DE4336921C2 - Method and device for automatic reasoning (inference) for rule-based fuzzy systems - Google Patents

Description

The invention relates to a method for automatic conclusion for rule-based fuzzy systems that are more transparent than the known methods and that Detection of inconsistencies, as well as devices designed for this.

In automation technology, the term fuzzy control refers to one on the The theory of unsharp sets based approach that offers the possibility Process knowledge, available in the form of linguistic IF-THEN rules, in one Control, control or monitoring system to bring. These systems arrange theirs Outputs to values based on their input variables, the IF-THEN rules as well depend on the design process. The initial values are determined in the steps of fuzzification, inference and defuzzification (see [1], or, for example, US Pat. No. 5,214,773). The inference is used for Processing the IF-THEN rules and consists of the sub-steps aggregation (Evaluation of the IF parts of the rules), inference core (actual closing) and Accumulation (merging of rules). For the inference core and defuzzification Various methods are known in the art, see [1]. These On the one hand, processes influence the transmission behavior of the fuzzy controller, on the other hand from an engineering point of view but also the transparency and systematics of the Draft.

With fuzzy control comes the requirement that the design process for the user is transparent, d. H. that the user based on results and intermediate results can understand clearly how they came about and what meaning on the basis of the expenditure, is of great importance. This transparency contributes to the manageability and systematization of the design and makes troubleshooting easier and the gradual improvement of dynamic Properties of the controller. In the present description, this is on the The method based on the invention explains the one compared to the known methods Achieved greater transparency and thus as well as through additional performance features goes beyond the state of the art.

The description of the method uses a fuzzy system with p input variables and N production rules of the form

IF "precondition i", THEN "conclusion i", i = 1. . . N,

and an output variable a. The consideration of only one output variable serves for clarity and does not limit the generality. In the case of several output variables, the production rules, each of which relates to one output variable, can be viewed separately. This does not change the basic procedure. Furthermore, let the output variable a together with M be linguistic values by the membership functions m _j (a), j = 1. . . M, defined.

Fuzzification and aggregation can be carried out using known methods [1]. The results of this give the degree m _Wi , 1 = 1. . . N, at which the preconditions of the N production rules are met.

For each production rule in the step of the inference kernel an individual error function e _i (a), i = 1. . . N, formed. It evaluates the values of the output variables a with regard to their suitability as the resulting output value, i.e. indicates which output values favor the rule under consideration.

The single error function is a fuzzy set, the degree of membership of a value to the fuzzy set "errors related to the single error function the ith production rule "the smaller the better the suitability of the value as Aus is current value. Thus, the values of a are considered suitable by the rule in question The smallest functional value.

According to [2], the semantics of an IF-THEN rule can be generalized Modus Ponens can be characterized as follows:

• - the greater the degree of fulfillment of the precondition, the more certain is the conclusion cure,
• - If the precondition is not completely fulfilled, the uncertainty is maximum, it is no conclusion possible.

According to this generalized Ponens mode, the individual error function can be defined as the amount of the difference between the degree of fulfillment m _Wi and the relevant membership function m _i (a) weighted with the degree of fulfillment m _{Wi of} the precondition:

e _i (a) = m _Wi · | m _Wi - m _i (a) |. (1)

The weighting in Eq. (1) with the factor m _Wi can be interpreted as an AND operation using the product operator. As an alternative to Eq. (1) the minimum operator can also be used for this AND operation. In this case the individual error function is obtained

e _i (a) = min (m _Wi , | m _Wi - m _i (a) |). (2)

The two definitions in Eq. (1) and (2) ensures that a production rule with a non-fulfilled precondition (m _Wi = 0) does not later contribute to decision-making. The individual error function then has the value zero for all a, where no special a favors. The amount | m _Wi - m _i (a) | ensures that the individual error function disappears for those values of the output variable a for which the membership function m _i (a) is equal to the degree of fulfillment of the precondition.

A general weighting g _{i can be} introduced in a generalization:

e _i (a) = g _Wi · m _Wi · | m _Wi - m _i (a) | (3)

respectively.

e _i (a) = g _i · min (m _Wi , | m _Wi - m _i (a) |). (4)

Here g _{i is} a weighting factor assigned to the i th production rule. It indicates the certainty with which the rule in question is valid at all. If g _i = 0, the rule in question is not included in the decision-making process, if g _i = 1, the rule is fully included.

The formation of the total of N individual functions represents the interference kernel. Nor sometimes you will get Eq. (3) or (4) as these are the generalized Consider mode ponens. It can be used in certain technical applications However, it also makes sense to drop the requirement of the Modus Ponens that at If the precondition is not met, no conclusion can be drawn. If the precondition is completely not fulfilled, one also concludes that full constant non-compliance with the conclusion, so the general individual error function

e _i (a) = g _i · | m _Wi - m _i (a) |. (5)

be formed. The admissibility of this implication rule must be determined by the Expert knowledge can be ensured.

The processing of the inference kernel by forming the individual error functions with the help one of the eq. (1) to (5) is not obvious and differs significantly from the known ones Procedure [1]. Claims 5 and 6 relate to this formation of single-fault radio ions.

The following is 1 by the example in Fig. Determining the individual errors radio tio _i (a) according to Eq. (3) clarified for g _i = 1. In the upper part of Fig. 1, a three corner membership function m _i (a) is shown. The precondition of the relevant production rule is met to the degree m _Wi . In the lower part of FIG. 1, the associated individual error function e _i (a) according to Eq. (3) applied. It can be seen that this single error function disappears for those abscissa values in which the tensile function m _i (a) assumes the value m _Wi . At these points, the requirement is met that the rule's conclusion is met to the same degree as the precondition.

In order to take into account the influence of all N production rules, the battery is used rule base error

formed as the sum of all N individual error functions. As a result, all N products rules of equal status according to the degree of fulfillment of the corresponding precondition conditions and the degree of validity of the respective rule for decision making. The logic operator for forming the rule base error is therefore an OR Operator, whereby by its realization by the usual sum in Eq. (6) values greater than 1 can also occur. If you are not used as an OR operator as in Eq. (6) the ordinary, but the algebraic or the limited sum [1], is so the value 1 is the upper bound of the rule base error. For further considerations the usual sum is calculated according to Eq. (6) used as a basis.

The rule base error can be understood as an abnormal fuzzy set. He evaluates the suitability of the values of the output variables a as the resulting output value of the Controllers with regard to the entire rule base, whereby again the evaluation all the more  the smaller the functional value of the rule base error is, the better. Through this interpreter animalability of the rule base error also contains more transparency significantly more information than the resulting fuzzy set with the known methods. This information can be used for several purposes. For one, it serves the base rule error as the basis for defuzzification; Rule base error secondary information to assess completeness and Freedom from contradiction of the knowledge base.

Since the rule base error e _RB (a) describes the suitability of the values of the output variables a as the output value of the controller, the value a _{R is determined} as the output value at which the rule base error becomes minimal. For this purpose, it is initially assumed in the following that the rule base error e _RB (a) assumes its absolute minimum only at a single point a = a _min , ie

In this case, the rule base error e _RB (a) is referred to as "uniquely decidable" and the resulting output value a _R is determined as

a _R = a _min . (8th)

The following simple example is considered to clarify the sequence of decision-making: Let us design a fuzzy controller with an input variable r and the output variable a. As shown in FIG. 2, the membership functions for the input and output variables each consist of three triangles, which represent the linear values "small", "medium" and "large". The membership functions are m _small (r), m _medium (r) and m _large (r) for the input variable r, and m _small (a), m _medium (a) and m _large (a) for the output variable a designated.

There are also three production rules:

Rule 1 (2, 3): IF "r small (medium, large)", THEN "a small (medium, large)".

The decision process is now considered in the event that the input value r = Is 30%. From the evaluation of the precondition of the rules one obtains

m _W1 = 0.4; m _W2 = 0.6; m _W3 = 0. (9)

With the help of Eq. (1) can now the individual error functions e₁ (a), e₂ (a) and e₃ (a) and according to Eq. (6) the rule base error e _RB (a) can be determined as the sum of the individual error functions. This is shown in Fig. 3. In this case, the rule base error e _RB (a) obviously assumes its absolute minimum at the point a _min = 30%. According to Eq. (8) the output value a _{R of} the fuzzy controller. If the output value a _{R is} calculated using the specified method for all values of the input variables r in the range 0. . . 100% through, you get the transmission characteristic of a P-controller with gain 1 as a result.

Claims 9 to 11 relate to the procedure if the rule base error e _RB (a) cannot be clearly determined according to inequality (7). In this case, conclusions can be drawn about contradictions or inconsistencies in the knowledge base. A method for checking the rule base for consistency is already known in the literature [3]. The basic idea is that two rules are contradictory if they activate different conclusions with the same or similar premise. Based on this basic idea, a measure of the inconsistency of rules is derived from the degrees of overlap of the membership functions of the IF parts and the THEN parts of the rules. This procedure has the following characteristics:

• - It is an off-line procedure, in which the rules in the off-line operation be examined for structural consistency; structural consistency means that rules based on the selected membership functions are basically consistent regardless of those used in online processing actually occurring conclusions in the form of sharp Baseline values; are decisive for the behavior of the fuzzy controller however, these actually occurring initial values, since only these actually be given to the process to be influenced. The actually occurring However, initial values depend on the properties of the operators Carrying out the procedural steps inference core, accumulation and Defuzzification that does not go into the procedure specified in [3].
• - The results of the procedure given in [3] may be irrelevant: Es is implicitly the whole of the sharp by the specified procedure Input variables spanned entrance space examined for inconsistencies. There however, the input variables of fuzzy controllers are often physical coupled size (e.g. state variables) can in many cases Subspaces of this entrance space are not at all of the trajectory that the  Input vector depending on the time it takes to be reached. are there are now inconsistencies in the rule base that make these inaccessible Affecting subspaces, then recognizing such inconsistencies is for the actual process influence irrelevant. It is regarding the Process influencing is not necessary to detect such inconsistencies.

In the following, a new procedure for the detection of contradictions or Inconsistencies described, which are in the approach, the implementation and in the Results completely different from the method given in [3]. This new one The method is based on the evaluation of the rule base error. It relates to it both on the input variables that actually affect online operation as well the operators actually used for the conclusions in each Procedural steps. A distinction must be made between the following cases:

Case 1: The absolute minimum of the rule base error e _RB (a) does not occur at a single abscissa value a _min , but is distributed over a connected area a₁aa₂, as is exemplarily shown in FIG. 4;
Case 2: the absolute minimum of the rule base error e _RB (a) occurs at several abscissa values a _mink , k = 1 that are not related. . . K, as exemplified in Fig. 5;
Case 3: any combination of cases 1 and 2.

The development of the course of the rule base error in case 1 can have different causes. This case occurs e.g. B. on if one of the membership functions of the output variable a constantly assumes the value 1 in this range, while all other membership functions disappear. In this case, the overlap of the linguistic values is not sufficient to unambiguously determine the initial value a _R. If a defuzzification is to be carried out on the basis of this course of the rule base error, in principle any of the values of a between a 1 and a 2 can be chosen as the starting value a _R , since these values of a between a 1 and a 2 are chosen as the starting value a _R , since these values are given by the rule base error are rated equally well. By specifying one of the values a 1 or a 2 or the arithmetic mean can be determined as the starting value a _R , for example. Possibly. In this case, additional criteria can be used for decision-making, such as B. minimizing the actuating energy.

Case 2 suggests contradictions in the rule base, since the rule base error more proposes more equal values than the output values of the fuzzy controller. This is shown by the continuation of the latter example; the rule base from the This example is modified as follows:

Rule 1: IF "r small", THEN "a small",
Rule 2. IF "r medium", THEN "a medium",
Rule 3: IF "r small", THEN "a large".

Rules 1 and 3 now contradict each other. The membership functions are further selected according to FIG. 2. The control basis error shown in FIG. 6 is now obtained for the input value r = 30%.

The contradiction in the rule base in the example leads to a course of the rule base error e _RB (a) after case 2. In this simple example, the contradiction is easily recognizable even without the evaluation of the rule base error. However, if the rule base is more extensive, the rule base error is suitable to detect unwanted contradictions in the rule base. If, in the example, you want to determine an initial value a _R despite the contradiction in the rule base, it may make more sense to choose one of the values a _min1 = 30% or a _min2 = 70% than the arithmetic mean, since the rule base error for this mean value takes on a larger value than in its absolute minimums.

In case 3, the procedure described can be combined for cases 1 and 2. The Statements made there apply analogously.

Claim 12 relates to a possibility for the cost-effective implementation of the method presented. The usual choice of membership functions is used as a polyline. Polygonal sections are linear functions that are completely determined by the abscissa and ordinate values of all corner points including the edge points. In the operating phase of the fuzzy controller, the individual error functions e _i (a), i = 1 must be in each cycle. . . N, the rule base error e _RB (a) and the resulting output value a _{R are} determined. The choice of membership functions as a polygon has the consequence that the one in Eq. (4) to (6) defined individual error functions e _i (a) and the rule base error e _RB (a) according to Eq. (7) as a polyline.

The determination of each individual error function can be carried out by first determining the abscissa values of all corner points. The individual error function in the representations according to Eq. (3) or (5) has only corner points on the abscissa values a, on which the membership function in question has corner points, and on the abscissa values a, on which the membership function assumes the value m _Wi . The single error function in the representation according to Eq. (4) has additional vertices on the abscissa values a, at which the amount | m _Wi - m _i (a) | assumes the value m _Wi . Only for the abscissa values of the corner points of the individual error function determined in this way must the respective Eq. (3), (4) or (5) can be used to determine the ordinate values of the vertices.

The determination of the rule base error as the sum of the N individual error functions can be carried out analogously. The abscissa values of all corner points of the rule base error are given by the total of the abscissa values of the corner points of all single steps learning functions. The sum of the individual error functions must only be based on these abscissa values NEN are formed. Also the absolute minimum of the rule base error for defuzzi Identification can be determined in a simple way: since a polyline is its absolute Minimum can only assume in its corner points, only the ordinate values the key points of the rule base error are compared.

The computing effort in this form of representation of membership and errors functions through their corner points is thus much less than with the discrete Interpolation point representation of the membership functions, as used in the MAX-MIN Focus method is common. This enables significantly shorter cycle times, especially the integration required with the focus method is no longer required.

Claims 13 and 14 relate to facilities in which the specified method is implemented.

literature

[1] Kahlert, J .; Frank, H .: Fuzzy logic and fuzzy control. Vieweg, Braunschweig, 1993.
[2] Bothe, H .: Fuzzy Logic - Introduction to theory and applications. Springer, Berlin, 1993.
[3] Pedrycz, W .: Fuzzy Control and Fuzzy Systems. Research Studies Press Ltd., Taunton (Sommerset), 2nd extended edition, 1993.

Claims (14)

1. Method for automatic conclusion (interference) for controllable fuzzy systems for regulating, controlling or monitoring technical processes or systems
with any number of inputs and any number Q of outputs, whereby for the qth output a _q with q = 1. . . Q any number N _q if-then rules of the form "if precondition i, then conclusion i" with i = 1. . . N _{q is} defined
characterized,
that in the process step of decision making (interference kernel) the respective degree of fulfillment m _{Wi of} the precondition i is linked to the fuzzy set m _i (a _q ) describing the conclusion i in such a way that this results in a function (designation: individual error function), which is a measure of the Suitability of the values of the output variable a _{q is used} as the output value with respect to the i-th rule under consideration, the suitability being better the smaller the function value at the point in question,
and that secondary information for assessing the knowledge base, ie the entirety of the linguistic variables and the if-then rules, is derived from the course of the individual error functions for inconsistencies. 2. The method according to claim 1, characterized in that in the process step of accumulation (merging rules) all N _q individual error functions are linked to a further function (designation: rule base error), which be a measure of the suitability of the values of the output variable a _q as the initial value regarding the entirety of all N _q if-then rules are used. 3. The method according to claim 2, characterized in that the derivation of the Secondary information to assess the knowledge base on inconsistencies from the Course of the linked individual error functions, i.e. out of the rule base error occurred. 4. The method according to claim 2 or 3, characterized in that in the procedural step of defuzzification (determining a sharp output value) from the rule base error or a linear combination of those values of the output variables a _q as the resulting output value a _{qr of} the q-th output of the fuzzy System is determined in which the rule base error becomes minimal. 5. The method according to any one of claims 1 to 4, that the N _q individual error functions are formed by a link from the degrees of fulfillment m _Wi the preconditions and the fuzzy sets m _i (a _q ), such that this link in the binary limit, the inhibition or which represents antivalence. 6. The method according to claim 5, characterized in that the N _q individual error functions (designation e _i (a _q ), i = 1... N _q ) are formed by one of the following links given under a) to e), wherein the factor g _i in c), d) and e) represents a weighting factor which is a measure of the reliability of the validity of the i-th if-then rule:

• a) e _i (a _q ) = m _Wi · | m _Wi - m _i (a _q ) |
• b) e _i (a _q ) = minimum (m _Wi , | m _Wi - m _i (a _q ) |)
• c) e _i (a _q ) = g _i · m _Wi · | m _Wi - m _i (a _q ) |
• d) e _i (a _q) = g _i min · (m _Wi, | m _Wi - m _i (a _q) |)
• e) e _i (a _q ) = g _i · | m _Wi - m _i (a _q ) |.

7. The method according to any one of claims 2 to 6, characterized in that in the accumulation the link to the rule base error e _RB (a _q ) is given by the usual sum or in the binary limit case by the OR link. 8. The method according to any one of claims 1 to 7, characterized in that an inconsistency of the knowledge base is concluded if and only if there is no value a _{qmin of} the output variable a _q for which applies. 9. The method according to claim 8, characterized in that one concludes exactly one or more contradictions in the rule base if the inequality specified in claim 8 is violated only by a finite number of values of the output variable a _q . 10. The method according to any one of claims 2 to 7, characterized in that precisely then concluded one or more contradictions in the rule base becomes when the rule base error besides an absolute minimum one or more owns local minima.   11. The method according to claim 8, characterized in that then precisely to an insufficiently precise specification of the linguistic variables or an insufficient coverage of the linguistic values of the linguisti rule is closed when the Un specified in claim 8 equation is violated by a countable infinite number of values. 12. The method according to any one of claims 1 to 11, characterized in that the fuzzy sets that define the linguistic values of the variables Polygons exist, and therefore only the Corners of these polygons, the corners of the individual error functions and the Key points of the rule base error are evaluated. 13. Device for performing the method according to one of claims 1 to 12, characterized in that at least partially with digital processors or other digital computing units. 14. Device for performing the method according to one of claims 1 to 12, characterized in that it is at least partially constructed with analog circuits.