A modal parameter identification method for structures with bounded and correlated uncertainties

Abstract

The present invention relates to a modal parameter identification method for a structure with bounded and correlated uncertainties, and belongs to the technical field of modal parameter identification and structural health monitoring. The method proposed in the present invention derives the propagation law of convex set uncertainty of structural parameters, and provides a general solution method for parameter identification under the influence of convex set uncertainty. The present invention takes into account the correlation between structural uncertainty factors, which is more scientific and more applicable than the identification method that considers the uncertainty of each factor separately; in addition, the present invention proposes a second-order convex set singular value decomposition and a second-order convex set eigenvalue decomposition algorithm. Compared with the first-order method, the second-order Taylor expansion of singular values ​​and eigenvalues ​​can obtain more accurate second-order interval estimation results of singular values ​​and second-order interval estimation results of eigenvalues, thereby obtaining more accurate estimation results of modal parameters such as natural frequency and damping ratio.

Description

Technical Field

[0001] The present invention is applicable to the problem of structural uncertainty identification, belongs to the fields of modal parameter identification, structural health monitoring, etc., and specifically relates to a modal parameter identification method for a structure with bounded and correlated uncertainties. Background Art

[0002] With the continuous advancement of modern engineering technology, the complexity and dynamic characteristics of buildings and infrastructure are becoming increasingly significant. The dynamic response of structures has become increasingly important for ensuring their safety and reliability. In recent years, the influence of natural factors such as earthquakes and wind loads has led to a sharp increase in the demand for structural health monitoring systems and damage identification technologies. Effective modal parameter identification not only helps engineers understand the behavior of structures under dynamic loads but also provides an important basis for structural maintenance and reinforcement. Therefore, accurately identifying the modal parameters of structures (such as frequency and damping ratio) has become a core task in structural dynamics research.

[0003] Traditional modal parameter identification methods, such as the ERA algorithm, perform well in multiple-input, multiple-output (MIMO) systems. ERA extracts the modal characteristics of a structure from input and output data by performing eigenvalue analysis on the system's state-space equations. However, ERA often suffers from reduced identification accuracy when processing high-dimensional data and in the presence of noise. Measurement noise can lead to the appearance of pseudo-modes, which in turn affects the accuracy of modal parameters. In addition, the ERA method has high requirements for matrix dimensions during the calculation process, and the consumption of computing resources becomes a major bottleneck when processing large structures.

[0004] To overcome these limitations, the ERA / DC method was developed. ERA / DC optimizes the algorithm's computational efficiency by introducing correlated Hankel matrices rather than generalized Hankel matrices. This method reduces computational complexity while also improving noise immunity. However, as the complexity of structural models increases, uncertainties in environmental and material parameters become increasingly prominent. These uncertainties can arise from a variety of factors, including variations in material properties, differences in construction techniques, and fluctuations in environmental conditions. These factors not only increase the difficulty of modal parameter identification but can also lead to inaccurate structural safety assessment results.

[0005] Against this backdrop, researchers have gradually recognized the shortcomings of traditional identification methods in addressing the impact of uncertainty on modal parameter identification. Many studies have shown that ignoring the correlations and uncertainties between parameters can lead to biased results. Therefore, establishing an effective mathematical model to describe these uncertainties is key to improving the accuracy of modal parameter identification. In recent years, convex set theory has demonstrated promising results in describing uncertainty, as it considers the correlations between uncertain parameters. By constructing an uncertainty model based on convex sets, researchers can more comprehensively consider the relationships between different parameters, thereby providing a more reliable basis for modal parameter identification.

[0006] While accounting for structural uncertainty, uncertainty persists throughout the identification process, impacting final accuracy. However, the perturbation method overcomes the limitations of traditional methods with high-dimensional data and uncertainty. This method not only focuses on the precise identification of modal parameters but also considers how these parameters vary under the influence of uncertainty. Only a small amount of information, such as the upper and lower bounds of the structural uncertainty, is required to further determine the identification interval, significantly reducing computational effort and time. Summary of the Invention

[0007] The purpose of the present invention is to overcome the problems of inaccurate identification caused by structural uncertainty and the correlation between uncertainty factors, and propose a modal parameter identification method for structures with bounded and correlated uncertainties.

[0008] The technical solution of the present invention is as follows: First, a convex set is used to quantify structural uncertainty. Correlations are added to the original independent uncertainty boundary pairs to generate new uncertainty boundaries. Next, a dynamic model of the structure is established to obtain the discretized state space equations, response functions, and generalized Hankel matrices, while simultaneously considering the propagation of convex set uncertainty. Then, based on perturbation theory, a second-order singular value decomposition method for convex sets is proposed. This method analyzes the propagation law of convex set uncertainty in the singular value decomposition process and obtains the singular value boundary estimation after the singular value decomposition. Finally, the system matrix required for identification is obtained, and a second-order eigenvalue decomposition is performed to obtain the eigenvalue boundary estimation. Then, based on the eigenvalue boundary estimation, the modal parameter interval is obtained.

[0009] The specific technical solution of the present invention is: a modal parameter identification method for a structure with bounded and correlated uncertainties, the method comprising the following steps:

[0010] Step 1: Determine the uncertainty parameters and their uncertainty ranges in the structural parameters;

[0011] Step 2: Construct the correlation matrix between the uncertainty parameters, and construct the uncertainty parameter expression in convex set form according to the uncertainty range of the parameters and the correlation matrix, which is the convex set uncertainty;

[0012] Step 3: Construct a dynamic model of the structure and calculate the partial derivatives after the convex set uncertainty propagation in the dynamic model;

[0013] Step 4: Discretize the state space of the dynamic model of the structure in step 3 to obtain a discrete-time, time-invariant, linear set of dynamic system state parameter equations. Then, obtain the system matrix, input matrix, and output matrix based on the obtained set of dynamic system state parameter equations, and calculate the partial derivatives of the system matrix and input matrix with respect to the uncertainty parameters and the uncertainty bounds.

[0014] Step 5: Calculate the response function under unit pulse excitation based on the system matrix, input matrix, and output matrix obtained in step 4. Then construct the generalized Hankel matrix based on the response function at each time interval and calculate the partial derivative of the generalized Hankel matrix with respect to the uncertainty parameters.

[0015] Step 6: Based on the partial derivative form of the generalized Hankel matrix with respect to the convex set uncertainty parameters obtained in step 5, introduce it into the second-order convex set singular value decomposition process of the generalized Hankel matrix to obtain the convex set uncertainty estimation form of the singular values ​​and the first-order and second-order partial derivative forms of the convex set uncertainty parameters, the convex set uncertainty estimation form of the left and right singular value vectors and the first-order and second-order partial derivative forms of the convex set uncertainty parameters;

[0016] Step 7: Obtain the generalized Hankel matrix after time lapse based on the generalized Hankel matrix obtained in step 5;

[0017] Step 8: Construct the structure identification system matrix based on the time-shifted generalized Hankel matrix obtained in step 7 and the convex set uncertainty estimation form of the singular values ​​and left and right singular vectors obtained in step 6, and calculate the partial derivative form of the structure identification system matrix with respect to the convex set uncertainty parameters;

[0018] Step 9: Perform a second-order convex set eigenvalue decomposition on the structure identification system matrix obtained in step 8 to obtain the convex set uncertainty estimation form of the complex eigenvalue and the first-order and second-order partial derivative forms of the complex eigenvalue with respect to the uncertainty parameters, including the convex set uncertainty estimation form of the real and imaginary parts of the complex eigenvalue and the partial derivative forms of the two with respect to the uncertainty parameters;

[0019] Step 10: Calculate the modal parameters of the structure based on the convex set uncertainty estimation form of the complex eigenvalues ​​obtained in step 9 and the first-order and second-order partial derivative forms of the uncertainty parameters, and obtain the uncertainty identification results of the natural frequency and damping ratio modal parameters of the structure to perform structural health detection, fault diagnosis and dynamic control.

[0020] The present invention has the beneficial effects:

[0021] 1. By constructing a correlation matrix between uncertainty parameters, the present invention can comprehensively consider the mutual influence between different uncertainty factors, making the identification of modal parameters more scientific.

[0022] 2. By introducing convex set theory, the present invention can effectively quantify and process structural uncertainties and their correlations, thereby reducing the deviation in the modal parameter identification process and providing more accurate modal parameter estimation.

[0023] 3. The present invention adopts perturbation theory, and only a small amount of information about the upper and lower bounds of uncertainty is needed to perform interval solutions for modal parameters, which significantly reduces the amount of calculation and the time required, and has more advantages in engineering applications.

[0024] 4. Based on perturbation theory, the present invention proposes a second-order singular value decomposition algorithm for convex sets. Through the second-order Taylor expansion, a more accurate singular value boundary estimation is obtained compared with the first-order singular value decomposition.

[0025] 5. This method is suitable for the identification of structural modal parameters in a variety of complex environments and conditions. Especially in the identification of large-scale structural modal parameters, it is more accurate than the identification method that does not consider the correlation. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 Schematic diagram of the method flow of the present invention;

[0027] Figure 2 This is a schematic diagram of a four-layer frame structure. DETAILED DESCRIPTION

[0028] In order to make the objectives, technical solutions, and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are intended only to illustrate the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other. To achieve the above-mentioned objectives, the present invention adopts the following technical solutions.

[0029] Figure 1 This is a flow chart of a modal parameter identification method for a structure with bounded and correlated uncertainties proposed by the present invention, comprising the following steps:

[0030] Step 1: Determine the uncertainty parameters and their uncertainty ranges in the structural parameters;

[0031] Step 2: Construct the correlation matrix between the uncertainty parameters, and construct the uncertainty parameter expression in convex set form according to the uncertainty range of the parameters and the correlation matrix, which is the convex set uncertainty;

[0032] Step 3: Construct a dynamic model of the structure and calculate the partial derivatives after the convex set uncertainty propagation in the dynamic model;

[0033] Step 4: Discretize the state space of the dynamic model of the structure in step 3 to obtain a discrete-time, time-invariant, linear set of dynamic system state parameter equations. Then, obtain the system matrix, input matrix, and output matrix based on the obtained set of dynamic system state parameter equations, and calculate the partial derivatives of the system matrix and input matrix with respect to the uncertainty parameters and the uncertainty bounds.

[0034] Step 5: Calculate the response function under unit pulse excitation based on the system matrix, input matrix, and output matrix obtained in step 4. Then construct the generalized Hankel matrix based on the response function at each time interval and calculate the partial derivative of the generalized Hankel matrix with respect to the uncertainty parameters.

[0035] Step 6: Based on the partial derivative form of the generalized Hankel matrix with respect to the convex set uncertainty parameters obtained in step 5, introduce it into the second-order convex set singular value decomposition process of the generalized Hankel matrix to obtain the convex set uncertainty estimation form of the singular values ​​and the first-order and second-order partial derivative forms of the convex set uncertainty parameters, the convex set uncertainty estimation form of the left and right singular value vectors and the first-order and second-order partial derivative forms of the convex set uncertainty parameters;

[0036] Step 7: Obtain the generalized Hankel matrix after time lapse based on the generalized Hankel matrix obtained in step 5;

[0037] Step 8: Construct the structure identification system matrix based on the time-shifted generalized Hankel matrix obtained in step 7 and the convex set uncertainty estimation form of the singular values ​​and left and right singular vectors obtained in step 6, and calculate the partial derivative form of the structure identification system matrix with respect to the convex set uncertainty parameters;

[0038] Step 9: Perform a second-order convex set eigenvalue decomposition on the structure identification system matrix obtained in step 8 to obtain the convex set uncertainty estimation form of the complex eigenvalue and the first-order and second-order partial derivative forms of the complex eigenvalue with respect to the uncertainty parameters, including the convex set uncertainty estimation form of the real and imaginary parts of the complex eigenvalue and the partial derivative forms of the two with respect to the uncertainty parameters;

[0039] Step 10: Calculate the modal parameters of the structure based on the convex set uncertainty estimation form of the complex eigenvalues ​​obtained in step 9 and the first-order and second-order partial derivative forms of the uncertainty parameters, and obtain the uncertainty identification results of the natural frequency and damping ratio modal parameters of the structure to perform structural health detection, fault diagnosis and dynamic control.

[0040] Furthermore, in step 1, the uncertainty of the structural parameters is determined as follows:

[0041] ,

[0042] in, represents the uncertainty vector, which is composed of uncertainty parameters composition, represents the lower and upper bounds of the uncertainty vector without considering the correlation between uncertainty parameters, represents the central value vector of the uncertainty vector, represents the radius vector of the uncertainty vector, Indicates the standard uncertainty range , the total number of uncertainty parameters is , each uncertainty parameter is expressed as a central value and radius The combination form of Indicates the The uncertainty parameters are related, superscript Indicates the parameter center value.

[0043] Furthermore, in step 2, the correlation matrix between the uncertainty parameters is constructed as follows:

[0044] ,

[0045] in, represents the correlation matrix between uncertainty parameters, Indicates the The uncertainty parameter is The covariance between the uncertainty parameters, Indicates the The uncertainty parameter is The correlation coefficient between the uncertainty parameters.

[0046] Furthermore, in step 2, the uncertainty parameters are expressed as a convex set form according to the parameter uncertainty and correlation matrix:

[0047] ,

[0048] in, represents a convex set.

[0049] Furthermore, in step 3, the dynamic model of the structure is constructed as:

[0050] ,

[0051] in, denote the displacement, velocity and acceleration vectors respectively, represents the external load, is the structural mass matrix, is the structural stiffness matrix, is the structural damping matrix, which obeys the Rayleigh damping form, that is, , is the Rayleigh damping coefficient, which is determined according to the set damping ratio.

[0052] Furthermore, in step 3, the structural mass matrix of the structural dynamics model is calculated For each uncertainty parameter Modified partial derivatives , structural stiffness matrix Partial derivatives with respect to uncertain parameters , structural damping matrix Partial derivatives with respect to uncertain parameters They are:

[0053] (1) Structural mass matrix Performing a first-order Taylor expansion, we get:

[0054] ,

[0055] in, ;

[0056] The above formula, Represents the mass matrix Partial derivatives of uncertain parameters;

[0057] (2) Structural stiffness matrix Performing a first-order Taylor expansion, we get:

[0058] ,

[0059] in, ;

[0060] The above formula, Represents the stiffness matrix For the first Partial derivatives of uncertain parameters;

[0061] (3) Structural damping matrix Performing a first-order Taylor expansion, we get:

[0062] ,

[0063] in, ;

[0064] The above formula, Represents the stiffness matrix For the first Partial derivatives of uncertain parameters.

[0065] Furthermore, in step 4, the state space of the constructed structural dynamics model is:

[0066] ,

[0067] in, represents the state variable, represents the output variable, represents the input variable, They represent the system matrix, input matrix, and output matrix in continuous time, respectively, and are calculated as follows:

[0068] ,

[0069] in, Represents the identity matrix.

[0070] Furthermore, in step 4, the discretized state space is:

[0071] ,

[0072] in, Represents the current sampling moment, They represent the system matrix and input matrix in discrete time, respectively, and are calculated as follows:

[0073] ,

[0074] in, is the sampling time interval, the discretized output matrix The representation is consistent with that in continuous time; the system matrix after discretization is and the input matrix For uncertain parameters The partial derivatives are:

[0075] ,

[0076] In a convex set, the system matrix There are the following relationships:

[0077] ,

[0078] in, represents a convex set, is the system matrix Middle Rank The convex set of column elements is calculated as follows:

[0079] ,

[0080] in, Representation matrix Middle Rank Nominal values ​​of column elements; orthogonal matrix and the diagonal matrix Obtained by Cholesky decomposition: ; represents the hyperellipsoid formed by the initial convex set uncertainty parameters; yes Bias vector for each uncertainty parameter:

[0081] ; express Convex set perturbation of ;

[0082] Through convex set uncertainty, we know The boundary of is calculated by the following formula:

[0083] ,

[0084] In step 4, the uncertainty range of the system matrix and the input matrix is:

[0085] .

[0086] Furthermore, in step 5, the system response function under unit pulse excitation is for:

[0087] ,

[0088] Construct the generalized Hankel matrix as:

[0089] ,

[0090] in, and denote the number of row blocks and column blocks of the generalized Hankel matrix respectively;

[0091] In step 5, the partial derivative of the generalized Hankel matrix with respect to the uncertainty parameter is calculated:

[0092] ,

[0093] in, .

[0094] Furthermore, in step 6, the generalized Hankel matrix considering parameter correlation is Perform a second-order singular value decomposition:

[0095] ,

[0096] in, denote the left singular value matrix and the right singular value matrix respectively, Respectively represent Column left singular value vector and Column right singular value vector:

[0097] ,

[0098] in, is the mode number of interest; 、 、 They represent the center value of the left singular value vector and the uncertainty range of the left singular value vector after the first-order and second-order Taylor expansion respectively; 、 、 They represent the center value of the right singular value vector and the uncertainty range of the right singular value vector after the first-order and second-order Taylor expansion respectively;

[0099] is the singular value matrix, Indicates the singular values; ,

[0100] Where, 、 、 They represent the central value of the singular value and the uncertainty range of the singular value after the first-order and second-order Taylor expansion respectively;

[0101] The radius after first-order Taylor expansion is calculated by the following formula:

[0102] ,

[0103] in, express Bias vector for each uncertainty parameter:

[0104] ,

[0105] express Bias vector for each uncertainty parameter:

[0106] ,

[0107] express Bias vector for each uncertainty parameter:

[0108] ,

[0109] The uncertainty range after the second-order Taylor expansion is calculated as follows:

[0110] ,

[0111] According to the propagation of uncertainty in convex sets, we have:

[0112] ,

[0113] in, They represent the partial derivative of the left singular value vector with respect to the uncertainty parameter and the partial derivative of the right singular value vector with respect to the interval uncertainty parameter, respectively, and are calculated by the following formula:

[0114] ,

[0115] in, Indicates the The left singular value vector elements, Indicates the The right singular value vector elements.

[0116] Furthermore, in step 7, the generalized Hankel matrix after time elapses is for:

[0117] .

[0118] Furthermore, in step 8, the structure identification system matrix constructed for:

[0119] ,

[0120] in, ;

[0121] Structural Identification System Matrix The partial derivative with respect to the uncertainty parameter is:

[0122] ,

[0123] Furthermore, in step 9, the structure identification system matrix Perform the second-order convex set eigenvalue decomposition as follows:

[0124] ,

[0125] in, is the eigenvalue vector, represents the complex eigenvalue matrix, Indicates the Item complex eigenvalue;

[0126] , ,

[0127] ,

[0128] Among them, the first-order partial derivative of the complex eigenvalue with respect to the uncertainty parameter and second-order partial derivatives It is calculated as follows:

[0129] ,

[0130] ,

[0131] Among them, the first-order partial derivative of the eigenvalue vector with respect to the uncertainty parameter is for:

[0132] ,

[0133] .

[0134] Furthermore, in step 10, the modal parameters of the structure calculated are:

[0135] natural circular frequency for: ,

[0136] in, They represent the central value of the natural frequency, the uncertainty range of the natural frequency, and the lower and upper bounds of the range, respectively. The expression is ;

[0137] The upper and lower bounds are determined as follows:

[0138] Will The expression is considered The quadratic function of Is a variable with a range, get exist The minimum and maximum values ​​within the range of variation are the lower and upper bounds, and the calculation process is as follows:

[0139] ,

[0140] ,

[0141] in, represents the uncertainty parameter of the i-th natural frequency with respect to the j-th uncertainty parameter uncertainty range;

[0142] Among them, the first-order and second-order partial derivatives of the natural circular frequency with respect to the uncertainty parameter are as follows:

[0143] ,

[0144] ,

[0145] in, Represents complex eigenvalues The partial derivatives of the real and imaginary parts of the uncertainty parameter are calculated as follows:

[0146] ,

[0147] ,

[0148] Damping ratio range for: ;

[0149] in, Respectively represent the central value of the damping ratio, the uncertainty range of the damping ratio, and the lower and upper bounds, The expression is ;

[0150] The upper and lower bounds are determined as follows:

[0151] Will The expression is considered The quadratic function of Is a variable with a range, get exist The minimum and maximum values ​​within the range of variation are the lower and upper bounds, and the calculation process is as follows:

[0152] ,

[0153] ,

[0154] in, represents the uncertainty parameter of the i-th natural frequency with respect to the j-th uncertainty parameter uncertainty range;

[0155] The first-order and second-order partial derivatives of the damping ratio with respect to the uncertainty parameter are as follows:

[0156] ,

[0157] .

[0158] Further, step 1: determine the uncertain parameters and parameter uncertainties in the structural parameters; specifically:

[0159] like Figure 2 As shown in the figure, a four-story frame structure is used as the basic example, assuming that the structural quality and stiffness Structural parameters such as are affected by processing conditions and working scenarios and are uncertain. They are expressed in the form of interval vectors as follows:

[0160]

[0161] in, Indicates the standard uncertainty range ; The uncertainty vectors representing the structural mass and structural stiffness are The lower and upper bounds of the uncertainty vector The uncertainty parameters composition, which can be expressed as the median and radius The number of uncertainty parameters is .

[0162] The material density of the structure is , Young's modulus is The cross-sectional areas of the four beams are , , , The cross-sectional area of ​​the cylinder is The mass and stiffness of the final model have an uncertainty of 2%.

[0163] In step 10, the uncertainty identification results of the modal parameters such as the natural frequency and damping ratio of the structure are shown in the following table:

[0164]

[0165] In this way, parameters such as the natural frequency of the structure are obtained. Based on the obtained modal parameters, a theoretical model can be provided for the design of the structural controller, and it can also be used as an important parameter for structural health monitoring.

[0166] In summary, the above are only preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for modal parameter identification of a structure with bounded and correlated uncertainties, characterized in that: The steps of the method include: Step 1: Determine the uncertainty parameters and their uncertainty ranges in the structural parameters; Step 2: Construct the correlation matrix between the uncertainty parameters, and construct the uncertainty parameter expression in convex set form according to the uncertainty range of the parameters and the correlation matrix, which is the convex set uncertainty; Step 3: Construct a dynamic model of the structure and calculate the partial derivatives after the convex set uncertainty propagation in the dynamic model; Step 4: Discretize the state space of the dynamic model of the structure in step 3 to obtain a discrete-time, time-invariant, linear set of dynamic system state parameter equations. Then, obtain the system matrix, input matrix, and output matrix based on the obtained set of dynamic system state parameter equations, and calculate the partial derivatives of the system matrix and input matrix with respect to the uncertainty parameters and the uncertainty bounds. Step 5: Calculate the response function under unit pulse excitation based on the system matrix, input matrix, and output matrix obtained in step 4. Then construct the generalized Hankel matrix based on the response function at each time interval and calculate the partial derivative of the generalized Hankel matrix with respect to the uncertainty parameters. Step 6: Based on the partial derivative form of the generalized Hankel matrix with respect to the convex set uncertainty parameters obtained in step 5, introduce it into the second-order convex set singular value decomposition process of the generalized Hankel matrix to obtain the convex set uncertainty estimation form of the singular values ​​and the first-order and second-order partial derivative forms of the convex set uncertainty parameters, the convex set uncertainty estimation form of the left and right singular value vectors and the first-order and second-order partial derivative forms of the convex set uncertainty parameters; Step 7: Obtain the generalized Hankel matrix after time lapse based on the generalized Hankel matrix obtained in step 5; Step 8: Construct the structure identification system matrix based on the time-shifted generalized Hankel matrix obtained in step 7 and the convex set uncertainty estimation form of the singular values ​​and left and right singular vectors obtained in step 6, and calculate the partial derivative form of the structure identification system matrix with respect to the convex set uncertainty parameters; Step 9: Perform a second-order convex set eigenvalue decomposition on the structure identification system matrix obtained in step 8 to obtain the convex set uncertainty estimation form of the complex eigenvalue and the first-order and second-order partial derivative forms of the complex eigenvalue with respect to the uncertainty parameters, including the convex set uncertainty estimation form of the real and imaginary parts of the complex eigenvalue and the partial derivative forms of the two with respect to the uncertainty parameters; Step 10: Calculate the modal parameters of the structure based on the convex set uncertainty estimation form of the complex eigenvalues ​​obtained in step 9 and the first-order and second-order partial derivative forms of the uncertainty parameters, and obtain the uncertainty identification results of the natural frequency and damping ratio modal parameters of the structure to perform structural health detection, fault diagnosis and dynamic control.

2. The method for modal parameter identification of a structure with bounded and correlated uncertainties according to claim 1, characterized in that: In step 1, the uncertainties in the structural parameters are determined as follows: ， in, represents the uncertainty vector, which is composed of uncertainty parameters composition, represents the lower and upper bounds of the uncertainty vector without considering the correlation between uncertainty parameters, represents the central value vector of the uncertainty vector, represents the radius vector of the uncertainty vector, Indicates the standard uncertainty range , the total number of uncertainty parameters is , each uncertainty parameter is expressed as a central value and radius The combination form of Indicates the The uncertainty parameters are related, superscript Indicates the parameter center value.

3. The method for modal parameter identification of a structure with bounded and correlated uncertainties according to claim 2, characterized in that: In step 2, the correlation matrix between uncertainty parameters is constructed as follows: ， in, represents the correlation matrix between uncertainty parameters, Indicates the The uncertainty parameter is The covariance between the uncertainty parameters, Indicates the The uncertainty parameter is The correlation coefficient between the uncertainty parameters.

4. The method for modal parameter identification of a structure with bounded and correlated uncertainties according to claim 3, characterized in that: In step 2, the uncertainty parameters are expressed as a convex set based on the parameter uncertainty and correlation matrix: ， in, represents a convex set.

5. The method for identifying modal parameters of a structure with bounded and correlated uncertainties according to claim 4, characterized in that: In step 3, the dynamic model of the structure is constructed as: ， in, denote the displacement, velocity and acceleration vectors respectively, represents the external load, is the structural mass matrix, is the structural stiffness matrix, is the structural damping matrix, which obeys the Rayleigh damping form, that is, , is the Rayleigh damping coefficient, which is determined according to the set damping ratio.

6. The method for identifying modal parameters of a structure with bounded and correlated uncertainties according to claim 5, characterized in that: In step 3, the structural mass matrix of the structural dynamics model is calculated For each uncertainty parameter Modified partial derivatives , structural stiffness matrix Partial derivatives with respect to uncertain parameters , structural damping matrix Partial derivatives with respect to uncertain parameters They are: (1) Structural mass matrix Performing a first-order Taylor expansion, we get: ， in, ; The above formula, Represents the mass matrix Partial derivatives of uncertain parameters; (2) Structural stiffness matrix Performing a first-order Taylor expansion, we get: ， in, , The above formula, Represents the stiffness matrix For the first Partial derivatives of uncertain parameters; (3) Structural damping matrix Performing a first-order Taylor expansion, we get: ， in, , The above formula, Represents the stiffness matrix For the first Partial derivatives of uncertain parameters.

7. The method for modal parameter identification of a structure with bounded and correlated uncertainties according to claim 6, characterized in that: In step 4, the state space of the constructed structural dynamics model is: ， in, represents the state variable, represents the output variable, represents the input variable, They represent the system matrix, input matrix, and output matrix in continuous time, respectively, and are calculated as follows: ， in, Represents the identity matrix.

8. The method for identifying modal parameters of a structure with bounded and correlated uncertainties according to claim 7, characterized in that: In step 4, the discretized state space is: ， in, Represents the current sampling moment, They represent the system matrix and input matrix in discrete time, respectively, and are calculated as follows: ， in, is the sampling time interval, the discretized output matrix The representation is consistent with that in continuous time; the system matrix after discretization is and the input matrix For uncertain parameters The partial derivatives are: ， In a convex set, the system matrix There are the following relationships: ， in, represents a convex set, is the system matrix Middle Rank The convex set of column elements is calculated as follows: ， in, Representation matrix Middle Rank Nominal values ​​of column elements; orthogonal matrix and the diagonal matrix Obtained by Cholesky decomposition: ; represents the hyperellipsoid formed by the initial convex set uncertainty parameters; yes Bias vector for each uncertainty parameter: ; express Convex set perturbation of ; Through convex set uncertainty, we know The boundary of is calculated by the following formula: ， In step 4, the uncertainty range of the system matrix and the input matrix is: 。 9. The method for identifying modal parameters of a structure with bounded and correlated uncertainties according to claim 8, characterized in that: In step 5, the system response function under unit pulse excitation is for: ， Construct the generalized Hankel matrix as: ， in, and denote the number of row blocks and column blocks of the generalized Hankel matrix respectively; In step 5, the partial derivative of the generalized Hankel matrix with respect to the uncertainty parameter is calculated: ， in, .

10. The method for modal parameter identification of a structure with bounded and correlated uncertainties according to claim 9, characterized in that: In step 6, the generalized Hankel matrix considering parameter correlation is Perform a second-order singular value decomposition: ， in, denote the left singular value matrix and the right singular value matrix respectively, Respectively represent Column left singular value vector and Column right singular value vector: ， in, is the mode number of interest; 、 、 They represent the center value of the left singular value vector and the uncertainty range of the left singular value vector after the first-order and second-order Taylor expansion respectively; 、 、 They represent the center value of the right singular value vector and the uncertainty range of the right singular value vector after the first-order and second-order Taylor expansion respectively; is the singular value matrix, Indicates the singular values; , where 、 、 They represent the central value of the singular value and the uncertainty range of the singular value after the first-order and second-order Taylor expansion respectively; The radius after first-order Taylor expansion is calculated by the following formula: ， in, express Bias vector for each uncertainty parameter: ， express Bias vector for each uncertainty parameter: ， express Bias vector for each uncertainty parameter: ， The uncertainty range after the second-order Taylor expansion is calculated as follows: ， According to the propagation of uncertainty in convex sets, we have: ， in, They represent the partial derivative of the left singular value vector with respect to the uncertainty parameter and the partial derivative of the right singular value vector with respect to the interval uncertainty parameter, respectively, and are calculated by the following formula: ， in, Indicates the The left singular value vector elements, Indicates the The right singular value vector elements.

11. The method for identifying modal parameters of a structure with bounded and correlated uncertainties according to claim 9, characterized in that: In step 7, the generalized Hankel matrix after time elapses is for: 。 12. The method for identifying modal parameters of a structure with bounded and correlated uncertainties according to claim 11, characterized in that: In step 8, the structure identification system matrix constructed for: ， in, ; Structural Identification System Matrix The partial derivative with respect to the uncertainty parameter is: 。 13. The method for identifying modal parameters of a structure with bounded and correlated uncertainties according to claim 12, characterized in that: In step 9, the structure identification system matrix Perform the second-order convex set eigenvalue decomposition as follows: ， in, is the eigenvalue vector, represents the complex eigenvalue matrix, Indicates the Item complex eigenvalue; ， ， ， Among them, the first-order partial derivative of the complex eigenvalue with respect to the uncertainty parameter and second-order partial derivatives It is calculated as follows: ， ， Among them, the first-order partial derivative of the eigenvalue vector with respect to the uncertainty parameter is for: ， 。 14. The method for identifying modal parameters of a structure with bounded and correlated uncertainties according to claim 13, characterized in that: In step 10, the modal parameters of the structure calculated are: natural circular frequency for: , in, They represent the central value of the natural frequency, the uncertainty range of the natural frequency, and the lower and upper bounds of the range, respectively. The expression is ; The upper and lower bounds are determined as follows: Will The expression is considered The quadratic function of Is a variable with a range, get exist The minimum and maximum values ​​within the range are the lower and upper bounds, and the calculation process is as follows: ， ， in, represents the uncertainty parameter of the i-th natural frequency with respect to the j-th uncertainty parameter uncertainty range; Among them, the first-order and second-order partial derivatives of the natural circular frequency with respect to the uncertainty parameter are as follows: ， ， in, Represents complex eigenvalues The partial derivatives of the real and imaginary parts of the uncertainty parameter are calculated as follows: ， ， Damping ratio range for: ; in, Respectively represent the central value of the damping ratio, the uncertainty range of the damping ratio, and the lower and upper bounds, The expression is ; The upper and lower bounds are determined as follows: Will The expression is considered The quadratic function of Is a variable with a range, get exist The minimum and maximum values ​​within the range are the lower and upper bounds, and the calculation process is as follows: ， ， in, represents the uncertainty parameter of the i-th natural frequency with respect to the j-th uncertainty parameter uncertainty range; The first-order and second-order partial derivatives of the damping ratio with respect to the uncertainty parameter are as follows: ， 。

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