Koopman Mode Research Articles

Representing turbulent statistics with partitions of state space. Part 2. The compressible Euler equations

This is the second part of a two-part paper. We apply the methodology of the first paper (Souza, J. Fluid Mech., vol. 997, 2024, A1) to construct a data-driven finite-volume discretization of the Liouville/Fokker–Planck equation of a high-dimensional dynamical system, i.e. the compressible Euler equations with gravity and rotation evolved on a thin spherical shell. We show that the method recovers a subset of the statistical properties of the underlying system, steady-state distributions of observables and autocorrelations of particular observables, as well as revealing the global Koopman modes of the system. We employ two different strategies for the partitioning of a high-dimensional state space, and explore their consequences.

Dynamic mode decomposition and short-time prediction of PM2.5 using the graph Neural Koopman network

Analyzing and accurately predicting the spatiotemporal dynamics of PM2.5 remain challenging. The existing spatiotemporal prediction approaches are associated with high model complexity and limited interpretability. Conventional methods combining Koopman theory and deep learning often neglect spatial correlations in spatiotemporal data. This study used the hourly PM2.5 dataset of the Beijing-Tianjin-Hebei region to reveal its spatiotemporal hierarchy using Koopman mode decomposition to identify the key dynamic modes. Furthermore, a Spatial Physics Constrained Learning (SPCL) model utilizing a graph representation learning method was proposed to combine the graph topological information of the PM2.5 spatial features with the Koopman feature function. The results showed that PM2.5 has growth, decay, and oscillation modes as well as daily, weekly, monthly, and yearly periods. SPCL achieved mean absolute error, root mean square error (RMSE), correlation r, and index of agreement values of 9.678, 13.922, 0.864, and 0.921, respectively. The average RMSE at 12 h improved by 16.1%, 12.7%, 0.9%, and 3.5% compared with using Long short-term Memory, Graph Convolutional Networks and Long Short-Term Memory Networks, Spatio-Temporal Graph Convolutional Networks, and Dynamic Spatiotemporal Graph Convolution Network, respectively. By discretizing the neural network hidden layers, the explanatory key of PM2.5 modes was elucidated, which demonstrated enhanced stability.

Analysis of Global and Key PM2.5 Dynamic Mode Decomposition Based on the Koopman Method

Understanding the spatiotemporal dynamics of atmospheric PM2.5 concentration is highly challenging due to its evolution processes have complex and nonlinear patterns. Traditional mode decomposition methods struggle to accurately capture the mode features of PM2.5 concentrations. In this study, we utilized the global linearization capabilities of the Koopman method to analyze the hourly and daily spatiotemporal processes of PM2.5 concentration in the Beijing–Tianjin–Hebei (BTH) region from 2019 to 2021. This approach decomposes the data into the superposition of different spatial modes, revealing their hierarchical spatiotemporal structure and reconstructing the dynamic processes. The results show that PM2.5 concentrations exhibit high-frequency cycles of 12 and 24 h, as well as low-frequency cycles of 124 and 353 days, while also revealing spatiotemporal modes of growth, recession, and oscillation. The superposition of these modes enables the reconstruction of spatiotemporal dynamics with a mean absolute percentage error (MAPE) of only 0.6%. Unlike empirical mode decomposition (EMD), Koopman mode decomposition (KMD) method avoids mode aliasing and provides a clearer identification of global and key modes compared to wavelet analysis. These findings underscore the effectiveness of KMD method in analyzing and reconstructing the spatiotemporal dynamics of PM2.5 concentration, offering new insights into the understanding and reconstruction of other complex spatiotemporal phenomena.

Koopman-inspired data-driven quantification of fluid–structure energy transfers

The linear-time-invariance notion to the Koopman analysis is a recent advance in fluid mechanics [Li et al., “The linear-time-invariance notion to the Koopman analysis: The architecture, pedagogical rendering, and fluid–structure association,” Phys. Fluids 34(12), 125136 (2022c) and Li et al., “The linear-time-invariance notion of the Koopman analysis—Part 2. Dynamic Koopman modes, physics interpretations and phenomenological analysis of the prism wake,” J. Fluid Mech. 959, A15 (2023a)], targeting the long-standing issue of correlating nonlinear excitation and response phenomena in fluid–structure interactions (FSI), or, in the simplified case, flow over rigid obstacles. Continuing the serial research, this work presents a data-driven, Koopman-inspired methodology to decouple nonlinear FSI by establishing cause-and-effect correspondences between structure surface pressure and the flow field. Exploiting unique features of the Koopman operator, the new methodology renders dynamic visualizations of in-sync, fluid–structure-coupled Koopman modes possible, fostering phenomenological analysis and statistical quantifications of FSI energy transfers. Instantaneous contribution contours and densities offer new angles to evaluate pathways of energy amplification and diminution. The methodology enables better descriptions and interpretations of phenomena occurring in the flow and on the boundary (walls) of an FSI domain and readily applies to a broad spectrum of engineering problems given its data-driven nature.

Another look at residual dynamic mode decomposition in the regime of fewer snapshots than dictionary size

Residual Dynamic Mode Decomposition (ResDMD) offers a method for accurately computing the spectral properties of Koopman operators. It achieves this by calculating an infinite-dimensional residual from snapshot data, thus overcoming issues associated with finite truncations of Koopman operators (e.g., Extended Dynamic Mode Decomposition), such as spurious eigenvalues. Spectral properties computed by ResDMD include spectra, pseudospectra, spectral measures, Koopman mode decompositions, and dictionary verification. In scenarios where the number of snapshots is fewer than the dictionary size, particularly for exact DMD and kernelized Extended DMD, ResDMD has traditionally been applied by dividing snapshot data into a training set and a quadrature set. We demonstrate how to eliminate the need for two datasets through a novel computational approach of solving a dual least-squares problem. We analyze these new residuals for exact DMD and kernelized Extended DMD, demonstrating ResDMD’s versatility and broad applicability across various dynamical systems, including those modeled by high-dimensional and nonlinear observables. The utility of these new residuals is showcased through three diverse examples: the analysis of a cylinder wake, the study of airfoil cascades, and the compression of transient shockwave experimental data. This approach not only simplifies the application of ResDMD but also extends its potential for deeper insights into the dynamics of complex 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.

Featurizing Koopman mode decomposition for robust forecasting.

This article introduces an advanced Koopman mode decomposition (KMD) technique-coined Featurized Koopman Mode Decomposition (FKMD)-that uses delay embedding and a learned Mahalanobis distance to enhance analysis and prediction of high-dimensional dynamical systems. The delay embedding expands the observation space to better capture underlying manifold structures, while the Mahalanobis distance adjusts observations based on the system's dynamics. This aids in featurizing KMD in cases where good features are not a priori known. We show that FKMD improves predictions for a high-dimensional linear oscillator, a high-dimensional Lorenz attractor that is partially observed, and a cell signaling problem from cancer research.

A local and hierarchical Koopman spectral analysis of fluid dynamics

AbstractA local and hierarchical Koopman spectral analysis is proposed to extend Koopman spectral analysis typically used in a linear system or an ergodic process to its application in general nonlinear dynamics. The continuous and analytic Koopman eigenfunctions and eigenvalues, derived from operator perturbation theory, are capable of dealing with a nonlinear transition process with mathematical rigorousness. A proliferation rule is identified to derive high‐order eigenvalues and eigenfunctions from lower‐order ones, thus various spectral patterns may be generated through recursive proliferations. The locally linear map around each state constructs base local Koopman eigenvalues and eigenfunctions from an algebraic eigenvalue problem, and high‐order ones are generated via the proliferation rule to express the systematic nonlinearity. The aforementioned hierarchy simplifies the Koopman spectral analysis and is verified by studying the development of Kármán vortex streets. The triangular chain and the lattice distribution of Koopman eigenvalues confirm the critical role of the proliferation rule and the hierarchy structure of Koopman eigenvalues. The local spectral analysis on the transition process shows that the periodic flow forms as the growth rates of the critical Koopman modes reduce to zero, and meanwhile, the Koopman modes at the same frequency superpose on each other to form the well‐known Fourier or Floquet modes, where the latter are the enhanced nonlinear motions due to the alignment of Koopman eigenvalues with the critical ones.

Koopman mode analysis on discovering distributed energy transfer of post-transient flutter of a bluff body

The flutter response of a rectangular cylinder model with an aspect ratio of 5 is investigated by direct numerical simulation for various Reynolds numbers (Re) and flow angles of attack (α). The results show that the lower α and Re, the larger the torsional amplitude at the critical reduced flow velocity (Ur) corresponding to the occurrence of flutter. High-order Koopman mode analysis method (HOKM) was developed to synchronously capture the inherent flow and structural modes. The element-mode-based energy transfer between cylinders and flow modes was used to reveal the underlying mechanism. HOKM analysis shows that only odd-order modes contribute to the occurrence of flutter, with an augmentation in Ur enhancing this phenomenon. By analyzing the work done in odd-order modes, it is found that the primary mode always occupies the main contribution to the work done. The leading-edge endpoint portion is always where the maximum work is done. An escalation in Re and α notably influences both the spatially relevant part (WΦ) and the temporally relevant part (Wc), while an increase in Ur predominantly impacts WΦ. This study offers universal guidance on the safety and protection of ocean engineering structures and flow control strategies.

Characterizing turbulence structures in convective and neutral atmospheric boundary layers via Koopman mode decomposition and unsupervised clustering

The atmospheric boundary layer (ABL) is a highly turbulent geophysical flow, which has chaotic and often too complex dynamics to unravel from limited data. Characterizing coherent turbulence structures in complex ABL flows under various atmospheric regimes is not systematically well established yet. This study aims to bridge this gap using large eddy simulations (LESs), Koopman theory, and unsupervised classification techniques. To this end, eight LESs of different convective, neutral, and unsteady ABLs are conducted. As the ratio of buoyancy to shear production increases, the turbulence structures change from roll vortices to convective cells. The quadrant analysis indicated that as this ratio increases, the sweep and ejection events decrease, and inward/outward interactions increase. The Koopman mode decomposition (KMD) is then used to characterize their turbulence structures. Our results showed that KMD can reveal non-trivial modes of highly turbulent ABL flows (e.g., transverse to the mean flow direction) and can reconstruct the primary dynamics of ABLs even under unsteady conditions with only ∼5% of the modes. We attributed the detected modes to the imposed pressure gradient (shear), Coriolis (inertial oscillations), and buoyancy (convection) forces by conducting novel timescale and quadrant analyses. We then applied the convolutional neural network combined with the K-means clustering to group the Koopman modes. This approach is displacement and rotation invariant, which allows efficiently reducing the number of modes that describe the overall ABL dynamics. Our results provide new insights into the dynamics of ABLs and present a systematic data-driven method to characterize their complex spatiotemporal patterns.

Data-driven modelling of brain activity using neural networks, diffusion maps, and the Koopman operator.

We propose a machine-learning approach to construct reduced-order models (ROMs) to predict the long-term out-of-sample dynamics of brain activity (and in general, high-dimensional time series), focusing mainly on task-dependent high-dimensional fMRI time series. Our approach is a three stage one. First, we exploit manifold learning and, in particular, diffusion maps (DMs) to discover a set of variables that parametrize the latent space on which the emergent high-dimensional fMRI time series evolve. Then, we construct ROMs on the embedded manifold via two techniques: Feedforward Neural Networks (FNNs) and the Koopman operator. Finally, for predicting the out-of-sample long-term dynamics of brain activity in the ambient fMRI space, we solve the pre-image problem, i.e., the construction of a map from the low-dimensional manifold to the original high-dimensional (ambient) space by coupling DMs with Geometric Harmonics (GH) when using FNNs and the Koopman modes per se. For our illustrations, we have assessed the performance of the two proposed schemes using two benchmark fMRI time series: (i) a simplistic five-dimensional model of stochastic discrete-time equations used just for a "transparent" illustration of the approach, thus knowing a priori what one expects to get, and (ii) a real fMRI dataset with recordings during a visuomotor task. We show that the proposed Koopman operator approach provides, for any practical purposes, equivalent results to the FNN-GH approach, thus bypassing the need to train a non-linear map and to use GH to extrapolate predictions in the ambient space; one can use instead the low-frequency truncation of the DMs function space of L2-integrable functions to predict the entire list of coordinate functions in the ambient space and to solve the pre-image problem.

Beyond expectations: residual dynamic mode decomposition and variance for stochastic dynamical systems

Koopman operators linearize nonlinear dynamical systems, making their spectral information of crucial interest. Numerous algorithms have been developed to approximate these spectral properties, and dynamic mode decomposition (DMD) stands out as the poster child of projection-based methods. Although the Koopman operator itself is linear, the fact that it acts in an infinite-dimensional space of observables poses challenges. These include spurious modes, essential spectra, and the verification of Koopman mode decompositions. While recent work has addressed these challenges for deterministic systems, there remains a notable gap in verified DMD methods for stochastic systems, where the Koopman operator measures the expectation of observables. We show that it is necessary to go beyond expectations to address these issues. By incorporating variance into the Koopman framework, we address these challenges. Through an additional DMD-type matrix, we approximate the sum of a squared residual and a variance term, each of which can be approximated individually using batched snapshot data. This allows verified computation of the spectral properties of stochastic Koopman operators, controlling the projection error. We also introduce the concept of variance-pseudospectra to gauge statistical coherency. Finally, we present a suite of convergence results for the spectral information of stochastic Koopman operators. Our study concludes with practical applications using both simulated and experimental data. In neural recordings from awake mice, we demonstrate how variance-pseudospectra can reveal physiologically significant information unavailable to standard expectation-based dynamical models.

Best practice guidelines for the dynamic mode decomposition from a wind engineering perspective

This paper distills the serial work of Li et al. (2022a, 2022b, 2022c, 2023b, 2023a, 2021, 2020a) into a practice guide for the Dynamic Mode Decomposition (DMD) with a particular emphasis on the quick learning and adaptation of the technique from a wind engineering perspective. The guide begins with discussions of the DMD's connections with the Koopman, Fourier & Laplace, the Proper Orthogonal Decomposition, and machine learning theories. Then, based on 1.5 × 106 core hours of parametric investigations on paradigmatic cases, recommendations that broadly apply to bluff-body wind engineering problems—especially regarding the input sequence—are formulated. In the temporal aspect, results highlight the DMD's preference for steady, stationary, equilibrium, or quasi-equilibrium systems, and suggest a sampling resolution 15–20 times the frequency of the dynamics of interest. Emphasis is also placed on the necessity of the Stabilization state for sampling convergence while avoiding the input of overdetermined systems. In the spatial aspect, high-dimensional, mean-subtracted data with minimal interpolation and noise proved to yield better Koopman models. Different input variables, though derived from the same system, may also result in non-unique Koopman modes depending on their nonlinear richness. Finally, two open-source MATLAB algorithms have been discussed and made open-source to facilitate the easy dissemination and usage for wind engineers. Via pedagogical examples, Algorithm 1 outlines a standard procedure to check for sampling convergence, and Algorithm 2 performs the similarity-expression DMD and visualizes the outputs.

Rigorous data‐driven computation of spectral properties of Koopman operators for dynamical systems

AbstractKoopman operators are infinite‐dimensional operators that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, Koopman operators can have continuous spectra and infinite‐dimensional invariant subspaces, making computing their spectral information a considerable challenge. This paper describes data‐driven algorithms with rigorous convergence guarantees for computing spectral information of Koopman operators from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), which provides the first scheme for computing the spectra and pseudospectra of general Koopman operators from snapshot data without spectral pollution. Using the resolvent operator and ResDMD, we compute smoothed approximations of spectral measures associated with general measure‐preserving dynamical systems. We prove explicit convergence theorems for our algorithms (including for general systems that are not measure‐preserving), which can achieve high‐order convergence even for chaotic systems when computing the density of the continuous spectrum and the discrete spectrum. Since our algorithms have error control, ResDMD allows aposteri verification of spectral quantities, Koopman mode decompositions, and learned dictionaries. We demonstrate our algorithms on the tent map, circle rotations, Gauss iterated map, nonlinear pendulum, double pendulum, and Lorenz system. Finally, we provide kernelized variants of our algorithms for dynamical systems with a high‐dimensional state space. This allows us to compute the spectral measure associated with the dynamics of a protein molecule with a 20,046‐dimensional state space and compute nonlinear Koopman modes with error bounds for turbulent flow past aerofoils with Reynolds number >105 that has a 295,122‐dimensional state space.

The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems

Koopman operators globally linearize nonlinear dynamical systems and their spectral information is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. However, Koopman operators are infinite dimensional, and computing their spectral information is a considerable challenge. We introduce measure-preserving extended dynamic mode decomposition (mpEDMD), the first Galerkin method whose eigendecomposition converges to the spectral quantities of Koopman operators for general measure-preserving dynamical systems. mpEDMD is a data-driven algorithm based on an orthogonal Procrustes problem that enforces measure-preserving truncations of Koopman operators using a general dictionary of observables. It is flexible and easy to use with any preexisting dynamic mode decomposition (DMD)-type method, and with different types of data. We prove convergence of mpEDMD for projection-valued and scalar-valued spectral measures, spectra, and Koopman mode decompositions. For the case of delay embedding (Krylov subspaces), our results include the first convergence rates of the approximation of spectral measures as the size of the dictionary increases. We demonstrate mpEDMD on a range of challenging examples, its increased robustness to noise compared to other DMD-type methods, and its ability to capture the energy conservation and cascade of a turbulent boundary layer flow with Reynolds number and state-space dimension .

Event Detection, Classification and Localization in an Active Distribution Grid Using Data-Driven System Identification, Weighted Voting and Graph

Event detection, classification, and localization (ED-C-L) in an active distribution network are challenging given system dynamics but critical for taking the control actions to mitigate the impact on the system operation. In this work, a new set of algorithms are developed for ED-C-L including, a) prony based dynamic window and event detection, b) data-driven system identification (DD-SI) using Koopman Mode analysis (KMA), physics-rule, and weighted-vote based assembling technique for event classification, and c) weighted graph and KMA based localization using distribution phasor measurement unit (D-PMU) data. DD-SI is utilized to construct a transient energy matrix (TEM) and capture the system dynamics using the temporal and spatial signature of events without using a labeled dataset. The proposed technique considers system dynamics for ED-C-L to achieve enhanced accuracy, compared to most of the existing data-driven techniques. A lab-scale real-time testbed is developed to generate D-PMU data utilizing a 24 bus distribution system simulated in the OPAL-RT/Hypersim for evaluating the performance of the proposed algorithm, and measured using selected metrics. Case scenarios consider utility-scale solar photovoltaic (PV) and battery energy storage system (BESS) integration at dispersed locations to validate proposed algorithms under dynamically changing operational scenarios. Further, the developed approach is applied on DPMU data from real distribution systems with DERs to showcase the superiority of the proposed approach to field systems.

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.

The linear-time-invariance notion of the Koopman analysis. Part 2. Dynamic Koopman modes, physics interpretations and phenomenological analysis of the prism wake

This serial work presents a linear-time-invariance (LTI) notion to the Koopman analysis, finding consistent and physically meaningful Koopman modes and addressing a long-standing problem of fluid mechanics: deterministically relating the fluid excitations and corresponding structure reactions. Part 1 (Li et al., Phys. Fluids, vol. 34, no. 12, p. 125136) developed the Koopman-LTI architecture and applied it to a pedagogical prism wake. By a systematic analytical procedure, the Koopman-LTI generated sampling-independent linear models that captured all the recurring dynamics embedded in the input data, finding six corresponding, orthogonal, and in-synch fluid–structure mechanisms. This Part 2 analyses the six modal duplets to underpin their physical implications, providing a phenomenological analysis of the subcritical prism wake. Visualizing the newly proposed dynamic Koopman modes, results show that two mechanisms at St1 = 0.1242 and St5 = 0.0497 describe shear layer dynamics, the associated Bérnard–Kármán shedding and turbulence production, which together overwhelm the upstream and crosswind walls by instigating a reattachment-type of reaction. The on-wind walls’ dynamical similarity renders them a spectrally unified fluid–structure interface. Another four harmonic counterparts, namely the subharmonic at St7 = 0.0683, the second harmonic at St3 = 0.2422, and two ultra-harmonics at St7 = 0.1739 and St13 = 0.1935, govern the downstream wall. Finally, this work discovered the vortex breathing phenomenon, describing the constant energy exchange in the wake's circulation-entrainment-deposition processes. With the Koopman-LTI, one may pinpoint the exact excitations responsible for a specific structure reaction, benefiting future investigations into fluid–structure interactions and nonlinear, stochastic systems.

Residual dynamic mode decomposition: robust and verified Koopmanism

Dynamic mode decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman operators. However, verifying the decomposition, equivalently the computed spectral features of Koopman operators, remains a significant challenge due to the infinite-dimensional nature of Koopman operators. Challenges include spurious (unphysical) modes and dealing with continuous spectra, which both occur regularly in turbulent flows. Residual dynamic mode decomposition (ResDMD), introduced by Colbrook & Townsend (Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems. 2021. arXiv:2111.14889), overcomes such challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems. ResDMD thus provides robust and verified Koopmanism. We implement ResDMD and demonstrate its application in various fluid dynamic situations at varying Reynolds numbers from both numerical and experimental data. Examples include vortex shedding behind a cylinder, hot-wire data acquired in a turbulent boundary layer, particle image velocimetry data focusing on a wall-jet flow and laser-induced plasma acoustic pressure signals. We present some advantages of ResDMD: the ability to resolve nonlinear and transient modes verifiably; the verification of learnt dictionaries; the verification of Koopman mode decompositions; and spectral calculations with reduced broadening effects. We also discuss how a new ordering of modes via residuals enables greater accuracy than the traditional modulus ordering (e.g. when forecasting) with a smaller dictionary. This result paves the way for more significant dynamic compression of large datasets without sacrificing accuracy.

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.