1,720,995 research outputs found
Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations
Full text linkThe relationship between Koopman mode decomposition, resolvent mode decomposition, and exact invariant solutions of the Navier-Stokes equations is clarified. The correspondence rests upon the invariance of the system operators under symmetry operations such as spatial translation. The usual interpretation of the Koopman operator is generalized to permit combinations of such operations, in addition to translation in time. This invariance is related to the spectrum of a spatiotemporal Koopman operator, which has a traveling-wave interpretation. The relationship leads to a generalization of dynamic mode decomposition, in which symmetry operations are applied to restrict the dynamic modes to span a subspace subject to those symmetries. The resolvent is interpreted as the mapping between the Koopman modes of the Reynolds stress divergence and the velocity field. It is shown that the singular vectors of the resolvent (the resolvent modes) are the optimal basis in which to express the velocity field Koopman modes where the latter are not a priori known
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Applications of Koopman Operator Theory to Highway Traffic Dynamics
Full text linkThe ever-increasing demands on transportation systems have led to the need for a robust and universal method for the analysis and forecasting of vehicular traffic systems. Traditional methods are mainly model-based, that is, the analysis is performed by investigating a mathematical model that represents the target dynamics of a traffic system. On the other hand, contemporary efforts have focused on utilizing artificial intelligence (AI) to model or forecast vehicular traffic dynamics. Despite these large efforts, there is still no single best-performing method for the analysis and forecasting of vehicular traffic dynamics. This is due to the very well known fact that the unpredictable behaviors involved in a traffic system, like human interaction and weather, leads to a very complicated high-dimensional nonlinear dynamical system. Therefore, it is difficult to obtain a mathematical or AI model that explains all events and time evolution of vehicular traffic dynamics. Even if such a model could be attained, it would not lead to a robust and universal way of traffic analysis and forecast, due to its need of extensive parameter tuning. Thus, in contrast to the model or AI-based approach, it is necessary to develop data-driven methods that can identify dynamically important spatiotemporal structures of traffic phenomena. In this thesis, we demonstrate how the Koopman operator theory can offer a model and parameter-free, data-driven approach to accurately analyzing and forecasting traffic dynamics. The Koopman operator theory framework is a rapidly developing theory in dynamical systems that offers powerful methods for analyzing complex nonlinear systems. The effectiveness of this framework is demonstrated by an application to the Next Generation Simulation (NGSIM) data set collected by the US Federal Highway Administration and the Performance Measurement System (PeMS) data set collected by the California Department of Transportation. By obtaining a Koopman mode decomposition (KMD) of the data sets, we are able to accurately reconstruct our observed dynamics, distinguish any growing or decaying modes, and obtain a hierarchy of coherent spatiotemporal patterns that are fundamental to the observed dynamics. Furthermore, it is demonstrated how the KMD can be utilized to accurately forecast traffic dynamics by obtaining a decomposition of a subset of the data, that is then used to predict a future subset of the data
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Mini-Workshop: Applied Koopmanism
Full text linkKoopman and Perron–Frobenius operators are linear operators that encapsulate dynamics of nonlinear dynamical systems without loss of information. This is accomplished by embedding the dynamics into a larger infinite-dimensional space where the focus of study is shifted from trajectory curves to measurement functions evaluated along trajectories and densities of trajectories evolving in time. Operator-theoretic approach to dynamics shares many features with an optimization technique: the Lasserre moment–sums-of-squares (SOS) hierarchies, which was developed for numerically solving non-convex optimization problems with semialgebraic data. This technique embeds the optimization problem into a larger primal semidefinite programming (SDP) problem consisting of measure optimization over the set of globally optimal solutions, where measures are manipulated through their truncated moment sequences. The dual SDP problem uses SOS representations to certify bounds on the global optimum. This workshop highlighted the common threads between the operator-theoretic dynamical systems and moment–SOS hierarchies in optimization and explored the future directions where the synergy of the two techniques could yield results in fluid dynamics, control theory, optimization, and spectral theory
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Theory and Applications of Pull-Back Operator Methods in Dynamical Systems
Full text linkDuring the 1930s, researchers realized that an abstract dynamical system induces a group of linear operators acting on the space of square-integrable functions. For measure-preserving systems, the induced operator is unitary and self-adjoint. As such, its spectrum is restricted to the unit circle and has been shown to encode many important statistical and geometric properties of the dynamical system. Since then, the induced Koopman group of operators' spectral properties have drawn an immense amount of research interest over the last decade. Due to the rise of computing capabilities and data availability, there has been an explosive amount of research into developing data-driven algorithms that can compute the spectrum numerically from data. In the first part of this dissertation, we demonstrate how Koopman operator methods can offer a model-free, data-driven approach to analyze and forecast highway traffic dynamics. By obtaining a decomposition of data sets collected by the Federal Highway Administration and the California Department of Transportation, we can reconstruct observed data, distinguish any growing or decaying patterns, and obtain a hierarchy of previously identified and never before identified spatiotemporal patterns. Furthermore, it is demonstrated how this methodology can be utilized to forecast highway network conditions. The developed forecasting scheme readily generalizes to the much-needed scenario of multi-lane highway networks without any loss to its performance or efficiency. Also, we do not rely on large historical training data nor parameter tuning or selection. Thereby providing a completely efficient and accurate method of analyzing and forecasting traffic patterns at the levels required by modern intelligent transportation systems.In the second part of this dissertation, we consider the equivalent induced linear operators acting on the space of sections of the tangent, cotangent, and tensor bundles of the state space. We begin by first demonstrating how these operators are indeed natural generalizations of Koopman operators acting on functions. The fundamental insight lies in understanding the connection between the differential geometric concept of pulling back objects (functions, vector fields, covector fields, tensor fields) under a diffeomorphism and how their pull-back relates to the Lie derivative of that object. We then draw connections between the various operators' spectrum and characterize the algebraic and differential topological properties of their spectrum. We describe these operators' discrete spectrum for linear dynamical systems and derive spectral type expansions for linear vector fields. The expansions derived resemble the familiar spectral expansion of functions under the Koopman operator. We define the notion of an "eigendistribution" and provide conditions for when an eigendistribution is integrable. We then demonstrate how to recover the foliations arising from their integral manifolds via the level sets of Koopman eigenfunctions. Many of the results presented in the second part of this dissertation stem from well-known differential geometric concepts. Prior work on such generalized operators on vector fields exists but has remained mostly unnoticed by the growing Koopman operator community. We conclude with an application to differential geometry where the well-known fact that the flows of commuting vector fields commute is generalized. Specifically, we show that the flows of two vector fields commute, subject to an appropriate rescaling of the flow time, if and only if one vector field is an eigensection of the other vector field. The eigenvalue prescribes the required time scaling, and we recover the original statement that the flows of commuting vector fields commute as a particular case of our result. We also apply our results to the study of a hyperbolic toral automorphism known as Arnold's Cat map. We demonstrate that the Lyapunov exponents are contained within the spectrum of the induced operator on vector fields, and we recover the stable and unstable foliations via the level sets of the joint eigenfunctions of the stable and unstable eigendistributions
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Koopman Operators and System Identification for Stochastic Systems
Full text linkThe use of the Koopman operator framework in dynamical systems has greatly expanded in recent years. Instead of considering the evolution of the state of a system, the Koopman semigroup tracks the evolution of observables on the state. Since the Koopman operator defined for an arbitrary dynamical system is linear, it allows us to use linear system theory and spectral methods to analyze nonlinear systems. This framework has also been extended to stochastic systems. Since the evolution of observables can only be defined probabilistically for random systems, stochastic Koopman operators are defined by taking the expectation of the future value of observables.In the first part of this thesis, we review the basic theory of random dynamical systems and stochastic Koopman operators. We can use these operators to represent a nonlinear RDS as an infinite dimensional linear operator. The basic theorems and definitions are given in this section, which will help form the foundation for the algorithms discussed in the second and third sections. Further, some simple examples are given for which the stochastic Koopman operator is well understood. These examples will recur as we use them to test the algorithms in the second section.The second section is devoted to the analysis of Dynamic Mode Decomposition (DMD) algorithms. DMD algorithms approximate a finite section of the (stochastic) Koopman operator using data from a trajectory. However, these methods are sensitive to noise, and will give a biased approximation if the observables contains randomness. To combat this, we introduce an new DMD variant which can approximate a finite section of the stochastic Koopman operator even when the data contains measurement noise. Further, we extend this algorithm for use with time delayed observables to create a variant of Hankel DMD which will converge for stochastic systems. We then demonstrate these algorithms on numerical examples.In the final section, we will discuss the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm for stochastic differential equations. The SINDy algorithm allows one to generate a representation of an ODE using a dictionary of functions and data from a trajectory. This algorithm has been extended to SDEs, but the accuracy is limited by the numerical approximations of the drift and diffusion functions. We demonstrate how we can use higher order approximations to these functions to generate a far more accurate representation of the SDE. We then test these approximations on several examples
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Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems
Full text linkThe dominating methodology used in the study of dynamical systems is the geometric picture introduced by Poincare. The focus is on the structure of the state space and the asymptotic behavior of trajectories. Special solutions such as fixed points and limit cycles, along with their stable and unstable manifolds, are of interest due to their ability to organize the trajectories in the surrounding state space.Another viewpoint that is becoming more prevalent is the operator-theoretic / functional-analytic one which describes the system in terms of the evolution of functions or measures defined on the state space. Part I of this doctoral dissertation focuses on the Koopman, or composition, operator that determines how a function on the state space evolves as the state trajectories evolve. Most current studies involving the Koopman operator have dealt with its spectral properties that are induced by dynamical systems that are, in some sense, stationary (in the probabilistic sense). The dynamical systems studied are either measure-preserving or initial conditions for trajectories are restricted to an attractor for the system. In these situations, only the point spectrum on the unit circle is considered; this part of the spectrum is called the unimodular spectrum. This work investigates relaxations of these situations in two different directions. The first is an extension of the spectral analysis of the Koopman operator to dynamical systems possessing either dissipation or expansion in regions of their state space. The second is to consider switched, stochastically-driven dynamical systems and the associated collection of semigroups of Koopman operators.In the first direction, we develop the Generalized Laplace Analysis (GLA) for both spectral operators of scalar type (in the sense of Dunford) and non spectral operators. The GLA is a method of constructing eigenfunctions of the Koopman operator corresponding to non-unimodular eigenvalues. It represents an extension of the ergodic theorems proven for ergodic, measure-preserving, on-attractor dynamics to the case where we have off-attractor dynamics. We also give a general procedure for constructing an appropriate Banach space of functions on which the Koopman operator is spectral. We explicitly construct these spaces for attracting fixed points and limit cycles. The spaces that we introduce and construct are generalizations of the familiar Hilbert Hardy spaces in the complex unit disc.In the second direction, we develop the theory of switched semigroups of Koopman operators. Each semigroup is assumed to be spectral of scalar-type with unimodular point spectrum, but possibly non-unimodular continuous spectrum. The functions evolve by applying one semigroup for a period of time, then switching to another semigroup. We develop an approximation of the vector-valued function evolution by a linear approximation in the vector space that the functions map into. A basis for this linear approximation is constructed from the vector-valued modes that are coefficients of the projections of the vector-valued observable onto scalar-valued eigenfunctions of the Koopman operator. The unmodeled modes show up as noisy dynamics in the output space. We apply this methodology to traffic matrices of an Internet Service Provider's (ISP's) network backbone. Traffic matrices measure the traffic volume moving between an ingress and egress router for the network's backbone. It is shown that on each contiguous interval of time in which a single semigroup acts the modal dynamics are deterministic and periodic with Gaussian or nearly-Gaussian noise superimposed.Part II of the dissertation represents a divergence from the first part in that it does not deal with the Koopman operator explicitly. In the second part, we consider the problem of using exponentially mixing dynamical systems to generate trajectories for an agent to follow in its search for a physical target in a large domain. The domain is a compact subset of the n-dimensional Euclidean space Rn. It is assumed that the size of the target is unknown and can take any value in some continuous range. Furthermore, it is assumed that the target can be located anywhere in the domain with equal probability.We cast this problem as one in the field of quantitative recurrence theory, a relatively new sub-branch of ergodic theory. We give constructive proofs for upper bounds of hitting times of small metric balls in Rn for mixing transformations of various speeds. The upper bounds and limit laws we derive say, approximately, that the hitting time is bounded above by some constant multiple of the inverse of the measure of the metric ball. From these results, we derive upper bounds for the expected hitting time, with respect to the range of target sizes [delta, V), to be of order O(-ln delta). First order, continuous time dynamics are constructed from discrete time mixing transformations and upper bounds for these hitting times are shown to be proportional to the discrete time case
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Koopman spectral analysis and study of mixing in incompressible flows
Full text linkThe main theme of this thesis is application of recently developed tools in dynamical systems theory in the study of incompressible fluid flows. These applications fall into two general categories: the first one is study of the flow evolution as an infinite-dimensional dynamical system and it is related to classical topics like flow stability and transition. The second area is the study of flow kinematics where tools of dynamical systems are used to study the trajectory of particles immersed within the flow, and includes topics like mixing enhancement, and prediction of pollution movement in the ocean or atmosphere.In studying flow dynamics, we utilize the Koopman operator framework for data-driven study of dynamical systems (introduced in chapter 1). The increasing popularity of this framework is due to a versatile combination of rigorous theory and data analysis algorithms which allows extraction of dynamic information from almost any type of data from a dynamical system. In chapter 2, we use the spectral properties of the Koopman operator, computed from data, to interpret the flow dynamics: we use the Koopman spectra to determine the attractor geometry, the Koopman eigenfunctions to map the state space linear coordinates, and the Koopman modes to characterize the unsteady motion in the flow domain.We also discuss the numerical computation of the Koopman spectral properties. As of now, this computation is mostly done through a class of numerical algorithms known as Dynamic Mode Decomposition (DMD). In chapter 3, we prove the convergence of a class of DMD algorithms, called Hankel-DMD, for systems with ergodic attractors. Our proofs are based on the fact that projections in the space of functions can be approximated via vector projections in DMD by the virtue of Birkhoff's ergodic theorem. This new result also provides insight on dynamics of chaotic systems with continuous spectrum and computation of Koopman eigenvalues for dissipative systems. We compare the performance of Hankel-DMD to the signal processing techniques used for fluid flows in chapter 2.One of the important questions in the study of flow kinematics is how to characterize the mixing in flows with aperiodic time dependence. This question has given rise to a variety of methodologies that strive to describe the mixing in a given aperiodic flow by detecting the coherent structures or other objects of special interest. In chapter 4, we study this problem from a different perspective, namely, weconsider how the mixing portrait is changed while the temporal regime of a bounded flow -the lid-driven cavity flow - changes from steady to aperiodic. We use the Koopman spectral properties of the flow (studied in chapter 2) and the so-called hypergraphs to isolate and characterize the effect of different elements from the flow dynamicson the mixing. For example, we will see how the interaction between vorticity distribution in the mean flow and non-zero Koopman frequencies determines the regions of slow mixing.In chapter 5, we report on application of hypergraphs combined with high-frequency radar data in the study of surface mixing in Santa Barbara channel (i.e the patch of Pacific Ocean between Santa Barbara coastline and Channel Islands) with special focus on prediction of oil slick movements in the aftermath of 2015 Refugio oil spill
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Koopman Representations in Control
Full text linkThe Koopman operator describes the time-evolution of scalar-valued functions under the action of a dynamical system. These functions are called observables, and their evolution is always linear, even if the underlying dynamical system is nonlinear. The linearity of the Koopman operator framework is attractive to both dynamical systems theorists who study the spectral properties of these operators as well as to control theorists who leverage linearity to simplify control design. Recently, the theory of Koopman representations has emerged, with researchers gradually exploring the benefits of alternate, potentially nonlinear ways of representing these systems. In this thesis, we explore three distinct ways of representing the Koopman operator and explore their application to control design.In the first part of this dissertation, we develop the mathematical underpinning of Koopman representation theory. The evolution from the state-space representation of a dynamical system to the Koopman operator is described, and its spectral content is explored. Next, nonlinear representations of the Koopman operator and its extension to systems with input is described. Finally, we introduce some of the numerical approximation schemes for the representations that are used in this paper. This chapter is meant to give the reader the mathematical background necessary to appreciate the results presented in the remaining chapters.In the second part of this dissertation, we demonstrate a linear representation of the Koopman operator which fully leverages spectral objects such as eigenfunctions and eigenvalues. The eigenfunctions are special observables which evolve under the action of the Koopman operator via multiplication by a complex scalar, the eigenvalues. This is analogous to the eigenvectors and eigenvalues of a linear transformation. A collection of eigenfunctions forms a finite-dimensional, linear representation of a dynamical system, and their evolution spans a Koopman-invariant subspace. Finding this finite-dimensional representation allows for the application of well-developed linear systems methodologies to nonlinear systems such as spectral analysis and linear optimal control methods. In this work, we introduce a deep learning architecture that learns the Koopman eigenfunctions of a dynamical system from data and constructs the resulting finite-dimensional, linear representation of the Koopman operator. In numerical examples, the eigenfunctions learned using this framework exhibit a predictive performance superior to popular fixed-basis methods such as Extended Dynamic Mode Decomposition (EDMD). Finally, we extend the architecture to controlled dynamical systems by simultaneously learning the eigenfunctions of the natural dynamics with special system-decoupling observables on the inputs. Numerical examples show that the linear predictors obtained in this way can be readily used to design controllers that act directly on the Koopman modes of the system. In the third part of this dissertation, we introduce our first use of the static Koopman operator in control. In our application, this is a linear operator which maps the space of functions of static poses of a soft robotic arm to the space of functions of the pressures in the arm's actuating muscles. We use static Koopman operator as a pregain term in our optimal control implementation alongside a traditional dynamic Koopman operator. Using both Koopman representations, we advance the modeling and control of soft robots into the inertial, nonlinear regime. We control motions of a soft, continuum arm with velocities 10x larger and accelerations 40x larger than those of previous work, and do so for high-deflection shapes with over 110 degrees of curvature. This work advances rapid modeling and control for soft robots from the realm of quasi-static to inertial, laying the groundwork for the next generation of compliant and highly dynamic robots.Lastly, in the fourth chapter, we introduce a nonlinear Koopman representation which leverages so-called input-parameterized Koopman eigenfunctions. In the control of systems with multiple fixed points, it is typical to use piecewise control methods and local Koopman models. In contrast, our input-parameterized eigenfunction representation is accurate globally and enables a finite dimensional model which can handle these control problems without ad-hoc piecewise methods. We illustrate this on the control between the basins of attraction of the Duffing oscillator with dissipation
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Robust Approximation of the Stochastic Koopman Operator
Full text linkWe analyze the performance of DMD-based approximations of the stochastic Koopman operator for random dynamical systems where either the dynamics or observables are affected by noise. Under certain ergodicity assumptions, we show that standard DMD algorithms converge provided the observables do not contain any noise and span an invariant subspace of the stochastic Koopman operator. For observables with noise, we introduce a new, robust DMD algorithm that can approximate the stochastic Koopman operator and demonstrate how this algorithm can be applied to Krylov subspace based methods using a single observable measured over a single trajectory. We test the performance of the algorithms over several examples
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