JPH032931A - Arithmetic system for fuzzy inference - Google Patents

Abstract

PURPOSE:To easily and quickly perform the process of the system with the small memory capacity despite a large number of elements of all spaces which define a fuzzy assembly by prescribing this fuzzy assembly with an alpha-level assembly. CONSTITUTION:An input part 1 is used together with a projection processing part 2, a mapping table 3, a Balpha memory 4, and an output 5 in the form of a series circuit. Then all subject fuzzy assemblies are prescribed with an alpha-level assembly. Thus the memory capacity can be reduced despite a large number of elements of all spaces which define a fuzzy assembly in comparison with a case where the fuzzy assembly is prescribed with a membership function. Then the processing quantity is reduced even in such a case where the fuzzy assembly is successively corrected by learning, etc.

Description

[Detailed description of the invention] [Purpose of the invention] (Industrial application field) The present invention relates to a fuzzy inference calculation method.

(Prior Art) Fuzzy inference has been applied in many fields, including various control fields, artificial intelligence, and decision support systems, and its effectiveness has been confirmed.

Particularly in the field of control, 'fuzzy control' enables automatic control of complex systems, which was previously impossible, and has had a great impact on industry.

The fuzzy inference method widely used in these applications is a conditional proposition IF A where the antecedent part is A and the consequent part is B.
By applying the fact A' to then B, the inference result B' is obtained by the inference shown in the following equation.

Conditional proposition: IF A then B Fact: A' Inference result = B' By the way, in fuzzy inference operations, conventionally, when the fuzzy set to be symmetrical is on the space of real numbers or natural numbers, the fuzzy set is called the member. It was defined using a ship function. -b-, fuzzy sets can also be defined by a set family of α-level sets according to the representation theorem.

α-level sets are a very effective form of expression when calculating fuzzy set operations and relational expressions.

In addition, considering the case where a fuzzy inference processing system is implemented using dedicated hardware using digital circuits, α −
Representation of level sets by set families is valid.

Below, we will provide additional details regarding this point.

First, let μA (u) be the membership function that defines the fuzzy set A on the space U, then the α-level set Aα of A is defined by the following equation.

Aα= (u l μA (Ll)≧α, u6U) α
((0, 1,) As is clear from the above equation, the α-level set is a crisp set (non-fuzzy set). Further, when A is a convex fuzzy set, Aα is a closed interval.

A convex fuzzy set is a fuzzy set that satisfies the membership function μ(U) or the following condition.

μ(λu+(1-λ)u)≧ rnin(μ(λu), μ((1-λ)U))1≧λ≧
O A function that satisfies the condition of equation (5) is called a quasi-concave function.

In the case of a convex fuzzy set, the α-level set is a closed interval, so it is expressed here as follows.

Aα! [u 1 , u ,] Based on the above definition, consider a case where a fuzzy inference processing system is implemented using dedicated hardware using a digital circuit.

Considering the membership function as the basis, the memory capacity required to store a 1F fuzzy set is equal to the number of elements in the total space N. This is because it is necessary to store the membership grade for each element of the entire space.

However, if the membership grade is expressed as an α-level set, the memory capacity will be 2M if the membership grade is expressed as M levels. That is, since a convex fuzzy set is assumed here, the a-level set becomes a closed interval, and it is sufficient to store the two values at the left end and right end. Therefore, if we consider recording an α-level set tα with λI for all membership grades, we will need a memory capacity twice as large as the number M of membership grades.

From the above, when writing the membership function as tα, if the number of elements in the total space is increased, the required memory capacity increases in proportion. On the other hand, the method of storing level sets is independent of the number of elements in the total space and depends only on the number of levels of membership greats. In order to increase the dynamic range of the target fuzzy set, it is advantageous to memorize level sets.

That is, if the following condition is satisfied, the memory capacity may be smaller if the fuzzy set is stored in a set family of level sets.

N >> 2M The small memory capacity not only causes a problem of the memory capacity itself, but also leads to a reduction in the repair 1L time when performing sequential repair iE of fuzzy sets.

A similar argument can be made when performing operations between fuzzy sets based on the expansion principle.

The α-level set is a very effective form of expression when calculating fuzzy set operations and relational expressions.

When operations between fuzzy sets based on the expansion principle are defined using membership functions, the membership grade must be determined for each element in the entire space and written into a memory that stores fuzzy sets. In a binary operation on a fuzzy set, when the number of elements in the total space is N, N min operations, a target binary operation, and writing to memory are required. The definition using the α-level set is convenient for digital processing because it involves calculation of numerical values that give the left and right ends of the closed interval corresponding to the α-level set.

In the binary operation of a convex fuzzy set, if the number of levels of membership grade is M, 2 to 1 target
All you need to do is operate on the terms and write them to memory. Therefore, in the fuzzy inference calculation based on the expansion principle, the larger the number of elements in the total space, the more advantageous it is to define the fuzzy set by the α-level set than by the membership function.

From the above, when a fuzzy inference engine is implemented using hardware using digital circuits, the fuzzy set is defined by α
- It is better to define by level set.

(Problem to be solved by the invention) Conventionally, membership functions are stored in ROM or RAM,
The desired membership grade was obtained by accessing the address corresponding to the element for which the membership grade was desired.

Therefore, when the number of elements in the total space that defines the fuzzy set is large, an address space proportional to the number of elements is required, resulting in a problem that a large memory capacity is required. This is not only a problem of memory capacity itself, but also a problem that if it is necessary to sequentially modify fuzzy sets by learning etc., the processing time will increase. In other words, all membership grades stored at all addresses corresponding to the entire space must be rewritten.

SUMMARY OF THE INVENTION Therefore, it is an object of the present invention to provide a fuzzy inference calculation method that can be easily and quickly processed with a small memory capacity even if the number of elements in the total space defining a fuzzy set is large.

[11vt formation of the invention] (Means for solving the problem) The present invention to achieve the above object is based on the fuzzy sets that define the antecedent and consequent parts, facts, and inference results of a conditional proposition, respectively. Let A'B' be A'B', and A is the logical product (AN
Let the fuzzy set ■Ak connected by D) be the fuzzy set Ak', and the fact A' is a fuzzy set Ak' (k-1, 2...
・In the fuzzy inference calculation method in which the fuzzy set 1'jAh' is obtained by combining K') with logical product (AND),
The total space defining the fuzzy set A of the antecedent part of the conditional proposition is Uk, the total space defining the fuzzy set B of the consequent part is V, and the α-level set of the fuzzy set Ak of the facts and the conditional proposition Direct product Ak'α・A2' α...Ak obtained from the space V that defines the fuzzy set of the consequent of
α・V and the fuzzy set n A of the antecedent part of the conditional proposition
b and the consequent part B of the conditional proposition, the fuzzy relation R of +1 term is logically ANDed with the a-level set Rα, and the set obtained by projecting this logical product onto the space V is Element u' = (u
', , u' 2 , ......uo) for the fuzzy set A of membership grade α°, and the inference result α for all elements U' that give this α°.
- Prepare a table that gives the level α whether the level set is an α-level set with the fuzzy set B of the consequent part, and use the α° as a key to draw the table to obtain 11t.
α′-level set B of B obtained from the level α
α′ is the a-level set B of the fuzzy set B′ of the inference result.
′ α.

(Operation) In the fuzzy inference calculation method of the present invention, all target fuzzy sets can be defined by α-level sets, and compared to the case where fuzzy sets are defined by membership functions, the entire space defining the fuzzy sets can be defined. Even when the number of elements is large, the required memory capacity is small, and even when the fuzzy set is successively modified by learning etc., the amount of processing is reduced.

(Example) Hereinafter, the present invention will be explained in detail using the drawings.

First, it is assumed that the fuzzy inference targeted here is based on extended fuzzy inference according to the following format.

Conditional proposition: l f [xisAk and xlsA
2ant1 ・-and x isA* ) ] th
en (YisB) fact: (x is
Ak') and (x isA2')
and-...and(x IsA *) Inference result: B' Here, Ah, Ah, B, and B' are all fuzzy sets, and the spaces Uk, V that define the fuzzy sets
is expressed by the following formula.

Further, the antecedent part is given by ■Ak, and is defined as follows, with 8 being the min operation.

The inference result B' is obtained by combining the +1-term fuzzy relationship table nA*' obtained from Ak and B.

Here, 11ax-1n synthesis is used for the synthesis calculation. In this case, the inference result B' can be obtained using the following equation.

...8uh, (Llm))8u++ (ub lj
2%+・, ljm; u): l'/u −
(3) The above fuzzy relationship R is defined by the following formula: R''''' μA+(u) △II A2 (u
) Δμ＾t(u) △・・・△μAk (u)
... (6) However, Δ represents the min operation.

Rt)-μA+(u)*μ^2(u)*μ^t(1
7) *・ * μA-(u) ... (7)
However, * represents an algebraic product.

FIG. 1 is a block diagram showing the configuration of a fuzzy inference calculation device according to an embodiment of the present invention.

As shown in the figure, the fuzzy inference calculation device shown in this example includes an input section 1, a projection processing section 2, a mapping table 3, and a
It is composed of a series circuit of an α-memory 4 and an output section 5.

The projection processing unit 2 uses direct viAk manually operated by the human power unit 1.
+ ' crXA2 (ZXAk a×a −
xAh' α×V and the α-level set R of the fuzzy relation R
The element 0°-(uol +
The membership grade α of the fuzzy set A of u 2 + ” u ′ @ ) is determined.

Mapping table 3 is α゜! j, element U′
The level a indicates which α-level set of the fuzzy set B of the consequent part is the α-level set of the inference result for .

The Bα-memory 4 is the B that is obtained from the level α′ obtained by drawing the mapping table 3 using α° as a key.
Give α. This Bα′ is the fuzzy set B of the inference result
' is α-level set B' α.

Figure 2 is an explanatory diagram of processing for fuzzy relation Rs;
The figure is an explanatory diagram of processing for fuzzy relation Rg, and Fig. 4~
Figure 7 shows the fuzzy relationships Rs and Rg.

This is a processing flowchart 1 corresponding to Rc and Rp, respectively.

First, an inference calculation method using the fuzzy relation Rs is shown in FIGS. 1, 2, and 4.

In the projection processing unit 2, α is determined in step 401 using the following equation.

a” mm in [μAh (uh,' ), μ^
m (uh+')] where u *+" u k,'
are the left end and right end of the closed interval giving the α-level set of the fuzzy set Ak', respectively.

In step 402, the mapping table 3 calculates α゜≠0
If α゜≠0, assign itself to the key α. That is, in the case of the fuzzy relation Rs,
The output from the mapping table 3 is α°.

Therefore, this α° is manually entered into the Ba-memory 4, and Ba0 is output at step 403. This Bα° is the α-level set B′ α of the fuzzy set B′ that gives the inference result.
becomes.

However, if α°-0 is determined in step 402, Ba-memory 4 is assumed to provide space V. K-
This process is illustrated in FIG. 2 for case 2.

In FIG. 2, the left and right ends of the level set of the fuzzy set shown in (a) and (b) are given to the projection processing unit 2, and when α゜≠0, the minimum value among these four end points is adopted, α゜ or V is Ba- through the mapping table 3.
It is output to memory 4, where B'α-Ba0 or B'
It is shown that α−φ is equal to j.

Next, the inference calculation method when using the fuzzy relation Rg is shown in FIGS. 1, 3, and 5.

The projection processing unit 2 calculates α using the following equation.

tl :in t n [u^* (u,'
>+uA* (u k,'), α in the mapping table 3 in the costep 501, as in the case of the fuzzy relation Rs. When ≠O, let α° itself be given to the key α. Therefore, this α° is input to the Ba-memory 4, and Ba0 is output. Step 503 Lever B
a.

becomes the α-level set B' α of the fuzzy set B' that gives the inference result. However, in the case of αo-0, the Ba-memory 4 determines this in step 502 and provides the space V in step 504.

FIG. 3 illustrates the processing for case 2-2. In the figure, each endpoint and a of the plurality of fuzzy sets shown in (a) and (b) are obtained in the projection processing unit 2, and the mapping table 3 and Ba-memory 4 calculate the result α of the rn i n operation.
This shows that when α ≠ 0, Ba is expressed as Ba-Ba, and when α ≠ 0, Ba is expressed as space ■.

The inference calculation method when using the fuzzy relation Rc is shown in FIGS. 1 and 6.

The projection processing unit 2 calculates α° using the following equation (steps 601 to 603).

Depending on whether the result is the empty set φ and the magnitude relationship between α and α, α° is calculated using the following formula (steps 701 to 705
).

... (8) Mapping table 3 follows the following formula (Step 11 formula%) Using α' given in the above formula, the α'-level set Bα' of B, where α is the level, is calculated using the fuzzy function that gives the inference result. α-level set B′α of set B′ (step 60
8 to 610) However, when α' is φ, the empty set φ is set to B'α.

Finally, FIG. 7 shows an inference calculation method using the fuzzy relation Rp.

Level α of A, α for wm l, - level set A
The logical product Aα1 nA' of the α1-level set A′α, for the level α1-1 of α, and A′
In the case of (10), where, al,, -tna x [μ^ (uhj')
, μ^ (uh, ′)]... (11). The Matuping fist table follows the following formula (S% formula%
) Based on α′ obtained by the above formula, B with α′ as the level
Let the α'-level set Bα of the inference result be the α-level set B'α of the fuzzy set B' with 5 inference results. However, α
is φ, let the empty set φ be B'a.

As explained above, in this example, the required memory capacity is small even when the number of elements in the total space that defines the fuzzy set is large, and even when the fuzzy set is sequentially modified by learning etc. Quantity decreases. In addition, operations between fuzzy sets based on the expansion principle are greatly facilitated and the operation speed is improved by defining fuzzy sets using level sets. Therefore, when an operation based on the expansion principle is introduced into a fuzzy inference engine, the representation form of the fuzzy set is unified and the overall processing efficiency is improved.

Furthermore, in this method, the inference KW is calculated independently for each α-level set.
, it is possible to obtain inference results for the required membership grade.

The present invention is not limited to the above-described embodiments, and can be implemented with arbitrary modifications without departing from the gist of the present invention.

〔Effect of the invention〕

As described above, since the present invention is a fuzzy inference calculation method as described in the claims, even if the number of elements in the total space defining a fuzzy set is large, it can be easily and quickly processed with a small memory capacity. I can do it.

[Brief explanation of drawings]

FIG. 1 is a block diagram showing a fuzzy inference device according to an embodiment of the present invention, FIGS. 2 and 3 are explanatory diagrams showing a processing method for fuzzy relationships Rs and Rg, and FIGS. Relationship Rs, Rg, Rc. It is a flowchart which shows the inference procedure of Rp. 1... Human power section 2... Projection processing section 3... Mapping table 4... Bα-memory 5... Output section

Claims (5)

[Claims] (1) Fuzzy sets that define the antecedent and consequent parts, facts, and inference results of a conditional proposition are A, B, A', and B', respectively.
and A is a set of K fuzzy sets A_k (k=1, 2, .
...K) is a fuzzy set ■A_k that is connected by logical product (AND), and the fact A' is a fuzzy set A of K' pieces.
_k' (k=1, 2,...K') is logically ANDed (A
In the fuzzy inference calculation method where the fuzzy set A_k' is a fuzzy set connected by ND), the total space defining the fuzzy set A of the antecedent part of the conditional proposition is U_k, and the total space defining the fuzzy set B of the consequent part of the conditional proposition is U_k. Let V be the direct product A_1′α×A_2′α×···×A_ obtained from the space V that defines the α-level set of the fuzzy set A_k′ of the facts and the fuzzy set of the consequent of the conditional proposition.
k'α×V and the fuzzy set of the antecedent part of the conditional proposition ■
K+1 obtained from A_k and the consequent part B of the conditional proposition
The set obtained by performing the logical product of the fuzzy relation R of the terms with the α-level set Rα and projecting this logical product onto the space V is the α of the inference results for the elements on the space U_k.
−Element u゜=(u゜_1
, u゜_2, ... u゜_k) for the fuzzy set A, and calculate the inference result α for all elements u′ that give this
- Prepare a table that gives the level α which α-level set of fuzzy set B of the consequent part the level set is, and obtain it from the level α′ obtained by searching the table using the α゜ as a key. a′-level set Bα′ of B
is the α-level set B′α of the fuzzy set B′ of the inference result.
A fuzzy inference calculation method characterized by the following. (2) In the fuzzy inference calculation method according to claim 1, membership functions of the fuzzy sets A_k and A_k' are defined as μ_A___k(u) and μ_A_k'(u), respectively.
When the left and right ends of the closed interval that gives the α-level set of A_k' are u■ and u■, respectively, then α゜ is expressed as α゜=m■n[μ_A_k(u■), μ_A_k(U■
)], and the table gives α゜ itself for α゜, and if α′ obtained by drawing this table using α゜ as a key is not 0, then α′ of B whose level is α′ is obtained. - A fuzzy inference operation characterized in that the level set Bα' is the α-level set B'α of the fuzzy set B' that gives the inference result, and when α' is 0, the entire space V is set as B'α. method. (3) In the fuzzy inference calculation method according to claim 1, membership functions of the fuzzy sets A_k and A_k' are μ_A(u) and μ_A_k'(u), respectively;
When the left and right ends of the closed interval giving the α-level set of the fuzzy set A_n' are u■ and u■, respectively, the above α゜ is α゜=m■n[μ_A_k(u■), μ_A_k(μ■
), α], and the table gives α° itself for α°, and if α′ obtained by drawing this table using α° as a key is not 0, then the value of B whose level is α′ is calculated by Let the α′-level set Bα′ be the α-level set B′α of the fuzzy set B′ that gives the inference result, and α
A fuzzy inference calculation method characterized in that when ' is 0, the entire space V is set to B'α. (4) In the fuzzy inference calculation method according to claim 1, an α-level set Aα of A and an α-level set A′ of A′ are provided.
Depending on whether or not the result of the logical product Aα■A′α of α is the empty set φ, the above α゜ can be changed to α゜=[α, Aα∩A′α≠φ [φ, Aα∩A′α = φ, and the above table gives α′, which is given by this α゜, when α′=[α, α≦α〜[φ, α>α゜ or α゜=φ. ,α′
is not 0, the α′-level set Bα′ of B whose level is α′ is set to α of the fuzzy set B′ that gives the inference result.
- level set B'α, and if α' is φ, then the empty set φ
A fuzzy inference calculation method characterized in that B'α is a fuzzy inference calculation method. (5) In the fuzzy inference calculation method according to claim 1, when α_m=max[μ_A(u■), μ_A(u■)], α_1−level set Aα for level α_1=1 of A
Depending on whether the result of the logical product Aα_1∩A'α_1 of α_1-level set A'α_1 with respect to level α_1=1 of _1 and A' is the empty set φ and the magnitude relationship between α_m and α, the above α° can be calculated as follows: α゜=[1, Aα_1∩A′α_1≠φ [a_m
, Aα_1∩A′α_1=φ and α_m≧α [φ, Aα_1∩A′α_1=φ and α_m<α, and the table above is based on this α゜, α′=[α/α゜, α When /α゜≦1 [φ, α′ given when α/α゜>1 or α゜=φ is given, and if α′ is not φ, α′ of B with α′ as the level - A fuzzy inference operation characterized in that the level set Bα′ is the α-level set B′α of the fuzzy set B′ that gives the inference result, and when α′ is φ, the empty set φ is set as B′α method.