On optimal design of experiments for static polynomial approximation of nonlinear systems

Models are of great importance for many purposes, including control design. However, most real systems are complex, frequently nonlinear and first principle models tend to be too complicated, or even unknown, for control-oriented modeling. Therefore, data-based models are often used; however, since most likely the true system is not an element of any assumed model class, the available model is an approximation of the real system. To identify nonlinear systems, universal approximations are often used, e.g., polynomial nonlinear models whose number of parameters rapidly increases with model complexity. Because of the high number of parameters to be identified and the presence of nonlinearity, the accurate choice of an appropriate excitation becomes essential and not trivial. The aim of the present paper is to analyze classical design of experiment (DOE) and present its limits in terms of prediction error, for the static polynomial setup under investigation. First, when the system belongs to the assumed model class, we suggest the use of a more suitable optimization criterion that we prove to be a generalization of the well-known V-optimality. Second, we show that if we design the excitation input based on a higher degree model than the one to be identified, it gives rise to a more efficient approximation.