CN115865390A - Nonlinear information physical system filtering method and system under FDI attack - Google Patents

Nonlinear information physical system filtering method and system under FDI attack Download PDF

Info

Publication number
CN115865390A
CN115865390A CN202210856707.7A CN202210856707A CN115865390A CN 115865390 A CN115865390 A CN 115865390A CN 202210856707 A CN202210856707 A CN 202210856707A CN 115865390 A CN115865390 A CN 115865390A
Authority
CN
China
Prior art keywords
filtering
representing
state
value
moment
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Pending
Application number
CN202210856707.7A
Other languages
Chinese (zh)
Inventor
缪克雷
陈友荣
延泽军
张旭东
金丹
吕晓雯
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Zhejiang Shuren University
Original Assignee
Zhejiang Shuren University
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Zhejiang Shuren University filed Critical Zhejiang Shuren University
Priority to CN202210856707.7A priority Critical patent/CN115865390A/en
Publication of CN115865390A publication Critical patent/CN115865390A/en
Pending legal-status Critical Current

Links

Images

Landscapes

  • Management, Administration, Business Operations System, And Electronic Commerce (AREA)

Abstract

The invention provides a filtering method of a nonlinear information physical system under FDI attack, which comprises the following steps: step 1, establishing a state space model of a nonlinear information physical system under FDI attack; step 2, setting an initial time k =1, and initializing model parameters; step 3, inputting a state filtering value and a filtering error covariance matrix at the moment k +1, and operating an unscented Gaussian and filtering method to realize the prediction of the state filtering value; step 4, calculating unknown parameters
Figure DDA0003753427250000011
Predicted value of (2)
Figure DDA0003753427250000012
Updating a filtering value and a filtering error covariance matrix at the moment of 5.K + 1; step 6, updating the unknown parameter estimated value at the moment k +1
Figure DDA0003753427250000013
Step 7, judging whether the moment k is smaller than the last moment N, if k is smaller than N, setting k = k +1, outputting a state estimation value at the moment of k +1, returning to the step 3, and if not, finishing the calculation; the method solves the problem that the state estimation of the nonlinear non-Gaussian information physical system under the FDI attack is difficult to accurately and efficiently realize by the current state estimation method aiming at the information physical system.

Description

Nonlinear information physical system filtering method and system under FDI attack
Technical Field
The invention relates to the technical field of information physical system security, in particular to a nonlinear information physical system filtering method and system under FDI attack.
Background
The Cyber Physical Systems (CPS) is used as a unified body of a computing process and a physical process, is a next-generation intelligent system integrating computing, communication and control, realizes interaction with the physical process through a human-computer interaction interface, controls a physical entity in a remote, reliable, real-time, safe and cooperative mode by using a networked space, and is widely applied to the fields of smart cities, smart power grids, industrial production and the like along with the development of network technology and computer technology, wherein the safe operation of the industrial cyber physical system is stable and is related to the national economic safety and the personal safety. The state estimation of the system by adopting a filtering method is one of hot research problems of an industrial information physical system. Kalman filtering is a common industrial information physical filtering method, and is an optimal filter under the assumption of a linear system and Gaussian distribution. However, it is difficult for a practical system to satisfy both the linear and gaussian assumptions. The extended Kalman filter can realize the state estimation of the nonlinear Gaussian system through the direct linear operation of the nonlinear system. However, extended kalman filtering is not suitable for highly nonlinear systems, and improved high-order extended kalman filters increase computational complexity. The kalman filter method assumes that the prior density of the states follows a gaussian distribution, and that the covariance matrices of the system process noise and the measurement noise are both known and follow a gaussian distribution. Therefore, the method based on the Kalman filtering is suitable for the industrial information physical system under the assumption of Gaussian distribution.
With the advance of informatization and industrialization deep fusion processes, industrial information physical systems mostly adopt a general protocol to be connected with a public network. Due to the existence of protocol bugs and the like, the system may be threatened by network attacks. False Data Injected (FDI) attacks are one of the common attacks in industrial cyber-physical systems, and attackers can implement attacks by injecting False Data into actuator signals of the system, disturb the normal operation of the system and are difficult to find. FDI attacks mainly affect the state estimation of the system. When an industrial cyber-physical system is subjected to FDI attack, the system may become non-Gaussian and no longer stable, which may cause a significant hidden danger to the operation of industrial production. At the moment, a divergent filtering result can be obtained by adopting the Kalman filtering method, so that the state estimation of the nonlinear non-Gaussian industrial information physical system under the FDI attack can not be well realized by directly using the Kalman filtering method.
Most industrial information physical systems suffering from FDI attacks are in a nonlinear non-Gaussian form, most methods aiming at the state estimation problem of the industrial information physical systems are suitable for linear Gaussian systems at present, and the problems of low precision, high calculation amount and the like exist.
Disclosure of Invention
Technical problem to be solved
Aiming at the defects of the prior art, the invention provides a nonlinear information physical system method and a nonlinear information physical system under FDI attack, which solve the problems that most methods for filtering aiming at the state estimation problem of the information physical system in the prior art are applicable to a linear Gaussian system, the precision is low, the calculated amount is high, and the like, and the methods are difficult to accurately and efficiently realize the state estimation of the nonlinear information physical system under FDI attack.
(II) technical scheme
In order to achieve the purpose, the invention is realized by the following technical scheme:
a nonlinear information physical system state estimation filtering method under FDI attack comprises the following steps:
step 1, establishing a state space model of a nonlinear information physical system under FDI attack;
step 2, setting an initial time k =1, and initializing model parameters;
step 3, inputting a state filtering value and a filtering error covariance matrix at the moment k +1, and operating an unscented Gaussian sum filtering method to realize the prediction of the state filtering value;
step 4, calculating unknown parameters
Figure SMS_1
Is greater or less than>
Figure SMS_2
Updating a filtering value and a filtering error covariance matrix at the moment of 5.K + 1;
step 6, updating the unknown parameter estimated value at the k +1 moment
Figure SMS_3
And 7, judging whether the moment k is smaller than the last moment N, if k is smaller than N, setting k = k +1, outputting the state estimation value at the moment k +1, returning to the step 3, and otherwise, finishing the calculation.
Preferably, the establishing of the state space model of the nonlinear information physical system under the FDI attack includes:
1.1 A discrete-time nonlinear information physical system model is expressed in the form of:
Figure SMS_4
wherein x is k ∈R m Representing the state vector of the system at time k, R tableRepresenting the real number field, m representing the dimension of the system state, z k ∈R n Representing the measurement vector of the system at time k, n representing the dimension of the measurement vector, u k ∈R m Represents the control input of the system at time k, f (x) k ) Represents a nonlinear state transfer function, h (x) k ) Representing a non-linear measurement function, n k Representing unknown non-Gaussian process noise, v k The measurement noise at the time k is represented by a covariance matrix L k Zero mean white gaussian noise, covariance matrix L k Is an n multiplied by n dimensional symmetric square matrix;
1.2 Considering that the system (1) is subject to the attack of the executor FDI, an executor FDI attack signal alpha is introduced ak The system model (1) is written in the form of the following model:
Figure SMS_5
wherein alpha is ak Representing an actuator FDI attack signal, and obeying non-Gaussian distribution;
1.3 Redefines the process noise of the system, writing equation (2) as the model:
Figure SMS_6
wherein
Figure SMS_7
Representing unknown sum non-gaussian process noise;
1.4 non-Gaussian noise to sum
Figure SMS_8
In a probability density function>
Figure SMS_9
Approximately expressed as the following gaussian mixture distribution:
Figure SMS_10
wherein N (| μ, Σ) represents a gaussian probability density function with mean μ and variance Σ; n is a radical of hydrogen n The number of gaussian mixtures is represented and,
Figure SMS_11
representing the weight of each gaussian distribution over the entire distribution.
Preferably, the setting of the initial time k =1 and the initialization of the model parameters include:
Figure SMS_12
Figure SMS_13
wherein, the first and the second end of the pipe are connected with each other,
Figure SMS_14
mean value, x, representing the state filtering expectation at the initial moment 0 Representing the system state at the initial moment; p 0|0 Representing an expected covariance matrix of the filtering error at the initial moment; e (.) represents a mathematical expectation function; () T Representing a transpose operation.
Preferably, the state filtering value and the filtering error covariance matrix at the time k are input at the time k +1, and an unscented gaussian filtering method is operated to realize the prediction of the state filtering value; the method comprises the following steps:
inputting the state filtering value at the k moment at the k +1 moment
Figure SMS_15
Sum filter error covariance matrix P k|k Running an unscented Gaussian sum filtering method to realize the prediction of a state filtering value;
3.1 Sigma point set generating 2m +1 state transfer functions
Figure SMS_16
The method comprises the following steps:
let i =0, repeat executing the following operations 2m +1 times:
if i =0, yield
Figure SMS_17
If i is more than or equal to 1 and less than or equal to m, get>
Figure SMS_18
If m +1 is not less than i and not more than 2m, obtaining
Figure SMS_19
Wherein
Figure SMS_20
Status vector representing the predicted time k +1 at time k>
Figure SMS_21
A state filter value representing the k time, m representing the dimension of the system state; parameter λ = α 2 (m + κ) -m; the parameter α represents a scaling factor; the parameter k represents a scaling parameter;
3.2 Generates 2m +1 state transfer function Sigma point set corresponding weight
Figure SMS_22
The method comprises the following steps: let i =0, repeat the following operations of 2m +1 times:
if i =0, yield
Figure SMS_23
If i is more than or equal to 1 and less than or equal to 2m, obtaining
Figure SMS_24
i＝i+1；
3.3 Sigma point redefining state transfer function
Figure SMS_25
The following were used:
Figure SMS_26
Figure SMS_27
wherein, the first and the second end of the pipe are connected with each other,
Figure SMS_28
sigma point representing the ith state transfer function; />
Figure SMS_29
Represents all->
Figure SMS_30
A linear weighted sum of;
3.4 Substituting the Sigma point of the state transfer function into the formula z k ＝h(x k )+v k And obtaining:
Figure SMS_31
wherein the content of the first and second substances,
Figure SMS_32
sigma Point, h (x) representing the ith measurement function k ) Representing a non-linear measurement function;
3.5 Calculated by the formula (10)
Figure SMS_33
Is based on the mean value->
Figure SMS_34
/>
Figure SMS_35
Wherein the content of the first and second substances,
Figure SMS_36
represents->
Figure SMS_37
The mean value of (a);
3.6 Computing a cross covariance matrix by equation (11)
Figure SMS_38
Figure SMS_39
Wherein
Figure SMS_40
Representing a cross covariance matrix for calculation of a common Kalman gain matrix;
3.7 Calculate a one-step prediction covariance matrix by equation (12)
Figure SMS_41
Figure SMS_42
Wherein
Figure SMS_43
Representing a one-step predictive covariance matrix for the computation of a common Kalman gain matrix, L k Measuring a covariance matrix of noise for the system;
3.8 Let the system (1) predict the distribution p (x) in one step of the state at time k k+1 |z 1:k ) The following gaussian mixture model:
Figure SMS_44
wherein
Figure SMS_45
For unknown parameters, m is the dimension of the system state.
Preferably, the calculating unknown parameters
Figure SMS_46
Is greater or less than>
Figure SMS_47
The method comprises the following steps:
4.1 Design unknown parameters P) k+1|k Conditional prior distribution P (P) k+1|k |z 1:k ) In the form:
Figure SMS_48
wherein the unknown parameters
Figure SMS_49
For calculation of a common gain matrix;
Figure SMS_50
an inverse Weissett distribution probability density function of a n x n dimensional matrix A; e (a) = T (T-n-1), E (·) denotes a desired function; exp { } denotes an exponential function; t represents a degree of freedom; gamma-shaped n (-) represents a gamma function; tr (.) represents a trace function;
4.2 Let the predicted value of the unknown parameter in the ith inverse Weisset distribution at the time k +1
Figure SMS_51
And &>
Figure SMS_52
The following were used:
Figure SMS_53
wherein, the first and the second end of the pipe are connected with each other,
Figure SMS_54
one of the parameters representing the inverse wexat distribution in expression (14), ρ represents a forgetting factor;
Figure SMS_55
wherein the content of the first and second substances,
Figure SMS_56
one of the parameters representing the inverse wexat distribution in formula (14);
4.3 Calculate unknown parameters
Figure SMS_57
Is greater than or equal to>
Figure SMS_58
/>
Figure SMS_59
Wherein
Figure SMS_60
Which can be used for the calculation of the status filter value update step.
Preferably, the updating of the filtered value and the filtering error covariance matrix at the time k +1 includes:
5.1 Calculating a common Kalman gain matrix through formula (18), calculating a filter value of the ith filter at the moment k +1 through formula (19), calculating a filter error covariance matrix of the ith filter at the moment k +1 through formula (20), calculating a weight of the ith filter at the moment k +1 through formula (21), and calculating a weight of the ith filter subjected to normalization processing at the moment k +1 through formula (22);
Figure SMS_61
Figure SMS_62
Figure SMS_63
Figure SMS_64
Figure SMS_65
wherein, the first and the second end of the pipe are connected with each other,
Figure SMS_66
representing a k +1 moment public Kalman gain matrix; />
Figure SMS_67
Represents the filtering value of the ith filter at the moment k + 1; />
Figure SMS_68
Representing a filtering error covariance matrix of the ith filter at the moment k + 1; />
Figure SMS_69
Represents the weight of the ith filter at the time k + 1; />
Figure SMS_70
Representing the weight of the ith filter subjected to normalization processing at the moment k + 1;
5.2 Calculate the filtered value at the time k +1 by equation (23)
Figure SMS_71
Calculating P by equation (24) k+1|k+1 A filtering error covariance matrix;
Figure SMS_72
Figure SMS_73
wherein, the first and the second end of the pipe are connected with each other,
Figure SMS_74
represents the filtered value at the moment k +1, and is a linear weighted sum of the filtered values of 2m +1 filters; p k+1|k+1 The covariance matrix of the filtering error at time k +1 is the linear weighted sum of the covariance matrices of the filtering errors of the filters 2m + 1.
Preferably, the unknown parameter estimated value at the k +1 moment is updated
Figure SMS_75
The method comprises the following steps:
6.1 Let system sum up processNoise(s)
Figure SMS_76
The posterior distribution of (a) is approximately represented by a mixed gaussian distribution as follows:
Figure SMS_77
where m is the dimension, parameter, of the system state
Figure SMS_78
Has a posterior probability density function of>
Figure SMS_79
Figure SMS_80
Representing a parameter>
Figure SMS_81
An estimated value of (d);
6.2 Approximation of the posterior joint probability distribution using a variational Bayesian method
Figure SMS_82
Namely: />
Figure SMS_83
Figure SMS_84
Wherein the content of the first and second substances,
Figure SMS_85
6.3 System state x) k+1 And unknown parameter estimates
Figure SMS_86
The joint posterior probability density function of (a) is approximated by the following equation:
p(x k+1 ,P k+1|k |z 1:k+1 )≈q(x k+1 )q(P k+1|k )， (27)
wherein z is 1:k+1 The measurement output from the time 1 to the time k +1 of the system is obtained;
6.4 Define the true posterior distribution p (x) k+1 ,P k+1|k |z 1:k+1 ) And approximate distribution q (x) k+1 )q(P k+1|k ) The KL divergence between is as follows:
Figure SMS_87
wherein ^ represents integral operation, and log () represents logarithm;
6.5 By minimizing equation (28), the following equation can be obtained:
logq(θ)＝E Ξ-θ [logp(Ξ,z 1:k+1 )]+C θ ， (29)
Figure SMS_88
wherein θ represents any element in the set xi; c θ Represents a constant related to θ;
6.6 Calculating the logarithmic expectation value of equation (26) from equation (29), we can obtain:
Figure SMS_89
wherein the content of the first and second substances,
Figure SMS_90
representation and parameter P k+1|k A related constant; tr (.) represents a trace function; is based on formula (31)>
Figure SMS_91
The following:
Figure SMS_92
6.7 Equation (32) is expressed as follows:
Figure SMS_93
Figure SMS_94
as can be seen from equation (31), the unknown parameter estimation value P k+1|k Approximation q (P) of probability density function k+1|k ) The product of probability density functions of distribution of 2m +1 inverse Weishate, namely
Figure SMS_95
Figure SMS_96
Figure SMS_97
Wherein q (P) k+1|k ) Representing the estimated value P of the unknown parameter k+1|k Approximation of the probability density function, q (P) k+1|k ) Is the product of 2m +1 inverse Weissett distributions;
Figure SMS_98
a posterior parameter representing the ith inverse weisset distribution; />
Figure SMS_99
A posterior parameter representing the ith inverse weisset distribution;
6.8 Derived from the nature of the distribution of formula (35), formula (36) and inverse weixate:
Figure SMS_100
wherein
Figure SMS_101
Represents the updated parameter at time k + 1.
The invention also provides a nonlinear information physical system state estimation system under FDI attack, which comprises:
a state space modeling module: the state space model is used for establishing a nonlinear information physical system under FDI attack;
an initialization setting module: the method is used for setting an initial time k =1 and initializing model parameters;
a state filtered value prediction module: the method is used for inputting a state filtering value and a filtering error covariance matrix at the moment k +1, and operating an unscented Gaussian sum filtering method to realize the prediction of the state filtering value;
a state estimation module: for calculating unknown parameters
Figure SMS_102
Is greater or less than>
Figure SMS_103
Updating a filtering value and a filtering error covariance matrix at the moment of k + 1;
updating unknown parameter estimation value at k +1 moment
Figure SMS_104
And judging whether the moment k is smaller than the last moment N, if the k is smaller than the last moment N, setting k = k +1, outputting a state estimation value at the moment of k +1, and otherwise, finishing the calculation.
The present invention also provides a computer-readable storage medium storing a computer program comprising program instructions which, when executed by a processor, perform a method for filtering a nonlinear cyber-physical system under FDI attack as described in any one of the preceding.
The invention also provides a nonlinear information physical system filtering terminal under FDI attack, which comprises: the device comprises an input device, an output device, a memory and a processor; the input device, the output device, the memory and the processor are connected with each other, wherein the memory is used for storing a computer program, the computer program comprises program instructions, and the protected processor is configured to call the program instructions to execute the nonlinear information physical system filtering method under the FDI attack.
(III) advantageous effects
The invention provides a nonlinear information physical system filtering method and system under FDI attack. The method has the following beneficial effects:
in the technical scheme of the invention, in order to realize the real-time state estimation of the nonlinear non-Gaussian information physical system under FDI attack, an unscented Gaussian and filtering method and a biased-variational Bayesian method are introduced, so that the state estimation precision is ensured, and the calculation complexity of the method is effectively reduced.
According to the invention, the FDI attack is modeled into the non-Gaussian signal, the FDI attack and the unknown process noise of the system are described into the total non-Gaussian process noise of the system without assuming that the FDI attack obeys a certain special distribution (such as Bernoulli distribution, normal distribution and the like), and then the nonlinear information physical system under the FDI attack is modeled into the nonlinear system with the unknown non-Gaussian process noise, so that the filtering problem of the nonlinear information physical system under the FDI attack can be converted into the filtering problem of the nonlinear non-Gaussian system with the unknown process noise.
In order to reduce the computational complexity, filtering is carried out by adopting an unscented Gaussian and filtering method, the computational complexity of the method can be effectively reduced, the unscented Gaussian and filtering method is popularized to a nonlinear system with unknown non-Gaussian process noise, and the estimation of unknown parameters can be realized by adopting a biased-variational Bayes method.
Drawings
FIG. 1 is a flow chart of a filtering method of a nonlinear information physical system under FDI attack according to the present invention;
fig. 2 is a structural diagram of a filtering system of a nonlinear information physical system under FDI attack according to the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention.
The embodiment of the invention provides a filtering method of a nonlinear information physical system under FDI attack, as shown in figure 1, comprising the following steps:
s1, establishing a state space model of a nonlinear information physical system under FDI attack;
s2, setting an initial time k =1, and initializing model parameters;
s3, inputting a state filtering value and a filtering error covariance matrix at the moment k +1, and operating an unscented Gaussian and filtering method to realize prediction of the state filtering value;
s4, calculating unknown parameters
Figure SMS_105
Is greater or less than>
Figure SMS_106
S5. Updating the filtering value and the filtering error covariance matrix at the moment of k + 1;
s6, updating the unknown parameter estimation value at the k +1 moment
Figure SMS_107
And S7, judging whether the moment k is smaller than the last moment N, if k is smaller than N, setting k = k +1, outputting a state estimation value at the moment k +1, returning to the step 3, and otherwise, finishing the calculation.
In one embodiment, the establishing a state space model of a nonlinear information physical system under FDI attack includes:
1.1 A discrete-time nonlinear information physical system model is expressed in the form of:
Figure SMS_108
wherein x k ∈R m Representing the state vector of the system at time k, R representing the real number domain, m representing the dimension of the system state, z k ∈R n Representing the measurement vector of the system at time k, n representing the dimension of the measurement vector, u k ∈R m To representControl input to the system at time k, f (x) k ) Representing a non-linear state transfer function, h (x) k ) Representing a non-linear measurement function, n k Representing unknown non-Gaussian process noise, v k The measurement noise at the time k is represented by a covariance matrix L k Zero mean white gaussian noise, covariance matrix L k Is an n x n dimensional symmetrical square matrix;
1.2 Considering that the system (1) is subject to the attack of the executor FDI, an executor FDI attack signal alpha is introduced ak The system model (1) is written in the form of the following model:
Figure SMS_109
wherein alpha is ak Representing an actuator FDI attack signal, and obeying non-Gaussian distribution;
1.3 Redefines the process noise of the system, writing equation (2) as the following model:
Figure SMS_110
wherein
Figure SMS_111
Representing unknown sum non-gaussian process noise;
1.4 non-Gaussian noise to sum
Figure SMS_112
Is based on the probability density function->
Figure SMS_113
Approximately expressed as the following gaussian mixture distribution:
Figure SMS_114
wherein N (| μ, Σ) represents a gaussian probability density function with mean μ and variance Σ; n is a radical of n The number of gaussian mixtures is represented and,
Figure SMS_115
representing the weight of each gaussian distribution over the entire distribution.
In one embodiment, the setting of initial time k =1 and the initialization of model parameters include:
Figure SMS_116
Figure SMS_117
wherein the content of the first and second substances,
Figure SMS_118
mean value, x, representing the expected state filtering at the initial time 0 Representing the system state at the initial moment; p 0|0 Representing an expected covariance matrix of the filtering error at the initial moment; e (.) represents a mathematical expectation function; () T Representing a transpose operation.
In one embodiment, the state filter value and the filter error covariance matrix at the time k +1 are input, and an unscented gaussian filtering method is operated to realize the prediction of the state filter value; the method comprises the following steps:
inputting the state filtering value at the k moment at the k +1 moment
Figure SMS_119
Sum filter error covariance matrix P k|k Running an unscented Gaussian sum filtering method to realize the prediction of a state filtering value;
3.1 Sigma point set generating 2m +1 state transfer functions
Figure SMS_120
The method comprises the following steps:
let i =0, repeat the following operations of 2m +1 times:
if i =0, yield
Figure SMS_121
If i is more than or equal to 1 and less than or equal to m, obtain>
Figure SMS_122
If m +1 is not less than i and not more than 2m, obtaining
Figure SMS_123
Wherein
Figure SMS_124
Status vector representing a prediction at time k +1>
Figure SMS_125
A state filter value representing the k moment, and m represents the dimension of the system state; parameter λ = α 2 (m + κ) -m; the parameter α represents a scaling factor; the parameter k represents a scaling parameter;
3.2 Generates 2m +1 state transfer function Sigma point set corresponding weight
Figure SMS_126
The method comprises the following steps: let i =0, repeat the following operations of 2m +1 times: />
If i =0, we obtain
Figure SMS_127
If i is more than or equal to 1 and less than or equal to 2m, obtaining
Figure SMS_128
i＝i+1；
3.3 Sigma point redefining state transfer function
Figure SMS_129
The following were used:
Figure SMS_130
Figure SMS_131
wherein, the first and the second end of the pipe are connected with each other,
Figure SMS_132
sigma point representing the ith state transfer function; />
Figure SMS_133
Represents all->
Figure SMS_134
A linear weighted sum of;
3.4 Substituting the Sigma point of the state transfer function into the formula z k ＝h(x k )+v k Obtaining:
Figure SMS_135
wherein the content of the first and second substances,
Figure SMS_136
sigma Point, h (x) representing the ith measurement function k ) Representing a non-linear measurement function;
3.5 Calculated by the formula (10)
Figure SMS_137
In (d) is based on the mean value>
Figure SMS_138
Figure SMS_139
Wherein the content of the first and second substances,
Figure SMS_140
represents->
Figure SMS_141
The mean value of (a);
3.6 Computing a cross covariance matrix by equation (11)
Figure SMS_142
Figure SMS_143
Wherein
Figure SMS_144
Representing a cross covariance matrix for calculation of a common Kalman gain matrix;
3.7 Calculate a one-step prediction covariance matrix by equation (12)
Figure SMS_145
Figure SMS_146
Wherein
Figure SMS_147
Representing a one-step predictive covariance matrix for the calculation of a common Kalman gain matrix, L k Measuring a covariance matrix of noise for the system;
3.8 Let the system (1) predict the distribution p (x) in one step of the state at time k k+1 |z 1:k ) The following gaussian mixture model:
Figure SMS_148
wherein
Figure SMS_149
For unknown parameters, m is the dimension of the system state.
In one embodiment, the calculating unknown parameters
Figure SMS_150
Is greater or less than>
Figure SMS_151
The method comprises the following steps:
4.1 Design unknown parameters P) k+1|k Conditional prior distribution P (P) k+1|k |z 1:k ) In the form:
Figure SMS_152
wherein the unknown parameters
Figure SMS_153
For calculation of a common gain matrix;
Figure SMS_154
an inverse Weissett distribution probability density function of a n x n dimensional matrix A; e (a) = T (T-n-1), E (·) denotes a desired function; exp { } denotes an exponential function; t represents a degree of freedom; gamma-shaped n (-) represents a gamma function; tr (.) represents a trace function;
4.2 Let the predicted value of the unknown parameter in the ith inverse Weisset distribution at the time k +1
Figure SMS_155
And &>
Figure SMS_156
The following were used:
Figure SMS_157
wherein, the first and the second end of the pipe are connected with each other,
Figure SMS_158
one of the parameters representing the inverse weisset distribution in expression (14), ρ represents a forgetting factor;
Figure SMS_159
wherein the content of the first and second substances,
Figure SMS_160
one of the parameters representing the inverse wexat distribution in formula (14);
4.3 Calculate unknown parameters
Figure SMS_161
Is greater than or equal to>
Figure SMS_162
Figure SMS_163
Wherein
Figure SMS_164
Which may be used for the calculation of the state filter value update step.
In one embodiment, the updating of the filtered value and the filtering error covariance matrix at the time k +1 includes:
5.1 A common Kalman gain matrix is calculated through a formula (18), a filter value of an ith filter at the moment k +1 is calculated through a formula (19), a filter error covariance matrix of the ith filter at the moment k +1 is calculated through a formula (20), the weight of the ith filter at the moment k +1 is calculated through a formula (21), and the weight of the ith filter subjected to normalization processing at the moment k +1 is calculated through a formula (22);
Figure SMS_165
Figure SMS_166
Figure SMS_167
Figure SMS_168
Figure SMS_169
wherein, the first and the second end of the pipe are connected with each other,
Figure SMS_170
representing a common Kalman gain matrix at time k +1；/>
Figure SMS_171
Represents the filtered value of the ith filter at the moment k + 1; />
Figure SMS_172
Representing a filtering error covariance matrix of the ith filter at the moment k + 1; />
Figure SMS_173
Represents the weight of the ith filter at the time k + 1; />
Figure SMS_174
Representing the weight of the ith filter subjected to normalization processing at the moment k + 1; />
5.2 Calculate the filtered value at the time k +1 by equation (23)
Figure SMS_175
Calculating P by equation (24) k+1|k+1 A filtering error covariance matrix;
Figure SMS_176
Figure SMS_177
wherein the content of the first and second substances,
Figure SMS_178
represents the filtered value at the moment k +1, and is a linear weighted sum of the filtered values of 2m +1 filters; p k+1|k+1 The covariance matrix of the filtering error at time k +1 is the linear weighted sum of the covariance matrices of the filtering errors of the filters 2m + 1.
In one embodiment, the updating of the unknown parameter estimated value at the time k +1
Figure SMS_179
The method comprises the following steps:
6.1 ) make systematic summation process noise
Figure SMS_180
The posterior distribution of (a) is approximately represented by a mixed gaussian distribution as follows:
Figure SMS_181
where m is the dimension, parameter, of the system state
Figure SMS_182
Has a posterior probability density function of
Figure SMS_183
Figure SMS_184
Represents a parameter->
Figure SMS_185
An estimated value of (d);
6.2 Approximation of the posterior joint probability distribution using a variational Bayesian method
Figure SMS_186
Namely:
Figure SMS_187
wherein, the first and the second end of the pipe are connected with each other,
Figure SMS_188
6.3 ) system state x k+1 And unknown parameter estimates
Figure SMS_189
The joint posterior probability density function of (a) is approximated by the following equation:
p(x k+1 ,P k+1|k |z 1:k+1 )≈q(x k+1 )q(P k+1|k )， (27)
wherein z is 1:k+1 For measuring the output of the system from 1 time to k +1 timeDischarging;
6.4 Define the true posterior distribution p (x) k+1 ,P k+1|k |z 1:k+1 ) And approximate distribution q (x) k+1 )q(P k+1|k ) The KL divergence between is as follows:
Figure SMS_190
wherein { [ integral ] represents an integral operation, and log () represents a logarithm;
6.5 By minimizing equation (28), the following equation can be obtained:
logq(θ)＝E Ξ-θ [logp(Ξ,z 1:k+1 )]+C θ ， (29)
Figure SMS_191
/>
wherein θ represents any element in the set Ξ; c θ Represents a constant related to θ;
6.6 Calculating the logarithmic expectation value of equation (26) from equation (29), we can obtain:
Figure SMS_192
wherein, C Pk+1|k Representation and parameter P k+1|k A related constant; tr (.) represents a trace function; in the formula (31)
Figure SMS_193
The following:
Figure SMS_194
6.7 Equation (32) is expressed as follows:
Figure SMS_195
as can be seen from equation (31), the unknown parameter estimation value P k+1|k Proximity of probability density functionSimilarity value q (P) k+1|k ) Is the product of the probability density functions of 2m +1 inverse Weissett distributions, i.e.
Figure SMS_196
Figure SMS_197
Figure SMS_198
Wherein q (P) k+1|k ) Representing the estimated value P of the unknown parameter k+1|k Approximation of the probability density function, q (P) k+1|k ) Is the product of 2m +1 inverse Weissett distributions;
Figure SMS_199
a posterior parameter representing the ith inverse weisset distribution; />
Figure SMS_200
A posterior parameter representing the ith inverse weisset distribution;
6.8 Derived from the nature of the distribution of formula (35), formula (36) and inverse weixate:
Figure SMS_201
wherein
Figure SMS_202
Represents the updated parameter at time k + 1.
As shown in fig. 2, an embodiment of the present invention further provides a nonlinear information physical system filtering system under FDI attack, including:
a state space modeling module: the state space model is used for establishing a nonlinear information physical system under FDI attack;
an initialization setting module: the method is used for setting an initial moment k =1 and initializing model parameters;
a state filtered value prediction module: the method is used for inputting a state filtering value and a filtering error covariance matrix at the moment k +1, and operating an unscented Gaussian sum filtering method to realize the prediction of the state filtering value;
a state estimation module: for calculating unknown parameters
Figure SMS_203
Is greater or less than>
Figure SMS_204
/>
Updating a filtering value and a filtering error covariance matrix at the moment of k + 1;
updating unknown parameter estimation value at k +1 moment
Figure SMS_205
And judging whether the time k is smaller than the last time N, if k is smaller than N, setting k = k +1, outputting the state estimation value at the time k +1, and otherwise, finishing the calculation.
An embodiment of the present invention further provides a computer-readable storage medium, where a computer program is stored, where the computer program includes program instructions, and the program instructions, when executed by a processor, perform the method for filtering a nonlinear cyber-physical system under FDI attack as described in any one of the foregoing.
The embodiment of the invention also provides a filtering terminal of a nonlinear information physical system under FDI attack, which comprises: the device comprises an input device, an output device, a memory and a processor; the input device, the output device, the memory and the processor are connected with each other, wherein the memory is used for storing a computer program, the computer program comprises program instructions, and the protected processor is configured to call the program instructions to execute the nonlinear information physical system filtering method under the FDI attack.
Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that changes, modifications, substitutions and alterations can be made in these embodiments without departing from the principles and spirit of the invention, the scope of which is defined in the appended claims and their equivalents.

Claims (10)

1. A nonlinear information physical system filtering method under FDI attack is characterized by comprising the following steps:
step 1, establishing a state space model of a nonlinear information physical system under FDI attack;
step 2, setting an initial time k =1, and initializing model parameters;
step 3, inputting a state filtering value and a filtering error covariance matrix at the moment k +1, operating an unscented Gaussian and filtering method, and predicting the state filtering value;
step 4, calculating unknown parameters
Figure QLYQS_1
Is greater or less than>
Figure QLYQS_2
Updating a filtering value and a filtering error covariance matrix at the moment of 5.K + 1;
step 6, updating the unknown parameter estimated value at the k +1 moment
Figure QLYQS_3
And 7, judging whether the moment k is smaller than the last moment N, if k is smaller than N, setting k = k +1, outputting the state estimation value at the moment k +1, returning to the step 3, and otherwise, finishing the calculation.
2. The method as claimed in claim 1, wherein the establishing of the state space model of the FDI-attacked nonlinear cyber-physical system comprises:
1.1 A discrete-time nonlinear information physical system model is expressed in the form of:
Figure QLYQS_4
wherein x is k ∈R m Representing the state vector of the system at time k, R representing the real number domain, m representing the dimension of the system state, z k ∈R n Representing the measurement vector of the system at time k, n representing the dimension of the measurement vector, u k ∈R m Represents the control input of the system at time k, f (x) k ) Represents a nonlinear state transfer function, h (x) k ) Representing a non-linear measurement function, n k Representing unknown non-Gaussian process noise, v k The measurement noise at the time k is represented by a covariance matrix L k Zero mean white gaussian noise, covariance matrix L k Is an n multiplied by n dimensional symmetric square matrix;
1.2 Considering that the system (1) is subject to an actuator FDI attack, an actuator FDI attack signal alpha is introduced ak The system model (1) is written in the form of the following model:
Figure QLYQS_5
wherein alpha is ak Representing an actuator FDI attack signal, and obeying non-Gaussian distribution;
1.3 Redefines the process noise of the system, writing equation (2) as the model:
Figure QLYQS_6
wherein
Figure QLYQS_7
Representing unknown sum non-gaussian process noise;
1.4 non-Gaussian noise to sum
Figure QLYQS_8
Is based on the probability density function->
Figure QLYQS_9
Approximately expressed as the following gaussian mixture distribution:
Figure QLYQS_10
wherein N (| μ, Σ) represents a gaussian probability density function with mean μ and variance Σ; n is a radical of n The number of gaussian mixtures is represented and,
Figure QLYQS_11
representing the weight of each gaussian distribution over the entire distribution.
3. The method for estimating the state of the nonlinear information physical system under the FDI attack as recited in claim 2, wherein the setting of the initial time k =1 and the initialization of the model parameters comprise:
Figure QLYQS_12
Figure QLYQS_13
wherein, the first and the second end of the pipe are connected with each other,
Figure QLYQS_14
mean value, x, representing the state filtering expectation at the initial moment 0 Representing the system state at the initial moment; p 0|0 Representing an expected covariance matrix of the filtering error at the initial moment; e (.) represents a mathematical expectation function; () T Representing a transpose operation.
4. The filtering method of the nonlinear informatics physical system under the FDI attack according to claim 3, wherein the state filtering value and the filtering error covariance matrix at the time k +1 are input, and an unscented Gaussian sum filtering method is operated to realize the prediction of the state filtering value; the method comprises the following steps:
inputting k time at k +1 timeState filtered value
Figure QLYQS_15
Sum filter error covariance matrix P k|k Running an unscented Gaussian sum filtering method to realize the prediction of a state filtering value;
3.1 Sigma point set for generating 2m +1 state transfer functions
Figure QLYQS_16
The method comprises the following steps:
let i =0, repeat the following operations of 2m +1 times:
if i =0, yield
Figure QLYQS_17
If i is more than or equal to 1 and less than or equal to m, obtain>
Figure QLYQS_18
If m +1 is not less than i and not more than 2m, obtaining
Figure QLYQS_19
i＝i+1；
Wherein
Figure QLYQS_20
Status vector representing the predicted time k +1 at time k>
Figure QLYQS_21
A state filter value representing the k time, m representing the dimension of the system state; parameter λ = α 2 (m + κ) -m; the parameter α represents a scaling factor; the parameter k represents a scaling parameter;
3.2 Generates 2m +1 state transfer function Sigma point set corresponding weight
Figure QLYQS_22
The method comprises the following steps: let i =0, repeat the following operations of 2m +1 times:
if i =0, yield
Figure QLYQS_23
If i is more than or equal to 1 and less than or equal to 2m, obtaining
Figure QLYQS_24
i＝i+1；
3.3 Sigma point redefining state transfer function
Figure QLYQS_25
The following were used:
Figure QLYQS_26
Figure QLYQS_27
wherein the content of the first and second substances,
Figure QLYQS_28
sigma point representing the ith state transfer function; />
Figure QLYQS_29
Represents all->
Figure QLYQS_30
A linear weighted sum of;
3.4 Substituting Sigma points of the state transfer function into the formula z k ＝h(x k )+v k Obtaining:
Figure QLYQS_31
wherein the content of the first and second substances,
Figure QLYQS_32
sigma Point, h (x) representing the ith measurement function k ) Representing a non-linear measurement function;
3.5 Calculated by the formula (10)
Figure QLYQS_33
In (d) is based on the mean value>
Figure QLYQS_34
/>
Figure QLYQS_35
Wherein, the first and the second end of the pipe are connected with each other,
Figure QLYQS_36
represents->
Figure QLYQS_37
The mean value of (a);
3.6 Computing a cross covariance matrix by equation (11)
Figure QLYQS_38
Figure QLYQS_39
Wherein
Figure QLYQS_40
Representing a cross covariance matrix for calculation of a common Kalman gain matrix;
3.7 Calculate a one-step prediction covariance matrix by equation (12)
Figure QLYQS_41
Figure QLYQS_42
Wherein
Figure QLYQS_43
Representing a one-step predictive covariance matrix byFrom the calculation of the common Kalman gain matrix, L k Measuring a covariance matrix of noise for the system;
3.8 Let the system (1) predict the distribution p (x) in one step of the state at time k k+1 |z 1:k ) The following gaussian mixture model:
Figure QLYQS_44
wherein
Figure QLYQS_45
For unknown parameters, m is the dimension of the system state.
5. The method as claimed in claim 4, wherein the calculating unknown parameters includes calculating the unknown parameters
Figure QLYQS_46
Is greater or less than>
Figure QLYQS_47
The method comprises the following steps:
4.1 Design unknown parameters P) k+1|k Conditional prior distribution P (P) k+1|k |z 1:k ) In the form:
Figure QLYQS_48
wherein the unknown parameters
Figure QLYQS_49
For calculation of a common gain matrix;
Figure QLYQS_50
an inverse Weissett distribution probability density function of a n x n dimensional matrix A; e (a) = T (T-n-1), E (·) denotes an expectation function; exp { } denotes an exponential function; t represents a degree of freedom; gamma-shaped n (-) represents a gamma function;tr (.) represents a trace function;
4.2 Let the predicted value of the unknown parameter in the ith inverse Weisset distribution at the time k +1
Figure QLYQS_51
And &>
Figure QLYQS_52
The following were used:
Figure QLYQS_53
wherein the content of the first and second substances,
Figure QLYQS_54
one of the parameters representing the inverse weisset distribution in expression (14), ρ represents a forgetting factor;
Figure QLYQS_55
wherein, the first and the second end of the pipe are connected with each other,
Figure QLYQS_56
one of the parameters representing the inverse wexat distribution in formula (14); />
4.3 Calculate unknown parameters
Figure QLYQS_57
Is greater than or equal to>
Figure QLYQS_58
Figure QLYQS_59
Wherein
Figure QLYQS_60
Which can be used for the calculation of the status filter value update step.
6. The method as claimed in claim 5, wherein the updating of the filter value and the filter error covariance matrix at the time k +1 comprises:
5.1 Calculating a common Kalman gain matrix through formula (18), calculating a filter value of the ith filter at the moment k +1 through formula (19), calculating a filter error covariance matrix of the ith filter at the moment k +1 through formula (20), calculating a weight of the ith filter at the moment k +1 through formula (21), and calculating a weight of the ith filter subjected to normalization processing at the moment k +1 through formula (22);
Figure QLYQS_61
Figure QLYQS_62
Figure QLYQS_63
Figure QLYQS_64
Figure QLYQS_65
wherein, the first and the second end of the pipe are connected with each other,
Figure QLYQS_66
representing a k +1 moment public Kalman gain matrix; />
Figure QLYQS_67
Represents the filtered value of the ith filter at the moment k + 1; />
Figure QLYQS_68
Representing a filtering error covariance matrix of the ith filter at the moment k + 1; />
Figure QLYQS_69
Represents the weight of the ith filter at the time k + 1; />
Figure QLYQS_70
Representing the weight of the ith filter subjected to normalization processing at the moment k + 1;
5.2 Calculate the filtered value at the time k +1 by equation (23)
Figure QLYQS_71
Calculating P by equation (24) k+1|k+1 A filtering error covariance matrix;
Figure QLYQS_72
Figure QLYQS_73
wherein the content of the first and second substances,
Figure QLYQS_74
represents the filtered value at the moment k +1, and is a linear weighted sum of the filtered values of 2m +1 filters; p is k+1|k+1 The covariance matrix of the filtering error at time k +1 is the linear weighted sum of the covariance matrices of the filtering errors of the filters 2m + 1.
7. The method as claimed in claim 6, wherein the updating of the unknown parameter estimated value at the time k +1 is performed by the method for filtering the nonlinear information physical system under FDI attack
Figure QLYQS_75
The method comprises the following steps:
6.1 ) make systematic summation process noise
Figure QLYQS_76
The posterior distribution of (a) is approximately represented by a mixed gaussian distribution as follows:
Figure QLYQS_77
where m is the dimension, parameter, of the system state
Figure QLYQS_78
Has an a posteriori probability density function of->
Figure QLYQS_79
Figure QLYQS_80
Representing a parameter>
Figure QLYQS_81
An estimated value of (d);
6.2 Approximation of the posterior joint probability distribution using a variational Bayesian method
Figure QLYQS_82
Namely:
Figure QLYQS_83
wherein the content of the first and second substances,
Figure QLYQS_84
6.3 ) system state x k+1 And unknown parameter estimates
Figure QLYQS_85
Is approximated by the following equation:
p(x k+1 ,P k+1|k |z 1:k+1 )≈q(x k+1 )q(P k+1|k )， (27)
wherein z is 1:k+1 The measurement output from the time 1 to the time k +1 of the system is obtained;
6.4 Define the true posterior distribution p (x) k+1 ,P k+1|k |z 1:k+1 ) And approximate distribution q (x) k+1 )q(P k+1|k ) The KL divergence between is as follows:
Figure QLYQS_86
wherein ^ represents integral operation, and log () represents logarithm;
6.5 By minimizing equation (28), the following equation can be obtained:
logq(θ)＝E Ξ-θ [logp(Ξ,z 1:k+1 )]+C θ ， (29)
Figure QLYQS_87
wherein θ represents any element in the set Ξ; c θ Represents a constant related to θ;
6.6 Calculating the logarithmic expectation value of equation (26) from equation (29), we can obtain:
Figure QLYQS_88
wherein the content of the first and second substances,
Figure QLYQS_89
representation and parameter P k+1|k A related constant; tr (.) represents a trace function; is based on formula (31)>
Figure QLYQS_90
The following were used:
Figure QLYQS_91
6.7 Equation (32) is expressed as follows:
Figure QLYQS_92
as can be seen from equation (31), the unknown parameter estimation value P k+1|k Approximation q (P) of a probability density function k+1|k ) Is the product of the probability density functions of 2m +1 inverse Weissett distributions, i.e.
Figure QLYQS_93
Figure QLYQS_94
Figure QLYQS_95
Wherein q (P) k+1|k ) Representing the estimated value P of the unknown parameter k+1|k Approximation of the probability density function, q (P) k+1|k ) Is the product of 2m +1 inverse Weissett distributions;
Figure QLYQS_96
a posterior parameter representing the ith inverse weishat distribution; />
Figure QLYQS_97
A posterior parameter representing the ith inverse weisset distribution;
6.8 Derived from the nature of the distribution of formula (35), formula (36) and inverse weixate:
Figure QLYQS_98
wherein
Figure QLYQS_99
Represents the parameter updated at time k + 1.
8. A nonlinear information physical system filtering system under FDI attack is characterized by comprising:
a state space modeling module: the state space model is used for establishing a nonlinear information physical system under FDI attack;
an initialization setting module: the method is used for setting an initial time k =1 and initializing model parameters;
a state filtered value prediction module: the method is used for inputting a state filtering value and a filtering error covariance matrix at the moment k +1, and operating an unscented Gaussian sum filtering method to realize the prediction of the state filtering value;
a state estimation module: for calculating unknown parameters
Figure QLYQS_100
In a manner that a prediction value is greater than or equal to>
Figure QLYQS_101
Updating a filtering value and a filtering error covariance matrix at the moment of k + 1;
updating unknown parameter estimation value at k +1 moment
Figure QLYQS_102
And judging whether the time k is smaller than the last time N, if k is smaller than N, setting k = k +1, outputting the state estimation value at the time k +1, and otherwise, finishing the calculation.
9. A computer-readable storage medium, characterized in that the computer-readable storage medium stores a computer program comprising program instructions which, when executed by a processor, perform a method for filtering a nonlinear cyber-physical system under FDI attack as recited in any one of claims 1 to 7.
10. A nonlinear information physical system state estimation filtering terminal under FDI attack is characterized by comprising: the device comprises an input device, an output device, a memory and a processor; the input device, the output device, the memory and the processor are connected with each other, wherein the memory is used for storing a computer program, the computer program comprises program instructions, and the protected processor is configured to call the program instructions to execute the nonlinear informatics physical system filtering method under FDI attack according to any one of claims 1-7.
CN202210856707.7A 2022-07-19 2022-07-19 Nonlinear information physical system filtering method and system under FDI attack Pending CN115865390A (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN202210856707.7A CN115865390A (en) 2022-07-19 2022-07-19 Nonlinear information physical system filtering method and system under FDI attack

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN202210856707.7A CN115865390A (en) 2022-07-19 2022-07-19 Nonlinear information physical system filtering method and system under FDI attack

Publications (1)

Publication Number Publication Date
CN115865390A true CN115865390A (en) 2023-03-28

Family

ID=85660328

Family Applications (1)

Application Number Title Priority Date Filing Date
CN202210856707.7A Pending CN115865390A (en) 2022-07-19 2022-07-19 Nonlinear information physical system filtering method and system under FDI attack

Country Status (1)

Country Link
CN (1) CN115865390A (en)

Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116753961A (en) * 2023-08-16 2023-09-15 中国船舶集团有限公司第七〇七研究所 Dynamic positioning ship high-speed tracking navigation method based on state observation

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116753961A (en) * 2023-08-16 2023-09-15 中国船舶集团有限公司第七〇七研究所 Dynamic positioning ship high-speed tracking navigation method based on state observation
CN116753961B (en) * 2023-08-16 2023-10-31 中国船舶集团有限公司第七〇七研究所 Dynamic positioning ship high-speed tracking navigation method based on state observation

Similar Documents

Publication Publication Date Title
Robertson et al. A moving horizon‐based approach for least‐squares estimation
Zhang et al. Bayesian Inference for State-Space Models With Student-$ t $ Mixture Distributions
Bailey et al. Chaos with confidence: asymptotics and applications of local Lyapunov exponents
CN115865390A (en) Nonlinear information physical system filtering method and system under FDI attack
Hu et al. Event-based input and state estimation for linear discrete time-varying systems
CN116149397B (en) Temperature self-adaptive control system for electronic grade hydrogen peroxide storage
Wills et al. Wiener system identification using the maximum likelihood method
Varziri et al. Selecting optimal weighting factors in iPDA for parameter estimation in continuous-time dynamic models
Beskos et al. Score-based parameter estimation for a class of continuous-time state space models
CN112948743B (en) Coal mine gas concentration deficiency value filling method based on space-time fusion
Huang et al. A federated dictionary learning method for process monitoring with industrial applications
Huang et al. A robust and efficient system identification method for a state-space model with heavy-tailed process and measurement noises
Sadhu et al. Central difference formulation of risk-sensitive filter
Ameen et al. A pseudo-spectral scheme for systems of two-point boundary value problems with file and right sided fractional derivatives and related integral equations
Yao et al. A network security situation prediction method through the use of improved TCN and BiDLSTM
Noack et al. Bounding linearization errors with sets of densities in approximate Kalman filtering
Detommaso et al. Stein variational online changepoint detection with applications to hawkes processes and neural networks
Han Risk sensitive FIR filters for stochastic discrete-time state space models
Yin et al. Tracking and identification of regime-switching systems using binary sensors
Harvey et al. Missing observations in dynamic econometric models: A partial synthesis
Yu et al. A parallel game model‐based intrusion response system for cross‐layer security in industrial internet of things
Coquelin et al. Particle filter-based policy gradient in POMDPs
Wang et al. Improving particle filter with better proposal distribution for nonlinear filtering problems
Techakesari et al. Practical stability of approximating discrete-time filters with respect to model mismatch
Nielsen et al. Estimating parameters in discretely, partially observed stochastic differential equations

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination