System identification in the presence of trends and outliers using sparse optimization

For an effective empirical system identification, structural disturbances made up of outliers, level shifts and piecewise linear offsets needs to be detected and taken into consideration. This has led to the introduction of numerous trend filtering methods for signal analysis.

The previously used approach which involves data preprocessing before system identification is known to be quite non-ideal due to the challenges it faces in separating structural disturbances present in data  from the effect of system inputs.

An ℓ₁ trend filtering method used to identify ARX and ARMAX system models simultaneously with structural disturbances was applied by  researchers at Åbo Akademi University in Finland. The research work published in Journal of Process Control was approached by the way of sparse optimization, which is solved by ℓ₁-regularization with an iterative reweighting.

The authors demonstrated the proposed method with simulated examples and experimental data from a pilot-plant distillation column. After several algorithms were set up for identification of ARX and ARMAX models, cross-validation was implemented to verify the selected model structure.

When observing the simulated examples of the ARX model, the prediction error of identified model and estimated structural disturbances were obtained. From the estimated model parameters and root mean square error values of the predicted model output, the estimates obtained while considering structural disturbances correlated well with those acquired when disturbances are known. When structural disturbances were ignored, system identification was erroneous and as a result, data preprocessing failed in supplying correct parameter estimates. When estimating trend-type disturbances simultaneously with the system parameters, the trends,  large outliers, as well as the system dynamics were correctly estimated. Model estimations which overlooked structural disturbances brought about a poorer parameter estimates which failed ultimately.

The method proposed by the authors in this study may also be applied to detection of structural disturbances such as trends and outliers in experimental data affected by an unknown dynamical system. By simultaneous application of detrending and system identification, the method shows a clear advantage over existing data preprocessing methods for detection of structural components in data sequences.