Identification of PWARX hybrid models with unknown and possibly different orders

We consider the problem of identifying the orders and the model parameters of PWARX hybrid models from noiseless input/output data. We cast the identification problem in an algebraic geometric framework in which the number of discrete states corresponds to the degree of a multivariate polynomial p and the orders and the model parameters are encoded on the factors of p. We derive a rank constraint on the input/output data from which one can estimate the coefficients of p. Given p, we show that one can estimate the orders and the parameters of each ARX model from the derivatives of p at a collection of regressors that minimize a certain objective function. Our solution does not require previous knowledge about the orders of the ARX models (only an upper bound is needed), nor does it constraint the orders to be equal. Also the switching mechanism can be arbitrary; hence the switches need not be separated by a minimum dwell time. We illustrate our approach with an algebraic example of a switching circuit and with simulation results in the presence of noisy data.