Recursive identification of switched ARX systems

We consider the problem of recursively identifying the parameters of a deterministic discrete-time Switched Auto-Regressive eXogenous (SARX) model, under the assumption that the number of models, the model orders and the mode sequence are unknown. The key to our approach is to view the identification of multiple ARX models as the identification of a single, though more complex, lifted dynamical model built by applying a polynomial embedding to the input/output data. We show that the dynamics of this lifted model do not depend on the value of the discrete state or the switching mechanism, and are linear on the so-called hybrid model parameters. Therefore, one can identify the parameters of the lifted model using a standard recursive identifier applied to the embedded input/output data. The estimated hybrid model parameters are then used to build a polynomial whose derivatives at a regressor give an estimate of the parameters of the ARX model generating that regressor. The estimated ARX model parameters are shown to converge exponentially to their true values under a suitable persistence of excitation condition on a projection of the embedded input/output data. Such a condition is a natural generalization of the well known result for ARX models. Although our algorithm is designed for perfect input/output data, our experiments also evaluate its performance as a function of the level of noise for different choices of the number of models and model orders. We also present an application to temporal video segmentation.