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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.

This paper presents a data-driven model-based control system for autonomous underwater vehicles (or AUVs) subject to input delays. This work is motivated by the input time delays that can arise in underwater robotics due to communication restrictions and sensor malfunctions. Such delays can highly degrade the performance of classical control structures resulting in unpredictable system behaviours. The proposed control architecture addresses such limitations. The approach incorporates a linear dynamic representation of the system obtained using the Koopman operator in an observer/state prediction formulation. The proposed control architecture is designed based on discrepancies between the data-driven estimation of the system’s behaviour and the actual AUV performance using chain predictors. The capabilities of the proposed approach are shown through experiments performed with a 4 degrees-of-freedom autonomous underwater vehicle. The results demonstrate stable behaviours without steady-state errors, even in the presence of long delay.

In the practical application of high-frequency surface wave radar (HFSWR), ionospheric clutter has become the principal obstacle to target detection. Since ionospheric clutter has nonlinear and time-varying characteristics, the current suppression methods still require improvement. In this letter, we proposed a method merging the deep neural networks (DNNs) and linear Koopman operator from the dynamic perspective, combining the powerful learning capacity of neural networks and the interpretability of Koopman theory, and applied linear Koopman mode to represent nonlinear ionospheric clutter in the state space based on time-delayed coordinates to achieve the finite-dimensional linear approximate reconstruction of ionospheric clutter. The validation results of the measured data reveal that the method has satisfactory modeling accuracy, which proves the effectiveness of the approach and lays the foundations for the subsequent breakthrough in new methods for ionospheric clutter suppression. [ABSTRACT FROM AUTHOR]

The estimation and reconstruction of complex fluids like wind turbine wake are challenging problems, which require approximating high-dimensional, nonlinear dynamic systems with a limited number of physical measurements. Current works mainly focus on the reduced-order approximation for fluid dynamics like dynamic mode decomposition, or linear stochastic estimation for static mapping. In this article, the problem of yaw-controlled wind turbine wake dynamics approximation and reconstruction are addressed. A Koopman-linear flow estimator is designed, which forms a control-oriented linear dynamic state-space model. The full wake flow is first approximated on its dynamic performances and then reconstructed from low-order physical states. The novelty mainly yields on the added yaw excitation that a control-oriented dynamic model is obtained. The Kalman filter is also involved that measured data in future time could be embedded while the measurement noise is filtered. At last, static and dynamic estimation tests verify the effectiveness of the proposed estimator. [ABSTRACT FROM AUTHOR]

This paper presents a model-free fault detection and isolation (FDI) method for nonlinear dynamical systems using Koopman operator theory and linear geometric technique. The key idea is to obtain a Koopman-based reduced-order model of a nonlinear dynamical system and apply the linear geometric FDI method to detect and isolate faults in the system. Koopman operator is an infinite-dimensional, linear operator which lifts the nonlinear dynamic data into an infinite-dimensional space where the correlations of dynamic data behave linearly. However, due to the infinite dimensionality of this operator, an approximation of the operator is needed for practical purposes. In this work, the Koopman framework is adopted toward nonlinear dynamical systems in combination with the linear geometric approach for fault detection and isolation. In order to demonstrate the efficacy of the proposed FDI solution, a mathematical nonlinear dynamical system, and an experimental three-tank setup are considered. Results show a remarkable performance of the proposed geometric Koopman-based fault detection and isolation (K-FDI) technique.

Continuous monitoring of the spatio-temporal dynamic behavior of critical infrastructure networks, such as the power systems, is a challenging but important task. In particular, accurate and timely prediction of the (electro-mechanical) transient dynamic trajectories of the power grid is necessary for early detection of any instability and prevention of catastrophic failures. Existing approaches for prediction of dynamic trajectories either rely on the availability of accurate physical models of the system, use computationally expensive time-domain simulations, or are applicable only at local prediction problems (e.g., a single generator). In this paper, we report the application of two broad classes of data-driven learning models – along with their algorithmic implementation and performance evaluation – in predicting transient trajectories in power networks using only streaming measurements and the network topology as input. One class of models is based on the Koopman operator theory which allows for capturing the nonlinear dynamic behavior via an infinite-dimensional linear operator. The other class of models is based on the graph convolutional neural networks which are adept at capturing the inherent spatio-temporal correlations within the power network. Transient dynamic datasets for training and testing the models are synthesized by simulating a wide variety of load change events in the IEEE 68-bus system, categorized by the load change magnitudes, as well as by the degree of connectivity and the distance to nearest generator nodes. The results confirm that the proposed predictive models can successfully predict the post-disturbance transient evolution of the system with a high level of accuracy.

Transient stability control of power systems is based on actions that are taken automatically following a disturbance to ensure that the system remains in synchronism. Examples of such measures include generator rejection and the insertion of dynamic braking resistors. Methods like these are designed to rapidly absorb excess energy or otherwise alter the generation-demand balance at key points in the system. While these methods are often effective, they lack the ability to inject real power to compensate for a deficit. Utility-scale inverter-based resources, particularly energy storage systems, enable bidirectional modulation of real power with the bandwidth necessary to provide synchronizing torque. These resources, and the control strategies they enable, have garnered substantial research interest. This paper provides a critical review of research on real power modulation strategies for transient stability control. The design of these control strategies is heavily informed by the methods used to assess changes in the transient stability margins. Rigorously assessing these changes is difficult because the dynamics of large-scale power systems are inherently nonlinear. The well-known equal-area criterion is physically intuitive, but conceptual extensions are necessary for multi-machine systems. So-called direct methods of transient stability analysis offer a more general alternative; however, these methods require many simplifying assumptions and have difficulty incorporating detailed system dynamics. In this paper, we discuss data-driven methods for offline stability assessment based on Koopman operator theory.

This paper proposes a data-centric model predictive control (MPC) for supplemental control of a DFIG-based wind farm (WF) to improve power system stability. The proposed method is designed to control active and reactive power injections via power converters to reduce the oscillations produced by the WF during disturbance conditions. Without prior knowledge of the system model, this approach utilizes the states measurements of the DFIG subsystem for control design. Therefore, a data-driven optimal controller with a decentralized feature is developed. The learning process is based on Koopman operator theory where the unknown nonlinear dynamics of the DFIG is reconstructed by lifting the nonlinear dynamics to a linear space with an approximate linear state evolution. Extended dynamic mode decomposition (EDMD) is then applied to determine the lifted-state space matrices for the proposed Koopman-based model predictive controller (KMPC) design. The effectiveness of the proposed scheme is tested on New England IEEE 68-bus 16-machine system under three-phase fault conditions. The results ascertain the effectiveness of the proposed scheme to enhance the system damping characteristics.

Dynamic state estimation (DSE) plays an important role in the real-time control and monitoring of distribution systems, which are high-dimensional space–time systems. The degree of nonlinearity of distribution system increases drastically with the availability of sustainable energy and the diversification of load types, thereby reducing the accuracy of the standard linear model. Under the existing schemes, the measurement noise may not follow a Gaussian distribution. Therefore, a Koopman operator-based Kalman particle filter (KKPF) is proposed herein for estimating the dynamic states of a distribution system. The KKPF performs data-driven dynamic state estimation based on the Koopman operator theory, which does not rely on the distribution system model and can be applied to high-dimensional systems. Furthermore, this method can be applied to dynamic systems and noise with both Gaussian and non-Gaussian distributions. The KKPF method provides accurate estimation results when only a small number of measurements are available. IEEE 141 system and an improved 141 system were used to test the performance of the proposed KKPF compared to the EKF, CKF, and PF. The test revealed that the proposed KKPF can obtain accurate results under high-dimensional and non-Gaussian noise environments.

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]

We propose a Koopman operator-based surrogate model for propagating parameter uncertainties in power system nonlinear dynamic simulations. First, we augment a priori known state-space model by reformulating parameters deemed uncertain as pseudo-state variables. Then, we apply the Koopman operator theory to the resulting state-space model and obtain a linear dynamical system model. This transformation allows us to analyze the evolution of the system dynamics through its Koopman eigenfunctions, eigenvalues, and modes. Of particular importance for this letter, the obtained linear dynamical system is a surrogate that enables the evaluation of parameter uncertainties by simply perturbing the initial conditions of the Koopman eigenfunctions associated with the pseudo-state variables. Simulations carried out on the New England test system reveal the excellent performance of the proposed method in terms of accuracy and computational efficiency. [ABSTRACT FROM AUTHOR]

As the wind power penetration increases, wind farms are required by the grid codes to provide frequency regulation services. This article develops a fully data-driven model predictive control (DMPC) scheme for the wind farm to provide temporal frequency support. The main technical challenge is the complexity and the nonlinearity of wind turbine dynamics that make the DMPC intractable. Based on Koopman operator (KO) theory, a specialized dynamic mode decomposition (SDMD) algorithm is proposed, which fits a global linear dynamic model of the wind turbines. The performance of learning dynamics is powered through integrating the physical knowledge of the wind turbine into the specialized observables of KO. To stabilize the rotor speeds in frequency regulation, the active power contribution is optimally dispatched in a moving horizon fashion. Simulation results show that the DMPC can efficiently learn and predict the wind turbine dynamics. During the frequency response process, the proposed method can effectively track the frequency support order specified by the utility grid operator while significantly stabilizing the rotor speeds. [ABSTRACT FROM AUTHOR]

Micro smart sensors and sensor networks are the key bases for building a fully visible, perceptible and controllable transparent substation in power grid. However, the massive deployment of micro smart sensors often leads to network congestion and directly affects the quality of information transmission. In practice, control design for micro sensor networks is often disturbed by factors such as nonlinearity, time delay and time-varying parameters, and the mathematical model is inaccurate. To solve these problems, this paper proposes a Koopman operator based data-driven congestion control scheme without using any system model information, based on model predictive control (MPC) and extended state observer (ESO). Firstly, according to the Koopman operator theory, a data-driven linear Koopman model for micro smart sensor networks is derived, only using the input/output data. Then an ESO based MPC scheme is designed to obtain the optimal packet dropping probability in the micro smart sensor node buffer area. Specifically, ESO performs real-time online estimation of the modeling errors, such as time-varying parameters, time delay and the external disturbances. Then the estimated modeling errors are viewed as total disturbances and are compensated in MPC, so that the queue length in the node buffer can quickly stabilize at the set value. Extensive simulations are conducted to verify the accuracy of Koopman model, and the effectiveness of the proposed control system design.

Automatic train operation systems of high-speed trains are critical to guarantee operational safety, comfort, and parking accuracy. However, implementing optimal automatic operation control is challenging due to the train’s uncertain dynamics and actuator saturation. To address this issue, this paper develops a data-driven Koopman model based predictive control method for automatic train operation systems. The proposed control scheme is designed within a data-driven framework. First, using operational data of trains and the Koopman operator, an explicit linear Koopman model is built to characterize the train dynamics. Then, a model predictive controller is designed based on the Koopman model under comfort and actuator constraints. Furthermore, an online update mechanism for the Koopman model is developed to cope with the changing dynamic characteristics of trains, which reduces the accumulation errors and improves control performance. Stability analysis of the closed-loop control system is provided. Comparative simulation results validate the effectiveness of the proposed control approach.

The study of cybersecurity incidents is an active research field. The purpose of this work is to determine accurate measures of cybersecurity incidents. An effective method to aggregate cybersecurity incident reports is defined to set these measures. As a result we are able to make predictions and, therefore, to deploy security policies. Forecasting time-series of those cybersecurity aggregates is performed based on Koopman’s method and Dynamic Mode Decomposition algorithm. Both techniques have shown to be accurate for a wide variety of dynamical systems ranging from fluid dynamics to social sciences. We have performed some experiments on public databases. We show that the measure of the risk trend can be effectively forecasted.

This paper proposes Koopman operator theory and the related algorithm dynamical mode decomposition (DMD) for analysis and control of signalized traffic flow networks. DMD provides a model-free approach for representing complex oscillatory dynamics from measured data, and we study its application to several problems in signalized traffic. We first study a single signalized intersection, and we propose applying this method to infer traffic signal control parameters such as phase timing directly from traffic flow data. Next, we propose using the oscillatory modes of the Koopman operator, approximated with DMD, for early identification of unstable queue growth that has the potential to cause cascading congestion. Then we demonstrate how DMD can be coupled with knowledge of the traffic signal control status to determine traffic signal control parameters that are able to reduce queue lengths. Lastly, we demonstrate that DMD allows for determining the structure and the strength of interactions in a network of signalized intersections. All examples are demonstrated using a case study network instrumented with high resolution traffic flow sensors. [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]

Recent studies have shown that the nonlinear dynamics of nano-positioning systems (e.g., piezo-electric actuators (PEAs)) can be accurately captured by recurrent neural networks (RNNs). One direct application of this technique is PEA system control for precision positioning: linearize the nonlinear RNN model and then apply model predictive control (MPC). However, due to the linearization approach commonly used (e.g., Taylor series), the control bandwidth and the control performance are quite limited as the obtained linear system is only guaranteed to be accurate within small neighborhood of the linearization point. To address this issue, we propose a Koopman operator-based approach for linearization and then use the obtained linear parameter-varying model for predictive control. This linearization scheme can significantly decrease the overall approximation error within the MPC prediction horizon, and thus, lead to improved tracking performance. The proposed approach was validated through two applications-trajectory tracking of PEA, and deformation control of polymers during atomic force microscope nano-indentation.

This paper develops a robust generalized maximum-likelihood Koopman operator-based Kalman filter (GM-KKF) to estimate the rotor angle and speed of synchronous generators. The approach is data driven and model independent. Its design phase is carried out offline and requires estimates of the synchronous generators’ rotor angle and speed, along with active and reactive power at the generators’ terminal; in real-time operation, only measurements of the rotor speed, active, and reactive power are used. We first investigate the probability distribution of the transformed dynamic states by means of Q–Q plots and verify that the states of the GM-KKF approximately follow a Student's t-distribution with 20 degrees of freedom when the initial state vector is normally distributed. Under this assumption, our filter presents high statistical efficiency. Numerical simulations carried out on the IEEE 39-bus test system reveal that the GM-KKF has a faster convergence rate than the non-robust Koopman operator-based Kalman filter thanks to the adoption of a batch-mode regression formulation. They also show that the computing time of the GM-KKF is roughly reduced by one-third as compared to the one taken by our previously developed robust GM-extended Kalman filter. [ABSTRACT FROM AUTHOR]

We perform modal analysis of short-term swing dynamics in multi-machine power systems. The analysis is based on the so-called Koopman operator, a linear, infinite-dimensional operator that is defined for any nonlinear dynamical system and captures full information of the system. Modes derived through spectral analysis of the Koopman operator, called Koopman modes, provide a nonlinear extension of linear oscillatory modes. Computation of the Koopman modes extracts single-frequency, spatial modes embedded in non-stationary data of short-term, nonlinear swing dynamics, and it provides a novel technique for identification of coherent swings and machines. [ABSTRACT FROM PUBLISHER]