Recursive Identification of Switched ARX Models with Unknown Number of Models and Unknown Orders

We consider the problem of recursively identifying the parameters of a switched ARX (SARX) model from input/output data under the assumption that the number of models, the model orders and the switching sequence are unknown. Our approach exploits the fact that applying a polynomial embedding to the input/output data leads to a lifted ARX model whose dynamics are linear on the so-called hybrid model parameters and independent of the switching sequence. In principle, one can use a standard recursive algorithm to identify such hybrid parameters. However, when the number of models and the model orders are unknown the embedded regressors may not be persistently exciting, hence the estimates of the hybrid parameters may not converge exponentially to a constant vector. Nevertheless, we show that these estimates still converge to a vector that depends continuously on the initial condition. By identifying the hybrid model parameters starting from two different initial conditions, we show that one can build two homogeneous polynomials whose derivatives at a regressor give an estimate of the parameters of the ARX model generating that regressor. After properly enforcing some of the entries of the hybrid model parameters to be zero, such estimates are shown to converge exponentially to the true ARX model parameters under suitable persistence of excitation conditions on the input/output data. Although our algorithm is designed for the case of perfect input/output data, our experiments also show its performance with noisy data.