Koopman Operator Theory Research Articles

YoMo: Yoshimura Continuum Manipulator for MR Environment.

Origami robots have garnered attention due to their versatile deformation and potential applications, particularly for medical applications. In this article, we propose a Yoshimura continuum manipulator (YoMo) that can achieve accurate control of the tip position for the magnetic resonance (MR) environment. The YoMo made of a single piece of paper is cable-actuated to generate the bending and shortening deformation. The paper-based YoMo attached to an arc frame can readily function under different orientations in the MR environment. The design and fabrication of YoMo were formulated according to the Yoshimura folding pattern. The kinematics model based on constant curvature assumption was derived as a benchmark method to predict the tip position of the YoMo. The Koopman operator theory was applied to describe the relationship between the tip position and the length change under different orientations. The linear quadratic regulator integrated into the Koopman-based model (K-LQR) was adopted to achieve the trajectory tracking. Comprehensive experiments were carried out to examine the proposed YoMo, its modeling and control methods. The performance of the YoMo including stiffness and workspace was characterized via a customized test setup. The Koopman-based method demonstrates the superiority over the constant curvature-based model to predict the tip position. The K-LQR control method was examined with different trajectories, and the impact of the orientation, speed, and different trajectories were taken into consideration. The results demonstrate the YoMo is capable of achieving trajectory tracking with satisfied accuracy, indicating its potential for medical applications in the MR environment.

A Koopman-based residual modeling approach for the control of a soft robot arm

Soft robots are challenging to model and control due to their poorly defined kinematics and nonlinear dynamics. Recently, Koopman operator theory has been shown capable of constructing control-oriented soft robot models from data. However, building these models requires extensive data collection and they do not necessarily generalize well outside of the training observations. This paper presents a more data-efficient and generalizable approach to soft robot modeling that first identifies a physics-based Koopman model then supplements it with a data-driven residual Koopman model. The resulting combined model is linear and thus compatible with real-time model-based control techniques such as Model Predictive Control (MPC). The efficacy of the approach is demonstrated on several simulated systems and on a real soft robot arm, where it is shown to generate models that are more accurate than purely physics-based models and require less data to construct than purely data-driven models. Using a model-based controller, the soft arm is able to successfully track end effector trajectories, perform a pick-and-place task, and write on a dry-erase board, showcasing the applicability of this framework to increase the capabilities of soft robotic systems.

Variability of SST through Koopman Modes

Abstract The majority of dynamical systems arising from applications show a chaotic character. This is especially true for climate and weather applications. We present here an application of Koopman operator theory to tropical and global sea surface temperature (SST) that yields an approximation to the continuous spectrum typical of these situations. We also show that the Koopman modes yield a decomposition of the datasets that can be used to categorize the variability. Most relevant modes emerge naturally, and they can be identified easily. A difference with other analysis methods such as empirical orthogonal function (EOF) or Fourier expansion is that the Koopman modes have a dynamical interpretation, thanks to their connection to the Koopman operator, and they are not constrained in their shape by special requirements such as orthogonality (as it is the case for EOF) or pure periodicity (as in the case of Fourier expansions). The pure periodic modes emerge naturally, and they form a subspace that can be interpreted as the limiting subspace for the variability. The stationary states therefore are the scaffolding around which the dynamics takes place. The modes can also be traced to the Niño variability and in the case of the global SST to the Pacific decadal oscillation (PDO). Significance Statement We compute the Koopman modes for the tropical and global SST, demonstrating that significant dynamical modes can be identified also in complex high-dimensional datasets with continuous spectra. The Koopman modes are then used to identify the stationary subspace, namely, the limit subspace for the invariant evolution of the system.

Soft Sensors for Industrial Processes Using Multi-Step-Ahead Hankel Dynamic Mode Decomposition with Control

In this paper, a novel data-driven approach for the development of soft sensors (SSs) for multi-step-ahead prediction of industrial process variables is proposed. This method is based on the recent developments in Koopman operator theory and dynamic mode decomposition (DMD). It is derived from Hankel DMD with control (HDMDc) to deal with highly nonlinear dynamics using augmented linear models, exploiting input and output regressors. The proposed multi-step-ahead HDMDc (MSA-HDMDc) is designed to perform multi-step prediction and capture complex dynamics with a linear approximation for a highly nonlinear system. This enables the construction of SSs capable of estimating the output of a process over a long period of time and/or using the developed SSs for model predictive control purposes. Hyperparameter tuning and model order reduction are specifically designed to perform multi-step-ahead predictions. Two real-world case studies consisting of a sulfur recovery unit and a debutanizer column, which are widely used as benchmarks in the SS field, are used to validate the proposed methodology. Data covering multiple system operating points are used for identification. The proposed MSA-HDMDc outperforms currently adopted methods in the SSs domain, such as autoregressive models with exogenous inputs and finite impulse response models, and proves to be robust to the variability of systems operating points.

Deep neural data-driven Koopman fractional control of a worm robot

This paper proposes an innovative approach for enhancing the locomotion control of a worm robot by integrating deep neural networks with Koopman operator theory and fractional order control techniques. Traditional control methods often struggle with the nonlinear dynamics and uncertainties inherent in such systems. The Koopman operator framework facilitates the representation of these dynamics in a linear and infinite-dimensional space, thereby enabling advanced control strategies. By leveraging deep neural networks, we approximate the Koopman operator efficiently, enhancing the system’s adaptability and performance. Additionally, fractional order control is introduced to provide robustness against uncertainties and disturbances, offering greater flexibility in capturing the complex dynamics of the worm robot. Extensive simulations validations demonstrate the effectiveness and efficiency of the proposed approach across various environmental conditions. Results show superior performance compared to traditional methods, affirming the viability of integrating deep neural networks with fractional order control for advanced robotic applications. This study contributes to the field by showcasing a novel methodology that not only achieves precise and robust control of worm robot locomotion but also sets a foundation for future research in adaptive and intelligent robotic systems.

Deep Koopman Operator-based degradation modelling

Developing reliable health indicators for industrial assets is essential for accurate condition monitoring, fault detection, and predicting the remaining useful lifetime. However, constructing such indicators is challenging, especially given the increasing complexity of industrial systems and the not well-understood degradation dynamics. Previously proposed autoencoder-based methods for unsupervised health indicator construction faced the difficulty of constraining the latent representation over the system’s lifetime to obtain trendable and prognosable health indicators. Koopman operator theory provides a natural solution to this challenge. In this work, we first demonstrate the successful application of the Deep Koopman Operator approach for learning the dynamics of industrial systems. This results in a latent representation that provides sufficient information for estimating the remaining useful life of the asset. Secondly, we propose a novel Koopman-Inspired Degradation Model for modelling the degradation of dynamical systems with control. The Koopman-based algorithms demonstrate superior or comparable performance with autoencoder-based approaches in predicting the remaining useful life of assets such as CNC milling machine cutters and Li-ion batteries.

Deep Embedding Koopman Neural Operator-Based Nonlinear Flight Training Trajectory Prediction Approach

Accurate flight training trajectory prediction is a key task in automatic flight maneuver evaluation and flight operations quality assurance (FOQA), which is crucial for pilot training and aviation safety management. The task is extremely challenging due to the nonlinear chaos of trajectories, the unconstrained airspace maps, and the randomization of driving patterns. In this work, a deep learning model based on data-driven modern koopman operator theory and dynamical system identification is proposed. The model does not require the manual selection of dictionaries and can automatically generate augmentation functions to achieve nonlinear trajectory space mapping. The model combines stacked neural networks to create a scalable depth approximator for approximating the finite-dimensional Koopman operator. In addition, the model uses finite-dimensional operator evolution to achieve end-to-end adaptive prediction. In particular, the model can gain some physical interpretability through operator visualization and generative dictionary functions, which can be used for downstream pattern recognition and anomaly detection tasks. Experiments show that the model performs well, particularly on flight training trajectory datasets.

Koopman Spectral Linearization vs. Carleman Linearization: A Computational Comparison Study

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

Data-driven acoustic control of a spherical bubble using a Koopman linear quadratic regulator.

Koopman operator theory has gained interest as a framework for transforming nonlinear dynamics on the state space into linear dynamics on abstract function spaces, which preserves the underlying nonlinear dynamics of the system. These spaces can be approximated through data-driven methodologies, which enables the application of classical linear control strategies to nonlinear systems. Here, a Koopman linear quadratic regulator (KLQR) was used to acoustically control the nonlinear dynamics of a single spherical bubble, as described by the well-known Rayleigh-Plesset equation, with several objectives: (1) simple harmonic oscillation at amplitudes large enough to incite nonlinearities, (2) stabilization of the bubble at a nonequilibrium radius, and (3) periodic and quasiperiodic oscillation with multiple frequency components of arbitrary amplitude. The results demonstrate that the KLQR controller can effectively drive a spherical bubble to radially oscillate according to prescribed trajectories using both broadband and single-frequency acoustic driving. This approach has several advantages over previous efforts to acoustically control bubbles, including the ability to track arbitrary trajectories, robustness, and the use of linear control methods, which do not depend on initial guesses.

Learning-based data-driven optimal deployment control of tethered space robot

To avoid complex constraints of the traditional nonlinear method for tethered space robot (TSR) deployment, a data-driven optimal control framework with an improve deep learning based Koopman operator is proposed in this work. In consideration of nonlinearity of tethered space robot dynamics, its finite dimensional global linear representation called lifted linear system is derived with the Koopman operator theory. A deep learning scheme is adopted to find the embedding functions associate with Koopman operator. And an auxiliary neural network is developed to encode the nonlinear control term of finite dimensional lifted system. Then a controllability constraint is considered for learning a controllable lifted linear system. Besides two loss functions that relate to reconstruction and prediction ability of lifted linear system are designed for training the deep neural network. With the learned lifted linear dynamics, Linear Quadratic Regulator (LQR) is applied to derive the optimal control policy for the tethered space robot deployment. Finally, simulation results verify the effectiveness of proposed framework and show that it could deploy tethered space robot more quickly with less swing of in-plane angle.

RETROFIT: Real-Time Control of Time-Dependent 3D Point Cloud Profiles

Abstract In modern manufacturing processes, ensuring the precision of 3D profiles of products is crucial. Nonetheless, achieving this accuracy is challenging due to the complex interactions between process inputs and the data structure of the 3D profile data. Our solution, a 3D profile-based control framework, addresses this challenge by actively adapting and controlling the manufacturing process to enhance 3D shape accuracy. 3D profile scans represent the ultimate measure of desired part quality. Therefore, utilizing them as the system responses for control purposes yields the most direct and effective feedback. We leverage recent advancements from Koopman operator theory to create an effective model-based control strategy. Initially, we estimate the process model by exploring the relationship between 3D profiles and heterogeneous process inputs. Then, we formulate an online model predictive control law. Challenges include dealing with unstructured, high-dimensional 3D point cloud data, capturing spatial and temporal structures, and integrating heterogeneous, high-dimensional process input data into the control model. To overcome these challenges, we introduce RETROFIT, a solution designed for the real-time control of time-dependent 3D point cloud profiles. Unlike traditional models, RETROFIT is not bound by linear assumptions and can handle unstructured 3D point cloud data directly. We demonstrate its effectiveness through a wire arc additive manufacturing case study, highlighting its potential to enhance 3D profile accuracy in manufacturing processes.

A Koopman operator-based prediction algorithm and its application to COVID-19 pandemic and influenza cases

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

Data-Driven Emergency Frequency Control for Multi-Infeed Hybrid AC-DC System

With the continuous development of large-scale complex hybrid AC-DC grids, the fast adjustability of HVDC systems is required by the grid to provide frequency regulation services. This paper develops a fully data-driven linear quadratic regulator (LQR) for the HVDC to provide temporal frequency support. The main technical challenge is the complexity and the nonlinearity of multi-infeed hybrid AC-DC (MIDC) systems dynamics that make the LQR intractable. Based on Koopman operator (KO) theory, a Koopman eigenpairs construction method is developed to fit a global linear dynamic model of MIDC systems. Once globally linear representation of uncontrolled system dynamics is obtained offline, the control term is constituted by the gradient of the identified eigenfunctions and the control matrix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol{B}$</tex-math></inline-formula> . In case that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol{B}$</tex-math></inline-formula> is unknown, we propose a method to identify it based on the verified Koopman eigenfunctions. The active power reference is optimized online for LCC-HVDC in a moving horizon fashion to provide frequency support, with only locally measurable frequency and transmission power. The robustness of the proposed control method against approximation errors of the linear representation in eigenfunction coordinates is analyzed. Simulation results show the effectiveness, robustness and adaptability of the proposed emergency control strategy.

Koopman Operator theory applied to Lambert’s problem with a spectral behavior analysis

Lambert’s problem has been long studied in the context of space operations; its solution enables accurate orbit determination and spacecraft guidance. This work presents an approximate-analytical solution to Lambert’s problem using the Koopman Operator (KO). In contrast to previous methods in the literature, the KO provides the analysis of a nonlinear system by seeking a transformation that embeds the nonlinear dynamics into a global linear representation. The new methodology to solve for Lambert solutions proposed in this paper considers the position of the system’s eigenvalues on the phase plane, evaluating accurate state transition polynomial maps for a computationally efficient propagation of the dynamics. The minimum-energy and minimum-Δv solutions to Lambert’s problem are found using the KO. The spectral behavior of the solutions is studied and a deeper understanding of the dynamical system is provided. The methodology and solutions are compared in accuracy and performance with other techniques found in the literature, highlighting the benefits of the newly developed analytical approach over classical numerical methodologies.

Analytical Large-Signal Modeling of Inverter-Based Microgrids With Koopman Operator Theory for Autonomous Control

Analytical Large-Signal Modeling of Inverter-Based Microgrids With Koopman Operator Theory for Autonomous Control

Constructing polynomial libraries for reservoir computing in nonlinear dynamical system forecasting.

Reservoir computing is an effective model for learning and predicting nonlinear and chaotic dynamical systems; however, there remains a challenge in achieving a more dependable evolution for such systems. Based on the foundation of Koopman operator theory, considering the effectiveness of the sparse identification of nonlinear dynamics algorithm to construct candidate nonlinear libraries in the application of nonlinear data, an alternative reservoir computing method is proposed, which creates the linear Hilbert space of the nonlinear system by including nonlinear terms in the optimization process of reservoir computing, allowing for the application of linear optimization. We introduce an implementation that incorporates a polynomial transformation of arbitrary order when fitting the readout matrix. Constructing polynomial libraries with reservoir-state vectors as elements enhances the nonlinear representation of reservoir states and more easily captures the complexity of nonlinear systems. The Lorenz-63 system, the Lorenz-96 system, and the Kuramoto-Sivashinsky equation are used to validate the effectiveness of constructing polynomial libraries for reservoir states in the field of state-evolution prediction of nonlinear and chaotic dynamical systems. This study not only promotes the theoretical study of reservoir computing, but also provides a theoretical and practical method for the prediction of nonlinear and chaotic dynamical system evolution.

Koopman operator learning using invertible neural networks

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.

Data-Driven Model Predictive Control for Uncalibrated Visual Servoing

This paper addresses the image-based visual servoing (IBVS) control problem with an uncalibrated camera, unknown dynamics, and constraints. A novel data-driven uncalibrated IBVS (UIBVS) strategy is proposed, incorporated with the Koopman-based model predictive control (KMPC) algorithm and the adaptive robust Kalman filter (ARKF). First, to alleviate the need for calibration of the camera’s intrinsic and extrinsic parameters, the ARKF with an adaptive factor is utilized to estimate the image Jacobian matrix online, thereby eliminating the laborious camera calibration procedures and improving robustness against camera disturbances. Then, a data-driven MPC strategy is proposed, wherein the unknown nonlinear dynamic model is learned using the Koopman operator theory, resulting in a linear Koopman prediction model. Only input–output data are used to construct the prediction model, and hence, the proposed approach is robust against model uncertainties. Furthermore, with a symmetric quadratic cost function, the proposed approach solves the quadratic programming problem online, and visibility constraints as well as joint torque constraints are taken into account. As a result, the proposed KMPC scheme can be implemented in real time, and the UIBVS performance degradation which arises from the control torque constraints can be avoided. Simulations and comparisons for a 2-DOF robotic manipulator demonstrate the feasibility of the proposed approach. Simulation results further validate that the computation time of the proposed approach is comparable to the one of kinematic-based methods.

Credit assignment for trained neural networks based on Koopman operator theory

Credit assignment for trained neural networks based on Koopman operator theory

Control of soft robots with inertial dynamics.

Soft robots promise improved safety and capability over rigid robots when deployed near humans or in complex, delicate, and dynamic environments. However, infinite degrees of freedom and the potential for highly nonlinear dynamics severely complicate their modeling and control. Analytical and machine learning methodologies have been applied to model soft robots but with constraints: quasi-static motions, quasi-linear deflections, or both. Here, we advance the modeling and control of soft robots into the inertial, nonlinear regime. We controlled motions of a soft, continuum arm with velocities 10 times larger and accelerations 40 times larger than those of previous work and did so for high-deflection shapes with more than 110° of curvature. We leveraged a data-driven learning approach for modeling, based on Koopman operator theory, and we introduce the concept of the static Koopman operator as a pregain term in optimal control. Our approach is rapid, requiring less than 5 min of training; is computationally low cost, requiring as little as 0.5 s to build the model; and is design agnostic, learning and accurately controlling two morphologically different soft robots. This work advances rapid modeling and control for soft robots from the realm of quasi-static to inertial, laying the groundwork for the next generation of compliant and highly dynamic robots.