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Abstract
Hysteresis and temperature drift deteriorate the demodulation performance of tunable Fabry-Perot (F-P) filters. This study addresses a novel adaptive weight least square support vector regression (AWLSSVR) to compensate for the hysteresis and temperature drift of F-P filters. The temperature drift of a referent fiber Bragg grating(FBG) is used to estimate the temperature drifts of other three sensing FBGs, and a novel adaptive weighting strategy with an asymmetric noise interval is proposed, to eliminate the effects of noise in the training dataset. The experimental results show that when the temperature-changing modes of the training and testing datasets were close to each other, the error of the proposed method is reduced to 8.7 pm, while the errors of the other three conventional methods based on LSSVR are more than 10.8 pm. Further, when the temperature-changing modes of the training and testing datasets were partly different, the error of the proposed method was reduced to 5.4 pm, while the errors of other methods were more than 11.9 pm. It was verified experimentally that the proposed AWLSSVR method is more accurate and robust than other versions of WLSSVR for training samples with noise, requires no additional hardware, and covers the entire C band.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Piezo-electrical transducer(PZT) driven tunable Fabry-Perot (F-P) filters are widely applied to FBG demodulation owing to its fast demodulation speed, wide spectral range, high resolution, and high capacity [1,2]. Under a certain applied voltage, the piezo-electrical response of PZT would change the cavity length of the F-P filter, and light with a certain wavelength will be transmitted through the F-P filter at maximum transmittance. However, hysteresis and temperature changes gradually change the piezo-electric constant and dielectric constant of the material [3,4]. Consequently, the wavelength-voltage relationship of the F-P filter continues to shift. In long-term monitoring, the demodulated wavelength may cause significant drift, greatly affecting the demodulation precision [5,6].
Scholars have proposed several wavelength reference methods to overcome the problems of hysteresis and temperature drift. Wang et al. designed a copper box with thermal stability for four reference FBGs, and the correlation between temperatures and wavelengths of the reference FBGs was calibrated before every scan of the F-P filter [7]. This method realized a high-precision dynamic calibration, but the large waste of bandwidth is an unavoidable issue. Additionally, the thermostat module increases the cost and complexity of the demodulation system. Further, the calibration method based on the F-P etalon has been widely employed in practical applications for decades [8,9]. However, F-P etalons are susceptible to unpredictable temperature variations [10]. Moreover, because the fundamental absorption lines of atoms and molecules are insensitive to the environmental temperature, this characteristic of gas is excellent for wavelength reference [11]. Absorption lines of acetylene (C2H2) [12] and hydrogen cyanide (HCN) [13] have been employed to calibrate the temperature drift of the etalon wavelength. Although investigations that only use absorption lines as wavelength references have been reported [14], the wavelength range is 3 nm only, which cannot satisfy the requirements of many practical applications. Fan et al. proposed a novel FBG demodulation method based on a self-marked HCN absorption spectrum as an absolute frequency reference [15]. The reference wavelength was determined by different wavelength intervals of the HCN absorption lines. In general, most methods use additional hardware to calibrate the reference wavelength. These hardware methods are complex and result in increased cost and volume.
With the rapid development of Artificial Intelligence (AI) [16,17], a multitude of approaches have emerged to build more accurate nonlinear models. The most commonly used methods for temperature compensation are support vector machine (SVM) [18–20] and neural networks [21,22]. Artificial neural networks (ANNs) have some shortcomings, such as overfitting and easy access to the local extremum. Deep learning techniques solve the problem of gradient vanishing and exploding problems of ANNs in large systems, but they require long computation time [23]. Contrary, the SVM, based on statistical theory and the structural risk minimization principle [24], outperforms ANN in terms of global optimization and generalization capability for small-sized datasets [25]. The SVM can be applied to nonlinear regression problems by solving a convex quadratic programming (QP) problem, and it shows good performance in temperature modeling. Some studies employ the SVM to solve temperature drift problem. Cheng et al. proposed a modeling method based on SVM with multiple temperature variable inputs for ring-laser gyroscopes, and the precision and generalization ability were improved [19]. Wang et al. introduced a combined kernel function with four parameters for SVM to compensate for the temperature drift of a fiber optical gyroscope [18].
As an extension of SVM, the least-squares SVM (LSSVM) overcomes the defect of slow training speed in it, by solving a linear equation set rather than a quadratic optimization problem [26]. Further, LSSVM has few parameters to be tuned [27], which means that it can achieve accurate regression more easily. Shi et al. proposed an improved least squares support vector regression (LSSVR) for gyroscope drift-error compensation [28], and the vector-based learning method was used to reduce those less important support vectors. Mao et al. proposed an adaptive weighting LSSVR with symmetric noise intervals to predict the hysteresis of piezo-electric actuators [29]. The effects of noise in the training dataset were eliminated, and the sample size was reduced at the same time by the weighting strategy. To make the soft sensor robust against data outliers and noise, Behnasr et al. proposed an iteratively weighted LSSVR that uses the Myriad weighting function [30]. However, the designed weights were usually too sensitive to the identified noise, and the noise was often strictly defined in a symmetric interval. In fact, the noise cannot be fully recognized. Because the noise identified by the algorithms may not be the real noise, the processing of the identified noises should be softer in algorithms.
In our previous studies, an AC-DC hysteresis compensation method for F-P filters to address the intrinsic nonlinear relationship between the displacement and voltage of the PZT [31]. Moreover, to compensate for the temperature drift of F-P filters, we employed polynomial fitting to characterize the relationship among the wavelength, voltage, and temperature [32]. However, these methods are only for short-term applications and cannot compensate for hysteresis and temperature drift at the same time. Recently, we proposed a LSSVR model based on integrated moving windows to compensate for wavelength drifts of FBGs [33] in a temperature-reducing mode. Unfortunately, this model was not suitable for a slowly changing temperature mode. Therefore, in this study, a novel AWLSSVR approach is proposed to compensate for the hysteresis and temperature drift simultaneously in a complex temperature-changing environment. The temperature drift of a referent FBG sensor is used to estimate the drifts of other sensing FBGs, and a novel adaptive weighting strategy with an asymmetric noise interval is proposed to eliminate the effects of noise in the training dataset.
2. Principle
2.1 Hysteresis and temperature drift of F-P filter
Based on the multiple-beam interference theory, the unit phase difference of a Fabry-Perot interferometer (FPI) is given by
where $\lambda$ is the beam wavelength of the beam, $n$ is the refractive index of the media between the two reflection surfaces of the FPI, $l$ is the separation of the reflection surfaces, for FPI, the two reflection surfaces are the two fiber ends, and $l$ is the length between them, $\theta$ is the angle of reflection, and $\Phi (\lambda )$ is the phase change owing to reflection. The transmission spectrum of an FPI reaches the maximum when the unit phase difference is equal to
where $k$ is an integer. Therefore, the transmission wavelength of an FPI can be deduced asIn practice, $l$ is much greater than $\lambda$, $\theta$ is close to $\frac {\pi }{2}$, and $\Phi (\lambda )\leq \pi$. Therefore, the transmission wavelength can be approximated as
The profile of the transmission spectrum of n FPI consists of a series of peaks with non-uniform separation. The range between two adjacent peaks is called the free spectral range (FSR), which is given by
From equations (4) and (5), it is observed that the transmission wavelengths and FSR of the F-P filter are determined only by the F-P cavity length. The temperature-induced medium refractive index change is in the order of 1$\times$10$^{-4}$ [34], which is negligible. Assuming $(T-T_{0}) \cdot \alpha \cdot l_{0}+D_{T}[V(t)]$ is $f[V(t),T]$, while voltage and temperature are both applied on the F-P filter, the cavity length can be expressed by
When the temperature and driving voltage conditions are given, a specific FBG has a specific detected voltage (position). Therefore, it can be inferred that, under the same condition, when the detected voltage (position) is given, the corresponding Bragg wavelength can be calculated. The relationship can be expressed as
where $\lambda _{B,p}$ is the detected Bragg wavelength of the gratings, $p$ is the grating number, and $V_{p}$ is the corresponding voltage of grating $p$. When the temperature is constant, the relationship between the output wavelength of the F-P filter and voltage is nonlinear, because of the characteristics of hysteresis and temperature drift of PZT. Moreover, under the same voltage, the relationship between the wavelength and temperature is also nonlinear.The compensation can be obtained from the actual detected Bragg wavelength of the gratings by subtracting the predictive drifts. Assuming that the actual detected Bragg wavelength of the grating is y and the predictive temperature drift is $y$ and the predictive temperature drift is $\hat {y}$, the compensated output of the tunable F-P filter by the LSSVR model is defined as
3. Adaptive weighting LSSVR with asymmetric interval
To obtain a robust estimation from noisy data, Suykens et al. proposed WLSSVR [35]. The model is expressed as follows:
Assuming a normal Gaussian distribution of $e_{k}$, $\hat {s}$ can be given by Equation (13).
where IQR is the interquartile range, that is, the difference between the $75^{th}$ percentile and $25^{th}$ percentiles.To obtain the values of $v_{k}$, the training dataset are used to train the classical LSSVR, and then, $\hat {s}$ is calculated from the $e_{k}$ distributions. From the statistical perspective of maximum likelihood estimation, the SSE cost function is optimal by assuming a normal Gaussian distribution. Therefore, Suykens et al. suggested using Eq. (12) to correct the assumption when the distribution of $e_{k}$ is not Gaussian. However, there are two important issues to be considered. Firstly, the three intervals in Eq. (12) are symmetrical, but this setting may be inconsistent with some real applications. Second, the core idea of WLSSVR is to determine the weights $v_{k}$ to achieve a robust and accurate regression model. For a sample datum, the larger the training error, the more likely it is to be outlier or noise. Normally, WLSSVR cannot completely distinguish sample points and noise points. Some samples that look like noise may be mixed in the identified noises. Therefore, adaptive weights are required for the controversial samples. The information included in these controversial samples should not be ignored. A new adaptive weighting strategy with an asymmetric noise interval was proposed as follows:
The adaptive weight function significantly affects the regression plane. Therefore, if the convergence speed is too high, useful samples are easily misidentified as noise. If the convergence speed is too slow, the noise points may be insufficiently punished. Therefore, this study introduces normalized $|e_{k}|$ to avoid the convergence too fast or too slow, and c3 to adjust the weight sensitivity to noise adaptively. Furthermore, the noise interval was set to be asymmetric, which is consistent to most practical applications. The new strategy carefully considers the weights and noise interval, which makes the proposed AWLSSVR more accurate and robust. The hyperparameters were determined using Bayesian optimization [36], which determines the value that minimizes the objective functions by building a surrogate reconstruction (probability model) based on the past evaluation results of the target.
The relationship between $v$ and $\frac {e_{k}}{\hat {s}}$ is shown in Fig. 1. It is shown that if the value of $\frac {e_{k}}{\hat {s}}$ is in the range of $c_{1}$ and $c_{2}$, $v$ is set to one, and if the value of $\frac {e_{k}}{\hat {s}}$ is smaller than $c_{1}$ or $\frac {e_{k}}{\hat {s}}$ is larger than $c_{2}$ , then $v$ is smaller. In most cases, $c_{1}$ and $c_{2}$ are in the range between -3 and 3, and this range is determined by the noise distribution of each model.
The proposed AWLSSVR approach is as follows:
1) Obtain the optimal hyperparameters $C$ and $\sigma$ of LSSVR by Bayesian Optimization.
2) LSSVR is applied to the model based on the training samples, and the fitting errors $e_{k}$ are obtained.
3) Calculate the weights vk and obtain for the optimal hyperparameters $C$, $\sigma$, $c_{1}$, $c_{2}$ and $c_{3}$ of AWLSSVR by Bayesian optimization.
4) Develop the AWLSSVR model with the weight $v_{k}$.
4. Application for modeling the hysteresis and temperature drift
4.1 Experimental setup
The experiments were performed on an FBG interrogation system with an F-P tunable filter. Figure 2 shows the system devices. The amplified spontaneous emission (ASE) light source outputs a broadband light that illuminates four FBGs with different Bragg wavelengths through a 3 dB coupler. The reflected light signal was received by a photodetector that transforms the intensity of the optical signal into a voltage amplitude. The voltage signal was sent to the data acquisition card. The data acquisition card receives the sensing signal from the photodetector and outputs a control voltage that drives the PZT in the F-P filter to work continuously. The FBGs were immersed in a water tank that provides a uniform and stable environment (18$^{\circ }$C) and the tunable F-P filter was placed in an electric oven. A calibrated thermistor was attached to the surface of the F-P filter for a numerical temperature reading.
The F-P tunable filter was controlled using a 1 Hz sawtooth voltage (2–4.5V), and the characteristic wavelengths of the FBGs illustrated in Table 1) were within the tuning range. In this case, the reflection peaks of each FBG were detected at different time positions during the tuning period. The physical perturbation sensed by FBGs were demodulated by analyzing the time position changes in each tuning period. The characteristic wavelengths of the four FBGs used in this experiment were measured using a high-resolution lightwave measurement system (HP 8164 B) and calculated using the centroid detection algorithm (CDA).
Table 1. Characteristic wavelengths of FBGs.
4.2 AWLSSVR model training and testing
To examine the accuracy and robustness of the proposed AWLSSVR algorithm, two different training datasets were used to estimate the wavelength drift of the same testing dataset. As shown in Fig. 3, 980 and 810 sample points of wavelength drift were used for the first and second training datasets, respectively. In the first training dataset, the temperature of the electric oven fluctuated approximately at 27.6$^{\circ }$C at the beginning. Subsequently, the temperature was decreased to 25.6 $^{\circ }$C and then increased to 27.3 $^{\circ }$C . Further, the second training dataset was part of the first training dataset. The initial fluctuating part of 4 hour in the first training dataset was eliminated, and the remaining smooth data were used as the second training dataset. The FBG0 sensor was chosen as the reference, and the other three FBGs were used for sensing. It was observed that as the temperature changed, the absolute wavelength drifts of the three sensing FBGs kept changing. The absolute wavelength drift increased with decreasing temperature, and the drift decreased with increasing temperature. The drift was relatively small at high temperatures, and the drift was relatively large at low temperatures. Furthermore, the drifting modes of the three FBGs were different. At the same temperature, the larger the characteristic wavelength of the FBG, the greater was the absolute wavelength drift.
Fig. 3. The first training dataset (a) The input temperature and the temperature drift of G0 (b) The output absolute wavelength drift of three sensing FBGs and the second training dataset (c) The input temperature and the temperature drift of G0 (d) The output absolute wavelength drift of three sensing FBGs.
In the AWLSSVR model, three factors including the temperature, the temperature change rate, and the wavelength drift of G0 were employed as features to estimate the absolute wavelength drift of sensing FBGs. All parameters used in the algorithms are listed in Table 2.
Table 2. Parameters employed in the proposed AWLSSVM algorithm.
As shown in Fig. 4, 750 sample points were used for the testing. No initial frequent fluctuations of temperature exist in the testing dataset. Therefore, the temperature-changing mode in the testing dataset was more similar to the second training dataset, but not similar to the first training dataset.
Fig. 4. The testing dataset (a) The input temperature and the temperature drift of G0 (b) The output absolute wavelength drift of sensing FBGs.
The prediction results of the two different training models are shown in Fig. 5. The results show that the proposed method achieved accurate regression in both experiments.
Fig. 5. Testing results of hysteresis and temperature drift of F-P filter with two kinds of training datasets. (a) FBG1 (b) FBG2 (c) FBG3.
To validate the proposed model, LSSVR, Suykens’ WLSSVR [32], and Mao’s WLSSVR [26] were further adopted to model the hysteresis and temperature drift of the F-P filter for comparison. The maximal absolute error (MAXE), root mean squared error (RMSE), and standard deviation (SD) were used to evaluate the robustness of the regression models, which are, respectively, expressed as:
where $y_{k}$ and $\hat {y}_{k}$ are true value and predicted value, respectively.The performance comparison of the four algorithms is shown in Fig. 6. In the first experiment, the model was trained by the first training dataset shown in Fig.3 (a)(b). The testing results of G0, G1 and G2 are G1(1), G2(1), and G3(1) respectively. Further, in the second experiment, the model was trained by the second training dataset shown in Fig.3 (c)(d). The testing results of G0, G1 and G2 are G1(2), G2(2), and G3(2), respectively. In the first experiment, the maximum of the MAXEs of the proposed method was 5.4 pm at G3 (1), while the results of the other methods were approximately 11.2-12.0 pm at G3 (1). The proposed method achieved approximately 51.8-55.0$\%$ smaller MAXE than other methods. Further, in the second experiment, the proposed method achieved a 19.8-34.6$\%$ smaller MAXE than other methods at G3(1). Moreover, the proposed method also outperforms other methods in terms of RMSE and SD, especially in the first experiment. The proposed method reduced the RMSE by 64.3-67.4$\%$ and the SD by 65.8-68.1$\%$ at G3 (1) compared to other methods in the first experiment.
Fig. 6. Performance comparisons of LSSVR, Suykens’ WLSSVR, Mao’s AWLSSVR, and the proposed method for hysteresis and temperature drift modeling. (a) MAXE (b) RMSE (c) SD.
The residual errors changing over time are shown in Fig. 7. In the first experiment, it is illustrated that from about the $4^{th}$ hour to the $12^{th}$ hour, the residual errors of the sensing FBGs in the proposed method are significantly smaller than those in other methods. The residual errors of the three FBGs in the constant temperature stage are relatively larger around the $8^{th}$ hour. The period from the $7.5^{th}$ to $9^{th}$ hour is a short constant-temperature stage between two cooling stages (from the beginning to about the $7.5^{th}$ hour and from about the $9^{th}$ to $12^{th}$ hour). Further, in the second experiment, the residual errors of the sensing FBGs in the proposed method are close to those in the WLSSVR by Suyken. In the fast-cooling stage from about the $2^{th}$ to $3^{th}$ hour, the residual errors of the sensing FBGs are relatively larger.
Fig. 7. The residual errors of three sensing FBGs in (a) First experiment and (b) Second experiment.
In general, the proposed method performed better than the other three methods, especially in the first experiment. It was verified experimentally that the proposed method was more robust and less affected by the difference between the training and testing datasets. The improved weighting scheme for the noise was softer and avoided being too sensitive to the noise identified.
5. Conclusion
To alleviate the hysteresis and temperature drift of FBG demodulation, this study employs the temperature drift of a referent FBG to characterize the temperature drift of the other sensing FBGs.A novel AWLSSVR algorithm is proposed to eliminate the noise effect in the modeling process. The asymmetric interval of noise and the LSSVR error normalization enhance the accuracy and robustness of the proposed AWLSSVR. Two training datasets are used to build models for comparing the performance of the four algorithms: LSSVR, Suykens’ WLSSVR, Mao’s AWLSSVR, and the proposed method. The experimental results are consistent with the theoretical analysis. The proposed method outperforms the other three methods in terms of both accuracy and robustness.
Recently, using the HCN gas method [15], the error of the temperature-induced wavelength drift was reduced to 2.6 pm and the standard deviation was 0.8 pm. Further, in a temperature-changing environment, with an auxiliary optical fiber Michelson interferometer (AFMI), the amplitude of the wavelength fluctuation is 3.5 pm and the standard deviation is 1.4 pm [37]. The performance of the methods based on additional hardware are slightly better than those of the proposed method based on LSSVR, although the cost and system complexity increased significantly. In contrast, the proposed method based on machine learning requires no additional hardware, and the compensating range covers the entire C band.
Funding
National Natural Science Foundation of China (61905139, 61935002).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Due to the nature of this research, participants of this study did not agree for their data to be shared publicly, so supporting data is not available.
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