Modeling of adaptive multi-output soft-sensors with applications in wastewater treatments

Given the multivariable coupling, strong nonlinearity and time-varying features in the wastewater treatment processes, adaptive strategies, including just-in-time learning (JITL), time difference (TD), and moving window (MW) methods have been chosen in this paper to enhance multi-output soft-sensor models to ensure online prediction for a variety of hard-to-measure variables simultaneously. In the proposed adaptive multi-output soft-sensors, multi-output partial least squares (MPLS), multi-output relevant vector machine (MRVM) and multi-output Gaussian process regression (MGPR) served as the multi-output models. The integration of adaptive strategies and multi-output models not only provides a solution for multi-output prediction, but also offers a potential to alleviate the degradation of multi-output soft-sensors. To further improve the adaptive ability, four adaptive soft-sensors, termed TD-MW, TD-JIT, JIT-MW, and TD-JIT-MW, have been proposed by mixing the three aforementioned adaptive strategies to upgrade multi-output softsensors. All the adaptive multi-output soft-sensors are analyzed and compared in terms of simulation data and practical industrial data, which exhibit stationary and nonstationary behaviors, respectively.