JPH04275633A - Construction system for membership function by neural network - Google Patents

Abstract

PURPOSE:To approximate a neuron characteristic function while minimizing the absolute value of the error against a membership function without performing learning by realizing the membership function in a fuzzy control with the neuron characteristic function. CONSTITUTION:The maximum gradient of the sigmoid function as a neuron characteristic function is set by obtaining the weight and threshold value with 1.3401 times of the gradient of the membership function large or small.

Description

[Detailed description of the invention]

0001

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for configuring membership functions using a neural network, which makes it possible to realize membership functions in fuzzy control using neuron characteristic functions. Fuzzy control has traditionally been applied in fields such as plant control where mathematical models cannot be established, but control using neural networks has recently attracted attention due to its development time, accuracy, and ease of adjustment. However, neural networks need to be adjusted to a neuron characteristic function that approximates the membership function through learning, which takes time, and the state of function approximation is determined by learning methods such as backpropagation. It is desirable to obtain neuron characteristic functions that approximate membership functions without being constrained by such learning methods.

0002

2. Description of the Related Art FIG. 8 shows an outline of conventional fuzzy control. Here, input variables X1 and X2 and output variable Y are used. Rule 1; if X1 is small
hen Y is large Rule 2; if X2 is large
If hen Y is small, the proportion (degree of belonging) of the actual value of the input variable Input variable X2
Rule 2 shows the proportion of the actual value of large that belongs to large.
Let the fitness of Furthermore, the membership functions of the output variable Y in each rule 1 and 2 are set to be large and small, respectively, as shown in FIGS. 8(c) and 8(d).

[0003] Therefore, the output of fuzzy control for arbitrary input values of input variables X1 and Figure 8 created by superimposing those multiplied by degrees
This is the center of gravity of the figure in (e).

0004

[Problems to be Solved by the Invention] Fuzzy control has the advantage of being able to control with a small number of rules that are easy to understand, but on the other hand, it has the problem that it is difficult to adjust the finished product. Therefore, we consider replacing the membership function with a neural network that can automatically adjust through learning.

FIG. 9 shows an example of representation of a membership function using one neuron. Note that FIG. 9 represents the membership function large in FIG. 8(c). This network takes the actual value of the input variable X1 as input, and outputs the degree of membership of the value of the input variable X1. The conventional method used to express membership functions in neurons is to first create many teacher signals using membership functions, and then use the teacher signals to make neurons learn, for example, by backpropagation. Ta.

FIG. 10 shows the input/output relationship when the input variable X is taken in steps of 0.05, and this is used as a teacher signal. This input/output relationship teaching signal is obtained from the membership function plotted in FIG. FIG. 12 shows the results of learning the teacher signal in FIG. 10 10,000 times. Figure 1
As shown in (11) and (12) of 2, w=17.740294 θ=7.982747 after 10000 learnings.

In FIG. 12, the other elements are as follows. (1) Indicates that the network structure is one layer of one neuron unit. (2) Indicates learning constant ε=5. (3) Indicates momentum α=0.4. (4) Indicates constant β=0.

(5) Indicates tolerance = 0.1. (6) Indicates the maximum number of times that learning can be performed, 10,000 times. (7) Range of output values given as sigmoid function 0
~1 is shown. (8) Indicates the table number of a random number table for initializing weights and thresholds (actually, a seed that provides a random number generation function).

(9) Indicates the value of the weight w of the neuro unit obtained through learning. (10) Shows the value of -θ of the neuro unit obtained through learning. (11) The presence or absence of weight learning of the neuro unit is indicated by 1 or 0. (12) The presence or absence of threshold learning of the neuro unit is indicated by 1 or 0. (13) Indicates that learning has been performed 10,000 times.

(14) Indicates how many times learning is to be monitored. When the learning results in FIG. 12 are graphed, it can be seen that the neuron characteristic function B is a good approximation to the membership function A, as shown in FIG. 13. By the way, the backpropagation method used for neuron learning searches for a solution that minimizes the sum of squared errors with respect to a teacher signal. However, when approximating a membership function, what humans desire is not necessarily the minimum sum of squared errors. For example, you may want to match the maximum value of the output displacement with respect to the input displacement to the slope of the membership function, in other words, you may want to clarify the upper limit of sensitivity, you may want to minimize the sum of errors, or you may want to minimize the maximum error. be.

However, in the backpropagation method,
There was a problem in that it was not possible to vary the meaning when approximating the neuron characteristic function to the membership function. Also, when performing approximation to minimize the sum of squares of errors using the backpropagation method, as many as 10,000 learning sessions were required. The present invention has been made in view of such conventional problems, and it is an object of the present invention to approximate the neuron characteristic function so as to minimize the sum of the absolute values of errors with respect to the membership function without the need for learning. The purpose of this paper is to provide a membership function construction method using a neural network that can perform the following functions.

0012

[Means for Solving the Problems] FIGS. 1 and 2 are diagrams explaining the principle of the present invention. In FIG. 1, the present invention first shows that FIG.
As shown in b), there are a, b(
However, when a<b) is a constant, when X≦a, Y=0 When a<X<b, Y=(X-a)/(ba-a) When b≦X, Y=1. This paper deals with a method of constructing a membership function using a neural network in which a certain membership function is approximated by a neuron 1 having a nonlinear characteristic function.

In the present invention, as shown in FIG. 1(c) for such a configuration method, the sum of the errors with the membership function, more precisely, the integral of the absolute value of the errors is minimized. Set the characteristic function of neuron 1 to . Specifically, the maximum slope of the characteristic function of neuron 1 is set to 1.3401, which is 1/(ba), which is the slope of the membership function. shall be.

In reality, the characteristic function of neuron 1 is defined as Y
= 1/{1+exp(-wX+θ)}, determine the connection weight w and threshold θ as w=5.3605/(ba-a) θ=2.6802(a+b)/(ba-a) The membership function is approximated by the characteristic function of the neuron.

Further, the characteristic function of neuron 1 is defined as Y=0.
It can also be expressed as 5+0.5tanh(wX-θ), so in this case, the connection weight and threshold θ are w=2.6802/(ba-a) θ=1.3401(a+b)/( By determining ba), the membership function is approximated by the characteristic function of the neuron.

Furthermore, the present invention, as shown in FIG. 2(b),
Between input b-a) b≦X
When , the membership function with the relationship Y=0 is approximated by neuron 1 with a nonlinear characteristic function. is 1.3401 times the slope of the membership function, -1/(ba).

In this case as well, when the characteristic function of neuron 1 is Y=1/{1+exp(-wX+θ)}, the connection weight w and threshold θ are w=-5.3605/(b-a). By determining θ=-2.6802(a+b)/(ba-a), the membership function is approximated by the characteristic function of the neuron.

Further, the characteristic function of neuron 1 is defined as Y=0.
5 + 0.5 tanh (w By determining ba), the membership function is approximated by the characteristic function of the neuron.

[0019]

[Operation] According to the membership function construction method using the neural network of the present invention having such a configuration, the maximum slope of the absolute value of the sigmoid function, which is a neuron characteristic function, is 1.3401 times the slope of the membership function. By determining and setting the weights w and θ such that θ is equal to θ, it is possible to uniquely obtain an approximation that minimizes the sum of errors of the neuron characteristic function with respect to the membership function, or more precisely, the integral of the absolute value of the errors.

[0020] Therefore, it is possible to perform an approximation suitable for the purpose that cannot be achieved using the backpropagation method, and furthermore, since there is no need to perform learning as in the backpropagation method, the amount of calculation can be reduced.

[0021]

Embodiment FIG. 3 is a block diagram of an embodiment of a neuron unit realizing a neuron used in the present invention. In FIG. 3, the neuro unit 10 has five inputs and one output as an example. The neuro unit 10 is provided with a multiplication processing section 12, and the multiplication processing section 22 multiplies the input signals X1 to X5 by internal connection weights w1, w2, . . . , w5, respectively. -5 is provided. Here, since the neuro unit 10 of the present invention approximates a membership function with one input and one output, it is sufficient to make only one input connection and weight valid and ignore the others. For example, only input signal X1 is used, and all other input signals X2 to X5 are fixed to 0.

13 is an accumulation processing section, and a multiplication processing section 12
All the multiplication results X1w1 to X5w5 output from are added. Reference numeral 14 denotes a threshold processing section, which performs nonlinear threshold processing on the cumulative value from the accumulation processing section 13 using, for example, a sigmoid function (neuron characteristic function) as an S-shaped function. In the present invention, approximation is performed to minimize the integral of the absolute value of the error of the sigmoid function with respect to the membership function. For example, the membership function la shown in Figure 1(b)
In the case of rge, that is, between input X and output Y, a, b
(However, when a<b) is a constant, when X≦a,
Y=0 a<X<b
When , Y=(X-a)/(ba-a) (
1) When b≦X,
When approximating a membership function with a relationship of Y=1 using the neuro unit 10, the weight is set so that the maximum value of the slope of the sigmoid function is 1.3401 times the slope of the membership function large, 1/(ba). Determine the values of w and threshold θ.

That is, the sigmoid function of the neuro unit 10 is expressed as Y=1/{1+exp(-wX+θ)}
(2
), the connection weight w and threshold θ are w=5.3605/(ba-a)

(3) θ=2.6802(a+b)/(ba-a
)
(4) is determined.

[0024] Furthermore, the sigmoid function of the neuro unit 10 is approximately as follows: Y=0.5+0.5tanh(wX-θ)
(5)
It can be expressed as In this case, the connection weight and threshold θ are w=2.6802/(ba-a)

(6) θ=1.3401(a+b)/(ba-a
)
(7) is determined.

On the other hand, the membership function sm shown in FIG.
In the case of all, that is, between input X and output Y, a, b(
However, when a<b) is a constant and X≦a,
Y=1 a<X<b
When , Y=-(X-a)/(ba-a) (8
) When b≦X,
When a membership function with a relationship of Y=0 is approximated by the neuro unit 10, the maximum slope of the absolute value of the characteristic function of the neuro unit 10 is the membership function small
The weight w and the threshold value θ are determined to be 1.3401 times the slope of −1/(ba).

That is, the sigmoid function of the neuro unit 10 is expressed as Y=1/{1+exp(-wX+θ)}
(9
), the connection weight w and threshold θ are w=-5.3605/(b-a)
(
10) θ=-2.6802(a+b)/(ba-a)
(11
).

[0027] Furthermore, the sigmoid function of the neuro unit 10 is approximately as follows: Y=0.5+0.5tanh(wX-θ)
(12), so in this case, the connection weight w and threshold θ are w=-2.6802/(b-a)
(
13) θ=-1.3401(a+b)/(ba-a)
(14
) can be determined.

Next, the reason why the integral of the absolute value of the error of the sigmoid function with respect to the membership function can be minimized by setting the maximum slope of the sigmoid function to 1.3401 times that of the membership function will be explained in detail. Now the membership function large

[0029]

[Math 1]

(15), and the sigmoid function is Y=
tanh(wX)

Consider approximation by (16). FIG. 4 shows a graph of the membership function of equation (15) and the sigmoid function of equation (16).

In FIG. 4, if s is the value of the X coordinate of the intersection of the first quadrant of the two functions, then s=tanh(ws), and the weight w is

[0032]

[Math 2]

(17). Therefore, half B(w) of the integral of the absolute value of the error for the membership function is

0
034]

[Math 3]

(18). Next, find the weight w that minimizes half B(w) of the integral of the error in equation (18). First, regarding the right side of equation (18),

[0036]

[Math 4]

[0037] Calculating dB(w)/dw from this results in the following.

[0038]

[Math 5]

[0039] Here, finding s such that dB(w)/dw=0 is as follows.

[0040]

[Math 6]

s that satisfies this is approximately s=0.79
99, therefore, from equation (17), the weight w is 4 decimal places.
If all digits are valid, w=1.3401 is obtained.

Here, the membership function large in FIG.
Let f(x) be,

[0043]

[Math 7]

[0044] The sigmoid function f'(X) is

[
0045

[Math. 8]

[0046] Therefore, assuming that both are approximately equal, the membership function f(x) is

[0047]

[Math. 9]

(19). Therefore, the weight w to minimize the integral of the absolute value of the error of the membership function expressed by tanh for the membership function large.
And the threshold value θ may be determined as follows: w=2.6802/(ba-a) θ=1.3401(a+b)/(ba).

Next, when the sigmoid function of the neuro unit 10 is set to Y=1/{1+exp(-wX+θ)} as shown in equation (2) above,
The same can be found, but to find it easier,

[0050]

[Math. 10]

Using this fact, from the result of the above equation (19), the weight w and threshold θ are determined to minimize the integral of the absolute value of the error of the membership function expressed by exp for the membership function large. teeth,

[0052]

[Formula 11] is obtained. Therefore, it is sufficient to determine w=5.3605/(ba-a) and θ=2.6802(a+b)/(ba-a).

In FIG. 5, for the membership function large, a=0.3 b=0.6, and in order to approximate this with the neuro unit 10 having a sigmoid function of exp in the above equation (2), the above (3)
) Based on equation (4), w=5.3605/(ba)=17.8683θ=2
．． 6802 (a + b) / (ba - a) = 8.406 is set, the membership function large and the sigmoid function are shown, and the approximate relationship of the sigmoid function that minimizes the integral of the absolute value of the error with respect to the membership function is the weight. This can be uniquely achieved by setting w and threshold value θ.

On the other hand, in the case of the membership function small shown in FIG. 2(b), that is, between the input X and the output Y,
When a, b (however, a<b) are constants, and X≦a,
Y=1 a<X<b
When , Y=-(X-a)/(ba-a) (20)
When b≦X,
A case will be described in which a membership function having a relationship of Y=0 is approximated by the neuro unit 10 having a nonlinear characteristic function. The sigmoid function of the threshold processing unit 14 in the neuro unit 10 that approximates this membership function small is f(X)=1/{1+exp(-wX+θ)}
(21). In order to approximate the sigmoid function so that the integral of the absolute value of the error with the membership function is minimized, the slope of the membership function small, -1/(ba), should be multiplied by 1.3401. .

Specifically, as the weight w of the sigmoid function expressed by exp in equation (21) and the threshold value θ, w=
-5.3605/(b-a)
(22)
θ=-2.6802(a+b)/(ba-a)
(23) may be set.

[0056] Also, if the membership function small is tan
Approximately using the sigmoid function of h, f(X)=0.5+0.5tanh(wX-θ)
(24), f(X)=0.5+0.5tanh(wX-θ)=1/
Utilizing the fact that {1+exp(-2wX+2θ)}, 2w=-5.3605/(ba-a) 2θ=-2.6802(a+b)/(ba-a) can be obtained. Therefore, the weight w and threshold θ are w=-2.6
802/(b-a)
(25) θ=-
1.3401(a+b)/(ba-a)
(26) may be set.

FIG. 6 shows the membership function larg according to the present invention.
A process flow diagram for determining a weight w and a threshold value θ for approximation to minimize the integral of the absolute value of the error of a sigmoid function for e and small is shown. In FIG. 6, first, a coefficient F is determined in step S1. The coefficient F is F=2 when using exp as a sigmoid function, tanh
When using , set F=1. Then step S2
reads the constants a and b that determine the attributes of the membership function. Next, a weight w is calculated in step S3, and a weight θ is further calculated in step S4, and the process ends.

One or more neural units that approximate a membership function using a sigmoid function according to the present invention are used in a layer that realizes a membership function of a neural network that realizes fuzzy control, and then the entire network is learned. By doing so, it is expected that learning efficiency will be improved. FIG. 7 is a block diagram of a specific embodiment of the neuro unit 10 shown in FIG. 3, and corresponding parts use the same numbers.

In FIG. 7, 12 is a multiplication type D/A converter, 13 is an accumulation processing section, and an analog adder 13a
and a holding circuit 13b. 14 is a threshold processing unit, 15
16 is an output holding section, 16 is an output switch section, 17 is an input switch section, 18 is a weight holding section, and 19 is a control circuit. The input switch section 17 receives the input signals X1 to X5 sequentially, and is controlled to be turned on in synchronization with the input timing of the input signals X1 to X5. Of course, since the neuro unit of the present invention has one input and one output, only the input signal X1 is valid and the other input signals X2 to X5 are 0.

Weight signals w1 to w5 are also transmitted in accordance with the input timings of input signals X1 to X5. Among these, the effective weight W1 is set to a value calculated according to the processing flow shown in FIG. Specifically, the control circuit 19 sequentially outputs the weight input control signals to the receiver 20.
The weight signals w1 to w5 are sequentially sent to the weight holding unit 18 via. As a result, multiplication values X1w1, X2w2, . . . X5w5 are sequentially calculated in the multiplication type D/A converter 12.

Since the holding circuit 13b of the accumulation processing section 13 is cleared to zero in the initial state, the first obtained multiplication value X1w1 is the zero of the holding circuit 13b and the analog adder 13a.
The obtained addition value X1w1 is held. Next, when the multiplication value X2w2 is input, the analog adder 13a
adds X1w1 held in the holding circuit 13b and newly inputted X2w2, and holds the added value (X1w1+X2w2) in the holding circuit. Similarly, the cumulative value Y is Y=X1
It is calculated as w1+X2w2+...X5w5.

When the accumulation processing in the accumulation processing section 13 is completed, the control circuit 18 outputs an arithmetic control signal, and in response to this, the threshold processing section 14 uses the value of the threshold θ calculated in the processing flow of FIG. The accumulated value Y is subjected to threshold processing, and the obtained output Y is temporarily held in the output holding unit 15. Subsequently, the control circuit 19 outputs an output control signal, the output switch 16 is turned on, and the output value Y held in the output holding section 15 is outputted to the outside.

In the above embodiment, the maximum slope of the absolute value of the neuron characteristic function is set to 1.3401 times the slope of the membership function, but if accuracy is not required, , 1.34 times, 1.3 times, etc., or if you want to increase the accuracy, you can further increase the number of significant figures, and the value is not limited to 1.3401.

[0064]

[Effects of the Invention] As explained above, according to the present invention, it is possible to obtain an approximation by a neural network of a membership function that has been determined by learning, without learning, by setting weights and thresholds, and further reduces errors. Since it is possible to perform an approximation that minimizes the sum of absolute values, it is possible to obtain the same level of accuracy as learning using a finite number of points.

[Brief explanation of the drawing]

[Figure 1] Diagram explaining the principle of the present invention (Part 1)

[Figure 2] Diagram explaining the principle of the present invention (Part 2)

[Fig. 3] Example configuration diagram of the neuro unit of the present invention

[Fig. 4] Explanatory diagram for proving the approximate relationship between the membership function and sigmoid function of the present invention

FIG. 5 is an explanatory diagram of the approximation results of the sigmoid function and membership function of the neurounit of the present invention

FIG. 6 is a processing flow diagram showing weight and threshold calculation processing according to the present invention.

[Figure 7] Specific example configuration diagram of the neuro unit in Figure 3

[Figure 8] Schematic diagram of conventional fuzzy control

[Figure 9] Explanatory diagram of membership function representation by neurons

[Figure 10] Explanatory diagram of the teacher signal obtained from the membership function

[Figure 11] Explanatory diagram of the membership function that determined the teacher signal in Figure 9

[Figure 12] Explanatory diagram showing the learning results of neurons

[Figure 1
3] Explanatory diagram showing the approximate state of the characteristic function by the membership function and learned neurons

[Explanation of symbols]

1: Neuron 10: Neuro unit 12: Multiplication processing section (multiplying AD converter) 13: Accumulation processing section 14: Threshold processing section 15: Output holding section 16: Output switch section 17: Input switch section 18: Weight holding section 19 :Control unit 20:Receiver

Claims (7)

[Claims] Claim 1: Between input X and output Y, a, b (however, a
<b) is a constant, when X≦a, Y=0 When a<X<b, Y=(X-a)/(ba-a) When b≦X, Y=1. In a method of constructing a membership function using a neural network in which the membership function is approximated by a neuron (1) having a nonlinear characteristic function, the integral of the absolute value of the error with the membership function is minimized. A method of configuring membership functions using a neural network, characterized in that a characteristic function of the neuron (1) is set. 2. In the method of configuring a membership function using a neural network according to claim 1, the maximum slope of the characteristic function of the neuron (1) is equal to the slope of the membership function, 1/(ba-a). A membership function construction method using a neural network, characterized in that the membership function is set to 1.3401 times as large as 1.3401. 3. In the method of configuring membership functions using a neural network according to claim 2, when the characteristic function of the neuron (1) is Y=1/{1+exp(-wX+θ)}, the connection The membership function is approximated by a neuron characteristic function by determining the weight w and the threshold value θ as w=5.3605/(ba-a) θ=2.6802(a+b)/(ba-a). A method for constructing membership functions using a neural network. 4. In the method of configuring a membership function using a neural network according to claim 2, when the characteristic function of the neuron (1) is Y=0.5+0.5tanh(wX−θ), the connection By determining the weight and threshold θ as w=2.6802/(ba-a) θ=1.3401(a+b)/(ba-a), it is possible to approximate the membership function with the characteristic function of the neuron. A method of constructing membership functions using a featured neural network. Claim 5: Between the input X and the output Y, a, b (however, a
<b) as a constant, when X≦a, Y=1 When a<X<b, Y=-(X-a)/(ba-a) b≦X
In a method of constructing a membership function using a neural network in which a membership function with a relationship of Y=0 is approximated by a neuron (1) having a nonlinear characteristic function, the characteristic function of the neuron (1) is A method for constructing a membership function using a neural network, characterized in that the maximum slope of the absolute value of is set to 1.3401 times the slope of the membership function, -1/(ba). 6. In the method of configuring a membership function using a neural network according to claim 5, when the characteristic function of the neuron (1) is Y=1/{1+exp(-wX+θ)}, the connection Approximate the membership function with a neuron characteristic function by determining the weight w and threshold θ as w=-5.3605/(ba-a) θ=-2.6802(a+b)/(ba-a) A method for constructing membership functions using a neural network. 7. In the method of configuring a membership function using a neural network according to claim 5, when the characteristic function of the neuron (1) is Y=0.5+0.5tanh(wX−θ), the connection By determining the weight and threshold θ as w=-2.6802/(ba-a) θ=-1.3401(a+b)/(ba-a), the membership function is approximated by the characteristic function of the neuron. A method of constructing membership functions using a neural network, which is characterized by the following.