Reduced-order Linear Model Research Articles

Domain Decomposition Strategy for Combining Nonlinear and Linear Reduced-Order Models

Increasingly ubiquitous reliance on expensive, high-fidelity numerical simulations has led to the emergence of reduced-order modeling as an effective method to predict high-dimensional, spatially discretized field outputs in an efficient manner. This study presents a method to construct parametric, nonintrusive reduced-order models (ROMs) by leveraging domain decomposition (DD) to enable using multiple dimension reduction (DR) techniques within different spatial subdomains of the field to construct a single predictive ROM. To enforce smoothness of solutions across subdomains, points at domain interfaces are reconstructed using the data-driven gappy proper orthogonal decomposition method. The method is assessed on three high-dimensional test cases with shocks, including one with two-dimensional turbulent flow around the RAE2822. Results show that fully nonlinear ROMs predict fields more accurately in the vicinity of discontinuous features compared to their linear counterparts, which are more accurate in regions that are far from these discontinuities. Furthermore, irrespective of the DR method, DD-based ROMs outperformed their global counterparts in all test cases. Finally, models that used a combination of DR methods simultaneously showed superior performance near shockwaves and a significant improvement in total error compared to existing approaches that employ a single DR method in all subdomains.

Learning Reduced-Order Linear Parameter-Varying Models of Nonlinear Systems

In this paper, we consider the learning of a Reduced-Order Linear Parameter-Varying Model (ROLPVM) of a nonlinear dynamical system based on data. This is achieved by a two-step procedure. In the first step, we learn a projection to a lower dimensional state-space. In step two, an LPV model is learned on the reduced-order state-space using a novel, efficient parameterization in terms of neural networks. The improved modeling accuracy of the method compared to an existing method is demonstrated by simulation examples.

A linear reduced-order model for the activated sludge process for the integration into a mixed-integer linear energy system optimisation model

Conventional wastewater treatment plants consume significant amounts of electricity. The constant aeration of the wastewater in order to foster the growth of microorganisms or the pumping of wastewater are two examples for energy-intensive processes within a plant. Case studies have shown that switching off blowers and inlet pumps for a certain period of time is possible without a loss in water quality. This yields a potential for wastewater treatment plants to provide demand response (DR) to the power system and thereby increase overall system flexibility. So far, the DR potential has only been quantified for individual plants, while the effects of large-scale DR provision by the wastewater treatment sector for the power system have not yet been studied. One reason for this is the lack of optimisation models which include both the wastewater treatment process and the power system operation in sufficient detail. Our model tackles this gap in the literature by providing a reduced-order linear biochemical model for the activated sludge process within a WWTP that can be incorporated into an operational energy system model. The results show that the effluent concentrations are predicted well by the linear reduced-order model in comparison to the results of the Standard Activated-Sludge model No. 1 (ASM1). Potential model applications are the variation of the airflow rate within a certain range and the variation of liquid influent flow rate to the system, which is a result of electricity load shedding of the inlet pumps and the blowers connected to the activated sludge tank.

Dynamic mode decomposition for data-driven analysis and reduced-order modeling of E × B plasmas: II. Dynamics forecasting

In part I of the article, we demonstrated that a variant of the dynamic mode decomposition (DMD) algorithm based on variable projection optimization, called optimized DMD (OPT-DMD), enables a robust identification of the dominant spatiotemporally coherent modes underlying the data across various test cases representing different physical parameters in an E × B simulation configuration. We emphasized that the OPT-DMD significantly improves the analysis of complex plasma processes, revealing information that cannot be derived using conventionally employed analyses such as the fast Fourier transform. As the OPT-DMD can be constrained to produce stable reduced-order models (ROMs) by construction, in this paper, we extend the application of the OPT-DMD and investigate the capabilities of the linear ROM from this algorithm toward forecasting in time of the plasma dynamics in configurations representative of the radial-azimuthal and axial-azimuthal cross-sections of a Hall thruster and over a range of simulation parameters in each test case. The predictive capacity of the OPT-DMD ROM is assessed primarily in terms of short-term dynamics forecast or, in other words, for large ratios of training-to-test data. However, the utility of the ROM for long-term dynamics forecasting is also presented for an example case in the radial-azimuthal configuration. The model’s predictive performance is heterogeneous across various test cases. Nonetheless, a remarkable predictiveness is observed in the test cases that do not exhibit highly transient behaviors. Moreover, in all investigated cases, the error between the ground-truth and the reconstructed data from the OPT-DMD ROM remains bounded over time within both the training and the test window. As a result, despite its limitation in terms of generalized applicability to all plasma conditions, the OPT-DMD is proven as a reliable method to develop low computational cost and highly predictive data-driven ROMs in systems with a quasi-periodic global evolution of the plasma state.

Reservoir computing reduced-order model based on particle image velocimetry data of post-stall flow

The present study proposes a reservoir computing reduced-order model (RCROM) of the post-stall flow around the National Advisory Committee for Aeronautics 0015 airfoil based on the time series velocity field, and the estimation accuracy of the RCROM is evaluated compared to that of a linear reduced-order model (LROM). The data were experimentally obtained by particle image velocimetry at a chord Reynolds number of 6.4 × 104 and an angle of attack of 18°. The low-dimensional description of the velocity field can be obtained by decomposing the velocity field with a proper orthogonal decomposition (POD) technique and by employing the leading POD mode coefficients as temporal variables of the data instead of the velocity field. Reservoir computing (RC) is adopted as a nonlinear function that predicts several steps ahead of the leading POD mode coefficients. The hyperparameters of RC are tuned by Bayesian optimization, and the optimized RCROM outperforms the LROM in terms of estimation accuracy. The estimation accuracy of the RCROM can be investigated under different numbers of the predicted dominant POD modes and prediction step conditions. As a result, the RCROM shows higher estimation accuracy than the LROM.

Model order reduction via substructuring for a nonlinear, differential‐algebraic machine tool model with moving loads

AbstractMachine tools are permanently exposed to complex static, dynamic and thermic loads. This often results in an undesired displacement of the tool center point (TCP), causing errors in the production process and thus limiting the achievable workpiece quality. Recent research efforts try to counter these effects by process‐parallel solutions utilizing machine internal data. Therefore, fast simulation models are a fundamental requirement. Here, model order reduction (MOR) becomes crucial. The resulting low‐dimensional models are required for various applications, e.g., in digital twins, the correction of thermally induced errors at the TCP during the production process as well as for lifetime calculations in predictive maintenance. In this contribution, we present a strategy of MOR for a coupled thermo‐mechanical model with a nonlinear subsystem and moving loads, using the example of a feed axis with nonlinear machine components. Applying tailored substructuring techniques, we are able to separate the linear and nonlinear system components. This allows to apply classic linear MOR methods to the much larger linear part, whereas the small nonlinear part is kept, and thus enables drastically reduced computing times. The relative movement is considered by a switched system approach. Transient thermo‐mechanical interactions of the feed axis are calculated in a final investigation, comparing the performance of the resulting reduced‐order model and the original one.

Wasserstein model reduction approach for parametrized flow problems in porous media

The aim of this work is to build a reduced order model for parametrized porous media equations. The main challenge of this type of problems is that the Kolmogorov width of the solution manifold typically decays quite slowly and thus makes usual linear model order reduction methods inappropriate. In this work, we investigate an adaptation of the methodology proposed in [Ehrlacher et al., Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces, ESAIM: Mathematical Modelling and Numerical Analysis (2020)], based on the use of Wasserstein barycenters [Agueh & Carlier, Barycenters in the Wasserstein Space, SIAM Journal on Mathematical Analysis (2011)], to the case of non-conservative problems. Numerical examples in one-dimensional test cases illustrate the advantages and limitations of this approach and suggest further research directions that we intend to explore in the future.

Optimal tuning of model predictive controllers for organic Rankine cycle systems recovering waste heat from heavy-duty vehicles

The organic Rankine cycle power system is an emerging technology, which is able to recover the waste heat from the diesel engine of heavy-duty trucks and thus increase the overall engine efficiency. One of the major technical challenges for the integration of the organic Rankine cycle unit on-board trucks are the broad and rapid fluctuations of the available waste heat, caused by the unsteady driving conditions of the truck. Model predictive control has shown to be a powerful tool to ensure safe operation and optimal performance of the organic Rankine cycle unit on-board trucks. This paper presents a novel systematic method for the tuning of model predictive controllers based on a multi-objective optimization routine using a fourth-order reduced linear model. The objectives of the optimization are the settling time due to a step change of the exhaust gas mass flow rate and the cumulative controller effort due to measurement noise. The results suggest that a trade-off exists between the two objectives. Among the controller design parameters, the input rate weight has the largest influence on the controller performance. Interestingly, the simplified optimization procedure based on the reduced-order linear model of the organic Rankine cycle unit can provide key information about the controller performance based on a more complex nonlinear model of the organic Rankine cycle unit when subjected to a realistic waste heat profile. The results indicate that the settling time due to a step change of the exhaust gas mass flow rate is a good indicator of the absolute mean square tracking error over the profile, and it should not exceed 15 s for an absolute mean square tracking error below 2 K. On the other hand, the cumulative controller effort due to measurement noise is strongly correlated to the cumulative controller effort over the profile, and it should stay below 0.5 %/s for a cumulative controller effort over the whole profile below 2 %/s. The presented method is a powerful tool to help the control designer to find the optimal design parameters of model predictive controllers in a systematic way, in contrast to the time-consuming, experience-based trial and error methods.

Linear Reduced-Order Model Predictive Control

Model predictive controllers use dynamics models to solve <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">constrained</i> optimal control problems. However, computational requirements for real-time control have limited their use to systems with low-dimensional models. Nevertheless, high-dimensional models arise in many settings, for example, discretization methods for generating finite-dimensional approximations to partial differential equations can result in models with thousands to millions of dimensions. In such cases, reduced-order models (ROMs) can significantly reduce computational requirements, but model approximation error must be considered to guarantee controller performance. In this article, a reduced-order model predictive control (ROMPC) scheme is proposed to solve robust, output feedback, constrained optimal control problems for high-dimensional linear systems. Computational efficiency is obtained by using projection-based ROMs, and guarantees on robust constraint satisfaction and stability are provided. The performance of the approach is demonstrated in simulation for several examples, including an aircraft control problem leveraging an inviscid computational fluid dynamics model with dimension 998 930.

A Physics-Guided Neural Network Dynamical Model for Droplet-Based Additive Manufacturing

This article develops a physics-guided data-driven model for the height evolution of parts printed in droplet-based additive manufacturing. The proposed model is a convolutional recurrent neural network (ConvRNN) whose structure is derived based on the physical understanding of mass conservation during the height evolution. Because of this physics-guided model structure, the model parameters obtained are invariant to the geometry of the printed part and thus portable from one geometry to another, the conditions on physical stability of the evolution translate directly to training stability of the neural network, and the data required to train this model are much less compared to a pure black-box model. These aspects of the model are validated experimentally on an inkjet 3-D printing setup. The proposed model outperforms a black-box off-the-shelf multilayer perceptron (neural network) by using about two orders of magnitude less data for training, at the same time delivering <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1.7\times $ </tex-math></inline-formula> smaller rms error on test data. The proposed model is also compared with a state-of-the-art reduced order linear model and shows <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1.4\times $ </tex-math></inline-formula> smaller rms error on test data. Finally, experimental results also underline that the model parameters learned are geometry invariant, that is, the model parameters trained on one geometry can be used to predict the height map evolution for other geometries without relearning.

SSAE and GRU Based Joint Modeling for Nonlinear Distributed Parameter Systems

The modeling and control issues for distributed parameter systems (DPSs) have received a great deal of attention. Because linear model order reduction (MOR) methods may ignore the nonlinear dynamics and lose some details, it is difficult to describe DPS accurately by common modeling methods. To effectively model such systems, a sparse stacked auto-encoder and gated recurrent unit (SSAE-GRU) model is proposed in this paper. Under the time/space separation theory, it is the mainstream way to perform MOR and identification of time series respectively. In the SSAE-GRU model, this practice is still adhered to but joint learning is recommended. SSAE can be used as an excellent MOR technique. A sparse activation strategy that is introduced makes its model space simple and easy to train. GRU has the ability to represent such complex temporal properties because the information stored by previous neurons can be transmitted to the current moment selectively. The joint training method allows them to be responsible and consider the connection between adjacent moments and spatial energy transfer overall. Then, we use L2 regularization in back-propagation to reduce the difficulty of model optimization and prevent overfitting. The modeling scheme is simulated on two typical chemical thermal processes. This article demonstrates the effectiveness of the proposed method as well as outstanding performance compared to existing methods.

Autoencoder-Based Reduced Order Observer Design for a Class of Diffusion-Convection-Reaction Systems

The application of autoencoders in combination with Dynamic Mode Decomposition for control (DMDc) and reduced order observer design as well as Kalman Filter design is discussed for low order state reconstruction of a class of scalar linear diffusion-convection-reaction systems. The general idea and conceptual approaches are developed following recent results on machine-learning based identification of the Koopman operator using autoencoders and DMDc for finite-dimensional discrete-time system identification. The resulting linear reduced order model is combined with a classical Kalman Filter for state reconstruction with minimum error covariance as well as a reduced order observer with very low computational and memory demands. The performance of the two schemes is evaluated and compared in terms of the approximated L2 error norm in a numerical simulation study. It turns out, that for the evaluated case study the reduced-order scheme achieves comparable performance with significantly less computational load.

System-Level Modelling and Simulation of a Multiphysical Kick and Catch Actuator System

This paper presents a system-level model of a microsystem architecture deploying cooperating microactuators. An assembly of a piezoelectric kick-actuator and an electromagnetic catch-actuator manipulates a structurally unconnected, magnetized micromirror. The absence of mechanical connections allows for large deflections and multistability. Closed-loop feedback control allows this setup to achieve high accuracy, but requires fast and precise system-level models of each component. Such models can be generated directly from large-scale finite element (FE) models via mathematical methods of model order reduction (MOR). A special challenge lies in reducing a nonlinear multiphysical FE model of a piezoelectric kick-actuator and its mechanical contact to a micromirror, which is modeled as a rigid body. We propose to separate the actuator–micromirror system into two single-body systems. This step allows us to apply the contact-induced forces as inputs to each sub-system and, thus, avoid the nonlinear FE model. Rather, we have the linear model with nonlinear input, to which established linear MOR methods can be applied. Comparisons between the reference FE model and the reduced order model demonstrate the feasibility of the proposed methodology. Finally, a system-level simulation of the whole assembly, including two actuators, a micromirror and a simple control circuitry, is presented.

Design and Investigation of a Nonlinear Damper Based on Energy Dissipation through Shock and Dry Friction to Suppress Critical Self-Excited Vibrations in Drilling Systems

In this paper, a passive damper based on energy dissipation through shock and dry friction (shock-friction damper) is investigated regarding its design and effectiveness for damping self-excited torsional vibrations similar to those occurring in deep drilling. The results are compared to the results of conventional friction dampers. The effectiveness of the damper for different operational drilling parameters that change during the drilling process, such as the weight on the bit and the rotary speed of the bit, is analyzed. Two linear reduced order models of a drill string that are based on a complex finite element model are set up. One is reduced using the component mode synthesis and one is reduced to the identified critical mode. A lumped mass represents the inertia of a forcedly connected nonlinear damper. A combined reduced order model of the complex system and the inertia damper is introduced to investigate its dynamic motion and stability. Particular focus is on the energy flow within the dynamic system and on the change of the dissipation energy in the contact. A semi-analytical solution is derived using the harmonic balance method that is used to investigate the damping effect for various designs and operational parameters. Herein, the modal properties as well as parameters of the damper are examined regarding the damping effect and the stability of the system. Finally, the capability of the mechanism to suppress the self-excitation due to the bit–rock interaction in a drilling system is discussed, and recommendations are made with respect to the design parameters and placement of the damper.

Combined hydro-wind frequency control scheme: Modal analysis and isolated power system case example

Wind speed fluctuations and the lack of Variable Speed Wind Turbines (VSWTs) synchronous inertia make difficult grid frequency control, mainly in isolated power systems with high penetration of wind energy. To solve this drawback, a combined wind-hydro frequency control scheme is proposed and evaluated. The VSWT rotational speed variations are included as an additional input of the hydro-power governor units, determining the hydro-power frequency control response in accordance with such VSWT rotational variations and grid frequency oscillations. This new combined control scheme is introduced in two cases: VSWTs providing or not inertial control. A modal analysis based on a linear reduced-order model is used to discuss the oscillation modes, participation factors and power system stability. An isolated power system located in El Hierro (Canary Island, Spain) is proposed as case example. Results, discussion and comparative to previous frequency control approaches based on twenty-two different schedules are included in the paper. The proposed combined frequency control scheme reduces both VSWTs rotational speed and frequency deviations, regardless VSWTs frequency contribution. Additionally, VSWTs inertial control becomes more effective under imbalances, mainly due to the minor rotational speed excursions.

Nonlinear vibration analysis of coated blisks in the presence of stiffness mistuning identification

Several papers have studied the modelling methods of coated blisks and proved the perfect damping performance of hard coatings. However, nonlinear strain dependence characteristics of the coatings have been rarely incorporated in opened studies of complicated coated blisks. In this paper, a linear reduced order model (ROM) of coated blisks is developed, where two reduction methods, Craig-Bampton (CB) method and the second modal analysis, are performed. By introducing the nonlinear strain dependence characteristics of the coatings in the linear ROM, a nonlinear ROM is presented and solved by an iteration method to obtain the forced responses of coated blisks with nonlinear strain dependence characteristics. Furthermore, the stiffness mistuning pattern and loss factor of uncoated blisks obtained by the developed experimental based identification process are introduced in the analysis model. The linear and nonlinear ROMs of coated blisks are validated by the experimental tests. Finally, the effect of nonlinear strain dependence characteristics on forced responses of coated blisks where the blisk substrate has different loss factors is analyzed.

Effects of nonlinear coatings on vibration characteristics of rotating blisks with mistuning features

Hard coatings have been investigated to control the vibration level of nonrotating blisks. Nevertheless, the rotational effects in previous investigations of coated blisks are neglected. In this work, a linear reduced order model on the basis of an improved cyclic symmetric analysis method and classical SNM (Subset of Nominal Modes) is present to simulate rotating blisks with linear coatings and mistuning features. Furthermore, the nonlinear reduced order model of rotating blisks incorporating nonlinear characteristics of coatings and mistuning features is developed. The nonlinear forced responses are solved by a developed accelerated iteration method. The effects of nonlinear coatings on vibration characteristics of rotating tuned blisks are investigated. Finally, deterministic and statistical analyses are performed to analyze the effects of the nonlinear characteristics of coatings on forced responses of rotating randomly mistuned blisks. The results indicate that considering the nonlinear characteristics leads to increasing reduction level of forced responses with the increment of excitation amplitude and the rotational effects play an important role on the vibration characteristics of the blisks.

Model Predictive Control of Material Volumes with Application to Vortical Structures

In this paper, a model predictive control algorithm for selectively steering material volumes in a boundary layer is proposed. Using direct numerical simulations of a laminar boundary layer with a Gaussian-distributed force field as the actuator, a reduced-order linear model of the wall-normal velocity dynamics on a control grid covering the neighborhood of the actuator is derived using sparsity-promoting dynamic mode decomposition with control. The spatial evolution of a target volume is probabilistically described by a Gaussian mixture model propagated using Taylor’s hypothesis. The identified linear model is then required to track a reference output that creates negative wall-normal velocity along the predicted trajectory. Under the model predictive control framework, an output tracking optimal control problem with input constraints is solved at each time step as new targets are taken into account on the fly, yielding an input signal that effectively drives the target volumes toward the wall while avoiding excessive actuation. The effectiveness of the control scheme is demonstrated by steering passive volumes as well as synthetically generated vortical structures, suggesting the potential of this method to be used in novel flow-control strategies that selectively displace vortical structures in boundary layers.

Quantitative evaluation of predictability of linear reduced-order model based on particle-image-velocimetry data of separated flow field around airfoil

A quantitative evaluation method for a reduced-order model of the flow field around an NACA0015 airfoil based on particle image velocimetry (PIV) data is proposed in the present paper. The velocity field data obtained by the time-resolved PIV measurement were decomposed into significant modes by a proper orthogonal decomposition (POD) technique, and a linear reduced-order model was then constructed by the linear regression of the time advancement of the POD modes or the sparsity promoting dynamic mode decomposition (DMD). The present evaluation method can be used for the evaluation of the estimation error and the model predictability. The model was constructed using different numbers of POD or DMD modes for order reduction of the fluid data and different methods of estimating the linear coefficients, and the effects of these conditions on the model performance were quantitatively evaluated. The results illustrates that forward (standard) model works the best with two to ten significant DMD modes selected by sparsity promoting DMD.

New Method Based on Model-Free Adaptive Control Theory and Kalman Filter for Multi-Product Unsteady Flow State Estimation

Abstract In this paper, a new methodology is proposed to realize real-time unsteady flow estimation for a multi-product pipeline system. Integrating transient flow model, adaptive control theory, and adaptive filter, this method is developed to solve the contradiction between the efficiency and accuracy in traditional model-based methods. In terms of improving computational efficiency, the linear flow model based on frequency response and difference transforming is established to replace the traditional nonlinear flow model for transient flow state estimation. To reduce the deviation between actual observations and linear model estimates, we first introduce a model-free adaptive control method as linear compensation of the reduced order unsteady flow state model. To overcome the interference of observation noise, the Kalman filter method is applied to the modified state space model to obtain the one-step-ahead transient flow estimation. The proposed method is applied to the transient flow state estimation of a multi-product pipeline system and compared with the model-based method and two data-driven methods. The proposed method can reduce the deviation of transient flow estimation between the reduced order linear model and the traditional nonlinear model to less than 0.5% under unforeseen conditions and shows strong robustness to noise interference and parameter drift.