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In this paper we present a time-domain notion of moments for a class of single-input, single-output nonlinear systems in terms of the evolution of the output of a generalized signal generator driven by the nonlinear system. We also define a new notion of moment matching and present a family of (nonlinear) parametrized reduced order models that achieve moment matching. We establish relations with existing notions of moment for nonlinear systems, showing that the newly derived and the existing families of reduced order models that achieve nonlinear moment matching, respectively, are equivalent. Furthermore, we compute the reduced order model that matches the moments at two chosen signal generators (one exciting the input of the system and another driven by the system), simultaneously. We also present a family of models computed on the basis of a nonlinear extension of the Petrov-Galerkin projection that achieve moment matching. Finally, we specialise the results to the case of nonlinear, input-affine systems.

a b s t r a c tIn this paper we present a version of balancing for nonlinear systems which is dissipative with respect to a general quadratic supply rate that depends on the input and the output of the system. We discuss an approach that allows us to apply the theory of balancing based upon Hankel singular value analysis. In order to do that we prove that the available storage and the required supply of the original system are the controllability and the observability functions of a modified, asymptotically stable, system. Then Hankel singular value theory can be applied and the axis singular value functions of the modified system equal the nonlinear extensions of ''similarity invariants'' obtained from the required supply and available storage of the original system. Furthermore, we also consider an extension of normalized comprime factorizations and relate the available storage and required supply with the controllability and observability functions of the factorizations. The obtained relations are used to perform model order reduction based on balanced truncation, yielding dissipative reduced order models for the original systems. A second order electrical circuit example is included to illustrate the results.

In this paper we present a time-domain notion of moments for a class of single-input, single-output nonlinear systems, affine in the input, in terms of the steady-state response of the output of a generalized signal generator driven by the nonlinear system. In addition, we define a new notion of moment matching and present the class of (nonlinear) parameterized reduced order models that achieve moment matching. Furthermore, we establish relations with existing notions of moment, showing that the families of reduced order models that achieve nonlinear moment matching are equivalent. Furthermore, we compute the reduced order model that matches moments at two sets of interpolation points, simultaneously, i.e., the number of interpolation points is twice the order of the model.