Suspension bridge structure system identification method based on observability and state estimation

Abstract

The application belongs to the technical field of bridge structure health monitoring, and discloses a suspension bridge structure system identification method based on observability and state estimation and a high-efficiency algorithm thereof; aiming at different component characteristics of the suspension bridge, the identifiable problem of the unit parameters is converted into the solvable problem of the mathematical equation by establishing the observability state equation of the suspension bridge, and the Jacobian matrix is constructed in combination with the high-efficiency algorithm (OM-SE) of state estimation to realize the explicit mapping between the measured data and the state variables, and the precision and the calculation efficiency of the parameter identification are improved. The whole stiffness matrix is established by the parabolic main cable unit, the Euler-Bernoulli beam unit and the direct stiffness method, the parameter inversion is carried out by using the double-layer iteration process, the method is suitable for the identification of the bending stiffness of the main beam and the axial stiffness of the cable of the long-span suspension bridge, and has been verified by the scale model test and the actual suspension bridge numerical case, the identification error is controlled within 5%, and the calculation efficiency is improved by more than 70%.

Description

Technical Field

[0001] This invention relates to the field of suspension bridge health monitoring and structural system identification technology, specifically to a method for identifying suspension bridge structural systems based on observability and state estimation, applicable to parameter inversion, health status assessment, and maintenance decisions for bridges with complex cable-stayed systems. Background Technology

[0002] As a key component of transportation infrastructure, bridges are typically designed for a service life of 50-120 years. However, during their service life, they are subject to the combined effects of natural and human factors such as wind loads, seismic forces, temperature changes, traffic loads, and overloading, leading to structural performance degradation, such as stiffness loss and bending deformation. As of 2025, my country had over 1.07 million highway bridges, with approximately 15% exhibiting structural defects or damage. The direct economic losses caused by bridge defects amount to billions of yuan annually.

[0003] Structural health monitoring (SHM) is a crucial means of ensuring the safe operation of bridges, and structural system identification (SSI), as the core technology of SHM, requires analyzing structural response data to infer unknown characteristic parameters (such as stiffness and damping) to provide a basis for health assessment. However, traditional methods for identifying the structural system of suspension bridges have the following significant problems: (1) Lack of identifiability determination: The "identifiability" of parameters cannot be determined at the structural unit level. When facing large and complex suspension bridges, the parameter identification process is cumbersome and inefficient, and it is easy to encounter situations where the solution is not unique or there is no solution.

[0004] (2) Low computational efficiency: Traditional parametric methods (such as the finite element model correction method) rely on a large number of iterative calculations and symbolic operations. Especially for complex structures such as suspension bridges containing parabolic line elements and Euler-Bernoulli beam elements, the computational cost is extremely high and it is difficult to meet the real-time monitoring needs of engineering.

[0005] (3) Insufficient accuracy and robustness: improper simplification of geometric nonlinearity (such as the parabolic shape of the main cable), and large identification error due to measurement noise and initial internal force deviation; some methods can only identify overall parameters and cannot accurately locate unit-level stiffness damage.

[0006] (4) Incomplete consideration of parameter sensitivity: The impact of key design parameters of suspension bridges (such as the sag-to-span ratio) on recognition accuracy was not systematically analyzed. Under different sag-to-span ratio conditions, the recognition results were unstable. Summary of the Invention

[0007] In view of the above-mentioned defects or improvement needs of the existing technology, the present invention provides a method for identifying suspension bridge structure systems based on observability and state estimation, thereby solving the technical problems of lack of identifiability determination, low computational efficiency and insufficient accuracy of traditional suspension bridge structure systems.

[0008] To achieve the above objectives, according to one aspect of the present invention, a method for identifying suspension bridge structural systems based on observability and state estimation is provided, comprising the following steps: S1: A structural system with a dedicated stiffness matrix is ​​constructed to address the characteristics of different components of the suspension bridge. The main cable adopts parabolic cable elements, the main beam and bridge tower adopt Euler-Bernoulli beam elements, and the suspension cables adopt truss elements. S2: Based on the stiffness matrices of the parabolic cable element, Euler-Bernoulli beam element, and truss element in step S1, the overall stiffness matrix is ​​assembled using the direct stiffness method. S3: Based on step S2, the stiffness matrix of each element in the local coordinate system is transformed to the global coordinate system, and assembled into a global stiffness matrix according to the nodal degrees of freedom, forming the force-displacement equilibrium equation of the suspension bridge structure. S4: By performing matrix block transformation and null space matrix analysis on the force-displacement equilibrium equation in step S3, the known and unknown quantities are separated to construct the observable state equation of the suspension bridge; at the same time, the null space matrix is ​​calculated, and the observable stiffness parameters are determined by judging the variables corresponding to the all-zero rows in the null space matrix. S5: Based on step S4, construct the Jacobian matrix, establish an explicit mapping relationship between the system observation data and the state variables and target variables, and determine the observability of the state variables through the null space matrix of the Jacobian matrix; S6: Based on step S5, the estimated value of the stiffness parameter is updated using a weighted least squares iterative algorithm until the preset convergence accuracy is met; S7: Based on the preset convergence accuracy in step S6, output the identification results, including the axial stiffness of the main cable and suspenders and the bending stiffness of the main beam and bridge tower, and evaluate the identification accuracy and reliability.

[0009] Preferably, the main cable in step S1 adopts a parabolic cable unit, the basic assumptions of which include: (1) The cable is made of homogeneous material, has a uniform cross section, and obeys Hooke's Law; (2) Neglecting the bending stiffness of the cable; (3) The sag-to-span ratio of the main cable is less than 1:8; (4) Neglect the cross-sectional changes caused by the tension of the cable; (5) The weight of the cable unit is distributed along the chord length and is perpendicular to the chord direction. The chord component of the cable weight is ignored.

[0010] Preferably, the equation of the main cable parabola in step S1 is:

[0011] in, Sagging degree x , l x , l z This represents the positional data of each point within a parabolic cable element.

[0012] Preferably, in step S1, the main beam and bridge tower adopt Euler-Bernoulli beam elements, considering bending-shear coupling, including axial stiffness and bending stiffness correlation coefficients. The stiffness matrix in the local coordinate system is:

[0013] in, L Let the length be the beam element length. A For cross-sectional area, E It is the elastic modulus.

[0014] Preferably, the slings in step S1 are constructed using truss elements, considering only axial stiffness. The stiffness matrix in the local coordinate system is:

[0015] in, L The length of the truss unit. A For cross-sectional area, E It is the elastic modulus.

[0016] Preferably, step S4 specifically includes the following steps: S41: Transform the identifiability of element parameters into the solvability of mathematical equations. Determine the parameters are fully observable when the null dimension of the stiffness matrix is ​​0, and observable only when the null dimension is non-zero. Establish the basic equations of observability state for suspension bridges. S42: Parameter identification is achieved by performing a block transformation on the basic equations in step S41; S43: Transform the equation identified in step S42 so that the unknown variables exist on the left side of the equation and the known quantities are separated to the right side of the equation; S44: Determine whether the equation transformed in step S43 has a solution using the null space matrix, i.e.: If the null space dimension of the stiffness matrix is ​​zero, then the observability equilibrium equations have a unique solution, corresponding to the complete observability of the structural parameters; conversely, when the stiffness matrix has a non-zero null space, its dimension reflects the number of degrees of freedom of the system.

[0017] Preferably, step S5 specifically includes the following steps: S51: An explicit mapping relationship between the system's observed data and the state and target variables was established using the Jacobian matrix. By solving the following weighted least squares formula, the most likely value of the state variable x was found, and its state estimation formula is as follows:

[0018] Where x is the state variable vector, z is the measurement vector, and R is the variance-covariance matrix. Indicates the error related to measurement. g This is a mapping relationship; S52: In step S51, there is a mapping relationship between the measurement vector z and the state variable vector x, and the mapping formula is as follows:

[0019] Where x is the state variable vector and z is the measurement vector. g For mapping relationship Indicates the error related to measurement; S53: Determine the solution for each iteration by iteratively solving the following system of linear equations. The optimal solution for the state variables;

[0020] in, For iteration counter, Indicates the first State variable estimates for the next iteration State estimation The measurement Jacobian matrix at the location, For state estimation The measurement vector predicted by the system model; The condition for a Jacobian matrix to be full rank is that the matrix Reversibility allows us to determine whether a system is observable or unobservable.

[0021] Preferably, the stiffness parameters in step S6 include the axial stiffness of the main cable and suspenders, as well as the bending stiffness of the main beam and main tower.

[0022] In summary, the technical solutions conceived by this invention have the following beneficial effects compared with the prior art: 1) The method of this invention directly realizes the identification of element-level parameters through the null space of stiffness matrix and Jacobian matrix, avoiding the "blind iteration" of traditional methods; 2) The OM-SE method replaces the symbolic recursion of the OM method with a numerical Jacobian matrix, which reduces the computation time by 74% compared with the OM method and by 85% compared with the finite element correction method, thus improving computational efficiency. 3) Supports multi-source data fusion, adapting to complex loads and environmental conditions; 4) The main cable adopts a parabolic model to consider geometric nonlinearity, which reduces the identification error by 37% compared with the linear modeling method; 5) Supports multi-condition loading and measurement point optimization. It has passed scale tests and actual bridge numerical verification and can be directly applied to the health monitoring of suspension bridges. 6) This invention can be implemented through MATLAB programming, and combined with finite element software such as MIDAS for data preprocessing and result verification; the system supports user-defined measurement sets, load conditions and accuracy requirements, and has good engineering applicability. Attached Figure Description

[0023] Figure 1 This is a flowchart of a method for identifying suspension bridge structural systems.

[0024] Figure 2 It is a cable element in parabolic form.

[0025] Figure 3 This is a finite element model diagram of the bridge.

[0026] Figure 4 This is a diagram showing the element numbering of a suspension bridge node.

[0027] Figure 5 This is a comparison curve of the axial stiffness of the main cable.

[0028] Figure 6 This is a comparison curve of the bending stiffness of the main beam.

[0029] Figure 7 This is a graph showing the influence of different sag-to-span ratios on the identification of the axial stiffness of the main cable in the OM method.

[0030] Figure 8 This is a graph showing the influence of different sag-span ratios on the identification of the axial stiffness of the main cable in the OM-SE method.

[0031] Figure 9 The graph shows the influence of different sag-span ratios on the identification of bending stiffness in the OM method.

[0032] Figure 10 The graph shows the influence of different sag-span ratios on the identification of bending stiffness in the OM-SE method. Detailed Implementation

[0033] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0034] Please see Figure 1 The present invention provides a method for identifying suspension bridge structural systems based on observability and state estimation, comprising the following steps: S1: A structural system with a dedicated stiffness matrix is ​​constructed to address the characteristics of different components of the suspension bridge. The main cable adopts parabolic cable elements, the main beam and bridge tower adopt Euler-Bernoulli beam elements, and the suspension cables adopt truss elements. S2: Based on the stiffness matrices of the parabolic cable element, Euler-Bernoulli beam element, and truss element in step S1, the overall stiffness matrix is ​​assembled using the direct stiffness method. S3: Based on step S2, the stiffness matrix of each element in the local coordinate system is transformed to the global coordinate system, and assembled into a global stiffness matrix according to the nodal degrees of freedom, forming the force-displacement equilibrium equation of the suspension bridge structure. S4: By performing matrix block transformation and null space matrix analysis on the force-displacement equilibrium equation in step S3, the known and unknown quantities are separated to construct the observable state equation of the suspension bridge; at the same time, the null space matrix is ​​calculated, and the observable stiffness parameters are determined by judging the variables corresponding to the all-zero rows in the null space matrix. S5: Based on step S4, construct the Jacobian matrix, establish an explicit mapping relationship between the system observation data and the state variables and target variables, and determine the observability of the state variables through the null space matrix of the Jacobian matrix; S6: Based on step S5, the estimated value of the stiffness parameter is updated using a weighted least squares iterative algorithm until the preset convergence accuracy is met; S7: Based on the preset convergence accuracy in step S6, output the identification results, including the axial stiffness of the main cable and suspenders and the bending stiffness of the main beam and bridge tower, and evaluate the identification accuracy and reliability.

[0035] Specifically, this invention proposes an observability structural system identification method for suspension bridges, which transforms the problem of element parameter identifiability into a problem of mathematical equation solvability. Through a rigorous parameterization process and optimization algorithm, it achieves accurate and rapid identification of stiffness parameters.

[0036] I. Observability Method: Step S4 specifically includes the following steps: S41: Observability method: Transform the identifiability of element parameters into the solvability of mathematical equations. By using the null space of the stiffness matrix, it is determined that the parameters are fully observable when the null space dimension is 0, and only the parameters corresponding to the zero row are observable when it is non-zero. The basic state equations for the observability of the suspension bridge are then established.

[0037] Taking structural mechanics analysis as an example, the force-displacement equilibrium relationship established based on the theory of material strength can be expressed in the following basic equation form:

[0038] in, Here is the stiffness matrix; It is a displacement column vector; It is a force column vector.

[0039] S42: Parameter identification is achieved by performing a block transformation on the basic equations in step S41:

[0040] in, Stiffness matrix The block matrix, the size of which is determined by... Sure; Displacement column vector The block vector; Vectors were listed for the force. The block matrix is ​​given by Table 1 and Table 0 below, where known and unknown quantities are represented respectively.

[0041] S43: Transform the equation identified in step S42 so that the unknown variables exist on the left side of the equation and the known quantities are separated to the right side of the equation; Transform Equation 1-2 so that the unknown variable exists on the left side of the equation, and the known quantity is separated to the right side:

[0042] in, It is a zero matrix, and its size corresponds to the corresponding element in the equation. It is an identity matrix, and its size corresponds to the corresponding element in the equation.

[0043]

[0044] in, for The null space matrix.

[0045] S44: Determine whether the equation transformed in step S43 has a solution using the null space matrix: That is, whether equation 1-3 has a solution is determined by whether equation 1-4 holds. If the equation has a solution, its general solution has the following form:

[0046] in, This is a particular solution to equation 1-3; This is the general solution of equation 1-3.

[0047] null space matrix of stiffness matrix The properties of the stiffness matrix determine the nature of the solutions to the structural equilibrium equations. If the null space dimension of the stiffness matrix is ​​zero, the equilibrium equations have a unique solution, corresponding to completely observable structural parameters. Conversely, when the stiffness matrix has a non-zero null space, its dimension reflects the number of degrees of freedom of the system.

[0048] The stiffness matrix method, a fundamental approach to structural analysis, is based on the coupling of nodal force-displacement relationships and material constitutive equations, and is a core algorithm in finite element analysis software. For a two-dimensional beam element structural system, it is assumed that it contains... If there are discrete nodes, their mechanical behavior can be described by the following equilibrium equation:

[0049] in, Here is the stiffness matrix; It is a displacement column vector. It is a force column vector.

[0050] In the analytical framework of the classical stiffness matrix method, material physical parameters (i.e., the characteristic parameters corresponding to beam elements) are usually preset. The known quantity is denoted by , while the variable to be determined is limited to the displacement vector. and load vector middle

[55] Once the boundary constraints and external load distribution of the structure are determined, the displacement vector can be... and load vector Some elements , Treating it as a known condition, the remaining components and This constitutes the set of unknown variables to be solved. Based on this, by performing appropriate matrix operations on equations 1-2 and 1-3, the original equilibrium equation 1-6 can be reformulated as equation 1-7 in the following form:

[0051] in, Let and be column vectors containing unknowns; Tong is a column vector containing observations; It is a coefficient matrix.

[0052] II. The state estimation method, at its core, establishes an explicit mapping relationship between system observation data and state and target variables using the Jacobian matrix. The goal of state estimation is to find the state variables by solving the following weighted least squares problem. The most likely value.

[0053] Step S5 specifically includes the following steps: S51: An explicit mapping relationship between the system's observed data and the state and target variables was established using the Jacobian matrix. By solving the following weighted least squares formula, the most likely value of the state variable x was found, and its state estimation formula is as follows:

[0054] Where x is the state variable vector, z is the measurement vector, and R is the variance-covariance matrix. Indicates the error related to measurement. g This is a mapping relationship; Against this backdrop The optimal solution corresponding to Formula 2-1.

[0055] S52: In step S51, there is a mapping relationship between the measurement vector z and the state variable vector x, and the mapping formula is as follows:

[0056] Where x is the state variable vector and z is the measurement vector. g For mapping relationship Indicates the error related to measurement; This indicates the error associated with the measurement.

[0057] These errors are typically assumed to follow a Gaussian distribution with a mean of zero, i.e. It also has a variance-covariance matrix. .

[0058] To solve the problem in Equation 2-1, the normal equation method is used.

[0059] S53: Determine the solution for each iteration by iteratively solving the following system of linear equations. The optimal solution for the state variables:

[0060] in, For iteration counter, Indicates the first State variable estimates for the next iteration State estimation The measurement Jacobian matrix at the location, For state estimation The measurement vector predicted by the system model.

[0061] According to Equation 2-3, the theoretical sufficient condition for the existence of a unique solution to Equation 2-1 for state estimation is that the system is deterministic and consistent, i.e., the Jacobian matrix is... Must be of full rank, rank is The condition for a Jacobian matrix to be full rank is that the matrix... Invertibility determines whether a system is observable or unobservable. Measuring the Jacobian matrix plays a crucial role in the observability of a system.

[0062] Furthermore, even if Formula 2-1 is at any point Even after linearization, the Jacobian matrix still maintains the structural relationship between measurements and state variables, and its differential form is:

[0063] in, Let be the differential measurement residual vector. For the differential change of the system state, This represents the differential change of the error.

[0064] III. State Estimation Model From the perspective of state estimation, various types of measurements related to the aforementioned variables must be considered. In this study, all possible measurement vectors in the zero-structure system can be represented as:

[0065] in, This indicates the measurement of external forces that are not constrained by boundary conditions. These represent displacement and rotation measurements constrained by boundary conditions. Typically, these values ​​are zero, but may be related to uncertainties in the structural foundation. This indicates displacement and rotation measurements that are not constrained by boundary conditions. Measurement of external forces constrained by boundary conditions.

[0066] Equations 1-7 only give the relationship between the state variables and the other variables in the structural system, without considering their relationship with the mechanical parameter vector. The relationship between them means that these mechanical parameters will be incorporated into the state estimation problem. Therefore, Equation 2-1 is changed to

[0067] Among them, parameters These constitute the decision variables in the optimization / estimation problem. In state estimation, this problem is often referred to as calibration. By differentiating Equation 1-2, we can obtain:

[0068] After transforming Equations 1-2 and 1-3, Equation 2-7 can be written as Equation 2-8:

[0069] From Equation 2-8, the relationships between unknown quantities and known quantities, as well as between unknown quantities and mechanical parameters, can be obtained as follows:

[0070] From Equation 2-4, we know that the measurement vector For state variables The differential is the Jacobian matrix. Therefore, the Jacobian matrix The format is as follows:

[0071] At the same time, as shown in Formula 2-4, the state variable The general solution is:

[0072] in, Jacobian matrix The null space matrix, The parameter vector. The null space matrix. When the inequality is zero, equation 2-12 has a unique solution. Therefore, the well-posedness of equation 2-6 depends on... Is it zero? Analyzing line by line, equation 2-12 can be expressed as:

[0073] for ,when Time state variables (or state variable) It is observable; otherwise, it is unobservable.

[0074] IV. Derivation of Bridge Element Model and Stiffness Matrix To address the different characteristics of suspension bridge components, a structural system with a dedicated stiffness matrix is ​​constructed. The main cable adopts parabolic cable elements, the main girder and bridge tower adopt Euler-Bernoulli beam elements, and the suspension cables adopt truss elements. The main cable adopts a parabolic cable unit, i.e., a parabolic model, whose basic assumptions include: (1) The cable is made of homogeneous material, has a uniform cross section, and obeys Hooke's Law; (2) Neglecting the bending stiffness of the cable; (3) The sag-to-span ratio of the main cable is less than 1:8; (4) Neglect the cross-sectional changes caused by the tension of the cable; (5) The weight of the cable unit is distributed along the chord length and is perpendicular to the chord direction. The chord component of the cable weight is ignored.

[0075] The equation of the main cable parabola is:

[0076] in, Sagging degree x , lx , l z This represents the positional data of each point within a parabolic cable element.

[0077] The main cable stiffness matrix (local coordinate system) is as follows: Parabolic cable The stiffness matrix at the node can be written in the following form:

[0078] in: The weight per unit stress-free length Stress-free length , For axial stiffness, H It is a horizontal tensile force.

[0079] The suspenders are constructed using truss elements. Considering only axial stiffness, the stiffness matrix in the local coordinate system is:

[0080] in, L The length of the truss unit. A For cross-sectional area, E It is the elastic modulus.

[0081] Main girder and bridge tower: Euler-Bernoulli beam elements are used, considering bending-shear coupling, including axial stiffness and bending stiffness correlation coefficients. The stiffness matrix in the local coordinate system is:

[0082] in, L Let the length be the beam element length. A For cross-sectional area, E It is the elastic modulus.

[0083] V. Assembly of the overall stiffness matrix Based on the stiffness matrices of the parabolic cable element, Euler-Bernoulli beam element, and truss element, the overall stiffness matrix is ​​assembled using the direct stiffness method. The process involves transforming the stiffness matrices of each element to the global coordinate system, superimposing them according to the nodal degrees of freedom to form the force-displacement equilibrium equations of the overall suspension bridge structure. Then, the element stiffness matrices in the local coordinate system are transformed to the global coordinate system of the suspension bridge, and the transformed element stiffness matrices are sequentially superimposed onto the stiffness matrix according to node numbers to fill the gaps. The overall stiffness matrix of the suspension bridge is obtained as shown in the following equation.

[0084] Where U is the nodal displacement vector, F is the nodal load vector, and K is the overall stiffness.

[0085] Based on the boundary constraints, nodal displacement components, and applied external loads of the structural system, the force-displacement equilibrium equations of the structure are established and solved. The resulting solution vector is the set of observable variables of the system. (Due to the matrix...) If the value is too large, it is represented by blocks. After separating the unknown and known quantities, a matrix is ​​obtained. sum matrix Observability is determined by solving the null space matrix V of [B]. If all rows of [V] are 0, then the corresponding variable is observable.

[0086] To address the issues of large computational load and cumbersome iteration in the OM method, a high-efficiency algorithm based on state estimation (OM-SE method) is introduced. This method establishes an explicit mapping between measurement data and state variable vectors through the Jacobian matrix and establishes its mathematical form through the stiffness method.

[0087] Based on the observability of state variables determined by the null space matrix of the Jacobian matrix, the estimated values ​​of stiffness parameters are updated using a weighted least squares iterative algorithm until the preset convergence accuracy is met. The stiffness parameters include the axial stiffness of the main cable and suspenders, as well as the bending stiffness of the main beam and main tower.

[0088] Based on the preset convergence accuracy, the identification results are output, including the axial stiffness of the main cable and suspenders and the bending stiffness of the main beam and bridge tower, and the identification accuracy and reliability are evaluated.

[0089] Another aspect of the present invention provides a high-performance algorithm (OM-SE method) for identifying suspension bridge structural systems based on observability and state estimation, comprising the following steps: 1) Construct an objective function using a state estimation model to minimize the measurement residuals; 2) The higher-order coupling terms of stiffness and displacement are preserved using the differential relationship of the Jacobian matrix; 3) Supports the fusion of data from multiple operating conditions and multiple measurement points to improve recognition accuracy.

[0090] The algorithm exhibits stable recognition accuracy within a sag-to-span ratio range of 1 / 9 to 1 / 12, making it suitable for health monitoring of long-span suspension bridges.

[0091] The main steps of the OM-SE method are: establishing the overall stiffness matrix K using the stiffness matrices of beam elements, cable elements, and rod elements, and forming the equilibrium equations. .in It is a displacement vector. For the load vector; set the initial values ​​of the material parameters. The initial displacements of each node are obtained using the known nodal loads, thus yielding... Constructing by block matrix matrix, Matrix and matrix Calculate the displacement matrix , ;combination , The matrix is ​​then deleting the rows corresponding to the constraints to generate the Jacobian matrix. Calculate the Jacobian matrix using the null space method. The null space matrix, where all rows of the null space matrix are zero, corresponds to the unknown parameters as observable variables; Observation matrix Does the data contain rows where all values ​​are 0? The corresponding unknown parameters are also considered as observable variables. Compare the current number of observables with the previous number. ,if and If they are equal, it means all unknowns have been observed, and the process ends. If they are not equal, the newly identified variables need to be added to the known state variables, and state estimation is used based on the computation set. The minimum error between the observed and the measured set, for observable parameters. and Perform calibration and update the initial values ​​until the residuals meet the accuracy requirements, indicating that all unknowns have been observed.

[0092] Taking a literature study on a single-span simply supported ground-anchored suspension bridge system as an example: 1) This single-span simply supported ground-anchored suspension bridge system has a main span of 1038 meters and exhibits the following typical characteristics: the main cable system has a rise-to-span ratio of 1 / 9, and the lateral spacing gradually changes from 23m at the anchorage to a standard spacing of 16m at mid-span; it employs a flat, streamlined steel box girder with excellent aerodynamic stability. A numerical model of this suspension bridge was constructed based on the Midas Civil 2019 platform. The parameterized characteristics of the model are shown in Table 1. 2) To verify the accuracy of the suspension bridge structural system identification method, a numerical model was used to analyze the parameter identification error of the suspension bridge structure under the condition of reduced member stiffness. The bridge's unit group includes the main girder, main cable, and suspension cable units. The sections and units of the components are numbered as follows: Figure 2-4 The mechanical parameters of each component should be referenced in the table and kept consistent. Component stiffness reduction condition: The bending stiffness of main beam units #9 and #24 decreases, and their bending stiffness is adjusted to... and The axial stiffness of cable element #40 decreased to its original value. According to the node number, a concentrated load is applied at node 10 of the main beam. These monitoring parameters are measurable variables. The actual mechanical parameters of each component are calculated using the systematic identification method of suspension bridge structure. The vertical displacement and rotation angle of the main beam node and the horizontal and vertical displacement of the main cable are selected as the main monitoring parameters. The systematic errors in the actual engineering measurement are not considered. The measurement results are shown in Tables 2 and 3.

[0093]

[0094]

[0095]

[0096] 3) The observability technique OM method was used to simulate the finite element model of the suspension bridge using MATLAB. A single-beam model was used to simulate the main beam and main tower, a parabolic model to simulate the cables, and rod elements to simulate the suspenders and main cables. First, the overall stiffness matrix was initialized. Then, the local stiffness matrices were systematically integrated into the corresponding positions of the global matrix using element positioning matrices, ultimately forming the complete structural stiffness matrix as follows:

[0097] The intrinsic elastic modulus of the suspension cable material. To account for the equivalent bulk density of sag effect, The stress-free length of the suspension cable. The angle of inclination between the cable axis and the horizontal plane. This represents the initial stress state of the suspension cable.

[0098] 4) Based on the position and mechanical parameters of each component, the overall stiffness matrix of the suspension bridge is obtained. Using a MATLAB-based suspension bridge structural system identification program, the stiffness matrix of the structure is embedded into this program to sequentially obtain the matrix. sum matrix And calculate the matrix null space matrix Determine the measurable unknown quantity and the number of iterations.

[0099] 5) Perform the OM-SE method to obtain the position and mechanical parameters of each component, similar to the OM method, to obtain the overall stiffness matrix of the suspension bridge. Then, use a high-efficiency performance identification program for suspension bridge structures written in MATLAB to embed the stiffness matrix of the structure into it. The matrices are obtained sequentially. sum matrix And calculate the Jacobian matrix. null space matrix Determine the measurable unknown quantity and the number of iterations.

[0100] 6) Both the OM method based on observability techniques and the OM-SE method based on state estimation can obtain complete data. and A comparative analysis of the observed data showed that for axial stiffness EA, the OM method identified an average absolute error of 5.72%; the OM-SE method identified an average absolute error of 4.12%; and the literature reported an axial stiffness identification error of 4.92%. This indicates that the OM-SE method is superior in calculating the axial stiffness of cables, and also superior in calculating the bending stiffness of main beams. EI The OM method identified an axial stiffness with an average absolute error of 7.16%; the OM-SE method identified an axial stiffness with an average absolute error of 4.54%; and the literature reported an axial stiffness identification error of 5.51%. This indicates that the OM-SE method is superior in calculating the bending stiffness of the main beam. Figure 5-10 As shown.

[0101] 7) The numerical case of the suspension bridge used the Observability Method (OM) and the Observability Method Based on State Estimation (OM-SE) for parameter identification, with computation times of 36 min and 8 min respectively. The OM method avoids the iterative process of FEMU through symbolic computation and recursive analysis, which significantly shortens the computation time. The OM-SE method replaces the symbolic recursion of OM with a numerical Jacobian matrix, which reduces the complexity of matrix decomposition and significantly improves efficiency.

[0102] 8) Through four types of sag-span ratios ( ), , , , The accuracy of suspension bridge identification methods was compared. For principal axial stiffness, the parameter identification deviation of the OM method showed a more significant upward trend with the increase of the rise-to-span ratio, with its increase being significantly greater than that of the OM-SE method. This indicates that the OM-SE method, through the introduction of state estimation technology, effectively reduces the impact of the rise-to-span ratio on the parameter identification accuracy, exhibiting better robustness. For beam bending stiffness, when the rise-to-span ratio increases, the identification deviation growth rate of the traditional OM method is about 2% higher than that of the OM-SE method. This quantitative difference demonstrates that the OM-SE method, based on state estimation improvement, can effectively suppress the interference of rise-to-span ratio changes on the parameter identification process, exhibiting more stable identification performance.

[0103] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for identifying suspension bridge structural systems based on observability and state estimation, characterized in that, Includes the following steps: S1: A structural system with a dedicated stiffness matrix is ​​constructed to address the characteristics of different components of the suspension bridge. The main cable adopts parabolic cable elements, the main beam and bridge tower adopt Euler-Bernoulli beam elements, and the suspension cables adopt truss elements. S2: Based on the stiffness matrices of the parabolic cable element, Euler-Bernoulli beam element, and truss element in step S1, the overall stiffness matrix is ​​assembled using the direct stiffness method. S3: Based on step S2, the stiffness matrix of each element in the local coordinate system is transformed to the global coordinate system, and assembled into a global stiffness matrix according to the nodal degrees of freedom, forming the force-displacement equilibrium equation of the suspension bridge structure. S4: By performing matrix block transformation and null space matrix analysis on the force-displacement equilibrium equation in step S3, the known and unknown quantities are separated to construct the observable state equation of the suspension bridge; at the same time, the null space matrix is ​​calculated, and the observable stiffness parameters are determined by judging the variables corresponding to the all-zero rows in the null space matrix. S5: Based on step S4, construct the Jacobian matrix, establish an explicit mapping relationship between the system observation data and the state variables and target variables, and determine the observability of the state variables through the null space matrix of the Jacobian matrix; S6: Based on step S5, the estimated value of the stiffness parameter is updated using a weighted least squares iterative algorithm until the preset convergence accuracy is met; S7: Based on the preset convergence accuracy in step S6, output the recognition result and evaluate the recognition accuracy and reliability.

2. The method for identifying suspension bridge structural systems based on observability and state estimation as described in claim 1, characterized in that, In step S1, the main cable adopts a parabolic cable unit, the basic assumptions of which include: (1) the cable material is homogeneous, with equal cross-section and conforms to Hooke's law; (2) the bending stiffness of the cable is ignored; (3) the sag-to-span ratio of the main cable is less than 1:8; (4) the cross-sectional changes caused by the tension of the cable are ignored; (5) the self-weight of the cable unit is distributed along the chord length, and the direction is perpendicular to the chord direction, ignoring the chord component of the cable self-weight.

3. The method for identifying suspension bridge structural systems based on observability and state estimation as described in claim 2, characterized in that, The equation of the parabola of the main cable in step S1 is: in, Sagging degree x , l x , l z This represents the positional data of each point within a parabolic cable element.

4. The method for identifying suspension bridge structural systems based on observability and state estimation as described in claim 1, characterized in that, In step S1, the main girder and bridge tower adopt Euler-Bernoulli beam elements, considering bending-shear coupling, including axial stiffness and bending stiffness correlation coefficients. The stiffness matrix in the local coordinate system is: in, L Let the length be the beam element length. A For cross-sectional area, E It is the elastic modulus.

5. The method for identifying suspension bridge structural systems based on observability and state estimation as described in claim 1, characterized in that, In step S1, the slings are constructed using truss elements, considering only axial stiffness. The stiffness matrix in the local coordinate system is: in, L The length of the truss unit. A For cross-sectional area, E It is the elastic modulus.

6. The method for identifying suspension bridge structural systems based on observability and state estimation as described in claim 1, characterized in that, Step S4 specifically includes the following steps: S41: Transform the identifiability of element parameters into the solvability of mathematical equations. Determine the parameters are fully observable when the null dimension of the stiffness matrix is ​​0, and observable only when the null dimension is non-zero. Establish the basic equations of observability state for suspension bridges. S42: Parameter identification is achieved by performing a block transformation on the basic equations in step S41; S43: Transform the equation identified in step S42 so that the unknown variables exist on the left side of the equation and the known quantities are separated to the right side of the equation; S44: Determine whether the equation transformed in step S43 has a solution using the null space matrix, i.e.: If the null space dimension of the stiffness matrix is ​​zero, then the observability equilibrium equations have a unique solution, corresponding to the complete observability of the structural parameters; conversely, when the stiffness matrix has a non-zero null space, its dimension reflects the number of degrees of freedom of the system.

7. The method for identifying suspension bridge structural systems based on observability and state estimation as described in claim 1, characterized in that, Step S5 specifically includes the following steps: S51: An explicit mapping relationship between the system's observed data and the state and target variables was established using the Jacobian matrix. By solving the following weighted least squares formula, the most likely value of the state variable x was found, and its state estimation formula is as follows: Where x is the state variable vector, z is the measurement vector, and R is the variance-covariance matrix. Indicates the error related to measurement. g This is a mapping relationship; S52: In step S51, there is a mapping relationship between the measurement vector z and the state variable vector x, and the mapping formula is as follows: Where x is the state variable vector and z is the measurement vector. g For mapping relationship Indicates the error related to measurement; S53: Determine the solution for each iteration by iteratively solving the following system of linear equations. The optimal solution for the state variables; in, For iteration counter, Indicates the first State variable estimates for the next iteration State estimation The measurement Jacobian matrix at the location, For state estimation The measurement vector predicted by the system model; The condition for a Jacobian matrix to be full rank is that the matrix Reversibility allows us to determine whether a system is observable or unobservable.

8. The method for identifying suspension bridge structural systems based on observability and state estimation as described in claim 1, characterized in that, The stiffness parameters in step S6 include the axial stiffness of the main cable and suspenders, as well as the bending stiffness of the main beam and main tower.

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