DE19640635C2 - Defuzzification procedure for signal processing fuzzy units - Google Patents

Description

The invention relates to a method for defuzzification for signal processing fuzzy units after the Preamble of claim 1.

With the procedure for defuzzification for signal processing fuzzy units The following can be achieved: Rules can be processed so that the effects simultaneously activated rules of the same effect are superimposed on each other and the effects of simultaneously activated counteracting rules mutually to be compensated. This is especially true for the processing of Er practical knowledge based on incomplete process knowledge. The The basic idea of the method is to defuzzify in a special way to be designed in such a way that, in contrast to conventional defuzzification regulations ten is not translation invariant. The process is technically easy to implement at for example, that conventional fuzzy units with a simple to kit module.  

Conventional fuzzy units with real-valued output variables are used for this motion-based design of non-linear map elements. You can use existing Use qualitative experience in a targeted manner to create inexpensive non-linear map controllers to find. Examples of such fuzzy units are fuzzy controllers for process control influencing or monitoring such as diagnosis or prognosis. This here Relevant prior art is described for example in the publications [1] Kiendl, H. and Fritsch, M .: Basic ideas from Fuzzy Control, series "Theory for An wender: Fuzzy Control ", Automation Technology 41 (1993), pp. A5-A8, [2] Kiendl., H .: Defuzzification process for signal processing fuzzy units and filter devices for this, patent specification DE 44 16 465 C1 and [3] Kiendl, H .: Process for production of manipulated variables at the output of a fuzzy controller and fuzzy controllers for this, Patent specification DE 43 08 083 C1 and in the publications [4] DE 44 27 020 C1, [5] DE 44 39 225 A1, [6] EP 05 09 796 A2 and [7] WO 96 04 601 A1. For the following description fuzzy controllers are used. The explanations are directly related to the whose fuzzy units are transferable.

To justify the practical utility of the inventive method first worked out that there is empirical knowledge that can be used with known fuzzy Controllers cannot be processed adequately. For this a temperature control system is included considered a PI controller. The output variable of the PI controller is ν. The PI controller a fuzzy controller is connected in parallel, which reacts to "additional measured variables" and so that it intervenes in certain situations. Its initial size u becomes ν added so that the quantity = ν + u acts on the controlled system. For example from the regulation of room temperature ϑ in a house with central heating went. The input variable of the controlled system is the position of a mixing valve, which determines how much of the hot water provided by the central heating system goes to the warm water circuit is added.

The following experience is used to construct the fuzzy controller: It is known that the heat requirement depends on the number of people who are in the room in question and also on how windy it is. To simplify the example extremely, only the two linguistic values average (D) and unusually large (u) are used for the number of people p and the wind force w. For the output variable u of the fuzzy controller, the linguistic values become negative small (NK), vanishing (Y), positive small (PK), positive medium (PM), positive large (PG) and positive very large (PSG) with the membership functions provided according to picture 1. Figure 2 shows an illuminating rule base for determining the output variable value u as a function of the number of people p and the wind force w. From this it follows that the correction intervention u should have a positive small value in the case of an unusually large number of people and an average number of people, the value should be negligibly small in the case of an unusually large number of people and an average number of people and finally the value should vanish in the case of an unusually large number of people and an average number of winds. The rule base according to Figure 2 consists of complete rules: Exactly one rule is provided for every possible combination of linguistic values of the two input variables. Such a rule base can be established if there are not too many input variables with not too many linguistic values. Otherwise the number of rules quickly becomes unacceptably large. In these cases, to reduce the number of rules, it is cheaper to work with rules whose premises do not access all input variables. For example, you can use the two incomplete rules

R ₁ : IF p = U THEN u = NK
R ₂ : IF w = U THEN u = PK

provide and set the default value u = 0 for u if neither of the two premises is fulfilled. With this rule base, in the two cases there is an unusually large number of people, combined with average wind force and unusually large wind force, combined with average number of people, the same behavior as with the rule base according to Figure 2. In the case of unusually large number of people, combined with unusually large wind force, the fuzzy controller provides the membership function

µ (u) = (µ ₁ ∧ µ _NK (u)) ∨ (µ ₂ ∧ µ _PK (u)),

where µ _{1 is} the degree of activation of the first and µ _{2 is} the degree of activation of the second rule, and where µ _NK (u) and µ _PK (u) are the membership functions that model the linguistic value NK and PK, respectively ( Figure 1). The resulting output value u _D depends on the choice of fuzzy operators and the defuzzification strategy. When using conventional defuzzification methods, as described in the publications [1] and [2], applies

-1 ≦ fu _D ≦ 1

in accordance with the effect of the rule highlighted by hatching in Figure 2

R ₃ : IF (p = U) ∧ (w = U) THEN u = V.

The oppositely acting rules R ₁ and R ₂ thus achieve the same effect as the rule R ₃ . To illustrate this, Figure 3 shows the output-side membership functions µ (u) and µ '(u), which are activated when rule R ₃ or the two rules R ₁ and R _{2 are} fully activated (regardless of the choice of the accumulation method ) results (left or right). With the usual defuzzification methods, the same output variable value u _D = 0 results in both cases.

In addition to the wind force w, instead of the number of people p, the fuzzy controller is now supplied with the outside temperature a with the linguistic values average (D) and unusually low (U) as the input variable. Figure 4 shows a qualitatively understandable rule base of complete rules for this. Instead, use the two incomplete rules that act in the same direction

R ₄ : IF a = U THEN u = PG
R ₅ : IF w = U THEN u = PK

with the agreement that the default value u = 0 should be set if the premise is not met by any rule. For the combinations of values (D, D), (U, D) and (D, U) on the input side, the same output variable values are obtained as with the control base according to Fig. 4. If the outside temperature is unusually low and the wind force is also unusually large, it delivers the rule highlighted by hatching in Figure 4

R ₆ : IF (a = U) ∧ (w = U) THEN u = PSG

correctly the linguistic output quantity value PSG. On the other hand, in this case the simultaneous activation of the two rules R ₄ and R ₅ does not result in a larger, when using the known defuzzification according to the focus method even a smaller output value, compared to the case where the outside temperature is unusually low and the wind force is average. To illustrate this fact, Figure 5 shows the output-side membership functions µ (u), µ '(u) and µ''(u), which change when rule R ₆ , the two rules R ₄ and R ₅ or only the one are fully activated Rule ₄ results (top, middle or bottom). With the customary defuzzification methods, this results in the widely differing output variable values u _D = 4, u _D '= 2 and u _D ''= 3. Obviously, the two rules R ₄ and R _{5 are} not adequately processed with conventional defuzzification methods.

The deeper reason for this deficiency lies in the fact that conventional defuzzification methods are translation-invariant: If u _{D is} the result of the defuzzification of a membership function µ (u), then when defuzzifying the function µ (u - d) results from shifting µ (u) around the piece d along the u-axis, the output value µ _D + d ( Figure 6). If, as desired, a value u _D ∈ [-1, + 1] is obtained by defuzzifying the membership function µ '(u) shown in Figure 3, then it is determined because of the translation invariance that the defuzzification of the Figure 5 shown membership function µ '(u) provides an undesired value u _D ∈ [1,3].

All of the publications cited above describe defuzzification processes which transla are invariant and thus have the deficiency explained above. So in the Document [1] the known translation invariant standard procedures (difficult point method, maximum method). The document [2] describes one generalized translation invariant defuzzification using the standard methods, like center of gravity method and maximum method, reproduced and stepless com Promisse in between enables. All variants described in publication [3] Hyperdefuzzification is based on an evaluation of the form of membership function and are therefore translation-invariant. The publication [4] describes a De Fuzzification device which, depending on selection signals, either defuzzifi decoration according to the maximum procedure or the priority procedure and there with is also translation invariant. The defuzzification in publication [5] is based on a determination of the area center of the area of the function, which is under the Graph of the membership function to be defuzzified lies and therefore translati is invariant. Defuzzification of the document [6] realizes the defuzzification based on the priority procedure in a special way and thus provides a trans lation invariant defuzzification. Finally, the publication [7] corresponds to Publication [4] and thus also provides a translation-invariant defuzzification.

The object of the invention is to remedy the defect described above.

The invention solves this problem by a method according to claim 1.

The method according to the invention is based on the idea of creating defuzzification methods with the desired properties in such a way that the defuzzification specification - in contrast to conventional regulations - is set up in such a way that it is not translation-invariant. The regulation provides one possibility for this

for defuzzifying a membership function µ (u) to be defuzzified in the definition interval [-u _max , u _max ]. Here c is a normalization factor. If - as is usually the case - µ (u) ≦ 1, the condition for c = 2 applies

| u _D | ≦ u _max (2)

Fulfills. Using some examples, Figure 7 shows that the above rule is not translationally invariant and works in the desired sense: Different membership functions µ (u) and the result u _{D of} their defuzzification according to the above rule (with u _max = 1 and c = 2). If, for example, the two output-side membership functions that lead to the output variable value u _D = 0.75 or 0.25 are superimposed as usual by a fuzzy OR operator (for example, the maximum operator) for accumulation, the membership function and its defuzzification are created the output quantity value u _D = 1. This means that the proposals for action in the same direction ("recommendations in the same direction") are overlaid as desired to reinforce each other. Conversely, it can be seen that the proposed actions of opposing rules ("opposing recommendations") - as also desired - are superimposed to compensate.

If singletons are used as output functions in the fuzzy controller, the same effect is achieved by the regulation

reached. Μ _K is the degree of activation of the kth rule, and u _k indicates the point at which the singleton belonging to rule R _k is located on the output side. If the restriction condition is | u _D | ≦ u _max , the result u _{D obtained} in the form

modify. To prevent the maximum values ± u _{max from being} reached by just a few contributions µ _k u _k , the above summation can be done with the Einstein sum standardized to a value u _max + ε

carry out. Figure 8 shows for u _max = 1, c = 1 and ε = 0.05 based on a few examples that this provision for defuzzification is not translation-invariant and works in the desired sense: like-minded recommendations are mutually reinforcing, opposing mutually compensating.

The defuzzification according to the invention clearly offers advantages if the Pro expert who formulated the rules, no absolute scale for the initial values te has in mind, but wants to express that under certain conditions the output value just set is to be increased or decreased. The  This case can occur in particular if the premises of the rules refer to un obtain different input variables. Then it seems more appropriate, like-minded Recommendations of simultaneously activated rules are superimposed on one another instead defuzzify them conventionally after accumulation.

In individual cases it has to be decided whether the rules are conventional or in the sense of a ver strengthening overlay of like-minded recommendations of simultaneously activated rules are to be processed. To assess the value of the De Fuzzification serves the following conclusion: With conventional fuzzy controllers in which one of the usual defuzzification methods described in publication [1] can be seen, in principle any non-linear map controller can be generated. Modes Defined controller structures with new methods for inference or defuzzification can therefore not lead to maps that can be compared with conventional fuzzy Controllers cannot be reached. Another question, however, is how good ever Fuzzy controller is suitable for making available experience knowledge to the target to find favorable maps. The inventive method for Defuzzification is better than traditional defuzzification methods net to use experience knowledge for the map generation, which is based on relative changes changes in output quantity values.

Adjustable compromises can be made between the conventional and the defuzzification method according to the invention. For this purpose, each rule R _{k is provided} with a factor λ _k with the meaning that the rule for λ _k = 1 is conventional and for λ _k = 0 in the sense according to the invention. The regulation provides for singletons on the output side

in the borderline cases that all λ _{k have} the value 1 or 0, a defuzzification process which corresponds to the usual emphasis method for singletons or the above-mentioned method ( 3 ) according to the invention for singletons. In the intermediate range 0 <λ _k <1 to compromise between these methods is obtained, each of the IMP EXP _k individually λ r together with R rules in accordance with their factor _k is considered.

It may make sense to choose the values λ _k such that the value of λ _k is smaller, the more general the premise of the rule R _{k in} question.

Even for continuous output-side membership functions, stepless transitions between the usual defuzzification according to the focus method and the method ( 1 ) according to the invention can be created using the factor λ _k .

The defuzzification ( 1 ) according to the invention can also be embedded in the concept of the inference filter described in the publication [2]: for example, the function is filtered through quadratic filtering

which arises from a continuous membership function µ (u) by a scale transformation, a function _T (u). Their maximum lies at the point u _D , which results from defuzzification of the original function µ (u) according to the usual focus method for continuous membership functions (F is the area under the function graph of µ (u).). _T (u) therefore represents the attractiveness function which is associated with the defuzzification rule (1). The functional value of this attractiveness function delivers for each potential output quantity value u the degree in which it is considered as a recommended output quantity value on the basis of the defuzzification concept according to the invention. If the value u _D , where _T (u) assumes the maximum, is not an output variable value (for example because negative rules in a two-strand fuzzy controller structure, as described in the publication [3], prohibit certain value ranges for the output variable ), then the course of _T (u) provides information about which other output value represents the best possible alternative.

The technical implementation of the defuzzification method according to the invention is very simple with the regulations (1) and (2) given in this description, since these are the usual regulations

for defuzzification of continuous membership functions or of membership Functionality in the form of singletons using the focus method except for the correspond to the missing denominator. Existing fuzzy units can therefore be easily built retrofit so that they defuzzify according to the inventive method carry out.

The formulas given here are now special configurations of the invention process. The basic idea of the method according to the invention is that deliberately avoiding conventional translational invariance reinforce the desired overlapping recommendations and the compensating overlay opposite recommendations can be achieved. This way of processing the Rules are not only for the use of fuzzy units for the purpose of re of interest, but also for data analysis, event detection planning, forecasting, image processing and for process monitoring. The desired effect that is achieved with the method according to the invention is also intended here using the example of Process quality assessment can be clarified again: If there are several positive ones So partial evaluations of the same process can be recorded, a ver strengthening overlay of these partial evaluations to an overall very positive Ge  overall evaluation would be desirable. On the other hand, it is desirable that compensate for sensible reviews.

For the sake of simplicity, the output variable value u ₀ = 0 was chosen as the "neutral value", which determines whether output variable values are to be regarded as the same or the opposite. It is clear that in the specific application the neutral output variable value can be different from zero, but this case can be traced back by transforming the output variable to the case that u ₀ = 0 is the neutral value.

Claims (8)

1. Method for defuzzifying a membership function µ (u) with µ (u) ≈ 0 defined on the output side in the interval u _min ≦ u ≦ u _max of signal-processing fuzzy modules with one or more input variables x ₁ , x ₂ . . ., x _m and an output variable u for generating a unique output variable value u _D for the purpose of influencing or observing a process by calculating the input variable values, the process having the property that two values u _D1 and u _D2 , one of which is larger and the other is smaller than a certain neutral value u _{0 which} is characteristic of the process, in the case of process influence act opposite to each other with regard to their directions of action and in the case of process observation tend to represent opposing observations and where the directions of action or the trends of the observations are in the same direction if the values u _D1 and u _{D2 are} either both smaller or both larger than the neutral value u ₀ , characterized in that the output variable value u _D as a function of µ (u) in the form u _D = u ₀ + u _{h is} generated, where u _{0 is} the neutral value already mentioned and the value u _h as the sum of Partial contributions u _{h, j is} formed, which are generated by dividing the interval u _min ≦ u <u _max into sufficiently small partial intervals I _j = u _j ≦ u <u _f + Δu so that the partial contributions u _{h, j} , if they do not have the value zero, for all cases in the same category of the two categories u ₀ <u _j and u _j + Δu <u ₀ each have the same sign and for two cases of different categories each have a different sign and that the amounts of these Partial contributions u _{h, j} are greater, the greater the value of the product | u _j - u ₀ |. u or | u ₀ - u _j - Δu | .u, where u is the mean of µ (u) in relation to those places of the interval I _j where µ (u) <0 applies. 2. The method according to claim 1 for the defuzzification of a defined in the interval [-u _max , u _max ] continuous membership functions µ (u), characterized in that the output variable value by the regulation
is formed, where c is a selectable normalization factor. 3. The method according to one or more of claims 1 and 2 for defuzzifying a continuous membership function µ (u) with a neutral value u ₀ = 0, for which µ (u) ≦ 1 applies, characterized in that the restriction condition | u _D | ≦ u _max is guaranteed by setting c = 2. 4. The method according to one or more of claims 1, 2 or 3, characterized in that the membership function µ (u) is transformed by an inference filter into a membership function (u), which can be determined at one of these claims u u _{D has} the absolute maximum. 5. The method according to claim 1 for defuzzifying membership functions µ (u) in the form of singletons, characterized in that the output values value u _D by the regulation
is defined, where c is a selectable standardization factor and r is the number of rules and where µ _{k is} the degree of activation of the kth rule R _k and u _k indicates the point at which the singleton belonging to rule R _k is located on the output side. 6. The method according to claim 5, characterized in that the Einstein sum for the summation
is used, where ε is a selectable parameter. 7. The method according to one or more of claims 1 to 2 or 4 to 6, characterized in that the fuzzy unit is followed by a limiting element, the output variable value by
given is. 8. The method according to one or more of claims 5 to 7, characterized in that each rule R _{k is provided} with a selectable factor λ _k and the output variable value according to the regulation
is determined.