180,911 research outputs found
Robust tube-based model predictive control with Koopman operators
No full textKoopman operators are of infinite dimension and capture the characteristics of nonlinear dynamics in a lifted global linear manner. The finite data-driven approximation of Koopman operators results in a class of linear predictors, useful for formulating linear model predictive control (MPC) of nonlinear dynamical systems with reduced computational complexity. However, the robustness of the closed-loop Koopman MPC under modeling approximation errors and possible exogenous disturbances is still a crucial issue to be resolved. Aiming at the above problem, this paper presents a robust tube-based MPC solution with Koopman operators, i.e., r-KMPC, for nonlinear discrete-time dynamical systems with additive disturbances. The proposed controller is composed of a nominal MPC using a lifted Koopman model and an off-line nonlinear feedback policy. The proposed approach does not assume the convergence of the approximated Koopman operator, which allows using a Koopman model with a limited order for controller design. Fundamental properties, e.g., stabilizability, observability, of the Koopman model are derived under standard assumptions with which, the closed-loop robustness and nominal point-wise convergence are proven. Simulated examples are illustrated to verify the effectiveness of the proposed approach.Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Robot Dynamic
Robust tube-based model predictive control with Koopman operators
No full textKoopman operators are of infinite dimension and capture the characteristics of nonlinear dynamics in a lifted global linear manner. The finite data-driven approximation of Koopman operators results in a class of linear predictors, useful for formulating linear model predictive control (MPC) of nonlinear dynamical systems with reduced computational complexity. However, the robustness of the closed-loop Koopman MPC under modeling approximation errors and possible exogenous disturbances is still a crucial issue to be resolved. Aiming at the above problem, this paper presents a robust tube-based MPC solution with Koopman operators, i.e., r-KMPC, for nonlinear discrete-time dynamical systems with additive disturbances. The proposed controller is composed of a nominal MPC using a lifted Koopman model and an off-line nonlinear feedback policy. The proposed approach does not assume the convergence of the approximated Koopman operator, which allows using a Koopman model with a limited order for controller design. Fundamental properties, e.g., stabilizability, observability, of the Koopman model are derived under standard assumptions with which, the closed-loop robustness and nominal point-wise convergence are proven. Simulated examples are illustrated to verify the effectiveness of the proposed approach
DeepKoCo: Efficient latent planning with a task-relevant Koopman representation
No full textThis paper presents DeepKoCo, a novel modelbased agent that learns a latent Koopman representation from images. This representation allows DeepKoCo to plan efficiently using linear control methods, such as linear model predictive control. Compared to traditional agents, DeepKoCo learns taskrelevant dynamics, thanks to the use of a tailored lossy autoencoder network that allows DeepKoCo to learn latent dynamics that reconstruct and predict only observed costs, rather than all observed dynamics. As our results show, DeepKoCo achieves a similar final performance as traditional model-free methods on complex control tasks, while being considerably more robust to distractor dynamics, making the proposed agent more amenable for real-life applications.Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Learning & Autonomous Contro
Koopman analysis of Burgers equation
No full textThe emergence of dynamic mode decomposition (DMD) as a practical way to attempt a Koopman mode decomposition of a nonlinear partial differential equation (PDE) presents exciting prospects for identifying invariant sets and slowly decaying transient structures buried in the PDE dynamics. However, there are many subtleties in connecting DMD to Koopman analysis, and it remains unclear how realistic Koopman analysis is for complex systems such as the Navier-Stokes equations. With this as motivation, we present here a full Koopman decomposition for the velocity field in the Burgers equation by deriving explicit expressions for the Koopman modes and eigenfunctions. As far as we are aware the first time this has been done for a nonlinear PDE. The decomposition highlights the fact that different observables can require different subsets of Koopman eigenfunctions to express them, and it presents a nice example in which (i) the Koopman modes are linearly dependent and so they cannot be fit a posteriori to snapshots of the flow without knowledge of the Koopman eigenfunctions, and (ii) the Koopman eigenvalues are highly degenerate, which means that computed Koopman modes become initial-condition-dependent. As a way of illustration, we discuss the form of the Koopman expansion with various initial conditions, and we assess the capability of DMD to extract the decaying nonlinear coherent structures in run-down simulations.</p
Global and Koopman modes analysis of sound generation in mixing layers
No full textIt is now well established that linear and nonlinear instability waves play a significant role in the noise generation process for a wide variety of shear flows such as jets or mixing layers. In that context, the problem of acoustic radiation generated by spatially growing instability waves of two-dimensional subsonic and supersonic mixing layers are revisited in a global point of view, i.e., without any assumption about the base flow, in both a linear and a nonlinear framework by using global and Koopman mode decompositions. In that respect, a timestepping technique based on disturbance equations is employed to extract the most dynamically relevant coherent structures for both linear and nonlinear regimes. The present analysis proposes thus a general strategy for analysing the near-field coherent structures which are responsible for the acoustic noise in these configurations. In particular, we illustrate the failure of linear global modes to describe the noise generation mechanism associated with the vortex pairing for the subsonic regime whereas they appropriately explain the Mach wave radiation of instability waves in the supersonic regime. By contrast, the Dynamic Mode Decomposition (DMD) analysis captures both the near-field dynamics and the far-field acoustics with a few number of modes for both configurations. In addition, the combination of DMD and linear global modes analyses provides new insight about the influence on the radiated noise of nonlinear interactions and saturation of instability waves as well as their interaction with the mean flow
Lie group valued Koopman eigenfunctions
No full text© 2023 IOP Publishing Ltd & London Mathematical Society cc-byEvery continuous-time flow on a topological space has associated to it a Koopman operator, which operates by time-shifts on various spaces of functions, such as C r , L 2, or functions of bounded variation. An eigenfunction of the vector field (and thus for the Koopman operator) can be viewed as an S 1-valued function, which also plays the role of a semiconjugacy to a rigid rotation on S 1. This notion of Koopman eigenfunctions will be generalized to Lie-group valued eigenfunctions, and we will discuss the dynamical aspects of these functions. One of the tools that will be developed to aid the discussion, is a concept of exterior derivative for Lie group valued functions, which generalizes the notion of the differential d f of a real valued function f. The extended notion of Koopman eigenfunctions utilizes a geometric property of usual eigenfunctions. We show that the generalization in a geometric sense can be used to reveal fundamental properties of usual Koopman eigenfunctions, such as their behaviour under time-rescaling, and as submersions
Lie group valued Koopman eigenfunctions
No full textEvery continuous-time flow on a topological space has associated to it a
Koopman operator, which operates by time-shifts on various spaces of functions,
such as , , or functions of bounded variation. An eigenfunction of
the vector field (and thus for the Koopman operator) can be viewed as an
-valued function, which also plays the role of a semiconjugacy to a rigid
rotation on . This notion of Koopman eigenfunctions will be generalized to
Lie-group valued eigenfunctions, and we will discuss the dynamical aspects of
these functions. One of the tools that will be developed to aid the discussion,
is a concept of exterior derivative for Lie group valued functions, which
generalizes the notion of the differential of a real valued function .
The extended notion of Koopman eigenfunctions utilizes a geometric property of
usual eigenfunctions. We show that the generalization in a geometric sense can
be used to reveal fundamental properties of usual Koopman eigenfunctions, such
as their behavior under time-rescaling, and as submersions
Exponentielle Familien, Suffizienz und das Theorem von Darmois-Koopman-Pitman
No full textIm Mittelpunkt dieser Arbeit steht die Analyse des klassischen Satzes von (Fisher-)Darmois-Koopman-Pitman. Drei Arbeiten von G. Darmois, B. O. Koopman und E. J. G. Pitman aus den Jahren 1935/36 haben diesen Satz unabhängig voneinander, aufbauend auf Arbeiten R. A. Fishers, publiziert. Die vorliegende Arbeit analysiert diese drei klassischen Arbeiten. Dabei wird deren Inhalt dargestellt und mit der statistischen Theorie aus heutiger Sicht verglichen. Zudem erfolgt ein Abriss der Theorie aus heutiger Sicht. Neben grundlegenden Begriffen der Maß- und Wahrscheinlichkeitstheorie werden statistische Grundlagen präsentiert. Exponentielle Verteilungsfamilien sowie suffiziente Statistiken werden ausführlich dargestellt. Des Weiteren erfolgt ein Abriss der Punktschätzung hinsichtlich der Begriffe Erwartungstreue, Fisher-Information und Maximum-Likelihood. Den Hauptteil der Arbeit bildet die Analyse der Arbeiten von G. Darmois, B. O. Koopman und E. J. G. Pitman, sowie der Vorarbeit von R. A. Fisher.This diploma thesis deals with the classical theorem of (Fisher-)Darmois-Koopman-Pitman. Three papers by G. Darmois, B. O. Koopman and E. J. G. Pitman from 1935/36 have published this theorem independently of one another, building on the works of R. A. Fisher. The present work analyzes these three classical works. Their content is presented and compared with contempoary statistical theory. An overview of the theory takes place from today’s perspective. In addition to basic concepts of measurement and probability theory, statistical foundations are presented. Exponential families as well as sufficient statistics are presented in detail. In addition, a brief presentation of point estimation is given in terms of expectation, Fisher information,
and maximum likelihood. The main part of the diploma thesis is the analysis of the work of G. Darmois, B. O. Koopman and E. J. G. Pitman, as well as the preliminary work of R. A. Fisher
Koopman mode expansions between simple invariant solutions
No full textA Koopman decomposition is a powerful method of analysis for fluid flows leading to an apparently linear description of nonlinear dynamics in which the flow is expressed as a superposition of fixed spatial structures with exponential time dependence. Attempting a Koopman decomposition is simple in practice due to a connection with Dynamic Mode Decomposition (DMD). However, there are non-trivial requirements for the Koopman decomposition and DMD to overlap which mean it is often difficult to establish whether the latter is truly approximating the former. Here, we focus on nonlinear systems containing multiple simple invariant solutions where it is unclear how to construct a consistent Koopman decomposition, or how DMD might be applied to locate these solutions. First, we derive a Koopman decomposition for a heteroclinic connection in a Stuart-Landau equation revealing two possible expansions. The expansions are centred about the two fixed points of the equation and extend beyond their linear subspaces before breaking down at a crossover point in state space. Well-designed DMD can extract the two expansions provided that the time window does not contain this crossover point. We then apply DMD to the Navier-Stokes equations near to a heteroclinic connection in low-Reynolds number () plane Couette flow where there are multiple simple invariant solutions beyond the constant shear basic state. This reveals as many different Koopman decompositions as simple invariant solutions present and again indicates the existence of crossover points between the expansions in state space. Again, DMD can extract these expansions only if it does not include a crossover point
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