JP2010071654A - Method and device for estimating causal physical quantity - Google Patents
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本発明は、原因物理量推定方法および装置に関し、さらに詳しくは、原因物理量に依存する結果物理量が与えられたときに煩雑な演算を要さずに結果物理量から原因物理量を推定することが出来る原因物理量推定方法および装置に関する。 The present invention relates to a causal physical quantity estimation method and apparatus, and more particularly, a causal physical quantity capable of estimating a causal physical quantity from a result physical quantity without requiring a complicated calculation when a result physical quantity depending on the causal physical quantity is given. The present invention relates to an estimation method and apparatus.
従来、測定用コイルで検出された磁界強度と信号源コイルの仮定の位置から計算された磁界強度の誤差が小さくなるように信号源コイルの仮定の位置を変更し、誤差が小さい信号源コイルの仮定の位置を求める位置推定方法が知られている（例えば、特許文献１参照。）。
上記従来の位置推定方法では、仮定の位置を変更するのに偏微分演算が必要になるなど、演算が煩雑になる問題点があった。
そこで、本発明の目的は、煩雑な演算を要さずに結果物理量から原因物理量を推定することが出来る原因物理量推定方法および装置を提供することにある。
The conventional position estimation method has a problem that the calculation is complicated, for example, partial differential calculation is required to change the assumed position.
Accordingly, an object of the present invention is to provide a causal physical quantity estimation method and apparatus capable of estimating a causal physical quantity from a result physical quantity without requiring a complicated calculation.
第１の観点では、本発明は、原因物理量ｘに依存する結果物理量ｙを与える順方向関数ｙ＝ｆ(ｘ)および結果物理量ｙを基に近似原因物理量ｘ’を推定する近似逆方向関数ｘ’＝ｇ’(ｙ)が既知であるとき、測定により得られた結果物理量ｙに対して
ｘ’＝ｇ’(ｙ)
ｘ1＝２・ｘ’−ｇ’(ｆ(ｘ’))
により第１推定原因物理量ｘ1を求めることを特徴とする原因物理量推定方法を提供する。
上記構成における数式の意味については実施例１で説明する。
上記第１の観点による原因物理量推定方法では、順方向関数ｙ＝ｆ(ｘ)の逆関数ｘ＝ｇ(ｙ)を用いるのではなく、近似逆方向関数ｘ’＝ｇ’(ｙ)を用いる。
例えば、ｙ＝ｘ2＋ｘという順方向関数ｆ(ｘ)を想定するとき、逆関数ｇ(ｙ)はｘ＝（−１±√{１＋４ｙ}）／２となり、演算が煩雑になる。これに対して、本発明では、例えば０＜ｘ≦０．１のときには、ｙ≒ｘであるから、ｘ’＝ｙという近似逆方向関数ｇ’(ｙ)を用いる。すると、上式は、
ｘ’＝ｙ
ｘ1＝２・ｘ’−（ｘ’2＋ｘ’）＝ｘ’−ｘ’2
となり、煩雑な演算を要さずに、結果物理量ｙから原因物理量ｘ1を推定することが出来る。
なお、順方向関数ｙ＝ｆ(ｘ)の逆関数ｘ＝ｇ(ｙ)を用いれば結果物理量ｙから原因物理量ｘを求めることが出来るが、物理現象においては、しばしば逆関数ｘ＝ｇ(ｙ)が非常に煩雑な演算になってしまう。このため、本発明では、逆関数ｘ＝ｇ(ｙ)を用いるのではなく、近似逆方向関数ｘ’＝ｇ’(ｙ)を用いる。
In a first aspect, the present invention provides a forward function y = f (x) that gives a result physical quantity y that depends on the cause physical quantity x and an approximate backward function x that estimates the approximate cause physical quantity x ′ based on the result physical quantity y. When '= g' (y) is known, for the result physical quantity y obtained by measurement, x '= g' (y)
x 1 = 2 · x′−g ′ (f (x ′))
The cause physical quantity estimation method characterized by obtaining the first estimated cause physical quantity x 1 is provided.
The meaning of mathematical expressions in the above configuration will be described in the first embodiment.
In the causal physical quantity estimation method according to the first aspect, the approximate backward function x ′ = g ′ (y) is used instead of the inverse function x = g (y) of the forward function y = f (x). .
For example, when a forward function f (x) of y = x 2 + x is assumed, the inverse function g (y) is x = (− 1 ± √ {1 + 4y}) / 2, and the calculation becomes complicated. On the other hand, in the present invention, when 0 <x ≦ 0.1, for example, y≈x, and therefore an approximate backward function g ′ (y) of x ′ = y is used. Then, the above formula is
x ′ = y
x 1 = 2 · x ′ − (x ′ 2 + x ′) = x′−x ′ 2
Thus, the causal physical quantity x 1 can be estimated from the result physical quantity y without requiring a complicated calculation.
If the inverse function x = g (y) of the forward function y = f (x) is used, the cause physical quantity x can be obtained from the resulting physical quantity y. However, in the physical phenomenon, the inverse function x = g (y ) Becomes a very complicated calculation. For this reason, in the present invention, instead of using the inverse function x = g (y), the approximate backward function x ′ = g ′ (y) is used.
第２の観点では、本発明は、原因物理量ｘに依存する結果物理量ｙを与える順方向関数ｙ＝ｆ(ｘ)および結果物理量ｙを基に近似原因物理量ｘ’を推定する近似逆方向関数ｘ’＝ｇ’(ｙ)が既知であり、ｉを２以上の整数とするとき、測定により得られた結果物理量ｙに対して
ｘ’＝ｇ’(ｙ)
ｘ1＝２・ｘ’−ｇ’(ｆ(ｘ’))
ｘi＝ｘ’＋ｘi-1−ｇ’(ｆ(ｘi-1)）
により第ｉ推定原因物理量ｘiを求めることを特徴とする原因物理量推定方法を提供する。
上記構成における数式の意味については実施例で説明する。
上記第２の観点による原因物理量推定方法では、順方向関数ｙ＝ｆ(ｘ)の逆関数ｘ＝ｇ(ｙ)を用いるのではなく、近似逆方向関数ｘ’＝ｇ’(ｙ)を用いる。
例えば、ｙ＝ｘ2＋ｘという順方向関数ｆ(ｘ)を想定するとき、逆関数ｇ(ｙ)はｘ＝（−１±√{１＋４ｙ}）／２となり、演算が煩雑になる。これに対して、本発明では、例えば０＜ｘ≦０．１のときには、ｙ≒ｘであるから、ｘ’＝ｙという近似逆方向関数ｇ’(ｙ)を用いる。すると、上式は、
ｘ’＝ｙ
ｘ1＝２・ｘ’−（ｘ’2＋ｘ’）＝ｘ’−ｘ’2
ｘ2＝ｘ’＋ｘ1−(ｘ1 2＋ｘ1)＝ｘ’−ｘ1 2
…
ｘi＝ｘ’＋ｘi-1−(ｘi-1 2＋ｘi-1）＝ｘ’−ｘi-1 2
となり、煩雑な演算を要さずに、結果物理量ｙから原因物理量ｘiを推定することが出来る。
In a second aspect, the present invention provides a forward function y = f (x) that gives a result physical quantity y that depends on the cause physical quantity x and an approximate backward function x that estimates the approximate cause physical quantity x ′ based on the result physical quantity y. When '= g' (y) is known and i is an integer equal to or greater than 2, for the result physical quantity y obtained by measurement, x '= g' (y)
x 1 = 2 · x′−g ′ (f (x ′))
x i = x ′ + x i−1 −g ′ (f (x i−1 ))
Providing cause physical quantity estimation method characterized by obtaining the i-th presumed cause physical quantity x i by.
The meaning of the mathematical expression in the above configuration will be described in an embodiment.
In the causal physical quantity estimation method according to the second aspect, the approximate backward function x ′ = g ′ (y) is used instead of the inverse function x = g (y) of the forward function y = f (x). .
For example, when a forward function f (x) of y = x 2 + x is assumed, the inverse function g (y) is x = (− 1 ± √ {1 + 4y}) / 2, and the calculation becomes complicated. On the other hand, in the present invention, when 0 <x ≦ 0.1, for example, y≈x, and therefore an approximate backward function g ′ (y) of x ′ = y is used. Then, the above formula is
x ′ = y
x 1 = 2 · x ′ − (x ′ 2 + x ′) = x′−x ′ 2
x 2 = x ′ + x 1 − (x 1 2 + x 1 ) = x′−x 1 2
...
x i = x ′ + x i−1 − (x i−1 2 + x i−1 ) = x′−x i−1 2
Next, without requiring complicated calculation, it is possible to estimate the cause physical quantity x i from the result the physical quantity y.
第３の観点では、本発明は、前記第２の観点による原因物理量推定方法において、予め設定した誤差閾値をＥとするとき、
｜ｘ’−ｇ’(ｆ(ｘ’))｜≦Ｅならｘ’だけを求め、
｜ｘ’−ｇ’(ｆ(ｘ’))｜＞Ｅかつ｜ｘ’−ｇ’(ｆ(ｘ1))｜≦Ｅならｘ1まで求め、
｜ｘ’−ｇ’(ｆ(ｘi-1))｜＞Ｅかつ｜ｘ’−ｇ’(ｆ(ｘi))｜≦Ｅならｘiまで求めることを特徴とする原因物理量推定方法を提供する。
上記第３の観点による原因物理量推定方法では、予め設定した誤差閾値Ｅ以下の誤差で原因物理量を求めることが出来る。
In a third aspect, the present invention relates to the causal physical quantity estimation method according to the second aspect, when a preset error threshold is E,
If | x′−g ′ (f (x ′)) | ≦ E, only x ′ is obtained.
| X'-g '(f ( x')) |> E and | x'-g '(f ( x 1)) | determined to if ≦ E x 1,
A causal physical quantity estimation method characterized by obtaining up to x i if | x′−g ′ (f (x i−1 )) |> E and | x′−g ′ (f (x i )) | ≦ E. provide.
In the causal physical quantity estimation method according to the third aspect, the causal physical quantity can be obtained with an error equal to or less than a preset error threshold E.
第４の観点では、本発明は、原因物理量ｘに依存する結果物理量ｙを与える順方向関数ｙ＝ｆ(ｘ)および結果物理量ｙを基に近似原因物理量ｘ’を推定する近似逆方向関数ｘ’＝ｇ’(ｙ)が既知であるとき、測定により得られた結果物理量ｙに対して
ｘ’＝ｇ’(ｙ)
ｘ1＝２・ｘ’−ｇ’(ｆ(ｘ’))
により第１推定原因物理量ｘ1を演算し出力する第１推定原因物理量演算手段を具備したことを特徴とする原因物理量推定装置を提供する。
上記第４の観点による原因物理量推定装置では、前記第１の観点による原因物理量推定方法を好適に実施できる。
In a fourth aspect, the present invention provides a forward function y = f (x) that gives a result physical quantity y that depends on the cause physical quantity x and an approximate backward function x that estimates the approximate cause physical quantity x ′ based on the result physical quantity y. When '= g' (y) is known, for the result physical quantity y obtained by measurement, x '= g' (y)
x 1 = 2 · x′−g ′ (f (x ′))
By providing cause physical quantity estimating apparatus characterized by comprising a first probable cause physical quantity calculation means for first calculates the probable cause physical quantity x 1 output.
In the causal physical quantity estimation device according to the fourth aspect, the causal physical quantity estimation method according to the first aspect can be suitably implemented.
第５の観点では、本発明は、原因物理量ｘに依存する結果物理量ｙを与える順方向関数ｙ＝ｆ(ｘ)および結果物理量ｙを基に近似原因物理量ｘ’を推定する近似逆方向関数ｘ’＝ｇ’(ｙ)が既知であり、ｉを２以上の整数とするとき、測定により得られた結果物理量ｙに対して
ｘ’＝ｇ’(ｙ)
ｘ1＝２・ｘ’−ｇ’(ｆ(ｘ’))
ｘi＝ｘ’＋ｘi-1−ｇ’(ｆ(ｘi-1)）
により第ｉ推定原因物理量ｘiを演算し出力する第ｉ推定原因物理量演算手段を具備したことを特徴とする原因物理量推定装置を提供する。
上記第５の観点による原因物理量推定装置では、前記第２の観点による原因物理量推定方法を好適に実施できる。
In a fifth aspect, the present invention provides a forward function y = f (x) that gives a result physical quantity y that depends on the cause physical quantity x and an approximate backward function x that estimates the approximate cause physical quantity x ′ based on the result physical quantity y. When '= g' (y) is known and i is an integer equal to or greater than 2, for the result physical quantity y obtained by measurement, x '= g' (y)
x 1 = 2 · x′−g ′ (f (x ′))
x i = x ′ + x i−1 −g ′ (f (x i−1 ))
A causal physical quantity estimation device comprising an i-th estimated causal physical quantity calculating means for calculating and outputting the i-th estimated causal physical quantity x i is provided.
In the causal physical quantity estimation device according to the fifth aspect, the causal physical quantity estimation method according to the second aspect can be suitably implemented.
第６の観点では、本発明は、原因物理量ｘに依存する結果物理量ｙを与える順方向関数ｙ＝ｆ(ｘ)および結果物理量ｙを基に近似原因物理量ｘ’を推定する近似逆方向関数ｘ’＝ｇ’(ｙ)が既知であり、ｉを２以上の整数とするとき、測定により得られた結果物理量ｙに対して
ｘ’＝ｇ’(ｙ)
により第０推定原因物理量ｘ’を演算し出力する第０推定原因物理量演算手段と、
ｘ1＝２・ｘ’−ｇ’(ｆ(ｘ’))
により第１推定原因物理量ｘ1を演算し出力する第１推定原因物理量演算手段と、
ｘi＝ｘ’＋ｘi-1−ｇ’(ｆ(ｘi-1)）
により第ｉ推定原因物理量ｘiを演算し出力する第ｉ推定原因物理量演算手段と、
予め設定した誤差閾値をＥとするとき、
｜ｘ’−ｇ’(ｆ(ｘ’))｜≦Ｅなら前記第０推定原因物理量演算手段だけを作動させ、
｜ｘ’−ｇ’(ｆ(ｘ’))｜＞Ｅかつ｜ｘ’−ｇ’(ｆ(ｘ1))｜≦Ｅなら前記第１推定原因物理量演算手段までを作動させ、
｜ｘ’−ｇ’(ｆ(ｘi-1))｜＞Ｅかつ｜ｘ’−ｇ’(ｆ(ｘi))｜≦Ｅなら前記第ｉ推定原因物理量演算手段までを作動させる演算制御手段とを具備したことを特徴とする原因物理量推定装置を提供する。
上記第６の観点による原因物理量推定装置では、前記第３の観点による原因物理量推定方法を好適に実施できる。
In a sixth aspect, the present invention provides a forward function y = f (x) that gives a result physical quantity y that depends on the cause physical quantity x and an approximate backward function x that estimates the approximate cause physical quantity x ′ based on the result physical quantity y. When '= g' (y) is known and i is an integer equal to or greater than 2, for the result physical quantity y obtained by measurement, x '= g' (y)
A zeroth estimated cause physical quantity calculating means for calculating and outputting the zeroth estimated cause physical quantity x ′,
x 1 = 2 · x′−g ′ (f (x ′))
A first estimated cause physical quantity calculating means for calculating and outputting a first estimated cause physical quantity x 1 by:
x i = x ′ + x i−1 −g ′ (f (x i−1 ))
And the i probable cause physical quantity calculation means for the i calculates the probable cause physical quantity x i output by,
When the preset error threshold is E,
If | x′−g ′ (f (x ′)) | ≦ E, only the 0th presumed cause physical quantity calculating means is operated,
If | x′−g ′ (f (x ′)) |> E and | x′−g ′ (f (x 1 )) | ≦ E, the first estimated cause physical quantity calculating means is operated.
If | x′−g ′ (f (x i−1 )) |> E and | x′−g ′ (f (x i )) | ≦ E, calculation control for operating up to the i th estimated cause physical quantity calculation means A causal physical quantity estimation apparatus characterized by comprising: means.
In the causal physical quantity estimation device according to the sixth aspect, the causal physical quantity estimation method according to the third aspect can be suitably implemented.
本発明の原因物理量推定方法および装置によれば、煩雑な演算を要さずに、結果物理量から原因物理量を推定することが出来る。 According to the causal physical quantity estimation method and apparatus of the present invention, it is possible to estimate the causal physical quantity from the result physical quantity without requiring a complicated calculation.
以下、図に示す実施の形態により本発明をさらに詳細に説明する。なお、これにより本発明が限定されるものではない。 Hereinafter, the present invention will be described in more detail with reference to embodiments shown in the drawings. Note that the present invention is not limited thereby.
図１は、実施例１に係る原因物理量推定装置１０を示す説明図である。
この原因物理量推定装置１０は、センサ装置Ｄから入力された結果物理量ｙから原因物理量ｘを推定し推定原因物理量ｘ’またはｘ1またはｘ2またはｘiを出力するコンピュータ１から構成される。
コンピュータ１は、原因物理量ｘに依存する結果物理量ｙを与える順方向関数ｙ＝ｆ(ｘ)と、結果物理量ｙを基に近似原因物理量ｘ’を推定する近似逆方向関数ｘ’＝ｇ’(ｙ)と、誤差閾値Ｅと、上限演算回数Ｉとを予めメモリに記憶している。
FIG. 1 is an explanatory diagram illustrating a cause physical
The causal physical
The
例えば、センサ装置Ｄは検出コイルにより磁界強度を計測する磁界強度計測装置であり、結果物理量ｙは磁界強度であり、原因物理量ｘは磁気発生源に対する検出コイルの空間位置である。 For example, the sensor device D is a magnetic field strength measuring device that measures the magnetic field strength with a detection coil. As a result, the physical quantity y is the magnetic field strength, and the cause physical quantity x is the spatial position of the detection coil with respect to the magnetic source.
図２は、コンピュータ１で実行される原因物理量推定処理を示すフロー図である。
FIG. 2 is a flowchart showing the cause physical quantity estimation process executed by the
ステップＳ１では、センサ装置Ｄから結果物理量ｙを取得する。
図３の（ａ）に示すように、原因物理量ｘと結果物理量ｙとは順方向関数ｙ＝ｆ(ｘ)の関係にある。
In step S1, the result physical quantity y is acquired from the sensor device D.
As shown in FIG. 3A, the cause physical quantity x and the result physical quantity y have a relationship of a forward function y = f (x).
ステップＳ２では、結果物理量ｙを基に近似逆方向関数ｘ’＝ｇ’(ｙ)を用いて第０推定原因物理量ｘ’を求める。
ｘ’＝ｇ’(ｙ)
図３の（ａ）に示すように、第０推定原因物理量ｘ’は原因物理量ｘの近似値になっている。
In step S2, the 0th estimated cause physical quantity x ′ is obtained using the approximate backward function x ′ = g ′ (y) based on the result physical quantity y.
x ′ = g ′ (y)
As shown in FIG. 3A, the 0th estimated cause physical quantity x ′ is an approximate value of the cause physical quantity x.
ステップＳ３では、第０推定原因物理量ｘ’を基に順方向関数ｙ＝ｆ(ｘ)を用いて第０計算結果物理量ｙ’を求める。
ｙ’＝ｆ(ｘ’)
図３の（ａ）に示すように、第０推定原因物理量ｘ’と第０計算結果物理量ｙ’とは順方向関数ｙ＝ｆ(ｘ)の関係にある。
In step S3, the 0th calculation result physical quantity y ′ is obtained using the forward function y = f (x) based on the 0th estimated cause physical quantity x ′.
y ′ = f (x ′)
As shown in FIG. 3A, the 0th estimated cause physical quantity x ′ and the 0th calculation result physical quantity y ′ are in a relationship of a forward function y = f (x).
ステップＳ４では、第０計算結果物理量ｙ’を基に近似逆方向関数ｘ’＝ｇ’(ｙ)を用いて第０計算原因物理量ｘ”を求める。
ｘ”＝ｇ’(ｙ’)＝ｇ’(ｆ(ｘ’))＝ｇ’(ｆ(ｇ’(ｙ)))
図３の（ａ）に示すように、第０計算原因物理量ｘ”は第０推定原因物理量ｘ’の近似値になっている。
In step S4, the zeroth calculation cause physical quantity x ″ is obtained using the approximate backward function x ′ = g ′ (y) based on the zeroth calculation result physical quantity y ′.
x ″ = g ′ (y ′) = g ′ (f (x ′)) = g ′ (f (g ′ (y)))
As shown in FIG. 3A, the 0th calculation cause physical quantity x ″ is an approximate value of the 0th estimation cause physical quantity x ′.
ステップＳ５では、第０推定原因物理量ｘ’と第０計算原因物理量ｘ”の間の誤差の大きさｅ＝｜ｘ’−ｘ”｜が誤差閾値Ｅ以下か否か判定する。
ここで、第０推定原因物理量ｘ’と第０計算原因物理量ｘ”の間の誤差ｘ’−ｘ”が原因物理量ｘと第０推定原因物理量ｘ’の間の誤差ｘ−ｘ’に略等しいと仮定する（この仮定が成り立つような物理現象に対して本発明は有用である）。つまり、
ｘ−ｘ’≒ｘ’−ｘ”
が成り立つとする。すると、｜ｘ’−ｘ”｜≦Ｅなら第０推定原因物理量ｘ’を必要な精度の原因物理量ｘとみなしてよいことになり、ステップＳ６へ進む。そうでないなら精度のより高い第１推定原因物理量ｘ1を求めるため、ステップＳ７へ進む。
In step S5, it is determined whether or not an error magnitude e = | x′−x ″ | between the 0th estimated cause physical quantity x ′ and the 0th calculated cause physical quantity x ″ is equal to or smaller than an error threshold E.
Here, the error x′−x ″ between the 0th estimated cause physical quantity x ′ and the 0th calculated cause physical quantity x ″ is substantially equal to the error xx ′ between the cause physical quantity x and the 0th estimated cause physical quantity x ′. (The present invention is useful for physical phenomena where this assumption holds). That means
xx′≈x′−x ″
Suppose that Then, if | x′−x ″ | ≦ E, the 0th estimated cause physical quantity x ′ may be regarded as the cause physical quantity x having the required accuracy, and the process proceeds to step S6. to determine the cause physical quantity x 1, the process proceeds to step S7.
ステップＳ６では、第０推定原因物理量ｘ’を出力する。そして、処理を終了する。 In step S6, the 0th estimated cause physical quantity x 'is output. Then, the process ends.
図４の（ｂ）に計算例を示す。
例えば、ｙ＝ｘ2＋ｘという順方向関数ｙ＝ｆ(ｘ)を想定する。そして、０＜ｘ≦０．１とし、ｘ’＝ｙを近似逆方向関数ｘ’＝ｇ’(ｙ)とする。すると、原因物理量ｘ＝０．１に対して結果物理量ｙ＝０．１１が計測される。
この結果物理量ｙ＝０．１１に対して第０推定原因物理量ｘ’は、
ｘ’＝ｙ＝０．１１
となる。この第０推定原因物理量ｘ’＝０．１１に対して第０計算結果物理量ｙ’は、
ｙ’＝ｘ’2＋ｘ’＝０．１２２１
となる。この第０計算結果物理量ｙ’＝０．１２２１に対して第０計算原因物理量ｘ”は、
ｘ”＝ｙ’＝０．１２２１
となる。誤差ｅは、
ｅ＝｜ｘ’−ｘ”｜＝０．０１２１
となる。そこで、例えばＥ＝０．０２ならばステップＳ６へ進み、推定原因物理量ｘ’＝０．１１を出力する。また、Ｅ＝０．０１ならばステップＳ７へ進む。
An example of calculation is shown in FIG.
For example, assume a forward function y = f (x) of y = x 2 + x. Then, 0 <x ≦ 0.1, and x ′ = y is an approximate backward function x ′ = g ′ (y). Then, the result physical quantity y = 0.11 is measured with respect to the cause physical quantity x = 0.1.
As a result, for the physical quantity y = 0.11 the 0th estimated cause physical quantity x ′ is
x ′ = y = 0.11
It becomes. For this 0th estimated cause physical quantity x ′ = 0.11, the 0th calculation result physical quantity y ′ is
y ′ = x ′ 2 + x ′ = 0.221
It becomes. For this 0th calculation result physical quantity y ′ = 0.1221, the 0th calculation cause physical quantity x ″ is
x ″ = y ′ = 0.1221
It becomes. The error e is
e = | x′−x ″ | = 0.0121
It becomes. For example, if E = 0.02, the process proceeds to step S6, and the estimated cause physical quantity x ′ = 0.11 is output. If E = 0.01, the process proceeds to step S7.
ステップＳ７では、第１推定原因物理量ｘ1を求める。
ｘ1＝２・ｘ’−ｘ”
これは、上記の式、
ｘ−ｘ’≒ｘ’−ｘ”
を変形し、
ｘ≒２・ｘ’−ｘ”＝ｘ1
としたものである。
すなわち、図４の（ａ）に示すように、第０推定原因物理量ｘ’と第０計算原因物理量ｘ”とから第１推定原因物理量ｘ1を求める。
In step S7, determining a first probable cause physical quantity x 1.
x 1 = 2 · x′−x ″
This is the above formula,
xx′≈x′−x ″
Transform
x≈2 · x′−x ″ = x 1
It is what.
That is, as shown in FIG. 4A, the first estimated cause physical quantity x 1 is obtained from the zeroth estimated cause physical quantity x ′ and the zeroth calculated cause physical quantity x ″.
図４の（ｂ）に計算例を示す。
例えば、ｘ’＝０．１１とｘ”＝０．１２２１から、ｘ1＝０．０９７９が求まる。
An example of calculation is shown in FIG.
For example, x 1 = 0.0979 is obtained from x ′ = 0.11 and x ″ = 0.221.
ステップＳ８では、第１推定原因物理量ｘ1を基に順方向関数ｙ＝ｆ(ｘ)を用いて第１計算結果物理量ｙ1を求める。
ｙ1＝ｆ(ｘ1)
図５の（ａ）に示すように、第１推定原因物理量ｘ1と第１計算結果物理量ｙ1とは順方向関数ｙ＝ｆ(ｘ)の関係にある。
In step S8, the first calculation result physical quantity y 1 is obtained using the forward function y = f (x) based on the first estimated cause physical quantity x 1 .
y 1 = f (x 1 )
As shown in FIG. 5A, the first estimated cause physical quantity x 1 and the first calculation result physical quantity y 1 are in a relationship of a forward function y = f (x).
ステップＳ９では、第１計算結果物理量ｙ1を基に近似逆方向関数ｘ’＝ｇ’(ｙ)を用いて第１計算原因物理量ｘ1’を求める。
ｘ1’＝ｇ’(ｙ1)
図５の（ａ）に示すように、第１計算原因物理量ｘ1’は第０推定原因物理量ｘ’の近似値になっている。
In step S9, the first calculation cause physical quantity x 1 ′ is obtained using the approximate backward function x ′ = g ′ (y) based on the first calculation result physical quantity y 1 .
x 1 '= g' (y 1 )
As shown in FIG. 5A, the first calculation cause physical quantity x 1 ′ is an approximate value of the zeroth estimation cause physical quantity x ′.
ステップＳ１０では、第０推定原因物理量ｘ’と第１計算原因物理量ｘ1’の間の誤差の大きさｅ＝｜ｘ’−ｘ1’｜が誤差閾値Ｅ以下か否か判定する。
ここで、第０推定原因物理量ｘ’と第１計算原因物理量ｘ1’の間の誤差ｘ’−ｘ1’が原因物理量ｘと第１推定原因物理量ｘ1の間の誤差ｘ−ｘ1に略等しいと仮定する（この仮定が成り立つような物理現象に対して本発明は有用である）。つまり、
ｘ−ｘ1≒ｘ’−ｘ1’
が成り立つとする。すると、｜ｘ’−ｘ1’｜≦Ｅなら第１推定原因物理量ｘ1を必要な精度の原因物理量ｘとみなしてよいことになり、ステップＳ１１へ進む。そうでないなら精度のより高い第２推定原因物理量ｘ2を求めるため、ステップＳ１２へ進む。
In step S10, it is determined whether or not an error magnitude e = | x′−x 1 ′ | between the 0th estimated cause physical quantity x ′ and the first calculated cause physical quantity x 1 ′ is equal to or smaller than an error threshold E.
Here, the error x-x 1 between errors x'-x 1 'is caused physical quantity x and the first probable cause physical quantity x 1 between the zeroth probable cause physical quantity x' and the first computation cause physical quantity x 1 ' It is assumed that they are approximately equal (the present invention is useful for physical phenomena for which this assumption holds). That means
x−x 1 ≒ x′−x 1 ′
Suppose that Then, if | x′−x 1 ′ | ≦ E, the first estimated cause physical quantity x 1 may be regarded as the cause physical quantity x with necessary accuracy, and the process proceeds to step S11. To determine the higher second probable cause physical quantity x 2 accuracy Otherwise, the process proceeds to step S12.
ステップＳ１１では、第１推定原因物理量ｘ1を出力する。そして、処理を終了する。 In step S11, it outputs a first probable cause physical quantity x 1. Then, the process ends.
図５の（ｂ）に計算例を示す。
例えば、第１推定原因物理量ｘ1＝０．０９７９に対して第１計算結果物理量ｙ1は、
ｙ1＝＝ｘ1 2＋ｘ1＝０．１０７４８４４
となる。この第１計算結果物理量ｙ1＝０．１０７４８４４に対して第１計算原因物理量ｘ1’は、
ｘ1’＝ｙ1＝０．１０７４８４４
となる。誤差ｅは、
ｅ＝｜ｘ’−ｘ1’｜＝０．００２５１５６
となる。そこで、例えばＥ＝０．００３ならばステップＳ１１へ進み、推定原因物理量ｘ1＝０．０９７９を出力する。また、Ｅ＝０．００２ならばステップＳ１２へ進む。
An example of calculation is shown in FIG.
For example, for the first estimated cause physical quantity x 1 = 0.0979, the first calculation result physical quantity y 1 is
y 1 == x 1 2 + x 1 = 0.1074844
It becomes. For this first calculation result physical quantity y 1 = 0.1074844, the first calculation cause physical quantity x 1 ′ is
x 1 '= y 1 = 0.1074844
It becomes. The error e is
e = | x′−x 1 ′ | = 0.0025156
It becomes. For example, if E = 0.003, the process proceeds to step S11, and the estimated cause physical quantity x 1 = 0.0979 is output. If E = 0.002, the process proceeds to step S12.
ステップＳ１２では、演算回数カウンタｉ＝２に初期化する。 In step S12, the operation counter is initialized to i = 2.
ステップＳ１３では、第ｉ推定原因物理量ｘiを求める。
ｘi＝ｘ’＋ｘi-1−ｘi-1’
この式は、図６に示すように、第（ｉ−１）推定原因物理量ｘi-1を基に順方向関数ｙ＝ｆ(ｘ)を用いて第（ｉ−１）計算結果物理量ｙi-1を求め、第（ｉ−１）計算結果物理量ｙi-1を基に近似逆方向関数ｘ’＝ｇ’(ｙ)を用いて第（ｉ−１）計算原因物理量ｘi-1’を求め、第０推定原因物理量ｘ’と第（ｉ−１）計算原因物理量ｘi-1’の間の誤差ｘ’−ｘi-1’が原因物理量ｘと第（ｉ−１）推定原因物理量ｘi-1の間の誤差ｘ−ｘi-1に略等しいと仮定する（この仮定が成り立つような物理現象に対して本発明は有用である）。つまり、
ｘ−ｘi-1≒ｘ’−ｘi-1’
が成り立つとする。この式を変形し、
ｘ≒ｘ’＋ｘi-1−ｘi-1’＝ｘi
としたものである。
すなわち、図７に示すように、第０推定原因物理量ｘ’と第（ｉ−１）推定原因物理量ｘi-1と第（ｉ−１）計算原因物理量ｘi-1’とから第ｉ推定原因物理量ｘiを求める。
In step S13, an i-th estimated cause physical quantity x i is obtained.
x i = x '+ x i-1 -x i-1 '
As shown in FIG. 6, this equation uses the forward function y = f (x) based on the (i−1) th estimated cause physical quantity x i−1 and uses the (i−1) th calculated result physical quantity y i. −1 , and based on the (i−1) -th calculation result physical quantity y i−1 , the approximate (i−1) -th calculation cause physical quantity x i−1 ′ is calculated using the approximate backward function x ′ = g ′ (y). look, 'and the (i-1) calculated cause physical quantity x i-1' zeroth probable cause physical quantity x error x'-x i-1 'causes the physical quantity x and the (i-1) between the probable cause It is assumed that the error between the physical quantities x i-1 is approximately equal to the error x−x i−1 (the present invention is useful for a physical phenomenon for which this assumption holds). That means
x−x i−1 ≒ x′−x i−1 ′
Suppose that Transform this equation,
x≈x ′ + x i−1 −x i−1 ′ = x i
It is what.
That is, as shown in FIG. 7, the i-th estimated cause physical quantity x ′, the (i−1) th estimated cause physical quantity x i−1, and the (i−1) th calculated cause physical quantity x i−1 ′. The cause physical quantity x i is obtained.
図８の（ａ）にｉ＝２の例を示す。また、図８の（ｂ）に計算例を示す。
例えば、第０推定原因物理量ｘ’＝０．１１、第１推定原因物理量ｘ1＝０．０９７９、第１計算原因物理量ｘ1’＝０．１０７４８４４から、第２推定原因物理量ｘ2＝０．１００４１５６が求まる。
FIG. 8A shows an example of i = 2. An example of calculation is shown in FIG.
For example, from the 0th estimated cause physical quantity x ′ = 0.11, the first estimated cause physical quantity x 1 = 0.0979, and the first calculated cause physical quantity x 1 ′ = 0.10074844, the second estimated cause physical quantity x 2 = 0. 1004156 is obtained.
ステップＳ１４では、図９の（ａ）に示すように、第ｉ推定原因物理量ｘiを基に順方向関数ｙ＝ｆ(ｘ)を用いて第ｉ計算結果物理量ｙiを求める。
ｙi＝ｆ(ｘi)
In step S14, as shown in FIG. 9 (a), obtaining the i-th calculation result physical quantity y i with i-th presumed cause physical quantity x i based on the forward function y = f (x).
y i = f (x i )
ステップＳ１５では、図９の（ａ）に示すように、第ｉ計算結果物理量ｙiを基に近似逆方向関数ｘ’＝ｇ’(ｙ)を用いて第ｉ計算原因物理量ｘi’を求める。
ｘi’＝ｇ’(ｙi)
In step S15, as shown in FIG. 9A, the i-th calculation cause physical quantity x i ′ is obtained using the approximate backward function x ′ = g ′ (y) based on the i-th calculation result physical quantity y i. .
x i '= g' (y i )
ステップＳ１６では、第０推定原因物理量ｘ’と第ｉ計算原因物理量ｘi’の間の誤差の大きさｅ＝｜ｘ’−ｘi’｜が誤差閾値Ｅ以下か否か判定する。図９の（ａ）に示すように、誤差ｘ’−ｘi’が原因物理量ｘと第ｉ推定原因物理量ｘiの間の誤差ｘ−ｘiに略等しいと仮定すれば、｜ｘ’−ｘi’｜≦Ｅなら第ｉ推定原因物理量ｘiを必要な精度の原因物理量ｘとみなしてよいことになり、ステップＳ１７へ進む。そうでないなら精度のより高い第２推定原因物理量ｘi+1を求めるため、ステップＳ１８へ進む。 In step S16, it is determined whether or not an error magnitude e = | x′−x i ′ | between the 0th estimated cause physical quantity x ′ and the i th calculated cause physical quantity x i ′ is equal to or smaller than an error threshold E. As shown in FIG. 9 (a), assuming error x'-x i 'is substantially equal to the error x-x i between cause physical quantity x and the i probable cause physical quantity x i, | x' If x i ′ | ≦ E, the i-th estimated cause physical quantity x i may be regarded as the cause physical quantity x with necessary accuracy, and the process proceeds to step S17. Otherwise, the process proceeds to step S18 in order to obtain the second estimated cause physical quantity x i + 1 with higher accuracy.
ステップＳ１７では、第ｉ推定原因物理量ｘiを出力する。そして、処理を終了する。 In step S17, the i-th estimated cause physical quantity x i is output. Then, the process ends.
図９の（ｂ）に計算例を示す。
例えば、第０推定原因物理量ｘ’＝０．１１、第２推定原因物理量ｘ2＝０．１００４１５６、第２計算原因物理量ｘ2’＝０．１１０４９８８９２７であり、誤差ｅ＝０．０００４９８８９２となる。そこで、例えばＥ＝０．０００５ならばステップＳ１７へ進み、推定原因物理量ｘ2＝０．１００４１５６を出力する。また、Ｅ＝０．０００４ならばステップＳ１８へ進む。
An example of calculation is shown in FIG.
For example, the 0th estimated cause physical quantity x ′ = 0.11, the second estimated cause physical quantity x 2 = 0.1004156, the second calculated cause physical quantity x 2 ′ = 0.11049889927, and the error e = 0.000498882. For example, if E = 0.0005, the process proceeds to step S17, and the estimated cause physical quantity x 2 = 0.1004156 is output. If E = 0.004, the process proceeds to step S18.
ステップＳ１８では、演算回数カウンタｉの値が上限演算回数Ｉに達していないならステップＳ１９へ進み、達しているならステップＳ２０へ進む。 In step S18, if the value of the calculation number counter i has not reached the upper limit calculation number I, the process proceeds to step S19, and if it has reached, the process proceeds to step S20.
ステップＳ１９では、演算回数カウンタｉの値を１だけインクリメントし、ステップＳ１３に戻る。
図１０はｉ＝３の計算例を示すもので、第０推定原因物理量ｘ’＝０．１１と第２推定原因物理量ｘ2＝０．１００４１５６と、第２計算原因物理量ｘ2’＝０．１１０４９８８９２７とから第３推定原因物理量ｘ3＝０．０９９９１６７０８が求まる。
In step S19, the value of the calculation number counter i is incremented by 1, and the process returns to step S13.
FIG. 10 shows a calculation example of i = 3, where the 0th estimated cause physical quantity x ′ = 0.11, the second estimated cause physical quantity x 2 = 0.1004156, and the second calculated cause physical quantity x 2 ′ = 0. From 11049889827, the third estimated cause physical quantity x 3 = 0.099916708 is obtained.
ステップＳ２０では、第Ｉ推定原因物理量ｘIを求めても必要な精度にならなかった旨の推定失敗メッセージを出力する。そして、処理を終了する。 At step S20, and outputs a first I probable cause physical quantity estimation failure message that did not even required precision seeking x I. Then, the process ends.
実施例１に係る原因物理量推定装置１０によれば、煩雑な演算を要さずに、結果物理量ｙから原因物理量ｘを推定することが出来る。
According to the causal physical
図１１に示すように、三次元空間に信号源三軸コイルＣ１，Ｃ２，Ｃ３と測定点三軸コイルｃ１，ｃ２，ｃ３を置き、信号源三軸コイルＣ１，Ｃ２，Ｃ３に電流Ｉを流し、測定点三軸コイルｃ１，ｃ２，ｃ３で磁界ｈを測定する。そして、信号源三軸コイルＣ１，Ｃ２，Ｃ３の中心を空間座標原点Ｏ(0,0,0)とし、測定点三軸コイルｃ１，ｃ２，ｃ３が第一象限にあるとしたときの測定点三軸コイルｃ１，ｃ２，ｃ３の位置Ｐ(x,y,z)を求める。このとき、位置Ｐ(x,y,z)が原因物理量であり、磁界ｈが結果物理量である。但し、磁界ｈの数値としては、データ処理を容易にするため、一般に「Ａ/ｍ」で表されている数値と比例関係にある別の数値を用いることとする。 As shown in FIG. 11, signal source triaxial coils C1, C2, and C3 and measurement point triaxial coils c1, c2, and c3 are placed in a three-dimensional space, and current I is passed through the signal source triaxial coils C1, C2, and C3. The magnetic field h is measured by the measurement point triaxial coils c1, c2, and c3. Then, the center of the signal source triaxial coils C1, C2, C3 is the spatial coordinate origin O (0, 0, 0), and the measurement point when the measurement point triaxial coils c1, c2, c3 are in the first quadrant. The position P (x, y, z) of the triaxial coils c1, c2, c3 is obtained. At this time, the position P (x, y, z) is the cause physical quantity, and the magnetic field h is the resulting physical quantity. However, as the numerical value of the magnetic field h, in order to facilitate data processing, another numerical value that is generally proportional to the numerical value represented by “A / m” is used.
信号源コイルＣ１にのみ電流Ｉを流したときに測定点三軸コイルｃ１，ｃ２，ｃ３で測定される磁界をｈ11，ｈ21，ｈ31とし、信号源コイルＣ２にのみ電流Ｉを流したときに測定点三軸コイルｃ１，ｃ２，ｃ３で測定される磁界をｈ12，ｈ22，ｈ32とし、信号源コイルＣ３にのみ電流Ｉを流したときに測定点三軸コイルｃ１，ｃ２，ｃ３で測定される磁界をｈ13，ｈ23，ｈ33とするとき、順方向関数ｈ11＝ｆ11(x,y,z)，ｈ21＝ｆ21(x,y,z)，ｈ31＝ｆ31(x,y,z)，ｈ12＝ｆ12(x,y,z)，ｈ22＝ｆ22(x,y,z)，ｈ32＝ｆ32(x,y,z)，ｈ13＝ｆ13(x,y,z)，ｈ23＝ｆ23(x,y,z)，ｈ33＝ｆ33(x,y,z)は例えば次式を用いる。但し、ａは、信号源三軸コイルＣ１，Ｃ２，Ｃ３の半径である。
上式の根拠は、新楽和夫，田辺行人，権平健一郎編「物理学公式」共立出版、昭和４５年、初版、第９４頁にある。 The grounds for the above formula are in Kazuo Shinraku, Yukito Tanabe, Kenichiro Gonpeira, “Physics Official” Kyoritsu Shuppan, 1965, first edition, page 94.
また、近似逆方向関数は例えば次式を用いる。
上式の根拠は、桂井誠著「基礎電磁気学」オーム社、平成１２年、第１版、第１１１頁にある。 The basis of the above formula is Makoto Katsui, “Basic Electromagnetics” Ohmsha, 2000, 1st edition, page 111.
本発明の原因物理量推定方法および装置は、例えば空間座標の原点に置いた磁気発生源から発生する磁気強度を検出コイルで結果物理量として計測し、その結果物理量を基に検出コイルの空間位置を推定するのに利用できる。また、ＣＴやＭＲＩなど一般に逆問題と呼ばれている問題を解いている装置に特に有効利用できる。 The causal physical quantity estimation method and apparatus of the present invention measures, for example, the magnetic intensity generated from a magnetic source placed at the origin of a spatial coordinate as a result physical quantity by a detection coil, and estimates the spatial position of the detection coil based on the result. Can be used to do. Further, it can be effectively used particularly for an apparatus that solves a problem generally called an inverse problem such as CT and MRI.
１ コンピュータ
１０ 原因物理量推定装置
1
Claims (6)
ｘ’＝ｇ’(ｙ)
ｘ1＝２・ｘ’−ｇ’(ｆ(ｘ’))
により第１推定原因物理量ｘ1を求めることを特徴とする原因物理量推定方法。 A forward function y = f (x) that gives a result physical quantity y depending on the cause physical quantity x and an approximate backward function x ′ = g ′ (y) that estimates an approximate cause physical quantity x ′ based on the result physical quantity y are known. When there is a physical quantity y obtained by measurement, x ′ = g ′ (y)
x 1 = 2 · x′−g ′ (f (x ′))
A causal physical quantity estimation method characterized in that the first estimated causal physical quantity x 1 is obtained by:
ｘ’＝ｇ’(ｙ)
ｘ1＝２・ｘ’−ｇ’(ｆ(ｘ’))
ｘi＝ｘ’＋ｘi-1−ｇ’(ｆ(ｘi-1)）
により第ｉ推定原因物理量ｘiを求めることを特徴とする原因物理量推定方法。 A forward function y = f (x) that gives a result physical quantity y depending on the cause physical quantity x and an approximate backward function x ′ = g ′ (y) that estimates an approximate cause physical quantity x ′ based on the result physical quantity y are known. Yes, when i is an integer of 2 or more, the result physical quantity y obtained by measurement is x ′ = g ′ (y)
x 1 = 2 · x′−g ′ (f (x ′))
x i = x ′ + x i−1 −g ′ (f (x i−1 ))
A causal physical quantity estimation method characterized in that the i-th estimated causal physical quantity x i is obtained by:
｜ｘ’−ｇ’(ｆ(ｘ’))｜≦Ｅならｘ’だけを求め、
｜ｘ’−ｇ’(ｆ(ｘ’))｜＞Ｅかつ｜ｘ’−ｇ’(ｆ(ｘ1))｜≦Ｅならｘ1まで求め、
｜ｘ’−ｇ’(ｆ(ｘi-1))｜＞Ｅかつ｜ｘ’−ｇ’(ｆ(ｘi))｜≦Ｅならｘiまで求めることを特徴とする原因物理量推定方法。
In the causal physical quantity estimation method according to claim 2, when the preset error threshold is E,
If | x′−g ′ (f (x ′)) | ≦ E, only x ′ is obtained.
| X'-g '(f ( x')) |> E and | x'-g '(f ( x 1)) | determined to if ≦ E x 1,
A cause physical quantity estimation method characterized by obtaining up to x i if | x′−g ′ (f (x i−1 )) |> E and | x′−g ′ (f (x i )) | ≦ E.
ｘ’＝ｇ’(ｙ)
ｘ1＝２・ｘ’−ｇ’(ｆ(ｘ’))
により第１推定原因物理量ｘ1を演算し出力する第１推定原因物理量演算手段を具備したことを特徴とする原因物理量推定装置。 A forward function y = f (x) that gives a result physical quantity y depending on the cause physical quantity x and an approximate backward function x ′ = g ′ (y) that estimates an approximate cause physical quantity x ′ based on the result physical quantity y are known. When there is a physical quantity y obtained by measurement, x ′ = g ′ (y)
x 1 = 2 · x′−g ′ (f (x ′))
Cause physical quantity estimating apparatus characterized by comprising a first probable cause physical quantity calculation means for first calculates the probable cause physical quantity x 1 output by.
ｘ’＝ｇ’(ｙ)
ｘ1＝２・ｘ’−ｇ’(ｆ(ｘ’))
ｘi＝ｘ’＋ｘi-1−ｇ’(ｆ(ｘi-1)）
により第ｉ推定原因物理量ｘiを演算し出力する第ｉ推定原因物理量演算手段を具備したことを特徴とする原因物理量推定装置。 A forward function y = f (x) that gives a result physical quantity y depending on the cause physical quantity x and an approximate backward function x ′ = g ′ (y) that estimates an approximate cause physical quantity x ′ based on the result physical quantity y are known. Yes, when i is an integer of 2 or more, the result physical quantity y obtained by measurement is x ′ = g ′ (y)
x 1 = 2 · x′−g ′ (f (x ′))
x i = x ′ + x i−1 −g ′ (f (x i−1 ))
Cause physical quantity estimating apparatus characterized by comprising a first i probable cause physical quantity calculation means for calculating and outputting a second i probable cause physical quantity x i by.
ｘ’＝ｇ’(ｙ)
により第０推定原因物理量ｘ’を演算し出力する第０推定原因物理量演算手段と、
ｘ1＝２・ｘ’−ｇ’(ｆ(ｘ’))
により第１推定原因物理量ｘ1を演算し出力する第１推定原因物理量演算手段と、
ｘi＝ｘ’＋ｘi-1−ｇ’(ｆ(ｘi-1)）
により第ｉ推定原因物理量ｘiを演算し出力する第ｉ推定原因物理量演算手段と、
予め設定した誤差閾値をＥとするとき、
｜ｘ’−ｇ’(ｆ(ｘ’))｜≦Ｅなら前記第０推定原因物理量演算手段だけを作動させ、
｜ｘ’−ｇ’(ｆ(ｘ’))｜＞Ｅかつ｜ｘ’−ｇ’(ｆ(ｘ1))｜≦Ｅなら前記第１推定原因物理量演算手段までを作動させ、
｜ｘ’−ｇ’(ｆ(ｘi-1))｜＞Ｅかつ｜ｘ’−ｇ’(ｆ(ｘi))｜≦Ｅなら前記第ｉ推定原因物理量演算手段までを作動させる演算制御手段とを具備したことを特徴とする原因物理量推定装置。 A forward function y = f (x) that gives a result physical quantity y depending on the cause physical quantity x and an approximate backward function x ′ = g ′ (y) that estimates an approximate cause physical quantity x ′ based on the result physical quantity y are known. Yes, when i is an integer of 2 or more, the result physical quantity y obtained by measurement is x ′ = g ′ (y)
A zeroth estimated cause physical quantity calculating means for calculating and outputting the zeroth estimated cause physical quantity x ′,
x 1 = 2 · x′−g ′ (f (x ′))
A first estimated cause physical quantity calculating means for calculating and outputting a first estimated cause physical quantity x 1 by:
x i = x ′ + x i−1 −g ′ (f (x i−1 ))
And the i probable cause physical quantity calculation means for the i calculates the probable cause physical quantity x i output by,
When the preset error threshold is E,
If | x′−g ′ (f (x ′)) | ≦ E, only the 0th presumed cause physical quantity calculating means is operated,
If | x′−g ′ (f (x ′)) |> E and | x′−g ′ (f (x 1 )) | ≦ E, the first estimated cause physical quantity calculating means is operated.
If | x′−g ′ (f (x i−1 )) |> E and | x′−g ′ (f (x i )) | ≦ E, calculation control for operating up to the i th estimated cause physical quantity calculation means And a causal physical quantity estimation device.
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