Nonlinear System Dynamic Parameter Identification Method Based on High-Order Dynamic Mode Decomposition

Abstract

The present invention relates to the field of dynamic parameter identification of nonlinear systems, and particularly to a method for identifying dynamic parameters of nonlinear systems based on high-order dynamic mode decomposition. The scheme includes: obtaining the time series of the nonlinear system through experiments or numerical simulations; determining the optimal time delay of the time series according to the mutual information method; according to the Takens embedding theorem, reconstructing the original time series into a high-order time series matrix through phase space; obtaining the similarity matrix of the high-order time series system matrix; obtaining the high-order dynamic modes of the nonlinear system; judging whether the characteristic spectrum of the high-order dynamic modes is located on or close to the unit circle in the complex plane, if not, returning to the step of reconstructing the original time series into a high-order time series matrix through phase space; if so, obtaining the modal frequency and damping ratio parameters of the nonlinear system. By introducing the mutual information method and the phase space reconstruction theory, the present invention optimizes the analysis process of the existing technology, and the present invention is applicable to the identification of dynamic parameters of nonlinear systems.

Description

Technical Field

[0001] The present invention relates to the field of identifying dynamic parameters of nonlinear systems, and particularly to a method for identifying dynamic parameters of nonlinear systems based on high-order dynamic modal decomposition. Background Art

[0002] In the field of nonlinear systems, nonlinear systems are often accompanied by complex random evolution phenomena that are difficult to understand. If time series can be used as samples and the essential characteristics, relevant action mechanisms, and spatio-temporal evolution laws of these complex phenomena can be intuitively displayed through a certain method or technical analysis, it is of great significance for studying nonlinear systems to understand transient changes and dynamic behaviors, and can also promote the analysis and calculation efficiency of nonlinear system dynamics. Therefore, the modal decomposition method based on feature extraction technology - the flow field reduced-order model has been proposed by researchers. Its essence is a data-driven technology, which decomposes a high-dimensional unsteady system into the superposition of dynamic modes or coherent structures on a low-dimensional coordinate system, so as to describe the spatio-temporal evolution of the nonlinear system in the low-dimensional space. The currently more commonly used flow field reduced-order method is the proper orthogonal decomposition method.

[0003] The proper orthogonal decomposition technology decomposes a nonlinear system into several spatially orthogonal modes, a process of converting a multi-dimensional isotropic random field into a set of uncorrelated one-dimensional spatial patterns. These patterns consist of a series of continuous random processes and are sorted according to the energy (i.e., eigenvalues) of each mode, so as to select the main modes of the random field.

[0004] Although the application scope of the proper orthogonal decomposition technology is very wide, due to the existence of the covariance matrix, the proper orthogonal decomposition analysis is only limited to the second-order characteristics of variables, and it cannot directly identify single-frequency dynamic coherent structures to explain the time evolution characteristics of random systems.

[0005] In addition, a nonlinear system is a random, complex high-dimensional dynamic system. The acquisition of the dynamic information of the system often directly depends on the time series of multiple variables. However, if the data is not fully extended to the dimension of the original dynamic system, direct modal analysis may not be able to discover the fuzzy dynamic characteristics hidden in the original data. Summary of the Invention

[0006] The purpose of the present invention is to overcome the shortcomings of the prior art and provide a method for identifying dynamic parameters of nonlinear systems based on high-order dynamic modal decomposition, which can well identify modal parameters such as the natural frequency and damping ratio of dynamic structures, and can separate nonlinear coupling signals to obtain single-frequency signals, having a good decoupling effect.

[0007] The present invention adopts the following technical solutions to achieve the above purpose. The present invention provides a method for identifying dynamic parameters of nonlinear systems based on high-order dynamic modal decomposition, including:

[0008] S1. Obtain the time series of the nonlinear system through experiments or numerical simulations;

[0009] S2. Determine the optimal time delay of the time series according to the mutual information method;

[0010] S3. According to the Takens embedding theorem, reconstruct the original time series into a high-order time series matrix through phase space;

[0011] S4. Obtain the similarity matrix of the high-order time series system matrix;

[0012] S5. Obtain the high-order dynamic modes of the nonlinear system;

[0013] S6. Judge whether the characteristic spectrum of the high-order dynamic mode is located on or close to the unit circle in the complex plane. If so, go to step S7; otherwise, return to step S3;

[0014] S7. Obtain the modal frequency and damping ratio parameters of the nonlinear system.

[0015] Furthermore, step S2 specifically includes:

[0016] Let the random variable X represent the wind pressure coefficient sequence K(t), and Y represent the lagging wind pressure coefficient sequence K(t + τ ), τ represent the time delay, then the specific process of the mutual information method is as follows:

[0017]

[0018] Among them, H ( X ) and H ( Y ) are the marginal entropies of X and Y respectively, H ( X , Y ) is the joint entropy of X and Y , MI ( X , Y ) are the mutual information coefficients of X and Y respectively, p ( x ) and p ( y ) are the marginal distributions of X and Y respectively, p ( x , y ) is X and YThe joint distribution of x, y is the specific value of the random variable X and Y .

[0019] Furthermore, step S3 specifically includes:

[0020] According to the Takens embedding theorem, the time series K ( N ) is constructed into a m -dimensional high-order time series matrix Q :

[0021]

[0022] where q j represents the matrix column vector at the j th moment, and the embedding dimension m is determined by whether the modulus of the high-order dynamic mode eigen-spectrum is equal to or close to 1.

[0023] Furthermore, step S4 specifically includes:

[0024] The time-shifted sequence matrices are respectively Q 1 = { q 1, q 2,..., q j-1} and Q 2 = { q 2, q 3,..., q j}. Based on the assumption of the Koopman operator theory, Q 1 and Q 2 have a linear mapping A , that is: , A represents the system matrix;

[0025] By performing singular value decomposition on the matrix Q 1, the orthogonal subspace of the similarity transformation can be obtained:

[0026]

[0027]

[0028] where U and V are unitary matrices, Σ is a diagonal matrix, H represents the conjugate transpose;

[0029] The similarity matrix à is obtained through the minimization problem of the Frobenius norm:

[0030]

[0031]

[0032] Matrix à is A a similarity transformation of

[0033] Furthermore, step S5 specifically includes:

[0034] The j th high-order dynamic mode φ j is a complex mode, expressed as: ;

[0035] Define the logarithm mapping of eigenvalues as the eigen-spectrum s j : ;

[0036] Wherein, λ j , w j are respectively the j th eigenvalue and eigenvector of Ã, and Δ t represents the time interval of the time series of the nonlinear system.

[0037] Furthermore, step S7 specifically includes:

[0038] The solution of the structural dynamic eigen-spectrum of the damped nonlinear system is:

[0039]

[0040] In the formula ω j , ζ j are respectively the natural frequency and the corresponding modal damping ratio;

[0041] Deform the solution of the structural dynamic eigenvalues of the damped nonlinear system to obtain the modal frequency f j and the modal damping ratio ζ j :

[0042]

[0043] In the formula, || represents the modulus of a complex number, and Re() represents the real part of a complex number.

[0044] The beneficial effects of the present invention are:

[0045] The present invention combines the mutual information method and the Takens embedding theorem with the dynamic modal decomposition method to form a high-order dynamic modal decomposition method, which reconstructs the phase space of the nonlinear system and expands the data dimension. It can mine the fuzzy dynamic characteristics hidden in the data set and make the dynamic modal decomposition mode obtained by decomposition more neutral and stable.

[0046] The present invention can well identify modal parameters such as the natural frequency and damping ratio of the dynamic structure through a high-order dynamic modal decomposition method, and can separate nonlinear coupled signals to obtain single-frequency signals, thus having a good decoupling effect.

[0047] The nonlinear system reconstructed by the high-order dynamic modal decomposition of the present invention is more capable of describing and fitting the local characteristics of the original nonlinear system, because the high-order dynamic modal decomposition directly reconstructs the nonlinear system instead of reconstructing the energy field. The low-frequency modes decomposed by the dynamic modal decomposition method contain most of the pulsating energy and explain the dominant frequency of the nonlinear system. Therefore, in revealing the dynamic mechanism and characteristics of the random field, the high-order dynamic modal decomposition method has more advantages. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 It is a flow chart of a method for identifying dynamic parameters of a nonlinear system based on high-order dynamic modal decomposition provided by the present invention;

[0049] Figure 2 It is a schematic diagram of measuring point arrangement provided by the implementation of the present invention;

[0050] Figure 3 is a schematic diagram of wave height response provided by the present invention;

[0051] Figure 4 It is a schematic diagram of the power spectrum provided by the present invention;

[0052] Figure 5 is a graph showing the variation of the mutual information coefficient with time delay provided by the present invention;

[0053] Figure 6 It is a schematic diagram of the characteristic spectrum of high-order dynamic modes after phase space reconstruction provided by the present invention;

[0054] Figure 7 is a graph of the first-order dynamic modal coefficient and the number of snapshots provided by the present invention;

[0055] Figure 8 is a graph of the power spectrum and frequency of the first-order modal coefficient provided by the present invention;

[0056] Figure 9 It is a curve diagram of the modal vibration shape and the degree of freedom number provided by the present invention. DETAILED DESCRIPTION

[0057] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.

[0058] The present invention provides a method for identifying dynamic parameters of a nonlinear system based on high-order dynamic mode decomposition, as Figure 1 shown, including:

[0059] S1. Obtain the time series of the nonlinear system through experiments or numerical simulations;

[0060] S2. Determine the optimal time delay of the time series according to the mutual information method;

[0061] Let the random variable X represent the wind pressure coefficient sequence K(t), and Y represent the lagged wind pressure coefficient sequence K(t + τ ), τ where represents the time delay, then the specific process of the mutual information method is as follows:

[0062]

[0063] Among them, H ( X ) and H ( Y ) are the marginal entropies of X and Y respectively, H ( X , Y ) is the joint entropy of X and Y , MI ( X , Y ) are the mutual information coefficients of X and Y respectively, p ( x ) and p ( y ) are the marginal distributions of X and Y respectively, p ( x , y ) is the joint distribution of X and Y , x, y is the specific value of the random variable X and Y .

[0064] S3. According to the Takens embedding theorem, reconstruct the original time series into a high-order time series matrix through phase space;

[0065] According to the Takens embedding theorem, the time series K ​​​​​( N ) is constructed as m a multi-dimensional high-order time series matrix Q :

[0066]

[0067] where q j represents the matrix column vector at the j th moment, and the embedding dimension m is determined by whether the modulus of the high-order dynamic mode eigenvalue spectrum is equal to or close to 1.

[0068] S4. Obtain the similarity matrix of the high-order time series system matrix;

[0069] The time-shifted sequence matrices are respectively Q 1 = { q 1, q 2,..., q j-1} and Q 2 = { q 2, q 3,..., q j}. Based on the assumption of the Koopman operator theory, Q 1 and Q 2 have a linear mapping A , that is: , A represents the system matrix;

[0070] By performing singular value decomposition on the matrix Q 1, the orthogonal subspace of the similarity transformation can be obtained:

[0071]

[0072]

[0073] where U and V are unitary matrices, Σ is a diagonal matrix, H represents the conjugate transpose;

[0074] The similarity matrix à is obtained through the minimization problem of the Frobenius norm:

[0075]

[0076]

[0077] The matrix à is A a similarity transformation of

[0078] S5. Obtain the high-order dynamic modes of the nonlinear system;

[0079] Step S5 specifically includes:

[0080] The j th high-order dynamic mode φ j is a complex mode, expressed as: ;

[0081] To facilitate the analysis of high-order dynamic modes, the logarithmic mapping of the eigenvalues of the damped nonlinear system is defined as the characteristic spectrum s j : ;

[0082] where λ j , w j are the j th eigenvalue and eigenvector of à respectively.

[0083] S6. Determine whether the characteristic spectrum of the high-order dynamic mode is located on or close to the unit circle in the complex plane. If so, go to step S7; otherwise, return to step S3;

[0084] S7. Obtain the modal frequency and damping ratio parameters of the nonlinear system.

[0085] The solution of the structural dynamic characteristic spectrum of the damped nonlinear system is calculated as:

[0086]

[0087] In the formula ω j , ζ j are the natural frequency and the corresponding modal damping ratio respectively;

[0088] The solution of the structural dynamic eigenvalue of the damped nonlinear system is deformed to obtain the modal frequency f j and the modal damping ratio ζ j :

[0089]

[0090] In the formula, || represents the modulus of a complex number, and Re() represents the real part of a complex number.

[0091] The following combines specific implementation cases and appendices Figures 2 - 9 to explain the present invention in detail.

[0092] (1) Experiment

[0093] In the embodiment of the present invention, a shaking table test of a tuned liquid damper with a grille is adopted for high-order dynamic modal method analysis. The tuned liquid damper is a dynamic vibration damper commonly used in the control of super high-rise buildings. The natural frequency, damping ratio and vibration mode of the tuned liquid damper system are important dynamic parameters for the design of the tuned liquid damper.

[0094] Since the liquid sloshing of the tuned liquid damper will exhibit obvious non-linear characteristics, the shaking table test method is more often used to obtain its performance parameters. The embodiment of the present invention adopts a shaking table test of the tuned liquid damper based on colored noise excitation. Figure 2 shows the schematic diagram of the measuring point arrangement of the tuned liquid damper model provided by the present invention. The net clear size L×B×H (length×width×liquid depth) of the tuned liquid damper model is 2.1m×0.64m×0.44m. The model scale ratio, frequency scale ratio and damping ratio scale ratio are 1:10, √10:1 and 1:1 respectively. Two gratings are arranged in total, and the consistency ratio S =0.55. Four digital wave gauges are used in the test to measure the wave height change of the liquid in the tuned liquid damper along the long side direction, the sampling frequency is 100Hz, and an acceleration sensor is used to measure the acceleration of the shaking table tabletop, and the sampling frequency is 25Hz. According to the wave theory, the frequency and modal damping ratio of the first-order mode of the liquid sloshing of the rectangular tuned liquid damper with a grille are 0.46Hz and 0.0477 respectively.

[0095] Figure 3 and Figure 4 respectively represent the wave height response and its power spectrum of the system wave height displacement meter. Figure 4 In f the abscissa S(f) represents the frequency, and the ordinate

[0096] (2) Construct a high-order time series matrix

[0097] By performing mutual information analysis on the test data, the delay time for phase space reconstruction of the data is obtained. Figure 5 represents the schematic diagram of the time delay when the mutual information coefficient decays from 1 to zero for the first time. The sampling times corresponding to the mutual information coefficient dropping to 0.05 are 9.02s respectively. At this time, it is considered that the data are independent of each other and used as the delay time for reconstructing the high-order time series matrix.

[0098] (3) High-order dynamic modal analysis

[0099] The high-order time series matrix is analyzed through steps S4-S5, such asFigure 6 As shown, the characteristic spectrum lies on or near the unit circle in the complex plane, indicating that the eigenvalues of the system matrix are stable or neutrally stable.

[0100] (4) First-order modal frequency and damping ratio of the tuned liquid damper

[0101] Through the analysis in step S7, as Figures 7 - 9 , the first-order frequency and damping ratio are obtained as 0.452 Hz and 0.0487 respectively, which are in good agreement with the results of the theoretical formula. It can be seen that the high-order dynamic modal method has good applicability for identifying the dynamic modal parameters of the nonlinear tuned liquid damper system. In addition, the high-order dynamic modal coefficient curve is a single-frequency decay curve. Combining with the power spectrum curve of the time mode shows that the high-order dynamic modal decomposition technology can well separate the coupled signals and play a decoupling role.

[0102] The above are only the preferred embodiments of the present invention. It should be understood that the present invention is not limited to the form disclosed herein, should not be regarded as excluding other embodiments, but can be used in various other combinations, modifications and environments, and can be changed within the scope of the concept described herein through the above teachings or the technology or knowledge in related fields. And the changes and modifications made by those skilled in the art without departing from the spirit and scope of the present invention should all be within the protection scope of the appended claims of the present invention.

Claims

1. A nonlinear system dynamic parameter identification method based on high-order dynamic modal decomposition, characterized in that: include: S1. Obtain the time series of nonlinear system through experiments or numerical simulation; S2, determining the optimal time delay of the time series according to the mutual information method; S3. According to Takkens embedding theorem, the original time series is reconstructed into a high-order time series matrix through phase space; S4, obtaining the similarity matrix of the high-order time series system matrix; The two time series matrices are Q 1={ q 1, q 2, ..., q j-1 }and Q 2={ q 2, q 3, ..., q j }, based on the Koopman operator theory assumption, Q 1 and Q 2There exists a linear mapping A, namely: , A represents the system matrix; Through the matrix Q 1 Perform singular value decomposition to obtain the orthogonal subspace of similarity transformation: in U and V is a unitary matrix, Σ is a diagonal matrix, H represents conjugate transpose; The similarity matrix is ​​obtained by minimizing the Frobenius norm à : matrix à yes A Similarity transformation of ; S5. Obtaining high-order dynamic modes of nonlinear systems; No. j Higher order dynamic modes φ j is a complex mode, expressed as: ; The logarithmic mapping of the eigenvalues ​​is defined as the eigenspectrum s j : ; in, λ j , w j They are respectively j eigenvalues ​​and eigenvectors, Δ t The time interval representing the time series of a nonlinear system; S6, judging whether the high-order dynamic modal characteristic spectrum is located at or close to the unit circle in the complex plane, if so, proceeding to step S7, otherwise returning to step S3; S7. Obtain the modal frequency and damping ratio parameters of the nonlinear system.

2. The method for identifying dynamic parameters of a nonlinear system based on high-order dynamic modal decomposition according to claim 1, characterized in that: Step S2 specifically includes: Let the random variable X Represents the wind pressure coefficient sequence K ( t ), Y Represents the lagged wind pressure coefficient series K ( t + τ ), τ Represents the time delay, then the specific process of the mutual information method is as follows: in, H ( X ), H ( Y ) are respectively X , Y The marginal entropy of H ( X , Y )for X , Y The joint entropy of MI ( X , Y ) are respectively X , Y The mutual information coefficient of p ( x ), p ( y ) are respectively X , Y The marginal distribution of p ( x , y )for X , Y The joint distribution of x, y is a random variable X , Y The specific value of .

3. The method for identifying dynamic parameters of a nonlinear system based on high-order dynamic modal decomposition according to claim 1, characterized in that: Step S3 specifically includes: According to Takkens embedding theorem, the time series K ( N ) is constructed as m dimensional high-order time series matrix Q : In the formula q j Indicates j The matrix column vector at each moment, embedding dimension m It is determined by whether the mode of the characteristic spectrum of the high-order dynamic mode is equal to or close to 1.

4. The method for identifying dynamic parameters of a nonlinear system based on high-order dynamic modal decomposition according to claim 1, characterized in that: Step S7 specifically includes: The solution of the structural dynamics characteristic spectrum of the damped nonlinear system is: In the formula ω j , ζ j are the natural frequencies and the corresponding modal damping ratios, respectively; Transform the solution of the structural dynamics eigenvalues ​​of the damped nonlinear system to obtain the modal frequencies f j and modal damping ratio ζ j : In the formula, || represents the modulus of the complex number, and Re() represents the real part of the complex number.

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