Moving horizon estimation for ARMAX processes with additive output noise

Abstract

Abstract Auto-Regressive-Moving-Average with eXogenous input (ARMAX) models play an important role in control engineering for describing practical systems. However, ARMAX models can be non-realistic in many practical contexts because they do not consider the measurement errors on the output of the process. Due to the auto-regressive nature of ARMAX processes, a measurement error may affect multiple data entries, making the estimation problem very challenging. This problem can be solved by enhancing the ARMAX model with additive error terms on the output, and this paper develops a moving horizon estimator for such an extended ARMAX model. In the proposed method, measurement errors are modeled as nuisance variables and estimated simultaneously with the states. Identifiability was achieved by regularizing the least-squares cost with the l2-norm of the nuisance variables, which leads to an optimization problem that has an analytical solution. For the proposed estimator, convergence results are established and unbiasedness properties are also proved. Insights on how to select the tuning parameter in the cost function are provided. Because of the explicit modeling of output noise, the impact of a measurement error on multiple data entries can be estimated and reduced. Examples are given to demonstrate the effectiveness of the proposed estimator in dealing with additive output noise as well as outliers.