Optimal feedback control of the incompressible Navier-Stokes-Equations using reduced order models

Pyta, Lorenz; Herty, Michaël; Abel, Dirk

doi:10.1109/cdc.2015.7402595

Cited by 2 publications

(3 citation statements)

References 15 publications

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“…Likewise, we have evenly sampled the set of control candidates on the parameterised control line (13) and on the control surface (see Remark. 1) for our method and the baseline, respectively.…”

Section: Resultsmentioning

confidence: 99%

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Streamline-Based Control of Underwater Gliders in 3D Environments

Lee

Yoo

et al. 2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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Autonomous underwater gliders use buoyancy control to achieve forward propulsion via a sawtooth-like, riseand-fall trajectory. Because gliders are slow-moving relative to ocean currents, glider control must consider the effect of oceanic flows. In previous work, we proposed a method to control underwater vehicles in the (horizontal) plane by describing such oceanic flows in terms of streamlines, which are the level sets of stream functions. However, the general analytical form of streamlines in 3D is unknown. In this paper, we show how streamline control can be used in 3D environments by assuming a 2.5D model of ocean currents. We provide an efficient algorithm that acts as a steering function for a single rise or dive component of the glider's sawtooth trajectory, integrate this algorithm within a sampling-based motion planning framework to support long-distance path planning, and provide several examples in simulation in comparison with a baseline method. The key to our method's computational efficiency is an elegant dimensionality reduction to a 1D control region. Streamlinebased control can be integrated within various sampling-based frameworks and allows for online planning for gliders in complicated oceanic flows.

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Section: Resultsmentioning

confidence: 99%

“…From a control perspective, the problem of navigating in flow field has been well-studied [10][11][12][13]. The most relevant to our problem is the minimum time feedback control [14] that solved for optimal feedback control law from the dynamic Hamilton Jacobi Bellman (HJB) equation.…”

Section: Related Workmentioning

confidence: 99%

Streamline-Based Control of Underwater Gliders in 3D Environments

Lee

Yoo

et al. 2019

2019 IEEE 58th Conference on Decision and Control (CDC)

View full text Add to dashboard Cite

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“…Many researchers have dedicated their work to developing optimal control methods based on POD for which convergence towards the true optimum can be proved, either using the singular values associated with the POD modes [Row05,HV05,TV09] or by trust-region approaches [Fah00]. An approach to closed-loop flow control using POD-based surrogate models has been developed in [PHA15]. A well-known drawback is that the number of required POD modes grows rapidly with increasing complexity of the system dynamics so that Galerkin models can become infeasible.…”

Section: Introductionmentioning

confidence: 99%

Koopman operator-based model reduction for switched-system control of PDEs

2019

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We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables. A numerical approximation of the Koopman operator therefore yields a linear system for the observation of an autonomous dynamical system. In our approach, by introducing a finite number of constant controls, the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem. This allows us to replace each of these systems by a K-ROM which can be solved orders of magnitude faster. By this approach, a nonlinear infinite-dimensional control problem is transformed into a low-dimensional linear problem. In situations where the Koopman operator can be computed exactly using Extended Dynamic Mode Decomposition (EDMD), the proposed approach yields optimal control inputs. Furthermore, a recent convergence result for EDMD suggests that the approach can be applied to more complex dynamics as well. To illustrate the results, we consider the 1D Burgers equation and the 2D Navier-Stokes equations. The numerical experiments show remarkable performance concerning both solution times and accuracy.

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Optimal feedback control of the incompressible Navier-Stokes-Equations using reduced order models

Cited by 2 publications

References 15 publications

Streamline-Based Control of Underwater Gliders in 3D Environments

Streamline-Based Control of Underwater Gliders in 3D Environments

Koopman operator-based model reduction for switched-system control of PDEs

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