Controllability and observability covariance matrices for the analysis and order reduction of stable nonlinear systems

This paper presents a framework for nonlinear systems analysis that is based upon controllability and observability covariance matrices. These matrices are introduced in the paper and it is shown that gramians for linear systems form special cases of the covariance matrices. The covariance matrices can be transformed via a balancing-like transformation and nonlinearity measures are defined based upon these transformed covariance matrices. Subsequently, the covariance matrices are used for reduction of the nonlinear model. It is shown that the model reduction procedure reduces to balanced model truncation for linear systems for impulse inputs. Furthermore, it is also shown that several model reduction procedures that were developed by other researchers, and assumed to be independent from one another, are related. The findings are illustrated with an example.