A method for identifying dynamic loads on nonlinear structures based on improved particle filtering

Abstract

The present invention discloses a method for identifying dynamic loads on nonlinear structures based on an improved particle filter. The method comprises the following steps: Step 1: establishing a state (displacement and velocity) transfer equation and an observation equation for a nonlinear structural system; Step 2: establishing a time-discretized state transfer equation and observation equation for the nonlinear system containing process noise; Step 3: given an initial value and variance value of a state vector, establishing an improved particle filter based on a particle filter and a weighted least squares method, wherein the weighted least squares method is used for load identification and the particle filter is used for state identification; Step 4: continuously identifying the load and system state using the improved particle filter based on the real-time measured dynamic acceleration response of the structure. This method is particularly suitable for dynamic load identification of nonlinear structures in the presence of non-Gaussian noise.

Description

Technical Field

[0001] The invention relates to a nonlinear structural dynamic load identification method based on improved particle filtering, and belongs to the technical field of structural dynamics. Background Art

[0002] Dynamic loads have a significant impact on the dynamic properties of structures. Generally, these are difficult to obtain through measurement methods. This is primarily due to the large size of force gauges, making them difficult to deploy on-site, and their influence on the original structural system characteristics, leading to significant deviations in the measured results. Consequently, a branch of dynamic load identification technology has been developed, also known as the second-class inverse problem of structural dynamics. This involves inversely determining the structural loads based on a limited number of measured responses and a system model.

[0003] For a long time, load identification research has focused on solving ill-posed problems in linear structural systems. These methods are deterministic and rarely consider noise errors in the model. However, nonlinear structural systems are more widely used in engineering practice and are more difficult to solve, so in-depth research is needed. Summary of the Invention

[0004] The purpose of the present invention is to provide a nonlinear structural dynamic load identification method based on improved particle filtering, which can effectively solve the problem of nonlinear system or structural dynamic load identification in the presence of non-Gaussian noise.

[0005] In order to solve the above problems, the present invention adopts the following solutions:

[0006] A nonlinear structural dynamic load identification method based on improved particle filtering includes the following steps:

[0007] S100. Establish the state transfer equation and observation equation of the nonlinear system:

[0008] For nonlinear structural systems Construct a state vector containing structure states and structures Establish state transfer equations and observation equations for nonlinear systems;

[0009] Among them, the state transfer equation is:

[0010]

[0011] Observation equation:

[0012] Where M is the mass matrix, p(t) represents the node displacement at time t, With respect to the speed at time t, They represent the acceleration with respect to time t, α=[α1 α2 … α α ] Trepresents the parameters of the structural system, is a nonlinear function of the system with respect to displacement, velocity and structural parameters, y(t) represents the measured response of the structural system, u(t) is the external load excitation, and B u is the position influence matrix of the external load vector, H and D represent the position influence matrices of the measured response with respect to the state and external load, respectively, and the superscript “T” indicates the transpose of the matrix or vector;

[0013] S200, establish the state transfer equation and observation equation for the time discretization of the nonlinear system containing process noise, as shown in the following formula:

[0014] z n+1 =f(z n ,u n )+w n ; n = 1, 2…m;

[0015] y n =h(z n )+D n u n +v n ;

[0016] Where, subscript n represents the nth sampling moment, m is a positive integer, and z n Represents the state vector at the nth sampling moment, u n represents the external load excitation at the nth sampling moment, y n represents the acceleration measurement response at the nth sampling moment, w n represents the system noise, whose mean and variance are assumed to be 0 and G respectively n ;v n represents the observation noise, whose mean and variance are assumed to be 0 and R respectively n .

[0017] S300, given an initial value and a variance value of a state vector, establishing an improved particle filter based on a particle filter and a weighted least squares method, wherein the weighted least squares method is used for load identification, and the particle filter method is used for state identification;

[0018] S400, based on the real-time measurement of the structural dynamic acceleration response, continuously identifies the load through an improved particle filter and the state {z 1|1 ,…,z n|n ,…,z m|m}.

[0019] Particle filtering is an approximate Bayesian filtering algorithm based on Monte Carlo simulation. It is particularly capable of handling nonlinearity and non-Gaussian noise in the system. It has been widely used in many fields such as visual tracking, communication, signal processing and control.

[0020] The present invention requires only individual acceleration measurement signals to jointly identify the state and load of nonlinear structural systems. The improved particle filter proposed in this invention incorporates minimum variance unbiased load estimation within the particle filter framework, making it suitable for load identification in nonlinear, non-Gaussian noise structural systems.

[0021] As a further improvement, in step S100, the acceleration measurement signal is used as the measurement signal, and the observation equation is:

[0022]

[0023] Among them H a Represent the position influence matrix of the measured displacement and acceleration signals respectively, then there is

[0024]

[0025] D=H a M -1 B u .

[0026] Further improvement, in step S200, f c is a time-continuous function.

[0027] Further improvement, in S300, the improved particle filter includes the following four steps:

[0028] S301, Initialization: Define vector The true value z n 、u n In the observation vector (y0,y1,y2,…,y n ), the state variance matrix is assumed to be Given the initial value of the state vector and variance P 1|0 , according to the prior probability density function Generate particle set Importance weights of all particles for

[0029] S302, load identification step:

[0030] The following formulas are included:

[0031]

[0032] Among them, D n is a constant, D n =D;

[0033] S303, measurement update step:

[0034] S3031. Calculate the importance weight: Take the proposed probability density function as p(z n+1 |z n ), then

[0035]

[0036] S3032, Resampling: The coefficient of variation of the importance weight is used as a measure of the degree of particle degradation. That is, when the coefficient of variation COV of the weight exceeds the set threshold, it indicates that the degree of uneven distribution of particles is high. Resampling technology is used to suppress the further degradation of system particles.

[0037] The coefficient of variation is as follows:

[0038]

[0039] Among them, std represents standard deviation and mean represents mean;

[0040] If COV is greater than the set threshold, the resampling algorithm is executed and the weights are set.

[0041] Otherwise, no resampling algorithm is performed and the weights are normalized

[0042]

[0043] The state minimum variance unbiased estimate is as follows:

[0044]

[0045] S304, time update step: at time n+1, predict the next state according to the state transition equation

[0046] Set n=n+1 and go to step S302 for loop iteration.

[0047] The beneficial effects of the present invention are as follows:

[0048] 1. The present invention only requires individual acceleration measurement signals to perform joint identification of the state and load of a nonlinear structural system.

[0049] 2. The improved particle filter proposed in the present invention incorporates the minimum variance unbiased estimation of the load under the framework of the particle filter, and is suitable for the load identification problem of nonlinear and non-Gaussian noise structure systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1Schematic diagram of a flow chart of a nonlinear structural dynamic load identification method based on improved particle filtering according to the present invention;

[0051] Figure 2 Schematic diagram of the structural model in Example 1 of the present invention (a four-story shear frame building with hysteresis characteristics);

[0052] Figure 3 This is the recognition result of the load u using the improved particle filter in Example 1 of the present invention;

[0053] Figure 4 This is a diagram of the recognition result using the improved particle filter in Example 1 of the present invention; wherein, Figure 4 (a) Comparison of the second-layer displacement recognition results and the true value; Figure 4 (b) Comparison of the second-layer speed recognition results and the true value; Figure 4 (c) Comparison between the identification results and the true value of the hysteresis displacement of the second layer.

[0054] Figure 5 is the identification result of the load u using the improved particle filter in Example 2 of the present invention; wherein, Figure 5 (a) Load identification results within 1 s; Figure 5 (b) is the load identification result diagram within 0.2-0.4s;

[0055] Figure 6 This is a diagram of the recognition result using the improved particle filter in Example 2 of the present invention; wherein, Figure 6 (a) Comparison of the second-layer displacement recognition results and the true value; Figure 6 (b) Comparison of the second-layer speed recognition results and the true value. DETAILED DESCRIPTION

[0056] In order to make the purpose and technical solution of the present invention clearer, the technical solution of the present invention will be clearly and completely described below in conjunction with the embodiments of the present invention.

[0057] Example 1:

[0058] In this example, the object is a nonlinear four-story shear beam building with typical hysteretic characteristics, such as Figure 2 The kinematic differential equation is as follows:

[0059]

[0060] in, is a nonlinear function in this embodiment, corresponding to r(t) is the Bouc-Wen type hysteresis displacement vector, as shown below

[0061] Among them, θ j ,β j ,γ j and ζ j is the Bouc-Wen type hysteresis parameter, j = 1, 2, 3, 4. Generally speaking, the hysteresis characteristics of a structure can be used as a basis for monitoring structural damage under dynamic excitation.

[0062] Parameter settings in this example: mass m of each layer j , stiffness k j and the damping parameter c j m1 = m2 = m3 = m4 = 300 kg, k1 = k2 = k3 = k4 = 20 N / m, and c1 = c2 = c3 = c4 = 1 Ns / m. The hysteresis parameters are related by θ1 = θ2 = θ3 = θ4 = 6, β1 = β2 = β3 = β4 = 0.8, γ1 = γ2 = γ3 = γ4 = 0.1, and ζ1 = ζ2 = ζ3 = ζ4 = 0.9. The external load u(t) acts on the fourth layer and is assumed to be a white noise random excitation. The time history response of this nonlinear structure is calculated using the Runge-Kutta method. The three acceleration responses of layers 1, 3, and 4 are measured, with 3% ambient noise added to the measured responses.

[0063] According to the recognition method of the present invention, the state vector in this example is Unknown loads are unknown quantities to be identified, and nonlinear structural parameters θ j ,β j ,γ j and ζ j is a known quantity. The time sampling interval is 0.5s, and the threshold of the importance weight coefficient of variation is 200%.

[0064] According to the above conditions, such as Figure 1 As shown, according to step S100, the structural state transfer equation can be obtained The observation equations y(t) are:

[0065]

[0066] Among them, the mass matrix M, damping matrix K and stiffness matrix C are:

[0067]

[0068] The position influence matrix of the external load vector and the position influence matrix of the measured acceleration signal are:

[0069] B u =[0 0 0 1] T ,H a =[1 0 1 1] T .

[0070] in addition, D=H a M -1 B u .

[0071] Step S200: Establish the state transfer equation and observation equation of the time discretization of the nonlinear system containing process noise, as shown in the following formula:

[0072] z n+1 =f(z n ,u n )+w n ; n = 1, 2…m;

[0073] y n =h(z n )+D n u n +v n ;

[0074]

[0075] Where, subscript n represents the nth sampling moment, m is a positive integer, and z n Represents the state vector at the nth sampling moment, u n represents the external load excitation at the nth sampling moment, y n represents the acceleration measurement response at the nth sampling moment, w n represents the system noise, whose mean and variance are assumed to be 0 and G respectively n ;v n represents the observation noise, whose mean and variance are assumed to be 0 and R respectively n .

[0076] S301, Initialization: Define vector The true value z n 、u n In the observation vector (y0,y1,y2,…,y n ), the state variance matrix is assumed to be Given the initial value of the state vector and variance P 1|0 , according to the prior probability density function Generate particle set Importance weights of all particles for

[0077] S302, load identification step:

[0078] The following formulas are included:

[0079]

[0080] Among them, D n is a constant, D n =D;

[0081] S303, measurement update step:

[0082] S3031. Calculate the importance weight: Take the proposed probability density function as p(z n+1 |z n ), then

[0083]

[0084] S3032, Resampling: The coefficient of variation of the importance weight is used as a measure of the degree of particle degradation. That is, when the coefficient of variation COV of the weight exceeds the set threshold, it indicates that the degree of uneven distribution of particles is high. Resampling technology is used to suppress the further degradation of system particles.

[0085] The coefficient of variation is as follows:

[0086]

[0087] Among them, std represents standard deviation and mean represents mean;

[0088] If COV is greater than the set threshold, the resampling algorithm is executed and the weights are set.

[0089] Otherwise, no resampling algorithm is performed and the weights are normalized

[0090]

[0091] The state minimum variance unbiased estimate is as follows:

[0092]

[0093] S304, time update step: at time n+1, predict the next state according to the state transition equation

[0094] Set n=n+1 and go to step S302 for loop iteration.

[0095] S400, based on the real-time measurement of the structural dynamic acceleration response, continuously identifies the load through an improved particle filter and the state {z 1|1 ,…,z n|n ,…,z m|m}.

[0096] Figure 3A comparison chart of the true value and the identified value of the load is shown, and it can be seen that the accuracy of load identification is very high. Figure 4 The recognition of displacement, velocity, and hysteresis displacement of the second layer is shown. It can be seen that the basic graphs of the true value and the recognition value are coincident, and the recognition accuracy is very high.

[0097] Example 2:

[0098] In this example, the object is a three-story nonlinear elastic Duffing shear beam building, and the differential equation of motion is as follows:

[0099]

[0100]

[0101] where p j is the displacement of the jth layer (j = 1, 2, 3), m1 = m2 = m3 = 1000 kg, c1 = c2 = c3 = 0.6 kNs / m, k1 = 120 kN / m, k2 = 120 kN / m, k3 = 60 kN / m, k4 = 200 kN / m, k5 = 200 kN / m, and k6 = -50 kN / m. For a purely elastic structure, k4 = k5 = k6 = 0, and the natural frequencies are ω1 = 0.73 Hz, ω2 = 1.74 Hz, and ω3 = 2.93 Hz, respectively. The corresponding third-order damping rates are ζ1 = 1.42%, ζ2 = 4.56%, and ζ3 = 5.08%. The acceleration responses of the three layers are the measured signals, with 3% ambient noise added to the measured responses.

[0102] According to the method of the present invention, the state vector in this example is The unknown load u is the unknown quantity to be identified. The time sampling interval is 0.002s, and the threshold of the importance weight coefficient of variation is 200%.

[0103] From the above conditions, we can know that:

[0104]

[0105] Then the differential equation of structural motion is:

[0106]

[0107] According to step S100, the structural state transfer equation can be obtained: The observation equations y(t) are:

[0108]

[0109] The position influence matrix of the external load vector and the position influence matrix of the measured acceleration signal are:

[0110] B u =[0 1 0] T , H a =[1 1 1] T .

[0111] in addition, D=H a M -1 B u .

[0112] The following steps S200, S300 and S400 are followed in sequence to continuously identify the load through the improved particle filter of the present invention. and the state {z 1|1 ,…,z n|n ,…,z m|m}. Figure 5 A comparison chart of the true value and the identified value of the load is shown, and it can be seen that the accuracy of load identification is very high. Figure 6 The displacement and velocity recognition of the second layer are shown. It can be seen that the basic graphs of the true value and the recognition value are coincident, and the recognition accuracy is very high.

[0113] Anything not specifically described in this invention is prior art or can be implemented using prior art. Furthermore, the specific embodiments described in this invention are merely preferred embodiments of the invention and are not intended to limit the scope of the invention. In other words, any equivalent variations and modifications made within the scope of the present invention should be considered within the technical scope of this invention.

Claims

1. A nonlinear structural dynamic load identification method based on improved particle filtering, characterized in that: The steps include: S100. Establish the state transfer equation and observation equation of the nonlinear system: For nonlinear structural systems Construct a state vector containing structure states and structures Establish state transfer equations and observation equations for nonlinear systems; Among them, the state transfer equation is: Observation equation: Where M is the mass matrix, p(t) represents the node displacement at time t, With respect to the speed at time t, They represent the acceleration with respect to time t, α=[α1 α2…α α ] T Represents the parameters of the structural system, is a nonlinear function of the system with respect to displacement, velocity and structural parameters, y(t) represents the measured response of the structural system, u(t) is the external load excitation, and B u is the position influence matrix of the external load vector, H and D represent the position influence matrices of the measured response with respect to the state and external load, respectively, and the superscript "T" indicates the transpose of the matrix or vector; S200, establish the state transfer equation and observation equation for the time discretization of the nonlinear system containing process noise, as shown in the following formula: z n+1 =f(z n ,u n )+w n ;n=1,2…m; y n =h(z n )+D n u n +v n ;; Where, subscript n represents the nth sampling moment, m is a positive integer, and z n Represents the state vector at the nth sampling moment, u n represents the external load excitation at the nth sampling moment, y n represents the acceleration measurement response at the nth sampling moment, w n represents the system noise, whose mean and variance are assumed to be 0 and G respectively n ;v n represents the observation noise, whose mean and variance are assumed to be 0 and R respectively n ; S300, given an initial value and a variance value of a state vector, establishing an improved particle filter based on a particle filter and a weighted least squares method, wherein the weighted least squares method is used for load identification, and the particle filter method is used for state identification; S400, based on the real-time measurement of the structural dynamic acceleration response, continuously identifies the load through an improved particle filter and the state {z 1|1 ,…,z n|n ,…,z m|m }.

2. The nonlinear structural dynamic load identification method based on improved particle filtering according to claim 1 is characterized in that: In step S100, the acceleration measurement signal is used as the measurement signal, and the observation equation is: Among them H a Represent the position influence matrix of the measured displacement and acceleration signals respectively, then there is D=H a M -1 B u 。 3. The nonlinear structural dynamic load identification method based on improved particle filtering according to claim 1 is characterized in that: In the step S200, 4. The nonlinear structural dynamic load identification method based on improved particle filtering according to claim 2 is characterized in that: In step S300, the improved particle filter includes the following four steps: S301, Initialization: Define vector The true value z n 、u n In the observation vector (y0,y1,y2,…,y n ), the state variance matrix is assumed to be Given the initial value of the state vector and variance P 1|0 , according to the prior probability density function Generate particle set Importance weights of all particles for S302, load identification step: The following formulas are included: Among them, D n is a constant, D n =D; S303, measurement update step: S3031. Calculate the importance weight: Take the proposed probability density function as p(z n+1 |z n ), then S3032, Resampling: The coefficient of variation of the importance weight is used as a measure of the degree of particle degradation. That is, when the coefficient of variation COV of the weight exceeds the set threshold, it indicates that the degree of uneven distribution of particles is high. Resampling technology is used to suppress the further degradation of system particles. The coefficient of variation is as follows: Among them, std represents standard deviation and mean represents mean; If COV is greater than the set threshold, the resampling algorithm is executed and the weights are set. Otherwise, no resampling algorithm is performed and the weights are normalized The state minimum variance unbiased estimate is as follows: S304, time update step: at time n+1, predict the next state according to the state transition equation Set n=n+1 and go to step S302 for loop iteration.

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