A remote state estimation method, system and storage medium based on neural network and nonlinear system under incomplete observation

Abstract

The present invention discloses a remote state estimation method, system and storage medium based on a neural network and a nonlinear system under incomplete observation, belonging to the field of network control. It aims to solve the problems that existing methods ignore the errors caused by linearization and the effects of signal attenuation and measurement outliers on the estimation accuracy of the state estimator. In the present invention, an amplify-and-forward repeater is arranged between the sensor and the estimator to amplify and forward the signal; and an outlier rejection algorithm is used to reject the outliers in the signal after forwarding; a neural network is introduced to approximate the nonlinear part in the system; the state estimation value, weight estimation value, upper bound of the state estimation error covariance matrix, upper bound of the weight estimation error covariance matrix, upper bound of the state covariance matrix and upper bound of the measurement covariance matrix at the previous moment are input; a recursive estimator based on the neural network is constructed, and an update model of the weights of the neural network is constructed; and remote state estimation of the nonlinear system based on the neural network is realized.

Description

Technical Field

[0001] The present invention belongs to the field of network control, and relates to a remote state estimation method, system and storage medium based on a neural network and a nonlinear system under incomplete observation. Background Art

[0002] The problem of state estimation for nonlinear systems, as a research hotspot in the fields of control engineering and signal processing, has always attracted the attention of many researchers. For different system models, they have developed a variety of efficient estimation algorithms. These algorithms often highly rely on a full understanding of the nonlinear characteristics when applied. However, in engineering practice, it is often extremely difficult to obtain complete information about the nonlinearity.

[0003] In addition, in practical engineering applications, the research on the problem of remote state estimation is more universal, and the main challenge it faces is to ensure the integrity of measurement information during the transmission process. And during the transmission process, incomplete observation phenomena such as measurement outliers and signal attenuation often occur, bringing certain difficulties to state estimation. On the one hand, the measurement signal may gradually reduce in power due to attenuation during the transmission process, thus seriously reducing the signal quality. On the other hand, due to the existence of unpredictable factors such as instrument failures, environmental mutations and communication pulse interferences, the measurements of sensors may deviate significantly from the normal range (referred to as measurement outliers). Summary of the Invention

[0004] The technical problem to be solved by the present invention is:

[0005] Existing methods ignore the errors caused by linearization and the occurrence of incomplete observation phenomena such as signal attenuation and measurement outliers, which affect the estimation accuracy of the state estimator.

[0006] The technical solution adopted by the present invention to solve the above technical problem:

[0007] The present invention provides a remote state estimation method for a nonlinear system based on a neural network and under incomplete observation, including the following steps:

[0008] Step 1, establish a dynamic mathematical model of the nonlinear system for long-distance transmission;

[0009] Step 2, set the initial values of the state and state estimation, the initial values of the state covariance matrix and the state estimation covariance matrix, the measurement initial value and the measurement covariance matrix initial value;

[0010] Step 3, set an amplify-and-forward repeater with known amplification factor and channel noise between the sensor and the estimator to amplify and forward the signal; and use an outlier rejection algorithm for the forwarded signal to reject outliers in the signal;

[0011] Step 4: Introduce the neural network to approximate the nonlinear part in the system, and set the initial value of weight estimation and the initial value of weight estimation covariance matrix;

[0012] Step 5: Input the state estimation value at time l at time k + 1 Weight estimation value Upper bound ∑ of state estimation error covariance matrix k , upper bound Γ of weight estimation error covariance matrix k , upper bound Ω of state covariance matrix k and upper bound Υ of measurement covariance matrix k ;

[0013] Step 6: Construct a neural network-based recursive estimator and construct an update model for the weights of the neural network; calculate the state estimation value, weight estimation value, upper bound matrix of state estimation error covariance, and upper bound matrix of weight estimation error covariance at time k + 1, solve the tuning parameters of the neural network and the gain matrix of the estimator, and realize the remote state estimation of the nonlinear system based on the neural network;

[0014] Step 7: Determine whether the value of k + 1 exceeds the total duration N. If it does not exceed, perform the operations in Step 5 and Step 6 at the next moment; otherwise, end.

[0015] Furthermore, for the dynamic mathematical model of the nonlinear system with long-distance transmission described in Step 1, its state space form is:

[0016]

[0017] where, represents the state vector of the networked system at time k, R represents the real number field, represents the n x -dimensional real space, is the measurement output of the system, w k is Gaussian noise with zero mean and variance Q k > 0, f(·) is a bounded nonlinear function, A k , C k and D k are known appropriate-dimensional matrices.

[0018] In Step 3, the amplify-and-forward repeater is used to transmit the measurement signal in the sensor-repeater channel, and the transmission model is defined as:

[0019]

[0020] where r k is the measurement signal received by the relay end, B1 is the diagonal matrix of the channel coefficients of the sensor-repeater channel, h1 is the transmission power; υ 1,kThe channel noise of the sensor-repeater channel, which is a Gaussian noise with zero mean and variance R l,k > 0;

[0021] The amplify-and-forward repeater amplifies and forwards the measurement signal in the repeater-estimator channel. The transmission model is defined as:

[0022]

[0023] where y k is the signal actually received by the estimator, a > 0 is the amplification factor, B2 is the diagonal matrix describing the channel coefficient of the repeater-estimator channel, h2 is the transmission power, and v 2，k is the channel noise of the repeater-estimator channel, which is a Gaussian noise with zero mean and variance R 2，k > 0;

[0024] According to (2) and (3), the value actually transmitted to the estimator after the measurement value passes through the amplify-and-forward repeater is:

[0025]

[0026] where

[0027] Furthermore, the process of using the outlier rejection algorithm to reject outliers in the signal described in step three includes the following process:

[0028] First, define a desired neighbor distance ξ k as:

[0029]

[0030] where δ k l is the proximity coefficient whose value satisfies , and s is the length of the sliding window;

[0031] And define the upper and lower bounds of the desired region according to the desired neighbor distance as follows:

[0032]

[0033] where is the value of the upper bound of the desired region, is the value of the lower bound of the desired region, and the desired region D k of the measurement value at time k λ d is the coefficient that determines the sensitivity of the outlier detection condition, and σ d，k is ||y k+1 - y kThe standard deviation of ||, l = {k - N, k - N + 1,..., k - 2});

[0034] Define a desired neighbor direction vector as:

[0035]

[0036] where m = 1, 2…n;

[0037] At the measurement time k≥N (N≥0) can be written as:

[0038]

[0039] For the m-th element in y k the upper and lower bounds of are defined as:

[0040]

[0041] where is the value of the upper bound of the desired direction vector region, is the value of the lower bound of the desired direction vector region, then the desired direction vector region at time k of the measurement value is is the coefficient that determines the sensitivity of the outlier detection condition, represents the standard deviation of;

[0042] First, calculate d at time k k = ||y k - y k-1 ||; If d k ∈D k it means that y k is a normal value and directly output y k ; Otherwise, it means that there is an outlier component in y k and it is necessary to further detect and process each component of y k ; Secondly, calculate the of each component at time k. If it means that is a normal component and directly output Otherwise, it means that is an outlier component and output

[0043] Let represent the actually received measurement, where represents the measurement value actually received from sensor m, and we can get:

[0044]

[0045] For the convenience of calculation, define the variable as:

[0046]

[0047] According to equations (5)-(9), the actual measurement value received by the estimator is obtained

[0048]

[0049] where

[0050] Furthermore, the neural network described in step four is:

[0051] f(x k ) = Uσ(x k ) + o k (11)

[0052] where U is the weight of the neural network, σ(x k ) is the activation function of the neural network, and o k is the approximation error;

[0053] Assume that U, σ(x k ) and o k satisfy the following conditions:

[0054]

[0055] where and are known positive scalars.

[0056] Furthermore, the neural network-based recursive estimator constructed in step six is:

[0057]

[0058] where is the estimate of x k at time k, is the estimate of U at time k, and K k is the gain of the estimator to be designed;

[0059] Define the cost function as:

[0060]

[0061] where

[0062] According to the cost function, the update model of the neural network weight is constructed as:

[0063]

[0064] wherein are the tuning parameters of the neural network.

[0065] Furthermore, the solving processes of the gain matrix of the estimator and the tuning parameters of the neural network in step six are as follows:

[0066] Define the weight estimation error and the state estimation error as follows:

[0067]

[0068] wherein

[0069]

[0070] Define the weight estimation error covariance matrix and the state estimation covariance matrix as follows:

[0071]

[0072] According to the results of (1) and (10), define the state covariance matrix and the measurement covariance matrix as follows:

[0073]

[0074] Given a positive scalar satisfying the initial conditions X0 ≤ Ω0, The solutions of the two recurrence equation matrices are the upper bounds Ω k+1 of the state covariance matrix and Υ k+1 of the measurement covariance matrix, respectively:

[0075]

[0076] wherein and I represents the identity matrix;

[0077] Given positive scalars μ i (i = 1, 2,..., 22), satisfying the initial conditions P0 ≤ Γ0, The solutions of the two recurrence equation matrices are the upper bounds Γ k+1 of the weight estimation error covariance matrix and ∑ k+1 of the state estimation error covariance matrix, respectively:

[0078]

[0079] wherein

[0080]

[0081] κ1 = (1μ1 + μ2 + μ3 + μ4 + μ5 + μ6),

[0082]

[0083] Furthermore, from formulas (24) and (25), we can obtain:

[0084]

[0085] Based on the results of formula (26) and (27), construct the estimated gain K k and the neural network tuning parameter matrix respectively as:

[0086]

[0087] where

[0088]

[0089]

[0090] The present invention provides a remote state estimation system for a non - linear system based on a neural network and under incomplete observation. The system has program modules corresponding to the steps of the method described in any one of the above - mentioned technical solutions, and when running, executes the steps in the above - mentioned remote state estimation method for a non - linear system based on a neural network and under incomplete observation.

[0091] The present invention provides a computer - readable storage medium. The computer - readable storage medium stores a computer program, and the computer program is configured to implement the steps in the above - mentioned remote state estimation method for a non - linear system based on a neural network and under incomplete observation when called by a processor.

[0092] Compared with the prior art, the beneficial effects of the present invention are:

[0093] The present invention proposes a remote state estimation method, system and storage medium for a non - linear system based on a neural network and under incomplete observation. It sets up an amplify - and - forward relay to improve the quality of measurement signals, uses an outlier rejection algorithm to improve the accuracy of measurement signals, and at the same time considers the errors and computational burden generated by linearization calculation, and introduces a neural network to approximate the non - linear part of the system. The constructed neural - network - based recursive state estimator can simultaneously ensure the performance index of minimizing both the weight estimation error covariance and the state estimation covariance. The estimation method in the present invention can more reflect the actual working mode of the remote state estimator for non - linear systems, and has higher estimation accuracy. This analysis method is convenient to solve and easy to implement. Brief Description of the Drawings

[0094] Figure 1 Flow chart of the remote state estimation method for a non - linear system based on a neural network and under incomplete observation in an embodiment of the present invention;

[0095] Figure 2 Flow chart of the operation of the amplify - and - forward repeater in an embodiment of the present invention;

[0096] Figure 3 Actual measurement values of the non - linear network system in an embodiment of the present invention

[0097] Figure 4 Actual measurement values of the non - linear network system in an embodiment of the present invention

[0098] Figure 5 Actual measurement values of the non - linear network system in an embodiment of the present invention

[0099] Figure 6 Curve of the state change of the non - linear network system in an embodiment of the present invention And the corresponding state estimation curve

[0100] Figure 7 Curve of the state change of the non - linear network system in an embodiment of the present invention And the corresponding state estimation curve

[0101] Figure 8 Curve of the state change of the non - linear network system in an embodiment of the present invention And the corresponding state estimation curve

[0102] Figure 9 Upper - bound curve and mean - square error curve of the state - estimation error covariance matrix in an embodiment of the present invention;

[0103] Figure 10 Estimation error curve in an embodiment of the present invention;

[0104] Figure 11 Curve of the norm of the approximation error of the non - linear function in an embodiment of the present invention. Detailed implementation manners

[0105] To enable those skilled in the art to better understand the solution of the present invention, the exemplary embodiments or examples of the present invention will be described below in conjunction with the accompanying drawings. Obviously, the described embodiments or examples are only part of the embodiments or examples of the present invention, rather than all of them. All other embodiments or examples obtained by those of ordinary skill in the art based on the embodiments or examples of the present invention without creative efforts shall fall within the scope of protection of the present invention.

[0106] To make the above objects, features and advantages of the present invention more obvious and understandable, the following detailed description will be given to the specific embodiments of the present invention in conjunction with the accompanying drawings.

[0107] Specific Embodiment 1: In combination with Figure 1 , the present invention provides a remote state estimation method for a nonlinear system based on a neural network and under incomplete observation, including the following steps:

[0108] Step 1: Establish a dynamic mathematical model of the nonlinear system for long-distance transmission;

[0109] Step 2: Set the initial values of the state and state estimation, the initial values of the state covariance matrix and the state estimation covariance matrix, the initial measurement value and the initial measurement covariance matrix;

[0110] Step 3: Set an amplification factor and an amplify-and-forward repeater with known channel noise between the sensor and the estimator to amplify and forward the signal; and use an outlier rejection algorithm to remove the outliers in the signal after forwarding;

[0111] Step 4: Introduce a neural network to approximate the nonlinear part in the system, and set the initial values of the weight estimation and the initial value of the weight estimation covariance matrix;

[0112] Step 5: Input the state estimation value at time k at time k + 1 weight estimation value upper bound ∑ of the state estimation error covariance matrix k upper bound Γ of the weight estimation error covariance matrix k upper bound Ω of the state covariance matrix k and upper bound Υ of the measurement covariance matrix k ;

[0113] Step 6: Construct a recursive estimator based on a neural network, and construct an update model for the weights of the neural network; calculate the state estimation value, weight estimation value, upper bound matrix of the state estimation error covariance, and upper bound matrix of the weight estimation error covariance at time k + 1, solve the tuning parameters of the neural network and the gain matrix of the estimator, and realize the remote state estimation of the nonlinear system based on the neural network;

[0114] Step 7: Determine whether the value of k + 1 exceeds the total duration N. If it does not exceed, perform the operations in Step 5 and Step 6 at the next moment; otherwise, end.

[0115] Specific Embodiment 2: The dynamic mathematical model of the non - linear system for long - distance transmission described in Step 1 has the following state - space form:

[0116]

[0117] where represents the state vector of the networked system at time k, R represents the real - number field, R nx represents n x dimensional real - number space, is the measurement output of the system, w k is Gaussian noise with zero mean and variance Q k > 0, f(·) is a bounded non - linear function, A k , C k and D k are known matrices of appropriate dimensions. Other parts of this embodiment are the same as those of Specific Embodiment 1.

[0118] Use and P0 to represent the initial value of the state and the initial error covariance matrix, so as to provide the initial value of the state at time step t = 0 and the quantification of the uncertainty of this estimated value. The two initial values are usually obtained according to prior information or previous measurements of the system, and the state estimator is initialized at the beginning stage of the filter algorithm.

[0119] Specific Embodiment 3: As Figure 2 shown, in Step 3, the amplification - and - forwarding repeater is used to transmit the measurement signal in the sensor - repeater channel. The transmission model is defined as:

[0120]

[0121] where r k is the measurement signal received by the relay end, B1 is the diagonal matrix of the channel coefficients of the sensor - repeater channel, h1 is the transmission power; v 1，k is the channel noise of the sensor - repeater channel, which is a Gaussian noise with zero mean and variance R 1，k > 0;

[0122] The amplification - and - forwarding repeater amplifies and forwards the measurement signal in the repeater - estimator channel. The transmission model is defined as:

[0123]

[0124] where y kFor the signal actually received by the estimator, a > 0 is the amplification factor, B2 is the diagonal matrix describing the channel coefficients of the repeater - estimator channel, h2 is the transmission power, and v 2，k is the channel noise of the repeater - estimator channel, which is a Gaussian noise with zero mean and variance R 2，k > 0;

[0125] According to (2) and (3), the value actually transmitted to the estimator after the measurement value passes through the amplify - and - forward repeater is:

[0126]

[0127] where Other parts of this implementation scheme are the same as those of the first specific implementation scheme.

[0128] Specific implementation scheme four: The process of using the outlier rejection algorithm to reject outliers in the signal in step three includes the following:

[0129] First, define a desired neighbor distance ξ k as:

[0130]

[0131] where δ k l is the proximity coefficient whose value satisfies and s is the length of the sliding window;

[0132] And define the upper and lower bounds of the desired region according to the desired neighbor distance as follows:

[0133]

[0134] where is the value of the upper bound of the desired region, is the value of the lower bound of the desired region, and the desired region D of the measurement value at time k k is λ d is the coefficient determining the sensitivity of the outlier detection condition, σ d，k is the standard deviation of ||y k+1 - y k ||, and l = {k - N, k - N + 1,..., k - 2}.

[0135] To increase the flexibility of the outlier rejection algorithm, two factors need to be carefully considered when determining λ d : the mutual interference between the filtering algorithm and the outlier rejection algorithm, and the necessary requirements for performance estimation in engineering.

[0136] Then, define a desired neighbor direction vector as:

[0137]

[0138] where m = 1, 2... n;

[0139] At the measurement time k ≥ N (N ≥ 0), can be written as:

[0140]

[0141] For the m-th element in y k the upper and lower bounds are respectively defined as:

[0142]

[0143] where is the value of the upper bound of the expected direction vector region, is the value of the lower bound of the expected direction vector region, then the expected direction vector region of the measurement value at time k is is the coefficient that determines the sensitivity of the outlier detection condition, denotes the standard deviation of

[0144] The core idea of the outlier rejection algorithm is to use the normal component corresponding to the previous moment of the measurement value to replace the abnormal component at the current moment. The specific process is as follows: First, calculate d at time k k = ||y k - y k-1 ||; if d k ∈ D k , it means that y k is a normal value, and directly output y k ; otherwise, it means that there is an outlier component in y k , and it is necessary to further detect and process each component of y k . Secondly, calculate the of each component at time k. If , it means that is a normal component, and directly output ; otherwise, it means that is an outlier component, and output

[0145] Let represent the actually received measurement, where represents the measurement value actually received from sensor m. According to the outlier rejection algorithm, we can get:

[0146]

[0147] For the convenience of calculation, define the variable as:

[0148]

[0149] According to equations (5)-(9), the actual measurement value received by the estimator is obtained

[0150]

[0151] where Other parts of this implementation scheme are the same as those of the first specific implementation scheme.

[0152] Specific implementation scheme five: According to the approximation property of the neural network, introduce a neural network to approximate the nonlinearity. The neural network described in step four is:

[0153] f(x k ) = Uσ(x k ) + o k (11)

[0154] where U is the weight of the neural network, σ(x k ) is the activation function of the neural network, and o k is the approximation error;

[0155] Use P0 U to represent the initial variance of the weight and the initial value of the weight estimation covariance matrix, so as to provide a quantification of the uncertainty of the weight estimation at time t = 0 and initialize the weight estimator at the beginning stage of the weight estimation algorithm.

[0156] Assume that U, σ(x k ) and o k satisfy the following conditions:

[0157]

[0158] where and are known positive scalars. Other parts of this implementation scheme are the same as those of the first specific implementation scheme.

[0159] Specific implementation scheme six: According to the optimization process from step one to step four, the following compact form of the nonlinear system is obtained:

[0160]

[0161] The neural network-based recursive estimator constructed in step six is:

[0162]

[0163] where is xk The estimate at time k is the estimate of U at time k, K k is the gain of the estimator to be designed;

[0164] According to the backpropagation property of the neural network, the cost function is defined as:

[0165]

[0166] where

[0167] According to the definition and properties of the cost function, the update model of the neural network weights is constructed as:

[0168]

[0169] where is the tuning parameter of the neural network. Other parts of this implementation scheme are the same as those of the first specific implementation scheme.

[0170] Specific implementation scheme seven: The solving process of the gain matrix of the estimator and the tuning parameter of the neural network in step six is as follows:

[0171] Define the weight estimation error and the state estimation error as:

[0172]

[0173] where

[0174]

[0175] Define the weight estimation error covariance matrix and the state estimation covariance matrix as:

[0176]

[0177] Design a neural network-based recursive estimator for a nonlinear system affected by incomplete observation phenomena, and ensure that both the state estimation error covariance and the weight estimation error covariance have an upper bound;

[0178] In addition, through the fully designed estimator gain and the tuning parameter of the neural network, minimize the upper bounds of both estimation error covariances;

[0179] According to the results of (1) and (10), define the state covariance matrix and the measurement covariance matrix as:

[0180]

[0181] Given a positive scalar satisfying the initial conditions X0 ≤ Ω0, The solutions of the two recurrence equation matrices are the upper bounds Ω of the state covariance matrix k+1 and the upper bounds Υ of the measurement covariance matrix k+1 respectively are:

[0182]

[0183] where and I represents the identity matrix;

[0184] Given positive scalars μ i (i = 1, 2,..., 22), satisfying the initial conditions P0 ≤ Γ0, the solutions of the two recurrence equation matrices, that is, the upper bounds Γ of the weight estimation error covariance matrix k+1 and the upper bounds Σ of the state estimation error covariance matrix k+1 respectively are:

[0185]

[0186] where

[0187]

[0188] κ1 = (1 + μ1 + μ2 + μ3 + μ4 + μ5 + μ6),

[0189]

[0190] Furthermore, from formulas (24) and (25), we can obtain:

[0191]

[0192] According to the results of (26) and (27), construct the estimation gain K k and the neural network tuning parameter matrix respectively are:

[0193]

[0194] where

[0195]

[0196] Other parts of this implementation scheme are the same as those of the sixth specific implementation scheme.

[0197] A remote state estimation method (algorithm) based on neural network and nonlinear system under incomplete observation proposed by the present invention is the underlying technical core of the present invention, and various products can be derived based on the said algorithm.

[0198] The method proposed based on the present invention uses a programming language to develop a remote state estimation system for a non-linear system based on a neural network and under incomplete observation. This system has program modules corresponding to the steps of the above technical solution, and when running, it executes the steps in the above remote state estimation method for a non-linear system based on a neural network and under incomplete observation.

[0199] Store the computer program of the developed system (software) on a computer-readable storage medium. The computer program is configured to implement the steps of the above remote state estimation method for a non-linear system based on a neural network and under incomplete observation when called by a processor. That is, the present invention is materialized on a carrier to become a computer program product.

[0200] The various embodiments of the systems and techniques described herein can be implemented in digital electronic circuit systems, integrated circuit systems, dedicated ASICs (application-specific integrated circuits), computer hardware, firmware, software, and / or combinations thereof. These various embodiments can include: implemented in one or more computer programs that can be executed and / or interpreted on a programmable system including at least one programmable processor. The programmable processor can be a dedicated or general-purpose programmable processor that can receive data and instructions from a storage system, at least one input device, and at least one output device, and transmit the data and instructions to the storage system, the at least one input device, and the at least one output device.

[0201] The computing programs (also referred to as programs, software, software applications, or code) in the present invention include machine instructions for a programmable processor, and these computing programs can be implemented using high-level procedural and / or object-oriented programming languages, and / or assembly / machine languages. As used herein, the terms "machine-readable medium" and "computer-readable medium" refer to any computer program product, device, and / or apparatus (e.g., disk, optical disk, memory, programmable logic device PLD) for providing machine instructions and / or data to a programmable processor, including a machine-readable medium that receives machine instructions as a machine-readable signal. The term "machine-readable signal" refers to any signal for providing machine instructions and / or data to a programmable processor.

[0202] Example 1

[0203] To prove the effectiveness of the method of the present invention, the following example is used to simulate and verify the method of the present invention.

[0204] The system parameters are selected as:

[0205]

[0206] Wherein

[0207]

[0208] The initial values of the state and the initial error covariance matrix are x0 = [10 60 19] T and P0 = 3I;

[0209] The relevant parameters of the neural network are selected as:

[0210]

[0211] The remaining parameters are selected as Q k = 0.01, s = 50, λ d = 1.2, B1 = 0.8, B2 = 0.005, a = 1.2, q1 = q2 = 1.5, R 1，k = R 2，k = 0.5. The root mean square error (MSE) is defined as: N = 300 is the number of independent experiments.

[0212] The simulation results are as Figures 3 to 11 shown. As can be seen from Figures 3 - 5 it, for the actual output of each component of the measurement value: when the curve value is 0, it means that the component is normal and the normal output; when the curve value is 1, it means that the component is a measurement outlier, and the value of the previous moment of this component is output. As can be seen from Figures 6 - 8 it, the blue curve is the actual state value of the system, and the red curve is the state value estimated by the neural network recursive estimator proposed by the present invention. It can be seen that the estimated curve can track the actual state curve, which is consistent with the theoretical analysis; as can be seen from Figure 9 it, the estimated error covariance matrix has an upper bound and the estimated error curve satisfies the mean square exponential boundedness; Figure 10 as can be seen from Figure 11 it, the error curve of the state estimation fluctuates gently near 0, which further shows that the estimator has good performance and accurate results. As can be seen from

[0213] The above has introduced in detail the remote state estimation method for a nonlinear system based on a neural network and incomplete observation provided by the present invention. Specific examples are used in this article to elaborate on the principle and implementation manner of the present invention. The description of the above embodiments is only used to help understand the method and its core idea of the present invention; at the same time, for those of ordinary skill in the art, according to the idea of the present invention, there will be changes in the specific implementation manner and application scope. In summary, the content of this specification should not be construed as a limitation to the present invention.

Claims

1. A remote state estimation method based on a neural network and a nonlinear system under incomplete observation, characterized in that It includes the following steps: Step 1: Establish a dynamic mathematical model of a non-linear system for long-distance transmission; Step 2: Set the initial values of the state and state estimation, the initial values of the state covariance matrix and the state estimation covariance matrix, the initial measurement value, and the initial measurement covariance matrix; Step 3: Set an amplification factor and an amplify-and-forward repeater with known channel noise between the sensor and the estimator to amplify and forward the signal; and use an outlier rejection algorithm to reject outliers in the signal after forwarding; Step 4: Introduce a neural network to approximate the non-linear part in the system, and set the initial values of the weight estimation and the initial value of the weight estimation covariance matrix; Step 5: Input the state estimate value at the k-th moment at the (k + 1)-th moment Weight estimate value Upper bound ∑ of the state estimation error covariance matrix k , upper bound Γ of the weight estimation error covariance matrix k , upper bound Ω of the state covariance matrix k and upper bound γ of the measurement covariance matrix k ; Step 6: Construct a recursive estimator based on a neural network, and construct an update model for the weights of the neural network; calculate the state estimation value, weight estimation value, upper bound matrix of the state estimation error covariance, and upper bound matrix of the weight estimation error covariance at time k + 1, solve the tuning parameters of the neural network and the gain matrix of the estimator, and realize the remote state estimation of the non-linear system based on the neural network; Step 7: Determine whether the value of k + 1 exceeds the total duration N. If it does not exceed, perform the operations of Step 5 and Step 6 at the next moment, otherwise end; The process of using the outlier rejection algorithm to reject outliers in the signal in Step 3 includes the following: Define a desired neighbor distance ξ k as follows: Among them, δ k l is the proximity coefficient whose value satisfies , s is the length of the sliding window, and y k is the value actually transmitted to the estimator; Define the upper and lower bounds of the expected region according to the expected neighbor distance, as follows: where is the value of the upper bound of the expected region, is the value of the lower bound of the expected region, then the expected region D of the measured value at time k k is λ d is the coefficient that determines the sensitivity of the wild value detection condition, σ d，k is the standard deviation of ||y k+1 -y k ||, l = {k - N, k - N + 1,..., k - 2}; Define a desired neighbor direction vector as follows: where m = 1, 2... n; At measurement time k≥N (N≥0), can be written as: For the n-th element in y k the upper and lower bounds of are defined as follows: Among them, is the value of the upper bound of the expected direction vector region, is the value of the lower bound of the expected direction vector region. Then, the expected direction vector region of the measurement value at time k is is the coefficient that determines the sensitivity of the outlier detection condition, denotes the standard deviation of First, calculate d at time k k = ||y k - y k-1 ||; if d k ∈ D k , it means that y k is a normal value, and directly output y k ; otherwise, it means that there is an outlier component in y k , and it is necessary to further detect and process each component of y k ; Secondly, calculate the of each component at time k. If , it means that is a normal component, and directly output Otherwise, it means that is an outlier component, and output Let represent the actual measurement value received by the estimator, where represents the measurement value actually received from sensor m, and we can obtain: For the convenience of calculation, define the variable as follows: According to equations (5)-(9), obtain the actual measurement value received by the estimator Among them The recursive estimator based on the neural network constructed in Step 6 is: Among them, A k , C k are known dimension-adaptive matrices, K k is the gain of the estimator to be designed, a > 0 is the amplification factor, h1 is the transmission power, h2 is the transmission power, B1 is the diagonal matrix of the channel coefficients of the sensor-relay channel, B2 is the diagonal matrix of the channel coefficients describing the relay-estimator channel, σ(x k ) is the activation function of the neural network; Define the cost function as: Among them Construct an update model for the weights of the neural network according to the cost function as: Among them are the tuning parameters of the neural network.

2. The remote state estimation method for a non-linear system based on a neural network and under incomplete observation according to claim 1, characterized in that For the dynamic mathematical model of the non-linear system for long-distance transmission in Step 1, its state space form is: wherein, represents the state vector of the networked system at time k, R represents the real number field, represents n x dimensional real space, is the measurement output of the system, w k is Gaussian noise with zero mean and variance Q k > 0, f(·) is a bounded nonlinear function, D k is a known matrix of appropriate dimension.

3. The remote state estimation method for a non-linear system based on a neural network and under incomplete observation according to claim 2, wherein In Step 3, use the amplify-and-forward repeater to transmit the measurement signal in the sensor-repeater channel, and define the transmission model as: where r k is the measurement signal received by the relay, and v 1,k is the channel noise of the sensor-relay channel, which is a Gaussian noise with zero mean and variance R 1,k > 0; The amplify-and-forward repeater amplifies and forwards the measurement signal in the repeater-estimator channel, and defines the transmission model as: where v 2,k is the channel noise of the repeater - estimator channel, which is a Gaussian noise with zero mean and variance R 2,k > 0; According to (2) and (3), the value actually transmitted to the estimator after the measurement value passes through the amplify-and-forward repeater is: Among them 4. The remote state estimation method for a non-linear system based on a neural network and under incomplete observation according to claim 3, characterized in that, The neural network in Step 4 is: f(x k ) = Uσ(x k ) + o k (11) where U is the weight of the neural network, and o k is the approximation error; Suppose U, σ(x k ) and o k satisfy the following conditions: wherein and are known positive scalars.

5. The remote state estimation method for a non-linear system based on a neural network and under incomplete observation according to claim 4, characterized in that, The solution process of the gain matrix of the estimator and the tuning parameters of the neural network in Step 6 is: Define the weight estimation error and the state estimation error respectively as: where Define the weight estimation error covariance matrix and the state covariance matrix respectively as: According to the results of (1) and (10), define the state covariance matrix and the measurement covariance matrix respectively as: Given positive scalars satisfying the initial condition X0 ≤ Ω0,[[]] the solutions of two recurrence equation matrices are the upper bounds Ω of the state covariance matrix k+1 and the upper bound γ of the measurement covariance matrix k+1 which are respectively:[[]] Among them and I represents the identity matrix; Given a positive scalar μ i (i = 1, 2, …, 22), satisfying the initial conditions P0 ≤ Γ0, The solutions of two recurrence equation matrices, that is, the upper bounds Γ k+1 of the weight estimation error covariance matrix and Σ k+1 of the state estimation error covariance matrix are respectively: where κ1 = (1 + μ1 + μ2 + μ3 + μ4 + μ5 + μ6), Furthermore, through formulas (24) and (25), it can be obtained that: Construct the estimated gain K according to the results of equations (26) and (27). k and the neural network tuning parameter matrix respectively as follows: where 6. A remote state estimation system based on a neural network and a nonlinear system under incomplete observation, characterized in that, This system has program modules corresponding to the steps of the method described in any one of the above claims 1 to 5, and executes the steps in the above-mentioned remote state estimation method of the non-linear system based on the neural network and incomplete observation when running.

7. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program, and the computer program is configured to implement the steps in the remote state estimation method of the non-linear system based on the neural network and incomplete observation described in any one of claims 1 to 5 when called by a processor.

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