Load identification method and system based on singular value adaptive improved regularization
By using the singular value adaptive improved regularization method, the noise sensitivity problem of ill-conditioned matrices in load identification is solved, thereby improving the stability and accuracy of load identification. This method is applicable to fields such as vibration testing of aerospace and mechanical equipment.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- SOUTHEAST UNIV
- Filing Date
- 2026-01-07
- Publication Date
- 2026-04-17
AI Technical Summary
In existing technologies for load identification, the singular values of ill-conditioned matrices decay rapidly, leading to increased noise sensitivity. Traditional regularization methods struggle to achieve a balance between information preservation and noise suppression, affecting the stability and accuracy of load identification.
An improved regularization method based on singular value adaptation is adopted. The adjustment exponent is calculated by singular value decomposition, and the optimal regularization parameter is selected by combining the generalized cross-validation function. An improved regularization objective function is constructed to achieve differentiated processing of singular values of different sizes.
It improves the stability and accuracy of load identification, enhances the ability to retain key structural information, reduces reliance on operator expertise, and is suitable for load identification in ill-conditioned systems.
Smart Images
Figure CN121880702A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of structural dynamic load identification, specifically a load identification method based on singular value adaptive improved regularization. Background Technology
[0002] In engineering fields such as aerospace structures and mechanical equipment vibration testing, it is often necessary to obtain external dynamic loads through the inversion of measured structural responses. However, the system matrix of the load identification problem often exhibits obvious ill-conditioned characteristics, such as rapid decay of singular values, large spectral distribution span, and high sensitivity to noise. In particular, when the singular value decay rate is fast, the noise sensitivity corresponding to the direction of small singular values increases significantly, making the solution process highly susceptible to interference from measurement errors, resulting in unstable and unreliable results.
[0003] To suppress noise amplification in ill-conditioned inverse problems, traditional methods commonly employ the Tikhonov regularization approach, introducing a uniform adjustment parameter to stabilize the solution. However, single-parameter regularization has limitations: different singular value directions exhibit vastly different sensitivities to noise; large singular value directions carry the majority of information but may be unnecessarily over-suppressed, while small singular value directions suffer the most severe noise amplification but may remain due to insufficient regularization strength. Since all singular value directions are treated with the same intensity, a suppression strategy that allocates noise as needed cannot be implemented, making it difficult to achieve an effective balance between "information preservation" and "noise suppression." This fails to meet the demands of modern engineering and scientific computing for high-stability, high-precision solutions. Therefore, this invention provides a load identification method and system based on singular value adaptive improved regularization. Summary of the Invention
[0004] The purpose of this invention is to provide a load identification method and system based on singular value adaptive improved regularization. By introducing an adaptive adjustment mechanism related to the size of singular values, it achieves enhanced suppression of small singular value directions and information protection of large singular value directions, thereby significantly improving the robustness and inversion accuracy of ill-conditioned matrix inversion, which has important engineering significance.
[0005] According to a first aspect of the present invention, in order to achieve the above-mentioned objective, the present invention provides the following technical solution: a load identification method based on singular value adaptive improved regularization, comprising the following steps: Singular value decomposition is performed on the pre-constructed system matrix, and the regulation index corresponding to each order is calculated based on the distribution characteristics of the singular values. The regulation index is used to apply differentiated suppression strength to singular values of different sizes. The range of regularization parameters is set, and the optimal regularization parameters are obtained by using the minimization criterion of the generalized cross-validation (GCV) function. By combining the adjustment index and regularization parameter for load identification, an improved regularization objective function is constructed and solved to obtain the optimal solution for load identification.
[0006] Furthermore, singular value decomposition is performed on the system matrix, and the regulation index corresponding to each order is calculated based on the characteristics of the singular value distribution, as follows: (21) Let H The singular value decomposition of the system identification matrix is as follows: (1) In the formula, U Let Σ be the left singular vector matrix, and Σ be the singular value matrix. VT It is a right singular vector matrix. σi , i =1,2,…,n is the nth i Singular values of order; (22) In order to achieve different levels of suppression for singular values of different sizes, a linear logarithmic mapping variable is introduced here. t j The details are as follows: (2) In the formula, σ j The system frequency response function matrix is the first... j The singular value of order is obviously And satisfy t j =0 corresponds to the direction of the maximum singular value. t j =1 corresponds to the direction of the minimum singular value; (23) Use this normalization index to define adaptive adjustment parameters β j : (3) In the formula, βj Corresponding to the j The adjustment parameters used for the singular values of order 1. β 0 represents the global adjustment coefficient, i.e., the suppression strength for the minimum singular value. Here, the condition number, which reflects the ill-conditioned nature of the matrix, is used to adjust the coefficient. β The value 0 is specifically: (4) In the formula, σ1 is the first singular value of the system frequency response function matrix, σ n The system frequency response function matrix is the first... n Singular value of order.
[0007] Furthermore, the range of regularization parameters is defined, and the optimal regularization parameters are selected using the minimization criterion of the generalized cross-validation function, as follows: (31) Assuming the initial state of the system is zero, the load identification model can be expressed as: Hx=y ,in x The load to be solved is... y Given the known response to the measurement, the optimization problem corresponding to the proposed improved regularization method is expressed as: (5) In the formula λ For regularization parameters, β j To adjust the exponent and control the suppression intensity for different singular values; σ 1 represents the first singular value of the system's frequency response function matrix; Solving equation (5) by combining equation (1) yields the expression for the identified load: (6) In the formula, It is a left singular value vector. It is a right singular value vector. For observation data y In the j The projection onto each basis vector; the superscript H denotes the conjugate transpose. (32) The generalized cross-validation method is used to select the regularization parameter. It automatically determines the optimal regularization parameter by minimizing the ratio between the residual term and the regularization term. The generalized cross-validation function adopts a simplified expression based on singular values as follows: (7) In the formula, This represents the "effective degrees of freedom" retained by the model. g j It is a filtering factor based on singular value decomposition, corresponding to the degree of retention of principal components; Based on the principle of minimizing the GCV function, the optimal regularization parameter is selected according to equation (7). λ opt : (8).
[0008] Furthermore, by combining the adjustment exponent and regularization parameter for load identification, an improved regularization objective function is constructed and solved to obtain the optimal solution for load identification, as follows: (41) Combine the adjustment exponents of each singular value direction calculated by equation (3) with the optimal regularization parameter obtained by equation (8) to perform load identification, and calculate the optimal solution of improved regularization according to the following formula: (9) In the formula, ximproved To improve the optimal solution of regularization, i.e. the result of load inversion.
[0009] According to a second aspect of the present invention, the present invention provides a load identification system based on singular value adaptive improved regularization, for implementing the load identification method based on singular value adaptive improved regularization described in the first aspect, comprising: The adjustment index calculation module is used to perform singular value decomposition on the pre-constructed system matrix and calculate the adjustment index corresponding to each order based on the singular value distribution characteristics. The adjustment index is used to apply differentiated suppression strength to singular values of different sizes. The regularization parameter filtering module is used to set the range of regularization parameters and uses the minimization criterion of the generalized cross-validation (GCV) function to filter out the optimal regularization parameters. The load identification module is used to identify loads by combining the adjustment exponent and regularization parameters, construct an improved regularization objective function and solve it to obtain the optimal solution for load identification.
[0010] According to a third aspect of the present invention, a terminal device is provided, including a memory, a processor, and a computer program stored in the memory and capable of running on the processor. The memory stores the computer program capable of running on the processor, and when the processor loads and executes the computer program, it employs the load identification method based on singular value adaptive improved regularization described in the first aspect.
[0011] According to a fourth aspect of the present invention, the present invention provides a storage medium comprising computer-executable instructions, which, when executed by a computer processor, are used to perform the load identification method based on singular value adaptive improved regularization as described in the first aspect.
[0012] According to a fifth aspect of the present invention, the present invention provides a computer program product comprising a computer program, which, when executed by a processor, is used to load and execute the load identification method based on singular value adaptive improved regularization as described in the first aspect.
[0013] Beneficial effects: 1. This invention effectively addresses the inherent shortcomings of the traditional Tikhonov regularization method in ill-conditioned matrix inversion by introducing a singular value adaptive adjustment mechanism. Traditional methods use a single regularization parameter to uniformly correct all singular value directions, while this invention achieves differentiated processing for singular values of different magnitudes, while enhancing the preservation of key structural information. It exhibits good adaptability and robustness, improving the stability and recognition accuracy of load inversion, and is suitable for solving load identification problems in ill-conditioned systems.
[0014] 2. This invention uses the minimization criterion of the generalized cross-validation (GCV) function to automatically determine the optimal regularization parameter, which can automatically adjust the global adjustment coefficient according to the specific ill-conditioning of the system matrix, demonstrating good adaptability and reducing the dependence on the professional experience of operators. Attached Figure Description
[0015] Figure 1 This is a flowchart illustrating the method described in this invention; Figure 2 These are the singular values and corresponding adjustment indices for the first 12 system orders of the rotor model; Figure 3 The time-domain plot shows the load identification results at a 5% noise level. Figure 4 The time-domain plot shows the load identification results at a 10% noise level. Figure 5 The time-domain plot shows the load identification results at a 15% noise level. Detailed Implementation The technical solutions of the embodiments of this disclosure will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this disclosure, and not all embodiments. Based on the embodiments of this disclosure, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this disclosure.
[0016] This embodiment employs a load identification simulation experiment on a Bo 105 rotor blade model using a singular value adaptive improved regularization-based load identification method. The rotor blade model adopts a NACA 23012 airfoil structure, with a blade length of 12.05 cm, an airfoil chord length of 2.61 cm, and is made of ABS plastic with a material density of 1040 kg / m³. 3 The elastic modulus is 2 GPa. The second harmonic excitation signal is set as the input load as shown below, with the specific expression as follows: (1) The sampling frequency was set to 2000Hz, the sampling duration to 1s, and the number of time-series sample points to 2000. The above-mentioned excitation signal was applied at the root of the blade and the response signal was measured at the tip of the blade. 5%, 10%, and 15% Gaussian white noise were applied to the response signal to simulate the real measurement environment.
[0017] This load identification method is implemented based on this simulation model.
[0018] Please see Figures 1-5 This invention provides a technical solution: a load identification method based on singular value adaptive improved regularization, comprising the following steps: S1. Perform singular value decomposition on the system matrix, and calculate the regulation exponents of each order based on the singular values of the system. This includes the following steps: (S11) Based on the known blade system matrix H Considering the first 6 modes, perform singular value decomposition on them: (2) In the formula, U Let Σ be the left singular vector matrix, and Σ be the singular value matrix. VT It is a right singular vector matrix. σi ( i =1,2,…,12) is the first i Singular values of order; (S12) Calculate the regulation index corresponding to each singular value direction based on the singular values of the first 12 system orders. The results are as follows: Figure 2 As shown, the calculation formula is as follows: (3) In the formula, σ j The system frequency response function matrix is the first... j Singular values of order, βj Corresponding to the j The adjustment parameters used for the singular values of order 1. β 0 represents the global adjustment coefficient, which is the suppression strength for the minimum singular value; S2. Define the range of regularization parameters and use the GCV function minimization criterion to select the optimal regularization parameters; specifically, this includes the following steps: (S21) Set the range of the regularization parameter λ to 10-20~1010, with 100 possible values, and iterate through the load calculation results corresponding to each regularization parameter λ: (4) In the formula, For the first i An improved regularization solution with a set of regularization parameter values, σ j The system frequency response function matrix is the first... j Singular values of order; It is a left singular value vector (data space basis vector). It is a right singular value vector (a basis vector in the model space). For observation data y In the j Projection onto each basis vector (superscript H indicates conjugate transpose); (S22) Calculate the GCV function value for each regularization parameter: (5) In the formula, This represents the "effective degrees of freedom" retained by the model.g j It is a filtering factor based on singular value decomposition, corresponding to the degree of retention of principal components; Based on the principle of minimizing the GCV function, the optimal regularization parameter is selected according to equation (5). λ opt : (6) S3. Combine the adjustment exponent and the optimal regularization parameter to identify the load and obtain the optimal solution for improved regularization; specifically including the following steps: (S31) Combine the adjustment exponents of each singular value direction calculated by equation (3) with the optimal regularization parameter obtained by equation (6), and substitute them into the known measurement response. y Load identification is performed, and the optimal solution for improved regularization is calculated according to the following formula: (7) In the formula, ximproved To improve the optimal solution of regularization, i.e. the result of load inversion.
[0019] Time-domain plots of load identification for rotor blade models at different noise levels are shown below. Figure 3-5 As shown in the figure, this method exhibits the best load identification accuracy under all three set noise levels. Figure 3 The image shows the time-domain results of load identification for a rotor blade model under 5% Gaussian white noise. Figure 4 The image shows the time-domain results of load identification for a rotor blade model under 10% Gaussian white noise. Figure 5 The time-domain results of load identification for the rotor blade model under 15% Gaussian white noise are shown.
[0020] In summary, this invention improves the traditional Tikhonov regularization method by introducing a singular value adaptive weighting mechanism. This mechanism can enhance the suppression of small singular value noise amplification through adaptive adjustment, while also enhancing the ability to retain key structural information. It has good adaptability and robustness, improves the stability and recognition accuracy of load inversion, and is suitable for solving the load recognition problem in ill-conditioned systems.
[0021] Example 2: This embodiment provides a load identification system based on singular value adaptive improved regularization, used to implement the load identification method based on singular value adaptive improved regularization described in Embodiment 1, including: The adjustment index calculation module is used to perform singular value decomposition on the pre-constructed system matrix and calculate the adjustment index corresponding to each order based on the singular value distribution characteristics. The adjustment index is used to apply differentiated suppression strength to singular values of different sizes. The regularization parameter filtering module is used to set the range of regularization parameters and uses the minimization criterion of the generalized cross-validation (GCV) function to filter out the optimal regularization parameters. The load identification module is used to identify loads by combining the adjustment exponent and regularization parameters, construct an improved regularization objective function and solve it to obtain the optimal solution for load identification.
[0022] Example 3: The present invention provides a terminal device, including a memory, a processor, and a computer program stored in the memory and capable of running on the processor. The memory stores the computer program capable of running on the processor. When the processor loads and executes the computer program, it employs the load identification method based on singular value adaptive improved regularization described in Embodiment 1.
[0023] It should be noted that the terminal device can be a computer device such as a desktop computer, a laptop computer, or a cloud server, and the terminal device includes, but is not limited to, a processor and a memory. For example, the terminal device may also include input / output devices, network access devices, and buses.
[0024] Furthermore, the processor can be a central processing unit (CPU). Of course, depending on the actual use, other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), off-the-shelf programmable gate arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. can also be used. The general-purpose processor can be a microprocessor or any conventional processor, etc., and this application does not limit it in this regard.
[0025] Example 4: The present invention provides a storage medium containing computer-executable instructions, which, when executed by a computer processor, are used to perform the load identification method based on singular value adaptive improved regularization described in Embodiment 1.
[0026] The computer program can be stored in a computer-readable medium. The computer program includes computer program code, which can be in the form of source code, object code, executable file, or certain middleware. The computer-readable medium includes any entity or device capable of carrying computer program code, recording media, USB flash drive, portable hard drive, magnetic disk, optical disk, computer memory, read-only memory (ROM), random access memory (RAM), electrical carrier signals, telecommunication signals, and software distribution media, etc. It should be noted that the computer-readable medium includes, but is not limited to, the above-mentioned components.
[0027] Example 5: The present invention provides a computer program product, which includes a computer program. When the computer program is executed by a processor, it is used to load and execute the load identification method based on singular value adaptive improved regularization described in Embodiment 1.
[0028] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus.
[0029] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.
[0030] In the description of this specification, references to terms such as "an embodiment," "example," "specific example," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of this disclosure. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.
Claims
1. A load identification method based on singular value adaptive improved regularization, characterized in that, Includes the following steps: Singular value decomposition is performed on the pre-constructed system matrix, and the regulation index corresponding to each order is calculated based on the distribution characteristics of the singular values. The regulation index is used to apply differentiated suppression strength to singular values of different sizes. The range of regularization parameters is set, and the optimal regularization parameters are obtained by using the minimization criterion of the generalized cross-validation (GCV) function. By combining the adjustment index and regularization parameter for load identification, an improved regularization objective function is constructed and solved to obtain the optimal solution for load identification.
2. The load identification method based on singular value adaptive improved regularization according to claim 1, characterized in that: The system matrix is subjected to singular value decomposition, and the regulation index corresponding to each order is calculated based on the characteristics of the singular value distribution, as follows: (21) Let H The singular value decomposition of the system identification matrix is as follows: (1) In the formula, U Let Σ be the left singular vector matrix, and Σ be the singular value matrix. VT It is a right singular vector matrix. σi , i =1,2,…,n is the nth i Singular values of order; (22) In order to achieve different levels of suppression for singular values of different sizes, a linear logarithmic mapping variable is introduced here. t j The details are as follows: (2) In the formula, σ j The system frequency response function matrix is the first... j The singular value of order is obviously And satisfy t j =0 corresponds to the direction of the maximum singular value. t j =1 corresponds to the direction of the minimum singular value; (23) Use this normalization index to define adaptive adjustment parameters β j : (3) In the formula, βj Corresponding to the j The adjustment parameters used for the singular values of order 1. β 0 represents the global adjustment coefficient, i.e., the suppression strength for the minimum singular value. Here, the condition number, which reflects the ill-conditioned nature of the matrix, is used to adjust the coefficient. β The value 0 is specifically: (4) In the formula, σ1 is the first singular value of the system frequency response function matrix, σ n The system frequency response function matrix is the first... n Singular value of order.
3. The load identification method based on singular value adaptive improved regularization according to claim 2, characterized in that: The range of regularization parameters is defined, and the optimal regularization parameters are selected by minimizing the generalized cross-validation function, as follows: (31) Assuming the initial state of the system is zero, the load identification model can be expressed as: Hx=y ,in x The load to be solved is... y Given the known response to the measurement, the optimization problem corresponding to the proposed improved regularization method is expressed as: (5) In the formula λ For regularization parameters, β j To adjust the exponent and control the suppression intensity for different singular values; σ 1 represents the first singular value of the system's frequency response function matrix; Solving equation (5) by combining equation (1) yields the expression for the identified load: (6) In the formula, It is a left singular value vector. It is a right singular value vector. For observation data y In the j Projection onto each basis vector; the superscript H denotes the conjugate transpose; (32) The generalized cross-validation method is used to select the regularization parameter. It automatically determines the optimal regularization parameter by minimizing the ratio between the residual term and the regularization term. The generalized cross-validation function adopts a simplified expression based on singular values as follows: (7) In the formula, This represents the "effective degrees of freedom" retained by the model. g j It is a filtering factor based on singular value decomposition, corresponding to the degree of retention of principal components; Based on the principle of minimizing the GCV function, the optimal regularization parameter is selected according to equation (7). λ opt : (8)。 4. The load identification method based on singular value adaptive improved regularization according to claim 3, characterized in that: Load identification is performed by combining the adjustment exponent and regularization parameter. An improved regularization objective function is constructed and solved to obtain the optimal solution for load identification, as follows: (41) Combine the adjustment exponents of each singular value direction calculated by equation (3) with the optimal regularization parameter obtained by equation (8) to perform load identification, and calculate the optimal solution of improved regularization according to the following formula: (9) In the formula, ximproved To improve the optimal solution of regularization, i.e. the result of load inversion.
5. A load identification system based on singular value adaptive improved regularization, used to implement the load identification method based on singular value adaptive improved regularization as described in any one of claims 1 to 4, characterized in that, include: The adjustment index calculation module is used to perform singular value decomposition on the pre-constructed system matrix and calculate the adjustment index corresponding to each order based on the singular value distribution characteristics. The adjustment index is used to apply differentiated suppression strength to singular values of different sizes. The regularization parameter filtering module is used to set the range of regularization parameters and uses the minimization criterion of the generalized cross-validation (GCV) function to filter out the optimal regularization parameters. The load identification module is used to identify loads by combining the adjustment exponent and regularization parameters, construct an improved regularization objective function and solve it to obtain the optimal solution for load identification.
6. A terminal device, comprising a memory, a processor, and a computer program stored in the memory and capable of running on the processor, characterized in that, The memory stores a computer program that can run on the processor. When the processor loads and executes the computer program, it employs the load identification method based on singular value adaptive improved regularization as described in any one of claims 1 to 4.
7. A storage medium containing computer-executable instructions, characterized in that, The computer-executable instructions, when executed by a computer processor, are used to perform the load identification method based on singular value adaptive improved regularization as described in any one of claims 1 to 4.
8. A computer program product, characterized in that, The computer program product includes a computer program that, when executed by a processor, is used to load and execute the load identification method based on singular value adaptive improved regularization as described in any one of claims 1 to 4.