A fast order-recursive algorithm for Toeplitz submatrix systems with applications to estimation of ARX systems

Abstract

The Levinson-type algorithms have not been applied to the linear minimum mean square error (LMMSE) estimation of parameters of an autoregressive system with exogenous inputs (ARX system) because the Yule-Walker equation in such a case is not a block-Toeplitz system, but is composed of block-Toeplitz submatrices. A new algorithm called the order-recursive algorithm (ORA) is developed to solve such systems, and it is applied to other LMMSE estimation problems of ARX systems. The resulting algorithm operates efficiently and recursively in the order of either the lagged output part or the exogenous input part. Meanwhile, it generates a set of LMMSE ARX models of different order as by-products. As a result, the ORA can be useful in many fields, including linear filtering of ARX and ARMA (autoregressive moving average) processes, system identification, model reduction, and adaptive control.<>