We introduce a novel interpretation of the Loewner and shifted Loewner matrices constructed from tangentatial interpolation data. This interpretation requires the development of new objects: the left-and right-Loewner matrices. Once the interpretation is introduced, a method for the on-line estimation of the Loewner and shifted Loewner matrices is presented.

We present a review of some recent contributions to the theory and application of nonlinear model order reduction by moment matching. The tutorial paper is organized in four parts: 1) Moments of Nonlinear Systems; 2) Playing with Moments: Time-Delay, Hybrid, Stochastic, Data-Driven and Beyond; 3) The Loewner Framework; 4) Applications to Optimal Control and Wave Energy Conversion.

We introduce a novel method of model reduction for nonlinear systems by extending the Loewner framework developed for linear time-invariant (LTI) systems. This objective is achieved by defining Loewner functions obtained by utilizing a state-space interpretation of the Loewner matrices. A Loewner equivalent model using these functions is derived. This allows constructing reduced order models achieving interpolation in the Loewner sense.

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