# Fuzzy controller

The fuzzy logic is preferably used when a technical process with several input and output variables with strongly changing parameters and non-linear subsystems is to be controlled without human intervention ( plant operator ) as possible .

The fuzzy set theory as a fuzzy set theory was developed in 1965 by Lotfi Zadeh at the University of California , Berkeley . In Japan , the fuzzy set theory ( fuzzy logic ) was successfully used as a fuzzy controller (fuzzy controller) in industrial processes in the 1980s and only found its way into Europe with numerous applications in the 1990s. Harro Kiendl is one of the German pioneers .

In the system-analytical sense, a fuzzy control system is a static, non-linear control system that uses sharp input variables of a complex process according to the rules of a rule base to create fuzzy control variables and sharp defuzzified value signals with which a satisfactory process result is achieved.

Overview of the fuzzified input variables and the output variables of the fuzzy controller.

The fuzzification of a sharp physical input variable (measured value) of a technical process is the quantification through fuzzy definitions with linguistic terms such as warm, cold, much, little, more, large, small. The so-called graphical trapezoidal or triangular fuzzy sets are used to determine the degrees of membership from the sharp input signals within the basic sets (variables) . The terms belonging to the variables of the basic sets correspond in simplified terms to the linguistic terms of the fuzzy sets. ${\ displaystyle \ mu _ {i}}$${\ displaystyle A, B, \ ldots}$ ${\ displaystyle a_ {i} (A), b_ {i} (B), \ ldots}$

The degrees of affiliation act on the rules of the rule base created with expert knowledge and from this, fuzzy manipulated variables and analog defuzzified sharp output signals are formed, which have a static effect on the actuators of a process.

The adaptation of a fuzzy controller without a mathematical model of the process is relatively unproblematic with the expert knowledge of a known process.

Fuzzy controllers refer to the procedures of the fuzzy controller, but are mostly functional modifications, simplifications or additions with the fuzzy logic.

The basic functional mechanism of a control is based on the feedback of the controlled variable and input of the inverted target / actual value comparison into the controller. Precise control with the static fuzzy controller is only possible in limited applications of non-linear types of controlled systems.

Optimal controls can be achieved by expanding the fuzzy controller with integral and differential components of the control deviation with the help of empirical settings. In comparison, when used as a controller on linear single-variable controlled systems, they have no functional advantages over classic PID controllers .

## Use of the fuzzy controller

A person is able to ensure optimal process management with the help of fuzzy process information through targeted interventions (manipulated variables) in the process flow. For example, after a learning procedure for equilibrium on the bicycle, it can generate speed-dependent centrifugal forces through targeted steering movements, which keep the rolling bicycle upright thanks to the human sense of balance.

The fuzzy controllers based on fuzzy logic (fuzzy: blurred, blurred) work in a comparable way with the help of expert knowledge. Here, too, fuzzy input variables are processed into a defined manipulated variable using a knowledge-based control algorithm.

Fuzzy systems refer to very special graphic processes with fuzzy linguistic terms of human ways of thinking in connection with simple logical equations (IF-THEN rule base) in order to create one or more manipulated variables from several fuzzy variables . Signal flow diagrams in block diagrams are required to understand how they work.

A fuzzy system can be viewed as a static non-linear transmission system. Sharp input variables also lead to sharp output variables.

Essential procedural steps in the conception of a fuzzy controller relate to the fuzzification of the linguistic variables to degrees of membership, the merging of the affiliations of the linguistic terms via the evaluation of the premises (min-max operators) with the help of the rule base (IF-THEN rules), the accumulation the manipulated variables and the defuzzification of the manipulated variables to a sharp analog signal (e.g. area centroid method). The manipulated variables of the fuzzy controller can act on the actuating devices of a technical system in order to achieve the desired technical behavior of the system. An analog defuzzified manipulated variable signal can be designed in connection with a given technical system as a controlled system to a control loop.

Understanding the very special technical terms is required to use the fuzzy controller.

• In fuzzy systems, the quantification of sharp physical input variables of a process takes place through fuzzy definitions with linguistic terms such as warm, cold, much, little, more, large, small.
• The fuzzification of the sharp input variables is done by means of triangular or trapezoidal so-called fuzzy sets based on the Gaussian normal distribution in affiliations of the corresponding basic set , e.g. B. a temperature range . ${\ displaystyle \ mu}$${\ displaystyle \ vartheta}$
• The various affiliations act in the inference on the IF-THEN rules of the rule base. (IF <condition> THEN <consequence>)
• Inference (meaning: processed knowledge) includes the rule base , the implication and the accumulation .
• The implication defines the access of the rule base to the control units of the process.
• In the subsequent accumulation (accumulation) the fuzzy manipulated variable (or several manipulated variables) is graphically mapped.
• The defuzzification forms a sharp value from the accumulation z. B. according to the area-center of gravity method, if required.

For the behavior of a technical system z. B. a linear or non-linear dynamic system with several inputs and outputs, no mathematically exact model of the transfer behavior is required, but expert knowledge to estimate the linguistic variables. With the fuzzy control system, very simple controls, e.g. B. Control of household washing machines, as well as control loops with complex industrial systems.

Fuzzy controllers are mostly used in non-linear multi-variable systems that have the following properties such as:

• Processes whose mathematical models are complex or difficult to describe.
• Processes with conventional procedures that require corrective interventions by human hands (plant operators).
• When a process can only be run manually.

The aim of using the fuzzy controller is to automate such processes. Applications of the fuzzy controller can be found in all areas of industry up to consumer articles such as:

• Control of rail vehicles or storage and retrieval systems where travel times, braking distances and position accuracy depend on the masses, conveying distances, rail adhesion values ​​and schedules. In general, these processes are multi-variable systems, the reference variables of which are program-controlled and regulated.
Scheme of an inverse pendulum on a carriage
• In the automotive industry, the control of the automatic transmission is successfully operated with the fuzzy logic.
• Simpler applications in private households can be found in washing machines and dishwashers.
• Controls in cameras,
• Typical mechanical fuzzy demonstration model of a control of an inverse pendulum with one degree of freedom in universities:
A mobile carriage of mass M is moved horizontally by a force F. The pendulum mass to be balanced reports the pendulum position through an angle of inclination θ . The angle of inclination and the angular velocity v θ are the input measured variables of the fuzzy controller. The controlled variable is θ = 90 °. The control range of the angle of inclination only applies to θ ≫0 ° and θ ≪ 180 ° deviation from the base point of the pendulum. If the location of the slide is not of interest, it is usually a fuzzy PD controller with 2 input variables and one output manipulated variable, which acts on the mechanical model of the pendulum as an unstable non-linear controlled system.

## Linguistic variables and fuzzy sets

### Fuzzy sets

Classic definition of quantity

In mathematics, a set is a combination of various defined objects. In classical set theory (sharp set) an element belongs to a set (true = logical 1) or not to a set (not true = logical 0).${\ displaystyle M}$${\ displaystyle x}$${\ displaystyle M}$${\ displaystyle M}$

The function defines the membership in the set , whether an element is included in the set . ${\ displaystyle x}$${\ displaystyle M}$${\ displaystyle x}$${\ displaystyle M}$

${\ displaystyle x (M) = {\ begin {cases} 1, & {\ text {if}} x \ in M ​​\\ 0, & {\ text {if}} x \ notin M \ end {cases}} }$

One differentiates:

List notation: M = {Element 1, Element 2, ...}

Descriptive set notation: M = { x | x has the properties E 1, E 2,…, E n}

Definition of the fuzzy set

Term definitions:

• Fuzzification is the conversion of a sharply defined input variable within the dimension of a basic set into one or more membership values.
• A linguistic variable usually represents a physical quantity in any system as a basic set. This variable is defined by linguistic terms (fuzzy sets) .${\ displaystyle A}$${\ displaystyle a (A)}$
• Colloquially, terms of linguistic variables are common for events or processes, which do not allow clear affiliations to a set, such as warm, large, thick, near, short, old, etc. These terms of subsets can be identified by the fuzzy sets e.g. B. can be defined in a technical process if expert knowledge is available.
• Linguistic terms (fuzzy sets) are fuzzy subsets of the basic set . If the basic amount refers z. B. on the heat in the dimension of temperature, then the linguistic terms of the subsets (fuzzy sets) are: [cold, warm, hot, very hot] or symbolic , which are transferred as fuzzy sets into a coordinate system using expert knowledge.${\ displaystyle a (A)}$${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle a_ {1}, a_ {2}, a_ {3} \ ldots}$
• The fuzzy sets are drawn as linguistic terms in the form of graphic symbols, mostly as triangles and trapezoids on the abscissa. See the diagram in the next section.${\ displaystyle a_ {1} \ ldots a_ {n}}$
• The linguistic variable is entered on the abscissa of a coordinate system and contains the dimension of a mostly physical quantity. A scaled value from the linguistic variable (basic set) corresponds to an input variable for fuzzification of the subset (fuzzy set) into a degree of membership .${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle e (A)}$${\ displaystyle \ mu}$
• The membership function (= membership) defines the corresponding degree of membership between 0 and 1 (= 0 and 100%) for a sharp output signal for the membership of the variables (basic set) on one of the terms (subsets) . B. .${\ displaystyle 0 <\ mu (A) <1}$${\ displaystyle A}$${\ displaystyle a_ {1} \ ldots a_ {n}}$ ${\ displaystyle \ mu}$${\ displaystyle 0 {,} 3: = 30 \, \%}$
• The degree of membership is the fuzzified sharp output variable if the sharp input variable corresponds to a value of the basic set (the abscissa). In the case of overlapping fuzzy sets, several fuzzy sets can also be hit by a signal of the input variable . For example . Different values ​​of the input signal can also produce the same values ​​of the degree of membership .${\ displaystyle \ mu \, (A) = \ mu _ {a1} \ ldots \ mu _ {an}}$${\ displaystyle e (A)}$${\ displaystyle e (A)}$${\ displaystyle \ mu _ {a2} = 0 {,} 2, \ \ mu _ {a3} = 0 {,} 5}$${\ displaystyle e (A)}$${\ displaystyle \ mu}$
• Expert knowledge is based on the experience of an expert who can assess the behavior of any linear or nonlinear technical system and can define linguistic variables as basic quantities.
• Linguistic variables are usually designated with capital letters A, B, C,…. The output variables of a fuzzy controller are designated with the last letters of the alphabet X, Y, Z.

The sharp separation of the classical sets with the assessment (true or not true) is abolished in the theory of fuzzy sets and replaced by so-called fuzzy sets as fuzzy subsets. Fuzzified sets create a mostly linear relationship to an output variable - the degree of membership - between a value of the basic set and the elements (subsets with the symbols of the fuzzy sets) of a basic set . ${\ displaystyle A}$${\ displaystyle a_ {1} \ ldots a_ {n}}$${\ displaystyle \ mu}$

A fuzzy set is thus determined by a graphic method using a coordinate system in which so-called fuzzy sets are drawn in as graphic elements such as triangles and trapezoids. Fuzzy sets also allow the description of non-linear relationships between input and output variables of a system (e.g. technical system) with the help of linguistic fuzzy terms such as [very little, little, more, much, ...], which are determined by expert knowledge.

Fuzzy subsets (linguistic terms) are - based on the graphic representation of the Gaussian bell curve for easier calculation - mostly triangular or trapezoidal functions. They are part of a basic set (linguistic variable) and are represented as graphic models in a coordinate system depending on the number of linguistic terms. The abscissa contains the scaled physical dimension of the basic set (linguistic variable) within which the terms are divided as fuzzy sets with their support points, mostly overlapping. The ordinate identifies all subsets (fuzzy sets) by the degree of membership . ${\ displaystyle A}$ ${\ displaystyle \ mu (A)}$

Depending on how many fuzzy sets a basic set contains, the degree of membership = 0 to 1 can be achieved depending on an input value of the basic set . It therefore depends on the value within the basic set which fuzzy set and how many fuzzy sets are activated (“firing”) and which degrees of membership result from this. ${\ displaystyle A}$${\ displaystyle e (A)}$${\ displaystyle \ mu (A)}$${\ displaystyle e (A)}$${\ displaystyle A}$${\ displaystyle \ mu (A)}$

If an input variable with a value of the basic set meets an overlapping fuzzy set, several values ​​of the degree of membership can result on the ordinate. Different values ​​of the input signal can produce the same values ​​of the degree of membership . ${\ displaystyle \ mu \, (A) = \ mu _ {a1} \ ldots \ mu _ {an}}$${\ displaystyle e (A)}$${\ displaystyle \ mu}$

There are different terms for the functions of the fuzzy subsets, all of which mean the same thing: fuzzy set, fuzzy element, term of a fuzzy variable, linguistic term.

### Linguistic variables of the fuzzy controller

A fuzzy variable refers to the basic set of a scaled physical quantity, the terms of which represent fuzzy sets and are graphically mapped in a coordinate system - as already explained.

A graphic model of the signal flows of the fuzzy controller consists of several input and output variables that are linked to one another by several working rules of the rule base. The sharp input signals of this model are fuzzified and the processed fuzzy output signals are defuzzified into manipulated variables.

Fuzzy variables are mostly physical quantities, such as B. the "temperature", whose linguistic terms can be defined as fuzzy sets such as "very cold", "cold", "warm", "very warm". Other fuzzy variables such as B. the "distance" can be described by the linguistic terms like "very close", "near", "far", "very far". A linguistic term is defined as a fuzzy subset over a basic set.

Abbreviations of standard terms are often used for any physical quantity such as "positive large", "positive medium", "positive small", "near zero", "negative large", "negative medium", negative small. There are already 7 terms which, in a single variable system, relate to an input-output characteristic that goes through the origin of the coordinate system from the positive to the negative range. If the process involves multi-variable systems, the input-output relationship of the fuzzy controller is determined by maps.

Between the two linguistic terms of a basic set of temperature such as “warm” and “very warm”, this property has a degree of truth of belonging between 0 and 1. These terms of the variable temperature correspond to human perceptions. Nevertheless, there is a certain arbitrariness in the definition and assignment of the fuzzy sets with their support points over the basic set of the variable “temperature”. Expert knowledge already plays a role here, because there is a known difference in the linguistic term heat, whether the term "very warm" is the bath water temperature or the tea-water temperature.

Representation of a fuzzy basic set A with 6 subsets a1… a6

Triangular and trapezoidal fuzzy sets are the most commonly used forms of a fuzzy subset of fuzzy variables because of their ease of calculation.

Singletons (line functions) correspond to sharp linguistic elements (terms), which each represent only one physical value and are mostly used to simplify defuzzification. If x 0 is a certain value of the basic set X, then a value other than zero is given for the degree of membership only at the point x = x 0 . For the conclusion of the THEN part of an IF-THEN rule, singletons can be used instead of triangular or trapezoidal fuzzy sets to simplify the computational evaluation. Details follow in the Inference chapter.

A number of 2 to 7 fuzzy sets of a basic set are often sufficient for describing a technical process of a physical signal variable.

For a better understanding, fuzzy sets are arranged graphically in a coordinate system on an abscissa with sharp values ​​of the physical quantity and assigned the degree of membership μ = 1 ≡ 100% for any basic quantities in the ordinate. The trapezoidal or triangular shapes of the fuzzy sets are often - but not always - designed symmetrically and overlap within a certain area in the base points, usually between 20 and 50% on the abscissa. If there is no overlap, in single-variable systems the output variable in this specific area has the degree of membership zero, as does the input-output characteristic.

By choosing the position of the fuzzy sets - the size of the overlap, the symmetry of the sets, the type of rule base - in single-variable systems, the input-output characteristic of the fuzzy controller can be varied within wide limits from linear to continuously non-linear and fractionally non-linear be designed.

The number of variables and the associated terms should be as small as possible because they mean a lot of computing work on a microcomputer, especially when calculating the degree of fulfillment and linking variables with the fuzzy rules.

### Definition of indexed fuzzy variables and their symbols

If one restricts oneself to the representations of the fuzzy variables and the most common fuzzy operators, the fuzzy logic is a very simple process, control technology and, with restrictions, control technology can be used successfully as a controlled system without precise knowledge of a dynamic multi-variable process. Most of this can be seen in the graphical function diagrams shown. In contrast, the mathematical description of the signaling functions of the sharp input signals up to the sharp signal output variables (manipulated variables) of the fuzzy controller is difficult.

The technical literature does not show a uniform naming of fuzzy variables, subsets, basic sets, as well as variable symbols and their indexes. In particular, publications from the university sector can cause less experienced interested parties to understand the fuzzy theory with multiple indexes of degrees of fulfillment, fuzzy quantities and rules.

To make it easier to understand, the following simplification of the naming of variables and their indexing is defined:

• Sharp control-related input and output signals
Sharp input and output signals are referred to as e1, e2… e n for the input signals and u1, u2… u n for the output signals in accordance with the generally introduced symbols of control engineering . The indexing n means a finite integer from 1… ≪ .
• Blurred signals
A fuzzified signal corresponds to a function of the degree of membership μ = f [e, fuzzy variable, fuzzy term (fuzzy set)] of a fuzzy subset (fuzzy set) of the fuzzy variables with the basic set taken as a result of a sharp input signal e that is assigned to the range of input signal values ​​e. To identify the affiliation, μ is indexed with the linguistic term of the subset or with an associated letter symbol.
Fuzzy signals result from the fuzzy logic processing procedure of all signal values ​​of the inference after the fuzzification and before the defuzzification. While the linguistic variable of a basic set represents a signal channel, for a given sharp input signal the activated associated linguistic term (fuzzy set) or the degree of membership of this term already means the fuzzy signal value.
• Fuzzy variables
The fuzzy variables for single and multiple variable systems relate to scaled basic sets and contain a limited number of membership functions (= terms, = elements, = fuzzy sets). Fuzzy variables are written with capital letters of the initial alphabet A, B, C…. designated. The fuzzy output variables can be entered with the capital letters of the end alphabet e.g. B. X, Y, Z or, in the case of an output variable with U, can be referred to as a control variable.
Fuzzy input variables are used to convert sharp input signals into fuzzy subsets (fuzzy sets). They are part of the premise (IF part of a work rule).
Fuzzy output variables are used to convert fuzzy partial manipulated variables after accumulation (summary) into clear manipulated variables. They are part of the conclusion (THEN part of a work rule).
• Fuzzy terms (fuzzy sets) as subsets of the fuzzy variables
The necessary indexing of fuzzy variables with the long linguistic terms of the fuzzy terms is unsuitable in equations. Instead, the symbols of the variable A are designated in the same order with a1, a2,…, a n , for variables B with b1, b2,…, b n . This also applies to other variable symbols!
• Affiliation μ of the fuzzy input variables
The degree of membership μ (A) or μ (B) with the maximum value 1 over a variable of the basic sets A, B, ... always relates to a term (fuzzy set) of a variable of a graphic fuzzy model. The graphic model shows the relationship between membership on the ordinate and the basic set on the abscissa in the coordinate system. The basic set corresponds to the dimension of the physical input variable. For example, the symbol ϑ applies to the scaling of the temperature of the basic quantity of the variable A on the abscissa.
μ a1 (ϑ) = 1 corresponds to the maximum degree of membership of a fuzzy set (subset) a1 of the variable A in the dimension of temperature.
• Affiliation μ as output variable (Mamdani method)
The graphic fuzzy model of the fuzzy output variables corresponds to that of the fuzzy input variables. The affiliation μ of a fuzzy output variable is determined as a conclusion from the premise result of the input variable. This requires knowledge of the fuzzy rules to be used and the fuzzy operators.
In the event that a fuzzy partial manipulated variable (fuzzy set) of the variable U (basic quantity) is u3 = "valve 2/3 open" as a fuzzy set, then the degree of membership means μ u3 (U) = 0.5, that the fuzzy subset has a trapezoidal shape with a height of 0.5 when using min or max operators of the premise evaluation.
• Degree of fulfillment and membership function of a fuzzy subset
While the membership function a1 (ϑ) is a function of a fuzzy set, the degree of membership is a certain value of the membership, e.g. B. μ a1 (A) = 0.5, which results from a sharp value of the basic set A according to an associated working rule of the rule base for a certain term a1.
This value of the degree of membership for a certain activated fuzzy set z. B. μ αi (A) = 0.5 is also referred to as “ degree of fulfillment of the premise” (if part of the rule) or “degree of fulfillment of the rule”. In the specialist literature, this concept of the degree of fulfillment is also referred to in simplified form as α (alpha section, alpha or H = trapezoidal height). However, α or H is always a value that relates to the affiliation μ or to the ordinate of the graphic fuzzy model.

### Calculation of the fuzzy sets with straight line equations

Common fuzzy sets as fuzzy subsets of a fuzzy variable

The graphic model of a fuzzy variable corresponds, using expert knowledge, to a division of normally overlapping fuzzy sets onto the basic set X on the abscissa in the coordinate system. The maximum height of all fuzzy sets of the variable X corresponds to the degree of fulfillment μ (X) = 1 of the ordinate. For the fuzzification of a sharp input variable e in the area of ​​the basic set of the variable X, the degree of membership or the degree of membership of the fuzzy sets can easily be represented graphically with a certain accuracy. Since the fuzzy controller is a microcomputer that can perform mathematical and logical operations, straight line equations must be used to calculate the degree of fulfillment of triangular or trapezoidal fuzzy sets.

It therefore depends on the size of the sharp input signal within the basic set which fuzzy set and whether it hits the rising or falling ramp and whether two fuzzy sets are hit in the case of overlapping fuzzy sets.

In the fuzzy set models shown in the graphic with the names of the key values ​​a, b, c, d and the degree of fulfillment μ (X) = 1, the variable x corresponds to the input variable e. The degree of fulfillment μ (X) = 1 is met depending on the size of the input signal in the area of ​​the basic set as often as many terms have been assigned to a variable.

For a fuzzy variable with several fuzzy sets, the straight line equations of the rise and fall must be calculated to determine the degree of fulfillment, depending on the sharp input variable and the position of the ramp. From this, when the degree of fulfillment is linked with other variables, the fuzzy output variables (fuzzy partial manipulated variables) are determined.

To calculate the membership function μ (X) for a given sharp value of the input variable e on the scaled abscissa of a basic set, the following linear equations of the ramps are given:

Straight line equation for the rise of the ramp:

${\ displaystyle \ mu (x, a, b, c) = {\ frac {xa} {ba}} \ qquad {\ text {for}} \ a \ leq x \ leq b}$

Straight line equation for the slope of the ramp:

${\ displaystyle \ mu (x, a, c, d) = {\ frac {dx} {dc}} \ qquad {\ text {for}} \ c \ leq x \ leq d}$

### Fuzzy operators

Operations with fuzzy sets are necessary when several linguistic statements of the IF-THEN rules have to be linked. In addition to fuzzy sets, fuzzy relations are an important part of the fuzzy set theory. The relationships of the same basic sets of neighboring fuzzy sets are described by fuzzy relationships using the most common operators such as fuzzy OR, fuzzy AND and fuzzy NOT functions. These relationships have a different meaning than those of Boolean algebra, because they must also be related to the fuzzy sets. The division of the truth content into any number of intermediate steps between true and false comes closer to the human way of thinking than bivalent logic.

AND-OR-NOT operators for linking membership functions (subsets)

The behavior of a fuzzy controller is determined by so-called working rules of the rule base, which consist of an IF part and a THEN part. The fuzzy subsets of the same or unequal basic sets are linked by means of the fuzzy operators. The 3 most frequently used operators for linking fuzzified subsets are as follows:

• AND operator
The MIN operator forms the set average of fuzzy sets. The AND link corresponds to the average of the two areas of the fuzzy sets.
Example of the linked sets μ x1 and μ x2 of the basic set X:
${\ displaystyle \ mu _ {x} (X) = \ min [\ mu _ {x1} (X), \ mu _ {x2} (X)] = [\ mu _ {x1} \ cap \ mu _ { x2}] (X)}$
Example min operator for μ x1 = 0.2 and μ x2 = 0.4:
${\ displaystyle \ mu _ {\ min} (X) = \ min [0 {,} 2 (X); 0 {,} 4 (X)] = 0 {,} 2 (X)}$
Other AND operators that are less used:
prod operator (algebraic product) and min-avr operator (combines the min operator with the arithmetic mean)
• The OR operator
The OR link between two fuzzy sets is represented as a union with the maximum operator. The max operator forms the set union of the fuzzy sets.
Example of the linked sets μ x1 and μ x2 of the basic set X:
${\ displaystyle \ mu _ {x} (X) = \ max [\ mu _ {x1} (X), \ mu _ {x2} (X)] = [\ mu _ {x1} \ cup \ mu _ { x2}] (X)}$
Example max operator for μ x1 = 0.2 and μ x2 = 0.4:
${\ displaystyle \ mu _ {\ max} (X) = \ max [0 {,} 2 (X); 0 {,} 4 (X)] = 0 {,} 4 (X)}$
Other OR operators that are less used:
probor operator (probabilistic OR algebraic sum) and max-avr operator (combines the max operator with the arithmetic mean)
• NOT function
The complement operator forms the complement μ x of the fuzzy set μ x1 of the basic set X with the following operation:
${\ displaystyle \ mu _ {x} (X) = 1- \ mu _ {x1} (X)}$

### Fuzzy relations

Relations are suitable for describing relationships between different variables and attributes. A fuzzy relation corresponds to a fuzzy set, the basic set of which is a Cartesian product of several basic sets. Multi-digit relations are relationships between sharp or fuzzy sets on different basic sets. A multi-digit relation is a subset of the Cartesian product set of the basic sets, a fuzzy relation is always a fuzzy set.

A two-digit fuzzy relation is a mapping of two basic sets z. B. of two variables:

${\ displaystyle A * B \ longrightarrow \ {0; 1 \}}$

The adjacent graphic "Rule activation with unequal basic quantities" shows as relation rule R the relation of the degrees of membership according to rule 1:

Rule activation with two fuzzy subsets (fuzzy sets) as a result of 2 sharp input variables e1 and e2

The terms a n and b n of these variables A and B indicate how strongly they are related to one another. In a relation matrix z. B. with the variables A, which corresponds to a basic amount of temperature and the variable B, which corresponds to a basic amount of pressure, the following relation of the terms a2 = "warm" and b2 = "medium pressure" can be formed, their assignment for a certain Rule base work rule is valid.

${\ displaystyle \ underbrace {{\ text {IF temperature is warm}} \ quad AND \ quad {\ text {pressure is medium}}} _ {\ text {IF A = ​​a2 AND B = b2}} \ quad \ underbrace {THEN \ quad {\ text {valve is 2/3 open}}} _ {\ text {THEN U = u3}}}$

The subsets of the terms “warm” and “medium” are represented as the relation of the basic sets of temperature and pressure. The min operator is used to determine the affiliations μ R of the AND operation of the premises of the IF part.

${\ displaystyle \ mu _ {R} (A, B) = \ min \ {\ mu _ {ai} (A), \ mu _ {bi} (B) \}}$

Numerical example of the fuzzy relation (premise relation of 2 fuzzy sets of unequal basic sets):

For a few at equal distances ( equidistance : = “equal distances on a scale”) with 5 active measuring points, selected sharp input signals e1 and e2 for the degrees of membership μ = {0; 0.5; 1.0; 0.5; 0} the subsets (fuzzy sets) a2 (A) and b2 (B) are activated. For the specified degrees of membership, the corresponding sharp values ​​of the input signals for temperature and pressure can be determined according to the graphic on the right using simple scaling. According to the table below, each value of the term a2 is associated with each value of the term b2 for a particular membership. This is the premise evaluation of the IF part of the associated work rule.

The scaled data of the base set of temperature of variable A are: {20; 40; 60; 80; 100} (ϑ)

The scaled data of the basic amount of pressure of variable B is: {4; 8th; 12; 16; 20} (P)

The terms are ANDed using the min operator according to rule 1 as shown in the graphic for the fuzzy variables.

The result of the table are the rule-activated membership values μ R of two fuzzy subsets a2 and b2, which were found using some values ​​of the two scaled basic sets.

Tabular representation of the degrees of membership (degrees of fulfillment) as fuzzy relations of the basic quantities of temperature and pressure with the subsets "temperature warm" and "medium pressure" according to the above rule:

 Variable pressure, term "medium" Variable Temp. Term "warm" 4 bar 8 bar 12 bar 16 bar 20 bar 20 ° C 0 0 0 0 0 40 ° C 0 0.5 0.5 0.5 0 60 ° C 0 0.5 1.0 0.5 0 80 ° C 0 0.5 0.5 0.5 0 100 ° C 0 0 0 0 0

The practical meaning of this table is related to determining the degrees of membership of the conclusion of the work rule. In order to implement a computer program for a fuzzy controller, the degrees of membership of each fuzzy set with the discrete values ​​of the basic sets must be determined using straight line equations. The min-max operators are suitable for the logical links between the subsets. From the summary of the conclusion of the THEN part of each rule, the function of the fuzzy part manipulated variable and, with the accumulation, the fuzzy manipulated variable is determined.

## Overview of the fuzzy controller

Principle of the fuzzy controller

The concept of the fuzzy controller according to Mamdani relates to the following sub-functions:

The knowledge base contains the entire expert knowledge that relates to the control of a non-linear process based on experience with linguistic terms and actions. With their knowledge, the prerequisite is created to convert the sharp system input signals into fuzzy sets and to formulate the fuzzy rules of the rule base.

Elements (terms, fuzzy sets) of fuzzy sets are described by a value pair, which is derived from the sharp value of the basic set z. B. the variable A and the degree of membership μ a (A). The sharp value of the basic set of the variable A can be a value of an input signal e.

The rule base contains the fuzzy logic control rules for determining the fuzzy output variables (fuzzy manipulated variables) from the fuzzy input variables.

Data origin for the design of the fuzzy controller:

• Survey or observation of the experts in their process operation,
• Analysis of the process and creation of simple process models,
• Deriving data from the simplified process model.

Project planning strategy:

• Define the model of the basic quantities of the inputs and outputs of the fuzzy controller.
• Arrange membership functions (fuzzy sets) within the basic set.
• Establish tax rules as a rule base.

Example of a linguistic control rule for a fuzzy controller with two input variables and one output variable:

${\ displaystyle {\ text {rule}} \ colon \ \ \ underbrace {IF \ \ \ underbrace {{\ text {inside temperature cool}} \ \ AND \ \ {\ text {outside temperature cold}}} _ {Pr {\ ddot {a}} miss \ (precondition)} \ \ THEN \ \ \ underbrace {\ text {heating to maximum}} _ {conclusion \ (conclusion)}} _ {implication \ (fuzzy rule)}}$

Summary of the functions of the fuzzy controller:

 Graphic fuzzy model of a fuzzy variable Each fuzzy variable is represented as a graphic model in which the so-called basic quantity is plotted as a scaled physical signal variable (e.g. a temperature range) on the abscissa in a coordinate system. Within the range of the basic set, the mostly triangular fuzzy subsets (fuzzy sets) with their support points on the abscissa are divided, mostly overlapping, according to expert knowledge. The ordinate is assigned the degree of membership of the individual fuzzy sets with the maximum value μ = 1 ≡ 100%. Fuzzification Sharp changing input signals meet different fuzzy sets depending on the size in the area of ​​the basic set and result in continuously increasing or decreasing degrees of membership between 0 and 1. The degrees of membership within the area of ​​a fuzzy set can be understood as fuzzified signals. In the case of overlapping fuzzy sets, one or two adjacent fuzzy sets are activated depending on the size of a given value of the input signal e. This results in z. B. for a variable A two degrees of membership μ a1 (A) and μ a2 (A). The result of mapping a sharp value of the input variable e to the membership functions (fuzzy sets) of a fuzzy variable is also referred to as the vector of the membership functions. Example: Fuzzy variable A, fuzzy sets a1 ... a5, fuzzy sets a3 and a4 active. ${\ displaystyle e \ {\ xrightarrow {\ text {fuzzified}}} \ \ mu _ {a} = \ {0 \ \ 0 \ \ 0 {,} 8 \ \ 0 {,} 3 \ \ 0 \}}$ Inference: For each sharp value of an input signal of a variable, the obtained degree of membership μ of the associated fuzzy set is checked for the truth content with so-called IF-THEN rules of the rule base and, if necessary, an assignment to the output variable is made as a fuzzy part in the THEN part of the rule - control size. Is a "precondition" according to the rule fulfilled in the IF part (premise evaluation), i. H. a fuzzy set is active for a sharp input signal, then the conclusion is made in the THEN part as an assignment to an output variable. According to the “Mamdani implication”, the truth content of the conclusion must not be greater than the truth content of the premise. The degrees of membership of several sharp signal inputs via fuzzy variables with different basic sets are linked with min-max operators. Then the assignment takes place to a fuzzy output variable or several output variables as fuzzy partial manipulated variables. The partial manipulated variables of an output variable are accumulated to form a fuzzy manipulated variable. Defuzzification: From the accumulated fuzzy manipulated variable, the sharp manipulated variable is calculated using various methods.

## Structure of the fuzzy controller

### Disambiguation

In the German specialist literature, a number of foreign terms have been used to describe fuzzy controllers, which are not all used in a uniform manner.

The most important terms:

 Rule base The rule base is a list of rules according to linguistic terms for single-variable and multivariable processes. It contains the fuzzy logic rules for converting the fuzzy variables of the input variables into fuzzy fuzzy output variables (fuzzy manipulated variables). This is done using the IF-THEN rules. The knowledge of the system behavior of the process is determined in the fuzzy controller by the following form of IF-THEN rules in the rule base: ${\ displaystyle {\ text {rule}} \ colon \ quad \ underbrace {IF \ <{\ text {precondition}}> \ quad THEN \ <{\ text {conclusion}}>} _ {\ text {Fuzzy implication }}}$ The IF part (premise) and THEN part (conclusion) can be linked in a fuzzy-logical manner in multi-variable processes with different fuzzy variables (fuzzy variables). Premise (Latin praemissa: = sentence sent in advance) = prerequisite, assumption. IF part of a rule. Conclusion (Latin: conclusio: conclusion, conclusion). THEN part of a rule. Implication (Latin implicare: = entangling) = linking of statements Collective term for a logical rule statement (premises → conclusions). A fuzzy-logical rule (= linguistic rule = control rule = production rule) is referred to as a fuzzy implication. Here: Execution of the IF-THEN link. Modus ponens (Latin: modus: procedure, condition, manner and ponere: to place, to set) Modus ponens is a final rule that allows further statements to be derived from a true statement. The conclusion is defined as follows: If the statement A → B is true and A is also true, then the statement B must also be true. Inference (Latin infero: bring in, deduce, close). Meaning in logic: the conclusion generated in a system of rules. Conversion of fuzzified input signals via the rule base to fuzzy control signals. Frequently used operators: Max-Min inference according to Mamdani. (Use: max operators and min operators) Aggregation (lat. Aggregatio: accumulation, union) = aggregation of data into larger units. Execution of all AND operations of the premise. Summary of several premise-fulfillment degrees of the rules. Accumulation (Latin: accumulare: accumulate, accumulate). OR operation of the implication results μ of all rules. (= Summary of the degree of fulfillment μ of several rules).

### Fuzzification

Typical applications of the fuzzy controller are usually a non-linear process with several input variables, for which the expert knowledge of a simplified model of the process is available.

The measuring range of a physical, sharp input signal is scaled to a suitable value range of the basic set of the linguistic variable (signal channel). According to expert knowledge, the basic set is divided into fuzzy subsets (fuzzy sets), usually overlapping. The maximum degree of membership for each fuzzy set corresponds to μ = 1. In multi-variable systems, several measured variables and manipulated variables of the fuzzy controller can occur.

The individual work steps are:

• Definition of the terms of a fuzzy variable: fuzzy names, number, type of fuzzy sets and their support points.
• Scaling of the ramp-shaped relationships of the terms to the degree of membership μ = 1 within the measuring range of the signal input variable. A subset of the linguistic terms is assigned to the measuring range of the input variable in such a way that the entire measuring range is divided over the selected number of terms.
When using triangular fuzzy sets, there are 3 support points, 2 base points within the measuring range of a signal input on the abscissa and a top point for degree of membership 1 on the ordinate. The fuzzy sets of linguistic variables (e.g. temperature and their terms: cold, warm, hot) are assigned to the sharp temperature measurement values, usually an electrical voltage. The graphical representation of a fuzzy variable as a fuzzy set usually takes place on the abscissa in the sharp values ​​of the physical quantity.
Each individual signal channel becomes a vector of fuzzy sets and degrees of membership.
• The fuzzy sets are often arranged symmetrically and overlap at the base points (support points). Overlaps of 20% to 50% are typical.
50% overlap means that the following fuzzy set begins to increase for μ = 0 in the maximum of the degree of membership μ = 1 of the previous fuzzy set.
• In the beginning and end of the entire fuzzy set, parts of trapezoidal fuzzy sets (indifference area) are mostly used to achieve limiting effects.
If a measured value is larger than the largest scaled linguistic term "very large", the degree of membership should certainly be 1. The same applies vice versa for the limitation of the smallest linguistic value “very small” or with negative values ​​of the basic set “very small negative” also with membership level 1. For an even smaller input measurement value than “very small” or “very small negative” remains the affiliation with 1.
• The arrangement of the fuzzy sets on the value range of the physical measured variable has a major influence on the shape of the input-output characteristic (transfer characteristic) in a single-variable system.
• Large overlapping of the support points of neighboring fuzzy sets on the abscissa means a smooth characteristic curve (approximation to linearity).
• 50% overlapping of neighboring triangular fuzzy sets - they intersect at μ = 0.5 - mean linear behavior of the input-output characteristic, provided there are no different evaluations with factors.
• Gapful fuzzy sets that do not overlap at the base points lead to a partial transfer characteristic with the value zero.
• Steeper ramps compared to neighboring fuzzy sets mean steeper part of the characteristic curve (greater part gain).

### Inference (database, rule base)

Fuzzy rules contain the expert knowledge of specialists on how a technical system should be operated based on experience. For the optimal function of the fuzzy controller, the choice of the arrangement of the terms (fuzzy sets) on the abscissa of the graphical model and the affiliation in the fuzzification is just as important as the establishment of the rules of the rule base in the inference unit.

The rule base contains the fuzzy logic rules for determining the fuzzy output variables (fuzzy manipulated variables) from the fuzzy input variables.

The basis of the fuzzy rules is the following form:

${\ displaystyle {\ text {rule}} \ colon \ quad IF \ <{\ text {precondition}}> \ quad THEN \ <{\ text {conclusion}}>}$

The application of the theory of fuzzy sets according to Zadeh - as fuzzy logical inference - means to draw a conclusion from the fulfillment of a certain condition. The condition and the conclusion with fuzzy sets can be represented as follows for so-called IF-THEN rules, which consist of an IF part and a THEN part.

The application of the implication (linking of statements) of a fuzzy working rule of the rule base corresponds to an IF-THEN link.

Premise evaluation for the same basic quantities

Example of the AND operation of the input variable A and terms (fuzzy sets) a1 and a2 for the fuzzy output variable U:

${\ displaystyle \ mu _ {u} (U) = \ min \ {\ mu _ {a1} (A), \ mu _ {a2} (A) \}}$

The minimum operator ensures that the implication cannot become greater than the evaluation of the premises. If, for example, the premise result is fulfilled with the degree of membership μ a1 (A) = 0.5, then the conclusion should also have the degree of membership μ u2 (U) = 0.5.

The membership function μ {a1 → a2} (A) is formed by the min operator simply by forming the minimum of the two degrees of fulfillment according to the above equation.

Example of a working rule with one input variable and one output variable:

${\ displaystyle {\ text {rule}} \ colon \ quad \ underbrace {\ IF \ quad \ underbrace {{\ text {Internal temperature is cool}} \} _ {\ text {Premise (precondition)}} \ quad THEN \ quad \ underbrace {\ {{\ text {Heating on medium}} \}} _ {\ text {Conclusion (conclusion)}}} _ {\ text {Implication (fuzzy rule)}}}$

The following statement applies to the logical connection of a working rule with symbols A and U for an input and an output variable:

${\ displaystyle {\ text {rule}} \ colon \ quad \ underbrace {\ IF \ quad \ underbrace {\ A = a1 \} _ {\ text {premise}} \ quad THEN \ quad \ underbrace {\ {U = u2 \}} _ {\ text {Conclusion}}} _ {\ text {Implication}} \ qquad {\ begin {cases} A = {\ text {input variable}} \\ U = {\ text {output variable}} \ \ {\ text {a1 and u2 are terms (fuzzy sets)}} \\\ end {cases}}}$

The input and output variables each refer to a certain basic amount. The terms a1 and u2 belonging to the variables A and U are fuzzy subsets according to the working rule.

For this rule, the premise has the property of A for a certain degree of membership μ a1 (A). The conclusion has the property of U for a certain degree of membership μ u2 (U).

According to the basic idea of ​​Mamdani (Mamdami implication), the truth content μ u2 of the conclusion must not be greater than the truth content μ a1 of the premise. Depending on the type of work rules, the min or max operators apply to linking degrees of fulfillment of the same basic quantities.

IF-THEN rule with several basic sets

Relationships between different basic sets are generally described as fuzzy relations.

Example of a rule with two input variables and one output variable:

${\ displaystyle {\ text {rule}} \ colon \ quad \ underbrace {{\ text {IF inside temperature cool}} \ quad AND \ quad {\ text {outside temperature cold}}} _ {\ text {IF part with 2 Premises}} \ quad \ underbrace {THEN \ quad {\ text {Heating max}}} _ {\ text {THEN part with conclusion}}}$

Example: Definition of a fuzzy rule with two linguistic input variables A and B and their terms a 1 ... a n and b 1 ... b n and a linguistic output variable U and terms u 1 ... u n :

${\ displaystyle {\ text {rule}} \ colon \ quad \ underbrace {\ IF \ quad \ underbrace {\ A = a2 \ quad AND \ quad B = b1 \} _ {\ {\ text {2 partial premises} }} \ quad THEN \ quad \ underbrace {\ {U = u3 \}} _ {\ text {Conclusion}}} _ {\ text {Implication}}}$

If the basic sets A and B of the partial premises are different, only the responses from fuzzy sets are linked to specific input values.

Fuzzy controller with two input variables for the basic quantities of temperature and pressure with three fuzzy sets each and one output variable U with four fuzzy sets

According to this rule, two fuzzy input variables must be processed with the variables A and B in AND operation in order to determine the degree of fulfillment of the output variables (conclusion) μ u (U). The Mamdani guideline already mentioned: "The truth content of the conclusion must not be greater than that of the premise" requires a small modification (i.e. cutting off, limiting) of the processing rule presented. The result fuzzy set of the conclusion is thus cut off (limited) at the level of the minimum degree of fulfillment of the AND-linked premises with the min operator.

The complete description of the problem to control a process with the working rules of the rule base can lead to a considerable number of rules, because there is an exponential relationship between the number of rules and the variables with the associated terms as follows:

${\ displaystyle {\ text {number of rules}} = {\ text {(number of terms)}} ^ {\ text {(number of variables)}}}$

After that, with 3 fuzzy input variables, each with 7 of the maximum recommended number of terms, you would arrive at a maximum number of 343 rules.

Therefore, the recommendation for setting up the rule base applies: The number of variables and the associated terms should be as low as possible because, especially when calculating the degree of fulfillment and linking variables with the fuzzy rules, a lot of computing work on a microcomputer - and not least one large programming effort - mean.

In the IF part of a rule, the terms of various fuzzy variables are linked with the fuzzy operators. The IF part can contain any logical combination with AND and OR operators of terms of different variables. The THEN part with the conclusion is usually a simple assignment of a linguistic value to a fuzzy output variable.

The AND connection of the fuzzy statement of the premise with the rule mentioned represents a two-digit fuzzy relation. The fuzzy relation results from the application of the min operator:

${\ displaystyle \ mu _ {R} (A, B) = \ min \ {\ mu _ {a1} (A), \ mu _ {b1} (B) \}}$

The connection of a rule with partial premises can also occur as an OR connection.

Example: partial premises in the OR link of a rule:

${\ displaystyle {\ text {rule}} \ colon \ quad \ IF \ quad \ underbrace {\ A = a1 \ quad OR \ quad B = b1 \} _ {\ {\ text {2 partial premises}}} \ quad THEN \ quad \ underbrace {\ {U = u2 \}} _ {\ text {conclusion}}}$

The OR connection of the fuzzy statement of the premise of this rule also represents a two-digit fuzzy relation. The fuzzy relation results from the use of the Max operator:

${\ displaystyle \ mu _ {R} (A, B) = \ max \ {\ mu _ {a1} (A), \ mu _ {b1} (B) \}}$

This rule of the OR link can also be split into two simple rules in the AND link:

{\ displaystyle {\ begin {aligned} {\ text {rule 1}} \ colon \ quad & IF \ quad A = a1 \ quad THEN \ quad U = u2 \\ {\ text {rule 2}} \ colon \ quad & IF \ quad B = b2 \ quad THEN \ quad U = u2 \ end {aligned}}}
Fuzzy controller with two input variables e1, e2 and singletons in the output variable.

Sub-steps of inference:

The inference unit of fuzzy controllers forms the linguistic conclusion with the fuzzy terms of the fuzzy variables (fuzzified input signals) via the linguistic rules (rule base) and consists of several inference processing sub-steps. The application of each active rule provides the resulting output fuzzy set on the basis of the inference scheme by transferring the degree of fulfillment of the rule to the respective fuzzy set of the conclusion.

The degree of fulfillment is defined in such a way that with the fuzzy AND link it is as large as the smallest degree of membership of the input variables. Similarly, the degree of fulfillment in a fuzzy OR link is defined in such a way that it is as large as the greatest degree of membership of the input variables. Fuzzy sets indicate the corresponding degree of fulfillment of a fuzzy logical statement for each sharp value of an input variable. You set up a "membership function".

The aim of evaluating the premise is to determine the degree to which each rule belongs. Both the premise and the conclusion are defined as fuzzy sets.

Let the variables A with the terms a 1 , a 2 to a i and B with the terms b 1 , b 2 to b i be given:

Depending on a sharp input signal, a certain fuzzy set a i of a fuzzy variable A will respond with a certain affiliation μ a . However, two or more neighboring fuzzy sets such as a 1 and a 2 can also address at the same time. This addressing of the rules is also referred to as “firing”. It depends entirely on the size of the sharp input signal e i which and how many fuzzy sets fire.

In multi-variable systems, for example with input variables A and B, several fuzzy sets can fire simultaneously depending on a sharp input signal e 1 of a basic set for A and another sharp input signal e 2 of a different basic set for B.

With the implication, the active (firing) fuzzy sets of the premise - depending on the AND or OR link - for each rule with the min or max operator have a limiting effect on the graphic model of the conclusion and the fuzzy set belonging to the rule a (triangular fuzzy set becomes trapezoidal).

Addressing several rules results in graphic partial fuzzy models of the output variable, which are summarized via the accumulation using the max operators.

### Defuzzification

The inference method supplies a fuzzy set of a linguistic output variable that results from the union of the individual output fuzzy sets. Various methods are known for selecting a sharp value from a fuzzy set.

• Maximum method (mean of maxima)
With this method and its variants, the precise manipulated variable sought is formed from the maximum of the accumulated membership functions.
Disadvantage: Only rules of the rule base that generate the maximum are taken into account. Depending on the size of the sharp input signals, jumps in the determined sharp manipulated variable are possible.
• Center of Gravity method
This method forms the center of gravity of the envelope curve of the accumulated membership functions μ (U) for the scaled basic set on the abscissa. From the graphical surface form of the membership functions, a numerical value is calculated as the abscissa value by forming the area center of gravity, which corresponds to the manipulated variable U S (indexing s for center of gravity method). Continuous changes in the sharp input variables e 1 to e n produce continuously sharp manipulated variables U S when this method is used .

In fuzzy controllers, methods for determining the area center of gravity U S are predominantly used for defuzzification . The relationship between the degree of membership μ (U) of the accumulated output variable U and the values ​​of the abscissa from the initial value U A to the end value U E applies .

Defuzzification according to the area center of gravity method with numerical calculation.
• The exact calculation for determining the center of gravity of an area U S is:
${\ displaystyle U_ {S} = {\ frac {\ int \ limits _ {U_ {A}} ^ {U_ {E}} U \ cdot \ mu (U) dU} {\ int \ limits _ {U_ {A }} ^ {U_ {E}} \ mu (U) dU}} \ qquad {\ begin {cases} U = {\ text {output variable}} \\ U_ {A} = {\ text {start of area for}} \ \ mu (U) = 0 \\ U_ {E} = {\ text {end of area for}} \ \ mu (U) = 0 \ end {cases}}}$
• Method of numerical calculation of the centroid of an arbitrary function μ (U):
The method of numerical calculation of the area's center of gravity is complex. Therefore, simplified methods are used in order to reduce the amount of computational work and memory requirements of a microcomputer.
${\ displaystyle U_ {S} \ approx {\ frac {\ sum \ limits _ {i = 1} ^ {n} \ Delta U_ {i} \ cdot \ mu _ {i}} {\ sum \ limits _ {i = 1} ^ {n} \ mu _ {i}}} \ qquad {\ begin {cases} \ Delta U_ {i} = {\ text {Values ​​of uniform distances on the U-axis of}} \ U_ {A} \ to \ U_ {E}, \\\ mu _ {i} = {\ text {Values ​​of membership (line functions) on the envelope,}} \\ n = {\ text {Number of}} \ \ Delta y_ { i} {\ text {values ​​(support points) from}} \ U_ {A} \ to \ U_ {E}. \\\ end {cases}}}$
From the example in the graphic opposite, U i = 120 ΔU elements were used for numerical calculation for the entered data . This results in the center of gravity with U S on the abscissa axis as a reference value:
${\ displaystyle U_ {S} \ approx {\ frac {\ Delta U_ {1} \ cdot \ mu _ {1} + \ Delta U_ {2} \ cdot \ mu _ {2} + \ cdots + \ Delta U_ { 120} \ cdot \ mu _ {120}} {\ mu _ {1} + \ mu _ {2} + \ cdots + \ mu _ {120}}} = {\ underline {\ underline {22.43}} }}$
• Simplification of the numerical calculation through a minimal number of support points
By means of the numerical calculation of the center of gravity as a sharp manipulated variable U S , relatively precise centers of gravity can be achieved even with a few U i values:
Example according to the data in the graphic opposite with 7 support points:
${\ displaystyle U_ {S} \ approx {\ frac {\ Delta U_ {1} \ cdot \ mu _ {1} + \ Delta U_ {2} \ cdot \ mu _ {2} + \ cdots + \ Delta U_ { 7} \ cdot \ mu _ {7}} {\ mu _ {1} + \ mu _ {2} + \ cdots + \ mu _ {7}}}}$
${\ displaystyle U_ {S} \ approx {\ frac {8 \ times 0 + 12 \ times 0 {,} 15 + 16 \ times 0 {,} 15 + 20 \ times 0 {,} 5 + 24 \ times 0 { ,} 75 + 28 \ cdot 0 {,} 5 + 32 \ cdot 0} {0 + 0 {,} 15 + 0 {,} 15 + 0 {,} 5 + 0 {,} 75 + 0 {,} 5 +0}} = {\ underline {\ underline {22 {,} 54}}}}$
The deviation of the calculation with 7 interpolation points from the relatively precise reference value is approx. 1% in this example and, with this simplification, is relatively precise; the deviations can be greater with other curve shapes.
• Simplification of the area center of gravity calculation with line functions (singletons)
Defuzzification according to the simplified area center of gravity method with singletons
A further simplification of a simple calculation as an approximation to the calculation of the area's center of gravity results when the symmetrical trapezoidal individual areas of the total area are defined by the center of gravity on the y-axis and the area height with μ i as values ​​of the membership function. This is based on rectangular elements. The height of the rectangles corresponds to the degree of fulfillment μ of the respective rule. With this approximation, the overlap of neighboring triangular-shaped fuzzy sets is taken into account too much, which leads to a small error.
Approximation of the calculation of the center of gravity through the moments of the trapezoidal individual surfaces:
${\ displaystyle U_ {S} \ approx {\ frac {\ sum \ limits _ {i = 1} ^ {n} U_ {si} \ cdot \ mu _ {i}} {\ sum \ limits _ {i = 1 } ^ {n} \ mu _ {i}}} = {\ frac {U_ {s1} \ cdot \ mu _ {1} + U_ {s2} \ cdot \ mu _ {2} + \ cdots} {\ mu _ {1} + \ mu _ {2} + \ cdots}} \ qquad {\ begin {cases} U_ {si} = {\ text {Center of gravity of the individual area}} \\\ mu _ {i} = {\ text {Trapezoidal height of the single area}} \\ n = {\ text {number of trapezoidal single areas}} \\\ end {cases}}}$
According to the graphic example with the 2 drawn line functions (singletons), the calculation of the center of gravity (trapezoidal height method) results as follows:
${\ displaystyle U_ {S} \ approx {\ frac {16 \ times 0 {,} 15 + 24 \ times 0 {,} 75} {0 {,} 15 + 0 {,} 75}} = {\ underline { \ underline {22 {,} 67}}}}$
This result means a deviation from the reference value 22.43 of approx. 1% and happens to be still a very good result with a simple calculation method.
This equation also corresponds to the application of the centroid method with line functions (singletons). Analytical calculations on the transfer behavior, in particular the use of the fuzzy controller as a controller in multi-variable systems, have shown that the simplification through fuzzy set as singletons can save expensive computing time without significantly impairing the transfer behavior of the fuzzy controller.

### Pros and cons of fuzzy controllers

Per

• The fuzzy controller as a map controller masters non-linear dynamic processes.
• A mathematical model of the process to be controlled or regulated is not required.
• A fuzzy model of the process is analyzed through observation and experience (plant operator, expert knowledge) from the process.
• Using linguistic terms for process states (“warm”, “cold”, “valve open”), production rules are created to control the process.
• As a single variable system, the size of the overlap of the fuzzy sets of a fuzzy variable can be used to model an input-output characteristic of linear, continuously non-linear, with dead zone and multi-point switching behavior.
• Fuzzy controllers can also be understood by interested laypeople.
• Robustness: Parameter deviations within the process and certain limits are tolerated without harmful effects.
• With some restrictions, the failure of individual signal generators cannot disrupt the entire process.
• Hardware and software costs can be minimized by using fuzzy chips.

Contra

• The fuzzy controller is a static system and is largely inferior to a linear standard controller for precise control due to the lack of dynamic components .
• To expand the fuzzy controller into a dynamic controller, integral or differential components of measured variables must be prepared as sharp input variables for a desired dynamic.
• The stability check of an exact regulation is difficult. If a mathematical model of the controlled system is available, a stability test can be carried out by means of numerical calculation.
• Depending on the number of variables (signal inputs) and the number of fuzzy sets used per variable and their exponential relationships to the number of production rules, the fuzzy rule base can grow confusing.
• The fuzzy controller cannot be used universally in control engineering, but mostly refers to the application of the control of a non-linear multivariable process that is difficult to describe mathematically.

## Fuzzy controller and fuzzy regulation

### Conventional regulations

Conventional control applications with linear control systems can be solved satisfactorily with standard controllers such as P , PI , PD and PID controllers. Because the P controller is a static controller and therefore cannot compensate for any delay components in the controlled system, it is rarely used. Its P gain must remain low for higher-order controlled systems, otherwise the control loop will become unstable. Low P-gain means a large control deviation.

Various linearization methods are known for non-linear controlled systems, in which a model of the controlled system is broken down into a static non-linearity and a dynamic, linear transmission system. The simplest controller design is for a demanding control using the simulation of the control loop with the numerical calculation over the discretized time.

### Fuzzy control

Block diagram of a single-loop control loop with a dynamic fuzzy controller

The typical application of the fuzzy controller are mostly non-linear multi-variable systems in which the controlled variables have to be fed back as input variables.

Single-variable fuzzy controllers have a typical non-linear input-output variable characteristic. Due to the different arrangement of the fuzzy sets (fuzzification) and the design of the rule base (inference), any basic behavior of the input-output characteristic can be defined, such as linear, non-linear, with dead zone and multi-point switching behavior. They have no dynamic components. As single-variable systems, they are therefore comparable in their behavior to a proportional controller (P controller), which is given any non-linear characteristic curve depending on the linguistic variables and their evaluation.

A fuzzy controller can be used as a static single variable controller directly for an unknown controlled system if the output variable of the process is fed back into the controller as a controlled variable. It is a static controller with a non-linear input-output characteristic curve that can be defined within wide limits. Without knowledge of the mathematical model of the controlled system, its application is limited, as it cannot compensate for any delaying components of the controlled system. Controlled systems with global I behavior can, however, be successfully controlled with a fuzzy controller.

The static fuzzy controller can be expanded with D and I behavior, but does not have any functional advantage over a conventional PID controller when a linear controlled system is specified. The mathematical model of the controlled system is required for optimal use in a control loop.

A fixed setpoint or the deviation from a fixed setpoint can be entered into a fuzzy controller with an appropriate design of the rule base, whereby a control loop is created in connection with the controlled system and the fed back controlled variable.

The actual application of the fuzzy controller as a map controller applies to the process of a non-linear multi-variable system with several input and output variables, the mathematical model of which is unknown or difficult to describe. Unfortunately, a mathematical fuzzy controller design strategy as in the case of linear systems does not (yet) exist.

### Fuzzy controller according to Mamdani and Sugeno

The basic structure of the fuzzy controller differs in two different concepts:

• Rational fuzzy controller according to Mamdani.
The controller structure consists of the following procedures: Fuzzification of sharp input signals in linguistic quantities, inference with the establishment of the rule base and defuzzification of the manipulated variable, usually with the area-center-of-mass method, into sharp values.
The controller structure of the premise is identical to the rational fuzzy controller. The differences between the Mamdani type and the Sugeno type are the rules of the rule base and the defuzzification of the precise manipulated variable. The sharp conclusions of the manipulated variable values ​​are made with the weighting of the rules of the rule base. The step of defuzzification from accumulated fuzzy sets does not exist. This is associated with a reduction in the mathematical complexity of the arithmetic operations without having to accept a significant deterioration in the exact data.
• Fuzzy controller type Mamdani with singletons
Fuzzy controllers of the Mamdani type with singletons of the output variables in the conclusion are clearer according to the graphical representation of the input and output variables, because the output variables and the input variables relate identically to the basic set. The output variable U with the terms u i as singletons lie within the basic set of the fuzzy manipulated variable on the abscissa, the degree of fulfillment μ αi shows the amplitude of the singletons.
The calculation of the manipulated variable with singletons of the Mamdani type is identical to the Sugeno type. The rules of the rule base of the two types of calculation are different. For the Sugeno type, the conclusion for each rule Ri relates to a product (factor k i to be specified ) * (input variable e i). This means that the manipulated variable range is to be determined by the factors k i.

#### Sugeno Takagi Kang regulator

(The order of the names is also carried out differently in the specialist literature, e.g. Takagi-Sugeno controller )

Symbol definition:
Variables: A,…, Z
terms; A = {a 1 , a 2 , ..., a n }; B = {b 1 , b 2 ,…, b n }
constants: k 1 , k 2 , k 3 as any numerical values> 0 within the stability range,
indexing: n = 1, 2,… ≪

One rule of the rule base for the Sugeno type is in the form of linguistic terms:

${\ displaystyle \ underbrace {{\ text {IF temperature is warm}} \ quad AND \ quad {\ text {pressure is medium}}} _ {\ text {IF A = ​​a AND B = b}} \ quad \ underbrace {THEN \ quad {\ text {valve = numerical value}}} _ {\ text {THEN U = f (e1, e2)}}}$

The if part of a rule of the rule base of the fuzzy controller is identical for the Mamdani type with the Sugeno type.

The THEN part of the Sugeno type differs from the THEN part of the Mamdani type in that no fuzzy set is determined in the conclusion, but from the sharp values ​​of the input variables e 1 ... e n are determined via the premise evaluation of the rules the corresponding degrees of fulfillment μ α are derived and weighted with constants k.

The calculation of the precise manipulated variable U S for a zero-order controller (= no dynamic components) of the Sugeno type is as follows:

${\ displaystyle U_ {S} = {\ frac {\ sum \ limits _ {i = 1} ^ {r} U_ {i} \ cdot \ mu _ {\ alpha i}} {\ sum \ limits _ {i = 1} ^ {r} \ mu _ {\ alpha i}}} = {\ frac {U_ {1} \ cdot \ mu _ {\ alpha 1} + U_ {2} \ cdot \ mu _ {\ alpha 2} + \ cdots} {\ mu _ {\ alpha 1} + \ mu _ {\ alpha 2} + \ cdots}} \ qquad {\ begin {cases} U_ {i} = {\ text {const. Manipulated variable value of the associated rule}} \\\ mu _ {\ alpha i} = {\ text {Degree of fulfillment according to the min operator}} \\ i = {\ text {1 ... r}}, \ {\ text {indexing of rules and degrees of fulfillment}} \\\ end {cases}}}$
Three fuzzy sets of a fuzzy controller with one input variable.

### Calculation example of a static controller with symmetrical non-linear characteristic

#### Use of a fuzzy controller

A non-linear static controller whose transfer characteristic has a lower gain in the vicinity of the control deviation than in the rest of the working range can control a controlled system with global I behavior better than a P controller.

Data:

• The controller has an input working range (basic quantity) for a control deviation e of ± 1.
• The 3 fuzzy sets are arranged symmetrically with the same gradients overlapping over the area of ​​the basic set.
• The manipulated variables U are weighted with k1 = 2.5, k2 = 0.4, k3 = 2.5.
• The degree of fulfillment μ αi result from the rules.
Establishing the rule base for a basic set with 3 linguistic subsets (terms):
{\ displaystyle {\ begin {aligned} {\ text {rule 1}} \ colon \ quad & IF \ quad e = {\ text {less than zero}} \ quad THEN \ quad U1 = 2 {,} 5 \ cdot e \ \ {\ text {rule 2}} \ colon \ quad & IF \ quad e = {\ text {close to zero}} \ qquad THEN \ quad U2 = 0 {,} 4 \ cdot e \\ {\ text {rule 3} } \ colon \ quad & IF \ quad e = {\ text {greater than zero}} \ quad THEN \ quad U3 = 2 {,} 5 \ cdot e \ end {aligned}}}
Input-output characteristic of a fuzzy controller type Sugeno

In the adjacent graphic of the controller fuzzy model of the terms, a value of the sharp input variable e = −0.8 is drawn. The terms a1 and a2 are activated. This results in the following degrees of membership for the basic set A:

${\ displaystyle \ mu _ {a} (A) = \ {\ mu _ {a1} = 0 {,} 8; \ \ mu _ {a2} = 0 {,} 2; \ \ mu _ {a3} = 0 \} (A)}$

The following equation applies to the calculation of the manipulated variable U S for the value of the input variable e = −0.8:

${\ displaystyle U_ {S} = {\ frac {U_ {1} \ cdot \ mu _ {\ alpha 1} + U_ {2} \ cdot \ mu _ {\ alpha 2}} {\ mu _ {\ alpha 1 } + \ mu _ {\ alpha 2}}} = {\ frac {e \ cdot k1 \ cdot \ mu _ {\ alpha 1} + e \ cdot k2 \ cdot \ mu _ {\ alpha 2}} {\ mu _ {\ alpha 1} + \ mu _ {\ alpha 2}}} = {\ frac {-0 {,} 8 \ cdot 2 {,} 5 \ cdot 0 {,} 8-0 {,} 8 \ cdot 0 {,} 4 \ cdot 0 {,} 2} {0 {,} 8 + 0 {,} 2}} = - 1 {,} 664}$

The aim of this calculation of the manipulated variable U S is to determine the representation of the transfer characteristic of the non-linear controller U S = f (e). This can be done in the form of a table by specifying data in the range of the basic quantity from −1 to 1 for the input variable e. The degrees of membership for each sharp value of the input variable are to be calculated using the straight line equations of the ramps.

The graphical representation of the characteristic shows a continuous non-linear course. Three identical evaluation factors k1 = k2 = k3 = 2.5 would result in a completely linear input-output characteristic within the limits established by the fuzzy sets.

The behavior of this controller on a linear controlled system with global I behavior is examined for the setpoint step change W (t) = 4 (= 100%) and a disturbance variable d (t) = −2 (= -50%).

Controlled system:

${\ displaystyle G (s) = {\ frac {(0 {,} 5 \ cdot s + 1) \ cdot (s + 1)} {3 \ cdot s}}}$

Fuzzy controller: Static single variable controller with three fuzzy sets according to the graphical fuzzy model, type Sugeno.

Result:

Setpoint jump with a Takagi-Sugeno controller on a linear controlled system with global I-behavior and a static disturbance variable.

Behavior of the non-linear controller according to the graph of the input-output characteristic shown:

• The speed at which the controlled variable increases until the setpoint is reached depends on the gain and the size of the limitation of the manipulated variable U.
• The decreasing controller gain around the range of approx. E = −0.2 ... 0.2 leads to a weaker and thus better damping of the transient behavior. This becomes all the more clear if the evaluation factors of the input variable e (t) were increased with k1 and k3.
• The point of attack of the disturbance variable d (t) = −2 occurs after the I element, before the two delay elements. The course of the controlled variable collapses temporarily by the value Δy = 1.3.

Conclusion:

• The use of the Mamdani fuzzy controller with singletons is clearer and easier to understand than the Sugeno method. No evaluation factors are required, but the manipulated variable range is determined with the basic set of output variables.
• Fuzzy controllers as static single variable controllers on a linear controlled system are unsuitable because the planning and programming effort is high. As a static controller, its use on linear controlled systems is very limited. No delay components of the controlled system can be compensated.
• Fuzzy controllers as static single variable controllers can be useful on a non-linear controlled system if the continuous course of the non-linearity of the system is to be compensated.
• Dynamic fuzzy controllers with PI, PD or PID behavior on linear control systems are more complex, but not better, than comparable linear standard controllers .
• Dynamic fuzzy controllers on unknown non-linear controlled systems such as For example, the “ inverse pendulum ” model, which is often used for study purposes, can be useful if the exact system model is mathematically difficult to grasp.

#### Realization of a fuzzy controller or fuzzy regulator

• The technical implementation of the fuzzy controller or a fuzzy controller as hardware requires a programmable microcomputer ( CPU ), because geometric routes (ramps) and logical functions have to be calculated.
• Fuzzy logic is also available commercially for programmable logic controllers .

## Individual evidence

1. With the search term "you tube inverted pendulum" on Google you can find interesting video sequences about the working methods of realized mechanical fuzzy models. See also the mechanical model of an upright double pendulum with chaotic behavior of the controlled system
2. Schirotzek / Scholz: Mathematics for engineers and natural scientists: Chapter: Basic concepts of set theory.
3. Prof. Ebrahim Mamdani, Imperial College of Science, Technology and Medicine, University of London, Electrical and Electronic Engineering.
E. Mamdani successfully applied the theory of fuzzy logic according to L. Zadeh to control a steam engine in the 1970s and thus paved the way for subsequent applications.
4. After reviewing about 20 lecture manuscripts from the environment of German universities with regard to the definition of the term "degree of fulfillment", some manuscripts set [degree of fulfillment = degree of membership] and others define analogously [degree of fulfillment = resulting degree of membership of linked fuzzy sets], which makes sense.
5. Holger Lutz, Wolfgang Wendt / pocket book of control engineering; Chapter: Application of Fuzzy Logic in Control Engineering; Subchapter: Fuzzy Relations
6. Gerd Schulz / Control Engineering 2: Chapter: Fuzzy logical reasoning; Subchapter: Fuzzy Relations
7. Holger Lutz, Wolfgang Wendt / Pocket Book of Control Engineering: Chapter: Fuzzy Regulations and Controls (Fuzzy Control)
8. Holger Lutz, Wolfgang Wendt / pocket book of control engineering; Chapter: Application of Fuzzy Logic in Control Engineering, Subchapter: Defuzzification.
9. Gerd Schulz / Control technology 2, multi-variable control, digital control technology, fuzzy control: Chapter: Basics of fuzzy quantities; Subchapter: Linguistic Variables and Terms - Fuzzification.
10. Holger Lutz, Wolfgang Wendt / pocket book of control engineering; Chapter: Fuzzy Regulation and Control; Sub-steps of the inference procedure
11. Gerd Schulz / Control technology 2, multi-variable control, digital control technology, fuzzy control; Chapter: Fuzzy logical reasoning; Subchapter: Fuzzy Inference
12. Holger Lutz, Wolfgang Wendt / pocket book of control engineering; Chapter: Application of Fuzzy Logic in Control Engineering, Subchapter: Defuzzification.
13. Gerd Schulz / Control technology 2, multi-variable control, digital control technology, fuzzy control; Chapter: Fuzzy logical reasoning; Subchapter: Defuzzification.
14. Gerd Schulz: Control engineering 2, multivariable control, digital control engineering, fuzzy control. Chapter: Basics of Fuzzy Control; Subchapter: Characteristic curves of fuzzy controllers.
15. Prof. Ebrahim Mamdani, Imperial College of Science, Technology and Medicine, University of London, Electrical and Electronic Engineering
16. One of the early fuzzy control applications by Dr. Sugeno was the automatic control of a small car with fuzzy systems with the Takagi-Sugeno model. The groundbreaking work had a tremendous impact on fuzzy control research and influenced applications in household appliances, automotive and process controls
17. Siemens fuzzy shell PROFUZZY. This software requires the target hardware of the Simatic S5 type.

## literature

• Winfried Schirotzek, Siegfried Scholz: Mathematics for engineers and scientists . 3. Edition. BG Teubner, Stuttgart / Leipzig 1999, ISBN 3-519-00271-X .
• Gerd Schulz: Control technology 2, multivariable control, digital control technology, fuzzy control . 2nd Edition. Oldenbourg Verlag Munich Vienna, Munich 2002, ISBN 3-486-58318-2 .
• Lefteri H. Tsoukalas, Robert E. Uhrig: Fuzzy and Neural Approaches in Engineering. Wiley-Interscience, 1996, ISBN 0-471-16003-2 .
• Holger Lutz, Wolfgang Wendt: Pocket book of control engineering: with MATLAB and Simulink. 11th edition. Verlag Europa-Lehrmittel, 2019, ISBN = 978-3-8085-5869-0.