Modal parameter estimation of turbulence response based on self-attention generative model

Abstract

Modal parameter estimation of turbulence response is an important aspect of flutter test data processing. Using the modal parameter estimation results of turbulence response and the damping extrapolation curve, the flutter boundary of aircrafts can be predicted. However, owing to the randomness of atmospheric turbulence excitation, modal parameter estimation of the turbulence response has a challenge. This study analyses the turbulence response and calculates the corresponding impulse response using the seq-to-seq self-attention generative network. The encoder performs feature compression of the turbulence response, whereas the multi-head self-attention structure in the middle layer extracts the time series features. Up-sampling is performed based on the decoder to obtain the impulse response, and modal parameter estimation of the turbulence response is achieved. The self-attention generative model is validated using simulation data and the turbulence response is obtained by the wind tunnel test and the flutter flight test. The results demonstrate the feasibility and engineering applicability of the seq-to-seq generative model.

Keywords

Deep learning turbulence response flutter flight test seq-to-seq network model U-Net structure generative model

1. Introduction

Atmospheric turbulence excitation is a wide-band random excitation. In the flutter flight testing of aircrafts, atmospheric turbulence is the primary excitation form. The structural response describes the modal information of the aeroelastic system and also includes the random noise of atmospheric turbulence. However, owing to the unmeasurable nature of random excitation, modal parameter estimation methods of random excitation are typically output-only. Stochastic Subspace Identification (SSI) is a modal parameter estimation method based on the state space model. The modal parameter estimation of the response signal of a highly flexible wing aeroelastic model in a wind tunnel test was studied using the SSI method (Y. Govers et al., 2019). The aircraft modal parameters were examined based on SSI in the flight tests (Jelicic et al., 2017) 0. The modal parameters of a flutter flight test aircraft based on Curve-fitting Frequency Domain Decomposition (CFDD) is carried out (Schulze et al., 2013). The modal parameter estimation of turbulence response based on CFDD involves multi-channel analysis of response signals. Considering the random and deterministic information contained in the turbulence response, the impulse response is calculated from the turbulence response based on a deep learning–based generative model (S. Duan et al., 2022).

The analysis of vibration signal based on deep learning is a current research hotspot. In the study of milling tool monitoring, a comparative analysis of handcrafted feature and deep learning autoencoder was carried out, and the features were used for fault diagnosis of vibration signals (Alabsi M et al., 2021). The noised vibration signal was processed by establishing a seq-to-seq deep learning network, and the denoised vibration signal was calculated (Du W et al., 2023). In the study of intelligent fault diagnosis of spindle-rolling bearings, an attention dense convolutional network was established to analysis of vibration signals, and the dual attention mechanism was used for channel attention analysis (Jiang S et al., 2021). A neural network consisting of self-attention and LSTM was established to analyze the noised vibration signal (Wang Y et al., 2022). Focused on source separation, a deep neural network using a time waveform generation was established (Defossez A et al., 2020), and time-frequency domain loss was obtained using short-time Fourier transform (STFT) to improve the accuracy of source separation. A waveform-based deep neural network model named Conv-TansNet achieved speech source separation (Kadıoğlu B et al., 2020), and proved that it is feasible for analysing time series signals based on the seq-to-seq generative model.

Aiming at the modal parameter estimation of turbulence response in flutter tests, the research progress of seq-to-seq generative models based on deep learning, the encoder extracts the feature of input turbulence response, multi-head self-attention analyses the time series features, and the impulse response signal is obtained through decoder and up-sampling. The generative model improves the accuracy of waveform generation through the U-Net structure and skip connection. The contributions of this work can be summarised as follows.

• Turbulence response is analysed based on multi-head self-attention generative model.

• U-Net structure and skip connection are structures under seq-to-seq network model.

• Compared with the SSI, there is no need to estimate the modal parameters through the stabilisation diagram, and the algorithm execution time is less.

• Compared with the generative model of LSTM, self-attention achieves feature extraction through fewer encoder and decoder layers.

• Self-attention generative model is verified using simulation data and flutter flight test data.

2. Related methods

In waveform-based seq-to-seq neural networks, the network structure includes an encoder structure for down-sampling and a decoder structure for up-sampling. Skip connection is performed between the features of the encoder and the decoder. Then, the network structure is designed as a U-Net to improve the accuracy of the waveform. Analysing the time series features of the encoder output through long short-term memory (LSTM) or self-attention is the primary purpose of the waveform-based seq-to-seq neural network. The modal parameter estimation of the turbulence response based on seq-to-seq generative model is performed. Based on the deep learning signal processing methods such as image and speech, vibration response signal of aeroelastic structure is processed by the deep learning methods.

2.1. Encoder and decoder

In the deep learning network of seq-to-seq, it is a common method to analyse the characteristics of the input signal by the encoder and improve the resolution through the decoder. In the research on the denoise and enhancement of the speech signal (Zheng C et al., 2021), the establishment of an encoder to input the noisy speech signal performed feature extraction and then calculated the denoised speech signal through the decoder. In the text-to-speech study (Miao C et al., 2021), the network structure was established for encoding and feature compression on the input text, and the decoder generated the speech signal of the waveform. Through the analysis of the encoder and decoder, it was possible to improve the accuracy of the waveform signal effectively. In a study on speech enhancement, the features were extracted by the encoder based on the time-frequency information of the speech signal (C. Zheng et al., 2021), and the time series analysis was performed on the features from the encoder. The decoder improved the resolution of the model output and realised speech enhancement of the input signal.

The input and output of the generative model are images or speech signals of the same resolution. As the input and output of the generative model contain a large amount of data, the feature extraction operation based on the convolutional network is conducive to reducing network parameters. The encoder reduces parameters by down-sampling and obtains the compression feature of the input signal. To obtain the output signal with the same resolution as the input signal, it is necessary to perform up-sampling through the convolution operation to improve the output signal’s resolution. The features are calculated using equations (1) and (2) in the encoder and decoder structure, respectively.

F_{e_{i}} = R e L U (W^{e_{i}} * X_{e})

(1)

F_{d_{i}} = R e L U (W^{d_{i}} * X_{d})

(2)

In the encoder,

F_{e_{i}}

is the feature output of the ith layer,

W^{e_{i}}

is the weight of the ith layer,

X_{e}

is the input of each layer, and the input is the original signal or the output feature of the previous layer according to different layers. In the decoder,

F_{d_{i}}

is the feature output of the ith layer,

W^{d_{i}}

is the weight of the ith layer, and

X_{d}

is the input of the ith layer, and according to the number index of the decoder layer, the input is the feature of the middle layer or the output of the previous decoder layer. The skip connections of the U-Net between the encoder and decoder make the input features of the decoder also include the output features of the encoder at the corresponding position layer.

2.2. Multi-head self-attention

Attention is a neural network model applied to time series analysis. Establishing a self-attention method to calculate different features can effectively improve the waveform accuracy. Focused on the seq-to-seq in machine translation, the analysis of sequence signals was performed based on the self-attention mechanism (Vaswani A et al., 2017). A post-self-attention augment network was established to analyse waveform sequence signals (Lu Z et al., 2022). Using the feature extraction of the time and time-frequency domain encoders in the speech signal waveform, time series features were analysed through the self-attention mechanism in the middle layer (Défossez A 2021). Speech denoising in the waveform domain with self-attention was studied, and time series analysis of the features via self-attention was performed in the network structure, which improved the accuracy of denoise (Kong Z et al., 2022).

A sliced attention-based neural network was established for music source separation and analysed the time spectrum of the extracted music signal (Li T et al., 2021). Studies have shown that analysing encoded features based on self-attention in the time series can achieve waveform domain generation. The self-attention method can obtain the corresponding query matrix Q, key matrix K, and value matrix V through preliminary features for encoder output features.

Q = F_{e} W^{Q}, K = F_{e} W^{k}, V = F_{e} W^{V}

(3)

where

F_{e}

is the feature output by the last layer of the encoder,

W^{Q}

is the weight of the query matrix,

W^{k}

is the weight of the key matrix, and

W^{V}

is the weight of the value matrix. Based on the Q, K, and V matrices calculated the different features using equation (3), and the calculation of self-attention features is carried out by equation (4).

A t t e n t i o n (Q, K, V) = s o f t \max (\frac{Q K^{T}}{\sqrt{d_{k}}}) V

(4)

where

\frac{1}{\sqrt{d_{k}}}

represents the scaling factor and softmax represents the activation function.

2.3. U-Net and skip connection

The U-Net structure in deep learning is mainly used in the analysis of the reconstructed signals of the generative model. As the resolutions of the input and output signals in the generative model are identical, the input signal feature by the encoder is typically extracted. Then, the resolution is improved by the up-sampling of the decoder. Therefore, the network structure is visually represented as the U-Net (Markovic D et al., 2022). The U-Net and skip connection structure in the system identification of bridge vibration signals proved the feasibility of the U-Net structure for vibration signal analysis (Z. Shang et al., 2022). In a study on blind source separation, feature analysis of the input speech signal through convolutional coding operations was performed, up-sampling made the output signal consistent with the original input signal. As the encoding and decoding structures were identical in the network structure, feature analysis was performed by skip connection to improve the computational accuracy of the waveform domain (Défossez A et al., 2019(a); Défossez A et al., 2019(b)). The skip connection method of the U-Net structure was studied and proposed to improve the skip connection from a channel-wise perspective with a transformer (H. Wang et al., 2022). A deep learning framework was developed to predict nonlinear stress distribution in microstructural representations of composite materials (Sepasdar R et al., 2022); the generator of the deep learning is composed by encoder and decoder connection through the U-Net.

3. Model

3.1. Problem analysis

Due to the complexity of aerodynamic and structural coupling in the flutter test signals, the basic principles of the turbulence response are verified through the random and deterministic simulation of atmospheric turbulence. Considering the randomness of turbulence excitation, the response signal is obtained through the convolution of random excitation and impulse response in the simulation test process. The purpose is to obtain the system response of random excitation. The randomness and determinism are described in turbulence response in flutter testing.

In the flutter test, the structure response signal excited by atmospheric turbulence is called the turbulence response. To realise the modal parameter estimation of the structural response, it is necessary to analyse the structural response signal based on random excitation. From the perspective of system identification, the impulse response of the structure is the most suitable form of system response. Due to the problems of test risk and equipment modification, the structural response of atmospheric turbulence excitation is a common response signal in the flutter test. At the same time, due to the randomness of atmospheric turbulence, the turbulence response contains both the randomness of the excitation and the determinism of the structure information. The relationship between the turbulence response and the impulse response can be described by equation (5).

x (t) * h (t) = y (t)

(5)

where

y (t)

is the turbulence response signal,

x (t)

is the atmospheric turbulence excitation, and

h (t)

is the impulse response.

The relationship between turbulence and impulse responses can be described as convolution. The measurable turbulence response can calculate the corresponding impulse response based on the generative model and then estimate the modal parameters based on the generated impulse response.

3.2. Architecture

The analysis of the impulse response based on the turbulence response can be described as the research of the seq-to-seq generative model of deep learning. Therefore, the network structure is the deep neural network of the generative model, including the encoder and decoder. The network structure includes encoder, multi-head self-attention, and decoder for the analysis of the input turbulence response signal. The specific structure will be further explained.

Encoder is a multi-layer feature extraction composed by one-dimensional convolution, LeakReLU activation function, and BatchNormal, where the convolution kernel size is 21, padding is 10, and stride is 1, and down-sampling is used to compress the feature. The encoder is composed of five layers of feature extraction structures.

Multi-head self-attention analyses the features outputted by the encoder through the self-attention method. Through a fully connection network, the Q, K, and V matrices are calculated from the features outputted by the encoder. Multi-head self-attention structure calculates the input feature (Q, K, and V matrices), and the number of heads is 6. Based on the attention mechanism without changing the dimension of the feature, the dimension of the attention feature outputted by self-attention is the same as the dimension of the input feature.

Decoder is similar to the encoder. It also performs feature analysis through a multi-layer structure (one-dimensional convolution, LeakReLU activation function, BatchNormal, and up-sampling). At the same time, the input of decoder is performed by skip connection. The skip-connected features are combined to improve the final output impulse response waveform further. Figure 1.

3.3. Loss function

The loss function is an important part of deep learning generative model research. Two loss functions in the time and time-frequency domains are used to optimise the generative waveform model. The time-domain loss function includes energy loss, time loss, and phase loss in this study.

The energy loss is calculated as

{l o s s}_{e} = M S E (y^{2} - {\hat{y}}^{2})

(6)

where

M S E

is the mean square error and y and

\hat{y}

represent the labelled and generated impulse response, respectively.

The time loss function is calculated as follows:

{l o s s}_{t} = M S E (y - \hat{y})

(7)

The phase loss function is calculated as follows:

{l o s s}_{p} = M S E (Δ y - Δ \hat{y})

(8)

where

Δ y

and

Δ \hat{y}

represent the first-order difference of time series. The loss calculation of the first-order difference improves the phase accuracy of the generated impulse response.

Thus, the time-domain loss function is expressed as follows:

l o s s_{T} = l o s s_{e} + l o s s_{t} + l o s s_{p}

(9)

The time-domain loss function includes three parts—energy loss, time loss, and phase loss. The final time-domain loss function is calculated using equation (9).

Spectral convergence and STFT magnitude loss are used to calculate the time-frequency domain loss.

{l o s s}_{s c} = \frac{‖ S T F T (y) - S T F T (\hat{y}) ‖}{{‖ S T F T (y) ‖}_{F}}

(10)

{l o s s}_{m a g} = M S E (S T F T (y) - S T F T (\hat{y}))

(11)

where

{‖ ‖}_{F}

represents the Frobenius norms function and STFT( ) represents the time-frequency domain analysis result obtained using the STFT function. The total time-frequency domain loss function is as follows.

l o s s_{S T F T} = l o s s_{s c} + l o s s_{m a g}

(12)

Based on the analysis results of the time-domain and time-frequency domain loss functions, the overall loss function for network optimisation of the generative model can be expressed as

l o s s = α_{1} l o s s_{T} + α_{1} l o s s_{S T F T}

(13)

where loss is the total loss function;

l o s s_{T}

and

l o s s_{S T F T}

are the loss functions in the time and time-frequency domains, respectively. Additionally,

α_{1}

and

α_{2}

are the hyperparameter weights of the time and time-frequency domain loss functions, and the value is set to 0.5.

4. Method validation

4.1. Dataset construction

Turbulence response signal can be obtained via measurement, but it is difficult to measure the impulse response of the structural system in the wind tunnel test and flutter flight test. There is a convolution relationship between turbulence response and the corresponding impulse response. Therefore, it is possible to construct a data set for training a model by mathematical simulation. Turbulence response of the data set was calculated to form the data set using equation (5). Impulse response is calculated as follows:

h (t) = \sum_{i}^{N} A_{i} e^{- ξ_{i} ω_{i}} \sin (ω_{i} t)

(14)

where N means that the order of the impulse response is 2, 3, and 4, respectively. Building a multi-order impulse response dataset can effectively avoid model overfit caused by the error between the simulation dataset and the engineering test data. In the i^th mode,

A_{i}

is the amplitude,

ξ_{i}

is the damping, and

ω_{i}

is the frequency. To prevent overfitting of the model optimisation process, the damping and frequency parameters are set by random selection within the specified range. Considering that the actual frequency range of flutter is mainly concentrated in the low-frequency band and the damping is generally large, the frequency range is 1–60 Hz, and the damping range is 0.005–0.2. The impulse response signal is calculated using equation (14), and turbulence response is calculated based on convolution random noise and impulse response in the simulation data set construction.

4.2. Simulation result verification

Impulse response calculation is performed based on the self-attention generative model in the simulation verification. Simulated turbulence response signal (top), real impulse response, and generated impulse response (bottom) are shown in Figure 2(a) (simulation signal #1), and Figure 2(b) is the spectrum of the time series in Figure 2(a).

The waveform of time series is shown that the generation effect of the fundamental frequency is the same as the real impulse response, but there is some noise caused by model optimisation. Due to the limitations of the simulation data set, the generated impulse responses have some spurious modes.

Figure 3(a) (simulated signal #2) contains the time series of the turbulence response (top) and the impulse response signal (bottom), including the real and the generated. Figure 3(b) is the spectrum waveform of simulated signal #2 and its impulse response.

Because the damping values corresponding to the two groups of frequencies in simulation signal #2 are relatively large, the comparison results of the impulse responses in the frequency domain are consistent. In addition, the test speed is often far from the flutter speed in the actual flutter test flight, and the damping of the structural response is large. Comparing the estimation results of the large damping modal parameters provides a basis for the subsequent modal parameter estimation based on the flutter flight test data.

Table 1 shows the modal parameter estimation results of the two sets of simulated signals. As the simulated signals have true values, the generative model analyses the turbulence response signal based on self-attention, and the corresponding impulse response signal is calculated. The matrix pencil method estimates modal parameters of generated impulse response (T.K. Sarkar and Pereira, 1995). Considering the waveform comparison of the time and frequency domain, the identification error of the frequency is small. Because the absolute value of the damping parameter is small, the damping error is relatively large. As a simulated signal, the signal noise and non-stationarity have minimal effect, the generated impulse response is more consistent with the real waveform comparison result, and the modal parameter estimation results are the same.

Table 1.

Modal parameter estimation result of simulation signal.

	True value		Proposed method
	Frequency (Hz)	Damping	Frequency (Hz)	Damping
Simulation signal #1	9.540	0.1063	9.462	0.1136
Simulation signal #1	20.053	0.0932	20.154	0.0925
Simulation signal #2	13.369	0.1133	13.371	0.1147
Simulation signal #2	19.537	0.1022	19.532	0.1031

4.3. Wind tunnel test data validation

The feasibility of the self-attention generative model is verified based on the simulation signal. On this basis, the verification is carried out based on the turbulence response acceleration signal of the wind tunnel test aeroelastic model (Figure 4). The acceleration sensor is located on the main beam of the aeroelastic model wing that facilitates data acquisition of structural response signals.

Figure 5(a) (top) is the time series of the wind tunnel test acceleration response signals (wind tunnel test data). Corresponding to Figure 5(a) (bottom) is the impulse response signal calculated by the self-attention generative model, and Figure 5(b) is the spectrum of the corresponding signal.

From the time-domain waveform of the generated impulse response, the generative model based on self-attention can basically realise the signal processing of the turbulence response. Judging from the corresponding results of the spectrum, the key modal frequencies of the turbulence response can be basically reconstructed.

Table 2 lists the modal parameter estimation results of the turbulence response. From the comparison, the analysis results of the main modal frequencies are consistent with the results of ground vibration testing (GVT) and SSI. Due to the absolute value of the modal damping, there is a certain error in the result. It is worth noting that the damping ratio is usually used in practical engineering, and the relationship between the damping ratio and damping is 100 times.

Table 2.

Modal parameter estimation result of wind tunnel test data.

GVT	SSI		Proposed method
Frequency (Hz)	Frequency (Hz)	Damping ratio	Frequency (Hz)	Damping ratio
5.612	5.630	7.364	—	—
18.235	18.215	3.865	18.621	9.606

4.4. Flutter flight test data validation

Based on the flutter flight test data of a certain aircraft, the impulse response generation analysis and verification of the self-attention generative model are conducted. As there is no real impulse response, Figure 6(a) shows the time series of turbulence response (top, flutter test signal) and the time series of generated impulse response (bottom). Figure 6(b) is the spectrum of two sets of time series.

As the turbulence response signal contains large damping, the impulse response wave decays quickly. The amplitude is stable after the decay is zero, and there is no other noise, which is more comparable to the ideal impulse response. Based on the comparison results of the spectrum, the turbulence response and the generated impulse response are relatively consistent for identifying fundamental frequencies. Due to the limitation of the model training and simulation data set, the model has a certain overfitting problem and a certain amount of noise in the generated impulse response spectrum.

Table 3 analyses the modal parameter estimation result of turbulence response (flutter flight test signal) based on the SSI method. The generated impulse response through the self-attention generative model is estimated the modal parameters based on the matrix pencil method. As there is no true value of modal parameters for comparison in the flutter test flight signal analysis, only the corresponding frequency information can be obtained for the GVT. Considering the comparison results of modal parameters of the SSI and the proposed method, the identification effect of the main mode of interest is superior. However, there are still cases where the identification error of individual modes of interest is relatively large.

Table 3.

Modal parameter estimation result of flutter test signal.

GVT	SSI		Proposed method
Frequency (Hz)	Frequency (Hz)	Damping ratio	Frequency (Hz)	Damping ratio
10.432	10.592	6.040	9.825	6.204
25.415	25.197	4.581	25.706	4.318
19	16.049	5.793	18.413	13.883

4.5. Comparison of the generative model of LSTM and self-attention

In the time series analysis of the generative model, its basic structure is often composed by the encoder and decoder in the seq-to-seq model. The main function of the encoder is to reduce the dimension of the original input long sequence. The middle layer is used to establish a time sequence feature analysis structure. Finally, the decoder performs up-sampling to meet a certain resolution requirement of the output time sequence. LSTM was used to analyse sequence signal features in the middle layer (S. Duan et al., 2022). In this study, multi-head self-attention is used for time series analysis. Still, considering the difference between the convolution layer number in the encoder and decoder, the layer number of features differs in the amount of features output by the encoder. In the generative model of LSTM, feature analysis is performed based on the 10 layers of convolution in the encoder, and the output features of the encoder are used for time series analysis. The generative self-attention model only passes five convolution layers, composed of the encoder, which is used to analyse the input turbulence response.

Figure 7(a) and 7(b) are the time and frequency domain waveforms of turbulence response and the generated impulse response based on the self-attention generative model, respectively. Figure 8(a) and 8(b) are the time and frequency domain waveform comparisons of the turbulence response and generated impulse response of the LSTM-based generative model, respectively.

From the waveforms of the impulse response signals generated by the same two set data, the waveform of self-attention is more similar to the waveform of the ideal impulse response. In contrast, the impulse response generated by LSTM has frequency separation at different times. From the frequency waveform, the waveform of the fundamental key frequency is consistent with the impulse response. In the follow-up research, the advantages of self-attention and LSTM can be considered fully to improve the accuracy of the generated impulse response.

5. Conclusion

Aiming at the randomness and deterministic information, the turbulence response is analysed based on the multi-head self-attention generative model, the corresponding impulse response is calculated, and the modal information is obtained based on the generated impulse response signal. As a seq-to-seq network model, the U-Net structure and skip connection on the encoder and decoder features can improve the accuracy of impulse response generation. Finally, based on the simulation data and turbulence response signals of the flutter test flight, the method is validated, thereby confirming the theoretical feasibility and engineering applicability of the developed method. It can be seen from the comparison of the modal parameter estimation results, the multi-head self-attention can calculate the impulse response signal relatively more accurately, and the waveform comparison between the time domain and the frequency domain can be seen to be closer to the form of the impulse response signal.

Supplemental Materials

Supplemental Materials - Modal parameter estimation of turbulence response based on self-attention generative model

Supplemental Materials for modal parameter estimation of turbulence response based on self-attention generative model by Shiqiang Duan, Hua Zheng, Jinge Yu, and Yafeng Wu in Journal of Vibration and Control

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This work is supported by the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant no. CX2021077) and the Fundamental Research Funds for the Central Universities (Grant no. 31020190MS702).Conflicts of Interest.

ORCID iD

Hua Zheng

Supplemental Material

The supplementary material file contains the flutter flight testing data, which we used to validate the proposed method (). The data were sourced from an acceleration sensor in an aeroelastic model subjected to physical testing. The data is given in .txt format. Supplemental material for this article is available online.

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