Reduced-order Linear Model Research Articles

Model-reduced gradient-based history matching

Gradient-based history matching algorithms can be used to adapt the uncertain parameters in a reservoir model using production data. They require, however, the implementation of an adjoint model to compute the gradients, which is usually an enormous programming effort. We propose a new approach to gradient-based history matching which is based on model reduction, where the original (nonlinear and high-order) forward model is replaced by a linear reduced-order forward model and, consequently, the adjoint of the tangent linear approximation of the original forward model is replaced by the adjoint of a linear reduced-order forward model. The reducedorder model is constructed with the aid of the proper orthogonal decomposition method. Due to the linear character of the reduced model, the corresponding adjoint model is easily obtained. The gradient of the objective function is approximated, and the minimization problem is solved in the reduced space; the procedure is iterated with the updated estimate of the parameters if necessary. The proposed approach is adjointfree and can be used with any reservoir simulator. The method was evaluated for a waterflood reservoir with channelized permeability field. A comparison with an adjoint-based history matching procedure shows that the model-reduced approach gives a comparable quality of history matches and predictions. The computational efficiency of the model-reduced approach is lower than of an adjoint-based approach, but higher than of an approach where the gradients are obtained with simple finite differences.

Evaluation of Aeroelastic Uncertainty Analysis Methods

DOI: 10.2514/1.47118 Flutter is a destructive and potentially explosive phenomenon that is the result of the simultaneous interaction of aerodynamic, elastic, and inertial forces. The nature of flutter mandates that flutter flight testing be cautious and conservative. Because of this, further investigation of uncertainty analysis with respect to the flutter problem is desiredandwarranted.Predictionof flutterinthetransonicregimerequirescomputationallyexpensivehigh-fidelity simulation models. Because of the computational demands, traditional uncertainty analysis is not often applied to transonic flutter prediction, resulting in reduced confidence in the results. The work described herein is aimed at exploring various methods to reduce the existing computational time limitations of traditional uncertainty analysis. Specifically,thecouplingofdesignofexperimentandresponsesurfacemethodsandtheapplicationofanalysisare appliedtoavalidatedaeroelasticmodeloftheAGARD445.6wing.Fromahigh-fidelitynonlinearaeroelasticmodel, a linear reduced-order model is produced that captures the essential dynamic characteristics. Using reduced-order models, the design of experiment, response surface methods, and -analysis approaches are compared with traditional Monte Carlo-based stochastic simulation. All of these approaches to uncertainty analysis have advantages and drawbacks. Results from these methods and their robustness are compared and evaluated. HE potentially explosive nature of flutter mandates that flutter flight testing be cautious and conservative. The development of a flutterstabilityparameterwasundertakenfourdecadesagotogivea more reliable technique to predict the onset of flutter. Other methods of predicting flutter such as damping versus velocity are known to have a number of shortcomings, one of which can be the sudden degradationindampingasillustratedintheexplosive flutterexample shown in Zimmerman and Weissenburger [1]. In this classic paper, the flutter stability parameter was developed using the classical two-dimensional, two-degree-of-freedom (pitch/plunge) aeroelastic wing model, and the technique was shown to be applicable for higher-degree-of-freedom analysis. The flutter stability parameter (flutter margin) is calculated using the decay rate and damped frequency. Traditionally, computational aeroelasticity for loads and flutter prediction combines a linear finite element formulation for the structure with linear aerodynamic methods. At the same time, the prediction of aerodynamic performance and control surface effectivenessaccountsfortheeffectsofthestructuralelasticdeformations on the external aerodynamics by means of correction factors applied to results obtained when the aircraft is assumed to be rigid. Both practices are well-established in aircraft design and give accurate, reliable, and rather inexpensive predictions for static and dynamic

Incorporation of Feedback Control into a High-Fidelity Aeroservoelastic Fighter Aircraft Model

Flight testing for aeroservoelastic clearance is an expensive and time consuming process. Large degree-of-freedom high-fidelity nonlinear aircraft models using computational fluid dynamics coupled with finite element models can be used for accurately predicting aeroelastic phenomena in all flight regimes including subsonic, supersonic, and transonic. With the incorporation of an active feedback control system, these high-fidelity models can be used to reduce the flight-test time needed for aeroservoelastic clearance. Accurate computational fluid dynamics/finite element models are computationally complex, rendering their runtime ill suited for adequate flight control system design. In this work, a complex, large-degree-of-freedom, transonic, inviscid computational fluid dynamics/finite element model of a fighter aircraft is fitted with a flight control system for aeroelastic oscillation reduction. A linear reduced-order model of the complete aeroelastic aircraft dynamic system is produced directly from the high-order nonlinear computational fluid dynamics/finite element model. This rapid runtime reduced-order model is used for the design of the flight control system, which includes models of the actuators and common nonlinearities in the form of rate limiting and saturation. The oscillation reduction controller is successfully demonstrated via a simulated flight test using the high-fidelity nonlinear computational fluid dynamics/finite element/flight control system model.

Modeling and control of tubular solid-oxide fuel cell systems. I: Physical models and linear model reduction

This paper describes the development of a transient model of an anode-supported, tubular solid-oxide fuel cell (SOFC). Physically based conservation equations predict the coupled effects of fuel channel flow, porous-media transport, heat transfer, thermal chemistry, and electrochemistry on cell performance. The model outputs include spatial and temporal profiles of chemical composition, temperature, velocity, and current density. Mathematically the model forms a system of differential-algebraic equations (DAEs), which is solved computationally. The model is designed with process-control applications in mind, although it can certainly be applied more widely. Although the physical model is computationally efficient, it is still too costly for incorporation directly into real-time process control. Therefore, system-identification techniques are used to develop reduced-order, locally linear models that can be incorporated directly into advanced control methodologies, such as model predictive control (MPC). The paper illustrates the physical model and the reduced-order linear state-space model with examples.

Modeling of an automotive fuel cell thermal system

This work develops an 8th order, non-linear thermal model of an automotive proton exchange membrane (PEM) fuel cell system. Subsystem models are developed from first principals where ever possible and validated against data from a physical system. The system model is validated against data from an automotive 120 kW fuel cell system, with good agreement. Next, a reduced order model is constructed from the full model and the performance of the two models are compared. The reduced order linear model provided an acceptable representation of the full non-linear model.

Effects of Canopy-Payload Relative Motion on Control of Autonomous Parafoils

An eight-degree-of-freedom model is developed that accurately models relative pitching and yawing motion of a payload with respect to a parafoil. Constraint forces and moments are found analytically rather than using artificial constraint stabilization. A turn rate controller common in precision placement algorithms is used to demonstrate that relative yawing motion of the payload can result in persistent oscillations of the system. A model neglecting relative payload yawing failed to predict the same oscillations. It is shown that persistent oscillations can be eliminated by reduction of feedback gains; however, resulting tracking performance is poor. A reduced-order linear model is shown to be able to adequately predict relative payload dynamics for the proposed turn rate controller on the full eight-degree-of-freedom system.

Effects of a Cracked Blade on Mistuned Turbine Engine Rotor Vibration

An efficient methodology for predicting the nonlinear forced vibration response of a turbine engine rotor with a cracked blade is presented and used to investigate the effects of the damage on the forced response. The influence of small random blade-to-blade differences (mistuning) and rotation on the forced response are also considered. Starting with a finite element model, a hybrid-interface method of component mode synthesis (CMS) is employed to generate a reduced-order model (ROM). The crack surfaces are retained as physical degrees of freedom in the ROM so that the forces due to contact in three-dimensional space can be properly calculated. The resulting nonlinear equations of steady-state motion are solved by applying an alternating frequency/time-domain method, which is much more computationally efficient than traditional time integration. Using this reduced-order modeling and analysis framework, the effects of the cracked blade on the system response of an example rotor are investigated for various mistuning levels and rotation speeds. First, the advantages of the selected hybrid-interface CMS method are discussed and demonstrated. Then, the resonant frequency shift associated with the stiffness loss due to the crack and the vibration localization about the cracked blade are thoroughly investigated. In addition, the results of the nonlinear ROMs are compared with those obtained with linear ROMs, as well as blade-alone ROMs. It is shown that several key system vibration characteristics are not captured by the simpler models, but that some insight into the system response can be gained from the blade-alone response predictions. Furthermore, it is demonstrated that while the effects of the crack often appear similar to those of mistuning, the effects of mistuning and damage can be distinguished by observing and comparing the response across multiple families of system modes.

Physically Based Model-Predictive Control for SOFC Stacks and Systems

This paper discusses model-predictive controllers (MPC) that can incorporate physical knowledge of fuel-cell behavior into real-time multiple-input-multiple-output (MIMO) process-control strategies. The controller development begins with a high-fidelity, transient, physical model that represents the physical and chemical processes responsible for fuel-cell function. However, because such large nonlinear models cannot be solved in real time as part of the con-troller logic, linear reduced-order state-space models are required. The model reduction is accomplished via a process called system identification. The controller is designed to interpret sensors in the context of the reduced-order model and determine optimal actua-tion sequences that cause the system to follow a desired output tra-jectory. The process is demonstrated for a tubular SOFC stack that could be used for auxiliary-power unit (APU) applications.

Modeling and Control of Distillation Column in a Petroleum Process

This paper introduces a calculation procedure for modeling and control simulation of a condensate distillation column based on the energy balance (L‐V) structure. In this control, the reflux rate L and the boilup rate V are used as the inputs to control the outputs of the purity of the distillate overhead and the impurity of the bottom products. The modeling simulation is important for process dynamic analysis and the plant initial design. In this paper, the modeling and simulation are accomplished over three phases: the basic nonlinear model of the plant, the full‐order linearised model, and the reduced‐order linear model. The reduced‐order linear model is then used as the reference model for a model‐reference adaptive control (MRAC) system to verify the applicable ability of a conventional adaptive controller for a distillation column dealing with the disturbance and the model‐plant mismatch as the influence of the plant feed disturbances.

Modal reduction of mathematical models of biological molecules

This paper reports a detailed study of modal reduction based on either linear normal mode (LNM) analysis or proper orthogonal decomposition (POD) for modeling a single α- d-glucopyranose monomer as well as a chain of monomers attached to a moving atomic force microscope (AFM) under harmonic excitations. Also a modal reduction method combining POD and component modal synthesis is developed. The accuracy and efficiency of these methods are reported. The focus of this study is to determine to what extent these methods can reduce the time and cost of molecular modeling and simultaneously provide the required accuracy. It has been demonstrated that a linear reduced order model is valid for small amplitude excitation and low frequency excitation. It is found that a nonlinear reduced order model based on POD modes provides a good approximation even for large excitation while the nonlinear reduced order model using linear eigenmodes as the basis vectors is less effective for modeling molecules with a strong nonlinearity. The reduced order model based on component modal synthesis using POD modes for each component also gives a good approximation. With the reduction in the dimension of the system using these methods the computational time and cost can be reduced significantly.

Flow Control in a Driven Cavity Incorporating Excitation Phase Differential

Control of fluid flow is particularly difficult because of inherent nonlinearity in the Navier‐Stokes equations and high dimensionality of typical approximations of these equations. A control design is demonstrated for flow restricted to creeping flow within a driven cavity. Such a restriction allows linear reduced-order models to be generated as state-space systems. In particular, these models are generated as subspaces of the flow where each model represents modes associated with phase differential between exogenous disturbances. A linear parametervarying controller is designed to account for the range of dynamics introduced by the phase differential among subspaces. The controller is introduced to the reduced-order models, which contain individual phase-differential subspaces, and the full-order model, which contains all phase-differential subspaces, for disturbance attenuation. These closed-loop simulations show that the gain-scheduled controller, designed for individual subspaces, is able to reduce the flow velocity along the centerline of the cavity significantly for the full-order flow.

DYNAMIC MODELLING FOR CONDITION MONITORING OF GAS TURBINES: GENETIC ALGORITHMS APPROACH

The problem of model-based condition monitoring of aero gas turbine engines is considered. Genetic algorithms are applied for the dynamic modelling of aero engines by estimating parameters of the linear reduced-order model. The use of genetic algorithms affords flexibility in the choice of performance metrics. Real engine data is used to investigate the performance genetic algorithms and this approach is compared with traditional modelling techniques used in industry.

Modeling, simulation and control of a scheibel liquid–liquid contactor: Part 1. Dynamic analysis and system identification

The liquid–liquid extraction process is well-known for its complexity and often entails intensive modeling and computational efforts to simulate its dynamic behaviour. However, rigorous mathematical models are usually impractical or are of limited usefulness for control system design. Therefore, there is a need to derive simpler models for this process. Reduced-order linear models can be derived through applying system identification on the input–output simulation data. As a first step, a rigorous model for dynamic simulation of an extraction process is developed. This model employs an improved detailed stage-wise mixing stage with backmixing and it takes into account the variation in hydrodynamics, mass transfer, and physical properties throughout the length of the extraction column. It also approximates end effects by incorporating two mixing stages at both ends in addition to calculation of mass transfer within calming zones through the use of a mass transfer weight factor. The model is validated with dynamic experimental data for a nine stage Scheibel extraction column of type I. The simulation model is shown to be accurate for prediction of process behaviour under different operating conditions. Dynamic analysis of the process is conducted on the developed rigorous simulation model. Then, system identification is applied to derive linear time-invariant reduced-order models, which relate the input process variables ( agitator speed, solvent feed flowrate and concentration, feed concentration and flowrate) to the output process variables ( raffinate concentration and extract concentration). The identified model predictions are found to be in a good agreement with the rigorous ones.

Optimal Neurocontroller for Nonlinear Benchmark Structure

A neurocontrol method is applied to the nonlinear benchmark control problem. A neurocontroller is trained based on a reduced-order linear design model, then it is used to control a nonlinear evaluation model. In training the controller, a sensitivity evaluation scheme is used and weights are updated by minimizing a cost function. Absolute accelerations directly measured from sensors are used as the feedback signals for the controller. Not only the current step acceleration, but delay signals of sensor readings, are used to enhance the training capability. Numerical examples show that the controlled responses are considerably reduced compared with the uncontrolled case. In conclusion, the possibility of the proposed control algorithm as a candidate for the controller of nonlinear building is shown.

Suppression of Transient Vibration for Geared Mechanical System with Backlash Using Model-Based Control

This paper deals with a control technique of eliminating the transient vibration of a geared mechanical system with backlash. This technique is based on a model-based control in order to establish the damping effect at the driven machine part. The control model of dynamical compensator is composed of a linear reduced-order model related to the velocity control loop and the delay element related to the backlash. The control model estimates a load speed converted to the motor shaft. The difference between the estimated load speed and the motor speed is calculated dynamically, and it is added to the velocity command to suppress the transient vibration of the load. This control model is easily obtained from design or experimental data and its algorithm can be easily installed in a DSP. This control technique is applied to a mechanical system composed of spur gears. Simulations and experiments show satisfactory control results.

OPTIMAL VIBRATION ESTIMATION OF A NON-LINEAR FLEXIBLE BEAM MOUNTED ON A ROTATING COMPLIANT HUB

To eliminate the need for sensor placement on rotating flexible beams such as turbine blades, helicopter rotors and like applications, a new approach has been developed based on the linear quadratic estimator (LQE) technique for estimating the vibration of any point on the span of a rotating flexible beam mounted on a compliant hub ( plant) in the presence of process and measurements noise. A non-linear model of the plant is utilized in this study to mimic the actual plant behavior. The corresponding plant dynamics of the LQE are in the form of a reduced order linear model constructed from the eigenvalues and eigenfuctions of a finite element dynamic model of the plant formulated in the state space. A virtual hub deflection (that mimics the actual measurement of the vertical hub deflection needed by the estimation process) is generated by the non-linear model of the plant. The LQE reconstructs the states of the plant, including transverse deflection of the beam at any point, from the measurements of the vertical deflection of the hub, assuming that it is the most accessible state for measurement. Estimated beam tip deflection obtained by the proposed technique is then compared to the tip deflection generated by the non-linear model and the results show good agreement.

The use of pulse responses and system reduction for 2D unsteady flows using the Euler equations

AbstractThis paper describes the application of the theory of pulse responses to a linearised Euler scheme on a moving grid. The impulse responses of such a linear system are memory functions or temporal representations of the manner in which, and the time over which a perturbation remains active in the response of the system. The exact response of the linear system to an arbitrary input can then be predicted via convolution. The linearised Euler scheme is derived from a full nonlinear Euler method based on Jameson’s cell-centred scheme which has been modified to make it time-accurate and has the necessary terms to account for grid motion. The discrete form of the full Euler equations are linearised about the corresponding nonlinear steady mean flow assuming the unsteadiness in the flow and grid are small. The resulting set of differential equations for the flow perturbations are solved at each time step using a dual-time scheme. These equations can be easily solved to obtain pulse responses, which capture the entire frequency content of the linear system. These pulse responses then provide a simple and efficient method for calculating the system response to many different inputs via convolution. A further use of the pulse responses is their potential to allow a reduced-order linear model to be constructed, which preserves some of the accuracyof the original system. Preliminary work carried out in this area is presented. Results of calculations for heave, pitch and ramp test cases are presented.

Performance Limitations of a Class of Two-Stage Electro-Hydraulic Flow Valves

AbstractBy examining the dynamics of a popular type of two-stage electronic proportional valve, this paper addresses its performance limitations, with both cautions in control implementation and suggestions in valve design. While several benefits do exist, there are limitations to the closed loop performance of the valve when it is included in a valve-controlled electro-hydraulic system. These limitations come from the structural feature that the pilot flow not only controls but also contributes to the total flow. Although for steady state performance this design gives a higher flow efficiency, for dynamic performance it results in zeros in the open loop transfer function, which will limit the closed loop bandwidth of a flow control system. A non-linear analytical model of this particular type of valve is derived first. It is then simplified as a reduced order linear model with the inherit system zeros illustrated. Validation of the analysis is obtained by experimental results on a testbed.

SUPPRESSION OF TRANSIENT VIBRATION FOR GEARED MECHANICAL SYSTEM WITH BACKLASH USING MODEL-BASED CONTROL

This paper deals with a control technique of eliminating the transient vibration of a geared mechanical system with backlash. This technique is based on a model-based control in order to establish the damping effect at the driven machine part. The control model of dynamical compensator is composed of a linear reduced-order model related to the velocity control loop and the delay element related to the backlash. The control model estimates a load speed converted to the motor shaft. The difference between the estimated load speed and the motor speed is calculated dynamically, and it is added to the velocity command to suppress the transient vibration of the load. This control model is easily obtained from design or experimental data and its algorithm can be easily installed in a DSP. This control technique is applied to a mechanical system composed of spur gears. Simulations and experiments show satisfactory control results.

An Arnoldi approach for generation of reduced-order models for turbomachinery

A linear reduced-order aerodynamic model is developed for aeroelastic analysis of turbomachines. The basis vectors are constructed using a block Arnoldi method. Although the model is cast in the time domain in state-space form, the spatial periodicity of the problem is exploited in the frequency domain to obtain these vectors efficiently. The frequency domain proper orthogonal decomposition is identified as a special case of the Arnoldi method. We show an application where the aerodynamic model is coupled with a simple structural model that has two degrees of freedom for each blade. The technique is applicable to viscous and three-dimensional problems as well as multi-stage problems with inlet and exit disturbance flows, although here results are presented for two-dimensional, inviscid flow through a 20-blade single-stage rotor. In this case, the number of states of the model is on the order of 10 per blade passage, making it appropriate for control applications.