Improved Structure Detection For Polynomial NARX Models Using a Multiobjective Error Reduction Ratio
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Improved Structure Detection For Polynomial NARX Models Using a Multiobjective Error Reduction Ratio Samir Angelo Milani Martins · Erivelton Geraldo Nepomuceno · Márcio Falcão Santos Barroso
Received: 18 February 2013 / Revised: 16 August 2013 / Accepted: 21 August 2013 / Published online: 7 September 2013 © Brazilian Society for Automatics–SBA 2013
Abstract This paper addresses the problem of structure detection for polynomial NARX models. It develops MERR, a multiobjective extension of a methodology well-known as the error reduction ratio (ERR). It is shown that it is possible to choose terms which take into account dynamics of prediction error and other types of affine information, such as fixed points or static curve. Two examples are included to illustrate the proposed methodology. A numerical example shows that the technique is able to reconstruct the structure of a system, known a priori. The identification of a pilot DC–DC buck converter shows that the proposed approach is capable to find models valid over a wide range of operation points. In this latter example, MERR is compared with ERR in two forms: (i) affine information is applied only in the structure selection for MERR and (ii) affine information is applied for structure selection for MERR and for parameter estimation for both MERR and ERR. In both comparisons, MERR presented nondominated solutions of Pareto set. Keywords Multiobjective system identification · NARX models · Structure detection
S. A. M. Martins (B) · E. G. Nepomuceno · M. F. S. Barroso Department of Electrical Engineering, Federal University of São João del-Rei, Frei Orlando Square, 170-Center, São João del-Rei, MG 36307-352, Brazil e-mail: [email protected] E. G. Nepomuceno e-mail: [email protected] M. F. S. Barroso e-mail: [email protected]
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1 Introduction Despite major advances in system identification, model structure detection is still a great challenge (Baldacchino et al. 2012). An approach used to identify black-box models, such as polynomial nonlinear auto-regressive with exogenous inputs (NARX) (Leontaritis and Billings 1985), consist in choosing a predefined number of model terms in a larger set of candidate terms (Korenberg et al. 1988; Mendes and Billings 2001). Once the model increases its maximum nonlinearity degree, the search space is enlarged, increasing the complexity of structure detection. Considering a nonlinear polynomial NARX model with 10 as its maximum degree and input/output maximum lag, it would yield to a set of more than 30 million candidate models in the search space. Among them, each regressor may represent a specific system behaviour or not, being classified either as genuine or spurious (Aguirre and Billings 1994, 1995). Traditionally, structure detection uses only real dynamic data, acquired from a test station (Ljung 1987). Piroddi and Spinelly (2003) have developed an error-based simulation technique for structure detection, using dynamic data whereas Korenberg et al. (1988) have used the prediction error. In Baldacchino et al. (2012)
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