，，

1.School of Aeronautic Science and Engineering，Beihang University，Beijing 100083，P.R.China；2.Ningbo Institute of Technology，Beihang University，Ningbo 315100，P.R.China

Abstract: The determination of the dynamic load is one of the indispensable technologies for structure design and health monitoring for aerospace vehicles. However，it is a significant challenge to measure the external excitation directly. By contrast，the technique of dynamic load identification based on the dynamic model and the response information is a feasible access to obtain the dynamic load indirectly. Furthermore，there are multi-source uncertainties which cannot be neglected for complex systems in the load identification process，especially for aerospace vehicles. In this paper，recent developments in the dynamic load identification field for aerospace vehicles considering multi-source uncertainties are reviewed，including the deterministic dynamic load identification and uncertain dynamic load identification. The inversion methods with different principles of concentrated and distributed loads，and the quantification and propagation analysis for multi-source uncertainties are discussed. Eventually，several possibilities remaining to be explored are illustrated in brief.

Key words：dynamic load identification；concentrated dynamic load；distributed dynamic load；stochastic load；probabilistic uncertainties；non-probabilistic uncertainties

0 Introduction

It is common knowledge that the development of aerospace vehicles is closely associated with the national economy and defense strategy. Aircraft structures are often burdened with various dynamic loads in the flight process，for instance，the thrust load in the launch phase，the pulse pressure in the transonic phase and the strong aerodynamic load in the return phase. With the rapid enhancement of flight speed，flight distance and maneuverability of aerospace vehicles，the dynamic loads have become increasingly severe and intricate. The acquirement of external exciting forces is a prerequisite for the delica?cy management such as flight control and health mon?itoring for aerospace vehicles. In practical engineer?ing，it is difficult to measure external loads straight?forwardly through force sensors，while structural re?sponses under the load effect，such das the displace?ment，acceleration and strain，may be achieved ef?fortlessly［1］. Therefore，it is a feasible approach to calculate external loads indirectly via measured dy?namic responses and structural characteristics in com?bination with remarkable inversion approaches. The concept of load identification originated in the avia?tion field in the 1970s. It is proposed to acquire the actual load to enhance the performance of aircrafts［2］.

In general，multi-source uncertainties are in?eluctable for the dynamic load identification of aero?space vehicles，which signifies the identified load may also be indeterminate owing to the transitivity of multi-source uncertainties. On the one hand，the dynamic load may be stochastic with the speedy change of service environment［3］. Under this circum?stance，the random external load and structural re?sponses must be quantified as stochastic processes.On the other hand，the intrinsic characteristics and measured responses may also be uncertain，caused by either static factors（e.g.，material dispersion，machining tolerance and modeling error）or timevarying parameters（e.g.，disturbance of boundary conditions and deviation of indstrument measure?ment）. In addition，these aforementioned uncertain?ties may be accumulated during the service pro?cess［4］of aerospace vehicles，and their cross-cou?pling effects will lead to numerous noise indepen?dent of the real loads，which hinders the precise identification of the external dynamic load.

Fig.1 Schematic diagram of dynamic load identification for aerospace vehicles

Fig.2 Dynamic load identification method

The schematic diagram of dynamic load identifi?cation for aerospace vehicles considering multisource uncertainties is demonstrated in Fig.1. In brief，it can be subdivided into two major catego?ries：（1）The establishment of the deterministic load identification model［5］；（2）The quantification and propagation analysis of multi-source uncertainties.How to handle the influence of uncertainties on the inversion model and how to identify the uncertain dy?namic load efficiently［6］have become hot-spot issues for many scholars［7-10］. Herein，the developed identi?fication methods for different dynamic loads and the uncertainty analysis methods for uncertain load are reviewed in this paper，which can be seen in Fig.2.It is worthwhile mentioning that only the identifica?tion of loading history is summarized，without the identification of loading position and loading direc?tion［11］.

1 Deterministic Load Identification Methods

Different from the forward problem of structur?al dynamics，the dynamic load identification，called as the second inverse problem，is more complicated than solving the quadratic differential equation. Giv?en the mathematical model，it can be divided into the frequency-domain-based method and the timedomain-based method［12］. It is noted that the timedomain-based method has attracted more attention，since it can reconstruct the dynamic time history forthrightly. In general，the dynamic load identifica?tion is implemented under the framework of the fi?nite element method（FEM）. The governing dy?namic equation of aerospace vehicles can be com?monly depicted as

whereM，CandK，stand for the mass，the damp?ing and the stiffness matrices，respectively；u(t)，(t) and?(t) are the displacement，the velocity and the acceleration responses，respectively；F(t)denotes the external force. Enlightened by the mod?al transformation，Eq.（1）can be rewritten as

whereq(t)，?(t) and?(t) are corresponding modal responses；Mp，CpandΚprepresent the related modal characteristic matrices；P(t) means the mod?al force. Decoupling Eq.（2），we can get some lin?ear differential equations，i.e.

wherermeans ther-th order equation，andmthe number of truncated modes.

Actually，for simple structures，the dynamics calculation may be analyzed in physical space as Eq.（1），while for large-scale structures，it should be performed in modal space to reduce the comput?ing effort. There are two categories to be discussed in this section：The concentrated load（single-point or multi-point） and the distributed load，among which the latter is completed by modifying the iden?tification algorithm of the concentrated load.

1.1 Concentrated dynamic load identification

There are many excellent methods dealing with the identification of dynamic load，including the di?rect inverse method，the regularization method，the Kalman filter method and the machine learning method［13］.

1.1.1 The direct inverse method

The direct inverse method aims to deconvolve this relationship between external load and system response，which is the most common approach in the early stage owing to its intuitionistic advan?tage［14-15］. In frequency domain，carrying out the Fourier transform on both sides of Eq.（2），yields

whereωis the frequency，Q(ω) andP(ω) are Fou?rier spectrums of the external load and the structur?al response，respectively. Introducing the matrixH(ω)= (-ω2Mp+iωCp+Κp)-1of frequency re?sponse function（FRF），the load spectrumP(ω)can be obtained by

where the superscript + denotes the pseudo-inver?sion. Bartlett and Flannelly［16］used measured accel?eration responses to identify vertical and lateral dy?namic loads on the helicopter hub center in frequen?cy domain initially. Hillary et al.［17］reconstructed the dynamic load of a cantilever structure using the FRF of different positions，and proved that strain re?sponses outperform acceleration responses. Hansen and Starkey［18-19］revealed the ill condition of the FRF near the resonance region，and the growing identification error with the increase of the number of loads. Doyle［20-22］investigated considerable re?searches to reconstruct the location and history of the impact load.

In contrast，the direct inversion method in time domain emerged relatively late. Refs.［23-25］pro?posed this method for discrete-time systems based on modal coordinate transformation with regard to the flight load of rockets. It is assumed that the load is regarded as a constant during the period [tk，tk+1]，i.e.，pr(t)=prk，t∈[tk，tk+1]. Based on the Durham integral and the vibration equation，the result of Eq.（3）can be obtained as

whereΦis the truncated modal matrix.

Suppose that the number of measured respons?es isnr，and the number of external concentrated load isnf. The following relationship should be satis?fied：nr≥m≥nf. Ref.［26］presented five methods to obtain the transfer function between force and strain response to reduce noise interference. Sandesh et al.［27］identified external excitations and interface forces with the iterative time-domain identification. Liu et al.［28］proposed a time-domain Galerkin method for dynamic load identification，which can overcome the influence of noise

1.1.2 The regularization method

The direct inverse method can be summarized asAx=y，x∈X，y∈Y，whereAis the operator，Xthe solution space，andYthe data space. Unfortu?nately，the inversion of operatorAis often ill-condi?tioned leading to unstable results. There is no deny?ing that the regularization method aims to find a sta?ble approximate solution to replace the exact solu?tion of the inverse problem，in which two issues should be discussed：（1）The construction of regu?larization operator；（2）The selection of regulariza?tion parameters. Tihonov regularization［29］is an out?standing method for dynamic load identification，whose basic idea can be described as follows. For a bounded linear operatorA，solvexα∈Xto mini?mize the Tikhonov functionalJα(x)，namely

whereA?is the adjoint matrix，andfα(σ2i) the Tik?honov filtration factor determined by regularization parameterαand the singular valueσ2i.

Other regularization methods，like the truncat?ed singular value decomposition（TSVD）［30］and the iterative regularization method［31］，have also been extensively employed. Given that improper se?lection of regularization parameters will lead to unac?ceptable results，some methods，including general?ized deviation criterion［32］，generalized cross-valida?tion （GCV） method［33］and L-curve criterion［34］have been established to determine the regulariza?tion parameter. Jacquelin et al.［35］introduced the reg?ularization algorithm to the impact load identifica?tion process，and discussed the influence of different regularization methods on the identification results.Wang et al.［36］studied the load identification on com?posite laminated cylindrical shells through Tikhonov regularization with a new regularized filter operator.Numerous researches［37-39］have confirmed the bene?fits of the regularization method.

1.1.3 The Kalman filter（KF）method

Kalman filter is a recursive algorithm originat?ed from the control field［40］，which is modeled on the basis of the state-space equation. Different from the traditional Kalman filter，this method for dynam?ic load identification can estimate the system state and the unknown input simultaneously. The discrete state-space equation of Eq.（1）can be expressed as

whereX(k) embodies the state vector，andZ(k)the observation vector.w(k) andv(k) are the pro?cess noise vector and measurement noise vector.A，BandCdenote the state transition matrix，the input matrix and the identity matrix，respectively.

The two-stage and two-step recursion meth?od［41-42］，combining the Kalman filter and the least square algorithm，is most widespread for dynamic load identification. The dynamic load may be esti?mated recursively by virtue of the gain matrix，up?date state and covariance matrix generated，which was described in Ref.［13］. In terms of the weight?ing coefficient of the recursive least-squares algo?rithm，the conventional weighting input estimation（WIE）［43］，the adaptive WIE［44］and the intelligent fuzzy WIE［45］have been presented successively. In addition，Gillijns et al.［46-47］proposed an unbiased minimum-variance input and state estimation for lin?ear discrete-time systems with acceleration and dis?placement responses. Hsieh et al.［48］extended the in?put and state estimation from one-step delay to multi-step delay. As for the nonlinear system，Ma et al.［49］contributed the extended Kalman filter（EKF）for nonlinear estimation，in which the firstorder Taylor expansion is used to linearize the non?linear model，and the standard Kalman filter algo?rithm is used to estimate the state and unknown load. Ref.［50］ used the unscented Kalman filter（UKF）to avoid the derivatives，Jacobians calcula?tion and linearization approximations of EKF.

Another feasible approach is to extend the un?known input vector to the state vector，then use the standard Kalman filter to estimate the extended state vector. Lourens et al.［51］proposed an augment?ed Kalman filter（AKF）technique for joint inputstate estimation based on reduced-order models and vibration data from a limited number of sensors.Ref.［52］investigated a multi-metric approach to en?hance the stability and accuracy of the force estima?tion by the AKF method.

1.1.4 The machine learning method

Substantially，the dynamic load identification can be regarded as an optimization problem. With the development of computer technology，some in?telligent optimization algorithms based on machine learning have been proposed，and been gradually in?tegrated into the field of dynamic load identification.The characteristics of a specific structure have been concealed in input-output samples，so the complex nonlinear relationship between the dynamic re?sponse and load may be reasoned and learned by ma?chine learning models.

As a classical intelligent optimization algo?rithm，the neural network was used for load identifi?cation initially，whose processor can be summarized as follows：（1）Determine the topological structure of the neural network；（2）Adjust the network pa?rameters in the training and learning process；（3）Obtain the load sequence by inputting the measured response. Cao et al.［53］simulated the strain-load rela?tionship of aircraft wings by the artificial neural net?work，and analyzed the influence of network struc?ture，training algorithm and learning speed. Trivailo et al.［54］predicted both high-frequency buffet and low-frequency manoeuvre loading through the El?man network to improve the fatigue monitoring ca?pability of F/A-18 Empennage. Zhou et al.［55］recon?structed the impact load of nonlinear structures us?ing the deep recurrent neural network，whose effec?tiveness was verified through an experiment of a composite plate. Compared with the traditional methods，the neural-network-based method may have a wider prospect owing to the higher identifica?tion accuracy and stronger anti-interference ability.However，there is no universal method to deter?mine the network structure. Under such circum?stances，the support vector machine has been intro?duced into the load identification，and its validity is also clarified in Refs.［56-57］Furthermore，the opti?mization algorithms like genetic algorithm have also been applied to the load identification. Yan et al.［58］established an objective function for load identifica?tion based on the minimum difference between the calculated response and the measured response.Thus，the inverse problem can be transformed into a forward problem of parameter optimization.

Much work so far has focused on the identifica?tion of concentrated dynamic load. In addition to the aforementioned methods， there have also been many remarkable methods with distinguished advan?tages，which are not be expound them in detail here?in.

1.2 Distributed dynamic load identification

The distributed dynamic load acting on continu?ous structures is more complicated than the concen?trated load. There are relatively few studies devoted to distributed dynamic load identification. Time vari?able and space variable are both involved for the dis?tributed dynamic load，which may be independent or coupled with each other.For the continuous struc?ture，the governing equation can be commonly de?picted as

wherex(x，y，z) represents the space variable.ρ，candEsignify the material density，damping coeffi?cient and elastic modulus；αandβare the structural parameters；andu(x，t) andf(x，t) are the dis?placement response the distributed dynamic load. It is known thatu(x，t) andf(x，t) are both continu?ous functions in time-space dimension. In general，only discrete responses of limited measuring points may be obtained，which increases the difficulty to reconstruct the distributed function. Thus，it is nec?essary to transform the infinite-dimensional function into a finite-dimensional subspace. Approximating the distributed load by a set of linearly independent basis functions is a promising choice. Several identi?fication methods by functional approximation will be discussed below.

1.2.1 The generalized orthogonal polynomials approximation

The generalized orthogonal polynomial is a common function approximation method in a specif?ic interval，such as Legendre orthogonal polynomi?als，Chebyshev orthogonal polynomials，Laguerre orthogonal polynomials and Hermite orthogonal polynomials. Due to the title of“the most economi?cal expansion”，Chebyshev orthogonal polynomials are usually used in distributed load identification.The coordinates of structures need to be projected to the standard interval[-1，1] firstly. The expres?sion of Chebyshev orthogonal polynomial［59］with weight functionis

By discretizing the dynamic response in time di?mension，the distributed load at timetican be fitted by one-dimensional orthogonal polynomials taking one-dimensional structures for example，namely

whereJis the order of a polynomial，andaj(ti) the coefficient of thej-th polynomial.

By virtue of the FEM，the distributed load will be considered as a series of discretized loads on each node of the structure，which can be expanded as

whereT(x) denotes the polynomial matrix corre?sponding to the node position，andA(ti) the coeffi?cient vector. On the basis of the concentrated load identification method reviewed in Section 1.1，the coefficients of orthogonal polynomials will be calcu?lated. Dessi［60］identified the distribution of a wave load acting on a slender floating body using the prop?er orthogonal decomposition and integral spline ap?proximation technique. Wang et al.［4，61］proposed the distributed dynamic load acting on continuous structures including the cantilever beam and cantile?ver plate，in which the spatial load is approximated by Chebyshev orthogonal polynomials in time histo?ry under the load assumption of piecewise format.

1.2.2 The basis function approximation

For the linear system，the distributed dynamic load and the structural response can be expanded by the given basis function［62］，namely

whereχi(x) denotes thei-th basis function only spanning in the forcing space，φi(x) the structural responses generated by the load basis functionχi(x)，andWi(ti) the common weighting coeffi?cient for dynamic loadF(x，ti) and structural re?sponseu(x，ti). That is，the dynamic responseφi(x) is produced by the forceχi(x) purely. The dynamic load may be a single harmonic or multi-har?monic cases. Thus，the determination of weighting coefficientWi(ti) is the key to the reconstruction of distributed load，which can be numerically obtained by conventional modal analysis. As the curve-fitting method

Li et al.［63］assumed that the time history and the distribution function of the load are independent.The spatial function of the distributed load and re?sponse are fitted by finite basis functions using poly?nomial the selection technique，and then the time history can be reconstructed based on the shape function method of moving least-square-fitting.Cameron et al.［64-65］conducted the identification of the distributed flight load acting along the span and chord direction of aircraft through a least-squares minimization of Fourier coefficients with database Fourier coefficients.

1.2.3 The time-space double deconvolution method

For the distributed loadf(x，t) with timespace decoupled characteristics，it can be expressed by the product of the distribution functionψ(x) and the time history functions(t)，namelyf(x，t)=ψ(x)s(t). The displacement responses can be ana?lyzed by

whereg(x|x'，t-τ) is the Green’s kernel function.[?0，?1] is the loading area. In general，the arbitrary response can be regarded as the superposition of the responses caused by all loads. By discretizing them in time and space dimension，Eq.（17）can be trans?formed as

In the following，three situations will be dis?cussed for Eq.（18）. For the identification of the time history functions(t)，the distribution functionψ(x) is assumed to be known in prior. Eq.（18）can be transformed asu=Ψ1S1， in whichu=is composed ofa ndS=[s(t1)s(t2)…s(tm)]Tis the time sequence to be iden?tified. For the identification of the distribution functionψ(x)，the time history functions(t) is as?sumed to be known in advance. Eq.（18） can be transformed asu=S2Ψ2， in whichS2is composed ofs the space sequence to be identified. For the identification of bothψ(x) ands(t)，an initialization assumption should be made in advance，then the two aforemen?tioned steps should be repeatedly. This process is named as double iterative optimization.

Under the guidance of this method， Liu et al.［66-67］studied the iterative identification method of line distributed load on a composite plate using the displacement response，which assumed that the time-domain and spatial-domain of the load may be separated. Jiang et al.［68］identified the distributed dy?namic load of a vibrating Euler-Bernoulli beam based on the mode-selection method using the con?sistent spatial expression. Li et al.［69］proposed a de?coupling strategy based on the Green’s function method and the orthogonal polynomial approxima?tion to identify the time history and special distribu?tion separately.

To achieve a better understanding of the identi?fication methods of deterministic concentrated/dis?tributed load，their merits and demerits are summa?rized in Table 1.

Table 1 The advantages and disadvantages of deterministic load identification

2 Uncertain Dynamic Load Identi?fication Methods

The dynamic load identification method de?scribed in Section 2 is carried out under the deter?ministic assumption of structural performance，mea?sured response and dynamic load. However，uncer?tain factors exist widely in all processes of load iden?tification，which leads to the deviation between the reconstructed results and the actual load. Thus，ex?ploring the influence of multi-source uncertainties on the identification of dynamic load is of great signifi?cance for guiding the design and analysis of aero?space vehicles. The dynamic equation considering multi-source uncertainties can be transformed as

whereα=[α1，α2，…，αq] denotes theq-dimensional uncertain parameters.

2.1 The identification of the stochastic dynamic load

The difference between the stochastic dynamic load and deterministic dynamic load lies in the un?certainties in time history and the correlation be?tween each load. Since the stochastic dynamic load cannot be expressed by an exact time function，their power spectrum（PS）characteristics in fre?quency domain based on the theory of probability statistics are always considered as the variables to be identified.

2.1.1 Coherence analysis for stochastic loads

Multi-point stochastic dynamic load will host the basis in this section. The coherence can be divid?ed into three categories：Complete coherence，par?tial coherence and complete incoherence，which re?flects the degree of linearity between two loads and used for the mean value analysis. Complete coher?ence means that the stochastic dynamic loads are ho?mologous，while complete incoherence signifies the cross-PS between any two loads is zero. The PS matrix ofn-point stochastic excitation is defined asSF(ω)=[Sfi fj]n×n，i，j=1，2，…，n，whose proper?ties vary with their coherence. Through the coher?ence analysis of three kinds of stochastic excita?tion［70］，it can be concluded that the nonnegative def?inite PS has a unified spectral decomposition formu?la，namely

whererindicates the rank ofSF(ω). When the multi-point stochastic dynamic loads are completely coherent，r=1. When they are completely incoher?ent，r=n. When they are partial coherent，1＜r＜n. In addition，the PS matrixSF(ω) and its spectral vectorliis one-to-one corresponding due to the uniqueness of the spectral decomposition formula，eigenvalues and eigenvectors，namely，theSF(ω)can be represented exclusively byli. This is the the?oretical source of the identification of the stochastic dynamic load.

2.1.2 The inverse pseudo excitation method（IPEM）

Based on the theory of stationary stochastic vi?bration，the PSSF(ω) can be got by the PSSU(ω)of response and the FRF matrixH(ω)，i.e.

However，direct inverse as Eq.（21）is faced with the ill-posed problem of FRF matrix. The IPEM is proposed by Lin［71-72］，which is simple and efficient for stochastic vibration. It decomposes the PSSU(ω)and constructs a virtual responseyias fol?lows

where the virtual responseyimay be assumed to be generated by the virtual excitationfi=liejwt. Com?biningfi=H+(ω)yi，the PSSF(ω) can be ob?tained as Eq.（20）.

Guo et al.［73］identified the power spectral densi?ty matrix of uncorrelated or partially correlated ran?dom excitation experimentally，and confirmed the efficiency of the IPEM. It is noted that the inverse operation of the FRF matrix still exists in the tradi?tional IPEM. Thus，some weighted techniques are introduced to change the condition number of the FRF matrix. Leclere et al.［74］reconstructed the inter?nal loads exciting the engine block via weighted pseudo-inverse of the transfer matrix during opera?tion to alleviate its ill-conditioning. Jia et al.［75-76］pro?posed a Tikhonov regularization approach based on error analysis and weighted total least squares meth?od，and provided a selection method and a concrete form of weighting matrix. For the distributed sto?chastic excitation，Granger et al.［77］adopted the Tik?honov regularization method with Newton iteration to reconstruct the distributed load on nonlinear struc?tures. Jiang et al.［78］identified the one-dimension dis?tributed stochastic load by the IPEM combined with generalized Fourier expansion and projection tech?nique.

2.2 The load identification under structural probabilistic uncertainties

This section aims at the identification of con?ventional loads with exact time functions for uncer?tain aerospace vehicles with random fluctuation.When the uncertainties with sufficient sample infor?mation，the quantification and propagation analysis based on the probabilistic model has been developed completely. The Monte Carlo simulation（MCS）［79］can obtain the statistical properties of uncertain load by generating enough samples according to the prob?ability density function（PDF）of uncertain parame?ters. However，it is not suitable for engineering ap?plication，and often be treated as a verification meth?od. Subsequently，some methods have been pro?posed as follows.

2.2.1 The matrix perturbation method

The probabilistic uncertain parameterαcan be defined as its meanαmplus a small random perturba?tion Δαr，namelyα=αm+Δαr. Based on the per?turbation theory［80］，the relationship between exter?nal dynamic load and responses considering probabi?listic uncertainties be written as

By comparing the coefficients on both sides of Eq.（23），it can be transformed into two kinds of de?terministic issues on the bases of Taylor series ex?p ansion，namely

In other words，the dynamic load can be identified by the calculation of the mean value of external load and its sensitivity with respect to each random parameter. The stochastic characteris?tics of identified dynamic load can be further ob?tained as

By far the most works have been devoted to the response analysis for stochastic structures，yet a few examples have been applied in the dynamic load identification. Considering the randomness of geo?metrical， physical and boundary property， Sun et al.［81］，He et al.［82］and Wang et al.［83］combined the perturbation theory and regularization method to evaluate the dynamic load of random structures us?ing noisy responses，whose accuracy and efficiency are explained by several numerical examples com?pared with the MCS. Particularly，the first-order matrix perturbation method is restricted to the situa?tion where the coefficient of variation of random pa?rameters is quite small. Then，the Neumann expan?sion［84］is usually used to obtain the higher-order sta?tistics information for the matrix perturbation meth?od.

2.2.2 The polynomial-chaos-expansion method

As mentioned above，the first-order perturba?tion method is not a perfect way for stochastic struc?tures［85］with large fluctuation. Moreover，it disre?gards the distribution form of random parameters，so the same results will be obtained no matter what the PDF is. In view of this，Liu et al.［86-88］proposed a novel uncertain load identification method for sto?chastic structures with unimodal and bounded PDF based on polynomial chaos expansion. Similar to the matrix perturbation method，it also converts the complex stochastic analysis to several deterministic problems. If the uncertain parameterαiis unimodal，it can be expressed as the function of a stochastic pa?rameterβiwithλ-PDF，i.e.

Then，the transfer matrixG(α) can be depict?ed by the function of stochastic parameterβi，as

The external loadF(α)can be expanded as the sum of a series of polynomial chaosφ

Inspired by the orthogonality of polynomials，the coefficientszi1…iqcan be solved by the determinis?tic load identification technology. Eventually，the statistical properties of uncertain loads can be de?rived from the definitions of mean value and covari?ance. Schoefs et al.［89］estimated the system charac?teristics of offshore platforms by polynomial chaos expansion，and then identified the periodic tidal loads. Wu et al.［90］reconstructed the responses and forces of a stochastic system using polynomial chaos expansion and the Karhunen?Loève expansion，and investigated the influence of different correlation lengths of random system parameters on identified results. In addition，several other excellent methods have been successively used to deal with load identi?fication for stochastic structures. For example，Ba?tou et al.［91］reconstructed stochastic loads of a nonlinear dynamical system considering model uncer?tainties and data uncertainties，in which the mean value and dispersion parameter of PS function are calculated using the computational stochastic model and experimental responses. Zhang et al.［92］present?ed a Bayesian approach for force reconstruction con?sidering measurement noise and model uncertainty，in which uncertain FRFs are settled by Monte Carlo Markov chain methods.

2.3 The load identification under structural non?probabilistic uncertainties

Subjected to measurement cost and technolo?gy，it is impossible to gain enough samples of uncer?tain parameters to determine their PDF. Given this，the non-probabilistic model is introduced to quantize uncertain parameters. The interval model based on interval mathematics is a general method，in which only the upper and lower boundaries are necessary.The uncertain interval parameter vector［93］can be defined as

2.3.1 The Taylor-series-expansion method

To facilitate the analysis and discussion，two variables are defined firstly，namely the interval me?dian，and the interval radiusαr=When the uncertain level of all parame?tersαiis small，the uncertain load can be expanded by the first-order Taylor?series at the interval medi?an，namely

Thus，the load boundaries can be approximat?ed by

Similar to the matrix perturbation method in probabilistic model， the Taylor?series?expansion method only needq+1 times deterministic invers?ing calculation to determine the load boundary，in?cluding the load identification at the interval median and the gradient calculation with respect to each pa?rameter.

The dynamic load identification of structures with interval uncertainties has been intensively in?vestigated. Liu et al.［94-96］explored a series of re?searches of load identification with regularization methods considering measurement noises and mod?el uncertainties，to reconstruct the time history of the load interval. Ahmari et al.［97］established an in?verse analysis scheme，in which the result of the impact location is in a rectangle，and the result of time history is bounding sinusoidal curves with de?viation. However， the Taylor-series-expansion method has significant advantages only when the uncertainty problem is linear or the uncertain level is small. Therefore，Wang et al.［61］applied the sub?interval technique in dynamic load identification to avoid interval extension. But it is important to de?termine the subinterval number for each variable to make a tradeoff between efficiency and conver?gence.

2.3.2 The methods based on the surrogate model

The key point of dynamic load identification under non-probabilistic uncertainties is to get its maximum and minimum value in the interval do?main of uncertain variables. In order to further re?duce the overestimation or underestimation of load interval caused by uncertainty propagation analysis，some surrogate models，such as the Kriging mod?el，polynomial response surface method and artifi?cial neural network，are used to approximate the re?lationship between uncertain loads and uncertain pa?rameters. The detailed construction ways have been summarized in Ref.［98］. Generally，the construc?tion of the surrogate model is regarded as the uncer?tainty propagation in the inner layer，and the optimi?zation algorithm is needed in the outer layer to find the extreme point of uncertain load at each sampling instant.

Compared with the numerical simulation of the original FEM，the uncertainty propagation analysis based on surrogate model can effectively reduce the computational cost and filter out the unwished noise. But its accuracy and efficiency are highly de?pendent on the selection of the surrogate model and its hyper-parameter，which do not have a universal solution for all problems so far. This method is mainly used in the forward analysis of dynamics，and is in its initial stage for inverse issues. Ref.［99］utilized the Chebyshev orthogonal polynomials to fit the relationship between uncertain load and interval parameters at zero-cut of fuzzy interval. The maxi?mum and minimum points of uncertain variables are searched in a dimension-wise manner，and the corre?sponding loads can be identified via calling inverse methods. However，it ignores the coupling effect between uncertain variables.

In order to fascinate the understanding of uncer?tainty propagation methods，some prominent fea?tures are listed in Table 2.

Table 2 The advantages and disadvantages of different uncertainty propagation methods

3 Discussion of Development

There are many excellent reviews in the litera?ture dealing with the basic concepts of dynamic load identification in recent decades. However，further effort is required to better deal with the uncertain dy?namic load identification of aerospace vehicles.

Firstly， the structural dynamic responses，which are measured by strain gauges，accelerome?ters or displacement detectors，are the basis of the load identification. Different types and locations of response signals correspond to different identified re?sults. The sensor deployment optimization strategy including their type，number and position should be involved to improve the load identification accuracy.Secondly，the intelligent composite with self-diag?nose and self-healing function has been widely ap?plied to aerospace vehicles. The piezoelectric ele?ment is a novel component for monitoring the intelli?gent composite. The future looks bright to investi?gate the relationship between the external load and piezoelectric response and develop additional identi?fication methods for smart structures. Eventually，aerospace vehicles are always operated in hyperther?mal environments，and the change of temperature will cause the fluctuation of the structural dynamic characteristics. Under the circumstance of the tem?perature effect，the responses caused by the dynam?ic load may be annihilated. Therefore，it is neces?sary to propose an effective method to separate the temperature effect from the sensor monitoring data，so as to realize the dynamic load identification of aerospace vehicles in the thermal-mechanical cou?pling environment.

4 Conclusions

It is an urgent need but still a significant chal?lenge for uncertain dynamic load identification of aerospace vehicles，which can be considered as an interdisciplinary subject between the inverse prob?lem of structural dynamics and the uncertainty analy?sis. This paper provides a taxonomy and a review of alternative identification methods for both determin?istic and indeterministic dynamic load，following their applicability and specialty. The forthcoming re?search trend is prospected finally，which aims at pro?viding promising applications in the development of aerospace vehicles.