A method for predicting thermal-dynamic dual-hysteresis coupling parameters of an electric spindle

Abstract

The present application relates to the technical field of precision manufacturing and numerical control machine tool, and more particularly to a kind of electric spindle thermal-mechanical double hysteresis coupling parameter prediction method.The steps are as follows:S1: collect multi-dimensional operating state signal and process;S2: the vibration time-frequency feature map, current spectrum and temperature gray image after processing are channel superimposed, and three-dimensional feature tensor is formed;S3: construct ResNet-Attention electric spindle thermal-mechanical double hysteresis coupling parameter time series prediction model, input fusion feature vector into trained prediction model, and obtain the optimal thermal-mechanical double hysteresis parameter time series prediction model;S4: the thermal-mechanical double hysteresis parameter time series prediction model is dynamically simulated to obtain predicted value.The electric spindle thermal-mechanical double hysteresis coupling parameter prediction method provided by the present application, by fusing multi-source heterogeneous data, fully excavates the internal relation and complementary information between different data sources, effectively improves the prediction accuracy of the model on electric spindle thermal-mechanical double hysteresis coupling parameter.

Description

Technical Field

[0001] The present invention relates to the technical field of precision manufacturing and numerically controlled machine tools, and in particular to a method for predicting thermal-dynamic dual hysteresis coupling parameters of an electric spindle. Background Art

[0002] During high-speed machining, electric spindles experience thermal deformation hysteresis (thermal hysteresis) and dynamic response delay (dynamic hysteresis) due to electromagnetic losses (Joule heating) and sudden changes in dynamic loads. This results in complex nonlinear characteristics of "thermal-dynamic" dual hysteresis coupling, significantly reducing the dynamic accuracy of high-precision spindles. The greatest difficulty in modeling the mechanism of this thermal-dynamic dual hysteresis problem lies in constructing a mechanistic model. This is manifested in the significant differences in the temporal and spatial resolution and physical dimensions of multi-source heterogeneous data, making it difficult to characterize the coupling effects using traditional single-source modeling. Furthermore, the nonlinear characteristics of thermal-dynamic hysteresis make it difficult for traditional mechanistic models to accurately describe the dynamic interactions of multi-physics coupling, particularly the historical state dependence and time-varying characteristics of dynamic parameters (such as thermal expansion coefficient, nonlinear stiffness, thermal resistivity, and electromagnetic loss coefficient). The rapid evolution of the thermal-dynamic state during high-speed electric spindle operation requires a high-frequency model update capability. Traditional offline calibration methods struggle to track parameter changes in real time, and the solution of high-dimensional nonlinear differential equations is inefficient.

[0003] In order to solve the problems of insufficient prediction accuracy and real-time bottleneck in the prediction of thermal-dynamic dual hysteresis coupling parameters of electric spindles caused by idealized model assumptions, parameter uncertainty and dynamic coupling complexity, data assimilation technology can provide more accurate prediction results by combining actual measurement data with mathematical models.

[0004] To this end, a method for predicting the thermal-dynamic dual hysteresis coupling parameters of an electric spindle is designed to provide a technical solution to the above technical problems. Summary of the Invention

[0005] Based on this, it is necessary to provide a method for predicting the thermal-dynamic dual hysteresis coupling parameters of an electric spindle in order to solve the technical problems raised in the above background technology.

[0006] In order to solve the above technical problems, the present invention adopts the following technical solutions:

[0007] A method for predicting the thermal-dynamic dual hysteresis coupling parameters of an electric spindle, comprising the following steps:

[0008] S1: The vibration sensor, temperature sensor, and current sensor installed on the electric spindle synchronously collect multi-dimensional operating status signals; the original signals collected by each sensor are pre-processed, converted into time-frequency domains, and normalized to obtain a standardized feature matrix;

[0009] S2: Superimpose the processed vibration time-frequency feature map, current spectrum map and temperature grayscale image to form a three-dimensional feature tensor and perform data enhancement;

[0010] S3: Construct a ResNet-Attention motorized spindle thermal-dynamic dual hysteresis coupled parameter timing prediction model, and input the fused feature vector into the trained prediction model to obtain the optimal thermal-dynamic dual hysteresis parameter timing prediction model;

[0011] S4: Dynamically simulate the thermal-dynamic dual hysteresis parameter timing prediction model through the deep residual network model within the preset time window to obtain the predicted value.

[0012] As a preferred embodiment of the method for predicting thermal-dynamic dual hysteresis coupling parameters of an electric spindle provided by the present invention, in step S1, the steps are as follows:

[0013] 1) The vibration signal is collected by the acceleration sensor installed on the electric spindle bearing seat, the current signal is collected by the current transformer connected in series to the electric spindle power supply circuit, and the infrared thermal image signal is collected by the infrared thermal imager facing the key heating parts of the electric spindle;

[0014] 2) The original resolution of the infrared image captured by the infrared thermal imager is 640 × 480 pixels. After grayscale conversion, it is compressed to 224 × 224 pixels using bilinear interpolation to match the dimensions of the ResNet network input layer.

[0015] 3) Perform continuous wavelet transform on the vibration signal to generate a time-frequency energy distribution map. The Morlet wavelet basis function is used, and the scale parameter is set to 50 levels of discretization. The time-frequency map is converted to a logarithmic scale and normalized to a grayscale range of 0-255. The image is resized to a resolution of 224×224 using bicubic interpolation.

[0016] 4) The three-phase current signals were subjected to Park vector transformation to generate a two-dimensional time domain trajectory. A sliding window Fourier transform was used with a window length set to five times the fundamental period. The spectrum was normalized using the Z-score method and the image size was adjusted to 224 × 224 resolution using the regional maximum sampling method.

[0017] As a preferred embodiment of the method for predicting thermal-dynamic dual hysteresis coupling parameters of an electric spindle provided by the present invention, in step S2, the steps are as follows:

[0018] The order of channel arrangement of the three-dimensional feature tensor is the temperature grayscale image in the first channel, the vibration time-frequency image in the second channel, and the current spectrum image in the third channel;

[0019] The three-dimensional feature tensor is enhanced by random rotation, mirror flipping, and adding Gaussian noise.

[0020] As a preferred embodiment of the method for predicting thermal-dynamic dual hysteresis coupling parameters of an electric spindle provided by the present invention, in step S3, the steps are as follows:

[0021] A. Use the pre-trained ResNet-50 network as the basic architecture, modify the number of input layer channels to adapt to the 3D fusion data volume, and freeze the first 10 convolution kernels to retain general feature extraction capabilities;

[0022] B. The adaptive feature fusion module Attention connects the outputs of each residual block. The adaptive feature fusion module includes a channel attention submodule and a spatial attention submodule;

[0023] C. Adjust the number of output layer nodes to match the time-varying parameters of the electric spindle. The output layer uses the Softmax activation function, and the output includes time-varying parameters such as the thermal expansion coefficient, nonlinear stiffness, thermal resistivity, and electromagnetic loss coefficient of the electric spindle.

[0024] D. Freeze the underlying network parameters of the pre-trained ResNet-50 and only fine-tune the newly added adaptive feature fusion module and the top output layer parameters.

[0025] As a preferred embodiment of the method for predicting thermal-dynamic dual hysteresis coupling parameters of an electric spindle provided by the present invention, in step S4, the steps are as follows:

[0026] By updating the state prediction value from arrive Members within the time window of the moment;

[0027] The current time point The model parameter results are used as state prediction values;

[0028] Determine the current time point Is there an observation value? If there is no observation value, go directly to the next time point , and store the data; if there are observations, generate a set for subsequent error analysis and Kalman gain calculation;

[0029] Perform error analysis on the generated set, calculate the Kalman gain and calculate the current time point The state analysis value is stored and the calculation result is used as the next time point Use until Always updated time.

[0030] As a preferred implementation of the method for predicting thermal-dynamic dual hysteresis coupling parameters of an electric spindle provided by the present invention, the observation value is the real-time operation data of the electric spindle obtained by sensor monitoring.

[0031] As a preferred embodiment of the method for predicting the thermal-dynamic dual hysteresis coupling parameters of an electric spindle provided by the present invention, the ensemble Kalman filter algorithm directly calculates the covariance matrix of the state variable and the observation variable. and the observation-forecast covariance matrix , the expression is as follows:

[0032] ;

[0033] ;

[0034] Among them, the true value of the state variable Use the ensemble mean instead.

[0035] It can be seen without a doubt that the above-mentioned technical solution of this application can definitely solve the technical problem to be solved by this application.

[0036] At the same time, through the above technical solutions, the present invention has at least the following beneficial effects:

[0037] 1. The present invention provides a method for predicting the thermal-dynamic dual hysteresis coupling parameters of an electric spindle. This method extracts thermal gradient evolution characteristics from infrared thermal images, separates high-frequency hysteresis modal components from vibration data, and analyzes load-thermal coupling state parameters from current signals to form a joint feature vector that characterizes thermal deformation hysteresis, dynamic response lag, and electromagnetic-thermal coupling effects. By fusing multi-source heterogeneous data and fully exploring the inherent connections and complementary information between different data sources, the model effectively improves the prediction accuracy of the thermal-dynamic dual hysteresis coupling parameters of the electric spindle, thereby effectively solving the problem of unifying the spatiotemporal benchmarks of temperature field, vibration field, and electromagnetic field data.

[0038] 2. The present invention solves the problems of large deviation and lack of timeliness in hysteresis parameter prediction caused by thermal hysteresis, dynamic hysteresis and their coupling effects under service conditions of the electric spindle, and provides a real-time and highly reliable solution for active control of the dynamic precision of the electric spindle.

[0039] 3. The present invention assimilates the actual measured observation values ​​of the sensor with the model prediction values ​​through the ensemble Kalman filter (EnKF) data assimilation algorithm, effectively utilizes the actual measurement data to realize the dynamic adjustment of the model, and corrects and optimizes the time-varying parameters of the electric spindle thermal-dynamic dual hysteresis coupling parameter prediction model in real time, thereby improving the prediction accuracy of the electric spindle thermal-dynamic dual hysteresis coupling parameter prediction model; in addition, this method ensures that the electric spindle thermal-dynamic dual hysteresis coupling parameter prediction model can flexibly respond to the rapid changes of the electric spindle time-varying parameters, so that the transient characteristics of the actual operating conditions can be captured, further enhancing the adaptability of the model. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following is a brief introduction to the drawings required for use in the description of the embodiments. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0041] Figure 1 It is a schematic diagram of the overall process of the present invention;

[0042] Figure 2 Construct a flow chart for the deep residual network model based on transfer learning of the present invention;

[0043] Figure 3 This is a data assimilation flow chart of the present invention. DETAILED DESCRIPTION

[0044] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0045] In order to enable those skilled in the art to better understand the solutions of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings.

[0046] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features and technical solutions therein may be combined with each other.

[0047] It should be noted that similar reference numerals and letters denote similar items in the following drawings, and therefore, once an item is defined in one drawing, it does not require further definition or explanation in subsequent drawings.

[0048] Reference Figure 1-Figure 3 , a method for predicting the thermal-dynamic dual hysteresis coupling parameters of an electric spindle, the steps are as follows:

[0049] The vibration sensor, temperature sensor and current sensor installed on the electric spindle synchronously collect multi-dimensional operating status signals; and the original signals collected by each sensor are pre-processed, converted into time-frequency domains and standardized to obtain a standardized feature matrix;

[0050] The vibration sensor, temperature sensor, and current sensor installed on the electric spindle are used to synchronously collect multi-dimensional operating status signals. The steps are as follows:

[0051] The vibration signal is collected by the acceleration sensor installed on the electric spindle bearing seat, the current signal is collected by the current transformer connected in series in the electric spindle power supply circuit, and the infrared thermal image signal is collected by the infrared thermal imager facing the key heating parts of the electric spindle.

[0052] The processed vibration time-frequency feature map, current spectrum map and temperature grayscale image are channel-superimposed to form a three-dimensional feature tensor, and data enhancement is performed;

[0053] The original signals collected by each sensor are preprocessed, converted into time-frequency domains, and standardized. The steps are as follows:

[0054] The original resolution of the infrared images captured by the infrared thermal imager is 640×480 pixels. After grayscale conversion, they are compressed to 224×224 pixels using bilinear interpolation to match the dimensions of the ResNet network input layer. Continuous wavelet transform (CWT) is performed on the vibration signal to generate a time-frequency energy distribution map. The Morlet wavelet basis function is used with a scale parameter set to 50 levels of discretization. The time-frequency map is converted to a logarithmic scale and normalized to a grayscale range of 0-255. The image size is adjusted to 224×224 using bicubic interpolation. Park vector transform is performed on the three-phase current signal to generate a two-dimensional time domain trajectory map. The sliding window Fourier transform (STFT) is used with a window length set to 5 times the fundamental wave period. The spectrum map is normalized using Z-score processing, and the image size is adjusted to 224×224 using regional maximum sampling.

[0055] The processed vibration time-frequency feature map, current spectrum map, and temperature grayscale image are superimposed channel by channel to construct a three-dimensional feature tensor. The channels are arranged in the following order: the first channel is the temperature grayscale image, the second channel is the vibration time-frequency map, and the third channel is the current spectrum map. This three-dimensional feature tensor is then augmented with data, including random rotation, mirror flipping, and the addition of Gaussian noise.

[0056] Construct a ResNet-Attention motorized spindle thermal-dynamic dual hysteresis coupling parameter timing prediction model, and input the fused feature vector into the trained prediction model to obtain the optimal thermal-dynamic dual hysteresis parameter timing prediction model. The steps are as follows:

[0057] Using a pre-trained ResNet-50 network as the basic framework, the ResNet-50 architecture can identify the dynamic characteristics of the electric spindle system. The adaptive feature fusion module, Attention, is applied to the hidden layer output of the ResNet-50 architecture. The adaptive feature fusion module, which includes channel attention submodules and spatial attention submodules, can capture features at different scales and assign higher weights to important features. This allows the construction of a ResNet-Attention electric spindle thermal-dynamic dual hysteresis coupling parameter timing prediction model for electric spindles.

[0058] By freezing the underlying network parameters of the pre-trained ResNet-50, only the newly added adaptive feature fusion module and the top-level output layer parameters are fine-tuned to adapt to the task of compensating the thermal-dynamic dual hysteresis parameters of the electric spindle. This ensures that the model has good generalization ability with limited training data, and the output includes time-varying parameters such as the thermal expansion coefficient, nonlinear stiffness, thermal resistivity, and electromagnetic loss coefficient of the electric spindle.

[0059] The thermal-dynamic dual hysteresis parameter time series prediction model is dynamically simulated within a preset time window using a deep residual network model to obtain predicted values. The parameter prediction values ​​are used to update the members within the time window from tN to t. A determination is made as to whether there is an observation value at the current time point t. If there is no observation value, the algorithm proceeds directly to the next time point t+1 and stores the data. If there is an observation value, the thermal-dynamic dual hysteresis parameter time series prediction model is adjusted using the Ensemble Kalman Filter (EnKF) assimilation algorithm and dynamically simulated and evaluated again until it meets the preset standards. The observation value is the real-time operation data of the electric spindle obtained by sensor monitoring. The steps are as follows:

[0060] Step 1: Use the state prediction value to update the members in the time window from tN to time t;

[0061] Step 2: Use the model parameter results at the current time point t as the state prediction value;

[0062] Step 3: Determine whether there is an observation value at the current time point t; if there is no observation value, directly proceed to the next time point t+1 and store the data; if there is an observation value, generate a set for subsequent error analysis and Kalman gain calculation;

[0063] Step 4: Perform error analysis on the generated set, calculate the Kalman gain and the state analysis value at the current time point t, and store the calculation result for use at the next time point t+1 until it is updated from time tN to time t.

[0064] The calculation formula of the Kalman gain matrix K is:

[0065] ;

[0066] in, is the error covariance matrix, which represents the uncertainty of the state estimate; is the observation matrix, which maps the state space to the observation space; is the observation error covariance matrix, which represents the uncertainty of the observation value; express time; superscript Represents the predicted value.

[0067] EnKF directly calculates the covariance matrix of state variables and observation variables and the observation-forecast covariance matrix , and its calculation formula is:

[0068] ;

[0069] ;

[0070] Among them, the true value of the state variable Using the set mean instead, assuming that A collection, in Initialize each set at any time, and the specific process of data assimilation is as follows:

[0071] Step 1: Calculation The state variable model predicted value at time :

[0072] ;

[0073] in, express Moment The state variable analysis value of a set; express Moment The predicted value of the state variable of a set; express Time has come The changing relationship of the state variables at each moment is generally a nonlinear model operator; is the model error, which has a mean of 0 and a covariance matrix of Gaussian distribution.

[0074] Step 2: Calculate the state prediction covariance , covariance between state variables and observed variables , observation-forecast covariance , and its calculation formula is:

[0075] ;

[0076] ;

[0077] ;

[0078] ;

[0079] in: express The average value of the predicted value of the state variable at each moment.

[0080] Step 3: Calculation The Kalman gain matrix at time t is calculated as:

[0081] ;

[0082] Step 4: Calculation Moment state variable analysis values ​​of a set :

[0083] ;

[0084] in: express Observation data at the moment.

[0085] By constructing a ResNet-Attention time-series prediction model for the thermal-dynamic dual hysteresis coupling parameters of an electric spindle, the team extracted thermal gradient evolution characteristics from infrared thermal images, separated high-frequency hysteresis modal components from vibration data, and analyzed load-thermal coupling state parameters from current signals. This model then formed a joint feature vector that characterizes thermal deformation hysteresis, dynamic response lag, and electromagnetic-thermal coupling effects. By fusing multi-source heterogeneous data and exploring the inherent connections and complementary information between these different data sources, the team effectively improved the prediction accuracy of the thermal-dynamic dual hysteresis coupling parameters of the electric spindle.

[0086] A pre-trained ResNet-50 network is used as the basic framework. Leveraging its ability to identify dynamic features of electric spindle systems, an adaptive feature fusion module, Attention, consisting of channel and spatial attention submodules, is applied to its hidden layer outputs. This module captures features at different scales and assigns higher weights to important features. Consequently, a ResNet-Attention time-series prediction model for electric spindle thermal-dynamic dual-hysteresis coupling parameters is constructed. By freezing the underlying network parameters and fine-tuning the top-level parameters, the model's generalization capability is ensured. The final model output includes time-varying parameters such as the electric spindle's thermal expansion coefficient, nonlinear stiffness, thermal resistivity, and electromagnetic loss coefficient. By fusing multi-source heterogeneous data and exploring the inherent connections and complementary information between different data sources, the prediction accuracy of the electric spindle's thermal-dynamic dual-hysteresis coupling parameters is effectively improved.

[0087] Through the Ensemble Kalman Filter (EnKF) data assimilation algorithm, the actual measured observation values ​​of the sensor are assimilated with the model prediction values, and the actual measured data are fully utilized to dynamically adjust the thermal-dynamic dual hysteresis coupling parameter prediction model of the electric spindle, and the time-varying parameters of the model are corrected and optimized in real time, thereby improving the prediction accuracy of the prediction model; at the same time, this method enables the thermal-dynamic dual hysteresis coupling parameter prediction model of the electric spindle to flexibly respond to the rapid changes of the time-varying parameters of the electric spindle, capture the transient characteristics under actual operating conditions, and further enhance the adaptability of the model.

[0088] The preferred embodiments of the present invention disclosed above are intended only to help illustrate the present invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the present invention to the specific embodiments described. Obviously, many modifications and variations are possible based on the contents of this specification. These embodiments are selected and described in detail in this specification to better explain the principles and practical applications of the present invention, thereby enabling those skilled in the art to better understand and utilize the present invention. The present invention is limited only by the claims and their full scope and equivalents.

Claims

1. A method for predicting thermal-dynamic dual hysteresis coupling parameters of an electric spindle, characterized in that: Here are the steps: S1: The vibration sensor, temperature sensor, and current sensor installed on the electric spindle synchronously collect multi-dimensional operating status signals; the original signals collected by each sensor are pre-processed, converted into time-frequency domains, and normalized to obtain a standardized feature matrix; S2: superimpose the processed vibration time-frequency feature map, current spectrum map and temperature grayscale image to form a three-dimensional feature tensor; S3: Construct a ResNet-Attention motorized spindle thermal-dynamic dual hysteresis coupled parameter timing prediction model, and input the fused feature vector into the trained prediction model to obtain the optimal thermal-dynamic dual hysteresis parameter timing prediction model; S4: Dynamic simulation is performed within a preset time window using the ResNet-Attention motorized spindle thermal-dynamic dual hysteresis coupling parameter timing prediction model to obtain the predicted value; In step S1, the steps are as follows: 1) The vibration signal is collected by the acceleration sensor installed on the electric spindle bearing seat, the current signal is collected by the current transformer connected in series to the electric spindle power supply circuit, and the infrared thermal image signal is collected by the infrared thermal imager facing the key heating parts of the electric spindle; 2) The original resolution of the infrared image captured by the infrared thermal imager is 640 × 480 pixels. After grayscale conversion, it is compressed to 224 × 224 pixels using bilinear interpolation to match the dimensions of the ResNet network input layer. 3) Perform continuous wavelet transform on the vibration signal to generate a time-frequency energy distribution map. The Morlet wavelet basis function is used, and the scale parameter is set to 50 levels of discretization. The time-frequency map is converted to a logarithmic scale and normalized to a grayscale range of 0-255. The image is resized to a resolution of 224×224 using bicubic interpolation. 4) Perform Park vector transformation on the three-phase current signals to generate a two-dimensional time domain trajectory. A sliding window Fourier transform is used with a window length set to five times the fundamental period. The spectrum is normalized using the Z-score, and the image size is adjusted to 224 × 224 resolution using the regional maximum sampling method. In step S3, the steps are as follows: A. Use the pre-trained ResNet-50 network as the basic architecture, modify the number of input layer channels to adapt to the 3D fusion data volume, and freeze the first 10 convolution kernels to retain general feature extraction capabilities; B. The adaptive feature fusion module Attention connects the outputs of each residual block. The adaptive feature fusion module includes a channel attention submodule and a spatial attention submodule; C. Adjust the number of output layer nodes to match the time-varying parameters of the electric spindle. The output layer uses the Softmax activation function, and the output includes the thermal expansion coefficient, nonlinear stiffness, thermal resistivity, and electromagnetic loss coefficient of the electric spindle. D. Freeze the underlying network parameters of the pre-trained ResNet-50 and only fine-tune the newly added adaptive feature fusion module and the top output layer parameters.

2. The method for predicting thermal-dynamic dual hysteresis coupling parameters of an electric spindle according to claim 1, characterized in that: In step S2, the steps are as follows: The order of channel arrangement of the three-dimensional feature tensor is the temperature grayscale image in the first channel, the vibration time-frequency image in the second channel, and the current spectrum image in the third channel; The three-dimensional feature tensor is enhanced by random rotation, mirror flipping, and adding Gaussian noise.

3. The method for predicting thermal-dynamic dual hysteresis coupling parameters of an electric spindle according to claim 1, characterized in that: In step S4, the steps are as follows: By updating the state prediction value from arrive Members within the time window of the moment; The current time point The model parameter results are used as state prediction values; Determine the current time point Is there an observation value? If there is no observation value, go directly to the next time point , and store the data; if there are observations, generate a set for subsequent error analysis and Kalman gain calculation; Perform error analysis on the generated set, calculate the Kalman gain and calculate the current time point The state analysis value is stored and the calculation result is stored as the next time point Use until Always updated time.

4. The method for predicting thermal-dynamic dual hysteresis coupling parameters of an electric spindle according to claim 3, characterized in that: The observed value is the real-time operation data of the electric spindle obtained by sensor monitoring.

5. The method for predicting thermal-dynamic dual hysteresis coupling parameters of an electric spindle according to claim 3, characterized in that: If there are observations, the thermal-dynamic dual hysteresis parameter time series prediction model is adjusted through the ensemble Kalman filter algorithm assimilation algorithm and the dynamic simulation and accuracy evaluation are re-performed until it meets the preset standards; The ensemble Kalman filter algorithm directly calculates the covariance matrix of the state variables and the observation variables and the observation-forecast covariance matrix Calculation, the expression is as follows: ； ； Among them, the true value of the state variable Use the ensemble mean instead, express The predicted value of the state variable at time , is the observation matrix, which maps the state space to the observation space.

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