Identification and data-driven model reduction of state-space representations of lossless and dissipative systems from noise-free data

We illustrate procedures to identify a state-space representation of a lossless- or dissipative system from a given noise-free trajectory; important special cases are passive- and bounded-real systems. Computing a rank-revealing factorization of a Gramian-like matrix constructed from the data, a state sequence can be obtained; state-space equations are then computed solving a system of linear equations. This idea is also applied to perform model reduction by obtaining a balanced realization directly from data and truncating it to obtain a reduced-order mode

Modeling and Simulation of Nonlinearly Loaded Electromagnetic Systems via Reduced Order Models - A Case Study: Energy Selective Surfaces

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Recurrences reveal shared causal drivers of complex time series

Many experimental time series measurements share unobserved causal drivers. Examples include genes targeted by transcription factors, ocean flows influenced by large-scale atmospheric currents, and motor circuits steered by descending neurons. Reliably inferring this unseen driving force is necessary to understand the intermittent nature of top-down control schemes in diverse biological and engineered systems. Here, we introduce a new unsupervised learning algorithm that uses recurrences in time series measurements to gradually reconstruct an unobserved driving signal. Drawing on the mathematical theory of skew-product dynamical systems, we identify recurrence events shared across response time series, which implicitly define a recurrence graph with glass-like structure. As the amount or quality of observed data improves, this recurrence graph undergoes a percolation transition manifesting as weak ergodicity breaking for random walks on the induced landscape -- revealing the shared driver's dynamics, even in the presence of strongly corrupted or noisy measurements. Across several thousand random dynamical systems, we empirically quantify the dependence of reconstruction accuracy on the rate of information transfer from a chaotic driver to the response systems, and we find that effective reconstruction proceeds through gradual approximation of the driver's dominant orbit topology. Through extensive benchmarks against classical and neural-network-based signal processing techniques, we demonstrate our method's strong ability to extract causal driving signals from diverse real-world datasets spanning ecology, genomics, fluid dynamics, and physiology.Comment: 8 pages, 5 figure

An informativity approach to the data-driven algebraic regulator problem

In this paper, the classical algebraic regulator problem is studied in a data-driven context. The endosystem is assumed to be an unknown system that is interconnected to a known exosystem that generates disturbances and reference signals. The problem is to design a regulator so that the output of the (unknown) endosystem tracks the reference signal, regardless of its initial state and the incoming disturbances. In order to do this, we assume that we have a set of input-state data on a finite time-interval. We introduce the notion of data informativity for regulator design, and establish necessary and sufficient conditions for a given set of data to be informative. Also, formulas for suitable regulators are given in terms of the data. Our results are illustrated by means of two extended examples

Metric for attractor overlap

We present the first general metric for attractor overlap (MAO) facilitating an unsupervised comparison of flow data sets. The starting point is two or more attractors, i.e., ensembles of states representing different operating conditions. The proposed metric generalizes the standard Hilbert-space distance between two snapshots to snapshot ensembles of two attractors. A reduced-order analysis for big data and many attractors is enabled by coarse-graining the snapshots into representative clusters with corresponding centroids and population probabilities. For a large number of attractors, MAO is augmented by proximity maps for the snapshots, the centroids, and the attractors, giving scientifically interpretable visual access to the closeness of the states. The coherent structures belonging to the overlap and disjoint states between these attractors are distilled by few representative centroids. We employ MAO for two quite different actuated flow configurations: (1) a two-dimensional wake of the fluidic pinball with vortices in a narrow frequency range and (2) three-dimensional wall turbulence with broadband frequency spectrum manipulated by spanwise traveling transversal surface waves. MAO compares and classifies these actuated flows in agreement with physical intuition. For instance, the first feature coordinate of the attractor proximity map correlates with drag for the fluidic pinball and for the turbulent boundary layer. MAO has a large spectrum of potential applications ranging from a quantitative comparison between numerical simulations and experimental particle-image velocimetry data to the analysis of simulations representing a myriad of different operating conditions.Comment: 33 pages, 20 figure