A Parameter Estimation Method and System for BWBN Model
Through the improved TMCMC algorithm and BWBN model, combined with adaptive sample weight adjustment and Longguta solution, the high-dimensional problem in BWBN model parameter estimation is solved, and more efficient and accurate parameter estimation is achieved, suitable for nonlinear dynamic response analysis of engineering structures.
Patent Information
- Application Number
- CN202210770940.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2022-06-30
- Publication Date
- 2025-08-01
- Estimated Expiration
- 2042-06-30
AI Technical Summary
The existing BWBN model parameter estimation methods are prone to falling into a single peak trap when estimating high-dimensional parameters, and the statistical estimation efficiency of traditional methods is low, making it difficult to effectively describe the nonlinear dynamic response of the engineering structure.
The improved TMCMC algorithm (iTMCMC) is used to adaptively adjust the sample weight of the Markov chain, combine the pinch function of the BWBN model to perform parameter estimation, and use the fourth-order Longguda to solve differential equations, establish a likelihood function and covariance matrix, and optimize the prior distribution of the parameters to be identified.
It improves the accuracy and efficiency of parameter estimation, reduces the deviation of the estimated value, and can effectively solve the multi-peak or extremely flat single-peak problems. It is suitable for multi-dimensional parameter estimation, and has higher versatility and applicability.
Smart Images
Figure CN115183969B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the field of vibration control, and particularly to a method and system for parameter estimation of a BWBN model. Background Art
[0002] With the development of modern industrialization and society, suppressing or at least attenuating undesirable vibrations that may affect a system or structure is of great significance for the safety of building bridges, the precision of processing, the working stability of equipment, and the riding comfort of vehicles. Vibration control is usually achieved using passive, semi-active, or active systems, each of which has considerable hysteretic performance. The Bouc-Wen-Baber-Noori (BWBN) model can be effectively used to analyze and describe the hysteretic performance of multiple systems and capture a series of hysteretic loop patterns.
[0003] The Bouc-Wen-Baber-Noori hysteretic model is further developed on the basis of the classical Bouc-Wen hysteretic model, considering strength, stiffness degradation, and pinching effects. Many engineering structures will enter an inelastic state under dynamic loads and exhibit hysteretic characteristics. Hysteretic characteristics are also called elastoplasticity. Hysteresis generally comes from the nonlinear characteristics of materials, the friction characteristics of contact surfaces, and the contact deformation between bonded surfaces, etc. Under the action of a load, there is a hysteretic relationship between the restoring force and displacement of the structure. When the load is periodic, due to the structure entering elastoplastic deformation, the displacement of plastic deformation cannot be fully restored, causing the curve to change along different paths and form a hysteretic loop. The actual hysteretic curve of the structure is very complex and difficult to apply to the analysis of the nonlinear characteristics of the structure. Therefore, it is necessary to establish a hysteretic model that is both easy to describe mathematically and can reflect the hysteretic characteristics of the structure to predict the nonlinear dynamic response of engineering structures.
[0004] The BWBN model improves the problem that the Bouc-Wen model cannot describe the pinched hysteresis phenomenon. At the same time, the problem is that the parameter dimension is large. To describe the hysteretic performance of the system using the BWBN model, it is necessary to determine multiple parameters in the model. Existing parameter estimation methods all require the use of different filters. The traditional MCMC method is prone to falling into a single-peak trap and has a risk of failure when facing high-dimensional parameter estimation. And TMCMC is suitable for problems with very few uncertain parameters. If the number of uncertain parameters is large, the results may have large deviations, and at the same time, the efficiency of statistical estimation is low. Summary of the Invention
[0005] The purpose of the present invention is to overcome the above-mentioned defects existing in the prior art and provide a method and system for parameter estimation of a BWBN model.
[0006] The purpose of the present invention can be achieved by the following technical solutions:
[0007] According to the first aspect of the present invention, a method for parameter estimation of a BWBN model is provided, including the following steps:
[0008] Obtain the measurement data of the experiment, where the measurement data includes force and displacement, determine the physical parameters of the experiment, establish a BWBN model, substitute the measurement data and physical parameters into the BWBN model, solve the BWBN model, and determine W undetermined parameters to be identified in the BWBN model;
[0009] Initialize each undetermined parameter to be identified, determine its maximum value, minimum value, and initial value, assume that each undetermined parameter satisfies a Gaussian distribution, calculate the likelihood function of each undetermined parameter respectively, obtain the initial prior distribution of each undetermined parameter, use the initial prior distribution as the prior distribution of the undetermined parameter, and use the initial value as the estimated value of the undetermined parameter;
[0010] Extract samples from the prior distribution of each undetermined parameter respectively and update the estimated value of each undetermined parameter based on the samples to complete one round of optimization. Repeat this step until the preset termination condition is met and stop the iteration;
[0011] Output the estimated value of each undetermined parameter.
[0012] Further, in the j-th round of optimization, samples are extracted from the prior distribution of the s-th undetermined parameter to be identified and the estimated value of the s-th undetermined parameter to be identified is updated based on the samples, where j = 1, 2,..., s = 1, 2,..., W. Specifically:
[0013] Extract N j,s samples from the prior distribution of the s-th undetermined parameter to be identified: N j,s is the sample extraction value of the s-th undetermined parameter to be identified in the j-th round of optimization;
[0014] Calculate the gradient coefficient of the j-th round of optimization. The calculation formula is as follows:
[0015]
[0016] where represents the sample variation coefficient of the s-th undetermined parameter to be identified in the j-th round of optimization;
[0017] Calculate the weighted coefficient of each sample respectively. The calculation formula is as follows:
[0018]
[0019] represents the weighted coefficient of the k-th sample of the s-th undetermined parameter to be identified in the j-th round of optimization, and D represents the measurement data of the experiment;
[0020] Calculate N j,sThe average value of the weighted coefficients of the samples is calculated as follows:
[0021]
[0022] where Se j,s represents the average value of the sample weighted coefficients of the sth parameter to be identified in the jth round of optimization;
[0023] Calculate the covariance matrix, and the calculation formula is as follows:
[0024]
[0025] where ∑ j,s represents the covariance matrix of the N j,s samples of the sth parameter to be identified in the jth round of optimization, represents the mean value of the N j,s samples of the sth parameter to be identified in the jth round of optimization, T represents matrix transpose, and ξ j represents the adaptive scaling factor preset in the jth round of optimization;
[0026] Randomly generate an index l from and select a sample from with probability from Take the Gaussian distribution centered on and the covariance matrix ∑ j,s as a proposed distribution of the assumed μ c to obtain a Markov chain starting from the initial sample distribution of μ c ; if where r is a sample from a uniform distribution from 0 to 1, then let Update the prior distribution and the initial value of the sth parameter to be identified, otherwise, repeat this step.
[0027] Furthermore, at the beginning of each round of optimization, determine the value of the adaptive scaling factor ξ j , and its calculation formula is:
[0028]
[0029]
[0030] where W represents the total number of parameters to be identified, p r represents the sample acceptance rate, t r represents the target acceptance rate, and Na represents the number of chains to be adjusted, that is, the number of Markov chains in the jth round of optimization .
[0031] Furthermore, the preset termination condition is the gradient coefficient q of the jth round of optimizationj Equal to 1, it is considered that the optimal estimated value of each parameter to be identified is found, and the iteration stops.
[0032] Furthermore, the physical parameters of the experiment include mass and initial stiffness, and the BWBN model is as follows:
[0033]
[0034]
[0035] v(t) = 1.0 + δ v ε(t)
[0036] η(t) = 1.0 + δ η ε(t)
[0037] Where m represents mass, x represents displacement, c represents damping coefficient, k e represents stiffness, z represents damping displacement, F(t) represents force, h(z) reflects the pinching effect, v(t) represents strength degradation, η(t) represents stiffness degradation, ε(t) represents hysteretic dissipated energy, n, a, β, γ, δ v , δ η are parameters to be identified, and represent the first derivative and second derivative of x, represents the first derivative of z.
[0038] Furthermore, the fourth-order Runge-Kutta method is used to solve the differential equation in the BWBN restoring force measurement model as follows:
[0039]
[0040] K1 = f(x n , y n )
[0041]
[0042]
[0043] K4 = f(x n + h, y n + hK3)
[0044] Where f() represents the differential equation of the BWBN model, x n is the displacement in the measurement data, y n = F(t) is the force in the measurement data, h is the Runge-Kutta time step, and K1, K2, K3, K4 are the transfer parameters in Runge-Kutta.
[0045] Furthermore, the principle of establishing the likelihood function is:
[0046]
[0047]
[0048] ln P3 = ln P1 + ln P2
[0049] where σ1 represents the unknown variance of the displacement prediction error, and σ2 represents the unknown variance of the total dissipated energy prediction error. represents the mean value of x, and N() represents the Gaussian distribution.
[0050] Furthermore, when calculating the likelihood function of each parameter to be identified, it is necessary to normalize the predicted error dissipated energy, and the calculation formula is:
[0051]
[0052] where L = ln P3, θ g is the adjacent composed of the parameters to be identified, the subscript g is the index of force and displacement in the measurement data, N j,s is the sample extraction value of the s-th parameter to be identified in the j-th round of optimization preset.
[0053] Furthermore, before calculating the likelihood function, it is necessary to determine the total dissipated energy from the hysteresis curve of the measurement data.
[0054] According to the second aspect of the present invention, there is provided a parameter estimation system for a BWBN model, including:
[0055] A data acquisition module for acquiring measurement data and physical parameters of an experiment, where the measurement data includes force and displacement;
[0056] A BWBN model module for establishing a BWBN model, substituting the measurement data and physical parameters into the BWBN model, solving the BWBN model, and determining W parameters to be identified in the BWBN model;
[0057] An initialization module for initializing each parameter to be identified, determining its maximum value, minimum value and initial value, assuming that each parameter to be identified satisfies a Gaussian distribution, calculating the likelihood function of each parameter to be identified respectively, obtaining the initial prior distribution of each parameter to be identified, using the initial prior distribution as the prior distribution of the parameter to be identified, and using the initial value as the estimated value of the parameter to be identified;
[0058] An iterative optimization module for performing at least one round of optimization operation and stopping the iteration when the preset termination condition is satisfied. In each round of optimization operation, samples are respectively extracted from the prior distribution of each parameter to be identified and the estimated values of each parameter to be identified are updated based on the samples;
[0059] An output module for outputting the estimated value of each parameter to be identified.
[0060] Compared with the prior art, the present invention has the following beneficial effects:
[0061] The present application introduces variable adaptive adjustment parameters, adjusts the sample weighting coefficient after each Markov chain to be adjusted, thereby adjusting the target distribution, improving the performance of the algorithm, reducing the bias in the estimated value, making the estimated deviation of the sample mean and variance small, accelerating the search speed, taking advantage of the advantages of the transitional Markov chain Monte Carlo method itself, effectively solving the sampling of multi-modal or extremely flat unimodal problems, and at the same time applying the pinching and squeezing function of the BWBN model, making the system have higher versatility. Description of the Drawings
[0062] Figure 1 is a flowchart of the present invention;
[0063] Figure 2 is the normalized energy consumption diagram in the embodiment;
[0064] Figure 3 is the probability distribution diagram of each parameter to be identified in the embodiment;
[0065] Figure 4 is the hysteresis curve diagram in the embodiment. Detailed Embodiment
[0066] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. This embodiment is implemented on the premise of the technical solution of the present invention, and gives the detailed implementation method and specific operation process, but the protection scope of the present invention is not limited to the following embodiments.
[0067] Embodiment 1:
[0068] In view of the defects of the parameter estimation method of the BWBN model in the prior art, the applicant points out that the iTMCMC algorithm adaptively adjusts the proposal distribution and adjusts the sample weights after each MCMC step, so it is superior to the original TMCMC method in terms of the deviation of the mean and standard deviation and the sampling of the effective sample number, and can be better applied to multi-dimensional parameter estimation problems. Therefore, the iTMCMC algorithm is improved and the iTMCMC algorithm is used for the parameter estimation of the BWBN model.
[0069] According to the first aspect of the present invention, there is provided a parameter estimation method for a BWBN model, as Figure 1 shown, including the following steps:
[0070] S1. Obtain the measurement data of the experiment. The measurement data includes the force F(t) and the displacement x. Determine the physical parameters of the experiment, establish the BWBN model, substitute the measurement data and the physical parameters into the BWBN model, solve the BWBN model, and determine the W undetermined parameters in the BWBN model;
[0071] Among them, in this embodiment, the measurement data is the data collected in the hysteretic force experiment, and numerical simulation data can also be used. It should be noted that the hysteretic images represented by the measurement data or the numerical simulation data need to conform to or approximate the BWBN model so that the BWBN model can be used to describe the measurement data.
[0072] The BWBN restoring force measurement model is a relatively mature mathematical model, and its principle and establishment process will not be elaborated here, and relevant practitioners can understand. The established BWBN restoring force measurement model is as follows:
[0073]
[0074]
[0075] v(t) = 1.0 + δ v ε(t)
[0076] η(t) = 1.0 + δ η ε(t)
[0077] Among them, m represents the mass, x represents the displacement, c represents the damping coefficient, k e represents the stiffness, z represents the damping displacement, F(t) represents the force, h(z) reflects the pinching effect, v(t) represents the strength degradation, η(t) represents the stiffness degradation, ε(t) represents the hysteretic dissipated energy, n, a, β, γ, δ v 、δ η are undetermined parameters, and represent the first derivative and the second derivative of x, represents the first derivative of z.
[0078] That is, the differential equation in the BWBN model. The physical parameters in the experiment include the mass and the initial stiffness, etc., which will not be elaborated here.
[0079] After substituting the known values and the measurement data into the BWBN model, it can be solved by the fourth-order Runge-Kutta method as follows:
[0080]
[0081] K1 = f(x n ,y n )
[0082]
[0083]
[0084] K4 = f(x n + h, y n + hK3)
[0085] where f() represents the differential equation of the BWBN model, x n is the displacement in the measurement data, y n = F(t) is the force in the measurement data, h is the Runge-Kutta time step, and K1, K2, K3, K4 are the transfer parameters in Runge-Kutta.
[0086] S2. Initialize each parameter to be identified, determine its maximum value, minimum value, and initial value. Assume that each parameter to be identified follows a Gaussian distribution, calculate the likelihood function of each parameter to be identified, obtain the initial prior distribution of each parameter to be identified, use the initial prior distribution as the prior distribution of the parameter to be identified, and use the initial value as the estimated value of the parameter to be identified;
[0087] After solving the differential equation with the fourth-order Runge-Kutta, it is equivalent to obtaining the representation of each parameter to be identified. Specifically, there are 17 undetermined parameters in the BWBN model, and 2 parameters are introduced to describe the input noise. Therefore, there are a total of 19 parameters to be identified. In other application scenarios, different numbers of parameters to be identified can be set according to needs, such as 7, 21, etc. According to experience, set their value ranges and initial values as shown in the following table:
[0088] Table 1 Initialization of Parameters to be Identified
[0089]
[0090]
[0091] Calculate the natural frequency of the system, assume that the unknown parameters to be identified follow a Gaussian distribution, calculate the likelihood function, and obtain its initial prior distribution; the specific establishment process of the likelihood function is as follows. Establish a Gaussian distribution N() with the mean of the displacement mean and the variance of the total energy dissipation prediction error, and establish the following likelihood function:
[0092]
[0093]
[0094] ln P3 = ln P1 + ln P2
[0095] where σ1 represents the unknown variance of the displacement prediction error, and σ2 represents the unknown variance of the total dissipation energy prediction error. denotes the average value of x, and N() denotes the Gaussian distribution;
[0096] Then, take the partial derivative to solve for the parameter values, and thus obtain the initial prior distribution of each parameter to be identified by calculating the likelihood function of each parameter to be identified.
[0097] When calculating the likelihood function of each parameter to be identified, it is necessary to normalize the predicted error dissipation energy. Figure 2 An example of the normalized energy dissipation diagram is given, and the calculation formula is:
[0098]
[0099] where L = ln P3, θ g is the adjacent composed of parameters to be identified, the subscript g is the index of force and displacement in the measured data, and N j,s is the sample extraction value of the s-th parameter to be identified in the j-th round of optimization preset.
[0100] Before calculating the likelihood function, it is necessary to determine the total dissipation energy from the hysteresis curve of the measured data, and its calculation method is common knowledge in the art and will not be elaborated here.
[0101] S3. Respectively extract samples from the prior distribution of each parameter to be identified and update the estimated values of each parameter to be identified based on the samples, complete one round of optimization, and repeat this step until the preset termination condition is met, and then stop the iteration;
[0102] Taking the extraction of samples from the prior distribution of the s-th parameter to be identified in the j-th round of optimization and updating the estimated value of the s-th parameter to be identified based on the samples as an example, the iterative optimization process is described. Among them, j = 1, 2,..., s = 1, 2,..., W, specifically:
[0103] (1) Extract N j,s samples from the prior distribution of the s-th parameter to be identified: N j,s is the sample extraction value of the s-th parameter to be identified in the j-th round of optimization preset. The sample extraction values of each parameter to be identified in each round of optimization can be different. In this embodiment, for the convenience of calculation, it is uniformly stipulated that N j,s = 1000;
[0104] (2) Calculate the gradient coefficient of the j-th round of optimization. The preset termination condition is that the gradient coefficient q j of the j-th round of optimization is equal to 1, indicating that the optimal estimated value of each parameter to be identified has been found. Therefore, in this step, if the calculated gradient coefficient is 0, stop the iteration, complete the optimization, and execute step S4. In addition, the maximum number of iterations can be set as the termination condition. The calculation formula of the gradient coefficient is as follows:
[0105] q0 = 0, q j = min(|CV(τ j,s ) - 1|, s = 1, 2,..., W)
[0106] where CV(τ j,s ) represents the coefficient of variation of the s-th parameter to be identified in the j-th round of optimization, and min(|CV(τ j,s ) - 1|, s = 1, 2,..., W) means calculating the coefficient of variation of the 19 parameters to be identified in the j-th round and selecting the coefficient of variation with the smallest difference from 1 as the gradual change coefficient in the j-th round of optimization. The coefficient of variation of samples is a commonly used technical means in this field, and its calculation formula and principle will not be elaborated here. Relevant practitioners can understand it;
[0107] (3) Calculate the weighting coefficient of each sample respectively, and the calculation formula is as follows:
[0108]
[0109] represents the weighting coefficient of the k-th sample of the s-th parameter to be identified in the j-th round of optimization, and D represents the measured data of the experiment;
[0110] (4) Calculate the average value of the weighting coefficients of N j,s samples, and the calculation formula is as follows:
[0111]
[0112] where Se j,s represents the average value of the sample weighting coefficients of the s-th parameter to be identified in the j-th round of optimization;
[0113] (5) Calculate the covariance matrix, and the calculation formula is as follows:
[0114]
[0115] where ∑ j,s represents the covariance matrix of the N j,s samples of the s-th parameter to be identified in the j-th round of optimization, represents the mean value of the N j,s samples of the s-th parameter to be identified in the j-th round of optimization, T represents matrix transpose, and ξ j represents the preset adaptive scaling factor in the j-th round of optimization;
[0116] (6) Randomly generate an index l from , and select a sample from with probability , will be used as A Gaussian distribution centered around and covariance matrix ∑ j,s As a hypothesized μ c For the proposed distribution, obtain μ c Start a Markov chain with the initial sample distribution; if Where r is a sample from a uniform distribution from 0 to 1, then let Update the prior distribution and initial value of the s-th parameter to be identified; otherwise, repeat this step.
[0117] Generate a uniform distribution from 0 to 1, mainly used to judge whether the probability distribution of this sampling is less than the distribution. If the probability of the uniform distribution is greater than the probability of the sampled sample, that is It indicates that the chain of this sampling is incorrect and can be discarded. Regenerate the index l. If It indicates that this sampling can be accepted, and then use μ c Update the prior distribution and initial value of the s-th parameter to be identified.
[0118] After completing the above steps (1)-(6), the update of the prior distribution and estimated value of the s-th parameter to be identified in the j-th round of optimization is completed. Then let s + 1 and execute steps (1)-(6) again to complete the optimization of the next parameter to be identified in the j-th round. After completing the optimization of all 19 parameters to be identified, since the preset termination condition is not met, let j + 1 and start the next round of iterative optimization.
[0119] In step (5), the adaptive scaling factor ξ j Is determined at the beginning of each round of optimization, and its calculation formula is:
[0120]
[0121]
[0122] Where W represents the total number of parameters to be identified, p r Represents the sample acceptance rate, t r Represents the target acceptance rate, and Na represents the number of chains to be adjusted, that is, in the j-th round of optimization The number of Markov chains.
[0123] S4. Output the estimated value of each parameter to be identified and the probability density function pdf. Among them, in this embodiment, the probability distribution of the parameter to be identified is as Figure 3 Shown.
[0124] In this embodiment, after multiple rounds of optimization, the estimated values and variance values of 19 parameters to be identified are obtained as shown in the following table:
[0125] Table 2 Estimated values of parameters to be identified
[0126]
[0127]
[0128] In this embodiment, after obtaining the estimated values of the various parameters in the BWBN model, they can be substituted back into the BWBN model. Then, the measured data and physical parameters of the experiment are substituted into the BWBN model again to solve for the hysteresis curve of the experiment. The simulated hysteresis curve diagram is as Figure 4 shown.
[0129] According to the second aspect of the present invention, there is provided a parameter estimation system for a BWBN model, including:
[0130] A data acquisition module for acquiring the measured data and physical parameters of the experiment, where the measured data includes force and displacement;
[0131] A BWBN model module for establishing a BWBN model, substituting the measured data and physical parameters into the BWBN model, solving the BWBN model, and determining W to-be-identified parameters in the BWBN model;
[0132] An initialization module for initializing each to-be-identified parameter, determining its maximum value, minimum value, and initial value, assuming that each to-be-identified parameter satisfies a Gaussian distribution, respectively calculating the likelihood function of each to-be-identified parameter, obtaining the initial prior distribution of each to-be-identified parameter, using the initial prior distribution as the prior distribution of the to-be-identified parameter, and using the initial value as the estimated value of the to-be-identified parameter;
[0133] An iterative optimization module for performing at least one round of optimization operations and stopping the iteration when the preset termination condition is satisfied. In each round of optimization operation, samples are respectively drawn from the prior distribution of each to-be-identified parameter and the estimated values of each to-be-identified parameter are updated based on the samples;
[0134] An output module for outputting the estimated values of each to-be-identified parameter.
[0135] The iTMCMC sampling method is based on Bayesian model updating technology. Applying the iTMCMC sampling method to estimate the parameters of the BWBN nonlinear hysteresis model can estimate the multi-dimensional parameter vector except for the non-sensitive hysteresis amplitude parameter A, and return the posterior distribution, estimated value, and variance of the parameters. Compared with the MCMC sampling method, this application adjusts the sample weights after each MCMC sampling step to reduce the average deviation of the model evidence estimation, applies a burn-in period in the MCMC step to improve the posterior approximation, and adaptively selects the proposed target distribution interval of the MCMC algorithm to achieve an acceptance rate close to the optimal. Therefore, this application overcomes the disadvantages of the existing algorithms while retaining the advantages of MCMC sampling.
[0136] Applying this application can bring the following advantages:
[0137] The effective number of independent samples sampled is large, the sampling efficiency is high, the estimation deviation of the mean value and the variance value is small, it can effectively solve the sampling of multi-peak or extremely flat single-peak problems, has a wide application range, can well simulate the hysteresis image with pinch hysteresis phenomenon, is easy to implement, does not require a filter to realize the parameter estimation affected by noise, can provide the probability distribution of the BWBN model parameters, and is of great significance for the evaluation of model selection.
[0138] The preferred specific embodiments of the present invention have been described in detail above. It should be understood that those of ordinary skill in the art can make many modifications and variations according to the concept of the present invention without creative work. Therefore, all technical solutions that can be obtained by those skilled in the art in the technical field of the present invention based on the concept of the present invention through logical analysis, reasoning or limited experiments should be within the protection scope determined by the claims.
Claims
1. A method for parameter estimation of a BWBN model, characterized in that Including the following steps: Obtain the measurement data of the experiment, where the measurement data includes force and displacement, determine the physical parameters of the experiment, establish a BWBN model, substitute the measurement data and physical parameters into the BWBN model, solve the BWBN model, and determine the W to-be-identified parameters in the BWBN model; Initialize each to-be-identified parameter respectively, determine its maximum value, minimum value and initial value, assume that each to-be-identified parameter follows a Gaussian distribution, calculate the likelihood function of each to-be-identified parameter respectively, obtain the initial prior distribution of each to-be-identified parameter, use the initial prior distribution as the prior distribution of the to-be-identified parameter, and use the initial value as the estimated value of the to-be-identified parameter; Draw samples from the prior distribution of each to-be-identified parameter respectively and update the estimated value of each to-be-identified parameter based on the samples to complete one round of optimization. Repeat this step until the preset termination condition is met and stop the iteration; Output the estimated value of each to-be-identified parameter; In the j-th round of optimization, draw samples from the prior distribution of the s-th to-be-identified parameter and update the estimated value of the s-th to-be-identified parameter based on the samples, where j = 1, 2,..., s = 1, 2,..., W. Specifically: Draw N samples from the prior distribution of the s-th parameter to be identified j,s samples: N j,s is the sample extraction value of the s-th parameter to be identified in the j-th round of optimization preset Calculate the gradient coefficient of the j-th round of optimization. The calculation formula is as follows: Among them, represents the sample coefficient of variation of the sth parameter to be identified in the jth round of optimization; Calculate the weighting coefficient of each sample respectively. The calculation formula is as follows: represents the weighting coefficient of the k-th sample of the s-th parameter to be identified in the j-th round of optimization, and D represents the measured data of the experiment; Calculate N j,s The average value of the weighted coefficients of the samples is calculated according to the following formula: Among them, Se j,s represents the average value of the sample weighting coefficients of the sth parameter to be identified in the jth round of optimization; Calculate the covariance matrix. The calculation formula is as follows: Among them, ∑ j,s represents the covariance matrix of N j,s samples of the sth parameter to be identified in the jth round of optimization, represents the mean value of N j,s samples of the sth parameter to be identified in the jth round of optimization, T represents matrix transpose, and ξ j represents the preset adaptive scaling factor in the jth round of optimization; Randomly generate an index l from and select a sample from with probability ; Take the Gaussian distribution centered at and the covariance matrix ∑ j,s as a proposed distribution for a hypothesized μ c to obtain a Markov chain starting with the initial sample distribution μ c ; if where r is a sample from a uniform distribution from 0 to 1, then let Update the prior distribution and initial value of the sth parameter to be identified; otherwise, repeat this step. At the start of each round of optimization, determine the value of the adaptive scaling factor ξ j whose calculation formula is as follows: Among them, W represents the total number of parameters to be identified, and p r represents the sample acceptance rate, and t r represents the target acceptance rate. Na represents the number of chains to be adjusted, that is, in the j-th round of optimization the number of Markov chains; The preset termination condition is that the gradient coefficient q in the j-th round of optimization j equals 1. It is considered that the optimal estimated value of each parameter to be identified is found, and the iteration is stopped.
2. The parameter estimation method of a BWBN model according to claim 1, wherein The physical parameters of the experiment include mass and initial stiffness. The BWBN model is as follows: v(t) = 1.0 + δ v ε(t) η(t) = 1.0 + δ η ε(t) where m represents mass, x represents displacement, c represents the damping coefficient, k e represents stiffness, z represents the damping displacement, F(t) represents force, h(z) reflects the pinching effect, v(t) represents the strength degradation, η(t) represents the stiffness degradation, ε(t) represents the hysteretic dissipated energy, n, a, β, γ, δ v and η are parameters to be identified, and represent the first and second derivatives of x, represents the first derivative of z.
3. The parameter estimation method of a BWBN model according to claim 2, wherein Use the fourth-order Runge-Kutta method to solve the differential equation in the BWBN restoring force measurement model as follows: K1 = f(x n , y n ) K4 = f(x n + h, y n + hK3) where f() represents the differential equation of the BWBN model, x n is the displacement in the measurement data, y n = F(t) is the force in the measurement data, h is the Runge-Kutta time step, and K1, K2, K3, and K4 are the transfer parameters in the Runge-Kutta method.
4. The parameter estimation method of a BWBN model according to claim 2, wherein The principle for establishing the likelihood function is: lnP3 = lnP1 + lnP2 where, σ1 represents the unknown variance of the displacement prediction error, and σ2 represents the unknown variance of the total dissipated energy prediction error. denotes the mean value of x, and N() represents the Gaussian distribution.
5. The parameter estimation method of a BWBN model according to claim 4, wherein When calculating the likelihood function of each to-be-identified parameter, it is necessary to normalize the predicted error dissipated energy. The calculation formula is: where L = lnP3, θ g is adjacent to the parameters to be identified, the subscript g is the index of force and displacement in the measured data, N j,s is the sample extraction value of the s-th parameter to be identified in the j-th round of optimization preset.
6. The parameter estimation method of a BWBN model according to claim 5, characterized in that Before calculating the likelihood function, it is necessary to determine the total dissipated energy from the hysteresis curve of the measurement data.
7. A parameter estimation system for a BWBN model, characterized in that, Based on the parameter estimation method of the BWBN model as described in any one of claims 1-6, including: A data acquisition module for obtaining the measurement data and physical parameters of the experiment, where the measurement data includes force and displacement; A BWBN model module for establishing a BWBN model, substituting the measurement data and physical parameters into the BWBN model, solving the BWBN model, and determining the W to-be-identified parameters in the BWBN model; An initialization module for initializing each to-be-identified parameter respectively, determining its maximum value, minimum value and initial value, assuming that each to-be-identified parameter follows a Gaussian distribution, calculating the likelihood function of each to-be-identified parameter respectively, obtaining the initial prior distribution of each to-be-identified parameter, using the initial prior distribution as the prior distribution of the to-be-identified parameter, and using the initial value as the estimated value of the to-be-identified parameter; An iterative optimization module for performing at least one round of optimization operation and stopping the iteration when the preset termination condition is met. In each round of optimization operation, draw samples from the prior distribution of each to-be-identified parameter respectively and update the estimated value of each to-be-identified parameter based on the samples; An output module for outputting the estimated value of each to-be-identified parameter.