Tunable Fabry-Perot (F-P) filters powered by piezoelectric ceramics are prone to hysteresis and temperature drift in fiber Bragg grating (FBG) sensing systems. The demodulated wavelength of tunable F-P filters may produce significant drift during long-term monitoring, which exerts a significant impact on the measurement accuracy of FBG sensing systems. Incorporating a hardware calibration module into the FBG sensing system, including the reference grating method, gas absorption line method, F-P etalon method, and composite wavelength reference method, is the current way of error compensation for the tunable filter. These techniques can successfully reduce the drift error of tunable filters, but typically increase the technical complexity, structural complexity, cost, and even unidentified problems. As a result, it is now practical and affordable to employ a software compensation technique to predict and correct the output drift error of the tunable filter induced by hysteresis and temperature fluctuations. Unfortunately, the output drift error trend of tunable filters over time cannot be accurately tracked by conventional offline models, which limits the model's capacity to make up for it. Therefore, based on least squares support vector machine (LSSVM) and numerous reference gratings, this study proposes a dynamic compensation approach for tunable filter demodulation errors.

Four FBGs (FBG0, FBG1, FBG2, and FBG3) are employed for the reference and sensing gratings in this study. Firstly, the experimental environment's direct temperature-related values are chosen to serve as the dynamic compensation model's input characteristics. The high association between the wavelength drift errors of each FBG in the tunable filter's output spectrum is also thoroughly taken into account in this study. The drift of the reference grating is adopted in this study as one of the input features of the dynamic compensation model to compensate for the absence of precise temperature information inside the F-P cavity. This study employs moving window technology to continuously update the input and output feature quantities of the model and rebuilds the error compensation model to realize real-time prediction and compensation of the most recent drift error of the filter, thus preventing the model performance from degrading. It also highlights how the dynamic model's performance is affected by the moving window's length, the number of reference gratings, and the characteristic wavelength's separation between the reference grating and the sensing grating. The aforementioned approach has been validated in several temperature variation modes.

Firstly, FBG3 positioned in the top of the FBG arrangement distribution receives error compensation (Table 2). In the cooling mode, the maximum absolute error after dynamic compensation reduces from 39.12 pm to 2.53 pm as the number of reference gratings increases. As the number of reference gratings rises in the cooling-heating mode, the maximum absolute error after dynamic compensation falls from 77.02 pm to 8.78 pm. Secondly, FBG2 at the center of the FBG arrangement distribution receives error compensation (Table 3). In the cooling mode, the maximum absolute error after dynamic compensation reduces from 33.65 pm to 3.63 pm as the number of reference gratings in the dynamic model input features rises. The maximum absolute error after dynamic compensation falls from 69.25 pm to 7.84 pm in the cooling-heating mode as the number of reference gratings grows. The aforementioned findings demonstrate that as the number of reference gratings grows, the dynamic model's compensation accuracy gradually increases. Additionally, the experimental findings regarding the characteristic wavelength's distance between the reference grating and the sensing grating indicate that, for the same number of reference gratings, the closer characteristic wavelengths of the reference grating and the sensing grating leads to better compensation capacity of models whose spectral position is adopted as the input feature.

Firstly, this paper adopts the moving window technique as the foundation for building the online drift soft compensation model to prevent performance degradation of the initial model. Then, the experiment builds a nonlinear model between the surface temperature of the filter and the output drift error using the spectral locations of several reference gratings as the input features. The effectiveness of the model is also discussed concerning the moving window's length, the variety of reference gratings, and the characteristic wavelength's distance between the reference and sensing gratings. The experimental results on two datasets with various patterns of temperature variation show that the model's compensation capacity grows as the window length and the number of reference gratings do. Additionally, when the number of reference gratings is the same,the characteristic wavelengths of the reference grating and the sensing grating are closer to one another,and the dynamic model's compensation capacity is greater. In addition to the current hardware compensation method, the online dynamic soft compensation method presented in this study offers a fresh idea for real-time dynamic compensation of F-P filters' output drift errors.