Nonlinear Model Reduction by Moment Matching

Abstract

Mathematical models are at the core of modern science and technology. An accurate description of behaviors, systems and processes often requires the use of complex models which are difficult to analyze and control. To facilitate analysis of and design for complex systems, model reduction theory and tools allow determining “simpler” models which preserve some of the features of the underlying complex description. A large variety of techniques, which can be distinguished depending on the features which are preserved in the reduction process, has been proposed to achieve this goal. One such a method is the moment matching approach. This monograph focuses on the problem of model reduction by moment matching for nonlinear systems. The central idea of the method is the preservation, for a prescribed class of inputs and under some technical assumptions, of the steady-state output response of the system to be reduced. We present the moment matching approach from this vantage point, covering the problems of model reduction for nonlinear systems, nonlinear time-delay systems, data-driven model reduction for nonlinear systems and model reduction for “discontinuous” input signals. Throughout the monograph linear systems, with their simple structure and strong properties, are used as a paradigm to facilitate understanding of the theory and provide foundation of the terminology and notation. The text is enriched by several numerical examples, physically motivated examples and with connections to well-established notions and tools, such as the phasor transform

Simulation-driven fixed-order controller tuning via moment matching

“…This family is similar to the one described by (17), for Ξ =Ξ, with the change of coordinates given by the matrix P . In order to determine the conditions for choosing the matrices [B ν , D ν ], while preserving the transient response of C −1 o , we thus solve the problem of identifying the matrices [A ν , B ν , C ν , D ν ] of the family of inverse controllers in (9).…”

“…with L ∈ R 1×ν , S ∈ R ν×ν a non-derogatory 1 matrix and the triple (L, S, ω(0)) minimal, see [15]. According to [17], the signal generator S in (11) is selected so that its characteristic polynomial p(S) is given by…”

“…To build the fixed-order model C −1 ν that matches the moments of C −1 o at S, it is sufficient to estimate Ξ. Nonetheless, since B ν and D ν are free parameters, we would like to seek the "best" model, in some sense to be defined, among the family of fixed order models (17). To this end, we estimate all the parameters [Ξ, B ν , D ν ] from data.…”

“…The family of fixed order controllers (17), with Ξ = Ξ and Ξ estimated as described in Section III-A, does not provide any guarantee on the preservation of the transient response of C −1 o . Therefore, the matrices [B ν , D ν ] are chosen so that the inverse fixed-order controller C −1 ν preserves the transient response of C −1 o .…”

“…Therefore, the matrices [B ν , D ν ] are chosen so that the inverse fixed-order controller C −1 ν preserves the transient response of C −1 o . According to [17], to accomplish this goal we consider the family of fixed-order inverse controllers in (9), which satisfies equation (15) and the approximated moment matching condition…”