In this paper, we propose a Koopman operator based approach to describe the nonlinear dynamics of a quadrotor on SE(3) in terms of an infinite-dimensional linear system which evolves in the space of observable functions (lifted space) and which is more appropriate for control design purposes. The major challenge when using the Koopman operator is the characterization of a set of observable functions that can span the lifted space. Recent methods either use tools from machine learning to learn the observable functions or guess a suitable set of observables that best describes the nonlinear dynamics. Instead of guessing or learning the observables, in this work we derive them in a systematic way for the quadrotor dynamics on SE(3). In addition, we prove that the proposed sequence of observable functions converges pointwise to the zero function, which allows us to select only a finite set of observable functions to form (an approximation of) the lifted space. Our theoretical analysis is also confirmed by numerical simulations which demonstrate that by increasing the dimension of the lifted space, the derived linear state space model can approximate the nonlinear quadrotor dynamics more accurately.
Koopman-based learning methods can potentially be practical and powerful tools for dynamical robotic systems. However, common methods to construct Koopman representations seek to learn lifted linear models that cannot capture nonlinear actuation effects inherent in many robotic systems. This paper presents a learning and control methodology that is a first step towards overcoming this limitation. Using the Koopman canonical transform, control-affine dynamics can be expressed by a lifted bilinear model. The learned model is used for nonlinear model predictive control (NMPC) design where the bil…
This paper presents a mathematic dynamic model of a quadrotor unmanned aerial vehicle (QUAV) by using the symbolic regression approach and then proposes a hierarchical control scheme for trajectory tracking. The symbolic regression approach is capable of constructing analytical quadrotor dynamic equations only through the collected data, which relieves the burden of first principle modeling. To facilitate position tracking control of a QUAV, the design of controller can be decomposed into two parts: a proportional-integral controller for the position subsystem is first designed to obtain the d…
This paper introduces an analytical framework for the derivation of hybrid equations of motion of a flexible quadrotor. This approach helps obtain rigid and elastic equations of motion simultaneously, in a decoupled form, which facilitates the controller design. A delay-dependent stability condition is obtained for the overall system dynamics, including the operator with reaction time delay, the adaptive controller and the flexible quadrotor dynamics. Two different adaptive controllers are implemented to address the uncertainties. It is demonstrated via simulations that the flexible arm tip os…
We consider the problem of safe control design for a class of nonlinear, control-affine systems subject to an unknown, additive, nonlinear disturbance. Leveraging recent advancements in the application of Koopman operator theory to the field of system identification and control, we introduce a novel fixed-time identification scheme for the infinitesimal generator of the infinite-dimensional, but notably linear, Koopman dynamical system analogous to the nonlinear system of interest. That is, we derive a parameter adaptation law that allows us to recover the unknown, residual nonlinear dynamics …
Modularized Koopman Bilinear Form (M-KBF) is presented to model and predict the transient dynamics of microgrids in the presence of disturbances. As a scalable data-driven approach, M-KBF divides the identification and prediction of the high-dimensional nonlinear system into the individual study of subsystems; and thus, alleviating the difficulty of intensively handling high volume data and overcoming the curse of dimensionality. For each subsystem, Koopman bilinear form is applied to efficiently identify its model by developing eigenfunctions via the extended dynamic mode decomposition method…
In this article we design a backstepping control law based on geometric principles to swing up a spherical pendulum mounted on a moving quadrotor. The available degrees of freedom in the control vector also permit us to position the plane of the quadrotor parallel to the ground. The problem addressed here is, indeed, novel and has many practical applications which arise during the transport of a payload mounted on top of a quadrotor. The modeling and control law are coordinate-free and thus avoid singularity issues. The geometric treatment of the problem greatly simplifies both the modeling an…
Low inertia, nonlinearity and a high level of uncertainty (varying topologies and operating conditions) pose challenges to microgrid (MG) systemwide operation. This paper proposes an online adaptive Koopman operator optimal control (AKOOC) method for MG secondary voltage and frequency control. Unlike typical data-driven methods that are data-hungry and lack guaranteed stability, the proposed AKOOC requires no warm-up training yet with guaranteed bounded-input-bounded-output (BIBO) stability and even asymptotical stability under some mild conditions. The proposed AKOOC is developed based on an …
A turbocharger plays an essential part in reducing emissions and increasing the fuel efficiency of road vehicles. The pulsating flow of exhaust gases, along with high heat exchange from the turbocharger casing, makes developing control-oriented models difficult. Several researchers have used maps provided by manufacturers to solve this problem. These maps often fail to incorporate any heat transfer effects and are unsuitable for wide operating regions. Also, with the availability of more and better sensor data, there is a need for a method that can exploit this to obtain a better predictive mo…
In this paper, we systematically derive a finite set of Koopman based observables to construct a lifted linear state space model that describes the rigid body dynamics based on the dual quaternion representation. In general, the Koopman operator is a linear infinite dimensional operator, which means that the derived linear state space model of the rigid body dynamics will be infinite-dimensional, which is not suitable for modeling and control design purposes. Recently, finite approximations of the operator computed by means of methods like the Extended Dynamic Mode Decomposition (EDMD) have sh…
Generalized bilinear Koopman realization from input-output data for multi-step prediction with metaheuristic optimization of lifting function and its application to real-world industrial system
Tokyo City University（东京都市大学）；Kyushu University（九州大学）；Hokkaido University（北海道大学）；ISUZU Advanced Engineering Center Ltd.（五十铃高级工程中心有限公司）
CKNet: A Convolutional Neural Network Based on Koopman Operator for Modeling Latent Dynamics from Pixels
College of Intelligence Science and Technology, National University of Defense Technology（国防科技大学智能科学与技术学院）；Unmanned System Technology Research Center, NIIDT（NIIDT无人系统技术研究中心）
Koopman operators are of infinite dimension and capture the characteristics of nonlinear dynamics in a lifted global linear manner. The finite data-driven approximation of Koopman operators results in a class of linear predictors, useful for formulating linear model predictive control (MPC) of nonlinear dynamical systems with reduced computational complexity. However, the robustness of the closed-loop Koopman MPC under modeling approximation errors and possible exogenous disturbances is still a crucial issue to be resolved. Aiming at the above problem, this paper presents a robust tube-based M…
Physics-informed Deep Mixture-of-Koopmans Vehicle Dynamics Model with Dual-branch Encoder for Distributed Electric-drive Trucks
School of Vehicle and Mobility, and State Key Laboratory of Intelligent Green Vehicle and Mobility, Tsinghua University, Beijing, China（车辆与移动系统学院和智能绿色车辆与移动系统国家重点实验室，清华大学，北京，中国）；KargoBot Inc., Beijing, China（KargoBot公司，北京，中国）
Differentiable Physics-Informed Adaptive Koopman Control for Stable Flight under Unknown Disturbances
Research Center for Intelligent Robotics, School of Astronautics, Northwestern Polytechnical University（西北工业大学航天学院智能机器人研究中心）；Department of Automation, Tsinghua University（清华大学自动化系）
MAKO: Meta-Adaptive Koopman Operators for Learning-based Model Predictive Control of Parametrically Uncertain Nonlinear Systems
Water Research Institute (NEWRI), Nanyang Technological University, Singapore（新跃大学水研究 institute（NEWRI））；School of Chemistry, Chemical Engineering and Biotechnology, Nanyang Technological University, Singapore（新跃大学化学、化工与生物技术学院）；Soft Robotics Lab, ETH Zurich, Switzerland（苏黎世联邦理工学院软机器人实验室）；Department of Civil and Environmental Engineering, National University of Singapore, Singapore（新加坡国立大学土木与环境工程系）
University of California, Santa Barbara（加州大学圣巴巴拉分校）
Soft robots promise improved safety and capability over rigid robots when deployed in complex, delicate, and dynamic environments. However, the infinite degrees of freedom and highly nonlinear dynamics of these systems severely complicate their modeling and control. As a step toward addressing this open challenge, we apply the data-driven, Hankel Dynamic Mode Decomposition (HDMD) with time delay observables to the model identification of a highly inertial, helical soft robotic arm with a high number of underactuated degrees of freedom. The resulting model is linear and hence amenable to contro…
In this paper, a unified batch-online learning approach is introduced to learn a linear representation of nonlinear system dynamics using the Koopman operator. The presented system modeling approach leverages a novel incremental Koopman-based update law that retrieves a mini-batch of samples stored in a memory to not only minimizes the instantaneous Koopman operator's identification errors but also the identification errors for the batch of retrieved samples. Discontinuous modifications of gradient flows are presented for the online update law to assure finite-time convergence under easy-to-ve…
Observer-Based Estimation and Hydrostatic Inertia Modeling for Cooperative Transport of Variable-Inertia Loads with Quadrotors
Norwegian University of Science and Technology（挪威科技大学）；Universidad de San Andrés (UdeSA)（圣安德烈斯大学）；CONICET（阿根廷国家科学研究委员会）；Centre for Automation and Robotics (CSIC-UPM)（自动化与机器人中心）
Deep Neural Koopman Operator-based Economic Model Predictive Control of Shipboard Carbon Capture System
Nanyang Technological University（南洋理工大学）；School of Chemistry, Chemical Engineering and Biotechnology（化学、化学工程与生物技术学院）；Nanyang Environment & Water Research Institute (NEWRI)（南洋环境与水研究所）
Control Forward-Backward Consistency: Quantifying the Accuracy of Koopman Control Family Models
Division of Engineering and Applied Science, California Institute of Technology（加州理工学院工程与应用科学系）；Department of Mechanical and Aerospace Engineering, University of California, San Diego（加州大学圣地亚哥分校机械与航空航天工程系）
Dept. of Electrical and Computer Engineering, University of California, Riverside（加州大学河滨分校电气与计算机工程系）；Dept. of Mechanical and Aerospace Engineering, University of California San Diego（加州大学圣地亚哥分校机械与航空航天工程系）；Dept. of Planning and Control, Zoox Inc.（Zoox公司规划与控制部）；Dept. of Mechanical Engineering, University of Michigan（密歇根大学机械工程系）；Dept. of Mechanical Engineering, Yale University（耶鲁大学机械工程系）；Dept. of Mechanical Engineering, Northwestern University（西北大学机械工程系）；Institute of AI Industry Research (AIR), Tsinghua University（清华大学人工智能产业研究院）
