1 User friendly Box-Jenkins identification using nonparametric noise models J. Schoukens, Y. Rolain, G. Vandersteen, R. Pintelon Abstract—The identification of SISO linear dynamic systems in the presence of output noise disturbances is considered. A ’nonparametric’ Box-Jenkins approach is studied: the para- metric noise model is replaced by a nonparametric model that is obtained in a preprocessing step, and this without any user interaction. The major advantage for the user is that i) one method can be used to replace the classical ARX, ARMAX, OE, and Box-Jenkins models; ii) no noise model order should be selected. This makes the identification much easier to use for a wider public; iii) a bias on the plant model does not create a bias on the noise model. The disadvantage of the proposed nonparametric approach is a small loss in efficiency with respect to the optimal parametric choice. These results are illustrated on a series of well selected problems. Index Terms—system identification, non-parametric noise models, Box-Jenkins I. I NTRODUCTION I N the classical time domain prediction error framework, a parametric plant- and noise model is estimated simul- taneously for the system given by y (t)= G 0 (q) u 0 (t)+ v (t) , (1) where q -1 is the backward shift operator, and with v (t) the disturbing noise modeled as filtered white noise: v (t)= H 0 (q) e (t). The plant and noise models are respectively given by G 0 (q)= B 0 (q) /A 0 (q) , and H 0 (q)= C 0 (q) /D 0 (q) , with A 0 ,B 0 ,C 0 ,D 0 polynomials in q. During the identifi- cation step, the noise model H (q,θ) acts as a parameter dependent filter on the residuals in the least squares cost function [2], [3] V N (θ)= 1 N N  t=1 ( H -1 (q,θ)[y (t) - G (q,θ) u 0 (t)] ) 2 , (2) which adds a frequency weighting to the cost function. The user has to select the model structure of both the plant model Vrije Universiteit Brussel, Department ELEC. This work was supported in part by the Fund for Scientific Research (FWO-Vlaanderen), by the Flemish Government (Methusalem), and by the Belgian Government through the Interuniversity Poles of Attraction (IAP VI/4) Program. G (q,θ) and the noise model H (q,θ). The choice of the noise model not only reflects the prior knowledge about the system, it also affects the complexity of the optimization problem to find the minimum of the cost function V N (θ). For example, choosing H (q,θ)=1/A (q,θ) expresses that the plant and the noise models have the same poles, resulting in an optimization problem that is linear-in-the-parameters. This is called the ARX model. In the ARMAX model, there is a larger flexibility in the noise model by adding also zeros to the noise model: H (q,θ)= C (q,θ) /A (q,θ), Now, a nonlinear optimization problem is faced to minimize the cost function. For the output error model (OE), the disturbing noise is assumed to be white: H (q,θ)=1, and in the Box- Jenkins model there is no relation between the plant and the noise model [2], [3]. It is clear that the Box-Jenkins model can cover the ARX, ARMAX, and OE situation, but it results also in a more difficult optimization problem to be solved. The user has to solve now a double model selection problem: the order of both the plant- and the noise the model should be selected. The noise model structure selection problem can be avoided if a good nonparametric noise model would be available. It can be used as a parameter independent weight- ing vector in the weighted least squares method. So only the plant model order has to be retrieved by the user. The numerical search procedure becomes also more robust so that the risk to end in local minima is reduced. This brings us to the contribution of this paper. When dealing with system identification we can consider on the one hand the classical prediction error framework that makes use of parametric noise models. It results in optimal estimates (consistent and efficient), provided that the user makes the correct choices for the plant- and noise model structure and order. However, if the user fails to do so, these highly desirable properties are lost and (large) errors can be created. On the other hand we have the nonparametric noise model approach, where no user interaction at all is requested to select the noise model-structure and -order. Only the plant model structure selection should be addressed. This results in a very user friendly modeling technique at a cost of a loss in efficiency. However, the risk to end up with poor models due to a bad user choice is strongly reduced. So there is a possibility to trade optimal, but high risk methods, for good (not optimal), but low risk methods. In this paper we will 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, December 12-15, 2011 978-1-61284-799-3/11/$26.00 ©2011 IEEE 2148