Data processing: measuring – calibrating – or testing – Measurement system – Statistical measurement
Reexamination Certificate
2000-11-03
2002-12-31
Hoff, Marc S. (Department: 2857)
Data processing: measuring, calibrating, or testing
Measurement system
Statistical measurement
C702S147000, C702S179000, C702S181000, C702S182000, C702S183000
Reexamination Certificate
active
06502061
ABSTRACT:
TECHNICAL FIELD
This invention relates to an information processing method and device, and a program supply medium. Particularly, it relates to an information processing method and device for estimating or measuring a parameter using observation data, and a program supply medium for supplying a processing program based on the estimation or measurement method.
BACKGROUND ART
In a system for measuring or estimating a parameter of a target system using observation data as an input, because of uncertainty (noise component) of input data or insufficiency of observation data for the parameter to be estimated, the value of the parameter to be estimated is often inappropriate with respect to the actual parameter.
In order to solve this problem, the Kalman filter or a filter for recursive estimation derived therefrom is used, which is described in R. E. Kalman, “A New Approach to Linear Filtering and Prediction Problems”, Trans. ASME-J. Basic Eng., pp.35-45, March 1960.
As the filter for recursive estimation, the Kalman filter is frequently used for a linear system. For an application with a nonlinear system, the extension of the Kalman filter or EKF is the best known filter. This extension of Kahnan filter (EKF) is described in A. Gelb, “Applied Optical Estimation”, The Analytic Science Corp., 1974. Also, there are employed the unscented filter, described in S. J. Julier, et al., “A New Extension of the Kalman Filter to Nonlinear Systems”, Proc. Of AeroSense: The 11th International Symposium on Aerospace/Defense Sensing, Simulation and Control”, and the pseudo Kalman filter, described in T. Vileville and P. Sanders, “Using Pseudo Kalman Filters in the Presence of Constraints, Application to Sensing Behaviours”, Technical Report 1669, INRIA-Sophia, Valbonne, France, 1992.
While the recursive estimation filter such as the Kalman filter is adapted for estimating the current state variable (parameter) using the current observation value, there is also known a smoother for estimating the past state variable using the current observation value. Hereinafter, the recursive estimation filter such as the Kalman filter will be first described and then the smoother will be described.
The Kalman filter has an efficient minimum square method mounted thereon and is capable of estimating the past, current and future states. It is also capable of estimating a state variable which lacks information and is capable of updating not only the state variable to be estimated but also the error covariance value representing the accuracy of estimation. Therefore, it is known as a robust estimation method.
The Kalman filter is one of the recursive estimation filters. The recursive estimation filter constantly updates the estimation value using only the latest data from continuously inputted observation values.
FIG. 1
shows the schematic structure of the recursive estimation filter. This recursive estimation filter includes a prediction section
131
for predicting a motion and an update section
132
for updating predicted data, and carries out estimation by alternately repeating prediction at the prediction section
131
and update at the update section
132
.
FIG. 2
is a block diagram showing the schematic structure of a Kalman filter
120
as one recursive estimation filter. First, a set of state variables to be estimated is described as a state vector x. If there are n units of state variables, the state vector x is the n-th order vector. The state vector x of the k-th cycle (one cycle corresponds to the observation cycle) is described as the state vector X
k
. The predication process for predicting the next state is expressed by the following equation (1). In the following equation, the vector is described with bold characters. The linear time differentiation of the vector is described with a dot (.) attached to its upper part, and the quadratic time differentiation is described with two dots (..) attached to its upper part.
X
k+1
=A
k
x
k
+W
k
(1)
In this case, the matrix A
k
is a state transition matrix of the k-th cycle, which is an n×n matrix. w
k
represents the prediction process noise generated in the prediction process and is assumed to be the normally distributed Gaussian noise.
The observation process is expressed by the following equation (2).
y
k
=H
k
x
k
+v
k
(2)
The covariance matrices of the process noise generated in the processes of the equations (1) and (2) are described as Q
k
and P
k
, respectively.
In the equation (2), y
k
represents the observation vector, which represents the vector of a set of observation values of the k-th cycle, and H
k
represents the observation matrix of the k-th cycle. The observation vector y
k
need not be of the same order as the state vector. If it is assumed that the observation vector y
k
is of the m-th order, the observation matrix H
k
is an m×n matrix. v
k
represents the observation noise generated in the observation process, which is assumed to be the Gaussian noise.
The Kalnan filter
120
repeats prediction and update so as to carry out estimation. In this case, the state vector estimated before update is described by the following formula (3) and the estimated state vector after update is described by the following formula (4).
{circumflex over (x)}
−
k
(3)
{circumflex over (x)}
k
(4)
The hat mark ({circumflex over ( )}) attached to the upper part in the formulas (3) and (4) indicates that it is an estimation value. In the following text, however, description of x
−
k
({circumflex over ( )}) and x
k
({circumflex over ( )}) is used.
The above-described P
k
is assumed to represent the estimated error covariance after update. The estimated error covariance before update is described by the following formula (5).
P
−
k
(5)
On the foregoing assumption, the processing at the prediction section
131
will be described. As shown in
FIG. 2
, in the processing e
1
at the prediction section
131
, the following prediction of the state is carried out on the basis of the equation (6).
{circumflex over (x)}
−
k
=A
k−1
{circumflex over (x)}
k−1
(6)
In the processing e
2
at the prediction section
131
, the following estimated error covariance P
−
k
is carried out on the basis of the equation (7).
P
−
k
=A
k−1
P
k−1
A
k−1
T
+Q
k−1
(7)
In this equation, A
k−1
T
represents the transported matrix of the state transition matrix A
k−1
.
Next, the processing at the update section
132
will be described. In the processing c
1
at the update section
132
, the Kalman gain K
k
is calculated on the basis of the following equation (8).
K
k
=P
−
k
H
k
T
(
H
k
P
−
k
Hk
T
+R
k
)
−1
(8)
In this equation, H
k
T
represents the transported matrix of the observation matrix H
k
and ( )
−1
represents the inverse matrix of the matrix within ( ).
In the processing c
2
at the update section
132
, estimation of the state variable is updated on the basis of the following equation (9) using the observation value inputted to the update section
132
.
{circumflex over (x)}={circumflex over (x)}
−
k
+K
k
(
y
k
−H
k
{circumflex over (x)}
−
k
) (9)
In the processing c
3
at the update section
132
, the estimated error covariance P
k
is updated on the basis of the following equation (10).
P
k
=(
I−K
k
H
k
)
P
−
k
(10)
In this equation, I represents the unit matrix of the n×n dimension.
With the Kalman filter
120
as described above, it is assumed that both the prediction process and the observation process are linear. Actually, however, these processes are often nonlinear. In order to cope with this, a derived filter such as the above-described EKF (extension of the Kalman filter) is proposed. The most frequently used EKF will now be described.
In the EKF, the prediction process is expressed by the following equation (11) (c
Bell Boyd & Lloyd LLC
Hoff Marc S.
Sony Corporation
Suarez Felix
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