On the construction and parameterization of interpolants in the Loewner framework

Abstract

We develop a general method for the construction of interpolants in the Loewner framework for nonlinear differential–algebraic systems. The approach involves building a family of systems preserving the properties of Loewner equivalence and matching of tangential data functions. It is shown that under mild conditions this family of systems parameterizes all interpolants of sufficiently large dimension matching the tangential data while possessing tangential generalized controllability and observability functions with full column and row rank Jacobians. As a result, this family of systems provides all possible degrees of freedom that an interpolant can have. The results are also discussed in the linear setting and, when taken in combination with existing results, provide a broader framework for the construction of linear interpolating systems.