Multi-innovation OIF Elman network modeling method based on measurement data

Abstract

The invention belongs to the technical field of system modeling, and relates to a multi-innovation OIF Elman network modeling method based on measurement data, which comprises the following steps of: constructing a mathematical model of an OIF Elman network, and obtaining a to-be-estimated parameter matrix of the network by defining a parameter matrix; a sliding window mechanism is introduced, a to-be-estimated parameter matrix is utilized to define a multi-information criterion function, and a parameter estimation sub-algorithm for estimating the OIF Elman network weight is constructed; establishing a dynamic factor adjustment strategy by adopting an exponential decay and minimum value limitation mode; in the parameter estimation sub-algorithm, a dynamic factor adjustment strategy is introduced, and a gradient parameter estimation algorithm is constructed; and estimating a parameter matrix of the OIF Elman network model through a gradient parameter estimation algorithm based on the acquired input and output measurement data. The method is suitable for online modeling, and is superior to the existing method in the aspect of modeling precision.

Description

Technical Field

[0001] The present invention belongs to the technical field of system modeling, and in particular relates to a multi-innovation OIF Elman network modeling method based on measurement data. Background Art

[0002] Accurately capturing and modeling system dynamics from sequential measurement data is crucial for applications such as condition monitoring, soft sensor modeling, and system identification. This requires the construction of mathematical models that can reflect time dependencies and effectively utilize historical information to improve the quality of decision-making based on measurement data. Unlike feedforward neural networks, recurrent neural networks (RNNs) retain internal states by introducing recursive connections, facilitating the storage and utilization of historical information when processing sequential measurement data. Among various recursive networks, the Elman network is a representative model. Its key feature is that it implements recursive feedback from hidden layers through successive layers, thereby capturing time dependencies. This structure enhances the model's ability to characterize dynamic patterns and is therefore widely used in tasks requiring the processing of time series measurement data and the characterization of system dynamics.

[0003] The classic Elman network has become the foundation for numerous subsequent structural variations designed to further enhance the network's dynamic modeling capabilities. Among these variations, the Output-Input Feedback (OIF) Elman network is of particular interest. This structure is developed based on the improved Elman network. The improved Elman network distinguishes itself from the original structure by introducing self-feedback connections with fixed gains in the context nodes, but its feedback path is limited to hidden layer nodes. Given that feedback information from each layer in a neural network can affect its ability to process dynamic information, the OIF Elman network further develops this structure by introducing feedback information from the output layer, effectively improving the model's sensitivity to historical states. Specifically, the OIF Elman network adds a second relay layer to store and feedback output layer information, forming a dual feedback structure that further enhances the network's ability to process dynamic information.

[0004] In machine learning, neural network weight training is essentially a process of fitting observational samples. In system identification, this process can be viewed as a parameter estimation problem based on measured data, with the core goal of building a highly accurate dynamic model. Parameter estimation is a key factor influencing network modeling performance. Accurate parameter estimation not only improves the model's adaptability to noise and uncertainty but also significantly enhances its generalization capabilities.

[0005] For Elman-type networks, the most widely used parameter estimation method is the gradient method, which typically updates parameter estimates in an iterative or recursive manner. In recent years, swarm intelligence algorithms have also been used to solve parameter estimation problems, including the cuckoo algorithm, the sparrow search algorithm, the particle swarm optimization algorithm, and the seagull optimization algorithm. These methods utilize a collaborative swarm search mechanism for parameter optimization, effectively exploring the solution space and providing new parameter estimation strategies for Elman-type networks.

[0006] Although a variety of parameter estimation algorithms have been proposed for Elman-type networks, parameter estimation methods specifically for OIF Elman networks are still relatively scarce. Most of the existing methods rely on the stochastic gradient (SG) algorithm, which is widely used in neural networks, but this algorithm does not fully consider the unique structural characteristics of OIF Elman networks. It is worth mentioning that Zhang et al. proposed a Chicken Swarm Optimization (CSO) algorithm for OIF Elman networks. This method simulates the hierarchical division of labor in the foraging behavior of chickens to perform a global optimization search for network parameters. However, the CSO algorithm is essentially an offline estimation method that relies on batch data processing, which may lead to high computational overhead. At the same time, its dynamic self-adaptation capabilities are limited and it is not suitable for online modeling scenarios where frequent parameter updates are required.

[0007] A method for online estimation of the state of health (SOH) of a new energy vehicle power battery (CN104035037A). In terms of model structure, the OIF Elman network in this method only adds a connecting layer to the traditional Elman network. However, the self-connection feedback gain factor (referred to as the self-feedback factor) of the connecting layer is a fixed value and is not updated in real time. In other words, the feedback mechanism of the network in this method is a fixed structure and lacks adaptive adjustment capabilities. In terms of algorithm, this method uses an adaptive learning rate momentum gradient descent backpropagation algorithm for parameter estimation, i.e., weight training. This algorithm is based on the traditional quadratic error criterion function and fails to fully reflect the dynamic change characteristics of the model error. It is prone to falling into local minima, resulting in low network parameter update efficiency or unstable training results.

[0008] Research on system identification method based on exponential criterion function (Lv Lei, master's thesis of Hubei University of Technology, 2021). In terms of model structure, this method studies the parameter estimation problem of two linear systems: controlled autoregressive system and finite impulse response moving average system. Its modeling form is linear difference equation; in terms of algorithm, the algorithm in this method is derived based on the exponential criterion function, and the proposed algorithm is only applicable to linear systems. Summary of the Invention

[0009] The object of the present invention is to overcome the deficiencies in the prior art and to provide a multi-innovation OIF Elman network modeling method based on measurement data.

[0010] In order to achieve the purpose of the present invention, the present invention is implemented by adopting the following technical solutions.

[0011] A multi-innovation OIF Elman network modeling method based on measurement data includes the following steps:

[0012] S1. Constructing a mathematical model of the OIF Elman network based on the OIF Elman network structure, performing system identification using the mapping relationship between the input and output of the mathematical model, and determining the parameters to be estimated in the mathematical model;

[0013] S2. Define a parameter matrix and a vector with the parameters to be estimated as elements to obtain the parameter matrix to be estimated of the OIF Elman network;

[0014] S3. Introducing a sliding window mechanism, using the parameter matrix to be estimated, defining a multi-innovation criterion function based on the measurement data and the weight decay factor, and constructing a parameter estimation sub-algorithm for estimating the weights of the OIF Elman network based on the multi-innovation criterion function;

[0015] S4. Using exponential decay and minimum value restriction, a dynamic factor adjustment strategy is established that can adaptively adjust the self-feedback factor and weight decay factor;

[0016] S5. In the parameter estimation sub-algorithm, a dynamic factor adjustment strategy is introduced to construct a gradient parameter estimation algorithm;

[0017] S6. Estimate the parameter matrix of the OIF Elman network model using a gradient parameter estimation algorithm based on the collected input and output measurement data.

[0018] Furthermore, the mathematical model is described as:

[0019] Hidden layer:

[0020]

[0021] Where: u q (k)∈R is the network input; j (k)∈R,χ cl (k)∈R,y c (k)∈R are the outputs of the network hidden layer, the first receiving layer and the second receiving layer respectively; are the weights connecting the first receiving layer and the hidden layer, the weights connecting the input layer and the hidden layer, and the weights connecting the second receiving layer and the output layer respectively; is the middle vector, defined as

[0022] f(·)∈R is the activation function of the hidden layer node, using the tanh function:

[0023]

[0024] The first receiving layer:

[0025] χ cl (k) = αχ cl (k-1)+χ l (k-1),k=1,2,…,n, (4)

[0026] Where: α∈R (0≤α<1) is the self-connection feedback gain factor of the first receiving layer, referred to as the self-feedback factor;

[0027] The second receiving layer and output layer:

[0028] y c (k) = γy c (k-1)+y(k-1), (5)

[0029]

[0030] Where: y(k)∈R is the network output; γ∈R (0≤γ<1) is the self-feedback factor of the second receiving layer; are the weights connecting the hidden layer and the output layer.

[0031] Furthermore, the parameter matrix to be estimated is:

[0032]

[0033] Where: are intermediate parameter matrices and vectors.

[0034] Furthermore, the multi-innovation criterion function is:

[0035]

[0036] Where: η>0 is the weight decay factor, which controls parameter shrinkage through L2 regularization; is the error between the network output at time k and the expected output, Output vector for the hidden layer of the network.

[0037] Furthermore, the dynamic factor adjustment strategy is expressed as:

[0038]

[0039] in: is the new information at time k; 0<β i <1, i=1, 2, 3 are adjustment coefficients; 0≤α min <1, 0≤γ min <1,η min >0 is the minimum value to ensure the lower limit of the factor; α min <α init <1,γ min <γ init <1,η init >η min is the initial value; α decay >0,γ decay >0,η decay >0 is the decay rate that controls the adaptation speed.

[0040] Furthermore, the gradient parameter estimation algorithm is expressed as:

[0041]

[0042] Where: ∈(k)≥0 is obtained by solving the optimization problem The resulting step length; It is a stacked innovation matrix consisting of parameter estimates and observation data obtained at time k-1.

[0043] Beneficial effects

[0044] The present invention constructs a multi-information criterion function based on measurement data and weight attenuation factors; proposes a dynamic factor adjustment strategy to achieve adaptive adjustment of the weight attenuation factor and the network self-feedback factor; and proposes a gradient parameter estimation algorithm based on dynamic factors and multi-information. The algorithm has a recursive nature and can achieve real-time updating of network parameters during the data acquisition process. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 Schematic diagram of the structure of the OIF Elman network of the present invention;

[0046] Figure 2 This is a graph showing changes in power load data over time according to the present invention;

[0047] Figure 3 MAPEs graph of the method under different (n, p) of the present invention on the validation data;

[0048] Figure 4 This is a comparison diagram of actual and predicted power loads of the present invention. DETAILED DESCRIPTION

[0049] The present invention will be further described with reference to the embodiments and the accompanying drawings.

[0050] Example 1, as Figure 1 A multi-innovation OIF Elman network parameter estimation method based on measurement data:

[0051] Step 1, problem description:

[0052] Consider a multi-input single-output OIF Elman network (output-input feedback Elman network), which consists of r input layer nodes, n hidden layer nodes and one output node, as shown in Figure 1 Its mathematical model is described as follows:

[0053] Hidden layer:

[0054]

[0055] Where: u q (k)∈R is the network input; j (k)∈R,χ cl (k)∈R,y c (k)∈R are the outputs of the network hidden layer, the first receiving layer and the second receiving layer respectively; are the weights connecting the first receiving layer and the hidden layer, the weights connecting the input layer and the hidden layer, and the weights connecting the second receiving layer and the output layer respectively; is the middle vector, defined as f(·)∈R is the activation function of the hidden layer node, and the tanh function is used here:

[0056]

[0057] The tanh function was chosen as the activation function because it maps the input to the range (-1, 1), which helps accelerate model convergence and improve training stability. Compared to the sigmoid function, its gradient is larger when the input is close to 0, which can more effectively transmit the error signal and alleviate the vanishing gradient problem.

[0058] The first receiving layer:

[0059] χ cl (k) = αχ cl (k-1)+χ l (k-1),l=1,2,…,n, (4)

[0060] Wherein: α∈R (0≤α<1) is the self-connection feedback gain factor of the first receiving layer, referred to as the self-feedback factor for short.

[0061] The second receiving layer and output layer:

[0062] y c(k) = γy c (k-1)+y(k-1), (5)

[0063]

[0064] Where: y(k)∈R is the network output; γ∈R (0≤γ<1) is the self-feedback factor of the second receiving layer; are the weights connecting the hidden layer and the output layer.

[0065] Equations (1)-(6) constitute the OIF Elman network. In machine learning, Equations (1)-(6) are regarded as expressions of the network, where n and r represent the number of nodes, are the network parameters to be trained. Unlike machine learning, in system identification, we regard the mapping relationship between the input and output of the mathematical model as the expression of the model, where n and r represent the model order. It is the parameter to be estimated in the model. The purpose of parameter estimation is to estimate the unknown parameters using measured input and output data.

[0066] Define the parameter matrix and vector:

[0067]

[0068] The parameters to be estimated of the OIF Elman network can be written as a parameter matrix:

[0069]

[0070] Step 2: Modeling method

[0071] The goal of this paper is to propose a novel parameter estimation algorithm for estimating the parameter matrix of the OIF Elman network model. The algorithm utilizes dynamic measurement data {u1(k), u2(k), ..., u r (k), y(k)} to model.

[0072] The output vector of the network hidden layer is defined as:

[0073]

[0074] The error between the output of the OIF Elman network at time k and the expected output is:

[0075]

[0076] The stochastic gradient algorithm widely used in neural networks is to minimize the criterion function To determine the optimal parameter estimates. Represents the parameter vector The estimation at time k. Then the parameter estimation recursive update formula of the stochastic gradient algorithm for the OIF Elman network can be expressed as:

[0077]

[0078] in: Represents the information matrix that combines the measured data with the parameter estimates at the previous moment.

[0079] Equations (9)-(11) use only a single innovation To update the parameter estimates of the OIF Elman network, Represents the estimated value of the hidden layer output at time k. This type of method can be classified as a single-information correction algorithm because it only relies on the latest information item for parameter update. However, this type of algorithm fails to make full use of historical information, which may lead to suboptimal parameter estimation. In addition, the self-feedback factors α and γ in the network are set to fixed values, further limiting the modeling ability of the network. In order to overcome these limitations and improve the accuracy of parameter estimation, the present invention derives a gradient parameter estimation algorithm based on dynamic factors and multiple innovations for the OIF Elman network from the perspective of innovation correction and dynamic penalty.

[0080] In order to improve the accuracy of parameter estimation, a dynamic window mechanism is introduced. Using the most recent p measurement data (p ≥ 1 represents the innovation length), a multi-innovation criterion function with a weighted decay factor is defined as follows:

[0081]

[0082] Where: η>0 is the weight decay factor, which controls parameter shrinkage through L2 regularization, thereby improving the generalization ability and stability of the model during the gradient-based optimization process.

[0083] Define the network input vector u(k) and the output vector χ of the receiving layer 1 c (k):

[0084] u(k):=[u1(k),u2(k),…,u r (k)] T ∈R r , (12)

[0085] χ c (k): = [χ c1 (k), χ c2 (k),…,χ cn (k)] T ∈R n . (13)

[0086] Define the stacking error matrix and the stacking output matrix:

[0087]

[0088] Υ c (p, k): = [χ c (k), χ c (k-1),…,χ c (k-p+1)] T ∈R p×n . (15)

[0089] Introducing the stacking matrix:

[0090]

[0091] Where: 0 is the intermediate vector matrix and

[0092]

[0093] beg About Omega 1 The first-order derivative of and using equations (14)-(16) can be obtained as the gradient matrix

[0094]

[0095] Similarly, we can get 2 ,ω 3 ,ω 4 The gradient vector and matrix of :

[0096]

[0097]

[0098] in: is the stacked input matrix of the network, the stacked output vector of the hidden layer and the stacked output vector of the following layer 2:

[0099] U(p,k):=[u(k),u(k-1),…,u(k-p+1)]T, (23)

[0100] Υ(p,k):=[x(k),χ(k-1),…,χ(k-p+1)]T, (24)

[0101] Y c (p, k): = [y c (k), y c (k-1),…,y c (k-p+1)] T . (25)

[0102] Based on equations (19)-(22), the stacked innovation matrix is ​​introduced

[0103]

[0104]

[0105]

[0106] About The gradient matrix can be calculated as follows:

[0107]

[0108] In addition, the present invention also develops a dynamic factor adjustment strategy, which uses exponential decay and minimum value constraints to ensure the adaptive adjustment of self-feedback factors α and γ and weight decay factor η. is the new information at time k. The corresponding update formulas for the self-feedback factor and weight attenuation factor are designed as follows:

[0109]

[0110] Where: 0<β i <1, i=1,2,3 is the adjustment coefficient; 0≤α min <1,0≤γ min <1,η min >0 is the minimum value to ensure the lower limit of the factor; α min <α init <1,γ min <γ init <1,η init >η min is the initial value; α decay >0,γ decay >0,η decay >0 is the decay rate that controls the adaptation speed.

[0111] make In the expression (1), j (k) is the estimate at time k. Based on the dynamic self-feedback factors in equations (28) and (29), χ in equation (4) cl (km) and y in formula (5) c The estimated distance (km) can be calculated using the following formula:

[0112]

[0113] Thus, we can obtain Υ(p, k) in formula (24) and Υ in formula (15): c (p, k), Y in formula (25)c (p, k), χ(km) in formula (8), χ in formula (13) c Estimated value (km):

[0114]

[0115] Based on the above derivation, use the negative gradient to search for minimization Using Equations (27) and (30), we obtain the Dynamic Factor and Multi-Innovation-based Gradient (DFMIG) parameter estimation algorithm for estimating the OIFElman network:

[0116]

[0117] Where: ∈(k)≥0 is obtained by solving the optimization problem The resulting step length; It is a stacked innovation matrix consisting of parameter estimates and observation data obtained at time k-1.

[0118] Example 1, as Figures 2 to 4 FIG4 shows an example of application of the method in power load forecasting.

[0119] Power load forecasting plays a key role in ensuring the safe and stable operation of the power system. Its accuracy directly affects the formulation of power generation plans, grid dispatch optimization, and power market trading decisions. At the same time, accurate load forecasting can promote the consumption of renewable energy, reduce system operating costs, and is an important technical support for the realization of smart grids and energy digital transformation. In this example, the performance of the proposed method is evaluated using the power load data of Wuxi, China collected from August 1 to 10, 2018. Figure 2 As shown, the dataset is sampled at 15-minute intervals, with 96 data points collected per day in megawatts (MW).

[0120] A sliding window mechanism is used to generate forecasts using data from five consecutive days prior to the forecast date and the time step immediately preceding the current time on the forecast date. The training, validation, and test sets are dynamically updated daily: in each iteration, the first six days of data are used for training (with the first five days as input and the sixth day as output), the seventh day's data is used for validation, and the eighth day's data is used for testing. Specifically, load data from August 8, 9, and 10 (i.e., days 8, 9, and 10) is used as the test set to evaluate forecast performance. For each day's forecast, the data from the previous five days is used as input to generate the forecast results, thereby verifying the algorithm's ability to handle sequence forecasting tasks.

[0121] Because the forecast input is based on the data from the previous five consecutive days, the network order r is fixed at 5. To determine the optimal values ​​for order n and innovation length p, this study evaluated the DFMIG algorithm using power load data from August 1st to 8th. All possible combinations of n and p ranging from 1 to 15 were tested, and the forecast performance was evaluated using the mean absolute percentage error (MAPE) calculated based on the validation data (i.e., the load data from August 7th). The definition of MAPE is as follows:

[0122]

[0123] Among them, L r represents the number of data points, y(k) and Represent the actual power load and the predicted power load respectively. By comparing the MAPE values ​​corresponding to different (n, p) combinations, the parameter combination that minimizes MAPE on the verification data is finally determined to be the optimal parameter set. Figure 3 As shown, the minimum MAPE value is obtained when n=5 and p=9, and this configuration is determined to be the optimal parameter combination.

[0124] The proposed DFMIG algorithm is used for power load forecasting, and the SG algorithm and CSO algorithm are used for comparative experiments. Among them, the CSO algorithm is an iterative algorithm, while the SG algorithm and the proposed algorithm are both recursive algorithms. In addition to MAPE, the evaluation indicators also include Root Mean Square Error (RMSE) and R 2 :

[0125]

[0126] Table 1 summarizes the RMSE, MAPE, and R of the three algorithms on the 8th, 9th, and 10th days. 2 value, Figure 4 The comparison of prediction results of three algorithms is shown.

[0127] Table 1. RMSE, MAPE, and R of power load forecasting 2 Performance Index Comparison

[0128]

[0129] from Figure 4It can be observed that the DFMIG algorithm proposed in this study can closely track the actual load curve in the three consecutive days of forecasting, and its performance is significantly better than the SG and CSO algorithms. In particular, on the 10th day, the CSO algorithm has a significant forecast deviation. The results in Table 1 show that the proposed algorithm shows the best performance in all three evaluation indicators and different test dates. In contrast, the CSO algorithm performs well on the 8th and 9th days, but its accuracy drops significantly on the 10th day, which is manifested by a sharp increase in RMSE and MAPE values ​​and a decrease in R 2 The value is significantly reduced. It should be noted that the CSO algorithm is designed for the OIF-Elman network model and is an offline estimation method. Its performance is highly dependent on the number of iterations. Although increasing the number of iterations can improve the prediction accuracy, it will bring huge computational overhead. In this simulation experiment, the average single-stage training time of the DFMIG algorithm is 0.0117 seconds, while that of the SG algorithm is 0.0027 seconds and that of the CSO algorithm is 2.2544 seconds. Although the SG algorithm and the proposed algorithm belong to the same recursive mechanism and perform relatively stably under the sliding window mechanism, its prediction accuracy is significantly lower than that of the DFMIG algorithm on all test days. Overall, the proposed DFMIG algorithm shows significant advantages in balancing prediction accuracy and computational efficiency.

[0130] This paper addresses the parameter estimation problem of the OIF Elman network within a dynamic system modeling framework based on measurement data, proposing a DFMIG algorithm. This algorithm expands the innovation correction framework by integrating multiple innovations with a penalty mechanism. It also introduces a dynamic adjustment strategy for weight decay and self-feedback factors, effectively improving the network's dynamic performance. Simulation results demonstrate the superior performance of the DFMIG-based OIF Elman network in measurement data modeling.

[0131] The preferred embodiments of the present invention are described above with reference to the accompanying drawings, but are not intended to limit the scope of the present invention. Any modifications, equivalent substitutions, and improvements made by those skilled in the art without departing from the scope and essence of the present invention should be within the scope of the present invention.

Claims

1. A multi-innovation OIF Elman network modeling method based on measurement data, characterized by: The steps include: S1. Based on the OIF Elman network structure, a mathematical model of the OIF Elman network is constructed, and the mapping relationship between the input and output of the mathematical model is used to perform system identification and determine the parameters to be estimated in the mathematical model; S2. Define a parameter matrix and a vector with the parameters to be estimated as elements to obtain the parameter matrix to be estimated of the OIF Elman network; S3. Introducing a sliding window mechanism, using the parameter matrix to be estimated to define a multi-innovation criterion function based on the measurement data and the weight decay factor, and constructing a parameter estimation sub-algorithm for estimating the weights of the OIF Elman network based on the multi-innovation criterion function; S4. Using exponential decay and minimum value restriction, a dynamic factor adjustment strategy is established that can adaptively adjust the self-feedback factor and weight decay factor; S5. In the parameter estimation sub-algorithm, a dynamic factor adjustment strategy is introduced to construct a gradient parameter estimation algorithm; S6. Estimate the parameter matrix of the OIF Elman network model using a gradient parameter estimation algorithm based on the collected input and output measurement data.

2. The multi-innovation OIF Elman network modeling method based on measurement data according to claim 1, characterized in that: The mathematical model is described as: Hidden layer: Where: u q (k)∈R is the network input; j (k)∈R,χ cl (k)∈R,y c (k)∈R are the outputs of the network hidden layer, the first receiving layer and the second receiving layer respectively; are the weights connecting the first receiving layer and the hidden layer, the weights connecting the input layer and the hidden layer, and the weights connecting the second receiving layer and the output layer respectively; is the middle vector, defined as f(·)∈R is the activation function of the hidden layer node, using the tanh function: The first receiving layer: x cl (k)=ah cl (k-1)+x l (k-1),l=1,2,…,n, (4) Where: α∈R(0≤α<) is the self-connection feedback gain factor of the first receiving layer, referred to as the self-feedback factor; The second receiving layer and output layer: y c (k)=γy c (k-1)+y(k-1), (5) Where: y(k)∈R is the network output; γ∈R (0≤γ<1) is the self-feedback factor of the second receiving layer; are the weights connecting the hidden layer and the output layer.

3. The multi-innovation OIF Elman network modeling method based on measurement data according to claim 1, characterized in that: The parameter matrix to be estimated is: Where: are intermediate parameter matrices and vectors.

4. The multi-innovation OIF Elman network modeling method based on measurement data according to claim 1, characterized in that: The multi-innovation criterion function is: Where: η>0 is the weight decay factor, which controls parameter shrinkage through L2 regularization; is the error between the network output at time k and the expected output, Output vector for the hidden layer of the network.

5. The multi-innovation OIF Elman network modeling method based on measurement data according to claim 1, characterized in that: The dynamic factor adjustment strategy is expressed as: in: is the new information at time k; 0<β i <1, i=1, 2, 3 are adjustment coefficients; 0≤α min <1, 0≤γ min <1,η min >0 is the minimum value to ensure the lower limit of the factor; α min <α init <1,γ min <γ init <1,η init >η min is the initial value; α decay >0,γ decay >0,η decay >0 is the decay rate that controls the speed of adaptation.

6. The multi-innovation OIF Elman network modeling method based on measurement data according to claim 1, characterized in that: The gradient parameter estimation algorithm is expressed as: Where: ∈(k)≥0 is obtained by solving the optimization problem The resulting step length; It is a stacked innovation matrix consisting of parameter estimates and observation data obtained at time k-1.

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