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Abstract
Although tunable Fabry-Perot (F-P) filters are widely acknowledged as fiber Bragg grating (FBG) demodulators, F-P filters exhibit drift error when subjected to ambient temperature and piezo-electrical transducer (PZT) hysteresis. To address the drift issue, the majority of the existing literature makes use of additional devices like the F-P etalon and gas chamber. In this study, a novel drift calibration method based on two-stage decomposition and hybrid modeling is proposed. The initial drift error sequences are broken down into three frequency components using the variational mode decomposition (VMD), and the medium-frequency components are further broken down using the secondary VMD. The initial drift error sequences are significantly simplified by the two-stage VMD. On this foundation, the long short-term memory (LSTM) network and polynomial fitting (PF) are used to forecast the low-frequency and high-frequency drift errors, respectively. The LSTM enables the prediction of intricate nonlinear local behaviors, while the PF method predicts the overall trend. The benefits of LSTM and PF can be effectively utilized in this manner. Compared to the single-stage decomposition, two-stage decomposition achieves superior results. The suggested method is an affordable and effective alternative to the current drift calibration techniques.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Because of their distinct benefits, such as its small size, lightweight, and immunity to electromagnetic interference, fiber Bragg grating (FBG) sensors are becoming more and more practical in a variety of industries, including aerospace, nuclear, civil engineering, petrochemical, and others [1,2]. Effectively determining the sensor's reflected wavelength and perturbation-induced wavelength shifts is the most important challenge of FBG interrogation [3]. Piezo-electrical transducer (PZT) driven tunable Fabry-Perot (F-P) filters are one of the most widely used types of tunable filters [4]. They are typically used to carry out high-resolution and fast wavelength interrogations. However, utilizing the F-P filter results in a brand-new issue: The filter's transmission wavelength is difficult to maintain due to dynamic hysteresis and the changing ambient temperature. The drift error may even result in a measurement failure in long-term FBG monitoring applications [5,6].
In the literature available, FBG referent gratings [7,8], F-P etalons [9,10], gas absorption line [5–12], and composite wavelength Refs. [13] have been proposed to solve the demodulation error of F-P filters. Reference grating method utilized few wavelength reference, which offers poor calibration. To counteract the effects of ambient temperature, reference FBGs must have stable center wavelengths. The F-P etalon approach allows for the introduction of numerous wavelength reference points spaced at equal intervals and enables a more accurate calibration at constant ambient temperature. Nevertheless, at various steady-state temperatures, the F-P etalon's central wavelength still shifts. The technique based on acetylene gas can provide temperature-insensitive wavelength reference points, but the coverage is limited, and peak extraction is difficult to implement. The temperature-stable gas serves as the reference for the composite wavelength reference method, but the temperature-changing situation has not been considered. Fan et. al. [5] found a way to deal with the temperature-changing situation: self-marked HCN gas. The self-marked HCN absorption lines serve as absolute wavelength references. Jiang et. al. [13] improved the temperature stability of the FBG demodulation system in a temperature-changing environment by combining F-P etalon and an acetylene gas chamber. Currently, most commercial calibration methods use either an etalon or a gas chamber, and high cost and large volume prevent the widespread application of FBG sensors and F-P filters.
Thanks to the boom of processing capabilities, artificial intelligence (AI) recently emerged as a practical and efficient solution for a number of challenging problems in the field of fiber sensing. The AI-based approach has been used to address overlapping [14,15], cross-sensitivity [16,17], and temperature drift [18,19]. Noting that each of the aforementioned models is based on a single model, which makes it difficult to generalize to large samples and easy to premature convergence. On the other hand, signal decomposition methods are usually combined with AI models. The goal of the signal decomposition is to decompose the unstable original data into a number of relatively stable subsequences. The prediction accuracy can be improved by applying separate modeling to various subsequences.
In some engineering fields, scholars have tended to carry out prediction of time series using signal decomposition and AI models. Dong et. al. [20] introduced a two-stage decomposition and long short-term memory (LSTM) network for predicting surface water quality. Due to the two-stage decompositions that mined the time series features buried in the high-frequency component, the prediction performance was improved twice. Shi et. al. [21] combined two-stage decomposition and LSTM for cutterhead torque, and the model predictive error is successfully reduced. Lin et. al. [22] employed ensemble empirical mode decomposition (EEMD), variational mode decomposition (VMD) and bidirectional LSTM (BILSTM) to effectively extracted the temporal correlation characteristic in each decomposed sequence. In the area of hybrid modeling, Li et. al. [23] combined single-stage decomposition and hybrid modeling with multivariable linear regression (MLR) and LSTM network. Ye et. al. [24] integrated LSTM network and support vector regression (SVR) to predict all components decomposed by EMD technique. Singular spectrum analysis (SSA) and VMD were utilized by Liu et. al. [25] to extract the trend information, and LSTM and extreme learning machine (ELM) were used to predict the low-frequency and high-frequency components, respectively. The model exhibited the best multistep prediction performance across all models, but it performs poorly in 1-step or 2-step prediction. In the abovementioned studies, the single decomposition is effective, and the addition of the secondary decomposition is suggested for further enhancing the model performance. On the other hand, the validity of the hybrid model has not been adequately examined in the majority of situations.
Based on the aforementioned problems, a novel drift calibration method using two-stage decomposition and hybrid modeling is proposed for Fabry-Perot filter. In our earlier research, the drift error of the F-P filter was studied using a single model (LSSVR) [26], while the integration method of multiple single models has also been proposed [27]. The drift error data has a significant time dependence, but the order of time series data is not taken into account by LSSVR. Therefore, two-stage decomposition and a hybrid model of LSTM network and polynomial fitting (PF) are proposed here. The contributions of the proposed calibration method are described as follows: (a) A second decomposition is introduced to further decompose the medium-frequency component acquired by the first VMD in order to address the failure problem of a single decomposition. (b) The LSTM network, a deep learning algorithm with satisfactory performance in long short dependencies, is utilized to complete the forecasting for the components including trend information. (c) For the low-frequency components created by the two decompositions, the forecasting is completed by the polynomial fitting that has quick prediction speed.
The order of the remaining text is as follows. The structure for the suggested calibration approach is presented in Section 2, along with a brief summary of each of the necessary component algorithms. Section 3 presents two case studies where the proposed model's and other models’ prediction outcomes are assessed. Section 4 brings this study to a close.
2. Methodology
2.1 VMD decomposition technique
The main goal of VMD is to design and resolve variational problems. It is an adaptive and entirely non-recursive modal variational and signal processing system [13]. The drift error of the F-P filter is decomposed here by VMD since it contains both trendiness and complicated randomness. The original signal S is first broken down into K components u to create the variational problem. The constrained variational expressions are:
Given a precision ε, iterate until Eq. (4) is satisfied.
2.2 Long short-term memory networks
In contrast to other artificial neural network technologies, LSTM is a unique type of recurrent neural network (RNN) that may be used to describe dynamic information of any duration in time series data. Through gated units, LSTM adds or removes past data information that is stored in memory cells, preventing gradient explosions and enabling long-term memory. The original learning samples are combined with the new learning mode and retrained to improve prediction accuracy, convergence speed, and stability. These features render LSTM ideal for processing the F-P filter's drift prediction. Figure 1 depicts the working process. LSTM networks introduce the idea of cellular states to consider the temporal correlations hidden in long-term states and avoid the issue of gradient disappearance [28].
The output gate, input gate, and oblivion gate are the three gates that the LSTM uses to regulate cell state. Oblivion gate is used to control whether to forget the hidden cell state of the previous layer, the mathematical expression is
where Xt is the current input, and h(t-1) is the hidden state at the previous moment, and Wx, Wh, bf are weights and biases. Input gate is in charge of the input sequence position. It consists of two sections that are composed and multiplied to update the state of the cell. The mathematical expression is where tanh is the activation function, and Wxc, Whc, and bc are weights and biases.Output gate consists of two sections and the mathematical expression is
where Wxo, Who, and bo are the weights and biases.The weights of the LSTM are updated by the adaptive moment estimation (Adam) optimization algorithm [29] in this study. The loss function is mean square error (MSE). Adam is a first-order optimization algorithm that iteratively updates the weights of neural networks based on training data.
2.3 VMD-VMD-PF-LSTM hybrid model
Ambient temperature and hysteresis both contribute to the F-P filter's drift error. A novel drift calibration method was proposed, which combined a two-stage decomposition technique and hybrid modeling. The hybrid model consists of PF and LSTM. Figure 2 depicts the precise workflow of the proposed VMD-VMD-PF-LSTM hybrid model. The following are the detailed descriptions:
Step 1: VMD was used to break down the original drift data into numerous modes, including three IMFs with varying frequencies and a residual component R. The first VMD reduced the volatility and complexity of the original data.
Step 2. The medium-frequency IMF2 obtained by step 1 was furtherly decomposed into IMF1’, IMF2’, IMF3’ and R’ with clearer inherent characteristics using VMD. The complex fluctuation patterns in the medium-frequency IMF2 are distinguished in this step.
Step 3. Polynomial fitting was used to predict the low-frequency modes in two decompositions, including the IMF1 and IMF1’ obtained in step 1 step 2, respectively.
Step 4. LSTM was utilized to predict the high- and medium-frequency components in two decompositions, including the IMF3 and R obtained in step 1, and IMF2’, IMF3’, and R obtained in step 2.
Step 5: After denormalizing each model's prediction outcomes, each outcome was integrated to have the final prediction result.
3. Experimental results and analysis
3.1 Experimental setup
A FBG sensing and demodulation system shown in Fig. 3 was put through tests. The amplified spontaneous emission (ASE) produces broadband light that illuminates FBGs. The spectrum reflected by FBGs is received by the F-P filter. The PZT in the F-P filter is driven by a sawtooth voltage. The F-P filter scans the reflected spectrum, which is then captured by a data acquisition card. All FBGs were submerged in water for a uniform and stable climate, and the F-P filter was placed in an electric oven for a temperature-changing environment. A thermistor was used to measure the temperature of the filter's outer layer.
The characteristic wavelengths of the FBGs utilized in the studies are displayed in Table 1. G0 is used as the reference grating, and G1 is used as the sensing grating. The gratings are put in a constant temperature water tank, and the temperature fluctuation of the water tank is less than ±0.1°C. Because FBG is simultaneously affected by strain and temperature, the reference grating method is adopted here for temperature compensation. Both the reference and sensing grating are in the same temperature field. Temperature compensation can be achieved by subtracting the wavelength shift caused by the temperature change from the total wavelength shift measured by the sensing grating. Examining the spectrum positions of gratings enables to determine the measurand. The FBG characteristic wavelengths were identified using HP 8164B, a framework for high-resolution lightwave estimation. The centroid detection algorithm (CDA) is used for peak detection.
Table 1. Characteristic wavelengths of FBGs
3.2 Modeling and testing
In the first case, the F-P filter was placed in a monotonic cooling environment. As depicted in Fig. 4, the temperature fluctuation and drift of the FBGs’ spectrum location (centroid index) were observed. It has been noted that the drift error gradually increased as the temperature dropped. The first 600 samples made up the training dataset, while the next 150 samples made up the testing dataset. The temperature and the temperature changing rate are used as features, and the F-P filter's drift is the model output.
To evaluate the drift error, a two-stage VMD decomposition was performed. Figure 5 displays the results. The original sequence was first decomposed into three subsequences with more noticeable fluctuation patterns, which displayed various patterns, trends, and changes. The medium-frequency IMF2 produced from the first VMD decomposition was then furtherly decomposed using the secondary VMD. It is evident that the IMF2 still includes various fluctuation patterns. Distinguish distinct data patterns sufficiently is essential for prediction accuracy.
Fig. 5. Decomposed results of original drift data by two-stage VMD decomposition (a) Raw data and primary decomposition (b) Secondary decomposition
In order to verify the validity and accuracy of the proposed VMD-VMD-PF-LSTM model, other models including individual models (LSSVR and LSTM), single-stage-decomposition-based models (VMD-LSTM and VMD-PF-LSTM), and the two-stage decomposition model (VMD-VMD-LSTM) are taken for comparison. Figure 6 shows the experimental results. As shown in Fig. 6 (a), the observation curve is the true wavelength drift, and it has been increasing over time. The final the true wavelength drift exceeds 240 pm. All six prediction models can predict the trend of the wavelength drift, but the prediction curves of LSSVR and LSTM have several abrupt changes. Since single-stage decomposition is introduced into LSTM, the performance of the VMD-LSTM model is not improved as expected. The VMD-PF-LSTM model has a similar result with the VMD-LSTM. After the secondary decomposition, the prediction performance is significantly improved. Both the VMD-VMD-PF-LSTM and VMD-VMD-LSTM are much better than the other models, and the hybrid model VMD-VMD-PF-LSTM achieves the best predictive performance among all the models. For clearly demonstrating the prediction performance of six models, Fig. 6 (b) shows prediction error obtained by subtracting the predicted value from the actual value. The maximum absolute error (MAXE) of the LSSVR, the LSTM, the VMD-LSTM, the VMD-PF-LSTM, the VMD-VMD-LSTM, the VMD-VMD-PF-LSTM is 33.1, 19.9, 20.2, 22.4, 14.2, 12.4 pm respectively.
Contrary to the general conclusion, single-stage VMD did not produce better results when predicting the drifts. Meanwhile, it is demonstrated that hybrid modeling does not make much difference when the signal decomposition is insufficient. The FBG demodulation systems are usually for quasi static measurement, and the ambient temperature and hysteresis are both changed extremely slowly. Therefore, a single decomposition is not sufficient for separate various fluctuation patterns. Aiming to solve the failure of single decomposition, the two-stage decomposition is introduced, and the medium frequency component is furtherly decomposed by VMD. The prediction performance is improved greatly.
To further verify the effectiveness of the proposed method, a cooling–stable-cooling dataset with the same dataset size (750 samples) was introduced to conduct the same experiment. Figure 7 displays the details of the second dataset. Comparing with the first dataset, the second dataset includes a stable phase between two cooling processes. Also, the initial stable temperature in the first dataset is not included in the second dataset. The experimental results are presented in Fig. 8.
The similar conclusions are drawn from the experimental results in Fig. 8. Figure 8 (a) shows wavelength drift prediction of the six models. Figure 8 (b) is the prediction error. It is obvious in Fig. 8 (b) that the dark blue curve (VMD-LSTM) and the light blue curve (VMD-PF-LSTM) is higher than other curves in most of the time. This phenomenon demonstrates that the LSTM models with single decomposition perform worse than other models. In terms of error statistics, the MAXE of the VMD-LSTM and the VMD-PF-LSTM is 22.6, 25.1 pm respectively, while the MAXE of the LSSVR, the LSTM, the VMD-VMD-LSTM, the VMD-VMD-PF-LSTM is 21.2, 17.4, 10.1, 7.7 pm, respectively. The VMD-VMD-PF-LSTM model performs best, which means the two-stage decomposition is effective in improving model accuracy. Besides, it is worth noting that the hybrid model of PF and LSTM plays a more important role here.
The MAXE and root mean squared error (RMSE) were used to evaluate the performance of the regression models, which are, respectively, expressed as:
where yk and ${\hat{y}_k}$ are true value and predicted value, respectively.Figure 9 demonstrates the error evaluation results in two datasets. In dataset 1, the MAXE and RMSE are slightly reduced by the addition of the hybrid of PF and LSTM. The MAXE and RMSE of the VMD-VMD-LSTM are 14.2 and 5.5 respectively, while the corresponding values of the VMD-VMD-PF-LSTM are 12.4 and 5.4 respectively. However, in dataset 2, the MAXE and RMSE are significantly reduced by the addition of the hybrid of PF and LSTM. The MAXE and RMSE of the VMD-VMD-LSTM are 10.1 and 4.8 respectively, while the corresponding values of the VMD-VMD-PF-LSTM are 7.7 and 3.3 respectively. The MAXE is reduced by 23.76% and the RMSE is reduced by 31.25%.
In summary, the proposed model performs best and the single LSTM model performs second better among all the models. The proposed VMD-VMD-PF-LSTM model reduces the MAXE of single LSTM model by 37.69% and 55.75% and RMSE by 31.65% and 59.26% in dataset 1 and dataset 2 respectively. This suggests that VMD-VMD-PF-LSTM model could better extract the nonlinearity of the drift error data and low bias of prediction. With the proposed VMD-VMD-PF-LSTM model, the wavelength demodulation accuracy is reduced from 91.7 pm to 12.4 pm in the case of dataset 1, while the ambient temperature changes over 0.58°C. In the case of dataset 2, the wavelength demodulation accuracy is reduced from 33 pm to 7.7 pm, while the ambient temperature changes over 0.1°C. Since the temperature of the filter is close to the ambient temperature during data sampling, the temperature range in experiments is relatively narrow. This work is an exploration of temperature drift compensation for tunable optical filters with AI models in a narrow temperature range.
4. Conclusion
A novel drift calibration method combining two-stage decomposition and hybrid modeling is proposed for F-P filter in this study. The new method integrates the benefits of signal decomposition and two prediction algorithms including LSTM and PF. VMD is used to decompose the original drift sequence, and then the secondary VMD is introduced to furtherly decompose the medium-frequency component obtained by the first VMD. The different fluctuation patterns are completely separated using the two-stage decomposition, which brings great advantages for prediction models. Additionally, the LSTM and PF algorithms are used to predict, respectively, the low and high-frequency components. The PF offers the rapid prediction of the general trend of the change, but the LSTM permits the prediction of complicated nonlinearity local behaviors. Therefore, the suggested hybrid prediction approach can efficiently utilize the benefits of PF and LSTM.
The individual prediction models, the single-stage-decomposition-based prediction models, and the two-stage-decomposition-based prediction models are all used as comparisons to examine the prediction performance of the suggested model. The experiments’ findings show that: (a) the proposed VMD-VMD-PF-LSTM model has the best prediction performance among all the involved models; (b) the two-stage decomposition plays a vital role for the drift error prediction due to the limited capability of single-stage decomposition; and (c) the hybrid modeling with PF and LSTM is valid for drift calibration of F-P filter.
Funding
National Natural Science Foundation of China (61905139, 61935002, 62005157); Science and Technology Commission of Shanghai Municipality (SKLSFO2021-03).
Disclosures
The authors declare that there are no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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