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The suspension system of the Electromagnetic Suspension (EMS) maglev train is crucial for ensuring safe operation. This article focuses on data-driven modeling and control optimization of the suspension system. By the Extended Dynamic Mode Decomposition (EDMD) method based on the Koopman theory, the state and input data of the suspension system are collected to construct a high-dimensional linearized model of the system without detailed parameters of the system, preserving the nonlinear characteristics. With the data-driven model, the LQR controller and Extended State Observer (ESO) are applied to optimize the suspension control. Compared with baseline feedback methods, the optimization control with data-driven modeling reduces the maximum system fluctuation by 75.0% in total. Furthermore, considering the high-speed operating environment and vertical dynamic response of the maglev train, a rolling-update modeling method is proposed to achieve online modeling optimization of the suspension system. The simulation results show that this method reduces the maximum fluctuation amplitude of the suspension system by 40.0% and the vibration acceleration of the vehicle body by 46.8%, achieving significant optimization of the suspension control. [ABSTRACT FROM AUTHOR]

In this paper, we aimed to identify the dynamics of a crude distillation unit (CDU) using closed-loop data with NARX−NN and the Koopman operator in both linear (KL) and bilinear (KB) forms. A comparative analysis was conducted to assess the performance of each method under different experimental conditions, such as the gain, a delay and time constant mismatch, tight constraints, nonlinearities, and poor tuning. Although NARX−NN showed good training performance with the lowest Mean Squared Error (MSE), the KB demonstrated better generalization and robustness, outperforming the other methods. The KL observed a significant decline in performance in the presence of nonlinearities in inputs, yet it remained competitive with the KB under other circumstances. The use of the bilinear form proved to be crucial, as it offered a more accurate representation of CDU dynamics, resulting in enhanced performance. [ABSTRACT FROM AUTHOR]

Nonlinearity presents a significant challenge in developing quantum algorithms involving differential equations, prompting the exploration of various linearization techniques, including the well-known Carleman Linearization. Instead, this paper introduces the Koopman Spectral Linearization method tailored for nonlinear autonomous ordinary differential equations. This innovative linearization approach harnesses the interpolation methods and the Koopman Operator Theory to yield a lifted linear system. It promises to serve as an alternative approach that can be employed in scenarios where Carleman Linearization is traditionally applied. Numerical experiments demonstrate the effectiveness of this linearization approach for several commonly used nonlinear ordinary differential equations. [ABSTRACT FROM AUTHOR]

Spatio-temporal dynamic mode decomposition (STDMD) is an extension of dynamic mode decomposition (DMD) designed to handle spatio-temporal datasets. It extends the framework so that it can analyze data that have both spatial and temporal variations. This facilitates the extraction of spatial structures along with their temporal evolution. The STDMD method extracts temporal and spatial development information simultaneously, including wavenumber, frequencies, and growth rates, which are essential in complex dynamic systems. We provide a comprehensive mathematical framework for sequential and parallel STDMD approaches. To increase the range of applications of the presented techniques, we also introduce a generalization of delay coordinates. The extension, labeled delay-embedding STDMD allows the use of delayed data, which can be both time-delayed and space-delayed. An explicit expression of the presented algorithms in matrix form is also provided, making theoretical analysis easier and providing a solid foundation for further research and development. The novel approach is demonstrated using some illustrative model dynamics. [ABSTRACT FROM AUTHOR]

For positioning and anti-swing control of bridge cranes, the active learning control method can reduce the dependence of controller design on the model and the influence of unmodeled dynamics on the controller's performance. By only using the real-time online input and output data of the bridge crane system, the active learning control method consists of the finite-dimensional approximation of the Koopman operator and the design of an active learning controller based on the linear quadratic optimal tracking control. The effectiveness of the control strategy for positioning and anti-swing of bridge cranes is verified through numerical simulations. [ABSTRACT FROM AUTHOR]

Dynamic Mode Decomposition with Control is a powerful technique for analyzing and modeling complex dynamical systems under the influence of external control inputs. In this paper, we propose a novel approach to implement this technique that offers computational advantages over the existing method. The proposed scheme uses singular value decomposition of a lower order matrix and requires fewer matrix multiplications when determining corresponding approximation matrices. Moreover, the matrix of dynamic modes also has a simpler structure than the corresponding matrix in the standard approach. To demonstrate the efficacy of the proposed implementation, we applied it to a diverse set of numerical examples. The algorithm's flexibility is demonstrated in tests: accurate modeling of ecological systems like Lotka-Volterra, successful control of chaotic behavior in the Lorenz system and efficient handling of large-scale stable linear systems. This showcased its versatility and efficacy across different dynamical systems. [ABSTRACT FROM AUTHOR]

We present novel methods for computing the online dynamic mode decomposition (online DMD) for streaming datasets. We propose a framework that allows incremental updates to the DMD operator as data become available. Due to its ability to work on datasets with lower ranks, the proposed method is more advantageous than existing ones. A noteworthy feature of the method is that it is entirely data-driven and does not require knowledge of any underlying governing equations. Additionally, we present a modified version of our proposed approach that utilizes a weighted alternative to online DMD. The suggested techniques are demonstrated using several numerical examples. [ABSTRACT FROM AUTHOR]

The accurate simulation of the dynamics of the anaerobic–anoxic–oxic (A2O) process in the biochemical reactions in wastewater treatment plants (WWTPs) is important for system prediction and optimization. Previous studies have used real-time monitoring data of WWTPs to develop data-driven predictive models, but these models cannot be used to provide mathematical analysis of A2O dynamic properties. In this study, we developed a new simulation and analysis method for determining A2O dynamics in biochemical reactions using deep learning and the Koopman operator to address the above problems. This method was validated through data from a real-world WWTP in east China and compared it with the traditional deep learning model. According to the results, the new method achieved high-accuracy prediction. Meanwhile, with the help of the Koopman operator, the new method was able to analyze the asymptotical stability and convergence behavior of the A2O process, which provides a brand-new perspective for the in-depth study of biochemical reactor dynamics. [ABSTRACT FROM AUTHOR]

This paper provides the theoretical foundation for the approximation of the regions of attraction in hyperbolic and polynomial systems based on the eigenfunctions deduced from the data-driven approximation of the Koopman operator. In addition, it shows that the same method is suitable for analyzing higher-dimensional systems in which the state space dimension is greater than three. The approximation of the Koopman operator is based on extended dynamic mode decomposition, and the method relies solely on this approximation to find and analyze the system's fixed points. In other words, knowledge of the model differential equations or their linearization is not necessary for this analysis. The reliability of this approach is demonstrated through two examples of dynamical systems, e.g., a population model in which the theoretical boundary is known, and a higher-dimensional chemical reaction system constituting an original result. [ABSTRACT FROM AUTHOR]

Since conventional eco approach systems can only achieve longitudinal automation, they may be disabled due to the impedance of the slow-moving vehicle. In addition, they could sacrifice a lot in travel time and it is hard to control the articulated truck with a complex dynamic model. This research presents an enhanced eco approach system for connected and automated articulated trucks, and is able to: (i) overtake slow-moving vehicles for sustainability and mobility; (ii) efficiently optimize the travel duration approaching a signalized intersection; (iii) achieve the trade-off between fuel saving and vehicle mobility; and (iv) improve computational efficiency and optimality for the articulated truck control. To achieve these features, the problem was formulated as an optimal control problem. A longitudinal and lateral coupled truck dynamic model was utilized for enabling the truck to own the automatic overtaking capability. The data-driven-based Koopman operator theory was adopted to globally linearize the truck dynamic model for reducing the computational burden while ensuring optimality. The optimal control problem is transformed from the time domain to the space domain in order for optimizing travel duration and considering the signal timing constraint. A quantitative evaluation was conducted to validate the performance of the Koopman system dynamics. In addition, the simulation experiment was designed to compare the proposed controller against human drivers and the conventional eco approach, which only has longitudinal automation. The results demonstrate that the proposed controller improves the fuel efficiency by 5.12–67.15%, and outperforms the two baseline controllers by 9.09–32.65% in terms of fuel saving. This range is caused by the different arrival times of the ego articulated truck. [ABSTRACT FROM AUTHOR]

Dynamic mode decomposition (DMD) is a data-driven, modal decomposition technique that describes spatiotemporal features of high-dimensional dynamic data. The method is equation-free in the sense that it does not require knowledge of the underlying governing equations. The main purpose of this article is to introduce new alternatives to the currently accepted algorithm for calculating the dynamic mode decomposition. We present two new algorithms which are more economical from a computational point of view, which is an advantage when working with large data. With a few illustrative examples, we demonstrate the applicability of the introduced algorithms. [ABSTRACT FROM AUTHOR]

In this paper, an efficient model predictive control (MPC) of velocity tracking of automated vehicles is proposed, in which a reference signal is given a priori. Five degree-of-freedom vehicle dynamics with nonlinear tires is chosen as the prediction model, in which coupling characteristics of longitudinal and lateral dynamics are taken into account. In order to balance computational burden and prediction accuracy, Koopman operator theory is adopted to transform the nonlinear model into a global linear model. Then, the global linear model is used in the design of MPC to reduce online computational burden and avoid solving nonconvex/nonlinear optimization problems. Furthermore, the effectiveness of Koopman operator in vehicle dynamics control is verified using a Matlab/Simulink environment. Validation results demonstrate that dynamic mode decomposition with control (DMDc) and extended dynamic mode decomposition (EDMD) algorithms are more accurate in model validation and dynamic prediction than local linearization, and DMDc algorithm has less computational burden on solving optimization problems than the EDMD algorithm. [ABSTRACT FROM AUTHOR]

A quantum model of a free-electron laser (FEL) is considered. Two different approaches for the exploration of the the FEL system are considered. In the first case, the Heisenberg equations of motion are mapped on the basis of the initial wave functions, which consists of the photon coherent states and many-dimensional electron coherent states. This mapping is an exact procedure, which makes it possible to obtain an exact equation of motion for the intensity of the laser field in a closed form. The obtained equation is controlled by a Koopman operator. The analytical expression for the evolution of the FEL intensity is obtained in the framework of a perturbation theory, which is constructed for a small time scale. The second way of the consideration is based on the construction of the many-dimensional path integrals for the evolution of the wave function. This method also makes it possible to estimate the time evolution and the gain of the FEL intensity. [ABSTRACT FROM AUTHOR]

Extended Dynamic Mode Decomposition (EDMD) allows an approximation of the Koopman operator to be derived in the form of a truncated (finite dimensional) linear operator in a lifted space of (nonlinear) observable functions. EDMD can operate in a purely data-driven way using either data generated by a numerical simulator of arbitrary complexity or actual experimental data. An important question at this stage is the selection of basis functions to construct the observable functions, which in turn is determinant of the sparsity and efficiency of the approximation. In this study, attention is focused on orthogonal polynomial expansions and an order-reduction procedure called p-q quasi-norm reduction. The objective of this article is to present a Matlab library to automate the computation of the EDMD based on the above-mentioned tools and to illustrate the performance of this library with a few representative examples. [ABSTRACT FROM AUTHOR]

Interconnected systems are widespread in modern technological systems. Designing a reliable control strategy requires modeling and analysis of the system, which can be a complicated, or even impossible, task in some cases. However, current technological developments in data sensing, processing, and storage make data-driven control techniques an appealing alternative solution. In this work, a design methodology of a decentralized control strategy is developed for interconnected systems based only on local and interconnection time series. Then, the optimization problem associated with the predictive control design is defined. Finally, an extension to interconnected systems coupled through their input signals is discussed. Simulations of two coupled Duffing oscillators, a bipedal locomotion model, and a four water tank system show the effectiveness of the approach. [ABSTRACT FROM AUTHOR]

We study numerical approaches to computation of spectral properties of composition operators. We provide a characterization of Koopman Modes in Banach spaces using Generalized Laplace Analysis. We cast the Dynamic Mode Decomposition-type methods in the context of Finite Section theory of infinite dimensional operators, and provide an example of a mixing map for which the finite section method fails. Under assumptions on the underlying dynamics, we provide the first result on the convergence rate under sample size increase in the finite-section approximation. We study the error in the Krylov subspace version of the finite section method and prove convergence in pseudospectral sense for operators with pure point spectrum. Since Krylov sequence-based approximations can mitigate the curse of dimensionality, this result indicates that they may also have low spectral error without an exponential-in-dimension increase in the number of functions needed. [ABSTRACT FROM AUTHOR]

Using both analytical and numerical techniques, we discuss wave solutions within the framework of an extended nonlinear Schrödinger equation with constant coefficients equipped with spatiotemporal dispersion, self-steepening effects, and a Raman scattering term. We present the exact traveling wave solution of the system in terms of Jacobi elliptic functions and mention some symmetry results as they relate to the resulting ordinary differential equation. A constructed bright soliton solution serves as the base to compare a numerical solution of the system using spectral Fourier methods with a precise statistical low-rank approximation using a data-driven approach aided by the Koopman operator theory. We found that the spatiotemporal feature added to the model serves as a regularizing tool that enables a precise reconstruction of the original solution. [ABSTRACT FROM AUTHOR]

There are multiple ways in which a stochastic system can be out of statistical equilibrium. It might be subject to time-varying forcing; or be in a transient phase on its way towards equilibrium; it might even be in equilibrium without us noticing it, due to insufficient observations; and it even might be a system failing to admit an equilibrium distribution at all. We review some of the approaches that model the effective statistical behavior of equilibrium and non-equilibrium dynamical systems, and show that both cases can be considered under the unified framework of optimal low-rank approximation of so-called transfer operators. Particular attention is given to the connection between these methods, Markov state models, and the concept of metastability, further to the estimation of such reduced order models from finite simulation data. All these topics bear an important role in, e.g., molecular dynamics, where Markov state models are often and successfully utilized, and which is the main motivating application in this paper. We illustrate our considerations by numerical examples. [ABSTRACT FROM AUTHOR]

A data-driven analysis method known as dynamic mode decomposition (DMD) approximates the linear Koopman operator on a projected space. In the spirit of Johnson–Lindenstrauss lemma, we will use a random projection to estimate the DMD modes in a reduced dimensional space. In practical applications, snapshots are in a high-dimensional observable space and the DMD operator matrix is massive. Hence, computing DMD with the full spectrum is expensive, so our main computational goal is to estimate the eigenvalue and eigenvectors of the DMD operator in a projected domain. We generalize the current algorithm to estimate a projected DMD operator. We focus on a powerful and simple random projection algorithm that will reduce the computational and storage costs. While, clearly, a random projection simplifies the algorithmic complexity of a detailed optimal projection, as we will show, the results can generally be excellent, nonetheless, and the quality could be understood through a well-developed theory of random projections. We will demonstrate that modes could be calculated for a low cost by the projected data with sufficient dimension. [ABSTRACT FROM AUTHOR]

We consider the Koopman operator theory in the context of nonlinear infinite-dimensional systems, where the operator is defined over a space of bounded continuous functionals. The properties of the Koopman semigroup are described and a finite-dimensional projection of the semigroup is proposed, which provides a linear finite-dimensional approximation of the underlying infinite-dimensional dynamics. This approximation is used to obtain spectral properties from the data, a method which can be seen as a generalization of the Extended Dynamic Mode Decomposition for infinite-dimensional systems. Finally, we exploit the proposed framework to identify (a finite-dimensional approximation of) the Lie generator associated with the Koopman semigroup. This approach yields a linear method for nonlinear PDE identification, which is complemented with theoretical convergence results. [ABSTRACT FROM AUTHOR]

We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems. Unlike Lyapunov-based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently using the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory can be understood through the lens of Perron–Frobenius and Koopman operators. This allows us to use data-driven methods to approximate these operators and combine them with the SOS techniques to establish a convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples. [ABSTRACT FROM AUTHOR]

Inferring the latent structure of complex nonlinear dynamical systems in a data driven setting is a challenging mathematical problem with an ever increasing spectrum of applications in sciences and engineering. Koopman operator-based linearization provides a powerful framework that is suitable for identification of nonlinear systems in various scenarios. A recently proposed method by Mauroy and Goncalves is based on lifting the data snapshots into a suitable finite dimensional function space and identification of the infinitesimal generator of the Koopman semigroup. This elegant and mathematically appealing approach has good analytical (convergence) properties, but numerical experiments show that software implementation of the method has certain limitations. More precisely, with the increased dimension that guarantees theoretically better approximation and ultimate convergence, the numerical implementation may become unstable and it may even break down. The main sources of numerical difficulties are the computations of the matrix representation of the compressed Koopman operator and its logarithm. This paper addresses the subtle numerical details and proposes a new implementation algorithm that alleviates these problems. [ABSTRACT FROM AUTHOR]

The extended dynamic mode decomposition algorithm is a tool for accurately approximating the point spectrum of the Koopman operator. This algorithm provides an approximate linear expansion of non-linear discrete-time systems, which can be useful for system analysis and controller design. The accuracy of this algorithm depends heavily on the availability of a set of basis functions that provide the ability to capture the nonlinear dynamics of the underlying system. Recently, the use of orthogonal polynomials, along with reduction techniques for the dimension and maximum order of the polynomial basis, have been successfully used to approximate nonlinear systems with the additional benefit of using smaller datasets. This paper expands the current methods for selecting the set of observables for nonlinear systems with periodic behavior, which is prone to a representation in terms of trigonometric functions. The benefit of working with orthogonal polynomials is preserved by embedding the trigonometric functions into the orthogonal basis. The algorithm is illustrated with the data-driven modelling of an inverted pendulum in simulation and real-life experiments. [ABSTRACT FROM AUTHOR]

In this paper, we address the data-driven modeling of a nonlinear dynamical system while incorporating a priori information. The nonlinear system is described using the Koopman operator, which is a linear operator defined on a lifted infinite-dimensional state-space. Assuming that the L 2 gain of the system is known, the data-driven finite-dimensional approximation of the operator while preserving information about the gain, namely L 2 gain-preserving data-driven modeling, is formulated. Then, its computationally efficient solution method is presented. An application of the modeling method to feedback controller design is also presented. Aiming for robust stabilization using data-driven control under a poor training dataset, we address the following two modeling problems: (1) Forward modeling: the data-driven modeling is applied to the operating data of a plant system to derive the plant model; (2) Backward modeling: L 2 gain-preserving data-driven modeling is applied to the same data to derive an inverse model of the plant system. Then, a feedback controller composed of the plant and inverse models is created based on internal model control, and it robustly stabilizes the plant system. A design demonstration of the data-driven controller is provided using a numerical experiment. [ABSTRACT FROM AUTHOR]

Hierarchical support vector regression (HSVR) models a function from data as a linear combination of SVR models at a range of scales, starting at a coarse scale and moving to finer scales as the hierarchy continues. In the original formulation of HSVR, there were no rules for choosing the depth of the model. In this paper, we observe in a number of models a phase transition in the training error—the error remains relatively constant as layers are added, until a critical scale is passed, at which point the training error drops close to zero and remains nearly constant for added layers. We introduce a method to predict this critical scale a priori with the prediction based on the support of either a Fourier transform of the data or the Dynamic Mode Decomposition (DMD) spectrum. This allows us to determine the required number of layers prior to training any models. [ABSTRACT FROM AUTHOR]