Koopman Operator Research Articles

Design of Sliding Mode Controllers using Reduced-order Koopman Mode Representations

The Koopman operator framework allows for a linear, but infinite-dimensional, representation of the dynamics of a non-linear system. The Koopman modes, or observables, and the resulting linear dynamics are derived purely using a data-driven framework, where the data are system outputs measured at discrete samples; improving accuracy of the Koopman representation requires a large number of such modes to be considered. Recent results consider the system input as well, in the derivation of the discrete linear dynamics, thus enabling the design of controllers. Sliding mode controllers (SMCs), including the discrete-time versions, can handle parameter uncertainties and variations and also ensure that the control objective is satisfied in finite time. In this paper, a discrete-time SMC is designed for the output control of a dynamic system approximated by fewer Koopman modes; the SMC is expected to handle uncertainties introduced by the ignored modes. Conditions are identified for the closed-loop system to be stable, with the occurrence of sliding mode.

HAVOK Model Predictive Control for Time-Delay Systems with Applications to District Heating

A computationally efficient Model-Predictive Control (MPC) approach is proposed for systems with unknown delay using only input/output data. We use the Koopman operator framework and the related Hankel Alternative View of Koopman (HAVOK) algorithm to identify a model in a basis of projected time-delay coordinates and demonstrate a novel MPC structure that reduces and bounds the computational complexity. The proposed HAVOK-MPC approach is validated experimentally on a laboratory-scale District Heating System (DHS), demonstrating excellent prediction and tracking performance while only requiring knowledge of a conservative upper bound on the system delay.

Learning globally linear predictors using deep Koopman embeddings with application to marine vehicles

Linearity of the model for controlled dynamical systems is a very desirable property because of its simplicity in the state prediction and control. Koopman operator theory provides a framework for global mapping of a nonlinear system into an equivalent linear system. The goal of this work is to exploit Koopman theory and modern machine learning techniques to find the linear system representation of the underlying nonlinear system for future state predictions. The model generated in this way is completely data driven and requires no a priori knowledge of the underlying dynamics system. The model is applied to two marine vehicles whose trajectories are generated using simulation and evaluated against common model identification techniques. The results show that proposed method is comparable to conventional identification methods and even outperforms them in cases when complex nonlinear dynamics, which is often neglected, becomes relevant.

An Online Data-Driven Method for Microgrid Secondary Voltage and Frequency Control With Ensemble Koopman Modeling

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 ensemble Koopman state space modeling with full basis functions that combines both linear and nonlinear bases without the need of event detection or switching. An iterative learning method is also developed to exploit model parameters, ensuring the effectiveness and the adaptiveness of the designed control. Simulation studies in the 4-bus (with detailed inner-loop control) MG system and the 34-bus MG system showed improved modeling accuracy and control, verifying the effectiveness of the proposed method subject to various changes of operating conditions even with time delay, measurement noise, and missing measurements.

Nonlinear Sampled-data Systems—A lifting framework

This short note gives a new framework for dealing with nonlinear sampled-data systems. We introduce a new idea of lifting, which is well known for linear systems, but not successfully generalized to nonlinear systems. This paper introduces a new lifting technique for nonlinear, time-invariant systems, which are different from the linear counterpart as developed in Bamieh et al. (1991); Yamamoto (1994) etc. The main difficulty is that the direct feedthrough term effective in the linear case cannot be generalized to the nonlinear case. Instead, we will further lift the state trajectory, and obtain an equivalent time-invariant discrete-time system with function-space input and output spaces. The basic framework, as well as the closed-loop equation with discrete-time controller, is given. As an application of this framework, we give a representation for the Koopman operator derived from the given original nonlinear system.

Safe Control Design for Unknown Nonlinear Systems with Koopman-based Fixed-Time Identification

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 in the system within a finite-time, independent of an initial estimate. We then use properties of fixed-time stability to derive an estimation error bound on the unknown dynamics as an explicit function of time, which allows us to synthesize a safe controller using control barrier function based methods. We conduct a quadrotor-inspired case study in support of our proposed method, in which we show that safe trajectory tracking is achieved despite unknown, nonlinear dynamics.

Data-Driven Modeling and Distributed Predictive Control of Mixed Vehicle Platoons

With the development of automatic driving technology and the internet of vehicles, platooning based on control of connected autonomous vehicles has become one of the most promising approaches to improve traffic efficiency. This paper studies the control problem of mixed vehicle platoons consisting of human-driven vehicles and connected autonomous vehicles. Firstly, we propose a data-driven method to model mixed vehicle platoons based on Koopman operator theory. This method gives a way to represent the mixed vehicle platoon by a linear model in a high-dimensional space, the approximation of which is obtained by a neural network framework. Secondly, we employ model predictive control (MPC) to address the platoon control problem of mixed vehicle platoons, where both centralized MPC and distributed MPC algorithms are designed. Finally, the effectiveness of the data-driven modeling method and the centralized/distributed MPC algorithms is verified by numerical simulations. It is revealed that the proposed data-driven DMPC algorithm exhibits comparable control performance with less computation cost compared with the centralized MPC algorithm, and it shows faster convergence speed than the nonlinear model based DMPC algorithm.

Koopman Operator Approximation under Negative Imaginary Constraints

Koopman Operator Approximation under Negative Imaginary Constraints

A Data-Driven Deep Neural Network for Modeling of Ionospheric Clutter in HFSWR

In the practical application of 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 paper, we proposed a method merging the deep neural networks 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 modelling accuracy, which proves the effectiveness of the approach and lays the foundations for the subsequent breakthrough in new methods for ionospheric clutter suppression.

Koopman Representation for Boolean Networks

Boolean networks, as logical dynamical systems whose system states are Boolean variables, arise from applications in biology, computer networks, and social networks etc. The representation and control of Boolean networks have attracted a lot of attention in recent years. On a parallel line of research, Koopman developed an operator view of nonlinear dynamical systems in early 1930s, which shows that using observable functions all nonlinear dynamics can be represented as (infinite dimensional) linear systems. In this paper, we present a framework for representing Boolean networks via Koopman approach. First of all, we construct addition and scalar multiplication operations over the set of logical functions over the binary field, defining a linear Boolean function space. Then associated with any Boolean mapping, the induced Koopman operator is defined as an operator over such Boolean function space. Next, we show that if there exists a set of logical functions that are invariant in the Koopman sense, a Boolean network can be represented by a finite-dimensional linear system. This Koopman perspective for Boolean networks is also extended to controlled Boolean networks.

Network Structure Identification Based on Measured Output Data Using Koopman Operators

This paper considers the identification problem of network structures of interconnected dynamical systems using measured output data. In particular, we propose an identification method based on the measured output data of each node in the network whose dynamic is unknown. The proposed identification method consists of three steps: we first consider the outputs of the nodes to be all the states of the dynamics of the nodes, and the unmeasurable hidden states to be dynamical inputs with unknown dynamics. In the second step, we define the dynamical inputs as new variables and identify the dynamics of the network system with measured output data using Koopman operators. Finally, we extract the network structure from the identified dynamics as the information transmitted via the network. We show that the identified coupling functions, which represent the network structures, are actually projections of the dynamical inputs onto the space spanned by some observable functions. Numerical examples illustrate the validity of the obtained results.

Automatic Tracking of Surgical Instruments with a Continuum Laparoscope Using Data‐Driven Control in Robotic Surgery

In existing surgery process, surgeons need to manually adjust the laparoscopes to provide a better field of view (FOV) during operation, which may distract surgeons and slow down the surgery process. Herein, a data‐driven control method that uses a continuum laparoscope to adjust the FOV by tracking the surgical instruments is presented. A Koopman‐based system identification method is first applied to linearize the nonlinear system. Shifted Chebyshev polynomials are used to construct observation functions that transfer low‐dimension observations to high‐dimension ones. The Koopman operator is approximated using a finite‐dimensional estimation method. An optimal controller is further developed according to the trained linear dynamic model. Furthermore, a learning‐based pose estimation framework is designed to detect keypoints on surgical instruments and provides visual feedback for the control system. Compared with other detection methods, the proposed scheme achieves a higher detection precision and provides more optional keypoints for tracking. Simulation and experiments validate the feasibility of the proposed control method. Experimental results show that the proposed method can automatically adjust the field of continuum laparoscope by tracking surgical instruments in real time and satisfy the clinical requirements.

Nonlinear set‐membership state estimation based on the Koopman operator

SummaryIn 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.

Parametric dynamic mode decomposition for reduced order modeling

Dynamic Mode Decomposition (DMD) is a model-order reduction approach, whereby spatial modes of fixed temporal frequencies are extracted from numerical or experimental data sets. The DMD low-rank or reduced operator is typically obtained by singular value decomposition of the temporal data sets. For parameter-dependent models, as found in many multi-query applications such as uncertainty quantification or design optimization, the only parametric DMD technique developed was a stacked approach, with data sets at multiple parameter values were aggregated together, increasing the computational work needed to devise low-rank dynamical reduced-order models. In this paper, we present two novel approach to carry out parametric DMD: one based on the interpolation of the reduced-order DMD eigen-pair and the other based on the interpolation of the reduced DMD (Koopman) operator. Numerical results are presented for diffusion-dominated nonlinear dynamical problems, including a multiphysics radiative transfer example. All three parametric DMD approaches are compared.

An operator methodology for the global dynamic analysis of stochastic nonlinear systems

In a global dynamic analysis, the coexisting attractors and their basins are the main tools to understand the system behavior and safety. However, both basins and attractors can be drastically influenced by uncertainties. The aim of this work is to illustrate a methodology for the global dynamic analysis of nondeterministic dynamical systems with competing attractors. Accordingly, analytical and numerical tools for calculation of nondeterministic global structures, namely attractors and basins, are proposed. First, based on the definition of the Perron-Frobenius, Koopman and Foias linear operators, a global dynamic description through phase-space operators is presented for both deterministic and nondeterministic cases. In this context, the stochastic basins of attraction and attractors’ distributions replace the usual basin and attractor concepts. Then, numerical implementation of these concepts is accomplished via an adaptative phase-space discretization strategy based on the classical Ulam method. Sample results of the methodology are presented for a canonical dynamical system.

Predictive Closed-Loop Remote Control Over Wireless Two-Way Split Koopman Autoencoder

Real-time remote control over wireless is an important yet challenging application in fifth-generation and beyond due to its mission-critical nature under limited communication resources. Current solutions hinge on not only utilizing ultrareliable and low-latency communication (URLLC) links but also predicting future states, which may consume enormous communication resources and struggle with a short prediction time horizon. To fill this void, in this article we propose a novel two-way Koopman autoencoder (AE) approach wherein: 1) a sensing Koopman AE learns to understand the temporal state dynamics and predicts missing packets from a sensor to its remote controller and 2) a controlling Koopman AE learns to understand the temporal action dynamics and predicts missing packets from the controller to an actuator co-located with the sensor. Specifically, each Koopman AE aims to learn the Koopman operator in the hidden layers while the encoder of the AE aims to project the nonlinear dynamics onto a lifted subspace, which is reverted into the original nonlinear dynamics by the decoder of the AE. The Koopman operator describes the linearized temporal dynamics, enabling long-term future prediction and coping with missing packets and closed-form optimal control in the lifted subspace. Simulation results corroborate that the proposed approach achieves a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$38\times $ </tex-math></inline-formula> lower mean squared control error at 0-dBm signal-to-noise ratio (SNR) than the nonpredictive baseline.

The Koopman Operator in Nonlinear Theory and its Application to Power System Engineering

1．はじめに

Hybrid Scheme of Kinematic Analysis and Lagrangian Koopman Operator Analysis for Short-Term Precipitation Forecasting

With the accumulation of meteorological big data, data-driven models for short-term precipitation forecasting have shown increasing promise. We focus on Koopman operator analysis, which is a data-driven scheme to discover governing laws in observed data. We propose a method to apply this scheme to phenomena accompanying advection currents such as precipitation. The proposed method decomposes time evolutions of the phenomena between advection currents under a velocity field and changes in physical quantities under Lagrangian coordinates. The advection currents are estimated by kinematic analysis, and the changes in physical quantities are estimated by Koopman operator analysis. The proposed method is applied to actual precipitation distribution data, and the results show that the development and decay of precipitation are properly captured relative to conventional methods and that stable predictions over long periods are possible.

Koopman and Perron-Frobenius operators on reproducing kernel Banach spaces.

Koopman and Perron-Frobenius operators for dynamical systems are becoming popular in a number of fields in science recently. Properties of the Koopman operator essentially depend on the choice of function spaces where it acts. Particularly, the case of reproducing kernel Hilbert spaces (RKHSs) is drawing increasing attention in data science. In this paper, we give a general framework for Koopman and Perron-Frobenius operators on reproducing kernel Banach spaces (RKBSs). More precisely, we extend basic known properties of these operators from RKHSs to RKBSs and state new results, including symmetry and sparsity concepts, on these operators on RKBS for discrete and continuous time systems.

데이터 기반의 자율 주행 차량 모델링: Koopman Operator 접근법과 그 응용

This paper presents a data-driven modeling method with the Koopman operator for automated vehicles. The lateral motion of a vehicle can be calculated using a linear motion model; however, owing to the complex interactions between the tires and road, a vehicle