Identification of deterministic switched ARX systems via identification of algebraic varieties

Abstract

We present a closed-form (linear-algebraic) solution to the identification of deterministic switched ARX systems and characterize conditions that guarantee the uniqueness of the solution. We show that the simultaneous identification of the number of ARX systems, the (possibly different) model orders, the ARX model parameters, and the switching sequence is equivalent to the identification and decomposition of a projective algebraic variety whose degree and dimension depend on the number of ARX systems and the model orders, respectively. Given an upper bound for the number of systems, our algorithm identifies the variety and the maximum orders by fitting a polynomial to the data, and the number of systems, the model parameters, and the switching sequence by differentiating this polynomial. Our method is provably correct in the deterministic case, provides a good sub-optimal solution in the stochastic case, and can handle large low-dimensional data sets (up to tens of thousands points) in a batch fashion. © Springer-Verlag Berlin Heidelberg 2005.