JPH06119177A - Non-fuzzifying method for fuzzy inference - Google Patents

Abstract

PURPOSE:To provide a non-fuzzifying method for fuzzy inference capable of shortening computing time under required arithmetic accuracy and a proper circuit scale. CONSTITUTION:Assuming singletons head-cut by a representative point and the grade of a consequent part for each of the membership functions of plural output labels as yi, gi (i: natural number >=2), (n) head-cut singletons g1, g2,..., gn (n: natural number >=2) are selected in sequence of higher value, and a centroid point or a definition point y0 that is a result of non-fuzzifying is computed based on approximate equation y0 1+g1(y2-y1)+g3(y3-y1)+...+gn(yn-y1).

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a defuzzification method in fuzzy inference used for controlling various home electric appliances and vehicles.

[0002]

2. Description of the Related Art Fuzzy control using fuzzy inference is being applied to a wide range of existing control such as control of various home appliances and vehicles.

The multiple fuzzy inference, which is the core of this fuzzy inference, is based on the degree of coincidence (grade) between the fuzzy concept (input label) and the fact (input data) included in the antecedent part for each of a plurality of rules. Of the fuzzy concept (output label) included in the consequent part of each rule by performing the "min-max operation" on the pre-stage part that calculates the grade of It consists of an intermediate stage where the membership function is truncated according to the calculated consequent grade, and a latter stage where a definite conclusion (output data) is obtained from the truncated output label membership function.

The latter part of the above fuzzy inference is called defuzzification, and the definite conclusion obtained by this is called the centroid or the definite point of the truncated membership function. There is. Conventionally, for this defuzzification, a barycentric method, a simplified barycentric method that simplifies the barycentric method, and the simplest maximum height method have been proposed.

That is, as shown in FIG. 2, for the three output labels B1, B2, B3, the fuzzy variable y
And the relationship between this fuzzy variable y and the fitness of each output label is the membership function μ _B1 (y), μ _B2 (y),
It is defined by μ _B3 (y) and each membership function is truncated by the consequent grades g ₁ , g ₂ , g ₃ calculated based on the min-max operation for the grade of the input label. To do. First, according to the strictest centroid method, the fuzzy variable y that should be determined as output data
The center of gravity y _{0 of} is given by the following equation. y ₀ = [Σ _{i = 1,2,3} Σ _y y · μ _Bi (y)] / [Σ _{i = 1,2,3} Σ _y μ _Bi (y)] (1) However, fuzzy variables y is a discrete value, and the symbol Σ _y
Means addition over the entire fuzzy variable y.

Next, according to the simplified center of gravity method, which is a simplification of the above-described center of gravity method, each membership function of FIG. 2 is, as shown in FIG. 1, only at the respective representative positions y ₁ , y ₂ , y ₃ . It is approximated by a singleton so that the unit quantity becomes 1.0. Each membership function approximated by this singleton is truncated by grades g ₁ , g ₂ , g ₃ of the consequent part, and the center of gravity yo is given by the following equation. y ₀ ≅Σ _{i = 1,2,3} (y _i · g _i ) / Σ _{i = 1,2,3} (g _i ) ... (2)

Further, according to the simplest maximum height method,
The center of gravity y ₀ is approximated by the representative position y ₁ of the largest one (g ₁ ) among the singletons truncated by the grade of the consequent part, as in the following equation. y ₀ ≈ y ₁ (3)

Conventionally, in the control using the fuzzy inference as described above, the center of gravity method is mainly used as a defuzzification method for low-speed control of home electric appliances, and the maximum height is sacrificed for accuracy in high-speed control. The law has been adopted.

[0009]

If the fuzzy control is to be applied to a technical field requiring relatively high speed such as traveling control of an automobile and suspension control,
It is necessary to dramatically reduce the conventional calculation time, typically by about three digits, while keeping the circuit scale within a reasonable range. This circuit size reduction and calculation time reduction are due to the pre-stage of calculating the input label grade, the intermediate stage of calculating the consequent grade based on the min-max calculation of the calculated input label grade, It is necessary to cut off the membership function of the output label according to the grade of the consequent part and calculate the representative point while harmonizing each defuzzification in the latter part.

The applicant of the present invention has made several patent applications before and after this patent application, the former part for calculating the grade of the input label, and the min-max of the grade of the input label calculated in this former part. An invention is disclosed in which the processing time of the intermediate stage of calculating the grade of the consequent part based on the calculation is greatly shortened under a small circuit scale. Therefore, for defuzzification which is the final stage of fuzzy inference, it is necessary to shorten the processing time and simplify the arithmetic circuit.

For the purpose of only shortening the defuzzification processing time, the maximum height method of the simplest algorithm may be adopted. However, in this maximum height method, averaging processing for already calculated multiple grades is omitted, so fluctuations in the control system cannot be buffered, and convergence to a steady value may be delayed or unstable. There are problems such as. That is, in the example of FIG. 2, g ₁ > g ₂ at some point.
However, at the next time, this magnitude relationship reverses and g ₁ <
If a fluctuation such as g ₂ exists in the control system, the representative point y ₀ reciprocates between y ₁ and y ₂ , which causes a problem such as an increase in overshoot or undershoot in control characteristics. . Therefore, a main object of the present invention is to shorten the processing time for defuzzification, which is the final stage of fuzzy inference, while realizing required processing accuracy and reduction in circuit scale.

[0012]

According to the defuzzification method of the present invention for solving the above-mentioned problems of the prior art, the representative point and the consequent part for each of the membership functions μ _Bi (y) of a plurality of output labels. If the singletons truncated according to the grade are y _i and g _i (i is a natural number greater than or equal to 2), then g (n is a natural number greater than or equal to 2) g in descending order from the truncated singleton ₁ , g ₂ ...
..Selecting g _n , the center of gravity y which is the result of defuzzification
₀ for y ₀ ≈ y ₁ + g ₂ (y ₂ −y ₁ ) + g ₃ (y ₃ −y ₁ ).
The calculation is based on the approximate expression of + ... + g _n (y _n −y ₁ ). Furthermore, as is often used in fuzzy inference, when the variable range of the output variable is set so that the label centered on zero (representative point) has the maximum grade, the representative point y of the maximum grade is near the steady state. _{Since 1} becomes zero, the above equation can be approximated as y ₀ ≈g ₂ y ₂ + g ₃ y ₃ + ... + g _n y _n .

[0013]

If the simplified center of gravity method exemplified in the equation (2) is expanded when there are a sufficient number of membership functions, the following equation is obtained. y ₀ ≅Σ _{i = 1,2,3 ・ ・ ・} (y _i · g _i ) / Σ _{i = 1,2,3 ・ ・ ・} (g _i ) ・ ・ ・ (4) Head depending on the grade of consequent part cut singleton leaving only (hereinafter referred to as "grade") n number of grade g ₁ to a large order of _{all, g 2, g 3 ··· g} n, if omit the other grades, the following We obtain an approximate expression like this. y ₀ ≈ (y ₁ · g ₁ + y ₂ · g ₂ + y ₃ · g ₃ + ... + y _n · g _n ) / (g ₁ + g ₂ + g ₃ ... + g _n ) ... (5) Next Then, the following equation is obtained by modifying the equation (5). y ₀ ≈y ₁ + [g ₂ (y ₂ −y ₁ ) + g ₃ (y ₃ −y ₁ ) + ... + g _n (y _n −y ₁ )] / (g ₁ + g ₂ + g ₃ ... + G _n ) ... (6)

Further, in the expression (6), the following approximate expression is obtained by approximating g ₁ + g ₂ + g ₃ ... + g _n ≈1 (7). y ₀ ≈y ₁ + g ₂ (y ₂ −y ₁ ) + g ₃ (y ₃ −y ₁ ) + ... + g _n (y _n −y ₁ ) ... (8)

Fuzzy inference employs a rough approximation in which the product of grades is selected by selecting the minimum grade, and the sum of grades is substituted by selecting the maximum grade. Considering the points, it is considered that the approximation of the expression (7) does not cause a problem that it is too coarse as compared with the accuracy of the approximation adopted in the entire fuzzy inference. Further, in the control using fuzzy inference, as the size of a certain grade approaches the upper limit value and all other grades tend to approach zero as the non-control system approaches the target value, ( It is considered that the approximation of the equation (7) does not cause much problem in terms of accuracy.

According to a preferred embodiment of the present invention, (8)
By setting n to 2 in the equation, the following equation is obtained. y ₀ ≈y ₁ + g ₂ (y ₂ −y ₁ ) ... (9) According to another preferred embodiment of the present invention, the following equation is obtained by setting n to 3 in equation (8). obtain. y ₀ ≈ y ₁ + g ₂ (y ₂ −y ₁ ) + g ₃ (y ₃ −y ₁ ) ... (10)

[0017]

EXAMPLE A simulation result based on the equation (9), which is an example of the present invention, will be described. That is, FIG.
As shown in FIG. 2, a cart 2 with wheels is placed on a horizontal rail 1, and a pin 3 is fixed to the central portion of the top of the cart,
A physical model in which a rod 4 having a uniform thickness is supported by the pin 3 so as to have a degree of freedom of rotation only around an axis passing through the pin 3 and perpendicular to the paper surface is assumed. Consider a control model in which a horizontal force F (bipolar) is applied to the carriage 2 to bring the rod 4 closer to a vertical state.

The inclination of the rod 4 from the vertical plane is θ, and the rate of change with time is dθ / dt. These are set as input fuzzy variables. For these input fuzzy variables, seven input labels, zero (ZR), large (± L) on both positive and negative sides, intermediate (± M), and small (± S) are set, and the membership function of each is shown. It is defined as shown in 4. In addition, the inclination θ from −60 ° to + 60 °
And the rate of change over time from −150 ° / s to + 150 ° / s are both represented by internal variables from 0 to 255.

On the output side, the differential force dF is set as a fuzzy variable, and this fuzzy variable is also
Similar to the input label, seven output labels of zero (ZR), large (± L) on both positive and negative sides, intermediate (± M), and small (± S) are set, and the membership function of each is shown in FIG. It is defined by a singleton as shown in. Note that −
Differential force d varying from 25Newtong to + 25Newtong
F is represented by an internal variable from 0 to 255.

Then, the following fuzzy rule as shown in FIG. 6 is set between the input label and the output label. If θ = ZR and dθ / dt = −L then dF = −L If θ = ZR and dθ / dt = −M then dF = −M If θ = ZR and dθ / dt = −S then dF = −S If θ = ZR and dθ / dt = ZR then dF = ZR If θ = ZR and dθ / dt = + S then dF = + S If θ = ZR and dθ / dt = + M then dF = + M If θ = ZR and dθ / dt = + L then dF = + L If θ = -L and dθ / dt = ZR then dF = -L If θ = -M and dθ / dt = ZR then dF = -M If θ = -S and dθ / dt = ZR then dF = −S If θ ＝ ＋ S and dθ / dt ＝ ZR then dF ＝ ＋ S If θ ＝ ＋ M and dθ / dt ＝ ZR then dF ＝ ＋ M If θ ＝ ＋ L and dθ / dt ＝ ZR then dF ＝ ＋ L If θ ＝ ＋ S and dθ / dt = -S then dF = + S If? = -S and d? / dt = + S then dF = -S The above fuzzy rule is illustrated in Fig. 6.

However, as can be seen from the above fuzzy rule, the information regarding the position of the truck is not taken into consideration in this simulation, and therefore, the control regarding the stop position of the truck in the steady state is not performed.

Further, the mass of the rod is m _r , the length is 2 L, the mass of the truck is m _c , the friction coefficient between the wheel and the rail 1 is u _c ,
Assuming that the friction coefficient between the pin 3 and the rod 4 is u _p , the constant of gravity is G, and the angle between the vertical plane and the rod 4 is θ, the equation of motion is
It is given by the following formula. d ² θ / dt ² = [G · sin θ- (dθ / dt ) (u p / m r L) + cos θ _{[-F-m r L × (dθ} / dt) 2 sin θ ± u c ] / ( m _c + m _r )] / L [4 / 3−m _r cos ² θ / (m _c + m _r )] (11) d ² x / dt ² = [F− ± u _c + m _r L [sin θ (dθ / dt) ² −cos θd ² θ / dt ² ]] / [m _c + m _r ] ... (12)

For the physical model shown in FIG. 3, θ, d
The initial values of θ / dt and F are 35 °, 0 ° / s, and
0 Newtong and the differential force dF which is the control amount for this
Is calculated by a fuzzy operation, a new F is calculated by adding dF to the current F, and θ and dθ with respect to F are calculated.
The simulation was performed by calculating / dt using the equations (11) and (12), and finding a new control amount dF for the θ and dθ / dt by fuzzy calculation. The result of this simulation is shown in FIG.

The solid line shown in FIG. 7 is the result when the approximate calculation method of the equation (9) of this embodiment is adopted, and the broken lines are Δ type, S type, and 7 type output labels as shown in FIG. This is the result when the most rigorous and complicated operation method of defining a Z-type membership function and performing complete centroid calculation according to equation (1) is adopted. The two-dot chain line in FIG. 7 is the result of applying the calculation method of the formula (2) to the one obtained by approximating the seven membership functions with the seven singletons and cutting them off. This is the result when the arithmetic method of the formula (2) is adopted only for the three largest ones out of the seven cut singletons. Furthermore, the alternate long and short dash line is the result when the maximum height method shown in equation (3) is adopted.

When the complete center of gravity calculation is performed according to the equation (1), the best result closest to the state of critical braking is obtained. The next best result in terms of less ripple due to overshoot and the like is obtained for the 3-point method and the 7-point method. However, a steady error occurs in the 3-point method and the 7-point method, and the reason is considered to be as follows.
That is, in the strict center of gravity method, the center of gravity is calculated by excluding the overlapping part of each membership function, whereas in the simple center of gravity method by the singleton, since the overlapping part of the membership function does not occur, such exclusion is performed. It depends on what you do not forget.

The second best result in that the ripple and the steady error are small is obtained for the two-point method of this embodiment. In particular, the point to be noted is that the steady-state error in the two-point method of this embodiment converges to the same value as when the strict centroid calculation is adopted. This is because the second largest grade, which appears around the maximum grade, changes frequently at each fuzzy operation (oscillates around the maximum grade), which is unique to the 3-point method and the 7-point method. It is considered that the steady error is compensated. That is, the disadvantage of the present embodiment that the ripple is somewhat large is sufficiently compensated by the advantage that the calculation time is extremely short. In addition,
It is the result based on the maximum height method shown by the one-dot chain line,
As far as this control model is concerned, it is not practical in that the ripple and the steady-state error are excessive.

[0027]

As described in detail above, according to the defuzzification method of the present invention, n grades g in descending order.
₁ , g ₂ ... G _n are selected, and y ₀ ≈y ₁ + g ₂ (y ₂
-Y ₁ ) + ... + g _n (y _n −y ₁ ) is used to calculate the barycentric point y ₀ , so as verified by simulation, the calculation speed remains accurate. With this, the effect that the scale of the arithmetic circuit can be significantly reduced is obtained.

[Brief description of drawings]

FIG. 1 is a conceptual diagram for explaining the principle of the present invention while comparing it with a conventional simple center of gravity method and a maximum height method.

FIG. 2 is a conceptual diagram for explaining a conventional centroid method.

FIG. 3 is a conceptual diagram showing the configuration of a controlled physical model used to confirm the effect of one embodiment of the present invention.

FIG. 4 is a conceptual diagram showing a membership function of an input label set for the model of FIG.

5 is a conceptual diagram showing a membership function of output labels set for the model of FIG.

6 is a block diagram showing a fuzzy rule set for the model of FIG. 3. FIG.

7 is a conceptual diagram showing simulation results obtained for the model of FIG.

[Explanation of symbols]

g _{1 to} g ₃ Consequent grade y _{1 to} y ₃ Center value of output label membership function μ _Bi (y) Output label membership function

Claims (6)

[Claims] 1. A membership function μ of a plurality of output labels.
For each _Bi (y), the representative position and the singleton truncated by the grade of the consequent part are y _i , g
_{When i} (i is a natural number of 2 or more), n pieces (where n is a natural number of 2 or more) in descending order from the truncated singletons g ₁ , g ₂ ...
..Selecting g _n , the center of gravity y which is the result of defuzzification
₀ for y ₀ ≈ y ₁ + g ₂ (y ₂ −y ₁ ) + g ₃ (y ₃ −y ₁ ).
A defuzzification method in fuzzy inference, which is characterized in that an operation is performed based on an approximate expression of + ... + g _n (y _n −y ₁ ). 2. The fuzzy according to claim 1, wherein n is 2 and the center of gravity y ₀ is calculated based on an approximate expression of y ₀ ≈y ₁ + g ₂ (y ₂ −y ₁ ). Defuzzification method in inference. 3. The n according to claim 1, wherein n is 3, and the center of gravity y ₀ is approximated to y ₀ ≈y ₁ + g ₂ (y ₂ −y ₁ ) + g ₃ (y ₃ −y ₁ ). A defuzzification method in fuzzy inference, which is characterized in that it is calculated based on 4. The method according to claim 1, wherein y ₁ is zero, and the center of gravity y ₀ is calculated based on an approximate expression of y ₀ ≈g ₂ y ₂ + g ₃ y ₃ + ... + g _n y _n A defuzzification method in fuzzy inference that is characterized by: 5. The defuzzification method in fuzzy inference according to claim 4, wherein n is 2 and the barycentric point y ₀ is calculated based on an approximate expression of y ₀ ≈g ₂ y ₂ . 6. The fuzzy inference according to claim 4, wherein n is 3 and the barycenter y ₀ is calculated based on an approximate expression of y ₀ ≈g ₂ y ₂ + g ₃ y _3. Fuzzification method.