System operation modal parameter estimation method based on compressed random subspace identification

Abstract

The invention discloses a vibration system operation modal parameter estimation method based on compressed random subspace identification, and the method comprises the steps: carrying out the compressed sampling of a monitoring signal of a vibration system through employing a general covariance sampling mode, and obtaining a signal sample; determining a sampling matrix and a sensing matrix based on the general covariance sampling mode; estimating a compressed covariance matrix by using the signal sample; reconstructing a complete covariance matrix based on the covariance matrix; and performing operation modal parameter estimation by using a covariance random subspace identification algorithm.

Description

Technical Field

[0001] The present invention relates to the technical field of operational modal parameter identification, and particularly to a method for estimating operational modal parameters of a vibration system based on compressed stochastic subspace identification. Background Art

[0002] Modal parameters are the key mechanical characteristics of a structure / system, which are only related to the physical parameters and mechanical models of the structure itself. Therefore, modal parameter identification is of great significance for structural health monitoring. The identification technology that extracts modal parameters only based on output data is called Operational Modal Analysis (OMA). Due to the fact that it is not necessary to measure the input excitation force, it is simple to operate and easy to measure, and operational modal analysis is widely used in the field of structural health monitoring. Stochastic subspace identification is a mainstream operational modal analysis method, which uniformly samples signals at a high rate that satisfies the Shannon-Nyquist sampling theorem. With the increase in the monitoring duration and the number of monitored objects, the amount of structural health monitoring data becomes extremely large, resulting in extremely high costs for data sampling, transmission, and storage, which limits the application of structural health monitoring.

[0003] The above information disclosed in the background art section is only used to enhance the understanding of the background of the present invention, and thus may include information that does not constitute the prior art known to those of ordinary skill in the art. Summary of the Invention

[0004] The present invention provides a method for estimating operational modal parameters of a vibration system based on compressed stochastic subspace identification, which can identify the operational mode of the system from non-uniform and undersampled monitoring data, reduce the signal sampling rate of system operational mode identification, reduce the costs of signal sampling, transmission, and storage, and can meet the requirements of long-term system health monitoring.

[0005] A method for estimating operational modal parameters of a vibration system based on compressed stochastic subspace identification includes:

[0006] Step 1: Compressively sample the monitoring signal of the vibration system using a general covariance sampling mode to obtain a signal sample;

[0007] Step 2: Determine a sampling matrix and a sensing matrix based on the general covariance sampling mode;

[0008] Step 3: Estimate the compressed covariance matrix using the signal sample;

[0009] Step 4: Reconstruct a complete covariance matrix based on the covariance matrix;

[0010] Step 5: Perform operational modal parameter estimation using a covariance stochastic subspace identification algorithm.

[0011] In the described method for estimating the operating modal parameters of a vibration system based on compressed random subspace identification, step 1 includes:

[0012] Let represent the upper frequency limit of the monitoring signal ; Let represent the sampling rate of the sampling channel, and determine the downsampling factor N that satisfies the condition according to Equation (1)

[0013] (1)

[0014] where: is the ceiling symbol; the downsampling factor N is an integer that satisfies the above equation,

[0015] Then design a compressed sampling pattern that satisfies Equation (2)

[0016] (2)

[0017] where: the compressed sampling pattern is a set of integers; if the compressed sampling pattern satisfies Equation (2), it is called a general covariance sampling pattern, is the set composed of elements less than L in the non - negative difference set of

[0018] (3)

[0019] where and represent any two elements in , taking all cases that satisfy

[0020] Finally, determine the nominal time unit T according to the sampling rate and the downsampling factor N: , use P channels with a sampling rate of , that is, the sampling period is NT, to uniformly sample the signal. Among them, the p - th channel starts with a delay relative to the first channel, and the delay time is , , P is equal to the number of elements in is equal to the p - th element in .

[0021] In the described method for estimating the operating modal parameters of a vibration system based on compressed random subspace identification, the delay time of the first channel is 0 as a reference, that is , and the sampling duration is then the sample sampled by the p-th channel is

[0022] (4).

[0023] In the vibration system operation mode parameter estimation method based on compressed random subspace identification, the steps include

[0024] determining the sampling matrix and the sensing matrix :

[0025] (5)

[0026] where: G is a set integer; is the identity matrix of size ; is the Kronecker product; is the binary matrix of size whose element at the p-th row and the -th column is 1, and the elements at the remaining positions are 0,

[0027] (6)

[0028] where: is the number of columns of the sensing matrix , , is the compressed covariance subspace basis matrix, which is obtained by compressing : , , represents the transpose of the matrix, is the complete covariance subspace basis matrix, whose matrix size is , the element at the -th row and the -th column of is 1, and the remaining elements are 0, g is a traversing integer that traverses from 1 to GN, is the vectorization operation of the matrix, which concatenates all columns of the matrix into a long vector.

[0029] In the vibration system operation mode parameter estimation method based on compressed random subspace identification, the step 2 includes estimating the compressed covariance matrix of the signal samples, (7),

[0030] where: Q is the number of snapshots;

[0031] , where \(q\) is an integer that traverses from 0 to \(Q - 1\); .

[0032] In the described method for estimating the operating modal parameters of a vibration system based on compressed random subspace identification, step 3 includes,

[0033] First, solve for the expansion coefficients of the covariance subspace ,

[0034] (8),

[0035] where: is the vector formed by concatenating all the columns of the matrix , represents the inverse of the matrix,

[0036] Based on , reconstruct the complete covariance matrix ,

[0037] (9)

[0038] where: represents the \(s\)-th element in

[0039] In the described method for estimating the operating modal parameters of a vibration system based on compressed random subspace identification, step 4 includes,

[0040] Construct two block Toeplitz matrices using the reconstructed covariance matrix: and

[0041] (10)

[0042] where: is the \(i\)-th element of the vector of length \(2GN - 1\) formed by concatenating the first column and the first row elements of , \(i\) is a given positive integer, , represents the floor function symbol,

[0043] Then perform singular value decomposition on

[0044] (11)

[0045] where is the diagonal matrix corresponding to the signal singular values, is the diagonal matrix corresponding to the noise singular values; and ​The left orthogonal matrices corresponding to the signal and noise singular values, respectively; and The right orthogonal matrices corresponding to the signal and noise singular values, respectively,

[0046] Calculate the observable matrix and the controllable matrix :

[0047] (12)

[0048] (13)

[0049] Calculate the state matrix and the output matrix

[0050] (14)

[0051] (15)

[0052] Where: is the matrix composed of the 1st to B rows in , and B is equal to the system degrees of freedom,

[0053] Perform eigenvalue decomposition on the state matrix to obtain the eigenvalues and eigenvectors , where B is the number of system modes,

[0054] Calculate the system natural frequency , damping ratio and mode shape

[0055] (16)

[0056] (17)

[0057] (18).

[0058] Compared with the prior art, the present invention has the following advantages: The present invention adopts compressive sampling to reduce the data sampling, transmission, and storage costs of structural health monitoring. The present invention adopts the general covariance mode to perform periodic non-uniform sampling on the signal. Compared with random sampling, the general covariance mode has a definite compressive sampling mode, and therefore, it is easy to be implemented by hardware. In terms of modal parameter identification, the present invention does not rely on the prior reconstruction of the signal sparsity, but directly estimates the signal covariance matrix through the least squares algorithm and realizes modal parameter identification based on the covariance matrix, with low computational complexity and high stability. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] By reading the detailed description in the following preferred specific embodiments, various other advantages and benefits of the present invention will become clear to those of ordinary skill in the art. The accompanying drawings of the specification are only for the purpose of showing the preferred embodiments and are not considered to be a limitation of the present invention. Obviously, the following-described drawings are only some embodiments of the present invention, and those of ordinary skill in the art can obtain other drawings based on these drawings without creative efforts. Moreover, throughout the drawings, the same reference numerals are used to represent the same components.

[0060] In the drawings:

[0061] Figure 1 is a schematic diagram of a 6-degree-of-freedom mass-spring-damper system in an embodiment of the present invention;

[0062] Figure 2 is a schematic diagram of periodic non-uniform sampling based on a general covariance model in an embodiment of the present invention;

[0063] Figure 3 is a schematic diagram of obtaining an acceleration sample sequence at the 4th node by compressive sampling with a general covariance model in an embodiment of the present invention;

[0064] Figure 4 is a stability diagram of the modal frequency identification result in an embodiment of the present invention;

[0065] Figure 5 is a schematic diagram of the modal shape identification result in an embodiment of the present invention;

[0066] Figure 6 is a schematic diagram of the step flow in an embodiment of the present invention;

[0067] Figure 7 is a schematic diagram of compressive sampling of monitoring signals with a general covariance sampling model in an embodiment of the present invention.

[0068] The present invention will be further explained below with reference to the accompanying drawings and embodiments. Specific Embodiments

[0069] The specific embodiments of the present invention will be described in more detail below with reference to the accompanying drawings. Although the specific embodiments of the present invention are shown in the drawings, it should be understood that the present invention can be implemented in various forms and should not be limited by the embodiments described herein. On the contrary, these embodiments are provided to enable a more thorough understanding of the present invention and to fully convey the scope of the present invention to those skilled in the art.

[0070] It should be noted that in the description of the specification and the claims, certain terms are used to refer to specific components. Those skilled in the art should understand that technicians may use different nouns to refer to the same component. The specification and the claims do not distinguish components based on the difference in nouns, but rather on the difference in the functions of the components. As mentioned throughout the specification and the claims, "comprising" or "including" is an open-ended term and should be interpreted as "including but not limited to". The subsequent description in the specification is the preferred implementation manner for implementing the present invention, but the description is for the purpose of the general principles of the specification and is not used to limit the scope of the present invention. The protection scope of the present invention shall be subject to what is defined by the appended claims.

[0071] To facilitate the understanding of the embodiments of the present invention, the following will further explain with specific embodiments in conjunction with the drawings, and each drawing does not constitute a limitation on the embodiments of the present invention.

[0072] As Figures 1 to 7 shown, the method for estimating the operating modal parameters of a vibration system based on compressive random subspace identification includes the following steps:

[0073] Step 1, performing compressive sampling on the monitoring signal of the vibration system using the general covariance sampling mode to obtain a signal sample;

[0074] Step 2, determining the sampling matrix and the sensing matrix based on the general covariance sampling mode;

[0075] Step 3, estimating the compressed covariance matrix using the signal sample;

[0076] Step 4, reconstructing the complete covariance matrix based on the covariance matrix;

[0077] Step 5, performing operating modal parameter estimation using the covariance random subspace identification algorithm.

[0078] In the preferred implementation manner of the method for estimating the operating modal parameters of a vibration system based on compressive random subspace identification, the said Step 1 includes:

[0079] Let represent the upper frequency limit of the monitoring signal ; let represent the sampling rate of the sampling channel, and determine the downsampling factor N that meets the conditions according to Equation (1)

[0080] (1)

[0081] Where: is the ceiling symbol; the downsampling factor N is an integer that meets the above formula,

[0082] Then, design a compressive sampling pattern that satisfies Equation (2).

[0083] (2)

[0084] where: the compressive sampling pattern is a set of integers; if the compressive sampling pattern satisfies Equation (2), it is called a general covariance sampling pattern, is the set composed of elements less than L in the non - negative difference set of

[0085] (3)

[0086] where and represent any two elements in and all cases satisfying

[0087] Finally, determine the nominal time unit T according to the sampling rate and the down - sampling factor N: , and use P channels with a sampling rate of , that is, a sampling period of NT, to uniformly sample the signal. The p - th channel starts with a delay relative to the first channel, and the delay time is , , P is equal to the number of elements in is equal to the p - th element in .

[0088] In the preferred embodiment of the vibration system operating mode parameter estimation method based on compressive random subspace identification, the delay time of the first channel is 0 as a reference, that is , the sampling duration is , then the sample sampled by the p - th channel is

[0089] (4).

[0090] In the preferred embodiment of the vibration system operating mode parameter estimation method based on compressive random subspace identification, step 2 includes

[0091] Determine the sampling matrix and the sensing matrix from the general covariance sampling pattern:

[0092] (5)

[0093] Wherein: G is a set integer; is the identity matrix of size ; is the Kronecker product; is the size of binary matrix, whose element at the p-th row and the -th column is 1, , and the elements at the remaining positions are 0,

[0094] (6)

[0095] Wherein: is the sensing matrix number of columns of, , is the compressed covariance subspace basis matrix, which is obtained by compression: , , denotes the transpose of the matrix, is the complete covariance subspace basis matrix, whose matrix size is , in the -th row and the -th column is 1, and the remaining elements are 0, g is a traversing integer, traversing from 1 to GN, is the vectorization operation of the matrix, which concatenates all columns of the matrix into a long vector.

[0096] In the preferred embodiment of the method for estimating the operating mode parameters of a vibration system based on compressed random subspace identification, step 3 includes estimating the compressed covariance matrix of the signal samples , (7),

[0097] Wherein: Q is the number of snapshots;

[0098] , q is a traversing integer, traversing from 0 to Q - 1; .

[0099] In the preferred embodiment of the method for estimating the operating mode parameters of a vibration system based on compressed random subspace identification, step 4 includes,

[0100] First, solve the expansion coefficients of the covariance subspace ,

[0101] (8),

[0102] Wherein: is obtained from the matrix The vector formed by concatenating all columns of represents the inverse of a matrix,

[0103] Based on , reconstruct the complete covariance matrix ,

[0104] (9)

[0105] Where: represents the s-th element in

[0106] In the preferred implementation of the vibration system operating mode parameter estimation method based on compressed random subspace identification described above, step 5 includes

[0107] Construct two block Toeplitz matrices using the reconstructed covariance matrix: and

[0108] (10)

[0109] Where: is the i-th element of the vector of length 2GN - 1 formed by concatenating the first column and the first row elements of , where i is a given positive integer , represents the floor function symbol

[0110] Then perform singular value decomposition on (11)

[0111] (11)

[0112] Where is the diagonal matrix corresponding to the signal singular values, is the diagonal matrix corresponding to the noise singular values; and are the left orthogonal matrices corresponding to the signal and noise singular values respectively; and are the right orthogonal matrices corresponding to the signal and noise singular values respectively,

[0113] Calculate the observable matrix and the controllable matrix :

[0114] (12)

[0115] (13)

[0116] Calculate the state matrix and the output matrix

[0117] (14)

[0118] (15)

[0119] Wherein: is a matrix composed of the first to B rows in , where B is equal to the degrees of freedom of the system,

[0120] Perform eigenvalue decomposition on the state matrix to obtain eigenvalues and eigenvectors , where B is the number of modes of the system,

[0121] Calculate the natural frequency of the system , damping ratio and mode shape

[0122] (16)

[0123] (17)

[0124] (18).

[0125] A system for estimating the operating modal parameters of a vibration system includes

[0126] a compressive sampling unit that performs compressive sampling on the monitoring signal of the vibration system using the general covariance sampling mode to obtain a signal sample;

[0127] a determination unit that determines a sampling matrix and a sensing matrix based on the general covariance sampling mode;

[0128] an estimation unit that estimates the compressed covariance matrix using the signal sample;

[0129] a reconstruction unit that reconstructs a complete covariance matrix based on the covariance matrix;

[0130] a modal parameter estimation unit that performs operating modal parameter estimation using the covariance random subspace identification algorithm.

[0131] In one embodiment, a method for estimating the operating modal parameters of a system based on compressive random subspace identification includes the following steps:

[0132] Step 1, select the general covariance sampling mode to perform compressive sampling on the monitoring signal;

[0133] Let represent the measured signal The upper frequency limit; let represent the sampling rate of the sampling channel. Determine the downsampling factor N that satisfies the condition according to the following formula

[0134] (1)

[0135] where: is the ceiling symbol; note that N is an integer that satisfies the above formula. The larger N is, the lower the compression ratio of the signal, but the higher the requirement for timing accuracy and the more sensitive to sampling jitter. In practical engineering, N needs to be selected as required according to the compression ratio and hardware conditions on the premise of satisfying Equation (1).

[0136] Then design a compressive sampling pattern that satisfies Equation (2)

[0137] (2)

[0138] where: is a set of integers; if satisfies Equation (2), it is called a general covariance pattern, is the set composed of elements less than L in the non - negative difference set of

[0139] (3)

[0140] where: and represent any two elements in and all cases that satisfy

[0141] Finally, determine the nominal time unit T according to the sampling rate of the channel and the downsampling factor N: . Use P channels with a sampling rate of , that is, a sampling period of NT, to uniformly sample the signal. Among them, the p - th channel starts with a delay relative to the first channel, and the delay time is , . P is equal to the number of elements in is equal to the p - th element in . Without loss of generality, let the delay time of the first channel be 0 as a reference, that is . Assume that the sampling duration is , then the sample sampled by the p - th channel is

[0142] (4).

[0143] In this exemplary instance Taking a six-degree-of-freedom mass-spring-damping system as the research object, the implementation process of the operational modal parameter estimation method proposed by the present invention is described.

[0144] The six-degree-of-freedom system adopted is as Figure 1 shown, and its mass matrix M0 and stiffness matrix K0 are respectively

[0145]

[0146] Where: ,

[0147]

[0148] In the simulation, the damping is assumed to be Rayleigh damping. Therefore, the damping matrix C0 is a linear combination of M0 and K0: , where . The input of the system is set as Gaussian random white noise.

[0149] According to the mass matrix M0, stiffness matrix K0, and damping matrix C0 of the simulation system, the theoretical modal frequencies of the simulation system are: 34.597, 73.334, 109.864, 143.887, 181.937, and 227.015 Hz; the theoretical modal damping ratios are: 0.1259%, 0.0773%, 0.0707%, 0.0729%, 0.0790%, and 0.0888%.

[0150] The highest frequency of the signal to be measured in this example Hz, and the channel sampling rate in the simulation is set to 40 Hz. Therefore, the downsampling factor To satisfy being an integer, in this exemplary example, take .

[0151] Then design a compressive sampling pattern that satisfies Equation (2) : . In this example, take . Then from and N, determine the nominal time unit T: seconds. Periodic non-uniform sampling of the acceleration data at each node is performed using 4 channels with a sampling rate of 40 Hz. Among them, the start times of the 1st, 2nd, 3rd, and 4th channels are delayed by 0T, 1T, 4T, and 6T respectively. The specific form is as Figure 2 shown. The sampling duration seconds, and the number of samples K collected by each channel = 400. The samples collected by the 1st to 4th channels are respectively

[0152]

[0153] The acceleration sample sequence at the 4th node obtained by general covariance pattern compressive sampling is as Figure 3 shown.

[0154] Step 2, determine the sampling matrix and the sensing matrix :

[0155] (5)

[0156] where: G is an artificially set integer; is the identity matrix of size ; is the Kronecker product; is the binary matrix of size whose element at the p-th row and the -th column is 1, , and the elements at the remaining positions are 0.

[0157] (6)

[0158] where: is the number of columns of the sensing matrix , . is the compressed covariance subspace basis matrix, which is obtained by compressing : , . denotes the transpose of the matrix. is the complete covariance subspace basis matrix, whose matrix size is , the element at the -th row and the -th column of is 1, and the remaining elements are 0, and g is a traversing integer that traverses from 1 to GN.

[0159] In this exemplary instance , G is taken as 40, is the identity matrix of ,

[0160] The sampling matrix is obtained from .

[0161] Generate the complete covariance subspace basis matrix according to the definition : The element in the -th row and -th column is 1, and the remaining elements are 0. g is an integer that traverses from 1 to 520.

[0162] Then, from calculate the compressed covariance subspace basis matrix , .

[0163] Finally, from obtain the sensing matrix .

[0164] Step 3: Estimate the compressed covariance matrix using the signal samples of compressed sampling ;

[0165] (7)

[0166] where: Q is the number of snapshots; , q is an integer that traverses from 0 to Q - 1;

[0167] In this exemplary instance , . , . From obtain the compressed covariance matrix .

[0168] Step 4: Reconstruct the complete covariance matrix based on the compressed covariance matrix;

[0169] First, solve the expansion coefficients of the covariance subspace

[0170] (8)

[0171] where: is the vector formed by concatenating all columns of the matrix , represents the inverse of the matrix.

[0172] Based on , reconstruct the complete covariance matrix

[0173] (9)

[0174] where: represents the s-th element in

[0175] In this exemplary instance . From obtain the expansion coefficients , and then reconstruct to obtain the complete covariance matrix .

[0176] Step 5, perform operational modal parameter estimation using the covariance random subspace identification algorithm

[0177] First, construct two block Toeplitz matrices using the reconstructed covariance matrix: and

[0178] (10)

[0179] where: is the i-th element of the vector of length 2GN - 1 formed by concatenating the first column and the first row elements of , i is a positive integer given manually, and it is recommended to choose , represents the floor symbol.

[0180] Then perform singular value decomposition on

[0181] (11)

[0182] where is the diagonal matrix corresponding to the signal singular values, is the diagonal matrix corresponding to the noise singular values; and are the left orthogonal matrices corresponding to the signal and noise singular values respectively; and are the right orthogonal matrices corresponding to the signal and noise singular values respectively.

[0183] Calculate the observable matrix and the controllable matrix :

[0184] (12)

[0185] (13)

[0186] Calculate the state matrix and the output matrix

[0187] (14)

[0188] (15)

[0189] where: is composed of ​The matrix formed by the 1st to B-th rows, where B is equal to the degrees of freedom of the system.

[0190] For the state matrix Perform eigenvalue decomposition to obtain the eigenvalues and eigenvectors , where B is the number of modes of the system.

[0191] Calculate the natural frequencies , damping ratios and mode shapes

[0192] (16)

[0193] (17)

[0194] (18)

[0195] In this exemplary instance , the degrees of freedom of the system B = 6, and the modal frequencies and damping calculated from equations (16) and (17) are shown in the following table

[0196]

[0197] where the true values are the theoretical values directly calculated based on the mass matrix M0, stiffness matrix K0, and damping matrix C0 of the simulation system. The estimated values are the modal parameters identified from the data obtained by compressive sampling in the present invention. The estimated values are very close to the true values, verifying the effectiveness of the operational modal parameter estimation method proposed in the present invention.

[0198] The stability diagram of the modal frequency identification results is as Figure 4 shown. Observing Figure 4 it can be seen that the proposed operational modal parameter identification method can accurately identify 6 modal frequencies. The modal mode shapes calculated from equation (18) are as Figure 5 shown. Observing Figure 5 it can be seen that the modal mode shapes estimated by the proposed operational modal parameter identification method are in good agreement with the true mode shapes, further verifying the effectiveness of the operational modal parameter estimation method proposed in the present invention.

[0199]

Application Example

[0200] In this exemplary instance , taking a 6-degree-of-freedom mass-spring-damper system as the research object, describe the implementation process of the operational modal parameter estimation method proposed in the present invention.

[0201] The 6-degree-of-freedom system adopted is as Figure 1 shown, and its mass matrix M0 and stiffness matrix K0 are respectively

[0202]

[0203] Among them: ,

[0204]

[0205] In the simulation, the damping is assumed to be Rayleigh damping. Therefore, the damping matrix C0 is a linear combination of M0 and K0: , where . The input of the system is set to Gaussian white noise.

[0206] According to the mass matrix M0, stiffness matrix K0, and damping matrix C0 of the simulation system, the theoretical modal frequencies of the simulation system are: 34.597, 73.334, 109.864, 143.887, 181.937, and 227.015 Hz; the theoretical modal damping ratios are: 0.1259%, 0.0773%, 0.0707%, 0.0729%, 0.0790%, and 0.0888%.

[0207] The highest frequency of the signal to be measured in this example Hz, and the channel sampling rate in the simulation is set to 40 Hz. Therefore, the downsampling factor To satisfy an integer, in this exemplary example, take .

[0208] Then design a compressive sampling pattern that satisfies Equation (2) : . In this example, take . Then from and N, determine the nominal time unit T: seconds. Periodic non-uniform sampling of the acceleration data at each node is performed using 4 channels with a sampling rate of 40 Hz. Among them, the start times of the 1st, 2nd, 3rd, and 4th channels are delayed by 0T, 1T, 4T, and 6T respectively. The specific form is as Figure 2 shown. The sampling duration seconds, and the number of samples K collected by each channel = 400. The samples collected by the 1st to 4th channels are respectively

[0209]

[0210] The acceleration sample sequence at the 4th node obtained by compressive sampling with the general covariance pattern is as Figure 3 shown.

[0211] In this exemplary instance , G is taken as 40, is The unit matrix. ,

[0212] From Obtain the sampling matrix .

[0213] Generate the complete covariance subspace basis matrix according to the definition : The element in the th row and th column is 1, and the rest of the elements are 0. g is a traversing integer, traversing from 1 to 520.

[0214] Then from Calculate the compressed covariance subspace basis matrix , .

[0215] Then from Obtain the sensing matrix .

[0216] In this exemplary instance , . , . From Obtain the compressed covariance matrix .

[0217] From Obtain the expansion coefficient , and then from Reconstruct to obtain the complete covariance matrix .

[0218] Finally, use the covariance random subspace identification algorithm to perform operational modal parameter estimation. In this exemplary instance instance , the system degree of freedom B = 6, and the modal frequencies and damping calculated by Eqs. (16) and (17) are shown in the following table

[0219]

[0220] Where the true values are the theoretical values directly calculated according to the mass matrix M0, stiffness matrix K0, and damping matrix C0 of the simulation system. The estimated values are the modal parameters identified from the data obtained by compressive sampling in the present invention. The estimated values are very close to the true values, verifying the effectiveness of the operational modal parameter estimation method proposed in the present invention.

[0221] The stability diagram of the modal frequency identification results is as Figure 4 shown. Observing Figure 4 it can be seen that the proposed operational modal parameter identification method can accurately identify 6 modal frequencies. The modal shapes calculated by Eq. (18) are asFigure 5 as shown. Observe Figure 5 It can be seen that the modal shapes estimated by the proposed operational modal parameter identification method are highly consistent with the true modal shapes, further verifying the effectiveness of the proposed operational modal parameter estimation method of the present invention.

[0222] In one embodiment, the vibration system is a mass-spring-damper system.

[0223] In one embodiment, if and , then . Therefore, when is the general covariance pattern.

[0224] Although the embodiments of the present invention have been described above in conjunction with the accompanying drawings, the present invention is not limited to the above specific embodiments and application fields. The above specific embodiments are merely illustrative and guiding, rather than restrictive. Those of ordinary skill in the art can also make many forms under the inspiration of this specification and without departing from the scope protected by the claims of the present invention, and these all belong to the scope of protection of the present invention.

Claims

1. A vibration system operation modal parameter estimation method based on compressed random subspace identification, characterized in that: The steps include: Step (1), using a general covariance sampling mode to compress and sample the monitoring signal of the vibration system to obtain a signal sample; Step (2), determining a sampling matrix and a perception matrix based on the general covariance sampling mode; Step (3), estimating a compressed covariance matrix using the signal samples; Step (4), reconstructing a complete covariance matrix based on the covariance matrix; Step (5), use the covariance random subspace identification algorithm to estimate the operational modal parameters.

2. The vibration system operation modal parameter estimation method based on compressed random subspace identification according to claim 1 is characterized in that: Preferably, the step (1) comprises: make Indicates monitoring signal The upper frequency limit of Represents the sampling rate of the sampling channel. According to formula (1), the downsampling factor N that meets the conditions is determined (1)， in: is the rounding symbol; the downsampling factor N is an integer that satisfies the above formula, Then design a compressed sampling mode that satisfies equation (2) , (2)， Among them: Compressed sampling mode is a set of integers; if the compressed sampling mode The condition satisfying formula (2) is called the universal covariance sampling mode. for The non-negative difference set of is a set of elements less than L, and its mathematical definition is as follows: (3)， in and express Any two elements in , take all the elements that satisfy situation, Finally, according to the sampling rate and the downsampling factor N determines the nominal time unit T: , using P sampling rates as , that is, the channel with a sampling period of NT uniformly samples the signal, where the pth channel is delayed relative to the first channel, and the delay time is , , P is equal to The number of elements in equal The pth element in .

3. The vibration system operation modal parameter estimation method based on compressed random subspace identification according to claim 2 is characterized in that: The delay time of the first channel is 0 for reference, that is, The sampling duration is , then the sample sampled by the pth channel is (4)。 4. The vibration system operation modal parameter estimation method based on compressed random subspace identification according to claim 2 is characterized in that: The step (2) comprises: The sampling matrix is ​​determined by the general covariance sampling mode and the perception matrix : (5)， Where: G is a set integer; For size The identity matrix of is the Kronecker product; For size The binary matrix of The elements of the column are 1, , the remaining position elements are 0, (6)， in: The perception matrix The number of columns, , is the compressed covariance subspace basis matrix, which is given by Compressed to get: , , represents the transpose of a matrix, is the complete covariance subspace basis matrix, whose matrix size is , Located in the middle Line The elements of the column are 1, and the rest are 0. g is a traversal integer, from 1 to GN. It is a vectorized operation on the matrix, which concatenates all the columns of the matrix into a long vector.

5. The vibration system operation modal parameter estimation method based on compressed random subspace identification according to claim 4 is characterized in that: The step (3) includes estimating the compressed covariance matrix of the signal samples , (7) Where: Q is the number of snapshots; , q is the integer to be traversed, from 0 to Q-1; .

6. The vibration system operation modal parameter estimation method based on compressed random subspace identification according to claim 5 is characterized in that: The step (4) comprises: First, solve the expansion coefficient of the covariance subspace , (8)， in: For the matrix The vector concatenated from all columns of represents the inverse of the matrix, based on , reconstructing the complete covariance matrix , (9)， in: express The sth element in .

7. The vibration system operation modal parameter estimation method based on compressed random subspace identification according to claim 5 is characterized in that: Step (5) includes, Construct two block Toeplitz matrices using the reconstructed covariance matrix: and , (10)， in: Reason The i-th element of a vector of length 2GN-1 is formed by concatenating the elements of the first column and the first row of , where i is a given positive integer. , Indicates the floor symbol, Then Perform singular value decomposition (11)， in is the diagonal matrix corresponding to the singular values ​​of the signal, is the diagonal matrix corresponding to the noise singular values; and are the left orthogonal matrices corresponding to the singular values ​​of signal and noise respectively; and are the right orthogonal matrices corresponding to the signal and noise singular values, respectively. Compute the observable matrix and controllable matrix : (12)， (13)， Calculate the state matrix And the output matrix , (14)， (15)， in: Is The matrix composed of rows 1 to B in the equation, B is equal to the system's degree of freedom, For the state matrix Perform eigenvalue decomposition to obtain eigenvalues and the eigenvector , where B is the number of modes of the system, Calculate the system natural frequency , Damping ratio and vibration mode , (16)， (17)， (18)。

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