A Gramian Based Approach to Nonlinearity Quantification and Model Classification

This paper introduces a new method for quantifying the nonlinearity of a process. The approach compares the controllability and observability gramians of a system that is linearized at its steady state operating point to empirical gramians, which are computed from data collected within an operating region of the nonlinear process. This comparison results in two measures for the nonlinearity of the input-output behavior of a process: one for the nonlinearity in the input-to-state behavior (controllability) and one for the nonlinearity in the state-to-output behavior (observability). This is important for controller/observer design, as well as model reduction, because it indicates when it is appropriate to replace a nonlinear model with a linearization around its operating point. In addition, severely nonlinear models can be identified as being Wiener-like models, Hammerstein-like models, or nonlinear in both input-to-state and state-to-output behavior. The proposed approach is illustrated with several examples comparing this method with nonlinearity measures used in the literature.