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

• System identification in the presence of structural disturbances is studied.
• Outliers, level shifts and trends are modeled as sparse signals.
• System model and disturbances are identified simultaneously by sparse optimization.
• Sparse optimization problem is solved by ℓ1-relaxation.
• The method is applied to simulated examples and pilot-plant distillation column data.

In empirical system identification, it is important to take into account the effect of structural disturbances, such as outliers and trends in the data, which might otherwise deteriorate the identification accuracy. A commonly used approach is to preprocess the data to remove outliers and trends, followed by system identification using the processed data. This approach is not optimal because before a system model is available it may not be possible to separate outliers and trends in the data from excitation by the system inputs. In this study a procedure is presented for simultaneous identification of ARX and ARMAX system models and unknown structural disturbances, consisting of outliers and piece-wise linear offsets or trends. This is achieved by introducing sparse representations of the disturbances, having only a few non-zero values. The system identification problem is formulated as a least-squares problem with a sparsity constraint. The sparse optimization problem is solved using ℓ1-regularization with iterative reweighting, which can be solved efficiently as a sequence of convex optimization problems. Simulated examples and experimental data from a pilot-plant distillation column are used to demonstrate that using the proposed method accurate system models can be identified from experimental data containing unknown trends and outliers.