CN102982196A - Time frequency domain time varying structure modal parameter identification method based on time varying common demominator model - Google Patents

Abstract

The invention relates to a time frequency domain time varying structure modal parameter identification method based on a time varying common demominator model and belongs to the technical field of structural dynamics. Firstly, structural dynamics response signals measured and obtained by aircraft or spacecraft structures with the time varying characteristics under the work situations are analyzed in a time frequency mode to obtain a time relative power spectral function of non-parametric evaluation corresponding to time varying structures. Then, the time varying common demominator model is used as a parametric model of the time varying structural dynamics to evaluate to-be-evaluated parameters of the time varying common demominator model through a least squares methods of the time domain. Finally, the evaluated to-be-evaluated parameters of the time varying common demominator model is utilized to calculate the model frequency and the model damping ratio corresponding to the time varying common demominator model. The time frequency domain time varying structure modal parameter identification method based on the time varying common demominator model is suitable for model parameters recognition of time varying structure in the field of aircraft and spacecraft engineering application and has the advantages of being simple and convenient to use. Furthermore, users are low in participation degree.

Description

Based on the time varying structure modal parameter identification method when becoming the time-frequency domain of common denominator model

Technical field

The present invention relates to a kind of based on the time varying structure modal parameter identification method when becoming the time-frequency domain of common denominator model, belong to the Structural Dynamics technical field.

Background technology

In the production and life of reality, many engineering structures show such time varying characteristic, the carrier rocket that liquid fuel reduces gradually in the vehicle-bridge system in encouraging such as train, the flight course, the aircraft under the aerodynamic force additive effect, the variable spacecraft of flexible extensible geometry, rotating machinery etc.

Space industry at home, the spacecraft of a new generations such as large space station, carrier rocket of new generation, large flexible expansion satellite has been put in the up-to-date space flight development plan of China, becomes the main direction of coming few decades China Aerospace device development.Be in operation existing and become factor when stronger without exception of the structure of large space station, carrier rocket of new generation, large flexibility expansion satellite, space articulation problem such as following large space station, the fuel mass of active service and following carrier rocket of new generation consumes fast, and the space development of large flexibility expandable type satellite etc.Therefore, as the time become important method and the approach that structural dynamic characteristics is analyzed, the time varying structure modal parameter identification research will become one of emphasis of Future Spacecraft Structural Dynamics research.The time varying structure modal parameter identification become model frequency, Mode Shape and the modal damping of structure can identification the time, these parameters have important physical significance, can for the time become the aspect such as structural design, monitoring structural health conditions, structure failure diagnosis, structural vibration control of structure application strong support is provided.

Difference according to the mathematical model that adopts is distinguished, and the method for varying structure modal parameter identification mainly contains four classes when existing:

The first kind is the time-varying modal parameter identification method based on online recursion technology that develops from traditional time constant modal parameters identification.

The basis of these class methods is traditional time constant modal parameters discrimination methods, difference is for be considered in each moment data sequentially, old data pass into silence gradually, and new data constantly add to come in, and the estimated value of modal parameter is corrected constantly at each.There is the defective of two aspects in these class methods: first, exist the On The Choice of observation data and forgetting factor (algorithm), need to do compromise between the two in accuracy of identification and tracking power, and also be difficult to solve for the relevant adaptability of choosing of different structure; The second, these class methods are from traditional Modal Parameters Identification, and therefore the response message of input and output two aspects of Structure of need is difficult to apply to as flying spacecraft etc. and can only obtain the modal parameters identification of output response signal.

Equations of The Second Kind is based on the in short-term Modal Parameters Identification of constant hypothesis.

These class methods are divided into the one by one little time period with data (structural response), and constant when the bar structure parameter is regarded as within each time period, then discre value in each section is processed with certain data processing technique (such as curve) and obtained the time dependent rule of modal parameter.Do not use the data message of preceding paragraphs when its feature is the modal parameter of a period of time after estimating, changing faster to parameter, structure is estimated accuracy to be improved must choose very short data segment.The method comprises now comparatively stochastic subspace identification method (N4SID) and time correlation autoregressive moving-average model (Time-dependent ARMA, the TARMA) method of the recursion of state-based spatial model commonly used.The time varying structure modal parameter identification method development time of these class methods is the longest, development also perfect.But some intrinsic problems have limited it and have further developed and use: the first, in short-term constant restriction of assumption the application of these class methods for fast change, mutation parameter identification aspect; Second, these class methods need that form is fixed, clear and definite mathematical model, such as state-space model, seasonal effect in time series autoregressive moving-average model etc., therefore, the problem of determining the order of model is very outstanding in identification, the uncertain false mode that will introduce without the physics meaning of model order causes identification result unavailable, and the problems such as judgement of model order Rational choice, false mode more need further further investigation; The 3rd, as two kinds of main flows based on the Modal Parameters Identification of constant hypothesis in short-term---the stochastic subspace identification method of recursion and time correlation autoregressive moving-average model exist some other problem separately: the accumulation subspace method of state-based spatial model will use QR to decompose inevitably, Eigenvalues Decomposition (EVD) or svd (SVD) technology, this will inevitably bring the complicacy on the method Numerical Implementation, for the heavy construction structure, especially to the problem of online and Fast Identification requirement is arranged, this also needs further to study; The discrimination method research of time-based series model all can not be avoided the design of parameter tracking algorithm, although various improved least square methods, various filtering method constantly propose, use different track algorithms but work as same model, and different model application identical algorithms result difference is very large.

The 3rd class is the time-varying modal parameter identification method of artificial neural network.

Artificial neural network has been widely used in the Nonlinear Systems Identification problem, and obtains good effect but most of research work also only is confined to time-invariant system, just is generalized in recent years time-varying system.The document of publishing that artificial neural network is used for time-varying modal parameter identification area research seldom, it mainly concentrates on the mechanism Journal of Sex Research for simple structure (system).Have also that algorithm is complicated, counting yield is low and the problem such as identification precision is poor for real labyrinth.

The 4th class methods are based on the time varying structure modal parameter identification method of the imparametrization time-frequency domain of time frequency analysis.

From the angle of signal analysis, the time to become the structure dynamic response signal of structure under working environment be nonstationary random signal.

Classical Fourier transform has become the most strong analytical approach of signal process field and instrument through the development in a century, and this mainly is to be determined by its orthogonality and distinct physical significance and quick succinct computing method.But, because Fourier transform is to the time quadrature, removed the time varying signal in the non-stationary signal, thereby to require signal be stably, to the time become non-stationary signal and be difficult to abundant portrayal.In order to satisfy the requirement to jump signal, Non-stationary Signal Analysis, nineteen forty-six, Gabor has proposed the windowed FFT analytical approach, also claim short time discrete Fourier transform (short-time Fourier transform, STFT), by choosing of suitable window function, just can realize time frequency analysis to a certain extent, but because temporal resolution and frequency resolution will be subject to the restriction of window function width, always can not arrive simultaneously the best.1948, Ville proposed famous Eugene Wigner-Weir distribution (Wigner-Ville distribution, WVD).It distributes as a kind of energy type time-frequency combination, has compared many advantageous properties with other time-frequency distributions, such as true marginality, weak supportive, translation invariance etc., is a very useful Non-stationary Signal Analysis instrument.Cross term appears because the Eugene Wigner of many signals-Weir distributes, can limit its effect in many occasions, so the researchist on this basis afterwards, multiple improved form has been proposed, such as exponential distribution, generalized exponential distribution, generalized bilinear time-frequency distributions etc., wherein the generalized bilinear time-frequency distributions is called again Koln class energy type time-frequency distributions.Afterwards on this basis, people have proposed again the methods such as Koln class time-frequency distributions, and these Time-Frequency Analysis Method are widely used and have obtained many gratifying results in the nonstationary random signal analysis.

Nearly ten years, since the advantage of time frequency analysis aspect the nonstationary random signal analysis, the research of change and Nonlinear Systems Identification when increasing researcher uses time frequency analysis to carry out.Time-Frequency Analysis Method to the time become and the nonlinear organization modal parameter carries out identification and also gradually becomes one of focus of Modal Parameter Identification research field.Ghanem in 2000 controls the differential equation with Structural Dynamics and launches at a series of wavelet basiss, replaces original physical responses with wavelet coefficient, and the modal parameter of system that adopted the method identification of finding the solution EXPANSION EQUATION FOR STEEL; Zhang in 2003 and Xu become when simple to one structural response Gabor conversion identification the model frequency of structure; The mode that Roshan-Ghias in 2007 adopts analytic derivation has been carried out WVD and SPWVD conversion to the response under a single-mode system and the system with 3 degrees of freedom free vibration, and has estimated model frequency and the damping ratio of system according to transformation results.

All right and wrong are parameterized for the varying structure modal parameter identification method during existing time-frequency domain based on time frequency analysis, although become the model frequency of structure when the method that has can be good at picking out, but the method for the imparametrization subjective consciousness that depends on the user and experience in various degree, and also there is not good way for the time varying structure modal damping ratio identification under the arbitrary excitation.

Summary of the invention

Varying structure modal parameter identification problem when the present invention is directed to aircraft and spacecraft, varying structure modal parameter identification method when having proposed to become when a kind of the time-frequency domain of common denominator model, its basic ideas are: the structure dynamic response signal that at first aircraft with time-varying characteristics or spacecraft structure is measured under working environment carries out time frequency analysis, obtain the time correlation power spectrum function to seasonable change structure that imparametrization is estimated, then take the time become the common denominator model as the time become the parameterized model of Structural Dynamics, become the solve for parameter of common denominator model when the least square method by time-frequency domain estimates, utilize at last estimate the time become the common denominator model solve for parameter calculate model frequency and damping ratios to seasonable change structure.

Specific implementation step of the present invention is as follows:

Step 1, the main frequency scope that becomes arrangement works state, interested time range according to by identification aircraft or spacecraft the time and become structure during by identification aircraft or spacecraft, as the about 0-20Hz of carrier rocket, the about 0-100Hz of satellite, the about 0-10Hz of Large Deployable satellite, the about 0-50Hz of aircraft wing etc., set required sampling time and the sample frequency of identification, and the structure dynamic response signal of identification structure is gathered.

Step 2 is chosen arbitrarily reference signal, and respectively each response signal is carried out time frequency analysis in conjunction with reference signal, the time correlation power spectrum function G of the signal that meets with a response from the response signal that step 1 gathers _k(t _τ, ω _f).Wherein, t _τExpression time-sampling point, ω _fExpression frequency sampling point, subscript τ=1,2 ..., N _τ, f=1,2 ..., N _f, N _τBe time-sampling number, N _fBe the frequency sampling number, k=1,2 ..., N _sN _r, N _sBe output number of responses, N _rFor the response signal reference is counted.

The present invention adopts smooth and pseudo Wigner-Ville distribution (Smooth Pseudo smooth and pseudo Wigner-Ville Distribution, SPWVD) calculate by the power spectrum function of identification structure response signal, implement simply, counting yield is high, and cross term in can suppressing preferably to distribute.

Step 3, according to sampling time and sample frequency and by identification aircraft or spacecraft the time become characteristics, it is as follows to become the common denominator model when setting up:

${\hat{G}}_k(t_{\mathrm{\τ}},{\mathrm{\ω}}_f)=\frac{B_k(t_{\mathrm{\τ}},{\mathrm{\ω}}_f)}{A(t_{\mathrm{\τ}},{\mathrm{\ω}}_f)}---\left(1\right)$

Wherein, divide submultinomial and denominator polynomial expression to be respectively:

Wherein,

For the time-frequency basis function (time polynomial and frequency polynomial expression are respectively i and j rank, i=0,1,2 ..., n _t, j=0,1,2 ..., n _ω, n _tBe time polynomial exponent number, n _ωBe frequency polynomial expression exponent number), molecule multinomial coefficient b _{K, i, j}With common denominator a _{I, j}Be written as vector form:

$B_{k,j}={[b_{k,0,j},b_{k,1,j},...,b_{k,n_t,j}]}^T,$ $A_j={[a_{0,j},a_{1,j},...,a_{n_t,j}]}^T---\left(4\right)$

Order:

${\mathrm{\β}}_k={[B_{k,0}^T,B_{k,1}^T,...,B_{k,n_{\mathrm{\ω}}}^T]}^T;$ $\mathrm{\α}={[A_0^T,A_1^T,...,A_{n_{\mathrm{\ω}}}^T]}^T---\left(5\right)$

Then have:

$\mathrm{\θ}={[{\mathrm{\β}}_1^T,...,{\mathrm{\β}}_k^T,...,{\mathrm{\β}}_{N_o{N}_i}^T,{\mathrm{\α}}^T]}^T---\left(6\right)$

Become the solve for parameter vector in the common denominator model when wherein, θ is.

Step 4 becomes the solve for parameter vector θ in the common denominator model when adopting the least-squares parameter method to obtain.

Detailed process is as follows:

The cost function of definition least square:

$l_{\mathrm{LS}}\left(\mathrm{\θ}\right)=\underset{k=1}{\overset{N_r{N}_s}{\mathrm{\Σ}}}\left[\begin{array}{cc}{\mathrm{\β}}_k^T& {\mathrm{\α}}^T\end{array}\right]\left[\begin{array}{cc}{P}_k& Q_k\\ Q_k^T& R_k\end{array}\right]\left[\begin{array}{c}{\mathrm{\β}}_k\\ \mathrm{\α}\end{array}\right]---\left(7\right)$

Wherein, $P_k=\mathrm{Re}\left({\mathrm{\Θ}}_k^H{\mathrm{\Θ}}_k\right),$ $Q_k=\mathrm{Re}\left({\mathrm{\Θ}}_k^H{\mathrm{\Ξ}}_k\right)$ With $R_k=\mathrm{Re}\left({\mathrm{\Ξ}}_k^H{\mathrm{\Ξ}}_k\right).$

W wherein _k(t _τ, ω _f) be weight function.

Utilize the common denominator parameter vector α ' to be estimated after following formula is tried to achieve constraint:

D′α′=b′ (10)

Wherein, $D^{\′}=D_{[1:(n_t+1)(n_{\mathrm{\ω}}+1)-\mathrm{1,1}:(n_t+1)(n_{\mathrm{\ω}}+1)-1]},$ $b^{\′}=-D_{[1:(n_t+1)(n_{\mathrm{\ω}}+1)-1,(n_t+1)(n_{\mathrm{\ω}}+1)]},$ $D=2{\mathrm{\Σ}}_{k=1}^{N_r{N}_s}(R_k-Q_k^H{P}_k^{-1}{Q}_k).$

Order $\mathrm{\α}=\left[\begin{array}{c}{\mathrm{\α}}^{\′}\\ 1\end{array}\right],$ And calculate molecular parameter to be estimated vector β by following formula _k:

${\mathrm{\β}}_k=P_k^{-1}{Q}_k\mathrm{\α}---\left(11\right)$

Thereby become the solve for parameter vector θ in the common denominator model when obtaining, and then become the common denominator model when definite.

Step 5, according to by the application needs of identification aircraft or spacecraft modal parameter, the time interval during such as the controlled frequency of carrier rocket vibration control system, satellite structure health monitoring requires etc., the given moment point t that needs the compute mode parameter _{τ '}, and utilize common denominator parameter vector α that step 4 obtains with timely change common denominator model, calculate given time t _{τ '}The model frequency f that lower identification obtains _rWith damping ratios ξ _r:

$f_r=\frac{\mathrm{Im}\left({\mathrm{\λ}}_r\right)}{2\mathrm{\π}},$ ${\mathrm{\ξ}}_r=\frac{\mathrm{Re}\left({\mathrm{\λ}}_r\right)}{\left|{\mathrm{\λ}}_r\right|}---\left(12\right)$

Wherein, Im and Re are respectively imaginary part and the real part of getting the bracket intermediate value, λ _rBe the t take α as coefficient _{τ '}The time inscribe the time become denominator polynomial expression A (t in the common denominator model _{τ '}, r root ω), ω are by the frequency variable of identification aircraft or spacecraft.

Beneficial effect

The present invention is from the angle of parameterized time-frequency domain, provides a kind of time varying structure modal parameter identification method of the parametrization time-frequency domain based on time frequency analysis, and physical significance is clear; Be applicable to the aerospace engineering application the time become the Modal Parameter Identification of structure, and required user's participation is lower, has to use simple and characteristics easily.

Description of drawings

Fig. 1 is the Three Degree Of Freedom spring-dampers-quality system in the embodiment;

When being Three Degree Of Freedom in the embodiment, Fig. 2 becomes the SPWVD of structural random response;

Fig. 3 is the modal parameter that identification obtains in the embodiment.

Embodiment

For better explanation objects and advantages of the present invention, become during below by the Three Degree Of Freedom under the arbitrary excitation when structure example comes aircraft under the analog operation state and spacecraft and become structure, the present invention is further specified.

Three Degree Of Freedom spring-dampers-the quality system of present embodiment, as shown in Figure 1.

The parameter of system with 3 degrees of freedom is k ₁=k ₂=k ₃=10 ⁵, c ₁=1.0, c ₂=0.5, c ₃=0.5, initial mass is m ₁(0)=0.2, m ₂(0)=0.1, m ₃(0)=0.1.The dynamic control equation of system is:

$M\left(t\right)\stackrel{\·\·}{x}+C\stackrel{\·}{x}+\mathrm{Kx}=f\left(t\right)---\left(13\right)$

Wherein M (t) for the time mass matrix that becomes, M (t)=M ₀(1-0.5t), M ₀Be the mass matrix of initial time, damping matrix and stiffness matrix are

$C=\left[\begin{array}{ccc}0.2& 0& 0\\ 0& 0.2& 0\\ 0& 0& 0.1\end{array}\right],$ $C=\left[\begin{array}{ccc}1.5& -0.5& 0\\ -0.5& 1.0& -0.5\\ 0& -0.5& 0.5\end{array}\right],$ $K=\left[\begin{array}{ccc}2& -1& 0\\ -1& 2& -1\\ 0& -1& 1\end{array}\right]\×{10}^5$

F (t) is the white-noise excitation of 100 units for the amplitude that is applied on the Three Degree Of Freedom.

Be actuated to emulation Gauss white noise on first degree of freedom of the system of acting in the present embodiment.System responses adopts Newmark-β numerical integration method to calculate (γ=0.5, β=0.1), and wherein integration step is 1/4096s.The signal sampling frequency is 2048Hz, and the sampling time is 1s.

In the present embodiment, based on the time when becoming the time-frequency domain of common denominator model the specific implementation step of varying structure modal frequency estimation as follows:

Step 1, three degree of freedom acceleration are the used response signal of identification, and sample frequency is 1024Hz, and the sampling time is 1s.

Step 2 with the acceleration responsive signal of first response point reference signal when carrying out time frequency analysis, and distributes by smooth and pseudo Wigner-Ville and to obtain the time correlation power spectrum function of each response signal and reference signal.40 times average smooth and pseudo Wigner-Ville distributes as shown in Figure 2.

Step 3 becomes the common denominator model when setting up, then count N to the time-sampling that regularly becomes in the common denominator model _τ=32, frequency sampling is counted N _f=256, time polynomial exponent number n _t=5 and frequency polynomial expression exponent number n _ω=32, and initialization solve for parameter vector θ.

Step 4 adopts least square method to estimate parameter vector θ.Wherein, W _k(t _τ, ω _f) be weight function, this example is made as 1.

Step 5 according to the common denominator parameter vector α among the parameter vector θ, calculates each structural modal frequency and damping ratios constantly.

The parameter identification that adopts this example out the time varying structure modal frequency and damping ratios as shown in Figure 3, wherein solid line is identifier, black circle is theoretical value.

This shows the model frequency that becomes structure when the present invention can be good at picking out, become the damping ratios of structure when picking out preferably.Since its only the response signal of Structure of need as input, therefore, be fit to duty the time become structure and carry out the model frequency identification.On the other hand, can obtain final model frequency and the identification result of damping ratios owing to user in whole process only need arrange a preliminary parameter, user's participation is lower, uses very simple and convenient.

Above-described specific descriptions; purpose, technical scheme and beneficial effect to invention further describe; institute is understood that; the above only is specific embodiments of the invention; be used for explaining the present invention, the protection domain that is not intended to limit the present invention, within the spirit and principles in the present invention all; any modification of making, be equal to replacement, improvement etc., all should be included within protection scope of the present invention.

Claims (2)

1. Based on the time varying structure modal parameter identification method when becoming the time-frequency domain of common denominator model, it is characterized in that: the specific implementation step is as follows:

   Step 1, the main frequency scope that becomes arrangement works state, interested time range according to by identification aircraft or spacecraft the time and become structure during by identification aircraft or spacecraft, set required sampling time and the sample frequency of identification, and the structure dynamic response signal of identification structure is gathered;

   Step 2 is chosen arbitrarily reference signal, and respectively each response signal is carried out time frequency analysis in conjunction with reference signal, the time correlation power spectrum function G of the signal that meets with a response from the response signal that step 1 gathers _k(t _τ, ω _f); Wherein, t _τExpression time-sampling point, ω _fExpression frequency sampling point, subscript τ=1,2 ..., N _τ, f=1,2 ..., N _f, N _τBe time-sampling number, N _fBe the frequency sampling number, k=1,2 ..., N _sN _r, N _sBe output number of responses, N _rFor the response signal reference is counted;

   Step 3, according to sampling time and sample frequency and by identification aircraft or spacecraft the time become characteristics, it is as follows to become the common denominator model when setting up:

   ${\hat{G}}_k(t_{\mathrm{\τ}},{\mathrm{\ω}}_f)=\frac{B_k(t_{\mathrm{\τ}},{\mathrm{\ω}}_f)}{A(t_{\mathrm{\τ}},{\mathrm{\ω}}_f)}---\left(1\right)$

   Wherein, divide submultinomial and denominator polynomial expression to be respectively:

   

   

   Wherein,

   

   Be the time-frequency basis function, time polynomial and frequency polynomial expression are respectively i and j rank, i=0, and 1,2 ..., nt, j=0,1,2 ..., n _ω, n _tBe time polynomial exponent number, n _ωBe frequency polynomial expression exponent number, molecule multinomial coefficient b _{K, i, j}With common denominator a _{I, j}Be written as vector form:

   $B_{k,j}={[b_{k,0,j},b_{k,1,j},...,b_{k,n_t,j}]}^T,$ $A_j={[a_{0,j},a_{1,j},...,a_{n_t,j}]}^T---\left(4\right)$

   Order:

   ${\mathrm{\β}}_k={[B_{k,0}^T,B_{k,1}^T,...,B_{k,n_{\mathrm{\ω}}}^T]}^T;$ $\mathrm{\α}={[A_0^T,A_1^T,...,T_{n_{\mathrm{\ω}}}^T]}^T---\left(5\right)$

   Then have:

   $\mathrm{\θ}={[{\mathrm{\β}}_1^T,...,{\mathrm{\β}}_k^T,...,{\mathrm{\β}}_{N_o{N}_i}^T,{\mathrm{\α}}^T]}^T---\left(6\right)$

   Become the solve for parameter vector in the common denominator model when wherein, θ is;

   Step 4 becomes the solve for parameter vector θ in the common denominator model when adopting the least-squares parameter method to obtain;

   Detailed process is as follows:

   The cost function of least square is:

   $l_{\mathrm{LS}}\left(\mathrm{\θ}\right)=\underset{k=1}{\overset{N_r{N}_s}{\mathrm{\Σ}}}\left[\begin{array}{cc}{\mathrm{\β}}_k^T& {\mathrm{\α}}^T\end{array}\right]\left[\begin{array}{cc}{P}_k& Q_k\\ Q_k^T& R_k\end{array}\right]\left[\begin{array}{c}{\mathrm{\β}}_k\\ \mathrm{\α}\end{array}\right]---\left(7\right)$

   Wherein, $P_k=\mathrm{Re}\left({\mathrm{\Θ}}_k^H{\mathrm{\Θ}}_k\right),$ $Q_k=\mathrm{Re}\left({\mathrm{\Θ}}_k^H{\mathrm{\Ξ}}_k\right)$ With $R_k=\mathrm{Re}\left({\mathrm{\Ξ}}_k^H{\mathrm{\Ξ}}_k\right);$

   

   

   W wherein _k(t _τ, ω _f) be weight function;

   Try to achieve common denominator parameter vector α ' to be estimated after the constraint according to following formula:

   D′α′＝b′ (10)

   Wherein, $D^{\′}=D_{[1:(n_t+1)(n_{\mathrm{\ω}}+1)-\mathrm{1,1}:(n_t+1)(n_{\mathrm{\ω}}+1)-1]},$ $b^{\′}=-D_{[1:(n_t+1)(n_{\mathrm{\ω}}+1)-1,(n_t+1)(n_{\mathrm{\ω}}+1)]},$ $D=2{\mathrm{\Σ}}_{k=1}^{N_r{N}_s}(R_k-Q_k^H{P}_k^{-1}{Q}_k);$

   Order $\mathrm{\α}=\left[\begin{array}{c}{\mathrm{\α}}^{\′}\\ 1\end{array}\right],$ And calculate molecular parameter to be estimated vector β by following formula _k:

   ${\mathrm{\β}}_k=P_k^{-1}{Q}_k\mathrm{\α}---\left(11\right)$

   Thereby become the solve for parameter vector θ in the common denominator model when obtaining;

   Step 5, according to by the application needs of identification aircraft or spacecraft modal parameter, the given moment point t that needs the compute mode parameter _τ', and utilize common denominator parameter vector α that step 4 obtains with timely change common denominator model, calculate given time t _τThe model frequency f that ' lower identification obtains _rWith damping ratios ξ _r:

   $f_r=\frac{\mathrm{Im}\left({\mathrm{\λ}}_r\right)}{2\mathrm{\π}},$ ${\mathrm{\ζ}}_r=\frac{\mathrm{Re}\left({\mathrm{\λ}}_r\right)}{\left|{\mathrm{\λ}}_r\right|}---\left(12\right)$

   Wherein, Im and Re are respectively imaginary part and the real part of getting the bracket intermediate value, λ _rBe the t take α as coefficient _τIn ' time, become denominator polynomial expression A (t in the common denominator model when inscribing _τ', r root ω), ω are by the frequency variable of identification aircraft or spacecraft.
2. According to claim 1 based on the time varying structure modal parameter identification method when becoming the time-frequency domain of common denominator model, it is characterized in that: adopt smooth and pseudo Wigner-Ville to distribute and calculate by the power spectrum function of identification structure response signal.

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