Fuzzy logic is a powerful, yet straightforward, problem-solving technique with widespread applicability, especially in the areas of control and decision making. In general, it is most useful in handling problems not easily definable by practical mathematical models. For instance, fuzzy logic has been employed in such tasks as managing stock-market portfolios and controlling subway systems.

Fuzzy derives much of its power from its ability to draw conclusions and generate responses based on vague, ambiguous, qualitative, incomplete, or imprecise information. In this respect, fuzzy-based systems have a reasoning ability similar to that of humans. In fact, the behavior of a fuzzy system is represented in a very simple and natural way. This allows quick construction of understandable, maintainable, and robust systems. In addition, a fuzzy approach generally requires much less memory and computing power than conventional methods, thereby permitting smaller and less expensive systems.

Lotfi Zadeh, a professor at the University of California at Berkeley, is the person most widely associated with fuzzy logic. In 1965 he presented the original paper formally defining fuzzy-set theory, from which fuzzy logic emerged. Zadeh extended traditional theory to resolve the paradoxes sometimes generated from the "nothing-or-all" classifications of Aristotelian logic. Traditionally, a logic premise has two extremes: either completely true or completely false. However, in the fuzzy world, a premise ranges in degree of truth from 0 to 100 percent, which allows it to be partially true and partially false.

By incorporating this "degree of truth" concept, fuzzy logic extends traditional logic in two ways. First, sets are labeled qualitatively (using linguistic terms such as "tall," "warm," "active," "nearby," and so on), and the elements of these sets are assigned varying degrees of membership. For instance, a 5'11" man and a 6'4" man may both be members of a set of "tall" men, although the 6'4" man has a higher degree of membership. Secondly, any action or output resulting from a premise being true executes to a strength reflecting the degree to which that premise is true.

As an example, imagine a fan motor, the speed of which is a function of temperature, as shown in Table 1. The current supplied to the fan motor is regulated by sets of temperature: cold, cool, warm, and hot. In this system, as the temperature gradually moves from warm to cool, the current gradually moves from 50 to 15. By continuously tracking inputs, outputs can avoid abrupt changes, even as inputs transcend set boundaries. Fuzzy-based systems are constructed so that generated outputs change in a smooth and continuous manner, regardless of inputs crossing set boundaries.

Temperature Fan Speed Relative Motor Current ------------------------------------------ Cold Off 0 Cool Slow 15 Warm Medium 50 Hot Fast 100

*Table 1:** Fan-speed control.*

### Organization of a Fuzzy System

Figure 1 illustrates the flow of data through a fuzzy system. System inputs undergo three transformations to become system outputs. First, a fuzzification process that uses predefined membership functions maps each system input into one or more degrees of membership. Then, the rules in the rule base (also predefined) are evaluated by combining degrees of membership to form output strengths. And lastly, the defuzzification process computes system outputs based on strengths and membership functions.

Fuzzification of Inputs. Fuzzification is the process of assigning or calculating a value to represent an input's degree of membership in one or more qualitative groupings, called "fuzzy sets." Figure 2 shows a system input, temperature, with fuzzy sets cold, cool, warm, and hot. Each temperature value has a degree of membership in each of these sets. The degree of membership is determined by a membership function, which is defined based on experience or intuition. (Figure 9 will illustrate the degree of membership calculation for a trapezoidal membership function.) It is accepted that membership functions change several times as the system is tuned to achieve desired responses to given inputs.

Generally, once the system is in operation, the membership functions do not change. Simple shapes such as trapezoids and triangles are often used to define membership in fuzzy sets, but any suitable function can be used. In addition, you must decide upon the number of fuzzy sets per system input.

In Figure 2, a fuzzy set labeled comfortable could be inserted between cool and warm. The number of fuzzy-set membership functions and the shapes you choose depend on such things as required accuracy, responsiveness and stability of the system, ease of implementation, manipulation, and maintenance, and so on. The trapezoidal and triangular membership functions are most common and have proven to be good compromises between effectiveness and efficiency. The fuzzy sets must span the X-axis covering the entire range, or universe of discourse, for a system input. Mapping to the Y-axis ranges from 0 to 1 and represents the degree to which an input value is a member of that particular fuzzy set. Overlapping between set boundaries is desirable and key to the smooth operation of the system. It permits membership in multiple--even seemingly contradictory--sets. In Figure 2, 63 degrees can be both cool and warm, but it is cool to a greater degree. An overlap of 25 percent between adjacent fuzzy sets is a general rule of thumb.

The fuzzification process permits a binding to take place between linguistic terms (cold, nearby, active, large, and so on) and membership functions, making the terms meaningful to a computer. As a result, a designer can express or modify the behavior of a system using such natural language, thus enhancing the possibility of clear and concise descriptions of complex tasks.

Evaluation of Rules. To govern the system's behavior, the designer develops a set of rules that have the form of If-Then statements. The If side of a rule contains one or more conditions, called "antecedents;" the Then side contains one or more actions, called "consequences." The antecedents of rules correspond directly to degrees of membership calculated during the fuzzification process.

For example, consider a potential rule from the stock-market system shown in Figure 3: If share price is decreasing And trading volume is heavy, Then order is sell. The two conditions "share price is decreasing" and "trading volume is heavy" are the rule's antecedents. Each antecedent has a degree-of-truth (membership) value assigned to it as a result of fuzzification. The action of the rule (or "fuzzy output") is to sell shares. During rule evaluation, strengths are computed based on antecedent values and then assigned to the rules' fuzzy outputs. Generally, a minimum function is used so that the strength of a rule is assigned the value of its weakest or least true antecedent. Other methods to compute rule strength can be used, such as multiplying antecedent values together. The action of selling shares is carried out to a degree that reflects the rule's strength. In other words, the amount of shares sold is based on the degree to which share price is decreasing and trading volume is heavy. Often, more than one rule applies to the same specific action, in which case the common practice is to use the strongest or most true rule; see Figure 4.

Rule 1: if A & B then Z & X Rule 2: if C & D then Z & Y Strength of Rule 1 = min (A,B) Strength of Rule 2 = min (C,D) X = Strength of Rule 1 Y = Strength of Rule 2 Z = max (Strength of Rule 1 Strength of Rule 2) = max (min(A,B), min(C,D))

*Figure 4:** Rule-evaluation computation.*

Defuzzification of Outputs. Even though the rule-evaluation process assigns strengths to each specific action, further processing, or "defuzzification," is required for two reasons. The first is to decipher the meaning of vague (fuzzy) actions, such as "order is sell," using membership functions. The second is to resolve conflicts between competing actions such as "order is sell" and "order is hold," which may have been triggered by certain conditions during rule evaluation. Defuzzification employs compromising techniques to resolve both the vagueness and conflict issues.

One common defuzzification technique, the "center-of-gravity method," consists of several steps. Initially, a centroid point on the X-axis is determined for each output membership function. Then, the membership functions are limited in height by the applied rule strength, and the areas of the membership functions are computed. Finally, the defuzzified output is derived by a weighted average of the X-axis centroid points and the computed areas, with the areas serving as the weights. The center-of-gravity method is illustrated in Figure 5.

Sometimes, "singletons" are used to simplify the defuzzification process; see Figure 6. A singleton is an output membership function represented by a single vertical line. Since a singleton intersects the X-axis at only one point, the center-of-gravity calculation reduces to just a weighted average calculation of X-axis points and rule strengths, with the rule strengths used as weights.

### Fuzzy Data Structures

To implement a fuzzy system in C, the following types of data must be accommodated:

- System inputs.
- Input membership functions.
- Antecedent values.
- Rules.
- Rule-output strengths.
- Output membership functions.
- System outputs.

Figure 7 illustrates an overall linked-list arrangement of system-input and membership-function nodes. The details of these structures are shown in Figure 8.

The system-input node is straight-forward and contains an input name, a membership-function pointer, and a next-input pointer. More interesting is the membership-function structure, which contains two X-axis points and two slope values that describe a trapezoidal membership function. This information is used to calculate antecedent values (degrees of membership), as shown in Figure 9 and Listing Three. The resulting antecedent value is stored in the "value" field of the membership-function structure. Rules can be represented by two sets of pointers; see Figure 10.

The first set indicates which antecedent values are used to determine the rule's strength, and the second set points to output locations where the strength is to be applied. Finally, a data arrangement similar to the input-data structure handles outputs and output membership functions; see Figure 11.

Listing One includes the C-code definition of these data structures. Articles by James M. Sibigtroth (see "References") explain the implementation of fuzzy systems at the assembly language level.

**Listing One**

/* General-purpose fuzzy inference engine supporting any number of system inputs and outputs, membership functions, and rules. Membership functions can be any shape defineable by 2 points and 2 slopes--trapezoids, triangles, rectanlges, etc. Rules can have any number of antecedents and outputs, and can vary from rule to rule. "Min" method is used to compute rule strength, "Max" for applying rule strengths, "Center-of-Gravity" for defuzzification. This implementation of Inverted Pendulum control problem has: System Inputs, 2 (pendulum angle and velocity); System Outputs, 1 (force supplied to base of pendulum); Membership Functions, 7 per system input/output; Rules, 15 (each with 2 antecedents & 1 output). If more precision is required, integers can be changed to real numbers.*/ #include <stdio.h> #define MAXNAME 10 /* max number of characters in names */ #define UPPER_LIMIT 255 /* max number assigned as degree of membership */ /* io_type structure builds a list of system inputs and a list of system outputs. After initialization, these lists are fixed, except for value field which is updated on every inference pass. */ struct io_type{ char name[MAXNAME]; /* name of system input/output */ int value; /* value of system input/output */ struct mf_type /* list of membership functions for */ *membership_functions; /* this system input/output */ struct io_type *next; /* pointer to next input/output */ }; /* Membership functions are associated with each system input and output. */ struct mf_type{ char name[MAXNAME]; /* name of membership function (fuzzy set) */ int value; /* degree of membership or output strength */ int point1; /* leftmost x-axis point of mem. function */ int point2; /* rightmost x-axis point of mem. function */ int slope1; /* slope of left side of membership function */ int slope2; /* slope of right side of membership function */ struct mf_type *next; /* pointer to next membership function */ }; /* Each rule has an if side and a then side. Elements making up if side are pointers to antecedent values inside mf_type structure. Elements making up then side of rule are pointers to output strength values, also inside mf_type structure. Each rule structure contains a pointer to next rule in rule base. */ struct rule_element_type{ int *value; /* pointer to antecedent/output strength value */ struct rule_element_type *next; /* next antecedent/output element in rule */ }; struct rule_type{ struct rule_element_type *if_side; /* list of antecedents in rule */ struct rule_element_type *then_side; /* list of outputs in rule */ struct rule_type *next; /* next rule in rule base */ }; struct rule_type *Rule_Base; /* list of all rules in rule base */

### Inverted-pendulum Example

Figure 12 shows a classic two-dimensional control problem known as the "inverted pendulum." The idea is to keep a pole vertically balanced. The pole is weighted at the top and attached at the bottom by a movable base. If the pole falls to the right or left, the base moves in the same direction to compensate. By monitoring the angle and angular velocity of the pendulum, a fuzzy system can determine the proper force to apply at the base to keep it balanced.

Figure 13 shows the fuzzy sets associated with the system inputs and output. The exact set of rules depends on the dynamics of the physical components, required robustness, and range of operating conditions.

Theoretically, the rule base in Figure 14 is sufficient to balance the pendulum, but other solutions exist. A general-purpose fuzzy inference engine like that in Listings One through Four can be applied to many applications. Listing One provides the header and data structures, Listings Two and Three present the major fuzzy processes, and Listing Four lists the math-support functions. The input-configuration files describing system input/output, the membership functions, and the rule base differ from application to application.

**Listing Two**

main() { initialize_system(); while(1){ get_system_inputs(); fuzzification(); rule_evaluation(); defuzzification(); put_system_outputs(); } }

**Listing Three**

/* Fuzzification--Degree of membership value is calculated for each membership function of each system input. Values correspond to antecedents in rules. */ fuzzification() { struct io_type *si; /* system input pointer */ struct mf_type *mf; /* membership function pointer */ for(si=System_Inputs; si != NULL; si=si->next) for(mf=si->membership_functions; mf != NULL; mf=mf->next) compute_degree_of_membership(mf,si->value); } /* Rule Evaluation--Each rule consists of a list of pointers to antecedents (if side), list of pointers to outputs (then side), and pointer to next rule in rule base. When a rule is evaluated, its antecedents are ANDed together, using a minimum function, to form strength of rule. Then strength is applied to each of listed rule outputs. If an output has already been assigned a rule strength, during current inference pass, a maximum function is used to determine which strength should apply. */ rule_evaluation() { struct rule_type *rule; struct rule_element_type *ip; /* pointer of antecedents (if-parts) */ struct rule_element_type *tp; /* pointer to consequences (then-parts) */ int strength; /* strength of rule currently being evaluated */ for(rule=Rule_Base; rule != NULL; rule=rule->next){ strength = UPPER_LIMIT; /* max rule strength allowed */ /* process if-side of rule to determine strength */ for(ip=rule->if_side; ip != NULL; ip=ip->next) strength = min(strength,*(ip->value)); /* process then-side of rule to apply strength */ for(tp=rule->then_side; tp != NULL; tp=tp->next) *(tp->value) = max(strength,*(tp->value)); } } /* Defuzzification */ defuzzification() { struct io_type *so; /* system output pointer */ struct mf_type *mf; /* output membership function pointer */ int sum_of_products; /* sum of products of area & centroid */ int sum_of_areas; /* sum of shortend trapezoid area */ int area; int centroid; /* compute a defuzzified value for each system output */ for(so=System_Outputs; so != NULL; so=so->next){ sum_of_products = 0; sum_of_areas = 0; for(mf=so->membership_functions; mf != NULL; mf=mf->next){ area = compute_area_of_trapezoid(mf); centroid = mf->point1 + (mf->point2 - mf->point1)/2; sum_of_products += area * centroid; sum_of_areas += area; } so->value = sum_of_products/sum_of_areas; /* weighted average */ } }

**Listing Four**

/* Compute Degree of Membership--Degree to which input is a member of mf is calculated as follows: 1. Compute delta terms to determine if input is inside or outside membership function. 2. If outside, then degree of membership is 0. Otherwise, smaller of delta_1 * slope1 and delta_2 * slope2 applies. 3. Enforce upper limit. */ compute_degree_of_membership(mf,input) struct mf_type *mf; int input; { int delta_1; int delta_2; delta_1 = input - mf->point1; delta_2 = mf->point2 - input; if ((delta_1 <= 0) || (delta_2 <= 0)) /* input outside mem. function ? */ mf->value = 0; /* then degree of membership is 0 */ else mf->value = min( (mf->slope1*delta_1),(mf->slope2*delta_2) ); mf->value = min(mf->value,UPPER_LIMIT); /* enforce upper limit */ } /* Compute Area of Trapezoid--Each inference pass produces a new set of output strengths which affect the areas of trapezoidal membership functions used in center-of-gravity defuzzification. Area values must be recalculated with each pass. Area of trapezoid is h*(a+b)/2 where h=height=output_strength=mf->value b=base=mf->point2-mf->point1 a=top= must be derived from h,b, and slopes1&2 */ compute_area_of_trapezoid(mf) struct mf_type *mf; { int run_1; int run_2; int base; int top; int area; base = mf->point2 - mf->point1; run_1 = mf->value/mf->slope1; run_2 = mf->value/mf->slope2; top = base - run_1 - run_2; area = mf->value * ( base + top)/2; return(area); }

Figures 14 and 15 contain the necessary information to implement the inverted-pendulum problem.

Rule 1: IF (angle is NL) AND (velocity is ZE) THEN (force is PL) Rule 2: IF (angle is ZE) AND (velocity is NL) THEN (force is PL) Rule 3: IF (angle is NM) AND (velocity is ZE) THEN (force is PM) Rule 4: IF (angle is ZE) AND (velocity is NM) THEN (force is PM) Rule 5: IF (angle is NS) AND (velocity is ZE) THEN (force is PS) Rule 6: IF (angle is ZE) AND (velocity is NS) THEN (force is PS) Rule 7: IF (angle is NS) AND (velocity is PS) THEN (force is PS) Rule 8: IF (angle is ZE) AND (velocity is ZE) THEN (force is ZE) Rule 9: IF (angle is ZE) AND (velocity is PS) THEN (force is NS) Rule 10: IF (angle is PS) AND (velocity is ZE) THEN (force is NS) Rule 11: IF (angle is PS) AND (velocity is NS) THEN (force is NS) Rule 12: IF (angle is ZE) AND (velocity is PM) THEN (force is NM) Rule 13: IF (angle is NM) AND (velocity is ZE) THEN (force is NM) Rule 14: IF (angle is ZE) AND (velocity is PL) THEN (force is NL) Rule 15: IF (angle is PL) AND (velocity is ZE) THEN (force is NL)

*Figure 14:** Inverted pendulum rule base.*

input: angle input: velocity output: force NL: 0 31 31 63 NL: 0 31 31 63 NL: 0 31 31 63 NM: 31 63 63 95 NM: 31 63 63 95 NM: 31 63 63 95 NS: 63 95 95 127 NS: 63 95 95 127 NS: 63 95 95 127 ZE: 95 127 127 159 ZE: 95 127 127 159 ZE: 95 127 127 159 PS: 127 159 159 191 PS: 127 159 159 191 PS: 127 159 159 191 PM: 159 191 191 223 PM: 159 191 191 223 PM: 159 191 191 223 PL: 191 223 223 255 PL: 191 223 223 255 PL: 191 223 223 255

*Figure 15:** Sample membership function input file.*

Figure 15 repeats the input/output and membership information shown Figure 13 in a format that can be easily parsed by an initialization routine. Such an initialization routine (not shown in the listings) sets up the required data structures, converting the four points describing a membership function into two points and two slopes; see Figure 16.

Generally, four points describe a trapezoid, but a triangle can be formed by making the two midpoints identical, as in Figure 16.

### Closing Remarks

The emergence of fuzzy logic is exciting because it is readily applicable to many problems too awkward to solve with conventional techniques. Any programmer can easily write code to implement a fuzzy inference engine like the one presented here. However, excellent fuzzy development tools exist which allow the designer to focus more on the application and behavior of the system and less on the implementation. These tools provide user-friendly, graphical interfaces with a rich set of support functions for analyzing, debugging, and simulating the system. Examples of such tools are: FIDE from Aptronix (San Jose, CA), CubiCalc from Hyperlogic (Escondido, CA), and TILShell from Togai InfraLogic (Irvine, CA). In addition, Motorola distributes free fuzzy development tools through their electronic BBS. "Freeware Data Services," at 512-891-3733 (in the subdirectory amcu/amcull).

Implementing the fuzzy engine manually, however, affords the ability to understand, optimize, or customize the fuzzy inference engine. This is especially important when experimenting with new fuzzy paradigms, such as rulebase hierarchies and adaptive or hybrid systems.

### References

Brubaker, David I. Introduction to Fuzzy Logic Systems. Menlo Park, CA: The Huntington Group, 1991.

Kosko, Bart. Neural Networks and Fuzzy Systems. Englewood Cliffs., NJ: Prentice-Hall, 1990.

Self, Kevin. "Designing with Fuzzy Logic." IEEE Spectrum (November, 1990).

Sibigtroth, James M. "Creating Fuzzy Micros." Embedded Systems Programming (December, 1991).

Sibigtroth, James M. "Implementing Fuzzy Expert Rules." AI Expert (April, 1992).

Williams, Tom. "Fuzzy Logic is Anything but Fuzzy." Computer Design (April, 1992).

*Greg is a member of the Motorola technical ladder and is currently merging fuzzy logic with microcontrollers. He has an MSEE from National Technological University and a BSEE from the University of Texas at Austin. Greg can be contacted at Motorola Advanced Microcontroller Division, 6501 William Cannon Drive West, Austin, Texas 78735-8598.*