CN108959808B - Optimized distributed state estimation method based on sensor network - Google Patents

Optimized distributed state estimation method based on sensor network Download PDF

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CN108959808B
CN108959808B CN201810813797.5A CN201810813797A CN108959808B CN 108959808 B CN108959808 B CN 108959808B CN 201810813797 A CN201810813797 A CN 201810813797A CN 108959808 B CN108959808 B CN 108959808B
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CN108959808A (en
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胡军
王志功
陈东彦
张红旭
于浍
李佳兴
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Harbin University of Science and Technology
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Abstract

An optimized distributed state estimation method based on a sensor network is used in the technical field of control systems and signal processing. The invention solves the problem that the existing state estimation method can not simultaneously process the state estimation of the sensor network with multiplicative noise and random nonlinear interference. The invention considers the influence of multiplicative noise and random nonlinear generation on the state estimation performance, obtains the distributed filtering method based on the Riccati chi-ti differential equation, achieves the purpose of resisting external disturbance, and compared with the state estimation method of the existing nonlinear time-varying system, the method can control the estimation error in a very small range, and can improve the estimation accuracy by more than 10 percent while being easy to solve. The invention can be applied to the technical field of control systems and signal processing.

Description

Optimized distributed state estimation method based on sensor network
Technical Field
The invention belongs to the technical field of control systems and signal processing, and particularly relates to an optimized distributed state estimation method based on a sensor network.
Background
Distributed filtering is an important research problem in control systems and signal processing, and is widely applied to signal processing tasks in the fields of aircraft formation, target tracking systems, environment and ecological monitoring, health monitoring, home automation, traffic control and the like.
For a sensor network with multiplicative noise and a phenomenon of random occurrence of nonlinear interference, the existing state estimation methods cannot simultaneously deal with the state estimation problem of a complex network with such a phenomenon, and therefore, the phenomena always affect the performance of the state estimation methods.
Disclosure of Invention
The invention aims to solve the problem that the existing state estimation method cannot simultaneously process the state estimation of a sensor network with multiplicative noise and random nonlinear interference phenomenon.
The technical scheme adopted by the invention for solving the technical problems is as follows:
an optimized distributed state estimation method based on a sensor network comprises the following specific steps:
establishing a dynamic model of a time-varying system with multiplicative noise and random nonlinear interference generation based on a sensor network;
step two, constructing a distributed filter equation of the dynamic model established in the step one, and performing state estimation on the dynamic model of the time-varying system by using the distributed filter equation;
step three, calculating an upper bound xi of a one-step prediction error covariance matrix of the dynamic model at the time kk+1|k；
Step four, demarcating xi on the one-step prediction error covariance matrix at the time k according to the dynamic model obtained in the step threek+1|kCalculating the gain matrix K of the ith sensor at the moment K +1 in the dynamic modelij,k+1I is 1,2, …, N is the number of sensors of the dynamic model, and j represents the sensor coupled to i;
step five, obtaining the gain matrix K of the ith sensor at the moment of K +1 in the step fourij,k+1Substituting the distributed filter equation in the step two to obtain the estimation of the ith sensor at the moment of k +1
Figure BDA0001739847150000011
Judging whether k +1 reaches the total time length M of the sensor network, if k +1 is less than M, executing a sixth step, and if k +1 is equal to M, finishing the state estimation of the time-varying system which is based on the sensor network and has multiplicative noise and random nonlinear interference;
step six, according to the gain matrix K of the ith sensor in the dynamic model calculated in the step four at the moment of K +1ij,k+1Calculating an upper bound xi of an estimation error covariance matrix of the dynamic model at the time k +1k+1|k+1；
And c, enabling k to be k +1, and executing the step two until k +1 is M.
The invention has the beneficial effects that: the optimized distributed state estimation method based on the sensor network simultaneously considers the influence of multiplicative noise and random nonlinear generation on the state estimation performance, obtains the distributed filtering method based on the Riccati difference equation, and achieves the purpose of resisting external disturbance.
Drawings
FIG. 1 is a flow chart of a method for optimized distributed state estimation based on a sensor network according to the present invention;
FIG. 2 is a state vector x for a time-varying systemkOf the first component
Figure BDA0001739847150000021
The actual state trajectory and its estimated comparison map;
FIG. 3 is a state vector x for a time-varying systemkThe second component of
Figure BDA0001739847150000022
The actual state trajectory and its estimated comparison map;
FIG. 4 is a diagram of 4 sensors in a state vector xkFirst component of
Figure BDA0001739847150000023
A lower estimation error contrast map;
FIG. 5 is a diagram of 4 sensors in state vector xkSecond component of
Figure BDA0001739847150000024
A lower estimation error contrast map;
FIG. 6 is a state vector x of a dynamic modelkThe solid line in the graph is the filtering mean square error MSE, and the broken line is the minimum upper bound.
Detailed Description
The technical solution of the present invention is further described below with reference to the accompanying drawings, but not limited thereto, and any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention shall be covered by the protection scope of the present invention.
The first embodiment is as follows: this embodiment will be described with reference to fig. 1. The method for estimating the optimized distributed state based on the sensor network in the embodiment comprises the following specific steps:
establishing a dynamic model of a time-varying system with multiplicative noise and random nonlinear interference generation based on a sensor network;
step two, constructing a distributed filter equation of the dynamic model established in the step one, and performing state estimation on the dynamic model of the time-varying system by using the distributed filter equation;
step three, calculating an upper bound xi of a one-step prediction error covariance matrix of the dynamic model at the time kk+1|k；
Step four, demarcating xi on the one-step prediction error covariance matrix at the time k according to the dynamic model obtained in the step threek+1|kCalculating the gain matrix K of the ith sensor at the moment K +1 in the dynamic modelij,k+1I is 1,2, …, N is the number of sensors of the dynamic model, and j represents the sensor coupled to i;
step five, increasing the ith sensor obtained in the step four at the moment of k +1Benefit matrix Kij,k+1Substituting the distributed filter equation in the step two to obtain the estimation of the ith sensor at the moment of k +1
Figure BDA0001739847150000031
Judging whether k +1 reaches the total time length M of the sensor network, if k +1 is less than M, executing a sixth step, and if k +1 is equal to M, finishing the state estimation of the time-varying system which is based on the sensor network and has multiplicative noise and random nonlinear interference;
step six, according to the gain matrix K of the ith sensor in the dynamic model calculated in the step four at the moment of K +1ij,k+1Calculating an upper bound xi of an estimation error covariance matrix of the dynamic model at the time k +1k+1|k+1；
And c, enabling k to be k +1, and executing the step two until k +1 is M.
The second embodiment is as follows: the embodiment further defines the method for estimating the optimized distributed state based on the sensor network according to the first embodiment, and the specific process of the first step is as follows:
establishing a dynamic model of a time-varying system with multiplicative noise and random occurrence nonlinearity based on a sensor network, wherein the state space form of the dynamic model is as follows:
Figure BDA0001739847150000032
Figure BDA0001739847150000033
wherein the content of the first and second substances,
Figure BDA0001739847150000034
is the state vector of the dynamic model at time k,
Figure BDA0001739847150000035
is the state vector of the dynamic model at time k +1,
Figure BDA0001739847150000036
is the real number domain of the state of the dynamic model, n is the dimension; y isi,kThe measurement output of the ith sensor of the dynamic model at the moment k, wherein i is 1,2, …, N, and N is the number of sensors of the dynamic model; a. thekIs a system matrix, αkIs the multiplicative noise of the dynamic model,
Figure BDA0001739847150000037
is a system disturbance matrix; beta is ai,kIs the multiplicative noise at time k for the ith sensor, Ci,kIs the measurement matrix of the ith sensor at time k,
Figure BDA0001739847150000038
the disturbance measurement matrix of the i sensors at the moment k is obtained; b iskIs a noise distribution matrix;
Figure BDA0001739847150000039
is the process noise of the dynamic model at time k,
Figure BDA00017398471500000310
is the real number domain of the process noise of the dynamic model, q is the dimension,
Figure BDA00017398471500000311
is the measurement noise of the ith sensor at time k;
αkgaussian distribution, beta, obeying zero mean, unit variancei,kGaussian distribution, alpha, obeying zero mean, unit variancekAnd betai,kIndependent of each other, omegakIs zero mean and variance Qk>White Gaussian noise of 0, QkIs the variance of the process noise, vi,kIs zero mean and variance Ri,k>0 white Gaussian noise, Ri,kIs to measure the variance of the noise.
f(xk) Is a non-linear function; xikIs a list of mutually independent random variables satisfying Bernoulli distribution, and a random variable xikProbability when 1, Prob { ξ }k1 and a random variable ξkProbability when 0 Prob { ξ }k0} each represents as follows:
Figure BDA0001739847150000041
wherein the content of the first and second substances,
Figure BDA0001739847150000044
is a known probability value, and
Figure BDA0001739847150000045
the nonlinear function f (-) is assumed to satisfy the following conditions of Lipschitz (Lipschitz):
‖f(a)-f(b)‖≤l‖a-b‖
where l is a known lipschitz constant, a and b are generalized arguments of the non-linear function, f (a) and f (b) represent the corresponding non-linear functions of the generalized arguments a and b, respectively, | · |, is a two-norm.
The third concrete implementation mode: the second embodiment further defines the method for estimating an optimized distributed state based on a sensor network described in the second embodiment, and the specific process of the second step in the second embodiment is as follows:
the distributed filter equation is constructed as follows:
Figure BDA0001739847150000042
Figure BDA0001739847150000043
wherein the content of the first and second substances,
Figure BDA0001739847150000046
is the state estimate of the ith sensor at time k,
Figure BDA0001739847150000047
is an estimate of the ith sensor at time k +1,
Figure BDA0001739847150000048
is a one-step prediction of the ith sensor at time k,
Figure BDA0001739847150000049
is independent variable of
Figure BDA00017398471500000410
A non-linear function of (d); kij,k+1Is the gain matrix at time k +1 for the ith sensor of the dynamic model,
Figure BDA00017398471500000411
a set representing all sensors coupled to the ith sensor; when j is at
Figure BDA00017398471500000413
Internal time, h ij1, otherwise hij＝0，hijRepresenting the connection relationship between the ith sensor and the jth sensor; y isj,k+1Is the measurement output of the j sensor at time k + 1;
Figure BDA00017398471500000412
is a one-step prediction of the j sensor at time k; cj,k+1Is the measurement matrix of the j sensor at time k + 1.
The fourth concrete implementation mode: the third embodiment further defines the method for estimating an optimized distributed state based on a sensor network described in the third embodiment, and the specific process of the third step in the present embodiment is as follows:
calculating an upper bound xi of a one-step prediction error covariance matrix of the dynamic model at the time k according to the following formulak+1|k：
Figure BDA0001739847150000051
Wherein ε isThe constant value of the current signal is known,
Figure BDA0001739847150000054
is formed by AkThe diagonal matrix is formed by the two groups of the diagonal matrix,
Figure BDA0001739847150000055
is that
Figure BDA0001739847150000056
Xi, xik|kIs the upper bound of the error covariance matrix of the dynamic model at time k;
Figure BDA0001739847150000057
is formed by
Figure BDA0001739847150000058
A composed diagonal matrix, XkIs xkThe covariance matrix at time k is,
Figure BDA0001739847150000059
is that
Figure BDA00017398471500000510
The transpose matrix of (a) is,
Figure BDA00017398471500000511
is formed by BkThe diagonal matrix is formed by the two groups of the diagonal matrix,
Figure BDA00017398471500000512
is the variance of the process noise of the augmented dynamic model at time k,
Figure BDA00017398471500000513
is that
Figure BDA00017398471500000514
The transposed matrix of (2); i is a unit matrix, and tr (×) is tracing on the corresponding matrix;
Figure BDA00017398471500000515
is an augmented systemThe matrix is a matrix of a plurality of matrices,
Figure BDA00017398471500000516
is the system disturbance matrix after the amplification,
Figure BDA00017398471500000517
is the noise distribution matrix after the amplification,
Figure BDA00017398471500000518
diagNis a diagonal matrix composed of N {. cndot.
The following notation is introduced:
Figure BDA00017398471500000519
Figure BDA00017398471500000520
fk＝colN{f(xk)}，
Figure BDA00017398471500000521
Kk＝{Kij,k}N×N，Hi＝diag{hi1I,…,hiNI}；Kkis Kij,kIn an expanded form, Kij,kIs a distributed filter gain matrix at the k moment after the ith sensor and the jth sensor are coupled;
the fifth concrete implementation mode: the present embodiment further defines the method for estimating an optimized distributed state based on a sensor network according to the fourth embodiment, and the specific process of step four in the present embodiment is as follows:
calculating a gain matrix K of the ith sensor in the dynamic model at the moment K +1 according to the following formulaij,k+1：
Figure BDA0001739847150000052
Wherein the content of the first and second substances,
Figure BDA00017398471500000522
representing a matrix dependent on a gain parameter
Figure BDA00017398471500000523
Extracting the corresponding sub-matrix from the data,
Figure BDA00017398471500000524
is expressed as
Figure BDA00017398471500000525
The matrix after removing the rows whose corresponding elements are all 0,
Figure BDA00017398471500000526
represents from
Figure BDA00017398471500000527
Removing corresponding matrixes after columns with all 0 elements and rows with all 0 elements; intermediate variables
Figure BDA00017398471500000528
And
Figure BDA00017398471500000529
the expression of (a) is as follows:
Figure BDA0001739847150000053
Figure BDA0001739847150000061
wherein xii,k+1|kIs the upper bound of the one-step prediction error covariance matrix for the ith sensor,
Figure BDA0001739847150000066
is composed of Ci,k+1A composed diagonal matrix, Ci,k+1Is the measurement matrix of the ith sensor at time k +1,
Figure BDA0001739847150000067
is that
Figure BDA0001739847150000068
The transposed matrix of (2); hiA diagonal matrix formed for the adjacency matrix;
Figure BDA0001739847150000069
is an intermediate variable, and
Figure BDA00017398471500000610
the expression of (a) is as follows:
Figure BDA0001739847150000062
Figure BDA0001739847150000063
wherein: xk+1Is xk+1The covariance matrix at time k +1,
Figure BDA00017398471500000611
is formed by
Figure BDA00017398471500000614
The diagonal matrix is formed by the two groups of the diagonal matrix,
Figure BDA00017398471500000620
is betai,k+1Perturbation matrix of, betai,k+1Is the multiplicative noise at time k +1 for the ith sensor;
Figure BDA00017398471500000612
is that
Figure BDA00017398471500000613
The transpose matrix of (a) is,
Figure BDA00017398471500000615
is the variance of the measurement noise of the augmented dynamic model at time k + 1.
Figure BDA00017398471500000616
R1,k+1Is the variance of the measurement noise of the 1 st sensor at time k + 1.
Finding xi in the third stepk+1|kThen, xii,k+1|kAnd is accordingly available.
The sixth specific implementation mode: in this embodiment, the method for estimating an optimized distributed state based on a sensor network described in the fifth embodiment is further limited, and in the sixth embodiment, the gain matrix K of each sensor in the dynamic model calculated according to the fourth step in the sixth step is calculatedij,k+1Calculating an upper bound xi of an estimation error covariance matrix of the dynamic model at the time k +1k+1|k+1The specific process comprises the following steps:
calculating the upper bound xi of the error covariance matrix of the dynamic model at the time k +1k+1|k+1The following formula is adopted:
Figure BDA0001739847150000064
wherein: gk+1Is the gain matrix at time k +1 of the augmented dynamic model,
Figure BDA00017398471500000617
is that
Figure BDA00017398471500000618
The transpose matrix of (a) is,
Figure BDA00017398471500000619
is Gk+1The transposed matrix of (2);
Figure BDA0001739847150000065
wherein: eiRepresenting a diagonal matrix made up of element 0 and element 1.
The seventh embodiment: the present embodiment further defines the optimized distributed state estimation method based on the sensor network described in the fourth, fifth, and sixth embodiments, and the theory described in the third, fourth, and fifth steps is:
and solving the minimum upper bound of the filtering error covariance matrix. Then ask xik+1|k+1So that P isk+1|k+1≤Ξk+1|k+1Wherein
Figure BDA0001739847150000077
Is the filter error covariance matrix at time k +1,
Figure BDA0001739847150000078
is the filtering error at the time instant k +1,
Figure BDA0001739847150000079
to the expectation of the element { · },
Figure BDA00017398471500000710
is composed of
Figure BDA00017398471500000711
The transposing of (1).
Because the filter error covariance matrix has uncertain items, the true value of the filter error covariance matrix cannot be obtained. Optimizing upper bound xi of filtering error covariance matrixk+1|k+1Can obtain the filter gain matrix K at the moment K +1ij,k+1。
Examples
The method of the invention is adopted for simulation:
system parameters:
Figure BDA0001739847150000071
Figure BDA0001739847150000072
C1,k＝[0.82 0.62]，C2,k＝[0.75 0.8]，
C3,k＝[0.74 0.75]，C4,k＝[0.75 0.7]，
Figure BDA0001739847150000073
Figure BDA0001739847150000074
in addition to this, the present invention is,
Figure BDA0001739847150000075
Figure BDA0001739847150000076
the state estimator effect:
FIG. 2 shows the state vector x of the dynamic modelkFirst component xkThe actual state trajectory and its estimated comparison map;
FIG. 3 shows the state vector x of the dynamic modelkThe second component xkThe actual state trajectory and its estimated comparison map;
FIG. 4 shows 4 sensors in the state vector xkFirst component x ofkA lower estimation error contrast map;
FIG. 5 shows 4 sensors in the state vector xkSecond component x ofkA lower estimation error contrast map;
FIG. 6 shows the state vector x of the dynamic modelkThe solid line in the graph is the filtering mean square error MSE, and the broken line is the minimum upper bound.

Claims (4)

1. An optimized distributed state estimation method based on a sensor network is characterized by comprising the following specific steps:
establishing a dynamic model of a time-varying system with multiplicative noise and random nonlinear interference generation based on a sensor network;
the specific process of the step one is as follows:
establishing a dynamic model of a time-varying system with multiplicative noise and random occurrence nonlinearity based on a sensor network, wherein the state space form of the dynamic model is as follows:
Figure FDA0003488187890000011
Figure FDA0003488187890000012
wherein x iskIs the state vector of the dynamic model at time k, xk+1Is the state vector of the dynamic model at time k +1, yi,kThe measurement output of the ith sensor of the dynamic model at the time k; a. thekIs a system matrix, αkIs the multiplicative noise of the dynamic model,
Figure FDA0003488187890000013
is a system disturbance matrix; beta is ai,kIs the multiplicative noise at time k for the ith sensor, Ci,kIs the measurement matrix of the ith sensor at time k,
Figure FDA0003488187890000014
is the disturbance measurement matrix of the ith sensor at the moment k; b iskIs a noise distribution matrix; omegakIs the process noise, v, of the dynamic model at time ki,kIs the measurement noise of the ith sensor at time k;
f(xk) Is a non-linear function; xikIs a list of mutually independent random variables satisfying Bernoulli distribution, and a random variable xikProbability when 1, Prob { ξ }k1 and a random variable ξkProbability when 0 Prob { ξ }k0} each represents as follows:
Figure FDA0003488187890000015
wherein the content of the first and second substances,
Figure FDA0003488187890000016
is a known probability value, and
Figure FDA0003488187890000017
the nonlinear function f (-) is assumed to satisfy the following condition of Liphoz:
||f(a)-f(b)||≤l||a-b||
wherein l is a known lipschitz constant, a and b are generalized independent variables of the nonlinear function, f (a) and f (b) respectively represent the nonlinear function corresponding to the generalized independent variables a and b, and | · | |, is a two-norm;
step two, constructing a distributed filter equation of the dynamic model established in the step one, and performing state estimation on the dynamic model of the time-varying system by using the distributed filter equation;
the specific process of the second step is as follows:
the distributed filter equation is constructed as follows:
Figure FDA0003488187890000021
Figure FDA0003488187890000022
wherein the content of the first and second substances,
Figure FDA0003488187890000023
is an estimate of the ith sensor at time k,
Figure FDA0003488187890000024
is the ith sensingThe estimate of the time at which the device is at time k +1,
Figure FDA0003488187890000025
is a one-step prediction of the ith sensor at time k,
Figure FDA0003488187890000026
is independent variable of
Figure FDA0003488187890000027
A non-linear function of (d); kij,k+1The gain matrix of the ith sensor of the dynamic model at the moment k + 1;
Figure FDA0003488187890000028
represents the set of all sensors coupled to i when j is at
Figure FDA0003488187890000029
Internal time, hij1, otherwise hij＝0，hijRepresenting the connection relationship between the ith sensor and the jth sensor; y isj,k+1Is the measurement output of the j sensor at time k + 1;
Figure FDA00034881878900000210
is a one-step prediction of the j sensor at time k; cj,k+1Is the measurement matrix of the j sensor at time k + 1;
step three, calculating an upper bound xi of a one-step prediction error covariance matrix of the dynamic model at the time kk+1|k；
Step four, demarcating xi on the one-step prediction error covariance matrix at the time k according to the dynamic model obtained in the step threek+1|kCalculating the gain matrix K of the ith sensor at the moment K +1 in the dynamic modelij,k+1I is 1,2, …, N is the number of sensors of the dynamic model, and j represents the sensor coupled to i;
step five, obtaining the gain matrix K of the ith sensor at the moment of K +1 in the step fourij,k+1Substituting the distributed filter in the second stepEquation to get the estimate of the ith sensor at time k +1
Figure FDA00034881878900000211
Judging whether k +1 reaches the total time length M of the sensor network, if k +1 is less than M, executing a sixth step, and if k +1 is equal to M, finishing the state estimation of the time-varying system which is based on the sensor network and has multiplicative noise and random nonlinear interference;
step six, according to the gain matrix K of the ith sensor in the dynamic model calculated in the step four at the moment of K +1ij,k+1Calculating an upper bound xi of an estimation error covariance matrix of the dynamic model at the time k +1k+1|k+1；
And c, enabling k to be k +1, and executing the step two until k +1 is M.
2. The optimized distributed state estimation method based on the sensor network according to claim 1, wherein the specific process of the third step is as follows:
calculating an upper bound xi of a one-step prediction error covariance matrix of the dynamic model at the time k according to the following formulak+1|k：
Figure FDA0003488187890000031
Where, ε is a known constant,
Figure FDA0003488187890000032
is formed by AkThe diagonal matrix is formed by the two groups of the diagonal matrix,
Figure FDA0003488187890000033
is that
Figure FDA0003488187890000034
Xi, xik|kIs the upper bound of the error covariance matrix of the dynamic model at time k;
Figure FDA0003488187890000035
is formed by
Figure FDA0003488187890000036
A composed diagonal matrix, XkIs xkThe covariance matrix at time k is,
Figure FDA0003488187890000037
is that
Figure FDA0003488187890000038
The transpose matrix of (a) is,
Figure FDA0003488187890000039
is formed by BkThe diagonal matrix is formed by the two groups of the diagonal matrix,
Figure FDA00034881878900000310
is the variance of the process noise of the augmented dynamic model at time k,
Figure FDA00034881878900000311
is that
Figure FDA00034881878900000312
The transposed matrix of (2); i is the identity matrix and tr (×) is the tracing of the corresponding matrix.
3. The optimized distributed state estimation method based on the sensor network according to claim 2, wherein the specific process of the step four is as follows:
calculating a gain matrix K of the ith sensor in the dynamic model at the moment K +1 according to the following formulaij,k+1：
Figure FDA00034881878900000313
Wherein the content of the first and second substances,
Figure FDA00034881878900000314
representing a matrix dependent on a gain parameter
Figure FDA00034881878900000315
Extracting the corresponding sub-matrix from the data,
Figure FDA00034881878900000316
is expressed as
Figure FDA00034881878900000317
The matrix after removing the rows whose corresponding elements are all 0,
Figure FDA00034881878900000318
represents from
Figure FDA00034881878900000319
Removing corresponding matrixes after columns with all 0 elements and rows with all 0 elements; intermediate variables
Figure FDA00034881878900000320
And
Figure FDA00034881878900000321
the expression of (a) is as follows:
Figure FDA00034881878900000322
Figure FDA00034881878900000323
wherein xii,k+1|kIs the upper bound of the one-step prediction error covariance matrix for the ith sensor,
Figure FDA00034881878900000324
is composed of Ci,k+1A composed diagonal matrix, Ci,k+1Is the measurement matrix of the ith sensor at time k +1,
Figure FDA00034881878900000325
is that
Figure FDA00034881878900000326
The transposed matrix of (2); hiA diagonal matrix formed for the adjacency matrix;
Figure FDA00034881878900000327
is an intermediate variable, and
Figure FDA00034881878900000328
the expression of (a) is as follows:
Figure FDA00034881878900000329
Figure FDA00034881878900000330
wherein: xk+1Is xk+1The covariance matrix at time k +1,
Figure FDA00034881878900000331
is formed by
Figure FDA00034881878900000332
The diagonal matrix is formed by the two groups of the diagonal matrix,
Figure FDA00034881878900000333
is betai,k+1Perturbation matrix of, betai,k+1Is the multiplicative noise at time k +1 for the ith sensor;
Figure FDA00034881878900000334
is that
Figure FDA00034881878900000335
The transpose matrix of (a) is,
Figure FDA00034881878900000336
is the variance of the measurement noise of the augmented dynamic model at time k + 1.
4. The method as claimed in claim 3, wherein the gain matrix K at time K of each sensor in the dynamic model calculated according to step four in step six is obtainedij,k+1Calculating an upper bound xi of an estimation error covariance matrix of the dynamic model at the time k +1k+1|k+1The specific process comprises the following steps:
calculating the upper bound xi of the error covariance matrix of the dynamic model at the time k +1k+1|k+1The following formula is adopted:
Figure FDA0003488187890000041
wherein: gk+1Is the gain matrix at time k +1 of the augmented dynamic model,
Figure FDA0003488187890000042
is that
Figure FDA0003488187890000043
The transpose matrix of (a) is,
Figure FDA0003488187890000044
is Gk+1The transposed matrix of (2);
Figure FDA0003488187890000045
wherein: eiRepresenting a diagonal matrix made up of element 0 and element 1.
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