Abstract

The optimal refined instrumental variable (RIV) method for the estimation of the Box-Jenkins (BJ) model is modified so that it functions as an optimal filter and state-estimation algorithm. In contrast to the previously developed minimal and non-minimal state space (NMSS) forms for an Auto-Regressive Moving Average with eXogenous variables (ARMAX) model, the new algorithm requires the introduction of a novel extended NMSS form. This facilitates representation of the more general noise component of the BJ model. The approach can be used for adaptive filtering and state variable feedback control.

Highlights

  • The interesting links between state and parameter estimation have been commented on ever since the publication of Kalman’s seminal work [1,2,3]

  • The refined instrumental variable (RIV) method of recursive parameter estimation is used to estimate the coefficients of this model [5] and following some manipulation, generate the optimal filtered output yk and states xk

  • While Taylor et al [7], and other prior work cited within, use a non-minimal statespace (NMSS) model for generalised digital control, including linear quadratic Gaussian (LQG) design with a Kalman filter, it is limited to the Auto-Regressive Moving Average eXogenous variables (ARMAX) model form

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Summary

Introduction

The interesting links between state and parameter estimation have been commented on ever since the publication of Kalman’s seminal work [1,2,3]. The refined instrumental variable (RIV) method of recursive parameter estimation is used to estimate the coefficients of this model [5] and following some manipulation, generate the optimal filtered output yk and states xk. The minimal canonical state space form utilised to implement the Kalman filter in [4], always yields an Auto-Regressive Moving Average eXogenous variables (ARMAX) model, i.e. similar to (1) but constrained by C(z−1) = A(z−1). This Letter develops a novel extended stochastic NMSS representation for the more general system in (1). This result completes the link between the latest RIVBJ estimation algorithm and adaptive optimal filtering and can be conveniently exploited for practical control system design

Background
Conclusions

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