CN109101759B - Parameter identification method based on forward and reverse response surface method - Google Patents
Parameter identification method based on forward and reverse response surface method Download PDFInfo
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Abstract
The invention discloses a parameter identification method based on a forward and reverse response surface method, which comprises the following steps: s1, constructing a positive response surface model; s2, constructing an inverse response surface model; s3, substituting the actual measurement structure response into the inverse response surface model to calculate an initial estimation value of the parameter to be identified; then taking the norm of the difference between the calculated value of the positive response surface model and the response value of the actually measured structure as an objective function, and taking the value range of the parameter to be identified and the norm of the difference between the parameter to be identified and the initial estimated value calculated by the inverse response surface model as constraint conditions, and establishing and solving an optimization inverse problem of the structure parameter identification; s4, determining statistical evaluation indexes of a response surface method, performing accuracy verification, and outputting a structural parameter identification result. The invention effectively solves the parameter estimation problem when the disease state exists among different groups of characteristic parameters in the parameter identification process, and effectively ensures the parameter identification precision.
Description
Technical Field
The invention relates to the technical field of parameter identification, in particular to a parameter identification method based on a forward and reverse response surface method.
Background
The dynamic model correction, parameter identification and damage identification of the structure are inverse problems of structural dynamics, the calculation equation set often has pathological conditions, and the measured data is easy to cause 'oscillation' of a solution. The solution of the inverse problem calculation equation set is non-unique due to the existence of the pathological condition, and in most cases, the optimal solution can be given only by an estimation method, and the magnitude of the estimation error is related to the pathological condition of the equation and the quantity and the accuracy of the measured information. How to accurately and effectively identify parameters of a structure under the conditions of pathological conditions of an inverse problem calculation equation set, limited measured information and noise pollution is a difficult problem to be broken through in the field of structural parameter identification.
In recent years, the problem of solving the system of pathological equations is not uncommon in the mathematical field, and a common approach is regularization. The selection of regularization parameters becomes a key for solving the problem of the pathological equation set, and common methods include a Morozov deviation rule, a generalized cross-checking method (GCV), an L curve method and the like. Although the regularization method is theoretically perfect, the resulting regularized solution balances between the least squares solution and the least norms solution, thereby controlling the degree of fit of the oscillations and corrections of the solution, which does not mean that the regularized solution has sufficient accuracy. The substitution model calculated by the positive response surface method is a dominant polynomial equation set, the method is simple and easy, the recognition accuracy is high on most problems, and the method has a great number of applications in the fields of model correction, parameter recognition and damage recognition. The positive response surface takes the parameter to be identified as a design parameter, the structural system response is taken as a characteristic parameter, the characteristic parameter is expressed as a polynomial function taking the design parameter as an independent variable, then test design is carried out, polynomial coefficients are solved, and a positive response surface model is established. And substituting the actual measurement response into the positive response surface model during parameter identification, establishing an inverse problem equation set and solving. When the characteristic parameters of different groups have pathological conditions, the result obtained by the optimization calculation of the inverse problem equation set established by the positive response surface method is always non-unique. The inverse response surface method is opposite to the positive response surface, the design parameters are expressed as polynomial functions taking the characteristic parameters as independent variables, the experimental design is solved with the polynomial undetermined coefficients, and an inverse response surface model is established. And when the parameters are identified, the actual response is directly substituted into the inverse response surface model to directly solve the parameters to be identified, and an inverse problem equation set is not required to be established. When the pathological states exist among the characteristic parameters of different groups, namely the pathological states exist in the equation set when the polynomial function coefficients are solved, the regularization method can be directly adopted for solving and obtaining the equation set. Therefore, if the advantages of the regularization method, the positive response surface method and the inverse response surface method can be combined, the positive response surface method and the inverse response surface method when the actual measurement data have a pathological state can be established, the fitting degree of solution oscillation and correction is controlled, meanwhile, the precision of the solution is improved, finally, the parameters of the structure are accurately and effectively identified, and the method has important significance for model correction, parameter identification, damage identification and health monitoring of the structure.
Disclosure of Invention
The invention aims to provide a parameter identification method based on a forward and reverse response surface method, which effectively solves the problem of parameter estimation when pathological conditions exist among different groups of characteristic parameters in the parameter identification process and effectively ensures the parameter identification precision.
In order to achieve the above purpose, the technical scheme adopted by the invention is as follows:
a parameter identification method based on a forward and reverse response surface method comprises the following steps:
s1, constructing a positive response surface model: firstly, defining a parameter to be identified of a structure and a response of a structural system, taking the parameter to be identified as a design parameter, taking the response of the structural system as a characteristic parameter, and expressing the characteristic parameter as a polynomial explicit function taking the design parameter as an independent variable; then, performing experimental design and regression fitting of polynomial coefficient to be determined, thereby establishing a positive response surface model of the structural system response;
s2, constructing an inverse response surface model: the method comprises the steps of (1) representing design parameters as polynomial explicit functions taking characteristic parameters as independent variables, and fitting polynomial undetermined coefficients by using a sick least square method according to test design results, so as to establish an inverse response surface model of structural system response;
s3, establishing and solving an optimization inverse problem of structural parameter identification: substituting the actual measurement structure response into the inverse response surface model to calculate an initial estimation value of the parameter to be identified; then taking the norm of the difference between the calculated value of the positive response surface model and the response value of the actually measured structure as an objective function, and taking the value range of the parameter to be identified and the norm of the difference between the parameter to be identified and the initial estimated value calculated by the inverse response surface model as constraint conditions, and establishing and solving an optimization inverse problem of the structure parameter identification;
s4, verifying the precision and outputting an identification result: and determining a statistical evaluation index of a response surface method, performing accuracy verification, and outputting a structural parameter identification result.
Further, the step S1 specifically includes the following steps:
s1.1, defining the parameter x to be identified 1 ，x 2 ，x 3 …x n Structural system response y for design parameters 1 ，y 2 ，y 3 …y n As the feature parameter, the feature parameter is expressed as a polynomial function having the design parameter as an argument:
wherein, beta is the undetermined coefficient of the polynomial function;
s1.2, performing test design, determining a parameter level of a design parameter as a sample point, outputting a corresponding characteristic parameter as a sample value, acquiring a sufficient number of sample data, substituting the sample data into an equation set for solving coefficients, and fitting coefficients of a polynomial function by adopting a least square regression;
s1.3, analyzing the significance of each parameter item to each response in the polynomial by adopting an F test method (ANOVA), specifically, carrying out F test on the parameter item A, wherein the statistic is that
In SS (x) A Is the sum of squares of the deviations caused by the A parameter term, SS e Is the sum of squares of the deviations f caused by errors A 、f e Degree of freedom for parameter terms and deviations, respectively, given the level of significance α, if F A ≥F 1-α (f A ，f e ) The parameter item A is considered significant, otherwise it is considered insignificantTherefore, parameter items with obvious influence on response in the response surface model are reserved, and parameter items with little influence on response are ignored;
s1.4, determining a positive response surface model of the structural system by using the screened parameter items and coefficients thereof.
Further, the step S2 specifically includes the following steps:
s2.1, according to the parameter x to be identified 1 ，x 2 ，x 3 …x n Structural system response y for design parameters 1 ，y 2 ，y 3 …y n As the feature parameter, the design parameter is expressed herein as a polynomial function having the feature parameter as an argument:
wherein alpha is the undetermined coefficient of the polynomial function;
s2.2, substituting the sample data obtained in the step S1.2 into the polynomial function established in the step 2.1 to generate a linear equation set for solving coefficients, determining regularization parameters by a generalized cross-checking method (GCV) or an L curve method, and determining an inverse response surface model of the structural system by adopting a regularized least square method or a regularized total least square method to regression fit the coefficient alpha of the polynomial function;
s2.3, substituting the actual measurement structure response into the inverse response surface model to calculate an initial estimation value of the parameter to be identified.
Further, the step S3 specifically includes the following steps:
s3.1, taking a norm of a difference between a positive response surface model calculated value and an actual measurement structure response value as an objective function:
wherein: r is the residual, R (p) = { f E }-{f A (p); p is a design parameter; { f E }{f A The analytical and experimental feature quantities;
s3.2, taking a value range of the parameter to be identified and a norm of a difference between the parameter to be identified and an initial estimated value calculated by the inverse response surface model as constraint conditions:
wherein: VLB and VUB are design spaces; p is a design parameter; p' is an initial estimation value obtained by calculation of the inverse response surface model; epsilon is a specified constant;
and S3.3, optimizing and solving an optimization inverse problem of structural parameter identification.
Further, the step S4 specifically includes the following steps:
s4.1, verifying the accuracy of the parameter identification result obtained by solving the step 3 by the following formula:
wherein: y is RS Representing a solution obtained by solving the inverse problem; y represents a system response truth value; y represents the average value of the response truth value; MSE is the mean square residual error;
s4.2, outputting identification parameters after the accuracy verification is qualified, and returning to the step S1 when the accuracy verification is not qualified, and improving the response surface model or increasing the test number of points.
The invention has the following beneficial effects:
the regularized least square method is adopted to fit a coefficient matrix of an inverse response surface, so that the problem of inverse parameter identification is solved, the estimated value of the identification parameter is obtained when the characteristic parameter has a pathological condition, the estimated value is used as a boundary condition, and the inverse problem of parameter identification is established by using a response surface method. The identification parameters with higher precision can be obtained through solving. The method can be used for identifying engineering structure parameters, provides accurate and effective analysis basis for model correction, parameter identification, damage identification and health monitoring of the structure, and has important theoretical significance and practical significance.
Drawings
Fig. 1 is a flowchart of a parameter identification method based on a forward and reverse response surface method according to an embodiment of the present invention.
Detailed Description
The present invention will be described in detail with reference to specific examples. The following examples will assist those skilled in the art in further understanding the present invention, but are not intended to limit the invention in any way. It should be noted that variations and modifications could be made by those skilled in the art without departing from the inventive concept. These are all within the scope of the present invention.
As shown in fig. 1, a parameter identification method based on a forward and reverse response surface method according to an embodiment of the present invention includes the following steps:
s1, constructing a positive response surface model
S1.1, defining the parameter x to be identified 1 ，x 2 ，x 3 …x n Structural system response y for design parameters 1 ，y 2 ，y 3 …y n As the feature parameter, the feature parameter is expressed as a polynomial function having the design parameter as an argument:
wherein, beta is the undetermined coefficient of the polynomial function;
s1.2, performing test design, determining a parameter level of a design parameter as a sample point, outputting a corresponding characteristic parameter as a sample value, acquiring a sufficient number of sample data, substituting the sample data into an equation set for solving coefficients, and fitting coefficients of a polynomial function by adopting a least square regression;
s1.3, analyzing the significance of each parameter item to each response in the polynomial by adopting an F test method (ANOVA), specifically, carrying out F test on the parameter item A, wherein the statistic is that
In SS (x) A Is the sum of squares of the deviations caused by the A parameter term, SS e Is the sum of squares of the deviations f caused by errors A 、f e Degree of freedom for parameter terms and deviations, respectively, given the level of significance α, if F A ≥F 1-α (f A ，f e ) If the parameter item A is considered to be significant, otherwise, the parameter item A is considered to be insignificant, so that the parameter item which has significant influence on the response in the response surface model is reserved, and the parameter item which has little influence on the response is ignored;
s1.4, determining the positive response surface model of the structural system by using the screened parameter items and coefficients thereof
S2, constructing an inverse response surface model
S2.1, according to the parameter x to be identified 1 ，x 2 ，x 3 …x n Structural system response y for design parameters 1 ，y 2 ，y 3 …y n As the feature parameter, the design parameter is expressed herein as a polynomial function having the feature parameter as an argument:
wherein alpha is the undetermined coefficient of the polynomial function;
s2.2, substituting the sample data obtained in the step S1.2 into the polynomial function established in the step 2.1 to generate a linear equation set for solving coefficients, determining regularization parameters by a generalized cross-checking method (GCV) or an L curve method, and determining an inverse response surface model of the structural system by adopting a regularized least square method or a regularized total least square method to regression fit the coefficient alpha of the polynomial function;
s2.3, substituting the actual measurement structure response into the inverse response surface model to calculate an initial estimation value of the parameter to be identified;
s3, establishing and solving an optimization inverse problem of structural parameter identification
S3.1, taking a norm of a difference between a positive response surface model calculated value and an actual measurement structure response value as an objective function:
wherein: r is the residual, R (p) = { f E }-{f A (p); p is a design parameter; { f E }{f A The analytical and experimental feature quantities;
s3.2, taking a value range of the parameter to be identified and a norm of a difference between the parameter to be identified and an initial estimated value calculated by the inverse response surface model as constraint conditions:
wherein: VLB and VUB are design spaces; p is a design parameter; p' is an initial estimation value obtained by calculation of the inverse response surface model; epsilon is a specified constant;
and S3.3, optimizing and solving an optimization inverse problem of structural parameter identification.
S4, verifying the precision and outputting the identification result
S4.1, verifying the accuracy of the parameter identification result obtained by solving the step 3 by the following formula:
wherein: y is RS Representing a solution obtained by solving the inverse problem; y represents a system response truth value;
represents the average of the response truth values; MSE is the mean square residual error;
s4.2, outputting identification parameters after the accuracy verification is qualified, and returning to the step 1 when the accuracy verification is not qualified, and improving the response surface model or increasing the test number of points.
The foregoing describes specific embodiments of the present invention. It is to be understood that the invention is not limited to the particular embodiments described above, and that various changes or modifications may be made by those skilled in the art within the scope of the appended claims without affecting the spirit of the invention. The embodiments of the present application and features in the embodiments may be combined with each other arbitrarily without conflict.
Claims (1)
1. The parameter identification method based on the forward and reverse response surface method is characterized by comprising the following steps of:
s1, constructing a positive response surface model: firstly, defining a parameter to be identified of a structure and a response of a structural system, taking the parameter to be identified as a design parameter, taking the response of the structural system as a characteristic parameter, and expressing the characteristic parameter as a polynomial explicit function taking the design parameter as an independent variable; then, performing experimental design and regression fitting of polynomial coefficient to be determined, thereby establishing a positive response surface model of the structural system response;
s2, constructing an inverse response surface model: the method comprises the steps of (1) representing design parameters as polynomial explicit functions taking characteristic parameters as independent variables, and fitting polynomial undetermined coefficients by using a sick least square method according to test design results, so as to establish an inverse response surface model of structural system response;
s3, establishing and solving an optimization inverse problem of structural parameter identification: substituting the actual measurement structure response into the inverse response surface model to calculate an initial estimation value of the parameter to be identified; then taking the norm of the difference between the calculated value of the positive response surface model and the response value of the actually measured structure as an objective function, and taking the value range of the parameter to be identified and the norm of the difference between the parameter to be identified and the initial estimated value calculated by the inverse response surface model as constraint conditions, and establishing and solving an optimization inverse problem of the structure parameter identification;
s4, verifying the precision and outputting an identification result: determining a statistical evaluation index of a response surface method, performing accuracy verification, and outputting a structural parameter identification result;
the step S1 specifically comprises the following steps:
s1.1, defining the parameter x to be identified 1 ，x 2 ，x 3 …x n Structural system response y for design parameters 1 ，y 2 ，y 3 …y n As the feature parameter, the feature parameter is expressed as a polynomial function having the design parameter as an argument:
wherein, beta is the undetermined coefficient of the polynomial function;
s1.2, performing test design, determining a parameter level of a design parameter as a sample point, outputting a corresponding characteristic parameter as a sample value, acquiring a sufficient number of sample data, substituting the sample data into an equation set for solving coefficients, and fitting coefficients of a polynomial function by adopting a least square regression;
s1.3, analyzing the significance of each parameter item in the polynomial to each response by adopting an F test method, specifically, performing F test on the parameter item A, wherein the statistic is that
In SS (x) A Is the sum of squares of the deviations caused by the A parameter term, SS e Is the sum of squares of the deviations f caused by errors A 、f e Degree of freedom for parameter terms and deviations, respectively, given the level of significance α, if F A ≥F 1-α (f A ，f e ) If the parameter item A is considered to be significant, otherwise, the parameter item A is considered to be insignificant, so that the parameter item which has significant influence on the response in the response surface model is reserved, and the parameter item which has little influence on the response is ignored;
s1.4, determining a positive response surface model of the structural system by using the screened parameter items and coefficients thereof;
the step S2 specifically includes the following steps:
s2.1, according to the parameter x to be identified 1 ，x 2 ，x 3 …x n Structural system response y for design parameters 1 ，y 2 ，y 3 …y n As the feature parameter, the design parameter is expressed herein as a polynomial function having the feature parameter as an argument:
wherein alpha is the undetermined coefficient of the polynomial function;
s2.2, substituting the sample data obtained in the step S1.2 into the polynomial function established in the step 2.1 to generate a linear equation set for solving coefficients, determining regularization parameters by a generalized cross test method or an L curve method, and determining an inverse response surface model of the structural system by adopting a regularized least square method or a regularized total least square regression to fit coefficients alpha of the polynomial function;
s2.3, substituting the actual measurement structure response into the inverse response surface model to calculate an initial estimation value of the parameter to be identified;
the step S3 specifically includes the following steps:
s3.1, taking a norm of a difference between a positive response surface model calculated value and an actual measurement structure response value as an objective function:
wherein: r is the residual, R (p) = { f E }-{f A (p); p is a design parameter; { f E }{f A The analytical and experimental feature quantities;
s3.2, taking a value range of the parameter to be identified and a norm of a difference between the parameter to be identified and an initial estimated value calculated by the inverse response surface model as constraint conditions:
VLB≤p≤VUB；
ε＞0
wherein: VLB and VUB are design spaces; p is a design parameter; p' is an initial estimation value obtained by calculation of the inverse response surface model; epsilon is a specified constant;
s3.3, optimizing and solving an optimization inverse problem of structural parameter identification;
the step S4 specifically includes the following steps:
s4.1, verifying the accuracy of the parameter identification result obtained by solving the step S3 by the following formula:
wherein: wherein: y is RS (j) Representing a solution obtained by solving the inverse problem once; y (j) represents the true value of the corresponding system response;
represents the average of the response truth values; n represents the number of checkpoints in the design space; MSE is the mean square residual error;
s4.2, outputting identification parameters after the accuracy verification is qualified, and returning to the step S1 when the accuracy verification is not qualified, and improving the response surface model or increasing the test number of points.
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