Hybrid Test Method for Model Updating Based on a Novel Constrained Volume Kalman Filter

Abstract

The present invention discloses a model updating hybrid test method based on a novel constrained cubature Kalman filter. This method adopts a multi-step symmetry sampling constraint strategy, adjusts the cubature points that violate the constraints along the direction of the line connecting them to their symmetric cubature points to the constraint boundary. At the same time, to ensure that the adjusted cubature points still have symmetry, the other symmetric cubature point also adopts the same sampling step. This constraint strategy not only maintains the symmetry of the cubature points of the cubature Kalman filter, but also adopts different step lengths for different groups of sampling points, maximizing the dispersion state of the cubature points of the cubature Kalman filter, solves the problem of the cubature Kalman filter algorithm without boundary constraints, and inherits the high-precision and high-stability characteristics of the cubature Kalman filter, which is conducive to the application and popularization of the model updating hybrid test technology.

Description

Technical Field

[0001] The present invention belongs to the field of seismic tests for structural engineering, and particularly relates to a model updating hybrid test method based on a novel constrained cubature Kalman filter. Background Art

[0002] The hybrid test method divides the prototype structure into two parts. Among them, the key components that the researchers are most concerned about or have strong nonlinearity are used as test substructures (physical tests are carried out in the laboratory); the remaining parts are used as numerical substructures (numerical simulations are performed using finite element model calculation software), and the substructures are connected through an online interaction method. This method combines substructure tests and numerical simulations, and can more realistically reflect the failure mechanism of the structure under seismic action. It is an effective means for structural seismic tests. This method effectively reduces costs, space, and time, and can be used to complete large-scale or even full-scale tests.

[0003] However, when numerical simulations are carried out for its numerical substructure, once most of the numerical substructure enters the nonlinear state, errors will occur in the assumed numerical model parameters. When the numerical calculation errors gradually increase with the progress of the test, it will inevitably lead to a significant reduction in the accuracy of the hybrid test, or even test failure. The model updating technology can correct the problem of errors in numerical model parameters and improve the test accuracy. The cubature Kalman filter algorithm is one of the commonly used parameter identification algorithms for model updating hybrid tests. However, when the state update value or sampling cubature points obtained by the cubature Kalman filter algorithm exceed a certain range, if the updated parameters are used for the numerical model, it will affect the stability and accuracy of the numerical model. Summary of the Invention

[0004] The purpose of the present invention is to provide a model updating hybrid test method based on a novel constrained cubature Kalman filter to solve the problems of the unbounded constraint of the cubature Kalman filter algorithm and the accuracy of model updating in hybrid tests.

[0005] The technical solution of the present invention:

[0006] A model updating hybrid test method based on a novel constrained cubature Kalman filter includes the following steps:

[0007] S1. Divide the overall structure into a numerical substructure and a test substructure, and establish the overall structure motion equation; among them, the numerical substructure is established using finite element software;

[0008] S2. Define the pre-identified model parameters, and establish the state equation and the observation equation of the novel constrained cubature Kalman filter, and give the initial state estimation mean and the initial state estimation covariance matrix , Process noise covariance matrix and observation noise covariance matrix ;

[0009] S3. Solve the overall structural motion equation using a numerical integration algorithm to obtain the structural command at the th step ( , , ), the loading command for the test substructure ( , , ), and the numerical substructure calculation command ( , , ); , and are the displacement vector, velocity vector, and acceleration vector of the structure respectively. The subscript represents the integration step, the subscript N represents the numerical substructure, and the subscript E represents the test substructure;

[0010] S4. Send the loading command for the test substructure at the th step ( , , ) to the loading system. The loading system completes the physical loading of the test substructure to obtain the reaction force of the test substructure at the th step ;

[0011] S5. Based on the loading command for the test substructure at the th step ( , , ) and its reaction force , use a new constrained volume Kalman filter to online identify the model parameters of the test substructure, and transmit the identified parameters to the numerical substructure to update the parameters in the numerical substructure that have the same model as the test substructure;

[0012] S6. Send the numerical substructure calculation command at the th step ( , , ) to the numerical substructure. After calculation, the reaction force of the numerical substructure at the th step can be obtained;

[0013] S7. Transmit the substructure reaction forces and obtained in S4 and S6 to the structural motion equation;

[0014] S8. Let = Repeat steps S3 - S7 until the test ends.

[0015] Furthermore, the specific method of using the new constrained volume Kalman filter to online identify the test sub - structure model parameters in step S5 is as follows:

[0016] Assume that the mean of the state prior estimate at the prediction step and the covariance matrix are respectively equal to the mean of the state estimate at the previous update step and the covariance matrix , that is ;

[0017] According to the mean of the state prior estimate at the prediction step and the covariance matrix , centered on symmetrically construct

[0018]

[0019] In the formula, is the th volume point at the th step, is the sampling step size of the th volume point at the n dimensional unit sphere and the intersection point set of the space coordinate axes; represents the th column of the point set (if , then j = 1, 2, 3, 4).

[0020] The weight corresponding to each volume point is:

[0021]

[0022] To ensure the symmetry strategy of volume point sampling, a multi - step - size sampling constraint strategy is adopted. Adjust the volume points that violate the constraints to the constraint boundary along the connection direction with their symmetric volume points. At the same time, to ensure that the adjusted volume points still have symmetry, the symmetric volume points also adopt the same sampling step size;

[0023] The sampling step size of the volume points of the new constrained volume Kalman filter is:

[0024]

[0025] The reasonable value range of the volume points is , then the constraint conditions are:

[0026]

[0027]

[0028]

[0029] The volume points pass through the observation equation and are obtained as , where is the system input;

[0030] After passing through the observation equation , the mean, covariance matrix, and cross-covariance matrix of the volume points are:

[0031]

[0032]

[0033]

[0034] Calculate the Kalman gain:

[0035]

[0036] Based on the observation quantity i at the -th step, use the Kalman update equation to calculate the transformed volume points:

[0037]

[0038] The obtained volume points are weighted and statistically processed to obtain the updated state estimation mean and covariance respectively as:

[0039]

[0040] .

[0041] Judge whether the updated mean satisfies the constraint boundary , that is:

[0042]

[0043] If the above constraint is satisfied, proceed to the next step; otherwise, constrain the volume points.

[0044] To consider the constraint boundary, the volume points that do not satisfy the constraint boundary On the boundary of the proximity projection constraint to obtain the constrained volume points . The mathematical expression is:

[0045]

[0046] where, represents the th element of the vector;

[0047] Replace the constrained volume points with the volume points before constraint . The updated state estimation mean and covariance obtained through weighted statistics are respectively:

[0048]

[0049] .

[0050] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0051] A model updating hybrid test method based on a novel constrained cubature Kalman filter proposed by the present invention adopts a multi-step symmetry sampling constraint strategy. This strategy not only maintains the symmetry of the sampling points of the cubature Kalman filter, but also adopts different step sizes for different groups of sampling points, maximizing the dispersion state of the sampling points of the cubature Kalman filter, solving the problem of the unbounded constraint of the cubature Kalman filter algorithm, and inheriting the high-precision and high-stability characteristics of the cubature Kalman filter, which is beneficial to the application and popularization of the model updating hybrid test technology. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 is the parameter identification flow chart of the novel constrained cubature Kalman filter;

[0053] Figure 2 is the structural diagram of a three-story buckling-restrained brace single-span plane frame;

[0054] Figure 3 is the schematic diagram of the model updating hybrid test taking the buckling-restrained brace frame structure as an example;

[0055] Figure 4 is the multi-step constraint strategy diagram of the novel constrained cubature Kalman filter during two-dimensional parameter identification;

[0056] Figure 5 is the parameter update constraint strategy diagram of the novel constrained cubature Kalman filter during two-dimensional parameter identification;

[0057] Figure 6 is the three-story buckling-restrained brace single-span plane steel frame diagram;

[0058] Figure 7 It is a schematic diagram of a model updating hybrid test taking the buckling-restrained braced frame structure as an example. Specific implementation manners

[0059] In order to make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other. Specific implementation manner one:

[0061] Referring to Figures 1 - 5 to illustrate this embodiment, a model updating hybrid test method based on a novel constrained volume Kalman filter. In this embodiment, a three-story frame structure equipped with buckling-restrained braces is taken as an example, as Figure 2 shown.

[0062] For the three-story frame structure equipped with buckling-restrained braces, the Bouc-Wen hysteretic model is used to simulate the force-displacement relationship of the buckling-restrained braces. The parameters to be identified are the constitutive parameters of the Bouc-Wen model: the model stiffness k and the parameters that affect the hysteretic curve of the Bouc-Wen model. The present invention uses MATLAB software to establish the numerical substructure and the overall motion equation; calculates the dynamic response of the structure through the overall structure motion equation, and obtains the displacement commands of the numerical substructure and the test substructure respectively; uses MATLAB software to carry out structural parameter identification, so as to realize the model updating hybrid test, as Figure 3 shown.

[0063] The specific steps are as follows:

[0064] Step 1: Take the buckling-restrained brace on the first floor as the test substructure, and the rest as the numerical substructure; use MATLAB to establish the numerical substructure model and the overall motion equation; the buckling-restrained braces on the other floors are used as the numerical substructures to be updated; its overall motion equation can be expressed as:

[0065]

[0066] In the formula, , are the mass matrix and damping matrix of the structure respectively, , and are the displacement vector, velocity vector and acceleration vector of the structure respectively, F is the restoring force of the specimen, is the seismic acceleration record; the subscript The subscript \(n\) represents the integration step; the subscript \(N\) represents the numerical substructure, and the subscript \(E\) represents the experimental substructure; is the position vector of the seismic force action.

[0067] Step 2: The model parameters are: the model stiffness and the parameters affecting the hysteresis curve of the Bouc-Wen model . Establish the state equation and the observation equation of the novel constrained volume Kalman filter, and give the mean value of the initial state estimate and the covariance matrix of the initial state estimate, the process noise covariance matrix and the observation noise covariance matrix . Among them, r is the BoucWen restoring force, z is the hysteretic displacement.

[0068] Step 3: Use the numerical integration algorithm to solve the overall structural motion equation, and the structural command at the th step (\({\bf{u}}_n\), , , ), the loading command of the experimental substructure (\({\bf{f}}_{E,n}\), , , ), and the calculation command of the numerical substructure (\({\bf{f}}_{N,n}\), , , ) can be obtained.

[0069] Step 4: Send the loading command of the experimental substructure at the th step (\({\bf{f}}_{E,n}\), , , ) to the loading system. The loading system completes the physical loading of the experimental substructure, and the reaction force of the experimental substructure at the th step can be obtained.

[0070] Step 5: Based on the loading command of the experimental substructure at the th step (\({\bf{f}}_{E,n}\), , , ) and its reaction force , use the novel constrained volume Kalman filter to online identify the model parameters of the experimental substructure, and transmit the identified parameters to the numerical substructure to update the parameters of the numerical substructure with the same model as the experimental substructure.

[0071] Step 6: The calculation command of the numerical substructure at the th step (\({\bf{f}}_{N,n}\), , , ) Transmitted to the numerical substructure, after calculation, the reaction force of the numerical substructure at the step can be obtained. .

[0072] Step 7: Transmit the substructure reaction forces obtained in Step 4 and Step 6 and to the structural motion equation.

[0073] Step 8: Let = , and repeat Steps 3 to 7 until the test ends.

[0074] The specific method for online identifying the test substructure model parameters using the new constrained volume Kalman filter in Step 5 is as follows:

[0075] Assume that the mean and covariance matrix of the state prior estimate at the prediction step are respectively equal to the mean and covariance matrix of the state estimate at the previous update step, that is , ;

[0076] According to the mean and covariance matrix of the state prior estimate at the prediction step, symmetrically construct around volume points:

[0077]

[0078] where is the th volume point at the th step, is the sampling step size of the th volume point at the n step, represents the set of intersection points of the -dimensional unit sphere and the spatial coordinate axes; If , then j = 1, 2, 3, 4).

[0079] The weight corresponding to each volume point is:

[0080]

[0081] To ensure the symmetry strategy of volume point sampling, a multi-step sampling constraint strategy is adopted. The volume points that violate the constraints are adjusted to the constraint boundary along the direction of the line connecting them to their symmetric volume points. At the same time, to ensure that the adjusted volume points still have symmetry, the same sampling step size is used for the symmetric volume points.

[0082] The volume point sampling step size of the new constrained volume Kalman filter is:

[0083]

[0084] When n = 2, the new constraint step size is as Figure 4 shown.

[0085] The reasonable value range of the volume points is , then the constraint conditions are:

[0086]

[0087]

[0088]

[0089] The volume points pass through the observation equation and obtain , where is the system input.

[0090] After passing through the observation equation , the mean, covariance matrix, and cross-covariance matrix of the volume points are:

[0091]

[0092]

[0093]

[0094] Calculate the Kalman gain:

[0095]

[0096] Based on the i step observation , the Kalman update equation is used to calculate the transformed volume points:

[0097]

[0098] The obtained volume points are weighted statistically to obtain the updated state estimation mean and covariance respectively as:

[0099]

[0100]

[0101] Determine whether the updated mean value satisfies the constraint bounds , that is:

[0102]

[0103] If the above - mentioned constraints are satisfied, proceed to the next step; otherwise, constrain the volume points.

[0104] To consider the constraint boundary, project the volume points that do not satisfy the constraint bounds onto the nearest projection constraint boundary to obtain the constrained volume points . The mathematical expression is:

[0105]

[0106] where represents the th element of the vector ;

[0107] Replace the unconstrained volume points with the constrained volume points , and the updated state - estimation mean value and covariance obtained through weighted statistics are respectively:

[0108]

[0109]

[0110] When n = 2, the state update values before and after the constraint are as Figure 5 shown.

[0111] Specific implementation method 2: Refer to Figure 1 and Figures 4 - 7 to describe this implementation method. A model - updating hybrid test method based on a novel constrained - volume Kalman filter. This implementation method takes a two - story and two - span reinforced - concrete frame structure as an example, as Figure 6 shown.

[0112] For the two - story and two - span reinforced - concrete frame structure, the test sub - structure selects the left - hand half - column of the bottom layer. The parameters to be identified are the Kent - Scott - Park (Concrete01) model parameters of the concrete material: the concrete yield stress and the yield strain , the concrete ultimate stress and the ultimate strain The invention uses MATLAB software to establish the overall motion equation and OpenSees software to establish numerical substructures; calculates the dynamic response of the structure through the overall structural motion equation, and obtains the displacement commands of the numerical substructure and the test substructure respectively; uses MATLAB software to carry out structural parameter identification, so as to realize hybrid testing, such as Figure 7 as shown

[0113] The specific steps are as follows:

[0114] Step 1: Use MATLAB to establish the overall motion equation; the overall motion equation can be expressed as:

[0115]

[0116] In the formula, and are the mass matrix and damping matrix of the structure respectively, and and are the displacement vector, velocity vector and acceleration vector of the structure respectively, is the reaction force of the specimen, is the seismic wave record; the subscript represents the integration step; the subscript N represents the numerical substructure, and the subscript E represents the test substructure; is the seismic force action position vector.

[0117] Step 2: The constitutive model parameters of the concrete material: the concrete yield stress and the yield strain , the concrete ultimate stress and the ultimate strain . Establish the state equation and the observation equation of the new constrained volume Kalman filter. The observation equation here is embodied in the form of a finite element model, that is Figure 7 in the auxiliary surrogate model, and give the initial state estimation mean and the initial state estimation covariance matrix , the process noise covariance matrix and the observation noise covariance matrix .

[0118] Step 3: Use the numerical integration algorithm to solve the overall structural motion equation, and the structural command at the th step can be obtained ( , , ), the loading command of the test substructure and the calculation command of the numerical substructure.

[0119] Step 4: Send the loading command for the test substructure at the th step to the loading system. The loading system completes the physical loading of the test substructure, and the reaction force of the test substructure at the th step can be obtained. th step .

[0120] Step 5: Based on the loading command for the test substructure at the th step and its reaction force , use a new constrained volume Kalman filter to online identify the model parameters of the test substructure, and transmit the identified parameters to the numerical substructure to update the parameters with the same model as the test substructure in the numerical substructure.

[0121] Step 6: Transmit the calculation command for the numerical substructure at the th step to the numerical substructure. After calculation, the reaction force of the numerical substructure at the th step can be obtained. th step .

[0122] Step 7: Transmit the reaction forces of the substructures obtained in Step 4 and Step 6 and to the structural motion equation.

[0123] Step 8: Let = , and repeat Steps 3 to 7 until the test ends.

[0124] The specific method for online identifying the model parameters of the test substructure using the new constrained volume Kalman filter in Step 5 is as follows:

[0125] Assume that the mean of the prior state estimate and the covariance matrix at the prediction step are respectively equal to the mean of the state estimate and the covariance matrix at the previous update step, that is , ;

[0126] According to the mean of the prior state estimate and the covariance matrix at the prediction step, symmetrically construct volume points centered around :

[0127]

[0128] where is the th step and the volume points, is the step of the th volume point, ; represents n the intersection point set of the d-dimensional unit sphere and the spatial coordinate axes; represents the th column of the point set (if , then , j = 1, 2, 3, 4).

[0129] The weight corresponding to each volume point is:

[0130]

[0131] To ensure the symmetry strategy of volume point sampling, a multi-step sampling constraint strategy is adopted. The volume points that violate the constraints are adjusted to the constraint boundary along the direction of the line connecting them to their symmetric volume points. At the same time, to ensure that the adjusted volume points still have symmetry, the same sampling step is used for the symmetric volume points.

[0132] The sampling step of the volume points of the new constrained volume Kalman filter is:

[0133]

[0134] The new constraint step when n = 2 is as Figure 4 shown.

[0135] The reasonable value range of the volume points is , then the constraint condition is:

[0136]

[0137]

[0138]

[0139] The volume points pass through the observation equation , and is obtained, where is the system input.

[0140] After passing through the observation equation , the mean, covariance matrix, and cross-covariance matrix of the volume points are:

[0141]

[0142]

[0143]

[0144] Calculate the Kalman gain:

[0145]

[0146] Based on the measurement i at the step, use the Kalman update equation to calculate the transformed cubature points:

[0147]

[0148] The obtained cubature points are weighted and statistically processed to obtain the updated state estimation mean and covariance respectively as follows:

[0149]

[0150]

[0151] Judge whether the updated mean satisfies the constraint bounds , that is

[0152]

[0153] If the above constraints are satisfied, proceed to the next step; otherwise, constrain the cubature points.

[0154] To consider the constraint boundaries, project the cubature points that do not satisfy the constraint bounds onto the nearest constraint boundary to obtain the constrained cubature points . The mathematical expression is:

[0155]

[0156] where represents the th element of the vector

[0157] Replace the constrained cubature points with the unconstrained cubature points , and the updated state estimation mean and covariance obtained by weighted statistics are respectively:

[0158]

[0159]

[0160] When n = 2, the state update values before and after constraint are as Figure 5 shown.

[0161] It should be noted that according to the needs of implementation, each step / component described in this application can be split into more steps / components, or two or more steps / components or partial operations of steps / components can be combined into new steps / components to achieve the purpose of the present invention.

[0162] Those skilled in the art can easily understand that the above are only preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention should all be included within the protection scope of the present invention.

Claims

1. A hybrid test method for model updating based on a constrained volume Kalman filter, characterized in that, Including the following steps: S1. Divide the overall structure into numerical substructures and experimental substructures, and establish the motion equation of the overall structure; S2. Determine the pre-identified model parameters, establish the state equation of the constrained cubature Kalman filter and the observation equation , and give the mean of the initial state estimate of the constrained cubature Kalman filter and the covariance matrix of the initial state estimate , the process noise covariance matrix and the observation noise covariance matrix ; S3. Use a numerical integration algorithm to solve the overall structural motion equation and obtain the structural response at the step, including the loading command for the experimental substructure and the calculation command for the numerical substructure; S4. Send the loading command for the -step test sub-structure to the loading system. The loading system completes the physical loading of the test sub-structure and obtains the reaction force of the -step test sub-structure . S5. Based on the step test substructure loading command and its reaction force , use the constrained volume Kalman filter to identify the test substructure model parameters, and transmit the identified parameters to the numerical substructure to update the parameters in the numerical substructure that have the same model as the test substructure; S6. Transmit the calculation command of the step numerical substructure to the numerical substructure, and calculate the reaction force of the step numerical substructure ; S7. Transfer the sub - structure reaction forces obtained in S4 and S6 and to the global structural motion equations; S8. Let = , repeat steps S3 - S7 until the test ends; Among them, the constrained volume Kalman filter is used to identify the model parameters of the experimental substructure, including: Assume that the prior mean estimate of the state at the prediction step and the covariance matrix are respectively equal to the state estimate mean and the covariance matrix at the previous update step, that is , ; Prior mean estimate of the state at the prediction step and covariance matrix , centered at symmetrically construct volume points: In the formula, is the th volume point in the th step, is the sampling step size of the th volume point in the n th step, represents the th column of the point set, j = 1, 2, 3, 4; The weight corresponding to each volume point is: The volume points pass through the observation equation , and obtain , where is the system input; After the observation equation The mean, covariance matrix, and cross-covariance matrix of the volume points are as follows: Calculate the Kalman gain: Based on the i observations of the ith step, the Kalman update equation is used to calculate the transformed volume points: The obtained volume points are used to obtain an updated state estimation mean through weighted statistics and covariance respectively as follows: 。 2. The model updating hybrid test method based on the constrained volume Kalman filter according to claim 1, wherein Using the constrained volume Kalman filter to identify the model parameters of the experimental substructure also includes: Determine whether the updated mean value satisfies the constraint bounds , that is: If the above formula constraints are met, proceed to the next step; otherwise, constrain the volume points; To consider the constraint boundary, the volume points that do not satisfy the constraint bounds are projected onto the nearest constraint boundary to obtain the volume points after constraint , and the mathematical expression is: Among them, represents the th l element of the vector; Replace the volume points before constraint with the volume points after constraint to obtain the updated state estimation mean and covariance through weighted statistics as follows: ​ 。 3. The model update hybrid test method based on a constrained volume Kalman filter according to claim 2, wherein To ensure the symmetry strategy of volume point sampling, a multi-step sampling constraint strategy is adopted. The volume points that violate the constraints are adjusted to the constraint boundary along the line connecting them to their symmetric volume points. At the same time, to ensure that the adjusted volume points still have symmetry, the same sampling step is also used for the other symmetric volume point. The specific form of the sampling step of the volume points of the constrained volume Kalman filter is: The reasonable value range of the volume points is , then the constraint condition is: 。 4. The model update hybrid test method based on a constrained volume Kalman filter according to any one of claims 1 to 3, characterized in that The motion equation of the overall structure is: In the formula, and are the mass matrix and damping matrix of the structure respectively, and and are the displacement vector, velocity vector and acceleration vector of the structure respectively, is the restoring force of the specimen, is the seismic acceleration, and the subscript represents the integration step, the subscript N represents the numerical substructure, and the subscript E represents the test substructure, is the position vector of the seismic force action.

5. The hybrid test method for model updating based on a constrained volume Kalman filter according to claim 1, wherein Use finite element model calculation software to establish numerical substructures.

6. The hybrid test method for model updating based on a constrained volume Kalman filter according to claim 5, wherein The finite element model calculation software includes OpenSees and MATLAB.

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