CN113886920A - Bridge vibration response data prediction method based on sparse Bayesian learning - Google Patents

Bridge vibration response data prediction method based on sparse Bayesian learning Download PDF

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CN113886920A
CN113886920A CN202111169728.3A CN202111169728A CN113886920A CN 113886920 A CN113886920 A CN 113886920A CN 202111169728 A CN202111169728 A CN 202111169728A CN 113886920 A CN113886920 A CN 113886920A
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王其昂
戴阳
马占国
叶继红
周兆康
孙宏点
唐佳祺
褚臻辉
焦桂月
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Abstract

The invention discloses a bridge vibration response data prediction method based on sparse Bayesian learning. The method comprises the following steps: s1, drawing a corresponding time domain graph according to the dynamic response data of the bridge structure, performing time domain analysis on the time domain graph, and selecting corresponding vibration response data sections according to different dynamic analysis methods; s2, sequencing the vibration data segments according to a time series analysis method, inputting the vibration data segments into a Sparse Bayesian Learning (SBL) algorithm, properly adjusting the base width parameters of Gaussian kernel functions in the sparse SBL, and initially determining an SBL regression model; s3, testing the fitting degree and sparsity of the initially determined SBL regression model, and properly adjusting the base width parameters until the final SBL regression model is determined; and S4, predicting vibration data at the next moment according to the final SBL regression model. The sparse Bayesian learning algorithm based on the Gaussian kernel function is suitable for strong nonlinear vibration data, uncertainty of the vibration data is considered in the prediction process, robustness of a mathematical model is good, and accuracy of a prediction result is high.

Description

Bridge vibration response data prediction method based on sparse Bayesian learning
Technical Field
The invention belongs to the technical field of structural health monitoring, and relates to a bridge vibration response data prediction method based on sparse Bayesian learning.
Background
Large bridge structures are an important part of the building infrastructure. However, large bridges in service are inevitably damaged due to long-term traffic load, aging of materials, and extreme weather effects such as earthquakes and typhoons. In order to avoid economic loss and casualties caused by damage to bridges, a structural health monitoring system is developing forward at an extremely fast speed as a powerful tool for ensuring structural reliability and safe operation. The structural health monitoring system can obtain various structural response data such as strain, acceleration, displacement and the like by installing various sensors on the bridge structure. The structural acceleration (vibration) response data can reflect basic dynamic parameters (namely, the structure natural vibration frequency, the vibration mode and the like) of the structure, and whether the operation of a structural system is safe and reliable can be judged through the parameters. Because the structure dynamic characteristics reflect the state of the structure, the structure dynamic characteristics in the damage identification field of the structure health monitoring often exist as necessary analysis information of the damage identification result, if the future dynamic characteristics of the structure are predicted through vibration data, the structure is subjected to damage identification analysis in advance, and then the purpose of knowing the state of the structure in advance according to the damage identification result obtained through prediction data can be achieved.
The vibration response data prediction method is classified into a structure model-based method and a non-structure model method according to whether structure model information is required. The method based on the structure model requires that the structure model can describe the information of the actual structure as accurately and comprehensively as possible, and the method generally has the problem that the definition of the parameters such as material parameters, boundary conditions and the like of the structure model is difficult to accurately simulate the actual structure. The method without the structural model does not need the structural model, and can predict data only by a series of actually measured vibration response data acquired by an actual structure.
However, in the field of civil engineering, the measured vibration data has uncertainty due to the influence of environmental factors on the data, the influence of sensor accuracy on the data, and the like, which makes the prediction method of the non-structural model challenging, and a method capable of quantifying the uncertainty of the vibration data is required. Meanwhile, the nonlinearity of the vibration response data of the bridge is very obvious, so a method capable of processing the nonlinearity problem is also needed. While solving the non-linearity problem, it is also necessary to prevent the over-fitting problem of the mathematical model, because once the over-fitting problem occurs, the robustness of the mathematical model is reduced, and the accuracy of the prediction data is further reduced. Therefore, it is necessary to develop a vibration response data prediction method that can simultaneously solve the problems that the structural model is difficult to repeatedly etch the actual model, the vibration data has uncertainty, the vibration data has nonlinearity, and the mathematical model is over-fitted.
Disclosure of Invention
Aiming at the existing problems, the invention provides a bridge vibration response data prediction method which is driven by pure data, has uncertainty quantification capability, capability of processing nonlinear problems and strong robustness.
The invention provides a bridge vibration response data prediction method based on sparse Bayesian learning, which specifically comprises the following steps:
s1, drawing a corresponding time domain graph according to the dynamic response data of the bridge structure; performing time domain analysis on the time domain diagram, and selecting corresponding vibration response data segments according to different dynamic analysis methods;
s2, sorting the vibration response data segments of S1 according to a time series analysis method
Figure BDA0003292410460000021
Inputting into a sparse Bayesian learning algorithm, wherein
Figure BDA0003292410460000022
In the form of a time series of,
Figure BDA0003292410460000023
the method comprises the steps of properly adjusting a fundamental width parameter gamma of a Gaussian kernel function in sparse Bayesian learning, calculating a weight parameter w of a sparse Bayesian learning regression model, and initially determining a sparse Bayesian learning regression model of a vibration response data section for vibration data, wherein N is the number of data points;
s3, according to the sparse Bayes learning regression model of S2, the fitting degree and the sparsity of the model are checked (the larger the fitting degree and the sparsity are, the higher the prediction precision is), if the fitting degree and the sparsity do not meet the requirements, S2 and S3 are repeated until the final sparse Bayes learning regression model is determined;
s4, calculating the predicted value of the data point at the (N + 1) th moment according to the final sparse Bayesian learning regression model determined in the S3;
further, the Gaussian kernel function in step S2 is
Figure BDA0003292410460000024
Where the parameter γ is the base width, two experiences in selecting the base width parameter are given here: (1) the fitting degree becomes smaller with the increase of the base width; (2) sparsity becomes larger as the base width increases.
Further, the satisfaction requirements of the fitting degree and the sparsity in step S3 are: (1) the decision coefficient of the sparse Bayesian learning regression model is greater than 0.98; (2) the sparsity of the weight parameter (the weight is obtained by taking the number of w as 0 in all the w numbers) is at least 70%.
Further, the input x for a given time N +1 when calculating the predicted value in step S4*Its predicted value t*Can be calculated by the following formula:
p(t*|t)＝∫p(t*|w，α，σ-2)p(w，α，σ-2|t)dwdαdσ-2
the invention has the beneficial effects that: the sparse Bayesian learning algorithm based on the Gaussian kernel function realizes the prediction work of the vibration response data of the bridge structure under the data-driven frame, and compared with a prediction method based on a model, the sparse Bayesian learning algorithm based on the Gaussian kernel function is simpler to operate and can realize the real-time prediction of the response data. Meanwhile, the method is particularly suitable for vibration data which is strong in nonlinearity. In addition, the method considers the uncertainty of data in the prediction process, has good robustness of a mathematical model and high precision of a prediction result, and can provide accurate and effective vibration data for the health monitoring of the bridge structure.
Drawings
FIG. 1 is a schematic view of an arrangement of an Tianjin permanent and bridge and an accelerometer;
FIG. 2 is a flow chart of the present invention;
FIG. 3 is a schematic time domain representation of the acceleration of the Tianjin permanent and bridge sections;
FIG. 4 is a Bayesian regression model of the finally determined sparse under the free attenuation data type;
FIG. 5 is a Bayesian regression model of the finally determined sparsity under the environment excitation data type;
FIG. 6 is a flowchart of the baseline width parameter tuning;
FIG. 7 is the final weight parameters of the sparse Bayesian regression model;
fig. 8 is a graph comparing a predicted value and an actual value.
Detailed Description
As shown in fig. 2, the bridge vibration response data prediction method based on sparse bayesian learning mainly includes the following steps:
s1, drawing a corresponding time domain graph according to the dynamic response data of the bridge structure; and performing time domain analysis on the time domain diagram, and selecting corresponding vibration response data segments according to different dynamic analysis methods.
S2, sorting the vibration response data segments of S1 according to a time series analysis method
Figure BDA0003292410460000031
Is inputted intoIn a sparse Bayesian learning algorithm, wherein
Figure BDA0003292410460000041
In the form of a time series of,
Figure BDA0003292410460000042
and (3) for vibration data, N is the number of data points, a base width parameter gamma of a Gaussian kernel function in sparse Bayesian learning is properly adjusted, a weight parameter w of a sparse Bayesian learning regression model is calculated, and the sparse Bayesian learning regression model of a vibration response data section is initially determined.
And S3, according to the sparse Bayes learning regression model of S2, testing the fitting degree and sparsity (the larger the fitting degree and sparsity are, the higher the prediction precision is), and if the fitting degree and sparsity do not meet the requirements, repeating S2 and S3 until the final sparse Bayes learning regression model is determined.
And S4, calculating the predicted value of the data point at the (N + 1) th moment according to the final sparse Bayesian learning regression model determined in the S3.
In the method for predicting the vibration response data of the bridge based on the sparse Bayesian learning, the Gaussian kernel function in the step S2 is
Figure BDA0003292410460000043
Where the parameter γ is the base width, two experiences in selecting the base width parameter are given here: (1) the fitting degree becomes smaller with the increase of the base width; (2) sparsity becomes larger as the base width increases.
In the method for predicting the bridge vibration response data based on the sparse Bayesian learning, the fitting degree and the sparsity in the step S3 meet the requirements: (1) the decision coefficient of the sparse Bayesian learning regression model is greater than 0.98; (2) the sparsity of the weight parameter (the weight is obtained by taking the number of w as 0 in all the w numbers) is at least 70%.
In the bridge vibration response data prediction method based on sparse Bayesian learning, the input x at the given N +1 moment is used for calculating the predicted value in the step S4*Its predicted value t*Can be calculated by the following formula:
p(t*|t)＝∫p(t*|w，α，σ-2)p(w，α，σ-2|t)dwdαdσ-2
in order to make the objects, technical solutions and advantages of the present invention more apparent. The present invention will be described in further detail with reference to the following drawings and specific examples. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Example (b): the bridge vibration response data prediction method based on sparse Bayesian learning is explained by taking Tianjin Yonghe and a bridge of the benchmark problem as an example. The Tianjin permanent and bridge is a prestressed concrete cable-stayed bridge with double towers, double cable surfaces and a continuous floating system, the south is connected with Tianjin, and the north is connected with Hangu. The girder is 510 meters long, and the full-bridge girder has 7 test sections, 2 acceleration sensor of every test section, totally 14 unipolar acceleration sensor, and acceleration information is abundant. Fig. 1 is a schematic diagram of the arrangement of Tianjin permanent and bridge and accelerometer.
As shown in the flow chart of the invention in fig. 2, the time domain diagram of the permanent and large bridges is firstly analyzed, the time domain diagram of the permanent and large bridges is partially shown in fig. 3, and the acceleration data can be divided into free attenuation data and environment excitation data from the aspect of amplitude. Furthermore, in the dynamic analysis method, the types of acceleration generally required are also free-form damping data and environmental excitation data, such as a classical power spectrum and a cross-power spectrum, and the types of acceleration data required to be analyzed are free-form damping data and environmental excitation data, respectively. The present embodiment performs prediction work on these two types of data to explain the present invention.
And intercepting a section of free attenuation data and a section of environment excitation data in the time domain graph as acceleration data of the embodiment to be brought into sparse Bayesian learning. To illustrate the sparse Bayesian learning algorithm, the acceleration data is set as
Figure BDA0003292410460000051
Set a time sequence of
Figure BDA0003292410460000052
N is the number of data points, i.e. the training sample set
Figure BDA0003292410460000053
Substituting sparse Bayesian learning, and establishing a mathematical model (SBL regression model) of the time series and the acceleration data:
Figure BDA0003292410460000054
in the above formula, w ═ w1，w2，...，wm]TIs a weight vector to be solved; epsilon is the model error, obeying a Gaussian distribution N (0, sigma)2) And are independently and equally distributed; phi (x) is a design matrix of NxM, and a specific expression is as follows:
Φ(x)＝[1，K(xn，x1)，...，K(xn，xm)]T (2)
in the above formula, K (x, x)i) Is a gaussian kernel function. The model of equation (1) has the ability to represent non-linear problems due to the presence of the kernel function. In addition, since epsilon to N (0, sigma)2) The acceleration data t therefore also obeys a gaussian distribution:
Figure BDA0003292410460000055
in the formula (3), Φ ═ Φ (x)1)，φ(x2)，...，φ(xn)]T。
W, σ of the above formula2If the direct maximum likelihood method is used for solving, the result is usually over-fitted, and in order to avoid the phenomenon, the prior distribution can be added to w according to the Bayesian viewpoint, and the distribution is as follows:
Figure BDA0003292410460000056
in formula (4), α ═ α1，α2，...，αm}TFor hyper-parameters, the effect is to prevent w from over-fitting, while promoting sparsity of w, taking into account conjugate prior distributionsConvenience, superparameter obeys gamma distribution.
Under the Bayes framework, hidden variables w, alpha and sigma need to be determined2According to Bayesian reasoning, the posterior distribution of hidden variables is:
P(w，α，σ-2|t)＝P(w|t，α，σ-2)P(α，σ-2|t) (5)
further, the left term on the left side of the equal sign of the formula (5) is:
Figure BDA0003292410460000061
in equation (7), w follows a gaussian distribution, i.e., w to N (μ, Σ), specifically:
∑＝(A+σ-2ΦTΦ)-1，μ＝σ-2∑ΦTt (7)
finally, the evidence function P (t | α, σ) can be maximized-2) In the iterative process, part of the hyper-parameters tend to be infinite, meaning that the corresponding weight values are 0, while other hyper-parameters tend to some nonzero values stably, and the corresponding weight values are called correlation vectors, so that the whole calculation model is also called a correlation vector machine, namely sparse Bayesian learning can also be called sparse Bayesian learning.
Fig. 4 and 5 are sparse bayesian learning regression models obtained by performing algorithm and parameter tuning (parameter tuning process, see fig. 6) on the free attenuation data and the environment excitation data, respectively. Fig. 7(a) and (b) are weight parameters of the sparse bayesian regression models of fig. 4 and 5, respectively.
As shown in fig. 4 and 5, the determination coefficients of the two regression models were 0.989 and 0.999, respectively. As shown in fig. 7, the sparsity ratios of the sparse bayesian learning regression models of the free attenuation data and the environmental excitation data can be calculated to be 78% and 74.5%, respectively. Both of the two sparse bayesian learning regression models satisfy the inspection requirements in fig. 6, so that the final sparse bayesian learning regression model can be determined.
Finally, the determined sparse Bayesian regression is utilizedAnd predicting the vibration response data by the model. Input x for a given time instant N +1*Its predicted value t*Can be calculated by the following formula:
p(t*|t)＝∫p(t*|w，α，σ-2)p(w，α，σ-2|t)dwdαdσ-2 (8)
in order to illustrate the high precision, strong robustness and high applicability of the prediction result, the prediction is performed 100 times continuously, and the prediction result is shown in fig. 8. Fig. 8(a) and (b) are graphs comparing predicted values and measured values of free attenuation data and environmental excitation data, respectively. It can be seen from fig. 8 that the method provided by the present invention has high prediction accuracy and strong robustness.
It will be appreciated by those skilled in the art that the examples set forth herein are intended to assist the reader in understanding the principles of the invention and are to be construed as being without limitation to such specifically recited examples and embodiments and that all changes, equivalents, and modifications that come within the spirit and scope of the invention are desired to be protected.

Claims (4)

1. A bridge vibration response data prediction method based on sparse Bayesian learning is characterized by specifically comprising the following steps:
s1, drawing a corresponding time domain graph according to the dynamic response data of the bridge structure; performing time domain analysis on the time domain diagram, and selecting corresponding vibration response data segments according to different dynamic analysis methods;
s2, sorting the vibration response data segments of S1 according to a time series analysis method
Figure FDA0003292410450000011
Inputting into a sparse Bayesian learning algorithm, wherein
Figure FDA0003292410450000012
In the form of a time series of,
Figure FDA0003292410450000013
adjusting a fundamental width parameter gamma of a Gaussian kernel function in sparse Bayesian learning, calculating a weight parameter w of a sparse Bayesian learning regression model, and initially determining a sparse Bayesian learning regression model of a vibration response data segment;
s3, according to the sparse Bayesian learning regression model of S2, the fitting degree and sparsity of the model are checked, if the fitting degree and sparsity do not meet the requirements, S2 and S3 are repeated until the final sparse Bayesian learning regression model is determined;
and S4, calculating the predicted value of the data point at the (N + 1) th moment according to the final sparse Bayesian learning regression model determined in the S3.
2. The method according to claim 1, wherein the Gaussian kernel function in step S2 is
Figure FDA0003292410450000014
The base width parameter γ is adjusted according to the following rule: (1) the fitting degree becomes smaller with the increase of the base width; (2) sparsity becomes larger as the base width increases.
3. The method of claim 1, wherein the fitting degree and sparsity in step S3 are satisfied by: (1) the decision coefficient of the sparse Bayesian learning regression model is greater than 0.98; (2) the sparsity of the weight parameters is at least 70%.
4. Method according to claim 1, characterized in that the input x for a given time instant N +1 when calculating the prediction value in step S4*Its predicted value t*Can be calculated by the following formula:
p(t*|t)＝∫p(t*|w,α,σ-2)p(w,α,σ-2|t)dwdαdσ-2
wherein w is a weight parameter, α is a hyper-parameter, σ2Is the variance of gaussian noise.
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