LTI modelling of NFIR systems: near-linearity and control, LS estimation and linearization

Abstract

Linear time-invariant (LTI) modelling of nonlinear finite impulse response (NFIR) systems is studied from a control point of view. Nearly linear NFIR systems and their control-relevant properties are analysed in detail. The main modelling interest is in the analysis of least squares (LS) LTI identification when the true system is an NFIR system, which is possibly nearly linear. Linearization is used for comparison purposes as the second LTI modelling technique. Nearly linear systems provide a natural generalization of LTI systems to include nonlinearities that allow globally good LTI approximations, while at the same time, such nonlinearities can have a very dramatic effect on the local characteristics of the system. Several control-oriented examples illustrate the possible weaknesses and strengths of the studied LTI modelling techniques. Linearization is found to be especially vulnerable to the presence of even very small, only locally significant, nonlinearities. LS estimation can largely avoid such difficulties, but input design becomes a more critical issue than in standard linear estimation theory. Certain counter-intuitive properties of commonly used input–output stability notions, such as ℓ 2 stability, are discussed via the concept of near-linearity.