Multivariable soft sensor with a predictor of mutually dependent errors applied to an industrial fractionator

This paper addresses the development of a multivariable soft sensor (SS) with a predictor designed to handle mutual dependencies within multivariate error series. Typically, the mutual influence in vector time series is characterized using cross-correlation. The proposed multivariable cross-correlated error predictor (MCCEP) framework effectively manages such dependencies and is compatible with any data-driven SS model. Forecasted error values are fed back into the SS output as corrections, refining the final predictions of quality indicators. The MCCEP model is constructed through statistical analysis to minimize the generalized variance – defined as the determinant of the covariance matrix – of multivariate forecast errors. Unlike conventional approaches such as bias update techniques, the MCCEP model is chosen from a broad class of predictors for multivariate linear processes, explicitly considering the dynamic relationships among the univariate components of the SS error process. For the n-dimensional case, it is analytically demonstrated that MCCEP minimizes the generalized variance of multivariate errors by leveraging the cross-correlation functions among the univariate components of the time series, thereby enhancing SS accuracy. Analytical methods for constructing MCCEP using the autocovariance generating function and the squared SS error coherence spectrum are developed. The framework’s superiority is highlighted through a case study involving an industrial fractionator, where the SS with MCCEP outperforms conventional SSs employing dynamic partial least squares and bias updates or developed sequentially without considering interdependencies among univariate components of multi-output model errors.