The Convergence of a Composite Regressor ARMAX Predictor

We are considering the problem of identifying an ARMAX system. Two standard models for producing predictive estimates for such a system are the equation error model and the ARMAX model. In this paper an algorithm called Composite Regressor ARMAX Predictor is presented. The new model is intermediary between the two standard models. It achieves convergence without needing the usual strictly positive real condition that the ARMAX model needs. Yet, it produces estimates which should have significantly less bias than the equation error model. This paper focuses on the proof of convergence of the Composite Regressor ARMAX method. The convergence analysis is greatly complicated by the presence of an extra noise term in the error equation. Because of this term, two modifications are made to the usual stochastic gradient parameter estimation algorithm The proof must go through a rather complicated argument which looks at a Lyapunov type function which, unfortunately, is not monotone non-increasing. Appropriate bounds are constructed to care for the intervals when the Lyapunov type function increases. The result is a proof of convergence with a bound on the size of the residual.