Random subspace modal parameter uncertainty quantization acceleration calculation method

Abstract

The invention discloses a random subspace modal parameter uncertainty quantitative accelerated calculation method based on data-driven random subspace recognition, and belongs to the technical field of structural health monitoring and operation modal analysis. The method comprises the following steps: collecting multi-channel response data and setting parameters; constructing a block Hankel matrix and calculating a sample correlation matrix, and extracting and multiplexing a group of invariant sub-block intermediate quantities corresponding to the data set from the correlation matrix; the SVD module is used for constructing a reference projection matrix and carrying out SVD decomposition to obtain a subspace base; according to the method, matrix-level first-order disturbance propagation is implemented based on block data in uncertainty quantization, and the variance / covariance of modal parameters is obtained by combining one-step eigenvector disturbance and non-iterative eigenvalue disturbance, so that repeated construction of a high-dimensional intermediate matrix and modal-by-modal disturbance solution in a traditional method are avoided, the calculation amount and storage requirements are remarkably reduced, and the method is suitable for large-scale popularization and application. The engineering expandability is improved; and meanwhile, the statistical consistency of modal parameters and uncertainty estimation of the modal parameters is kept.

Description

Technical Field

[0001] This invention relates to the fields of structural health monitoring and operational modal analysis, specifically to an accelerated calculation method for quantifying the uncertainty of random subspace modal parameters based on Data-SSI, which is suitable for scenarios where modal parameters are quickly identified from structural response data and their uncertainty is quantified. Background Technology

[0002] In engineering applications such as structural health monitoring (SHM), damage identification, vibration control, and model correction, the core foundational work is to accurately identify modal parameters (natural frequencies, damping ratios, and mode shapes) from structural response data. Since environmental excitation inputs such as wind, traffic flow, and crowds are often impossible to measure simultaneously on-site, "output-only" runtime modal analysis (OMA) has become the mainstream technical approach.

[0003] Random Subspace Identification (SSI), as one of the core methods of OMA, treats the structure as a discrete-time random state-space system. It extracts modal parameters by subspace projection of the output data and estimation of the system matrix. It offers advantages such as greater robustness to noise and finite data length, good numerical stability, and scalability to multi-channel and mid-to-high-order systems. Data-SSI (SSI-data), an important variant of SSI, directly constructs a block Hankel matrix (or equivalent projection matrix) from the time-domain output. It then obtains the extended observable matrix and state sequence through SVD / EVD, and regresses to estimate the system matrices A and C. Finally, it obtains the modal parameters from the eigenvalue decomposition of A.

[0004] In real-world applications, factors such as finite data length, measurement noise, environmental non-stationarity, and modeling errors can lead to statistical fluctuations in modal parameter identification results. In particular, the damping ratio is more sensitive to noise and limited samples, making it difficult to support reliable engineering decisions based on a single identification value. Therefore, uncertainty quantification (UQ) becomes a critical step, requiring reliability measures such as the variance / covariance of modal parameters, confidence intervals, and correlations between different modes.

[0005] Existing modal parameter uncertainty quantification schemes are mainly divided into three categories: resampling-based (such as Bootstrap), Bayesian-based (such as BayesianOMA / SSI), and sensitivity / covariance propagation-based. Among them, sensitivity / covariance propagation-based methods do not require a large amount of repeated identification and have interpretable outputs, making them the preferred direction for engineering applications. However, current implementations have significant technical bottlenecks:

[0006] Excessive high-dimensional intermediate quantities lead to a surge in computational and storage burdens: The computation chain of Data-SSI requires the construction of large-scale block Hankel matrices and the performance of projection, EVD / SVD, pseudo-inverse, and multiple matrix multiplication operations; uncertainty analysis also requires the handling of high-dimensional intermediate quantities such as observable matrices and their pseudo-inverses, projection operators, and Jacobian correlation matrices. The dimensions of these intermediate quantities increase rapidly with the number of sensors, window depth, system order, and data length, resulting in a significant increase in memory usage and a substantial increase in runtime.

[0007] The computation of per-modal eigenvector perturbation leads to deterioration in time complexity: Traditional implementations require decomposing the eigenvector perturbation into solving each column of eigenvectors one by one. Each mode involves spectral gap correlation solution terms, pseudo-inverses and multiple matrix multiplications. When the system order increases or the number of tracking modes increases, the computational load increases linearly or even superlinearly. In high-order models or scenarios with a large number of data blocks, time consumption becomes the dominant factor, making it difficult to deploy in engineering.

[0008] Redundant computation is unavoidable: In the UQ framework of block Hankel, each data block needs to regenerate the correlation matrix, projection matrix and subsequent chain perturbation terms. However, a large number of intermediate quantities can be derived from the full Hankel matrix and remain unchanged for a given dataset. Traditional implementations lack a unified and reusable intermediate representation, resulting in redundant operations such as repeated inverse / pseudo-inverse and repeated matrix multiplication, which further increases time and memory overhead. Summary of the Invention

[0009] This invention provides a method and apparatus for accelerating the calculation of uncertainty quantification of random subspace modal parameters based on Data-SSI. The core idea is to construct a reusable and invariant intermediate quantity system, construct an equivalent system matrix, and use matrix-level one-step perturbation calculation to shorten the perturbation propagation chain, thereby achieving rapid quantification of modal parameter uncertainty without changing the Data-SSI identification results and the statistical meaning of uncertainty.

[0010] To achieve the above objectives, the present invention provides the following technical solution:

[0011] A method for accelerating the computation of uncertainty quantification of random subspace modal parameters includes the following steps:

[0012] S1: Acquire multi-channel output response sequences and set identification parameters, including reference / past window depth, data length, number of blocks, target system order, and sampling interval;

[0013] S2: Construct a block Hankel matrix H according to the Data-SSI method, wherein the block Hankel matrix H includes a reference block and a future block;

[0014] S3: Calculate the sample correlation matrix For the sample correlation matrix Perform predefined slicing to obtain at least one slice containing to The reusable invariant intermediate quantity, wherein the slicing method of the invariant intermediate quantity is as follows:

[0015]

[0016] in, For the number of channels, Reference / past window depth;

[0017] S4: Construct a reference projection matrix based on the invariant intermediate quantity. For the reference projection matrix Perform eigenvalue decomposition to obtain the eigenvector matrix. and eigenvalue matrix The first n principal components are selected for the order of the target system;

[0018] S5: Constructing the matrix of the equivalent system and The equivalent system matrix satisfies:

[0019]

[0020]

[0021] S6: The equivalent system matrix is ​​equivalent to the original system matrix of Data-SSI in terms of modal parameters.

[0022] S6: Divide the block Hankel matrix H along the column direction into... Sub-block Calculate the correlation matrix corresponding to each sub-block. And obtain sub-block perturbation The perturbation of each invariant intermediate quantity is obtained by slicing according to step S3. to ;

[0023] S7: Based on the above The perturbation of the projection matrix is ​​calculated using the chain rule and the inverse matrix perturbation formula. :

[0024]

[0025]

[0026] Then, the eigenvector perturbation is obtained in one step using a matrix-level one-step formula:

[0027]

[0028] in For Hadamard product, This is the spectral gap matrix constructed from eigenvalue intervals;

[0029] S8: By and Obtain the equivalent system matrix perturbation For equivalent system matrices Performing a first-order perturbation expansion yields the following for each block:

[0030] Its structure is as follows: Depend on , Composed of a few items; Depend on and Composed of multiple parts; perturbation using an inverse matrix: ;

[0031] S9: Based on , By using Jacobi linear mapping or block sample variance estimation, the variance, covariance, or confidence interval of the modal parameters can be obtained, thus completing the quantification of modal parameter uncertainty.

[0032] As a further technical solution of the present invention: In step S3, the parent correlation matrix The construction can be carried out in any of the following ways:

[0033] I. Calculation of the explicit construct block Hankel matrix H ;

[0034] II. Obtaining the outer product of column vectors by streaming accumulation The complete block Hankel matrix H is not stored.

[0035] III. Calculate the sample cross-correlation function under different lags. Assemble to obtain equivalent ;

[0036] IV. Obtain an approximate matrix H̃ by randomly projecting the block Hankel matrix H. Then, use H̃H̃... ^T approximate .

[0037] As a further technical solution of the present invention: in step S4, the eigenvalue decomposition of the reference projection matrix P0 is performed in any of the following ways:

[0038] I. Directly perform EVD decomposition;

[0039] II. Transformation into a generalized eigenvalue problem Solve the problem by avoiding explicit inversion;

[0040] III. Use subspace iteration or exponential iteration methods to solve only the first n principal eigenvectors;

[0041] IV. Perform SVD decomposition on the projection matrix or its factors to obtain an equivalent subspace basis.

[0042] As a further technical solution of the present invention: in step S5, the construction of the equivalent system matrix can also adopt any of the following similarity transformation forms:

[0043] I. , where T is a diagonal scaling matrix, whitening matrix, or balance transformation matrix;

[0044] II. Based on equivalent state sequences Obtained through least squares , ;

[0045] III. Construct an equivalent system matrix in the form of a predictor using a forward innovation model.

[0046] As a further technical solution of the present invention: in step S7, the feature vector is perturbed. The calculation can also be replaced by any of the following methods:

[0047] I. Transform it into solving a Sylvester or Lyapunov linear equation for the antisymmetric matrix Ω, satisfying ΔU=UΩ;

[0048] II. Propagation of subspace perturbations Subsequent approval structure ;

[0049] III. First-order sensitivity is obtained using automatic differentiation or matrix backpropagation, combined with... calculate .

[0050] As a further technical solution of the present invention: in step S6, the sub-block The construction adopts any of the following block partitioning strategies:

[0051] I. Overlapping blocks, with adjacent sub-blocks sharing some data;

[0052] II. Move the window into blocks, and move it along the time axis with a fixed window length to form sub-blocks;

[0053] III. Block bootstrapping method: Extract blocks from the time series and splice them together to form pseudo-sample sub-blocks;

[0054] IV. Jackknife variants of block partitioning with random column or row deletion.

[0055] As a further technical solution of the present invention: In step S9, the Jacobian linear mapping method satisfies:

[0056]

[0057] in:

[0058]

[0059]

[0060] in:

[0061]

[0062] The block sample variance estimation method satisfies:

[0063]

[0064] Where J is the Jacobian matrix. For modal parameters, This represents the number of blocks.

[0065] As a further technical solution of the present invention: for the invertible matrices C1, C2, and C3 in the invariant intermediate quantities, Cholesky decomposition or QR decomposition is used instead of explicit inversion, and the inverse matrix function is subsequently realized by solving linear equations.

[0066] This technique proposes an accelerated computation method for quantifying the uncertainty of mode parameters in random subspaces, which has the following advantages and benefits:

[0067] (1) Significantly reduce computational and storage requirements: By constructing a reusable and invariant intermediate variable system, C h Only one calculation is needed, and C1…C6 are obtained through index slicing, avoiding the repeated construction and storage of a large number of high-dimensional intermediate matrices in traditional methods; matrix-level one-step eigenvector perturbation replaces modal-by-modal solution, making the computational complexity increase more smoothly with the system order, which is particularly advantageous in high-order models and multi-block scenarios.

[0068] (2) Improved engineering scalability: This invention is applicable to long-term data (large N) and multi-sensor (large n) applications. y ), higher-order models (large n) or large data blocks (large n) b In this scenario, the increase in runtime and memory usage is significantly lower than that of traditional methods, which can meet engineering needs such as long-term monitoring batch processing and online real-time processing.

[0069] (3) Maintain statistical consistency: Only change the calculation path and the organization of intermediate variables, without changing the identification mechanism of Data-SSI and the statistical definition of uncertainty quantification. The output modal parameters and their variance / covariance are consistent or highly consistent with the traditional implementation. The coefficient of variation (cov) and confidence interval of the natural frequency and damping ratio of each mode are highly consistent with the traditional Data-SSI and empirical benchmark.

[0070] (4) Reduced implementation complexity and improved numerical stability: The explicit construction of observable matrices and their pseudo-inverses and the propagation of multi-level perturbations are avoided, and the number of chain expansion terms is reduced; the robustness to ill-conditioned matrices is improved by replacing explicit inversion with Cholesky / QR decomposition and solving generalized eigenvalue problems.

[0071] (5) Flexible adaptation to various engineering scenarios: Supports multiple variants such as streaming computing, overlapping block partitioning, and moving window block partitioning, and can be extended to systems such as covariance-driven SSI and weighted SSI, adapting to different engineering needs such as stationary / non-stationary data and online / offline processing. Attached Figure Description

[0072] Figure 1 : A flowchart comparing the traditional method and the method of this invention;

[0073] Figure 2 Different number of blocks Comparison of coefficient of variation (cov) and confidence intervals for lower modal parameters;

[0074] Figure 3 Different number of blocks The following is a comparison chart of the time consumption of traditional Data-SSI and the method of this invention. Detailed Implementation

[0075] The present invention will be further described below with reference to the embodiments. It should be noted that these are merely examples and descriptions of the inventive concept. Those skilled in the art can make various modifications or additions to the specific embodiments described or use similar methods to replace them, as long as they do not deviate from the inventive concept or exceed the scope defined in the claims, they should all be considered to fall within the protection scope of the present invention.

[0076] like Figure 1-3 As shown, a method for accelerating the computation of uncertainties in random subspace modal parameters is presented, as detailed below:

[0077] Step S1: Acquire multi-channel output and set recognition parameters (corresponding to "Datacollection" on the left of the figure).

[0078] collection Channel output response sequence (e.g., acceleration), settings:

[0079] Reference / Past Window Depth

[0080] Data length

[0081] Number of blocks (Used for uncertainty estimation)

[0082] Target system order (Can be selected from stability diagrams or energy criteria)

[0083] Sampling interval

[0084] Step S2: Construct the block Hankel matrix (Corresponding to the formula box in the lower left corner of the image)

[0085] Constructed using the Data-SSI method:

[0086]

[0087] in For reference (past) blocks, For the future block.

[0088] Step S3: Calculate the parent correlation matrix in one step and extract invariant intermediate quantities (corresponding to "improvemethods" in the figure).

[0089] First calculate:

[0090]

[0091] Then to By performing fixed slices, we obtain invariant intermediate quantities (as shown in the figure). ):

[0092]

[0093] Key points: For the same batch of data, once Calculation complete. It is obtained entirely from index slices, and the values ​​remain unchanged, allowing for repeated reuse in identification and uncertainty propagation.

[0094] Step S4: Construct the reference projection matrix from the invariant intermediate quantities and calculate EVD (corresponding to the traditional path). (alternative implementation)

[0095] Under UPC weights (W_1=W_2=I), the reference projection matrix can be written as:

[0096]

[0097] right Perform eigenvalue decomposition:

[0098]

[0099] in These are the eigenvectors and eigenvalue matrices, respectively (the first n principal components are selected for the system order).

[0100] Step S5: Construct the equivalent system matrix (avoid explicitly constructing the observable matrix) )

[0101] The "equivalent system" concept adopted in this invention: no need to restore the complete system. With the original Directly construct a pair of system matrices equivalent to the modal parameters. , (Corresponding to the final output on the right side of the flowchart) , (basic quantity)

[0102]

[0103]

[0104] Equivalence meaning: and They are equivalent in the sense of modal parameters (eigenvalues ​​and mode shape mapping), and therefore sufficient for mode and uncertainty propagation.

[0105] 3. Uncertainty Analysis and Disturbance Propagation (corresponding to the flowchart below) )

[0106] Step S6: Block Construction And obtained (This corresponds to the blockHankel approach in your paper)

[0107] Will Divided along the column direction into Sub-block (each piece contains) List, and do as needed. Scaling to maintain consistent convergence of relevant quantities, computation for each block:

[0108]

[0109]

[0110] Then, using the same slicing method as S3, we directly obtain:

[0111]

[0112] The corresponding flowchart below " ".

[0113] Step S7: Replace mode-by-mode perturbation with matrix-level "one-step" perturbation calculation (eliminating traditional mode-by-mode perturbation). )

[0114] First by Constructing projection matrix perturbation (Chain rule + inverse matrix perturbation):

[0115]

[0116]

[0117] Then, the matrix-level perturbation formula is used to obtain the result in one step. (In the traditional method above the flowchart) (alternative)

[0118]

[0119] in For Hadamard product, Constructed from eigenvalue intervals (commonly defined as: diagonal values ​​are 0, off-diagonal elements are 0 and 0). Proportional; used to map symmetric perturbations to orthogonal eigenvector perturbations. The key point is that instead of solving each mode column by column, the main time-consuming steps in the traditional path are replaced entirely.

[0120] Step S8: Directly from and Obtain the equivalent system matrix perturbation (Corresponding to the output on the right side of the flowchart)

[0121] For equivalent system matrix Each block can be obtained by performing a first-order perturbation expansion:

[0122]

[0123] Its structure is as follows: Depend on , It is composed of a few items; Depend on and It is composed of multiple parts. Inverse matrix perturbation is also used here:

[0124]

[0125] Therefore, only rely on It can be accomplished by perturbing it. The pursuit of.

[0126] Step S9: Obtain the covariance / standard deviation from the block sample statistics (output uncertainty index)

[0127] For each block k, get , Then, the uncertainty of the modal parameters can be obtained in two equivalent ways:

[0128] 1. Jacobi linear mapping method (suitable for alignment with traditional SSI-UQ form):

[0129]

[0130] in:

[0131]

[0132]

[0133] in:

[0134] ;

[0135] 2. Block sample variance estimation method (more direct implementation):

[0136]

[0137] The final output includes: frequency, damping ratio, standard deviation of mode shape (real / imaginary part), covariance, confidence interval, etc.

[0138] Figure 1 The traditional path above requires calculation at each level. And then The present invention is in Figure 1 The following achieves two-point substitution:

[0139] use The invariant intermediate system replaces a large number of observable matrices / pseudo-inverse correlation intermediates: Only count once. We only perform index slicing and then proceed directly to identification and perturbation propagation.

[0140] Using matrix level One-step formula replaces modal approach Avoiding the main bottlenecks of mode-by-mode, significantly improving high-order systems and large-scale... Scalability in various scenarios.

[0141] Achievable technical effects: (and) Figure 1 middle" (corresponding to the short chain)

[0142] Reduced computational cost: no longer requires explicit construction And its pseudo-inverse perturbation, and no longer seeking mode-by-mode. The overall chain has changed from a "long chain" to a "short chain".

[0143] Reduced storage requirements: High-dimensional intermediate quantities are significantly reduced, with the majority of storage space retained. and a small amount This reduces memory pressure.

[0144] Efficiency and scalability improvements: in long sequences Multi-channel Higher-order model n or large number of blocks The performance is more stable.

[0145] The statistical significance remains unchanged: only the calculation path and the organization of intermediate variables are rearranged, without changing the identification mechanism and uncertainty definition of Data-SSI / UPC. The output modal parameters and variance estimates are consistent with or highly consistent with traditional methods.

[0146] Experimental results:

[0147] To verify the invention under different numbers of blocks The stability and efficiency improvements were demonstrated by comparing the uncertainty characterization results and computation time. Figure 2 The coefficient of variation (cov) and its confidence interval for the natural frequency and damping ratio under each mode are given. It can be seen that the cov of the method in this invention is consistent with or highly close to the traditional Data-SSI and empirical / Monte Carlo benchmarks, indicating that optimization does not weaken the statistical consistency of uncertainty estimation; simultaneously, as... As the confidence interval bandwidth increases, it exhibits a typical trade-off between the number of blocks and the block length.

[0148] Figure 3 Different numbers of data blocks are given The following is a comparison of the time consumption of the two algorithms. With... The time consumption of traditional Data-SSI increases even faster due to the increase in frequency; this invention significantly reduces the overall time consumption by reusing invariant intermediate quantities and shortening the disturbance propagation link, and in addition... The advantages are more pronounced when the scale is larger, demonstrating better scalability.

[0149] The above is an exemplary description of the invention. Obviously, the specific implementation of the invention is not limited to the above-described manner. Any non-substantial improvement made using the inventive concept and technical solution of the invention, or the direct application of the inventive concept and technical solution to other situations without modification, is within the protection scope of the invention.

Claims

1. A random subspace modal parameter uncertainty quantification accelerated computation method, characterized in that, The method comprises the following steps: S1: collecting a multi-channel output response sequence, setting identification parameters, the identification parameters comprising a reference / past window depth, a data length, a block number, a target system order and a sampling interval; S2: constructing a block Hankel matrix H in a Data-SSI manner, the block Hankel matrix H comprising a reference block and a future block; S3: Compute the mother correlation matrix , and perform a predefined slice on the mother correlation matrix to obtain at least one reusable invariant intermediate quantity containing to , the slice manner of the invariant intermediate quantity being: wherein, is the number of channels, is the reference / past window depth; S4: constructing a reference projection matrix based on the invariant intermediate quantity , performing eigen decomposition on the reference projection matrix to obtain an eigenvector matrix and an eigenvalue matrix , and selecting the first n principal components for the target system order ; S5: constructing an equivalent system matrix and which satisfies: The equivalent system matrix is equivalent to the Data-SSI original system matrix in the sense of modal parameters. S6: divide the block Hankel matrix H into sub-blocks along the column direction , calculate the correlation matrix of each sub-block , and obtain the sub-block perturbation , and obtain the sub-block perturbation of each invariant intermediate quantity in the slicing manner of step S3 to ;​ S7: based on the , the projection matrix perturbation is calculated by chain rule and inverse matrix perturbation formula : The eigenvector perturbation is obtained by using the matrix step formula once again: wherein is the Hadamard product, is a spectral gap matrix constructed from the eigenvalue spacings; S8: by With get equivalent system matrix perturbation , equivalent system matrix do first-order perturbation expansion to get each block: Its structure is: by , a few items combined; by and combined; use inverse matrix perturbation: ; S9: based on , , the variance, covariance or confidence interval of the modal parameters is obtained through Jacobian linear mapping or block sample variance estimation method, and the uncertainty quantification of the modal parameters is completed.

2. The random subspace modal parameter uncertainty quantification accelerated computation method according to claim 1, characterized in that: In step S3, the sample correlation matrix is constructed in any one of the following ways: I. Computing after explicit construction of the building block Hankel matrix H ; II. The flow accumulation column vector outer product yields without storing the full block Hankel matrix H; III. Calculate the sample cross-correlation function for different lags , and assemble the equivalent ; IV. Randomly project the block Hankel matrix H to obtain an approximate matrix H, by HHH ^T approximate .

3. The stochastic subspace modal parameter uncertainty quantification accelerated computation method according to claim 1, characterized in that: In step S4, the characteristic decomposition of the reference projection matrix P0 adopts any one of the following manners: I. directly performing EVD decomposition; II. Conversion to generalized eigenvalue problem solving, avoiding explicit inversion; III. using a subspace iteration or power iteration method to solve only the first n principal characteristic vectors; IV. performing SVD decomposition on the projection matrix or a factor thereof to obtain an equivalent subspace basis.

4. The stochastic subspace modal parameter uncertainty quantification accelerated computation method according to claim 1, characterized in that: In step S5, the construction of the equivalent system matrix can also adopt any one of the following similar transformation forms: I, where T is a diagonal scaling matrix, a whitening matrix, or a balancing transform matrix; II. Based on Equivalent State Sequence , by least square , ; III. using a forward innovation model to construct an equivalent system matrix in the form of a predictor.

5. The stochastic subspace modal parameter uncertainty quantification accelerated computation method according to claim 1, characterized in that: In step S7, the eigenvector perturbation The calculation of the eigenvector perturbation can also be replaced by any of the following: I. transforming into a Sylvester or Lyapunov type linear equation for solving an antisymmetric matrix Ω, satisfying ΔU=UΩ; II. Propagating subspace perturbations , subsequently passing through configuring ; III. First order sensitivity is obtained using automatic differentiation or matrix backpropagation, combined with computing .

6. The stochastic subspace modal parameter uncertainty quantification accelerated computation method according to claim 1, characterized in that: In step S6, the sub-block is constructed using any of the following partitioning strategies: I. overlapping blocks, adjacent sub-blocks sharing part of the data; II. moving window block, moving along the time axis to form sub-blocks with a fixed window length; III. block bootstrap method, extracting blocks from the time sequence and splicing to form pseudo-sample sub-blocks; IV. Jackknife variant block with random column or row deletion.

7. The stochastic subspace modal parameter uncertainty quantification accelerated computation method according to claim 1, wherein: In step S9, the Jacobian linear mapping manner satisfies: wherein: wherein: The block sample variance estimation method satisfies: where J is the Jacobian matrix, is the modal parameter, is the number of blocks.

8. The stochastic subspace modal parameter uncertainty quantification accelerated computation method of claim 1, wherein: For the reversible matrices C1, C2 and C3 in the invariant intermediate quantity, Cholesky decomposition or QR decomposition is used instead of explicit inversion, and the inverse matrix function is realized by solving a linear equation subsequently.

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