CN107515980A - Two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct - Google Patents
Two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct Download PDFInfo
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Abstract
The present invention relates to a kind of Optimal Deployment Method of sensor, and in particular to two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct, belongs to sensor optimization topology field.Two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct, including：(1) layout the transposed matrix Ψ of the modal strain matrix of position to the corresponding candidate extracted from FEM modelTPivot in a column QR decomposition is carried out, determines the initial sensor arrangement set of m Line independent, wherein, m is positive integer；(2) reconstruction accuracy criterion and information redundance criterion are established, then establishes sensor placement Optimized model, uses the mode progressively to add up to continue to optimize iteration to determine final sensor placement.
Description
Technical field
The present invention relates to a kind of Optimal Deployment Method of sensor, and in particular to two step sequences of structure-oriented deformation reconstruct
Strain transducer Optimal Deployment Method, belong to sensor optimization topology field.
Background technology
Structural deformation reconstructs the technical problem to be faced in addition to FEM updating, is exactly the optimization cloth of sensor
Office.Sensor optimization layout plays a part of taking over from the past and setting a new course for the future in structural deformation reconstruct, belongs to NP hard combinatorial optimization problems.
Rational sensor placement can not only optimize number of sensors, and can collect most comprehensive reconfiguration information.At certain
Under a little particular job environment, because the limitation of financial cost and structure in itself can only select a small amount of or certain amount of sensing
Device, and excessive sensor can bring the problems such as system failure rate increase, data storage and analysis difficulty.Therefore, pass
The purpose of sensor optimization layout is that the constraints according to system, and searching is N number of to survey M in the free degree (M ＜ N) individual sensor
Optimal measuring point scheme, obtain most abundant information with abundant reaction structure power performance.
Mainly include traditional derivation algorithm and the random major class of class derivation algorithm two than more typical sensor placement method.Pass
Derivation algorithm of uniting includes KEM method, Guyan "flop-out" methods, QR decomposition methods, minimum MAC methods, comentropy etc..Random class, which solves, to be calculated
Method includes genetic algorithm, particle cluster algorithm, wolf pack algorithm, monkey group's algorithm etc..
X.H.Zhang etc. optimizes layout to displacement and the two kinds of sensor of strain, progressively deletes in certain freedom
Degree causes the sensor of the minimum opening position of object function (mark of deformation reconstructed error covariance) after deleting, given until reaching
Threshold value.This method is in " X.H.Zhang, et al.Integrated optimal placement of displacement
transducers and strain gauges for better estimation of structural response
[J].International Journal of Structural Stability and Dynamics,2011,11(3):48-
Reported in 51. ".
The estimation error minimum method (Estimation error minimization, EEM) of the introductions such as Chen Wei is to utilize
The single displacement transducer optimization layout that X.H.Zhang thought is carried out.It is mainly used in relatively simple structure, is effective independence
The expansion of method, there is the essence of KEM method, the phenomenon of sensor aggregation occur when number of probes is more than mode number,
Serious redundancy occurs in the information that sensor obtains.This method is in " Chen Wei, Zhao Wenguang, Zhu Hongping, et al.Optimal
sensor placement for structural response estimation[J].Journal of Central
South University,2014,21(10):3993-4001. " in reported.
The above-mentioned method on sensor optimization layout has the following disadvantages：
1st, relatively simple structure is primarily adapted for use in, sensor aggregation occurs when number of probes is more than mode number
Serious redundancy occurs in phenomenon, the information that sensor obtains.
2nd, existing sensor optimization layout method is for the large and complex structure with a large amount of frees degree, due to final layout
Sensor it is relatively fewer, it is necessary to ideal allocation plan can just be obtained by deleting most of free degree, therefore, calculate suitable
It is time-consuming.
3rd, existing sensor placement method is primarily to the purpose of modal idenlification, Damage Assessment Method, health monitoring,
The method for being related to structure-oriented deformation reconstruct is also less.
Although some investigators introduce some new thoughts or criterion, most methods are all in these methods
On the basis of for its defect carry out improvement.Current research work lays particular emphasis on the optimization layout of displacement transducer, but strains
Sensor has the characteristics that light structure, resolution ratio and high sensitivity, and sensors with auxiliary electrode should have prior engineer applied
Value, but the Research Literature on strain transducer layout but compares shortcoming.
In view of the above-mentioned problems, the strain transducer Optimal Deployment Method of the main research structure-oriented deformation reconstruct of the present invention,
The final layout position of sensor will not only can guarantee that less reconstructed error, and be obtained as far as possible with less sensor
More structural informations, i.e., the information that each sensor is surveyed not redundancy as far as possible.For it is above-mentioned the problem of, the present invention be based on modal method
Deformation re-construction theory, consider shadow of the information redundance to sensor placement of influence and sensor of the test error to reconstruction accuracy
Ring, it is proposed that two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct.
The content of the invention
Goal of the invention：The present invention has made improvements in view of the above-mentioned problems of the prior art, i.e., the invention discloses towards
Structural deformation reconstruct two step sequence strain transducer Optimal Deployment Methods, its can not only effectively realize number of sensors and
The optimization of position and operation efficiency is greatly improved.
Technical scheme：Two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct, including：
(1) the corresponding candidate extracted from FEM model is layouted the transposed matrix of the modal strain matrix of position
ΨTPivot in a column QR decomposition is carried out, determines the initial sensor arrangement set of m Line independent, wherein, m is positive integer；
(2) reconstruction accuracy criterion and information redundance criterion are established, then establishes sensor placement Optimized model, using by
The cumulative mode of step continues to optimize iteration to determine final sensor placement.
Further, step (1) includes：
(11) the corresponding candidate extracted from FEM model layouts the modal strain matrix Ψ of position, and should to mode
Become matrix progress transposition and obtain transposed matrix ΨT, modal strain matrix Ψ expression formula is as follows：
Wherein：
M is the mode number of interception；
ndBeing layouted for candidate, position is maximum to number；
(12) to transposed matrix ΨTPivot in a column QR decomposition is carried out, obtaining R matrixes, (R matrixes are exactly according to Line independent rule
To ΨTRow rearranged after matrix, according to the preceding m the elements in a main diagonal arranging situation of R matrixes, can screen
Go out ΨTCorresponding m column vector in matrix), according to R matrixes from transposed matrix ΨTIn filter out m condition preferably and it is linear solely
Vertical column vector, position is as initial sensor placement position corresponding to the column vector.
Further, step (2) includes：
(21) reconstruction accuracy criterion is established, being found out from remaining position candidate makes the covariance matrix mark of reconstructed error minimum
Sensing station
According to modal superposition principle
Strain displacement transfer equation can be obtained：
In formula：
Q (t) is real displacement；
ε (t) is logarithmic strain；
qm(t) it is modal coordinate；
Φ is modal displacement matrix, and its expression formula is：
M is to intercept mode number, ndBeing layouted for candidate, position is maximum to number；
Ψ is modal strain matrix, and its expression formula is：
M is to intercept mode number, ndBeing layouted for candidate, position is maximum to number；
ΨM(d) submatrix formed by arranging the row of strain transducer position correspondence in modal strain matrix；
ΦsThe submatrix being made up of the row corresponding to interested position in modal displacement matrix,
εm(t) it is measuring strain, its expression formula should be：
εm(t)=ΨM(d)qm(t)+e(t) (4)
In formula, e (t) is test noise；
So obtain：
Deformation evaluated error is obtained according to formula (3) and formula (6)：
Assuming that measurement error in formula (4) is independently of each other and Normal Distribution, i.e.,Then estimation misses
Difference covariance expression formula be：
In formula, E () is it is expected, E (e (t) eT(t) it is) covariance matrix of measurement noise, and it has assumed that measurement noise
It is independent mutually, then：
Formula (8) is substituted into formula (7), estimation error covariance matrix Δ can be obtained
In formula, each diagonal element in Δ matrix represents corresponding deformation variance of estimaion error, on diagonal most
Big numerical value represents maximum evaluated error, and the mark trace (Δ) of matrix represent all position deformation estimations variance and；
Due toFor constant, therefore its influence to reconstructing error covariance can not be considered, averaged power spectrum error can represent
For：
In formula, trace () be matrix mark, NsFor the number of sensors being laid out in set；
(22) information redundance criterion is established, is deleted in layouting position from remaining candidate larger with known sensor redundancy
Position；
In order that sensor obtains, structural information is relatively independent, avoids the information redundancy between sensor, it is specified that k-th of time
The layout modal strain information matrix of position correspondence of choosing is：
In formula, ΨkFor the modal strain vector of k-th position correspondence that can layout, i.e. modal strain matrix Ψ row k,
And AkFor m × m matrix poised for battle；
The modal strain information matrix that two candidates with close structural multidate information layout is also close, modal strain letter
The difference of breath matrix can be measured by their certain norm difference：
Dki(d)=| | Ak-Ai(d)|| (12)
In formula, Dki(d) layouting the mode of position and i-th of position correspondence of placement sensor set for k-th of candidate should
Become the space length difference of information matrix；
In order to modal strain information between preferably more each sensor difference degree, it is necessary to which coefficient of variation of adjusting the distance enters
Row Regularization, information redundancy degree is represented with equation below between sensor：
Due to | | Ak-Ai(d)||≤||Ak||+||Ai(d) | | set up for any mode strain information matrix, so
0≤R of distance difference coefficientki(d)≤1；
By Rki(d) with redundancy threshold values R0Contrasted, rejecting is less than R0Value, that is, complete to layout position from remaining candidate
Middle deletion and the larger position of known sensor redundancy；
(23) sensor placement Optimized model is established, elects the minimum position of current mark as newly-increased sensor placement position
With the minimum object function of reconstructed error, it is less than with sensor placement information redundance peace reconstructed error default
Threshold values is constraints, establishes following mathematical modeling：
In formula,
D is the sensing station of optimization；
For the threshold values of averaged power spectrum error；
Layouted for k-th of candidate the minimal redundancy angle value of position；
ΓfeasFor sensor position candidate feasible zone；
R0For redundancy threshold values；
(24) optimize iteration and judge whether to meet end condition
Using reconstruction accuracy criterion and information redundance criterion, deleted and known sensor redundancy in layouting position from candidate
Larger position is spent, finds out the sensing station for making the covariance matrix mark of reconstructed error minimum, to sensing station constantly more
New optimization, and judge whether to meet final decision condition, so as to export optimization placement scheme.
Further, the end condition in step (24) has following two：
(241) predetermined deformation reconstruction accuracy requirement whether is met；
(242) the maximum sensor number S of system permission whether is reachedM。
Further, A in step (22)kAnd Ai(d) when orthogonal, Rki(d) modal strain that=1, now sensor includes
Information is irredundant, Ak=Ai(d) when, Rki(d) the modal strain information that=0, now sensor includes is identical.
Beneficial effect：Two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct disclosed by the invention
Have the advantages that：
1. two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct are proposed, can be simultaneously to passing
Sensor quantity and position optimize, and are contrasted with the estimation error minimum method (S-EEM) based on strain, with the obvious advantage.
2. the re-construction theory based on modal method, consider influence and sensor redundancy degree pair of the test error to reconstruction accuracy
The influence of sensor placement.
3. Level Multiple Degree of Freedom Structures is directed to, a small amount of sensor placement problem, two step sequence strain transducer Optimal Deployment Methods
Computational efficiency can be greatly enhanced, rejects redundancy.
Brief description of the drawings
Fig. 1 is the stream of two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct disclosed by the invention
Cheng Tu；
Fig. 2 is the antenna array FEM model schematic diagram that ANSYS is established；
Fig. 3 a are that the two step sequence strain transducers reconstructed by structure-oriented deformation disclosed by the invention optimize layout side
Sensor on method optimization aft antenna front is layouted position；
Fig. 3 b are that the sensor on S-EEM methods optimization aft antenna front is layouted position；
Fig. 4 is that two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct and S-EEM methods calculate
Time with number of sensors change curve；
Fig. 5 is the two step sequence strain transducer Optimal Deployment Methods and S-EEM methods of structure-oriented deformation reconstruct
SMAC with number of sensors change curve；
Fig. 6 is two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct and the bar of S-EEM methods
Number of packages with number of sensors change curve.
Embodiment：
The embodiment of the present invention is described in detail below.
As shown in figure 1, two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct, including：
(1) the corresponding candidate extracted from FEM model is layouted the transposed matrix of the modal strain matrix of position
ΨTPivot in a column QR decomposition is carried out, determines the initial sensor arrangement set of m Line independent, wherein, m is positive integer；
(2) reconstruction accuracy criterion and information redundance criterion are established, then establishes sensor placement Optimized model, using by
The cumulative mode of step continues to optimize iteration to determine final sensor placement.
Further, step (1) includes：
(11) the corresponding candidate extracted from FEM model layouts the modal strain matrix Ψ of position, and should to mode
Become matrix progress transposition and obtain transposed matrix ΨT, modal strain matrix Ψ expression formula is as follows：
Wherein：
M is the mode number of interception；
ndBeing layouted for candidate, position is maximum to number；
(12) to transposed matrix ΨTPivot in a column QR decomposition is carried out, obtaining R matrixes, (R matrixes are exactly according to Line independent rule
To ΨTRow rearranged after matrix, according to the preceding m the elements in a main diagonal arranging situation of R matrixes, can screen
Go out ΨTCorresponding preceding m column vector in matrix), according to R matrixes from transposed matrix ΨTIn filter out m condition preferably (i.e. transposition
Matrix ΨTThe norm of all row according to from big to small arrange after preceding m arrange) and Line independent column vector, the column vector pair
The position answered is as initial sensor placement position.
Further, step (2) includes：
(21) reconstruction accuracy criterion is established, being found out from remaining position candidate makes the covariance matrix mark of reconstructed error minimum
Sensing station
According to modal superposition principle
Obtain strain displacement transfer equation：
In formula：
Q (t) is real displacement；
ε (t) is logarithmic strain；
qm(t) it is modal coordinate；
Φ is modal displacement matrix, and its expression formula is：
M is to intercept mode number, ndBeing layouted for candidate, position is maximum to number；
Ψ is modal strain matrix, and its expression formula is：
M is to intercept mode number, ndBeing layouted for candidate, position is maximum to number；
ΨM(d) submatrix formed by arranging the row of strain transducer position correspondence in modal strain matrix；
ΦsThe submatrix being made up of the row corresponding to interested position in modal displacement matrix,
εm(t) it is measuring strain, its expression formula should be：
εm(t)=ΨM(d)qm(t)+e(t) (4)
In formula, e (t) is test noise；
So obtain：
Deformation evaluated error is obtained according to formula (3) and formula (6)：
Assuming that measurement error in formula (4) is independently of each other and Normal Distribution, i.e.,Then estimation misses
Difference covariance expression formula be：
In formula, E () is it is expected, E (e (t) eT(t) it is) covariance matrix of measurement noise, and it has assumed that measurement noise
It is independent mutually, then：
Formula (8) is substituted into formula (7), estimation error covariance matrix Δ can be obtained
In formula, each diagonal element in Δ matrix represents corresponding deformation variance of estimaion error, on diagonal most
Big numerical value represents maximum evaluated error, and the mark trace (Δ) of matrix represent all position deformation estimations variance and；
Due toFor constant, therefore its influence to reconstructing error covariance can not be considered, averaged power spectrum error can represent
For：
In formula, trace () be matrix mark, NsFor the number of sensors being laid out in set；
(22) information redundance criterion is established, is deleted in layouting position from remaining candidate larger with known sensor redundancy
Position；
In order that sensor obtains, structural information is relatively independent, avoids the information redundancy between sensor, it is specified that k-th of time
The layout modal strain information matrix of position correspondence of choosing is：
In formula, ΨkFor the modal strain vector of k-th position correspondence that can layout, i.e. modal strain matrix Ψ row k,
And AkFor m × m matrix poised for battle；
The modal strain information matrix that two candidates with close structural multidate information layout is also close, modal strain letter
The difference of breath matrix can be measured by their certain norm difference：
Dki(d)=| | Ak-Ai(d)|| (12)
In formula, Dki(d) layouting the mode of position and i-th of position correspondence of placement sensor set for k-th of candidate should
Become the space length difference of information matrix；
In order to modal strain information between preferably more each sensor difference degree, it is necessary to which coefficient of variation of adjusting the distance enters
Row Regularization, information redundancy degree is represented with equation below between sensor：
Due to | | Ak-Ai(d)||≤||Ak||+||Ai(d) | | set up for any mode strain information matrix, so
0≤R of distance difference coefficientki(d)≤1；
By Rki(d) with redundancy threshold values R0Contrasted, rejecting is less than R0Value, that is, complete to layout position from remaining candidate
Middle deletion and the larger position of known sensor redundancy；
(23) sensor placement Optimized model is established, elects the minimum position of current mark as newly-increased sensor placement position
With the minimum object function of reconstructed error, it is less than with sensor placement information redundance peace reconstructed error default
Threshold values is constraints, establishes following mathematical modeling：
In formula,
D is the sensing station of optimization；
For the threshold values of averaged power spectrum error；
Layouted for k-th of candidate the minimal redundancy angle value of position；
ΓfeasFor sensor position candidate feasible zone；
R0For redundancy threshold values；
(24) optimize iteration and judge whether to meet end condition
Using reconstruction accuracy criterion and information redundance criterion, deleted and known sensor redundancy in layouting position from candidate
Larger position is spent, finds out the sensing station for making the covariance matrix mark of reconstructed error minimum, to sensing station constantly more
New optimization, and judge whether to meet final decision condition, so as to export optimization placement scheme.
Further, the end condition in step (24) has following two：
(241) predetermined deformation reconstruction accuracy requirement whether is met；
(242) the maximum sensor number S of system permission whether is reachedM。
Further, A in step (22)kAnd Ai(d) when orthogonal, Rki(d) modal strain that=1, now sensor includes
Information is irredundant, Ak=Ai(d) when, Rki(d) the modal strain information that=0, now sensor includes is identical.
Reference picture 2, it is the antenna array FEM model schematic diagram that ANSYS is established.The front thickness is 6mm, and profile is
Symmetrical octagon.Panel modulus of elasticity is 70GPa, Poisson's ratio 0.3, density 10044Kg/m3.Utilize ANSYS finite element analyses
Software shell163 units are modeled, and remove face plate edge and 1264 nodes for being unable to install sensor at loudspeaker mounting hole
Position, 3379 both candidate nodes of common residue.Optical fiber grid region length is about 10~15mm, draws close two node in square net
Distance be 36mm, the needs of installing space can be met.
Reference picture 3a and Fig. 3 b are two step sequence strain transducers of structure-oriented deformation reconstruct disclosed by the invention respectively
Sensor on Optimal Deployment Method and S-EEM methods optimization aft antenna front is layouted position.S-EEM methods are layouted near symmetrical
Ground is distributed in panel cantilever end and strains larger opening position, occurs the phenomenon of sensor integrated distribution at lower edges, and with
This phenomenon of increase of number of probes is more serious, and the two step sequence strain transducers optimization layout of structure-oriented deformation reconstruct
The sensor optimization placement scheme spatial distribution of method is more dispersed, and panel cantilever end, centre and edge are distributed, information
Redundancy phenomena unobvious.Comparatively, the cloth of two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct
Space of points distribution ornamental is more reasonable, avoids information redundancy phenomenon caused by being concentrated because of sensor placement position.
Reference picture 4, it is the two step sequence strain transducer Optimal Deployment Methods and S-EEM methods of structure-oriented deformation reconstruct
Calculate change curve of the time with number of sensors.The obvious two step sequences strain that can be seen that structure-oriented deformation reconstruct passes
The computational efficiency of sensor Optimal Deployment Method is far above S-EEM methods, and the calculating time reduces 100 times or so than S-EEM method.
Therefore, for large space Level Multiple Degree of Freedom Structures, two step sequence methods relative S-EEM methods under equal reconstruction accuracy are more efficient,
It is a kind of simple and direct effective sensor layout method.
Reference picture 5, it is the two step sequence strain transducer Optimal Deployment Methods and S-EEM methods of structure-oriented deformation reconstruct
Corresponding SMAC off diagonal elements maximum with number of sensors change curve.When number of sensors is less, towards knot
Configuration becomes two step sequence strain transducer Optimal Deployment Methods of reconstruct and the SMAC off diagonal element maximums of S-EEM methods
It is more or less the same, there is unexpected increased phenomenon in S-EEM methods when number of sensors reaches 13, almost tend to be steady afterwards,
And the SMAC off-diagonal element maximums of two step sequence method sensor placements are more stable, and the SMAC when number of sensors is larger
Off-diagonal element maximum is always below S-EEM methods.Obviously, in the case where number of sensors is more, two step sequence methods compared with
S-EEM methods are more reasonable, can ensure the larger modal vector space angle of cut.
Reference picture 6, it is the two step sequence strain transducer Optimal Deployment Methods and S-EEM methods of structure-oriented deformation reconstruct
The conditional number of transition matrix with number of sensors change curve.When number of probes is less, structure-oriented deformation reconstruct
Two step sequence strain transducer Optimal Deployment Methods are more slightly higher than the transition matrix conditional number of S-EEM method, with number of sensors
Increase, the conditional number of the transition matrix of two methods is on a declining curve, but two step sequences of structure-oriented deformation reconstruct should
Become sensor optimization layout method when number of sensors is more than 9, the conditional number of transition matrix is significantly lower than S-EEM methods.Though
Right S-EEM methods are integrally on a declining curve, but have wave phenomenon when arranging some number of sensors.Obviously, structure-oriented
Two step sequence strain transducer Optimal Deployment Methods of deformation reconstruct are in terms of conditional number compared with S-EEM method sensor placement schemes
More reasonable and change effect is also more stable.
Embodiments of the present invention are elaborated above.But the present invention is not limited to above-mentioned embodiment,
In art those of ordinary skill's possessed knowledge, it can also be done on the premise of present inventive concept is not departed from
Go out various change.
Claims (5)
1. two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct, it is characterised in that including：
(1) layout the transposed matrix Ψ of the modal strain matrix of position to the corresponding candidate extracted from FEM modelTCarry out
Pivot in a column QR is decomposed, and determines the initial sensor arrangement set of m Line independent, wherein, m is positive integer；
(2) reconstruction accuracy criterion and information redundance criterion are established, then establishes sensor placement Optimized model, use is progressively tired out
The mode added continues to optimize iteration to determine final sensor placement.
2. two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct as claimed in claim 1, it is special
Sign is that step (1) includes：
(11) the corresponding candidate extracted from FEM model layouts the modal strain matrix Ψ of position, and to modal strain square
Battle array carry out transposition obtains transposed matrix ΨT, modal strain matrix Ψ expression formula is as follows：
Wherein：
M is the mode number of interception；
ndBeing layouted for candidate, position is maximum to number；
(12) to transposed matrix ΨTPivot in a column QR decomposition is carried out, obtains R matrixes, according to R matrixes from transposed matrix ΨTIn filter out
M condition preferably and Line independent column vector, position is as initial sensor placement position corresponding to the column vector.
3. two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct as claimed in claim 1, it is special
Sign is that step (2) includes：
(21) reconstruction accuracy criterion is established, the biography for making the covariance matrix mark of reconstructed error minimum is found out from remaining position candidate
Sensor position
According to modal superposition principle
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In formula：
Q (t) is real displacement；
ε (t) is logarithmic strain；
qm(t) it is modal coordinate；
Φ is modal displacement matrix, and its expression formula is：
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</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>&phi;</mi>
<mn>12</mn>
</msub>
</mtd>
<mtd>
<msub>
<mi>&phi;</mi>
<mn>22</mn>
</msub>
</mtd>
<mtd>
<mo>...</mo>
</mtd>
<mtd>
<msub>
<mi>&phi;</mi>
<mrow>
<mi>m</mi>
<mn>2</mn>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>...</mo>
</mtd>
<mtd>
<mo>...</mo>
</mtd>
<mtd>
<mo>...</mo>
</mtd>
<mtd>
<mo>...</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>&phi;</mi>
<mrow>
<mn>1</mn>
<msub>
<mi>n</mi>
<mi>d</mi>
</msub>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>&phi;</mi>
<mrow>
<mn>2</mn>
<msub>
<mi>n</mi>
<mi>d</mi>
</msub>
</mrow>
</msub>
</mtd>
<mtd>
<mo>...</mo>
</mtd>
<mtd>
<msub>
<mi>&phi;</mi>
<mrow>
<msub>
<mi>mn</mi>
<mi>d</mi>
</msub>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
M is to intercept mode number, ndBeing layouted for candidate, position is maximum to number；
Ψ is modal strain matrix, and its expression formula is：
M is to intercept mode number, ndBeing layouted for candidate, position is maximum to number；
ΨM(d) submatrix formed by arranging the row of strain transducer position correspondence in modal strain matrix；
ΦsThe submatrix being made up of the row corresponding to interested position in modal displacement matrix,
<mrow>
<msubsup>
<mi>&Psi;</mi>
<mi>M</mi>
<mo>+</mo>
</msubsup>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&Psi;</mi>
<mi>M</mi>
<mi>T</mi>
</msubsup>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
<msub>
<mi>&Psi;</mi>
<mi>M</mi>
</msub>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<msubsup>
<mi>&Psi;</mi>
<mi>M</mi>
<mi>T</mi>
</msubsup>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mi>T</mi>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>&Phi;</mi>
<mi>s</mi>
</msub>
<msubsup>
<mi>&Psi;</mi>
<mi>M</mi>
<mo>+</mo>
</msubsup>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mo>;</mo>
</mrow>
εm(t) it is measuring strain, its expression formula should be：
εm(t)=ΨM(d)qm(t)+e(t) (4)
In formula, e (t) is test noise；
So obtain：
Deformation evaluated error is obtained according to formula (3) and formula (6)：
Assuming that measurement error in formula (4) is independently of each other and Normal Distribution, i.e.,Then evaluated error
Covariance expression formula is：
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<mi>E</mi>
<mrow>
<mo>(</mo>
<mi>&delta;</mi>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
<msup>
<mi>&delta;</mi>
<mi>T</mi>
</msup>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>E</mi>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
<mi>e</mi>
<mo>(</mo>
<mi>t</mi>
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<msup>
<mi>e</mi>
<mi>T</mi>
</msup>
<mo>(</mo>
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<mi>T</mi>
<mi>T</mi>
</msup>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<mi>T</mi>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mi>E</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>e</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<msup>
<mi>e</mi>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<msup>
<mi>T</mi>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, E () is it is expected, E (e (t) eT(t) it is) covariance matrix of measurement noise, and it has assumed that measurement noise is only mutually
It is vertical, then：
Formula (8) is substituted into formula (7), estimation error covariance matrix Δ can be obtained
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<mi>&Delta;</mi>
<mo>=</mo>
<mi>E</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>&delta;</mi>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<msup>
<mi>&delta;</mi>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<mi>T</mi>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mi>E</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>e</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<msup>
<mi>e</mi>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<msup>
<mi>T</mi>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mi>&Sigma;</mi>
</mrow>
<mo>)</mo>
</mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mi>&Sigma;</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<msubsup>
<mi>e</mi>
<mi>&epsiv;</mi>
<mn>2</mn>
</msubsup>
<mi>T</mi>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<msup>
<mi>T</mi>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, each diagonal element in Δ matrix represents corresponding deformation variance of estimaion error, the maximum number on diagonal
Value represents maximum evaluated error, and the mark trace (Δ) of matrix represent all position deformation estimations variance and；
Due toFor constant, therefore its influence to reconstructing error covariance can not be considered, averaged power spectrum error can be expressed as：
<mrow>
<msubsup>
<mi>e</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
<mi>g</mi>
</mrow>
<mn>2</mn>
</msubsup>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>t</mi>
<mi>r</mi>
<mi>a</mi>
<mi>c</mi>
<mi>e</mi>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
<msup>
<mi>T</mi>
<mi>T</mi>
</msup>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
<mo>)</mo>
</mrow>
</mrow>
<msub>
<mi>N</mi>
<mi>s</mi>
</msub>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, trace () be matrix mark, NsFor the number of sensors being laid out in set；
(22) information redundance criterion is established, is deleted and the larger position of known sensor redundancy in layouting position from remaining candidate
Put；
In order that sensor acquisition structural information is relatively independent, the information redundancy between sensor is avoided, it is specified that k-th of candidate's cloth
Point position correspondence modal strain information matrix be：
<mrow>
<msub>
<mi>A</mi>
<mi>k</mi>
</msub>
<mo>=</mo>
<msubsup>
<mi>&Psi;</mi>
<mi>k</mi>
<mi>T</mi>
</msubsup>
<msub>
<mi>&Psi;</mi>
<mi>k</mi>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>11</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, ΨkFor the modal strain vector of k-th position correspondence that can layout, i.e. modal strain matrix Ψ row k, and AkFor
M × m matrix poised for battle；
The modal strain information matrix that two candidates with close structural multidate information layout is also close, modal strain information square
The difference of battle array can be measured by their certain norm difference：
Dki(d)=| | Ak-Ai(d)|| (12)
In formula, Dki(d) the modal strain letter of position and i-th of the position correspondence of placement sensor set of being layouted for k-th of candidate
Cease the space length difference of matrix；
In order between preferably more each sensor modal strain information difference degree, it is necessary to adjust the distance coefficient of variation carry out just
Then change is handled, and information redundancy degree is represented with equation below between sensor：
<mrow>
<msub>
<mi>R</mi>
<mrow>
<mi>k</mi>
<mi>i</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mo>|</mo>
<mo>|</mo>
<msub>
<mi>A</mi>
<mi>k</mi>
</msub>
<mo>-</mo>
<msub>
<mi>A</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mo>|</mo>
<mo>|</mo>
</mrow>
<mrow>
<mo>|</mo>
<mo>|</mo>
<msub>
<mi>A</mi>
<mi>k</mi>
</msub>
<mo>|</mo>
<mo>|</mo>
<mo>+</mo>
<mo>|</mo>
<mo>|</mo>
<msub>
<mi>A</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mo>|</mo>
<mo>|</mo>
</mrow>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>13</mn>
<mo>)</mo>
</mrow>
</mrow>
Due to | | Ak-Ai(d)||≤||Ak||+||Ai(d) | | set up for any mode strain information matrix, so range difference
Different 0≤R of coefficientki(d)≤1；
By Rki(d) with redundancy threshold values R0Contrasted, rejecting is less than R0Value, that is, complete to delete in layouting position from remaining candidate
Except the position larger with known sensor redundancy；
(23) sensor placement Optimized model is established, elects the minimum position of current mark as newly-increased sensor placement position
With the minimum object function of reconstructed error, pre-set threshold value is less than with sensor placement information redundance peace reconstructed error
For constraints, following mathematical modeling is established：
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<mi>F</mi>
<mi>i</mi>
<mi>n</mi>
<mi>d</mi>
<mo>:</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>d</mi>
<mo>=</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<mrow>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo>,</mo>
<mi>z</mi>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mi>T</mi>
</msup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>M</mi>
<mi>i</mi>
<mi>n</mi>
<mo>:</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>t</mi>
<mi>r</mi>
<mi>a</mi>
<mi>c</mi>
<mi>e</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>T</mi>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<msup>
<mi>T</mi>
<mi>T</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>S</mi>
<mo>.</mo>
<mi>t</mi>
<mo>.</mo>
</mrow>
</mtd>
<mtd>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msubsup>
<mi>e</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
<mi>g</mi>
</mrow>
<mn>2</mn>
</msubsup>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mo>&le;</mo>
<mrow>
<mo>&lsqb;</mo>
<msubsup>
<mi>e</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
<mi>g</mi>
</mrow>
<mn>2</mn>
</msubsup>
<mo>&rsqb;</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msubsup>
<mi>R</mi>
<mi>k</mi>
<mi>min</mi>
</msubsup>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mo>&GreaterEqual;</mo>
<msub>
<mi>R</mi>
<mn>0</mn>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>d</mi>
<mo>&Element;</mo>
<msub>
<mi>&Gamma;</mi>
<mrow>
<mi>f</mi>
<mi>e</mi>
<mi>a</mi>
<mi>s</mi>
</mrow>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>14</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula,
D is the sensing station of optimization；
For the threshold values of averaged power spectrum error；
Layouted for k-th of candidate the minimal redundancy angle value of position；
ΓfeasFor sensor position candidate feasible zone；
R0For redundancy threshold values；
(24) optimize iteration and judge whether to meet end condition
Using reconstruction accuracy criterion and information redundance criterion, deleted in layouting position from candidate with known sensor redundancy compared with
Big position, the sensing station for making the covariance matrix mark of reconstructed error minimum is found out, sensing station is constantly updated excellent
Change, and judge whether to meet final decision condition, so as to export optimization placement scheme.
4. two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct as claimed in claim 3, it is special
Sign is that the end condition in step (24) has following two：
(241) predetermined deformation reconstruction accuracy requirement whether is met；
(242) the maximum sensor number S of system permission whether is reachedM。
5. two step sequence strain transducer Optimal Deployment Methods of structure-oriented deformation reconstruct as claimed in claim 3, it is special
Sign is, A in step (22)kAnd Ai(d) when orthogonal, Rki(d) the modal strain information that=1, now sensor includes is irredundant,
Ak=Ai(d) when, Rki(d) the modal strain information that=0, now sensor includes is identical.
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| CN109766617A (en) * | 2019-01-02 | 2019-05-17 | 大连理工大学 | A kind of displacement field reconstructing method based on strain transducer |
| CN109870134A (en) * | 2019-03-22 | 2019-06-11 | 西安交通大学 | A kind of contactless dynamic strain field measurement method of rotating vane and its system |
| CN110059373A (en) * | 2019-04-01 | 2019-07-26 | 南京航空航天大学 | Wing strain field reconstructed distribution formula optical fiber calculation method based on modal superposition principle |
| CN110611522A (en) * | 2019-09-20 | 2019-12-24 | 广东石油化工学院 | PLC signal reconstruction method and system using multiple regular optimization theory |
| CN112749470A (en) * | 2019-10-31 | 2021-05-04 | 北京华航无线电测量研究所 | Optimal fitting method for structural deformation sensor layout |
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| CN108875178A (en) * | 2018-06-04 | 2018-11-23 | 大连理工大学 | For reducing the probabilistic sensor arrangement method of distinguishing structural mode |
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| CN110059373A (en) * | 2019-04-01 | 2019-07-26 | 南京航空航天大学 | Wing strain field reconstructed distribution formula optical fiber calculation method based on modal superposition principle |
| CN110611522A (en) * | 2019-09-20 | 2019-12-24 | 广东石油化工学院 | PLC signal reconstruction method and system using multiple regular optimization theory |
| CN112749470A (en) * | 2019-10-31 | 2021-05-04 | 北京华航无线电测量研究所 | Optimal fitting method for structural deformation sensor layout |
| CN116132882A (en) * | 2022-12-22 | 2023-05-16 | 苏州上声电子股份有限公司 | Method for determining installation position of loudspeaker |
| CN116132882B (en) * | 2022-12-22 | 2024-03-19 | 苏州上声电子股份有限公司 | Method for determining installation position of loudspeaker |
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