arXiv subjects

Clarence W. Rowley

Publications and source records attributed to Clarence W. Rowley.

46 records · Page 3Linked to original sources

arXiv2014

A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

The Koopman operator is a linear but infinite dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system, and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a "black box" integrator. We will show that this approach is, in effect, an extension of Dynamic Mode Decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the "stochastic Koopman operator" [1]. Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data, and two that show potential applications of the Koopman eigenfunctions.

math.DS↗

arXiv2014

Dynamic Mode Decomposition for Large and Streaming Datasets

We formulate a low-storage method for performing dynamic mode decomposition that can be updated inexpensively as new data become available; this formulation allows dynamical information to be extracted from large datasets and data streams. We present two algorithms: the first is mathematically equivalent to a standard "batch-processed" formulation; the second introduces a compression step that maintains computational efficiency, while enhancing the ability to isolate pertinent dynamical information from noisy measurements. Both algorithms reliably capture dominant fluid dynamic behaviors, as demonstrated on cylinder wake data collected from both direct numerical simulations and particle image velocimetry experiments

physics.flu-dyn↗

arXiv2014

Toward compressed DMD: spectral analysis of fluid flows using sub-Nyquist-rate PIV data

Dynamic mode decomposition (DMD) is a powerful and increasingly popular tool for performing spectral analysis of fluid flows. However, it requires data that satisfy the Nyquist-Shannon sampling criterion. In many fluid flow experiments, such data are impossible to capture. We propose a new approach that combines ideas from DMD and compressed sensing. Given a vector-valued signal, we take measurements randomly in time (at a sub-Nyquist rate) and project the data onto a low-dimensional subspace. We then use compressed sensing to identify the dominant frequencies in the signal and their corresponding modes. We demonstrate this method using two examples, analyzing both an artificially constructed test dataset and particle image velocimetry data collected from the flow past a cylinder. In each case, our method correctly identifies the characteristic frequencies and oscillatory modes dominating the signal, proving the proposed method to be a capable tool for spectral analysis using sub-Nyquist-rate sampling.

physics.flu-dyn↗

arXiv2014

State-space model identification and feedback control of unsteady aerodynamic forces

Unsteady aerodynamic models are necessary to accurately simulate forces and develop feedback controllers for wings in agile motion; however, these models are often high dimensional or incompatible with modern control techniques. Recently, reduced-order unsteady aerodynamic models have been developed for a pitching and plunging airfoil by linearizing the discretized Navier-Stokes equation with lift-force output. In this work, we extend these reduced-order models to include multiple inputs (pitch, plunge, and surge) and explicit parameterization by the pitch-axis location, inspired by Theodorsen's model. Next, we investigate the na\"ıve application of system identification techniques to input--output data and the resulting pitfalls, such as unstable or inaccurate models. Finally, robust feedback controllers are constructed based on these low-dimensional state-space models for simulations of a rigid flat plate at Reynolds number 100. Various controllers are implemented for models linearized at base angles of attack $α_0=0^\circ, α_0=10^\circ$, and $α_0=20^\circ$. The resulting control laws are able to track an aggressive reference lift trajectory while attenuating sensor noise and compensating for strong nonlinearities.

physics.flu-dyn↗

arXiv2013

On Dynamic Mode Decomposition: Theory and Applications

Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. However, existing DMD theory deals primarily with sequential time series for which the measurement dimension is much larger than the number of measurements taken. We present a theoretical framework in which we define DMD as the eigendecomposition of an approximating linear operator. This generalizes DMD to a larger class of datasets, including nonsequential time series. We demonstrate the utility of this approach by presenting novel sampling strategies that increase computational efficiency and mitigate the effects of noise, respectively. We also introduce the concept of linear consistency, which helps explain the potential pitfalls of applying DMD to rank-deficient datasets, illustrating with examples. Such computations are not considered in the existing literature, but can be understood using our more general framework. In addition, we show that our theory strengthens the connections between DMD and Koopman operator theory. It also establishes connections between DMD and other techniques, including the eigensystem realization algorithm (ERA), a system identification method, and linear inverse modeling (LIM), a method from climate science. We show that under certain conditions, DMD is equivalent to LIM.

math.NA↗

arXiv2012

Stirring faces: mixing in a quiescent fluid

This fluid dynamics video depicts the mixing that occurs as a two-dimensional flat plate plunges sinusoidally in a quiescent fluid. Finite-time Lyapunov exponents reveal sets that are attracting or repelling. As the flow field develops, strange faces emerge.

physics.flu-dyn↗

arXiv2009

Reduced order models for control of fluids using the Eigensystem Realization Algorithm

In feedback flow control, one of the challenges is to develop mathematical models that describe the fluid physics relevant to the task at hand, while neglecting irrelevant details of the flow in order to remain computationally tractable. A number of techniques are presently used to develop such reduced-order models, such as proper orthogonal decomposition (POD), and approximate snapshot-based balanced truncation, also known as balanced POD. Each method has its strengths and weaknesses: for instance, POD models can behave unpredictably and perform poorly, but they can be computed directly from experimental data; approximate balanced truncation often produces vastly superior models to POD, but requires data from adjoint simulations, and thus cannot be applied to experimental data. In this paper, we show that using the Eigensystem Realization Algorithm (ERA) \citep{JuPa-85}, one can theoretically obtain exactly the same reduced order models as by balanced POD. Moreover, the models can be obtained directly from experimental data, without the use of adjoint information. The algorithm can also substantially improve computational efficiency when forming reduced-order models from simulation data. If adjoint information is available, then balanced POD has some advantages over ERA: for instance, it produces modes that are useful for multiple purposes, and the method has been generalized to unstable systems. We also present a modified ERA procedure that produces modes without adjoint information, but for this procedure, the resulting models are not balanced, and do not perform as well in examples. We present a detailed comparison of the methods, and illustrate them on an example of the flow past an inclined flat plate at a low Reynolds number.

math.OC↗

arXiv2009

Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators

We present an estimator-based control design procedure for flow control, using reduced-order models of the governing equations, linearized about a possibly unstable steady state. The reduced models are obtained using an approximate balanced truncation method that retains the most controllable and observable modes of the system. The original method is valid only for stable linear systems, and we present an extension to unstable linear systems. The dynamics on the unstable subspace are represented by projecting the original equations onto the global unstable eigenmodes, assumed to be small in number. A snapshot-based algorithm is developed, using approximate balanced truncation, for obtaining a reduced-order model of the dynamics on the stable subspace. The proposed algorithm is used to study feedback control of 2-D flow over a flat plate at a low Reynolds number and at large angles of attack, where the natural flow is vortex shedding, though there also exists an unstable steady state. For control design, we derive reduced-order models valid in the neighborhood of this unstable steady state. The actuation is modeled as a localized body force near the leading edge of the flat plate, and the sensors are two velocity measurements in the near-wake of the plate. A reduced-order Kalman filter is developed based on these models and is shown to accurately reconstruct the flow field from the sensor measurements, and the resulting estimator-based control is shown to stabilize the unstable steady state. For small perturbations of the steady state, the model accurately predicts the response of the full simulation. Furthermore, the resulting controller is even able to suppress the stable periodic vortex shedding, where the nonlinear effects are strong, thus implying a large domain of attraction of the stabilized steady state.

math.OC↗

arXiv2007

Modeling of Transitional Channel Flow Using Balanced Proper Orthogonal Decomposition

We study reduced-order models of three-dimensional perturbations in linearized channel flow using balanced proper orthogonal decomposition (BPOD). The models are obtained from three-dimensional simulations in physical space as opposed to the traditional single-wavenumber approach, and are therefore better able to capture the effects of localized disturbances or localized actuators. In order to assess the performance of the models, we consider the impulse response and frequency response, and variation of the Reynolds number as a model parameter. We show that the BPOD procedure yields models that capture the transient growth well at a low order, whereas standard POD does not capture the growth unless a considerably larger number of modes is included, and even then can be inaccurate. In the case of a localized actuator, we show that POD modes which are not energetically significant can be very important for capturing the energy growth. In addition, a comparison of the subspaces resulting from the two methods suggests that the use of a non-orthogonal projection with adjoint modes is most likely the main reason for the superior performance of BPOD. We also demonstrate that for single-wavenumber perturbations, low-order BPOD models reproduce the dominant eigenvalues of the full system better than POD models of the same order. These features indicate that the simple, yet accurate BPOD models are a good candidate for developing model-based controllers for channel flow.

math.OC↗

arXiv2007

Snapshot-based Balanced Truncation for Linear Time-periodic Systems

We introduce an algorithm based on a method of snapshots for computing approximate balanced truncations for discrete-time, stable, linear time-periodic systems. By construction, this algorithm is applicable to very high-dimensional systems, even with very high-dimensional outputs (or, alternatively, very high-dimensional inputs). An example is shown to validate the method.

math.OC↗