Nonlinear system global linearization modeling method, system, terminal and medium based on lie derivative and sparse identification

Abstract

The application relates to the field of nonlinear system modeling, and specifically provides a nonlinear system global linearization modeling method, system, terminal and medium based on Lie derivatives and sparse identification, the method comprising the following steps: obtaining an explicit nonlinear dynamic equation of a target system; based on the dynamic equation, constructing a candidate observation function library directly associated with the physical mechanism of the system by calculating Lie derivatives of system state variables or output variables; screening and reducing the dimension of the candidate observation function library by using a sparse identification method to obtain a low-dimensional observation function set; and based on the screened observation function and system operation data, identifying a limited-dimensional global linear model of the target system in the dimensioned observation space by using a data-driven algorithm. The application improves online solving efficiency and control real-time performance, and realizes efficient and reliable multi-target collaborative optimization control of systems such as wind driven generators.

Description

Technical Field

[0001] This invention relates to the field of nonlinear system modeling, specifically to a global linearization modeling method, system, terminal, and medium for nonlinear systems based on Lie derivatives and sparse identification. Background Technology

[0002] In advanced control of complex nonlinear systems such as wind turbines, control performance depends on the accuracy and computational efficiency of the internal predictive model. Existing modeling methods are mainly divided into two categories: locally linearized models and nonlinear models. Locally linearized models refer to Taylor expansions of nonlinear equations around a specific operating point. These models are simple in structure and computationally efficient, but are only effective within a small disturbance range. When operating conditions change significantly, the model accuracy drops sharply, making it difficult to cover fatigue load dynamics across all operating conditions, thus limiting the controller's performance over a wide operating range. Nonlinear models include physics-based high-fidelity models or data-driven black-box models. While these models can characterize global nonlinear characteristics, they are complex, highly nonlinear, and difficult to directly integrate with mature and efficient linear control theories and optimization algorithms, resulting in a heavy computational burden.

[0003] Koopman operator theory offers a new approach to the aforementioned problems. It maps the system to an infinite-dimensional linear space through a nonlinear observation function, thereby describing global nonlinear dynamics while preserving the linear structure. However, the key to the success of this method lies in the design of the observation function. Existing observation function design methods have significant shortcomings. For example, the general basis function method requires constructing a high-dimensional dictionary of hundreds of dimensions to obtain sufficient accuracy, leading to the "curse of dimensionality," high model redundancy, large computational cost, and a lack of physical interpretability. While the physical embedding method introduces domain knowledge to improve interpretability, it usually simply retains all derivation terms or neural network outputs without a systematic screening and compression mechanism. The dictionary size still has room for optimization, and the design process relies on expert experience and a lot of trial and error, lacking a unified and reproducible design criterion. Summary of the Invention

[0004] To address the aforementioned issues, this invention provides a global linearization modeling method, system, terminal, and medium for nonlinear systems based on Lie derivatives and sparse identification. The observation function is generated through sparse screening of a physical-guided candidate library of Lie derivatives, possessing strong interpretability and generalization ability. It eliminates reliance on expert experience and can directly empower control algorithms such as linear MPC and LQR, improving online solution efficiency and control real-time performance, and enabling efficient and reliable multi-objective collaborative optimization control for systems such as wind turbines.

[0005] In a first aspect, the technical solution of the present invention provides a global linearization modeling method for nonlinear systems based on Lie derivatives and sparse identification, comprising the following steps:

[0006] Obtain the explicit nonlinear dynamic equations of the target system, and based on the dynamic equations, construct a candidate observation function library that is directly related to the physical mechanism of the system by calculating the Lie derivatives of the system state variables or output variables.

[0007] A sparse identification method is used to screen and reduce the dimensionality of the candidate observation function library to obtain a low-dimensional observation function set;

[0008] Based on the filtered observation functions and system operation data, a data-driven algorithm is used to identify the finite-dimensional global linear model of the target system in the upgraded observation space.

[0009] Secondly, the technical solution of the present invention provides a global linearization modeling system for nonlinear systems based on Lie derivatives and sparse identification, comprising:

[0010] The candidate observation function library construction module is used to obtain the explicit nonlinear dynamic equations of the target system. Based on the dynamic equations, it constructs a candidate observation function library that is directly related to the physical mechanism of the system by calculating the Lie derivatives of the system state variables or output variables.

[0011] The observation function filtering module is used to filter and reduce the dimensionality of the candidate observation function library using a sparse identification method to obtain a low-dimensional observation function set.

[0012] The global linear model generation module is used to identify the finite-dimensional global linear model of the target system in the upgraded observation space based on the filtered observation functions and system operation data using data-driven algorithms.

[0013] Thirdly, the technical solution of the present invention provides a terminal, comprising:

[0014] The memory is used to store the global linearization modeling program for nonlinear systems based on Lie derivatives and sparse identification;

[0015] The processor is used to implement the steps of the global linearization modeling method for nonlinear systems based on Lie derivatives and sparse identification when executing the global linearization modeling program for nonlinear systems based on Lie derivatives and sparse identification.

[0016] Fourthly, the technical solution of the present invention provides a computer-readable storage medium storing a global linearization modeling program for nonlinear systems based on Lie derivatives and sparse identification. When the global linearization modeling program for nonlinear systems based on Lie derivatives and sparse identification is executed by a processor, it implements the steps of the global linearization modeling method for nonlinear systems based on Lie derivatives and sparse identification as described above.

[0017] As can be seen from the above technical solutions, this application has the following advantages:

[0018] (1) The constructed model is strictly linear in the observation space. At the same time, since the observation function originates from the global dynamics of the system, the linear model can approximate the original nonlinear system with high accuracy over a wide range of operating conditions, overcoming the narrow applicability of the traditional local linearization method.

[0019] (2) By constructing a physical candidate library based on Lie derivatives and subsequent sparse identification and screening, a low-dimensional set of observation functions can be automatically obtained. Compared with the method that requires hundreds of general basis functions, the state dimension of the final linear model is extremely low, which greatly reduces the online solution scale and computation time of subsequent model predictive control (MPC) and other optimization problems, and improves the real-time performance of the control algorithm.

[0020] (3) The candidate library of observation functions is directly generated from the system dynamic equations through Lie derivatives, and has a clear physical meaning. The sparse screening process further preserves the most critical dynamic features. The physical guidance and data verification approach makes the final model not only highly interpretable, but also shows good generalization prediction ability for working conditions outside the training data distribution.

[0021] (4) The entire observation function design process does not require manual trial and error based on expert experience, which is convenient for promotion and application. The resulting low-dimensional, linear state-space model can be directly used as the internal prediction model of a linear model predictive controller (MPC) or a linear quadratic regulator (LQR), thereby enabling efficient and reliable multi-objective collaborative optimization control of systems such as wind turbines. Attached Figure Description

[0022] To more clearly illustrate the technical solution of this application, the accompanying drawings used in the description will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0023] Figure 1 This is a schematic diagram of a global linearization modeling method for nonlinear systems based on Lie derivatives and sparse identification, provided as an embodiment of the present invention.

[0024] Figure 2 The pitch angle is the predicted result under different observation functions.

[0025] Figure 3 The results show the predicted rotor speed under different observation functions.

[0026] Figure 4 This represents the predicted generator speed under different observation functions.

[0027] Figure 5 The results show the predicted spindle torque under different observation functions.

[0028] Figure 6 The predicted thrust at the top of the tower is given under different observation functions.

[0029] Figure 7 The pitch angle is the predicted result for different training set sizes.

[0030] Figure 8 This represents the predicted rotor speed for different training set sizes.

[0031] Figure 9 This represents the predicted generator speed under different training set sizes.

[0032] Figure 10 The results show the prediction of the spindle torque under different training set sizes.

[0033] Figure 11 The predicted thrust at the top of the tower is given for different training set sizes.

[0034] Figure 12 This is a schematic block diagram of a global linearization modeling system for nonlinear systems based on Lie derivatives and sparse identification, provided for an embodiment of the present invention.

[0035] Figure 13 This is a schematic diagram of the structure of a terminal provided in an embodiment of the present invention. Detailed Implementation

[0036] To make the purpose, features, and advantages of this application more apparent and understandable, specific embodiments and accompanying drawings will be used to clearly and completely describe the technical solution protected by this application. Obviously, the embodiments described below are only some embodiments of this application, and not all embodiments. Based on the embodiments in this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0037] Unless otherwise defined, all technical and scientific terms used in this application have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used in this application and in the specification of this invention is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention.

[0038] Figure 1 This is a schematic flowchart illustrating a global linearization modeling method for nonlinear systems based on Lie derivatives and sparse identification, provided as an embodiment of the present invention. Figure 1The executing entity can be a global linearization modeling system for nonlinear systems based on Lie derivatives and sparse identification. The global linearization modeling method for nonlinear systems based on Lie derivatives and sparse identification provided in this embodiment of the invention is executed by a computer device; correspondingly, the global linearization modeling system for nonlinear systems based on Lie derivatives and sparse identification runs on the computer device. Depending on different requirements, the order of the steps in this flowchart can be changed, and some steps can be omitted.

[0039] like Figure 1 As shown, the method includes the following steps.

[0040] S1. Obtain the explicit nonlinear dynamic equations of the target system. Based on the dynamic equations, construct a library of candidate observation functions directly related to the physical mechanism of the system by calculating the Lie derivatives of the system's state variables or output variables.

[0041] S2 uses a sparse identification method to screen and reduce the dimensionality of the candidate observation function library, resulting in a low-dimensional observation function set.

[0042] S3, based on the filtered observation function and system operation data, uses a data-driven algorithm to identify the finite-dimensional global linear model of the target system in the upgraded observation space.

[0043] As a refinement and extension of the specific implementation of the above embodiments, in order to fully explain the specific implementation process of this embodiment, the following will provide possible embodiments to describe the specific implementation of the above steps in a non-limiting manner.

[0044] This embodiment uses a simplified model of the NREL5MW wind turbine as an example. It will model the aerodynamics, drive train, generator, tower and pitch system of the wind turbine. The following is a description of each part of the model.

[0045] (1) Aerodynamics

[0046] The aerodynamic component is the main source of nonlinearity in the wind turbine model, primarily including aerodynamic torque. , can be represented as:

[0047] (2.1)

[0048] (2.2)

[0049] (2.3)

[0050] (2.4)

[0051] in It is air density. It is the impeller radius. It is the equivalent wind speed at the generator rotor. It is the tip speed ratio. It is the propeller pitch angle. It is the generator rotor speed. It is the power coefficient, and its explicit formula is obtained through fitting.

[0052] (2) Transmission chain

[0053] The low-speed shaft dynamics are characterized using a single mass block model, and the rotor inertia is... and generator inertia Combined into equivalent inertia Its equation of motion can be expressed as:

[0054] (2.5)

[0055] (2.6)

[0056] (2.7)

[0057] in It's the gearbox ratio. It is the generator speed. This refers to the generator's electromagnetic torque. Main shaft torque. It can be calculated using the following formula:

[0058] (2.8)

[0059] (3) Generator

[0060] In steady-state analysis, the extremely short dynamic transient process of the system is often ignored, and the generator output active power can be expressed as:

[0061] (2.9)

[0062] in This refers to generator efficiency. It is assumed that during generator power control, the generator efficiency has been... By performing modeling compensation, the actual output power of the generator can be considered equal to the reference value of the active power of the wind turbine:

[0063] (2.10)

[0064] (4) Tower

[0065] The tower section is also a significant source of nonlinearity, primarily aerodynamic thrust. , can be represented as:

[0066] (2.11)

[0067] in It is the thrust coefficient, which is related to the power coefficient. Existence Relationship .

[0068] (5) Pitch system

[0069] Under simplified modeling assumptions, when the dynamic characteristics and nonlinear factors of the pitch actuator are ignored, and the inertia of the blade system is incorporated into the overall design of the wind turbine controller, the pitch angle θ can be adjusted using a gain-dispatch PI control algorithm based on speed deviation. The input to this controller is the filtered deviation signal between the generator speed ωf and the rated speed ωrated, and its control law can be expressed as:

[0070] (2.12)

[0071] (2.13)

[0072] (2.14)

[0073] (2.15)

[0074] in, and These are the proportional and integral gain constants of the PI controller. It is a correction factor. It is the maximum rate of change of the pitch angle. It is the generator speed. Measure the time constant of the filter. and All are gain constants.

[0075] This embodiment constructs a candidate observation function library based on the nonlinear dynamics model of the wind turbine and its Lie derivative information, and selects the combination of observation functions that has the greatest impact on the system dynamics through a sparse identification method. Finally, based on the selected observation vectors, a data-driven method is used to identify the global linearized model of the wind turbine.

[0076] The following is an introduction to Koopman operator theory. Koopman operator theory provides a linear operator framework for the analysis and prediction of nonlinear systems by characterizing the evolution of dynamical systems in the observation function space. Consider a discrete-time dynamical system... Introducing the observation function space Its elements are scalar functions defined on the state space. Koopman operator Defined as:

[0077] (3.1)

[0078] Therefore, the value of any observation function after one step of system evolution satisfies:

[0079] (3.2)

[0080] As can be seen, the Koopman operator acts directly on the observation function, rather than the state itself, describing the update pattern of the observation function over time using a linear operator. In prediction problems, the Koopman operator provides a linear prediction mechanism based on the observation function. Let's construct a vector-valued observation function. If there exists a finite-dimensional matrix Approximately satisfy Then it can be done through iteration. Achieve multi-step prediction of the observation function:

[0081] (3.3)

[0082] If the original state or output variables of interest It can be represented as A linear combination of these can map the prediction results in the observation space back to the state space or output space, thereby obtaining a prediction of the future behavior of the system.

[0083] In this embodiment, step S1, based on the dynamic equations, constructs a candidate observation function library directly related to the system's physical mechanism by calculating the Lie derivatives of the system's state variables or output variables. Specifically, this includes:

[0084] S1.1, Suppose that the dynamics of the target system are determined by a continuous-time vector field. Description, in which For state vectors, To control the input vector;

[0085] S1.2, for any smooth scalar function Its along the system vector field The first-order Lie derivative is defined as the function in Directional derivative in the direction:

[0086]

[0087] That The Lie derivative is given by the recursive relation and is expressed as:

[0088]

[0089] S1.3, combine each state component and its corresponding Lie derivatives of each order to form a set of candidate functions;

[0090] S1.4, using functions from the candidate function set as primitives, generates a vector-form candidate observation function library, represented as:

[0091]

[0092] in, The number of candidate functions.

[0093] In this embodiment, the Lie derivatives constituting the candidate function set only include the first-order and second-order Lie derivatives corresponding to the state components.

[0094] Specifically, under the condition that the nonlinear dynamic equations of the wind turbine are known, this embodiment constructs a set of candidate observation functions with clear physical meaning and consistent with the system vector field, providing a structured candidate set covering the dominant dynamic mode, so that subsequent screening can be carried out under physical constraints, thereby avoiding the problems of disordered expansion, redundant accumulation and insufficient interpretability that are common in high-dimensional state-input scenarios of general basis function dictionaries.

[0095] This embodiment organizes the candidate library construction into a reproducible process: using the wind turbine dynamics vector field as the generation core, Lie derivatives are recursively calculated for the prediction target to generate derivative chain features, and feature concatenation and unified library representation are performed on this basis, providing a physically reasonable function basis and a computable search space for subsequent sparse screening. Assume the system, given control input... The continuous-time dynamic form under action is For the nonlinear dynamics model of the wind turbine, the state vector and control vector are respectively expressed as:

[0096] (3.4)

[0097] To construct observation functions closely related to the system's physical structure, this embodiment builds a candidate observation function library based on Lie derivatives. For any smooth scalar function... Its along the system vector field The first-order Lie derivative is defined as the function in Directional derivative in the direction:

[0098] (3.5)

[0099] Furthermore, its The Lie derivative is given by the recursive relation and is expressed as:

[0100] (3.6)

[0101] Thus forming The derivative chain feature family starting from the first order. The key to using Lie derivatives in the construction of the observation dictionary lies in the fact that first-order terms directly correspond to the explicit dynamic update relationship of the system, while second-order and higher-order terms further introduce the local sensitivity and coupling effect of the vector field to the state, ensuring that the candidate features are consistent with the dynamic mechanism at the generation level. Based on this idea, this embodiment takes the key state variables of the wind turbine system as the basic prediction targets, recursively generates finite-order Lie derivative features for them, and uses them as an important component of the candidate observation function. Using the pitch angle... For example, its first-order Lie derivative is:

[0102] (3.7)

[0103] Its second-order Lie derivative is the second derivative of the first-order Lie derivative along the same vector field:

[0104] (3.8)

[0105] For ease of understanding, the following is an explanation of... The first and second Lie derivatives are further formalized and illustrated. Let... Then the first-order Lie derivative can be written as:

[0106] (3.9)

[0107] Based on this, the second-order Lie derivative Involving The second-order and mixed partial derivatives of each state variable can be written as:

[0108] (3.10)

[0109] Equation (3.10) shows that the second-order Lie derivative not only includes the quadratic term of the state derivative, but also includes the term derived from the state derivative. The introduced sensitivity propagation term can more completely characterize the coupling and nonlinear interaction mechanisms between wind turbine states. The aforementioned first and second-order partial derivatives can be calculated from the explicit expression of the wind turbine nonlinear model in the specific implementation, and serve as an important component in constructing the candidate observation function library.

[0110] To achieve a balance between model complexity and representational capability, this embodiment only calculates up to the second-order Lie derivative during the candidate library construction phase. When more detailed capture of higher-order nonlinear effects is required, the order of the Lie derivative can be extended according to equation (3.6), and the size of the final retained terms can be automatically controlled by the subsequent screening mechanism. Similarly, the state variables can be... We construct its first and second Lie derivatives respectively, thus obtaining derivative-type candidate observation functions directly related to the system dynamics. The original state and its first and second Lie derivatives are denoted as:

[0111] (3.11)

[0112] The candidate observation function library can then be represented as:

[0113] (3.12)

[0114] Each of them From set The elements constitute the model. Through the above method, the candidate observation functions are explicitly embedded with the physical structure and dynamic characteristics of the wind turbine model during the construction phase, forming a physically reasonable and structured set of function bases for subsequent sparse identification.

[0115] In this embodiment, step S2 uses a sparse identification method to screen and reduce the dimensionality of the candidate observation function library to obtain a low-dimensional observation function set, specifically including:

[0116] S2.1, based on the training dataset Based on the candidate observation function library, construct the candidate observation matrix and the target state matrix. The candidate observation matrix is ​​represented as follows:

[0117]

[0118] In the formula, Number of samples; target state matrix This is the state sequence for the next time step;

[0119] S2.2, by solving the band The regularized least squares problem yields the initial coefficient matrix. The least squares problem is represented as:

[0120]

[0121] In the formula, For regularization parameters; Denotes the Frobenius norm;

[0122] S2.3, for the initial coefficient matrix Thresholding is performed, represented as:

[0123] ,like

[0124] in, The preset threshold;

[0125] S2.4, iteratively perform regularization regression and thresholding operations until the support set of the coefficient matrix is ​​stable;

[0126] S2.5, based on the final obtained sparse coefficient matrix Extract the candidate functions corresponding to the non-zero columns to form a low-dimensional observation function set. , where r is the number of observation functions retained after filtering.

[0127] Specifically, candidate observation database The design emphasizes coverage and mechanistic consistency, thus often including redundant terms; simultaneously, different candidate functions exhibit significantly different explanatory power for system evolution. To obtain compact observation vectors suitable for Koopman operator training, this section further introduces the idea of ​​sparsity identification, transforming the observation function selection problem into a problem of identifying the sparse structure of the coefficient matrix, and stably eliminating candidate functions with smaller contributions through regularization and thresholding operations. To balance sparsity with prediction accuracy and generalization ability, grid search is further employed to systematically scan relevant hyperparameters. This process ultimately outputs several sets of low-dimensional observation function combinations, which can be directly used for training the global linearization model of subsequent data-driven algorithms.

[0128] Given a sequence of training data The upgraded feature matrix is ​​constructed using equation (3.12):

[0129] (3.13)

[0130] Then, a training objective matrix consistent with the modeling objective is selected. , indicating that in the training samples The set of states obtained under the action is usually taken as follows:

[0131] (3.14)

[0132] That is, the "first-order shift" state matrix obtained from discrete sampling is used to characterize the system at the input. From the action Evolved to The dynamic mapping. Establishing a linear representation relationship:

[0133] (3.15)

[0134] If the j-th candidate observation function ψj is... If the explanation contribution is small, then The j-th column will tend to zero, thus achieving automatic elimination and subset selection of the observation function. To ensure numerical stability when the candidate library has a high dimension, this embodiment uses a... Regularized regression serves as a fundamental step in coefficient estimation:

[0135] (3.16)

[0136] in This is used to alleviate multicollinearity and suppress noise amplification. Based on this, an element-wise thresholding operation is introduced on the coefficient matrix obtained from the regression to explicitly form a sparse support set:

[0137] ,like (3.17)

[0138] The process is iteratively executed within the "regression-threshold" framework until the support set converges. This iterative process can be viewed as a gradual pruning of the candidate pool: each round of thresholding removes candidates with smaller contributions from the support set, and subsequent regression steps only re-estimate the coefficients on the retained terms, thus gradually forming a stable sparse structure.

[0139] This embodiment employs a sparse identification method to screen and reduce the dimensionality of the candidate observation function library, obtaining a low-dimensional observation function set. It also includes adjusting the regularization parameter using grid search or cross-validation methods. With threshold parameter Joint optimization is performed to select the parameter combination that has the lowest prediction error on the validation set and the fewest number of observation functions, and a screening process is performed based on the optimized parameters.

[0140] Specifically, due to hyperparameters and These factors jointly determine the sparsity strength and fitting ability; this embodiment uses grid search for system configuration. Specifically, a preset candidate set { }and{ }, for each group ( , Repeat the iterative process of equations (3.16)–(3.17) to obtain the corresponding coefficient matrix. ( , The system analyzes the observation function combination that strikes a good balance between prediction accuracy and observation dimensionality by comprehensively evaluating the validation error and sparsity metrics. The resulting observation vector will then serve as the input space for subsequent Koopman operator training, enabling the automated acquisition of a low-dimensional, compact observation space from a high-coverage candidate library.

[0141] In this embodiment, step S3, based on the filtered observation function and system operation data, uses a data-driven algorithm to identify the finite-dimensional global linear model of the target system in the upgraded observation space, specifically including:

[0142] S3.1, Based on the filtered set of observation functions , training dataset Mapping to an r-dimensional observation space yields the observation state sequence. ;

[0143] S3.2, Constructing the data matrix of the r-dimensional observation space , and control input matrix ;

[0144] S3.3, based on Determine the projection matrix ,in And calculate the dimensionality-reduced observation sequence. and data matrix , ;

[0145] S3.4 employs an extended dynamic pattern decomposition algorithm with control input. In the dimensional observation space, the Koopman matrix is ​​identified by solving the following regularized least squares problem. :

[0146]

[0147] in, The state matrix, For the input matrix, To control the dimensions of the input, For regularization parameters;

[0148] S3.5, Identify the output matrix using the least squares method. This makes the system's physical output satisfy The finite-dimensional global linear model of the target system is obtained, expressed as:

[0149]

[0150] in, To predict the number of targets.

[0151] Specifically, based on the established observation vectors, wind turbine operation data is used to identify finite-dimensional Koopman operators through EDMDC, thereby obtaining a linear state-space model with control input in the observation space. Let the sampling time be... The collected discrete time series data are Based on this, state transition data pairs are constructed with the input sequence. , , The observation function obtained based on step S2. Define a vector-valued observation mapping:

[0152] (3.18)

[0153] Using the observation mapping (3.18), the training data is mapped to an r-dimensional observation space:

[0154] (3.19)

[0155] Construct the corresponding data matrix:

[0156] (3.20)

[0157] (3.21)

[0158] To obtain a more compact and numerically stable linearized model, this embodiment projects the observation vector onto a model composed of... Dominant dimensional subspace ( Let the projection matrix be... ,but

[0159] , , (3.22)

[0160] And after dimensionality reduction Establish a linear model in the 3D observation space:

[0161] (3.23)

[0162] in The state matrix, The input matrix is ​​denoted as .

[0163] make

[0164] (3.24)

[0165] Equation (3.23) can then be written in matrix form. .

[0166] By solving the following... The regularized least squares problem can identify the Koopman matrix. :

[0167] (3.25)

[0168] Depend on The block structure can be obtained and .

[0169] To return to the physical quantity output, a linear output mapping is introduced:

[0170] (3.26)

[0171] in This can be taken as the physical state of the wind turbine. or a subset thereof. Matrix Obtained through least squares reconstruction:

[0172] (3.27)

[0173] In summary, this embodiment yields a finite-dimensional Koopman global linearization model for prediction and control:

[0174] (3.28)

[0175] The following section utilizes NREL5MW wind turbine simulation data generated by the SimWindFarm toolbox to construct and verify the accuracy of the global linearization model of the wind turbine. The main parameters of the NREL5MW wind turbine used in this embodiment are shown in Table 1. The simulation data reading time step is set to 1 second, and the overall dataset is divided into a training set and a validation set. The validation set uses operating condition data that does not overlap with the training set. When constructing identification samples based on a sliding time window, each data window reads five state variables at time k to construct the output data matrix Y, and five state variables and two control input variables at time k-1 to construct the input data matrices X and U. Based on this, using the constructed training dataset, combined with the selected set of observation functions and the proposed finite-dimensional approximation method of the Koopman operator, data-driven modeling of the wind turbine system is performed, thereby identifying the global linearization model of the NREL5MW wind turbine.

[0176] Table 1: Main parameters of NREL 5 MW wind turbine

[0177]

[0178] To evaluate the effectiveness of the proposed observation function in global linearization modeling of wind turbines, this embodiment compares it with three commonly used general basis function observation functions: Polyharmonic Radial Basis Functions (RBF), Polynomial Basis Functions, and Random Fourier Features (RFF). All four observation functions are used for model identification based on the same training dataset, which contains 2000 data windows, i.e., 2000 sets of data. Samples were then used. Subsequently, the identified linear models were validated using the same validation data, and the root mean square error percentage (RMSE%) of the output was used as the model accuracy evaluation metric. Specifically, for a given output component j, let the set of sampling time indices corresponding to the truncated data interval used to calculate the error be . The sample size is Then we have:

[0179] (4.1)

[0180] (4.2)

[0181] (4.3)

[0182] The trained model was used for prediction validation and comparison. The comparison results are as follows: Figures 2 to 6 As shown in Table 2, Figures 2 to 6 The solid blue line, dashed dark green line, dashed light green line, dashed red line, and dashed purple line correspond to the prediction results of the true value, the polyharmonic radial basis function, the stochastic Fourier feature, the proposed method, and the multinomial basis function, respectively. It can be seen that the model dimension corresponding to the proposed observation function is only 7 dimensions, and it achieves prediction accuracy superior to or close to that of the general basis function framework across all output channels. In fatigue load-related... and The proposed method yields the lowest RMSE% among all methods, significantly outperforming the other three types of observation functions. While the Polyharmonic framework shows a slight advantage for certain variables, the RMSE% of the proposed method is only slightly different, indicating that its predictive performance is essentially on the same order of magnitude. In summary, the constructed global linearization model, while maintaining extremely low state dimensionality, can achieve high-precision predictions of key fatigue loads and major operating states. This characteristic makes it promising for significantly reducing the computational complexity of online optimization and control law solving in practical wind turbine control applications, greatly shortening algorithm solution time, and achieving high prediction accuracy while ensuring computational efficiency.

[0183] Table 2: Comparison of RMSE% of output variables under different observation functions

[0184]

[0185] To further analyze the impact of the proposed observation function selection method on the modeling data requirements, this embodiment also examines the effect of training set size variation on model accuracy under the proposed observation function. Four training set sizes (500, 1000, 2000, and 3000 data windows) were selected, and model identification was performed while maintaining other settings consistent. The prediction performance and RMSE% variation of the models obtained under different training set sizes were compared. The comparison results are as follows: Figures 7 to 11 As shown in Table 3, Figures 7 to 11In the diagram, the solid blue line, dashed dark green line, dashed light green line, dashed red line, and dashed purple line correspond to the true values, training set sizes of 500, 1000, 2000, and 3000, respectively. As the training set size increases from 500 to 2000 data windows, the RMSE% of the model on each output channel generally shows a monotonically decreasing trend, particularly at the pitch angle. and fatigue load-related variables and The error decay was particularly significant, indicating that appropriately increasing the number of samples helps improve the identification accuracy of the globally linearized model. However, when the training set was further increased to 3000 data windows, the RMSE% of most output channels rebounded to varying degrees, indicating that under the current observation function and model structure, too many training samples did not continue to bring accuracy gains, but may have led to a slight deterioration in prediction performance due to factors such as noise accumulation or overfitting. Overall, when an appropriate training set size is selected, such as 2000 data windows, the model achieves the best overall prediction performance on each key variable, achieving a relatively ideal trade-off between data utilization efficiency and identification accuracy.

[0186] Table 3: Comparison of RMSE% of each output variable of the observation function obtained by this method under different training set sizes

[0187]

[0188] The foregoing has described in detail an embodiment of a global linearization modeling method for nonlinear systems based on Lie derivatives and sparse identification. Based on the global linearization modeling method for nonlinear systems based on Lie derivatives and sparse identification described in the above embodiment, this invention also provides a global linearization modeling system for nonlinear systems based on Lie derivatives and sparse identification corresponding to this method.

[0189] Figure 12 This invention provides a schematic block diagram of a global linearization modeling system for nonlinear systems based on Lie derivatives and sparse identification. In this embodiment, the global linearization modeling system 1200 for nonlinear systems based on Lie derivatives and sparse identification can be divided into multiple functional modules according to its functions. A module, as referred to in this invention, is a series of computer program segments that can be executed by at least one processor and perform a fixed function, and is stored in memory.

[0190] The candidate observation function library construction module 1210 is used to obtain the explicit nonlinear dynamic equations of the target system. Based on the dynamic equations, it constructs a candidate observation function library that is directly related to the physical mechanism of the system by calculating the Lie derivatives of the system state variables or output variables.

[0191] The observation function screening module 1220 is used to screen and reduce the dimensionality of the candidate observation function library using a sparse identification method to obtain a low-dimensional observation function set.

[0192] The global linear model generation module 1230 is used to identify the finite-dimensional global linear model of the target system in the upgraded observation space based on the filtered observation function and system operation data using a data-driven algorithm.

[0193] The global linearization modeling system for nonlinear systems based on Lie derivatives and sparse identification in this embodiment is used to implement the aforementioned global linearization modeling method for nonlinear systems based on Lie derivatives and sparse identification. Therefore, the specific implementation of this system can be found in the embodiment section of the global linearization modeling method for nonlinear systems based on Lie derivatives and sparse identification above. Thus, its specific implementation can be referred to the description of the corresponding embodiments, and will not be elaborated here.

[0194] Furthermore, since the global linearization modeling system for nonlinear systems based on Lie derivatives and sparse identification in this embodiment is used to implement the aforementioned global linearization modeling method for nonlinear systems based on Lie derivatives and sparse identification, its function corresponds to that of the above method, and will not be repeated here.

[0195] Figure 13 This is a schematic diagram of a terminal 1300 provided in an embodiment of the present invention, including: a processor 1310, a memory 1320, and a communication unit 1330. The processor 1310 is used to implement the process steps of the above-described embodiment of the global linearization modeling method for nonlinear systems based on Lie derivatives and sparse identification when implementing the global linearization modeling program for nonlinear systems based on Lie derivatives and sparse identification stored in the memory 1320.

[0196] This invention also provides a computer storage medium, which may be a magnetic disk, optical disk, read-only memory (ROM), or random access memory (RAM), etc. The computer storage medium stores a global linearization modeling program for nonlinear systems based on Lie derivatives and sparse identification. When executed by a processor, this program implements the process steps of the aforementioned embodiment of the global linearization modeling method for nonlinear systems based on Lie derivatives and sparse identification.

[0197] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A global linearization modeling method for nonlinear systems based on Lie derivatives and sparse identification, characterized in that, Includes the following steps: Obtain the explicit nonlinear dynamic equations of the target system. Based on the dynamic equations, construct a candidate observation function library directly related to the physical mechanism of the system by calculating the Lie derivatives of the system state variables or output variables. The target system is a wind turbine system, and the explicit nonlinear dynamic equations are any one of the following: the aerodynamic torque equation, the transmission chain motion equation, the tower aerodynamic thrust equation, and the pitch system equation of the wind turbine system. A sparse identification method is used to screen and reduce the dimensionality of the candidate observation function library to obtain a low-dimensional observation function set; Based on the filtered observation functions and system operation data, a data-driven algorithm is used to identify the finite-dimensional global linear model of the target system in the upgraded observation space. Specifically, based on the dynamic equations, a candidate observation function library directly related to the system's physical mechanism is constructed by calculating the Lie derivatives of the system's state variables or output variables. This library includes: Suppose that the dynamics of the target system are given by a continuous-time vector field. Description, in which For state vectors, To control the input vector; For any smooth scalar function Its along the system vector field The first-order Lie derivative is defined as the function in Directional derivative in the direction: That The Lie derivative is given by the recursive relation and is expressed as: The state components and their corresponding Lie derivatives of each order are combined to form a set of candidate functions; Using functions from the candidate function set as primitives, a vector-form candidate observation function library is generated, represented as: in, The number of candidate functions; Among them, the Lie derivatives that constitute the candidate function set only include the first-order Lie derivatives and second-order Lie derivatives corresponding to the state components; Specifically, a sparse identification method is used to screen and reduce the dimensionality of the candidate observation function library, resulting in a low-dimensional observation function set, which includes: Based on training dataset Based on the candidate observation function library, construct the candidate observation matrix and the target state matrix. The candidate observation matrix is ​​represented as follows: In the formula, The number of samples; By solving the band The regularized least squares problem yields the initial coefficient matrix. The least squares problem is represented as: In the formula, For regularization parameters; Denotes the Frobenius norm; For the initial coefficient matrix Thresholding is performed, represented as: ,like in, The preset threshold; Iteratively perform regularized regression and thresholding operations until the support set of the coefficient matrix is ​​stable; Based on the final sparse coefficient matrix Extract the candidate functions corresponding to the non-zero columns to form a low-dimensional observation function set. , where r is the number of observation functions retained after filtering.

2. The global linearization modeling method for nonlinear systems according to claim 1, characterized in that, Target state matrix This is the state sequence for the next time step.

3. The global linearization modeling method for nonlinear systems according to claim 1, characterized in that, A sparse identification method is used to filter and reduce the dimensionality of the candidate observation function library, resulting in a low-dimensional observation function set, which also includes: Regularization parameters can be adjusted using grid search or cross-validation methods. With threshold parameter Joint optimization is performed to select the parameter combination that has the lowest prediction error on the validation set and the fewest number of observation functions, and a screening process is performed based on the optimized parameters.

4. The global linearization modeling method for nonlinear systems according to claim 3, characterized in that, Based on the filtered observation functions and system operation data, a data-driven algorithm is used to identify the finite-dimensional global linear model of the target system in the upgraded observation space, specifically including: Based on the filtered set of observation functions , training dataset Mapping to an r-dimensional observation space yields the observation state sequence. ; Constructing a data matrix for an r-dimensional observation space , and control input matrix ; based on Determine the projection matrix ,in And calculate the dimensionality-reduced observation sequence. and data matrix , ; An extended dynamic pattern decomposition algorithm with control input is employed. In the dimensional observation space, the Koopman matrix is ​​identified by solving the following regularized least squares problem. : in, The state matrix, For the input matrix, To control the dimensions of the input, For regularization parameters; The output matrix is ​​identified using the least squares method. This makes the system's physical output satisfy The finite-dimensional global linear model of the target system is obtained, expressed as: in, To predict the number of targets.

5. A global linearization modeling system for nonlinear systems based on Lie derivatives and sparse identification, characterized in that, include: The candidate observation function library construction module is used to obtain the explicit nonlinear dynamic equations of the target system. Based on the dynamic equations, by calculating the Lie derivatives of the system state variables or output variables, a candidate observation function library directly related to the physical mechanism of the system is constructed. The target system is a wind turbine system, and the explicit nonlinear dynamic equations are any one of the following: the aerodynamic torque equation, the transmission chain motion equation, the tower aerodynamic thrust equation, and the pitch system equation of the wind turbine system. The observation function filtering module is used to filter and reduce the dimensionality of the candidate observation function library using a sparse identification method to obtain a low-dimensional observation function set. The global linear model generation module is used to identify the finite-dimensional global linear model of the target system in the up-dimensional observation space based on the filtered observation functions and system operation data using data-driven algorithms. Specifically, based on the dynamic equations, a candidate observation function library directly related to the system's physical mechanism is constructed by calculating the Lie derivatives of the system's state variables or output variables. This library includes: Suppose that the dynamics of the target system are given by a continuous-time vector field. Description, in which For state vectors, To control the input vector; For any smooth scalar function Its along the system vector field The first-order Lie derivative is defined as the function in Directional derivative in the direction: That The Lie derivative is given by the recursive relation and is expressed as: The state components and their corresponding Lie derivatives of each order are combined to form a set of candidate functions; Using functions from the candidate function set as primitives, a vector-form candidate observation function library is generated, represented as: in, The number of candidate functions; Among them, the Lie derivatives that constitute the candidate function set only include the first-order Lie derivatives and second-order Lie derivatives corresponding to the state components; Specifically, a sparse identification method is used to screen and reduce the dimensionality of the candidate observation function library, resulting in a low-dimensional observation function set, which includes: Based on training dataset Based on the candidate observation function library, construct the candidate observation matrix and the target state matrix. The candidate observation matrix is ​​represented as follows: In the formula, The number of samples; By solving the band The regularized least squares problem yields the initial coefficient matrix. The least squares problem is represented as: In the formula, For regularization parameters; Denotes the Frobenius norm; For the initial coefficient matrix Thresholding is performed, represented as: ,like in, The preset threshold; Iteratively perform regularized regression and thresholding operations until the support set of the coefficient matrix is ​​stable; Based on the final sparse coefficient matrix Extract the candidate functions corresponding to the non-zero columns to form a low-dimensional observation function set. , where r is the number of observation functions retained after filtering.

6. A terminal, characterized in that, include: The memory is used to store the global linearization modeling program for nonlinear systems based on Lie derivatives and sparse identification; A processor, configured to implement the steps of the global linearization modeling method for nonlinear systems based on Lie derivatives and sparse identification as described in any one of claims 1 to 4 when executing the global linearization modeling program for nonlinear systems based on Lie derivatives and sparse identification.

7. A computer-readable storage medium, characterized in that, The readable storage medium stores a global linearization modeling program for nonlinear systems based on Lie derivatives and sparse identification. When the processor executes the global linearization modeling program for nonlinear systems based on Lie derivatives and sparse identification, it implements the steps of the global linearization modeling method for nonlinear systems based on Lie derivatives and sparse identification as described in any one of claims 1 to 4.

Images