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Abstract
In this paper, an effective wavelength detection approach based on long short-term memory (LSTM) network is proposed for fiber Bragg grating (FBG) sensor networks. The FBG sensor network utilizes a model-sharing mechanism, where the whole spectral wavelength is divided into several shareable regions and spectral overlap is allowed in each region. LSTM, a representative recurrent neural network in deep learning, is applied to learn the features directly from the spectra of FBGs and build the wavelength detection model. By feeding the spectra sequentially into the well-trained model, the Bragg wavelengths of FBGs can be quickly determined under overlap. The obtained LSTM model can be repeatedly used without re-training to improve the multiplexing capability. The results demonstrate that the LSTM-based method can realize high-accuracy and high-speed wavelength detection in the spectral overlapping situations. The proposed approach offers a flexible tool to enhance the sensing capacity of FBG sensor networks.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Fiber Bragg gratings (FBGs) have emerged as key optical components with their prominent characteristics such as low cost, miniature size, electromagnetic immunity, and high flexibility. FBG sensors are widely used to measure strain, temperature, vibration, acceleration and many other physical quantities, and they have enabled massive applications in structural health monitoring, aerospace, electric power industry, civil engineering, etc. [1]. One of the desirable advantages of FBG sensors is their wavelength-encoding and multiplexing capability. Wavelength division multiplexing (WDM), as a mainstream technique for multiplexing FBG sensors, is commonly employed to implement the multi-point distributed sensor network. However, the multiplexing capacity of a WDM FBG sensor network is limited by the available broad source bandwidth [2]. Each FBG in the sensor network is assigned to a unique spectral operational region and the spectra between adjacent FBGs are not permitted to overlap. This conventional requirement may lead to inefficient use of the spectral bandwidth resource and seriously restricts the multiplexing numbers of FBG sensors.
Recently, the spectra overlapping multiplexing have been applied to improve the capacity of a WDM FBG sensor network, where the operational regions or the spectra of FBGs within the network are allowed to overlap [3,4]. However, the overlapping spectra of FBGs may also bring the crosstalks of central wavelengths and make the conventional peak detection method invalidate. Hence, the primary task and prerequisite of realizing spectral multiplexing is to accurately interrogating the Bragg wavelength of FBG from the overlapping spectra.
Several advanced wavelength detection methods for solving the spectral overlap have been proposed. The optimization-based method is one typical method, which is based on the utilization of evolutionary algorithms and the transformation of wavelength detection issue into an optimization model. Shi et al. applied genetic algorithm (GA) technique to detect the Bragg wavelengths when the spectra of the FBGs were partially or completely overlapped [5]. Liang et al. employed particle swarm optimizer (PSO) equipped with a dynamic neighborhood and a tree structure resulting in reducing time cost [6]. Differential evolution (DE) algorithm, which has strong global search capability and quick convergence, was utilized to detect overlapping spectra of FBG to improve the detection performance [7,8]. However, the drawback of the above methods is that the detection speed is restricted by the increased running time. When the number of FBG sensors increases, they suffer from a relatively long computation time for the heuristic search procedure.
Under overlapping conditions, several efficient detection methods have been proposed. Triana et al. proposed the spectrally encoded FBG sensors [9–11]. The optical orthogonal codes were used to shape the spectral responses of FBGs and the wavelength position for each sensor was determined by the auto-correlation product. However, the method was only for the super-imposed or super-structured FBGs. Stewart et al. proposed the cross-correlation algorithm to predict Bragg wavelength locations for the spectral profiles of two FBGs [12]. The method provided rapid processing benefits, but the uncertainties would inevitably lead to uncorrected solutions. For the purpose of effectively saving the source bandwidth, the multi-objective optimization technique was employed to determine the optimal assignment of the overlapping operational region for each FBG in the network [13]. The detection accuracy of overlapping spectra can greatly affect both the utilization fo bandwidth and the multiplexing capability.
To enhance both the detection speed and detection accuracy, the machine learning based methods have been introduced to tackle the spectral overlap issue. The wavelength detection can be considered as a nonlinear regression problem and the regression model is constructed by learning from the spectrum samples using learning algorithms, such as support vector regression [14] and extreme learning machine [15]. Most recently, deep learning is emerging as a state-of-the-art machine-learning tool in various complex data analysis. Unlike other machine learning methods with the shallow architectures, deep learning is composed of multiple processing layers that can learn representations of data with multiple levels of abstraction [16–18]. Among many methodological variants of deep learning, recurrent neural network (RNN) has achieved impressive performance in various challenging areas.
In this paper, we present a deep network called Long Short-Term Memory (LSTM) for the wavelength detection of spectrally overlapping FBG sensor network. LSTM, proposed by Hochreiter and Schmidhuber [19], is a variation of RNN architecture that is particularly suitable to operate over long input sequences. We leverage LSTM to learn the feature of the spectrum profile and build the detection model of FBG sensor networks. LSTM can automatically and efficiently extract robust features from the reflection spectra using a convolutional network. After the completion of the model training, the central wavelength of each FBG sensor can be identified from the arbitrary overlapping spectra in the measurement. To improve the scalability and reusability, the proposed method takes a model-sharing mechanism. In the network, every two FBG sensors share a same spectral operating region and only one model is established for a pair of FBG sensors. For the whole FBG sensor network, the well-trained wavelength detection model is directly shared and reused by each FBG sensor pair. This means that the model can be repeatedly used without redoing the training procedure even when the number of sensors increases. The proposed method makes an improvement in processing speed, detection accuracy and multiplexing capability.
2. Principles of model-sharing FBG sensor network
The schematic diagram of the proposed FBG sensor network is shown in Fig. 1. The network structure is a series-parallel WDM FBG network. It consists of a broadband source, a 3dB optical coupler, a 1 * 2 splitter, an optical spectrum analyzer (OSA) and a personal computer (PC). The intelligent spectral analysis and the deep learning model is performed by the PC. To improve the multiplexing capacity and more fully utilize the available spectral bandwidth, a sharing mechanism is proposed. The total spectral bandwidth of the proposed FBG network is divided into several shareable regions and then each shareable region contains two or more FBG sensors to be multiplexed. Here, we design two FBG as a FBG pair to share a working region, and the Bragg wavelengths and ranges of the wavelength shift of the two FBGs are the same. In the same shareable region, the spectra of FBGs are allows to overlap, but different shareable regions are remain independent without crosstalk. In the mechanism, the bandwidth saving and multiplexing capability depend on the number of FBG sensors in a shareable region. Here, if we use one FBG pair in each region, the spectral bandwidth usage can be reduced by half compared with conventional WDM FBG network. That means the FBG network can multiplex double FBG sensors when the bandwidth of source is same.
Fig. 1 Schematic diagram of the proposed FBG sensor network. OSA: optical spectrum analyzer; PC: personal computer.
In the proposed network structure, the spectra of the FBG pair can be overlapped in the shareable region. According to the overlapping degree of the spectra, there are three types of overlapping scenarios: partially overlapping spectra, non-overlapping spectra, and completely overlapping spectra. In the case of partially overlapping spectra, the combined spectra has two reflection peaks. When the spectra are completely overlapped, the reflection peaks are overlapped and there is only one peak. The conventional peak detection method is hard to dealing with the crosstalk of spectra and can not accurately distinguish the peak wavelength for each FBG. In the case of non-overlapping spectra, through the spectra are separate, the mismatch of Bragg wavelength for the two FBGs may easily occur. Because the FBG pair share the same spectral region and the Bragg wavelength shift of each sensor is different. The location of Bragg wavelength can be any one in the FBG pair and results can be easily confused. Therefore, the issues of crosstalk and mismatch make the wavelength detection a challenging task for this FBG sensor network. In the paper, we will apply the deep learning model to overcome the difficulties.
For a n-pair FBG sensor network, the entire reflection spectrum R(λ) is the sum of all the reflection spectra of the FBGs belonging to the network, and can be expressed as
where λBj is the Bragg wavelength of FBGj, n is the number of FBG pairs in the network, and γ(.) is spectrum function of a single FBG. The spectral bandwidth can be divided into n regions. Let Rj denote the measured spectrum of the jth FBG pair. Then, R(λ) can be rewritten as where Γj(λ, λB1, λB2) is the theoretical function of the reflection spectrum for the jth FBG pair. Essentially, the wavelength detection problem is to find the solution of λB1 and λB2 from the measured reflection spectrum Rj(λ). Hence, if we can obtain the inverse function , λB1 and λB2 can be given byFrom Eq. (3), the wavelength detection problem can be considered as a construction of the inverse function of the reflection spectrum. It provides an effective way to handle the crosstalk of overlapping spectra during wavelength demodulation process. However, the inverse function cannot show the explicit formula, and it is difficult to be directly solved by the numerical methods. Therefore, we utilize the machine learning tool to train the inverse function model. Generally, it is need to train n models for every two FBGs or train a model for all FBGs. The former way requires n model training procedures with high computing resources. But in the latter method, the training time may increases with number of FBG sensors. Taking into account the similarity of spectral shape of different pairs of FBGs in the shareable region, the proposed method share a same wavelength detection model for each FBG pair in the whole FBG network.
Assume that jth FBG pair is considered as the reference, which can be arbitrarily chosen in a n-pair FBG sensor network. The corresponding wavelength region is denoted as [aj, bj]. Let denote the standardized wavelength detection model of jth FBG pair. the corresponding Bragg wavelengths of kth FBG pair in the network can be computed as
where the [ak, bk] is the wavelength region of kth FBG pair. The FBG model Θ can be applied to different FBG pair in the network.In the proposed method, LSTM network is employed to build the model Θ. To realize FBG interrogation, the wavelength sequential features of reflection spectrum are considered in the process of model training. First, the reflection spectrum of the FBG sensor network is equally serialized to m variables: x1, x2, . . ., xm. Each segmented spectrum sequence is fed into the memory block of LSTM, as shown in Fig. 2. The training dataset is constructed as follows:
where Xk ∈ Rl is the spectra of the reference FBG pair, and l is the number of sampling points. Yk = (λB1, λB2)k is the two corresponding Bragg wavelengths of the reference FBG pair. In the training process, the large amount of training set can be generated through the theoretical model of the spectrum. In the on-line detection process, to reduce the influence of noise, the actual reflection spectra measured from OSA need to be preprocessed by Savitzky-Golay FIR smoothing filter, and then be serialized according the shareable regions. The well-trained LSTM can output the reference values of Bragg wavelengths of FBGs, and the final Bragg wavelengths can be calculated using Eq. (4). Only one shareable LSTM model of FBG pair is applied during the on-line detection process. The details of LSTM are described in the following section.3. Long short-term memory network
Long short-term memory (LSTM) network is an advanced variant of Recurrent Neural Networks (RNNs) that can learn long term dependencies [20–22]. Different from traditional feed-forward neural network, LSTM is a sequence-based model that can connect previous information to the present task and are utilized to handle sequential data. It can receive a time series and expand it into a series of interconnected standard neurons. In order to address the long-term dependency problem, the LSTM network that can remember information for long periods of times is explicitly designed by Hochreiter et al. [19]. It is now widely used in many subsequent applications and works tremendously well on a large variety of problems.
Here, the wavelength detection can be considered a sequential learning problem, as shown in Fig. 2. Given an input FBG spectrum sequence X = {x1, x2, . . ., xm}, a LSTM model computes the hidden vector hm and the output vector Y = {λB1, λB2} given by:
where W denotes weight matrices, b denotes bias vectors, and H(.) is the recurrent hidden layer function.LSTM has the form of a chain of repeating modules of neural network called memory block as shown in Fig. 3. The memory block contains three multiplicative gates. The flow of information into and out of the block is guarded by the input and output gates. And then, in order to provide a way to reset partial memory, the forget gate is added. The details of three gates are introduced as follows.
3.1. Forget gate
The input of the forget gate is a vector concatenation of t − 1 sequence step output ht−1 and t sequence step input xt. The output of forget gate ft can be computed as:
where Wfx and Wfh are weight matrices of the neural network for the corresponding inputs. bf is the bias of the neural network. σ represents the Sigmoid activation function, given by:The outputs of sigmoid function range from 0 to 1, which plays a role of soft switch to decide which information should pass the gate. A value of zero means the signal is blocked by the gate, while a value of one means all of the input should be let through. In other words, the output 0 of the forget gate makes the memory block to completely forget the information, and 1 makes the memory cell to completely store the information.
3.2. Input gate
The input gate also looks at ht−1 and xt, and produces input gate it and input node gt that obtained by:
where Wix refers to the weight between the input gate and the input at current time step. Wih is the weight between the input gate and the output of the memory block in the last sequence step. The input gate is similar to forget gate, which decides the information to be updated. The definition of Wgx and Wgh in input node is similar to the input gate. The activation function of input node is ϕ represents the ’tanh’ function:Instead of using Sigmoid function, the input node uses ’tanh’ function to create a vector of new candidate values that could be added to the memory block state. The memory block state can be updated by:
where ⊙ stands for an element-wise multiplication. The old state st−1 is multiplied by forget gate ft, forgetting the information that decided to forget earlier. The new candidate values it ⊙ gt are added, scaled by the network decided to update the block state.3.3. Output gate
Finally, the output of a memory block is generated based on the memory block state and the output gate ot.
where Wox is the weight between the output gate and the input at current sequence step. Woh is the weight between the output gate and the output of the memory block in the last sequence step. st is the memory block state at current sequence step.The memory block state should first activated by ’tanh’ function. Then, the output gate ot is utilized to decide what parts of the block state should be maintained. This procedure repeats over sequences.
To train a LSTM recurrent network, Back-Propagation Through Time (BPTT), a gradient based training approach, is applied. The BPTT treats the full sequence as one training example and performs the back propagation algorithm on the LSTM. For back propagation, the root-mean-square (RMS) error is utilized as the loss function. Stochastic Gradient Descent (SGD) is employed to learn the weights and biases of LSTM.
4. Results
The experimental setup conducted for a typical n-pair FBG sensor network is presented in Fig. 1. In the FBG sensor system, the broadband source with FWHM of 50 nm and power of 100mW was used to illuminate the two FBG sensor link through a 3 dB coupler. The FWHM of the used FBG sensor is 0.2nm. An OSA (AQ6370D) with 10 pm resolution was used to collect the spectra of FBGs and connected to a PC (Intel Core i7-6850K CPU, single NVIDIA GeForce GTX 1080 Ti GPU, 32GB RAM), where the LSTM model was trained for wavelength detection. The GPU acceleration is used in the high-configuration PC to shorten the off-line training time. But once the training process is finished, the well-trained LSTM model can be directly applied for the on-line detection. The trained model can achieve a millisecond response level not requiring a high-configuration computer, and it also can be easily ported to the embedded system. The reflectivity of the FBG has approximately a Gaussian shape, given by
Where Ipeak is the peak reflectivity of the FBG in the WDM network. In each FBG pair, the peak reflectivity of FBG1 is ∼3dB lower than that of FBG2. The root-mean-square (RMS) error of the Bragg wavelength was employed as the metric to evaluate the performance of the proposed approach.4.1. Network training
First, we need to train a reusable wavelength detection model for a FBG pair. The range of the wavelength shift of the FBG pair was from 1550nm to 1552nm, and the FBG spectum was sampled by 200 points. For network training, the training samples were by the theoretical spectral values using (16) with specified parameters. The testing samples with size of 1000 were collected from OSA by randomly changing the applied strains. Before training, the spectra were normalized to [0, 1] by using linear scaling normalization, and the corresponding Bragg wavelengths of FBG1 and FBG2 were normalized to [0, 2]. The spectral sequence data is set as the model input, and the Bragg wavelengths of FBG1 and FBG2 are used as two-dimensional outputs for learning by LSTM. The LSTM network is implemented in TensorFlow with Kares API. The maximum epoch was set to be 2000.
For training LSTM, the number of hidden units that can affect the detection accuracies is first considered in the experiment. To determine the appropriate number of hidden units, we trained LSTM networks with number of hidden units in {10, 20, 50, 100, 150, 200, 300, 400, 500, 600, 700, 800}. Here, 10000 training samples and 1000 testing samples were used. The comparison of the RMS errors, training time and testing time of the LSTM models with different number of hidden units is shown in Fig. 4 It can be seen that the increase of the number of hidden units decreases the RMS error sharply and then slightly increases to a saturated value at 400. But meanwhile the model with large number of hidden units requires more training time and testing time, and the increase in time cost is almost linear. Compared with the training time, the testing time is extremely short. For a 800-layer LSTM, the total testing time of 1000 detection samples is only 0.6581s. The increase of the number of hidden units can improve the generalization performance, but it also leads to the increased complexity of the training model. Here, we select 400 as the optimal number of hidden units for the LSTM model, which provided a satisfactory trade-off between accuracy and time.
Fig. 4 RMS error, training time and testing time of LSTM models with different number of hidden units.
Then, we investigated the effect of different numbers of training samples on the model training when the number of hidden units is fixed to 400. The training sample size was varied from 1000 to 15000 with the interval of 1000. The tests were conducted on the same testing samples. The results are given in Fig. 5. It shows that as the training sample size increases, the RMS error decreases and then tends to saturation at 10000 sample size. The training time is linearly increased, but the testing time is nearly unchanged at around 0.16s. The change of training sample size can affect the detection accuracy and training efficiency. But the model structure remains unchanged and thus it will not affect the detection efficiency.
Fig. 5 RMS error, training time and testing time of LSTM models with different training sample sizes
The LSTM model for a FBG pair was trained with 400 hidden units by using 10000 randomly generated training samples. The training procedure is shown in Fig. 6. Fig. 6 shows the variation of RMS error for training samples and testing samples from 200 to 2000 epochs, respectively. With the increase of the epoch, both the training error and the testing error decrease sharply. When epoch exceeds 725, the training error declines in small amount and the testing error fluctuates between 0.5pm and 1pm. After training, the well-trained LSTM model can be used as the shareable model of wavelength detection for each FBG pair in the whole FBG sensor network.
4.2. Model testing
To validate the validity of the LSTM model, the detection tests were conducted on the 1000 testing spectra of two FBGs. Fig. 7 and Table 1 illustrate the wavelength detection results of four typical cases in the tests. Fig. 7 (a) and Fig. 7 (c) are the cases that the spectra of FBGs are partially overlapped. In (a), the Bragg wavelength difference of FBG1 and FBG2 Δλ is 0.244 nm, and it still includes two spectral peaks that correspond to two separate FBGs. It can be seen that the proposed model can exactly match the Bragg wavelength to the corresponding FBG sensor. In (c), the Δλ is −0.204nm, which is smaller than case (a). The shape of combined spectrum is characterized as asymmetrical unimodal spectrum, where the peak of FBG1 is not obvious and merged. The difficulty of this case is to accurately identify the Bragg wavelength of FBG1. From Fig. 7 (b), it shows that the model can detect the Bragg wavelengths even when the spectra are completely overlapped. Fig. 7 (d) is the case of non-overlapping spectra, where the two reflection spectra are independent as the normal situation. The detection RMS errors for the four cases are 0.334 pm, 0.492pm, 0.267pm, and 0.244pm, respectively. The mean RMS error for 1000 tests is 0.507pm, and the RMS errors are less than 1pm for 95.4% of all tests.
Table 1. Results of Four Testing Cases
The Bragg wavelength difference Δλ is employed to measure the overlap degree of spectra of FBG1 and FBG2. Fig. 8 gives the characteristic of RMS error when Δλ is range of −1.2nm to 1.2nm. The smaller value of Δλ means higher overlap degree of spectra and more difficult to identify. It can be seen that the detection accuracies of the LSTM model slightly fluctuate in a rational range and the RMS errors are lower than 0.5pm when Δλ < −0.1 and Δλ > 0.1. When the Δλ is within [−0.1,0.1], the serious overlap and crosstalk of spectra lead to the decline of the detection accuracy. The RMS errors of 3% samples are higher than 2pm mainly because the Bragg wavelength of FBG1 may be easily incorrectly identified as that of FBG2. This kind of false detection can be resolved by integrating the swap operator into the algorithm. From the detailed figure in Fig. 8, most samples still maintain relatively high accuracy. Hence, it indicates that the LSTM is capable of handling arbitrary overlapping spectra for each FBG pair with high detection accuracy.
4.3. Performance evaluation
Then, we applied the well-trained LSTM model to demodulate the spectrum for a n-pair FBG sensor network. The tests were run with different numbers of FBG sensors in the network: 2n = 2, 6, 10, 20, 30, 40, 50, 60. The spectra of the sensor network were measured when applying different strains. The total spectral bandwidth was divided into n working regions for each pair of FBG, and the spectrum of each region was fed into the same reusable LSTM model in parallel. The mean values of RMS error and testing time of 100 tests were used to evaluate the proposed method. The results are shown in Table. 2. When the number of FBG sensor increases from 2 to 60, the mean RMS errors are lower than 2pm. In terms of the detection speed, each detection is completed within milliseconds, which is rarely influenced by the number of FBG. For 60-FBG sensor network, the mean RMS error and testing time are 1.685pm and 15ms, respectively.
Table 2. Mean RMS Error and Testing Time of the Wavelength Detection for Different Numbers of FBGs.
The Fig. 9 and Fig. 10 show the original spectrum and output spectrum of LSTM when the number of FBG is 10 and 60, respectively. It can be observed that the Bragg wavelengths of all FBGs can be separated from the combined spectra when the spectra of the sensors are partially or completely overlapped. The obtained RMS errors are 0.679pm for 10 FBGs and 0.784pm for 60 FBGs. The main reason for the efficient FBG interrogation is the model-sharing mechanism utilized in the detection method where the wavelength detection for each FBG pair in the network is shared only one LSTM model and all the computations are handled in parallel. Additionally, the proposed method can be extended to large-scale FBG sensor network with high accuracy and high speed.
4.4. Comparison
To further evaluate the performance of LSTM, the proposed method is compared with extreme learning machine (ELM), least squares support vector regression (LS-SVR), and particle swarm optimizer (PSO). The parameters of the sensor network and the environment experimental environment were kept the same. The number of hidden neurons of ELM was 4000. For PSO, the acceleration constant was set at 2. The population size of PSO was 20, and the maximum generation was 200. The results of mean RMS error of running 20 times are obtained for different algorithms. Fig. 11 shows the cumulative percentile of the RMS errors for the four methods. It can be seen that the cumulative percentile curve of LSTM is higher than other algorithms. In the test, 95.3% of the accuracies using LSTM are below 1pm. The cumulative percentiles of errors of below 1pm are 69.1%, 50.6%, and 25.8% for ELM, LSSVR, and PSO, respectively. The mean RMS errors are 0.507pm, 1.16pm, 1.62pm, and 2.41pm for LSTM, ELM, LSSVR, and PSO, respectively. Compared with previous machine learning algorithms, LSTM outperforms ELM and LSSVR with a higher cumulative percentile of low-error detection. Through considering the wavelength sequential features, LSTM is capable of learning high-level representations and long term dependencies across the spectra of FBGs. The use of deep network architecture enhances the learning ability and obtains high detection accuracy. Compared with PSO algorithm, the well-trained LSTM can avoid the uncertainty and randomness of evolutionary algorithm with superior stability. The LSTM based wavelength detection method can help in improving both the accuracy and the generalization ability of model.
5. Conclusion
This paper develops an efficient wavelength detection technique for FBG sensor network using a deep learning method, long short-term memory. To improve the multiplexing capacity in the limited bandwidth, the spectra of FBGs is allowed to overlap and every two FBGs share a same spectral operating region in the network. The LSTM model is trained by learning from the spectra of FBGs in a shareable region, and can be directly extended to identify the Bragg wavelength of FBGs for the whole sensor network. A major advantage of the proposed method is that, when the number of FBG increases, there is no need to retrain a new model and still maintains a high level of accuracy and efficiency. In the results, the LSTM model obtains mean RMS error of 1.685pm for wavelength detection of a 60 FBG sensor network when partial or completed overlap of spectra occurs. Compared with previously reported methods, the LSTM achieves comparable performance in terms of the detection accuracy. The proposed method offers a promising alternative for increasing the number of FBGs in the sensor network.
Funding
National Natural Science Foundation of China (61703105, 61703106); Natural Science Foundation of Fujian Province of China (2017J01500); Key Natural Foundation for Young Scholars of Fujian Province (JZ160415); Research Program of Distinguished Young Talents of Fujian Province (601934); Qishan Talent Support Program of Fuzhou University (XRC-1623); Research Foundation of Fuzhou University (XRC-17011).
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