On System Gains, Nonlinearity Measures, and Linear Models for Nonlinear Systems

This article is concerned with the assessment and derivation of (best) linear models for nonlinear systems. We introduce a general framework reminiscent of uncertainty descriptions in linear robust control theory in order to define a family of six model quality indices for linear models of nonlinear systems. Each quality index corresponds to the gain of an error system, where the nonlinear system can be represented as the interconnection of a nominal linear model and this error system. For an analysis in a certain region of operation, these gains are considered over a specified signal set. A best linear model is a linear model that minimizes the quality index and the minimal quality index serves at the same time as a measure of nonlinearity of the nonlinear system. We show that the model quality indices and nonlinearity measures can be defined under very weak assumptions, and we give conditions that guarantee the existence of optimal linear models. We discuss the structure of the problem of computing the nonlinearity measures. We show that the local linearization of a state space system is a best linear model for the limiting case of a vanishing operating range for one of the measures. We show the relation between the steady-state behavior of a system and its gain and nonlinearity measures. Furthermore we give linear models and upper bounds for nonlinearity measures for systems composed of a linear dynamic and a nonlinear static part. In the case of scalar (SISO) systems, these bounds are given by the sector bounds of the nonlinearity. Finally, it is shown that a lower bound on the nonlinearity measures can be derived using harmonic analysis. Several small examples serve to illustrate the results and the underlying ideas.