State smoothing by sum-of-norms regularization

The presence of abrupt changes, such as impulsive disturbances and load disturbances, make state estimation considerably more difficult than the standard setting with Gaussian process noise. Nevertheless, this type of disturbances is commonly occurring in applications which makes it an important problem. An abrupt change often introduces a jump in the state and the problem is therefore readily treated by change detection techniques. In this paper, we take a rather different approach. The state smoothing problem for linear state space models is here formulated as a least-squares problem with sum-of-norms regularization, a generalization of the ℓ₁-regularization. A nice property of the suggested formulation is that it only has one tuning parameter, the regularization constant which is used to trade off fit and the number of jumps.