Aero-engine system identification method based on koopman neural network
Patent Information
- Application Number
- CN202410836307.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2024-06-26
- Publication Date
- 2026-10-30
- Estimated Expiration
- 2044-06-26
AI Technical Summary
然而，目前基于Koopman理论进行航空发动机系统的辨识仍鲜有研究
[0061]本发明的有益效果：本发明针对航空发动机系统辨识领域缺乏对非线性动态进行全局线性化描述的问题，提出了一种融合深度学习与Koopman算子理论的航空发动机系统辨识方法，实现了航空发动机系统的准确辨识及非线性系统的全局线性化表述。本发明是将Koopman算子理论引入航空发动机系统辨识领域当中，通过神经网络将系统状态映射到不变子空间中，实现了航空发动机系统的关键参数动态特性的精确描述。
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Figure CN118861802B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of model parameter identification technology, and in particular to an identification method for aero-engine systems based on a Koopman deep neural network. Background Technology
[0002] Aero-engines are core components of the aviation industry, and their performance parameters determine the flight capability, economy, and environmental adaptability of aircraft systems. Accurate modeling of aero-engine systems is a crucial issue in the aviation industry. The basic idea of the Koopman operator is to elevate the nonlinear system to a linear space for study. Its theory provides a data-driven modeling method for nonlinear systems, enabling analysis and solution through data methods to address the characterization problem of nonlinear systems. As a nonlinear dynamics modeling tool, the Koopman operator maps the nonlinear evolution of a system to a linear evolution process in an infinite-dimensional space, allowing the analysis and control of complex nonlinear systems using the theory and tools of linear dynamics. This enables efficient modeling and prediction of dynamic systems within a data-driven framework, making it particularly suitable for modeling and controlling complex systems. Furthermore, it does not require a priori physical models, thus possessing broad application prospects.
[0003] During actual flight, engine systems experience various operating conditions, such as climb, cruise, descent, and takeoff. Therefore, an aero-engine system is a nonlinear system with multiple operating points. Current research on linear system identification methods for aero-engine systems largely focuses on linearization around a specific operating point, neglecting the global implications. These methods require linearization at every sampled operating point, making them suitable only for nonlinear systems operating within a small range. They are unsuitable for controlling high-dimensional, complex nonlinear systems and struggle to guarantee global characteristics. The Koopman operator, however, is a global linearization method that elevates the state of a nonlinear system to a linear space, enabling precise description of the nonlinear dynamic characteristics across the entire flight envelope. Nevertheless, research on aero-engine system identification based on Koopman theory remains scarce.
[0004] Traditional EDMD-based Koopman methods rely on manually selecting lifting functions, which makes it difficult to guarantee practical results. Using neural networks instead of manually selected state lifting functions can effectively improve the accuracy of Koopman operator approximation. Summary of the Invention
[0005] The objective of this invention is to realize a high-dimensional global linearization representation of nonlinear systems based on Koopman operator theory and deep learning algorithms, thereby designing a system identification method capable of estimating the output of aero-engine systems within a large envelope and multiple operating conditions.
[0006] A method for identifying aero-engine systems based on a Koopman neural network, comprising the following steps:
[0007] Step 1: Determine the form of the aero-engine system model and rewrite the model based on the input fuel quantity and state observation values of the aero-engine system;
[0008] Step 1.1: The discrete state-space model of the aero-engine system is as follows:
[0009] x k+1 =f(x) k ,u k )
[0010] Where k represents the sampling time of the aero-engine system; x k =[x 1k ,x 2k ,…,x nk [This refers to the state observations of an aero-engine system, u] k =[u 1k ,u 2k ,…,u mk ] represents the input fuel quantity of the aero-engine system, and f is the state transition function.
[0011] Step 1.2: Divide the input and output data of the aero-engine system into training and test sets. Normalize the training set data using the L2 norm normalization method;
[0012]
[0013] Here, ||·||2 is used to calculate the L2 norm of the corresponding vector. x k and u k The result of normalization.
[0014] The normalized equations for the aero-engine system are as follows:
[0015]
[0016] The final dataset obtained is:
[0017]
[0018] Among them, X i Y represents the i-th sample in the training set; i Indicates with X i The corresponding i target samples, where m represents the total number of samples; U i Let X be the corresponding i-th input. i Y i and U i Specifically, it is expressed as follows:
[0019]
[0020] In the formula This represents the state observation value corresponding to the (j-2)th time step under the i-th sample. This represents the input value corresponding to the (j-2)th time step for the i-th sample.
[0021] The aero-engine system equations for this dataset can be rewritten as:
[0022] Y i =f(X) i U i )
[0023] Given a dataset S, the goal is to determine the system state transition function f while preserving the global linearity of the aero-engine system in the higher-dimensional space.
[0024] Step 2: Construct the Koopman operator dimensionality increase and decrease functions using a variational autoencoder, determine suitable loss functions and optimizers, and iteratively optimize the parameters of the deep neural network model to obtain the optimal dimensionality increase and decrease functions.
[0025] The core idea of Koopman operator theory is to map state observations to a higher-dimensional linear invariant subspace through an increasing-dimensional function, thereby linearizing the aero-engine system. Finally, the high-dimensional state is reduced to obtain the identification result. Compared to traditional methods, neural networks offer greater flexibility, adaptability, generalization ability, and parallel computing power in fitting increasing-dimensional and reducing-dimensional functions. They can more effectively capture complex nonlinear relationships, adapt to different types of data and problems, and achieve end-to-end learning. Therefore, this invention uses a neural network to fit increasing-dimensional and reducing-dimensional functions.
[0026] The principle of transforming the original nonlinear system into a linear system through an up-dimensional function is as follows: Figure 2 As shown in the figure, K represents the state transition matrix in the linear space, Ψ is the dimension-upgrading function, and R and H are the nonlinear aero-engine system space and the linear invariant subspace after dimension-upgrading, respectively.
[0027] Step 2.1: Determine the basic structure, number of network layers, and number of neurons per layer of the neural network used for dimensionality increase and decrease to obtain a deep neural network model.
[0028] The deep neural network model described includes an encoder and decoder for a Variational Autoencoder (VAE); VAEs offer significant advantages in generative models. First, they achieve unsupervised learning by learning the latent distribution of the data, enabling the generation of new samples that are similar to but not identical to the original data. Second, VAEs introduce continuity and smoothness into the latent space, making interpolation and manipulation within it easier, thus providing greater flexibility for sample generation and modification.
[0029] The Variational Autoencoder (VAE) is used to fit the dimensionality increase and decrease functions. The VAE encoder consists of three fully connected layers: one input layer, one output layer, and one hidden layer. The input layer has 2 neurons, the output layer has 64 neurons, and the hidden layer has 40 neurons. The vector generated by the VAE encoder is composed of the mean vector of the latent space in the first half and the variance vector in the second half. The dimensionality-increased latent state is then obtained through sampling. The VAE decoder has the same structure as the encoder, but its input layer has 32 neurons, its output layer has 2 neurons, and each hidden layer has 20 neurons. After training and optimization, the resulting VAE encoder and decoder represent the optimal dimensionality increase and decrease functions.
[0030] The expression for forward propagation of a neuron is:
[0031] h1 = tanh(W0I + b0)
[0032] h i+1 =tanh(W i h i+1 +b i )
[0033] Ψ(x)=tanh(W3h3+b3)
[0034] Among them W i ∈R 1XD ,h i ∈R 1 ,b i ∈R 1 Let i = 1, 2, 3 be the weights, hidden layer outputs, and biases of the i-th layer of the neural network, respectively; tanh be the hyperbolic tangent function; Ψ be the dimension-increasing function; and I be the network input.
[0035] Step 2.2 Determine the optimizer, activation function, and loss function of the deep neural network model.
[0036] The deep neural network model uses the ReLU activation function, the Adam optimizer, and the loss function is expressed as:
[0037]
[0038] In the formula, X p U represents the state observations of the system at time p. p Let ψ and ψ represent the input of the system at time p; -1 λ1, λ2, and λ3 represent the nonlinear functions fitted by the encoder network and decoder network, respectively; λ1, λ2, and λ3 are the weight coefficients of the corresponding terms, A is the state transition matrix, and B is the input matrix.
[0039] Step 2.3 updates the parameters of the deep neural network model from Step 2.1 using backpropagation to obtain the optimal dimensionality increase and dimensionality decrease functions. The specific parameter update formula is as follows:
[0040]
[0041] In the formula W m The weights between layer m and layer m-1; b represents the weights between the updated m-th and m-1-th layers; m The bias is between layer m and layer m-1; α represents the learning rate; and These are the deviation corrections for the accumulated weight and bias gradients during the training of the loss function; and These are the deviation corrections for the weights and the squared cumulative gradients of the bias gradients, respectively; ε is the smoothing term.
[0042] Step 3: Use the optimized deep neural network model from Step 2 to increase the dimensionality of the system state, obtaining a linear state space representation. Calculate the state transition matrix A and input matrix B using the least squares method, and then estimate the nonlinear state transition function and system output value in the original state space.
[0043] Step 3.1 Use the deep neural network from Step 2 to increase the dimensionality of the system state, and obtain the linear state space representation after dimensionality increase.
[0044] The process of dimensionality enhancement using neural networks can be represented as:
[0045] Y plift =ψ(Y p )
[0046] X plift =ψ(X) p )
[0047] Where Y p Y is the system output at time p. plift ,X plift For system state Y p ,Xp The dimensionality-upgraded state after neural network mapping. ψ is the dimensionality-upgrade function, which is the encoder of the variational autoencoder (VAE) in step 2.
[0048] The linear state-space equation can be expressed as:
[0049] Y plift =AX plift +BU p
[0050] Y p =ψ -1 (Y plift )
[0051] Where ψ -1 This is the dimensionality reduction function, which is the decoder of the variational autoencoder (VAE) in step 2.
[0052] Step 3.2: Use the least squares method to obtain the state transition matrix A and input matrix B in the upgraded state, and then determine the nonlinear state transition function f in the original state space.
[0053]
[0054] The nonlinear state transition function f in the original state space can be determined as:
[0055] f(X p )=ψ -1 +(A(ψ(X p )))+BU p )
[0056] Step 3.3: Based on the state estimate of the system at time p-1 and input value U p-1 The estimated system state at time p is obtained as follows:
[0057]
[0058] Step 3.4 uses root mean square error and mean absolute error as accuracy evaluation indicators to determine whether the deep neural network model with embedded Koopman operator meets the design accuracy requirements.
[0059] Root mean square error (NRMSE) and coefficient of determination (R²) 2 The calculation formulas for ) are as follows:
[0060]
[0061] The beneficial effects of this invention are as follows: Addressing the lack of a globally linearized description of nonlinear dynamics in the field of aero-engine system identification, this invention proposes an aero-engine system identification method integrating deep learning and Koopman operator theory. This achieves accurate identification of aero-engine systems and a globally linearized representation of nonlinear systems. This invention introduces Koopman operator theory into the field of aero-engine system identification, mapping the system state to an invariant subspace through a neural network, thereby achieving a precise description of the dynamic characteristics of key parameters of the aero-engine system. Attached Figure Description
[0062] Figure 1 This is a flowchart of an aero-engine system identification method based on the Koopman neural network.
[0063] Figure 2 This is a schematic diagram of the Koopman operator theory.
[0064] Figure 3 A schematic diagram of inputting fuel quantity (PRBS signal) for a single operating point.
[0065] Figure 4 This is the estimation result of the high-pressure rotor speed of an aero-engine model under single-operating-point PRBS signal excitation.
[0066] Figure 5 The results represent the estimation of the low-pressure rotor speed of an aero-engine model under single-operating-point PRBS signal excitation.
[0067] Figure 6 The results show the estimation of the high-pressure turbine outlet temperature of an aero-engine model under single-operation-point PRBS signal excitation.
[0068] Figure 7 This is the estimation result of the high-pressure compressor outlet pressure of an aero-engine model under single-operation-point PRBS signal excitation.
[0069] Figure 8 Schematic diagram for inputting fuel quantity (PRBS signal) at multiple operating points.
[0070] Figure 9 The results show the estimation of the high-pressure rotor speed of an aero-engine model under multi-operating-point PRBS signal excitation.
[0071] Figure 10 The results show the estimation of the low-pressure rotor speed of an aero-engine model under multi-operating-point PRBS signal excitation.
[0072] Figure 11 The results show the estimation of the high-pressure turbine outlet temperature of an aero-engine model under multi-operating-point PRBS signal excitation.
[0073] Figure 12The results show the estimation of the high-pressure compressor outlet pressure of an aero-engine model under multi-operating-point PRBS signal excitation. Detailed Implementation
[0074] A linearization and identification method for aero-engine systems based on Koopman neural networks, the process of which is as follows: Figure 1 As shown, it is implemented in the following steps:
[0075] Step 1: Determine the form of the aero-engine system model and rewrite the model based on the input fuel quantity and state observation values of the aero-engine system;
[0076] Step 1.1: The discrete state-space model of the aero-engine system is as follows:
[0077] X k+1 =f(X) k ,u k )
[0078] Where k represents the sampling time of the aero-engine system; x k =[x 1k ,x 2k ,…,x nk [This refers to the state observations of an aero-engine system, u] k =[u 1k ,u 2k ,…,u mk ] represents the input fuel quantity of the aero-engine system, and f is the state transition function.
[0079] Step 1.2: Divide the input and output data of the aero-engine system into training and test sets. Normalize the training set data using the L2 norm normalization method;
[0080]
[0081] Here, ||·||2 is used to calculate the L2 norm of the corresponding vector. x k and u k The result of normalization.
[0082] The normalized equations for the aero-engine system are as follows:
[0083]
[0084] The final dataset obtained is:
[0085]
[0086] Among them, X i Y represents the i-th sample in the training set; iIndicates with X i The corresponding i target samples, where m represents the total number of samples; U i Let X be the corresponding i-th input. i Y i and U i Specifically, it is expressed as follows:
[0087]
[0088] In the formula This represents the state observation value corresponding to the (j-2)th time step under the i-th sample. This represents the input value corresponding to the (j-2)th time step for the i-th sample.
[0089] The aero-engine system equations for this dataset can be rewritten as:
[0090] Y i =f(X) i U i )
[0091] Given a dataset S, the goal is to determine the system state transition function f while preserving the global linearity of the aero-engine system in the higher-dimensional space.
[0092] Step 2: Construct the Koopman operator dimensionality increase and decrease functions using a variational autoencoder, determine suitable loss functions and optimizers, and iteratively optimize the parameters of the deep neural network model to obtain the optimal dimensionality increase and decrease functions.
[0093] The core idea of Koopman operator theory is to map state observations to a higher-dimensional linear invariant subspace through an increasing-dimensional function, thereby linearizing the aero-engine system. Finally, the high-dimensional state is reduced to obtain the identification result. Compared to traditional methods, neural networks offer greater flexibility, adaptability, generalization ability, and parallel computing power in fitting increasing-dimensional and reducing-dimensional functions. They can more effectively capture complex nonlinear relationships, adapt to different types of data and problems, and achieve end-to-end learning. Therefore, this invention uses a neural network to fit increasing-dimensional and reducing-dimensional functions.
[0094] The principle of transforming the original nonlinear system into a linear system through an up-dimensional function is as follows: Figure 2 As shown in the figure, K represents the state transition matrix in the linear space, Ψ is the dimension-upgrading function, and R and H are the nonlinear aero-engine system space and the linear invariant subspace after dimension-upgrading, respectively.
[0095] Step 2.1: Determine the basic structure, number of network layers, and number of neurons per layer of the neural network used for dimensionality increase and decrease to obtain a deep neural network model.
[0096] The deep neural network model described includes an encoder and decoder for a Variational Autoencoder (VAE); VAEs offer significant advantages in generative models. First, they achieve unsupervised learning by learning the latent distribution of the data, enabling the generation of new samples that are similar to but not identical to the original data. Second, VAEs introduce continuity and smoothness into the latent space, making interpolation and manipulation within it easier, thus providing greater flexibility for sample generation and modification.
[0097] The Variational Autoencoder (VAE) is used to fit the dimensionality increase and decrease functions. The VAE encoder consists of three fully connected layers: one input layer, one output layer, and one hidden layer. The input layer has 2 neurons, the output layer has 64 neurons, and the hidden layer has 40 neurons. The vector generated by the VAE encoder is composed of the mean vector of the latent space in the first half and the variance vector in the second half. The dimensionality-increased latent state is then obtained through sampling. The VAE decoder has the same structure as the encoder, but its input layer has 32 neurons, its output layer has 2 neurons, and each hidden layer has 20 neurons. After training and optimization, the resulting VAE encoder and decoder represent the optimal dimensionality increase and decrease functions.
[0098] The expression for forward propagation of a neuron is:
[0099] h1 = tanh(W0I + b0)
[0100] h i+1 =tanh(W i h i+1 +b i )
[0101] Ψ(x)=tamh(W3h3+b3)
[0102] Among them W i ∈R 1XD ,h i ∈R 1 ,b i ∈R 1 Let i = 1, 2, 3 be the weights, hidden layer outputs, and biases of the i-th layer of the neural network, respectively; tanh be the hyperbolic tangent function; Ψ be the dimension-increasing function; and I be the network input.
[0103] Step 2.2 Determine the optimizer, activation function, and loss function of the deep neural network model.
[0104] The deep neural network model uses the ReLU activation function, the Adam optimizer, and the loss function is expressed as:
[0105]
[0106] In the formula, X p U represents the state observations of the system at time p. p Let ψ and ψ represent the input of the system at time p; -1 λ1, λ2, and λ3 represent the nonlinear functions fitted by the encoder network and decoder network, respectively; λ1, λ2, and λ3 are the weight coefficients of the corresponding terms, A is the state transition matrix, and B is the input matrix.
[0107] Step 2.3 updates the parameters of the deep neural network model from Step 2.1 using backpropagation to obtain the optimal dimensionality increase and dimensionality decrease functions. The specific parameter update formula is as follows:
[0108]
[0109] In the formula W m The weights between layer m and layer m-1; b represents the weights between the updated m-th and m-1-th layers; m The bias is between layer m and layer m-1; α represents the learning rate; and These are the deviation corrections for the accumulated weight and bias gradients during the training of the loss function; and These are the deviation corrections for the squared cumulative weight and bias gradients, respectively; ε is the smoothing term.
[0110] Step 3: Use the optimized deep neural network model from Step 2 to increase the dimensionality of the system state, obtaining a linear state space representation. Calculate the state transition matrix A and input matrix B using the least squares method, and then estimate the nonlinear state transition function and system output value in the original state space.
[0111] Step 3.1 Use the deep neural network from Step 2 to increase the dimensionality of the system state, and obtain the linear state space representation after dimensionality increase.
[0112] The process of dimensionality enhancement using neural networks can be represented as:
[0113] Y plift =ψ(Y p )
[0114] X plift =ψ(X) p )
[0115] Where Y p Y is the system output at time p. plift ,X plift For system state Yp ,X p The dimensionality-upgraded state after neural network mapping. ψ is the dimensionality-upgrade function, which is the encoder of the variational autoencoder (VAE) in step 2.
[0116] The linear state-space equation can be expressed as:
[0117] Y plift =AX plift +BU p
[0118] Y p =ψ -1 (Y plift )
[0119] Where ψ -1 This is the dimensionality reduction function, which is the decoder of the variational autoencoder (VAE) in step 2.
[0120] Step 3.2: Use the least squares method to obtain the state transition matrix A and input matrix B in the upgraded state, and then determine the nonlinear state transition function f in the original state space.
[0121]
[0122] The nonlinear state transition function f in the original state space can be determined as:
[0123] f(X p )=ψ -1 (A(ψ(X p ))+BU p )
[0124] Step 3.3: Based on the state estimate of the system at time p-1 and input value U p-1 The estimated system state at time p is obtained as follows:
[0125]
[0126] Step 3.4 uses root mean square error and mean absolute error as accuracy evaluation indicators to determine whether the deep neural network model with embedded Koopman operator meets the design accuracy requirements.
[0127] Root mean square error (NRMSE) and coefficient of determination (R²) 2 The calculation formulas for ) are as follows:
[0128]
[0129] The following embodiments are used to verify the beneficial effects of the present invention:
[0130] The NASA-developed JT9D turbofan engine system was selected to acquire the data required for aero-engine system identification. The state variable x... k The high-pressure rotor speed, low-pressure rotor speed, and high-pressure compressor outlet pressure of the aero-engine are set as x. k =[x 1k ,x 2k ,x 3k ], Input variable u k Sets the input fuel quantity for the aircraft engine.
[0131] Pseudo-binary random signal (PRBS) is used as the input excitation to obtain identification data. The characteristic curves of single-operating-point and multi-operating-point PRBS are shown below. Figure 3 and 8 As shown.
[0132] For PRBS, the accuracy of the proposed method is verified by using the first half of the data as the training set and the second half as the test set. Under a single operating point, the data sampling time is initially set to 0.04s, and the first 30 seconds of sampling data from the aero-engine model are used to identify system parameters. The identified parameters are then substituted into the input data for the latter 30 seconds to obtain the predicted values of the aero-engine output data. These predicted values are then compared with the actual output values of the aero-engine system in the latter 30 seconds to conduct a prediction experiment for system identification. Under multiple operating points, the first 90 seconds are used as sampling data, and the latter 90 seconds as test data, with other conditions remaining unchanged. Experiments are conducted under both single and multiple operating points. The comparison results under the single operating point condition are as follows: Figures 4 to 7 As shown, the comparison results under multiple operating point conditions are as follows: Figures 9 to 12 As shown.
[0133] Compared to traditional least squares identification methods, the method proposed in this invention achieves a certain degree of improvement in identification accuracy. This is achieved by using normalized root mean square error (NRMSE) and coefficient of determination (R²). 2 As a measure of estimation accuracy, NRMSE and R 2 The calculation method is as follows:
[0134]
[0135] Where y i For the true value, For predicted values, This is the average of the predicted values.
[0136] The estimation accuracy of the two methods at a single operating point with an input fuel quantity of 1.9 kg / s is compared in Table 1 below:
[0137] Table 1. Comparison of Single-Workpoint Methods
[0138]
[0139] The estimation accuracy of the two methods at multiple operating points with input fuel quantities of 0.883, 1.8988, and 0.35 kg / s is compared in Table 2 below:
[0140] Table 2. Comparison of Multi-Operating-Point Methods
[0141]
[0142] The experimental results show that the Koopman predicted curve and the actual curve have a high degree of overlap, and the normalized root mean square error between the Koopman predicted value and the actual value is very small. This demonstrates that the proposed method effectively avoids overfitting, the predicted system parameters have high accuracy, and it effectively suppresses the influence of noise.
Claims
1. A method for identifying aero-engine systems based on a Koopman neural network, characterized in that, The steps are as follows: Step 1: Determine the form of the aero-engine system model and rewrite the model based on the input fuel quantity and state observation values of the aero-engine system; Step 1.1: The discrete state-space model of the aero-engine system is as follows: x k+1 =f(x k ,u k ) Where k represents the sampling time of the aero-engine system; x k =[x 1k ,x 2k ,…,x nk [This refers to the state observations of an aero-engine system, u] k =[u 1k ,u 2k ,…,u mk ] represents the input fuel quantity of the aero-engine system, and f is the state transition function; Step 1.2: Divide the input and output data of the aero-engine system into training and test sets; normalize the training set data using the L2 norm normalization method; Where, ||·||2 is used to calculate the L2 norm of the corresponding vector; x k and u k The result of normalization; The normalized equations for the aero-engine system are as follows: The final dataset obtained is: Among them, X i Y represents the i-th sample in the training set; i Indicates with X i The corresponding i target samples, where m represents the total number of samples; U i For the corresponding i-th input; where X i Y i and U i Specifically, it is expressed as follows: In the formula This represents the state observation value corresponding to the (j-2)th time step under the i-th sample; This represents the input value corresponding to the (j-2)th time step for the i-th sample; The aero-engine system equations for this dataset can be rewritten as: Y i =f(X i ,U i ) Given a dataset S, the goal is to determine the system state transition function f while preserving the global linearity of the aero-engine system in the higher-dimensional space. Step 2: Construct the Koopman operator dimensionality increase and decrease functions using a variational autoencoder, determine suitable loss functions and optimizers, and iteratively optimize the parameters of the deep neural network model to obtain the optimal dimensionality increase and decrease functions. Step 2.1: Determine the basic structure, number of layers, and number of neurons per layer of the neural network used for dimensionality increase and decrease to obtain the deep neural network model; The deep neural network model described includes an encoder and a decoder for a variational autoencoder (VAE). The encoder and decoder of a Variational Autoencoder (VAE) are used to fit the dimensionality increase and decrease functions. The VAE encoder consists of three fully connected layers: one input layer, one output layer, and one hidden layer. The input layer has 2 neurons, the output layer has 64 neurons, and the hidden layer has 40 neurons. The vector generated by the VAE encoder is composed of the mean vector of the latent space in the first half and the variance vector in the second half. The latent state after dimensionality increase is obtained through sampling. The VAE decoder has the same structure as the encoder, but the decoder has 32 neurons in the input layer, 2 neurons in the output layer, and 20 neurons in each hidden layer. The encoder and decoder of the VAE obtained after training and optimization are the optimal dimensionality increase and decrease functions. The expression for forward propagation of a neuron is: h1 = tanh(W0I + b0) h i+1 = tanh(W i h i+1 +b i ) Ψ(x)=tanh(W3h3+b3) Among them W i ∈R 1XD ,h i ∈R 1 ,b i ∈R 1 Let i = 1, 2, 3 be the weights, hidden layer outputs, and biases of the i-th layer of the neural network, respectively; tanh is the hyperbolic tangent function; Ψ is the dimension-increasing function; and I is the network input. Step 2.2 Determine the optimizer, activation function, and loss function of the deep neural network model. The deep neural network model uses the ReLU activation function, the Adam optimizer, and the loss function is expressed as: In the formula, X p U represents the state observations of the system at time p. p Let ψ and ψ represent the input of the system at time p; -1 λ1, λ2, and λ3 represent the nonlinear functions fitted by the encoder network and decoder network, respectively; λ1, λ2, and λ3 are the weight coefficients of the corresponding terms, A is the state transition matrix, and B is the input matrix; Step 2.3 updates the parameters of the deep neural network model from Step 2.1 using backpropagation to obtain the optimal dimensionality increase and dimensionality decrease functions. The specific parameter update formula is as follows: In the formula W m The weights between layer m and layer m-1; b represents the weights between the updated m-th and m-1-th layers; m The bias is between layer m and layer m-1; α represents the learning rate; and These are the deviation corrections for the accumulated weight and bias gradients during the training of the loss function; and These are the deviation corrections for the weights and the squared cumulative bias gradients, respectively; ε is the smoothing term. Step 3: Use the optimized deep neural network model from Step 2 to increase the dimensionality of the system state, obtaining a linear state space representation. Calculate the state transition matrix A and input matrix B using the least squares method, and then estimate the nonlinear state transition function and system output value in the original state space. Step 3.1 Use the deep neural network from Step 2 to increase the dimensionality of the system state, and obtain the linear state space representation after dimensionality increase; The process of dimensionality enhancement using neural networks can be represented as: AND plift =ψ(Y p ) X pli =ψ(X p ) Where Y p Y is the system output at time p. plift ,X pli For system state Y p ,X p The dimensionality-upgraded state after neural network mapping; ψ is the dimensionality-upgraded function, which is the encoder of the variational autoencoder (VAE) in step 2; The linear state-space equation can be expressed as: Y plift =AX plift +BU p AND p =ψ -1 (AND plift ) Where ψ -1 This is the dimensionality reduction function, which is the decoder of the variational autoencoder (VAE) in step 2; Step 3.2: Use the least squares method to obtain the state transition matrix A and input matrix B in the upgraded state, and then determine the nonlinear state transition function f in the original state space. The nonlinear state transition function f in the original state space can be determined as: f(X p )=ψ -1 (A(ψ(X p ))+BU p ) Step 3.3: Based on the state estimate of the system at time p-1 and input value U p-1 The estimated system state at time p is obtained as follows: Step 3.4 uses root mean square error and mean absolute error as accuracy evaluation indicators to determine whether the deep neural network model with embedded Koopman operator meets the design accuracy requirements. Root mean square error (NRMSE) and coefficient of determination (R²) 2 The calculation formulas for ) are as follows: