Fuzzy Controllers CMOS Implementation

Fuzzy Controllers CMOS Implementation V. VARSHAVSKY 1, V. MARAKHOVSKY2, I. LEVIN3, N. KRAVCHENKO4 1 Advanced Logic Projects Inc., JAPAN, 2 The University of Aizu, JAPAN 3,4 Bar Ilan University, ISRAEL Abstract: - The subject of the study is hardware design of fuzzy controllers as CMOS analog devices on the base of controller descriptions in the view of multi-valued logical functions. A functional completeness of summing amplifier with saturation in an arbitrary-valued logic is proven that gives a theoretical background for analog implementation of fuzzy devices. Compared with the traditional approach based on explicit fuzzification, fuzzy inference, and defuzzification procedures analog fuzzy implementation has the advantages of higher speed, lower power consumption, smaller die area and more. The paper illustrates a design example for real industrial fuzzy controller and provides SPICE simulation results of its functioning. Key-Words: - Fuzzy Logic, Fuzzy Controller, Multi-Valued Logic Functions, Functional Completeness, Summing Amplifier with Saturation. 1 Introduction combination of input analog variables. On the Wide spread of the fuzzy control and high base of a fuzzy rules system and of fuzzy effectiveness of its applications in a grate inference rules it is possible to receive the set extend is determined by formalization of weighted values of the output linguistic opportunities of designer “fuzzy” (flexible”) variable. Using these values, membership representations about necessary behavior of a functions of the output variable, and one of the controller. These representations usually are several known methods of defuzzification it is formulated in the view of logical (fuzzy) rules possible to form the value of the analog output under linguistic variables of a type “If A then variable. The defuzzification procedure also B”. The linguistic variables themselves are includes digital-analog transformation. quality characteristics of input signals of types At present the most wide-spread way of “warm”, “cool”, “high”, “low”, “fast”, “slow” fuzzy logic control implementation is using and so on. the programmable fuzzy controllers, which are As a rule, or at least in a grate part of available on the market together with the applications, a fuzzy controller is a means of computer aided programming. transformer of input analog signals into an However, in spite of the implementation analog output signal. evidence and fuzzy controllers’ accessibility A linguistic variable is a “subjective” this approach to implementation possesses characteristic of an input analog variable and some disadvantages, e.g. such as high cost and the input variable transformation is given by low throughput (that is especially important membership functions determining for each when fuzzy control in the control contour is value of the input variable the set of weighted used) etc. values corresponding linguistic variables. This In this work we are going to show that for a procedure is called a fuzzification and it sufficient wide set of problems a fuzzy contains as its composite part the analog- controller can be implemented as a rather digital transformation. simple CMOS device that is used as embedded A set of combinations of weighted linguistic system or IP core. That is the basic idea of our variables corresponds to each value suggestion?  A fuzzy controller is a deterministic device, representation of the output linguistic variable for which one and only one value of the output on chosen value combinations of multi-valued analog variable corresponds to each value input variables. combination of the input analog variables. It In [4 – 6] we have shown that the analog means that the fuzzy controller should realize threshold function having the fallowing view an analog function Y = f ( x1 , x 2 ,..., x n ) 1.  n + k if ∑ω j ⋅ x j ≤ −k There are two important questions:  j =1 1. How to transit from standard specification  n n of a fuzzy logic function to the specification of y( X ) = − ∑ω j ⋅ x j if k > ∑ω j ⋅ x j > -k (1) corresponding analog function?  j =1 j =1 2. How to transit from an analog function  n specification and/or from standard − k if ∑ω j ⋅ x j ≥ +k specification of a fuzzy logic function to  j =1 corresponding CMOS implementation? conforms together with constants a First of all, let us address to membership functionally complete system in the (2k+1)- functions. In most cases [1 – 3], membership valued logic. This function can be functions have a triangle or trapeze view (see implemented on the base of summing fig.1). amplifier with saturation [5]. α 1 A B C D E F G 2 Summing Amplifier as a Multi- Valued Logical Element Summing amplifier’s behavior, accurate to the T members of the infinitesimal order that is Figure 1. Types of membership functions. determined by the amplifier’s gain factor in disconnected condition (fig.2), is described as In fig.1 linguistic points A and B are “cold”, follows: C – “fresh”, D and E – “worm”, F and G – “hot”. These points determine the connection  n R V V of the linguistic variables with values of the Vdd if ∑ 0 (V j − dd ) ≤ − dd  j=1 R j 2 2 analog variable T (in our case T is V n R V temperature). Relatively these points and Vout =  dd − ∑ 0 (V j − dd ) in other cases (2) similar points for other input variables we can  2 j =1 R j 2 compose a table of fuzzy rules connecting  V n R V value combinations of input linguistic 0 if dd ≤ ∑ 0 (V j − dd ) variables with values of the output linguistic  2 j=1 R j 2 variable. where Vdd – the supply voltage, V j – the On the base of membership functions we voltage on jth input, R j – the resistance of jth can put into accordance to the input and output linguistic variables a set of integer numbers input, R0 – the feedback resistance, and Vdd/2 splitting by appropriate way all diapason of – the midpoint of the supply voltage. changing of corresponding analog variables. n R0 Vdd Then the table of fuzzy rules will to determine Dependence of Vout on ∑R j =1 ⋅ (V j − 2 ) is j by obvious way the function of multi-valued logic, values of which define the digit shown in fig.3,a. Let us split the source voltage Vdd on m = 2k+1 voltage levels. Then replacing the input 1 We shall notice that in suppressing majority of voltages V j by m-valued logical variables publications on fuzzy controllers this function is given as a response surface and practically without exception this surface has a piece-linear view.  2 ⋅V j − Vdd arbitrary-valued logic, if any function of this xj = ⋅ k and the output voltage Vout logic can be represented as superposition of Vdd basic operations. by m-valued variable y and designating There are some known functionally R0 / R j = ω j the system (2) can be represented completed sets of functions. It is clear, that for as (1). Graphical view of (1) is shown in proving functional completeness of some new fig.3,b. function it is sufficient to show that the functions of the known functionally completed set can be represented as superposition of the considered function. One of functionally completed functions in m-valued logic is the Webb’s function [7]: w( x, y ) = [max( x, y ) + 1]mod m . (3) Therefore, for proving functional completeness of threshold operation in multi- valued logic it is sufficient to show how the Webb’s function can be represented through this operation. First, let us represent the function Figure 2. Summing amplifier: general max( x1 , x 2 ) by threshold functions. To do this structure (a); CMOS implementation using let us consider the function f a ( x ) diagram, symmetrical invertors (b). such as a) Vout a if a≥x f a ( x ) = max( x, a ) =  . (4) Vdd x if x>a This function diagram is shown in fig.4,a. a) y b) y k k Vdd / 2 fa (x) a a − Vdd / 2 n R0 Vdd ∑R (V j − 2 ) -k k -k k b) y j =1 j x x k -k -k n a -maj(x,-a,-k) a ∑ω x j =1 j j Figure 4. Diagrams of f a (x ) (a) and −k k − maj ( x,−a ,−k ) (b) functions. −k The − maj ( x,− a,− k ) function diagram is Figure 3. Summing amplifier’s behavior: shown in fig.4,b. Actually, as far as x < a within voltage coordinates (a); within multi- x − a − k < −k and −maj( x,−a,−k ) = −k . Note valued variables coordinates (b). that for all x values, Later on, we will call the functional f a ( x ) − maj ( x,−a,−k ) = a − k element, whose behavior is determined by the as it follows from fig.4, hence system (1), a multi-valued threshold element. f a ( x ) = −maj[maj ( x,−a,−k ), a,−k ] . (5) In the simplest case when ω j = 1, j = 1, 2, 3 , we Taking into consideration that − maj ( a, b, c) = will call it the majority element and designate maj ( − a ,−b,− c ) , it follows from (5) that as maj ( x1 , x 2 , x3 ) . max( x1 , x 2 ) = maj(maj( − x1 , x 2 , k ),− x 2 , k ) . (6) The basic operation (or set of basic operations) is called functionally completed in  Now let us consider the function ( x + 1) mod m Defuzzifier Inference Fuzzifier X A B Y Fuzzy representation by threshold functions. To analog digital digital analog make it clear let us turn to the sequence of pictures fig.5. a) y b) y k-1 k-1 Figure 6. Fuzzy device structure. Fuzzy Inference block based on the fuzzy rules generates a set of weighted linguistic x x variables values B = {b1 , b2 ,..., bk } . 1-k 1-k Defuzzifier converts a set of weighted ( x + 1) mod m ϕ 1 ( x ) = −maj ( x,1,0) linguistic (digital) variables B = {b1 , b2 ,..., bk } c) y d) y into a set of output analog variables k-1 k-1 Y = { y1 , y 2 ,..., y k } . As a rule, fuzzifier and defuzzifier are implemented as AD and DA (analog-digital x x and digital-analog) converters, i.e. by 1-k 1-k hardware implementation. Fuzzy inference is ϕ 2 ( x ) = −maj ( x,1 − k ,−k ) ϕ 3 ( x ) = k ⋅ maj (ϕ 2 ( x ), k ,0) usually implemented as microprocessor Figure 5. Implementation of software. ( x + 1) mod m function. On the other hand, there is the set of output analog variables, which values unambiguously From fig.5 it is easy to see that corresponds to each set of input analog ( x + 1) mod m = ϕ 1 ( x ) + 2ϕ 3 ( x ) variable values; hence a fuzzy device could be and obviously, this function can be specified as a functional analog of signal implemented on threshold elements too. converter Hence, the functional completeness of the Y ( X ) = { y1 ( X ), y 2 ( X ),..., y k ( X )} summing amplifier in arbitrary-valued logic is and its output Y determines a system of n- shown. dimensional surfaces. In cases of sufficient It is naturally that the proof procedure of simple membership functions (in known functional completeness does not give publications such functions are in majority), information about methods of effective for fuzzy controller implementations as analog synthesis. The methods of synthesizing devices it is sufficient to provide a piecewise- circuits in the proposed base are to be linear connection between a couples of points developed in future. However, as it will be calculated as adjacent values of a multi-valued shown below, for a number of real circuits the logic function. proposed base allows designing simple and Let m = 2k + 1 linguistic variable aj values efficient circuits. correspond to analog variable xj. Then basing on fuzzy rules system, we can specify a system of m-valued logic functions, as 3 Fuzzy Devices as Multi-Valued follows: and Analog Circuits B ( a1 , a 2 ,...a n ) = {b1 ( A), b2 ( A),..., bk ( A)} . (7) Conventional implementation of fuzzy devices Note that most publications describing fuzzy usually has the structure shown in fig.6. controllers contain the tables, specifying fuzzy Analog variables X = {x1 , x 2 ,..., x n } go the controllers’ behavior as (7). fuzzy device input. Fuzzifier converts a set of Let us consider an example. This example is analog variables x j into that of weighted taken from [9]: “Design of a Rule-Based Fuzzy Controller for the Pitch Axis of an linguistic (digital) variables A = {a1 , a 2 ,..., a n } . Unmanned Research Vehicle”.  The fuzzy control rules for the considered Vout = Vdd − Vin corresponds to it in the space device depend on the error value of summing amplifiers’ output voltages. Thus e = ref − output and changing of error CMOS circuit containing 12 transistors and 5 old e − new e resistors, which implements our function, is ce = . Fuzzifier brings seven sampling period shown in fig.8. levels for each of input linguistic variables Output (NB – negative big; NM – negative middle; 3 NS – negative small; ZO – zero; PS – positive 2 1 e-ce small; PM – positive middle; PB – positive -6 -5 -4 -3 -2 -1 big). The output has the same seven -1 1 2 3 4 5 6 gradations. The corresponding 49 fuzzy rules -2 -3 are represented in Table 1. Figure 7. Graphical representation of the Table of Fuzzy Rules. Table 1 function specified by table 2. e NB NM NS ZO PS PM PB NB ZO PS PM PB PB PB PB NM NS ZO PS PM PB PB PB NS NM NS ZO PS PM PB PB ce ZO NB NM NS ZO PS PM PB PS NB NB NM NS ZO PS PM Figure 8. CMOS implementation of the fuzzy PM NB NB NB NM NS ZO PS controller specified by table 2. PB NB NB NB NB NM NS ZO Let us split evenly the source voltage (e.g. 4 Example of a Fuzzy Controller 3.5V) onto seven logical levels corresponding Hardware Implementation to linguistic levels. Then table 2 will represent Let us consider a demo procedure of a real table 1 as the function of seven-valued logic. fuzzy controller design as an analog hardware device. The description of the controller is The Seven-Valued Function. Table 2 taken from [9]. It provides controlling of a car e seat and able to perform subtle attitude control -3 -2 -1 0 1 2 3 -3 0 1 2 3 3 3 3 without giving an unpleasant feeling to -2 -1 0 1 2 3 3 3 passengers. ce -1 -2 -1 0 1 2 3 3 The initial fuzzy controller specification is 0 -3 -2 -1 0 1 2 3 1 -3 -3 -2 -1 0 1 2 shown in table 3. 2 -3 -3 -3 -2 -1 0 1 Table of specification. Table 3 3 -3 -3 -3 -3 -2 -1 0 V ZR PM PL PVL It is seen from table 2 that the function is X dX symmetric with respect to “North-West – ZR * ZR ZR ZR ZR ZR ZR ZR PS PM South-East” diagonal and depends on e − ce . RM PL ZR PS PM PM This kind of dependence is shown in fig.7. RL ZR ZR PS PM PM It apparently follows from comparison of PL ZR PS PL PL fig.3 and fig.7 that in order to reproduce the function specified by table 2 it is sufficient to This table has three input variables: X – angle have one two-input summing amplifier and of steering wheel rotation, dX − derivative one inverter. from X, and V – car velocity. Let linguistic Note that inversion of logic variables lying values of the variables correspond to the within − k ÷ + k interval is the operation of voltages completely located in the interval 0– diametric negation x = − x ; the operation 3.5V as it is shown in table 4.  Correspondence linguistic V F1 variable values to voltages. Table 4 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 X ZR=0 RM=1.75V RL=3.5V -3 -3 -3 -3 -3 -3 -1 -3 -3 -3 -3 -3 -1 -1 dX ZR=0 PL=3.5V V ZR=0 PM=(3.5/3)V PL=(7/3)V PVL=3.5V X -3 -1 1 3 -3 -3 -3 -3 -3 -3 -3 -1 -1 1 -1 -1 1 1 + -3 -3 -1 -1 1 1 1 Output ZR=0 PS=(3.5/3)V PM=(7/3)V PL=3.5V -3 -1 3 3 -3 -1 -1 1 1 1 1 Let us split evenly the voltage interval 0 – F3 F2 3.5V onto 7 logical levels, designate these 0 0 0 0 0 0 0 -3 -3 -3 -3 -3 -3 -3 levels with integer numbers from -3 to +3, and 0 0 0 0 0 0 0 -3 -3 -3 -3 -3 -3 -1 0 0 0 0 0 0 0 -3 -3 -3 -3 -3 -1 -1 transform table 3 into table 5 substituting 2 . 0 linguistic values of the variable with + 3 0 0 0 3 0 3 0 3 3 3 3 3 3 = -3 -3 -3 -3 -1 -3 -1 -1 1 1 1 3 3 3 corresponding logical values. 0 0 0 0 3 3 3 -3 -3 -1 -1 3 3 3 0 0 0 0 3 3 3 -3 -1 -1 1 3 3 3 Multi-valued logical function F. Table 5 V Figure 10. Description of the residual -3 -1 1 3 X dX function F2 ( X , V ) . -3 * -3 -3 -3 -3 -3 -3 -3 -1 1 0 3 -3 -1 1 1 As it is seen from fig.9, F1 ( X ,V ) can be 3 -3 -3 -1 1 1 supplementary defined up to a symmetrical 3 -3 -1 3 3 function with respect to its arguments and For value combinations of the input reduced to the function of one variable variables, which are absent in the table 5 (i.e., ( X + V ) with the graph shown in fig.11. for situations which are not defined by fuzzy F1 ( X + V ) rules), values of the output function can be 3 supplementary defined up to any accepted 2 values. 1 Now let us present the table 5 in the form 1 2 3 of decomposition: -3 -2 -1 0 (X +V ) / 2 if dX = −3( ZR) then F = F1 ( X ,V ) else -1 3 + dX -2 F = F1 ( X ,V ) + ( F2 ( X ,V ) − F1 ( X ,V )). -3 6 This decomposition provides piece-linear approximation of F on the interval of dX Figure 11. The function F1 ( X + V ) . changing from -3 to 3 and when dX = 3, This function has two zones of changing the F = F2 ( X , V ) . Definitions of the residual values and can be implemented on three functions F1 (θ ,V ) and F2 (θ ,V ) are given in amplifiers. Really fig.9 and fig.10 respectively. β1 = S (6 ⋅ X + 6 ⋅ V − 3), V β 2 = S (6 ⋅ X + 6 ⋅ V − 15), F1 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3 F1 = S ( 13 β 1 + 13 β 2 + 1), -3 -3 -3 -3 -3 -3 -1 where S is a function implemented by one -3 -3 -3 -3 -3 -1 -1 -3 -3 -1 1 -3 -3 -3 -3 -1 -1 1 summing amplifier. X From fig.10 it is possible to see that the -3 -3 -3 -1 -1 1 1 -3 -3 -1 -1 1 1 1 residual function F2 ( X ,V ) can be presented as -3 -1 1 1 -3 -1 -1 1 1 1 1 a sum of the same symmetrical function Figure 9. Description of the residual F1 ( X ,V ) and 23 of the correcting function function F1 ( X , V ) . F3 ( X , V ) : F2 ( X ,V ) = F1 ( X ,V ) + 23 ⋅ F3 ( X ,V ) .  Let us split the matrix of the correcting The implementation of the rule 3 can has a function F3 ( X , V ) on two sub-matrixes: view: F3 F4 F5 β 8 = S [ 12 ( dX − 3)], − β 8 = S ( β 8 ) , 0000000 0000000 0000000 γ 3 ( X ,V ) = S ( β 8 + F3 ( X ,Y )), 0000000 0000000 0000000 0000000 0000000 0000000 . γ 4 ( X ,V ) = S ( − β 8 + F3 ( X , Y )), 0 0 33333 = 0 033333 + 00 0 00 00 0 0 00 333 0 00 00 0 0 00 0 0333 S [ (γ 3 ( X ,V ) + γ 4 ( X ,V ) + F3 ( X ,V ))] . 2 3 0 0 00 333 0 00 00 0 0 00 0 0333 0 0 00 333 0 00 00 0 0 00 0 0333 This implementation demands five summing amplifiers and its output amplifier can be It is obvious that F3 = F4 + F5 . The function combined with the output amplifier of the F1 F4 can be defined in the form of the rule 1: implementation to produce the output signal of If ( X = 0) & (V ≥ −1) then F4 = 3 else F4 = 0 . the controller: As it is not difficult to check that this rule can F = F1 ( X + V ) + be implemented on five summing amplifiers: S [ 23 (γ 3 ( X ,V ) + γ 4 ( X ,V ) + F3 ( X ,V ))] = β 3 ( X ) = S (6 X − 3), S [ 13 ( β 1 + β 2 + 3) + 23 (γ 3 ( X ,V ) + β 4 ( X ) = S ( −6 X − 3), γ 4 ( X ,V ) + F3 ( X ,V ))]. β 5 (V ) = S ( −6V − 9), The final controller implementation circuit is γ 1 ( X ,Y ) = S [ 12 ( β 3 ( X ) + β 4 ( X ) + β 5 (V ) − 9)] , shown in fig.12. F4 ( X ,V ) = S [γ 1 ( X ,V ) − 3] . X 6 S1 β1 6 The function F5 is determined by the 1 0 S2 following rule 2: V 6 β2 6 If ( X ≥ 1) & (V ≥ 1) then F5 = 3 else F5 = 0 , 0 1 S3 S4 1/3 which can be implemented using four dX 1/2 β8 β8 S5 PWR 1/3 S17 1 1 1/3 2/3 F summing amplifiers: 1/2 1 γ 2/3 0 S7 4 β 6 ( X ) = S ( −6 X + 3) , S6 γ 2/3 -X 3 1 1 β 7 (V ) = S (6V − 3) , 1 S8 γ 2 ( X ,V ) = S [ 12 ( β 6 ( X ) + 3) + 12 ( β 7 (V ) + 3)], 1 -V F5 ( X ,V ) = S [γ 2 ( X ,V ) + 12 ( β 7 (V ) + 3)] . S9 β3 This implementation provides piecewise-linear 6 1 coupling the function F5 with the function F4 0 S10 S12 β4 1/2 γ S16 6 1 1/2 1 by the coordinate X on the interval 0,1 . 1 3/2 1 F3 0 1/2 1/2 The output amplifiers of the circuits S11 6 β5 1/2 implementing the function F4 and F5 can be 3 0 combined to get the implementation of the S13 β6 PWR 6 S15 γ function F3 : 1 1/2 2 1 F3 ( X ,V ) = F4 ( X ,V ) + F5 ( X ,V ) = 6 S14 β7 1/2 S (γ 1 ( X ,V ) − 3 + γ 2 ( X ,V ) + ( β 8 (V ) + 3)) = 1 1 2 0 S (γ 1 ( X ,V ) + γ 2 ( X ,V ) + 12 β 8 (V ) − 23 ). Figure 12. The controller implementation. Now we are able to present the controller In this implementation the elements S7 and output value in the form: S8 are used for inverting signs of the F = F1 ( X + V ) + (rule 3) arguments X and V. where rule 3 is Functioning of the controller has been if dX = 3 then 23 F3 ( X,V ) else ( dX = -3) 0. checked with SPICE simulation (MSIM 8). MOSIS BSIM3v3.1, 7 level models of 0.4µm transistors has been used. The source voltage synthesizing circuits in the offered base. in the experiments was 3.5V. Though the given examples show possible Results of SPICE simulation of the circuit in high efficiency and effectiveness of the fig.12 are represented in fig.13. In this figure implementation offered this is true only in the response surfaces in the coordinates X, V regard to specific examples, chosen in special are shown for the cases dX=0V (fig.13,a) and way. dX=3.5V (fig.13,b). Techniques of synthesizing fuzzy devices in the offered base and the problems of implementability under the conditions of real production should be resolved on further work stages. References: [1] An Introduction to Fuzzy Logic Applications in Intelligent Systems, by Ronald R. Yager, Lotfi A. Zadeh (Editor), Kluwer International Series in Engineering and Computer Science, 165, Jan. 1992, 356 p. [2] Fuzzy Logic Technology and Applications, by Robert J. Marks II (Editor), IEEE Technology Update Series, Selected Conference Papers, 1994, 575 p. [3] Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems, by George J. Klir (Editor), Bo Yuan (Editor), Selected Papers by Lotfi A. Zadeh (Advances in Fuzzy Systems - Applications and Theory), World Scientific Pub Co.; Vol. 6, June 1996, 826 p. [4] V. Varshavsky, V. Marakhovsky, I. Levin, and N. Kravchenko, Fuzzy Device, New Japanese Patent Application No. 2003-190073, filed to Japan’s Patent Office, July 2nd, 2003. [5] V. Varshavsky, V. Marakhovsky, I. Levin, and Figure 13. The response surfaces for the cases N. Kravchenko, Summing Amplifier as a dX=0V (a) and dX=3.5V (b). Multi-Valued Logical Element For Fuzzy Control, WSEAS Transactions on Circuit and SPICE simulation results confirm logical Systems, Issue 3, Vol. 2, July 2003, pp. 625 – correctness of the controller behavior. 631. [6] V. Varshavsky, I. Levin, V. Marakhovsky, A. Ruderman, and N. Kravchenko, CMOS Fuzzy 5 Conclusions Decision Diagram Implementation, WSEAS In the above examples of controllers, push- Transactions on Systems, Issue 2, Vol. 3, April 2004, pp. 615 – 631. pull summing amplifiers are used. The [7] Post E., Introduction to a general theory of summing amplifier however can be of another elementary propositions, Amer. J. Math., 43, type, e.g. differential type or any other types 1921, pp. 163-185. of operational amplifiers. [8] Fuzzy Control Systems, Abraham Kandel The proof of functional completeness shown (Edited by), Lotfi A. Zadeh (Foreword by), in section 2 provides the possibility of CRC Press LLC, Sept. 1993, pp. 81 – 86. implementing arbitrary function of multi- [9] Mori Takakazu, Hamada Eiichi, Nishiyama valued logic in the base of summing Kunio, Controller for Vehicle Seat, Toyota amplifiers. However, none mentioned above Motor Corp., Patent of Japan No. 05-085235, answers the question concerning the efficiency Date of publication: 06.04.1993 of such implementation and the techniques of