Structural load identification method based on orthogonal polynomial fitting and double kalman filter

Abstract

The application discloses a structure load identification method based on orthogonal polynomial fitting and double Kalman filtering, which comprises the following steps: 1, using orthogonal polynomials to fit unknown loads applied to the structure; 2, constructing a state space equation group of fitting coefficients, and using a standard Kalman filtering method to identify the fitting coefficients of the polynomials; 3, constructing a state space equation group of the structure state, and using the Kalman filtering method again to identify the structure state. The application is aimed at the low-frequency drift problem of the identification load and displacement when the traditional Kalman filtering method based on the least square method only uses acceleration measurement signals to identify the load and state, and the double Kalman filtering method based on the orthogonal polynomial fitting can effectively solve the above low-frequency drift problem. The essence is that a mathematical constraint of the orthogonal polynomial fitting is introduced in the identification process, so that the identification result tends to be stable, and the engineering practical application is facilitated.

Description

Technical Field

[0001] This invention belongs to the field of structural load identification technology, specifically relating to a structural load identification method based on orthogonal polynomial fitting and dual Kalman filtering. Background Technology

[0002] External loads on a structure play a crucial role in structural optimization design, fault diagnosis, and health monitoring. Since external loads are difficult to measure directly with instruments, many deterministic load inverse methods have been developed, also known as the inverse problem of the second kind in structural dynamics, primarily including frequency domain methods and time domain methods.

[0003] However, deterministic inverse methods cannot take into account structural model errors. In recent years, uncertainty methods for load identification have become a research hotspot. References [1, 2] propose a Bayesian theory-based load identification method in the frequency domain and time domain, respectively, and derive adaptive regularization parameters. This type of method takes into account structural model errors. Another type of uncertainty method can continuously identify unknown loads and structural states. Reference [3] proposes a method based on GDF. [4] A load / response joint identification method was proposed, and modal reduction was used to reduce the computational burden. Based on the standard Kalman spectroscopy, this method derived a least-squares unbiased estimate of the load and state. (Lourens et al.) [5、6] A load / state joint identification method based on enhanced Kalman filtering (AKF) is proposed. This method treats the load and structural state as an enhanced state and uses standard Kalman filtering for identification. Its accuracy is greatly affected by the load covariance. (Azam et al.) [7、8] A load / state joint identification method based on DKF is proposed. This method uses a random walk model for the unknown load and constructs its state-space equations. A two-step Kalman filter is then used for continuous identification of the load and state. However, when the above method uses only acceleration measurement signals for system identification, spurious low-frequency drift in load and displacement occurs. This is because the acceleration signal is insensitive to the quasi-static components of the input load. Although regularization methods and some signal post-processing methods can solve this problem, these methods are not suitable for real-time inversion.

[0004] Existing literature:

[0005] [1]Zhang E, Antoni J, Feissel P.Bayesian force reconstruction with anuncertain model[J]. Journal of Sound and Vibration, 2012,331:798-814.

[0006] [2] Wang Ting, Wan Zhimin, Zheng Weiguang. Structural Time-domain Load Identification Based on Gibbs Sampling. Journal of Vibration and Shock, 2018, 37(02): 85-90.

[0007] [3] Lourens E, Papadimitriou C, Gillijins S, Reynders E, De Roeck G, Lombaert G. Joint Input-Response Estimation for Structural Systems Based on Reduced-order Models and Vibration Date from a Limited Number of Sensors. Mechanical Systems and Signal Processing, 2012, 29: 310-327.

[0008] [4] Gillijins S, De Moor B. Unbiased Minimum-variance Input and State Estimation for Linwar Discrete-time Systems with Direct Feedthrough. Automatica, 2007, 43: 934-937.

[0009] [5] Lourens E, Reynders E, De Roech G, Degrande G, Lombaert. An Augmented Kalman Filter for Force Identification in Structural Dynamics. Mechanical Systems and Signal Processing, 2012, 27: 446-460.

[0010] [6] Naets F, Cuadrado J, Desmet W. Stable Force Identification in Structural Dynamics Using Kalman Filtering and Dummy-measurements. Mechanical Systems and Signal Processing, 2015, 50-51: 235-248.

[0011] [7]Azam S, Chatzi E, Papadimitriou C.Adual Kalman filter approach forstate estimation via output-only acceleration measurements. Mechanical Systems and Signal Processing, 2015, 60-61:866-886.

[0012] [8] Azam S, Chatzi E, Papadimitriou C, Smyth A. Experimental validation of the Kalman-type filters for online and real-time state and input estimation. Journal of Control, 2015, 23(15): 1-26. Summary of the Invention

[0013] The purpose of this invention is to provide a structural load identification method based on orthogonal polynomial fitting and dual Kalman filtering, which solves the technical problem in the prior art of how to accurately identify structural parameters by measuring the response through sensors when the external load is unknown.

[0014] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0015] The structural load identification method based on orthogonal polynomial fitting and dual Kalman filtering includes the following steps:

[0016] Step 1: Use orthogonal polynomials to fit the unknown loads applied to the structure;

[0017] Step 2: Construct the state-space equation system of the fitting coefficients and use the standard Kalman filter to identify the fitting coefficients of the polynomials;

[0018] Step 3: Construct the state-space equation set of the structural state, and use the Kalman filter method again to identify the structural state.

[0019] Further optimization, step 1 includes the following steps:

[0020] Step 1.1: Using the finite element method, the differential equations of motion for a dynamic system with n degrees of freedom are expressed as follows:

[0021]

[0022] In the above formula, M represents the system mass matrix, C represents the system damping matrix (viscous damping), and C is a symmetric matrix; K represents the system stiffness matrix, p(t), The vectors represent the displacement, velocity, and acceleration responses of the nodes on the structure, respectively; u(t) represents the external load excitation vector applied to the structure, and B... u This represents the positional influence matrix of the external load vector;

[0023] Step 1.2: The state-space method is used to describe the structural dynamic system. The state vector X(t) of the structure is represented by the structural displacement and velocity vectors, i.e. Then the state transfer equation

[0024]

[0025] in,

[0026]

[0027] I represents the identity matrix;

[0028] If only acceleration signals are used as measurement data, then the state observation equation of the structure is:

[0029] y(t)=Hx(t)+Du(t) (4)

[0030] In the above equation, y(t) is the equation for the measured acceleration response, H is the position influence matrix of the measured acceleration response, and D is the load transfer matrix of the measured acceleration response.

[0031] H = [-H0M] -1 K -H0M -1 C],D=[H0M -1 B u (5)

[0032] Step 1.3: Considering system process noise and measurement noise, the sampling frequency is set to 1 / Δt. Formulas (2) and (4) are written in the following time-discrete form:

[0033] x k+1 =Ax k +Bu k +w k ;k=1,2…T (6)

[0034] y k =Hx k +Du k +v k ;k=1,2…T (7)

[0035] In the above formula, the subscript k represents the k-th sampling time, xk Let x represent the state vector at time k. k+1 y represents the state vector at k time points; k The equation representing the measured acceleration response at time k; u k This represents the external load excitation vector applied to the structure; w k v k Let G and R represent the system noise and observation noise at time k, respectively. Both are uncorrelated Gaussian white noise, with zero mean and variances G and R, respectively.

[0036] In the above formula, matrices A and B are respectively:

[0037]

[0038] Step 1.4: Based on the theory of orthogonal polynomial fitting approximation curves, the structural load is fitted using orthogonal polynomials. Then:

[0039] u(t)=Φ(t)θ+η (9)

[0040] In the above formula, Φ(t) is the orthogonal polynomial basis function, θ is the fitting coefficient, and η is the fitting error, assumed to be Gaussian white noise with zero mean and variances G. η To represent; substituting formula (9) into formulas (6) and (7), the state transfer equation and state observation equation are rewritten in the following polynomial form:

[0041] x k+1 =Ax k +BΦ k θ+Bη k +w k (10)

[0042] y k =Hx k +DΦ k θ+Dη k +v k (11).

[0043] Once the orthogonal polynomial type is determined, it can be seen from formulas (10) and (11) that the load identification of the structure is transformed into the identification of the fitting coefficients, and the load value can be calculated by formula (9).

[0044] Further optimization, step 2 specifically includes the following steps:

[0045] Step 2.1: Construct the state-space equations for the fitting coefficients: The fitting coefficients are expressed as follows using a random walk model:

[0046]

[0047] In the above formula, θ k θ represents the fitting coefficient at time k. k+1 Represents the fitting coefficient at time k+1. The process noise representing the fitting coefficients is assumed to have a mean of 0 and a variance of G. θ The probability distribution of Gaussian white noise;

[0048] Then the state observation equation (11) is rewritten as:

[0049] y k =DΦ k θ k +Hx k +Dη k +v k (13)

[0050] Step 2.2: The state-space equations about the fitting coefficients θ are formed by formulas (12) and (13), and then the standard Kalman filter is used to identify the fitting coefficients. The standard Kalman filter equations include a state prediction step and a time update step. The state prediction step is to predict the state using the data of the current state based on the mathematical model identified by least squares. The time update step is to correct the state again using the measurement information at the current time, as follows:

[0051] State prediction step:

[0052] θ k|k-1 =θ k-1|k-1 (14)

[0053]

[0054] In the above formula, θ k|k-1 Let θ represent the prior value of the fitting coefficient θ at time k. k-1|k-1 This represents the posterior value of θ at time k-1. This represents the covariance value of the fitting coefficient θ at time k-1. This represents the covariance of the posterior values ​​of the fitted coefficient θ at time k-1; Let represent the gain matrix with respect to the fitting coefficients θ at time k;

[0055] Time update steps:

[0056]

[0057] θ k|k Let θ represent the posterior value of θ at time k;

[0058] Solving the above formula yields the load fitting coefficients.

[0059] Further optimization, as can be seen from formulas (14)-(18), requires structural state values, i.e., x in formula (17), to solve for the load fitting coefficients. k Therefore, in step 3, the Kalman filter method is used again to identify the structural state, and its filtering equation is as follows:

[0060] State prediction step:

[0061] x k|k-1 =Ax k-1|k-1 +BΦ k θ k|k (19)

[0062]

[0063] K k =P k|k-1 H T (H·P k|k-1 ·H T +DG η D T +R k ) -1 (twenty one)

[0064] Where, x k|k-1 Let x represent the prior value of x at time k. k-1|k-1 P represents the posterior value of x at time k-1. k|k-1 K represents the covariance of the fitting coefficient x at time k-1. k Let x represent the gain matrix with respect to the fitting coefficients x at time k.

[0065] Time update steps:

[0066] x k|k =x k|k-1 +K k (y k -Hx k|k-1 -D k Φ k θ k|k ) (twenty two)

[0067] P k|k =P k|k-1 -K k HP k|k-1 (twenty three)

[0068] Given initial value: θ 0|0 , G θ G η , R, x 0|0 P 0|0By combining the above dual Kalman filtering method (14)-(22), the load fitting coefficients and structural state vector can be continuously obtained. Then, the structural load value can be obtained by referring to the following formula.

[0069] u k =Φ k θ k (twenty four).

[0070] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0071] This invention employs orthogonal polynomial fitting and Kalman filtering to fit unknown structural loads using orthogonal polynomials, thus imposing mathematical constraints on the loads and enabling the identification of the structural system using only acceleration measurement signals. The structural load identification problem is transformed into the identification of orthogonal polynomial fitting coefficients. Two consecutive Kalman filters are applied to continuously identify the fitting coefficients and structural states (displacement and velocity), effectively solving the problem of low-frequency drift of loads and displacements during the identification process and facilitating practical engineering applications. Attached Figure Description

[0072] Figure 1 This is a planar truss structure diagram;

[0073] Figure 2 This is a comparison chart of the actual and identified values ​​of the load in Example 1;

[0074] Figure 3 These are the actual and identified displacement values ​​for degree 19 in Example 1;

[0075] Figure 4 The actual and identified values ​​of velocity for degree 19 in Example 1;

[0076] Figure 5 This is a comparison chart of the actual and identified values ​​of the load under different noise levels in Example 1 during the 20-20s interval;

[0077] Figure 6 This is a comparison chart of the actual and identified load values ​​during the 20-20s interval in Example 1;

[0078] Figure 7 The true displacement and velocity values ​​identified in Example 1 are: (a) the true displacement value identified; (b) the true velocity value and the identified value.

[0079] Figure 8 This is a comparison chart of the actual and identified values ​​of the load in Comparative Example 1;

[0080] Figure 9 The true values ​​and identified values ​​of displacement and velocity for degree 19 in Comparative Example 1; (a) True value and identified value of displacement; (b) True value and identified value of velocity;

[0081] Figure 10 This is a comparison chart of the actual and identified values ​​of the load in Comparative Example 2;

[0082] Figure 11 The true values ​​and identified values ​​of displacement and velocity for the 19 degrees of freedom in Comparative Example 2; (a) True values ​​and identified values ​​of displacement; (b) True values ​​and identified values ​​of velocity;

[0083] Figure 12 This is a comparison chart of the actual and identified values ​​of the load in Example 2;

[0084] Figure 13 These are the actual and identified displacement values ​​for degree 23 in Example 2;

[0085] Figure 14 These are the actual and identified displacement values ​​for degree 23 in Example 2;

[0086] Figure 15 A comparison chart of the actual and identified values ​​of the load;

[0087] Figure 16 The true values ​​and identified values ​​of displacement and velocity for degree 23 in Comparative Example 3; (a) True value and identified value of displacement; (b) True value and identified value of velocity. Detailed Implementation

[0088] To make the objectives and technical solutions of this invention clearer, the technical solutions of this invention will be clearly and completely described below in conjunction with the embodiments of this invention.

[0089] The feasibility of the proposed solution is verified by using a planar truss as the numerical simulation object. Figure 1 As shown, a) is a schematic diagram of the planar truss structure; b) is a finite element model of the truss and a schematic diagram of the sensor arrangement. The planar truss contains 31 bar elements and 17 nodes. The length of the horizontal bar element is 2m, and the length of the 45° inclined bar element is... Furthermore, all the rod elements have the same cross-sectional dimensions, with a cross-sectional area of ​​8.95 × 10⁻⁶. -5 m 2 The elastic modulus is 2×10 7 Pa, density is 7.85 × 10⁻⁶. 3 kg / m 3 .

[0090] In this example, each rod element is modeled as a lumped mass model, containing two nodes, each with two degrees of freedom in the lateral and vertical directions. The structural system damping adopts a viscous damping model C = δM + βK, where the damping coefficients are δ = 0.1523 and β = 4.6203 × 10⁻⁶. -4 .

[0091] Example 1:

[0092] In this embodiment, the dynamic external load acts on node 9 as a vertical force. A sinusoidal excitation u = 40sin(2πt) is used, and nodes 1 and 17 are fixed constraints. Each node has two degrees of freedom, labeled sequentially according to node number. Except for nodes 1 and 17, which are fixed constraints and have no degree of freedom numbers, there are a total of 30 degrees of freedom. Only acceleration measurement signals are used, with the measured degrees of freedom numbers being 4, 6, 8, 12, 16, and 18, a total of 6. 5% background noise is incorporated into the measurement signals.

[0093] A 10th-order Lelande orthogonal polynomial is used to fit the external load (the 10th-order Lelande orthogonal polynomial is a classic orthogonal polynomial fitting method, which is essentially taking the basis functions of a 10th-order Lelande orthogonal polynomial, and belongs to existing technology). The dual Kalman filtering method based on orthogonal polynomials described in this invention is applied for joint identification of the load / state. The identified load, displacement, and velocity are as follows: Figure 2 , 3 As shown in Figure 4, Figure 2 This is a comparison chart of the actual and identified values ​​of the load in Example 1; Figure 3 The true and identified values ​​of displacement for a degree of freedom of 19; Figure 4 The values ​​are the actual and identified values ​​of velocity with 19 degrees of freedom. It can be seen that there is no low-frequency drift phenomenon in the load and displacement, and the identification results are relatively accurate. The relative error rate of the load is 1.83%, and the relative error rate of the displacement is 0.92%.

[0094] 1) Add 2%, 5%, and 10% white noise to the measurement signal respectively to verify the accuracy of the recognition. Figure 5 The figure shows a comparison of the actual and recognized values ​​of the load at different noise levels over 20-20 seconds. It can be seen that even with 10% background noise, the method described in this invention can still achieve good recognition results.

[0095] 2) Four acceleration measurement signals were used for system identification, with degrees of freedom of 4, 15, 16, and 18 respectively. The identification results are as follows: Figure 6 , 7 As shown, Figure 6 A comparison chart of the actual and identified values ​​of the load during the 20-20s interval; Figure 7 The true values ​​of displacement and velocity are identified for a system with 19 degrees of freedom; (a) the true displacement value identified; (b) the true velocity value and the identified value. For clarity, only the identification results within the 20-22s timeframe are shown. It can be seen that the accuracy is slightly lower than that of the response to 6 acceleration measurements, but still relatively accurate.

[0096] Comparative Example 1:

[0097] Chinese patent application, application number: 2021103547265, titled "A Joint Identification Method for Structural State / Parameter / Load Based on Extended GDF," is an earlier application by the applicant. This method is used to perform joint load / state identification on the loads applied in Example 1, and the results are as follows: Figure 8 , Figure 9 As shown, Figure 8 A comparison chart of the actual and identified values ​​of the load; Figure 9 The true values ​​of displacement and velocity are identified for a system with 19 degrees of freedom; (a) true displacement value and identified value; (b) true velocity value and identified value. It can be seen that the identified displacement value is very accurate and basically coincides with the true value curve, but the identified load and displacement show obvious low-frequency drift.

[0098] Comparative Example 2:

[0099] Existing papers, Azam S, Chatzi E, Papadimitriou CA, dual Kalman filter approach for state estimation via output-only acceleration measurements. Mechanical Systems and Signal Processing, 2015, 60-61: 866-886; and Azam S, Chatzi E, Papadimitriou C, Smyth A. Experimental validation of the Kalman-type filters for online and real-time state and input estimation. Journal of Control, 2015, 23(15): 1-26, disclose a load / state joint identification method using the DKF method.

[0100] The DKF method was used for joint load / state identification, and the results are as follows: Figure 10 , Figure 11 As shown, Figure 10 A comparison chart of the actual and identified values ​​of the load; Figure 11 The actual displacement and velocity values ​​are identified for a system with 19 degrees of freedom; (a) actual displacement value and identified value; (b) actual velocity value and identified value. It can be seen that the identified load and displacement also exhibit significant low-frequency drift. This is because the GDF and DKF methods rely solely on the acceleration measurement signal, which has intrinsic instability during identification.

[0101] Example 2:

[0102] The external load is a double sine curve, u = 40sin(1.6πt) + 40sin(2.4πt), and other conditions are the same as in Example 1.

[0103] A 10th-order Lelande orthogonal polynomial was used to fit the external load, and the load / state joint identification was performed using the method described in this invention. The results are as follows: Figure 12 , 13 As shown in Figure 14, Figure 12 This is a comparison chart of the actual and identified values ​​of the load in Example 2; Figure 13 These are the actual and identified displacement values ​​for degree 23 in Example 2; Figure 14 The figures above show the actual and identified displacement values ​​for degree 23 in Example 2. It is clear from the figures that the low-frequency drift phenomenon has been greatly reduced, and the identification accuracy has been significantly improved.

[0104] As can be seen from Example 1 and Comparative Example 1, the dual Kalman filtering method based on orthogonal polynomial fitting can solve the problem of low-frequency drift in the traditional Kalman filtering method based solely on least squares. This is because the introduction of orthogonal polynomial fitting increases the mathematical constraints on the load, making the recognition result region stable.

[0105] Comparative Example 2:

[0106] The external load is the double sine function curve in Example 2. The GDF method is used for joint load / state identification, and the identification result is as follows: Figure 15 , 16 As shown in Comparative Example 1, the conclusion is that a significant low-frequency drift phenomenon was observed in the identified load and displacement. Figure 15 This is a comparison chart of the actual and identified values ​​of the load. Figure 16 The results show the true and identified values ​​of displacement and velocity for a degree of freedom of 23. Among them, (a) shows the true and identified values ​​of displacement; and (b) shows the true and identified values ​​of velocity.

[0107] In summary, this invention addresses the low-frequency drift problem in load and displacement identification that easily occurs when traditional Kalman filtering methods based on least squares rely solely on acceleration measurement signals for joint load / state identification. It proposes a dual Kalman filtering method based on orthogonal polynomial fitting, which effectively solves this low-frequency drift problem. Essentially, it introduces a mathematical constraint of orthogonal polynomial fitting into the identification process, stabilizing the identification results. Numerical examples verify the feasibility and accuracy of this method, using Lelande orthogonal polynomials for single-sine and double-sine external load fitting. Furthermore, the effectiveness was verified under different noise levels and different numbers of acceleration measurement signals.

[0108] Impact loads are common in real-world engineering. In addition, curve fitting includes Chebyshev orthogonal polynomials, Gegenbauer orthogonal polynomials, and cubic spline curves. Essentially, the load can still be represented by the product of the basis function and the fitting coefficients. Therefore, the load / state joint identification can be implemented using the method proposed in this invention.

[0109] Unless otherwise specified, all embodiments in this invention are existing technologies or can be implemented using existing technologies. Furthermore, the specific implementation examples described in this invention are merely preferred embodiments and are not intended to limit the scope of this invention. That is, all equivalent changes and modifications made to the content of the claims of this invention should be considered within the technical scope of this invention.

Claims

1. A structural load identification method based on orthogonal polynomial fitting and dual Kalman filtering, characterized in that, Includes the following steps: Step 1: Use orthogonal polynomials to fit the unknown loads applied to the structure; Step 2: Construct the state-space equation system of the fitting coefficients and use the standard Kalman filter to identify the fitting coefficients of the polynomials; Step 3: Construct the state-space equation set of the structural state, and use the Kalman filter method again to identify the structural state; Step 1 includes the following steps: Step 1.1: Using the finite element method, the differential equations of motion for a dynamic system with n degrees of freedom are expressed as follows: （1） In the above formula, M represents the system mass matrix, C represents the system damping matrix (viscous damping) and is a symmetric matrix; K represents the system stiffness matrix. , , The vectors represent the displacement, velocity, and acceleration responses of the nodes on the structure, respectively; u(t) represents the external load excitation vector applied to the structure, and B... u This represents the positional influence matrix of the external load vector; Step 1.2: The state-space method is used to describe the structural dynamic system. The state vector X(t) of the structure is represented by the structural displacement and velocity vectors, i.e. Then the state transfer equation : （2） in, （3） I represents the identity matrix; If only acceleration signals are used as measurement data, then the state observation equation of the structure is: （4） In the above formula, The equation for measuring acceleration response is given, where H is the position influence matrix for measuring acceleration response, and D is the load transfer matrix for measuring acceleration response. （5） Step 1.3: Considering system process noise and measurement noise, set the sampling frequency to 1 / △ t, Formulas (2) and (4) can be written in the following time-discrete form: （6） （7） In the above formula, the subscript k represents the k-th sampling time, x k Let x represent the state vector at time k. k+1 y represents the state vector at k time points; k The equation representing the measured acceleration response at time k; u k This represents the external load excitation vector applied to the structure; w k v k Let G and R represent the system noise and observation noise at time k, respectively. Both are uncorrelated Gaussian white noise, with zero mean and variances G and R, respectively. In the above formula, matrices A and B are respectively: （8） Step 1.4: Based on the theory of orthogonal polynomial fitting approximation curves, the structural load is fitted using orthogonal polynomials. Then: （9） In the above formula, These are orthogonal polynomial basis functions. These are the fitting coefficients; The fitting error is assumed to be Gaussian white noise with a mean of zero and variances of [missing information]. To represent; substituting formula (9) into formulas (6) and (7), the state transfer equation and state observation equation are rewritten in the following polynomial form: （10） （11） Once the orthogonal polynomial type is determined, it can be seen from formulas (10) and (11) that the load identification of the structure is transformed into the identification of the fitting coefficients, and the load value can be calculated by formula (9).

2. The structural load identification method based on orthogonal polynomial fitting and dual Kalman filtering according to claim 1, characterized in that, Step 2 specifically includes the following steps: Step 2.1: Construct the state-space equations for the fitting coefficients: The fitting coefficients are expressed as follows using a random walk model: （12） In the above formula, This represents the fitting coefficient at time k. Represents the fitting coefficient at time k+1. The process noise representing the fitting coefficients is assumed to have a mean of 0 and a variance of . The probability distribution of Gaussian white noise; Then the state observation equation (11) is rewritten as: （13） Step 2.2: Use formulas (12) and (13) to compose the fitting coefficients. The state-space equations are obtained, and then the standard Kalman filter is used to identify the fitting coefficients. The standard Kalman filter equations include a state prediction step and a time update step, as detailed below: State prediction step: （14） （15） （16） In the above formula, Represents the fitting coefficients at time k. The prior value, Represents the time at time k-1 The posterior value, This indicates the fit coefficients at time k-1. The covariance of prior values, This indicates the fit coefficients at time k-1. The covariance of the posterior values; Represents the fitting coefficients at time k. The gain matrix; Time update steps: （17） （18） Represents time k The posterior value; Solving the above formula yields the load fitting coefficients.

3. The structural load identification method based on orthogonal polynomial fitting and dual Kalman filtering according to claim 2, characterized in that, As can be seen from formulas (14)-(18), the solution of the load fitting coefficients also requires structural state values. Therefore, in step 3, the Kalman filter method is used again to identify the structural state, and its filtering equation is as follows: State prediction step: （19） （20） （21） in, Represents time k The prior value, Represents the time at time k-1 The posterior value, This indicates the fit coefficients at time k-1. covariance value, Represents the fitting coefficients at time k. The gain matrix; Time update steps: （22） （23） Given initial values: , , , R , By combining the above dual Kalman filtering method (14)-(22), the load fitting coefficients and structural state values ​​x can be continuously obtained. Then, the structural load values ​​can be obtained by referring to the following formula. （24）。

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