Model quality in identification of nonlinear systems

In this note, the problem of the quality of identified models of nonlinear systems, measured by the errors in simulating the system behavior for future inputs, is investigated. Models identified by classical methods minimizing the prediction error, do not necessary give "small" simulation error on future inputs and even boundedness of this error is not guaranteed. In order to investigate the simulation error boundedness (SEB) property of identified models, a Nonlinear Set Membership (NSM) method recently proposed by the authors is taken, assuming that the nonlinear regression function, representing the difference between the system to be identified and a linear approximation, has gradient norm bounded by a constant /spl gamma/. Moreover, the noise sequence is assumed unknown but bounded by a constant /spl epsiv/. The NSM method allows to obtain validation conditions, useful to derive "validated regions" within which to suitably choose the bounding constants /spl gamma/ and /spl epsiv/. Moreover, the method allows to derive an "optimal" estimate of the true system. If the chosen linear approximation is asymptotically stable (a necessary condition for the SEB property), in the present note a sufficient condition on /spl gamma/ is derived, guaranteeing that the identified optimal NSM model has the SEB property. If values of /spl gamma/ in the validated region exist, satisfying the sufficient condition, the previous results can be used to give guidelines for choosing the bounding constants /spl gamma/ and /spl epsiv/, additional to the ones required for assumptions validation and useful for obtaining models with "low" simulation errors. The numerical example, representing a mass-spring-damper system with nonlinear damper and input saturation, demonstrates the effectiveness of the presented approach.

[1]  M. B. Matthews,et al.  The identification of nonlinear discrete-time fading-memory systems using neural network models , 1994 .

[2]  W. Rugh.  Linear System Theory , 1992 .

[3]  Yves Rolain,et al.  Fast approximate identification of nonlinear systems , 2003, Autom..

[4]  Marios M. Polycarpou,et al.  High-order neural network structures for identification of dynamical systems , 1995, IEEE Trans. Neural Networks.

[5]  Carlo Novara,et al.  Set Membership identification of nonlinear systems , 2004, Autom..

[6]  Kumpati S. Narendra,et al.  Identification and control of dynamical systems using neural networks , 1990, IEEE Trans. Neural Networks.

[7]  Eduardo D. Sontag,et al.  NEURAL NETS AS SYSTEMS MODELS AND CONTROLLERS , 1992 .

[8]  Susanne Ernst,et al.  Identification with Dynamic Neural Networks - Architectures, Comparisons, Applications , 1997 .

[9]  Snehasis Mukhopadhyay,et al.  Neural Networks for System Identification , 1997 .

[10]  J. Suykens,et al.  Partially linear models and least squares support vector machines , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[11]  Niels Kjølstad Poulsen,et al.  Neural Networks for Modelling and Control of Dynamic Systems: A Practitioner’s Handbook , 2000 .

[12]  Lennart Ljung,et al.  Nonlinear black-box modeling in system identification: a unified overview , 1995, Autom..

[13]  Rahmat Shoureshi,et al.  Neural networks for system identification , 1989, IEEE Control Systems Magazine.

[14]  Heinz Unbehauen,et al.  Structure identification of nonlinear dynamic systems - A survey on input/output approaches , 1990, Autom..

[15]  Lennart Ljung,et al.  Estimating Linear Time-invariant Models of Nonlinear Time-varying Systems , 2001, Eur. J. Control.

[16]  Jonathan R. Partington,et al.  On linear models for nonlinear systems , 2003, Autom..