Nonlinear Moment Matching-Based Model Order Reduction

Abstract

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.

Data-Driven Model Reduction of Monotone Systems by Nonlinear DC Gains

“…and note that each element of Γ and ∂Π(x c )/∂x c is nonnegative by (37) and (40). Then, after recalling that monotonicity of the high-order system is equivalent to (5), it is readily verified that also the reduced-order model satisfies (5). Thus, the reduced-order dynamics is monotone, where it is also noted that the output equation of (38) satisfies (3).…”

“…According to (5), if the inequalities (35) hold, then this error system is monotone with the state z and external inputs x r and u. From Proposition 3.3, its infinity induced norm is |h(w(a)) − h r (w r (a))| ∞ /a.…”

“…Model order reduction of nonlinear dynamical systems has been studied in several fields. In systems and control, theoretical frameworks, especially those built upon balancing and moment matching, have been developed; see, e.g., [1]- [5], for the original works. Often, nonlinear model reduction approaches rely on solutions to nonlinear partial differential equations (PDEs), which makes their application to middleand large-scale systems extremely challenging.…”

An Approach to Compute Low-Order H∞ Controllers

“…Moment matching is described in (Ionescu & Astolfi, 2011), (Ionescu & Astolfi, 2016). One of the first methods which achieves moment matching is Asymptotic Waveform Evaluation (AWE), first proposed in (Pillage & Rohrer, 1990).…”

Empirical Differential Balancing for Nonlinear Systems

“…Besides balancing, also moment matching (Antoulas (2005)) is a well known tool for model reduction. For nonlinear control systems this method has only been recently developed, see Astolfi (2010); Ionescu and Astolfi (2016). However, there still remains the problem that solutions to nonlinear PDE (partial differential equations) are required for both balanced truncation and moment matching.…”