Invariant spectral foliations with applications to model order reduction and synthesis
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ORIGINAL PAPER
Invariant spectral foliations with applications to model order reduction and synthesis Robert Szalai
Received: 10 December 2019 / Accepted: 12 August 2020 © The Author(s) 2020
Abstract The paper introduces a technique that decomposes the dynamics of a nonlinear system about an equilibrium into low-order components, which then can be used to reconstruct the full dynamics. This is a nonlinear analogue of linear modal analysis. The dynamics is decomposed using Invariant Spectral Foliation (ISF), which is defined as the smoothest invariant foliation about an equilibrium and hence unique under general conditions. The conjugate dynamics of an ISF can be used as a reduced order model. An ISF can be fitted to vibration data without carrying out a model identification first. The theory is illustrated on a analytic example and on free-vibration data of a clamped-clamped beam. Keywords Model order reduction · Invariant foliation · Non-linear system identification 1 Introduction In this paper, we highlight how invariant foliations [15,29] of dynamical systems can be used to derive reduced order models (ROM) either from data or physical models. We consider dynamics about equilibria only. We assume a deterministic process, such that future states of the system are fully determined by initial conditions. An invariant foliation is a decompoR. Szalai (B) Department of Engineering Mathematics, University of Bristol, Bristol, UK e-mail: [email protected]
sition of the state space into a family of manifolds, called leaves, such that the dynamics brings each leaf into another (see Fig. 1). If a leaf is brought into itself, then it is also an invariant manifold. A foliation is generally characterised by its co-dimension, which equals the number of parameters needed to describe the family of leaves so that it covers the state space. The dynamics that maps one leaf of an invariant foliation into another leaf has the same dimensionality as the co-dimension of the foliation. We call this mapping the conjugate dynamics, which is lower dimensional than the dynamics of the underlying system and therefore suitable to be used as an ROM. Such ROM treats all initial conditions within one leaf equivalent to each other and characterises the dynamics of the whole system. In contrast, the conjugate dynamics (ROM) on an invariant manifold captures the dynamics only on a low-dimensional subset of the state space. The conjugate dynamics on an invariant manifold, however, describes the exact evolution of initial conditions taken from the invariant manifold, while the conjugate dynamics on an invariant foliation is imprecise about the evolution, it can only tell which leaves a trajectory goes through. This ambiguity about the state has some advantages: for all initial conditions there is a leaf and a valid reduced dynamics. In contrast, when using invariant manifolds, the initial condition must come from the invariant manifold in order to have a valid prediction. Multiple foliations can act as a coordinate system about the equilibri
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