A state estimation method for discrete neural networks with random uncertainty under polling protocol

Abstract

The present invention discloses a discrete neural network state estimation method with random uncertainty under a polling protocol, the method comprising the following steps: step 1, establishing a model of a discrete time-varying neural network system with random nonlinearity and random sensor saturation; step 2, introducing a polling protocol to schedule data transmission; step 3, constructing a time-varying state estimator; step 4, providing two performance indicators that the error system simultaneously satisfies; step 5, obtaining the main lemma of the acceptable accuracy of the estimation algorithm; step 6, using an inequality processing method, given a gain matrix, obtaining an error system formula that simultaneously satisfies H ∞ Performance indicators and variance constraints. This method proposes a new probability-dependent time-varying state estimation algorithm, which can be implemented based on the solution of certain matrix inequalities. It directly analyzes the estimation error system of the same order as the original system, reducing the amount of calculation and computational complexity, and presents a variance-constrained state estimation algorithm in the non-augmented case.

Description

Technical Field

[0001] The present invention relates to a state estimation method for a neural network, and in particular to a variance-constrained H state estimation method for a discrete neural network with random nonlinearity and random sensor saturation under a polling protocol. ∞ State estimation method. Background Art

[0002] Discrete neural networks (CNNs) have the advantage of extracting and detecting information from complex or imprecise data. Currently, there is considerable interest in the theoretical research and algorithm development of CNNs, primarily in areas such as pattern recognition, fault detection, and fragile state estimation. These advantages have enabled the successful application of CNNs in diverse interdisciplinary fields, including physics, engineering, and biology, to realize a variety of complex neural network functions. State estimation of CNNs is a crucial component of this research and a key area of ​​research breakthroughs and challenges.

[0003] Neural networks have garnered widespread attention due to their ability to overcome the shortcomings of traditional artificial intelligence in information processing, such as pattern recognition and signal reception. They also offer extremely fast optimization processes and can function reliably even in the presence of noise or other interference in the data. Estimating the states of neurons in a neural network is essential for achieving desired performance. Since it is generally difficult to obtain complete information about the states of all neurons, effective state estimation for discrete neural networks is of great significance.

[0004] Extensive research has been conducted in the literature on state estimation problems affected by saturation in network communications or transmission processes. Indeed, sensor saturation is a common nonlinear problem in practical systems. Ignoring this phenomenon when designing state estimation algorithms can degrade the performance of the estimation error system and even lead to instability. Summary of the Invention

[0005] To address the state estimation problem in dynamic networks with sensor saturation, current research focuses primarily on discrete time-invariant neural networks. This paper proposes a state estimation method for discrete neural networks with random uncertainty under a time-varying polling protocol. This method proposes a new probability-dependent time-varying state estimation algorithm, implemented based on the solutions of certain matrix inequalities. It directly analyzes the estimated error system of the same order as the original system, reducing the computational effort and complexity, and presents a variance-constrained state estimation algorithm for the non-augmented case.

[0006] The purpose of the present invention is achieved through the following technical solutions:

[0007] A variance-constrained state estimation method for a discrete neural network under a polling protocol includes the following steps:

[0008] Step 1: Establish a model of a discrete time-varying neural network system with random nonlinearity and random sensor saturation:

[0009]

[0010] Where: x k is the state vector of the discrete uncertain neural network at time k; A k represents the self-feedback diagonal matrix at time k; α n,k represents x at time k n,k Feedback weight; B 1k 、B 2k is the connection weight matrix at time k; y k is the output measured at time k; z k represents the controlled output of the system at time k; v 1k and v 2k is zero mean at time k and has a covariance of Q k >0 and R k Gaussian white noise > 0; f(x k )、g(x k ) is the nonlinear activation function at time k; B 1k 、B 2k 、D k 、H k is a real-valued matrix known at time k; ΔA k is the parameter uncertainty at time k; random variable α k and β k Describe the random nonlinearity and random sensor saturation phenomena at time k respectively;

[0011] Step 2: Introduce the polling protocol to schedule data transmission according to the model established in step 1:

[0012]

[0013] in: is the measurement output of the i-th sensor at time k, mod() represents the remainder function of division;

[0014] Define the update matrix as φ i =diag{δ(i-1)I, δ(i-2)I,…, δ(im)I}, δ(·)∈{0,1}, then there is:

[0015]

[0016] Where: k=mod(ki,m)+1∈{1,2,...,m} describes the number of nodes that can occupy the network channel. represents the update matrix at time k, and m represents the total number of sensors;

[0017] Step 3: Based on the measurement output information of the model established in step 2, construct the following time-varying state estimator:

[0018]

[0019] in: is the estimated state vector of the discrete uncertain neural network at time k; K k Describe the estimator gain matrix at time k; is a known positive scalar; Represents the estimated output value; H k represents the known real-valued matrix at time k; Represents the output value of the polling protocol at time k;

[0020] Step 4: Obtain the design state estimator formula and give the error system that satisfies the following two performance indicators:

[0021] (1) For a given disturbance attenuation level, the matrix and γ>0 is given, for the initial state Obey the following H ∞ Performance indicators:

[0022]

[0023] in: J1 represents the first performance indicator, Z k represents the estimated error of the controlled output, γ represents a known positive scalar, e0 represents the initial error, E φ 、 represents a known positive definite matrix;

[0024] (2) Define the covariance matrix:

[0025]

[0026] The estimated error covariance satisfies:

[0027]

[0028] Where: k is a series of pre-given estimation accuracy matrices, J2 represents the second performance indicator, 0≤k<N, N represents the maximum time taken;

[0029] Step 5. Find the acceptable accuracy of the estimation algorithm. Main lemma:

[0030] Lemma 1: If the activation function satisfies the sector boundedness condition, the following form can be obtained:

[0031]

[0032]

[0033]

[0034]

[0035] Among them: U 1k 、U 2k and V 1k 、V 2k is a real matrix of appropriate dimension known at time k; e k Represents the error between the actual value and the estimated value at time k, which is given by represents the activation function at time k;

[0036] Lemma 2: If the activation functions f(s) and g(s) satisfy the sector boundedness, it can be deduced that:

[0037]

[0038]

[0039] Among them: s represents the variable of the activation function, ρ, are random numbers (0,1) that are not correlated with each other;

[0040] Lemma 3: The saturation function σ(τ) satisfies the following theorem:

[0041]

[0042] in: represents a constant and tr() finds the trace of a matrix;

[0043] Step 6: Using the inequality processing method, given the gain matrix, the error system is obtained and satisfies H ∞ Performance indicators and variance constraints, the specific steps are as follows:

[0044] Step 6.1. Construct the following time-varying state estimator to satisfy H ∞ Sufficient conditions for performance constraints:

[0045] Consider a discrete uncertain neural network with sensor failure: given the gain matrix K k , for γ>0, the matrix E φ ≥0 and In the initial conditions Under the condition that there exists a positive definite matrix {S k} 1≤k≤N+1 and T k The following inequality is satisfied:

[0046]

[0047]

[0048]

[0049]

[0050]

[0051]

[0052]

[0053]

[0054]

[0055]

[0056] in: Represents the updated matrix of the measurement output, tr() represents the trace of the matrix, λ is a positive scalar;

[0057] Step 6.2: Satisfy the estimation error covariance performance constraint by solving the following recursive linear matrix inequality:

[0058] Considering a discrete time-varying neural network with random nonlinearity and random sensor saturation: Given a gain matrix K k , in the initial condition Under the condition that there exists a positive definite matrix {X k} 1≤k≤N+1 satisfy:

[0059] X k+1 ≥Ψ(X k ),

[0060] in:

[0061]

[0062]

[0063]

[0064]

[0065]

[0066]

[0067]

[0068]

[0069] in: represents the updated matrix of the measurement output, and tr() represents finding the trace of the matrix;

[0070] Then we get: N+1 represents the upper limit of time;

[0071] Step 63: By solving step 61 and step 62, the error system satisfies both the variance constraint and H ∞ Sufficient conditions for performance indicators.

[0072] Compared with the prior art, the present invention has the following advantages:

[0073] 1. This invention proposes a variance constraint H under a polling protocol for discrete time-varying neural network control systems. ∞ The state estimation method takes into account the influence of randomly occurring sensor saturation and randomly occurring nonlinearity. Compared with the existing minimum estimation of error covariance, the variance-constrained state estimation strategy of the present invention provides a more relaxed technology by introducing a given upper limit constraint to reflect the allowable accuracy of the proposed state estimation method, and gives a variance-constrained state estimation algorithm in the non-augmented case.

[0074] 2. The present invention uses Lyapunov's stability theorem to provide a design method for state estimation of discrete time-varying neural network control systems in the form of linear matrix inequalities. The proposed state estimation method under variance constraint has time-varying characteristics.

[0075] 3. The method of the present invention is not only applicable to processing the estimation problem of neural networks, but also applicable to online applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0076] Figure 1 It is a flow chart of a variance-constrained state estimation method for a discrete neural network under a polling protocol of the present invention;

[0077] Figure 2 is the controlled output z k and its estimation

[0078] Figure 3 is the output estimation error

[0079] Figure 4 It is e 1,k Upper bounds on the error variance and actual error covariance;

[0080] Figure 5 It is e 2,k Upper bounds on the error variance and the actual error covariance. DETAILED DESCRIPTION

[0081] The technical solution of the present invention is further described below with reference to the accompanying drawings, but is not limited thereto. Any modification or equivalent replacement of the technical solution of the present invention that does not depart from the spirit and scope of the technical solution of the present invention should be included in the scope of protection of the present invention.

[0082] The present invention provides a variance-constrained state estimation method for discrete neural networks under a polling protocol. The method is aimed at a class of discrete uncertain neural networks and studies the variance constraint H of discrete neural networks with random sensor saturation and random nonlinearity under a cyclic protocol. ∞ In the state estimation problem, randomly occurring sensor saturation and nonlinear phenomena are each modeled by a random variable obeying a Bernoulli distribution with a known probability. A saturation function is introduced to reduce the negative impact of measurement outliers on estimation performance. To alleviate unnecessary network congestion in the communication channel, a cyclic protocol is introduced to specify which network node has access to the network channel at each time step. The main purpose of this invention is to design a time-varying finite field state estimator. The goal is to design a time-varying state estimation method that provides sufficient conditions in the presence of randomly varying nonlinearities, randomly occurring sensor saturation, and cyclic protocols, while achieving bounded error variance and a pre-set H. ∞ Performance indicators. Figure 1 As shown, the specific steps include:

[0083] Step 1: Establish a model of a discrete time-varying neural network system with random nonlinearity and random sensor saturation:

[0084]

[0085] in: is the state vector of the discrete uncertain neural network at time k; n represents the n-dimensional vector; the self-feedback diagonal matrix at time k is A k =diag{α 1,k ,α 2,k ,…α n,k} indicates; α n,k represents x at time k n,k Feedback weight; B 1k 、B 2kis the connection weight matrix at time k; To measure the output at time k, y m,k represents the measurement output of the mth dimension at time k, where m represents the dimension of the measurement output; represents the controlled output of the system at time k, r represents the r-dimensional vector; v 1k and v 2k is zero mean at time k and has a covariance of Q k >0 and R k Gaussian white noise > 0; f(x k )、g(x k ) is the nonlinear activation function at time k; B 1k 、B 2k 、D k 、H k is a real-valued matrix known at time k; ΔA k is the parameter uncertainty at time k, satisfying ΔA=M k F k N k , F k The unknown matrix at time k satisfies F k F k T <I,M k 、N k represents the dimension-appropriate matrix at time k; random variable α k and β k Describe the random nonlinearity and random sensor saturation at time k respectively; the nonlinear excitation functions f(s) and g(s) satisfy f(0) = 0, g(0) = 0 and the activation function satisfies the following sector bounded conditions:

[0086]

[0087]

[0088] Among them: U 1k 、U 2k 、V 1k and V 2k is a real-valued matrix of appropriate dimension, s represents the variable of the activation function;

[0089] Using Bernoulli distribution random variable α k and β k Describe the nonlinearity of random changes and the phenomenon of missing measurements at time k respectively, and satisfy:

[0090]

[0091]

[0092] Where: αk ∈[0,1] and β k ∈[0,1] is a known constant, is a positive scalar.

[0093] The saturation function σ(·) satisfies the following conditions:

[0094]

[0095] Where: i (ξ i )=sign(ξ i )min{ξ i,max ,|ξ i |}, where i∈{1 2 … m}, ξ i,max represents the saturation level, m represents the dimension, |ξ i | represents modulo, and the saturation function satisfies the following conditions:

[0096]

[0097] in: is a positive scalar, satisfying

[0098] Step 2: Based on the model established in step 1, a polling protocol is introduced to schedule data transmission:

[0099]

[0100] in: is the measurement output of the i-th sensor at time k, mod() represents the remainder function of division;

[0101] Define the update matrix as φ i =diag{δ(i-1)I,δ(i-2)I,…,δ(im)I},δ(·)∈{0,1}. In addition, we can also obtain:

[0102]

[0103] in: ξk=mod(ki,m)+1∈{1,2,…,m} describes the number of nodes that can occupy the network channel, and m represents the total number of sensors;

[0104] Step 3: Based on the measurement output information of the model established in step 2, construct the following time-varying state estimator;

[0105]

[0106] in: is the estimated state vector of the discrete uncertain neural network at time k; n represents the n-dimensional vector; the self-feedback diagonal matrix at time k is A k =diag{α 1,k ,α 2,k ,…α n,k} indicates; α 1,k represents x at time k 1,k Feedback weight; B 1k 、B 2k is the connection weight matrix at time k; K k Describe the estimator gain matrix at time k; is a known positive scalar; f(x k )、g(x k ) is the nonlinear estimated activation function at time k; Represents the estimated output value; H k represents the known real-valued matrix at time k; represents the output value of the polling protocol at time k; D k is a known real-valued matrix.

[0107] Step 4: Obtain the design state estimator formula and give the error system that satisfies the following two performance indicators:

[0108] (1) For a given disturbance attenuation level, the matrix and γ>0 is given, for the initial state Obey the following H ∞ Performance indicators:

[0109]

[0110] in: J1 represents the first performance indicator, Z k represents the estimated error of the controlled output, γ represents a known positive scalar, e0 represents the initial error, E φ 、 represents a known positive definite matrix;

[0111] (2) Define the covariance matrix:

[0112]

[0113] The estimated error covariance satisfies:

[0114]

[0115] Where: k (0≤k<N) is a series of pre-given estimation accuracy matrices, J2 represents the second performance indicator, and N represents the maximum time obtained.

[0116] Step 5. Find the acceptable accuracy of the estimation algorithm. Main lemma:

[0117] Lemma 1: If the activation function satisfies the sector boundedness condition, the following form can be obtained:

[0118]

[0119]

[0120]

[0121]

[0122] Among them: e k Indicates the error between the actual value and the estimated value at time k, U 1k 、U 2k 、V 1k 、V 2k Is the matrix known at time k, given by represents the activation function at time k;

[0123] Lemma 2: If the activation functions f(s) and g(s) satisfy the sector boundedness, it can be deduced that:

[0124]

[0125]

[0126] Among them: s represents the variable of the activation function, ρ, are random numbers that are independent of each other (0,1).

[0127] Lemma 3: The saturation function σ(τ) satisfies the following theorem:

[0128]

[0129] in represents a constant and

[0130] Step 6: Using the inequality processing method, given the gain matrix, the error system is obtained and satisfies H ∞ Performance metrics and variance constraints.

[0131] Step 6.1. Construct the following time-varying state estimator to satisfy H ∞ Sufficient conditions for performance constraints:

[0132] Consider a discrete uncertain neural network with sensor failure: given the gain matrix K k , for γ>0, the matrix E φ ≥0 and In the initial conditions Under the condition that there exists a positive definite matrix {S k} 1≤k≤N+1 and T k The following inequality is satisfied:

[0133]

[0134]

[0135]

[0136]

[0137]

[0138]

[0139]

[0140]

[0141]

[0142]

[0143] in: Represents the updated matrix of the measurement output, tr() represents the trace of the matrix, λ is a positive scalar;

[0144] Step 6.2: Satisfy the estimation error covariance performance constraint by solving the following recursive linear matrix inequality:

[0145] Considering a discrete time-varying neural network with random nonlinearity and random sensor saturation: Given a gain matrix K k , in the initial condition Under the condition that there exists a positive definite matrix {X k} 1≤k≤N+1 satisfy:

[0146] X k+1 ≥Ψ(X k ),

[0147] in:

[0148]

[0149]

[0150]

[0151]

[0152]

[0153]

[0154]

[0155]

[0156] in: represents the updated matrix of the measurement output, and tr() represents finding the trace of the matrix;

[0157] Then we get: N+1 represents the upper limit of time;

[0158] Step 63: By solving step 61 and step 62, the error system satisfies both the variance constraint and H ∞ Sufficient conditions for performance indicators.

[0159] Example:

[0160] This embodiment uses a DC motor model as an example to verify the effectiveness of the control method designed by the present invention. Considering a discrete neural network system, the following simulation is performed:

[0161] The parameter matrices of the discrete neural DC motor with random sensor saturation and random nonlinearity are:

[0162]

[0163]

[0164] G k =[-0.1 -0.16*sin(3k)], C k =[-0.1 -0.28*sin(2k)],

[0165] π=0.8, d=3,ρ=0.5,

[0166]

[0167] The activation function is as follows:

[0168]

[0169] Where: x k =(x 1,k x 2,k ) Tis the state vector of the neuron. Let γ = 0.3, N = 90, and the weight matrix be and R k =1, and the initial state x k,0 =[-0.56 -0.15] T , by solving the inequality and iterating, we can get K k And the simulation results are shown in Table 1.

[0170]

[0171]

[0172] Simulation diagram Figure 2-Figure 5 As shown. Among them, Figure 2 、 Figure 3 、 Figure 4 、 Figure 5 Denote the upper bounds of the controlled output and its estimated value, the output estimation error, the error variance, and the actual error covariance respectively. The simulation results demonstrate the variance constraint H of the discrete neural network with random sensor saturation and random nonlinearity under the polling protocol. ∞ Feasibility and effectiveness of state estimation state methods.

Claims

1. A discrete neural network state estimation method with random uncertainty under a polling protocol, characterized by The method comprises the following steps: Step 1: Establish a model of a discrete time-varying neural network system with random nonlinearity and random sensor saturation: Where: x k is the state vector of the discrete uncertain neural network at time k; A k represents the self-feedback diagonal matrix at time k; B 1k 、B 2k is the connection weight matrix at time k; y k is the output measured at time k; z k represents the controlled output of the system at time k; v 1k and v 2k is zero mean at time k and has a covariance of Q k >0 and R k Gaussian white noise > 0; f(x k )、g(x k ) is the nonlinear activation function at time k; B 1k 、B 2k 、D k 、H k is a real-valued matrix known at time k; ΔA k is the parameter uncertainty at time k; random variable α k and β k Describe the random nonlinearity and random sensor saturation phenomena at time k respectively; Step 2: Introduce the polling protocol to schedule data transmission according to the model established in step 1: in: is the measurement output of the i-th sensor at time k, mod() represents the remainder function of division; Define the update matrix as φ i =diag{δ(i-1)I, δ(i-2)I,…,δ(im)I}, δ(·)∈{0,1}, then there is: Where: k =mod(ki,m)+1∈{1,2,...,m} describes the number of nodes that can occupy the network channel. represents the update matrix at time k, and m represents the total number of sensors; Step 3: Based on the measurement output information of the model established in step 2, construct the following time-varying state estimator: in: is the estimated state vector of the discrete uncertain neural network at time k; K k Describe the estimator gain matrix at time k; is a known positive scalar; Represents the estimated output value; H k represents the known real-valued matrix at time k; Represents the output value of the polling protocol at time k; Step 4: Obtain the design state estimator formula and give the error system that satisfies the following two performance indicators: (1) For a given disturbance attenuation level, the matrix E φ >0, and γ>0 is given, for the initial state e0, Obey the following H ∞ Performance indicators: in: J1 represents the first performance indicator, represents the estimated error of the controlled output, γ represents a known positive scalar, e0 represents the initial error, E φ 、 represents a known positive definite matrix; (2) Define the covariance matrix: The estimated error covariance satisfies: Where: k is a series of pre-given estimation accuracy matrices, J2 represents the second performance indicator, 0≤k<N, N represents the maximum time taken; Step 5. Find the acceptable accuracy of the estimation algorithm. Main lemma: Lemma 1: If the activation function satisfies the sector boundedness condition, the following form can be obtained: Among them: U 1k 、U 2k and V 1k 、V 2k is a real matrix of appropriate dimension known at time k; e k Represents the error between the actual value and the estimated value at time k, which is given by represents the activation function at time k; Lemma 2: If the activation functions f(s) and g(s) satisfy the sector boundedness, it can be deduced that: Among them: s represents the variable of the activation function, ρ, are random numbers (0,1) that are not correlated with each other; Lemma 3: The saturation function σ(τ) satisfies the following theorem: in: represents a constant and tr() finds the trace of a matrix; Step 6: Using the inequality processing method, given the gain matrix, the error system is obtained and satisfies H ∞ Performance indicators and variance constraints, the specific steps are as follows: Step 6.

1. Construct the following time-varying state estimator to satisfy H ∞ Sufficient conditions for performance constraints: Consider a discrete uncertain neural network with sensor failure: given the gain matrix K k , for γ>0, the matrix E φ ≥0 and In the initial conditions Under the condition that there exists a positive definite matrix {S k } 1≤k≤N+1 and T k The following inequality is satisfied: in: Represents the updated matrix of the measurement output, tr() represents the trace of the matrix, λ is a positive scalar; Step 6.2: Satisfy the estimation error covariance performance constraint by solving the following recursive linear matrix inequality: Considering a discrete time-varying neural network with random nonlinearity and random sensor saturation: Given a gain matrix K k , in the initial condition Under the condition that there exists a positive definite matrix {X k } 1≤k≤N+1 satisfy: X k+1 ≥Ψ(X k ), in: in: represents the updated matrix of the measurement output, and tr() represents finding the trace of the matrix; Then we get: N+1 represents the upper limit of time; Step 63: By solving step 61 and step 62, the error system satisfies both the variance constraint and H ∞ Sufficient conditions for performance indicators.

2. The discrete neural network state estimation method with random uncertainty under the polling protocol according to claim 1 is characterized in that The ΔA k Satisfying ΔA=M k F k N k , F k is an unknown matrix at time k, satisfying F k F k T <I,M k 、N k represents the dimension-appropriate matrix at time k.

3. The discrete neural network state estimation method with random uncertainty under the polling protocol according to claim 1 is characterized in that The f(s) and g(s) satisfy f(0)=0, g(0)=0, and satisfy the following sector bounded conditions: Among them: U 1k 、U 2k 、V 1k and V 2k is a real-valued matrix of appropriate dimension, and s represents the variable of the activation function.

4. The discrete neural network state estimation method with random uncertainty under the polling protocol according to claim 1 is characterized in that The α k and β k Describe the phenomena of random variation, nonlinearity, and missing measurements, respectively, and satisfy: Where: α k ∈[0,1] and β k ∈[0,1] is a known constant.

5. The discrete neural network state estimation method with random uncertainty under the polling protocol according to claim 1 is characterized in that The saturation function σ(·) satisfies the following conditions: Where: i (ξ i )=sign(ξ i )min{ξ i,max ,|ξ i |}, where i∈{12…m}, ξ i,max represents the saturation level, m represents the dimension, |ξ i | represents modulo, and the saturation function satisfies the following conditions: in: is a positive scalar, satisfying

Images