Fuzzy control system

The fuzzy set theory and fuzzy set theory was established in 1965, developed by Lotfi Zadeh at the University of California, Berkeley. In Japan, the fuzzy set theory ( fuzzy logic) has been successfully used as a fuzzy controller ( fuzzy controller ) in the 1980s in industrial processes and held only in the 1990s in Europe with numerous applications feeder.

The principle of the fuzzy controller is to capture sharp physical input signals of a technical process and to evaluate by means of linguistic terms from the expert knowledge of membership functions and logical if-then - operations and to make the transition from linguistic variables to sharp manipulated variables. Here, with the help of graphical usually triangular fuzzy set models of the fuzzy sets are the standard methods of " fuzzification ", " inference with rule base " and " defuzzification " apply.

The basic idea of the fuzzy controller refers to the integration of expert knowledge with linguistic terms, by the fuzzy controller is more or less modeled with empirical methodology optimally to a non-linear process with multiple inputs and outputs, without the mathematical model of the process present.

The typical application of the fuzzy controller is due to the lack of dynamic properties of the rule base is preferably a method of controlling a technical process. Therefore, the adaptation of the fuzzy controller with the expert knowledge of a known process with no mathematical model is also relatively unproblematic.

Fuzzy controllers can be equally interpreted in conjunction with an industrial process as a controller or control unit.

Exact arrangements with the static control performance of the fuzzy controller are possible only in limited use cases of non -linear processes species. The basic functional mechanism of the scheme is based on feedback of the controlled variable and input the inverted setpoint-actual value comparison in the controller. It is at the discretion of whether the fuzzy controller can be classified as a nonlinear controller.

Fuzzy controller are based on the method of fuzzy controller, but are mostly functional modifications, simplifications or additions to the fuzzy logic. Optimal rules can be achieved with integrals and differential proportions of the input signals with the help of empirical settings by extensions of the fuzzy controller. You have to compare it in use as a regulator of linear SISO control loops no functional advantages over the traditional PID controllers.

4.1 disambiguation
4.2 fuzzification
4.3 Inference ( data base, rule base )
4.4 defuzzification
4.5 Pros and Cons fuzzy controller

5.1 Conventional schemes
5.2 Fuzzy Control
5.3 Fuzzy controller of Mamdani and Sugeno 5.3.1 Sugeno - Takagi - sound controller

5.4.1 Application of a fuzzy controller
5.4.2 Implementation of a fuzzy controller and fuzzy controller

Application of the fuzzy controller

A person is able to ensure, by means of fuzzy process information about specific interventions ( control variables ) in the process flow optimal process control. It can generate on the bike through targeted steering motions speed-sensitive adjusting centrifugal forces which maintain thanks to the human vestibular rolling bicycle, for example, according to a learning procedure for the balance.

The on the fuzzy logic (fuzzy: out of focus, blurred ) based fuzzy controllers operate with the help of expert knowledge in a similar way. Here, too, the input variables for a fuzzy knowledge control algorithm is processed at a defined control value.

Fuzzy controllers are generally used in non-linear multivariable systems which have the following features like:

Processes whose mathematical models to describe time-consuming or difficult,
Processes with conventional methods that require corrective of human intervention ( plant operator ),
If a process can only be moved manually.

The objective of the use of the fuzzy controller is to automate such processes. Applications of the fuzzy controller can be found in all sectors of industry to consumer goods such as:

Control of railway vehicles or storage and retrieval systems in which travel times, distances and position accuracies depend on the masses, conveying paths, rail adhesion values and schedules. In general, it is in these processes to multivariable systems whose control variables are program - controlled and regulated.
In the automotive industry, the control of the automatic transmission is operated with the fuzzy logic successfully.
Simpler applications in private households can be found in washing machines and dishwashers.
Controls in cameras,

In colleges typical mechanical fuzzy demonstration model of a control of an inverted pendulum with one degree of freedom:

Linguistic variables and fuzzy sets

Fuzzy sets

In classical set theory is one of an item to an amount within certain limits. If x is a subset of the ground set X so x ∈ X, then the membership μ of this element of the crowd is always 1 log or Log 0 or true or not true.

This sharp separation is removed in the theory of fuzzy sets and replaced by so-called fuzzy sets as fuzzy partial quantities. It allows the continuous signal technical processing of linguistic terms by the membership degree μ with intermediate values within the membership of 0 and 1

A fuzzy partial amount of x1 ... xn ( fuzzy set, membership function ) of a base set X is defined by their affiliation ( = Membership = membership) μ (X ) is defined. The membership is not decided with true or false, but for example true for a given sharp input signal e ( X) the affiliation of the variable X (basic amount ) to one of the terms x1 ... xn ( = partial quantities, = membership functions ) with the membership degree μ (x ) = 0.3 ( ≡ 30 %) of.

Fuzzy subsets - the graphical representation of the Gaussian bell curve for ease of predictability, inspired - mostly triangular or trapezoidal functions. They are part of a basic amount and X can be represented as a graphical model in a coordinate system. The abscissa contains the physical dimension of the scaled base set within which the subsets are usually divided as fuzzy sets with their bases overlapping. The ordinate indicates all subsets ( fuzzy sets ) by the membership degree μ. All maxima of the fuzzy sets are = associated with the degree of membership μ (X) 1 ≡ 100 %.

The fuzzy subsets of the universal set X (eg heat in the dimension of temperature) are functions (eg, cold, warm, hot) in the form of linguistic terms or can be described with symbols x1, x2, x3.

There are various terms of the functions of the fuzzy subsets that mean all the same thing: membership function, fuzzy set, fuzzy membership, term of a fuzzy variable, linguistic Term

The membership functions ( fuzzy sets ) x1 ... xn (n = maximum number ) show the progression of the fuzzy subsets; they are symmetrically disposed within the base amount usually, but need not, and can also have different gradients.

The degree of membership μ (X) corresponds to a value that cuts through a certain sharp value of the input variable e within the basic set X the function of a fuzzy sets or more fuzzy sets. Depending on how much fuzzy sets contains a basic quantity as often as the basic set of performance level μ (X ) = 0 is reached to 1 depending on a value. So it depends on the value of e within the basic set X from which fuzzy set and how many fuzzy sets can be activated ( "fire" ) be and what degrees of membership arising from it.

The fuzzy fuzzy logic allows the description of non-linear relationships between input and output variables by means of linguistic terms and work rules ( rule base ) that can be realized by expert as follows.

IF - THEN rule with an input variable

IF - THEN rule with two input variables and AND operation

Colloquially are about events or transactions terms of the linguistic variables common which allow no clear affiliation to a lot of how warm, large, thick, close, short, old, etc. These terms of quantities can by the fuzzy sets, for example, in define a technical process, if expert knowledge is present.

Linguistic variables of the fuzzy controller

A fuzzy variable refers to the basic amount of a scaled physical quantity whose terms represent fuzzy sets and graphically in a coordinate system - is shown - as already explained.

A graphical model of the signal flows of the fuzzy controller comprises a plurality of input and output variables which are linked to one another by a plurality of work rules of the rule base. The sharp input signals of this model are fuzzifiert and the processed fuzzy outputs are defuzzified to control values.

Fuzzy variables are usually physical quantities, such as the temperature, the linguistic terms as fuzzy sets as " very cold ", " cold", "warm ", "very warm" can be defined. Other fuzzy variables such as the " distance " to the linguistic terms such as " very close ", " close ", " far ", are "very far ". A linguistic term is defined as a fuzzy subset of a universal set.

Often abbreviations of standard terms are used for arbitrary physical quantities, such as " positive large", " positive medium", " positive small ", " close to zero ", " negative large ", " negative medium " negative small. These are already 7 terms that in a single-variable refer to an input-output characteristic curve that goes from positive to negative in the area through the origin of the coordinate system. If it is in the process to multivariable 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 ever a truth degree of membership between 0 and 1 These terms of variable temperature correspond to human emotions. Nevertheless, there is a certain arbitrariness in the definition and assignment of fuzzy sets with their bases on the basic amount of the variable " temperature". Here already the expert knowledge plays a role, because it is known in the linguistic term heat a difference whether it is " very warm " is the bath water temperature or the tea -water temperature during the Term.

Triangular and trapezoidal fuzzy sets are due to the simple predictability the most commonly used forms of fuzzy subset of fuzzy variables.

Singletons (line functions) correspond to sharp linguistic elements ( terms ), which only ever represent a physical value and are generally used for convenience in defuzzification. If x0 is a certain value of the universal set X, then it is only at the point x = x0 given for the membership degree of a non-zero value. For the conclusion of the THEN part of an IF - THEN rule can be used instead of triangular or trapezoidal fuzzy sets to simplify the computational evaluation of singletons. Details follow in the chapter inference.

For the description of a technical process of a physical signal size a number 2-7 fuzzy sets a basic amount are often sufficient.

Fuzzy sets are graphically arranged for better understanding in a coordinate system on an abscissa with sharp values of the physical size and for any basic quantities in the ordinate the degree of membership μ = 1 ≡ 100 % allocated. The trapezoidal or triangular shapes of the fuzzy sets are often - but not always - symmetrically designed and overlap within a certain range in the base points usually between 20% to 50 % on the abscissa. If no overlap exists in Eingrößensystemen to its original size in this particular area of the membership degree zero and also the input - output characteristic.

By choosing the position of the fuzzy sets - the size of the intersection (overlap ), the symmetry of the sets, the nature of the rule base - can at Eingrößensystemen the input-output characteristic of the fuzzy controller in a wide range of linear and continuous non-linear and broken non-linear be designed.

The number of variables and the associated terms should be as low as possible, because they mean in particular the calculation of the degree of fulfillment and relationships of variables with the fuzzy rules a lot of computation on a micro computer.

Definition of indexed fuzzy variables and their symbols

If we restrict ourselves to the representations of fuzzy variables and common fuzzy operators, it is in the fuzzy logic a fairly simple process control technology and restricted control technology without exact knowledge of a dynamic multi-variable process applied as controlled system with success. This is mostly evident in the illustrated graphic function diagrams. However, the mathematical description of the technical functions of the sharp signal input signals up to the sharp signal outputs ( manipulated variables ) of the fuzzy controller is difficult.

The literature shows no consistent naming of fuzzy variables, subsets, basic quantities and variable symbols and their indexing. In particular, publications from academia can cause comprehension difficulties of fuzzy theory untrained interested in multiple indexing of fulfillment degrees, fuzzy sets and rules.

For easier understanding, the following simplifying the naming of variables and their indexing is defined:

Sharp technical control input and output signals

Fuzzy variables

Fuzzy terms ( fuzzy sets ) as subsets of the fuzzy variables

Membership μ of the fuzzy input variables

Membership μ as an output variable ( Mamdani method )

Degree of fulfillment and membership function of a fuzzy subset

Calculation of the fuzzy sets with linear equations

The graphical model of a fuzzy variable corresponds to using expert knowledge of a division of usually overlapping fuzzy sets on the ground 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, the ordinate. For the fuzzification of a sharp input variable e in the basic set of variables X, the degree of membership or the membership degrees of fuzzy sets with a certain accuracy can be easily graphically represent. Since it is in the fuzzy controller is a microcomputer can perform mathematical and logical operations, linear equations must be used for the calculations of the degree of fulfillment of triangular or trapezoidal fuzzy sets.

So it depends on the size of the sharp input signal within the basic quantity which fuzzy set and whether it meets the sloping ramp or sloping ramp, and whether overlapping fuzzy sets two fuzzy sets are taken.

In the illustrated in the graph fuzzy-set models with the terms of the basic parameters a, b , c, d and the degree of fulfillment μ (X ) = 1, the size equivalent to x of the input variable e The performance level μ (X) = 1 in depending on the size of the input signal as often met in the base set, as far as terms of a variable are assigned.

For a fuzzy variable with several fuzzy sets has the linear equations of the rise and fall are calculated for determining the degree of fulfillment as a function of the sharp input size and the position of the ramp. From the determination of the fuzzy output variable is in linking the degrees of satisfaction with other variables ( fuzzy part manipulated variables ).

For the calculation of the membership function μ (X) for a given sharp value of the input variable e on the abscissa scaled a basic amount of the following linear equations of the ramps are met:

Linear equation for the rise of the ramp:

Linear equation for the decay of the ramp:

Fuzzy operators

Operations with fuzzy sets are required if multiple linguistic statements of the IF - THEN rules must be linked. Besides fuzzy sets fuzzy relations are an important area of fuzzy set theory. The relations of the same basic quantities of adjacent fuzzy sets are described by fuzzy relations of the most common operators such as fuzzy OR, fuzzy AND and fuzzy NOT functions. These relationships have a different meaning than the Boolean algebra, since they must also have a reference to the fuzzy sets. The subdivision of the truth content in any number of intermediate steps between true and false is the human way of thinking closer than the two-valued logic.

The behavior of the fuzzy controller is determined by the so-called work rules of the rule base, which consist of an IF part and a THEN part. The combination of fuzzy subsets of equal or unequal basic quantities by means of fuzzy operators. The three most commonly used operators to combine fuzzified subsets are as follows:

AND operator

The OR operator

NOT function

Fuzzy relations

Relations are useful for describing correlations between different variables and attributes. A fuzzy relation is equivalent to a fuzzy set whose basic set is a Cartesian product of several basic quantities. Multi-character relations are relations between sharp or fuzzy sets on different basic quantities. A multi- ary relation is a subset of the Cartesian product set of basic quantities, a fuzzy relation is always a fuzzy set.

A 2-digit fuzzy relation is a picture of two basic quantities such as of two variables:

The graph on the " rule activation with unequal base levels " shows as a relational rule R is the relation of membership degrees according to the rule 1:

The terms an and bn of these variables A and B indicate how strongly they are related to each other. In a relation matrix for example with the variables A, which corresponds to a basic amount of the temperature and the variable B, which corresponds to a basic amount of pressure that can be the following relation of the terms, a2 = " warm" and form b2 = " print medium" whose mapping for a specific work rule of rule base is valid.

The subsets of the terms "warm" and "medium" are temperature and pressure shown as the ratio of the basic quantities. To determine the affiliations uR of the AND operation of the premises of the IF part of the min - operator is applied.

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

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

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

The scaled data of the basic amount of stress of variable B are { 4; 8; 12; 16; 20} (P)

The ANDing of terms via the min operator in accordance with Rule 1, according illustrated graphics of the fuzzy variables.

The result of the table are the rule- activated membership values uR of two fuzzy subsets a2 and b2, which were found on some of the values of the two scaled basic quantities.

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

The practical significance of this table is based on the determination of Zuhörigkeitsgrade the conclusion of the work rule. For the realization of a computational program for a fuzzy controller, the membership degrees of each fuzzy set with the discrete values of the basic quantities must be determined by linear equations. For the logic operations of the subsets, the min -max operators are. From the summary of the conclusion of the THEN part of each rule, the function of the fuzzy part - manipulated variable and the accumulation is determined, the fuzzy control variable.

Overview diagram of the fuzzy controller

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

The knowledge base contains all the expert knowledge that relates the basis of experience with linguistic terms and actions to control of a non-linear process. With their knowledge, laying the basis for implementing the sharp system input signals into fuzzy sets and formulate fuzzy rules of the rule base.

Elements ( terms, fuzzy sets ) of fuzzy sets are described by a pair of values , which consists of the sharp value of the fundamental quantity, for example the variable A and the membership degree microamps (A) exists. The sharp value of the fundamental quantity 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 control variables ) from the fuzzy input variables.

Origin of data for the design of the fuzzy controller:

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

Project planning strategy:

Specify model of the basic quantities of the inputs and outputs of the fuzzy controller.
Membership functions ( fuzzy sets ) within the basic quantity order.
Set up tax rules as a rule base.

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

Summary of Features of the fuzzy controller:

Graphical fuzzy model of a fuzzy variable

Fuzzification

Inference:

Defuzzification:

Construction of the fuzzy controller

Disambiguation

In the German literature, a number of foreign terms for the description of the fuzzy controller have established that not all will be used consistently.

The most important terms:

Rule base

Premise (Latin praemissa: = advance swift rate) = Prerequisite, acceptance.

Conclusion (Latin conclusio: inference, conclusion ).

Implication ( lat. implicare: = Ver - wrapping ) = relate answers

Modus ponens (Latin: mode: procedure, condition, manner and ponere: put, set )

Inference ( lat. infero: carrying in, conclude, close).

Aggregation (Latin aggregations: accumulation, association ) = Summary of data into larger units.

Accumulation (Latin accumulare: accumulation, accumulate ).

Fuzzification

In typical applications of the fuzzy controller is usually a non-linear process having a plurality of inputs for which there is the expert knowledge of a simplified model of the process.

The measuring range of a sharp physical input signal is scaled to an appropriate range of values of the basic amount of the linguistic variable (signal channel). The basic amount is usually divided overlapped by the expert knowledge into fuzzy subsets ( fuzzy sets). The maximum degree of membership for each fuzzy set corresponds to μ = 1 In multivariable systems, several measured variables and manipulated variables of the fuzzy controller may occur.

The individual steps are:

Definition of the terms of a fuzzy variable: fuzzy name, number, type of fuzzy sets and their bases.
Scaling the ramped relations of terms to the membership degree μ = 1 within the Messbrereiches the signal input size. The linguistic terms are each assigned with a subset of the range of the input variable in such a way that the entire measuring range with the selected number of terms is divided.

The fuzzy sets are often arranged symmetrically and overlap in the base points ( vertices ). Are typical overlap of 20% to 50%.

The start and end of the whole fuzzy set parts are generally trapezoidal fuzzy sets ( indifference area) used to achieve boundary effects.

The arrangement of the fuzzy sets on the range of values of the physical measurement variable at a single-variable system have great influence on the shape of the input - output characteristic (characteristic curve ).

Large overlaps the bases of adjacent fuzzy sets on the abscissa means smooth characteristic curve ( approach to linearity).
50% overlap of adjacent triangular fuzzy sets - they intersect at μ = 0.5 - mean linear behavior of the input - output characteristic, unless valuations are available with different factors.
Lückende fuzzy sets that do not overlap in the base points, lead to a partial transfer characteristic with the value zero.
Steeper ramps from adjacent fuzzy sets mean steeper part - characteristic curve (larger partial reinforcement ).

Inference ( data base, rule base )

Fuzzy rules include the expertise of professionals, as the basis of experience, a technical system is to be run. For optimal function of the fuzzy controller, the choice of arrangement of terms ( fuzzy sets ) on the abscissa of the graphical model and the membership in the fuzzification is just as important as defining the rules of the rule base in the inference unit.

The rule base contains the fuzzy logic rules to determine the fuzzy output variables ( fuzzy control variables ) from the fuzzy input variables.

Basis of the fuzzy rules is the following form:

The application of the theory of fuzzy sets by Zadeh - as a fuzzy logical reasoning - said to draw conclusions from the fulfillment of a certain condition, a conclusion. The condition and the conclusion of the fuzzy sets can be displayed on the so-called IF-THEN rules, consisting of an IF part and a THEN part as follows.

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

Premise evaluation for the same basic quantities

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

With the minimum operator is assured that the implication can not be greater than the evaluation premise. For example, if the premise result with the degree of membership μa1 (A) = 0.5 are met, is also the conclusion of the degree of membership μu2 (U ) = 0.5 exhibit.

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

Example of a work rule with an input variable and an output variable:

For the logical combination of a work rule with symbols A and U for an input and an output variable, the statement is:

The input and output variables relate to each have a certain basic amount. The variables A and U associated terms a1 and u2 are fuzzy subsets according to labor rule.

To this rule has the premise to a certain degree of membership μa1 (A) the property of A. The conclusion having a certain degree of membership μu2 (U ) has the property of U.

According to the basic idea of Mamdani ( Mamdami implication ) the veracity μu2 the conclusion must not be greater than the truth of the premise μa1. For the linking of fulfillment degrees of the same basic quantities apply depending on the type of work rules, the min or max operators.

IF - THEN rule with several basic quantities

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

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

Example: Definition of a fuzzy rule with two linguistic input variables A and B and their terms a1 ... an and b1 ... bn and a linguistic output variable U and terms u1 ... un:

At different basic sets A and B of part - premises only the answers of fuzzy sets to be linked to specific input values .

Under this rule, two fuzzy input variables with the variables A and B must be processed in AND operation to determine the degree of fulfillment of the output variables ( conclusion ) uU (U). The aforementioned Mamdani - directional sentence: "The truth of the conclusion must not be greater than that of the premise " requires a small modification ( ie cutting, limiting ) the processing rule shown. This (limited) the result fuzzy set of the conclusion in the amount of the minimum performance level of the AND link of the premises with the min - cut operator.

The complete description of the problem of controlling a process with the work rules of the rule base can lead to a considerable number of rules, as there is a correlation between exponetieller the number of rules and the variables with the respective terms as follows:

Then one would already a 3 - input fuzzy variables with 7 the recommended maximum number of terms of a maximum number of rules 343.

Therefore, the recommendation is to set up the rule base: The number of variables and the associated terms should be as low as possible because they particularly when calculating the degree of fulfillment and relationships of variables with the fuzzy rules a lot of computation on a micro computer - one and last but not least large programming effort - mean.

In the IF part of a rule, the terms of different fuzzy Veriablen be linked with the fuzzy operators. The IF part can contain any logical link with AND and OR operators of terms of different variables. The THEN part with the conclusion is usually simply by assigning a linguistic value to a fuzzy output variable.

The AND operation of the fuzzy statement with the premise of that rule represents a two-digit fuzzy relation represents the fuzzy relation is obtained by applying the min operator:

The combination of a rule with partial premises may also occur as an OR operation.

Example: part of premises in OR operation of a rule:

The OR of the fuzzy statement of the premise of this rule is also called a two-digit fuzzy relation represents the fuzzy relation is obtained by applying the max - operator:

This rule of the OR function can be split on two simple rules in AND operation:

Some steps in the inference:

The inference unit of fuzzy controllers together with the fuzzy terms of the fuzzy variables ( fuzzified input signals) of the linguistic rules (rule -based) the linguistic conclusion consists of a plurality of inference processing steps. The application of each active rule returns based on the Inferenzschemas the resulting output fuzzy set, by transferring the degree of fulfillment of the rule to the respective fuzzy set of conclusions.

The performance level is defined such that it is so large in the fuzzy AND operation, as the smallest degree of membership of the input variables. Similarly, the degree of satisfaction with a fuzzy OR operation is defined such that it is as large as the largest degree of membership of the input quantities. Fuzzy sets indicate the corresponding degree to which a fuzzy logic statement for each sharp value of an input variable. They represent a "membership function".

The aim of the analysis is the premise to determine the degree of membership for each rule. Both the premise and the conclusion are defined as fuzzy sets.

Let the variable A with the terms a1, a2 with the terms b1, b2 given by ai and bi to B:

Depending on a sharp input signal, a certain fuzzy set ai a fuzzy variable A with a certain affiliation microamps will appeal. But it can address two or more adjacent fuzzy sets such as A1 and A2 at the same time. This response of the rules is "fire" also referred to. It depends entirely on the size of the sharp input signal ei from which and how many fuzzy sets fire.

For multivariable systems, for example with the input variables A and B can each fire several fuzzy sets simultaneously in response to a sharp input signal e1 a basic amount for A and another sharp Eingangssinals e2 another basic quantity for B.

Depending on AND or OR logic - - With the implication of the active ( firing ) Fuzzy sets the premise effect for each rule with the min or max operator a limiting effect on the graphical model of the conclusion and the associated per rules fuzzy set a ( triangular fuzzy set is trapezoidal).

By addressing multiple rules to graphical part -fuzzy models of the output variable, which are summarized on the accumulation by the max - operators arise.

Defuzzification

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

Maximum method (mean of maxima)

Centroid method ( Center of Gravity )

Wherein fuzzy controllers predominantly method for the determination of the area center of gravity of U.S. can be used for defuzzification. For the relationship of the membership degree μ applies (U) of the accumulated output variables U to the values of the abscissa from the initial value to the final value UA UE.

The exact calculation for the determination of the center of gravity of a plane U.S. is:

Method of numerical calculation of the centroid of an arbitrary function μ (U):

Simplify the numerical calculation by a minimum number of vertices

Simplification of the surfaces of gravity calculation with line features ( singletons )

Pros and Cons fuzzy controller

Per

The fuzzy controller as a map regulator controlled nonlinear dynamical processes.
A mathematical model of the controlled or process to be controlled is not required.
A fuzzy model of the process is analyzed by observation and experience ( plant operators, expert knowledge) from the process.
By means of linguistic terms for process states (" warm", " cold ", " valve open" ) are created production rules to control the process.
As a single-variable, by the amount of overlap of the fuzzy sets of a fuzzy variable, an input - output characteristic of a linear, continuous non-linear, can be modeled with the dead zone and multi-point switching.
Fuzzy controller can also be understood by interested laymen.
Robustness: parameter deviations within the process and certain limits be tolerated without adverse effects.
With restrictions on the failure of individual signal generators can not interfere with the whole process.
When using fuzzy Ships hardware and software costs can be minimized.

Contra

The fuzzy controller is a static system and is largely inferior to a standard linear regulator for precise control because of the lack of dynamic proportions.
To extend the fuzzy controller to a dynamic controller integral or differential amounts of measured variables must be treated as a sharp input variables for a desired dynamics.
The stability test of an exact control is difficult. in the presence of a mathematical model of the controlled system can be carried out by means of numerical calculation of a stability test.
Depending on the number of variables (signal inputs) and the number of fuzzy sets used per variable and its exponential relationship with the number of production rules can lead to a confusing increase in the fuzzy rule base.
The fuzzy controller is not universally used in control engineering, but it usually refers to the application of controlling a non- linear multi-variable process that is mathematically difficult to describe.

Fuzzy controller and fuzzy control

Conventional schemes

Conventional control applications with linear processes are using the standard controllers such as P, P, PD and PID controllers satisfying to solve. Because the P-controller is a static controller and thus can not compensate for a delay shares of the controlled system, it is not often used. His P- gain must remain low for controlled systems of higher order, otherwise it leads to instability of the system. Low P gain is large deviation.

For nonlinear systems different linearization method (see Articles systems theory: A nonlinear transmission system ) are known in which a model of the controlled system is decomposed into a static nonlinearity in a dynamic, linear transmission system. The simplest controller is designed for a sophisticated control by means of the simulation of the control loop with the numerical computation of the discretized time.

Fuzzy control

The typical application of the fuzzy controllers are usually non-linear multi-variable systems in which the control variables have to be fed back as inputs.

SISO fuzzy controllers have a typical non-linear input - output magnitude characteristic. Due to the different arrangement of the fuzzy sets ( fuzzification ) and the design of the rule base ( inference ), any principal behavior of the input-output characteristic can be defined as linear, nonlinear, with dead zone and multi-point switching behavior. You have no dynamic components. As a single variable, they are therefore comparable in behavior with a proportional controller ( P controller ), the function of the linguistic variables and their assessment any non-linear characteristic is given.

A fuzzy controller can be used as a static SISO controller directly to an unknown control system, when the output of the process is fed back as controlled variable in the controller. It is here, this is a static control with a determined within wide limits nonlinear input-output characteristic. Without knowledge of the mathematical model of the controlled system its application is limited because it can not compensate for delaying the shares of the controlled system. However, processes with global I - behavior can be successfully controlled by a fuzzy controller.

The static fuzzy controller can be complemented by extension with D- and I- behavior, but in default of a linear controlled system no functional advantage over a conventional PID controller. For optimal use in a closed loop mathematical model of the controlled system is necessary.

A fuzzy controller can also be a fixed reference value, or the deviation is input from a fixed reference value with a corresponding configuration of the rule base, whereby a loop is created in connection with the control line and the recycle control variable.

The actual application of the fuzzy controller as a map controller applies the process of a nonlinear multivariable system with several input and output variables, whose mathematical model is unknown or difficult to describe. A mathematical fuzzy controller design strategy as in the linear systems does not exist (yet ).

Fuzzy controller of Mamdani and Sugeno

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

Rational fuzzy controller Mamdani.

Functional fuzzy controller by Sugeno. (see also Takagi Sugeno controller)

Fuzzy Controller Type Mamdani with singletons

Sugeno - Takagi - sound controller

( The order of the names is also done differently in the literature, such as Takagi - Sugeno controller)

Symbol definition: Variables: A. ... Z terms; A = { a1, a2, ... an}; B = { b1, b2, ... bn} Constants k1, k2, k3 as the arbitrary numerical values > 0, within the stability range, Index: n = 1, 2, ... ∞ <<

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

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

The THEN part Sugeno type differs from the THEN part of the type Mamdani fact that is determined in the conclusion no fuzzy set, but e1 ... en of the sharp values of the input variables on the premise evaluation of the relevant rules derived satisfaction degrees μα and weighted with constant k.

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

Sample calculation of a static control with symmetrical non-linear

Application of a fuzzy controller

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

Data:

The controller has an input workspace (basic amount ) for a deviation of ± 1 s
3, the fuzzy sets can be arranged symmetrically with the same gradient of the area of the basic amount of overlap.
The control variables U are measured with k1 = 2.5, k2 = 0.4, k3 = 2.5.
The degree of fulfillment μαi arise over the rules.

In the graph on the controller's fuzzy model of the terms a sharp value of the input variable e = -0.8 is located. There are the terms a1 and a2 enabled. This result for the basic set A, the following degrees of membership:

For the calculation of the manipulated variable U.S. for the value of the input variable e = -0.8, the equation is:

The purpose of this calculation of the manipulated variable is the U.S. representation of the transfer characteristic of the nonlinear controller U.S. = f (s) to be determined. This can be done in tabular form by progressively adding data in the base set from -1 to 1 are specified for the input variable e. The degrees of membership for each sharp value of the input variable to be calculated, using the straight-line equations of the ramps.

The graphical representation of the characteristic curve shows a continuous non- linear. Three equal weighting factors k1 = k2 = k3 = 2.5 a completely linear input - output characteristic would result within the limits established by the fuzzy sets boundaries.

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

Controlled system:

Fuzzy controllers: Static SISO controller with three fuzzy sets according Pictured graphical Sugeno type fuzzy model.

Result:

Behavior of the nonlinear controller according illustrated graphics of the input - output characteristic:

The rate of rise of the controlled variable to reach the target value is U. Depending on the gain and the size of the limiting of the manipulated variable
The decreasing controller gain to the range of approximately e = -0.2 ... 0.2 leads to a weaker and thus better damping of the transient response. This becomes more apparent when the assessment factors of the input variable e (t ) would be increased with k1 and k3.
The point of application of the disturbance d (t) = -2 is performed after the I-element, before the two delay elements. The course of the controlled variable temporarily suspends the value Dy a = 1.3.

Conclusion:

The application of the fuzzy controller Mamdani with singletons is clearer and easier to understand compared to the process Sugeno. There are no weighting factors needed but the controller output range is set with the basic set of output variables.
Fuzzy controller as a static Eingrößenregler on a linear controlled system are impractical because of the planning and programming effort is high. As a static controller be used on linear processes is very limited. You can not delay the shares of the controlled system can be compensated.
Fuzzy controller as static Eingrößenregler may be useful to a non-linear controlled system, if the continuous curve of the non-linearity of the track to be compensated.
Dynamic fuzzy controller with PI or PD or PID control of linear controlled systems are more expensive but not better than comparable standard linear regulator.
Dynamic fuzzy control of unknown non -linear processes such as the model used for study purposes like " inverted pendulum" can be useful if the exact system model is mathematically difficult to detect.

Implementation of a fuzzy controller and fuzzy controller

The technical implementation of the fuzzy controller and a fuzzy controller as a hardware requires a programmable microcomputer ( CPU), because geometrical distances ( ramps) and logical functions must be calculated.
Fuzzy logic as integrated circuits together with software offered commercially.

Singleton (mathematics)

358002