Deformation monitoring sensor optimization arrangement method based on Fisher information matrix

Abstract

The invention discloses a Fisher information matrix-based deformation monitoring sensor optimization arrangement method. The method comprises the steps of defining observation variables and unknown parameters; establishing a relation matrix; calculating a probability density function and a quality function of the observation variable; calculating a Fisher information matrix according to the deformation reconstruction equation; the traces of the Fisher information matrix are maximized to achieve optimal arrangement of the sensors. According to the method, the relation between the Fisher information matrix and the deformation reconstruction precision is deduced from the basic principle of an inverse finite element, optimal arrangement of the sensor is completed by maximizing the trace of the Fisher information matrix represented by the overall stiffness matrix, and efficient optimal arrangement of the sensor is achieved.

Description

Technical Field

[0001] The present invention belongs to the technical field of aerospace structural health monitoring, and particularly relates to an optimized layout method for deformation monitoring sensors based on the Fisher information matrix. Background Art

[0002] An airborne conformal antenna is an antenna array that conforms to the shape of an aircraft platform. The telecommunications components and the fuselage structural components share functions. It has the characteristics of not affecting the aerodynamic shape of the aircraft, not increasing the radar cross-section of the aircraft, maximizing the aperture of the antenna array, and having strong target recognition and detection capabilities. It is widely used in various advanced aircraft and vehicles. During the service life of an aircraft, the complex time-varying environment will cause unpredictable deformations in the structure, and complex thermal couplings will occur with the aircraft frame, skin or other structures of the fuselage, resulting in various complex deformations. Such deformations will cause the positions of the antenna array elements to shift, leading to problems such as a decrease in gain, poor directivity, and beam pointing deviation, seriously restricting the positioning accuracy and resolution of the conformal array antenna, and even seriously affecting the normal operation of the airborne conformal antenna. To ensure the normal operation of the airborne conformal antenna, it is necessary to correct the array amplitude-phase error caused by structural deformation and perform feedback control on the element beams, and real-time and reliable deformation monitoring is the basis for correction and compensation.

[0003] In the field of aerospace, resistance strain gauges or fiber Bragg grating sensors are widely used because of their advantages such as light weight and small parameters. They can be directly arranged on the surface of complex structures to directly collect their strain information, and combined with deformation reconstruction algorithms, real-time online deformation monitoring of the structure can be achieved, while meeting the design requirements of aerospace structure lightweighting. Among the existing deformation reconstruction algorithms, the inverse finite element method has received extensive attention because of its advantages such as not requiring material parameters and load information and high reconstruction accuracy.

[0004] The deformation reconstruction accuracy of the inverse finite element method highly depends on the number and layout positions of the sensors. The existing optimization layout methods often directly give the sensor layout depending on engineering experience, or use intelligent algorithms to solve the optimal layout. The existing methods cannot give the principle of optimized sensor layout, and the sensor optimization layout based on intelligent algorithms is time-consuming. Summary of the Invention

[0005] Aiming at the deficiencies of the above-mentioned existing technologies, the purpose of the present invention is to provide an optimized layout method for deformation monitoring sensors based on the Fisher information matrix to solve the problems of unclear optimization layout principle and large calculation amount in the existing sensor optimization layout based on inverse finite element deformation reconstruction technology.

[0006] To achieve the above purpose, the technical solution adopted by the present invention is as follows:

[0007] An optimization layout method for deformation monitoring sensors based on the Fisher information matrix of the present invention is as follows:

[0008] 1) Define the observation variable F and the unknown parameter U;

[0009] 2) Establish the relationship matrix K;

[0010] 3) Calculate the probability density function and the quality function of the observation variable F;

[0011] 4) Calculate the Fisher information matrix according to the deformation reconstruction equation;

[0012] 5) Maximize the trace of the Fisher information matrix to achieve the optimal layout of the sensors.

[0013] Further, the step 1) specifically includes:

[0014] Divide the structure into inverse elements, and establish an element-class load vector F for each inverse element. e The element-class load vector is obtained from the measured values of the strain sensors; then assemble the element-class load vectors to obtain the observation variable F, and the expression is as follows:

[0015]

[0016] Where n e is the total number of inverse elements; T e is the transformation matrix that transforms the local coordinates into the global coordinates;

[0017] The unknown parameter U is the displacement vector of the structure, which is used for deformation monitoring of the structure.

[0018] Further, in the step 2), establishing the relationship matrix according to the variational principle specifically includes:

[0019] According to the division of the inverse elements, establish the relationship matrix K between the observation variable F and the unknown parameter U based on the least-squares variational method, and establish the error functional about the theoretical strain and the measured strain, and the expression is as follows:

[0020] Φ e (U e ) = w m ||e(U e ) - e ε || 2 + w b ||k(U e ) - k ε || 2 + w g ||g(U e ) - g ε || 2(2)

[0021] where Φ e is the element error function; U e is the element nodal displacement; e ε , k ε and g ε are the measured values of the plane strain vector, curvature vector and transverse shear strain vector respectively; e(U e ), k(U e ) and g(U e ) are the theoretical values of the plane strain vector, curvature vector and transverse shear strain vector respectively; w m , w b and w g are all weights of the error function, which are respectively: w m = w b = 1, w g = 1×10 -3 ;

[0022] Performing the least squares on the error functional and minimizing it gives the following expression:

[0023]

[0024] After simplification, it gives:

[0025] K e U e = F e (4)

[0026] where K e is the element class stiffness matrix, and formula (4) gives the relationship between the observed variables and the unknown parameters within an inverse element.

[0027] Furthermore, the specific steps for establishing the relationship matrix K in step 2) include:

[0028] After establishing the element class stiffness matrix K e for each inverse element, assembling the element class stiffness matrices gives the relationship matrix K, and the expression is as follows:

[0029]

[0030] where α e represents the weight. When no sensor is arranged within the inverse element, that is, when the strain information of this inverse element is not observed, the value of α e is 1×10 -5 ; otherwise, the value is 1;

[0031] After assembly, the following formula is obtained:

[0032] KU = F (6)

[0033] Among them, the relationship matrix K represents the relationship between the observed variable F and the unknown parameter U. Equation (6) is called the deformation reconstruction equation in inverse finite element.

[0034] Furthermore, calculating the probability density function of the observed variable F in step 3) specifically includes:

[0035] For a structure divided into n e inverse elements, there are n degrees of freedom correspondingly. Among them, F is an observed variable with a length of n, K is an n×n relationship matrix, and U is an unknown parameter vector with a length of n; the i-th element F i in the observed variable F is expressed as:

[0036]

[0037] where K ij represents the element in the i-th row and j-th column of the relationship matrix K, and U j represents the j-th element of the vector U; since the observed variable F is calculated from the measured strain, therefore, considering the observation / measurement error, the observed variable F can be written as:

[0038] F = F t + δ (8)

[0039] where F t represents the observed variable without the influence of measurement noise; δ represents the Gaussian white noise generated by the measurement error, with a mean of 0 and a variance of σ F 2 ; then the probability density function f(F|U) of the observed variable F is:

[0040]

[0041] Furthermore, calculating the quality function of the observed variable F in step 3) specifically includes:

[0042] Define the likelihood function log f(F|U) as follows:

[0043]

[0044] The quality function is defined as the derivative of log f(F|U) with respect to U, and the expression is as follows:

[0045]

[0046] Furthermore, step 4) specifically includes:

[0047] In inverse finite element, the deformation reconstruction equation is:

[0048] KU = F(12)

[0049] The expression of the Fisher information matrix P is as follows:

[0050]

[0051] Among them, E(·) represents taking the expectation of a random variable. After substituting the probability density function of the observed variable F, the expression of the matrix P is obtained as follows:

[0052]

[0053] Since the variance σ F 2 is a constant, after ignoring the constant term, the Fisher information matrix is equivalent to:

[0054] P = K T K(15)

[0055] Among them, maximizing the Fisher information matrix P can minimize the calculation error of the unknown parameter U, making the solution of the vector U more accurate; the trace of the Fisher information matrix P is used to measure the size of the matrix. When the trace of the matrix P is larger, the deformation reconstruction accuracy is higher.

[0056] Furthermore, the specific steps of step 5) include:

[0057] A sensor layout corresponds to a relationship matrix K and a corresponding information matrix P. Taking maximizing the trace of the information matrix P as the optimization objective function, a genetic algorithm is used to optimize the sensor layout.

[0058] Advantages of the present invention:

[0059] The present invention derives the relationship between the Fisher information matrix and the deformation reconstruction accuracy from the basic principle of inverse finite element, providing a theoretical basis for the optimization of sensor layout in inverse finite element. The present invention completes the optimization of sensor layout by maximizing the trace of the Fisher information matrix represented by the overall class stiffness matrix, avoiding a large number of matrix inversion operations during the optimization process and realizing efficient optimization of sensor layout. Brief description of the drawings

[0060] Figure 1 is a flowchart of the method of the present invention.

[0061] Figure 2 is a schematic diagram of the homogeneous plate structure used in the embodiment.

[0062] Figure 3 is a schematic diagram of the division of inverse elements.

[0063] Figure 4 is a schematic diagram of the initial sensor layout.

[0064] Figure 5 Schematic diagram of the intermediate result for optimizing the layout.

[0065] Figure 6 Schematic diagram of the optimized sensor layout.

[0066] Figure 7a Schematic diagram of the displacement result of the finite element simulation

[0067] Figure 7b Schematic diagram of the result of inverse finite element deformation reconstruction.

[0068] Figure 7c Schematic diagram for comparing the reconstruction results.

[0069] Figure 7d Schematic diagram of the reconstruction error. Specific implementation manner

[0070] For the convenience of those skilled in the art to understand, the present invention will be further described below in conjunction with the embodiments and the accompanying drawings. The content mentioned in the implementation manner does not limit the present invention.

[0071] Refer to Figure 1 As shown, a method for optimizing the layout of deformation monitoring sensors based on the Fisher information matrix of the present invention is as follows:

[0072] 1) Define the observation variable F and the unknown parameter U;

[0073] Specifically, the definition of the observation variable and the unknown parameter in step 1) specifically includes:

[0074] For example, a cantilever plate structure as shown in Figure 2 is adopted. The observation variable F refers to the variable measured by the strain sensor; the inverse elements of the structure are divided, and 98 inverse elements are divided, as shown in Figure 3 and a unit-class load vector F e is established for each inverse element. The unit-class load vector is calculated from the measurement values of the strain sensors; then the unit-class load vectors are assembled to obtain the observation variable F:

[0075]

[0076] where the total number of inverse elements is 98, and T e is the transformation matrix that transforms the local coordinates into the global coordinates;

[0077] The unknown parameter U is the displacement vector of the structure, which is used for deformation monitoring of the structure.

[0078] 2) Establish the relationship matrix K;

[0079] Specifically, the establishment of the relationship matrix according to the variational principle in step 2) specifically includes:

[0080] Based on the division of inverse elements, establish the relationship K between the observation variable F and the unknown parameter U according to the least - squares variational principle, and establish the error functional regarding the theoretical strain and the measured strain as follows:

[0081] Φ e (U e ) = w m ||e(U e ) - e ε || 2 + w b ||k(U e ) - k ε || 2 + w g ||g(U e ) - g ε || 2

[0082] Among them, Φ e is the element error function; U e is the element node displacement; e ε , k ε and g ε are the measured values of the plane strain vector, curvature vector, and transverse shear strain vector respectively; e(U e ), k(U e ), and g(U e ) are the theoretical values of the plane strain vector, curvature vector, and transverse shear strain vector respectively; w m , w b and w g are all the weights of the error function, which are respectively: w m = w b = 1, w g = 1×10 -3 ;

[0083] Perform the least - squares on the error functional and make it minimum to obtain:

[0084]

[0085] After simplification, it is obtained:

[0086] K e U e = F e

[0087] Among them, K e is the element - like stiffness matrix, and the above formula gives the relationship between the observation variable and the unknown parameter within an inverse element.

[0088] Specifically, the establishment of the relationship matrix K in step 2) specifically includes:

[0089] For each inverse element, establish the element class stiffness matrix K e After that, assemble the element class stiffness matrix to obtain the relationship matrix K, as follows:

[0090]

[0091] where α e represents the weight. When no sensor is arranged in the inverse element and the strain information of the element is not observed, the value of α e is 1×10 -5 ; otherwise, the value is 1;

[0092] After assembly, we get:

[0093] KU = F

[0094] where the matrix K represents the relationship between the observed variable F and the unknown parameter U, and this equation is called the deformation reconstruction equation in the inverse finite element.

[0095] 3) Calculate the probability density function and quality function of the observed variable F;

[0096] Specifically, the calculation of the probability density function of the observed quantity F in step 3) specifically includes:

[0097] For a structure divided into 98 inverse elements, there are 720 degrees of freedom. Among them, F is an observed variable with a length of 720, K is a 720×720 relationship matrix, and U is an unknown parameter vector with a length of 720; the i-th element F i in the observed variable F can be expressed as:

[0098]

[0099] where K ij represents the element in the i-th row and j-th column of the K matrix, and U j represents the j-th element of the vector U; since the vector F is calculated from the measured strain, therefore, considering the observation / measurement error, the observed variable F can be written as:

[0100] F = F t + δ

[0101] where F t represents the observed variable without the influence of measurement noise; δ represents the Gaussian white noise generated by the measurement error, with a mean of 0 and a variance of σ F 2 ; then the probability density function of the observed quantity F is:

[0102]

[0103] Specifically, calculating the quality function of the observable F in step 3) specifically includes:

[0104] Define the likelihood function as follows:

[0105]

[0106] The quality function is defined as the derivative of log f(F|U) with respect to U, and its expression is as follows:

[0107]

[0108] 4) Calculate the Fisher information matrix according to the deformation reconstruction equation;

[0109] Specifically, calculating the Fisher information matrix according to the deformation reconstruction equation in step 4) specifically includes:

[0110] In the inverse finite element, the deformation reconstruction equation is:

[0111] KU = F

[0112] The expression of the Fisher information matrix is:

[0113]

[0114] where E(·) represents taking the expectation of the random variable; after substituting the probability density function of the observed variable F, the matrix P is obtained as:

[0115]

[0116] Since the variance σ F 2 is a constant, after ignoring the constant term, the Fisher information matrix can be equivalently:

[0117] P = K T K

[0118] Maximizing the Fisher information matrix P can minimize the calculation error of the unknown parameter U, making the solution of the vector U more accurate; the trace of the Fisher information matrix P is used to measure the size of the matrix. When the trace of the matrix P is larger, the deformation reconstruction accuracy is higher.

[0119] 5) Maximize the trace of the Fisher information matrix to achieve the optimal layout of the sensors;

[0120] Specifically, step 5) specifically includes:

[0121] A sensor layout corresponds to a relationship matrix K and a corresponding information matrix P. An intelligent algorithm, such as a genetic algorithm, is used to optimize the sensor layout with the maximization of the trace of the information matrix P as the optimization objective function, and then a sensor layout with high reconstruction accuracy is obtained.

[0122] Compare the deformation reconstruction results under different sensor layouts. Figure 4 The deformation reconstruction error corresponding to the shown sensor layout is 74.162%, and the trace of its Fisher information matrix is 140.021. Figure 5 The reconstruction error under the shown sensor layout is 20.172%, and the trace of the information matrix P is 166.375. Figure 6 The reconstruction error corresponding to the shown optimal sensor layout is 0.808%, and the trace of the Fisher information matrix is 176.149. The comparison chart of the reconstruction errors is as Figure 7a - Figure 7d shown.

[0123] The present invention completes the layout optimization of the sensor by calculating the trace of the Fisher information matrix, without solving the inverse of the matrix during the optimization process, and the operation speed is relatively fast.

[0124] The specific application ways of the present invention are numerous. The above description is only the preferred embodiment of the present invention. It should be pointed out that for those of ordinary skill in the art of this technology, several improvements can be made without departing from the principle of the present invention, and these improvements should also be regarded as the protection scope of the present invention.

Claims

1. A method for optimizing the arrangement of deformation monitoring sensors based on Fisher information matrix, characterized in that: Here are the steps: 1) Define observed variables F and unknown parameters U; 2) Establish a relationship matrix K; 3) Calculate the probability density function and quality function of the observed variable F; 4) Calculate the Fisher information matrix according to the deformation reconstruction equation; 5) Maximize the trace of the Fisher information matrix to achieve optimal placement of sensors.

2. The method for optimizing the arrangement of deformation monitoring sensors based on Fisher information matrix according to claim 1 is characterized in that: The step 1) specifically includes: Divide the structure into inverse units and establish a unit class load vector F for each inverse unit e , element class load vector F e It is obtained from the measurement value of the strain sensor; then the unit load vector is assembled to obtain the observed variable F, which is expressed as follows: Among them, n e is the total number of inverse units; T e is the transformation matrix, which transforms local coordinates into global coordinates; The unknown parameter U is the displacement vector of the structure, which is used to monitor the deformation of the structure.

3. The method for optimizing the arrangement of deformation monitoring sensors based on Fisher information matrix according to claim 1 is characterized in that: The step 2) of establishing a relationship matrix according to the variational principle specifically includes: According to the division of the inverse unit, the relationship matrix K between the observed variable F and the unknown parameter U is established based on the least squares variation method, and the error functional between the theoretical strain and the measured strain is established. The expression is as follows: Φ e (U e )=w m ||e(U e )-e ε || 2 +w b ||k(U e )-k ε || 2 +w g ||g(U e )-g ε || 2 (2) Among them, Φ e is the unit error function; U e is the unit node displacement; e ε , k ε and g ε are the measured values ​​of the plane strain vector, curvature vector and transverse shear strain vector respectively; e(U e )、k(U e ) and g(U e ) are the theoretical values ​​of plane strain vector, curvature vector and transverse shear strain vector respectively; w m 、w b and w g are the weights of the error function; Taking the least square method for the error functional and minimizing it, we get the following: After simplification, we get: K e U e =F e (4) Among them, K e is the element-like stiffness matrix, and formula (4) gives the relationship between the observed variables and the unknown parameters in an inverse element.

4. The method for optimizing the arrangement of deformation monitoring sensors based on Fisher information matrix according to claim 3 is characterized in that: The step 2) of establishing the relationship matrix K specifically includes: For each inverse element, establish the element class stiffness matrix K e After that, the unit class stiffness matrix is ​​assembled to obtain the relationship matrix K, as follows: Among them, α e represents the weight. When no sensor is arranged in the inverse unit, that is, when the strain information of the inverse unit is not observed, α e The value of is 1×10 -5 ; Otherwise, the value is 1; After assembly, the following formula is obtained: KU=F (6) Among them, the relationship matrix K represents the relationship between the observed variables F and the unknown parameters U. Formula (6) is called the deformation reconstruction equation in the inverse finite element.

5. The method for optimizing the arrangement of deformation monitoring sensors based on Fisher information matrix according to claim 1 is characterized in that: The probability density function of the observed variable F is calculated in step 3), specifically including: For the n e The structure of the inverse unit corresponds to n degrees of freedom, where F is the observation variable of length n, K is the n×n relationship matrix, and U is the unknown parameter vector of length n; the i-th element F in the observation variable F i It is expressed as: Among them, K ij represents the i-th row and j-th column element of the relationship matrix K, U j represents the jth element of vector U; since the observed variable F is calculated from the measured strain, the observed variable F can be written as: F=F t +d (8) Among them, F t represents the observed variable without measurement noise; δ represents the Gaussian white noise generated by measurement error, with mean 0 and variance σ F 2 ; Then the probability density function f(F|U) of the observed variable F is:

6. The method for optimizing the arrangement of deformation monitoring sensors based on Fisher information matrix according to claim 1 is characterized in that: The quality function of the observed variable F is calculated in step 3), specifically including: The likelihood function log f(F|U) is defined as follows: The quality function is defined as the derivative of log f(F|U) with respect to U, and is expressed as follows:

7. The method for optimizing the arrangement of deformation monitoring sensors based on Fisher information matrix according to claim 6 is characterized in that: The step 4) specifically includes: In the inverse finite element, the deformation reconstruction equation is: KU=F (12) The expression of Fisher information matrix P is: Among them, E(·) represents the expectation of the random variable. After substituting the probability density function of the observed variable F, the matrix P is obtained as follows: Since the variance σ F 2 is a constant, so after ignoring the constant term, the Fisher information matrix is ​​equivalent to: P=K T K (15) Among them, maximizing the Fisher information matrix P can minimize the calculation error of the unknown parameter U, making the solution of the vector U more accurate; the trace of the Fisher information matrix P is used to measure the size of the matrix. When the trace of the matrix P is larger, the deformation reconstruction accuracy is higher.

8. The method for optimizing the arrangement of deformation monitoring sensors based on Fisher information matrix according to claim 1 is characterized in that: The step 5) specifically includes: A sensor layout corresponds to a relationship matrix K and a corresponding information matrix P. The sensor layout is optimized using a genetic algorithm, with the maximization of the trace of the information matrix P as the optimization objective function.

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