JP2001273278A - Device and method for optimization - Google Patents

Abstract

PROBLEM TO BE SOLVED: To obtain a selection function included within the range of the spread of an error or the like to an entry having this spread. SOLUTION: An optimum area (area held between an upper limit function M+(x) and a lower limit function M-(x)) is set by the entry having the spread of the error or the like. An inequality is established between this optimum area and an optimum function containing a non-determined parameter. This inequality is approximated to finite determination expressions and by solving these expressions concerning the parameter, a solution area is found. By selecting one point inside this solution area, one optimum function f(x) included within the optimum area can be obtained.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method and an apparatus which can be used, for example, for designing a digital filter and obtain a compatible function used for the design.

[0002]

2. Description of the Related Art For example, in a conventional digital filter design, a curve representing the amplitude characteristic of a filter is approximated by an optimization method, and a transfer function is calculated based on the approximated curve. As an example of the optimization method, a sequence of points representing ideal characteristics is prepared, a function including a plurality of parameters for approximating the characteristic is set, a distance between the function and the sequence of points is obtained, and this distance is squared. There is known a method of obtaining a desired approximate curve by determining a parameter so that the total sum of the calculated values is minimized.

[0003]

However, such a conventional method merely represents input data as a fixed point sequence having no width and calculates the distance between the input data and a target function. Has a certain degree of spread due to errors or the like, this spread cannot be handled.

The present invention solves the above-mentioned problems, and addresses various industrially applicable problems such as the structure and shape of a physical system, system identification, graph display, data interpolation, signal prediction model, and pattern recognition. It is another object of the present invention to provide a suiting apparatus and a suiting method that can handle input data or the like having a spread which has been difficult in the past.

[0005]

According to the present invention, there is provided a suiting apparatus comprising: means for setting an appropriate area corresponding to at least an allowable range of physical characteristics; An aptitude function determining means for obtaining a function by solving simultaneous inequalities, wherein the aptitude function determining means sets the aptitude function as a family of functions having parameters and obtains the parameters by solving the simultaneous inequalities. And In addition, the suitability optimization method according to the present invention includes a first step of setting a suitability region corresponding to at least a permissible range of a physical property, and a suitability inequality at least approximately included in the suitability region within the suitability region. And a second step of determining by solving the suitability function as a family of functions having parameters, and determining the parameters by solving simultaneous inequalities.

In this specification, the physical characteristics mean required characteristics, required shapes, measured values, communication signals, pattern signals, and the like. Also, in this specification, the optimization method is a function (hereinafter, referred to as an aptitude function) that at least approximately passes through a range corresponding to the spread of the input (hereinafter, referred to as an aptitude area), and more specifically, this aptitude function. A method to derive the parameter values of a function.

[0007] This suitability optimization method requires an enormous amount of calculation even when the suitability function is expressed in a simple form. Therefore, it is impossible to obtain a solution by manual calculation, and it can be implemented only by using a digital computer or the like. The optimization device refers to a device for performing the optimization method. That is, it refers to a digital computer or the like on which the suiting method is performed.

Hereinafter, a typical suiting method will be described using vector notation. A typical example of the suitability of the method, a suitable function including a parameter placed and f (x), in the range D of the multidimensional ^{space, M - (x) <f} (x) <M + (x) (∀x∈D) (1) It is represented by a method of finding a parameter value that satisfies (1). (1)
The formula is called a decision formula for suitability, and f is such that the formula (1) is satisfied.
(x) is called the aptitude function. The M ⁺ (x) the maximum function constrains the upper limit of suitability function f (x), M constrain the lower limit ^- (x) a lower bound function, upper bound function M ⁺ (x) and lower bound function M ^- by (x) The entire constrained range is represented by the appropriate region T (= ｛(x, y) | x
∈D, M ^- a ^{(x) <y <M +} (x)}) is called. Upper limit function M
⁺ (X), lower bound function M ^- (x), respectively + ∞, may take a value of -∞.

Equation (1) is represented by a finite number of inequalities, and a range in which a parameter of f (x) that satisfies all of the finite number of inequalities is called a solution region. This solution area is obtained by transforming the appropriate area into the parameter space. Therefore, one point in the solution area corresponds to one aptitude function taking a value in the aptitude area.

In the suiting apparatus and the suiting method of the present invention, the suitability function f (x) is not limited to a function having a real value, but may be a functional, an operator, or the like. Thus, a recurrence formula of a sequence may be used. Also D
May be a discrete set of points, a continuous range, or unconnected.

If the parameters are common, there may be a plurality of functions to be optimized. In this case, a different suitability region may exist for each function desired to be suitability.

[0012]

EXAMPLES The present invention will be described below with reference to examples. First,
Embodiments 1 to 4 explain the suiting apparatus and the suiting method of the present invention, and Embodiment 5 and the following describe the application of this apparatus and method to various technical and industrial systems.

[Embodiment 1] Here, the solution of the suitability function by the suitability optimization method will be described in the following order. 1. 1. Reduction of suitability method problem to simultaneous inequalities Solving simultaneous inequalities Determining the degree of the fitness function f (x)

1. Reduction of Suitability Method Problem to Simultaneous Inequalities The suitability optimization method will be described with reference to FIG. In this embodiment
And each point sequence P_j= (X_j, y_j) (j = 1,2, ..., n) is
For example, it is generated based on input data. Control aptitude function
The range D on the x-axis to be reduced is ｛x₁, x_Two,. . . , x_n｝
You. Each point P_iConsidering the spread due to the error of
_iAt this point P_iThe upper limit
Number M⁺(x_i) And the lower limit function M^-(x
_i) Is determined. Such a setting is used for each point sequence P_iAgainst
Done. That is, the point sequence P corresponding to the upper limit function⁺ ₁
= (X₁, y⁺ ₁), P⁺ _Two= (X_Two, y⁺ _Two),. . . , P⁺
_n = (X_n , y⁺ _n ) And the point sequence P corresponding to the lower limit function^-
₁= (X₁, y^- ₁), P^- _Two= (X_Two, y^- _Two),. . . ,
P^- _n= (X_n, y^- _n) Is determined. Where y⁺ _i 
Is M⁺(x_i) And y^- _iIs M^-(x_i). This
An appropriate area T is determined by the entire sequence of these points.

It is assumed that the aptitude function can be expressed by a first-order rational function (2) with respect to x. _{f (x) = (a 0} + a 1 x) / (b 0 + b 1 x) (∀x∈D) (2) _{where, a 0, a 1, b} 0, b 1 is a real number parameters.
The equation (2) is substituted into the equation (1), and the denominator (b ₀ +
Paying the denominator under the condition that b ₁ x _i ) is positive,
Equation (1) is equivalent to the simultaneous inequalities (3), (4),
It is transformed into (5). _{-A 0 -a 1 x i + b} 0 M + (x i) + b 1 x i M + (x i)> 0 (3) a 0 + a 1 x i -b 0 M - (x i) -b 1 x ^{_{i M - (x i)>}} 0 (4) b 0 + b 1 x i> 0 (5)

The simultaneous inequalities (3), (4) and (5) are
Parameter a to display as the dot product of₀, a₁,
b₀, b₁A four-dimensional parameter with dimensions equal to the number of
The interval is set. Vector in this parameter space
X, η⁺(x_i), η^-(x _i ) And η⁰(x_i)
Put it like the formula below. X = (a₀, a₁, b₀, b₁) (6) η⁺(x_i) = (−1, −x_i, M⁺(x_i), x_iM⁺(x_i)) (7) η^-(x_i) = (1, x_i, −M^-(x_i), − X_iM^-(x_i)) (8) η⁰(x_i) = (0, 0, 1, x_i) (9) X is a parameter vector, η⁺(x_i) (Hereafter, the upper limit
Is called the upper bound function M⁺(x_i)
Vector, η^-(x_i) (Hereinafter called the lower limit vector
B) is the lower limit function M^-(x_i) Vector representing the restriction
And η⁰(x_i) Is necessary because equation (2) does not diverge.
This is a vector representing a necessary condition.

Using these equations, simultaneous inequality (3),
The deciding formulas of the optimization method represented by (4) and (5) are the following inequalities (10), (11), and (12) represented by inner products with the parameter vector X, respectively. _{^{(X, η + (x i}} ))> 0 (10) (X, η - (x i))> 0 (11) (X, η 0 (x i))> 0 (12)

Thus, the inequalities (10), (11) and
And (12) is the input point sequence P on the range D _iAgainst
Given the problem, the problem of finding the suitability function is finite
It is expressed by a simultaneous inequality, and the solution can be actually obtained.
For example, point sequence P_iIs 500, the inequality (1
The numbers of 0), (11) and (12) are each 500
is there.

When a finite number of inequalities (10), (11) and (12) obtained for all input point sequences Pi are simultaneously solved for the parameter vector X, the region where the parameter vector X exists is This is the solution area S. The solution area S is generally a convex cone and is represented by the following equation (13). S = {s ₁ X ₁ + s ₂ X ₂ +... + S _m X _m | s ₁ , s ₂ ,. . . , s _m > 0｝ (13) X ₁ = (a ₀₁ , a ₁₁ , b ₀₁ , b ₁₁ ), X ₂ = (a ₀₂ ,
_{a 12, b 02, b 12} ) ,. . . , X _m = (a _0m , a _1m , b _0m ,
Each vector of b _1m ) represents a vertex of this solution area S. One point X _S = (a _0s , a _1s , b _0s ,
The suitability function f (x) defined by b _1s ) always takes a value in the suitability region T.

2. Solving Simultaneous Inequalities In many cases, it is sufficient to obtain one suitability function f (x), or it is sufficient to obtain the partial solution S1 of the solution area S. Therefore, for example, the existence of the solution area S is determined and the solution of only the partial solution S1 is determined, and the calculation can be speeded up.

Referring to FIGS. 2 and 3, an example of the determination of the existence of the solution area S using the linear programming and the solution of the partial solution S1 will be described below. Here, for the sake of simplicity, η ⁺ (used in the above inequalities (10), (11), and (12)
x _i), η ^- represented by (x _i), η ⁰ (the corresponding vector to all vectors of _{x i) η j (j =} 1,2, ..., 3n). As a result, the inequalities (10), (11), and (12) are represented by the following inequality (14), which is equivalent to the inequalities (10), (11), and (12). _{(X, η j)> 0} (j = 1,2, ..., 3n) (14) where _{^{η i = η + (x i}} ) (i = 1,2, ..., n) (15 ) η _{i + n} = η ^- placing a _{(x i) (16) η} i + 2n = η 0 (x i) (17). Also, the display by the component of η _j is (η _j0 , η _j1 ,
η _j2 , η _j3 ).

In the inequality (14), the following inequality (18) holds even when the parameter vector X is multiplied by α. (αX, η _j ) = α (X, η _j )> 0 (18) Then, X is multiplied by (1 / a ₀ ), and X ″ = (1 / a ₀ ) X = (1, a ₁ / a ₀₎ , b ₀ / a ₀ , b ₁ / a ₀ ) without loss of generality. Then, the above equation (14) expressed by the inner product of the four-dimensional vector gives the three-dimensional vector X ′, η ′ The following inequality (19) expressed by the inner product of _j
Is reduced to 0 <(X ″, η _j ) = η _j0 + (X ′, η ′ _j ) (19) where X ′ and η ′ _j are represented by X ′ = (X ′ ⁽¹⁾ , X ′ ⁽²⁾ , ^{X '(3)) = (} a 1 / a 0, b 0 / a 0, b 1 / a 0) η' j = (η j1, η j2, placed and eta _j3).

The possible area S 'of the inequality (19) is given by the following equation (2)
0). S '= ｛s₁X '₁+ S_TwoX '_Two+ ... + s_nX '_n  | S₁, s_Two,. . . , s_n> 0, s₁+ S_Two+ ... + s_n= 1｝ (20) Each X '_k(k = 1,2, ..., n) are the possible areas S ′
Is a three-dimensional vector representing the vertices of.

The evaluation function is given by V ₁ =
X ′ ⁽¹⁾ (= a ₁ / a ₀ ), and V ₁ is _first maximized by linear programming. First, the existence of a feasible solution is determined to determine whether the feasible region S 'is an empty set.
If the possible area S ′ is an empty set, the solution area S is also an empty set, and no suitability function exists. When the possible area S 'is not an empty set, the following processing is continued.

The contour plane where V ₁ is constant is H1 in FIG.
A plane perpendicular to the axis of X ′ ⁽¹⁾ as represented by H2 or the like. Region S that maximizes the V ₁ from 2 'point of X' _3. Similarly, V ₂ = −X ′ ⁽¹⁾ (= −a ₁ / a ₀ ) V ₃ = X ′ ⁽²⁾ (= b ₀ / a ₀ ) V ₄ = −X ′ ⁽²⁾ (= −b ₀ / A ₀ ) V ₅ = X ′ ⁽³⁾ (= b ₁ / a ₀ ) V ₆ = −X ′ ⁽³⁾ (= −b ₁ / a ₀ ), and each possible area S ′ is maximized. X ′ ₇ , X ′ ₁ , X ′ ₅ , X ′ ₁₄ , X ′ ₁₅ are obtained. In this case, S '1 = {(s 1 X' 3 + s 2 X '7 + s 3 X' 1 + s 4 X '5 + s 5 X' 14 + s 6 X '15) | s 1, s 2,. . . , s ₆ > 0, s ₁ + s ₂ +... + s ₆ = 1｝ (21) If S ′ 1 is a partial area included in the possible area S ′ as shown in FIG. I understand.

It is not necessary to limit the evaluation function Vi (i = 1, 2,...) To such a method, and a sufficiently large area can be obtained.

Considering the parameter space, the partial solution S1 is
It is represented by equation (22). S1 = ｛α (1, s₁X '_Three+ S_TwoX '₇+ S_ThreeX '₁+ S_FourX '_Five  + S_FiveX '₁₄+ S₆X '_Fifteen) | Α> 0, s₁, s_Two,. . . , s₆> 0, s₁+ S_Two+ ... + s₆= 1｝ (22) One of the parameter vectors existing in this partial solution S1 X_s1= (A_0s1, a_1s1, b_0s1, b_1s1) Into equation (2), one aptitude function f
(x) is obtained.

The determination of the existence of the solution area S and the solution of the partial solution S1 are not limited to this method, and a method such as computational geometry may be used. In addition, it is also possible to obtain the entire solution domain S instead of the partial solution S1 by this technique such as computational geometry (Literature, Introduction to Computational Geometry: FP Preparer, MI Schemos, Takao Asano , Translated by Tetsuo Asano,
Published by Soken Publishing Company on July 1, 1992). Further, a solution method using a penalty function or the like or an approximate solution method can be used. The solution by the penalty function is a method of reducing an inequality problem to an optimization problem. That is, this is to construct a penalty function that takes a large function value in a range that does not satisfy the inequality and a small function value in a range that satisfies the inequality, and then finds a point that minimizes this penalty function. (Literature, Nonlinear Programming (Part II, Chapter 10): Hiroshi Konno, Hiroshi Yamashita, Nisshin Giren,
The first print was issued in 1978, and the sixth print in 1990).

If it is sufficient to find only one suitability function, for example, one slacks variable Y is introduced, the inequality (19) is set as 0 <(X ′, η _j ) + η _j0 −Y, and Y May solve a linear programming problem that maximizes
If a solution exists and 0 <Y, then X 'at that time
Satisfies equation (19), so that one suitability function is obtained. There are a number of ways to reduce to linear programming depending on how slack variables are introduced.

The rational function (2) is defined as a set of two linearly independent functions (p ₀ (x), p ₁ (x),
. . . , p _M (x)), (q ₀ (x), q ₁ (x), ..., q
_N (x) the following rational function f consisting ratio of linear combinations _{of) (x) = (a 0} p 0 (x) + a 1 p 1 (x) + ··· + a M p M (x)) / ( b ₀ q ₀ (x) + b ₁ q ₁ (x) +... + b _N q _N (x)) (23) It is easy to extend the term approach. Here, a ₀ , a ₁ ,. . . ,
a _M , b ₀ , b ₁ ,. . . , b _N are real parameters, x is an n-dimensional vector, and each function p ₀ (x), p ₁ (x),. . . , p
_M (x), q ₀ (x), q ₁ (x),. . . , Q _N (x) is a piecewise continuous function of the D, for example, ^{x n,} cos x, e x and step functions and the like.

3. Method for Determining Order of Suitability Function f (x) FIG. 4 shows an example of a procedure for obtaining the suitability function with the minimum order.
It is a flowchart. If the fitness function f (x) is
Assume that it is a rational function expressed by the ratio of N-th order polynomial of x
I do. In step 111, the upper limit function M⁺(x_i) And below
Limit function M^-(x _i) Is entered. That is, each point sequence P
_i= (X_i, y_i), For example, the upper limit
Function M⁺(x₁), M⁺(x_Two) ,. . . , M⁺(x_n) But
1.05y₁, 1.04y_Two,. . . , 1.06y_n, And the lower limit function M^-
(x₁), M^-(x_Two) ,. . . , M ^-(x_n) Is 0.94y₁,
0.97y_Two,. . . , 0.92y_nIt is determined. Step 11
In 2, the initial value of the order N is set to 1.

In step 113, the inequality (X, η _j )> 0 is constructed by the above-described method. Here, η _j is the same as that used in equation (14).
In step 114, the solution area S of the parameter vector X
Is determined.

If the solution area S does not exist, the order N of the denominator and numerator of the rational function f (x) is increased by one in step 121. In step 122, this rational function f
It is determined whether the order N of (x) has exceeded the maximum achievable order _Nmax . If the maximum order is not exceeded,
Return to step 113.

This procedure is repeated, and if it is determined in step 114 that the solution area S exists, step 1
At 15, a solution of the partial solution S1 of the solution area S is obtained, and a family of the suitability function f (x) is obtained. In step 116, if necessary, a partial solution S1, that is, an appropriate suitability function f (x) is selected from the family of suitability functions f (x), and is displayed on, for example, a display screen. Thereafter, the program ends.

In step 122, if the order N is
Large order N_maxIf it is determined that
At 123, for example, “N” is displayed on the display screen._max 
The following orders are not suitable. " Steps
In 124, the upper limit function M⁺(x _i) And the lower limit function M^-(
x_iWait for the input as to whether or not to redo the setting of).

The upper limit when it is desired to generate an approximate curve following the maximum degree N _max function M ⁺ (x _i) and lower bound function M ^- (
It is necessary to reduce the approximation accuracy by changing the setting of x _i ).
In this case, returning to step 111, the upper limit function M ⁺ (x
_i) a lower bound function M ^- setting (x _i) is changed. For example, each point sequence _{P i = (x i, y} i) tolerance of approximately ± 5%
To about ± 10%. Thereafter, in step 112, the order N of the rational function f (x) is set to 1, and the processing from step 113 to step 122 is repeated. If it is determined in step 114 that the solution area S exists, steps 116 and 117 are executed, and
The suitability function f (x) is displayed on the display screen.

On the other hand, in step 124, for example,
The suitability function f (x) obtained by further expanding the suitability region T is
If the desired specification cannot be expressed, the upper limit function M
⁺(x _i) And the lower limit function M^-(x_i) Settings are changed
Then, this program ends.

The above-described method shows an example of the simplest algorithm for simplification of the description. For example, if an algorithm such as a known bisection method is used, the order can be determined more quickly. . The order N obtained by the above search is the minimum order of at least one of the numerator and the denominator. Therefore, the order of the numerator and denominator can be minimized by fixing one order to N and changing the other order up to N to determine the existence of the solution region S. In this way, it is sufficient to simply perform a one-dimensional search without performing a two-dimensional search for the entire combination of the order of the numerator and denominator, and the efficiency is high.

As described above, according to the suiting method of this embodiment, the following effects can be obtained. That is, since the input data is given in a range, it is possible to handle input data having a spread which has been difficult in the past, and an aptitude function is obtained not as a unique function but as a family. Therefore, when the suitability function f (x) of the present embodiment is used, for example, as a transfer function indicating a circuit configuration of an analog system described later, deviations and aging of circuit components and members fall within the range of the solution area S. By doing so, the transfer function obtained can satisfy the required specifications for a long time regardless of its aging.

In addition, even if the function is difficult to handle by the conventional method such as the optimization method, an appropriate function can be obtained. For example, in the design of an IIR digital filter whose square amplitude characteristic is expressed in the form of a rational function, a desired transfer function can be obtained by using the suiting method of this embodiment. In the optimization method, the evaluation function becomes a very complicated function, and it is extremely difficult to obtain the optimum value by a numerical solution such as Newton's method due to global non-convergence.

Further, an aptitude function f taking a value in the aptitude area T
(x) is obtained with the minimum order. The upper bound function M ⁺ optionally (x _i) and lower bound function M ^- settings (x _i), for example by changing interactively on a display,
Realistically satisfies the specification, that is, the maximum feasible order N
Within _max , the fitness function f (x) is obtained with the smallest order in the fitness region T.

For example, if the suiting method of this embodiment is used for interpolation, the following effects can be obtained. Input point sequences with errors and widths can be interpolated. A rational function can be used as the interpolation formula to be set. As a result, a sharp suitability function without overshoot or swelling can be obtained. Can be represented by one fitness function without using piecewise polynomials. It is possible to calculate how many approximations to use. Since the interpolation formula is obtained as a range, fine adjustment can be made within this range. Here, the interpolation formula indicates a function for interpolating data such as the shape of the system and the graph display.

If the suiting method of this embodiment is applied to a CAD system or the like, it can be used for graph display, shape display, and the like.

Further, in the suiting apparatus and the suiting method of this embodiment, if the parameters are common, a plurality of suiting functions may be set. In this case, different suitability regions may exist in each suitability function. For example, in this embodiment, placing condition required for rational function does not diverge (5) g (x) = b 0 + b 1 x (24), both f (x) and g (x) It can be considered that it has become aptitude. The formula for determining suitability in this case is (1)
And g (x)> 0 (25) (see equation (5)).

[Embodiment 2] Here, the case where the suitability function is a function other than the rational function will be described. When the solution area S of the parameter can be represented by a convex area, a function other than the rational function such as Expression (23) can be used as the suitability function. That is, the parameter of the family of the function is X = (a ₀ ,
a ₁ ,. . . , A _m), ⁺ the upper bound function M (x), the lower bound function M ^- (x), and placing the suitability function f (X, x) and, in response to the above-described (1), M ^- (x) <f (X, x) <M ⁺ (x) (∀x∈D) (26)

[0046] _{Here, B 1 (X, x)} = f (X, x) -M - (x) (27) B 2 (X, x) = -f (X, x) + M + (x) ( 28), the decision equation (26) is given by the following simultaneous inequality (2)
9) and (30). 0 <B ₁ (X, x) (29) 0 <B ₂ (X, x) (∀x∈D) (30)

When each inequality is evaluated for each input point sequence P _i as in the first embodiment, a finite number of inequalities are obtained. If a partial solution is obtained by simultaneously combining these inequalities, a family of suitability functions f (X, x) is obtained. For this purpose, for example, a method in which the portion using the linear programming in the first embodiment is replaced with a nonlinear programming is conceivable. That is, a plurality of evaluation functions are obtained, a feasible solution that maximizes each of the evaluation functions is obtained, and their convex hulls are set to S1. Since it is assumed that the solution region S is a convex region, this S1 is a partial decomposition of a convex hull shape approximating S from the inside. As a result, it is possible to optimize a function that represents a physical system or the like that cannot be represented by a rational function. Even when the solution area cannot be represented by a convex hull, one solution can be obtained by solving equations (29) and (30).

[0048] [Example 3] In Example 3, D is a continuous range limit function M ⁺ (x) and lower bound function M ^- for suitability method when (x) is piecewise continuous function, lower bound function M ^- (x) is explained taking the following order as an example. 1. 1. Reduction of the problem of continuous suitability using convex hulls into simultaneous inequalities 2. Example of configuration method of generated convex hull Other Techniques In this embodiment, the suitability function f (x) is expressed in the form of a rational function (2). Further, as shown in FIG. 5, lower bound function M ^- a (x) is represented by the broken line in FIG.

1. Reduction of Continuous Suitability Method Problem Using Convex Hull into Simultaneous Inequalities The following inequality (31) is obtained by the same method as that used in the first embodiment to derive expression (11). ^{(X, η - (x)} )> 0 (∀x∈D) (31) The lower limit vector eta ^- (x) similar to the equation (8) lower bound function M
^- are as defined from (x), lower bound function M ^- in which (x) is converted into the parameter space.

In the case of this embodiment, since x changes continuously, there are an infinite number of simultaneous inequalities of the equation (31), which cannot be handled in a device such as a digital computer. Therefore, Equation (31) is approximated by a finite number of simultaneous inequalities by a method using a convex hull described below.

The section D has an appropriate size corresponding to the required accuracy.
Finite interval [x₁, x_Two], [x_Two, x _Three],. . . , [x
_n-1, x_n].

[0052] In FIG. 6, the lower limit function of FIG. 5 M ^- (x)
And the point _{^{M - (x i), M}} - (x i + 1) is, the lower limit vector eta each ^{- (x), η - (} x i), η - are converted into (x i _{+ 1).} The interval _{[x i, x i + 1} ] M corresponding to an arbitrary point y on the ^- (y) is, the lower limit vector eta ^- corresponds to (y).

First, referring to FIGS. 5 and 6, the lower limit vector η
^- the (y) described a method for approximated by the convex hull body. Certain parameter vector X ₀ is a lower limit vector _{^{η - (x i), η}} - (
to _{x i + 1), (X} 0, η - (x i) - assumed that satisfies the)> 0 (33))> 0 ((x i + 1 32) (X 0, η) . η ^- (x _i) and η ^- (x
_{i + 1)} for any vector ξ on a line segment L connecting the line segments of the official ^{_{ξ∈ {s 1 η - (x}} i) + s 2 η - (x i + 1) | s 1, s 2> 0, s ₁ + s ₂ = 1｝ (34) Thus (X _0, ξ) is _{(X 0, ξ) = s} 1 (X 0, η - (x i)) - represented as _{((x i + 1) X} 0, η) (35) + s 2 . By the way, since s ₁ and s ₂ > 0, (3
2) and (33), (X ₀ , ξ)> 0 (36) holds. But the interval _{[x i, x i + 1} ] points in the y, i.e. the line segment L is not on the lower limit vector eta ^- relates (y), always ^{_{(X 0, η - (y}} ))> 0 (y∈ [x _i , x _{i + 1} ]) (37). eta ^- (y) is because they represent the lower limit function, any point of the rational function f (x) in the interval of FIG. _{5 [x i, x i +} 1], lower bound function M ^- above the (x) Is not necessarily located at

[0054] Therefore always (37) so that equation holds, the lower limit vector eta ^- approximated by ^- a (y) convex hull C. Here, the convex hull is defined as an area in which a line segment connecting any two points inside the convex hull is always included in the inside.

Convex hull C^-Is the interval [x_i, x_{i + 1}Any on
Lower limit vector η^-(y), that is, η^-(y) ∈C^- (38) is made to always hold. Here we generate the convex
Call it a hull. At this time, the generated convex hull C^-Is, for example,
Expression (39) is expressed (shaded portion in FIG. 6). C^-= ｛S₁η^-(x_i) + S_Twoη^-(x_{i + 1}) + S_Threeη^- _i1+ S_Fourη^- _i2  + S_Fiveη^- _i3| S₁, s_Two, s_Three, s_Four, s_Five> 0, s₁+ S_Two+ S_Three+ S_Four+ S_Five= 1｝ (39) This generated convex hull C^-The method of determining will be described in detail later.

Next, a method of approximating an infinite number of inequalities (37) with a finite number of inequalities on the section [x _i , x _{i + 1} ] will be described. The inequality (37) is replaced by the following inequality (40). ^{(X, C -)> 0} (∀η - ∈C - (X, η -)> 0) (40)

According to the definition of the convex hull, the inequality (40) is
This is equivalent to the following simultaneous inequalities (41) to (45). _{^{(X, η - (x i}} ))> 0 (41) (X, η - (x i + 1))> 0 (42) (X, η - i1)> 0 (43) (X, η - i2 )> 0 (44) (X, η ^- _i3 )> 0 (45)

By a calculation similar to the calculations of the equations (32) to (36), the parameter vector X, which is the solution of this simultaneous inequality, is given by (X, η ′)> 0 for any element η ′ of C ^−. Fulfill. Since equation (38) holds, equation (37) holds. Considering this in the original graph, the interval [x _i ,
x _{i + 1} ], the fitness function f (x) corresponding to X is
It indicates that located above (x) ^- always lower bound function M in FIG.

If inequalities are established for each of the divided sections in the same manner as in the inequalities (41) to (45), the existence determination of the solution area and the solution are performed by the solution of the simultaneous inequalities similar to the first embodiment. The aptitude function is required.

As described above, in the present embodiment, the following effects are obtained in addition to the effects obtained in the first embodiment. First, the upper limit function M ⁺ (x) and the lower limit function M
^- (x) even when the given piecewise continuous function,
By reducing the inequality (31) to a finite number of inequalities using the generated convex hull, it becomes possible to calculate the suitability.

[0061] Since with generation convex hull body, the lower limit function as Q in FIG. 5 M ^- (x) suitability below function f (x) that pops is prevented. Thus, even when the suitability function is a high-order rational function and the function value vibrates sharply in response to a change in a variable, it is ensured that the suitability function is reliably included in the continuous suitability region T.

Furthermore, the generated convex hull can absorb errors, deviations, and the like of the upper limit function or the lower limit function, and such a function is used in the seventh embodiment.

The sections are determined so that they do not overlap each other, and may have different sizes. For example, it is preferable to finely divide a part that requires particularly high accuracy, such as a severe vibration of the function. Also, the upper limit function M ⁺ (x)
The lower bound function M ^- (x) is not a need to break with the same period may be set according to the respective requirements.

2. Examples of Methods for Configuring Generated Convex Hull Referring to FIGS. 7 to 10, some algorithms for configuring a generated convex hull C that covers a part of the curve η (x) will be described. Here, the parameter space is illustrated two-dimensionally for simplicity, but is actually a space having a dimension equal to the number of parameters of the suitability function f (x). Curve η (x)
, For example the lower limit vector of Example 3 eta ^- corresponds to (y).

I) Method A As shown in FIG. 7, points on the curve η (x) or points η ₁ , η ₂ ,. . . , η _m , and a convex hull C ′ (hatched portion in FIG. 7, m = 3 in FIG. 7) having these as vertexes is generated. As shown in FIG. 7, the convex hull C 'does not always include the curve η (x). Therefore, as shown in FIG. 8, the inner point η ₀ of the convex hull C ′ is taken, and the convex hull C ′ is expanded around the inner point η ₀ . η ′ _i (α _i ) = α _i (η _i− η ₀ ) + η ₀ (46) where i = 1,2, ..., m α = (α ₁ , α ₂ , ..., α _m ) Where α _i is the dilation parameter. This η ' ₁ (
α ₁ ), η ' ₂ (α ₂ ),. . . , η ′ _m (α _m ) is defined as C (α) (hatched portion in FIG. 8).

Each α _i is gradually increased, and the convex hull C
When (α) completely covers the curve η (x), the increase is stopped. The convex hull C (α) at this time is defined as a generated convex hull C.

In addition to increasing α _i , α _i covering the curve η (x) is obtained by combining increase and decrease.
Can be determined faster. For example, each α _i is changed while decreasing the width to be changed for each step, and it is determined whether or not the convex hull C (α) includes the curve η (x) for each step. If within the predetermined accuracy, stop this change. The convex hull C (α
) Is a generated convex hull C.

A plurality of inner points η ₀ may exist.

Ii) Method B As shown in FIG. 9, as the first convex hull C ', an appropriate shape that can include a curve, for example, a polyhedron including both ends of the curve η (x), or a combination of a plurality of these is used. Determine things. The convex hull C ′ is expanded in the same manner as in the method A to obtain a curve η (x).
Is included.

Iii) Method C As shown in FIG. 10, one point η ₀ near the curve η (x) is taken. The distance between this η ₀ and each point on the curve η (x) is calculated,
Let the largest one be R. A convex polyhedron C circumscribing a sphere having a center η ₀ and a radius R is created and defined as a generated convex hull C.

The method of generating the generated convex hull is not limited to the above methods A to C, and a conventionally known method in computational geometry or the like can be adopted.

The present construction method relates to a convex hull that approximates a curve. However, this method is also applicable to a case where x has a high dimension, that is, a case where η (x) represents a curved surface or the like. It is possible.

3. Other approaches the upper bound function M ⁺ (x) and lower bound function M ^- Method (x) to obtain a finite number of inequalities in the case of piecewise continuous function is not limited to the above-described method, for example, the upper bound function M ⁺ (x)
The lower bound function M ^- (x) may be a method such that is approximated by a dense sequence of points. In some cases, the overhang can be removed by slightly modifying the fitness function as needed. For example, a proper function f (x) is a rational formula of x, the interval [x _i, x _{i + 1]} if it exceeds the upper bound function M ⁺ (x) in, E (x) = M
⁺ (x) -f (x) has an even number of real zeros in this interval.
Therefore, the parameters of f (x) are expressed by the zeros of E (x), and the real zeros are defined as two sets, and the positions are slightly corrected so that each set becomes a complex conjugate set of imaginary zeros. And
By reconstructing the parameter of f (x) from these, it is possible to remove only the protrusion while keeping the form of the suitability function substantially.
This approach is useful not only when D is on the real axis,
For example, the present invention can be applied to a one-dimensional region, such as a part of a unit circumference on a complex plane, in a similar manner.

Embodiment 4 FIG. 11 shows Embodiment 4 of the present invention.
3 shows a schematic configuration of an apparatus for performing the suiting method. With reference to FIG. 11, the suiting apparatus of the present invention will be described.

The device 11 is, for example, a digital computer, and includes a storage device 12, an arithmetic device 13, and an input device 1.
4. It has an output device 15 and a control device 16, which are interconnected by bus lines 17,18. Storage device 12 includes a region or an upper limit function storage unit 21 stores the upper bound function M ⁺ a (x), lower bound function M ^- storing the area, that is the lower limit function storage unit 22 for storing (x), the product convex hull member A region, ie, a generated convex hull storage unit 23, a region for storing a simultaneous inequality solving program, ie, a simultaneous inequality solving program unit 24, a region for storing a generated convex hull configuration program, ie, a generated convex hull configuration program unit 25, And the like, that is, a control program section 26 for storing a control program. Arithmetic unit 13
Is a CPU or the like. The input device 14 is, for example, a keyboard,
A mouse, numerical file, digitizer or light pen. The output device 15 is, for example, a display, a numerical file, a plotter or a printer. The control device 16 controls each device for executing the program.

FIG. 12 shows an example of data displayed on the screen of the display of the output device 15. Based on the point sequence given via the input device 14, the upper limit function M ⁺
(x) and lower bound function M ^- setting a (x). The suitability function f (x) obtained by the suitability method is as follows.
It is expressed by the following rational function (47) expressed by the following polynomial. _{f (x) = (A 0} + A 1 x + A 2 x 2 + A 3 x 3 + A 4 x 4) / (B 0 + B 1 x + B 2 x 2 + B 3 x 3 + B 4 x 4) (47) where A _0, A ₁ , A ₂ , A ₃ , A ₄ , B ₀ , B ₁ , B ₂ , B ₃ , B ₄
This is a real parameter.

Equation (47) shows the square amplitude characteristic of a digital bandpass filter. The fact that the transfer function of the digital filter is obtained from equation (47) showing the square amplitude characteristic will be described in detail in the fifth embodiment.

As described above, according to the suiting apparatus of the fourth embodiment, the same effect as that of the third embodiment can be obtained. In the suiting apparatus of the fourth embodiment, the upper limit and the lower limit of the suitability area are visually displayed on the screen of the display, and the suitability function f (x) is superimposed and displayed on the displayed suitability area. Have. Therefore, the spread of the input data and the correspondence between the spread of the input data and the processing results of the optimization method can be directly understood visually, so that when the optimization method is used in a design system, for example, the design becomes easy, and The effect that the change and the design change are performed efficiently can be obtained.

[Embodiment 5] Here, a case will be described in which the suiting method of the present invention is applied to the design of a digital filter as an example of a digital signal processing system. Digital filters are classified into IIR filters and FIR filters. In the conventional IIR filter design, it is difficult to handle only the rectangular characteristics, and many calculation methods for determining the ripple characteristics and delay characteristics of each band must be used depending on the case. In addition, the FIR filter has a problem that the characteristic is deteriorated by the window function. Further, as a problem common to both, in general, it is not guaranteed that the characteristics fall within an expected range, it is impossible to compensate for deterioration of characteristics due to quantization errors of coefficients, and the order of the system can achieve required characteristics. It will be larger than the minimum order,
There was a problem.

The above-mentioned problem can be solved by using the suitability optimization method of the present invention, and the digital filter can be freely adjusted with a desired characteristic.
It can be designed with the minimum order so as to fall within the desired characteristic range. Hereinafter, the design of the digital filter will be described in the following order. For simplicity, the sampling period T is standardized to 1, but can be easily generalized to other periods. Further, the phase characteristics and the argument arg are displayed in an unwrapped manner, and represent a continuous multiple rotation angle exceeding the range of 0 to 2π. 1. 1. Amplitude characteristic design of a digital filter having a rational function transfer function 2. Characteristic design of digital filter having polynomial transfer function and all-pole transfer function (1) Simultaneous design of amplitude and phase design (2) Phase characteristic design (3) Amplitude characteristic design of perfect linear phase FIR filter All-pass filter design

1. Design of amplitude characteristics of digital filter having transfer function of rational function form First, a digital filter whose transfer function is expressed in the form of a rational function is designed by ignoring phase characteristics and designating only amplitude characteristics. The range _Do requiring the characteristic is set as _Do = {0 ≦ ω ≦ π} (48). Although the order of the transfer function can be set arbitrarily, the case where the numerator and denominator are each a quartic equation will be described.

The transfer function H (z) is expressed as H (z) = (a_o+ a₁z^-1+ a_Twoz^-2+ a_Threez^-3+ a_Fourz^-Four) / (B_o+ b₁z^-1+ b_Twoz^-2+ b_Threez^-3+ b_Fourz^-Four) = P (z) / Q (z) (49), and the numerator and denominator are P (z) and Q, respectively, as described above.
(z). Equation (49) represents the transfer function of the IIR filter.
However, in the case of the FIR filter, Q (z) = b _oWhen
It is good.

For convenience of calculation, the amplitude characteristic is shown as a squared amplitude characteristic.
I will decide. The squared amplitude characteristic of the filter is expressed by H (z)
z = e^jwAnd substitute H (e^jw) H^*(e
^jw)^*Represents a complex conjugate). H (e^jw) H^*(e^jw ) = H (e^jw) H (e^-jw) = (A_o+ A₁cosω + A_Twocos^Twoω + A_Threecos^Threeω + A_Fourcos^Fourω) / (B_o+ B₁cosω + B_Twocos^Twoω + B_Threecos^Threeω + B_Fourcos^Fourω) (50) The squared amplitude characteristic is expressed as a function of cos ω as in the above equation.
It is. The numerator and denominator are p (cosω) and q (cosω), respectively.
 And f (cosω) = p (cosω) / q (cosω).
A_o, A₁, A_Two, A_Three, A_FourIs a_o, a₁,
a_Two, a_Three, a_FourOf the second order, B_o,
B₁, B_Two, B_Three, B_FourIs b_o, b ₁, b_Two, b
_Three, b_FourAs a second order polynomial. Here,
Is cos^kω is described as a basis function.
It is not limited to taking, for example, cos kω
(k = 0,1, ..., 4) as the basis function₀Suitability on
Can also be performed.

[0084] range D limit function representing a requested range of _o on the square amplitude characteristic M ⁺ (cosω), lower bound function M ^- give (cos .omega). Since cos omega on D _o takes a value from 1 to -1, x
= Cos ω, D = {− 1 ≦ x ≦ 1}, and the suitability is determined by the following decision formula. ^{M + (x) <f (} x) = p (x) / q (x) <M - (x) (x∈D) (51) 0 <q (x) (x∈D) (52) where The condition that q (x) is positive is a condition necessary for the filter to be stable. From the above equation, the parameters A _o , A ₁ , A ₂ ,
A ₃ , A ₄ and B _o , B ₁ , B ₂ , B ₃ , B ₄ are determined. From the square amplitude characteristic p (x) / q (x) thus obtained, a corresponding transfer function P (z) / Q (z) is obtained as follows.

First, the denominator Q (z) of the transfer function is calculated from the denominator q (x).
The conversion to is performed as follows. q (x) is factored in the range of real numbers by numerical solution of a higher-order algebraic equation, and
Quadratic factor q _i (x) = α _i + β _ix and quadratic factor q _j (x) = α
_{j + β j x + γ j} x 2 (1 ≦ i <j ≦ 4) q (x) by the product of _{= q 1 q 2 ··· q m} (m ≦ 4) expressed as (53). Here, each q _k is set to be positive on D.
Let Q be the first-order transfer function whose square amplitude characteristic is q _i (cosω)
_i , and a second-order transfer function having a square amplitude characteristic of q _j (cos ω) is Q _j . These are easily obtained by solving simultaneous quadratic algebraic equations. At this time, Q (z) = Q ₁ (z) Q ₂ (z)... Q _m (m ≦ 4) (54) is the denominator of the transfer function to be obtained. P (z) is similarly obtained. Each q _k, Q corresponds to p _k, P _{k (k} =
1, 2,... M), and Q _k is selected from among them, so as to satisfy the filter stability condition (H (z) has a pole inside the unit circle). This choice is always possible by imposing the condition 0 <q (x). If the selected Q _k and P _k are different, the phase characteristics will be different. Conversely, by appropriately selecting Q _k and P _k , a desired phase characteristic can be realized. It goes without saying that the above method can be applied to higher-order filters. Further, the method of factorizing q (x) and converting it to Q (z) as described above is merely an example of a method for obtaining a transfer function, and is for simplifying the calculation. You may ask for it.
For example, b _o ,..., B ₄ can be directly obtained from B _o ,..., B ₄ by a numerical solution of an appropriate high-dimensional algebraic equation.

2. Characteristic design of a digital filter having a polynomial transfer function and an all-pole transfer function Here, an FIR filter in which the transfer function H (z) is a polynomial of z ^{−1, and} a polynomial of H ⁻¹ (z) that is z ⁻¹ A design of a certain all-pole filter will be described. (1) Design of Simultaneous Designation of Amplitude and Phase First, a design in which both amplitude and phase are designated will be described. The range of optimization is defined as D _{0 in} equation (48). Let the transfer function of the FIR filter be H (z) = a _o + a ₁ z ^-1 + a ₂ z ^-2 +... + _An z ^-n (55). M ⁺ (ω
^{), M - (ω),} θ + (ω), θ - put the (ω). These specific inputs may be made by drawing a graph using a mouse or a keyboard on a CRT screen, for example, or by directly inputting mathematical expressions. Of course, file input may be used. Furthermore, Mc (ω) = (M + (ω) + M - (ω)) / 2 ( intermediate value of the amplitude of the upper and lower limits) (56) θc (ω) = (θ + (ω) + θ - (ω) ) / 2 (intermediate value the upper and lower limits of the phase) (57) ΔM (ω) = (M + (ω) -M - (ω)) / 2 (1/2 of the amplitude tolerance) (58) [Delta] [theta] ( ^{ω) = (θ + (ω} ) -θ - (ω)) / 2 (1/2 value of the phase _{tolerance) (59) I H (ω} ) = Im (H (e jw)) ( imaginary transfer function (60) R _H (ω) = Re (H ( ^ejw )) (real part of transfer function) (61) The appropriate area in this case is the shaded area in FIG. 13, and if Δθ is not too large, the appropriate area can be approximated by a rectangular area having vertices at points c ₁ , c ₂ , c ₃ , and c ₄ in FIG. This approximation is not limited to a rectangle, but may be a polygon or a circle. Δ
When θ is large or when it is desired to increase the approximation accuracy, a convex hull having more vertices may be used. This aptitude area is C
Assuming that (ω), the formula for determining the suitability is H ( ^ejw ) ∈C (ω) (62). That is, it is sufficient that H (e ^jw ) is within C (ω).

For convenience of calculation, the appropriate area C (ω) is now shown in FIG.
As shown in Fig. 4, it is rotated by -θc and moved on the real axis.
, H (e^jw ) Is divided into a real part and an imaginary part and expressed as a real number.
That is, b (Mc (ω) −ΔM (ω)) <R_H(ω) cos θc (ω) + I_H(ω) sinθc (ω) <b (Mc (ω) + ΔM (ω)) (63) −b (Mc (ω) sinΔθ (ω)) <− R_H(ω) sinθc (ω) + I_H(ω) cosθc (ω) <b (Mc (ω) sinΔθ (ω)) (64) b is a parameter for adjusting the shape of the equation.
After calculating the function, it is set to 1. Further, X = (a_o, a₁,. . . , a_n, b) (65) ξc (ω) = (1, cos ω,..., cos (n−1) ω, cos nω, 0) (66) ξs (ω) = (0, −sin ω,. ..., -sin (n-1) ω, -sin nω, 0) (67)_H(ω) = (X, ξs (ω)) (68) R_H(ω) = (X, ξc (ω)) (69) Therefore, the decision formula is b (Mc (ω) -ΔM (ω)) <(X, cos θc (ω)) c (ω) + sin θc (ω) ξs (ω)) <b (Mc (ω) + ΔM (ω)) (70) −b (Mc (ω) sinΔθ (ω)) <(X, −sin θc (ω) ξc (ω) + cos θc (ω) ξs (ω) <b (Mc (ω) sinΔθ (ω)) (71) e_{n + 2} = (0,0, ..., 0,1) and η⁺(ω) = (Mc (ω) + ΔM (ω)) e_{n + 2}  − (Cosθc (ω) ξc (ω) + sin θc (ω) ξs (ω)) (72) η^-(ω) = − (Mc (ω) −ΔM (ω)) e_{n + 2}  + cos θc (ω) ξc (ω) + sin θc (ω) ξs (ω) (73) φ⁺(ω) = Mc (ω) sinΔθ (ω) e_{n + 2}  − (−sin θc (ω) ξc (ω) + cos θc (ω) ξs (ω)) (74) φ^-(ω) = Mc (ω) sinΔθ (ω) e_{n + 2}  + (− Sin θc (ω) ξc (ω) + cos θc (ω) ξs (ω)) (75) As a result, the decision expression is eventually 0 <(X, η⁺(ω)) (76) 0 <(X, η^-(ω)) (77) 0 <(X, φ⁺(ω)) (78) 0 <(X, φ^-(ω)) (79). After that, D₀And create a generated convex hull to optimize
Is performed, the point X in the solution area is directly used as the transfer function (55).
Since these are parameters of the equation, both the amplitude and phase characteristics
An FIR filter that falls within the range specified in (1) is obtained.

In the above embodiment, the appropriate area C (ω) is set by the upper limit function and the lower limit function as shown in equations (63) and (64). However, the present invention is not limited to this.
As shown by C ′ (ω) in FIG. 3, a direct convex hull may be used. (Corresponding to claim 4)

For an all-pole filter, the reciprocal of the point in the above-described suitability area C (ω) is regarded as a new suitability area and H ⁻¹
(z) may be optimized as in the case of FIR. In particular,
For example, a convex hull including regions having vertices at points 1 / c ₁ , 1 / c ₂ , 1 / c ₃ , and 1 / c ₄ may be set as a new appropriate region. However, in this case, in order for H (z) to be stable, it is necessary that all zeros of H ⁻¹ (z) exist within the unit circle.
When ω → 2π, the constraint that θ _C (ω) −θ _C (0) = 0 must be satisfied (principle of declination).

Although the present design method exemplifies the design of a one-dimensional FIR filter for simplicity, it is extremely easy to extend the present design method to a multidimensional FIR filter.

(2) Phase Characteristics Design Here, the amplitude characteristics are ignored. Since only the phase is set, in FIG. 13, M ⁻ = 0 and M ⁺ = + ∞,
Is basically the same as in the previous section. When rotated by -θc the H (e ^jw) similarly to the previous section are moved on the real axis as shown in FIG. 15, the phase determination formula of H (e ^jw) is, - tanΔθ (ω) <Im (exp (-jθc (ω)) H (e ^jw )) / Re (exp (-jθc (ω)) H (e ^jw )) <tanΔθ (ω) (80) 0 <Re (exp (-jθc (ω)) H (e ^jw )) (81) (provided that Δθ <π / 2). Here Re (exp (-jθc (ω) ) H (e jw)) = (X, cos θc (ω) ξc (ω) + sin θc (ω) ξs (ω)) (82) Im (exp (- jθc (ω)) H (e ^jw )) = (X, −sinθc (ω) ξc (ω) + cos θc (ω) ξs (ω)) (83) (X, ξc (ω), ξs ( ω) (65),
(See equations (66) and (67)). Next, η ⁺ (ω) = (cosθc (ω) tanΔθ (ω) + sinθc (ω)) ξc (ω) + (sinθc (ω) tanΔθ (ω) −cosθc (ω)) ξs (ω) (84 ) ^- (Ω) = (cosθc (ω) tanΔθ (ω)-sinθc (ω)) ξc (ω) + (sinθc (ω) tanΔθ (ω) + cosθc (ω)) ξs (ω) (85) η ⁰ (ω) = cosθc (ω) ξc (ω) + sinθc (ω) (s (ω) (86) The decision formula is as follows: 0 <(X, η ⁺ (ω)) (87) 0 <( ^{X, η - (ω))} (88) 0 <(X, η 0 (ω)) ( can be expressed as 89). In the following, X can be determined using the suiting method according to the present invention in the same manner as in the preceding section.

The phase characteristics of the all-pole type filter can be similarly optimized, but the same constraint as in the preceding paragraph must be satisfied in order to satisfy the stability of the filter.

(3) Amplitude of perfect linear phase FIR filter
Design The transfer function of even order of the perfect linear FIR filter is
(Odd order cases can be applied with slight modifications.
). H (z) = a_o+ a₁z^-1+ ・ ・ ・ + A_nz^-n  +_{an + 1} z^{-(n + 1)}+ a_nz^{-(n + 2)}+ ・ ・ ・ + A_oz^-2n(90) In this case, according to equations (60) and (61), R_H(ω) = a_o(1 + cos 2nω) + a₁(cosω + cos (2n-1) ω) + ・ ・ ・ + a_{n + 1}cos (n + 1) ω (91) I_H(ω) =-a_o(0 + sin 2nω)-a₁(sinω + sin (2n-1) ω)-・ ・ ・-a_{n + 1}sin (n + 1) ω (92), X = (a_o, a₁,. . . , a_{n + 1}, b) (93) ξc (ω) = (1 + cos2nω, cos ω + cos (2n-1) ω,..., cos (n + 1) ω, 0) (94) ξs (ω) = ( -sin2nω, -sinω-sin (2n-1) ω, ..., -sin (n + 1) ω, 0) (95)_H(ω) = (X, ξc (ω)) (96) I_H(ω) = (X, ξs (ω)) (97) After that, (1) simultaneous design of amplitude and phase
Find X and determine transfer function in the same way as described in section
(See equations (70) to (79)). However
In this case, the phase characteristic is completely linear,
Upper limit function θ⁺(ω), lower limit function θ^-(ω) is too much
Take a non-negative positive number Δ (ω)⁺(ω) = − nω + Δ (ω) (98) θ^-(ω) = − nω−Δ (ω) (99) Or θ_C= −nω and (7
8), the condition of formula (79) may be removed and optimization may be performed.
No. Also, simply θ⁺, Θ^-From (98) and (99)
Even if Δ is small enough, “(1) Shake
Design by specifying width and phase simultaneously ''
A linear phase FIR filter can be obtained. In addition,
The amplitude term of the transfer function is calculated in advance, and the
You may perform sexualization.

3. Design of All-Pass Filter The design of the phase characteristic of the FIR filter has been described in the section “(2) Design of phase characteristic”, but the design can be applied to design an all-pass filter. n the transfer function of the following all-pass filter H (z), when put between _{P (z) = a o +} a 1 z -1 + ··· + a n z -n (100), H (z) = z ⁻ⁿ P (z ⁻¹ ) / P (z) (101), and its phase characteristic is arg H (e ^jw ) = − nω−2 arg P (e ^jw ) (102). The upper and lower limit functions of the phase characteristic are θ ⁺
(Ω), θ ^- determined equation and (omega) ^{is, θ - (ω) <-} nω-2 argP (e jw) <θ + because becomes (ω) (103), φ + (ω) = ( ^{-nω-θ - (ω))} / 2 - if put a (104) φ (ω) = (-nω-θ + (ω)) / 2 (105), φ + (ω), φ - (ω) As the upper limit function and the lower limit function of argP (e ^jw ) may be optimized by the method described in “(2) Phase characteristic design”. Here, as a stability condition of the ^{filter, φ + (ω), φ} - (ω) is, P (z)
It is necessary to determine as the zero point is present in all the unit ^{_{circle, φ C = (φ + +}} φ -) / 2 and putting, omega → 2 [pi
In this case, the condition φ _C (ω) −φ _C (0) = − 2πn must be satisfied. With this condition, the order n of the filter determines the rough shape of the characteristic. However, if only the difference from the linear phase is set as the design target of the characteristic, the order of the filter can be selected even under this condition.

By combining the all-pass filter designed in this way and the IIR filter designed by designating only the amplitude characteristic, an IIR filter designating both the amplitude characteristic and the phase characteristic can be constructed. . The phase characteristic of the filter that satisfies only the amplitude characteristic may be subtracted from the designated phase characteristic to obtain the phase characteristic of the all-pass filter.

As described above, the following effects can be obtained by applying the suitability optimization method of the present invention to the design of a digital filter. That is, the amplitude and phase characteristics can be designed freely and easily. In addition, the characteristics can be kept within the required range, and the minimum order that can realize the characteristics can be strictly calculated, so that the number of elements can be minimized. The amplitude characteristic can be designed for both FIR and IIR filters, and can be easily designed to always satisfy the stability condition of the filter. Since the FIR filter can be designed directly without the intervention of a window function, the characteristics do not deteriorate. Further, since the solution region has a wide range, the solution region is stable against various deviations such as a quantization error, and can be optimized and optimized for other design conditions by utilizing the spread of the solution region.

In the description so far, a digital signal processing system such as a digital filter capable of achieving the required characteristics by providing the required characteristics has been demanded. By reversing this concept, this system can be identified by giving the characteristics of the actual system instead of the required characteristics. For example, a frequency characteristic of a certain digital signal processing system is measured, and a predetermined measurement error is added to the measurement result to set a predetermined characteristic region. This setting can be automatically set in advance if the range of the measurement error is known. Next, the target system (such as a transfer function) can be identified by performing the same suitability as in the above embodiment so as to fall within the characteristic region. Such a method or apparatus can be used, for example, for structural analysis of a digital signal processing system.

In the above embodiment, the method of setting the lower limit function and the upper limit function of the amplitude characteristic and the phase characteristic of the filter and the method of obtaining the transfer function from the optimized amplitude characteristic and the phase characteristic are merely examples. II as follows
The frequency characteristics of the R filter can also be designed. Embodiment 1 Taking the IIR filter of the term as an example, the required frequency characteristic is represented by a target function g (ω), and the system error tolerance M
(ω), the norm of the complex number z is defined as {z} = max (Imz,
Rez), ‖P (e ^jw ) −g (ω) Q (e ^jw )
‖ <B ₀ M (ω) to the determined type, the parameter vector _{X = (b 0, b 1} , .., b 4, a 0, .., a 4) may be performed suitability as. In this case, in order to stabilize the filter, the second embodiment should be used. It is necessary to add a conditional expression relating to Q to some extent to the above-mentioned determination expression by the method of the term (2) or the like.

Embodiment 6 Next, the suiting method of the present invention is applied to the design in the frequency domain of a linear analog system (for example, an analog filter, an amplifier circuit, a control device for controlling a robot, etc.). An example is shown below. Conventional analog system design methods are mainly based on rectangular frequency characteristics and the design method is very complicated.It is difficult to design flexible characteristics. However, there are problems that a large number of parts are required, that there is no guarantee that the characteristics are within the expected range, and that characteristic deterioration due to various deviations and errors cannot be compensated.

The use of the suiting method of the present invention can solve all of the above problems. Hereinafter, the description will be made in the following order as in the case of the digital filter. 1. 1. Amplitude characteristics design of analog system having rational transfer function 2. Characteristic design of an analog system having a polynomial transfer function and an all-pole transfer function Phase correction system

1. Design of Amplitude Characteristics of Analog System Having Transfer Function of Rational Function Type First, an analog system whose transfer function is expressed in the form of a rational function is designed by ignoring phase characteristics and designating only amplitude characteristics. The order of the transfer function can be set arbitrarily,
Hereinafter, the case of the quartic equation will be described. Transfer function H (s)
Is given by H (s) = (a _o + a ₁ s + a ₂ s ² + a ₃ s ³ + a ₄ s ⁴ ) / ( _bo + b ₁ s + b ₂ s ² + b ₃ s ³ + b ₄ s ⁴ ) = P (s) / Q (s) (106), and the numerator and denominator are P (s) and Q, respectively, as described above.
(s). Q (s) = b for a polynomial transfer function
_o

For convenience of calculation, the amplitude characteristic is shown as a squared amplitude characteristic.
I will decide. The squared amplitude characteristic of the system is H (s)
Substituting s = jω, H (jω) H^*(jω)
Defined by (^*Represents a complex conjugate). H (jω) H^*(jω) = H (jω) H (−jω) = (A_o+ A₁ω^Two+ A_Twoω^Four+ A_Threeω⁶+ A_Fourω⁸) / (B_o+ B₁ω^Two+ B_Twoω^Four+ B_Threeω⁶+ B_Fourω⁸) (107) The squared amplitude characteristic is ω^TwoExpressed as a function of
You. This numerator and denominator are p (ω^Two), Q (ω^Two)
And f (ω^Two) = P (ω^Two) / Q (ω^Two) far. A
_o, A₁, A_Two, A_Three, A_FourIs a_o, a₁, a
_Two, a_Three, a_FourOf the second order, B_o, B
₁, B_Two, B_Three, B_FourIs b_o, b ₁, b_Two, b
_Three, b_FourAs a second order polynomial.

[0103] upper bound function representing the request area of the square amplitude characteristic M ⁺ (ω ^2), lower bound function M ^- give (omega ^2), performs suitability by x = omega ² Distant following equation. ^{M + (x) <f (} x) = p (x) / q (x) <M - (x) (0 ≦ x <+ ∞) (108) 0 <q (x) (0 ≦ x <+ ∞ (109) In the above equation, the condition that q (x) is positive is a condition necessary for the system to be stable. In practice, it is only necessary to make the aptitude appropriate for a wide range of x. However, the behavior of the transfer function at infinity can be controlled as follows. That is, at s '= 1 / s, the point at infinity s' = 0 is introduced,
Expression (106) is expressed using s ′. And x '= 1 /
At ω ² , a decision formula similar to the formulas (108) and (109) is made. These are in the range of (0 ≦ x ′ <1 + ε), and the expressions (108) and (109) are (0
Within the range of ≦ x <1 + ε), the entire composition is simultaneously optimized. Here, ε is a small positive number. In this way, the infinite region of the half line can be reduced to the finite region of the semicircle on the Riemann sphere, and the suitability on the finite region can be achieved. From the above formula, the parameters A _o , A ₁ , A ₂ , A ₃ ,
A ₄ and B _o , B ₁ , B ₂ , B ₃ , B ₄ are determined. From the square amplitude characteristic p (ω ² ) / q (ω ² ) thus obtained, the transfer function P (s) / Q (s) can be obtained in the same manner as in the case of the digital filter described above. It is necessary to select Q (s) such that the real part of the pole of H (s) becomes negative as a stability condition of the system.

2. Polynomial type transfer function, the characteristic design of the case when, where characteristic design of analog systems having a total-pole transfer function is a polynomial of the transfer function H (s) is s and H ^-1 (s) is a polynomial of s explain. First, a design in which both the amplitude and the phase are designated will be described, and then a design of only the phase characteristics will be described. (1) Design for Simultaneous Designation of Amplitude and Phase This design is almost the same as that of the digital filter.
_H (ω) = Im (H (jω)) and R _H (ω) = Re (H (jω)) only (see equations (60) and (61)). These include n
When the is slightly changed by the modulo 4, n from expressed even if any a slight modification will be described when it is a multiple of 4, the transfer function H (s) _{H (s) = a o +} a is a _{1 s + ··· + a n s} n (110), X = (a o, a 1, ... a n, b) (111) ξc (ω) = (1,0, -ω 2 ^{, 0, ..., ω n,} 0) (112) ξs (ω) = (0, ω, 0, -ω 3, ..., if put and _{0,0) (113), I H} (ω ) = (X, ξs (ω)) (114) R _H (ω) = (X, ξc (ω)) (115) The transfer function can be obtained in exactly the same manner as in “(1) Simultaneous design of amplitude and phase of digital filter”.

In the case of an all-pole type in which H ⁻¹ (s) is a polynomial of s, it can be obtained in the same manner as described above. Of course H (s
) Is stabilized so that the real part of the zero of H ^-1 (s) becomes all negative. That is,
When ω → + ∞, θ (ω) −θ (−ω) = πn must be satisfied.

(2) Phase Characteristic Design Using the equations (111) to (115) and the equations (84) to (89), the analog filter can be designed in the same manner as in “(2) Phase characteristic design” of the digital filter. Phase characteristics can be designed.

3. Phase Correction System The phase correction system is an analog system in which the amplitude characteristic is constant regardless of the frequency and only the phase is corrected. This corresponds to an all-pass filter in a digital filter. The transfer function of order n phase correction system H (s), when put between _{P (s) = a o +} a 1 s + a 2 s 2 ··· + a n s n (116), H (s) = P (−s) / P (s) (117), and its phase characteristic is arg H (jω) = arg (P ² (−jω) / (P (jω) P (−jω))) = −2 argP (jω) (118) The upper and lower limit functions of the phase characteristic are θ ^{+
(Ω), θ - (ω} ) and a decision equation, θ ^- (ω) ^<- because comprising ^{2 argP (jω) <θ +} (ω) (119), φ + (ω) = - θ - ( ω) / 2 (120) φ - (ω) = - θ + (ω) / 2 (121) ^{Distant, φ + (ω), φ} - (ω) limit function of argP (j [omega]), as a lower bound function The suitability may be determined by the method described in “(2) Phase characteristic design”. Here, as a stability condition of the ^{filter, φ + (ω), φ} - (ω) is required to satisfy the constraints on the zeros of P (s). Similarly to the digital filter, by combining a system designed by designating only the amplitude characteristic and a phase correction system, it is possible to configure an analog system having a rational function type transfer function designating both the amplitude and the phase. it can.

As described above, the same effects as those obtained in the design of the digital filter can be obtained in the design of the analog system. Furthermore, the above method
As in the case of digital signal processing systems, the required characteristics can be substituted for the characteristics of the actual analog system and used to identify this system.

The method described in the sixth embodiment can be applied to all objects that can be represented by a time-invariant linear system. Control of a chemical plant is given as an example other than the physical system. In chemical plant control, for example, the pressure, temperature, or the concentration of the produced substance with respect to the amount of raw material input per unit time in the reaction tank may be expressed as a linear system, and this may be Laplace transformed to obtain a transfer function. Hereinafter, according to the sixth embodiment, it is possible to determine various coefficients of the system such as the heating amount and the pressurizing amount which can be set artificially, that is, to design a plant control system.

[Embodiment 7] Next, an identification apparatus for identifying a physical or chemical system using the suiting method of the present invention will be described. This identification device can be used, for example, for voice and image pattern recognition, vibration analysis, and information compression by a linear predictor. Although the system identification has been described in the fifth and sixth embodiments, in the above-described embodiment, the system structure is identified based on the frequency characteristics. Here, an identification method for a discrete series signal, for example, a time series signal such as an impulse response characteristic is dealt with.

In general, as shown in FIG. 16, when a signal x from a signal source 101 is received and a target system 102 outputs y, as shown in FIG. A model of the system 102 is determined. x and y are continuous or discrete signals. In the case of a continuous signal, the signal is often sampled and discretized, and the output from the target system 102 sampled at the nth time is supplied to y _n and the target system 102. input when was the x _n of the system _{y n = Σa i f i (} y n-1, y n-2, ..., y nM, x n, x n-1, ..., x nN ) (122) and the parameters a ₀ , a ₁ ,. . . , a _L is determined. Σ in the equation (122) represents the sum of 0 to L for i. M and N are the order of the system and the signal y _nM
Is a signal that is traced back by M from the nth time, and x _nN is N
Represents a signal that goes back only by There is also an identification device that identifies a system based on only the output y from the target system 102 without using the signal x from the signal source 101. In this case, Expression (122) becomes an expression that does not include x _i .

[0112] Conventional identification _{methods, V = Σ (y n -Σa} i f i (y n-1, ..., y nM, x n, ..., x nN)) is denoted by ² (123) Thus, the parameters are determined so that V is minimized. In Equation (123), Σ represents the sum of 0 to k for n and the sum of 0 to L for i. However, this method is based on the premise that the target system satisfies statistical properties that are convenient to handle, and this method only minimizes the sum of squares V of the identification error,
There may be a momentary large error between the obtained model and the response characteristics of the true system, and there is a problem that the accuracy of identification is unknown. By using the suitability optimization method of the present invention, the identification error can be systematically evaluated at each moment, so that an appropriate identification device that takes account of the identification error can be obtained. Hereinafter, an identification apparatus using the suitability optimization method according to the present invention will be described. 1. Identification of discrete linear system 2. Identification of a class of nonlinear systems A description will be given in the order of identification of linear analog systems.

1. Identification of discrete linear system_y= b₀y_n+ b₁y_n-1+ ・ ・ ・ + B_My_nM  + (a₀, x_n) + (a₁, x_n-1) + ・ ・ ・ + (A_N, x_nN) + c (124). Where a_k, x_kIs an L-dimensional vector,
Each parenthesized superscript is a component of each vector.
Minute. b0 is a parameter for shaping the expression
Set 1 later on the meter. S_yIs linear (1
22) shows a form in which the right side of equation (22) is shifted to the left side.
Is typically 0 and is called the prediction residual.
You. Let η be the measured response_n= (1, y_n, y_n-1,. . . , y_nM, x_n ⁽¹⁾, x_n ⁽²⁾,. . . , x_n ^(L), x_n-1 ⁽¹⁾,. . . , x_nN ^(L)) (125). The response has errors and fluctuations,_nWhen
Then, η_n+ Δη_n∈C_n  Is constructed such that always holds, and it is expressed as C_n= ｛S₁η_n ⁽¹⁾+ s_Twoη_n ⁽²⁾+ ... + s_Kη_n ^(K)│ 0 <s_k, s₁+ s_Two+ ・ ・ ・ + S_K= 1｝ (126). Where η_n ⁽¹⁾, η_n ⁽²⁾,. . . , η
_n ^(K)Is the generated convex hull C_nIs the vertex of Also, X = (c, b₀, b₁,. . . , b_M, a₀ ⁽¹⁾, a₀ ⁽²⁾,. . . , a₀ ^(L), a₁ ⁽¹⁾,. . . , a_N ^(L)) (127). The allowable identification error (allowable error range)
Number M⁺(n), lower limit function M^-(n), the decision formula
Is b₀M^-(n) <S_y(C_n) <B₀M⁺(n) (n = 0,1,2, ..., n_max) (128) That is, b₀M^-(n) <(X, η_n ^(k)) <B₀M⁺(n) (k = 1,2, ..., K, n = 0,1,2, ..., n_max) (129). y_nN is the impulse response characteristic,
_maxAs | y_n| When n approaches 0 to some extent
However, if it becomes smaller, an error will be generated as the convex hull C
_nCan also be absorbed. After this, the hands already explained
The solution area S is obtained by performing optimization by the method. This C_nIs an example
3 is slightly different from the generated convex hull. But e₀= (0,
1,0,0,. . , 0) and C⁺ _n= -C_n+ E₀M⁺(n) (130) C^- _n= C_n-E₀M^-(n) (131) If equation (129) is given, 0 <(X, C⁺ _n) (132) 0 <(X, C^- _n(133) and the generated convex hull and the decision formula in the same format as in the third embodiment.
Can be expressed by The solution area S remains the proper identification parameter.
Data set. Errors and fluctuations in the input to the identification system
, Even if there are_nAs long as it is included in
System identification error is within the tolerance
Strong robustness can be obtained.

In this case, the actual configuration of the generated convex hull is, for example, η _n ⁽¹⁾ , η _n
⁽²⁾ ,. . . , η _n ^(k) , this can be achieved by setting a slightly larger convex hull including all of them as C _n .
The degree of increase is determined in consideration of factors that cause deviations such as system nonlinearity, aging, and load fluctuation. Statistical techniques may be employed. When the error is known in advance, the generated convex hull can be automatically configured according to the error. It is also possible to construct a generated convex hull for each set of measured values of the response to different input signals, and to optimize the entire convex hull.

In the above example, the input data is generated by the convex hull C
_{Although expressed in} the form of _n , it is not always necessary to generate a convex hull. That is, the system can be identified by setting C _n = {η _n } ignoring the error.

FIG. 17 is a block diagram of an identification device using the above-described identification method.
The system is provided with a region determining means 105 for providing a region required for aptitude to the signals x and y from the second device, and a characteristic determining unit 106 for determining characteristics of the system so as to fall within the region. The area determining means 105 gives the input signal an upper and lower limit function and a generated convex hull as in equation (126), and the characteristic determining means 106 calculates the system parameters based on equation (129).

[0117] 2. Form to be identified based nonlinear system of a class in advance select _{f i S y = Σa i f} i (y n ', x n') shall be assumed (134). Σ represents the sum of 0 to N for i. Here xn is a vector, subscript on dated parentheses shall representing each component of the respective _{vector, y n '= (y n} , y n-1, ..., y nM) x n' = ^{_{(x n (1), x}} n (2), ..., x n (L), x n-1 (1), ..., x nN (L)) and (135), f _0,. . . , f _N are a set of linearly independent functions.
Note that f _i (y ′, x ′) may be non-linear with respect to the arguments x ′, y ′. _{η n = (f 1 (y} n ', x n'), f 2 (y n ', x n'), ..., f n (y n ', x n')) (136) Distant eta _{If n} is regarded as measured data and X = (a ₀ , a ₁ ,..., a _N ) (137), the method of the preceding section can be applied almost as it is.

3. Identification of a linear analog system In the case of a continuous system, the error generated at the time of discretization is represented by a convex hull C
Absorb _n and tolerance. This will be described with a simple analog system. S _y = by _n + a ₀ (dx / dt) + a ₁ (d ² x / dt ² ) (138) Assuming that a ₀ and a ₁ are determined. By discretizing the _{differential, dx / dt = (x n} -x n-1) / Δt + Δ 1 (139) d 2 x / dt 2 = (x n -2x n-1 + x n-2) / Δt 2 + Δ 2 (140). Δt is a sampling interval, and Δ ₁ and Δ ₂ are discretization errors. If accuracy is required, higher order discretization is performed until Δ ₁ and Δ ₂ become sufficiently small. _{x n '= (x n -x} n-1) / Δt (141) x n "= (x n -2x n-1 + x n-2) / Δt 2 Putting the _{(142), S y = by} n + a 0 (x n '+ Δ 1) + a 1 (x n "+ Δ 2) to become (143). Here, it is assumed that no only error discretization. In this case, the maximum value and the minimum value that can be taken by Δ ₁ and Δ ₂ are regarded as the error range of the input data x _n ′, x _n ”and the generated convex hull C _n
May be configured. After that, the suitability described in the previous section is performed.
When the solution area S becomes an empty set due to large Δ ₁ and Δ ₂ , the accuracy of discretization may be increased to some extent, or the order of the assumed system may be increased. Further, the above method can be applied to a system including a time delay such as integration.

[0119] Alternatively, the input signal remains continuous first ^{η - (t) = (-} M - (t), dx (t) / dt, d 2 x (t) / dt 2) (144) η + (t) = (M ⁺ (t), − (dx (t) / dt), − (d ² x (t) / dt ² )) (145) and X = (b, a ₀ , a ₁ ) decision equation ^{0 <(X, η - (} t)) (146) 0 <(X, η + (t)) (147) vertical and then if proper space is continuous by the procedure of example 3 May be performed.

[0120] The system S _y thus obtained it is clear that no characteristic deterioration due to discretization. That continuous input x (t), always bM respect ^{y (t) - (t)} <S y (x (t), y (t)) <bM + (t) (148) is established.

The above-mentioned identification method can be used for the impulse response design of the system. In other words, a system design can be performed by giving a required characteristic of a time series instead of the measured value of the target system in the above-mentioned identification method, and giving a design allowable range as an allowable error range instead of the identification error. Here the design tolerance range _{S y ((128), (} 148) type)
Alternatively, it can be given in the form of a range C _n (equation (126)) of η _n .

In the case of a digital filter, the method described in the section “1. Identification of discrete linear system” in the seventh embodiment can be applied as it is. S _y becomes an FIR filter when M = 0,
When M ≧ 1, it represents an IIR filter, and can be used for either filter. The required characteristics are expressed by η _n and the convex hull C _n is formed in consideration of the quantization error and the like. It is also possible to simply set C _n = {η _n } (149). Thus, the solution area S obtained by the optimization becomes the appropriateness parameter, that is, the coefficient of the filter.

In the case of an analog system, if the method of “3. Identification of linear analog system” of the seventh embodiment is applied, a linear analog system (for example, a control device for controlling an analog filter, an amplifier circuit, a robot, etc.) Etc.) in the time domain. The difference from the digital filter is that a discretization error is inevitably added to the input of the required characteristic.

According to the identification device described above, an allowable identification error range is given and identification is performed within the range, so that the identification error can be evaluated, and an appropriate identification device considering the identification error can be obtained. And the following effects can be obtained. That is, the discretization error of the continuous system can be absorbed, the robust identification can be performed robustly against the fluctuation of the target system, noise, and the like. It is possible to estimate the order of the target system by determining the existence of the solution region S while changing the order of the model system in the same manner as in the flowchart of FIG. 4. Other conditions using the spread of the solution region S (For example, power consumption).

Although the present embodiment relates to a system in which an input discrete sequence is represented by one subscript n, it can be applied to a multidimensional system having an input discrete sequence represented by many subscripts. It is very easy to extend the present embodiment.

[Embodiment 8] Next, another embodiment using the suiting method of the present invention will be described. In a system in which an analog system, an A / D converter, a D / A converter, and the like coexist, amplitude and phase characteristics are variously affected by the analog system and the signal converter (aperture effect, group delay distortion, and the like). I would like to correct these with a digital filter to achieve the desired characteristics. However, in such a case, a conventional optimization method or the like cannot realize a flexible characteristic, or even if it can, a high-order digital filter is required. So, in general,
Measures have been taken to add a mechanism for correction separately from the basic structure of the device, such as multi-rate filters and a device for the analog system. By using the suiting method of the present invention, it is possible to design a digital filter with a flexible characteristic and a small order, thereby making it possible to equalize a characteristic distorted by the intervention of an analog system or the like.

FIG. 18 shows an example in which the characteristic correction device of the present invention is connected to an analog system. The signal from the analog system 201 is converted into a digital signal by an A / D converter 202, and the characteristic correction device 203 Sent to The characteristic correction device 203 includes the digital filter described in the fifth and seventh embodiments. The characteristic correction is performed as follows. (1) First, the correction device 203 is set to a state of simply passing a signal. (2) A reference signal is sent to the analog system 201, and the characteristics of the analog system 201 and the A / D converter 202 are measured from the output corresponding to the reference signal. At this time, the correction device 203
May be used as an identification device to perform characteristic measurement. (3) The difference between the characteristic before correction and the desired characteristic is set as the characteristic of the digital filter 203. The characteristic design of the filter in the time domain can be performed by the method described in the section of “1. (4) Regarding the characteristic design of the filter in the frequency domain, the configuration of the digital filter is determined depending on whether the correction target is one of the amplitude characteristic and the phase characteristic, or both. If only the amplitude characteristic is used, any of IIR and FIR filters can be used. When only the phase characteristic is used, an all-pass filter or an FIR filter is used. When both amplitude and phase are corrected, IIR and all-pass filter, FIR and all-pass filter, FIR
There is only an option.

In the above example, the characteristic correction device 203 is connected to the subsequent stage of the analog system 201. However, the characteristic correction device may be arranged before the analog system to perform correction. That is, digital input → characteristic correction device → D / A converter → analog system → analog output. When a digital system and an analog system are mixed, a correction device may be connected in the middle of the entire system. For example, analog input → analog system 1
→ A / D converter → characteristic correction device → D / A converter → analog system 2 → analog output. That is, since the transfer function of the entire system is represented by the product of the transfer functions of the components, the correction device may be arranged at any of the front, rear, and middle stages of the system.

By performing the characteristic correction as described above, the degree of freedom in designing an analog system is greatly increased. In some cases, the configuration of the analog system can be considerably simplified. Further, the above correction can be automatically performed using a microprocessor or the like.

[Embodiment 9] Next, a description will be given of an embodiment in which the suiting is performed while the suiting area is intentionally reduced in the parameter space. This is effective, for example, when designing a circuit so that a required specification can always be satisfied even if a component of the electronic circuit or the like changes with time or temperature. This embodiment is also effective in digital computers for errors such as loss of digits in numerical calculations.

The problem of the original optimization method is as follows: 0 <(X, η _i ) (i = 1, 2,..., M) (150) X = (a ₁ , a ₂ ,. _n , b) _Let η _i = (η _i1 , η _i2 ,..., η _in , η _{in + 1} ). η _i is a vector representing an upper limit function, a lower limit function, and the like, and the appropriate region is expressed as η _{i in the} parameter space. b is a parameter for adjusting the form of the equation, and is set to 1 later.

The change δ (∈R ^{n + 1} ) of X is always included in the convex hull Δ. That is, X + δ∈X + bΔ (151) Here delta is, delta = | can be expressed as _{{s 1 Δ 1 + s 2} Δ 2 + ··· + s L Δ L s 1 + ··· + s L = 1, Δ j ∈R n + 1} Therefore, if 0 <(X + bΔ _j , η _i ) (j = 1, 2,..., L, i = 1, 2,..., M) (152), then equation (151) is satisfied. Equation (150) holds for any X + δ. Equation (152) can be transformed into (X, η _i ) + b (Δ _j , η _i ) (153), so that η _j ′ = (η _i1 , η _i2 , ..., η _in , η _{in + 1} + min (Δ _j , η _i )) (i = 1, 2,..., M) (1 ≦ j ≦ L), Equation (152) gives 0 <(X, η _i ′) ( i = 1,2, ... m) (154). This is not the only formulation, but various formulations are possible depending on which component of η _i the change of X is to be brought.

In this way, the equation (154) can be expressed as the problem of the optimization method.
Considering all of the elements Xs in the solution domain of equation (154),
0 <(Xs + δ, η_i) (I = 1,2, ..., m), and it can be seen that it is robustly stable. Original question
If the title is for a polynomial suitability function, then η_i
From η_i'To just modify the upper bound function to min (Δ _j, η
_i) (Where 1 ≦ j ≦ L).
To be as large as it is, that is, the suitability of the original problem
It corresponds to directly reducing the area. In the general case
However, this modification reduces the convex hull representing the appropriate area.
Become.

The properties handled in the suiting apparatus and the suiting method of the present invention are not limited to physical properties, but may be information processing techniques such as information compression or chemical properties.

[0135]

As described above, according to the optimizing apparatus and the optimizing method of the present invention, the physical system structure, shape, system identification, graph display, data interpolation, signal prediction model, pattern recognition, etc., can be used in industry. With respect to various problems that can be used, it is possible to obtain an effect that it is possible to handle an input or the like having a spread which has been difficult in the past.

[Brief description of the drawings]

FIG. 1 is a diagram showing an upper limit function, a lower limit function, and an aptitude function according to the first embodiment of the present invention.

FIG. 2 is a schematic view of a possible area S ′ according to the first embodiment of the present invention.

FIG. 3 is a schematic view of a partial area S′1 of a possible area S ′ according to the first embodiment of the present invention.

FIG. 4 is a flowchart according to the first embodiment of the present invention.

FIG. 5 is a diagram illustrating the vicinity of a lower limit function according to a third embodiment of the present invention.

FIG. 6 is a diagram illustrating a generated convex hull C ⁻ wrapping a lower limit function M ⁻ (x) according to the third embodiment of the present invention.

FIG. 7 is a diagram showing an initial state of a process of forming a generated convex hull by the method A according to the third embodiment of the present invention.

FIG. 8 is a diagram showing an end state of a process of forming a generated convex hull by the method A.

FIG. 9 is a diagram showing a process of forming a generated convex hull by a method B.

FIG. 10 is a diagram showing a process of configuring a generated convex hull by a method C.

FIG. 11 is a schematic diagram showing an apparatus for performing the suiting method according to the fourth embodiment of the present invention.

FIG. 12 is a diagram of a square amplitude characteristic obtained in Embodiment 4 of the present invention.

FIG. 13 is a diagram illustrating a design of a digital filter that simultaneously specifies an amplitude characteristic and a phase characteristic according to a fifth embodiment of the present invention.

14 is a diagram in which an appropriate area C (ω) in FIG. 13 is moved on a real axis.

FIG. 15 is a diagram illustrating a design of a digital filter in which only a phase characteristic is designated.

FIG. 16 is a block diagram of a conventional identification device.

FIG. 17 is a block diagram of an identification device according to a seventh embodiment of the present invention.

FIG. 18 is a block diagram of a characteristic correction device according to an eighth embodiment of the present invention.

[Explanation of symbols]

T proper area f (x) suitability function P ⁺ ₁ , P ⁺ ₂ ,. . . , P ⁺ _n lower limit function points P ^- ₁ , P ^- ₂ ,. . . , P ^- _n upper limit function point D range

Claims (27)

[Claims] 1. An optimization method for determining a parameter of a system having physical characteristics by a computer system comprising an input unit, a storage unit, and an arithmetic unit, wherein the data of the physical characteristics is input by the input unit,
A first step of setting an appropriate area corresponding to the permissible range of the physical characteristic in the storage means, and using the arithmetic means, an appropriate function included at least approximately within the appropriate area is calculated by the simultaneous inequality Second obtained by solving
Wherein the second step sets the suitability function as a family of functions having parameters corresponding to the parameters of the system, and a condition that the suitability function is included in the suitability region. Setting a simultaneous function inequality for the parameters of the suitability function, solving the simultaneous inequalities to find a solution area of the parameters of the suitability function, even if there is a deviation in the system, the parameters of the suitability function And selecting a solution such that is within the range of the solution region. 2. The suiting method according to claim 1, wherein the suitability region is defined by an upper limit function for limiting an upper limit of the suitability region and a lower limit function for limiting a lower limit of the suitability region. 3. The suiting method according to claim 1, wherein the suitability area is defined by an objective function and an allowable error range. 4. The method according to claim 1, wherein the appropriate area is defined by at least one convex hull. 5. The suiting method according to claim 1, wherein the suitability area is defined by a lower limit or an upper limit on a point sequence. 6. The method according to claim 1, wherein the suitability function represents a function for interpolating data. 7. The method of claim 1, wherein the suitability function represents a response characteristic of a digital signal processing system. 8. The method of claim 1, wherein said suitability function represents a response characteristic of an analog system. 9. A method for measuring a characteristic of a system having an analog system and at least one of an A / D converter and a D / A converter, and obtaining a difference between the measured characteristic and a desired characteristic. A characteristic correction method for correcting the difference by a digital signal processing system, wherein input means,
The parameters of the digital signal processing system are stored using a computer having a storage unit and a calculation unit.The opening is input as physical characteristics by the input unit, and an appropriate area corresponding to an allowable range of the physical characteristics is input. Setting in the storage means, and using the arithmetic means, by solving a simultaneous inequality, an aptitude function determination step of obtaining an aptitude function at least approximately included in the aptitude area, designing the aptitude function determination step Is
Setting the aptitude function as a family of functions having parameters corresponding to the parameters of the digital signal processing system, and setting a condition that the aptitude function is included in the aptitude region as a simultaneous inequality for the parameters of the aptitude function. And setting a solution area of the parameters of the suitability function by solving the simultaneous inequalities, even if there is a deviation in any of the systems, the parameters of the suitability function is in the range of the solution area Selecting a solution to fit. 10. A system which is connected to a system having an analog system and at least one of an A / D converter and a D / A converter, receives a measurement result of the characteristics of the system, and receives the characteristics of the system and the desired characteristics. A characteristic correction device for correcting the difference between the two by a digital signal processing system, wherein the characteristic correction device is configured using a computer, and the difference is input as a physical characteristic, and an allowable range of the physical characteristic is input. Means for setting a suitable area on the memory corresponding to, and suitable function determining means for obtaining a suitable function included at least approximately in the suitable area by solving simultaneous inequalities, the suitable function determining means, First means for setting an aptitude function as a family of functions having parameters corresponding to the parameters of the digital signal processing system; A second means for setting the condition that the suitability function is included in the suitability region by a simultaneous inequality for the parameters of the suitability function, and a third means for solving the simultaneous inequality to obtain a solution region for the parameters of the suitability function And a fourth means for selecting a solution such that the parameters of the suitability function fall within the range of the solution area even when there is a deviation in any one of the systems. apparatus. 11. The suiting method according to claim 1, wherein the physical property is a discrete sequence signal. 12. The method according to claim 11, wherein said suitability function represents a discrete system or an analog system. 13. An optimization method for determining parameters of a system having physical characteristics by a computer system comprising input means, storage means, and arithmetic means, comprising the steps of: inputting physical characteristics which are discrete sequence signals; Setting an aptitude area corresponding to the allowable range on a memory, and an aptitude function determination step of finding an aptitude function at least approximately included in the aptitude area by solving simultaneous inequalities, the aptitude function determination step , Approximating the constraint that the aptitude function is included in the aptitude area by a generated convex hull, and setting it as a family of functions having parameters corresponding to the parameters of the system. Steps constituting the generated convex hull so as to include a vector represented in the parameter space of the function. Setting a constraint that the aptitude function is included in the aptitude area by a simultaneous inequality with a vertex of the generated convex hull as a constraint; and solving the simultaneous inequality and solving the parameter of the aptitude function. Optimizing method. 14. A computer system for determining a parameter of a system having physical characteristics, wherein the data of the physical characteristics is inputted, and an allowable range of the physical characteristics is approximated by an upper limit or a lower limit on a point sequence. Means for setting an area on a memory, and an aptitude function obtained by solving a simultaneous inequality equation while setting an aptitude function at least approximately included in the range of the aptitude area as a family of functions having parameters. Determining means, wherein the suitability function determining means sets the suitability function as a family of functions having parameters corresponding to the parameters of the system, and the suitability function is included in the suitability region. A second means for setting the condition as a simultaneous inequality for the parameters of the suitability function, and solving the simultaneous inequality Third means for obtaining a solution area of a parameter of the suitability function, and fourth means for selecting a solution such that the parameter of the suitability function falls within the range of the solution area even when the system has a deviation. And an aptitude area display means for visually displaying the aptitude area, and an aptitude function display means for displaying the aptitude function superimposed on the aptitude area displayed by the aptitude area display means, Function display means, while the suitability function obtained once is displayed by the suitability function display means,
Alternatively, a computer system which displays the absence of an aptitude function and interactively accepts a point sequence forming an aptitude area and a change in part or all of an upper limit value or a lower limit value of the point sequence. 15. An aptitude optimizing method for determining and displaying parameters of a system having physical characteristics by a computer system comprising input means, storage means, arithmetic means and display means, the input means comprising: Enter the data,
Setting an appropriate area on the memory that approximates the allowable range of the physical property with an upper limit or a lower limit on a point sequence;
An aptitude function that is at least approximately included within the range of the aptitude area using the arithmetic means, is set as a family of functions having parameters, and an aptitude function determination step of determining the parameters by solving simultaneous inequalities. The suitability function determining step includes: setting the suitability function as a family of functions having parameters corresponding to the parameters of the system; and a condition that the suitability function is included in the suitability region, Setting with simultaneous inequalities for the parameters of
Solving the simultaneous inequalities to find a solution area of the parameters of the suitability function, and, even when there is a deviation in the system, selecting a solution so that the parameters of the suitability function fall within the range of the solution area. Including an aptitude area display step of visually displaying the aptitude area, and an aptitude function display step of superimposing and displaying the aptitude function on the aptitude area displayed by the aptitude area display step, The function display step displays the suitability function once obtained, or displays the absence of the suitability function, and shows a sequence of points constituting the suitability region and a part or all of the upper limit value or lower limit value of the sequence of points. A method for optimizing, characterized by interactively accepting a change of a character. 16. A step of inputting data of physical characteristics, setting an appropriate area in the memory in which an allowable range of the physical properties is approximated by an upper limit or a lower limit on a point sequence, and setting the appropriate area within the appropriate area. An aptitude function included at least approximately, as a family of functions having parameters, an aptitude function determination step of obtaining the parameters by solving simultaneous inequalities, and an aptitude area display step of visually displaying the aptitude area And a program for executing an aptitude function display step of superimposing and displaying the aptitude function on the aptitude area displayed by the aptitude area display step, wherein the aptitude function determination step includes: Setting as a family of functions having parameters corresponding to the parameters of the system; and Setting the condition of being included in the aptitude region by simultaneous inequalities for the parameters of the aptitude function, solving the simultaneous inequalities to obtain a solution area of the parameters of the aptitude function, and when the system has a deviation. And selecting a solution so that the parameters of the suitability function fall within the range of the solution area, and the suitability function displaying step includes displaying the suitability function once obtained, or To show the absence of
A storage device characterized by interactively accepting a point sequence forming an appropriate area and a change in part or all of an upper limit value or a lower limit value of the point sequence. 17. A computer system for determining parameters of a system having physical characteristics, comprising: means for inputting data of the physical characteristics and setting an appropriate area corresponding to an allowable range of the physical characteristics in a memory; An aptitude function determining means for obtaining an aptitude function at least approximately included in the aptitude area by solving an inequality, wherein the aptitude function determining means converts the aptitude function into a function having a parameter corresponding to a parameter of the system. A first means for setting the family as a family, a second means for setting the condition that the aptitude function is included in the aptitude area by a simultaneous inequality for the parameters of the aptitude function, and solving the simultaneous inequality And third means for obtaining a solution area of a parameter of the suitability function, wherein the suitability function protrudes from the suitability area. In this case, means for correcting the aptitude function by correcting the position of the zero point of the function of the difference between the boundary of the appropriate area near the protruding portion and the obtained aptitude function to remove the overhang is provided. A computer system characterized by the above-mentioned. 18. An optimization method for determining a parameter of a system having physical characteristics by a computer system including an input unit, a storage unit, and an arithmetic unit, wherein the data of the physical characteristics is input by the input unit;
Setting an appropriate area in the memory corresponding to the permissible range of the physical characteristics, and an appropriate function determining step of obtaining an appropriate function at least approximately included in the appropriate area by solving simultaneous inequalities using the arithmetic means. The suitability function determining step, setting the suitability function as a family of functions having parameters corresponding to the parameters of the system, and a condition that the suitability function is included in the suitability region, Setting a simultaneous inequality for the parameters of the fitness function;
Solving the simultaneous inequalities to obtain a solution area of the parameters of the suitability function, and using the arithmetic means, when the suitability function protrudes from the suitability area, the appropriate area of the suitability area near the protruding part is included. A method for optimizing, comprising a step of correcting the aptitude function by removing a protrusion by correcting a position of a zero point of a function of a difference between the boundary and the obtained aptitude function. 19. A step of inputting data of physical characteristics, setting an appropriate area corresponding to an allowable range of the physical properties in a memory, and solving an simultaneous inequality to an appropriate function included at least approximately in the appropriate area. A program for executing a fitness function determining step for determining the fitness function, wherein the fitness function determining step sets the fitness function as a family of functions having parameters corresponding to the parameters of the system. And setting the condition that the aptitude function is included in the aptitude area by simultaneous inequalities for the parameters of the aptitude function, and obtaining the solution area of the parameters of the aptitude function by solving the simultaneous inequalities And, when the fitness function is out of the fitness area, the A program for executing a step of correcting the aptitude function and removing the overflow by correcting the position of the zero point of the function of the difference between the boundary of the aptitude region near the minute and the obtained aptitude function is stored. Storage device. 20. A computer system for determining parameters of a system having physical characteristics, comprising: means for inputting data of the physical characteristics and setting an appropriate area corresponding to an allowable range of the physical characteristics in a memory; An aptitude function determining means for obtaining an aptitude function approximately included in the aptitude area by solving an inequality, wherein the aptitude function determination means generates a constraint that the aptitude function is included in the aptitude area. , And the suitability function is set as a family of functions determined by parameters corresponding to the parameters of the system, and the domain of the physical property is set to a section having a size corresponding to a desired accuracy. Means for dividing, so that a vector representing the restriction of the suitability area in the parameter space of the suitability function is included on each section. Means for configuring the generated convex hull, means for setting a constraint that the aptitude function is included in the aptitude area by a simultaneous inequality with a vertex of the generated convex hull as a constraint, and solving the simultaneous inequality Means for obtaining a solution area of the parameter of the suitability function. 21. An optimizing method for determining a parameter of a system having physical characteristics by a computer system comprising input means, storage means, and arithmetic means, wherein data of the physical characteristics is input, and the data is set to an allowable range of the physical characteristics. Setting a corresponding aptitude area in a memory, and an aptitude function determination step of finding an aptitude function approximately included in the aptitude area by solving a simultaneous inequality, wherein the aptitude function determination step is such that the aptitude function is Approximating the constraint of being included in the suitability region by a generated convex hull, and setting the suitability function as a family of functions determined by parameters, and defining the physical property domain according to desired accuracy. Dividing the appropriate area into a section having an appropriate size in the parameter space of the appropriate function. Configuring the generated convex hull so as to include a vector represented by the following equation; and setting a constraint that the aptitude function is included in the aptitude region by a simultaneous inequality using the vertex of the generated convex hull as a constraint. And a step of solving the simultaneous inequalities to obtain a solution area of the parameters of the suitability function. 22. A step of inputting data of physical characteristics, setting an appropriate area corresponding to an allowable range of the physical properties in a memory, and solving an simultaneous inequality to obtain an appropriate function included approximately in the appropriate area. A fitness function determining step to determine and a storage device for executing a program for executing, the fitness function determining step approximates a constraint that the fitness function is included in the fitness area with a generated convex hull, And setting the suitability function as a family of functions determined by parameters, dividing the domain of the physical property into sections of a size corresponding to a desired accuracy, and on each section, Configuring the generated convex hull so as to include a vector representing the restriction of the suitability region in the parameter space of the suitability function; and Setting a constraint of being included in the suitability region by a simultaneous inequality with the vertex of the generated convex hull as a constraint; and solving the simultaneous inequality to obtain a solution region of a parameter of the suitability function. A storage device characterized by the above-mentioned. 23. A computer system for determining parameters of a system having physical characteristics, comprising: means for inputting data of the physical characteristics and setting an appropriate area corresponding to an allowable range of the physical characteristics in a memory; An aptitude function determining means for obtaining an aptitude function at least approximately included in the aptitude area by solving an inequality, wherein the aptitude function determining means converts the aptitude function to a function having a parameter corresponding to a parameter of the system. A first means for setting as a family of; a second means for setting a condition that the suitability function is included in the suitability region by a simultaneous inequality for a parameter of the suitability function;
Third means for solving the simultaneous inequality to obtain a solution area of the parameter of the suitability function, and converting a predetermined change amount of the parameter of the suitability function into a change amount of the suitability area in the parameter space, A computer system comprising an appropriate area reducing unit for reducing an appropriate area in the parameter space by an amount corresponding to a change amount of the appropriate area. 24. An optimization method for determining a parameter of a system having physical characteristics by a computer system comprising an input unit, a storage unit, and an arithmetic unit, wherein the input unit inputs the data of the physical characteristics,
Setting an appropriate area in the memory corresponding to the permissible range of the physical characteristics, and an appropriate function determining step of obtaining an appropriate function at least approximately included in the appropriate area by solving simultaneous inequalities using the arithmetic means. The suitability function determining step, the step of setting the suitability function as a family of functions having parameters corresponding to the parameters of the system, and a condition that the suitability function is included in the suitability region, Setting a simultaneous function inequality for the parameters of the suitability function, and solving the simultaneous inequality to obtain a solution area of the parameters of the suitability function, and a predetermined change amount of the parameters of the suitability function in a parameter space. The suitable area is converted into the appropriate area by the amount corresponding to the change in the appropriate area. A method for optimizing, comprising a step of reducing in a parameter space. 25. A step of inputting data of physical characteristics, setting an appropriate area corresponding to an allowable range of the physical properties in a memory, and solving an simultaneous inequality to an appropriate function included at least approximately in the appropriate area. A program for executing a fitness function determining step of determining the fitness function, wherein the fitness function determining step sets the fitness function as a family of functions having parameters corresponding to system parameters; and Setting the condition that is included in the aptitude region by a simultaneous inequality for the parameters of the aptitude function, and solving the simultaneous inequality to obtain a solution area of the parameters of the aptitude function, Converts a specified change in function parameters into a change in the appropriate area in the parameter space , Memory device by the amount suitability area corresponding to the variation of the suitability region, characterized in that the program for performing the steps of reducing in the parameter space is stored. 26. A computer system for determining parameters of a system having physical characteristics, comprising: means for inputting data of the physical characteristics and setting an appropriate area corresponding to an allowable range of the physical characteristics in a memory; An aptitude function determining means for obtaining an aptitude function at least approximately included in the aptitude area by solving an inequality, wherein the aptitude function determining means converts the aptitude function to a function having a parameter corresponding to a parameter of the system. A first means for setting as a family of; a second means for setting a condition that the suitability function is included in the suitability region by a simultaneous inequality for a parameter of the suitability function;
Third means for solving the simultaneous inequalities to find a solution area of the parameters of the suitability function, and selecting a solution so that the parameters of the suitability function fall within the range of the solution area even when the system has a deviation. Computer system comprising: 27. A first step of inputting data of physical characteristics and setting an appropriate area corresponding to an allowable range of the physical properties in a memory, and an aptitude function approximately included in the appropriate area. A storage device storing a program for performing a second step of using a processor to determine by solving a system of inequalities, said second step corresponding to said fitness function corresponding to a system parameter Setting the family as a family of functions having the parameters described above, setting the condition that the suitability function is included in the suitability region by a parameter simultaneous inequality of the suitability function, and solving the simultaneous inequality to solve the suitability function. Determining a solution region of the parameter;
Selecting a solution such that the parameters of the suitability function fall within the range of the solution area even when the system has a deviation.