vout on ⎪− k if ∑ ω j ⋅ x j ≥ +k ⎩ j =1 ∑j=1 RR0 ⋅ (V j − V2dd ) is shown in Fig.3 (a). n Graphical view of (2) is shown in Fig.3 (b). j ISSN: 1109-9445 Issue 6, Volume 4, June 2007 111  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 Later on, we will call the functional element, f a ( x ) = −maj( x,− a,−k ) + a + k whose behavior is determined by the system (2), a multi-valued threshold element. When as it follows from Fig.4, hence ω j = 1, j = 1, 2, 3 , we will call it a majority element f a ( x ) = −maj ( −maj ( x,− a,−k ), a, k ) . (5) and designate as maj ( x1 , x 2 , x3 ) . Taking into consideration − maj (a , b, c ) = maj ( − a,−b,− c ) , 2.2 Functional Completeness of the it follows from (5) that Threshold Element The basic operation (or a set of basic operations) is max( x1 , x 2 ) = maj ( maj ( x1 ,− x 2 ,− k ),− x 2 ,− k ) . (6) called functionally completed in arbitrary-valued logic, if any function of this logic can be represented Now let us consider the representation of the as superposition of the basic operations. function y = ( x + 1) mod m , x ≥ 0, 0 ≤ y ≤ m-1 through There are some known functionally complete sets threshold functions. First of all we designate of functions. It is clear, that for proving the m = 2k + 1 and change the beginning of coordinates functional completeness of a certain new function it so that the function will have a form is sufficient to show that every function of the y = ( x + k + 1) mod( 2 k +1) − k , x ≥ − k , - k ≤ y ≤ + k . To known functionally complete set can be represented implement this function on threshold elements let us as a superposition of the considered function. One turn to the sequence of pictures in Fig.5. of functionally complete functions in m-valued logic is the Webb’s function [8]: y y a) k-1 b) k-1 w( x , y ) = [max( x, y ) + 1]mod m . (3) Therefore, for proving functional completeness of the threshold operation in multi-valued logic it is x x sufficient to show how the Webb’s function can be 1-k 1-k represented through this operation [5 – 7]. y = ( x + k + 1) mod( 2 k +1) − k ϕ1 ( x ) = −maj( x,1,0) First, let us represent the function max( x1 , x 2 ) c) y d) y by threshold functions. To do this let us consider the k-1 k-1 function f a ( x ) , such as ⎧a if a≥x x x f a ( x ) = max( x , a ) = ⎨ , ⎩x if x>a (4) 1-k 1-k | x |≤ k , | a |≤ k . ϕ 2 ( x ) = − maj( x,1 − k ,− k ) ϕ 3 ( x ) = k ⋅ maj(ϕ 2 ( x ), k ,0) The diagram of this function is shown in Fig. 5 Implementation of the function Fig.4(a). y = ( x + k + 1) mod( 2 k +1) − k . a) y b) y It is easy to see that k k fa (x) ( x + k + 1) mod( 2 k +1) − k = ϕ1 ( x ) + 2ϕ 3 ( x ) a a k k and obviously, this function can also be -k x -k x implemented on threshold elements as -k -k y = maj ( maj ( x ,1,0), k ⋅ maj ( maj ( x ,1 − k ,− k ), −k ,0), a -maj(x,-a,-k) a k ⋅ maj (maj ( x,1 − k , −k ), − k ,0)). Fig. 4 Diagrams of the functions a) f a (x ) and Hence, the functional completeness of the b) − maj ( x ,− a,−k ) . summing amplifier in arbitrary-valued logic is shown. The proof procedure of functional The − maj ( x,−a ,− k ) function diagram is shown completeness naturally does not give information in Fig.4(b). Actually, as far as x<a about methods of effective synthesis. Some methods x − a − k < −k and − maj( x,−a,−k ) = −k . Note of a circuit design in the proposed basis will be that for all values of x , developed later. ISSN: 1109-9445 Issue 6, Volume 4, June 2007 112  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 2.3 Fuzzy Devices as Multi-Valued and controllers’ behaviour as (7) and a plenty of Analog Circuits publications contain piecewise-linear Conventional implementation of fuzzy devices approximations of the corresponding surfaces. usually has the structure shown in Fig.6. Analog The apparent conclusion can be made from the variables X = {x1 , x 2 ,..., x n } enter the fuzzy device things mentioned above: if a fuzzy controller is represented as (7), it can be implemented as input. Fuzzifier converts a set of analog variables superposition of multi-valued threshold elements. In x j into sets of weighted linguistic (digital) this case, owing linear behavior of the threshold variables A = {a1 , a 2 ,..., a n } . element in the zone between the saturation levels ((2) and Fig.3 (b)), natural piecewise linear approximation appears between the discrete points Defuzzifier of specification. Inference Fuzzifier X A B Y Fuzzy analog digital digital analog In the last subsection of this section some illustrations will be given to show that for a number of real applications the offered approach can Fig.6 Conventional structure of a fuzzy device provides simple and efficient circuits of controllers. implementation. Fuzzy Inference block generates based on the 2.4 Fuzzy Controller Implementations as fuzzy rules a set of weighted linguistic variables Circuits from Threshold Elements values B = {b1 , b2 ,..., bk } . Defuzzifier converts sets of weighted linguistic (digital) variables B = {b1 , b2 ,..., bk } into a set of 2.4.1 Example 1 output analog variables Y = { y1 , y 2 ,..., y k } . Let us consider the example, which is taken from [9, pp. 81 – 86]: “Design of a Rule-Based Fuzzy As a rule, fuzzifier and defuzzifier include AD Controller for the Pitch Axis of an Unmanned and DA (analog-digital and digital-analog) Research Vehicle”. converters and are implemented on both levels The fuzzy control rules for the considered device (hardware and software). Fuzzy inference is usually depend on the error value e = ref − output and implemented on the level of microprocessor software. old e − new e changing of error ce = . Fuzzifier It is easy to see that each set of values of output sampling period analog variables unambiguously corresponds to gives seven linguistic variables for each of input some set of input analog variable values; hence a analog variables (NB – negative big; NM – negative fuzzy device could be specified as a functional middle; NS – negative small; ZO – zero; PS – analog of a signal converter positive small; PM – positive middle; PB – positive Y ( X ) = { y1 ( X ), y 2 ( X ),..., y k ( X )} big). The output has the same seven gradations. and its output Y determines a system of n- Corresponding 49 fuzzy rules are represented in dimensional surfaces. In cases of sufficient simple Table 1. membership functions (in known publications such Table 1: Table of Fuzzy Rules functions are in majority), for fuzzy controller e implementations as analog devices it is sufficient to NB NM NS ZO PS PM PB provide a piecewise-linear approximation between a NB ZO PS PM PB PB PB PB couples of points calculated as adjacent values of a NM NS ZO PS PM PB PB PB multi-valued logic function. NS NM NS ZO PS PM PB PB ce ZO NB NM NS ZO PS PM PB Let m = 2k + 1 linguistic variables a j ( a j ∈ A ) PS NB NB NM NS ZO PS PM correspond to values of analog variable xj PM NB NB NB NM NS ZO PS ( x j ∈ X ). Then basing on a system of fuzzy rules, PB NB NB NB NB NM NS ZO we can specify a system of m-valued logic Let us split evenly the source voltage (e.g. 3.5V) functions, as follows: onto seven logical levels corresponding to linguistic B( A) = {b1 ( A), b2 ( A),..., bk ( A)} . (7) levels and enumerate them with integer numbers Note that most publications describing fuzzy from -3 to +3. Then Table 2 will represent Table 1 controllers contain tables specifying fuzzy as the function of seven-valued logic. ISSN: 1109-9445 Issue 6, Volume 4, June 2007 113  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 Table 2: The Seven-Valued Function them is the operational force sensor Fh ; the other is e “the environmental force sensor” ω . Each of input 0 -3 -2 -1 0 1 2 3 and output variables of the manipulator controller is -3 0 1 2 3 3 3 3 represented with three linguistic variables – S -2 -1 0 1 2 3 3 3 (small), M (middle) and B (big). The controller has -1 -2 -1 0 1 2 3 3 five fuzzy rules, as it follows: ce 0 -3 -2 -1 0 1 2 3 If ω = S then Output=B; 1 -3 -3 -2 -1 0 1 2 If ω = B then Output=S; 2 -3 -3 -3 -2 -1 0 1 If ω = M and Fh = S then Output=S; 3 -3 -3 -3 -3 -2 -1 0 If ω = M and Fh = M then Output=M; It is seen from Table 2 that the function is If ω = M and Fh = B then Output=B; symmetric with respect to “North-West – South- The controller Output is three-valued logic East” diagonal and its values can be calculated as function specified in Table 3. e − ce . This dependency is shown in Fig.7. Table 3: The ternary function 3 Output Fh 2 −1 0 +1 1 -6 -5 -4 -3 -2 -1 e-ce −1 +1 +1 +1 1 2 3 4 5 6 -1 ω 0 −1 0 +1 -2 -3 +1 −1 −1 −1 Fig.7 Graphical representation of the function specified by Table 2. It can be simply proved by trivial substitution that Output = maj ( 2ω , − Fh , 0) and СMOS It apparently follows from comparison of Fig.3 implementation coincides with the circuit shown in (b) and Fig.7 that in order to reproduce the function Fig.8, if make substitutions Ve = VFh , Vce = Vω and specified by Table 2 it is sufficient to have one two- input summing amplifier and one one-input change the weight of the input Vω to 2. amplifier that will be called inverter. Note that inversion of logic variables lying within − k ÷ + k interval is the operation of 2.4.3 Example 3. Fuzzy Controller for Washing diametric negation x = − x ; the operation Machine This example is taken from Aptronix Incorporated Vout = Vdd − Vin corresponds to it in the terms of (http://www.aptronix.com/fuzzynet). summing amplifier’s input and output voltages. Thus CMOS circuit containing 12 transistors and 5 A. Controller specification resistors, which implements our function, is shown Input variables: in Fig.8. Dirtiness of clothes: Large (L), Medium (M), and S1 Small (S); Vdd-Ve S2 1 1 Vout Type of dirtiness: Greasy (G), Medium (M), and Not 1 Greasy (NG). Ve Vce Output variable is washing time (minutes): Very Fig.8 Implementation of the fuzzy controller Long (VL), Long (L), Medium (M), Short (S), and specified by Table 2. Very Short (VS). Fuzzy rules are represented in Table 4. 2.4.2 Example 2 Table 4: Matrix of linguistic variables This example is taken from [9, pp. 168 – 172]: Dirtiness of clothes Wash. time “Manipulator for Man-Robot Cooperation (Control S M L Method of Manipulator/Vehicle System with Fuzzy NG VS S M Inference)”. Type of dirt. M M M L In the considered example the experimental G L L VL manipulator has two force/torque sensors. One of ISSN: 1109-9445 Issue 6, Volume 4, June 2007 114  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 According to our approach Table 4 can be ( wash. time) = S (ϕ 3 (Y ) − 0.5 ⋅ X − 2 − ϕ 2 ) . transformed into the table of multi-valued logic variables (Table 5). Now let us introduce the function: 0, − 2, − 2 Table 5: Matrix of multi-valued variables Dirtiness of clothes (Y) ϕ 4 ( X , Y ) = S ( X + Y + 4) = − 2, − 2, − 2 (11) Wash. time − 2, − 2, − 2 −2 0 +2 −2 −2 −1 0 Type of dirt. and form the second intermediate sum: 0 0 0 +1 (X) +2 +1 +1 +2 + 1, 0, 0 In this table the output variable Wash. time has 5 0.5 ⋅ ϕ 4 ( X , Y ) + 1 = 0, 0, 0 = −ϕ 2 . (12) logical levels but input variables X and Y have only 0, 0, 0 three. Because of change ranges of the output and Finally input variables should be the same in the Table 5 logical levels of input variables X and Y are −2, 0, ( wash time ) = S [ϕ 3 (Y ) − 0.5 ⋅ X − 1 + and +2. 0.5 ⋅ ϕ 4 ( X ,Y )] = S 4 [ S1 (0.5Y − 2) + (13) 0.5 ⋅ S 2 ( X + Y + 4) + 0.5 ⋅ S 3 ( X ) − 1] B. Functional decomposition Let us represent the washing time matrix as a sum of two matrices: In Fig.9 the CMOS implementation of the wash time ϕ1 ϕ2 expression (13) is presented. The circuit is implemented as the superposition of four multi- − 2 −1 0 −1 −1 0 −1 0 0 valued threshold elements. (8) 0 0 +1 = 0 0 +1 + 0 0 0 +1 +1 + 2 +1 +1 + 2 0 0 0 S1 Y 0.5 or ( wash. time) = S ( −ϕ1 − ϕ 2 ) were S is the 1 S4 function of summing amplifier with saturation. S2 1 Vout 0.5 Let us take into consideration a function of one Y 1 0.5 X 1 0.5 variable PWR 2 ϕ 3 (Y ) = S (0.5 ⋅ Y − 2) = + 2 + 2 + 1 . (9) S3 In (9) Y corresponds to the dirtiness of clothes and X 1 varies from –2 to +2 as follows Fig.9 CMOS implementation of fuzzy controller Y =| −2 0 + 2 | . for washing machine. Now the following intermediate sum is introduced: The result of the SPICE simulation of the circuit in Fig.9 is shown in Fig.10 in the form of response + 1, + 1, 0 surface. ϕ 3 (Y ) − 0.5 ⋅ X − 2 = 0, 0, − 1 . (10) In Fig.10 all variables are represented in voltages. The correspondence of logical values to voltages is − 1, − 1, − 2 shown in Table 6. It is easy to see that the controller Here X corresponds to the type of dirtiness and output signal represented by the surface in Fig.10 varies also from –2 to 2 as follows has linear approximation between adjacent logical levels. −2 Table 6: Correspondence voltages to logical levels X = 0 . -2 -1 0 1 2 +2 0V 0.875V 1.75V 2.625V 3.5V From (8) and (10) it is easy to see that (10) is − ϕ1 and ISSN: 1109-9445 Issue 6, Volume 4, June 2007 115  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 note that equations (14) and (15) represent multi- valued functions in a piecewise-constant manner. An example of a 7-valued function is given in Fig.11(a). a) b) 3 3 2 2 1 1 1 2 3 1 2 3 −3 −2 −1 −3 −2 −1 −1 −1 −2 −2 −3 −3 Fig.10 Results of SPICE simulation for the Fig.11 Example of a seven-valued function: controller in Fig.9. a) piecewise-constant representation; b) piecewise-linear representation. 3 Universal Method of Implementing Taking into account fuzzification and Fuzzy Inference Rules defuzzification procedures in fuzzy logic, It was shown in 2.2 and [5, 6] that a summing corresponding multi-valued logic function should amplifier with saturation is a functionally complete has at least piecewise-linear approximation between element in any multi-valued logic (of an arbitrary adjacent logical levels. Fig.11(b) gives an example value). Thus it may serve as a basis for hardware of such a representation of the function with evenly implementation of fuzzy devices. distributed logical values of the input and the output The study subject is design techniques for analog in the range of corresponding voltages. CMOS circuits implementing fuzzy controller multi-valued functions. Without departing from the general character of 3.1 Masking Inputs of Summing Amplifiers the study, let us suppose that the logic has odd value Let us rewrite the definition (2) of the inverting ( m = 2k + 1 and − k ≤ x j ≤ + k ) . Let’s also assume summing amplifier with saturation in the following L that X = {x1 , x 2 , , x n } is a set of input multi- form: ⎧ n valued variables and y = F ( X ) is the output ⎪ + k if ∑ i =1 α i ⋅ xi + β ≤ − k variable. Then for a function of multi-valued logic it ⎪ n ⎪ is possible to build an analog of the Shannon’s S ( A ⋅ X ; β ) = ⎨ − ∑α i ⋅ xi − β in other cases (16) decomposition in the binary logic: ⎪ i =1 ⎪ n y i = Fi ( X ) = ⎪ −k if + k ≤ ∑α i ⋅ xi + β ⎩ i =1 +k (14) = U [if x j = α then y = F ( x j = α , X \ x j )]. where A = {α1 ,α 2 ,..., α n } is a set of weight α =−k coefficients, X = {x1 , x2 ,..., xn } is a set of analog or Equation (14) can be further expanded so that it would be possible to build an realizing circuit using multi-valued variables, β is a constant symbolizing the variables exclusion method. To this effect, we a threshold, and ± k is a saturation value (in the need a sub-circuit implementing the function: case of m-valued logic, m = 2k + 1 ). Let us introduce a masking function M α (x ) by if Z = A then y = F ( Z = A, X \ Z ) (15) the next way: where Z ⊂ X and A is a value combination of the variable set Z .2 ⎧+ k if x ≤ α −1 ⎪ Having a basic element (sub-circuit realizing M α ( x ) = ⎨k (α − x ) if α −1< x < α +1 (17) (15)), we can implement a fuzzy device directly ⎪− k if α +1 ≤ x ⎩ according to the system of fuzzy rules. However, where α ( − k ≤ α ≤ k ) is a fixed value of the 2 It is possible to add else in (15) that can be defined by variable x . It can be easily seen that when x = α , circuit requirements. ISSN: 1109-9445 Issue 6, Volume 4, June 2007 116  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 a) M (x) b) M α (α ) = 0 . Fig.12 illustrates an example of the -1 3 -M-1 (x) 3 function M −1 ( x ) for m = 7 . 2 2 S(-M -1(x);2) S(M -1 (x);2) 3 M-1(x) 1 1 2 -3 -2 -1 1 2 3 x -3 -2 -1 1 2 3 x -1 -1 1 1 2 3 -2 -2 −3 −2 −1 x -3 -3 −1 c) y −2 3 −3 2 Fig.12 M −1 ( x ) diagram for m = 7 . 1 -3 -2 -1 1 2 3 x Taking into account that the source voltage -1 Vdd has the logical value equal to +k and the ground -2 -3 potential Vgnd has the logical value equal to –k, the mask-function can be easily implemented on bases Fig.14 Implementation example of the rule (19). of summing amplifier as ⎧ S ( kx; − α ⋅ Vdd ) if α < 0 3.2 Mask-Functions of Other Types ⎪ To decrease the number of variables, which an m- M α ( x ) = ⎨ S ( kx ) if α = 0 (18) ⎪ S ( kx; α ⋅ V ) if α > 0 valued logical function depends on, by one using the ⎩ gnd analog of Shannon’s decomposition (14) we need to where x ( Vgnd ≤ x ≤ Vdd ) is measured in voltages. implement m rules of the type (19) and to find m components F ( x = α , Y ), -k ≤ α ≤ + k of the Using the mask-function M α (x ) it is possible to decomposition. Sometimes the number of rules can implement the rule be reduced, if the function F ( x , Y ) doesn’t change if x = α then y = F ( x = α ,Y ) else y = 0 , (19) on some interval of changing logical values of the variable x. A single rule can correspond to such which extracts the value of the function interval of the variable x and the conditional part of F ( x = α , Y ) in the point x = α , using the circuit this rule can have one of three forms: α ≤ x ≤ β , from summing amplifiers shown in Fig.13. x ≤ β , α ≤ x where − k ≤ α < β ≤ + k . For the condition α ≤ x ≤ β let us construct the following F(x=α ,Y) mask-function: 1 1 y Mα (x) 1 1 1 M α ,β ( x ) = 1 ⎧+ k if x ≥ β + 1 ⎪ (21) 1 1 -Mα(x) ⎨ − M α −1 ( x ) − M β +1 ( x ) if α − 1 < x < β + 1 ⎪ Fig.13 Implementation of the rule (19). ⎩− k if x ≤ α − 1 This implementation can be written in analytical It is easy to see (Fig.15(b)) that on the interval form as α ≤ x ≤ β this function takes the value 0. In the case when α = −k or β = k , this mask- y = S{S [ M α ( x ); F ( x = α ,Y )]; S [ S ( M α ( x )); (20) function will have one of the forms: F ( x = α ,Y )]; F ( x = α ,Y )}. ⎧+ k if x ≥ β + 1 For example, in the case when α = −1 , M −k , β ( x ) = ⎨ or (22) F ( x = −1, Y ) = 2 , and m = 7, the behavior of the ⎩+ k − M β +1 ( x ) if x < β + 1 circuit in Fig.13 can be represented by Fig.14. ⎧− M α -1 ( x ) − k if α − 1 < x Analyzing the implementation of the rule (19) it M α ,+k ( x ) = ⎨ (23) ⎩− k if x ≤ α − 1 is possible to see that in it the condition x = α is realized as the condition M α ( x ) = 0 . and represents conditions x≤β or α≤x respectively. ISSN: 1109-9445 Issue 6, Volume 4, June 2007 117  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 Mask-functions (21), (22), and (23) can be 3.3 An Application Example of Interval implemented on bases of summing amplifiers as Masking M α ,β ( x ) = S [ M α −1 ( x ); M β +1 ( x )] , (24) For further explanation of the matter discussed in 3.2, let us recall an example from [4, pp. 123 – 128] M −k ,β ( x ) = S [Vgnd ; M β +1 ( x )] , (25) “A Fuzzy Logic Force Controller for a Stepper M α ,k ( x ) = S [ M α −1 ( x );Vdd ] . (26) Motor Robot”. The fuzzy controller implements the function of Let us look how the masking can be performed two analog variables: position error and force error, for a wider scope of the variable changes, such as: which will be designate as x and y respectively. if α ≤ x ≤ β then y = F (α ≤ x ≤ β ,Y ) = Φ (Y ) Each of the variables x and y is represented with (27) else y = 0. 7 linguistic variables: NL, NM, NS, ZE, PS, PM, Using the mask-function M α ,β ( x ) it is possible PL, and their membership functions are shown in Fig.16. to transform the rule (27) into the following form: NL NM NS ZE PS PM PL if M α,β ( x ) = 0 then y = F (α ≤ x ≤ β ,Y ) = 1 (28) degree of membership = Φ (Y ) else y = 0. 0.8 The rule (28) can be implemented with the circuit 0.6 shown in Fig.13, if to change in it the inputs M α ( x ) 0.4 and F ( x = α , Y ) with the inputs M α ,β ( x ) and 0.2 Φ (Y ) respectively. This implementation is 0 represented analytically as -3 -2 -1 0 1 2 3 universe of discourse y = S{S [ M λ ,δ ( x ); Φ (Y )]; S [ S ( M λ ,δ ( x )); (29) Fig.16 Fuzzy sets for force error and position Φ (Y )]; Φ (Y )}. error inputs. The sequence of pictures in Fig.15 illustrates the The Inference Engine Rule Matrix for the output implementation of the rule linguistic variable from the cited work looks as it is if − 2 ≤ x ≤ +1 then y = −2 else y = 0 shown in Table 7. for the case of ( m = 9 )-valued logic. Table 7: Rule matrix of the inference engine position error (x) a) 4 b) 4 NL NM NS ZE PS PM PL 3 M+2(x) 3 NL NM NL NL NL NL NL NM 2 2 NM NS NM NM NM NM NM NS M-2,+1(x) = 1 S(M-3(x);M+2(x)) 1 force NS ZE NS NS NS NS NS ZE error ZE ZE ZE ZE ZE ZE ZE ZE -4 -3 -2 -1 1 2 3 4 x -4 -3 -2 -1 1 2 3 4 x -1 -1 (y) PS ZE PS PS PS PS PS ZE -2 -2 PM PS PM PM PM PM PM PS M-3(x) -3 -3 PL PM PL PL PL PL PL PM -4 -4 Let us transform the Table 7 into the Table 8 c) y 4 d) 4 taking into account that we are going to produce 3 3 fuzzy inference calculating values of the 2 2 corresponding multi-valued logic function. 1 1 Table 8 comprises only two different columns -4 -3 -2 -1 -1 1 2 3 4 x -4 -3 -2 -1 -1 1 2 3 4 x defining two functions depending on the variable -2 -2 force error (Table 9). S(S(M-2,+1(x));-2) S(M-2,+1(x);-2) S(S(S(M (x));-2);S(M (x);-2);-2) Fig.17 illustrates graphs of these functions. It is -3 -2,+1 -2,+1 -4 -4 easy to see that the function F1 ( y ) looks like mask- Fig.15 Example of mask-functions application. function M −1,1 ( y ) but has different slops of the lines. By analogy with (17), (18), (21), (24), Fig.15(a), and Fig.15(b), it is possible to construct ISSN: 1109-9445 Issue 6, Volume 4, June 2007 118  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 the function F1 ( y ) in accordance with graphics in if NM ≤ x ≤ PM then Output = F2 ( y ) or Fig.18. else Output = F1 ( y ); Table 8: Matrix of the multi-valued logic function if − 2 ≤ x ≤ +2 then Output = F2 ( y ) position error (x) (31) else Output = F1 ( y ). -3 -2 -1 0 1 2 3 -3 -2 -3 -3 -3 -3 -3 -2 It is possible to split the rule (31) into two rules -2 -1 -2 -2 -2 -2 -2 -1 and represent them as: force -1 0 -1 -1 -1 -1 -1 0 error 0 0 0 0 0 0 0 0 if M -2,+2 ( x ) = 0 then Output = F2 ( y ) (32) (y ) 1 0 1 1 1 1 1 0 else Output = 0; 2 1 2 2 2 2 2 1 3 2 3 3 3 3 3 2 if M -2, +2 ( x ) ≠ 0 then Output = 0 (33) else Output = F1 ( y ). Table 9: Two different functions of the force error force error (y) The rule (32) can be implemented in accordance -3 -2 -1 0 1 2 3 with (29) and (30) and (24) as F1 ( y ) -2 -1 0 0 0 1 2 ⎧ M −2, +2 ( x ) = S 4 ( M −3 ( x ); M +3 ( x )), F2 ( y ) -3 -2 -1 0 1 2 3 ⎪ ⎪− M −2, +2 ( x ) = S 5 ( M −2, +2 ( x )), ⎨ ⎪Φ 1 = S11{S 6 [ M −2, +2 ( x ); F2 ( y )]; S 7 [− M −2, +2 ( x ); a) b) ⎪ F2 ( y )]; F2 ( y )}. F1(y) 3 F2(y) 3 ⎩ 2 2 It is easy to check that the rule (33) can be 1 1 implemented in accordance with the structural -3 -2 -1 1 2 3 y -3 -2 -1 1 2 3 y scheme shown in Fig.13, in which the output -1 -1 amplifier has the weight equal to 2 of the input -2 -2 F1 ( y ) : -3 -3 Φ 2 = − S10 {S 9 [ M −2, +2 ( x ); F1 ( y )]; S8 [ − M −2, +2 ( x ); Fig.17 Components of the function defined by F1 ( y )];2 F1 ( y )}. Table 8 and decomposed relative to variable x. Finally the output of the controller can be a) b) 3 F1 ( y ) = S 3 ( 23 S1 ; 23 S 2 ) 3 calculated as S 2 ( 32 y; 32 Vgnd ) Output = Φ 1 + Φ 2 . 2 2 1 1 For producing this summation it is possible to use -3 -2 -1 1 2 3 y -3 -2 -1 1 2 3 y summing amplifier S11 -1 -1 -2 -2 Output = S11 ( −Φ1 ;−Φ 2 ). S1 ( 23 y; 23 Vdd ) -3 -3 Fig.19 illustrates the structural scheme of the controller implementation with elements containing Fig.18 Constructing of the function F1 ( y ) . designations of input weights. As a result, the functions F1 ( y ) and F2 ( y ) can Vdd M −3( x) 3 S4 S6 1 S11 be implemented as x 3 1 1 1 Vout 1 1 1 3 S5 S7 1 F1 ( y ) = S 3 [ 23 S 2 ( 23 y ; 32 V gnd ); 23 S1 ( 23 y ; 32 Vdd )], 3 M +3 ( x) 1 (30) 1 1 F2 ( y ) = y. S1 S3 S8 3/2 2/3 1 S10 y 3/2 2/3 1 1 It is seen from the Table 7 and Table 8 that the S2 S9 1 2 3/2 1 behaviour of the controller's output in the 3/2 1 decomposition by variable x has the form: Fig.19 Structural diagram of the controller. ISSN: 1109-9445 Issue 6, Volume 4, June 2007 119  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 The controller circuit has been constructed from into component matrices with relatively simple three-stage push-pull CMOS operational amplifiers configuration of elements allocation, for which with 1-MegOhm resistors in the feedback (Fig.2 rather simple implementations can be find. The (b)). It’s functioning has been checked with SPICE topologies of valuable elements inside of such simulation (MSIM 8). MOSIS BSIM3v3.1, level 7 component matrices can be specified as model of 0.4μm transistors has been used. In this symmetrical, diagonal, matrixes with linear paper, all other SPICE simulation experiments with configurations of elements, with elements located designed circuits of controllers have been executed along rows and columns, matrices containing single under the same conditions. valuable element, and others. In the experiments with the controller presented The best way to introduce particular design in Fig.19, source voltage was 3.5V, input variable x methods is to show possible matrix decomposition changed linearly from 0V to 3.5V, input variable y into a set of implementable matrices on bases of a changed discreetly in accordance with its logical real design example. values and kept constant value within one cycle of x Let us take the description of the rather complex changing. fuzzy controller from the patent [10] of Toyota For the controller constructed from 3-stage Motors Corporation. The controller calculates a elements results of SPICE simulation are shown in regeneration time decision coefficient R on the base Fig.20. It is possible to see that the functioning of on a differential pressure coefficient K p and total the controller is correct (logical values of the circuit fuel consumption Q f . The set of fuzzy rules in output depend on the logical values of the input variables in accordance with Table 7 and 8). terms of linguistic variables is represented in Table 10. Transformations of the input and output analog signals are performed in accordance with corresponding membership functions. Table 10: Fuzzy rules for regeneration time decision coefficient R. Kp NB NM NS ZO PS PM PB NB NB NB NM NS ZO PB PB NM NB NM NS NS ZO PB PB NM NM NS ZO ZO PS PB Q f NS ZO NM NS ZO ZO PS PM PB PS NS ZO ZO PS PS PM PB PM ZO ZO PS PM PM PM PB Fig.20 SPICE simulation results for the controller PB PS PS PM PM PM PB PB constructed from 3-stage summing amplifiers. Analysis of the membership functions in [10] of linguistic variables representing input and output 4 Particular Methods of Fuzzy analog signals shows that the linguistic variables Inference Implementation having maximum weight are evenly distributed The universal method of implementing multi-valued within the change ranges of corresponding analog logic functions proposed in the previous section can signals. It means that without losing the accuracy of be always used but often can give inappropriate representation, these linguistic variables can be results due to its universality. For this reason some replaced with logical values as it is shown in Table particular design methods for fuzzy inference part of 11. controllers were developed. These methods utilize Table 11: The 7-valued logical function specific properties of certain multi-valued logical x\y -3 -2 -1 0 1 2 3 function descriptions corresponding to sets of fuzzy -3 -3 -3 -1 -1 0 3 3 inference rules. -2 -3 -2 -1 -1 0 3 3 According to the approach described above, an -1 -2 -2 -1 0 0 1 3 initial set of fuzzy rules is represented in the form of 0 -2 -1 0 0 1 2 3 a matrix or matrices defining multi-valued logical 1 -1 0 0 1 1 2 3 functions. As a rule these matrices cannot be 2 0 0 1 2 2 2 3 directly implemented. They must be decomposed 3 1 1 2 2 2 3 3 ISSN: 1109-9445 Issue 6, Volume 4, June 2007 120  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 In this table, signals Q f and K p are changed with To implement the function f1 ( z ) let us represent 7-valued logic variables x and y respectively. it as a sum of 5 subfunctions α j ( z ) shown in Fig.22. 4.1 Extracting a Symmetrical Component 3 Matrix 2 Let the Table 11 of the controller is represented as initial matrix M, which, in its turn, can be 1 represented as sum of two component matrixes (M1 α3 α4 α5 and M2). -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 M z=(x+y)/2 α1 -1 − 3 − 3 − 2 −1± 0 + 3 + 3 α2 − 3 − 2 −1−1± 0 + 3 + 3 -2 − 2 − 2 −1± 0 ± 0 +1+ 3 -3 − 2 −1± 0 ± 0 +1+ 2 + 3 = Fig.22 Five components of the function f1 ( z ) . −1± 0 ± 0 +1+1+ 2 + 3 ± 0 ± 0 +1+ 2 + 2 + 2 + 3 It is easy to see, that 5 +1+1+ 2 + 2 + 2 + 3 + 3 f1 ( z ) = ∑j α j ( z ) . (34) M1 M2 =1 − 3− 3− 2 − 2 −1± 0 ± 0 ≤ 0 ≤ 0 ± 0 +1+1+ 3 + 3 Let us consider formation of α j (z ) using − 3− 2 − 2 −1± 0 ± 0 +1 ≤ 0 ± 0 +1± 0 ± 0 + 3 ≥ 2 summing amplifiers on the example of α1 ( z ) . For − 2 − 2 −1± 0 ± 0 +1+1 ± 0 ± 0 ± 0 ± 0 ± 0 ± 0 ≥ 2 this let us address to Fig.23. − 2 −1± 0 ± 0 +1+1+ 2 + ± 0 ± 0 ± 0 ± 0 ± 0 +1 ≥ 1 3 −1± 0 ± 0 +1+1+ 2 + 2 ±0±0±0±0±0±0 ≥1 β1 2 ± 0 ± 0 +1+1+ 2 + 2 + 3 ± 0 ± 0 ± 0 +1± 0 ± 0 ≥ 0 ± 0 +1+1+ 2 + 2 + 3+ 3 +1± 0 +1± 0 ± 0 ≥ 0 ≥ 0 1 δ1 Matrix M1 corresponds to a symmetrical component -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 and M2 corresponds to nonsymmetrical residual z=(x+y)/2 component. -1 Matrix M1 is symmetrical relative to the side diagonal. Its components can be represented as a -2 function f1 ( z ) of one variable z = ( x + y ) / 2 . After -3 performing the linear approximation between Fig.23 Representation of the function α1 ( z ) . adjacent logical levels this function will has the form shown in Fig.21. The function β1 ( z ) in Fig.23 can be implemented as 3 f1(z) β1 ( z ) = − S ( k1 ⋅ z + a1 ) . For z = −2.25 , β1 = 0 then a1 = 2.25 ⋅ k1 . Taken into account that 2 k1 = 6 / 0.5 = 12 , we receive a1 = 27 and 1 β1 ( x, y ) = − S (6 ⋅ x + 6 ⋅ y + 27) , z=(x+y)/2 δ 1 ( x , y ) = 16 ⋅ β 1 ( x , y ) . -3 -2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 Finally -1 α1 ( x , y ) = δ 1 ( x , y ) − 0.5 = 16 ⋅ β1 ( x , y ) − 0.5 . (35) -2 In the same way it is possible to find -3 α 2 ( x, y ) = 13 β 2 ( x, y ) − 1 = Fig.21 Graph of the function f1 ( z ) . − 13 S (3 ⋅ x + 3 ⋅ y + 6) − 1; ISSN: 1109-9445 Issue 6, Volume 4, June 2007 121  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 α 3 ( x, y ) = 16 β 3 ( x , y ) + 0.5 = logical value less than a and more than b respectively. Let us split the matrix M2 in two − 16 S (6 ⋅ x + 6 ⋅ y − 3) + 0.5; matrices (M3 and M4) and try to implement the α 4 ( x , y ) = 16 β 4 ( x , y ) + 0.5 = matrix M3. The matrix M4 is a new residual matrix. (36) Let us address to Fig.24. It is easy to see that the − 16 S (6 ⋅ x + 6 ⋅ y − 15) + 0.5; matrix M3 consists of elements with two different α 5 ( x, y ) = 16 β 5 ( x, y ) + 0.5 = values, which can be separated with help of two − 16 S (6 ⋅ x + 6 ⋅ y − 27) + 0.5. parallel lines: x − 3 y + 8 = 0 and x − 3 y + 8 = 1 . A new variable is introduced The function f1 ( z ) can be calculated in w = x − 3y + 8 . (38) accordance with (34) on one summing amplifier. Finally taking into account mutual compensation of Value of the variable w in the point with constants, we have coordinates ( x , y ) is proportional to the distance of f1 ( x, y ) = S{− 16 ⋅ [ β1 ( x, y ) + 2 ⋅ β 2 ( x, y ) + this point from the line. In all points lying on and up (37) of the line, w ≤ 0 , and in all points lying on and β 3 ( x, y ) + β 4 ( x, y ) + β 5 ( x, y )]}. down of the dashed line ( x − 3 y + 7 = 0 ), w ≥ 1 . Thus, the implementation of the function f1 ( x, y ) , which represents the matrix M1, consists -3 of six summing amplifiers. -2 -1 x-3y+8=0 4.2 Extracting a Matrix with Elements Separated by a line -3 -2 -1 0 1 2 3 y This method is applicable for realization of matrices 1 composed from two types of elements, which can be separated with a line. 2 After extracting the symmetrical component the residual matrix is M2. 3 x M2 Fig.24 Separating valuable elements of ≤ 0 ≤ 0 0 +1+1 + 3 + 3 the matrix M3. ≤ 0 0 +1 0 0 + 3 ≥ 2 It is easy to see, that the matrix M3 representing 0 0 0 0 0 0≥2 the function f 2 ( x, y ) can be implemented as 0 0 0 0 0 +1 ≥ 1 = f 2 ( x, y ) = S{S [3( − x + 3 y − 8)] − 3} . (39) 0 0 0 0 0 0 ≥1 0 0 0 +1 0 0 ≥ 0 In this implementation all valuable matrix elements are equal to “3”. +1 0 +1 0 0 ≥ 0 ≥ 0 M3 M4 0 0 0 0 0 + 3+ 3 ≤ 0 ≤ 0 0 +1+1≥ 0 ≥ 0 4.3 Extracting a Matrix with Rectangular Configuration of Valuable Elements 0 0 0 0 0 + 3+ 3 ≤ 0 0 +1 0 0 ≥ 0 ≥ 0 Let us introduce a Pyramid Function that is the 0 0 0 0 0 0+3 0 0 0 0 0 0≥0 function, which corresponds to a matrix with a 0 0 0 0 0 0+3 + 0 0 0 0 0 +1 ≥ 0 single valuable element and represents a rule of the type 0 0 0 0 0 0+3 0 0 0 0 0 0≥0 if ( x = a ) & ( y = b) then f ( x, y ) = c 0 0 0 0 0 0 0 0 0 0 +1 0 0 ≥ 0 (40) else f ( x, y ) = 0. 0 0 0 0 0 0 0 +1 0 +1 0 0 ≥ 0 ≥ 0 This function is shown in Fig.25. This matrix has some elements of types ≤ a The Pyramid Function has some fixed value c and ≥ b . This means that instead of values a ( −k ≤ c ≠ 0 ≤ + k ) at the point (a , b) and at the rest and b of the elements it is possible to substitute any of the space bordered by points neighboring to ISSN: 1109-9445 Issue 6, Volume 4, June 2007 122  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 (a , b) this function is zero. The transition from c to γ (x) zero is linear. Neighborhood is defined by k coordinate increments Δ x = ±1 and Δ y = ±1 . f(x,y) x -k a-1 a a+1 k y x -k Fig.27 Graph of the function γ (x ) . The function γ ( x , y ) equals to “0” when Fig.25 A pyramid function. ( x = a ) & ( y = b) and equals to “+ k ” for all other Let's turn to Fig.26 to construct the pyramid matrix points. function. Now it is easy to construct the pyramid function with height “c” presented in Fig.25: Ma+1(x) -Ma-1(x) c k f ( x, y ) = S [γ ( x , y ) − k ] . (45) k A pyramid of an arbitrary height is obtained by simple input gain factor scaling of the next amplifier. The pyramid sign can be elementarily a-1 a+1 x changed at the stage of component mask-function -k a k constructions. We anticipate some complications in the case when it is needed to receive good "sewing" pyramids with already implemented functions. The -k pyramid function f ( x, y ) (45) of Fig.25 type has Fig.26 Component functions of the pyramid intersections with the flats y = {b, b ± 5, b ± 1} and projection onto the flat y = b ; c = k . with the flats x = {a , a ± 0.5, a ± 1} shown in Fig.28. The figure shows two component mask-functions a) f b) f y=b x=a M a +1 ( x ) and − M a −1 ( x ) those are implemented for y = b ± 0.5 x = a ± 0 .5 (2 ⋅ k + 1) -valued logic ( − k ≤ x ≤ + k ) as: y = b±1 x = a ±1 M a +1 ( x ) = S [k ⋅ ( x − a − 1)], a x b y (41) − M a −1 ( x ) = S [k ⋅ ( − x + a − 1)]. Similarly component functions of the pyramid Fig.28 Graph of the pyramid function a) by projection onto the flat x = a for the case c = k can coordinate x and b) by coordinate y. be constructed as: M a +1 ( y ) = S [k ⋅ ( y − b − 1)], When “sewing” a pyramid function with other (42) − M a −1 ( y ) = S [k ⋅ ( − y + b − 1)]. functions to get monotonous piece-linear It is easy to check that the function approximation between adjacent logical values the view of the pyramid function by each of its γ ( x ) = S [ M a +1 ( x ) − M a −1 ( x ) − 2k ] (43) coordinate can be changed to one of variants shown has the form shown in Fig.27; γ (x ) equals to “0” in Fig.29. when x = a and equals to “ + k ” for all other To construct a pyramid function with graphs by integer argument values. coordinates x and y of the Fig.29(a) type (center In a similar manner, we can construct the trapeze), it is sufficient to substitute in the functions following function of 2 variables: (41) – (45) instead of original variables x, y new γ ( x, y ) = S [ M a +1 ( x ) − M a −1 ( x ) + variables z = x + y , w = x − y and instead of points (44) M b+1 ( y ) − M b−1 ( y ) − 4k ]. a, b points c = a + b , d = a − b respectively. ISSN: 1109-9445 Issue 6, Volume 4, June 2007 123  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 a) f b) f c) f It is easy to see that the function y=b y=b y=b f ( x, y ) = ϕ [ x, ξ ( y )] + ψ [ x, ξ ( y )] has the right-hand y = b ± 0.5 y = b ± 0.5 y = b ± 0.5 y = b ±1 y = b ±1 y = b ±1 trapeze form along the axis x (Fig.29(b)). If the a x a x a x function ξ ( y ) has a triangle form, ξ (b) = ± k , and ξ ( y < b − 1) = ξ ( y > b + 1) = 0 , the function f ( x, y ) is a pyramid function. The way of constructing the Fig.29 Possible graphs of a pyramid function by function f ( x, y ) with left-hand trapeze form along one of coordinates: a) center trapeze, b) right trapeze, c) left trapeze. the axis x (Fig.29(c)) now is obvious. It is not difficult to check that the functions It means the transition to the pyramid function ϕ ( x , ξ ( y )) and ψ ( x, ξ ( y )) can be implemented as shown in Fig.30, which is implemented by analogy ϕ [ x, ξ ( y )] = S [ M a −1 ( x ) + 2k − ϕ ( y )] , (47) f(x,y) ψ [ x, ξ ( y )] = S [ − M a +1 ( x ) + 2k − ϕ ( y )] + y (48) 2k − ϕ ( y ). z The pyramid function approach is not limited to rules with point condition (40) and may be extended x to rules with interval condition of the type if (a1 ≤ x ≤ a 2 and b1 ≤ y ≤ b2 ) (49) w then f ( x, y ) = k else f ( x, y ) = 0. Fig.30 A pyramid in coordinates z = x + y Interval conditions can be implemented by simple and w = x − y . changing constants in mask-functions. The rule (49) represents matrices with rectangular configurations with (41) – (45) as of valuable elements. Such matrices can be implemented as truncated pyramids. f ( x, y ) = S{S [ S (k ( x + y − a − b − 1)) + Note that the function similar to (44) may be S (k ( − x − y + a + b − 1)) + constructed for an arbitrary number of variables. (46) Implementation of a pyramid function of two S (k ( x − y − a + b − 1)) + variables requires 6 amplifiers. Introducing each S (k ( − x + y + a − b − 1)) − 4k ] − k }. additional variable requires two additional amplifiers. For implementing a function that has graphs along one of its variables (e.g. x) of the right trapeze type (Fig.30(b)) or left trapeze type (Fig.30(c)), let 4.4 Extracting a Matrix with Valuable us introduce two intermediate functions ϕ [ x , ξ ( y )] Elements Laying on a Diagonal and ψ [ x, ξ ( y )] shown in Fig.31. Let us split the matrix M4 on two matrices (M5 and M6) and try to implement the matrix M5. Matrix M6 a) ϕ (x,ξ(y)) b) ψ (x,ξ(y)) ξ(y)=0 is the next residual matrix. k k M4 0< ξ(y)<k ≤ 0 ≤ 0 0 +1+1≥ 0 ≥ 0 a-1 a ξ(y)=k x ξ(y)=k x ≤ 0 0 +1 0 0 ≥ 0 ≥ 0 -k k -k a-1 a a+1 k 0 0 0 0 0 0≥0 0< ξ(y)<k 0 0 0 0 0 +1 ≥ 0 = -k ξ(y)=0 0 0 0 0 0 0≥0 -k Fig.31 Intermediate functions a) ϕ [ x, ξ ( y )] , 0 0 0 +1 0 0≥0 b) ψ [ x, ξ ( y )] . +1 0 +1 0 0≥0≥0 ISSN: 1109-9445 Issue 6, Volume 4, June 2007 124  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 M5 M6 Fig.30. Substitution of the variables z and w in 0 0 0 0 0 0 0 ≤ 0 ≤ 0 0 +1+1 ≥ 0 ≥ 0 formula (46) instead of x+ y and x− y 0 0 0 0 0 0 0 ≤ 0 0 +1 0 0 ≥ 0 ≥ 0 respectively, a = +1 , b = +1 , k = 3 , and changing the sign of the function gives 0 0 0 0 0 0 +1 0 0 0 0 0 0≥0 0 0 0 0 0 +1 0 + 0 0 0 0 0 0≥0 f 32 ( z, w) = S{S [ β 6 ( z ) + δ 6 ( z ) + 0 0 0 0 0 0 0 0 0 0 0 0 0≥0 β 7 ( w) + δ 7 ( w) + 12] + 3} 0 0 0 +1 0 0 0 0 0 0 0 0 0≥0 where β 6 ( z ), δ 6 ( z ) are already implemented and 0 0 +1 0 0 0 0 +1 0 0 0 0≥0≥0 β 7 ( w) = S (3w + 3), δ 7 ( w) = S ( −3w + 3). In its turn, the matrix M5 can be composed from two matrices Finally, the function f 3 ( x, y ) corresponding to the M 51 M 52 matrix M5 can be implemented as 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 3 ( x, y ) = 13 S{β 6 ( x, y ) + δ 6 ( x, y ) + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0+3 0 0 0 0 0 0 0 S [ β 6 ( x, y ) + δ 6 ( x, y ) + (50) 1 1 β 7 ( x, y ) + δ 7 ( x, y ) + 12] + 9} + 1 0 0 0 0 0+3 0 + 0 0 0 0 0 0 0 3 3 0 0 0 0+3 0 0 0 0 0 0 -3 0 0 where 0 0 0+3 0 0 0 0 0 0 0 0 0 0 β 6 ( x, y ) = S (3x + 3 y − 3), 0 0+3 0 0 0 0 0 0 0 0 0 0 0 δ 6 ( x, y ) = S ( −3x − 3 y + 9), and represented as the sum of corresponding β 7 ( x, y ) = S (3x − 3 y + 3), functions f 3 ( x, y ) = 13 [ f 31 ( x, y ) + f 32 ( x, y )] . δ 7 ( −3x + 3 y + 3). In the matrix M51, elements with the value “+3” lay on the line x + y − 2 = 0 . This matrix can be described by the function of one variable 4.5 Implementation of the Matrix M6 f 31 ( z = x + y ) , which is defined by the rule Let us split the matrix M6 on two matrices (M7 and if z = 2 then f 31 ( z ) = 3 else f 31 ( z ) = 0. M8) and try to implement the matrix M7. The new residual matrix is M8, all elements of which are The function f 31 ( z ) can be constructed as it is defined. shown in Fig.32. M6 a) β6 , δ6 b) α6 , f31 3 3 ≤ 0 ≤ 0 0 +1+1≥ 0 ≥ 0 2 β6 δ6 2 ≤ 0 0 +1 0 0 ≥ 0 ≥ 0 f31 1 1 0 0 0 0 0 0≥0 z=x+y 2 z=x+y 0 0 0 0 0 0≥0 = -6 -4 -2 0 2 4 6 -6 -4 -2 0 4 6 0 0 0 0 0 0≥0 -1 -1 α6 0 0 0 0 0 0≥0 -2 -2 +1 0 0 0 0 ≥ 0 ≥ 0 -3 -3 Fig.32 Functional representations of the M7 M8 matrixes M5. 0 0 0 +1+1+1+1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 +1 0 0 0 0 It is easy to see from this figure that 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f 31 ( z ) = α 6 ( z ) + 3, α 6 ( z ) = S [ β 6 ( z ) + δ 6 ( z ) + 6], 0 0 0 0 0 0 0+ 0 0 0 0 0 0 0 β 6 ( z ) = S (3z − 3), δ 6 ( z ) = S ( −3z + 9). 0 0 0 0 0 0 0 0 0 0 0 0 0 0 For good “sewing” the function f 32 with f 31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 and the function f 3 with f1 the function f 32 has to 0 0 0 0 0 0 0 +1 0 0 0 0 0 0 be implemented as pyramid function of variables z = ( x + y ) / 2 and w = ( x − y ) / 2 as it is shown in ISSN: 1109-9445 Issue 6, Volume 4, June 2007 125  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 The matrix M7 has one rectangular component with (46) but the following approach, which is and can be represented as the function f 4 ( x, y ) that shown in Fig.34, gives better implementation. is defined by the rule a) −Μ3(x) b) −Μ−3(y)−3 if ( x = −3 ) & ( y ≥ 0 ) then f 4 ( x, y ) = 1 3 3 else f 4 ( x, y ) = 0. 2 2 By analogy with constructing formulas (41) – (45) 1 1 1 2 3 -3 -2 -1 let us compose two auxiliary functions: γ 1 ( x ) for x y -3 -2 -1 -1 -1 1 2 3 the condition x = −3 and γ 1 ( y ) for the condition -2 -2 y ≥ 0 . These functions are represented in Fig.33 -3 -3 and can be constructed as Fig.34 Auxiliary functions: a) − M 3 ( x ) and γ 1 ( x ) = S [ − M −3 ( x )], γ 1 ( y ) = S [ M −1 ( y ) + 3] . b) M − 3 ( y ) − 3 . a) γ1(x) b) γ1(y) 3 3 Now it is not difficult to construct the function 2 2 1 1 f 5 ( x, y ) = 13 {S [− M −3 ( y ) − 3 − M 3 ( x )] − M 3 ( x )} x -3 -2 -1 y and finally the function f 5 is implemented as -3 -2 -1 -1 1 2 3 -1 1 2 3 -2 -2 f 5 ( x, y ) = 13 S [ S ( −3 y − 9) + S ( −3x + 9) − 3] + -3 -3 1 (52) S ( −3x + 9). Fig.33 Two auxiliary functions: 3 a) γ 1 ( x ) for the condition x = −3 and As experiments showed, the monotonic b) γ 1 ( y ) for the condition y ≥ 0 . piecewise-linear approximation between the logical level in the point (-2,-1) and logical levels in the adjacent points of the functions f1 ( x, y ) and Based on the functions γ 1 ( x ) and γ 1 ( y ) it is possible to construct the function of two variables f 4 ( x, y ) will be obtained, if the pyramid function γ 1 ( x, y ) = S [− M −3 ( x ) + M −1 ( y ) + 3]. f6 in the point (-2,-1) is implemented in Taking into account that for k = 3 accordance with the formula (46). - M −3 ( x ) = S ( −3x − 9), M −1 ( y ) = S (3 y + 3) f 6 ( x, y ) = 13 S{S [ S (3x + 3 y + 6) + it is not difficult to find ( −3 x − 3 y − 12) + S (3 x − 3 y ) + f 4 ( x , y ) = S [ S ( −3x − 9) + S (3 y + 3) + 3] + 1 . (51) 1 3 S ( −3x + 3 y − 6) − 12] − 3}. The residual matrix M 8 has only two nonzero After some transformations providing the elements with coordinates ( x = +3, y = −3) and possibility to save one summing amplifier this ( x = −2, y = −1) . Let us designate the functions, function looks as follows with help of which the controller function can be f 6 ( x, y ) = 13 S [ S ( −3x − 3 y − 6) + corrected in these points, as f 5 and f 6 , respectively. Depending on coordinates of a S (3x + 3 y + 12) + S ( −3x + 3 y ) + (53) valuable element and conditions of good “sewing” S (3x − 3 y + 6) + 12] + 1. its function with already implemented fragments different implementation methods can be used. The function f 5 ( x, y ) is equal to 0 everywhere 4.6 Implementation of the Controller except the point ( x = +3, y = −3) , at which Correctness of the designed controller and its f 5 ( x, y ) = 1 . For monotonic piecewise-linear functioning has been checked with SPICE simulation (MSIM 8). In simulation experiments connection of this function with the function MOSIS BSIM3v3.1 level 7 models of 0.4μm f1 ( x, y ) , graphs of the function f 5 ( x, y ) along transistors have been used. each of its arguments must have the form of Fig.29 As building blocks for the controller circuit two (a) or (b). The function f 5 ( x, y ) can be types of summing amplifiers were used (ordinary implemented as pyramid function in accordance and powerful). They are built on bases of three- ISSN: 1109-9445 Issue 6, Volume 4, June 2007 126  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 Fig.36 The controller schematic for experiments. stage push-pull CMOS operational amplifier3 with Analytical description of the controller schematic 1.5-MegOhm resistor in the feedback. Examples of can be derived on bases of the function such summing amplifiers are shown in Fig.35. implementations (35) – (37), (39), (50) – (53) and Transistors in this figure are marked with two has the following form numbers, which designate transistor dimensions (length and width). f1 ( x, y ) = S 9 [ 16 S 3 (6 x + 6 y + 27) + The controller schematic for experiments is 1 3 S 4 (3 x + 3 y + 6) + 16 S 5 (6 x + 6 y − 3) + represented in Fig.36. Two powerful elements PHS1 1 S 6 (6 x + 6 y − 15) + 16 S 7 (6 x + 6 y − 27)]; and PHS2 of the controller produce signals -y and -x 6 respectively. Other elements are ordinary. f 2 ( x, y ) = S9 ( S8 ( −3 x + 9 y − 24) − 3) ; f 3 ( x, y ) = 13 S12 {S10 (3x + 3 y − 3) + 3 The operational amplifier of this type is the simplest S11 ( −3 x − 3 y + 9) + S15 [ S10 (3x + 3 y − 3) + and is chosen only with the purpose of simplifying SPICE simulation. ISSN: 1109-9445 Issue 6, Volume 4, June 2007 127  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 the functioning of the controller is correct because of logical values of the circuit output depend on the logical values of the input variables in accordance with the Table 11. Fig.37 The response surface of the controller. Fig.35 Operational amplifier with feedback: a) ordinary, b) powerful. Moreover, the controller output signal has monotonic piecewise-linear approximation between S11 ( −3 x − 3 y + 9) + S14 (3 x − 3 y + 3) + adjacent logical levels. Thus the designed controller S13 ( −3 x + 3 y + 3) + 12] + 9} + 1; can be used as an analog device, which has analog f 4 ( x, y ) = 13 S18 [ S 16 ( −3 x − 9) + S17 (3 y + 3) + 3] + 1 ; inputs and produces an analog output signal. f 5 ( x, y ) = 13 S 21[ S19 ( −3x + 9) + S 20 ( −3 y − 9) − 3] + 13 S19 ( −3x − 9); 5 Transformation of Analog Signals f 6 ( x, y ) = S 26 [ S 23 ( −3x − 3 y − 6) + 1 3 into Multi-valued Logic Variables S 22 (3x + 3 y + 12) + S 25 ( −3x + 3 y ) + In previous sections functions of multi-valued logic were specified by tables of fuzzy rules over S 24 (3x − 3 y + 6) + 12] + 1; linguistic variables. It was implicitly assumed that 6 values of analog signals, which corresponded to F ( x, y ) = ∑j =1 f j ( x, y ) = S 28 [S 27 ( S9 + 13 S12 + linguistic variables, were evenly distributed in the range of voltages representing analog signals. 1 S18 + 13 S19 + 13 S 21 + 13 S 26 + 3)]. 3 Otherwise by artificial means the number of Enumeration of summing amplifiers in this linguistic variables can be increased that leads to description corresponds to enumeration of elements growing the implementation complexity. in the controller circuit. The controller contains 28 In this section a procedure of transforming input amplifiers and 86 resistors. Resistor values have analog variables into multiple-valued logic variables been calculated as R j = R0 / w j where w j is logical with evenly distributed logical levels is suggested. This procedure in some sense is analogous to the weight of the jth element input signal. procedure of fuzzification in fuzzy controllers. In experiments, source voltage was 3.5V. Input Because of using a fuzzy control description for variable x changed linearly from 0V to 3.5V, input implementation of controllers as multi-valued variable y changed discreetly and kept constant logical functions the same term “fuzzification” for value within one cycle of x changing. The voltage the suggested procedure of logical levels range was evenly divided onto seven logical levels equalization for input variables will be used. so that the logical levels “-3” and “+3” The same procedure is supposed to be used when corresponded to voltages Vgnd and Vdd respectively. output multi-valued variables with evenly Results of SPICE simulation of the controller distributed logical levels demand backward schematic are represented in Fig.37. This figure has transformation to an analog form with not evenly been constructed by using GNUplot and illustrates distributed voltages corresponding to logical levels. the response surface in the coordinates X, Y. In this case the term “defuzzification procedure” Analyzing the surface it is possible to conclude that will be applied. ISSN: 1109-9445 Issue 6, Volume 4, June 2007 128  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 5.1 Fuzzification Procedures OVERLAP (N&Z, Z&P, typically about 50% of Let us examine more attentively the fuzzification width but can be less)”. 4 procedure for the case of linear membership Fig.38 illustrates the features of the triangular functions or membership functions, which membership function, which is used in the sufficiently simply can be represented as piecewise- following example. linear, and propose sufficiently simple universal The procedure of fuzzification and constructing method. Here the standard determination of a corresponding diagram is examined on an example membership function is used. The membership of the Container Crane fuzzy Controller, function determines the weight of the corresponding membership functions for which are given in linguistic variable b for each value of an analog Fig.39.5 variable X: It is assumed, without disrupting the generality wb = F (b, X ); 0 ≤ wb ≤ 1 . of reasoning, that with changing the angle within The simplest example of membership functions is the limits ( − 90o ÷ +90o ) and the distance in the given in Fig.38. limits (-10 ÷ +30) yards the corresponding analog voltages vary within the range (0÷3.5) V. The source voltage of the controller circuit is also 3.5V. Fig.38 The simplest type of membership functions. “The membership function is a graphical representation of the magnitude of participation of Fig.39 Membership functions for the Container each input. It associates a weighting with each of the Crane Fuzzy Controller: a) for the angle inputs that are processed, define functional overlap and b) for the distance. between inputs, and ultimately determines an output response. The rules use the input membership Table 12 determines the function of fuzzification for values as weighting factors to determine their the piecewise-linear membership functions of the influence on the fuzzy output sets of the final output variable angle shown in Fig.39(a). It contains conclusion. Once the functions are inferred, scaled, linguistic variables, corresponding to them angle and combined, they are defuzzified into a crisp values and voltages, and also logical values evenly output which drives the system. There are different distributed within the voltage range. Linearity the memberships functions associated with each input membership functions gives the possibility to and output response. Some features to note are: connect the points of logical values by straight lines. SHAPE - triangular is common, but bell, The corresponding fuzzification (equalization) trapezoidal, haversine and, exponential have been function is given in Fig.40. In this figure the used (More complex functions are possible but variations of voltages from the average require greater computing overhead to implement.); (equilibrium) point of summing amplifier are plotted HEIGHT or magnitude (usually normalized to 1); along the axes. WIDTH (of the base of function); SHOULDERING (locks height at maximum if an 4 Citation is taken from “Fuzzy Logic – an Introduction”, outer function. Shouldered functions evaluate as 1.0 part 4, by Steven D. Kaehler, past their center); http://www.seattlerobotics.org/encoder/mar98/fuz/fl_part CENTER points (center of the member function 4.html shape); 5 http://www.fuzzytech.com/e/e_a_pfd.html ISSN: 1109-9445 Issue 6, Volume 4, June 2007 129  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 Table 12: Angle membership functions. ϕ1 (α ) = −0.5S ( 4.5α ); neg_big neg_small zero pos_small pos_big ϕ 2 (α ) = − S (1.125α + 2.19); -90°÷-60° -20° 0° 20° 60°÷90° (0÷0.58)V 1.361V 1.75V 2.139V (2.917÷3.5)V ϕ 3 (α ) = − S (1.125α − 2.19); (54) -2 -1 0 +1 +2 F1 (α ) = S ( −ϕ 1 (α ) − ϕ 2 (α ) − ϕ 3 (α )) = 0V 0.875V 1.75V 2.625V 3.5V S ( 12 S ( 4.5α ) + S (1.125α + 2.19) + S (1.125α − 2.19)). Vout -1.75V 1.75V or +2(pos_big) Now let us show how to construct and implement the fuzzification function for the input variable distance. As can be inferred from Fig.39(b), the 0.875V or +1(pos_small) membership functions are characterized, first, by asymmetry of the measured distance ((-10 ÷ +30) Vin -1.75V yards) and, second, by the explicit asymmetry of the -1.167V 0 (zero) 1.167V linguistic variable positions along the distance axis. -1.75V -1V -0.39V 0.39V 1V 1.75V It assumed that the complete range of the measured -0.875V or -1(neg_small) distance corresponds to the complete range of the supply voltages (0V ÷ 3.5V) or (-1.75V ÷ +1.75V) in deviations from the middle point of amplifiers. For -1.75V or -2(neg_big) this case, the fuzzification function is determined by Table 13. Fig.40 Piecewise-linear function for fuzzification of the variable angle. Table 13: Distance membership functions neg_close zero close medium far For implementation of the function ≤ -5 yards 0 yards 3 yards 10 yards ≥ 20 yards Vout = F1 (Vin ) shown in Fig.40 three auxiliary ≤ 0.4375V 0.875V 1,1375V 1.75V ≥ 2.625V -2 -1 0 +1 +2 functions should be introduced. These functions are 0V 0.875V 1.75V 2.625V 3.5V represented in Fig.41. Their sum with saturation on the levels ± 1.75V determines the fuzzificated input In this table the linguistic variable close corresponds function for the controller fuzzy inference part, to value “log.0” and the linguistic variable zero which, as it has been already proved in previous corresponds to the value “log.-1”. The balance point sections, can be implemented as a multi-valued of the amplifier input voltage corresponds to logic function. linguistic variable medium. Corresponding function Vout (Vin ) = F2 (Vin ) is 1.75V ϕ 2 (α ) given in Fig.42. For implementation of this function 0.875V ϕ1 (α ) Vout -1.75V +2 (far) 1.75V -1.167V -0.389V 0.389V 1.167V -1.75V -1V 1V 1.75V +1 (medium) 0.875V -0.875V Vin -1.75V -1.3125V -0.875V 0 (close) -1.75V -1V -0.6175V 0V 0.875V 1V 1.75V ϕ 3 (α ) -1.75V -0.875V -1 (zero) Fig.41 Component functions for the function represented in Fig.40. -1.75V -2 (neg_close) Fig.42 Piecewise-linear fuzzification function for In Fig.41 the angle α and functions ϕ j (α ) are the variable distance. represented in positive and negative voltages. These component functions and the fuzzifier output it is necessary to realize four auxiliary functions, function F1 (α ) can be implemented by the whose sum with saturation on the levels ±1.75V will following way: give the desired result. The auxiliary functions are given in Fig.43. Their values and value of the ISSN: 1109-9445 Issue 6, Volume 4, June 2007 130  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito Manuscript received Nov. 12, 2007; revised Feb. 18, 2007 variable d (distance) are represented in negative and a) −ϕ1 positive voltages. 4.25 1.75V ψ j (d ) α 1.125 −ϕ2 0.5 F1 (α) 1 1.25 ψ 1 (d ) Vdd 1 1.125 −ϕ3 0.875V 1.25 0.4375V ψ 2 (d ) d b) -1.3125V -0.875V d 2 −ψ 1 -1.75V -1V -0.6175V 0V 0.875V 1V 1.75V 2 -0.4375V -0.875V 13.6 −ψ2 1 Vdd 5.8 0.25 F2 (d) 5.67 −ψ3 0.25 -1.75V 1 1 1 −ψ4 ψ j (d ) 1.75V 1 Fig.44 Fuzzifiers of the variables a) angle and 0.875V b) distance. ψ 3 (d ) 0.4375V -1.3125V -0.6175V 0.875V 1.75V d Summing amplifiers used in the schematics are -1.75V -1V 0V 1V constructed on the bases of three-stage push-pull -0.4375V CMOS operational amplifier in accordance with -0.875V Fig.35(a). ψ 4 (d ) -1.75V Fig.43 Component functions for the function represented in Fig.42. The ways of forming the component functions given in Fig.43 and the function F2 (d ) are shown below: ψ 1 (d ) = − S ( 2 ⋅ d + 3.5); ψ 2 (d ) = −0.25S (13.6 ⋅ d + 10.14); ψ 3 ( d ) = −0.25S (5.67 ⋅ d + 1.75); ψ 4 (d ) = − S ( d − 1.75); (55) F2 ( d ) = S ( −ψ 1 ( d ) −ψ 2 ( d ) −ψ 3 ( d ) −ψ 4 (d )) = S [ S ( 2 ⋅ d + 3.5) + 0.25S (13.6 ⋅ d + 10.14) + 0.25S (5.67 ⋅ d + 1.75) + S ( d − 1.75)]. 5.2 Fuzzifier Implementations For the completion of the fuzzifier design it only remains to determine the values of the input resistances of summing amplifiers and to conduct SPICE simulation for checking correctness of the implementations (54) and (55). These implementations are represented graphically in Fig.45 Fuzzifier schematics a) for the angle and Fig.44(a) and Fig.44(b) respectively. Their b) for the distance. schematics, which have been used for SPICE simulations, are shown in Fig.45 (a) and Fig.45 (b). Results of SPICE simulation of the fuzzifiers for variables angle and distance are shown in Fig.46 and Fig.47 respectively. ISSN: 1109-9445 Issue 6, Volume 4, June 2007 131  WSEAS TRANSACTIONS on ELECTRONICS V. Varshavsky, V. Marakhovsky, I. Levin, H. Saito simulation using the summation operation with restrictions. In all examples of controllers presented in the paper, the push-pull summing amplifier containing three CMOS invertors is used. Obviously this amplifier circuit is the simplest among operational amplifiers of other types but unfortunately it has the worst characteristics. It was chosen only by two reasons: first, to simplify SPICE simulation of designed controllers and second, to show that using even such primitive and imperfect building block gives rather appropriate characteristics of designed Fig.46 Output of the fuzzifier shown in Fig.45(a) controllers. Certainly in real projects of controllers it and derived by SPICE simulation. is better to use another types of operational amplifiers, e.g., a differential amplifier. Someone may object that summing amplifiers in all examples of controllers designed with help of suggested methodology contain resistors of large values and it is very difficult to implement these resistors in CMOS VLSI technology. Indeed it is correct. In our case p-well resistors (1-10K Ohms/sq.) or pinch resistors (5-20K Ohms/sq.) can be used. These resistors are compatible with CMOS technology but occupy very large die area, possess bad accuracy, and have big temperature and voltage coefficients. By these reasons the possibility of creating a dynamical model of the summing Fig.47 Output of the fuzzifier shown in Fig.45(b) amplifier with saturation using capacitors instead of and derived by SPICE simulation. resistors has been considered. This consideration gave positive results and perhaps will be published It is easy to see that the simulation plots are in the future. exactly the same as it is required for fuzzification of The proposed methodology has been applied for the input variables angle (Fig.40) and distance designing several devices specified as fuzzy (Fig.42). This proves the correctness of the fuzzifier controllers, showed high efficiency and gave very implementations. economical implementations. Techniques of It should be noted that in the case of software synthesizing fuzzy devices in the offered base implementation of the fuzzification and should get further developing and problems of defuzzification functions, their component functions implementability under the conditions of real may be chosen not only piecewise-linear but production should be resolved in the nearest future. providing any reasonable approximations. References: 6 Conclusion [1] Implementation of Fuzzy Logic, Texas It was shown that all parts of fuzzy controllers can Instruments, SPRA028, January 1993. be effectively implemented on bases of summing [2] An Introduction to Fuzzy Logic Applications in amplifiers with saturation in accordance with the Intelligent Systems, by Ronald R. Yager, Lotfi proposed methodology. This methodology is A. Zadeh (Editor), Kluwer International Series oriented to hardware implementation of fuzzy in Engineering and Computer Science, 165, Jan. controllers as analog devices. In comparison with 1992, 356 p. software implementation of the controllers, their [3] Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems, by hardware implementation have advantages of better George J. Klir (Editor), Bo Yuan (Editor), response time and reliability, low power Selected Papers by Lotfi A. 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