Koopman Representation Research Articles

Linear operator theory of phase mixing

ABSTRACT We study solutions of the collisionless Boltzmann equation (CBE) in a functional Koopman representation. This facilitates the use of linear spectral techniques characteristic of the analysis of Schrödinger-type equations. For illustrative purposes, we consider the classical phase mixing of a non-interacting distribution function in a quartic potential. Solutions are determined perturbatively relative to a harmonic oscillator. We impose a form of coarse-graining by choosing a finite-dimensional basis to represent the distribution function and time evolution operators, which sets a minimum length-scale on phase space structure. We observe a relationship between the dimension of the representation and the multiplicity of the harmonic oscillator eigenvalues. System dynamics are understood in terms of degenerate subspaces of the linear operator spectra. Each subspace is associated with a mode of the harmonic oscillator, the first two being bending and breathing structures. The quartic potential splits the degenerate eigenvalues within each subspace. This facilitates the formation of spiral structure as deformations from the harmonic oscillator modes. We ultimately argue that this construction provides a promising avenue for study of self-interacting systems experiencing phase mixing, which is an outstanding problem in the context of the Gaia DR2 vertical phase space spirals.

Koopman-Based MPC With Learned Dynamics: Hierarchical Neural Network Approach.

This article presents a data-driven control strategy for nonlinear dynamical systems, enabling the construction of a Koopman-based linear system associated with nonlinear dynamics. The primary idea is to apply the deep learning technique to the Koopman framework for globally linearizing nonlinear dynamics and impose a Koopman-based model predictive control (MPC) approach to stabilize the nonlinear dynamical systems. In this work, we first generalize the Koopman framework to nonlinear control systems, enabling comprehensive linear analysis and control methods to be effective for nonlinear systems. We next present a hierarchical neural network (HNN) approach to deal with the crucial challenge of the finite-dimensional Koopman representation approximation. In particular, a scale-invariant constrained network in the HNN includes four modules, in which a predictor module and a linear module can accurately approximate the finite Koopman eigenfunctions and Koopman operator, respectively, thus forming the lifted linear system. Then, we design the Koopman-based MPC scheme for controlling nonlinear systems with constraints by adopting the modified MPC with a saturation-like function on the lifted linear system. Importantly, the Koopman-based MPC enjoys higher computational efficiency compared to the classical linear MPC and nonlinear MPC methods. Finally, a physical experiment on an overhead crane system is provided to demonstrate the effectiveness of the proposed data-driven control framework.

Koopman-Operator Control Optimization for Relative Motion in Space

A high-order optimal control strategy implemented in the Koopman operator framework is proposed in this work. The new technique exploits the Koopman representation of the solution of the equations of motion to develop an energy-optimal inverse control methodology. The operator theory can reformulate a nonlinear dynamic system of finite dimension into a linear system with an infinite number of dimensions. As a result, the state of any nonlinear dynamics is represented as a linear combination of high-order orthogonal polynomials, which creates the state transition polynomial map of the solution. Because the optimal control technique can be reduced to a two-point boundary value problem, the Koopman map is used to connect the state and control variables in time, such that optimal values are obtained through map inversion and polynomial evaluation. The new technique is applied to rendezvous applications in space, where the relative motion between two satellites is modeled with a high-order polynomial series expansion of the Lagrangian of the system, such that the Clohessy–Wiltshire equations represent the reduction of the high-order model to a linear truncation. The proposed numerical applications are analyzed to show the robustness and limitations of the novel technique.

Data-driven safe control via finite-time Koopman identifier

In this paper, a novel data-driven invariant-based safe control scheme for control of a nonlinear vehicle is developed. First, a novel incremental finite-time Koopman representation-based identifier is introduced to learn approximated lifted-states linear system model of system from a limited budget of incrementally sampled data points which not only minimises the instantaneous Koopman representation's identification errors but also the identification errors for a batch of past samples collected in a history stack. A great advantage of the presented identifier is that the uncertainty of the identified system is also quantified and updated over time on a slower time scale. Using set-theoretic tools, then, the quantified uncertainty will be leveraged in conjunction with the learned model in order to impose robust contractivity on the safe set, which guarantees its safety and stability. The proposed approach significantly reduces conservativeness through quantifying and updating the uncertainty level of the learned model over time.

A type I conjecture and boundary representations of hyperbolic groups

AbstractWe establish new results on the weak containment of quasi‐regular and Koopman representations of a second countable locally compact group associated with nonsingular ‐spaces. We deduce that any two boundary representations of a hyperbolic locally compact group are weakly equivalent. We also show that nonamenable hyperbolic locally compact groups with a cocompact amenable subgroup are characterized by the property that any two proper length functions are homothetic up to an additive constant. Combining those results with the work of Ł. Garncarek on the irreducibility of boundary representations of discrete hyperbolic groups, we deduce that a type I hyperbolic group with a cocompact lattice contains a cocompact amenable subgroup. Specializing to groups acting on trees, we answer a question of C. Houdayer and S. Raum.

Learning Stable Models for Prediction and Control

In this article, we demonstrate the benefits of imposing stability on data-driven Koopman operators. The data-driven identification of stable Koopman operators (DISKO) is implemented using an algorithm [1] that computes the nearest <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stable</i> matrix solution to a least-squares reconstruction error. As a first result, we derive a formula that describes the prediction error of Koopman representations for an arbitrary number of time steps, and which shows that stability constraints can improve the predictive accuracy over long horizons. As a second result, we determine formal conditions on basis functions of Koopman operators needed to satisfy the stability properties of an underlying nonlinear system. As a third result, we derive formal conditions for constructing Lyapunov functions for nonlinear systems out of stable data-driven Koopman operators, which we use to verify stabilizing control from data. Finally, we demonstrate the benefits of DISKO in prediction and control with simulations using a pendulum and a quadrotor and experiments with a pusher-slider system. The paper is complemented with a video: <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://sites.google.com/view/learning-stable-koopman</uri> .

On the spectral theory of groups of automorphisms of S-adic nilmanifolds

AbstractLet $S=\{p_1, \ldots , p_r,\infty \}$ for prime integers $p_1, \ldots , p_r.$ Let X be an S-adic compact nilmanifold, equipped with the unique translation-invariant probability measure $\mu .$ We characterize the countable groups $\Gamma $ of automorphisms of X for which the Koopman representation $\kappa $ on $L^2(X,\mu )$ has a spectral gap. More specifically, let Y be the maximal quotient solenoid of X (thus, Y is a finite-dimensional, connected, compact abelian group). We show that $\kappa $ does not have a spectral gap if and only if there exists a $\Gamma $ -invariant proper subsolenoid of Y on which $\Gamma $ acts as a virtually abelian group,

Extending the extended dynamic mode decomposition with latent observables: the latent EDMD framework

Bernard O Koopman proposed an alternative view of dynamical systems based on linear operator theory, in which the time evolution of a dynamical system is analogous to the linear propagation of an infinite-dimensional vector of observables. In the last few years, several works have shown that finite-dimensional approximations of this operator can be extremely useful for several applications, such as prediction, control, and data assimilation. In particular, a Koopman representation of a dynamical system with a finite number of dimensions will avoid all the problems caused by nonlinearity in classical state-space models. In this work, the identification of finite-dimensional approximations of the Koopman operator and its associated observables is expressed through the inversion of an unknown augmented linear dynamical system. The proposed framework can be regarded as an extended dynamical mode decomposition that uses a collection of latent observables. The use of a latent dictionary applies to a large class of dynamical regimes, and it provides new means for deriving appropriate finite-dimensional linear approximations to high-dimensional nonlinear systems.

Exotic ideals in free transformation group C∗$\mathbf {C}^*$‐algebras

AbstractLet be a discrete group acting freely via homeomorphisms on the compact Hausdorff space and let be the completion of the convolution algebra with respect to a ‐norm . A non‐zero ideal is exotic if . We show that exotic ideals are present whenever is non‐amenable and there is an invariant probability measure on . This fact, along with the recent theory of exotic crossed product functors, allows us to provide answers to two questions of K. Thomsen.Using the Koopman representation and a recent theorem of Elek, we show that when is a countably‐infinite group having property (T) and is the Cantor set, there exists a free and minimal action of on and a ‐norm on such that contains the compact operators as an exotic ideal. We use this example to provide a positive answer to a question of A. Katavolos and V. Paulsen.The opaque and grey ideals in have trivial intersection with , and a result from Exel and Pitts [Characterizing groupoid ‐algebras of non‐Hausdorff étale groupoids, Lecture Notes in Mathematics, vol. 2306, Springer, Cham, 2022] shows that they coincide when the action of is free; however, the problem of whether these ideals can be non‐zero was left unresolved. We present an example of a free action of on a compact Hausdorff space along with a ‐norm for which these ideals are non‐trivial, in particular, they are exotic ideals.

Koopman interpretation and analysis of a public-key cryptosystem: Diffie-Hellman key exchange

The security of public-key cryptosystems relies on computationally hard problems, that are classically analyzed by number theoretic methods. In this paper, we introduce a new perspective on cryptosystems by interpreting the Diffie-Hellman key exchange as a nonlinear dynamical system. Employing Koopman theory, we transfer this dynamical system into a higher-dimensional space to analytically derive a purely linear system that equivalently describes the underlying cryptosystem. In this form, analytic tools for linear systems allow us to reconstruct the secret integers of the key exchange by simple manipulations. Moreover, we provide an upper bound on the minimal required lifting dimension to obtain perfect accuracy. To demonstrate the potential of our method, we relate our findings to existing results on algorithmic complexity. Finally, we transfer this approach to a data-driven setting where the Koopman representation is learned from data samples of the cryptosystem.

Design of Sliding Mode Controllers using Reduced-order Koopman Mode Representations

The Koopman operator framework allows for a linear, but infinite-dimensional, representation of the dynamics of a non-linear system. The Koopman modes, or observables, and the resulting linear dynamics are derived purely using a data-driven framework, where the data are system outputs measured at discrete samples; improving accuracy of the Koopman representation requires a large number of such modes to be considered. Recent results consider the system input as well, in the derivation of the discrete linear dynamics, thus enabling the design of controllers. Sliding mode controllers (SMCs), including the discrete-time versions, can handle parameter uncertainties and variations and also ensure that the control objective is satisfied in finite time. In this paper, a discrete-time SMC is designed for the output control of a dynamic system approximated by fewer Koopman modes; the SMC is expected to handle uncertainties introduced by the ignored modes. Conditions are identified for the closed-loop system to be stable, with the occurrence of sliding mode.

The spectral form of Koopman representations of the group of measurable functions with values in the circle

We compute the spectral form of the Koopman representation induced by a natural boolean action of $L^0(\lambda , \mathbb T)$ identified earlier by the authors. Our computation establishes the sharpness of the constraints on spectral forms of Koopman repre

Finite Dimensional Koopman Form of Polynomial Nonlinear Systems*

The Koopman framework is a popular approach to transform a finite dimensional nonlinear system into an infinite dimensional, but linear model through a lifting process using so-called observable functions. While there is an extensive theory on infinite dimensional representations in the operator sense, there are few constructive results on how to select the observables to realize them. When it comes to the possibility of finite Koopman representations, which are highly important from a practical point of view, there is no constructive theory. Hence, in practice, often a data-based method and ad-hoc choice of the observable functions is used. When truncating to a finite number of basis, there is also no clear indication of the introduced approximation error. In this paper, we propose a systematic method to compute the finite dimensional Koopman embedding of a specific class of polynomial nonlinear systems in continuous-time, such that the embedding can fully represent the dynamics of the nonlinear system without any approximation.

Koopman operator for time-dependent reliability analysis

Time-dependent structural reliability analysis of nonlinear dynamical systems is non-trivial; subsequently, scope of most of the structural reliability analysis methods is limited to time-independent reliability analysis only. In this work, we propose a Koopman operator based approach for time-dependent reliability analysis of nonlinear dynamical systems. Since the Koopman representations can transform any nonlinear dynamical system into a linear dynamical system, the time evolution of dynamical systems can be obtained by Koopman operators seamlessly regardless of the nonlinear or chaotic behavior. Despite the fact that the Koopman theory has been in vogue a long time back, identifying intrinsic coordinates is a challenging task; to address this, we propose an end-to-end deep learning architecture that learns the Koopman observables and then use it for time marching the dynamical response. Unlike purely data-driven approaches, the proposed approach is robust even in the presence of uncertainties; this renders the proposed approach suitable for time-dependent reliability analysis. We propose two architectures; one suitable for time-dependent reliability analysis when the system is subjected to random initial condition and the other suitable when the underlying system have uncertainties in system parameters. The proposed approach is robust and generalizes to unseen environment (out-of-distribution prediction). Efficacy of the proposed approached is illustrated using four numerical examples. Results obtained indicate supremacy of the proposed approach as compared to purely data-driven auto-regressive neural network, long short- term memory network, and DeepONet.

Generic measure preserving transformations and the closed groups they generate

We show that, for a generic measure preserving transformation T, the closed group generated by T is not isomorphic to the topological group \(L^0(\lambda , {{\mathbb {T}}})\) of all Lebesgue measurable functions from [0, 1] to \({\mathbb {T}}\) (taken with pointwise multiplication and the topology of convergence in measure). This result answers a question of Glasner and Weiss. The main step in the proof consists of showing that Koopman representations of ergodic boolean actions of \(L^0(\lambda , {\mathbb T})\) possess a non-trivial spectral property not shared by all unitary representations of \(L^0(\lambda , {{\mathbb {T}}})\). The main tool underlying our arguments is a theorem on the form of unitary representations of \(L^0(\lambda , {{\mathbb {T}}})\) from our earlier work.

The Koopman representation for self-similar groupoid actions

We introduce the [Formula: see text]-algebra [Formula: see text] generated by the Koopman representation [Formula: see text] of an étale groupoid [Formula: see text] acting on a measure space [Formula: see text]. We prove that for a level transitive self-similar action [Formula: see text] with [Formula: see text] finite and [Formula: see text] constant, there is an invariant measure [Formula: see text] on [Formula: see text] and that [Formula: see text] is residually finite dimensional with a normalized self-similar trace. We also discuss [Formula: see text]-fold similarities of Hilbert spaces in connection to representations of the graph algebra [Formula: see text] and self-similar representations of [Formula: see text] in connection to the Cuntz–Pimsner algebra [Formula: see text].

Distributed Koopman-Based Control of Improved Swing Equation

This paper presents a data-driven control for a set of synchronous generators based on the improved swing equation model by using the Koopman operator. First, the nonlinear dynamic of the generators is represented by a linear model in lifted space using extended dynamic mode decomposition (EDMD). Then, a linear predictor is built and used for the design of a model predictive control following three strategies: decentralized, non-cooperative, and cooperative. The Koopman representation of the generator is used to carry out an analysis, determining its eigenvalues. Several interconnected generators are controlled showing the performance of the interconnected system including voltage faults.

О купмановском представлении гамильтоновых потоков в бесконечномерных пространствах с инвариантной мерой

We study Hamiltonian flows in a real separable Hilbert space endowed with a symplectic structure. Measures on a Hilbert space that are invariant with respect to flows of completely integrable Hamiltonian systems and allow one to describe Hamiltonian flows in a phase space in terms of unitary groups in the space of functions squarely integrable with respect to an invariant measure are studied. The properties of the Koopman representation are described using the example of the Hamiltonian of a countable set of noninteracting harmonic oscillators. A spectral analysis of the generator of Koopman group for such a Hamiltonian is carried out. An invariant subspace of strong continuity of Koopman unitary group is described in terms of spectrum of the generator.

Derivative-Based Koopman Operators for Real-Time Control of Robotic Systems

This article presents a generalizable methodology for data-driven identification of nonlinear dynamics that bounds the model error in terms of the prediction horizon and the magnitude of the derivatives of the system states. Using higher order derivatives of general nonlinear dynamics that need not be known, we construct a Koopman-operator-based linear representation and utilize Taylor series accuracy analysis to derive an error bound. The resulting error formula is used to choose the order of derivatives in the basis functions and obtain a data-driven Koopman model using a closed-form expression that can be computed in real time. Using the inverted pendulum system, we illustrate the robustness of the error bounds given noisy measurements of unknown dynamics, where the derivatives are estimated numerically. When combined with control, the Koopman representation of the nonlinear system has marginally better performance than competing nonlinear modeling methods, such as SINDy and NARX. In addition, as a linear model, the Koopman approach lends itself readily to efficient control design tools, such as linear–quadratic regulator, whereas the other modeling approaches require nonlinear control methods. The efficacy of the approach is further demonstrated with simulation and experimental results on the control of a tail-actuated robotic fish. Experimental results show that the proposed data-driven control approach outperforms a tuned proportional–integral–derivative controller and that updating the data-driven model online significantly improves performance in the presence of unmodeled fluid disturbance. This article is complemented with a video available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://youtu.be/9_wx0tdDta0</uri> .

The Koopman Representation and Positive Rokhlin Entropy

Abstract In this paper, we study connections between positive entropy phenomena and the Koopman representation for actions of general countable groups. Following the line of work initiated by Hayes for sofic entropy, we show in a certain precise manner that all positive entropy must come from portions of the Koopman representation that embed into the left-regular representation. We conclude that for actions having completely positive outer entropy, the Koopman representation must be isomorphic to the countable direct sum of the left-regular representation. This generalizes a theorem of Dooley–Golodets for countable amenable groups. As a final consequence, we observe that actions with completely positive outer entropy must be mixing, and when the group is non-amenable they must be strongly ergodic and have spectral gap.