CN107515980B - Two-step sequence strain sensor optimized layout method for structural deformation reconstruction - Google Patents
Two-step sequence strain sensor optimized layout method for structural deformation reconstruction Download PDFInfo
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Abstract
The invention relates to an optimized layout method of a sensor, in particular to a two-step sequence strain sensor optimized layout method oriented to structural deformation reconstruction, belonging to the optimized layout technology of the sensorThe field of operation. The two-step sequence strain sensor optimized layout method for structural deformation reconstruction comprises the following steps: (1) transpose matrix psi of modal strain matrix corresponding to candidate stationing position extracted from finite element modelTPerforming row principal component QR decomposition, and determining m linearly independent initial sensor arrangement sets, wherein m is a positive integer; (2) and establishing a reconstruction precision criterion and an information redundancy criterion, then establishing a sensor layout optimization model, and continuously optimizing iteration in a step-by-step accumulation mode to determine the final sensor layout.
Description
Technical Field
The invention relates to an optimized layout method of sensors, in particular to a two-step sequence strain sensor optimized layout method oriented to structural deformation reconstruction, and belongs to the technical field of optimized layout of sensors.
Background
The technical problem to be faced by the structural deformation reconstruction is the optimized layout of the sensor except for the finite element model correction. The optimized layout of the sensor plays a role in starting and starting in structural deformation reconstruction and belongs to the NP hard combination optimization problem. The reasonable sensor layout can not only optimize the number of sensors, but also collect the most comprehensive reconstruction information. In some special working environments, only a small or specific number of sensors can be selected due to the economic cost and the structural limitation, and the excessive sensors can cause problems of increased system failure rate, difficult data storage and analysis and the like. Therefore, the purpose of optimizing the layout of the sensors is to find an optimal measuring point scheme of M (M < N) sensors in N measurable degrees of freedom according to the constraint conditions of the system, and obtain the most abundant information to fully reflect the dynamic performance of the structure.
The typical sensor layout method mainly includes two main categories of traditional solving algorithm and random solving algorithm. The traditional solving algorithm comprises an effective independence method, a Guyan reduction method, a QR decomposition method, a minimum MAC method, information entropy and the like. The random solving algorithm comprises a genetic algorithm, a particle swarm algorithm, a wolf swarm algorithm, a monkey swarm algorithm and the like.
X.h.zhang, etc. optimally lays out the two types of displacement and strain sensors, gradually deleting the sensor at the position where the objective function (trace of deformation reconstruction error covariance) is minimum after deletion of a certain degree of freedom until a given threshold is reached. This method is reported in "X.H.Zhang, et al.Integrated optical placement of display transducers and structures for testing of Structural response [ J ]. International Journal of Structural Stability and Dynamics,2011,11(3): 48-51".
The minimum Error Estimation Method (EEM) introduced by chen et al is a single displacement sensor optimized layout using the idea of x.h.zhang. The method is mainly used for a relatively simple structure, is an expansion of an effective independent method, has the essence of the effective independent method, and can generate the phenomenon of sensor aggregation when the number of the sensors is greater than the modal number, and the information acquired by the sensors can generate serious redundancy. The method is reported in Chenwei, Zhao Wen, Zhu hong Ping, et al, optimal sensor placement for structural response evaluation [ J ]. Journal of Central South University,2014,21(10):3993-4001 ].
The above-described method for optimizing the layout of the sensors has the following disadvantages:
1. the method is mainly suitable for a relatively simple structure, the phenomenon of sensor aggregation can occur when the number of the sensors is larger than the modal number, and the information acquired by the sensors can have serious redundancy.
2. For a large complex structure with a large number of degrees of freedom, the conventional sensor optimization layout method needs to delete most of the degrees of freedom to obtain a more ideal configuration scheme because the number of sensors to be finally laid out is relatively small, and therefore calculation is time-consuming.
3. The existing sensor layout method mainly aims at the purposes of modal identification, structural damage identification, health monitoring and the like, and the method for reconstructing structural deformation is less.
Although some researchers introduced some new ideas or guidelines, most methods are improvements based on the methods aiming at the defects. The current research work focuses on the optimized layout of the displacement sensor, but the strain sensor has the characteristics of light structure, high resolution, high sensitivity and the like, and the sensor has more important engineering application value, but research documents about the layout of the strain sensor are relatively deficient.
Aiming at the problems, the invention mainly researches an optimal layout method of the strain sensor for structural deformation reconstruction, and the final layout position of the sensor not only can ensure smaller reconstruction error, but also needs to obtain as much structural information as possible by fewer sensors, namely the information measured by each sensor is as non-redundant as possible. Aiming at the problems, the invention provides a two-step sequence strain sensor optimal layout method for structural deformation reconstruction, based on a modal deformation reconstruction theory, considering the influence of a test error on reconstruction precision and the influence of information redundancy of a sensor on sensor layout.
Disclosure of Invention
The purpose of the invention is as follows: the invention aims at solving the problems in the prior art, namely the invention discloses a two-step sequence strain sensor optimization layout method for structural deformation reconstruction, which can effectively realize the optimization of the number and the positions of sensors and greatly improve the operation efficiency.
The technical scheme is as follows: the two-step sequence strain sensor optimized layout method for structural deformation reconstruction comprises the following steps:
(1) transpose matrix psi of modal strain matrix corresponding to candidate stationing position extracted from finite element modelTPerforming row principal component QR decomposition, and determining m linearly independent initial sensor arrangement sets, wherein m is a positive integer;
(2) and establishing a reconstruction precision criterion and an information redundancy criterion, then establishing a sensor layout optimization model, and continuously optimizing iteration in a step-by-step accumulation mode to determine the final sensor layout.
Further, the step (1) comprises:
(11) extracting a modal strain matrix psi corresponding to the candidate stationing position from the finite element model, and transposing the modal strain matrix to obtain a transposed matrix psiTThe expression of the modal strain matrix Ψ is as follows:
m is the intercepted modal number;
ndnumbering the candidate stationing positions at the maximum;
(12) to the transposed matrix ΨTCarrying out column principal component QR decomposition to obtain an R matrix (the R matrix is psi according to a linear independent ruleTThe psi can be screened out according to the arrangement of the first m main diagonal elements of the R matrixTCorresponding m column vectors in the matrix) from the transposed matrix Ψ according to the R matrixTAnd (5) screening m linear independent column vectors with good personality, wherein the positions corresponding to the column vectors are used as the initial sensor layout positions.
Further, the step (2) comprises:
(21) establishing a reconstruction accuracy criterion, and finding out a sensor position with a minimum covariance matrix trace of a reconstruction error from the remaining candidate positions
According to the principle of modal superposition
The strain-displacement transformation equation can be obtained:
in the formula:
q (t) is the true displacement;
ε (t) is the true strain;
qm(t) is the modal coordinate;
Φ is a modal displacement matrix, whose expression is:
m is the number of interception modes, ndNumbering the candidate stationing positions at the maximum;
Ψ is a modal strain matrix expressed as:
m is the number of interception modes, ndNumbering the candidate stationing positions at the maximum;
ΨM(d) arranging a sub-matrix formed by rows corresponding to the positions of the strain sensors in the modal strain matrix;
Φsa sub-matrix composed of rows corresponding to the interest positions in the modal displacement matrix,
εm(t) is the measured strain, and the expression should be:
εm(t)＝ΨM(d)qm(t)+e(t) (4)
wherein e (t) is test noise;
so as to obtain:
and obtaining a deformation estimation error according to the formula (3) and the formula (5):
it is assumed that the measurement errors in equation (4) are independent of each other and follow a normal distribution, i.e.
The covariance expression of the estimation error is:
wherein E (. cndot.) is desired, E (E (t) ET(t)) is the covariance matrix of the measurement noise, and assuming that the measurement noise is independent of each other:
by substituting equation (8) for equation (7), the covariance matrix Δ of the estimation error can be obtained
Wherein each diagonal element in the delta matrix represents the variance of the corresponding deformation estimation error, the maximum number on the diagonal represents the maximum estimation error, and trace (delta) of the matrix represents the sum of the variances of all position deformation estimates;
due to the fact that
Being constant and therefore not considering its effect on the covariance of the reconstruction error, the average estimation error can be expressed as:
where trace (. cndot.) is the trace of the matrix, NsThe number of sensors in the set which is laid out;
(22) establishing an information redundancy criterion, and deleting positions with higher redundancy than the known sensor from the remaining candidate stationing positions;
in order to make the structural information acquired by the sensors relatively independent and avoid information redundancy among the sensors, a modal strain information matrix corresponding to the kth candidate stationing position is defined as follows:
in the formula, ΨkA is the modal strain vector corresponding to the k-th distributable point position, i.e. the k-th row of the modal strain matrix ΨkIs a m x m array matrix;
the modal strain information matrixes of two candidate distribution points with similar structure dynamic information are also similar, and the difference of the modal strain information matrixes can be measured by certain norm difference of the modal strain information matrixes:
Dki(d)＝||Ak-Ai(d)|| (12)
in the formula, Dki(d) The spatial distance difference of a modal strain information matrix corresponding to the kth candidate stationing position and the ith position of the arranged sensor set is obtained;
in order to better compare the difference degree of modal strain information among the sensors, the distance difference coefficient needs to be normalized, and the information redundancy degree among the sensors is expressed by the following formula:
since | | | Ak-Ai(d)||≤||Ak||+||Ai(d) The distance coefficient of difference is more than or equal to Rki(d)≤1；
R is to beki(d) And a redundancy threshold R0Comparing, and removing the residue less than R0The value of (a), namely, finishing deleting the position with larger redundancy than the known sensor from the rest candidate stationing positions;
(23) establishing a sensor layout optimization model, and selecting the position with the minimum current trace as the newly-added sensor layout position
Taking the minimum reconstruction error as an objective function and taking the redundancy of the sensor layout information and the average reconstruction error smaller than a preset threshold as constraint conditions, establishing the following mathematical model:
in the formula (I), the compound is shown in the specification,
d is the optimized sensor position;
Γfeasa feasible field for a sensor candidate position;
R0a redundancy threshold;
(24) performing optimization iteration and judging whether the termination condition is met
And deleting the position with higher redundancy than the known sensor from the candidate point distribution positions by utilizing a reconstruction precision criterion and an information redundancy criterion, finding out the sensor position which enables the covariance matrix trace of the reconstruction error to be minimum, continuously updating and optimizing the sensor position, and judging whether a final judgment condition is met, thereby outputting an optimized layout scheme.
Further, the termination conditions in step (24) are two of the following:
(241) whether the preset deformation reconstruction precision requirement is met or not;
(242) whether the maximum number S of sensors allowed by the system is reachedM。
Further, A in step (22)kAnd Ai(d) When being orthogonal, Rki(d) When the modal strain information contained by the sensor is not redundant, 1k＝Ai(d) When R iski(d) At 0, the modal strain information contained by the sensor is identical.
Has the advantages that: the two-step sequence strain sensor optimized layout method facing structural deformation reconstruction disclosed by the invention has the following beneficial effects:
1. the two-step sequence strain sensor optimization layout method for structural deformation reconstruction is provided, the number and the positions of the sensors can be optimized simultaneously, and the method is compared with a strain-based error estimation minimization method (S-EEM), so that the advantages are obvious.
2. Based on the reconstruction theory of the modal method, the influence of the test error on the reconstruction precision and the influence of the sensor redundancy on the sensor layout are considered.
3. Aiming at the multi-degree-of-freedom structure and the problem of small quantity of sensor layout, the two-step sequence strain sensor optimized layout method can greatly improve the calculation efficiency and eliminate redundant information.
Drawings
FIG. 1 is a flow chart of a two-step sequence strain sensor optimized layout method for structural deformation reconstruction disclosed in the present invention;
FIG. 2 is a schematic diagram of an antenna array finite element model established by ANSYS;
FIG. 3a is a diagram of sensor placement positions on an antenna array surface optimized by the two-step sequence strain sensor optimization layout method for structural deformation reconstruction disclosed in the present invention;
FIG. 3b is the sensor placement position on the antenna array surface after optimization by the S-EEM method;
FIG. 4 is a graph of the change of the calculation time with the number of sensors in a two-step sequence strain sensor optimization layout method and an S-EEM method for structure deformation reconstruction;
FIG. 5 is a graph of SMAC versus sensor number for a two-step sequence strain sensor optimization layout method and an S-EEM method for structure deformation reconstruction;
FIG. 6 is a graph of condition number versus sensor number for a two-step sequence strain sensor optimization layout method and an S-EEM method oriented to structural deformation reconstruction.
The specific implementation mode is as follows:
the following describes in detail specific embodiments of the present invention.
As shown in fig. 1, the two-step sequence strain sensor optimized layout method for structure deformation reconstruction includes:
(1) transpose matrix psi of modal strain matrix corresponding to candidate stationing position extracted from finite element modelTPerforming row principal component QR decomposition, and determining m linearly independent initial sensor arrangement sets, wherein m is a positive integer;
(2) and establishing a reconstruction precision criterion and an information redundancy criterion, then establishing a sensor layout optimization model, and continuously optimizing iteration in a step-by-step accumulation mode to determine the final sensor layout.
Further, the step (1) comprises:
(11) extracting a modal strain matrix psi corresponding to the candidate stationing position from the finite element model, and transposing the modal strain matrix to obtain a transposed matrix psiTThe expression of the modal strain matrix Ψ is as follows:
m is the intercepted modal number;
ndnumbering the candidate stationing positions at the maximum;
(12) to the transposed matrix ΨTCarrying out column principal component QR decomposition to obtain an R matrix (the R matrix is psi according to a linear independent ruleTThe psi can be screened out according to the arrangement of the first m main diagonal elements of the R matrixTThe corresponding first m column vectors in the matrix), from the transposed matrix Ψ according to the R matrixTScreening out m characteristic patternsGood (i.e. transposed matrix Ψ)TThe norm of all columns is arranged in the first m columns from large to small) and linearly independent column vectors, and the corresponding positions of the column vectors are used as initial sensor layout positions.
Further, the step (2) comprises:
(21) establishing a reconstruction accuracy criterion, and finding out a sensor position with a minimum covariance matrix trace of a reconstruction error from the remaining candidate positions
According to the principle of modal superposition
Obtaining a strain displacement conversion equation:
in the formula:
q (t) is the true displacement;
ε (t) is the true strain;
qm(t) is the modal coordinate;
Φ is a modal displacement matrix, whose expression is:
m is the number of interception modes, ndNumbering the candidate stationing positions at the maximum;
Ψ is a modal strain matrix expressed as:
m is the number of interception modes, ndNumbering the candidate stationing positions at the maximum;
ΨM(d) arranging a sub-matrix formed by rows corresponding to the positions of the strain sensors in the modal strain matrix;
Φsa sub-matrix composed of rows corresponding to the interest positions in the modal displacement matrix,
εm(t) is the measured strain, and the expression should be:
εm(t)＝ΨM(d)qm(t)+e(t) (4)
wherein e (t) is test noise;
so as to obtain:
and obtaining a deformation estimation error according to the formula (3) and the formula (5):
it is assumed that the measurement errors in equation (4) are independent of each other and follow a normal distribution, i.e.
The covariance expression of the estimation error is:
wherein E (. cndot.) is desired, E (E (t) ET(t)) is the covariance matrix of the measurement noise, and assuming that the measurement noise is independent of each other:
by substituting equation (8) for equation (7), the covariance matrix Δ of the estimation error can be obtained
Wherein each diagonal element in the delta matrix represents the variance of the corresponding deformation estimation error, the maximum number on the diagonal represents the maximum estimation error, and trace (delta) of the matrix represents the sum of the variances of all position deformation estimates;
due to the fact that
Being constant and therefore not considering its effect on the covariance of the reconstruction error, the average estimation error can be expressed as:
where trace (. cndot.) is the trace of the matrix, NsThe number of sensors in the set which is laid out;
(22) establishing an information redundancy criterion, and deleting positions with higher redundancy than the known sensor from the remaining candidate stationing positions;
in order to make the structural information acquired by the sensors relatively independent and avoid information redundancy among the sensors, a modal strain information matrix corresponding to the kth candidate stationing position is defined as follows:
in the formula, ΨkA is the modal strain vector corresponding to the k-th distributable point position, i.e. the k-th row of the modal strain matrix ΨkIs a m x m array matrix;
the modal strain information matrixes of two candidate distribution points with similar structure dynamic information are also similar, and the difference of the modal strain information matrixes can be measured by certain norm difference of the modal strain information matrixes:
Dki(d)＝||Ak-Ai(d)|| (12)
in the formula, Dki(d) The spatial distance difference of a modal strain information matrix corresponding to the kth candidate stationing position and the ith position of the arranged sensor set is obtained;
in order to better compare the difference degree of modal strain information among the sensors, the distance difference coefficient needs to be normalized, and the information redundancy degree among the sensors is expressed by the following formula:
since | | | Ak-Ai(d)||≤||Ak||+||Ai(d) The distance coefficient of difference is more than or equal to Rki(d)≤1；
R is to beki(d) And a redundancy threshold R0Comparing, and removing the residue less than R0The value of (a), namely, finishing deleting the position with larger redundancy than the known sensor from the rest candidate stationing positions;
(23) establishing a sensor layout optimization model, and selecting the position with the minimum current trace as the newly-added sensor layout position
Taking the minimum reconstruction error as an objective function and taking the redundancy of the sensor layout information and the average reconstruction error smaller than a preset threshold as constraint conditions, establishing the following mathematical model:
in the formula (I), the compound is shown in the specification,
d is the optimized sensor position;
Γfeasa feasible field for a sensor candidate position;
R0a redundancy threshold;
(24) performing optimization iteration and judging whether the termination condition is met
And deleting the position with higher redundancy than the known sensor from the candidate point distribution positions by utilizing a reconstruction precision criterion and an information redundancy criterion, finding out the sensor position which enables the covariance matrix trace of the reconstruction error to be minimum, continuously updating and optimizing the sensor position, and judging whether a final judgment condition is met, thereby outputting an optimized layout scheme.
Further, the termination conditions in step (24) are two of the following:
(241) whether the preset deformation reconstruction precision requirement is met or not;
(242) whether the maximum number S of sensors allowed by the system is reachedM。
Further, A in step (22)kAnd Ai(d) When being orthogonal, Rki(d) When the modal strain information contained by the sensor is not redundant, 1k＝Ai(d) When R iski(d) At 0, the modal strain information contained by the sensor is identical.
Referring to fig. 2, a schematic diagram of an antenna front finite element model established by ANSYS is shown. The thickness of the array surface is 6mm, and the appearance is a symmetrical octagon. The elastic modulus of the panel is 70GPa, the Poisson ratio is 0.3, and the density is 10044Kg/m3. Modeling is carried out by using an ANSYS finite element analysis software shell163 unit, 1264 node positions where the sensors cannot be installed in the panel edge and the loudspeaker installation hole are removed, and 3379 candidate nodes are remained. The length of the optical fiber grid region is about 10-15 mm, the distance between two adjacent nodes in the scribed square grid is 36mm, and the requirement of installation space can be met.
Referring to fig. 3a and fig. 3b, the sensor placement positions on the antenna array surface after the optimization of the two-step sequence strain sensor optimization layout method and the S-EEM method oriented to structural deformation reconstruction disclosed by the invention are respectively shown. The S-EEM method is characterized in that distribution points are approximately symmetrically distributed at the position where the cantilever end of the panel has large strain, the phenomenon that sensors are distributed in a concentrated mode occurs at the upper edge and the lower edge, the phenomenon is more serious along with the increase of the number of the sensors, the sensor optimization layout scheme of the two-step sequence strain sensor optimization layout method for structural deformation reconstruction is relatively dispersed in spatial distribution, the cantilever end, the middle and the edges of the panel are distributed, and the information redundancy phenomenon is not obvious. In contrast, the two-step sequence strain sensor optimized layout method for structural deformation reconstruction has more reasonable point distribution spatial distribution observability, and avoids the information redundancy phenomenon caused by centralized layout positions of the sensors.
Referring to fig. 4, a graph of the change of the calculation time of the two-step sequence strain sensor optimization layout method and the S-EEM method with the change of the number of sensors oriented to the structural deformation reconstruction is shown. Obviously, the calculation efficiency of the two-step sequence strain sensor optimized layout method for structural deformation reconstruction is far higher than that of the S-EEM method, and the calculation time is reduced by about 100 times compared with that of the S-EEM method. Therefore, aiming at a large space multi-degree-of-freedom structure, the two-step sequence method is more efficient than the S-EEM method under the same reconstruction precision, and is a simple and effective sensor layout method.
Referring to fig. 5, a graph of variation of maximum value of SMAC off-diagonal elements with the number of sensors is shown for a two-step sequence strain sensor optimization layout method and an S-EEM method for structure deformation reconstruction. When the number of the sensors is small, the difference between the maximum value of the SMAC off-diagonal elements of the two-step sequence strain sensor optimization layout method for structural deformation reconstruction and the maximum value of the SMAC off-diagonal elements of the S-EEM method is not large, the S-EEM method is suddenly increased when the number of the sensors reaches 13, the maximum value of the SMAC off-diagonal elements of the two-step sequence sensor layout is stable, and the maximum value of the SMAC off-diagonal elements is lower than that of the S-EEM method when the number of the sensors is large. Obviously, under the condition of a large number of sensors, the two-step sequence method is more reasonable than the S-EEM method, and can ensure a larger modal vector space intersection angle.
Referring to fig. 6, a graph of condition numbers of transformation matrices of a two-step sequence strain sensor optimization layout method and an S-EEM method for structure deformation reconstruction as a function of the number of sensors is shown. When the number of the sensors is less, the two-step sequence strain sensor optimization layout method for structural deformation reconstruction is slightly higher than the conversion matrix condition number of the S-EEM method, and the condition numbers of the conversion matrixes of the two methods are in a descending trend along with the increase of the number of the sensors, but when the number of the sensors is more than 9, the condition number of the conversion matrix is obviously lower than that of the S-EEM method. Although the S-EEM method generally tends to be down, there are fluctuations in the number of sensors deployed. Obviously, the two-step sequence strain sensor optimized layout method for structural deformation reconstruction is more reasonable in condition number and more stable in chemical effect compared with the sensor layout scheme of the S-EEM method.
The embodiments of the present invention have been described in detail. However, the present invention is not limited to the above-described embodiments, and various changes can be made within the knowledge of those skilled in the art without departing from the spirit of the present invention.
Claims (4)
1. The two-step sequence strain sensor optimization layout method oriented to structural deformation reconstruction is characterized by comprising the following steps:
(1) transpose matrix psi of modal strain matrix corresponding to candidate stationing position extracted from finite element modelTPerforming row principal component QR decomposition, and determining m linearly independent initial sensor arrangement sets, wherein m is a positive integer;
(2) establishing a reconstruction precision criterion and an information redundancy criterion, then establishing a sensor layout optimization model, and continuously optimizing iteration in a step-by-step accumulation mode to determine the final sensor layout, wherein:
the step (2) comprises the following steps:
(21) establishing a reconstruction accuracy criterion, and finding out a sensor position with a minimum covariance matrix trace of a reconstruction error from the remaining candidate positions
According to the principle of modal superposition
The strain-displacement transformation equation can be obtained:
in the formula:
q (t) is the true displacement;
ε (t) is the true strain;
qm(t) is the modal coordinate;
Φ is a modal displacement matrix, whose expression is:
m is the number of interception modes, ndNumbering the candidate stationing positions at the maximum;
Ψ is a modal strain matrix expressed as:
m is the number of interception modes, ndNumbering the candidate stationing positions at the maximum;
ΨM(d) arranging a sub-matrix formed by rows corresponding to the positions of the strain sensors in the modal strain matrix;
Φsa sub-matrix composed of rows corresponding to the interest positions in the modal displacement matrix,
εm(t) is the measured strain, and the expression should be:
εm(t)＝ΨM(d)qm(t)+e(t) (4)
wherein e (t) is test noise;
so as to obtain:
and obtaining a deformation estimation error according to the formula (3) and the formula (5):
it is assumed that the measurement errors in equation (4) are independent of each other and follow a normal distribution, i.e.
The covariance expression of the estimation error is:
wherein E (. cndot.) is desired, E (E (t) ET(t)) is the covariance matrix of the measurement noise, and assuming that the measurement noise is independent of each other:
by substituting equation (8) for equation (7), the covariance matrix Δ of the estimation error can be obtained
Wherein each diagonal element in the delta matrix represents the variance of the corresponding deformation estimation error, the maximum number on the diagonal represents the maximum estimation error, and trace (delta) of the matrix represents the sum of the variances of all position deformation estimates;
due to the fact that
Being constant and therefore not considering its effect on the covariance of the reconstruction error, the average estimation error can be expressed as:
where trace (. cndot.) is the trace of the matrix, NsThe number of sensors in the set which is laid out;
(22) establishing an information redundancy criterion, and deleting positions with higher redundancy than the known sensor from the remaining candidate stationing positions;
in order to make the structural information acquired by the sensors relatively independent and avoid information redundancy among the sensors, a modal strain information matrix corresponding to the kth candidate stationing position is defined as follows:
in the formula, ΨkA is the modal strain vector corresponding to the k-th distributable point position, i.e. the k-th row of the modal strain matrix ΨkIs a m x m array matrix;
the modal strain information matrixes of two candidate distribution points with similar structure dynamic information are also similar, and the difference of the modal strain information matrixes can be measured by certain norm difference of the modal strain information matrixes:
Dki(d)＝||Ak-Ai(d)|| (12)
in the formula, Dki(d) The spatial distance difference of a modal strain information matrix corresponding to the kth candidate stationing position and the ith position of the arranged sensor set is obtained;
in order to better compare the difference degree of modal strain information among the sensors, the distance difference coefficient needs to be normalized, and the information redundancy degree among the sensors is expressed by the following formula:
since | | | Ak-Ai(d)||≤||Ak||+||Ai(d) The distance coefficient of difference is more than or equal to Rki(d)≤1；
R is to beki(d) And a redundancy threshold R0Comparing, and removing the residue less than R0The value of (a), namely, finishing deleting the position with larger redundancy than the known sensor from the rest candidate stationing positions;
(23) establishing a sensor layout optimization model, and selecting the position with the minimum current trace as the newly-added sensor layout position
Taking the minimum reconstruction error as an objective function and taking the redundancy of the sensor layout information and the average reconstruction error smaller than a preset threshold as constraint conditions, establishing the following mathematical model:
in the formula (I), the compound is shown in the specification,
d is the optimized sensor position;
Γfeasa feasible field for a sensor candidate position;
R0a redundancy threshold;
(24) performing optimization iteration and judging whether the termination condition is met
And deleting the position with higher redundancy than the known sensor from the candidate point distribution positions by utilizing a reconstruction precision criterion and an information redundancy criterion, finding out the sensor position which enables the covariance matrix trace of the reconstruction error to be minimum, continuously updating and optimizing the sensor position, and judging whether a final judgment condition is met, thereby outputting an optimized layout scheme.
2. The two-step sequence strain sensor optimized layout method for structural deformation reconstruction as claimed in claim 1, wherein the step (1) comprises:
(11) extracting a modal strain matrix psi corresponding to the candidate stationing position from the finite element model, and transposing the modal strain matrix to obtain a transposed matrix psiTThe expression of the modal strain matrix Ψ is as follows:
m is the intercepted modal number;
ndnumbering the candidate stationing positions at the maximum;
(12) to the transposed matrix ΨTCarrying out column principal component QR decomposition to obtain an R matrix, and carrying out psi from the transposed matrix according to the R matrixTAnd (5) screening m linear independent column vectors with good personality, wherein the positions corresponding to the column vectors are used as the initial sensor layout positions.
3. The two-step sequence strain sensor optimized layout method for structure deformation reconstruction as claimed in claim 1, characterized in that the termination conditions in step (24) are two of the following:
(241) whether the preset deformation reconstruction precision requirement is met or not;
(242) whether the maximum number S of sensors allowed by the system is reachedM。
4. The two-step sequence strain sensor optimized layout method for structure deformation reconstruction as claimed in claim 1, wherein in step (22), AkAnd Ai(d) When being orthogonal, Rki(d) When the modal strain information contained by the sensor is not redundant, 1k＝Ai(d) When R iski(d) At 0, the modal strain information contained by the sensor is identical.
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