Low-order Model Research Articles

Online Identification and Detection of Manual Control Adaptation with Recursive Time Series

An online pilot manual control behavior identification method, based on recursive low-order time-series model estimation, is presented and validated using experimental data. Eight participants performed compensatory tracking tasks with time-varying vehicle dynamics, where, at an unpredictable moment during a run, a sudden degradation in dynamics could occur. They were instructed to push a button when they detected a change in dynamics. Two methods to automatically detect the moment when pilot adaptation occurs from online estimated parameter traces are discussed. Results show that pilots are more accurate in detecting changes than either algorithm. But when the algorithms are correct, they are often quicker to detect pilot adaptation than pilots themselves. The presented techniques have potential but need improvements.

Design and Analysis of a Liftfan for eVTOL Aircraft

Abstract Ducted liftfans provide greater hovering efficiency to eVTOL aircraft than open propellers of equal disk area. Liftfans are designed with minimised length to limit propulsor weight added and drag incurred during forward flight. Three challenges posed by the liftfan propulsor configuration are addressed in this paper. First, instead of a single-row propeller, liftfans require the design of a fully integrated stage. To maximise the hovering figure of merit, a non-dimensional measure of power consumption, the preliminary and 3D design variables of the rotor and splittered-diffuser stator rows are optimised simultaneously using 3D CFD. The resulting prototype liftfan design is validated through experimental testing. Second, despite efficiency improvements during hover, liftfans sustain additional inlet distortion and drag during forward flight due to their surrounding nacelles. A low-order model is developed to investigate the maximum range for different fan designs with varying diffuser area ratios and forward flight tilt angles. Design selections are validated using full annulus 3D CFD and experimental wind tunnel tests. Third, as the electric motors of liftfans are mounted within the hub, fan-driven active cooling is necessary to prevent overheating. The stagnation pressure losses incurred by cooling airflow must be minimised without impeding heat transfer. Low-order models are developed to predict motor heat transfer and loss and to guide the design of a mixed-flow cooling fan. Experimental testing and 3D CFD simulations confirm that the addition of forward sweep and adoption of a high blade count can reduce cooling fan losses.

A Reduced-Order Model of Lithium–Sulfur Battery Discharge

This paper examines the problem of modeling lithium–sulfur (Li-S) battery discharge dynamics. The importance of this problem stems from the attractive specific energy levels achievable by Li-S batteries, which can be particularly appealing for applications such as aviation electrification. Previous research presents different Li-S battery models, including “zero-dimensional” models that neglect diffusion while using the laws of electrochemistry to represent reduction–oxidation (redox) rates. Zero-dimensional models typically succeed in capturing key features of Li-S battery discharge, including the high plateau, low plateau, and dip point visible in the discharge curves of certain Li-S battery chemistries. However, these models’ use of one state variable to represent the mass of each active species tends to furnish high-order models, with many state variables. This increases the computational complexity of model-based estimation and optimal control. The main contribution of this paper is to develop low-order state-space model of Li-S battery discharge. Specifically, the paper starts with a seventh-order zero-dimensional model of Li-S discharge dynamics, analyzes its discharge behavior, constructs phenomenological second- and third-order models capable of replicating this behavior, and parameterizes these models. The proposed models succeed in capturing battery discharge behavior accurately over a wide range of discharge rates. To the best of our knowledge, these are two of the simplest published models capable of doing so.

The applicability limits of the lowest-order substitute model for a cantilever beam system hard-impacting a moving base.

This paper examines the circumstances under which a one-degree-of-freedom approximate system can be employed to predict the dynamics of a cantilever beam comprising an elastic element with a significant mass and a concentrated mass embedded at its end, impacting a moving rigid base. A reference model of the system was constructed using the finite element method, and an approximate lowest-order model was proposed that could be useful in engineering practice for rapidly ascertaining the dynamics of the system, particularly for predicting both periodic and chaotic motions. The number of finite elements in the reference model was determined based on the calculated values of natural frequencies, which were found to correspond to the values of natural frequencies derived from the application of analytical formulas. The precision of the parameter identification and the outcomes yielded by the substitute model were validated through the calculation of the regions of stable periodic solutions using the analytical Peterka method. Subsequently, the qualitative and quantitative limits of the substitute model's applicability were determined. The quantitative limits were delineated through the utilization of Lyapunov exponents and characteristics associated with the energy dissipation due to impacts and the average number of impacts per excitation period. These characteristics provide a foundation for the introduction of global distance measures of the dynamic behavior of diverse systems within a specified range of the control parameter.

A study on the representation of local effects in low order models for thin-walled rod members

This work aims to explore alternatives to represent local effects, such as local buckling and plasticity, in low order models for thin-walled rod members. The discussion is carried out on a theoretical-numerical level, and illustrative examples are provided. The techniques that are explored stem from two different approaches: (i) direct enrichment of the rod kinematics and (ii) multiscale methods. Direct enrichment of low order kinematics usually leads to models with optimal computational cost, while still at the downside of having lower-order (still limited) kinematics. Models derived from the Generalized Beam Theory (GBT) are an example of such an approach. Multiscale methods, in turn, rely on results of higher-order theories (e.g., shells and 3D solids) to improve the performance of the lower-order model. The associated computational cost and accuracy vary widely with the imposed coupling level between the different scales. It is possible to have models ranging from full coupling at run-time – the so-called strong coupling multiscale method – to no coupling at all – the higher-order models are used only to compute meaningful mechanical quantities that are passed on to the low order model at some point. The work is an on-going development of a PhD research by the first author, and the results provided so far are only partial.

Aerodynamic Nonlinearity Effects on Flight Dynamics of a Very Flexible Flying Wing

High-aspect-ratio (AR) wings are the prominent feature of very flexible aircraft, and greater AR designs are being explored for enhanced flight efficiency in civil aviation. However, the impact of higher flexibility on the aeroelastic behavior of fixed-wing aircraft still needs to be fully understood. This paper assesses the effects of aerodynamic nonlinearities from flow separation on the maneuvering and gust response of a high-aspect-ratio flying wing using low-order models. Three aerodynamic formulations are evaluated: a linear model without stall effects, a piecewise linear model with steady-stall characteristics, and a nonlinear model accounting for dynamic-stall phenomena. All models perform similarly during maneuvering oscillations near stall conditions, but the dynamic-stall model foresees flight dynamic instabilities poststall, which are not seen in other models given the same maneuver input. Under a limited number of discrete and continuous gust disturbances, the model without stall effects behaves nonconservatively, whereas the steady-stall model is more conservative compared to the dynamic-stall model, with respect to the prediction of flight instabilities. These outcomes highlight the importance of accounting for aerodynamic nonlinearities in understanding the flight dynamics of very flexible aircraft under moderate-to-high angle-of-attack excitations.

A low-order modeling approach for analyzing the performance of coaxial, counter-rotating ocean current turbines: The equivalent single rotor model

A low-order modeling approach for analyzing the performance of coaxial, counter-rotating ocean current turbines: The equivalent single rotor model

An experimental study on thermoacoustic instabilities of non-premixed combustion using internal flue gas recirculation

Internal flue gas recirculation (IFGR) is a commonly used low-NOx combustion technique in industrial boilers. This study experimentally investigated the nonlinear behavior of the self-excited thermoacoustic instability in non-premixed combustion using IFGR. CH* chemiluminescence imaging and Proper orthogonal decomposition (POD) analysis were conducted to study the characteristics of flame dynamics. The results showed that, as global equivalence ratio φg decreased, the combustion system sequentially exhibited combustion noise, 90 Hz limit cycle oscillation, dual-frequency transition, and 80 Hz limit cycle oscillation, accompanied by frequency shift and mode transition phenomena. For the 90 Hz and 80 Hz oscillation modes, the first mode of POD exhibited axial oscillation patterns and global flashing oscillation patterns, respectively. Finally, combustion chamber temperature measurement and low-order thermoacoustic network model was used to analyze the mechanisms behind mode transition and frequency shift phenomena. The results showed that the decrease in combustion chamber temperature, which was caused by φg decrease, led to a reduction in the frequency of the first natural acoustic mode, ultimately resulting in the mode transition. IFGR altered the range of φg where thermoacoustic oscillations occur. The phenomenon of frequency shift was primarily caused by the decrease in convective time delay τconv. This study further explores the influence of IFGR and φg on thermoacoustic oscillation modes and flame dynamics, which contributes to the development of stable, low-NOx non-premixed combustors.

A shortcut model for batch distillation columns using pertinent effective factors

A shortcut model is a low-order model that provides a simplified representation of a complex process, thereby enabling faster and more efficient analysis, design and optimization without the computational intensity of a full-scale model. For continuous distillation columns, the Kremser-Edmister approach allows building reliable shortcut models by defining the effective absorption and stripping factors as average values between the top and bottom trays of the column. This paper shows that, unfortunately, this approach does not extend to batch distillation because the effective factors computed from the top and bottom trays can be quite inaccurate. Next, the paper proposes a novel column segmentation and a new approach for computing pertinent effective absorption and stripping factors. The resulting shortcut model is shown to accurately reproduce the dynamic behavior of a rigorous tray-by-tray model and to significantly reduce the computational time necessary to solve a minimal-time optimization problem subject to recovery and purity constraints.

Parameter-Estimation-Based Gain-Scheduling Control of Weight on Bit in Drilling Process With Uncertain Penetration Resistance Coefficient.

It is inevitable to encounter through different formations in the drilling process for deep exploration, and the penetration resistance coefficient (PRC) is an uncertain parameter related to lithology. In this article, a parameter-estimation-based gain-scheduling controller is developed to eliminate undesired system performance deterioration due to the uncertain parameter. First, a drill-string axial finite element model with the uncertain PRC is established, and a control-oriented low-order model is derived via mode selection. A gain-scheduling controller is computed based on the quadratic stability condition of the closed-loop system, which can cope with the system's parameter uncertainty using variable low-frequency gain. An adaptive observer is designed to estimate the unmeasurable scheduling variable. Field data from a geothermal drilling well is obtained to validate our model. According to this drilling well scenario, both numerical and experimental results illustrate the effectiveness of our method. The explicit relationship between the controller gain and the uncertain parameter is presented. It is found that the performance of the closed-loop system is more sensitive to the controller gain when drilling in soft formations, requiring more attention in such scenarios.

Point-enhanced convolutional neural network: A novel deep learning method for transonic wall-bounded flows

Low order models can be used to accelerate engineering design processes. Ideally, these surrogates should meet the conflicting requirements of large design space coverage, high accuracy and fast evaluation. Within the context of aerospace applications at transonic conditions, this can be challenging due to the associated non-linearity of the flow regime. Different methods have been investigated in the past to predict the flow-field around shapes such as airfoils or cylinders. However, they usually have reduced spatial resolution, limiting the prediction capabilities within the boundary layer which is of interest for transonic wall-bounded flows. This work proposes a novel Point-Enhanced Convolutional Neural Network (PCNN) method that combines the advantages of the well-established PointNet and convolutional neural network approaches. The PCNN model has relatively low memory requirements in the training process, preserves the spatial correlation in the domain and has the same resolution as a traditional computational method. The architecture is used for the flow-field prediction of civil aero-engine nacelles in which it is demonstrated that the flow features of peak isentropic Mach number (Mis), pre-shock isentropic Mach number and shock location (X/Lnac) are captured within ΔMis = 0.02, ΔMis=0.04, ΔX/Lnac=0.007, respectively. The PCNN model successfully predicts the integral parameters of the boundary layer, in which the incompressible displacement thickness, momentum thickness and shape factor are typically within 5% of the CFD. Overall, the PCNN method is demonstrated for transonic wall-bounded flows for a range of flow physics that include shock waves and shock-induced separation.

Assessment of the performance and robustness of a new PID tuning technique for AVR systems

ABSTRACT In this work, a hybrid PID controller, combining Zeigler–Nichols (ZN) theory and the dominant pole placement, has been developed for the automatic voltage regulator (AVR) systems. Furthermore, the reduced order models (ROMs) of AVR systems, namely, AVR-1, AVR-2, and AVR-3, have been established using Particle Swarm Optimization (PSO), Big Bang Big Crunch (BBBC), and Pade’s approximation, respectively. Then, the proposed PID controller has been applied to both the original and developed lower order models of AVR to check the efficacy and robustness of the developed hybrid controller. It is shown that the proposed hybrid PID controller provides a very small value of the percentage peak overshoot (%M p ) and settling time (t s ). Also, the gain margin (GM) and phase margin (PM) are found to be in the desired range. Finally, the sensitivity analysis has also been performed to check the robustness of the proposed controller. The sensitivity function, complementary sensitivity function, and various performance indices have been evaluated for the proposed controller based on Bode’s Integral (BI), Tyreus–Lyuben (T–L), and Internal Model Control (IMC) approaches. The comparative studies of the results show that the proposed controller has smaller values of performance indices and sensitivity function.

Reduction of vortex-induced vibrations of a cantilevered hydrofoil with passive piezoelectric shunt

This work first investigates the ability of a low order fluid-structure model to fit the vortex-induced vibrations (VIV) observed on a truncated hydrofoil in a hydrodynamic tunnel. A particular VIV area is scrutinized, for which a hydrodynamic excitation mechanism due to a Karman-type vortex wake organization successively locks the first torsional and second bending mode of the cantilevered hydrofoil. Coupling two structure oscillators with a Van der Pol wake oscillator satisfactorily reproduces the amplitude response and the lock-in frequency. In order to build a low order model allowing to optimize control strategy, a fourth degree of freedom corresponding to the electric circuit of a resonant piezoelectric shunt has been added. Composed of an inductance and a resistance connected to a piezoelectric patch, the passive shunt was tuned to minimize the vibration amplitude in the frequency lock-in range. Model predictions are finally compared with experimental results.

Robustness and reliability of state-space, frame-based modeling for thermoacoustics

The Galerkin modal expansion is a well-known method used to develop reduced order models for thermoacoustics. A known issue is the appearance of Gibbs-type oscillations on velocity fluctuations at the interface between subdomains and at boundary conditions. Recent work of Laurent et al. (2019) [20] and Laurent et al. (2021) [23] have shown that it is possible to overcome this issue by using an over-completed frame, instead of a Galerkin modal basis. However, the low-order modeling based on this frame modal expansion may generate spurious modes. In this paper, the origin of these non-physical modes is identified and a method is proposed to automatically remove them from the outcome. By preventing any interaction between the physical and non-physical components, the proposed methodology drastically improves the robustness and reliability of the frame modal expansion modeling for thermoacoustics.

Floating particles transport through the free surface vortex technique: A novel numerical study to assess the interaction among different scales of vortex structures

Fat, oil, and grease (FOG) floating particles in the sump of sewage pumping stations will accumulate together to form rigid layers, resulting in failure for pump device. To overcome this, the free surface vortex (FSV) technique has been considered and applied to transport floating particles toward the submerged suction pump inlet. This paper investigates the potential of vortices as a means of downward motion of FOG. The entrainment capacity of FSV is investigated by numerical simulations using a coupled level-set and volume-of-fluid method. Two coherent structures are decomposed by proper orthogonal decomposition: FSV represented by the first two orders with high energy content and spiral vortex bands represented by low energy and high order models. The extracted ridges of the finite-time Lyapunov exponent (FTLE) delineate different regions of the flow field and effectively capture the evolution of Lagrangian coherent structures. The floating particles in the sump are first caught by the dividing line formed by the FTLE ridges, mixed in the entrainment zone, and then merged into the vortex. The enstrophy production term dominates the development of vorticity. Subject to the influence of flow velocity gradients, both radial and tangential vortices undergo a transition into axial vortices. This transformation enhances the vortex's capacity to entrain particles within the vortex core area, leading to their rapid inward spiraling toward the vortex center and eventual expulsion due to the vortex's entrainment effect.

Limit cycle oscillations in the zonal-flow-catalyzed interactions of ion-temperature-gradient turbulence

Limit-cycle oscillations are studied for ion temperature gradient turbulence, which, in the absence of large diamagnetic (mean) shear flows, saturates through energy transfer from unstable modes to large-scale stable modes via zonal-flow intermediary modes. Oscillations of zonal flow and turbulence levels are strongly constrained by the reactive, largely non-dissipative character of the zonal flows. Since existing predator–prey models for observed oscillations in experiments do not include energy transfer through zonal flows to stable modes, low-order fluid models with this physics are constructed and investigated. A simple three-wave truncation produces low-amplitude zonal flows that slowly oscillate around a zero mean, with turbulence oscillations between coupled wavenumbers that exceed linear frequencies by orders of magnitude. This inconsistency with experimental observations is caused by the weak non-linear drive of zonal flows in three-wave systems and the lack of multiple-wavenumber turbulent interactions. A more comprehensive model that preserves multiple wavenumber interactions within the context of conservative zonal-flow-mediated energy transfer to stable modes accurately reflects observed dynamics when the phase between stable and unstable modes is occasionally randomized.

An Image Compensation-Based Range–Doppler Model for SAR High-Precision Positioning

The range–Doppler (R–D) model is extensively employed for the geometric processing of synthetic aperture radar (SAR) images. Refining the sensor motion state and imaging parameters is the most common method for achieving high-precision geometric processing using the R–D model, comprising a process that involves numerous parameters and complex computations. In order to reduce the specialization and complexity of parameter optimization in the classic R–D model, we introduced a novel approach called ICRD (image compensation-based range–Doppler) to improve the positioning accuracy of the R–D model, implementing a low-order polynomial to compensate for the original imaging errors without altering the initial positioning parameters. We also designed low-order polynomial compensation models with different parameters. The models were evaluated on various SAR images from different platforms and bands, including spaceborne TerraSAR-X and Gaofen3-C images, manned airborne SAR-X images, and unmanned aerial vehicle-mounted miniSAR-Ku images. Furthermore, image positioning experiments involving the use of different polynomial compensation models and various numbers and distributions of ground control points (GCPs) were conducted. The experimental results demonstrate that geometric processing accuracy comparable to that of the classical rigorous positioning method can be achieved, even when applying only an affine transformation model to the images. Compared to classical refinement models, however, the proposed image-compensated R–D model is much simpler and easy to implement. Thus, this study provides a convenient, robust, and widely applicable method for the geometric-positioning processing of SAR images, offering a potential approach for the joint-positioning processing of multi-source SAR images.

Validation of a Quadrupole Model of Sound Radiation of a Turbulent Jet Based on Multi-Microphone Acoustic Measurements

A low-order model of quadrupole sound sources in a turbulent jet has been developed using the acoustic analogy method. Multimicrophone acoustic measurements of jet sound radiation are used to estimate the model parameters and validate it. Based on measurements carried out in different zones of the sound field, estimates of the size of the effective localization region of sound sources are made and the boundaries of the zone of dominance of quadrupole sound radiation over pseudo-sound fluctuations are determined. The proposed model can be used in practical estimates of the spectral and correlation characteristics of the far and near sound field of the jet.

Large eddy simulation of highly underexpanded sonic jets from elliptical nozzles

The injection of high-pressure jets into quiescent air poses significant challenges in fluid dynamics, pertinent to various industrial engineering applications. This study used large eddy simulations on a massively parallel computational framework, employing a grid of over 600 million nodes, to investigate the behavior of highly underexpanded sonic jets from elliptical nozzles at a nozzle pressure ratio of 15. Three elliptical nozzles, with aspect ratios of 1.5, 2.2, and 3.0, each having a sectional area equivalent to that of a circular jet with a diameter of D=1 mm, were analyzed. The aim was to clarify the gasdynamic and mixing characteristics of these jets to guide the design of next-generation injectors. A detailed analysis of the flow provided insights into the mechanisms of turbulence generation and Reynolds stress anisotropy. This was achieved using the componentality contour approach and a modified barycentric color mapping scheme, offering valuable reference data for developing lower-order models. The results indicate a non-axisymmetric radial expansion of the jet boundary in all elliptical injectors, leading to an axis switch phenomenon. The use of elliptical orifices was found to reduce jet penetration, mitigating issues such as fuel impingement in small engine combustion chambers and promoting improved air–fuel mixing quality.

The effect of hydrogen enrichment on the conical premixed methane–air flame response and thermoacoustic modes coupling

This paper investigates the thermoacoustic dynamic responses of hydrogen-enriched laminar premixed conical methane–air flames under dual-mode coupling. Utilizing the level set method and the G-equation model, one meticulously derives the flame describing function (FDF) in relation to variations in hydrogen enrichment levels (ηH). This precisely derived FDF is then integrated into the low-order network model of the Rijke tube to analyze the thermoacoustic unstable modes of the system. Finally, dual-frequencies incoming flow velocity perturbations are reintroduced as inputs to obtain the flame response under thermoacoustic modes coupling. Results show that while keeping the unstretched steady flame aspect ratio constant, an increase in ηH not only raises the FDF’s cut-off frequency but also enhances the FDF gain within the nonlinear frequency region, leading to more unstable modes, especially high frequency modes within the Rijke tube system. Furthermore, the flame response is further altered under the excitation of dual unstable modes. When both modes are within the linear frequency region of the FDF, the flame response is co-controlled by the two modes, predominantly by the mode with a larger velocity perturbation amplitude, with weaker modes coupling leading to the additional frequency perturbation having a suppressive effect on the flame response at the original frequency. Conversely, when one or two modes are within the nonlinear frequency region of the FDF, the flame response is dominated by the lower-frequency mode, with higher nonlinear modes coupling allowing the additional frequency perturbation to both promote and suppress the response at the original frequency and also couple to produce a significant difference frequency response. Novelty and significance The novelty of this paper lies in the determination of the flame describing function (FDF) with varying hydrogen enrichment levels and the stability of thermoacoustic systems, and more significantly, conducting an in-depth and comprehensive investigation into the nonlinear dynamic responses of hydrogen-enriched flames to different types of dual-mode coupling perturbations. The dual-mode perturbations result in either attenuation or amplification of the flame response, depending on whether the corresponding frequencies lie within the linear or nonlinear frequency regions of the FDF, which innovatively incorporates the influence of the FDF phase and examines the responses at the difference frequencies generated by the coupling. This lays a foundation for further exploration into the mechanisms of modes coupling in hydrogen-enriched and other thermoacoustic systems.