[go: up one dir, main page]
More Web Proxy on the site http://driver.im/

CN110414150B - Tensor subspace continuous system identification method of bridge time-varying system - Google Patents

Tensor subspace continuous system identification method of bridge time-varying system Download PDF

Info

Publication number
CN110414150B
CN110414150B CN201910697806.3A CN201910697806A CN110414150B CN 110414150 B CN110414150 B CN 110414150B CN 201910697806 A CN201910697806 A CN 201910697806A CN 110414150 B CN110414150 B CN 110414150B
Authority
CN
China
Prior art keywords
matrix
time
tensor
varying
subspace
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Active
Application number
CN201910697806.3A
Other languages
Chinese (zh)
Other versions
CN110414150A (en
Inventor
张二华
单德山
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Sichuan Highway Planning Survey and Design Institute Ltd
Original Assignee
Sichuan Highway Planning Survey and Design Institute Ltd
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Sichuan Highway Planning Survey and Design Institute Ltd filed Critical Sichuan Highway Planning Survey and Design Institute Ltd
Priority to CN201910697806.3A priority Critical patent/CN110414150B/en
Publication of CN110414150A publication Critical patent/CN110414150A/en
Application granted granted Critical
Publication of CN110414150B publication Critical patent/CN110414150B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization

Landscapes

  • Engineering & Computer Science (AREA)
  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Mathematical Physics (AREA)
  • Pure & Applied Mathematics (AREA)
  • Mathematical Analysis (AREA)
  • Mathematical Optimization (AREA)
  • Computational Mathematics (AREA)
  • Data Mining & Analysis (AREA)
  • Theoretical Computer Science (AREA)
  • Computing Systems (AREA)
  • Algebra (AREA)
  • Databases & Information Systems (AREA)
  • Software Systems (AREA)
  • General Engineering & Computer Science (AREA)
  • Complex Calculations (AREA)

Abstract

The invention discloses a tensor subspace continuous system identification method of a bridge time-varying system, which comprises the steps of collecting bridge time-varying signals; dividing the time-varying signal with the signal time course T into N time windows according to the incremental step L to obtain a Hankel matrix of the N time windows; establishing a mathematical model, X, of a matrix solution of a tensor subspace systemk＝UkSkVT+Wk(ii) a Solving a tensor subspace system matrix to obtain the frequency f of the ith order vibration mode of the kth time windowi，kDamping ratio ζi，kAnd all order mode vectors phik. According to the method, the time dimension is introduced, the two-dimensional matrix is expanded to the three-dimensional tensor, the time-varying Hankel tensor is established, the system matrix fast estimation based on tensor operation is realized based on tensor expansion and tensor fast parallel decomposition theories, the calculation efficiency and the identification result precision can be improved, the modal parameter information of the system is easier to identify, and the core technical support is provided for bridge real-time health monitoring.

Description

Tensor subspace continuous system identification method of bridge time-varying system
Technical Field
The invention relates to the field of identification of a time-varying nonlinear system of bridge engineering, in particular to a tensor subspace continuous system identification method of a time-varying system of a bridge.
Background
Traditional bridge detection is generally carried out manual inspection on a bridge structure by a bridge detection engineer, the method is time-consuming and mostly needs to interrupt traffic, detection evaluation results of the method are different from person to person, damage to a hidden part of the structure due to manual inspection is generally difficult to discover, and the requirement for rapid development of modern traffic is difficult to meet. The modern developed structural damage diagnosis technology, namely the intelligent health monitoring technology, is a simple and economic method for evaluating the safety state of the structure.
In recent years, with the vigorous development of structural health diagnosis technology and the wide application of the structural health diagnosis technology to large-span bridges, bridge system identification research based on health monitoring actual measurement signals is in the spotlight. The system identification method based on the measured signals is easy to obtain the modal information of the bridge with high precision under various excitation environments of the bridge, and becomes a key technology for evaluating the state of the bridge. The subspace system identification method is less in human intervention, good in robustness of identification results and increasingly emphasized in bridge structure system identification, but the subspace system identification method also has the defects of low calculation efficiency and incapability of meeting the requirement of continuous identification of long-term states of bridges. In addition, the vibration response of the bridge is caused to present the characteristics of nonlinearity, non-stationarity and strong noise due to the complex environment in which the bridge is located, and the characteristics directly influence the accuracy of the identification result of the subspace system identification method.
Existing subspace system identification methods can be generally categorized into three types: a random subspace system identification method, a recursive random subspace system identification method, and a deterministic random subspace system identification method. The random subspace method generally identifies the modal parameters by using the dynamic response with complete structure, and the identification result is difficult to accurately reflect the real change rule of the modal parameters of the time-varying nonlinear structure; the recursive random subspace system identification method can reflect the time-varying rule of the structural modal parameters, but the identification result has low precision and is difficult to be used for identifying the time-varying modal parameters of large-span complex bridges; the deterministic random subspace identification method is high in accuracy, but the requirement on the completeness of actual bridge excitation and response data is high.
Disclosure of Invention
The invention aims to solve the problems that when the existing subspace system identification method in the prior art is applied to a bridge time-varying system, the calculation efficiency is low, the identification result precision is low, the bridge modal information cannot be accurately reflected, the bridge detection requirement is difficult to meet and the like, and provides a tensor subspace continuous system identification method of the bridge time-varying system.
In order to achieve the above purpose, the invention provides the following technical scheme:
a tensor subspace continuous system identification method of a bridge time-varying system comprises the following steps:
the method comprises the following steps: collecting bridge time-varying signals;
step two: and (3) introducing a time dimension, and expanding the basic theory of subspace identification, namely matrix operation, to three-dimensional tensor calculation. Dividing the time-varying signal with the signal time course T into N time windows according to the increasing step length L (the length of each window is L), and constructing a time-varying three-dimensional Hankel tensor according to the signal acquisition time sequenceX∈R2i×j×N，
For a determination system where both input and output signals are known, the following slice format is represented along the time dimension:
Figure BDA0002148312280000021
for a stochastic system where the input signal is unknown, it is expressed in the time dimension in slice form as follows:
X(：，：，1)＝[YY2i，j，1]，…，X(：，：，k)＝[YY2i，j，k]，…，X(：，：，N)＝[YY2i，j，N]
wherein X (: k) is the Hankel matrix, UU of the k-th time window2i，j，kInput Hankel matrix, YY, for the kth time window2i，j，kOutputting a Hankel matrix for the kth time window;
step three: establishing a mathematical model for tensor subspace system matrix solution;
in the existing matrix subspace system identification theory, firstly, test signals which are not polluted by noise are considered, and a Hankel matrix X is formed, wherein X belongs to RM×NM < N, defining the rank of the matrix (X) ═ r ≦ M,
Figure BDA0002148312280000031
the SVD decomposition is performed as follows:
Figure BDA0002148312280000032
wherein the matrix U belongs to RM×M，V∈RN×NAre respectively as
Figure BDA0002148312280000033
The left singular matrix and the right singular matrix after SVD are orthogonal matrix, Us、UnIs a subset of the number U of bits,
Figure BDA0002148312280000034
Figure BDA0002148312280000035
Vs、Vnis a subset of V and is,
Figure BDA0002148312280000036
Figure BDA0002148312280000037
∑s＝diag(σ1，…，σr)，σi(i ═ 1,2, …, M) is the eigenvalue in the diagonal matrix, σ1≥σ2≥…≥σr≥σr+1＝…＝σM＝0。
From the above formula, one can obtain:
Figure BDA0002148312280000038
this can be inferred from the fact that the non-zero elements in the above formula can be represented as:
Figure BDA0002148312280000039
as can be seen from the above formula,
Figure BDA00021483122800000310
is the left singular vector of
Figure BDA00021483122800000311
The feature vector of (2).
By adding the white noise term psi to X, the measured signal contaminated by noise can be expressed as:
Y＝X+ψ
when the amount of data is sufficiently large, the following relationship exists
Figure BDA0002148312280000041
In the formula, E represents the mathematical expectation,
Figure BDA0002148312280000042
representing the variance of the white noise term psi, IM∈RM×MIs an identity matrix.
It can be deduced that:
Figure BDA0002148312280000043
is provided with
Figure BDA0002148312280000044
Ir∈Rr×rAs an identity matrix, we can obtain:
Figure BDA0002148312280000045
wherein,
Figure BDA0002148312280000046
compared with
Figure BDA0002148312280000047
In a negligibly small amount, S ═ diag (S)1，…，sM)，si> 0, the expression is as follows:
Figure BDA0002148312280000048
according to the theoretical derivation, the left singular matrix U and the diagonal matrix S of the actual measurement signal contain structural time-varying dynamic fingerprint information, and for a time-varying nonlinear system, the left singular matrix U and the diagonal matrix S reflect the time-varying characteristics of the system.
Based on the analysis, from the perspective of tensor decomposition, for the Hankel tensor established in the second step, after tensor decomposition, the left singular matrix and the diagonal matrix should have time-varying characteristics, so that a mathematical model for tensor subspace system matrix solution in the following form can be established.
Xk＝UkSkVT+Wk
In the formula, XkIs a simplified expression form of X (: k), UkLeft singular matrix, S, for the k-th time windowkIs a non-negative diagonal matrix of the kth time window, V is a right singular matrix which does not change with time, WkThe fitted residual sum of squares for the kth time window;
the mathematical model established in the invention is consistent with the tensor parallel factor model in form, and the formula decomposition is not unique as known from the tensor parallel factor decomposition theory. Therefore, for the practical problem to be solved in the present invention, U in the above formula is usedkMake constraints and define Uk＝QkH，Qk∈Rr×r、H∈Rr×rAre all orthogonal matrices, i.e. Uk TUk＝IkSo as to obtain the compound with the structure,
Xk≈QkHSkVT
step four: solving a tensor subspace system matrix;
after a mathematical model of tensor subspace decomposition is established, tensor subspace system matrix estimation is carried out according to the following steps:
inputting: three-dimensional time-varying Hankel tensor and an order upper boundary value RR of a stable graph system.
Step 4.1: for the order n of the steady graph system (n is a positive integer, and 1 < n < RR), the pair
Figure BDA0002148312280000051
Performing principal component analysis to obtain its bearing matrix as the initialization matrix of matrix V, H, S1，…，SNInitializing into an identity matrix;
step 4.2: to pair
Figure BDA0002148312280000052
Performing SVD to obtain
Figure BDA0002148312280000053
Qk＝RkTk TIn the formula, Tk，Δk，RkRespectively a left singular matrix, a diagonal matrix and a right singular matrix after SVD decomposition;
step 4.3: updating H and S by adopting a trilinear alternating least square algorithm1，…，SN，
Figure BDA0002148312280000054
Figure BDA0002148312280000055
Figure BDA0002148312280000056
Wherein,. indicates a Khatri-Rao product, X(i)(i ═ 1,2,3) is Hankel tensorXTensor expansions along three dimensions;
step 4.4: using convergence formula to update H, S1，…，SNCarrying out convergence judgment, returning to the step 4.3 if the judgment result is that the convergence is not realized, otherwise, entering the step 4.5;
step 4.5: according to the updated H, S1，...，SNRecalculating Qk＝RkTk T，Uk＝QkH；
Step 4.6: judging whether the system order n is equal to an upper bound value RR, if n is less than the upper bound value RR, repeating the steps 4.1 to 4.5 until the system order reaches RR, if n is equal to RR, stopping calculation, and outputting Uk，SkCalculation of VFruit;
step five: performing modal parameter identification to obtain the frequency f of the ith order vibration mode of the kth time windowi，kDamping ratio ζi，kAnd all order mode vectors phikRespectively as follows:
Figure BDA0002148312280000061
Figure BDA0002148312280000062
φk＝Ckψk
in the formula, λi，kIs the characteristic value of ith order vibration mode in the kth time window, Re represents the mathematical operation of the real part, Ck＝Γk(1: l, l is the number of rows, gamma)k＝UkSk 1/2，ψkIs the complex eigenvector of the kth time window.
Preferably, in the second step:
for a deterministic system, at time t +1, the new test signal forms the following row vector space:
Figure BDA0002148312280000063
then for the determination system, the Hankel matrix at time t +1 is:
Figure BDA0002148312280000071
for a random system, at time t +1, the new test signal forms a row vector space as follows:
Figure BDA0002148312280000072
for a random system, the Hankel matrix at time t +1 is:
[YY2i，j，t+1]＝[YY′2i，j，tφy(t+1)]
of formula (II) UU'2i，j，t、YY′2i，j，tInput matrix UU respectively representing time t2i，j，tOutput matrix YY2i，j，tThe matrix after the leftmost column is removed.
Preferably, in said step 4.4, define
Figure BDA0002148312280000073
If σ3 old-σ3 new＞εσ3 oldIf yes, judging that the convergence is not achieved, returning to the step 4.3, otherwise, entering the step 4.5, wherein c is a constant, and epsilon is a convergence coefficient.
Preferably, in the fifth step, an observation matrix Γ is definedk＝UkSk 1/2Then, the system state matrix a and the system output matrix C can be calculated by the following formula:
Figure BDA0002148312280000074
Ck＝Γk(1：l，：)
in the formula,
Figure BDA0002148312280000075
is expressed as gammakThe matrix after the first row is removed, kΓis expressed as gammakThe matrix after the last l rows is removed,
Figure BDA0002148312280000076
represents the Moore-Penrose pseudo-inverse of the matrix,
and for the system state matrix A, carrying out eigenvalue decomposition,
Ak＝ψkΛkψk -1
in the formula, ΛkIs a diagonal matrix, Λk＝diag(μi，k)，μi，kIs a complex characteristic value, psi, of the ith order vibrational mode in the kth time windowkIs the complex eigenvector of the kth time window,
obtaining the characteristic value lambda of the ith order vibration mode in the kth time window of the discrete systemi，k，
Figure BDA0002148312280000081
In the formula, Δ t represents a signal sampling time interval.
Compared with the prior art, the invention has the beneficial effects that:
compared with the existing identification method of the bridge subspace system, the identification method of the bridge subspace system is based on the subspace system identification theory of two-dimensional matrix operation, introduces the time dimension, expands the two-dimensional matrix to the three-dimensional tensor, and establishes the time-varying Hankel tensor. Based on tensor expansion and tensor rapid parallel decomposition theory, the system matrix rapid estimation based on tensor operation is realized. And establishing a high-precision rapid tensor subspace system identification theory and method suitable for bridges under different excitation environments by combining a stable graph method. Compared with the existing sliding window subspace system identification method, the tensor subspace system identification method established by the invention has the advantages that the calculation efficiency is higher, the calculation result is less influenced by noise, the stable polar axis in the stable graph is clearer and more distinguishable, and more order modal parameter information of the system is easier to identify. Therefore, by adopting the identification method, the calculation efficiency and the identification result precision can be obviously improved, and more orders of modal parameter information of the system can be easily identified, so that a core technical support is provided for the real-time health monitoring of the bridge.
Description of the drawings:
FIG. 1 is a conventional sliding window subspace system identification process.
Fig. 2 is an identification process of the tensor subspace continuous system identification method of the time-varying bridge system according to the present invention.
Fig. 3 is a vertical layout view of an acceleration sensor of a curved cable-stayed model bridge in embodiment 1 of the invention.
Fig. 4 is a plan layout view of an acceleration sensor of a curved cable-stayed model bridge in embodiment 1 of the invention.
Fig. 5 is a vertical layout view of the acceleration sensor of the curved cable-stayed model bridge in the bridge tower in embodiment 1 of the invention.
Fig. 6 is a schematic diagram of a stable graph generated by using the SSI sliding window method in embodiment 1 of the present invention.
FIG. 7 is a schematic diagram of the stability chart generated by the TSI method in example 1 of the present invention.
Fig. 8 is a graph comparing the frequency measured by the SSI sliding window method and the frequency identified by the TSI method in example 1 of the present invention.
FIG. 9 is a graph comparing the damping ratio measured by the SSI sliding window method and the damping ratio measured by the TSI method in example 1 of the present invention.
Fig. 10 is a comparison graph of the SSI sliding window method and the TSI method for identifying the first 3-order mode at the time when t is 0.0-8.0 s in example 1 of the present invention.
Fig. 11 is a comparison graph of the SSI sliding window method and the TSI method for identifying the first 3 order mode at the time when t is 8.0 to 8.4s in example 1 of the present invention.
Fig. 12 is a comparison graph of the SSI sliding window method and the TSI method for identifying the first 3 order mode at the time when t is 8.4-8.8 s in example 1 of the present invention.
Fig. 13 is a comparison graph of the SSI sliding window method and the TSI method for identifying the first 3 order mode at the time when t is 8.8 to 9.2s in example 1 of the present invention.
Fig. 14 is a comparison graph of the SSI sliding window method and the TSI method for identifying the first 3 order mode at the time when t is 9.2 to 9.6s in example 1 of the present invention.
Fig. 15 is a comparison graph of the SSI sliding window method and the TSI method for identifying the first 3 order mode at the time when t is 9.6 to 10.0s in example 1 of the present invention.
Fig. 16 is a graph comparing the frequency measured by the CSI sliding window method and the frequency identified by the TSI method in example 2 of the present invention.
Fig. 17 is a graph comparing the damping ratio measured by the CSI sliding window method and the frequency identified by the TSI method in example 2 of the present invention.
Fig. 18 is a comparison graph of the CSI sliding window method and the TSI method for identifying the first 3 order mode types at the time t is 1-6 s in example 2 of the present invention.
Fig. 19 is a comparison graph of the CSI sliding window method and the TSI method for identifying the first 3 order mode at the time t is 7-12 s in example 2 of the present invention.
Fig. 20 is a comparison graph of the CSI sliding window method and the TSI method for identifying the first 3 order mode at the time t is 13-18 s in example 2 of the present invention.
Fig. 21 is a comparison graph of the CSI sliding window method and the TSI method for identifying the first 3 order mode at the time t is 19-24 s in example 2 of the present invention.
Fig. 22 is a comparison graph of the CSI sliding window method and the TSI method for identifying the first 3 order mode at the time t is 25-30 s in example 2 of the present invention.
Fig. 23 is a comparison graph of the CSI sliding window method and the TSI method for identifying the first 3 order mode at the time t is 31-36 s in example 2 of the present invention.
FIG. 24 is a comparison graph of the first 3-order mode identified by the CSI sliding window method and the TSI method at the time t is 37-43 s in example 2 of the present invention.
Detailed Description
The present invention will be described in further detail with reference to test examples and specific embodiments. It should not be understood that the scope of the above-described subject matter is limited to the following examples, and any technique that can be implemented based on the teachings of the present invention is within the scope of the present invention.
A tensor subspace continuous system identification method of a bridge time-varying system comprises the following steps:
the method comprises the following steps: a signal acquisition device is installed on the bridge, and is used for sampling and transmitting a time-varying signal obtained by sampling to an upper computer;
step two: and (3) introducing a time dimension, and expanding the basic theory of subspace identification, namely matrix operation, to three-dimensional tensor calculation. In order to keep consistent with the tensor expression form, an input Hankel matrix and an output Hankel matrix at the t moment in the sliding window subspace are determined and written into UU2i，j，t、YY2i，j，t(for stochastic systems, input matrix UU2i，j，tUnknown, existence of only output momentsArray YY2i，j，t) The first corner mark 2i represents the row number of the Hankel matrix at the time t, the second corner mark j represents the column number of the Hankel matrix, and the third corner mark t represents the current time.
Let the new input signal column vector at time t +1 be ut+1The column vector of the output signal is yt+1At time t +1, the new test signal forms a row vector space as shown in equations (1) and (2) for the deterministic system and the stochastic system, respectively:
Figure BDA0002148312280000111
Figure BDA0002148312280000112
for a determination system, the row vector space of the formula (1) is attached to the rightmost column of the Hankel matrix at the t moment, and the original UU is attached to the right column of the Hankel matrix at the t moment2i，j，t、YY2i，j，tFor a random system, the row vector space of the formula (2) is attached to the rightmost column of the Hankel matrix at the t moment, and the original YY is removed2i，j，tThe left-most column of the input and output Hankel matrix at the t +1 moment is removed, namely the Hankel matrix at the t +1 moment and the Hankel matrix at the t moment are kept the same in dimension, the Hankel matrix at the t +1 moment of the system and the random system is determined to be respectively expressed as a formula (3) and a formula (4):
Figure BDA0002148312280000121
[YY2i，j，t+1]＝[YY′2i，j，tφy(t+1)] (4)
of formula (II) UU'2i，j，t、YY′2i，j，tInput matrix UU respectively representing time t2i，j，tOutput matrix YY2i，j，tThe matrix after the leftmost column is removed.
Dividing a time-varying signal having a time duration T into N time windows in incremental steps L (each window having a length L), the time-varying signal being divided into N time windowsCollecting time is sequential, and a time-varying three-dimensional Hankel tensor is constructedX∈R2i×j×N，
For a determination system where both input and output signals are known, the following slice format is represented along the time dimension:
Figure BDA0002148312280000122
for a stochastic system where the input signal is unknown, it is expressed in the time dimension in slice form as follows:
X(：，：，1)＝[YY2i，j，1]，…，X(：，：，k)＝[YY2i，j，k]，…，X(：，：，N)＝[YY2i，j，N] (6)
wherein X (: k) is the Hankel matrix, UU of the k-th time window2i，j，kInput Hankel matrix, YY, for the kth time window2i，j，kOutputting a Hankel matrix for the kth time window;
step three: establishing a mathematical model for tensor subspace system matrix solution;
based on the time-varying 3-dimensional Hankel tensor, a system matrix is estimated, and the problem to be solved is to analyze the mathematical essence of the system matrix, namely to establish a corresponding mathematical model for solving. For this purpose, the time-varying property of the system matrix of the nonlinear system is first analyzed by theoretical derivation, on the basis of which a mathematical model of the system matrix estimate is established.
In the identification method of the original matrix subspace system, firstly, test signals which are not polluted by noise are considered, and a Hankel matrix X is formed, wherein X belongs to RM×NM < N, defining the rank of the matrix (X) ═ r ≦ M,
Figure BDA0002148312280000131
the SVD decomposition is performed as follows:
Figure BDA0002148312280000132
wherein the matrix U belongs toRM×M，V∈RN×NAre respectively as
Figure BDA0002148312280000133
The left singular matrix S and the right singular matrix after SVD are orthogonal matrix, Us、UnIs a subset of the number U of bits,
Figure BDA0002148312280000134
Figure BDA0002148312280000135
Vs、Vnis a subset of V and is,
Figure BDA0002148312280000136
Figure BDA0002148312280000137
∑s＝diag(σ1，…，σr)，σi(i ═ 1,2, …, M) is the eigenvalue in the diagonal matrix, σ1≥σ2≥…≥σr≥σr+1＝…＝σM＝0。
From the above formula, one can obtain:
Figure BDA0002148312280000138
this can be inferred from the fact that the non-zero elements in the above formula can be represented as:
Figure BDA0002148312280000139
as can be seen from the above formula,
Figure BDA00021483122800001310
has a left singular vector U of
Figure BDA00021483122800001311
The feature vector of (2).
By adding the white noise term psi to X, the measured signal contaminated by noise can be expressed as:
Y＝X+ψ (10)
when the amount of data is sufficiently large, the following relationship exists
Figure BDA0002148312280000141
In the formula, E represents the mathematical expectation,
Figure BDA0002148312280000142
representing the variance of the white noise term psi, IM∈RM×MIs an identity matrix.
It can be deduced that:
Figure BDA0002148312280000143
is provided with
Figure BDA0002148312280000144
Ir∈Rr×rAs an identity matrix, we can obtain:
Figure BDA0002148312280000145
wherein,
Figure BDA0002148312280000146
compared with
Figure BDA0002148312280000147
In a negligibly small amount, S ═ diag (S)1，…，sM)，si> 0, the expression is as follows:
Figure BDA0002148312280000148
according to the theoretical derivation, the left singular matrix U and the diagonal matrix S of the actual measurement signal contain structural time-varying dynamic fingerprint information, and for a time-varying nonlinear system, the left singular matrix U and the diagonal matrix S reflect the time-varying characteristics of the system.
Based on the analysis, from the perspective of tensor decomposition, for the Hankel tensor established in the second step, after tensor decomposition, the left singular matrix and the diagonal matrix should have time-varying characteristics, so that a mathematical model for tensor subspace system matrix solution in the following form can be established.
Xk＝UkSkVT+Wk (15)
In the formula, XkIs a simplified expression form of X (: k), UkLeft singular matrix, S, for the k-th time windowkIs a non-negative diagonal matrix of the kth time window, V is a right singular matrix which does not change with time, WkThe fitted residual sum of squares for the kth time window;
the mathematical model established in the invention is consistent with the tensor parallel factor model in form, and the formula decomposition is not unique as known from the tensor parallel factor decomposition theory. Therefore, for the practical problem to be solved in the present invention, U in the above formula is usedkMake constraints and define Uk＝QkH，Qk∈Rr×r、H∈Rr×rAre all orthogonal matrices, i.e. Uk TUk＝IkSo as to obtain the compound with the structure,
Xk≈QkHSkVT (16)
step four: solving a tensor subspace system matrix;
after a mathematical model of tensor subspace decomposition is established, tensor subspace system matrix estimation is carried out according to the following steps:
inputting: three-dimensional time-varying Hankel tensor and an order upper boundary value RR of a stable graph system.
Step 4.1: for the order n of the steady graph system (n is a positive integer, and 1 < n < RR), the pair
Figure BDA0002148312280000151
Performing principal component analysis to obtain its bearing matrix asInitialization matrix of matrix V, H, S1，...，SNInitializing into an identity matrix;
step 4.2: to pair
Figure BDA0002148312280000152
Performing SVD to obtain
Figure BDA0002148312280000153
Qk＝RkTk TIn the formula, Tk，Δk，RkRespectively a left singular matrix, a diagonal matrix and a right singular matrix after SVD decomposition;
step 4.3: updating H and S by adopting a trilinear alternating least square algorithm1，…，SN，
Figure BDA0002148312280000154
Figure BDA0002148312280000161
Figure BDA0002148312280000162
Wherein,. indicates a Khatri-Rao product, X(i)(i ═ 1,2,3) is Hankel tensorXTensor expansions along three dimensions;
step 4.4: using convergence formula to update H, S1，…，SNCarrying out convergence judgment, and defining:
Figure BDA0002148312280000163
if σ3 old-σ3 new＞εσ3 oldIf yes, judging that the convergence is not reached, returning to the step 4.3, otherwise, entering the step 4.5, wherein c is a constant, and epsilon isA convergence factor.
Step 4.5: according to the updated H, S1，…，SNRecalculating Qk＝RkTk T，Uk＝QkH；
Step 4.6: judging whether the system order n is equal to an upper bound value RR, if n is less than the upper bound value RR, repeating the steps 4.1 to 4.5 until the system order reaches RR, if n is equal to RR, stopping calculation, and outputting Uk，SkThe calculation result of V;
and outputting a result: u corresponding to different order of stable graph systemk，Sk，V。
Step five: performing modal parameter identification
Defining an observation matrix Γk
Γk＝UkSk 1/2 (18)
According to the state space theory in the control theory, the system state matrix a and the system output matrix C can be calculated by the following formula:
Figure BDA0002148312280000164
Ck＝Γk(1：l，：) (20)
in the formula,
Figure BDA0002148312280000171
is expressed as gammakThe matrix after the first row is removed, kΓis expressed as gammakThe matrix after the last l rows is removed,
Figure BDA0002148312280000172
represents the Moore-Penrose pseudo-inverse of the matrix,
and for the system state matrix A, carrying out eigenvalue decomposition,
Ak＝ψkΛkψk -1 (21)
in the formula, ΛkIs a diagonal matrix, Λk＝diag(μi，k)，μi，kIs a complex characteristic value, psi, of the ith order vibrational mode in the kth time windowkIs the complex eigenvector of the kth time window,
the relation between the system state matrix A of the continuous system and the discrete system can be deduced to obtain the characteristic value lambda of the ith order vibration mode in the kth time window of the discrete systemi，k，
Figure BDA0002148312280000173
In the formula, Δ t represents a signal sampling time interval.
Characteristic value lambdai，kAnd system circular frequency omegai，kAnd damping ratio ζi，kThere is the following relationship between:
Figure BDA0002148312280000174
obtaining the frequency f of the ith order vibration mode of the kth time windowi，kDamping ratio ζi，kAnd all order mode vectors phikRespectively as follows:
Figure BDA0002148312280000175
Figure BDA0002148312280000176
φk＝Ckψk
in the formula, λi，kIs the characteristic value of ith order vibration mode in the kth time window, Re represents the mathematical operation of the real part, Ck＝Γk(1: l, l is the number of rows, gamma)k＝UkSk 1/2，ψkIs the complex eigenvector of the kth time window.
Fig. 1 is a process of identifying a conventional sliding window subspace system, and fig. 2 is a process of identifying a tensor subspace continuous system identification method of a bridge time varying system according to the present invention. Assuming that the number of time windows divided by the test signal is K, the maximum system order in the stable graph is RR, and for the sliding window subspace system identification method, data in each window needs to be subjected to a complete subspace system identification process based on matrix operation, and the total SVD (singular value decomposition) required is K.RR/2; the tensor subspace system identification method provided by the invention is characterized in that a time-varying 3-dimensional Hankel tensor is constructed through K time windows, in the system identification process, only in the stable graph analysis process, SVD decomposition is carried out once in each order, and the total SVD decomposition times in the whole system identification process are RR/2 times. SVD is one of the time-consuming and serious steps in the identification process of the traditional subspace system, and singular values are changed due to the fact that the SVD is easily influenced by noise in the identification process, so that the identification result of the final modal parameters is inaccurate, and the stable points are scattered on a stable graph and the stable axis is twisted.
In conclusion, compared with the traditional sliding window subspace system identification method, the tensor subspace continuous system identification method avoids the matrix QR decomposition process, reduces SVD decomposition times, has higher calculation efficiency, has less noise influence on a calculation result, has clearer and more distinguishable stable axes, and can easily identify modal parameter information of more orders of the system, thereby providing core technical support for the subsequent bridge real-time health monitoring.
Example 1
A curve cable-stayed model bridge with the curvature radius of 27.5m is selected for an impact load test in a perfect state, and a cable force sensor, a transverse acceleration sensor, a vertical acceleration sensor and a longitudinal acceleration sensor are arranged on the curve cable-stayed model bridge, as shown in figures 3-5. The signal sampling frequency is 256Hz, the initial modal information of the cable-stayed bridge is obtained based on the first 8s vibration test signal, and the new modal information of the cable-stayed bridge is obtained at every subsequent interval of 0.4 s. The curve cable-stayed model bridge is identified by respectively adopting the tensor subspace system identification method established by the invention and the traditional sliding window random subspace system identification method, so that the performance of the tensor subspace continuous identification method in the invention is compared with that of the traditional sliding window subspace system identification method.
Fig. 6-7 show the stable diagram generated by the conventional random sliding window Subspace system Identification method (hereinafter abbreviated as SSI sliding window method, SSI being storage Subspace Identification) and the Tensor Subspace continuous Identification method (hereinafter abbreviated as TSI method, TSI being sensor Subspace Identification) in the present invention. It can be seen that in the conventional random sliding window subspace system identification method, all stable axes of frequency, damping ratio and vibration mode at different time intervals are prone to be missed, for example, when t is 0-8 s and t is 8-8.4 s, the second-order vibration mode is unstable; when t is 9.2-9.6 s, the first-order mode is unstable, which directly results in that the SSI sliding window method based on the stability graph loses frequency, damping ratio and mode information of the corresponding order when t is 0-8 s, t is 8-8.4 s and t is 9.2-9.6 s. In the tensor subspace continuous identification method provided by the text, all stable axes of the frequency, the damping ratio and the vibration mode are basically fixed in different time periods, namely, in different time periods, the TSI method can obtain the stable frequency, the damping ratio and the vibration mode information of the system, and the number of the stable axes is more than that of the SSI sliding window method. On the other hand, the SSI sliding window method takes 337.49s to obtain the result, while the TSI method only takes 280.31s (the computer CPU which takes time for the calculation is I7-6700K, and the memory is 16G). The analysis shows that compared with the traditional SSI sliding window method, the TSI method established by the method can greatly improve the calculation efficiency and can obtain more system modal information.
TABLE 1 comparison of the time-varying frequency identification results of SSI method and TSI method (Unit: Hz)
Figure BDA0002148312280000191
Figure BDA0002148312280000201
TABLE 2 comparison of the results of time-varying damping ratio identification of the SSI method and TSI method (unit:%)
Figure BDA0002148312280000202
TABLE 3 comparison of the identification results of the time-varying normalized vibration mode MAC (modal assessment criterion) between the SSI method and the TSI method
Time period 1 st order 2 order 3 order
0～8s 0.998 0.998
8～8.4s 0.999 0.996
8.4～8.8s 0.999 0.998 0.995
8.8～9.2s 0.999 0.998 0.995
9.2～9.6s 0.999 0.997
9.6～10.0s 0.999 0.998 0.998
As can be seen from table 1 and fig. 8, in terms of system frequency parameters, the deviation between the TSI method identification result and the conventional SSI sliding window method identification result is maximally-0.84%, which is less than 1%, indicating that the frequency identification result precision of the TSI method is consistent with that of the SSI sliding window method while the calculation efficiency is improved.
As can be seen from table 2 and fig. 9, in terms of the system damping ratio parameters, the deviation between the recognition result of the TSI method and the recognition result of the conventional SSI sliding window method is large, and is at most-43.56%, but the damping ratio recognized by the SSI sliding window method itself has large fluctuation and low accuracy, so the damping ratio recognition accuracy of the TSI method is not comparable to the recognition accuracy of the conventional SSI method.
In fig. 10-15, the three panels of each figure are a comparison of the 1 st order mode, a comparison of the 2 nd order mode, and a comparison of the 3 rd order mode, respectively.
As can be seen from table 3 and fig. 10 to fig. 15, in terms of system vibration mode parameters, the deviation between the recognition result of the TSI method and the recognition result of the conventional SSI sliding window method is accurate to a percentile, which indicates that the vibration mode recognition result precision of the TSI method is the same as that of the SSI sliding window method while the calculation efficiency is greatly improved.
Example 2
A concrete bridge tower actual measurement vibration signal under excitation of white noise of a cable-stayed bridge vibration table test is selected, the signal sampling frequency is 256Hz, and the signal acquisition time is 47 s. And acquiring initial modal information of the cable-stayed bridge tower based on the first 6s vibration test signals, and acquiring new modal information of the cable-stayed bridge tower every 5s subsequently. The TSI method and the short-time sliding window determination Subspace system Identification method (CSI sliding window method, CSI is Combined Subspace Identification) established in the text are respectively adopted to identify the cable-stayed bridge tower, so that the performance of the TSI method compared with the performance of the CSI method is compared.
As shown in fig. 16 and 17, the stability graphs generated by the CSI sliding window method and the TSI method are respectively shown, it can be known that the system modal information identified by the CSI sliding window method under different time ranges is less "missing" compared with the TSI method, and only when t is 7-12 s, the third-order damping ratio and the mode shape are unstable; when t is 25-30 s, the second-order damping ratio and the mode shape are unstable, which directly results in that the frequency, the damping ratio and the mode shape information of the corresponding order are lost when t is 7-12 s and t is 25-30 s in the CSI sliding window method based on the stable graph. However, the method for determining the TSI proposed herein has stable axes including the system frequency, the damping ratio and the mode shape, which are substantially stable in different time periods, and the number of stable axes is more than that of the CSI sliding window method. On the other hand, the CSI sliding window method takes 179.28s to obtain the result, while the TSI method only takes 61.79s, and the time consumption is reduced by about 2/3. The analysis shows that, for a deterministic system, the TSI method established by the method can greatly improve the calculation efficiency and can obtain more system modal information compared with the traditional CSI sliding window method.
The comparison graphs of the frequency and the damping ratio identified by the CSI sliding window method and the TSI method are shown in FIGS. 16 and 17, and the comparison graphs of the first 3-order vibration modes identified by the CSI sliding window method and the TSI method are shown in FIGS. 18-24.
TABLE 4 comparison of the time-varying frequency identification results of the CSI sliding window method and the TSI method (unit: Hz)
Figure BDA0002148312280000221
TABLE 5 comparison of time-varying damping recognition results for CSI sliding window method and TSI method (unit:%)
Figure BDA0002148312280000222
TABLE 6 comparison of time-varying modal confidence criterion MAC value recognition results for CSI sliding window method and TSI method
Time period/s 1 st order 2 order 3 order
0～6 0.999 0.999 0.999
7～12 0.999 0.998
13～18 0.999 0.997 0.995
19～24 0.999 0.999 0.999
25～30 0.999 0.999
31～36 0.999 0.999 0.996
37～43 0.999 0.995 0.994
As can be seen from fig. 16 and table 4, in the aspect of system frequency identification, the maximum deviation between the TSI method identification result and the conventional CSI sliding window method identification result is-3.41%, which indicates that the frequency identification result precision of the TSI method is consistent with that of the CSI sliding window method while the calculation efficiency is greatly improved.
As can be seen from fig. 17 and table 5, in the aspect of system damping ratio identification, the deviation between the TSI method identification result and the conventional CSI sliding window method identification result is large, and is 60.81% at most, however, the damping ratio identified by the CSI sliding window method itself has large fluctuation and low accuracy, and therefore, the damping ratio identification accuracy of the TSI method is not comparable to the conventional CSI method identification accuracy.
In fig. 18-24, the three small graphs of each graph are respectively the comparison graph of the 1 st order bridge mode shape, the comparison graph of the 2 nd order bridge mode shape and the comparison graph of the 3 rd order bridge mode shape.
As can be seen from fig. 18 to 24 and table 6, in the aspect of system vibration mode identification, the deviation between the TSI method identification result and the conventional CSI sliding window method identification result is accurate to a percentile at different sensor positions, which indicates that the vibration mode identification result accuracy of the TSI method is the same as that of the CSI sliding window method while the calculation efficiency is greatly improved.
The above embodiments are only used for illustrating the invention and not for limiting the technical solutions described in the invention, and although the present invention has been described in detail in the present specification with reference to the above embodiments, the present invention is not limited to the above embodiments, and therefore, any modification or equivalent replacement of the present invention is made; all such modifications and variations are intended to be included herein within the scope of this disclosure and the appended claims.

Claims (4)

1. A tensor subspace continuous system identification method of a bridge time-varying system is characterized by comprising the following steps:
the method comprises the following steps: acquiring a bridge time-varying signal by using a signal acquisition device, wherein the signal acquisition device comprises a transverse acceleration sensor, a vertical acceleration sensor and a longitudinal acceleration sensor;
step two: dividing the time-varying signal with the signal time course T into N time windows according to the increasing step length L, and constructing a time-varying three-dimensional Hankel tensor according to the signal acquisition time sequenceX∈R2i×j×N，
For a determination system where both input and output signals are known, the following slice format is represented along the time dimension:
Figure FDA0003056747770000011
for a stochastic system where the input signal is unknown, it is expressed in the time dimension in slice form as follows:
X(:,:,1)＝[YY2i,j,1],…,X(:,:,k)＝[YY2i,j,k],…,X(:,:,N)＝[YY2i,j,N]
wherein X (: k) is the Hankel matrix, UU of the k-th time window2i,j,kInput Hankel matrix, YY, for the kth time window2i,j,kOutputting a Hankel matrix for the kth time window;
step three: establishing a mathematical model for tensor subspace system matrix solution;
Xk＝UkSkVT+Wk
in the formula, XkIs a simplified expression form of X (: k), UkLeft singular matrix, S, for the k-th time windowkIs a non-negative diagonal matrix of the kth time window, V is a right singular matrix which does not change with time, WkThe fitted residual sum of squares for the kth time window;
for U in the above formulakMake constraints and define Uk＝QkH，Qk∈Rr×r、H∈Rr×rAre all orthogonal matrices, r is XkRank of (1), i.e. Uk TUk＝IkSo as to obtain the compound with the structure,
Xk≈QkHSkVT
step four: solving a tensor subspace system matrix;
step 4.1: for the order n of the steady graph system, n is a positive integer and 1<n<RR, RR is the upper bound of the system order of the stability map, pair
Figure FDA0003056747770000021
Performing principal component analysis to obtain its bearing matrix as the initialization matrix of matrix V, H, S1，…,SNInitializing into an identity matrix;
step 4.2: to pair
Figure FDA0003056747770000022
Performing SVD to obtain
Figure FDA0003056747770000023
Qk＝RkTk TIn the formula, Tk，Δk，RkRespectively a left singular matrix, a diagonal matrix and a right singular matrix after SVD decomposition;
step 4.3: updating H and S by adopting a trilinear alternating least square algorithm1，…,SN，
Figure FDA0003056747770000024
Figure FDA0003056747770000025
Figure FDA0003056747770000026
Wherein,. indicates a Khatri-Rao product, X(i)(i ═ 1,2,3) is Hankel tensorXTensor expansions along three dimensions;
step 4.4: using convergence formula to update H, S1，…,SNCarrying out convergence judgment, returning to the step 4.3 if the judgment result is that the convergence is not realized, otherwise, entering the step 4.5;
step 4.5: according to the updated H, S1，…,SNRecalculating Qk＝RkTk T，Uk＝QkH；
Step 4.6: judging whether the system order n is equal to an upper bound value RR, if n is less than the upper bound value RR, repeating the steps 4.1 to 4.5 until the system order reaches RR, if n is equal to RR, stopping calculation, and outputting Uk,SkThe calculation result of V;
step five: performing modal parameter identification to obtain the frequency f of the ith order vibration mode of the kth time windowi,kDamping ratio ζi,kAnd all order mode vectors phikRespectively as follows:
Figure FDA0003056747770000031
Figure FDA0003056747770000032
φk＝Ckψk
in the formula, λi,kIs the characteristic value of ith order vibration mode in the kth time window, Re represents the mathematical operation of the real part, Ck＝Γk(1: l, l is the number of rows, gamma)k＝UkSk 1/2，ψkIs the complex eigenvector of the kth time window.
2. The method for identifying the tensor subspace continuous system of the time-varying bridge system according to claim 1, wherein in the second step:
for a deterministic system, at time t +1, the new test signal forms the following row vector space:
Figure FDA0003056747770000033
then for the determination system, the Hankel matrix at time t +1 is:
Figure FDA0003056747770000034
for a random system, at time t +1, the new test signal forms a row vector space as follows:
Figure FDA0003056747770000035
for a random system, the Hankel matrix at time t +1 is:
[YY2i,j,t+1]＝[YY'2i,j,t φy(t+1)]
of formula (II) UU'2i,j,t、YY'2i,j,tInput matrix UU respectively representing time t2i,j,tOutput matrix YY2i,j,tThe matrix after the leftmost column is removed.
3. The method for identifying the tensor subspace continuum system of the time-varying bridge system as claimed in claim 1, wherein in said step 4.4, definition is made
Figure FDA0003056747770000041
If σ3 old-σ3 new>εσ3 oldIf yes, judging that the convergence is not achieved, returning to the step 4.3, otherwise, entering the step 4.5, wherein c is a constant, and epsilon is a convergence coefficient.
4. The method for identifying the tensor subspace continuous system of the time-varying bridge system according to any one of claims 1 to 3, wherein in the fifth step, an observation matrix Γ is definedk＝UkSk 1/2Then, the system state matrix a and the system output matrix C can be calculated by the following formula:
Figure FDA0003056747770000042
Ck＝Γk(1:l,:)
in the formula,
Figure FDA0003056747770000043
is expressed as gammakThe matrix after the first row is removed, kΓis expressed as gammakThe matrix after the last l rows is removed,
Figure FDA0003056747770000044
represents the Moore-Penrose pseudo-inverse of the matrix,
and for the system state matrix A, carrying out eigenvalue decomposition,
Ak＝ψkΛkψk -1
in the formula, ΛkIs a diagonal matrix, Λk＝diag(μi,k)，μi,kIs a complex characteristic value, psi, of the ith order vibrational mode in the kth time windowkIs the complex eigenvector of the kth time window,
obtaining the characteristic value lambda of the ith order vibration mode in the kth time window of the discrete systemi,k，
Figure FDA0003056747770000051
In the formula, Δ t represents a signal sampling time interval.
CN201910697806.3A 2019-07-30 2019-07-30 Tensor subspace continuous system identification method of bridge time-varying system Active CN110414150B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN201910697806.3A CN110414150B (en) 2019-07-30 2019-07-30 Tensor subspace continuous system identification method of bridge time-varying system

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN201910697806.3A CN110414150B (en) 2019-07-30 2019-07-30 Tensor subspace continuous system identification method of bridge time-varying system

Publications (2)

Publication Number Publication Date
CN110414150A CN110414150A (en) 2019-11-05
CN110414150B true CN110414150B (en) 2021-06-22

Family

ID=68364625

Family Applications (1)

Application Number Title Priority Date Filing Date
CN201910697806.3A Active CN110414150B (en) 2019-07-30 2019-07-30 Tensor subspace continuous system identification method of bridge time-varying system

Country Status (1)

Country Link
CN (1) CN110414150B (en)

Families Citing this family (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111276183B (en) * 2020-02-25 2023-03-21 云南大学 Tensor decomposition processing method based on parameter estimation
CN111757250B (en) * 2020-05-27 2022-04-29 重庆邮电大学 Channel state information positioning fingerprint construction method based on tensor decomposition
CN116202760B (en) * 2023-05-05 2023-08-18 赛腾机电科技（常州）有限公司 Singular value decomposition method and system for third-order tensor for mechanical fault diagnosis

Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN1997999A (en) * 2004-03-29 2007-07-11 彼德·T·杰尔曼 System and method for determining elasticity of material
CN102520071A (en) * 2011-12-20 2012-06-27 江苏方天电力技术有限公司 Transmission tower modal parameter identification method based on improved subspace algorithm
CN107729592A (en) * 2017-08-14 2018-02-23 西安理工大学 Traced back the Time variable structure Modal Parameters Identification of track based on broad sense subspace
CN108318129A (en) * 2018-02-01 2018-07-24 石家庄铁道大学 The true and false discriminating method of bridge structure modal parameter and terminal device
CN109242007A (en) * 2018-08-24 2019-01-18 华中科技大学鄂州工业技术研究院 A kind of cross-domain isomery big data multi-angle of view clustering method and device based on tensor

Family Cites Families (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CA2983880A1 (en) * 2015-05-05 2016-11-10 Kyndi, Inc. Quanton representation for emulating quantum-like computation on classical processors
CN109669414B (en) * 2018-12-13 2020-10-09 宁波大学 Dynamic process monitoring method based on autocorrelation characteristic decomposition

Patent Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN1997999A (en) * 2004-03-29 2007-07-11 彼德·T·杰尔曼 System and method for determining elasticity of material
CN102520071A (en) * 2011-12-20 2012-06-27 江苏方天电力技术有限公司 Transmission tower modal parameter identification method based on improved subspace algorithm
CN107729592A (en) * 2017-08-14 2018-02-23 西安理工大学 Traced back the Time variable structure Modal Parameters Identification of track based on broad sense subspace
CN108318129A (en) * 2018-02-01 2018-07-24 石家庄铁道大学 The true and false discriminating method of bridge structure modal parameter and terminal device
CN109242007A (en) * 2018-08-24 2019-01-18 华中科技大学鄂州工业技术研究院 A kind of cross-domain isomery big data multi-angle of view clustering method and device based on tensor

Non-Patent Citations (3)

* Cited by examiner, † Cited by third party
Title
Fundamental Tensor Operations for Large-Scale Data Analysis in Tensor Train Formats;Peter VAN OVERSCHEE等;《Multidimensional Systems & Signal Processing》;20141231;全文 *
Michal Abrahamowicz等.Spline modelling of cumulative and non-linear effects of time-varying risk factors on the hazard.《Session Eropeen Meting of Statistitien》.2009, *
陈栋军.确定—随机子空间模态参数识别研究.《中国优秀硕士学位论文全文数据库 工程科技Ⅱ辑》.2016, *

Also Published As

Publication number Publication date
CN110414150A (en) 2019-11-05

Similar Documents

Publication Publication Date Title
Loh et al. Time domain identification of frames under earthquake loadings
CN110414150B (en) Tensor subspace continuous system identification method of bridge time-varying system
CN107256204B (en) Experimental device and method for multipoint vibration response frequency domain prediction based on transfer function
Sinha et al. Simplified models for the location of cracks in beam structures using measured vibration data
Devriendt et al. Operational modal analysis in the presence of harmonic excitations by the use of transmissibility measurements
Zhu et al. A sensitivity-based structural damage identification method with unknown input excitation using transmissibility concept
CN106878375B (en) A kind of cockpit pollutant monitoring method based on distributed combination sensor network
Meruane Model updating using antiresonant frequencies identified from transmissibility functions
Mao et al. The construction and comparison of damage detection index based on the nonlinear output frequency response function and experimental analysis
CN112949131B (en) Probability damage positioning vector method for continuous bridge cluster damage diagnosis
CN115200700B (en) Modal parameter identification method based on Welch method and covariance random subspace method
Sjövall et al. Component system identification and state-space model synthesis
CN107679328A (en) A kind of optimal sensor method for arranging of systematic parameter identification
CN108846200B (en) Quasi-static bridge influence line identification method based on iteration method
Hu et al. Model order determination and noise removal for modal parameter estimation
CN108664452B (en) Method and system for determining number of indoor personnel
Naseralavi et al. Elaborate structural damage detection using an improved genetic algorithm and modal data
Chen et al. Inverse damage prediction in structures using nonlinear dynamic perturbation theory
Arici et al. Statistical significance of modal parameters of bridge systems identified from strong motion data
CN114357778B (en) Cable force self-adaptive identification method in long-term monitoring scene
Meo et al. Damage assessment on plate-like structures using a global-local optimization approach
CN111428342B (en) Random dynamic load identification method based on frequency domain spectrum decomposition
Suwała Nonparametric identification of structural modifications in Laplace domain
Feizi et al. Identifying damage location under statistical pattern recognition by new feature extraction and feature analysis methods
Srinivas et al. Studies on methodological developments in structural damage identification

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
TA01 Transfer of patent application right
TA01 Transfer of patent application right

Effective date of registration: 20191127

Address after: 610000 7-10 Floor, Block B, Building No. 4, 200 Tianfu Fifth Street, Chengdu High-tech Zone, Sichuan Province

Applicant after: Sichuan Highway Planning, Survey and Design Research Institute Co., Ltd.

Address before: 330013 Jiangxi city in Nanchang Province Economic Development Zone East Shuanggang Street No. 808

Applicant before: East-China Communication Univ

GR01 Patent grant
GR01 Patent grant