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In this article, we propose a physics‐informed learning‐based Koopman modeling approach and present a Koopman‐based self‐tuning moving horizon estimation design for a class of nonlinear systems. Specifically, we train Koopman operators and two neural networks—the state lifting network and the noise characterization network—using both data and available physical information. The first network accounts for the nonlinear lifting functions for the Koopman model, while the second network characterizes the system noise distributions. Accordingly, a stochastic linear Koopman model is established in the lifted space to forecast the dynamic behaviors of the nonlinear system. Based on the Koopman model, a self‐tuning linear moving horizon estimation (MHE) scheme is developed. The weighting matrices of the MHE design are updated using the pretrained noise characterization network at each sampling instant. The proposed estimation scheme is computationally efficient, as only convex optimization needs to be solved during online implementation, and updating the weighting matrices of the MHE scheme does not require re‐training the neural networks. We verify the effectiveness and evaluate the performance of the proposed method via the application to a simulated chemical process. [ABSTRACT FROM AUTHOR]

This paper presents a data-driven Koopman modeling framework for globally linearizing highly nonlinear dynamical systems in lifted infinite-dimensional state space. In this framework, three data-driven models are proposed and identified to approximate the infinite-dimensional linear Koopman operator through a method called extended dynamic mode decomposition (EDMD). The implementation of EDMD requires a data set of snap pairs and a dictionary of scalar observables, which affects the accuracy of data-driven modeling. Five basis functions are compared and discussed to illustrate their suitable application scenarios. To verify the Koopman data-driven modeling framework, we apply it to the modeling of soft robotic systems, which is thought to be an extremely difficult task due to the large and continuous deformation of soft materials. Results demonstrate that Koopman linear, bilinear, and nonlinear models for both two-dimensional (2D) and three-dimensional (3D) soft robots are superior to the existing state-space modeling approach by achieving less normalized root mean square error (NRMSE). Among the three Koopman models, the constructed nonlinear model has higher performance than the bilinear model, followed by the linear one. Furthermore, the Monomial and Hermite basis functions are the optimal choices for constructing the Koopman linear, bilinear, and nonlinear models for the investigated 2D and 3D soft robots as they have the same structure when the degree of basis function is chosen less than four. Although the Fourier basis function is expected to outperform the Hermite and Gaussian basis functions in modeling systems with oscillatory motion, it is not a preferable selection in the high dimensional soft robotic modeling in our case due to its computational complexity and tendency to be non-convergent. The Sparse Fourier basis function cannot be regarded as a good choice for modeling soft robots, as it is only suitable for generating models with sparse data. It is worth noting that our findings can lay a solid foundation for the dynamics analysis and precise control of highly nonlinear dynamical systems, like soft robots in the future. [ABSTRACT FROM AUTHOR]

This paper proposes Koopman operator-based Stochastic Model Predictive Control (K-SMPC) for enhanced lateral control of autonomous vehicles. The Koopman operator is a linear map representing the nonlinear dynamics in an infinite-dimensional space. Thus, we use the Koopman operator to represent the nonlinear dynamics of a vehicle in dynamic lane-keeping situations. The Extended Dynamic Mode Decomposition (EDMD) method is adopted to approximate the Koopman operator in a finite-dimensional space for practical implementation. We consider the modeling error of the approximated Koopman operator in the EDMD method. Then, we design K-SMPC to tackle the Koopman modeling error, where the error is handled as a probabilistic signal. The recursive feasibility of the proposed method is investigated with an explicit first-step state constraint by computing the robust control invariant set. A high-fidelity vehicle simulator, i.e., CarSim, is used to validate the proposed method with a comparative study. From the results, it is confirmed that the proposed method outperforms other methods in tracking performance. Furthermore, it is observed that the proposed method satisfies the given constraints and is recursively feasible.

The analysis of a timeseries can provide many new perspectives if it is accompanied by the assumption that the timeseries is generated from an underlying dynamical system. For example, statistical properties of the data can be related to measure theoretic aspects of the dynamics, and one can try to recreate the dynamics itself. The underlying dynamics could represent a natural phenomenon or a physical system, where the timeseries represents a sequence of measurements. In this paper, we present a completely data-driven framework to identify and model quasiperiodically driven dynamical systems (Q.P.D.) from the timeseries it generates. Q.P.D. are a special class of systems that are driven by a periodic source with multiple base frequencies. Such systems abound in nature, e.g., astronomy and traffic flow. Our framework reconstructs the dynamics into two components - the driving quasiperiodic source with generating frequencies; and the driven nonlinear dynamics. We make a combined use of a kernel-based harmonic analysis, kernel-based interpolation technique, and Koopman operator theory. Our framework provides accurate reconstructions and frequency identification for three real-world case studies. [ABSTRACT FROM AUTHOR]

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 the emergency rescue and disposal of social public emergencies, supply transportation effectively provides a strong supply foundation and realistic conditions. The trajectory tracking control of emergency supplies transportation robot is the key technology to ensure the timeliness of transportation. In this article, the emergency supplies transportation robot is taken as the research object, based on Koopman operator theory, combined with radial basis function (RBF) neural network disturbance observer and adaptive prediction horizon event‐triggered model predictive control (APET‐MPC) algorithm to investigate the purely data‐driven trajectory tracking control problem of emergency supplies transportation robot when the model parameters and models are unknown. Firstly, the Koopman operator is used to establish a high‐dimensional linear model of the robot. Secondly, the RBF neural network disturbance observer is designed to estimate the disturbance during the robot operation and compensate it to the controller. Thirdly, APET‐MPC is used to optimize the trajectory tracking control of the emergency supplies transportation robot to reduce computational complexity. Finally, the performance of the proposed trajectory tracking controller is verified by Carsim/ Simulink joint simulation. The simulation results show that the model established by Koopman operator theory can achieve the high accuracy approximation of the robot. Compared with the MPC trajectory tracking controller, the APET‐MPC trajectory tracking controller based on RBF neural network disturbance observer (RBF‐APET‐MPC) improves the tracking accuracy of the robot and reduces the total triggering times of the system by more than 50%. [ABSTRACT FROM AUTHOR]

The adaptive spectral Koopman (ASK) method was introduced to numerically solve autonomous dynamical systems that laid the foundation for numerous applications across different fields in science and engineering. Although ASK achieves high accuracy, it is computationally more expensive for multidimensional systems compared with conventional time integration schemes like Runge--Kutta. In this work, we combine the sparse grid and ASK to accelerate the computation for multidimensional systems. This sparse-grid-based ASK (SASK) method uses the Smolyak structure to construct multidimensional collocation points as well as associated polynomials that are used to approximate eigenfunctions of the Koopman operator of the system. In this way, the number of collocation points is reduced compared with using the tensor product rule. We demonstrate that SASK can be used to solve ordinary differential equations (ODEs) and partial differential equations (PDEs) based on their semidiscrete forms. Numerical experiments are illustrated to compare the performance of SASK and state-of-the-art ODE solvers. [ABSTRACT FROM AUTHOR]

Soft robots, as a new type of robots, are highly flexible and deformable, which can adapt to the needs of tasks in complex environments. However, due to the complex nonlinear behaviors of soft materials and the unpredictable motion of actuators, accurate modeling and development of suitable controllers for soft robots are currently the main challenges for real applications. In this paper, we propose and compare two different modeling approaches to estimating the shape deformation of a two-dimensional pneumatic soft robot (2D-PSR): an analytical model based on the motion and air pressure dynamics and a data-driven model using Koopman operator theory and finite-dimensional approximate realization of the extended dynamic mode decomposition algorithm. The Unscented Kalman Filter (UKF) is applied to the two models to estimate the system state and filter the noises from sensors, respectively. Subsequently, based on the established models, the linear quadratic regulator (LQR) is designed to realize the precise trajectory tracking control of the 2D-PSR under two typical input signals. Both simulation and experimental results show that the proposed LQR control schemes with UKF designed based on the analytical model (A-FL-UKF-LQR) and Koopman linear model (K-UKF-LQR) can achieve the expectations in terms of tracking accuracy and robustness, in which the K-UKF-LQR framework outperforms the A-FL-UKF-LQR to a certain degree. [ABSTRACT FROM AUTHOR]

This paper presents a data-driven fault detection and isolation (FDI) for a multirotor system using Koopman operator and Luenberger observer. Koopman operator is an infinite-dimensional linear operator that can transform nonlinear dynamical systems into linear ones. Using this transformation, our aim is to apply the linear fault detection method to the nonlinear system. Initially, a Koopman operator-based linear model is derived to represent the multirotor system, considering factors like non-diagonal inertial tensor, center of gravity variations, aerodynamic effects, and actuator dynamics. Various candidate lifting functions are evaluated for prediction performance and compared using the root mean square error to identify the most suitable one. Subsequently, a Koopman operator-based Luenberger observer is proposed using the lifted linear model to generate residuals for identifying faulty actuators. Simulation and experimental results demonstrate the effectiveness of the proposed observer in detecting actuator faults such as bias and loss of effectiveness, without the need for an explicitly defined fault dataset. [ABSTRACT FROM AUTHOR]

Accurate mechanical system models are crucial for safe and stable control. Unlike linear systems, Lagrangian systems are highly nonlinear and difficult to optimize because of their unknown system model. Recent research thus used deep neural networks to generate linear models of original systems by mapping nonlinear dynamic systems into a linear space with a Koopman observable function encoder. The controller then relies on the Koopman linear model. However, without physical information constraints, ensuring control consistency between the original nonlinear system and the Koopman system is tough, as the learning process of the Koopman observation function is unsupervised. This paper thus proposes a two-stage learning algorithm that uses structural subnetworks to build a physics-informed network topology to simultaneously learn the Koopman observable functions and the system energy representation. In the Koopman matrix learning session, a quadratic-constrained optimization problem is solved to ensure that the Koopman representation satisfies the energy difference matching hard constraint. The proposed energy-preserving deep Lagrangian Koopman (EPDLK) framework effectively represents the dynamics of the Lagrangian system while ensuring control consistency. The effectiveness of EPDLK is compared with those of various Koopman observable function construction methods in multistep prediction and trajectory tracking tasks. EPDLK achieves better control consistency by guaranteeing energy difference matching, which facilitates the application of the control law generated on the Koopman system directly to the original nonlinear Lagrangian system. [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]

The data-driven approaches based on the Koopman operator theory have promoted the analysis and control of the nonlinear dynamics by providing an equivalent Koopman-based linear system associated with nonlinear systems. To facilitate the use of the Koopman framework for nonlinear systems with control inputs and to improve the prediction accuracy of the Koopman approximation, this work proposes a deep learning-based bilinear Koopman modeling framework. In this framework, we first deploy a deep neural network structure consisting of a lifting network, a control network, a linear layer, and a recovery network to fulfill the identification of the bilinear Koopman realization. During the neural network training process, the model uncertainty naturally arises from the data-driven setting variation. Then, to represent the impact of this implicit uncertainty, we integrate a variable parameter into the output of the control network to identify a relatively accurate model, thereby enhancing the prediction ability of the learned model. The non-convex property caused by the bilinear term is resolved using a linear approximation. After that, we apply a Koopman-based model predictive control scheme to the identified bilinear model with the parameter estimation to realize the control of the nonlinear dynamical system. [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]

This work describes a new model structure developed for prediction in Model Predictive Control (MPC). The model has a multi-model structure in which independent sub-models are employed for the consecutive sampling instants. The model lifts process states into a high-dimensional space in which a linear process description is applied. Depending on the influence of the manipulated variables on lifted states, three general model versions are described and model identification algorithms are derived. As a result of the multi-model structure, model parameters are found analytically from computationally uncomplicated least squares problems using the Extended Dynamic Mode Decomposition algorithm, but the evolution of states over the horizon used in MPC is taken into account. Next, the MPC algorithm for the described model is derived. It requires solving online simple quadratic optimisation tasks. The effectiveness of three considered model configurations and three versions of the lifting functions is examined for a nonlinear DC motor benchmark. Their impact on model accuracy, complexity, possible control accuracy and MPC calculation time is thoroughly discussed. Finally, a more complex polymerisation reactor process is considered to showcase the practical applicability of the presented approach to modelling and MPC. [ABSTRACT FROM AUTHOR]

Abstract This paper introduces a novel data‐driven approach to develop a fault‐tolerant model predictive controller (MPC) for non‐linear systems. By adopting a Koopman operator‐theoretic perspective, the proposed method leverages historical data from the system to construct a data‐driven model that captures the non‐linear behaviour and fault characteristics. The fault influence is addressed through an online estimation of a time‐varying Koopman predictor, which allows for adjusting the MPC control law to counteract the fault effects. This estimation is performed in a higher dimensional Koopman feature space, where the dynamics behave linearly. As a result, the non‐linear fault‐tolerant MPC optimization problem can be replaced with a more practical and feasible linear time‐varying one using the approximated Koopman predictor. Moreover, by incorporating the online update procedure, the time‐varying Koopman predictor can represent the dynamics of the faulty system. Hence, the controller can adapt and compensate for the faults in real‐time, integrating the fault diagnosis module in the MPC framework and eliminating the need for a separate fault detection unit. Finally, the efficacy of the proposed approach is demonstrated through case study results, which highlight the ability of the controller to mitigate faults and maintain desired system behaviour.

Abstract Future state prediction for nonlinear dynamical systems is a challenging task. Classical prediction theory is based on a, typically long, sequence of prior observations and is rooted in assumptions on statistical stationarity of the underlying stochastic process. These algorithms have trouble predicting chaotic dynamics, “Black Swans” (events which have never previously been seen in the observed data), or systems where the underlying driving process fundamentally changes. In this paper we develop (1) a global and local prediction algorithm that can handle these types of systems, (2) a method of switching between local and global prediction, and (3) a retouching method that tracks what predictions would have been if the underlying dynamics had not changed and uses these predictions when the underlying process reverts back to the original dynamics. The methodology is rooted in Koopman operator theory from dynamical systems. An advantage is that it is model-free, purely data-driven and adapts organically to changes in the system. While we showcase the algorithms on predicting the number of infected cases for COVID-19 and influenza cases, we emphasize that this is a general prediction methodology that has applications far outside of epidemiology.

Modeling of structural nonlinear dynamic behavior is a central challenge in civil and mechanical engineering communities. The phase space embedding of response time series has been demonstrated to be an efficient coordinate basis for data-driven approximation of the modern Koopman operator, which can fully capture the global evolution of nonlinear dynamics by a linear representation. This study demonstrates that linear and nonlinear structural dynamic vibrations can be represented by a universal forced linear model in a finite dimension space projected by time-delay coordinates. Compared with the existing methods, the proposed approach improves the performance of finite linear representation of nonlinear structural dynamics on two essential issues including the robustness to measurement noise and applicability to multidegree-of-freedom (MDOF) systems. For linear structures, the dynamic mode shapes and the corresponding natural frequencies can be accurately identified by using the time-delay dynamic mode decomposition (DMD) algorithm with acceleration response data experimentally measured from an 8-story shear-type linear steel frame. Modal parameters extracted from the time-delay DMD matched well with those identified from traditional modal identification methods, such as frequency domain decomposition (FDD) and complex mode indicator function (CMIF). In addition, numerical and experimental studies on nonlinear structures are conducted to demonstrate that the finite-dimensional DMD based on the discrete Hankel singular value decomposition (SVD) coordinate is highly symmetrically structured, and is able to accurately obtain a linear representation of structural nonlinear vibration. The resulting linearized data-driven equation-free model can be used to accurately predict the responses of nonlinear systems with limited training data sets. [ABSTRACT FROM AUTHOR]

This paper introduces a kernel-based approach for constructing approximations of the Koopman operators for semiflows in discrete time and orbital Koopman operators for continuous time semiflows. The primary advantage of the proposed construction is that the approximations follow certain rates of convergence which are dependent on how data samples fill certain subsets of the state space. In particular, we derive the rate of convergence for two scenarios: (1) the data samples Ξ are dense in a compact state space X, and (2) the data samples Ξ are dense in a limiting set Ω contained in an Euclidean space. Two general classes of Koopman operator approximations are considered in this paper, referred to as projection-based approximation and data-driven approximation. Projection-based approximations assume that the underlying dynamics governing the discrete or continuous time semiflows is known. On the other hand, data-driven approximations rely samples of the semiflow states to approximate the Koopman operator. In both types of approximations, the regularity of the underlying set and the smoothness of the space of functions on which the Koopman operator acts determine the rates of approximations. In the strongest error bounds derived in the paper, it is shown that the error in approximation of the Koopman operator decays like O (h Ω n , Ω p) , where h Ω n , Ω is the fill rate of the samples Ω n in the limiting set Ω and p is an exponent related to the choice of the kernel and the smoothness of functions on which the Koopman operator acts. Such error bounds are obtained when either the limiting subset Ω = X , when it is a proper subset Ω ⊂ X that is sufficiently regular, or when it is a type of smooth manifold Ω = M . [ABSTRACT FROM AUTHOR]

This paper introduces a novel data‐driven approach to develop a fault‐tolerant model predictive controller (MPC) for non‐linear systems. By adopting a Koopman operator‐theoretic perspective, the proposed method leverages historical data from the system to construct a data‐driven model that captures the non‐linear behaviour and fault characteristics. The fault influence is addressed through an online estimation of a time‐varying Koopman predictor, which allows for adjusting the MPC control law to counteract the fault effects. This estimation is performed in a higher dimensional Koopman feature space, where the dynamics behave linearly. As a result, the non‐linear fault‐tolerant MPC optimization problem can be replaced with a more practical and feasible linear time‐varying one using the approximated Koopman predictor. Moreover, by incorporating the online update procedure, the time‐varying Koopman predictor can represent the dynamics of the faulty system. Hence, the controller can adapt and compensate for the faults in real‐time, integrating the fault diagnosis module in the MPC framework and eliminating the need for a separate fault detection unit. Finally, the efficacy of the proposed approach is demonstrated through case study results, which highlight the ability of the controller to mitigate faults and maintain desired system behaviour. [ABSTRACT FROM AUTHOR]

Invariant measures encode the long-time behaviour of a dynamical system. In this work, we propose an optimization-based method to discover invariant measures directly from data gathered from a system. Our method does not require an explicit model for the dynamics and allows one to target specific invariant measures, such as physical and ergodic measures. Moreover, it applies to both deterministic and stochastic dynamics in either continuous or discrete time. We provide convergence results and illustrate the performance of our method on data from the logistic map and a stochastic double-well system, for which invariant measures can be found by other means. We then use our method to approximate the physical measure of the chaotic attractor of the Rössler system, and we extract unstable periodic orbits embedded in this attractor by identifying discrete-time periodic points of a suitably defined Poincaré map. This final example is truly data-driven and shows that our method can significantly outperform previous approaches based on model identification. [ABSTRACT FROM AUTHOR]

Future state prediction for nonlinear dynamical systems is a challenging task. Classical prediction theory is based on a, typically long, sequence of prior observations and is rooted in assumptions on statistical stationarity of the underlying stochastic process. These algorithms have trouble predicting chaotic dynamics, "Black Swans" (events which have never previously been seen in the observed data), or systems where the underlying driving process fundamentally changes. In this paper we develop (1) a global and local prediction algorithm that can handle these types of systems, (2) a method of switching between local and global prediction, and (3) a retouching method that tracks what predictions would have been if the underlying dynamics had not changed and uses these predictions when the underlying process reverts back to the original dynamics. The methodology is rooted in Koopman operator theory from dynamical systems. An advantage is that it is model-free, purely data-driven and adapts organically to changes in the system. While we showcase the algorithms on predicting the number of infected cases for COVID-19 and influenza cases, we emphasize that this is a general prediction methodology that has applications far outside of epidemiology. [ABSTRACT FROM AUTHOR]

The Dynamic Mode Decomposition (DMD) is a versatile and increasingly popular method for data driven analysis of dynamical systems that arise in a variety of applications in, e.g., computational fluid dynamics, robotics or machine learning. In the framework of numerical linear algebra, it is a data driven Rayleigh-Ritz procedure applied to a DMD matrix that is derived from the supplied data. In some applications, the physics of the underlying problem implies hermiticity of the DMD matrix, so the general DMD procedure is not computationally optimal. Furthermore, it does not guarantee important structural properties of the Hermitian eigenvalue problem and may return non-physical solutions. This paper proposes a software solution to the Hermitian (including the real symmetric) DMD matrices, accompanied with a numerical analysis that contains several fine and instructive numerical details. The eigenpairs are computed together with their residuals, and perturbation theory provides error bounds for the eigenvalues and eigenvectors. The software is developed and tested using the LAPACK package. [ABSTRACT FROM AUTHOR]

The Dynamic Mode Decomposition (DMD) is a method for computational analysis of nonlinear dynamical systems in data driven scenarios. Based on high fidelity numerical simulations or experimental data, the DMD can be used to reveal the latent structures in the dynamics or as a forecasting or a model order reduction tool. The theoretical underpinning of the DMD is the Koopman operator on a Hilbert space of observables of the dynamics under study. This paper describes a numerically robust and versatile variant of the DMD and its implementation using the state-of-the-art dense numerical linear algebra software package LAPACK. The features of the proposed software solution include residual bounds for the computed eigenpairs of the DMD matrix, eigenvectors refinements and computation of the eigenvectors of the Exact DMD, compressed DMD for efficient analysis of high dimensional problems that can be easily adapted for fast updates in a streaming DMD. Numerical analysis is the bedrock of numerical robustness and reliability of the software, that is tested following the highest standards and practices of LAPACK. Important numerical topics are discussed in detail and illustrated using numerous numerical examples. [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]

In this article, we consider a state estimation problem for large‐scale nonlinear processes in the absence of first‐principles process models. By exploiting process operation data, both process modeling and state estimation design are addressed within a distributed framework. By leveraging the Koopman operator concept, a parallel subsystem modeling approach is proposed to establish interactive linear subsystem process models in higher‐dimensional subspaces, each of which correlates with the original nonlinear subspace of the corresponding process subsystem via a nonlinear mapping. The data‐driven linear subsystem models can be used to collaboratively characterize and predict the dynamical behaviors of the entire nonlinear process. Based on the established subsystem models, local state estimators that can explicitly handle process operation constraints are designed using moving horizon estimation. The local estimators are integrated via information exchange to form a distributed estimation scheme, which provides estimates of the unmeasured/unmeasurable state variables of the original nonlinear process in a linear manner. The proposed framework is applied to a chemical process and an agro‐hydrological process to illustrate its effectiveness and applicability. Good open‐loop predictability of the linear subsystem models is confirmed, and accurate estimates of the process states are obtained without requiring a first‐principles process model. [ABSTRACT FROM AUTHOR]

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.

We present a flexible data-driven method for dynamical system analysis that does not require explicit model discovery. The method is rooted in well-established techniques for approximating the Koopman operator from data and is implemented as a semidefinite program that can be solved numerically. Furthermore, the method is agnostic of whether data are generated through a deterministic or stochastic process, so its implementation requires no prior adjustments by the user to accommodate these different scenarios. Rigorous convergence results justify the applicability of the method, while also extending and uniting similar results from across the literature. Examples on discovering Lyapunov functions, performing ergodic optimization, and bounding extrema over attractors for both deterministic and stochastic dynamics exemplify these convergence results and demonstrate the performance of the method. [ABSTRACT FROM AUTHOR]

Koopman operators linearize nonlinear dynamical systems, making their spectral information of crucial interest. Numerous algorithms have been developed to approximate these spectral properties, and dynamic mode decomposition (DMD) stands out as the poster child of projection-based methods. Although the Koopman operator itself is linear, the fact that it acts in an infinite-dimensional space of observables poses challenges. These include spurious modes, essential spectra, and the verification of Koopman mode decompositions. While recent work has addressed these challenges for deterministic systems, there remains a notable gap in verified DMD methods for stochastic systems, where the Koopman operator measures the expectation of observables. We show that it is necessary to go beyond expectations to address these issues. By incorporating variance into the Koopman framework, we address these challenges. Through an additional DMD-type matrix, we approximate the sum of a squared residual and a variance term, each of which can be approximated individually using batched snapshot data. This allows verified computation of the spectral properties of stochastic Koopman operators, controlling the projection error. We also introduce the concept of variance-pseudospectra to gauge statistical coherency. Finally, we present a suite of convergence results for the spectral information of stochastic Koopman operators. Our study concludes with practical applications using both simulated and experimental data. In neural recordings from awake mice, we demonstrate how variance-pseudospectra can reveal physiologically significant information unavailable to standard expectation-based dynamical models. [ABSTRACT FROM AUTHOR]

Koopman operators are infinite‐dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra and infinite‐dimensional invariant subspaces, making computing their spectral information a considerable challenge. This paper describes data‐driven algorithms with rigorous convergence guarantees for computing spectral information of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), which provides the first scheme for computing the spectra and pseudospectra of general Koopman operators from snapshot data without spectral pollution. Using the resolvent operator and ResDMD, we compute smoothed approximations of spectral measures associated with general measure‐preserving dynamical systems. We prove explicit convergence theorems for our algorithms (including for general systems that are not measure‐preserving), which can achieve high‐order convergence even for chaotic systems when computing the density of the continuous spectrum and the discrete spectrum. Since our algorithms have error control, ResDMD allows aposteri verification of spectral quantities, Koopman mode decompositions, and learned dictionaries. We demonstrate our algorithms on the tent map, circle rotations, Gauss iterated map, nonlinear pendulum, double pendulum, and Lorenz system. Finally, we provide kernelized variants of our algorithms for dynamical systems with a high‐dimensional state space. This allows us to compute the spectral measure associated with the dynamics of a protein molecule with a 20,046‐dimensional state space and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number >105 that has a 295,122‐dimensional state space. [ABSTRACT FROM AUTHOR]

Abstract Non‐linearity is an inherent feature of practical systems. Although there have been significant advances in the control of nonlinear systems, the proposed methods often require considerable computational resources or rely on local linearization around equilibrium points. The Koopman operator is an infinite‐dimensional linear operator that fully captures a system's non‐linear dynamics. However, one of the major problems is identifying a Koopman finite dimensional linear model for a nonlinear system. Initiated by the Willems’ fundamental Lemma, a class of data‐driven control methods has been developed for linear systems without the need to identify the system's matrices. Motivated by these two ideas, a data‐driven Koopman‐based predictive control scheme for non‐linear systems is proposed for unknown disturbed non‐linear systems utilising a finite‐length dataset. Then, considering the uncertainty in the Koopman state variables, a robust data‐driven Koopman predictive control structure is presented. Also, the results led to the design of a data‐driven Koopman predictive control strategy with terminal components to ensure the closed‐loop stability of nonlinear systems. The proposed scheme is tested on the distributed‐parameter model of the ACUREX solar collector field (located at Almería, Spain) to regulate the field outlet temperature around a desired value. Finally, simulation results show the effectiveness of the proposed approach.

The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space H with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators using a dense locally convex subspace X of H and its dual space X'. The three topological spaces X ⊂ H ⊂ X' are called the rigged Hilbert space or the Gelfand triplet. In this paper, the generalized spectra of the Perron--Frobenius operators of the one-sided and twosided shifts of finite type (symbolic dynamical systems) are determined. A one-sided subshift of finite type which is conjugate to the multiplication with the golden ratio on [0, 1] modulo 1 is also considered. A new construction of the Gelfand triplet for the generalized spectrum of symbolic dynamical systems is proposed by means of an algebraic procedure. The asymptotic formula of the iteration of Perron--Frobenius operators is also given. The iteration converges to the mixing state whose rate of convergence is determined by the generalized spectrum. [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]

Lyapunov functions play a vital role in the context of control theory for nonlinear dynamical systems. Besides its classical use for stability analysis, Lyapunov functions also arise in iterative schemes for computing optimal feedback laws such as the well-known policy iteration. In this manuscript, the focus is on the Lyapunov function of a nonlinear autonomous finite-dimensional dynamical system which will be rewritten as an infinite-dimensional linear system using the Koopman or composition operator. Since this infinite-dimensional system has the structure of a weak-* continuous semigroup, in a specially weighted Lp-space one can establish a connection between the solution of an operator Lyapunov equation and the desired Lyapunov function. It will be shown that the solution to this operator equation attains a rapid eigenvalue decay which justifies finite rank approximations with numerical methods. The potential benefit for numerical computations will be demonstrated with two short examples. [ABSTRACT FROM AUTHOR]