On Gaussian Process Based Koopman Operators

Lian, Yingzhao; Jones, Colin N.

doi:10.1016/j.ifacol.2020.12.217

Cited by 8 publications

(5 citation statements)

References 19 publications

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“…The second most common research area found applicable to this category were studies pertaining to motor control, which is especially relevant to UAVs and robots, but potentially also to other types of vehicles when examining them from a subsystem perspective. This was achieved in [93] using Gaussian process-based Koopman operator in robust controller design, and in [46] using EDMD for current control for the synchronous operation of motors. Finally, a power management study used Stochastic Adversarial Koopman Operator with Auxillary Neural Network for the quick learning of reduced order models that measure the state of charge of Lithium-ion batteries.…”

Section: A General Studies Applicable To Vehiclesmentioning

confidence: 99%

Vehicular Applications of Koopman Operator Theory—A Survey

2023

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Koopman operator theory has proven to be a promising approach to nonlinear system identification and global linearization. For nearly a century, there had been no efficient means of calculating the Koopman operator for applied engineering purposes. The introduction of a recent computationally efficient method in the context of fluid dynamics, which is based on the system dynamics decomposition to a set of normal modes in descending order, has overcome this long-lasting computational obstacle. The purely data-driven nature of Koopman operators holds the promise of capturing unknown and complex dynamics for reduced-order model generation and system identification, through which the rich machinery of linear control techniques can be utilized. Given the ongoing development of this research area and the many existing open problems in the fields of smart mobility and vehicle engineering, a survey of techniques and open challenges of applying Koopman operator theory to this vibrant area is warranted. This review focuses on the various solutions of the Koopman operator which have emerged in recent years, particularly those focusing on mobility applications, ranging from characterization and component-level control operations to vehicle performance and fleet management. Moreover, this comprehensive review of over 100 research papers highlights the breadth of ways Koopman operator theory has been applied to various vehicular applications with a detailed categorization of the applied Koopman operator-based algorithm type. Furthermore, this review paper discusses theoretical aspects of Koopman operator theory that have been largely neglected by the smart mobility and vehicle engineering community and yet have large potential for contributing to solving open problems in these areas.

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Section: A General Studies Applicable To Vehiclesmentioning

confidence: 99%

Vehicular Applications of Koopman Operator Theory—A Survey

2023

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“…However, they are geared towards approximating the spectrum of the operator rather than learning LTI predictors. Other approaches use various different learning architectures [34,35], considering joint learning of feature maps and operator approximations, commonly relaxing (KI) to an additional penalty in the objective (OR). Such approaches are dominated by autoencoder architectures [36][37][38][39] that are often lower-dimensional but introduce a nonlinear feature-to-output map in place of (2b).…”

Section: Related Workmentioning

confidence: 99%

“…For example, computing Fourier averages [59], we can determine the phases ω j of complex-conjugate pairs µ j,± = |µ j | e ±iωj and sample the modulus from a predefined distribution μj,± ∼ p(B 1 (0)). Alternative options could include using PCA of the data to extract the dominant spectrum [60], or using subspace identification techniques to extract the most observable spectral components [35]. Remark 2.…”

Section: Selecting Eigenvaluesmentioning

confidence: 99%

Diffeomorphically Learning Stable Koopman Operators

Bevanda

Beier

Sebastian

et al. 2022

IEEE Control Syst. Lett.

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Many machine learning approaches for decision making, such as reinforcement learning, rely on simulators or predictive models to forecast the time-evolution of quantities of interest, e.g., the state of an agent or the reward of a policy. Forecasts of such complex phenomena are commonly described by highly nonlinear dynamical systems, making their use in optimization-based decision-making challenging. Koopman operator theory offers a beneficial paradigm for addressing this problem by characterizing forecasts via linear dynamical systems. This makes system analysis and long-term predictions simple -involving only matrix multiplications. However, the transformation to a linear system is generally non-trivial and unknown, requiring learning-based approaches. While there exists a variety of approaches, they usually lack crucial learning-theoretic guarantees, such that the behavior of the obtained models with increasing data and dimensionality is often unclear. We address the aforementioned by deriving a novel reproducing kernel Hilbert space (RKHS) that solely spans transformations into linear dynamical systems. The resulting Koopman Kernel Regression (KKR) framework enables the use of statistical learning tools from function approximation for novel convergence results and generalization risk bounds under weaker assumptions than existing work. Our numerical experiments indicate advantages over state-of-the-art statistical learning approaches for Koopman-based predictors.

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“…The uncertain terms are f `= 37.5 Nm s rad and f c = 30 Nm s rad . We consider a tracking problem, which translates in a quadratic cost function with `t(x t , u t ) = kx t x ref,t k (10,1), Q f = diag(10, 100) and R = 10 3 . We set the sampling period to = 10 3 s 1 .…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…recognized in the field of controls for its flexibility in modelling complex unknown dynamics [9]. GPs have also been recently exploited in the quantification of model uncertainty [10]. In the field of optimal control, GPs have been exploited as a valid alternative to the prominent approach of modeling uncertainty as a stochastic disturbance.…”

Section: Introductionmentioning

confidence: 99%

Learning-driven Nonlinear Optimal Control via Gaussian Process Regression

Sforni

Notarnicola

Notarstefano

2021

2021 60th IEEE Conference on Decision and Control (CDC)

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In this paper we propose a novel numerical strategy to solve nonlinear optimal control problems for dynamical systems with partially unknown dynamics. The goal is to explore feasible system trajectories minimizing a given finite-time performance criterion. We suppose to be able to actuate an input sequence on the real system, but only an inaccurate description of the dynamics is available for the control design. The proposed learning-driven optimal control strategy combines a trajectory optimization procedure with a Gaussian process regression to iteratively enrich the model and perform the optimization steps. Thanks to this combined scheme, the strategy is able to explore the trajectory manifold while minimizing the cost function. To corroborate the theoretical results, numerical simulations on the optimal control of a pendulum are shown.

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On Gaussian Process Based Koopman Operators

Cited by 8 publications

References 19 publications

Vehicular Applications of Koopman Operator Theory—A Survey

Vehicular Applications of Koopman Operator Theory—A Survey

Diffeomorphically Learning Stable Koopman Operators

Learning-driven Nonlinear Optimal Control via Gaussian Process Regression

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