Route to Chaos in the Fluidic Pinball

Deng, Nan; Pastur, Luc; Morzyński, Marek; Noack, Bernd R.

doi:10.1115/fedsm2018-83359

Cited by 4 publications

(10 citation statements)

References 23 publications

(29 reference statements)

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“…As our final example, we use a flow control problem of high complexity. In particular, we consider the fluidic pinball [32], which shows chaotic behavior already at moderate Reynolds numbers. The fluidic pinball is a configuration with three cylinders placed on the edges of an equidistant triangle with edge length 1.5R, where R is the cylinder radius, cf.…”

Section: The Fluidic Pinballmentioning

confidence: 99%

“…We now study the performance of the behavior of the K-ROM-based controller for different Reynolds numbers. In [32] it was argued that the fluidic pinball without control possesses a quasi-periodic solution for 90 < Re < 120, and that it behaves chaotically for Re ≥ 120. Consequently, it becomes more and more challenging to construct accurate surrogate models with increasing Reynolds number.…”

Section: The Fluidic Pinballmentioning

confidence: 99%

“…Embedded in an MPC framework, we show that the affine interpolation approaches yields remarkable results for several nonlinear control systems. As examples, we consider several systems of increasing complexity, from the Duffing oscillator over the Burgers equation to the fluidic pinball [32], a fluid flow problem with chaotic behavior governed by the 2D incompressible Navier-Stokes equations.…”

Section: Introductionmentioning

confidence: 99%

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Data-Driven Model Predictive Control using Interpolated Koopman Generators

Peitz,

Otto,

Rowley

2020

Preprint

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In recent years, the success of the Koopman operator in dynamical systems analysis has also fueled the development of Koopman operator-based control frameworks. In order to preserve the relatively low data requirements for an approximation via Dynamic Mode Decomposition, a quantization approach was recently proposed in [1]. This way, control of nonlinear dynamical systems can be realized by means of switched systems techniques, using only a finite set of autonomous Koopman operatorbased reduced models. These individual systems can be approximated very efficiently from data. The main idea is to transform a control system into a set of autonomous systems for which the optimal switching sequence has to be computed. In this article, we extend these results to continuous control inputs using relaxation. This way, we combine the advantages of the data efficiency of approximating a finite set of autonomous systems with continuous controls. We show that when using the Koopman generator, this relaxation -realized by linear interpolation between two operators -does not introduce any error for control affine systems. This allows us to control high-dimensional nonlinear systems using bilinear, low-dimensional surrogate models. The efficiency of the proposed approach is demonstrated using several examples with increasing complexity, from the Duffing oscillator to the chaotic fluidic pinball.

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Section: The Fluidic Pinballmentioning

confidence: 99%

Section: The Fluidic Pinballmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Data-Driven Model Predictive Control using Interpolated Koopman Generators

Peitz,

Otto,

Rowley

2020

Preprint

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“…Figure 6: Setup of the fluidic pinball according to [28] To demonstrate the effectiveness of the K-ROM MPC approach, we again use a flow control problem, but of higher complexity. We now consider the fluidic pinball [28], which is a configuration with three cylinders placed on the edges of an equidistant triangle with edge length 1.5R, where R is the cylinder radius, cf. Fig.…”

Section: Model Predictive Controlmentioning

confidence: 99%

“…Embedded in an MPC framework, we observe that the approach also yields remarkable results for nonlinear control dependencies. As an example, we consider the fluidic pinball [28], a fluid flow problem governed by the 2D incompressible Navier-Stokes equations which shows chaotic behavior.…”

Section: Introductionmentioning

confidence: 99%

Controlling nonlinear PDEs using low-dimensional bilinear approximations obtained from data

Peitz

2018

Preprint

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In a recent article, we presented a framework to control nonlinear partial differential equations (PDEs) by means of Koopman operator based reduced models and concepts from switched systems [1]. The main idea was to transform a control system into a set of autonomous systems for which the optimal switching sequence has to be computed. These individual systems can be approximated very efficiently by reduced order models obtained from data, and one can guarantee equality of the full and the reduced objective function under certain assumptions. In this article, we extend these results to continuous control inputs using convex combinations of multiple Koopman operators corresponding to constant controls, which results in a bilinear control system. Although equality of the objectives can be carried over when the PDE depends linearly on the control, we show that this approach is also valid in other scenarios using several flow control examples of varying complexity.

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Deep Model Predictive Control with Online Learning for Complex Physical Systems

Bieker,

Peitz,

Brunton

et al. 2019

Preprint

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The control of complex systems is of critical importance in many branches of science, engineering, and industry. Controlling an unsteady fluid flow is particularly important, as flow control is a key enabler for technologies in energy (e.g., wind, tidal, and combustion), transportation (e.g., planes, trains, and automobiles), security (e.g., tracking airborne contamination), and health (e.g., artificial hearts and artificial respiration). However, the high-dimensional, nonlinear, and multiscale dynamics make real-time feedback control infeasible. Fortunately, these high-dimensional systems exhibit dominant, low-dimensional patterns of activity that can be exploited for effective control in the sense that knowledge of the entire state of a system is not required. Advances in machine learning have the potential to revolutionize flow control given its ability to extract principled, low-rank feature spaces characterizing such complex systems. We present a novel deep learning model predictive control (DeepMPC) framework that exploits low-rank features of the flow in order to achieve considerable improvements to control performance. Instead of predicting the entire fluid state, we use a recurrent neural network (RNN) to accurately predict the control relevant quantities of the system. The RNN is then embedded into a MPC framework to construct a feedback loop, and incoming sensor data is used to perform online updates to improve prediction accuracy. The results are validated using varying fluid flow examples of increasing complexity.

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Route to Chaos in the Fluidic Pinball

Cited by 4 publications

References 23 publications

Data-Driven Model Predictive Control using Interpolated Koopman Generators

Data-Driven Model Predictive Control using Interpolated Koopman Generators

Controlling nonlinear PDEs using low-dimensional bilinear approximations obtained from data

Deep Model Predictive Control with Online Learning for Complex Physical Systems

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