Multivariate Dynamic Mode Decomposition and Its Application to Bearing Fault Diagnosis

Fault diagnosis of multivariate signals is a significant area of engineering research. Feature information is collected from multiple dimensions. This improves the accuracy and robustness of mechanical fault diagnosis. Dynamic mode decomposition (DMD) is a method that extracts the basic structure and mode of motion, describing the behavior of dynamic systems. It decomposes a high-dimensional nonlinear dynamic system into single-frequency DMD modes. With the advent of the big data era, the interest in tensor analysis has increased. Tensor train (TT) decomposition represents the tensor structure through a series of low-rank tensor products. To extract the fault features of multivariate signals, a low-rank TT DMD (LRTTDMD) model is proposed in this article. The multivariate signals are first converted into a data tensor using phase space reconstruction. Subsequently, the data tensor is transformed into TT format using TT decomposition while ensuring a certain level of error. The tensor operation is introduced into the original DMD to process the data tensor of TT format and extract the fault feature. The construction of tensor low-rank extraction framework enhances the robustness of the algorithm to noise. Finally, the effectiveness and feasibility of LRTTDMD are validated using simulation signals and laboratory self-test bearing fault signals.

The objective of this study is to develop a reduced order model (ROM) that accurately predicts the key features of the wake generated by a wind turbine, based on the results of a computational fluid dynamics (CFD) simulation with the actuator line model (ALM). The simulation of a three-bladed up-wind wind turbine was performed using Reynolds-averaged Navier-Stokes (RANS) equations and the finite volume method. The two-equation turbulence model K-ωSST was used to approximate the Reynolds stresses and the Actuator Line Method (ALM) was used to simulate the effect of the wind turbine rotor on the flow field, representing the rotor blades as a distribution of forces acting on the fluid. Data from the RANS simulation was collected to be used by the Dynamic Mode Decomposition (DMD) algorithm. Velocity field was defined as the object of study for the construction of the reduced order model (ROM). From the implementation of the DMD algorithm, the eigenvalues and energy content were obtained for the field of interest. From these results, the main modes and their spectrum were calculated. The velocity field reconstruction was performed using the DMD algorithm, and the RMSE was computed relative to the simulation and experimental data, yielding a good approximation. Due to the turbulence model used, very stable results were obtained in the reduced order model, showing great capability to predict the fields of interest in time.

Abstract. We introduce the optimized dynamic mode decomposition (DMD) algorithm for constructing an adaptive and computationally efficient reduced-order model and forecasting tool for global atmospheric chemistry dynamics. By exploiting a low-dimensional set of global spatio-temporal modes, interpretable characterizations of the underlying spatial and temporal scales can be computed. Forecasting is also achieved with a linear model that uses a linear superposition of the dominant spatio-temporal features. The DMD method is demonstrated on 3 months of global chemistry dynamics data, showing its significant performance in terms of computational speed and interpretability. We show that the presented decomposition method successfully extracts and forecasts chemical patterns for leading chemical indicators, including nitric oxide, ozone, nitrogen dioxide, hydroxyl radical, isoprene, and carbon monoxide. Moreover, the DMD algorithm allows for rapid reconstruction of the underlying linear model, which can then easily accommodate non-stationary data and changes in the dynamics.

Analysis of aggregated load consumption forecasting in short, medium and long term horizons using Dynamic Mode Decomposition

Dynamic mode decomposition (DMD) provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of temporal, or spatio-temporal, data. A variety of regression techniques have been developed for producing the linear model approximation whose solutions are exponentials in time. For spatio-temporal data, DMD provides low-rank and interpretable models in the form of dominant modal structures along with their exponential/oscillatory behaviour in time. The majority of DMD algorithms, however, are prone to bias errors from noisy measurements of the dynamics, leading to poor model fits and unstable forecasting capabilities. The optimized DMD algorithm minimizes the model bias with a variable projection optimization, thus leading to stabilized forecasting capabilities. Here, the optimized DMD algorithm is improved by using statistical bagging methods whereby a single set of snapshots is used to produce an ensemble of optimized DMD models. The outputs of these models are averaged to produce a bagging, optimized dynamic mode decomposition (BOP-DMD). BOP-DMD improves performance by stabilizing and cross-validating the DMD model by ensembling; it also robustifies the model and provides both spatial and temporal uncertainty quantification (UQ). Thus, unlike currently available DMD algorithms, BOP-DMD provides a stable and robust model for probabilistic, or Bayesian, forecasting with comprehensive UQ metrics. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

Trading strategy based on dynamic mode decomposition: Tested in Chinese stock market

Dynamic mode decomposition (DMD) is a popular technique for modal decomposition, flow analysis, and reduced-order modeling. In situations where a system is time varying, one would like to update the system's description online as time evolves. This work provides an efficient method for computing DMD in real time, updating the approximation of a system's dynamics as new data becomes available. The algorithm does not require storage of past data and computes the exact DMD matrix using rank-1 updates. A weighting factor that places less weight on older data can be incorporated in a straightforward manner, making the method particularly well suited to time-varying systems. A variant of the method may also be applied to online computation of “windowed DMD,” in which only the most recent data are used. The efficiency of the method is compared against several existing DMD algorithms: for problems in which the state dimension is less than about 200, the proposed algorithm is the most efficient for real-time computation, and it can be orders of magnitude more efficient than the standard DMD algorithm. The method is demonstrated on several examples, including a time-varying linear system and a more complex example using data from a wind tunnel experiment. In particular, we show that the method is effective at capturing the dynamics of surface pressure measurements in the flow over a flat plate with an unsteady separation bubble.

The identification of coherent structures using proper orthogonal decomposition and dynamic mode decomposition

Operating convergent–divergent nozzles at low pressure ratios can lead to flow separation. Certain nozzle contours are known to produce a discrete acoustical tone under such conditions that are believed to be generated by a phenomenon known as transonic resonance, which involves a standing pressure wave situated between the separation shock and the nozzle exit plane. This paper reports the findings of a dynamic mode decomposition analysis of a perturbed axisymmetric unsteady Reynolds-averaged Navier–Stokes simulation of a nozzle flow known to exhibit this phenomenon. Two cases of different pressure ratios were studied in depth. The results show that the two cases differ in the shape of the standing pressure wave. The lower pressure ratio produces a standing pressure wave, whereas the higher pressure ratio produces a wave. In both cases, dynamic mode decomposition modes that match the standing pressure wave shape were found to be the least damped and the most energetic modes of the modes produced by the dynamic mode decomposition algorithm. The frequency of the mode for the lower pressure ratio matched the experimentally observed transonic tone frequency extraordinarily well, whereas the higher pressure ratio case was in fair agreement. The dynamic mode decomposition algorithm captured the transonic frequency for five additional pressure ratios.

We establish the convergence of a class of numerical algorithms, known as dynamic mode decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act on data coming from observables on a state space, arranged in Hankel-type matrices. The proofs utilize the assumption that the underlying dynamical system is ergodic. This includes the classical measure-preserving systems, as well as systems whose attractors support a physical measure. Our approach relies on the observation that vector projections in DMD can be used to approximate the function projections by the virtue of Birkhoff's ergodic theorem. Using this fact, we show that applying DMD to Hankel data matrices in the limit of infinite-time observations yields the true Koopman eigenfunctions and eigenvalues. We also show that the singular value decomposition, which is the central part of most DMD algorithms, converges to the proper orthogonal decomposition of observables. We use this result to obtain a representation of the dynamics of systems with continuous spectrum based on the lifting of the coordinates to the space of observables. The numerical application of these methods is demonstrated using well-known dynamical systems and examples from computational fluid dynamics.

Multivariate signal (MS) analysis, especially the assessment of its information transmission (for example, from the perspective of network science), is the key to our understanding of various phenomena in biology, physics and economics. Although there is a large amount of literature demonstrating that MS can be decomposed into space-time-frequency domain information, there seems to be no research confirming that multivariate information transmission (MIT) in these three domains can be quantified. Therefore, in this study, we attempted to combine dynamic mode decomposition (DMD) and parallel communication model (PCM) together to realize it. We first regarded MS as a large-scale system and then used DMD to decompose it into specific subsystems with their own intrinsic oscillatory frequencies. At the same time, the transition probability matrix (TPM) of information transmission within and between MS at two consecutive moments in each subsystem can also be calculated. Then, communication parameters (CPs) derived from each TPM were calculated in order to quantify the MIT in the space-time-frequency domain. In this study, multidimensional electroencephalogram (EEG) signals were used to illustrate our method. Compared with traditional EEG brain networks, this method shows greater potential in EEG analysis to distinguish between patients and healthy controls. This study demonstrates the feasibility of measuring MIT in the space-time-frequency domain simultaneously. This study shows that MIT analysis in the space-time-frequency domain is not only completely different from the MS decomposition in these three domains, but also can reveal many new phenomena behind MS that have not yet been discovered.

Understanding sloshing modes in cylindrical tanks with internal components is essential for risk mitigation. Moreover, the complex geometries make theoretical analysis difficult. While dynamic mode decomposition (DMD) has been widely adopted in fluid dynamics, it is meaningful to study its application in sloshing. Unlike conventional flow modes, DMD may face unique challenges due to the discontinuous free surface dynamics. To address this issue, this study selected a container with cylindrical components as the research object. The DMD method was employed to extract sloshing modes from numerical simulations using the Volume of Fluid method. The DMD accuracy is validated through comparison with simulation results. The Froude number (Fr) is used to assess DMD's robustness across different sloshing regimes. Results show that DMD method can effectively capture the sloshing modes. In linear sloshing, the overall modes are governed by the liquid-wall impact. When excited at the first order natural frequency, a nonlinear wave collision pattern is found and generates acceptable errors. Under the second order natural frequency excitation, the errors are sharply increased. Additionally, the relationship between DMD and Fr is independent of the tank filling rate. When Fr < 0.198, the DMD calculation result is reliable. For 0.198 ≤ Fr ≤ 0.368, DMD maintains high reconstruction accuracy, but the prediction error increases by 134.6%. When Fr > 0.368, both prediction and reconstruction errors surge by 128.3% and 127.5%. This work proposes a novel and effective framework in tank's sloshing analysis using DMD. The results provide valuable guidance for tank design, especially with complex geometries and internal components.

We investigate the characteristic of large-scale coherent motions over a large horizontal domain using the Dynamic Mode Decomposition (DMD) spectral analysis algorithm applied on measurements from two long-range pulsed lidars. We show the results and advantages of this methodology on six cases representative of three thermal stratification conditions at two heights relevant for wind energy: near-neutral, unstable and stable stratification at 50m and 200m above ground level. For these cases the DMD algorithm show three types of structures: streaks near the surface for near-neutral for neutral stratification, large-scale convective rolls for the unstable cases and sheet-like rotational patches for stable conditions. The DMD algorithm also shows the stationary effects of the terrain on the flow at 50m above ground level, within the atmospheric surface layer. The possibility of isolating terrain effects from coherent motions makes DMD attractive for studying complex atmospheric flow phenomena as well as to have more realistic input for wind farm flow simulations.

Analysis of characteristic wake flow modes on a generic transonic backward-facing step configuration

A note on the use of Dynamic Mode Decomposition in mechanics