CN106372649B - Integrated value Kalman filtering method based on quantization - Google Patents

Integrated value Kalman filtering method based on quantization Download PDF

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CN106372649B
CN106372649B CN201610694999.3A CN201610694999A CN106372649B CN 106372649 B CN106372649 B CN 106372649B CN 201610694999 A CN201610694999 A CN 201610694999A CN 106372649 B CN106372649 B CN 106372649B
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CN106372649A (en
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许大星
王海伦
柴国飞
陈佳泉
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HANGZHOU CNDE TECHNOLOGY Co.,Ltd.
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Quzhou University
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Abstract

The invention discloses a quantization-based integrated value Kalman filtering method, and compared with the prior art, the method has the problem of remote state estimation of a dynamic random system in a wireless sensor with limited bandwidth. The raw measurements are quantized into messages that are transmitted from the local sensors to the remote estimator. The information contained by the quantization strategy is used to represent the region of the set of raw measurements and to give the tightest ellipsoid approximation of this region. Further, a quantization-based ensemble kalman filtering method is proposed. In computer simulation, three algorithms are compared and analyzed, and the effectiveness of the algorithm is illustrated.

Description

Integrated value Kalman filtering method based on quantization
Technical Field
The invention relates to the technical field of microelectronics, in particular to a quantization-based integrated Kalman filtering method.
Background
With the development of microelectronic technology, wireless communication technology and embedded technology, wireless sensor networks have been rapidly developed. The wireless sensor network is a distributed sensor network, and the distributed sensors are low in price, flexible in position movement, variable in network setting and high in fault tolerance, so that the wireless sensor network is widely applied to various fields such as national defense and military, intelligent home, biomedical, environment monitoring, space exploration, industrial commerce and the like.
Since the energy of the sensors of the wireless sensor network is limited and the energy of the sensors is provided by batteries, it is very costly and sometimes difficult to replace the batteries, and it is sometimes even impossible to replace the batteries due to the working environment of the sensors, i.e. the end of the life of the sensors is indicated by the exhaustion of the batteries. The related art document indicates that the power consumption for transmitting one bit of data is 800 times the power consumption for a processor to execute one instruction. Therefore, the method for quantizing the measurement information is an effective method, the measured data can be processed in the network, the data transmission quantity is reduced, and the energy can be effectively saved. The limitation of communication resources is also a main problem faced by the wireless sensor network, data obtained by the sensor end often cannot be directly transmitted to the estimation end through the wireless communication network, each sensor can only transmit a limited number of bits at a time, and thus the observed value must be quantized before transmission. Therefore, the quantification problem is always a research hotspot. Another document proposes an optimal distributed estimator of random parameters and provides a scheduling method for allocating bit numbers to sensor nodes. Another document also proposes an adaptive quantization strategy under multi-dimensional observation, so as to design a centralized multi-sensor fusion estimation method under the condition of limited bandwidth. Another document employs a uniformly quantized strategy to reduce traffic by reducing packet length.
At present, all estimation values finally given by quantization-based estimation are point values, however, due to the existence of quantization errors, actual observed values are uncertain, and estimation by using quantized single-point values is obviously unreasonable. Another document proposes for the first time a set-valued kalman filtering method for a convex set of initial state estimate distributions, but this method still only considers the measurement of a single point value. Another document presents a state estimation method by relaxing the uniqueness false of the posterior probability distribution when considering the measured values as ensemble values, and further generalizes the method to the multi-sensor fusion problem. The recent document assumes that the measurement set can be represented by ellipsoids, and the sum operation property between the ellipsoids is used to provide a calculation method of a state estimation mean set, namely an ensemble Kalman filtering method. The authors detail that the fusion method is independent of the sensor data fusion order and demonstrates the progressiveness of estimating the mean set size.
Disclosure of Invention
It is an object of the present invention to provide a quantization-based ensemble kalman filtering method to solve the above problems.
The invention realizes the purpose through the following technical scheme:
the invention comprises a quantization strategy and a value Kalman filtering method;
quantization strategy:
suppose that the number of quantization bits of quantizer i has been allocatedIs preferably 1iBits, i.e. messages mi(k) Has a 1iBit, according to hypothesis 2, results
Figure GDA0002490343870000021
And, the quantizer i is in the interval
Figure GDA0002490343870000022
Share the same thing
Figure GDA0002490343870000023
A quantization point
Figure GDA0002490343870000024
The quantization points are uniformly or non-uniformly distributed in the quantization interval, i.e. satisfy
Figure GDA0002490343870000025
Figure GDA0002490343870000026
Commonly used quantization methods are generally probability-based, assuming that components are measured
Figure GDA0002490343870000027
Then y isi(k) Is quantized to according to the following formula
Figure GDA0002490343870000031
Or
Figure GDA0002490343870000032
Figure GDA0002490343870000033
Figure GDA0002490343870000034
Ensemble kalman filtering:
since the original measured value is quantized, the remote estimation end does not know the original observed value y (k), but can find out the interval range where each measured component is located according to the quantization rule formula (5), and by using the quantization rule information of each quantizer, a set-value kalman filtering method is provided:
assume that the measurement set yd (k) is a set of ellipses and can be parameterized as:
Yd(k):＝(c(k),Y(k)) (8)
where c (k) is the center of the ellipse and Y (k) is a matrix representing the ellipse; the trace Tr (Y (k)) of the matrix Y (k) represents the size of the ellipse set Yd (k);
introduction 1: (c)1,X1)，(c2,X2) Representing two sets of ellipses, then
Figure GDA0002490343870000035
Wherein p > 0 and p ═ TrX1)1/2/(TrX2)1/2So that the matrix (1+ p)-1)X1+(1+p)X2) The trace of (2) is minimal;
assume that the sensor measures the raw measurement value y (k) ═ y at time k1(k),y2(k),...,ym(k)]TEach component y thereofi(k) After passing through the quantization rule of the quantizer i, a quantization message m is formedi(k) Then, the information is transmitted to a remote estimation end through a wireless communication channel, and a quantization rule can be known in advance at the estimation end, so that the range of an interval where an actual measurement value is located can be judged according to the received information and the quantization rule;
if the remote end receives the message
Figure GDA0002490343870000036
i＝1,2,...,m，j∈{1,2，...,TiThe quantization rule information of the quantizer can be used to determine the actual measurement component yi(k) Located in a section
Figure GDA0002490343870000037
Within the interval
Figure GDA0002490343870000038
As a central point, to
Figure GDA0002490343870000039
Is the interval radius; since the quantization points are known in advance by the quantization rule, the respective measurement component yi(k) At the central point c of the intervali(k) And a section radius ri(k) Real-time online calculation can be performed;
when the estimation end receives all the messages m quantized by the quantizeri(k) In the meantime, by using the information of the quantization rule, the range of the interval where each actually measured component is located can be calculated, now assume ci(k) And ri(k) 1, 2.. m is already calculated, and for the ensemble kalman filtering method, an important step is to give an ellipsoid description of the observation ensemble, and an ellipsoid description method of the measurement value y (k) is given below;
the above gives the respective measured component y at the estimation endi(k) The range of the interval where the actual measurement value y (k) is located is defined, and the collection area of the actual measurement value y (k) is not given
Ω:＝{y(k)∈Rm|||yi(k)-ci(k)||≤ri(k),i＝1,2,...,m}，
Is a m-dimensional cuboid, and each side has a length of 2ri(k) The region, i.e. the region of the set of actual observations, now aims to find a tightest external ellipsoid containing Ω; definition of
Figure GDA0002490343870000041
Is a minimal ellipsoid containing omega and satisfies
Figure GDA0002490343870000042
Wherein c (k) ═ c1(k),c2(k),...,cm(k)]，
Figure GDA0002490343870000043
diag represents taking diagonal matrix operation;
for m 2, Ω and
Figure GDA0002490343870000044
the relationship between the two is shown in fig. 3, in fact, the center of the ellipse is the center point of the rectangle, that is, the center value of each section, and the ellipse matrix is closely related to the radius of the section;
due to convexity, the m-dimensional cuboid has at least 2 vertexes in the ellipsoid
Figure GDA0002490343870000045
On the boundary of (2), other vertices or contained in
Figure GDA0002490343870000046
In or at
Figure GDA0002490343870000047
On the boundary of (1); so the value of (k) can be calculated as follows
Figure GDA0002490343870000048
Wherein the content of the first and second substances,
Figure GDA0002490343870000049
thus, at each sampling instant, the tightest ellipsoid set region in which the observation is contained is (c (k),2(k)Y(k))。
the invention has the beneficial effects that:
the invention relates to a quantization-based integrated Kalman filtering method, which solves the problem of remote state estimation of a dynamic random system in a wireless sensor with limited bandwidth compared with the prior art. The raw measurements are quantized into messages that are transmitted from the local sensors to the remote estimator. The information contained by the quantization strategy is used to represent the region of the set of raw measurements and to give the tightest ellipsoid approximation of this region. Further, a quantization-based ensemble kalman filtering method is proposed. In computer simulation, three methods were compared and analyzed, demonstrating the effectiveness of the methods presented herein.
Drawings
FIG. 1 is a diagram of the distribution structure of the quantizer of the present invention;
FIG. 2 is a diagram of quantization intervals of the present invention
Fig. 3 shows that, in the present invention, when m is 2,
Figure GDA0002490343870000055
a minimum ellipse graph containing omega;
FIG. 4 is a state estimation interval graph of the present invention;
fig. 5 is a state estimation error interval graph of the present invention.
Detailed Description
The invention will be further described with reference to the accompanying drawings in which:
problem description:
description of the system:
consider the following class of linear time invariant systems
x(k+1)＝Ax(k)+w(k) (1)
y(k)＝Cx(k)+v(k) (2)
Wherein k is a time index,
Figure GDA0002490343870000051
is a vector of the states of the system,
Figure GDA0002490343870000052
is the observation vector for state x (k),
Figure GDA0002490343870000053
and
Figure GDA0002490343870000054
respectively, system process noise and observation noise, state transferBoth matrix a and observation matrix C have the appropriate dimensions.
The following are some reasonable necessary assumptions:
assume that 1: w (k), v (k) are zero-mean white Gaussian noises independent of each other, have variances of Q (k) and R (k), respectively, and satisfy
Figure GDA0002490343870000061
The mean value of the initial state x (0) is x0Variance is P0And independently of w (k), v (k).
Suppose 2 that the bandwidth of the wireless communication channel is L bits under the bandwidth constraint and that the wireless communication channel from the sensor to the fusion center is ideal with no bit errors.
Assume that 3:
Figure GDA0002490343870000062
wherein y isi(k) Is the i-th element of the observation vector y (k),U iand
Figure GDA0002490343870000063
is the observed value yi(k) And is a known constant.
Problem establishment:
considering the estimation problem as shown in fig. 1, due to limited communication resources and complex network environment, we often face the problem of limited bandwidth of the wireless sensor network. On the process side, the sensor can only transmit limited bit data to the remote estimation side, so the measured values have to be quantized before transmission. Measuring component yi(k) Is quantized by a corresponding quantizer i into a quantized message mi(k) And then transmitted to the remote estimation end through a wireless communication network. Finally, the estimator gives an estimate of the state under the optimal criterion based on the received quantized message.
Due to the actual observed value yi(k) When passing through the quantizer, the quantizer quantizes the observed values in a certain interval into a fixed value m approximately according to a corresponding quantization rulei(k) Also, it isIs the received quantized message m (k) m ═ m1(k),m2(k),...,mm(k)]TNot the original observation y (k). If we know the quantization rule, we can determine the range of the set of actual observed values in combination with the quantization message m (k), and in practice, the quantization rule is often obtained in advance. If it is not reasonable for the remote estimator to give an estimate alone after receiving the message, because the observed value is uncertain and is in a range of sets rather than a single point value, in particular if the quantization error is large, a single estimate may give a larger estimation error. It is therefore a natural idea that when the observations are in one set, the estimator gives an estimate that is also in one set.
The aim of the method is to explore the quantization rule information of a quantizer and provide an ensemble-valued Kalman filtering method.
The invention comprises a quantization strategy and a value Kalman filtering method;
quantization strategy:
each measured component is quantized into a message by a corresponding quantizer, and under the condition that the total channel bandwidth bit number L is certain, different bit numbers are allocated to each quantizer to have different estimation performanceiBits, i.e. messages mi(k) Has a 1iBit, according to hypothesis 2, results
Figure GDA0002490343870000071
The quantization intervals of the individual quantizers are as shown in FIG. 2, with quantizer i in the interval
Figure GDA0002490343870000072
Share the same thing
Figure GDA0002490343870000073
A quantization point
Figure GDA0002490343870000074
The quantization points are uniformly or non-uniformly distributed in the quantization interval, i.e. satisfy
Figure GDA0002490343870000075
Figure GDA0002490343870000076
Commonly used quantization methods are generally probability-based, assuming that components are measured
Figure GDA0002490343870000077
Then y isi(k) Is quantized to according to the following formula
Figure GDA0002490343870000078
Or
Figure GDA0002490343870000079
Figure GDA00024903438700000710
Figure GDA00024903438700000711
Note 1: the quantization points shown in formula (4) can be arbitrarily distributed in the interval by some rules
Figure GDA00024903438700000712
And (4) the following steps. When in use
Figure GDA00024903438700000713
The quantization strategy degenerates to the usual uniform quantization. This document is based on the more general quantization strategy as shown in equations (4), (5), (6), (7).
Ensemble kalman filtering:
since the original measured value is quantized, the remote estimation end does not know the original observed value y (k), but can find out the interval range where each measured component is located according to the quantization rule formula (5), and by using the quantization rule information of each quantizer, a set-value kalman filtering method is provided:
assume that the measurement set Yd (k) is an ellipse set and can be parameterized as
Yd(k):＝(c(k),Y(k)) (8)
Where c (k) is the center of the ellipse and Y (k) is a matrix representing the ellipse. The trace Tr (Y (k)) of the matrix Y (k) represents the size of the ellipse set Yd (k);
introduction 1: (c)1,X1)，(c2,X2) Representing two sets of ellipses, then
Figure GDA0002490343870000081
Wherein p > 0 and p ═ TrX1)1/2/(TrX2)1/2So that the matrix (1+ p)-1)X1+(1+p)X2) The trace of (2) is minimal;
note 2: since the two sets of ellipses may no longer be the set of ellipses after the sum operation, the exact set of ellipses and the analytical expression of the operation cannot be maintained. The sum of the ellipses is calculated with the approximated outer ellipses using lemma 1. Set of ellipses (c) in formula (9)1+c2,(1+p-1)X1+(1+p)X2) Is composed of a set of ellipses (c)1,X1) And (c)2,X2) The tightest ellipse.
Assume that the sensor measures the raw measurement value y (k) ═ y at time k1(k),y2(k),...,ym(k)]TEach component y thereofi(k) After passing through the quantization rule of the quantizer i, a quantization message m is formedi(k) Then transmitted to remote estimation end through wireless communication channel, and the quantization rule can be known in advance at estimation end, so that it can be judged according to received message and quantization ruleCutting off the range of the actual measurement value;
if the remote end receives the message
Figure GDA0002490343870000082
i＝1,2,...,m，j∈{1,2，…,TiThe quantization rule information of the quantizer can be used to determine the actual measurement component yi(k) Located in a section
Figure GDA0002490343870000083
Within the interval
Figure GDA0002490343870000084
As a central point, to
Figure GDA0002490343870000085
Is the interval radius. Since the quantization points are known in advance by the quantization rule, the respective measurement component yi(k) At the central point c of the intervali(k) And a section radius ri(k) Real-time online computing is possible.
Note 3: when the quantization strategy of quantizer i is uniform quantization, i.e.
Figure GDA0002490343870000091
Then the center point of the interval
Figure GDA0002490343870000092
Radius of interval
Figure GDA0002490343870000093
Namely, the central point of the interval is the value of the received quantized message, the radius of the interval is a fixed constant, and repeated calculation is not needed.
When the estimation end receives all the messages m quantized by the quantizeri(k) In the meantime, by using the information of the quantization rule, the range of the interval where each actually measured component is located can be calculated, now assume ci(k) And ri(k) I is 1,2, …, m is calculated, and for the ensemble kalman filtering method, an important step is to give an ellipsoid description of the observation set, belowThe surface gives the ellipsoid description of the measured values y (k).
The above gives the respective measured component y at the estimation endi(k) The range of the interval where the actual measurement value y (k) is located is defined, and the collection area of the actual measurement value y (k) is not given
Ω:＝{y(k)∈Rm|||yi(k)-ci(k)||≤ri(k),i＝1,2,…,m}，
It is a m-dimensional cuboid, each side of which is 2r longi(k) This region, i.e. the collection of actual observations, now aims to find the tightest outer ellipsoid containing Ω. Definition of
Figure GDA0002490343870000094
Is a minimal ellipsoid containing omega and satisfies
Figure GDA0002490343870000095
Wherein c (k) ═ c1(k),c2(k),…,cm(k)]，
Figure GDA0002490343870000096
diag denotes a take diagonal matrix operation.
For m 2, Ω and
Figure GDA0002490343870000097
the relationship between the two is shown in fig. 3, in fact, the center of the ellipse is the center point of the rectangle, i.e. the center value of each section, and the ellipse matrix is closely related to the section radius.
Due to convexity, the m-dimensional cuboid has at least 2 vertexes in the ellipsoid
Figure GDA0002490343870000098
On the boundary of (2), other vertices or contained in
Figure GDA0002490343870000099
In or at
Figure GDA00024903438700000910
On the boundary of (c). So the value of (k) can be calculated as follows
Figure GDA0002490343870000101
Wherein the content of the first and second substances,
Figure GDA0002490343870000102
thus, at each sampling instant, the tightest ellipsoid set region in which the observation is contained is (c (k),2(k)Y(k))。
note 4: and acquiring an m-dimensional cuboid region omega of the measurement value set by exploring information in the quantization rule, further acquiring an external ellipsoid most closely comprising the region omega by using a formula (10), and finally acquiring an estimation set of the state by using a collection value Kalman filtering method. In fact, the set of state estimates is very useful, not only to give a range of variation of the actual state, but also to provide reference information to the design of the quantizer. For example, when there is quantization error influence, the user may know the upper and lower bounds of the actual state change, and may allocate the bit number of each measurement component by referring to the size of the state set according to the actual need.
The specific detailed ensemble kalman filtering method is shown in table 1:
TABLE 1
Figure GDA0002490343870000103
Figure GDA0002490343870000111
Simulation example:
in this section, a simulation example of target tracking is used to verify the effectiveness of the method presented herein. A pseudo measurement noise based kalman filtering method and a raw measurement based kalman filtering method are used for comparison with the method presented herein. In statistical sense, using mean estimationTwo norm of error
Figure GDA0002490343870000112
To indicate the accuracy of the estimation of the state. t represents the number of simulation steps, x (k) is the actual state,
Figure GDA0002490343870000113
is the estimated state.
For convenience of description, the following filtering methods are simply labeled:
the method comprises the following steps: integrated value Kalman filtering method based on quantization
The method 2 comprises the following steps: kalman filtering method based on pseudo measurement noise
The method 3 comprises the following steps: kalman filtering method using raw measurement values
Consider the following linear second order system:
Figure GDA0002490343870000114
Figure GDA0002490343870000115
covariance matrices of process noise and observation noise are respectively
Figure GDA0002490343870000116
R (k) 0.2. Initial value x0＝[0.01 0.01]TInitial covariance matrix
Figure GDA0002490343870000117
The bandwidth L of the wireless communication channel is 16, and for simplicity of simulation the example uses a uniform quantization strategy, the simulation results are shown in fig. 4, fig. 5 and table 2.
TABLE 2
Statistical results of estimation errors
Figure GDA0002490343870000121
Since the estimator cannot obtain the original measured value after the measured component values are quantized by the quantizer, the interval in which the original measured value is located can be calculated by using information included in the quantization strategy at each sampling instant. In this case, we cannot say which point in the estimated mean set has the smallest estimation error, but we can use the center value of the set value estimator as the point value estimation result. From the estimation curve of the state of fig. 4, the estimation interval value of method 1 covers the actual state at any time, the three methods can basically track the actual state 1 and state 2 of the target, and from the state estimation error interval graph of fig. 5 and table 2, the average estimation error of method 3 is the smallest, which is mainly because the measured value used in method 3 is the original measured value, and the original measured value is not possible under the condition of limited bandwidth, and is mainly used for comparison in the method. The estimation error of method 2 is significantly larger than that of method 1, mainly because method 2 assumes uncorrelated quantization noise white gaussian noise and uses the upper bound of pseudo noise covariance as measurement noise covariance, and this excessively demanding assumption and excessively conservative noise processing inevitably affect the accuracy of the method, whereas method 1 not only uses the received message value, but also uses the additional information contained in the quantization strategy, which is the advantage and motivation of using the ensemble kalman filtering method.
And (4) conclusion:
the invention discusses the quantization-based collective Kalman state estimation problem under the condition of a wireless sensor network with bandwidth constraint for the first time. After the number of quantizer bits and the quantization rule of each measurement component are given, the information of the quantization rule is explored by using the quantized information to give the hyper-rectangular body set region of the actual measurement value. And (3) using an approximate external ellipsoid to contain the hyper-cuboid aggregation region, and providing a solution method of the tightest ellipsoid, thereby providing an ensemble Kalman filtering method based on quantization. And finally, comparing the state estimation interval of the Kalman filtering method based on the covariance matrix, the state estimation interval of the Kalman filtering statistical state estimation interval based on the Monte Carlo and the state estimation interval of the ensemble Kalman filtering method by using a computer simulation example. The effectiveness of the proposed method is demonstrated.
The foregoing shows and describes the general principles and features of the present invention, together with the advantages thereof. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.

Claims (1)

1. A quantization-based ensemble Kalman filtering method is characterized in that: the method comprises a quantization strategy and a value Kalman filtering method;
quantization strategy:
let the quantization bit number of quantizer i be already allocated and be liBits, i.e. messages mi(k) Has a 1iBit, the bandwidth of the wireless communication channel set under the bandwidth constraint is L bits, and the wireless communication channel from the sensor to the fusion center is ideal, free of bit errors, resulting in
Figure FDA0002493373240000011
And, the quantizer i is in the interval
Figure FDA0002493373240000012
Share the same thing
Figure FDA0002493373240000013
A quantization point
Figure FDA0002493373240000014
The quantization points are uniformly or non-uniformly distributed in the quantization interval, i.e. satisfy
Figure FDA0002493373240000015
Figure FDA0002493373240000016
The quantization being based on probability, with measured components
Figure FDA0002493373240000017
Then y isi(k) Is quantized to according to the following formula
Figure FDA0002493373240000018
Or
Figure FDA0002493373240000019
Figure FDA00024933732400000110
Figure FDA00024933732400000111
Ensemble kalman filtering:
after the original measured value is quantized, the remote estimation end does not know the original measured value y (k), but can provide an integrated value Kalman filtering method by exploring and utilizing the quantization rule information of each quantizer according to the interval range where each measured component is known in the quantization rule formula (3):
let the measurement set yd (k) be an ellipse set and can be parameterized as:
Yd(k)：＝(c(k)，Y(k)) (6)
where c (k) is the center of the ellipse and Y (k) is a matrix representing the ellipse; the trace Tr (Y (k)) of the matrix Y (k) represents the size of the ellipse set Yd (k);
(c1,X1)，(c2,X2) Representing two sets of ellipses, then
Figure FDA0002493373240000021
Wherein p > 0 and p ═ TrX1)1/2/(Tr X2)12So that the matrix (1+ p)-1)X1+(1+p)X2The trace of (2) is minimal;
the sensor measures the raw measured value y (k) ═ y at time k1(k),y2(k),...,ym(k)]TEach component y thereofi(k) After passing through the quantization rule of the quantizer i, a quantization message m is formedi(k) Then, the information is transmitted to a remote estimation end through a wireless communication channel, and a quantization rule can be known in advance at the estimation end, so that the range of an interval where an actual measurement value is located can be judged according to the received information and the quantization rule;
if the remote end receives the message
Figure FDA0002493373240000022
i＝1,2,...,m，j∈{1,2，...,TiThe actual measurement component y can be judged by using the quantization rule information of the quantizeri(k) Located in a section
Figure FDA0002493373240000023
Within the interval
Figure FDA0002493373240000024
As a central point, to
Figure FDA0002493373240000025
Is the interval radius; since the quantization points are known in advance by the quantization rule, the respective measurement component yi(k) At the central point c of the intervali(k) And a section radius ri(k) Real-time online calculation can be realized;
when the estimation end receives all the messages m quantized by the quantizeri(k) In the meantime, by using the information of the quantization rule, the range of the interval where each actually measured component is located can be calculated, and c is now seti(k) And ri(k) I 1,2, m is calculated and for the setThe important step of the value Kalman filtering method is to give an ellipsoid description of an observation set, and an ellipsoid description method of a measured value y (k) is given below;
the above gives the respective measured component y at the estimation endi(k) The range of the interval where the actual measurement value y (k) is located is defined, and the collection area of the actual measurement value y (k) is not given
Ω:＝{y(k)∈Rm|||yi(k)-ci(k)||≤ri(k),i＝1,2,...,m}，
Is a m-dimensional cuboid, and each side has a length of 2ri(k) This region, i.e. the region of the set of actual measured values, now aims to find the tightest outer ellipsoid containing Ω; definition of
Figure FDA0002493373240000026
Is a minimal ellipsoid containing omega and satisfies
Figure FDA0002493373240000031
Wherein c (k) ═ c1(k),c2(k),...,cm(k)]，
Figure FDA0002493373240000032
diag represents taking diagonal matrix operation;
for the case of m being 2, actually, the center of the ellipse is the center point of the rectangle, that is, the center value of each section, and the ellipse matrix is closely related to the radius of the section;
due to convexity, the m-dimensional cuboid has at least 2 vertexes in the ellipsoid
Figure FDA0002493373240000033
On the boundary of (2), other vertices or contained in
Figure FDA0002493373240000034
In or at
Figure FDA0002493373240000035
On the boundary of (1); therefore, the value of (k) is calculated as follows
Figure FDA0002493373240000036
Wherein the content of the first and second substances,
Figure FDA0002493373240000037
thus, at each sampling instant, the measurement values are contained in the tightest ellipsoid set of regions (c (k),2(k)Y(k))。
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