1,019 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
From Foucauldian Biopower to Energopower and Infopower:An Interview with Dominic Boyer and Colin Koopman
No full textKirsten Hasberg talks to Dominic Boyer, anthropologist and author of Energopolitics: Wind and Power in the Anthroprocene, and to Colin Koopman, philosopher and author of How We Became our Data: A Genealogy of the Informational Person. Their books published in mid-2019 put forward novel conceptualizations of Foucauldian biopower, which they term infopower and energopower, respectively. Criss-crossing between philosophical conceptualizations and concrete problems like the struggles of renewable energy communities (Boyer) and the influence of economic thinking on datafication (Koopman), the conversations show how Foucauldian concepts are relevant to today's power struggles inherent to the energy transition and the digital transformation.Kirsten Hasberg talks to Dominic Boyer, anthropologist and author of Energopolitics: Wind and Power in the Anthroprocene, and to Colin Koopman, philosopher and author of How We Became our Data: A Genealogy of the Informational Person. Their books published in mid-2019 put forward novel conceptualizations of Foucauldian biopower, which they term infopower and energopower, respectively. Criss-crossing between philosophical conceptualizations and concrete problems like the struggles of renewable energy communities (Boyer) and the influence of economic thinking on datafication (Koopman), the conversations show how Foucauldian concepts are relevant to today's power struggles inherent to the energy transition and the digital transformation
Applying Koopman Methods for Nonlinear Reachability Analysis
No full textIn this thesis we investigate the possibilities for applying Koopman methods for reachability analysis. Reachability analysis is a verification process used to determine that a dynamical system starting in an initial set X0 cannot reach a certain set of dangerous states D within a time interval [0,T]. Koopman methods seem promising, because they predict nonlinear behaviour using linear techniques. However they have not been widely applied to reachability analysis.We describe three different Koopman methods: data-driven, Polyflow and Carleman. We use the Polyflow method combined with ideas from several other methods to create a new reachability tool: PolyReach. Next, we analyse the performance of PolyReach by comparing it with a state-of-the-art reachability algorithm Flow* on various nonlinear systems. Finally, we summarize the strengths and weaknesses of the PolyReach tool and discuss ideas for further improvement.Mechanical Engineering | Systems and Contro
Dynamic wind turbine wake reconstruction: A Koopman-linear flow estimator
No full textA challenging topic arising in dynamic wind turbine wake is modeling, especially the low-order approximation. The central problem is the fact that it has high-dimensional and nonlinear wake characteristics. In this paper, a Koopman-linear flow estimator is designed according to the Koopman operator theory. Different from the conventional flow reconstruction with the linear stochastic estimation method, a dynamic state-space model with physical states is constructed. The wake dynamics are approximated using a limited number of measurable physical parameters by the dynamic part; then, the full wake flow is reconstructed from the low-order states by the estimation part. The flow estimator is designed into three different forms following Extended Dynamic Mode Decomposition (EDMD) method. Each form has its unique advantages. Precisely, probe sensors are placed in the studied space and provide direct information of the wake, and a few in-directly physical parameters are also included. Nonlinear integer programming is further adopted using a heuristic optimization algorithm, by which the sensor configurations are optimized. Comparisons with the standard Dynamic Mode Decomposition (DMD)-based wake model are adopted in time domain and frequency domain to verify the effectiveness of the proposed flow estimators. The results show acceptable accuracy in typical modeling cases and maintain good estimation accuracy when the measurement noises are involved. Finally, the proposed Koopman-linear flow estimator is compared with related stochastic estimation methods, in which the connections of the proposed estimator with stochastic ones are also discussed.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.Team Jan-Willem van Wingerde
Representer Theorem for Learning Koopman Operators
No full textIn this work, we consider the problem of learning the Koopman operator for discrete-time autonomous systems. The learning problem is formulated as a generic constrained regularized empirical loss minimization in the infinite-dimensional space of linear operators. We show that a representer theorem holds for the introduced learning problem under certain but general conditions, which allows convex reformulation of the problem in a specific finite-dimensional space without any approximation and loss of precision. We discuss the inclusion of various forms of regularization and constraints in the learning problem, such as the operator norm, the Frobenius norm, the operator rank, the nuclear norm, and the stability. Subsequently, we derive the corresponding equivalent finite-dimensional problem. Furthermore, we demonstrate the connection between the proposed formulation and the extended dynamic mode decomposition. We present several numerical examples to illustrate the theoretical results and verify the performance of regularized learning of the Koopman operators.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.Team Khosrav
Output Regulation of Nonlinear Systems in a Koopman Operator Framework
No full textThis thesis considers the problem of nonlinear output regulation in a Koopman operator framework. The goal of output regulation is to asymptotically track a reference and/or simultaneously reject a disturbance signal, both generated by some external autonomous system called the exosystem. The nonlinear output regulation problem is solvable if and only if a set of partial differential equations (PDE) are satisfied. From the solution, a feedback law can be obtained that achieves output regulation. However, solving the PDE is difficult. In this thesis, we instead aim to construct a feedback law by utilizing the Koopman operator instead.The Koopman operator associated with a state-space model of a (nonlinear) dynamical system describes the evolution of functions of the states, called observable functions, by propagating the state forward in time according to the flow of the system, and evaluating this at each possible observable function. The space of observables is an infinite-dimensional vector field. Therefore, the Koopman operator is infinite-dimensional and linear. The Koopman operator of an autonomous system associated with a nonlinear control system provides a bilinear description of the control system instead. The use of the Koopman operator to tackle the output regulation problem has not been done before in the literature. We identify conditions under which the Koopman operator can be used to rephrase the nonlinear output regulation problem as a bilinear output regulation problem. We then show when the bilinear output regulation problem is solved using linear dynamic error feedback. In particular, a Lyapunov-based approach is used to characterize a set of initial conditions for which the output is regulated. Finally, to verify the results, a numerical example is presented.Mechanical Engineering | Systems and Contro
Koopman Subspace Identification in the Presence of Measurement Noise
No full textThe ability to compute models that correctly predict the trajectories of a nonlinear system can become a significant challenge in systems and control. The introduction of Koopman operator theory helped to deal with this challenge. The Koopman operator is a composition operator that globally describes a nonlinear system in an infinite-dimensional linear framework. To implement this theory, the usual approach is to approximate the Koopman operator through data-driven methods. These algorithms use measurements of the nonlinear system to compute the approximated operator. Generally, noise can be present in real-world scenarios. Noisy measurements can have a considerable deteriorating effect on the data-driven approximation of Koopman operators. The approximation of this operator in presence of noisy training data is a necessary step for its implementation to a wider spectrum of real-world applications. Many robust numerical methods were designed to solve this issue. Koopman subspace identification (KSI) is a promising approach. As the name suggests, this algorithm employs subspace identification modeling to compute the matrix approximation of the Koopman operator. In this work, we test KSI against other state-of-the-art techniques. Additionally, we improve its performance in predicting the state trajectories of the nonlinear system in presence of noisy measurements. To this end, we propose a reducing-order routine that computes the most robust model against measurement noise. Furthermore, a randomized singular value decomposition is adopted to reduce computational times. The improved KSI is then compared against the other state-of-the-art algorithms in the presence of noisy data sets. We will show that the upgraded KSI outperforms most of the other techniques.Mechanical Engineering | Systems and Contro
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
State space modelling in macroeconomics and finance using SsfPack in S+Finmetrics
No full textAbstract This article surveys some common state space models used in macroeconomics and finance and shows how to specify and estimate these models using the SsfPack library of algorithms implemented in the S-PLUS module S+FinMetrics. Examples include recursive regression models, time varying parameter models, exact autoregressive moving average models and calculation of the Beveridge-Nelson decomposition, unobserved components models, stochastic volatility models, and term structure models. Introduction The first version of SsfPack appeared in 1998 and was developed further during the years that the last author was working with Jim Durbin on their 2001 textbook on state space methods. The fact that SsfPack functions are now a part of the S-PLUS software is partly due to Jim Durbin. He convinced Doug Martin that SsfPack would be very beneficial to S-PLUS. Indeed the persuasive arguments of Jim Durbin has initiated the development of SsfPack functions for S-PLUS as part of the S+FinMetrics module. It is therefore an honour for us, the developers of SsfPack for S+FinMetrics, to contribute to this volume with the presentation of various applications in economics and finance that require the use of SsfPack for S+FinMetrics in empirical research. State space modelling in economics and finance has become widespread over the last decade. Textbook treatments of state space models are given in Harvey (1989, 1993), Hamilton (1994), West and Harrison (1997), Kim and Nelson (1999), Shumway and Stoffer (2000), Durbin and Koopman (2001) and Chan (2002). However, until recently there has not been much flexible software for the statistical analysis of general models in state space form
PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator
No full textPyKoopman is a Python package for the data-driven approximation of the
Koopman operator associated with a dynamical system. The Koopman operator is a
principled linear embedding of nonlinear dynamics and facilitates the
prediction, estimation, and control of strongly nonlinear dynamics using linear
systems theory. In particular, PyKoopman provides tools for data-driven system
identification for unforced and actuated systems that build on the
equation-free dynamic mode decomposition (DMD) and its variants. In this work,
we provide a brief description of the mathematical underpinnings of the Koopman
operator, an overview and demonstration of the features implemented in
PyKoopman (with code examples), practical advice for users, and a list of
potential extensions to PyKoopman. Software is available at
http://github.com/dynamicslab/pykoopmanComment: 16 page
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