Closed-Loop Interpolation by Moment Matching for Linear and Nonlinear Systems

The relaxation of strong stability conditions on the system to be interpolated is one of the open problems in interconnection-based interpolation by moment matching. To address this issue, we revisit the interconnection-based notion of moment by introducing an output-based signal generator, called generalized signal generator. This generator, which models the desired interpolation points, is designed to drive the state of the system to the invariant manifold defining the moments. This characterization of moments yields a closed-loop scheme built from the output of the underlying system. Leveraging this scheme, we characterize all systems achieving moment matching from a prescribed generalized signal generator designed to interpolate the underlying model. Furthermore, we show that the generalized signal generator can be employed in the construction of parameterized models that preserve the Lur'e structure with the same static nonlinearity. Finally, we validate the closed-loop moment matching scheme on a Chua's circuit, showing how an electronic circuit that exhibits chaotic behavior can be interpolated by an interpolant possessing a unique limit cycle.