Asynchronous sampling rate distributed optimization state estimation method under random topology

Abstract

This invention discloses a distributed optimization state estimation method for asynchronous sampling rates under stochastic topology. The method is as follows: 1. Establish an asynchronous sampling rate time-varying nonlinear dynamic model with different system state update rates and measurement sampling rates; 2. Transform the dynamic model into a single-rate time-varying nonlinear dynamic model; 3. Design a state estimator under stochastic topology; 4. Calculate the state estimator at q... k The gain matrix K of the estimator at time step i (q k ) and G i (q k ); V. K i (q k ) and G i (q k Substituting this into the estimator, we obtain q. k+1 VI. Calculate the upper bound of the one-step prediction error covariance matrix; VII. [The text abruptly ends here, likely due to an incomplete sentence or missing information.] i (q k ) and G i (q k Substitute the upper bound of the one-step prediction error covariance matrix into the formula to calculate q. k+1 The minimum upper bound of the one-step prediction error covariance matrix at time step q; let q k =q k+1 Execute step three until q is satisfied. k+1 =K. This invention solves the problem that existing state estimation methods cannot simultaneously handle distributed optimization state estimation problems with stochastic nonlinearity and asynchronous sampling rates under stochastic topologies.

Description

Technical Field

[0001] This invention relates to a distributed optimization state estimation method, specifically a distributed optimization state estimation method with asynchronous sampling rate under a random topology. Background Technology

[0002] Multi-rate sampling and sensor networks are important research areas in networked control systems, and have been widely applied in fields such as the electrolytic aluminum industry, smart grids, and structural health monitoring.

[0003] In previous research on state estimation in networked systems, researchers generally assumed that all components sampled at the same rate. However, since networked systems typically contain various components with different physical properties, it is difficult to sample different components at the same rate. Therefore, implementing multi-rate sampling is beneficial for improving system performance while minimizing resource consumption. Furthermore, the Gilbert-Elliott channel plays a crucial role in state estimation research with asynchronous sampling rates under stochastic topologies, because in real-world networked systems, sensor nodes and their neighbors do not necessarily cooperate according to a fixed network topology. Distributed optimization state estimation methods combining stochastic topologies and asynchronous sampling rates will play a significant role in improving network random packet loss and sensor node rearrangement. Summary of the Invention

[0004] The purpose of this invention is to provide a distributed optimization state estimation method for asynchronous sampling rate under random topology, which solves the problem that existing estimation methods cannot comprehensively consider the random topology and asynchronous sampling rate in state estimation.

[0005] The objective of this invention is achieved through the following technical solution:

[0006] A distributed optimization state estimation method with asynchronous sampling rate under random topology includes the following steps:

[0007] Step 1: Establish a time-varying nonlinear dynamic model with asynchronous sampling rates where the system state update rate and measurement sampling rate are different.

[0008] x(s k+1 )=A(s k )x(s k )+f(x(s k ),η(s k ))+B(s k )w(s k )

[0009] y i (q k ) = Ci (q k )x(q k )+D i (q k )v i (q k )

[0010] Where x(s) k ) for s k The state variables of a time-varying system; x(s) k+1 ) for s k+1 The state variables of a time-varying system; y i (q k Let ) be the node i in q k The measurement output of the time system; A(s) k ) for s k The system matrix of the time-series system; B(s) k ) for s k The noise figure matrix of the time-matter system; C i (q k Let ) be the node i in q k Measurement matrix of the time system; D i (q k Let ) be the node i in q k The noise distribution matrix is ​​measured at any given time; w(s) k ) is s k The system at time v is Gaussian white noise; i (q k Let ) be the node i in q k Time measurement using Gaussian white noise; f(x(s) k ),η(s k )) is a random nonlinear function;

[0011] Step 2: Transform the dynamic model from Step 1 into a single-rate time-varying nonlinear dynamic model:

[0012]

[0013] In the formula,

[0014]

[0015]

[0016]

[0017]

[0018]

[0019]

[0020] Where f(x(q) k ),η(q k )) is a stochastic nonlinear function, where I represents the identity matrix, Π represents the quadrature symbol, diag{·} represents a diagonal matrix composed of "·" elements, and col{·} represents a column vector composed of "·" elements; h = s k+1 -s k It is the state update cycle; bh = q k+1 -q k It is the uniform sampling period of the sensor;

[0021] Step 3: For the time-varying nonlinear dynamic model in Step 2, a state estimator under a stochastic topology is designed using the Gilbert-Elliott channel. The specific steps are as follows:

[0022] Step 3.1 Assume {τ} ij (q k )} k≥0 It is a Markov chain used to describe the communication link between sensor nodes i and j, where 1≤i≤N, 1≤j≤N, N is the number of sensor nodes, and τ ij (q k ) = 1 indicates that sensor node i can be in q k The output measurement of node j is received at any time, while τ ij (q k If ) = 0, then the opposite is true; τ is defined as follows: ii (q k ) = 1; the set of transition probabilities is as follows:

[0023] Prob{τ ij (q k )=1|τ ij (q k-1 )=1}=α ij ,

[0024] Prob{τ ij (q k )=0|τ ij (q k-1 )=1}=1-α ij ,

[0025] Prob{τ ij (q k )=1|τ ij (q k-1 )=0}=β ij ,

[0026] Prob{τ ij (q k )=0|τ ij (qk-1 )=0}=1-β ij ,

[0027] Where α ij ,β ij ∈[0,1], with an initial distribution of Prob{τ ij (q0)=0}=α0 and Prob{τ ij (q0)=1}=1-α0, α0∈[0,1], Prob{·} represents the probability of corresponding to “·”;

[0028] Step 3.2, Given y(q) k )=col{y1(q k )y2(q k ),...,y N (q k )}have

[0029]

[0030] In the formula:

[0031]

[0032]

[0033]

[0034] Step 33: Based on the given network topology, for node i, the actual input to the corresponding estimator is as follows:

[0035]

[0036] Γ i (q k )=diag{τ i1 (q k )I,τ i2 (q k )I,...,τ iN (q k )I}

[0037] Steps 3 and 4: Construct the following estimator:

[0038]

[0039] In the formula, For node i in q k+1 The estimated value of time, K i (q k ) and G i (q k ) is the estimator gain matrix to be determined;

[0040] Step 4: Calculate the estimator in q k The gain matrix K of the estimator at time step i (q k ) and G i (q k ):

[0041]

[0042]

[0043]

[0044] In the formula,

[0045] Υ 1i (q k )=(1+ε1 -1 (q k ))Ξ i (q k )+(1+ε1(q k ))X(q k )

[0046]

[0047]

[0048]

[0049]

[0050] Among them Υ 1i (q k ), Υ 2i (q k ), Υ 3i (q k ), Υ 4i (q k ) and Υ 5i (q k All of them are real-valued matrices; Represents the block matrix [G] i1 … G in ]; and They are removing Υ 4i (q k The zero column and removing Υ 3i (q k The resulting matrix is ​​simplified by using the zeroth row and zeroth column of the matrix; This indicates that from the matrix Z = [Z1 Z2 … Z…] m All of the above satisfy h il The submatrix Z extracted from l that is not equal to 0l ;X(s) k+1 ) is the upper bound matrix of the second-order moment of the state of node i; Ξ i (q k Let ) be the node i in q k The upper bound of the one-step prediction error covariance matrix at time t;

[0051] Step 5: Place K i (q k ) and G i (q k Substituting this into the estimator from step three, we obtain q. k+1 State estimator at time 1

[0052]

[0053] In the formula, and These are nodes i in q. k+1 Time and q k State estimation variables at time 1;

[0054] Step 6: Based on the state estimator designed in Step 3, calculate the upper bound of the one-step prediction error covariance matrix. The specific steps are as follows:

[0055] Step 61: Based on the system's state equations, calculate the upper bound matrix X(s) of the second-order moments of node i's state. k+1 ):

[0056]

[0057] Where tr{·} represents the trace of the variable "·"; A T (s k ) and B T (s k ) respectively represent matrix A(s) k ) and B(s k The transpose of Ω; i (s k ) and Φ i (s k Let be a known matrix of appropriate dimension, and let a be a known positive integer;

[0058] Step 62: Calculate the upper bound Ξ of the one-step prediction error covariance matrix according to the following formula. i (q k+1 ):

[0059]

[0060]

[0061]

[0062]

[0063]

[0064] Among them, Ξ i (q k+1 ) is q k+1 Upper bound of the time-estimation error covariance matrix; X(q) k ) is q k The state covariance matrix of the system at time ε1(q) k ) is a positive scalar; ε1 -1 (q k ) is ε1(q k The inverse of ) K i T (q k )and They are respectively K i (q k )and The superscript "T" indicates the transpose of the matrix; the superscript "-1" indicates the inverse of the matrix.

[0065] Step 7: Estimator gain matrix K i (q k ) and G i (q k Substituting the upper bound of the one-step prediction error covariance matrix, calculate q. k+1 The minimum upper bound of the one-step prediction error covariance matrix at time step q; let q k =q k+1 Proceed to step three until q is satisfied. k+1 =K.

[0066] Compared with the prior art, the present invention has the following advantages:

[0067] 1. This invention considers the impact of random topology and random nonlinearity on the performance of system state estimation. In addition, the distributed optimization state estimation method has a recursive form, so the method has the advantages of being easy to solve and implement.

[0068] 2. This invention solves the problem that existing state estimation methods cannot simultaneously handle distributed optimization state estimation problems with stochastic nonlinearity and asynchronous sampling rates under random topology structures, thereby improving the accuracy of estimation performance for such problems. As can be seen from the simulation diagram, with the increase in the connection probability α between nodes... ij and β ij As α increases, the state estimation fits better and better, indicating that α ij and βij The larger the value, the better the performance of the asynchronous sampling rate distributed optimization state estimation method under random topology, which further verifies the feasibility and effectiveness of the distributed optimization state estimation method proposed in this invention. Attached Figure Description

[0069] Figure 1 This is a flowchart of the asynchronous sampling rate distributed optimization state estimation method under random topology structure according to the present invention.

[0070] Figure 2 It is α ij =α0=β ij The actual state trajectory x1(q) of the sensor network constructed when β0 = 0.9 k ) and x2(q k The state estimation trajectory after passing through the first estimator designed in this invention. and (Scenario 1)

[0071] Figure 3 It is α ij =α0=β ij The actual state trajectory x1(q) of the sensor network constructed when β0 = 0.9 k ) and x2(q k The state estimation trajectory after passing through the second estimator designed in this invention. and (Scenario 1)

[0072] Figure 4 It is α ij =α0=β ij The actual state trajectory x1(q) of the sensor network constructed when β0 = 0.9 k ) and x2(q k The state estimation trajectory after passing through the third estimator designed in this invention. and (Scenario 1)

[0073] Figure 5 It is α ij =α0=β ij When β0 = 0.9, the upper bound of the error covariance matrix and the mean square error of the last three nodes after passing through the estimator designed in this invention (Case 1).

[0074] Figure 6 It is α ij =α0=β ij The actual state trajectory x1(q) of the sensor network constructed when β0 = 0.1 k ) and x2(q kThe state estimation trajectory after passing through the first estimator designed in this invention. and (Scenario 2)

[0075] Figure 7 It is α ij =α0=β ij The actual state trajectory x1(q) of the sensor network constructed when β0 = 0.1 k ) and x2(q k The state estimation trajectory after passing through the second estimator designed in this invention. and (Scenario 2)

[0076] Figure 8 It is α ij =α0=β ij The actual state trajectory x1(q) of the sensor network constructed when β0 = 0.1 k ) and x2(q k The state estimation trajectory after passing through the third estimator designed in this invention. and (Scenario 2)

[0077] Figure 9 It is α ij =α0=β ij When β0 = 0.1, the upper bound of the error covariance matrix and the mean square error of the last three nodes after passing through the estimator designed in this invention (case two). Detailed Implementation

[0078] The technical solution of the present invention will be further described below with reference to the accompanying drawings, but it is not limited thereto. Any modifications or equivalent substitutions to the technical solution of the present invention that do not depart from the spirit and scope of the technical solution of the present invention should be covered within the protection scope of the present invention.

[0079] This invention provides a distributed optimization state estimation method with asynchronous sampling rate under random topology, such as... Figure 1 As shown, the method includes the following steps:

[0080] Step 1: Establish a time-varying nonlinear dynamic model with different asynchronous sampling rates for system state update rate and measurement sampling rate.

[0081] In this step, the state space description of the asynchronous sampling rate time-varying nonlinear dynamic model is as follows:

[0082] x(s k+1 )=A(s k )x(s k )+f(x(s k),η(s k ))+B(s k )w(s k )

[0083] y i (q k ) = C i (q k )x(q k )+D i (q k )v i (q k )

[0084] Where x(s) k ) for s k The system state variables of a time-varying system; x(s) k+1 ) for s k+1 The state variables of a time-varying system; y i (q k Let ) be the node i in q k The measurement output of the time system; A(s) k ) for s k The system matrix of the time-series system; B(s) k ) for s k The noise figure matrix of the time-matter system; C i (q k Let ) be the node i in q k Measurement matrix of the time system; D i (q k Let ) be the node i in q k The noise distribution matrix is ​​measured at any given time; w(s) k ) is s k The expected value at any given time is zero, and the variance is W(s). k The system contains Gaussian white noise; v i (q k Let ) be the node i in q k The expected value at any given time is zero, and the variance is R. i (q k Measurement of Gaussian white noise; f(x(s) k ),η(s k )) is a random nonlinear function satisfying f(0,η(s) k )) = 0 and the following properties:

[0085] E{f(x(sk),η(sk))|x(sk)}=0

[0086]

[0087] Where a is a known positive integer; x T (s k) is x(s k The transpose of f; T (x(s j ),η(s j f(x(s)) is f(x(s) j ),η(s j The transpose of )); E{·} represents the expectation of the corresponding matrix; Ω i (s k ) and Φ i (s k (i = 1, 2, ..., a) is a known matrix with appropriate dimensions; δ(·) is the Kronecker function; ∑ is the summation symbol.

[0088] Step 2: Transform the dynamic model from Step 1 into a single-rate time-varying nonlinear dynamic model.

[0089] Since the system's state update period and the sensor's sampling period can differ, the multi-rate system is iteratively converted into a single-rate system for later analysis.

[0090]

[0091] In the formula,

[0092]

[0093]

[0094]

[0095]

[0096]

[0097]

[0098] Where f(x(q) k ),η(q k )) is a stochastic nonlinear function, where I represents the identity matrix, Π represents the quadrature symbol, diag{·} represents a diagonal matrix composed of "·" elements, and col{·} represents a column vector composed of "·" elements. h=s k+1 -s k It is a state update cycle and h > 0; bh = q k+1 -q k b is the uniform sampling period of the sensor and b is a positive integer; and we assume that there are a total of N sensor nodes, where N is a positive integer.

[0099] Step 3: For the time-varying nonlinear dynamic model in Step 2, a state estimator under a stochastic topology is designed using the Gilbert-Elliott channel.

[0100] Because this sensor network has a time-varying topology, which is connected via Gilbert-Elliott channels and controlled by a set of Markov chains, we first assume {τ} ij (q k )} k≥0 (1≤i,j≤N) is a Markov chain used to describe the communication link between sensor nodes i and j, where τ ij (q k ) = 1 indicates that sensor node i can be in q k The output measurement of node j is received at any time, while τ ij (q k If τ = 0, then the opposite is true. Furthermore, it is stipulated that τ... ii (q k = 1. The set of transition probabilities is as follows:

[0101] Prob{τ ij (q k )=1|τ ij (q k-1 )=1}=α ij ,

[0102] Prob{τ ij (q k )=0|τ ij (q k-1 )=1}=1-α ij ,

[0103] Prob{τ ij (q k )=1|τ ij (q k-1 )=0}=β ij ,

[0104] Prob{τ ij (q k )=0|τ ij (q k-1 )=0}=1-β ij ,

[0105] Where α ij ,β ij ∈[0,1], with an initial distribution of Prob{τ ij (q0)=0}=α0 and Prob{τ ij (q0)=1}=1-α0,α0∈[0,1];Prob{·} represents the probability of corresponding to “·”.

[0106] Given y(q) k )=col{y1(q k )y2(qk ),...,y N (q k )}have

[0107]

[0108] In the formula:

[0109]

[0110]

[0111]

[0112] Given the network topology, for node i (1≤i≤N), the actual input to its estimator is as follows:

[0113]

[0114] Γ i (q k )=diag{τ i1 (q k )I,τ i2 (q k )I,...,τ iN (q k )I}

[0115] Finally, the following estimator is constructed:

[0116]

[0117] In the formula, For node i in q k+1 The estimated value of time, K i (q k ) and G i (q k ) is the estimator gain matrix to be determined.

[0118] Step 4: Calculate the estimator in q k The gain matrix K of the estimator at time step i (q k ) and G i (q k ).

[0119] In this step, q is calculated according to the following formula. k The gain matrix K of the estimator at time step i (q k ) and G i (q k ):

[0120]

[0121]

[0122]

[0123] In the formula,

[0124] Υ 1i (q k )=(1+ε1 -1 (q k ))Ξ i (q k )+(1+ε1(q k ))X(q k )

[0125]

[0126]

[0127]

[0128]

[0129]

[0130]

[0131] Among them Υ 1i (q k ), Υ 2i (q k ), Υ 3i (q k ), Υ 4i (q k ) and Υ 5i (q k All of them are real-valued matrices; Represents the block matrix [G] i1 … …G in ]; and They are removing Υ 4i (q k The zero column and removing Υ 3i (q k The resulting matrix is ​​simplified by using the zeroth row and zeroth column of the matrix; This indicates that from the matrix Z = [Z1 Z2 … Z…] m All of the above satisfy h il The submatrix Z extracted from l that is not equal to 0 l ;X(s) k+1 ) is the upper bound matrix of the second-order moment of the state of node i; Ξ i (qk Let ) be the node i in q k The upper bound of the one-step prediction error covariance matrix at time t; the other matrices are the same as above.

[0132] Step 5: Place K i (q k ) and G i (q k Substituting this into the estimator from step three, we obtain q. k+1 State estimator at time 1

[0133] Based on the estimator gain matrix K obtained in step four i (q k ) and G i (q k ), calculate state estimation variables

[0134]

[0135] In the formula, and These are nodes i in q. k+1 Time and q k State estimate variables at time; K i (q k ) and G i (q k ) is the estimator gain to be determined.

[0136] Step 6: Based on the state estimator designed in Step 3, calculate the upper bound of the one-step prediction error covariance matrix.

[0137] First, based on the system's state equations, calculate the upper bound matrix X(s) of the second-order moments of node i's state. k+1 ):

[0138]

[0139] Where tr{·} represents the trace of the variable "·"; A T (s k ) and B T (s k ) respectively represent matrix A(s) k ) and B(s k The transpose of ).

[0140] The upper bound Ξ of the one-step prediction error covariance matrix is ​​calculated using the following formula. i (q k+1 ):

[0141]

[0142]

[0143]

[0144] Among them, Ξ i (q k+1 ) is q k+1 Upper bound of the time-estimation error covariance matrix; X(q) k ) is q k The state covariance matrix of the system at time ε1(q) k ) is a positive scalar; ε1 -1 (q k ) is ε1(q k The inverse of ) K i T (q k )and They are respectively K i (q k )and The transpose of the matrix; the superscript "T" indicates the transpose of the matrix; the superscript "-1" indicates the inverse of the matrix; the other matrices are the same as above.

[0145] Step 7: Take q from step 4 k The gain matrix K of the estimator at time step i (q k ) and G i (q k Substituting the upper bound of the one-step prediction error covariance matrix from step three, calculate q. k+1 The minimum upper bound of the one-step prediction error covariance matrix at time step q; let q k =q k+1 Proceed to step three until q is satisfied. k+1 =K.

[0146] Substituting the estimator gain matrix into the upper bound of the one-step prediction error covariance Ξ i (q k+1 Find an expression that satisfies: The one-step prediction error covariance matrix, For node i in q k+1 The time estimation error, It is e i (q k E{·} is the transpose of ∠E(·), where E{·} represents the expected value of the random variable ·.

[0147] Example:

[0148] The system parameters set for simulation using the method described in this invention are as follows:

[0149]

[0150]

[0151] C1(q k )=[2+3cos(q k ) 2+3sin(q k )],

[0152] C2(q k )=[2+2sin(q k ) 3+3cos(q k )],

[0153] C3(q k )=[2+1sin(q k 3+6cos(q) k )],

[0154] D1(q k )=12+2sin(q k ),

[0155] D2(q k )=10+2sin(q k ),

[0156] D3(q k )=12+2cos(q k ),

[0157]

[0158] Where η1(s) k ) and η2(s k ) is zero-mean white Gaussian noise with a variance of 0.002, a = 1, Π1(s k = [0.10.2] T [0.1 0.2], Φ1(s k = diag{0.09, 0.25}.

[0159] Other simulation parameters and initial values ​​are selected as follows:

[0160] W(s k ) = 0.1, R1(s k ) = 0.06, R2(s k R3(s) = 0.08 k )=0.07,ε1(q k ) = 1.5, b = 2, h = 1.

[0161] Simulation results: Figure 2 , Figure 3 , Figure 4 The following are given when α ij =α0=0.9,β ij When β0 = 0.9, the actual state trajectories of the three nodes in the sensor network and the state estimation trajector after the estimation by the estimator designed in this invention are compared. It can be seen from the curve comparison in the figure that the estimator designed in this invention can obtain accurate state estimation trajectories. Figure 5 In the figure, MSE represents the mean squared error of the estimate, and Upperbound represents the upper bound of the mean squared error of the estimate. The figures show the values ​​when α... ij =α0=0.9,β ij The comparison diagram shows the mean square error of the estimation of the three nodes of the sensor network after passing through the estimator designed in this invention when β0 = 0.9, and the corresponding upper bound. The experimental results verify the effectiveness of the proposed method. Figure 6 , Figure 7 , Figure 8 The following are given when α ij =α0=0.1,β ij When β0 = 0.1, the actual state trajectories of the three nodes in the sensor network under another connection probability are compared with the state estimation trajector after the estimation by the estimator designed in this invention. Figure 9 In the figure, MSE represents the mean squared error of the estimation, and the upper bound represents the upper limit of the mean squared error of the estimation. The figures show the values ​​when α... ij =α0=0.1,β ij A comparison of the mean square error of the estimation of the three nodes of the sensor network after passing through the estimator designed in this invention when β0 = 0.1, and the corresponding upper bound. Figure 6 , Figure 7 , Figure 8 Mid-curve and Figure 2 , Figure 3 , Figure 4 By comparing the curves, it can be seen that when the connection probability α... ij and β ij After the addition, the state estimation effect of the estimator designed in this invention is significantly improved, which is consistent with the actual situation. The experimental results verify the effectiveness of the proposed method.

Claims

1. A distributed optimization state estimation method with asynchronous sampling rate under random topology, characterized in that... The method includes the following steps: Step 1: Establish a time-varying nonlinear dynamic model with asynchronous sampling rates where the system state update rate and measurement sampling rate are different; Step 2: Transform the dynamic model from Step 1 into a single-rate time-varying nonlinear dynamic model; Step 3: For the time-varying nonlinear dynamic model in Step 2, a state estimator under a stochastic topology is designed using the Gilbert-Elliott channel. The specific steps are as follows: Step 3.1 Assumption It is used to describe sensor nodes and A Markov chain for communication links between them, where N is the number of sensor nodes. Represents sensor nodes It is possible Receive node at all times The output measurement, and Conversely; regulations The set of transition probabilities is as follows: in The initial distribution is and , , Represents the corresponding " The probability of "". Step 32, Given have In the formula: in For nodes exist Measurement output of the time system; For nodes exist Measurement matrix of the time system; For nodes exist Measure the noise distribution matrix at any time; For nodes exist Time measurement using Gaussian white noise; Step 33: Based on the given network topology, for the nodes... The actual input to the corresponding estimator is as follows: Steps 3 and 4: Construct the following estimator: In the formula, For nodes exist The estimated value at time, and It is the estimator gain matrix to be determined; Step 4: Calculate the estimator in Estimator gain matrix at time and ; Step 5: and Substituting into the estimator in step three, we obtain State estimator at time 1 ; Step 6: Based on the state estimator designed in Step 3, calculate the upper bound of the one-step prediction error covariance matrix; Step 7: Estimator gain matrix and Substituting the upper bound of the one-step prediction error covariance matrix, calculate The minimum upper bound of the one-step prediction error covariance matrix at time step; let Proceed to step three until the condition is met. .

2. The asynchronous sampling rate distributed optimization state estimation method under random topology according to claim 1, characterized in that... In step one, the time-varying nonlinear dynamic model of the asynchronous sampling rate is as follows: in, for State variables of a time-varying system; for State variables of a time-varying system; for The system matrix of the time-series system; for The noise figure matrix of the time-matter system; yes The system at any given time is subject to Gaussian white noise. It is a random nonlinear function.

3. The asynchronous sampling rate distributed optimization state estimation method under random topology according to claim 2, characterized in that... The satisfy and the following properties: in Given a positive integer; yes Transpose of; yes Transpose of; This represents the expectation of the corresponding matrix; It is the Kronecker function; It is the summation symbol.

4. The asynchronous sampling rate distributed optimization state estimation method under random topology according to claim 2, characterized in that... In step two, the time-varying nonlinear dynamic model for a single rate is as follows: In the formula, in Represents the identity matrix. Represents the quadrature symbol. Represents the element " The diagonal matrix formed by these elements Represents the element " A column vector formed by "; It is the state update cycle; It is the uniform sampling period of the sensor.

5. The asynchronous sampling rate distributed optimization state estimation method under random topology according to claim 1, characterized in that... In step four, the estimator gain matrix is ​​calculated as follows. and : In the formula, in , , , and They are all real-valued matrices; Represents a block matrix ; and They are to remove Zero column and remove The resulting matrix is ​​simplified by using zero rows and zero columns; Indicates from matrix All of the above satisfy of Extracted submatrix ; For nodes The upper bound matrix of the second-order moments of the state; For nodes exist The upper bound of the one-step prediction error covariance matrix at time t.

6. The asynchronous sampling rate distributed optimization state estimation method under random topology according to claim 1, characterized in that... In step five, the state estimation variables are calculated according to the following formula. : In the formula, and These are nodes exist Time and State estimation variables at time 1.

7. The asynchronous sampling rate distributed optimization state estimation method under random topology according to claim 1, characterized in that... The specific steps of step six are as follows: Step 61: Calculate the nodes based on the system's state equations. Upper bound matrix of the second moment of state : in Indicates the variable " Seeking traces; and Represent matrices respectively and Transpose of; and Given a matrix of appropriate dimension, Given a positive integer; Step 62: Calculate the upper bound of the one-step prediction error covariance matrix according to the following formula. : in, for The upper bound of the time-estimation error covariance matrix; for The state covariance matrix of the system at any given time; It is a positive scalar; for The reverse; , and They are respectively , and transpose; superscript " " indicates taking the transpose of the matrix; superscript " " indicates taking the inverse of the matrix.

8. The asynchronous sampling rate distributed optimization state estimation method under random topology according to claim 1, characterized in that... In step seven, the estimator gain matrix is ​​substituted into the upper bound of the one-step prediction error covariance. An expression that satisfies: , The one-step prediction error covariance matrix, For nodes exist The time estimation error, yes transpose, Represents a random variable The mathematical expectation of “”.

Images