arXiv subjects

Joel A. Rosenfeld

Publications and source records attributed to Joel A. Rosenfeld.

At least 19 recordsLinked to original sources

arXiv2026

Integral Formulation of QENDy for Robust Nonlinear System Identification

This manuscript proposes an integral formulation of the newly defined quadratic embedding method for identifying nonlinear systems (QENDy). In the original algorithm, trajectory data points along with their time derivatives are used. Methods for calculating time derivatives make the algorithm sensitive to noise. Our integral formulation does not use the time derivatives. This results in a more robust method to learn the dynamics.

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arXiv2026

Supporting Lemmas for RISE-based Control Methods

A class of continuous controllers termed Robust Integral of the Signum of the Error (RISE) have been published over the last decade as a means to yield asymptotic convergence of the tracking error for classes of nonlinear systems that are subject to exogenous disturbances and/or modeling uncertainties. The development of this class of controllers relies on a property related to the integral of the signum of an error signal. A proof for this property is not available in previous literature. The stability of some RISE controllers is analyzed using differential inclusions. Such results rely on the hypothesis that a set of points is Lebesgue negligible. This paper states and proves two lemmas related to the properties.

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arXiv2026

Control Occupation Kernel Regression for Nonlinear Control-Affine Systems

This manuscript presents an algorithm for obtaining an approximation of a nonlinear high order control affine dynamical system. Controlled trajectories of the system are leveraged as the central unit of information via embedding them in vector-valued reproducing kernel Hilbert space (vvRKHS). The trajectories are embedded as the so-called higher order control occupation kernels which represent an operator on the vvRKHS corresponding to iterated integration after multiplication by a given controller. The solution to the system identification problem is then the unique solution of an infinite dimensional regularized regression problem. The representer theorem is then used to express the solution as finite linear combination of these occupation kernels, which converts an infinite dimensional optimization problem to a finite dimensional optimization problem. The vector valued structure of the Hilbert space allows for simultaneous approximation of the drift and control effectiveness components of the control affine system. Several experiments are performed to demonstrate the effectiveness of the developed approach.

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arXiv2025

On Dynamic Mode Decomposition of Control-affine Systems

This paper builds on the theoretical foundations for dynamic mode decomposition (DMD) of control-affine dynamical systems by leveraging the theory of vector-valued reproducing kernel Hilbert spaces (RKHSs). Specifically, control Liouville operators and control occupation kernels are used to separate the drift dynamics from the input dynamics. A provably convergent finite-rank estimation of a compact control Liouville operator is obtained, provided sufficiently rich data are available. A matrix representation of the finite-rank operator is used to construct a data-driven representation of its singular values, left singular functions, and right singular functions. The singular value decomposition is used to generate a data-driven model of the control-affine nonlinear system. The developed method generates a model that can be used to predict the trajectories of the system in response to any admissible control input. Numerical experiments are included to demonstrate the efficacy of the developed technique.

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arXiv2024

On Convergent Dynamic Mode Decomposition and its Equivalence with Occupation Kernel Regression

This paper presents a new technique for norm-convergent dynamic mode decomposition of deterministic systems. The developed method utilizes recent results on singular dynamic mode decomposition where it is shown that by appropriate selection of domain and range Hilbert spaces, the Liouville operator (also known as the Koopman generator) can be made to be compact. In this paper, it is shown that by selecting appropriate collections of finite basis functions in the domain and the range, a novel finite-rank representation of the Liouville operator may be obtained. It is also shown that the model resulting from dynamic mode decomposition of the finite-rank representation is closely related to regularized regression using the so-called occupation kernels as basis functions.

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arXiv2024

Dynamic Mode Decomposition with Control Liouville Operators

This paper builds the theoretical foundations for dynamic mode decomposition (DMD) of control-affine dynamical systems by leveraging the theory of vector-valued reproducing kernel Hilbert spaces (RKHSs). Specifically, control Liouville operators and control occupation kernels are introduced to separate the drift dynamics from the input dynamics. A given feedback controller is represented through a multiplication operator and a composition of the control Liouville operator and the multiplication operator is used to express the nonlinear closed-loop system as a linear total derivative operator on RKHSs. A spectral decomposition of a finite-rank representation of the total derivative operator yields a DMD of the closed-loop system. The DMD generates a model that can be used to predict the trajectories of the closed-loop system. For a large class of systems, the total derivative operator is shown to be compact provided the domain and the range RKHSs are selected appropriately. The sequence of models, resulting from increasing-rank finite-rank representations of the compact total derivative operator, are shown to converge to the true system dynamics, provided sufficiently rich data are available. Numerical experiments are included to demonstrate the efficacy of the developed technique.

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arXiv2023

The kernel perspective on dynamic mode decomposition

This manuscript revisits theoretical assumptions concerning dynamic mode decomposition (DMD) of Koopman operators, including the existence of lattices of eigenfunctions, common eigenfunctions between Koopman operators, and boundedness and compactness of Koopman operators. Counterexamples that illustrate restrictiveness of the assumptions are provided for each of the assumptions. In particular, this manuscript proves that the native reproducing kernel Hilbert space (RKHS) of the Gaussian RBF kernel function only supports bounded Koopman operators if the dynamics are affine. In addition, a new framework for DMD, that requires only densely defined Koopman operators over RKHSs is introduced, and its effectiveness is demonstrated through numerical examples.

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arXiv2022

Carleman Lifting for Nonlinear System Identification with Guaranteed Error Bounds

This paper concerns identification of uncontrolled or closed loop nonlinear systems using a set of trajectories that are generated by the system in a domain of attraction. The objective is to ensure that the trajectories of the identified systems are close to the trajectories of the real system, as quantified by an error bound that is prescribed a priori. A majority of existing methods for nonlinear system identification rely on techniques such as neural networks, autoregressive moving averages, and spectral decomposition that do not provide systematic approaches to meet pre-defined error bounds. The developed method is based on Carleman linearization-based lifting of the nonlinear system to an infinite dimensional linear system. The linear system is then truncated to a suitable order, computed based on the prescribed error bound, and parameters of the truncated linear system are estimated from data. The effectiveness of the technique is demonstrated by identifying an approximation of the Van der Pol oscillator from data within a prescribed error bound.

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arXiv2022

Occupation Kernel Hilbert Spaces for Fractional Order Liouville Operators and Dynamic Mode Decomposition

This manuscript gives a theoretical framework for a new Hilbert space of functions, the so called occupation kernel Hilbert space (OKHS), that operate on collections of signals rather than real or complex numbers. To support this new definition, an explicit class of OKHSs is given through the consideration of a reproducing kernel Hilbert space (RKHS). This space enables the definition of nonlocal operators, such as fractional order Liouville operators, as well as spectral decomposition methods for corresponding fractional order dynamical systems. In this manuscript, a fractional order DMD routine is presented, and the details of the finite rank representations are given. Significantly, despite the added theoretical content through the OKHS formulation, the resultant computations only differ slightly from that of occupation kernel DMD methods for integer order systems posed over RKHSs.

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arXiv2021

The Occupation Kernel Method for Nonlinear System Identification

This manuscript presents a novel approach to nonlinear system identification leveraging densely defined Liouville operators and a new "kernel" function that represents an integration functional over a reproducing kernel Hilbert space (RKHS) dubbed an occupation kernel. The manuscript thoroughly explores the concept of occupation kernels in the contexts of RKHSs of continuous functions, and establishes Liouville operators over RKHS where several dense domains are found for specific examples of this unbounded operator. The combination of these two concepts allow for the embedding of a dynamical system into a RKHS, where function theoretic tools may be leveraged for the examination of such systems. This framework allows for trajectories of a nonlinear dynamical system to be treated as a fundamental unit of data for nonlinear system identification routine. The approach to nonlinear system identification is demonstrated to identify parameters of a dynamical system accurately, while also exhibiting a certain robustness to noise.

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arXiv2021

Singular Dynamic Mode Decompositions

This manuscript is aimed at addressing several long standing limitations of dynamic mode decompositions in the application of Koopman analysis. Principle among these limitations are the convergence of associated Dynamic Mode Decomposition algorithms and the existence of Koopman modes. To address these limitations, two major modifications are made, where Koopman operators are removed from the analysis in light of Liouville operators (known as Koopman generators in special cases), and these operators are shown to be compact for certain pairs of Hilbert spaces selected separately as the domain and range of the operator. While eigenfunctions are discarded in the general analysis, a viable reconstruction algorithm is still demonstrated, and the sacrifice of eigenfunctions realizes the theoretical goals of DMD analysis that have yet to be achieved in other contexts. However, in the case where the domain is embedded in the range, an eigenfunction approach is still achievable, where a more typical DMD routine is established, but that leverages a finite rank representation that converges in norm. The manuscript concludes with the description of two Dynamic Mode Decomposition algorithms that converges when a dense collection of occupation kernels, arising from the data, are leveraged in the analysis.

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arXiv2021

An occupation kernel approach to optimal control

In this effort, a novel operator theoretic framework is developed for data-driven solution of optimal control problems. The developed methods focus on the use of trajectories (i.e., time-series) as the fundamental unit of data for the resolution of optimal control problems in dynamical systems. Trajectory information in the dynamical systems is embedded in a reproducing kernel Hilbert space (RKHS) through what are called occupation kernels. The occupation kernels are tied to the dynamics of the system through the densely defined Liouville operator. The pairing of Liouville operators and occupation kernels allows for lifting of nonlinear finite-dimensional optimal control problems into the space of infinite-dimensional linear programs over RKHSs.

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arXiv2021

Theoretical Foundations for the Dynamic Mode Decomposition of High Order Dynamical Systems

Conventionally, data driven identification and control problems for higher order dynamical systems are solved by augmenting the system state by the derivatives of the output to formulate first order dynamical systems in higher dimensions. However, solution of the augmented problem typically requires knowledge of the full augmented state, which requires numerical differentiation of the original output, frequently resulting in noisy signals. This manuscript develops the theory necessary for a direct analysis of higher order dynamical systems using higher order Liouville operators. Fundamental to this theoretical development is the introduction of signal valued RKHSs and new operators posed over these spaces. Ultimately, it is observed that despite the added abstractions, the necessary computations are remarkably similar to that of first order DMD methods using occupation kernels.

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arXiv2021

Dynamic Mode Decomposition for Continuous Time Systems with the Liouville Operator

Dynamic Mode Decomposition (DMD) has become synonymous with the Koopman operator, where continuous time dynamics are examined through a discrete time proxy determined by a fixed timestep using Koopman (i.e. composition) operators. Using the newly introduced "occupation kernels," the present manuscript develops an approach to DMD that treats continuous time dynamics directly through the Liouville operator, which can include Koopman generators. This manuscript outlines the technical and theoretical differences between Koopman based DMD for discrete time systems and Liouville based DMD for continuous time systems, which includes an examination of these operators over several reproducing kernel Hilbert spaces (RKHSs). While Liouville operators are modally unbounded, this manuscript introduces the concept of a scaled Liouville operator, which for many dynamical systems yields a compact operator over the exponential dot product kernel's native space. Hence, norm convergence of the DMD procedure is established when using scaled Liouville operators, which is a decided advantage over Koopman based DMD methods.

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arXiv2021

Liouville Operators over the Hardy Space

The role of Liouville operators in the study of dynamical systems through the use of occupation measures have been an active area of research in control theory over the past decade. This manuscript investigates Liouville operators over the Hardy space, which encode complex ordinary differential equations in an operator over a reproducing kernel Hilbert space.

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arXiv2021

Motion Tomography via Occupation Kernels

The goal of motion tomography is to recover a description of a vector flow field using information on the trajectory of a sensing unit. In this paper, we develop a predictor corrector algorithm designed to recover vector flow fields from trajectory data with the use of occupation kernels developed by Rosenfeld et al.. Specifically, we use the occupation kernels as an adaptive basis; that is, the trajectories defining our occupation kernels are iteratively updated to improve the estimation on the next stage. Initial estimates are established, then under mild assumptions, such as relatively straight trajectories, convergence is proven using the Contraction Mapping Theorem. We then compare to the established method by Chang et al. by defining a set of error metrics. We found that for simulated data, which provides a ground truth, our method offers a marked improvement and that for a real-world example we have similar results to the established method.

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arXiv2020

The Gradient descent method from the perspective of fractional calculus

Motivated by gradient methods in optimization theory, we give methods based on $ψ$-fractional derivatives of order $α$ in order to solve unconstrained optimization problems. The convergence of these methods is analyzed in detail. This paper also presents an Adams-Bashforth-Moulton (ABM) method for the estimation of solutions to equations involving $ψ$-fractional derivatives. Numerical examples using the ABM method show that the fractional order $α$ and weight $ψ$ are tunable parameters, which can be helpful for improving the performance of gradient descent methods.

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arXiv2018

Reachable Set Estimation and Safety Verification for Piecewise Linear Systems with Neural Network Controllers

In this work, the reachable set estimation and safety verification problems for a class of piecewise linear systems equipped with neural network controllers are addressed. The neural network is considered to consist of Rectified Linear Unit (ReLU) activation functions. A layer-by-layer approach is developed for the output reachable set computation of ReLU neural networks. The computation is formulated in the form of a set of manipulations for a union of polytopes. Based on the output reachable set for neural network controllers, the output reachable set for a piecewise linear feedback control system can be estimated iteratively for a given finite-time interval. With the estimated output reachable set, the safety verification for piecewise linear systems with neural network controllers can be performed by checking the existence of intersections of unsafe regions and output reach set. A numerical example is presented to illustrate the effectiveness of our approach.

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