A note on model reduction by moment matching for nonlinear systems A. Astolfi ∗ ∗ EEE Dept, Imperial College London, London, UK, and DISP, Universit` a di Roma “Tor Vergata”, Rome, Italy a.astolfi@ic.ac.uk Abstract: The model reduction problem by moment matching for (single-input, single-output) nonlinear systems is studied. A new family of reduced order models, achieving moment matching, is presented. It is shown that this new family is a natural nonlinear enhancement of the family of models obtained, in the linear case, using Krylov projections. Finally, connections between this novel family of models and the family proposed in Astolfi [2010] are discussed. The theory is illustrated by means of a simple example. Copyright c  2010 IFAC Keywords: Model reduction, moment matching, projection 1. INTRODUCTION The model reduction problem, for linear and nonlinear systems, has been widely studied in recent years. While for linear systems the theory is well-understood and several successful applications of model reductions have been reported, the nonlinear theory (and in particular the one relying on the notion of moment) has only recently been developed. Model reduction is a fundamentally important problem in the analysis and design of control systems, and re- duced order models constitute the starting point of several design and analysis procedures. The problem has been addressed from several perspectives exploiting the notions of Hankel operators, the theory of balanced realizations, interpolation theory and the notion of projections, see for example Scherpen [1993], Scherpen and van der Schaft [1994], Scherpen [1996], Scherpen and Gray [2000], Gray and Scherpen [2001], Willcox and Peraire [2002], Lall et al. [2003], Lall and Beck [2003], Fujimoto and Scherpen [2005], Gray and Scherpen [2005], Krener [2006], Verriest and Gray [2006], Fujimoto and Tsubakino [2008], Nillson and Rantzer [2010]. In this paper we focus on model reduction problems rooted in interpolation theory. In particular we consider model reduction problems based on the notion of moment, i.e. the reduced order model is required to interpolate the system to be reduced at some specific points: the moments of the model and of the system have to coincide at the selected points. Model reduction theory by moment matching for linear systems has been widely studied. An excellent overview of existing results is given in Antoulas [2005] (see also the references therein). A nonlinear counterpart of model reduction theory by moment matching has been given in the recent papers Astolfi [2007, 2010]. In what follows we intend to contribute to the nonlinear theory, and in particular to provide a nonlinear enhance- ment of the so-called projection-based model reduction theory. To this end, we revisit the linear theory provid- ing novel interpretations which allow the development of nonlinear counterparts. We also note that the nonlinear enhancement of the notion of Krylov projection provides a natural nonlinear version of the so-called Petrov-Galerkin projection method, see Antoulas [2005]. The paper is organized as follows. Section 2 provides some background material. In Section 3 the equivalence between two families of reduced order models for linear systems is established. This result is extended to nonlinear systems in Section 4, in which also a nonlinear enhancement of the so-called projection-based model reduction theory is established. Finally, Section 5 and Section 6 contain an illustrative example and some summarizing comments. Notation. Throughout the paper we use standard nota- tion. IR, IR n and IR n×m denote the set of real numbers, of n-dimensional vectors with real components, and of n × m-dimensional matrices with real entries, respectively. I C denotes the set of complex numbers. σ(A) denotes the spectrum of the matrix A ∈ IR n×n . Finally, ∅ denotes the empty set, I the identity matrix and 1 a column vector with all entries equal to one. 2. PRELIMINARIES This section is based upon the results in Antoulas [2005] and Astolfi [2010]. 2.1 Linear systems Consider a linear, single-input, single-output, minimal, continuous-time system described by equations of the form ˙ x = Ax + Bu, y = Cx, (1) with x(t) ∈ IR n , u(t) ∈ IR, y(t) ∈ IR, A ∈ IR n×n , B ∈ IR n and C ∈ IR 1×n constant matrices, and the associated transfer function 8th IFAC Symposium on Nonlinear Control Systems University of Bologna, Italy, September 1-3, 2010 978-3-902661-80-7/10/$20.00 © 2010 IFAC 1244 10.3182/20100901-3-IT-2016.00186