Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator

Peitz, Sebastian; Klus, Stefan

doi:10.1007/978-3-030-35713-9_10

Cited by 10 publications

(12 citation statements)

References 48 publications

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“…We observe excellent performance for monomials with degree three or larger. The addition of roots of x is not beneficial at all, and in particular, smaller dictionaries are favorable in terms of the data requirements, which is what one would expect and which was also reported in [30]. Note that this concerns only the one-step prediction error, as these dictionaries can (by construction) not yield lifted states z which remain within the Figure 2: Boxplot of the relative one-step prediction error over 10 training runs and 10 5 test points in each run for a dictionary of monomials up to degree at most five.…”

Section: Application To the Duffing Equation (Ode)supporting

confidence: 58%

Finite-data error bounds for Koopman-based prediction and control

Nüske¹,

Peitz²,

Philipp³

et al. 2021

Preprint

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The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems in recent years, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still quite scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points; for both ordinary and stochastic differential equations. Moreover, we extend our analysis to nonlinear control-affine systems using either ergodic trajectories or i.i.d. samples. Here, we exploit the linearity of the Koopman generator to obtain a bilinear system and, thus, circumvent the curse of dimensionality since we do not autonomize the system by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the proposed approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled. 1 F. Philipp was funded by the Carl Zeiss Foundation within the project DeepTurb-Deep Learning in und von Turbulenz. M. Schaller was funded by the DFG (project numbers 289034702 and 430154635).

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Section: Application To the Duffing Equation (Ode)supporting

confidence: 58%

Finite-data error bounds for Koopman-based prediction and control

Nüske¹,

Peitz²,

Philipp³

et al. 2021

Preprint

Self Cite

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“…The same approach was also used in combination with MPC in [52]. This state augmentation significantly increases the data requirements (all combinations of states and control inputs should be covered), such that an alternative transformation was proposed in [53,54] by restricting u(t) to a finite set of inputs {u 1 , . .…”

Section: Controlmentioning

confidence: 99%

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

Klus

Nüske

Peitz

et al. 2020

Physica D: Nonlinear Phenomena

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160

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We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems. arXiv:1909.10638v1 [math.DS] 23 Sep 2019Most of the aforementioned techniques turn out to be strongly related, with the unifying concept being Koopman operator theory [16,17,18]. In what follows, we will focus mainly on the generator of the Koopman operator and its properties and applications. SINDy [5] constitutes a milestone for data-driven discovery of dynamical systems. Because of the close relationship between the vector field of a deterministic dynamical system and its Koopman generator, SINDy is a special case of the framework we will introduce in this study. In [19,20], the authors presented an extension of SINDy to determine eigenfunctions of the Koopman generator. The discovered eigenfunctions are then used for control, resulting in the so-called KRONIC framework. Another extension of SINDy was derived in [21], allowing for the identification of parameters of a stochastic system using Kramers-Moyal formulae.A different avenue towards system identification was taken in [22,23]. Here, the Koopman operator is first approximated with the aid of EDMD, and then its generator is determined using the matrix logarithm. Subsequently, the right-hand side of the differential equation is extracted from the matrix representation of the generator. The relationship between the Koopman operator and its generator was also exploited in [24] for parameter estimation of stochastic differential equations.A method for computing eigenfunctions of the Koopman generator was proposed in [25], where the diffusion maps algorithm is used to set up a Galerkin-projected eigenvalue problem with orthogonal basis elements. Two efficient methods for computing the generator of the adjoint Perron-Frobenius operator based on Ulam's method and spectral collocation were presented in [26]. Provided that a model of the system dynamics is available, the computation of trajectories can be replaced by evaluations of the right-hand side of the system, which is often orders of magnitude faster.The purpose of this study is to present a general framework to compute a matrix approximation of the Koop...

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“…In [136], the MPC problem for the sequence of control inputs is transformed into a time switching optimization problem assuming a fixed sequence of consecutive discrete control inputs. The model can further be modified as a bilinear continuous, piecewise-smooth variant with constant system matrices, which does not suffer from the curse of dimensionality and allows for continuous control inputs via linear interpolation [137].…”

Section: Time-delay Coordinates and Dmdcmentioning

confidence: 99%

Data-Driven Approximations of Dynamical Systems Operators for Control

Kaiser

Kutz

Brunton

2020

Lecture Notes in Control and Information Sciences

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The Koopman and Perron Frobenius transport operators are fundamentally changing how we approach dynamical systems, providing linear representations for even strongly nonlinear dynamics. Although there is tremendous potential benefit of such a linear representation for estimation and control, transport operators are infinite-dimensional, making them difficult to work with numerically. Obtaining low-dimensional matrix approximations of these operators is paramount for applications, and the dynamic mode decomposition has quickly become a standard numerical algorithm to approximate the Koopman operator. Related methods have seen rapid development, due to a combination of an increasing abundance of data and the extensibility of DMD based on its simple framing in terms of linear algebra. In this chapter, we review key innovations in the data-driven characterization of transport operators for control, providing a high-level and unified perspective. We emphasize important recent developments around sparsity and control, and discuss emerging methods in big data and machine learning.

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Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator

Cited by 10 publications

References 48 publications

Finite-data error bounds for Koopman-based prediction and control

Finite-data error bounds for Koopman-based prediction and control

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

Data-Driven Approximations of Dynamical Systems Operators for Control

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