Recursive prediction error parameter estimator for non-linear models

This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. A recursive prediction error parameter estimation algorithm is derived for systems which can be represented by the NARMAX (non-linear ARMAX) model. A convergence analysis is presented using the differential equation approach, and the new concept of m-invertibility is introduced. The analysis shows that while a highly non-linear process model may be used to capture the non-linearity of the system it is advisable to fit a simple noise model. The results of applying the algorithm to both simulated and real data are included. I. Introduction Recursive identification of parameters in linear models is now a well-established field. Several methods of analysing recursive estimators have been proposed and an elegant cohesive theory has been developed (Ljung and Soderstrom 1983). In many practical applications, however, non-linear models may be required to achieve an acceptable predictive accuracy. Subject to some mild assumptions the NARMAX model (Billings and Leontaritis 1981, Leontaritis and Billings 1985) can be used as a basis for identification of such systems, and several of the basic principles of linear recursive identification can with obvious interpretations be applied to this model (Billings and Leontaritis 1982, Billings and Voon 1984, Fnaiech and Ljung 1987). In the present study a recursive prediction error estimator (RPEM) is derived for the polynomial NARMAX model. In order to apply the differential equation approach of convergence analysis developed by Ljung, the filter that generates the prediction should be exponentially stable and for the NARMAX model this coincides with the stability of the noise model. Whilst this is relatively easy to analyse when the noise model is linear, the new concept of m-invertibility is introduced for the general case of non-linear noise models. This leads to a convergence analysis of the RPEM for polynomial NARMAX …