arXiv subjects

Eurika Kaiser

Publications and source records attributed to Eurika Kaiser.

26 records · Page 2Linked to original sources

arXiv2017

Optimized Sampling for Multiscale Dynamics

The characterization of intermittent, multiscale and transient dynamics using data-driven analysis remains an open challenge. We demonstrate an application of the Dynamic Mode Decomposition (DMD) with sparse sampling for the diagnostic analysis of multiscale physics. The DMD method is an ideal spatiotemporal matrix decomposition that correlates spatial features of computational or experimental data to periodic temporal behavior. DMD can be modified into a multiresolution analysis to separate complex dynamics into a hierarchy of multiresolution timescale components, where each level of the hierarchy divides dynamics into distinct background (slow) and foreground (fast) timescales. The multiresolution DMD is capable of characterizing nonlinear dynamical systems in an equation-free manner by recursively decomposing the state of the system into low-rank spatial modes and their temporal Fourier dynamics. Moreover, these multiresolution DMD modes can be used to determined sparse sampling locations which are nearly optimal for dynamic regime classification and full state reconstruction. Specifically, optimized sensors are efficiently chosen using QR column pivots of the DMD library, thus avoiding an NP-hard selection process. We demonstrate the efficacy of the method on several examples, including global sea-surface temperature data, and show that only a small number of sensors are needed for accurate global reconstructions and classification of El Niño events.

math.DS↗

arXiv2017

Dynamic mode decomposition for compressive system identification

Dynamic mode decomposition has emerged as a leading technique to identify spatiotemporal coherent structures from high-dimensional data, benefiting from a strong connection to nonlinear dynamical systems via the Koopman operator. In this work, we integrate and unify two recent innovations that extend DMD to systems with actuation [Proctor et al., 2016] and systems with heavily subsampled measurements [Brunton et al., 2015]. When combined, these methods yield a novel framework for compressive system identification [code is publicly available at: https://github.com/zhbai/cDMDc]. It is possible to identify a low-order model from limited input-output data and reconstruct the associated full-state dynamic modes with compressed sensing, adding interpretability to the state of the reduced-order model. Moreover, when full-state data is available, it is possible to dramatically accelerate downstream computations by first compressing the data. We demonstrate this unified framework on two model systems, investigating the effects of sensor noise, different types of measurements (e.g., point sensors, Gaussian random projections, etc.), compression ratios, and different choices of actuation (e.g., localized, broadband, etc.). In the first example, we explore this architecture on a test system with known low-rank dynamics and an artificially inflated state dimension. The second example consists of a real-world engineering application given by the fluid flow past a pitching airfoil at low Reynolds number. This example provides a challenging and realistic test-case for the proposed method, and results demonstrate that the dominant coherent structures are well characterized despite actuation and heavily subsampled data.

eess.SY↗

arXiv2017

Linear genetic programming control for strongly nonlinear dynamics with frequency crosstalk

We advance Machine Learning Control (MLC), a recently proposed model-free control framework which explores and exploits strongly nonlinear dynamics in an unsupervised manner. The assumed plant has multiple actuators and sensors and its performance is measured by a cost functional. The control problem is to find a control logic which optimizes the given cost function. The corresponding regression problem for the control law is solved by employing linear genetic programming as an easy and simple regression solver in a high-dimensional control search space. This search space comprises open-loop actuation, sensor-based feedback and combinations thereof, thus generalizing former MLC studies. This methodology is denoted as linear genetic programming control (LGPC). Focus of this study is the frequency crosstalk between unforced unstable oscillation and the actuation at different frequencies. LGPC is first applied to the stabilization of a forced nonlinearly coupled three-oscillator model comprising open- and closed-loop frequency crosstalk mechanisms. LGPC performance is then demonstrated in a turbulence control experiment, achieving 22% drag reduction for a simplified car model. For both cases, LGPC identifies the best nonlinear control achieving the optimal performance by exploiting frequency crosstalk. Our control strategy is suited to complex control problems with multiple actuators and sensors featuring nonlinear actuation dynamics.

physics.flu-dyn↗

arXiv2016

Sparsity enabled cluster reduced-order models for control

Characterizing and controlling nonlinear, multi-scale phenomena play important roles in science and engineering. Cluster-based reduced-order modeling (CROM) was introduced to exploit the underlying low-dimensional dynamics of complex systems. CROM builds a data-driven discretization of the Perron-Frobenius operator, resulting in a probabilistic model for ensembles of trajectories. A key advantage of CROM is that it embeds nonlinear dynamics in a linear framework, and uncertainty can be managed with data assimilation. CROM is typically computed on high-dimensional data, however, access to and computations on this full-state data limit the online implementation of CROM for prediction and control. Here, we address this key challenge by identifying a small subset of critical measurements to learn an efficient CROM, referred to as sparsity-enabled CROM. In particular, we leverage compressive measurements to faithfully embed the cluster geometry and preserve the probabilistic dynamics. Further, we show how to identify fewer optimized sensor locations tailored to a specific problem that outperform random measurements. Both of these sparsity-enabled sensing strategies significantly reduce the burden of data acquisition and processing for low-latency in-time estimation and control. We illustrate this unsupervised learning approach on three different high-dimensional nonlinear dynamical systems from fluids with increasing complexity, with one application in flow control. Sparsity-enabled CROM is a critical facilitator for real-time implementation on high-dimensional systems where full-state information may be inaccessible.

physics.data-an↗

arXiv2016

Drag reduction of a car model by linear genetic programming control

We investigate open- and closed-loop active control for aerodynamic drag reduction of a car model. Turbulent flow around a blunt-edged Ahmed body is examined at $Re_{H}\approx3\times10^{5}$ based on body height. The actuation is performed with pulsed jets at all trailing edges combined with a Coanda deflection surface. The flow is monitored with pressure sensors distributed at the rear side. We apply a model-free control strategy building on Dracopoulos & Kent (Neural Comput. & Applic., vol. 6, 1997, pp. 214-228) and Gautier et al. (J. Fluid Mech., vol. 770, 2015, pp. 442-457). The optimized control laws comprise periodic forcing, multi-frequency forcing and sensor-based feedback including also time-history information feedback and combination thereof. Key enabler is linear genetic programming as simple and efficient framework for multiple inputs (actuators) and multiple outputs (sensors). The proposed linear genetic programming control can select the best open- or closed-loop control in an unsupervised manner. Approximately 33% base pressure recovery associated with 22% drag reduction is achieved in all considered classes of control laws. Intriguingly, the feedback actuation emulates periodic high-frequency forcing by selecting one pressure sensor in the optimal control law. Our control strategy is, in principle, applicable to all multiple actuators and sensors experiments.

physics.flu-dyn↗

arXiv2016

Chaos as an Intermittently Forced Linear System

Understanding the interplay of order and disorder in chaotic systems is a central challenge in modern quantitative science. We present a universal, data-driven decomposition of chaos as an intermittently forced linear system. This work combines Takens' delay embedding with modern Koopman operator theory and sparse regression to obtain linear representations of strongly nonlinear dynamics. The result is a decomposition of chaotic dynamics into a linear model in the leading delay coordinates with forcing by low energy delay coordinates; we call this the Hankel alternative view of Koopman (HAVOK) analysis. This analysis is applied to the canonical Lorenz system, as well as to real-world examples such as the Earth's magnetic field reversal, and data from electrocardiogram, electroencephalogram, and measles outbreaks. In each case, the forcing statistics are non-Gaussian, with long tails corresponding to rare events that trigger intermittent switching and bursting phenomena; this forcing is highly predictive, providing a clear signature that precedes these events. Moreover, the activity of the forcing signal demarcates large coherent regions of phase space where the dynamics are approximately linear from those that are strongly nonlinear.

math.DS↗

arXiv2016

Cluster-based control of nonlinear dynamics

The ability to manipulate and control fluid flows is of great importance in many scientific and engineering applications. Here, a cluster-based control framework is proposed to determine optimal control laws with respect to a cost function for unsteady flows. The proposed methodology frames high-dimensional, nonlinear dynamics into low-dimensional, probabilistic, linear dynamics which considerably simplifies the optimal control problem while preserving nonlinear actuation mechanisms. The data-driven approach builds upon a state space discretization using a clustering algorithm which groups kinematically similar flow states into a low number of clusters. The temporal evolution of the probability distribution on this set of clusters is then described by a Markov model. The Markov model can be used as predictor for the ergodic probability distribution for a particular control law. This probability distribution approximates the long-term behavior of the original system on which basis the optimal control law is determined. The approach is applied to a separating flow dominated by the Kelvin-Helmholtz shedding.

physics.flu-dyn↗

arXiv2014

Cluster-based reduced-order modelling of a mixing layer

We propose a novel cluster-based reduced-order modelling (CROM) strategy of unsteady flows. CROM combines the cluster analysis pioneered in Gunzburger's group (Burkardt et al. 2006) and and transition matrix models introduced in fluid dynamics in Eckhardt's group (Schneider et al. 2007). CROM constitutes a potential alternative to POD models and generalises the Ulam-Galerkin method classically used in dynamical systems to determine a finite-rank approximation of the Perron-Frobenius operator. The proposed strategy processes a time-resolved sequence of flow snapshots in two steps. First, the snapshot data are clustered into a small number of representative states, called centroids, in the state space. These centroids partition the state space in complementary non-overlapping regions (centroidal Voronoi cells). Departing from the standard algorithm, the probabilities of the clusters are determined, and the states are sorted by analysis of the transition matrix. Secondly, the transitions between the states are dynamically modelled using a Markov process. Physical mechanisms are then distilled by a refined analysis of the Markov process, e.g. using finite-time Lyapunov exponent and entropic methods. This CROM framework is applied to the Lorenz attractor (as illustrative example), to velocity fields of the spatially evolving incompressible mixing layer and the three-dimensional turbulent wake of a bluff body. For these examples, CROM is shown to identify non-trivial quasi-attractors and transition processes in an unsupervised manner. CROM has numerous potential applications for the systematic identification of physical mechanisms of complex dynamics, for comparison of flow evolution models, for the identification of precursors to desirable and undesirable events, and for flow control applications exploiting nonlinear actuation dynamics.

physics.flu-dyn↗