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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 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]

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.

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. [ABSTRACT FROM AUTHOR]

Summary: In this study, the Koopman operator is applied to solve the challenging nonlinear set‐membership (SM) state estimation problem. The basic idea is to lift the nonlinear system into a linear one with a higher dimension, and then linear SM estimation methods can be adopted. The Koopman operator is an infinite‐dimensional linear decomposition of nonlinear dynamics. This linearized system can fully describe the state evolution of the nonlinear dynamics with input, output, and noises. To apply SM state estimation algorithms, a finite‐dimensional approximation is obtained by using the data‐driven method. Meanwhile, the statistical properties of the approximation errors are also utilized to reduce the conservativeness of the filter. The probability distributions of approximation errors are described by sample particles. Coupled with the unknown but bounded noises, the linearized system is associated with two types of uncertainties. To this end, the merging SM and stochastic strategy is adopted to construct a class of confidence state set whose level approaches 1. After that, the optimal filter gain is given to minimize the size of the confidence state set. Finally, three numerical examples are given to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this work, a predictive control framework is presented for feedback stabilisation of nonlinear systems. To achieve this, we integrate Koopman operator theory with Lyapunov-based model predictive control (LMPC). The main idea is to transform nonlinear dynamics from state-space to function space using Koopman eigenfunctions -- for control affine systems this results in a bilinear model in the (lifted) function space. Then, a predictive controller is formulated in Koopman eigenfunction coordinates which uses an auxiliary Control Lyapunov Function (CLF) based bounded controller as a constraint to ensure stability of the Koopman system. Remarkably, the feedback control design proposed in this work remains completely data-driven and does not require any explicit knowledge of the original system. Furthermore, due to the bilinear structure of the Koopman model, seeking a CLF is no longer a bottleneck for LMPC. Benchmark numerical examples demonstrate the utility of the proposed feedback control design. [ABSTRACT FROM AUTHOR]

This paper proposes a data-driven actuator fault detection and isolation approach for the general class of nonlinear systems. The proposed method uses a deep neural network architecture to obtain an invariant set of basis functions for the Koopman operator to form a linear Koopman predictor for a nonlinear system. Then, the obtained linear model is used for fault detection and isolation purposes without relying on prior knowledge about the underlying dynamics. Moreover, a recursive method is proposed for fault detection and isolation that is entirely data-driven with the key feature of global validity for the system's whole operating region due to the Koopman operator's global characteristic. Finally, the approach's efficacy is demonstrated using two simulations on a coupled nonlinear system and a two-link manipulator benchmark. • A novel data-driven DMD-based Koopman FDI (K-FDI) method is presented for detecting and isolating actuator faults. • A new Koopman-based state-preserving DNN architecture is proposed for actuated systems. • The nonlinear FDI problem is formulated as linear FDI using Koopman operator. • The presented Koopman FDI method is extended to be computed incrementally for real-time situation. [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]

This paper introduces a novel and efficient data-driven approach for approximating a finite discrete-time bilinear model of control affine nonlinear dynamical systems with a tunable parameter that balances model dimension and prediction accuracy. An approximation of the Koopman operator based on the evolutions of the nonlinear system measurements used to lift a control-affine nonlinear system to a higher dimensional model. However, higher dimensional spaces can result in a long learning time and the curse of dimensionality in control analysis. The proposed approach addresses these challenges by introducing a convex optimization which identifies informative observable functions. This technique allows for the adjustment of a parameter to strike a balance between model dimension and accuracy in prediction. The main contribution of this study is to introduce a reduced dimensional bilinear model for a nonlinear complex system. This achievement is made possible by implementing convex sparse optimization, enabling the exploration of informative estimated Koopman eigenfunctions while minimizing the number of system measurements required. The optimization problem is solved using the alternating direction method of multipliers. The effectiveness of the proposed method is evaluated on three different nonlinear systems: a numerical nonlinear system, a Van der Pol oscillator, and a Duffing oscillator. In the last simulation, an estimation of the Koopman linear model is considered as a special case, and the policy iteration algorithm is employed to evaluate optimal control designed for different reduced-dimensional models. [ABSTRACT FROM AUTHOR]

This paper develops a real-time identification algorithm for structures of network systems. We have already proposed a structure identification method for nonlinear network systems with nonlinear couplings by combining Koopman operator theory and a sparse identification method. However, the proposed method is only applicable for offline use under the assumption that the state of an isolated system is available. To relax this assumption, this paper shows that the dynamical model of isolated systems can be extracted from measured data of interconnected systems under some weak restrictions on network structures. We also modify the identification algorithm to obtain real-time identification results with fewer calculation resources and analyse the convergence of the proposed methods. Numerical examples demonstrate that the proposed method can identify network structures in real-time and detect changes in networks correctly. [ABSTRACT FROM AUTHOR]

The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points, for both ordinary and stochastic differential equations while using either ergodic trajectories or i.i.d. samples. We illustrate these bounds by means of an example with the Ornstein–Uhlenbeck process. Moreover, we extend our analysis to (stochastic) nonlinear control-affine systems. We prove error estimates for a previously proposed approach that exploits the linearity of the Koopman generator to obtain a bilinear surrogate control system and, thus, circumvents the curse of dimensionality since the system is not autonomized by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the bilinear approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled. [ABSTRACT FROM AUTHOR]

Concentrated Solar Power plants (CSP) have the energy storage capability to generate electricity when sunlight is scarce. However, due to the highly non-linear dynamics of these systems, a simple linear controller will not be able to overcome the variable dynamics and multiple disturbance sources affecting it. In this paper, a deep Model Predictive Control (MPC) based on the Koopman operator is proposed and applied to control the Heat Transfer Fluid (HTF) temperature of a distributed-parameter model of the ACUREX solar collector field located at Almería, Spain. The Koopman operator is an infinite-dimensional linear operator that fully captures a system's non-linear dynamics through the linear evolution of functions of the state-space. However, one of the major problems is identifying a Koopman linear model for a non-linear system. Koopman eigenfunctions are involved in converting a non-linear model to a Koopman-based linear model. In this paper, a deep Long Short-Term Memory (LSTM) autoencoder is designed to calculate Koopman eigenfunctions of the solar collector field. The Koopman linear model is then used to design a linear MPC with terminal components to ensure closed-loop stability guarantees. Simulation results are utilized to show the satisfactory tracking performance of the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose a data-driven approach for uncertainty moment propagation through the non-linear power system dynamics. The proposed approach relies on the linear representation of a nonlinear system using the Perron-Frobenius & Koopman operators for propagating the moments. Data from non-linear simulations is used to approximate the linear operators by matrices, thereby enabling moment propagation by matrix multiplication. Results for a large-scale system are presented to demonstrate the accuracy and speed-up of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel approach for the data-driven characterization of power system dynamics. The developed method of Extended Subspace Identification (ESI) is suitable for systems with output measurements when all the dynamics states are not observable. It is particularly applicable for power systems dynamic identification using Phasor Measurement Units (PMUs) measurements. As in the case of power systems, it is often expensive or impossible to measure all the internal dynamic states of system components such as generators, controllers and loads. PMU measurements capture voltages, currents, power injection and frequencies, which can be considered as the outputs of system dynamics. The ESI method is suitable for system identification, capturing nonlinear modes, computing participation factor of output measurements in system modes and identifying system parameters such as system inertia. The proposed method is suitable for measurements with a noise similar to realistic system measurements. The developed method addresses some of the known deficiencies of existing data-driven dynamic system characterization methods. The approach is validated for multiple network models and dynamic event scenarios with synthetic PMU measurements. [ABSTRACT FROM AUTHOR]

Vibration responses of structures with inherent nonlinear behaviors can degrade the performance of linear theory based damage detection methods. This paper integrates the phase space reconstruction and Koopman operator to provide a linear representation of strongly nonlinear systems. Similar to the modal analysis of linear systems, the linearized model allows for handling nonlinear vibration responses as a superposition of the discovered nonlinear coordinate basis. This property provides opportunities to identify the structural condition change of structures with initial nonlinearity. The eigen-frequencies extracted from the Koopman operator are served as damage features. The performance of using the eigen-frequencies from dynamic mode decomposition for nonlinear structural damage detection is compared with the natural frequencies obtained from fast Fourier transformation and the time-frequency analysis method to emphasize the superiority of the proposed approach. Two experimental structures exhibiting inherent nonlinearity, namely, a magneto-elastic system and a precast segment beam, are employed to demonstrate the feasibility and effectiveness of using the proposed method for identifying condition change of nonlinear structures. Results demonstrate that the presented nonlinearity linearization framework and the damage feature defined in this study are suitable for reliably identifying the occurrence of structural damage and condition change in structures with inherent nonlinearities. [ABSTRACT FROM AUTHOR]

Dynamic mode decomposition is a relatively recent development in the field of modal decompositions, however, it is a commonly used technique for analyzing the dynamics of nonlinear systems. Its success is due to the fact that it is an equation-free, data-driven method capable of providing accurate assessments of the spatiotemporal coherent structures in a complex system, or short-time future estimates. The aim of this work is to present a new approach for computing the dynamic mode decomposition. We show that our algorithm is closely related to the currently accepted algorithm. In fact, the two approaches produce the same DMD eigenvalues, it is only the DMD modes that differ. We demonstrate the new approach on two examples model systems. [ABSTRACT FROM AUTHOR]

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

Time-delay embedding and dimensionality reduction are powerful techniques for discovering effective coordinate systems to represent the dynamics of physical systems. Recently, it has been shown that models identified by dynamic mode decomposition on time-delay coordinates provide linear representations of strongly nonlinear systems, in the so-called Hankel alternative view of Koopman (HAVOK) approach. Curiously, the resulting linear model has a matrix representation that is approximately antisymmetric and tridiagonal; for chaotic systems, there is an additional forcing term in the last component. In this paper, we establish a new theoretical connection between HAVOK and the Frenet–Serret frame from differential geometry, and also develop an improved algorithm to identify more stable and accurate models from less data. In particular, we show that the sub- and super-diagonal entries of the linear model correspond to the intrinsic curvatures in the Frenet–Serret frame. Based on this connection, we modify the algorithm to promote this antisymmetric structure, even in the noisy, low-data limit. We demonstrate this improved modelling procedure on data from several nonlinear synthetic and real-world examples. [ABSTRACT FROM AUTHOR]

The motivation of our research is to establish a Laplace-domain theory that provides principles and methodology to analyze and synthesize systems with nonlinear dynamics. A semigroup of composition operators defined for nonlinear autonomous dynamical systems--the Koopman semigroup and its associated Koopman generator--plays a central role in this study. We introduce the resolvent of the Koopman generator, which we call the Koopman resolvent, and provide its spectral characterization for three types of nonlinear dynamics: ergodic evolution on an attractor, convergence to a stable equilibrium point, and convergence to a (quasi-)stable limit cycle. This shows that the Koopman resolvent provides the Laplace-domain representation of such nonlinear autonomous dynamics. A computational aspect of the Laplace-domain representation is also discussed with emphasis on nonstationary Koopman modes. [ABSTRACT FROM AUTHOR]

Koopman operator with numerical approximation method for modelling nonlinear systems has become a popular data-driven approach in the past five years. However, when the system contains time-varying parameters, the data-driven Koopman operator-based model produces deviations between the nominal model and the true one. It affects the control performance when it serves as the nominal model in Model Predictive Control (MPC). To solve this issue, the Koopman operator-based model learned by using a multi-step prediction autoencoder and the strategy of online updating model are proposed. In the first step, a neural network with encoder-decoder structure is trained to search for the optimal lifted functions of the corresponding nonlinear system with fixed parameters. In the second step, an online updating strategy is presented to update the Koopman operator-based model using the collected data, when the nominal model is applied in the closed loop operation of MPC. In order to reduce the additional time consumption on updating the nominal model, the matrix inversion lemma is applied and it leads the strategy of updating to be presented in a recursive form. In general, this paper proposed an iterative MPC algorithm for online updating the Koopman operator-based model in a recursive form based on the optimal lifted function using autoencoder. Numerical simulations on three nonlinear cases with time-varying parameters show good performances on tracking control using the proposed Koopman operator-based model using autoencoder and online updating strategy. • A method based on autoencoder to learn the lifted functions was presented to obtain the approximation of Koopman operator. • The learned nominal model shows good prediction on evolution behaviors of nonlinear systems. • Adaptive model predictive control (MPC) was proposed to address the model mismatch on parameter-varying nonlinear systems. • The proposed Koopman operator-based Adaptive MPC can achieve high computational efficiency and good control performances. [ABSTRACT FROM AUTHOR]

The Koopman operator allows for handling nonlinear systems through a globally linear representation. In general, the operator is infinite-dimensional – necessitating finite approximations – for which there is no overarching framework. Although there are principled ways of learning such finite approximations, they are in many instances overlooked in favor of, often ill-posed and unstructured methods. Also, Koopman operator theory has long-standing connections to known system-theoretic and dynamical system notions that are not universally recognized. Given the former and latter realities, this work aims to bridge the gap between various concepts regarding both theory and tractable realizations. Firstly, we review data-driven representations (both unstructured and structured) for Koopman operator dynamical models, categorizing various existing methodologies and highlighting their differences. Furthermore, we provide concise insight into the paradigm's relation to system-theoretic notions and analyze the prospect of using the paradigm for modeling control systems. Additionally, we outline the current challenges and comment on future perspectives. [ABSTRACT FROM AUTHOR]

In recent years, the success of the Koopman operator in dynamical systems analysis has also fueled the development of Koopman operator-based control frameworks. In order to preserve the relatively low data requirements for an approximation via dynamic mode decomposition, a quantization approach was recently proposed in [S. Peitz and S. Klus, Automatica J. IFAC, 106 (2019), pp. 184-191]. This way, control of nonlinear dynamical systems can be realized by means of switched systems techniques, using only a finite set of autonomous Koopman operator-based reduced models. These individual systems can be approximated very efficiently from data. The main idea is to transform a control system into a set of autonomous systems for which the optimal switching sequence has to be computed. In this article, we extend these results to continuous control inputs using relaxation. This way, we combine the advantages of the data efficiency of approximating a finite set of autonomous systems with continuous controls, as the data requirements increase only linearly with the input dimension. We show that when using the Koopman generator, this relaxation--realized by linear interpolation between two operators--does not introduce any error for control affine systems. This allows us to control high-dimensional nonlinear systems using bilinear, low-dimensional surrogate models. The efficiency of the proposed approach is demonstrated using several examples with increasing complexity, from the Duffing oscillator to the chaotic fluidic pinball. [ABSTRACT FROM AUTHOR]

The Koopman framework proposes a linear representation of finite-dimensional nonlinear systems through a generally infinite-dimensional globally linear embedding. Originally, the Koopman formalism has been derived for autonomous systems. In applications for systems with inputs, generally a linear time invariant (LTI) form of the Koopman model is assumed, as it facilitates the use of control techniques such as linear quadratic regulation and model predictive control. However, it can be easily shown that this assumption is insufficient to capture the dynamics of the underlying nonlinear system. Proper theoretical extension for actuated continuous-time systems with a linear or a control-affine input has been worked out only recently, however extensions to discrete-time systems and general continuous-time systems have not been developed yet. In the present paper, we systematically investigate and analytically derive lifted forms under inputs for a rather wide class of nonlinear systems in both continuous and discrete time. We prove that the resulting lifted representations give Koopman models where the state transition is linear, but the input matrix becomes state-dependent (state and input-dependent in the discrete-time case), giving rise to a specially structured linear parameter-varying (LPV) description of the underlying system. We also provide error bounds on how much the dependency of the input matrix contributes to the resulting representation and how well the system behavior can be approximated by an LTI Koopman representation. The introduced theoretical insight greatly helps for performing proper model structure selection in system identification with Koopman models as well as making a proper choice for LTI or LPV techniques for the control of nonlinear systems through the Koopman approach. [ABSTRACT FROM AUTHOR]

In Koopman operator theory, a finite-dimensional nonlinear system is transformed into an infinite but linear system using a set of observable functions. However, manually selecting observable functions that span the invariant subspace of the Koopman operator based on prior knowledge is inefficient and challenging, particularly when little or no information is available about the underlying systems. Furthermore, current methodologies tend to disregard the importance of the invertibility of observable functions, which leads to inaccurate results. To address these challenges, we propose the so-called FlowDMD, aka Flow-based Dynamic Mode Decomposition, that utilizes the Coupling Flow Invertible Neural Network (CF-INN) framework. FlowDMD leverages the intrinsically invertible characteristics of the CF-INN to learn the invariant subspaces of the Koopman operator and accurately reconstruct state variables. Numerical experiments demonstrate the superior performance of our algorithm compared to state-of-the-art methodologies. [ABSTRACT FROM AUTHOR]

Precise and timely fault diagnosis is crucial in many practical systems and control processes. Particularly due to the increasing amount of available data collected by sensors, data-driven fault diagnosis has been a hot research topic in the prognosis and health management of industrial systems. In this paper, a reliable, accurate, and interpretable data-driven fault detection and isolation method is proposed for the general class of nonlinear systems. The proposed approach is based on the integration of the bilinear Koopman model realization, deep learning, and a bilinear parity-space framework. This work leverages the potential of neural networks to investigate lifting functions and bilinear Koopman realization simultaneously. Furthermore, to enhance the stability of the realized model, the input-to-state stability constraint is enforced on the training algorithm, ensuring the realized model is integral input-to-state stable. This method does not require any prior knowledge regarding the system dynamics and only uses the data collected from the normal operation of the system. In addition, it is capable of diagnosing all types of sensor and actuator faults, whether they are additive or multiplicative. The effectiveness of the proposed method is demonstrated experimentally using a laboratory setup of the three-tank system. Lastly, a comparison is conducted to demonstrate the advantages of the proposed method over another recent data-driven fault diagnosis method. [ABSTRACT FROM AUTHOR]

This paper presents a data-driven approach to approximate the dynamics of a nonlinear time-varying system (NTVS) by a linear time-varying system (LTVS), which results from the Koopman operator and deep neural networks. Analysis of the approximation error between states of the NTVS and the resulting LTVS is presented. Simulations on a representative NTVS show that the proposed method achieves small approximation errors, even when the system changes rapidly. Furthermore, simulations in an example of quadcopters demonstrate the computational efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper describes a method for learning low-dimensional approximations of nonlinear dynamical systems, based on neural network approximations of the underlying Koopman operator. Extended Dynamic Mode Decomposition (EDMD) provides a useful data-driven approximation of the Koopman operator for analyzing dynamical systems. This paper addresses a fundamental problem associated with EDMD: a trade-off between representational capacity of the dictionary and overfitting due to insufficient data. A new neural network architecture combining an autoencoder with linear recurrent dynamics in the encoded state is used to learn a low-dimensional and highly informative Koopman-invariant subspace of observables. A method is also presented for balanced model reduction of overspecified EDMD systems in feature space. Nonlinear reconstruction using partially linear multikernel regression aims to improve reconstruction accuracy from the low-dimensional state when the data has complex but intrinsically low-dimensional structure. The techniques demonstrate the ability to identify Koopman eigenfunctions of the unforced Duffing equation, create accurate low-dimensional models of an unstable cylinder wake flow, and make short-time predictions of the chaotic Kuramoto--Sivashinsky equation. [ABSTRACT FROM AUTHOR]

In this study, we propose a new data augmentation technique, Koopman-mixup (K-mixup), to improve the learning stability and final performance of offline reinforcement learning (RL) algorithms. K-mixup learns a Koopman invariant subspace to incorporate mixup augmentation, commonly used for classification tasks, into an RL framework. Mixup augmentation itself is known to be incompatible with RL because RL generally uses nonlinearly propagating state-based sequential inputs, whereas mixup relies on linear interpolation between a pair of inputs. To resolve the problem, Koopman embedding is used to convert a nonlinear system to a linear system, allowing successful mixup on arbitrary data pairs in any dataset. We evaluate the performance of K-mixup on several OpenAI Gym benchmark control simulations and compare it with the performance of other data augmentation methods. The comparison shows that only the proposed K-mixup consistently outperforms the base offline RL algorithm (CQL). [ABSTRACT FROM AUTHOR]

The majority of systems of practical interest are characterized by nonlinear dynamics. This renders the control and optimization of such systems a complex task owing to their nonlinear behaviour. Standard methods of dealing with nonlinear systems such as linearizing around a fixed point may not be an effective strategy for many systems, thus requiring an alternative approach. For this reason, we propose a novel deep learning framework to discover a transformation of a nonlinear dynamical system to an equivalent higher dimensional linear system using data generated from identification experiments. We demonstrate that the resulting learned linear representation accurately captures the dynamics of the original system for a wide range of conditions defined by the training dataset used. As a result of this, we show that the learned linear model can subsequently be used for the successful control of the original system. We demonstrate this by applying the proposed framework to three examples; a benchmark example from the literature, and two practical, complex nonlinear dynamical systems from the chemical engineering domain: the Continuous Stirred Tank Reactor (CSTR) system and finally the four-tank system. • Koopman operator theory and deep learning are merged to linearize nonlinear dynamics. • The learned linear models accurately capture the dynamics of the original system. • The learned linear representations can be used for tractable model-based control. • The proposed framework is applied to two practical nonlinear process systems. [ABSTRACT FROM AUTHOR]

Abstract: According to the Hartman–Grobman Theorem, a nonlinear system can be linearized in a neighborhood of a hyperbolic stationary point. Here, we extend this linearization around stable (unstable) equilibria or periodic orbits to the whole basin of attraction, for both discrete diffeomorphisms and flows. We discuss the connection of the linearizing transformation to the spectrum of Koopman operator. [Copyright &y& Elsevier]

We consider C 1 dynamical systems having an attracting hyperbolic fixed point or periodic orbit and prove existence and uniqueness results for C k (actually C loc k , α) linearizing semiconjugacies – of which Koopman eigenfunctions are a special case – defined on the entire basin of attraction. Our main results both generalize and sharpen Sternberg's C k linearization theorem for hyperbolic sinks, and in particular our corollaries include uniqueness statements for Sternberg linearizations and Floquet normal forms. Using our main results we also prove new existence and uniqueness statements for C k Koopman eigenfunctions, including a complete classification of C ∞ eigenfunctions assuming a C ∞ dynamical system with semisimple and nonresonant linearization. We give an intrinsic definition of "principal Koopman eigenfunctions" which generalizes the definition of Mohr and Mezić for linear systems, and which includes the notions of "isostables" and "isostable coordinates" appearing in work by Ermentrout, Mauroy, Mezić, Moehlis, Wilson, and others. Our main results yield existence and uniqueness theorems for the principal eigenfunctions and isostable coordinates and also show, e.g., that the (a priori non-unique) "pullback algebra" defined in Mohr and Mezić (2016) is unique under certain conditions. We also discuss the limit used to define the "faster" isostable coordinates in Wilson and Ermentrout (2018), Monga et al. (2019) in light of our main results. • Koopman eigenfunctions provide exact linear representations of nonlinear systems. • Eigenfunction existence can be deduced from classical linearization results. • Uniqueness was known only for analytic eigenfunctions of some analytic systems. • We obtain new uniqueness and existence results for C k eigenfunctions. • More generally, we obtain such results for C loc k , α linearizing semiconjugacies. [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]

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]

This paper considers the observability of nonlinear systems from a Koopman operator theoretic perspective – and in particular – the effect of symmetry on observability. We first examine an infinite-dimensional linear system (constructed using independent Koopman eigenfunctions) and relate its observability properties to the observability of the original nonlinear system. Next, we derive an analytic relation between symmetry and nonlinear observability; it is shown that symmetry in the nonlinear dynamics is reflected in the symmetry of the corresponding Koopman eigenfunctions, as well as presence of repeated Koopman eigenvalues. We then proceed to show that loss of observability in symmetric nonlinear systems can be traced back to the presence of these repeated eigenvalues. In the case where we have a sufficient number of measurements, the nonlinear system remains unobservable when these functions have symmetries that mirror those of the dynamics. The proposed observability framework provides insights into the minimum number of measurements needed to make an unobservable nonlinear system, observable. The proposed results are then applied to a network of nano-electromechanical oscillators coupled via a symmetric interaction topology. [ABSTRACT FROM AUTHOR]