Koopman Operator Research Articles

A study on data-driven identification and representation of nonlinear dynamical systems with a physics-integrated deep learning approach: Koopman operators and nonlinear normal modes

Identifying the intrinsic coordinates or modes of dynamical systems is essential to understand, analyze, and characterize the underlying complex dynamical behaviors. This is a critical challenge for nonlinear dynamical systems because the modal transformation, which is universal for linear systems, no longer applies. Koopman operators and nonlinear normal modes (NNMs) are two main frameworks aiming to provide a general representation of nonlinear dynamics. In this study, we investigate their abilities on representing nonlinear dynamical systems. Addressing the challenge that the closed-form models or equations of dynamical systems are generally unknown realistically, we first present a physics-integrated deep learning-based data-driven framework to identify the eigenfunction of Koopman operators and the nonlinear modal transformation function of NNMs, respectively. We then evaluate their representation abilities by their reconstruction and predictive accuracy of the nonlinear dynamical response using the identified Koopman modes and NNMs. We conduct numerical experiments on Duffing systems with varying nonlinearity levels and dimensions, and observe that NNMs achieve higher representation accuracy and better computational efficiency than Koopman operators using the same dimension of intrinsic coordinates or modes.

Learning-based robust model predictive control with data-driven Koopman operators

Learning-based robust model predictive control with data-driven Koopman operators

Dynamic mode decomposition type algorithms for modeling and predicting queue lengths at signalized intersections with short lookback

This article explores a novel data-driven approach based on recent developments in Koopman operator theory and dynamic mode decomposition (DMD) for modeling signalized intersections. On signalized intersections, vehicular flow and queue formation have complex nonlinear dynamics, making system identification, modeling, and controller design challenging. We employ a DMD-type approach to transform the original nonlinear dynamics into locally linear infinite-dimensional dynamics. The data-driven approach relies entirely on spatio-temporal snapshots of the traffic data. We investigate several key aspects of the approach and provide insights into the usage of DMD-type algorithms for application in adaptive signalized intersections. To validate the obtained linearized dynamics, we perform prediction of the queue lengths at the intersection and compare the results with the benchmark methods such as ARIMA and long short term memory (LSTM). The case study involves intersection pressure and queue lengths at two Orlando area signalized intersections during the morning and evening peaks. It is observed that DMD-type algorithms are able to capture complex dynamics with a linear approximation to a reasonable extent. The merits include faster computation times and significantly less requirement for a “lookback” (training) window.

Data-driven modeling for damping and positioning control of gantry crane

Flexible structural vibrations of large gantry cranes are a well-known issue. Several control strategies were proposed for this problem, which require a detailed dynamical model. Since the modeling of large-scaled structures, as well as of significant nonlinear friction effects is complicated and time consuming, a direct data-driven dynamic reconstruction based on the Koopman operator theory is proposed. Such identification results in a higher order linear state-space model, which reflects the nonlinear behavior in the broad dynamical range of the system and does not depend on a specific operating point. In this paper, a data-driven linear model identification is investigated for a simulation model, as well as for an experimental laboratory crane. The damping and positioning control dynamics based on the identified model were studied and it was shown, that the resulting performance is comparable to model-based nonlinear approaches.

Approximating the Stationary Bellman Equation by Hierarchical Tensor Products

We treat infinite horizon optimal control problems by solving the associated stationary Hamilton-Jacobi-Bellman (HJB) equation numerically to compute the value function and an optimal feedback law. The dynamical systems under consideration are spatial discretizations of non linear parabolic partial differential equations (PDE), which means that the HJB is non linear and suffers from the curse of dimensionality. Its non linearity is handled by the Policy Iteration algorithm, where the problem is reduced to a sequence of linear, hyperbolic PDEs. These equations remain the computational bottleneck due to their high dimensions. By the method of characteristics these linearized HJB equations can be reformulated via the Koopman operator in the spirit of dynamic programming. The resulting operator equations are solved using a minimal residual method. To overcome numerical infeasability we use low rank hierarchical tensor product approximation/tree-based tensor formats, in particular tensor trains (TT tensors), and multi-polynomials, together with high dimensional quadrature, e.g. Monte-Carlo. By controlling a destabilized version of viscous Burgers and a diffusion equation with unstable reaction term numerical evidences are given.

Koopman analysis of the periodic Korteweg-de Vries equation.

The eigenspectrum of the Koopman operator enables the decomposition of nonlinear dynamics into a sum of nonlinear functions of the state space with purely exponential and sinusoidal time dependence. For a limited number of dynamical systems, it is possible to find these Koopman eigenfunctions exactly and analytically. Here, this is done for the Korteweg-de Vries equation on a periodic interval using the periodic inverse scattering transform and some concepts of algebraic geometry. To the authors' knowledge, this is the first complete Koopman analysis of a partial differential equation, which does not have a trivial global attractor. The results are shown to match the frequencies computed by the data-driven method of dynamic mode decomposition (DMD). We demonstrate that in general, DMD gives a large number of eigenvalues near the imaginary axis and show how these should be interpreted in this setting.

Nonlinear model predictive control for autonomous vehicle drifting

In this paper, nonlinear model predictive control (NMPC) is proposed for autonomous vehicle drifting, that is, stabilizing the vehicle at a desired unstable equilibrium point. Firstly, a three-degree-of-freedom vehicle model with a nonlinear tire model is introduced, and the equilibrium points are calculated. The relationship between the desired unstable equilibrium point and the lateral stability region is analyzed based on the phase plane method. Secondly, NMPC is designed to force vehicle states to stay around the desired unstable equilibrium point, that is, to keep the vehicle in sustained drifting. The terminal region and terminal constraint of NMPC are determined off-line to guarantee stability. Thirdly, Koopman operator theory and dynamic mode decomposition with control are introduced to obtain an approximately linear model, by which the nonlinear optimization problem is converted to a quadratic programming problem. Finally, comparative experiments are conducted by simulation, in which various model uncertainties are considered. The effectiveness of the proposed approach to achieve sustained autonomous drifting and to ensure vehicle safety is illustrated, and the efficient implementation of the proposed approach is also shown.

A purely data-driven framework for prediction, optimization, and control of networked processes

Networks are landmarks of many complex phenomena where interweaving interactions between different agents transform simple local rule-sets into nonlinear emergent behaviors. While some recent studies unveil associations between the network structure and the underlying dynamical process, identifying stochastic nonlinear dynamical processes continues to be an outstanding problem. Here, we develop a simple data-driven framework based on operator-theoretic techniques to identify and control stochastic nonlinear dynamics taking place over large-scale networks. The proposed approach requires no prior knowledge of the network structure and identifies the underlying dynamics solely using a collection of two-step snapshots of the states. This data-driven system identification is achieved by using the Koopman operator to find a low-dimensional representation of the dynamical patterns that evolve linearly. Further, we use the global linear Koopman model to solve critical control problems by applying to model predictive control (MPC)–typically, a challenging proposition when applied to large networks. We show that our proposed approach tackles this by converting the original nonlinear programming into a more tractable optimization problem that is both convex (quadratic programming) and with far fewer variables.

Lie group valued Koopman eigenfunctions

Every continuous-time flow on a topological space has associated to it a Koopman operator, which operates by time-shifts on various spaces of functions, such as C r , L 2, or functions of bounded variation. An eigenfunction of the vector field (and thus for the Koopman operator) can be viewed as an S 1-valued function, which also plays the role of a semiconjugacy to a rigid rotation on S 1. This notion of Koopman eigenfunctions will be generalized to Lie-group valued eigenfunctions, and we will discuss the dynamical aspects of these functions. One of the tools that will be developed to aid the discussion, is a concept of exterior derivative for Lie group valued functions, which generalizes the notion of the differential of a real valued function f. The extended notion of Koopman eigenfunctions utilizes a geometric property of usual eigenfunctions. We show that the generalization in a geometric sense can be used to reveal fundamental properties of usual Koopman eigenfunctions, such as their behaviour under time-rescaling, and as submersions.

Stable data‐driven Koopman predictive control: Concentrated solar collector field case study

AbstractNon‐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.

Model-Predictive Control Design for Power System Oscillation Damping via Excitation – A Data-Driven Approach

This paper presents a novel power system oscillation damping controller design based on a data-driven model-predictive control (MPC) approach. The system dynamics are extracted from the measurements and applied to the synthesis of control actions in an equation-free manner, without requiring explicit knowledge of the underlying system model. Dynamic mode decomposition with control (DMDc), originating from the Koopman operator theory, connects the collected data with the nonlinear dynamical system modeling. The discrepancy between the model identified by DMDc and the real plant is labeled as a total disturbance. It is estimated in real time by an extended state observer (ESO) and mitigated via a linear MPC, assuming the estimate to be constant over the prediction horizon. The overall control framework is referred to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ESO-Koopman-MPC (EKM)</i> . Two <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">EKM</i> -based oscillation damping controllers are proposed - a standalone PSS and an integrated AVR+PSS. The controller operation is illustrated on the single machine infinite bus system (SMIB), Kundur two-area system and IEEE 39-bus system, and the performance is compared with conventional power system stabilizers (CPSSs) under various test scenarios.

Integration of Physics- and Data-Driven Power System Models in Transient Analysis After Major Disturbances

The article explores the analysis of transient phenomena in large-scale power systems subjected to major disturbances from the aspect of interleaving, coordinating, and refining physics- and data-driven models. Major disturbances can lead to cascading failures and ultimately to the partial power system blackout. Our primary interest is in a framework that would enable coordinated and seamlessly integrated use of the two types of models in engineered systems. Parts of this framework include: 1) optimized compressed sensing, 2) customized finite-dimensional approximations of the Koopman operator, and 3) gray-box integration of physics-driven (equation-based) and data-driven (deep neural network-based) models. The proposed three-stage procedure is applied to the transient stability analysis on the multimachine benchmark example of a 441-bus real-world test system, where the results are shown for a synchronous generator with local measurements in the connection point.

A deep Koopman operator‐based modelling approach for long‐term prediction of dynamics with pixel‐level measurements

AbstractAlthough previous studies have made some clear leap in learning latent dynamics from high‐dimensional representations, the performances in terms of accuracy and inference time of long‐term model prediction still need to be improved. In this study, a deep convolutional network based on the Koopman operator (CKNet) is proposed to model non‐linear systems with pixel‐level measurements for long‐term prediction. CKNet adopts an autoencoder network architecture, consisting of an encoder to generate latent states and a linear dynamical model (i.e., the Koopman operator) which evolves in the latent state space spanned by the encoder. The decoder is used to recover images from latent states. According to a multi‐step ahead prediction loss function, the system matrices for approximating the Koopman operator are trained synchronously with the autoencoder in a mini‐batch manner. In this manner, gradients can be synchronously transmitted to both the system matrices and the autoencoder to help the encoder self‐adaptively tune the latent state space in the training process, and the resulting model is time‐invariant in the latent space. Therefore, the proposed CKNet has the advantages of less inference time and high accuracy for long‐term prediction. Experiments are performed on OpenAI Gym and Mujoco environments, including two and four non‐linear forced dynamical systems with continuous action spaces. The experimental results show that CKNet has strong long‐term prediction capabilities with sufficient precision.

Data-Driven model identification and efficient MPC via quasi-linear parameter varying representation for ORC waste heat recovery system

Organic Rankine Cycle (ORC) stands out in low-grade waste heat recovery (WHR) technology for its significant performance. Considering the system’s coupling dynamics, model predictive control (MPC) has adopted its wide application in realizing reasonable adjustment of the ORC based WHR system and has been proved to be effective. MPC is usually applied under the premise of having established a relatively accurate model, thus achieving a satisfactory control performance. However, the popular first principles model of the ORC system turns out to be high-dimensional and will result in computationally costly during typical nonlinear MPC adoption. While model linearization around operating points would enable the employment of linear MPC and reduce online calculation amount to some extent, the obtained local model loses global validity, leading to possible unstable control performance. To address these problems, a practical input–output data-driven quasi-linear parameter varying (QLPV) model is constructed for the ORC based WHR system by introducing the Koopman operator to ensure the global control effect. The corresponding MPC algorithm is thus presented via QLPV representation which solves the constructed constrained optimization problem iteratively in the form of a series of quadratic programming (QP) problems at each time step. Moreover, considering the possible lack of adequate training data covering the important dynamics of the ORC based WHR systems in practical application, an online updating mechanism that involves recursive equations is proposed to realize prediction accuracy improvement. Simulations on prediction, setpoint tracking and disturbance rejection are performed to verify the established model accuracy and the control effectiveness of the proposed strategy.

Data assimilation in operator algebras

We develop an algebraic framework for sequential data assimilation of partially observed dynamical systems. In this framework, Bayesian data assimilation is embedded in a nonabelian operator algebra, which provides a representation of observables by multiplication operators and probability densities by density operators (quantum states). In the algebraic approach, the forecast step of data assimilation is represented by a quantum operation induced by the Koopman operator of the dynamical system. Moreover, the analysis step is described by a quantum effect, which generalizes the Bayesian observational update rule. Projecting this formulation to finite-dimensional matrix algebras leads to computational schemes that are i) automatically positivity-preserving and ii) amenable to consistent data-driven approximation using kernel methods for machine learning. Moreover, these methods are natural candidates for implementation on quantum computers. Applications to the Lorenz 96 multiscale system and the El Niño Southern Oscillation in a climate model show promising results in terms of forecast skill and uncertainty quantification.

Computationally Efficient Hierarchical Model Predictive Control Via Koopman Operator

Abstract Combined powertrain and velocity optimization can achieve significant energy efficiency improvements. However, due to the multitime scales in the system, the optimization is performed hierarchically and by separating time scales. To enforce state constraints, iteration between controller is introduced, for example, using Lagrange multipliers as metric for constraint violation. In this paper, an extension of the Koopman operator theory is presented with to obtain a data-driven approximation of the multipliers' behavior hence eliminating the need for iterations. Because the evolution of the Lagrange multipliers is the result of a fast dynamics optimization problem, and not the response of a nonlinear dynamical system, a novel technique in which the Lagrange multipliers are interpreted as a dynamic system is presented here. The approximate Koopman linear system is then derived using extended dynamic mode decomposition and it is integrated with the slow dynamic optimization. Results show that the Koopman augmented controller, which is solved as one single optimization, meets state and input constraints and achieves similar energy savings compared to an iterative approach.

Sorting-free Hill-based stability analysis of periodic solutions through Koopman analysis

In this paper, we aim to study nonlinear time-periodic systems using the Koopman operator, which provides a way to approximate the dynamics of a nonlinear system by a linear time-invariant system of higher order. We propose for the considered system class a specific choice of Koopman basis functions combining the Taylor and Fourier bases. This basis allows to recover all equations necessary to perform the harmonic balance method as well as the Hill analysis directly from the linear lifted dynamics. The key idea of this paper is using this lifted dynamics to formulate a new method to obtain stability information from the Hill matrix. The error-prone and computationally intense task known by sorting, which means identifying the best subset of approximate Floquet exponents from all available candidates, is circumvented in the proposed method. The Mathieu equation and an n-DOF generalization are used to exemplify these findings.

Evaluation of the Regions of Attraction of Higher-Dimensional Hyperbolic Systems Using Extended Dynamic Mode Decomposition

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.

An Improved Koopman-MPC Framework for Data-Driven Modeling and Control of Soft Actuators

The challenge of achieving precise control of soft actuators with strong nonlinearity is mainly due to the difficulty of deriving models suitable for model-based control techniques. Fortunately, Koopman operator provides a data-driven method for constructing control-oriented models of nonlinear systems to achieve model predictive control (MPC). It is called the Koopman-MPC framework, which is theoretically effective for soft actuators. Nevertheless, in this framework, a critical challenge is to select correct basis functions for Koopman-based modeling. Furthermore, there is room for improvement in control performance. To overcome these problems, this letter presents an improved Koopman-MPC framework to efficiently implement model-based control techniques for soft actuators. Firstly, we propose a systematic method for selecting the basis functions, which extends the measurement coordinates with derivative and time-delay coordinates and uses the spares identification of nonlinear dynamics (SINDy) algorithm. Secondly, an incremental model predictive control with dynamic constraints (IMPCDC) is developed based on the Koopman model. Finally, several comparative experiments are conducted to verify the utility of the improved Koopman-MPC framework for data-driven modeling and control of soft actuators.

Koopman-Operator-Based Robust Data-Driven Control for Wheeled Mobile Robots

The Koopman operator theory offers a way to construct explicit control-oriented high-dimensional linear dynamical models for the original nonlinear systems, solely using the input–output data of the dynamical system. The modeling accuracy of the Koopman model largely depends on the basis functions (lifting functions), dimensionality, and data quality. However, there has not been a systematic way to solve the problems mentioned above. In this article, a Koopman-operator-based robust data-driven control framework is proposed for wheeled mobile robots, via incorporating tools from control theory, to solve the problem of modeling errors of the Koopman model. By employing an extended state observer, the modeling errors of the Koopman model, including unknown external disturbances, are online estimated and compensated in the control signal in real time. Then, sliding-mode control is used to synthesize the controller. Importantly, the method of virtual control input is proposed, to cope with the model errors arising from the rotational motion of all the mobile robots. Besides, stability analysis is conducted, and the optimal dimensionality of the Koopman model is experimentally selected. Finally, experimental tests on an omnidirectional mobile robot are carried out to verify the effectiveness of the proposed control scheme, in terms of tracking performance and robustness.