Model Order Reduction for Quantum Molecular Dynamics
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Mathematics and Computing↗
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Large space system control technology model order reduction study
The mission control requirements for large space structures are presented as well as some of the problems associated with the active control of large structures in space. A generic model is introduced that contains most of the desired features in the modeling and control problem for large space structures.
Skelton, R. E.↗
The spatial order reduction problem and its effect on adaptive control of distrirbuted parameter systems
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Balas, M. J.↗
Stochastic response analysis, order reduction, and output feedback controllers for flexible spacecraft
Real disturbances and real sensors have finite bandwidths. The first objective of this paper is to incorporate this finiteness in the 'open-loop modal cost analysis' as applied to a flexible spacecraft. Analysis based on residue calculus shows that among other factors, significance of a mode depends on the power spectral density of disturbances and the response spectral density of sensors at the modal frequency. The second objective of this article is to compare performances of an optimal and a suboptimal output feedback controller, the latter based on 'minimum error excitation' of Kosut. Both the performances are found to be nearly the same, leading us to favor the latter technique because it entails only linear computations. Our final objective is to detect an instability due to truncated modes by representing them as a multiplicative and an additive perturbation in a nominal transfer function. In an example problem it is found that this procedure leads to a narrow range of permissible controller gains, and that it labels a wrong mode as a cause of instability. A free beam is used to illustrate the analysis in this work.
Hablani, H. B.↗
Model order reduction applied to a hot-bench simulation of an aeroelastic wind-tunnel model
Simulations of an aeroelastically scaled wind-tunnel model were developed for hot-bench testing of a digital controller. The digital controller provided active flutter-suppression, rolling-maneuver-load alleviation, and plant estimation. To achieve an acceptable time scale for the hot-bench application, the mathematical model of the wind-tunnel model was reduced from 220 states to approximately 130 states while assuring that the required accuracy was preserved for all combinations of 10 inputs and 56 outputs. The reduction was achieved by focussing on a linear, aeroelastic submodel of the full mathematical model and by applying a method based on the internally balanced realization of a dynamic system. The error-bound properties of the internally balanced realization significantly contribute to its utility in the model reduction process. The reduction method and the results achieved are described.
Buttrill, Carey S.↗
Operator Inference-based Model Order Reduction of Thermal Protection System Finite Element Simulations
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Tencer, John Thomas [Sandia National Laboratories ↗
Boundary Corrections for Kernel Approximation to Differential Operators
The kernel-based approach to operator approximation for partial differential equations has been shown to be unconditionally stable for linear PDEs and numerically exhibit unconditional stability for non-linear PDEs. These methods have the same computational cost as an explicit finite difference scheme but can exhibit order reduction at boundaries. In previous work on periodic domains, order reduction was addressed, yielding high-order accuracy. The issue addressed in this work is the elimination of order reduction of the kernel-based approach for a more general set of boundary conditions. Further, we consider the case of both first and second order operators. To demonstrate the theory, we provide not only the mathematical proofs but also experimental results by applying various boundary conditions to different types of equations. The results agree with the theory, demonstrating a systematic path to high order for kernel-based methods on bounded domains.
97 MATHEMATICS AND COMPUTING↗
A stiff order condition theory for Runge–Kutta methods applied to semilinear ODEs
Classical convergence theory of Runge–Kutta methods assumes that the time step is small relative to the Lipschitz constant of the ordinary differential equation (ODE). For stiff problems, that assumption is often violated, and a problematic degradation in accuracy, known as order reduction, can arise. Methods with high stage order, e.g., Gauss–Legendre and Radau, are known to avoid order reduction, but they must be fully implicit. For the broad class of semilinear ODEs, which consist of a stiff linear term and non-stiff nonlinear term, we show that weaker conditions suffice. Here, our new semilinear order conditions are formulated in terms of orthogonality relations and can be enumerated by rooted trees. Finally, we prove global error bounds that hold uniformly with respect to stiffness of the linear term.
Mathematics and Computing↗
Computational methods of robust controller design for aerodynamic flutter suppression
The development of Riccati iteration, a tool for the design and analysis of linear control systems is examined. First, Riccati iteration is applied to the problem of pole placement and order reduction in two-time scale control systems. Order reduction, yielding a good approximation to the original system, is demonstrated using a 16th order linear model of a turbofan engine. Next, a numerical method for solving the Riccati equation is presented and demonstrated for a set of eighth order random examples. A literature review of robust controller design methods follows which includes a number of methods for reducing the trajectory and performance index sensitivity in linear regulators. Lastly, robust controller design for large parameter variations is discussed.
Anderson, L. R.↗
Order-Reduced Solution of the Nonlinear High-Fidelity Generalized Method of Cells Micromechanics Relations
The High-Fidelity Generalized Method of Cells (HFGMC) is one technique for accurately simulating nonlinear composite material behavior. The HFGMC uses a higher-order approximation for the subcell displacement field that allows for a more accurate determination of the subcell stressstrain fields at the cost of some computational efficiency. In order to reduce computational costs associated with the solution of the ensuing system of simultaneous equations, the HFGMC global system of equations for doubly-periodic repeating unit cells with nonlinear constituents was reduced in size through the use of a Petrov-Galerkin-based Proper Orthogonal Decomposition order-reduction scheme. A number of cases were presented that address the computational feasibility of using order-reduction techniques to solve solid mechanics problems involving complex microstructures.
order-reduction↗
Structural mode significance using INCA
Structural finite element models are often too large to be used in the design and analysis of control systems. Model reduction techniques must be applied to reduce the structural model to manageable size. In the past, engineers either performed the model order reduction by hand or used distinct computer programs to retrieve the data, to perform the significance analysis and to reduce the order of the model. To expedite this process, the latest version of INCA has been expanded to include an interactive graphical structural mode significance and model order reduction capability.
Bauer, Frank H.↗
Nth order redundancy reduction.
Curve fitting for optimum redundance reduction on sampled data, describing polygonal and approximate polygonal techniques
Wilkins, L. C.↗
Controller reduction by preserving impulse response energy
A model order reduction algorithm based on a Krylov recurrence formulation is developed to reduce order of controllers. The reduced-order controller is obtained by projecting the full-order LQG controller onto a Krylov subspace in which either the controllability or the observability grammian is equal to the identity matrix. The reduced-order controller preserves the impulse response energy of the full-order controller and has a parameter-matching property. Two numerical examples drawn from other controller reduction literature are used to illustrate the efficacy of the proposed reduction algorithm.
Craig, Roy R., Jr.↗
Turbulent drag reduction for external flows
Paper presents a review and summary of turbulent drag reduction approaches applicable to external flows. Because relatively recent and exhaustive reviews exist for laminar flow control and polymer (hydrodynamic) drag reduction, the paper focuses upon the emerging areas of nonplanar geometry and large eddy alteration. Turbulent control techniques for air generally result in modest (but technologically significant) drag reductions (order of 20 percent or less) whereas hydrodynamic approaches can yield drag reductions the order of 70 percent. Paper also includes suggestions for alternative concepts and optimization of existing approaches.
Bushnell, D. M.↗
Turbulent drag reduction for external flows
A summary of turbulent drag reduction approaches applicable to external flows is given. Because relatively recent and exhaustive reviews exist for laminar flow control and polymer (hydrodynamic) drag reduction, the focus here is upon the emerging areas of nonplanar geometry and large-eddy alteration. Turbulent control techniques for air generally result in modest (but technologically significant) drag reductions (order of 20 percent or less), whereas hydrodynamic approaches can yield drag reductions the order of 70 percent. Suggestions are included for alternative concepts and optimization of existing approaches.
Bushnell, D. M.↗
On the control of elastic vehicles - Model simplification and stability robustness
Quantitative criteria are presented for model simplification, or order reduction, such that the reduced order model may be used to synthesize and evaluate a control law, and the stability and stability robustness obtained using the reduced-order model will be preserved when controlling the full-order system. The error introduced due to model simplification is treated as modeling uncertainty, and some of the results from multivariable robustness theory are brought to bear on the model simplification problem. A numerical procedure developed previously is shown to lead to results that meet the necessary criteria. The procedure is applied to reduce the model of a flexible aircraft. Also, the importance of the control law itself, in meeting the modeling criteria, is underscored. An example is included that demonstrates that an apparently robust control law actually amplifies modest modeling errors in the critical frequency region, and leads to undesirable results. The cause of this problem is identified to be associated with the canceling of lightly-damped transmission zeroes in the plant.
Schmidt, David K.↗
Design, test, and evaluation of three active flutter suppression controllers
Three control law design techniques for flutter suppression are presented. Each technique uses multiple control surfaces and/or sensors. The first method uses traditional tools (such as pole/zero loci and Nyquist diagrams) for producing a controller that has minimal complexity and which is sufficiently robust to handle plant uncertainty. The second procedure uses linear combinations of several accelerometer signals and dynamic compensation to synthesize the model rate of the critical mode for feedback to the distributed control surfaces. The third technique starts with a minimum-energy linear quadratic Gaussian controller, iteratively modifies intensity matrices corresponding to input and output noise, and applies controller order reduction to achieve a low-order, robust controller. The resulting designs were implemented digitally and tested subsonically on the active flexible wing wind-tunnel model in the Langley Transonic Dynamics Tunnel. Only the traditional pole/zero loci design was sufficiently robust to errors in the nominal plant to successfully suppress flutter during the test. The traditional pole/zero loci design provided simultaneous suppression of symmetric and antisymmetric flutter with a 24-percent increase in attainable dynamic pressure. Posttest analyses are shown which illustrate the problems encountered with the other laws.
Adams, William M., Jr.↗
S-OPT: A Points Selection Algorithm for Hyper-Reduction in Reduced Order Models
While projection-based reduced order models can reduce the dimension of full order solutions, the resulting reduced models may still contain terms that scale with the full order dimension. Hyper-reduction techniques are sampling-based methods that further reduce this computational complexity by approximating such terms with a much smaller dimension. The goal of this work is to introduce the points selection algorithm developed by Shin and Xiu as a hyper-reduction method. The selection algorithm was originally proposed as a stochastic collocation method for uncertainty quantification. Since the algorithm aims at maximizing a quantity $\mathcal{S}$ that measures both the column orthogonality and the determinant, we refer to the algorithm as S-OPT. Numerical examples are provided to demonstrate the performance of S-OPT and to compare its performance with a gappy proper orthogonal decomposition (POD) algorithm. Here, we found that using the S-OPT algorithm is shown to predict the full order solutions with higher accuracy than gappy POD especially when the number of sampling points is small, although we note that S-OPT shows slow asymptotic convergence with respect to the number of samples for some applications, e.g., Lagrangian hydrodynamics.
97 MATHEMATICS AND COMPUTING↗