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 γ. Moreover, the noise sequence is assumed unknown but bounded by a constant ε. The NSM method allows to obtain validation conditions, useful to derive "validated regions" within which to suitably choose the bounding constants γ and ε. 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 γ is derived, guaranteeing that the identified optimal NSM model has the SEB property. If values of γ in the validated region exist, satisfying the sufficient condition, the previous results can be used to give guidelines for choosing the bounding constants γ and ε, 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.