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We present an abstract approximation framework for estimation of parameters in nonlinear nonautonomous distributed systems. Specific examples involving linear spline approximations for retarded delay equations and cubic spline approximations for parabolic partial differential equations are discussed and shown to be included as special cases of our general framework.
A key challenge in designing the new High Speed Civil Transport (HSCT) aircraft is determining a good match between the airframe and engine. Multidisciplinary design optimization can be used to solve the problem by adjusting parameters of both the engine and the airframe. Earlier, an example problem was presented of an HSCT aircraft with four mixed-flow turbofan engines and a baseline mission to carry 305 passengers 5000 nautical miles at a cruise speed of Mach 2.4. The problem was solved by coupling NASA Lewis Research Center's design optimization testbed (COMETBOARDS) with NASA Langley Research Center's Flight Optimization System (FLOPS). The computing time expended in solving the problem was substantial, and the instability of the FLOPS analyzer at certain design points caused difficulties. In an attempt to alleviate both of these limitations, we explored the use of two approximation concepts in the design optimization process. The two concepts, which are based on neural network and linear regression approximation, provide the reanalysis capability and design sensitivity analysis information required for the optimization process. The HSCT aircraft optimization problem was solved by using three alternate approaches; that is, the original FLOPS analyzer and two approximate (derived) analyzers. The approximate analyzers were calibrated and used in three different ranges of the design variables; narrow (interpolated), standard, and wide (extrapolated).
Computational implementation of feedback control laws for linear hereditary systems requires the approximation of infinite dimensional feedback operators with finite dimensional operators. The dense subspaces of K-polygonal functions in reproducing kernel Hilbert spaces, RKH spaces, suggest finite dimensional approximations of the matrix representations of the control operators. A convergence theorem is developed for the approximations and the numerical implementation of the approximations is discussed.
The purpose of this paper is to show how the important problems of linear system theory can be solved concisely for a particular class of linear systems, namely block circulant systems, by exploiting the algebraic structure. This type of system arises in lumped approximations to linear partial differential equations. The computation of the transition matrix, the variation of constants formula, observability, controllability, pole allocation, realization theory, stability and quadratic optimal control are discussed. In principle, all questions which are solved here could also be solved by standard methods; the present paper clearly exposes the structure of the solution, and thus permits various savings in computational effort.
The efficiency gains obtained using higher-order implicit Runge-Kutta schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier-Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at each timestep are presented. The first algorithm (NMG) is a pseudo-time-stepping scheme which employs a non-linear full approximation storage (FAS) agglomeration multigrid method to accelerate convergence. The other two algorithms are based on Inexact Newton's methods. The linear system arising at each Newton step is solved using iterative/Krylov techniques and left preconditioning is used to accelerate convergence of the linear solvers. One of the methods (LMG) uses Richardson's iterative scheme for solving the linear system at each Newton step while the other (PGMRES) uses the Generalized Minimal Residual method. Results demonstrating the relative superiority of these Newton's methods based schemes are presented. Efficiency gains as high as 10 are obtained by combining the higher-order time integration schemes with the more efficient nonlinear solvers.
A two-time scale approximation for the linear quadratic optimal output feedback regulator program is examined. Necessary conditions for optimality, as well as an algorithm for computing locally near-optimal gains are derived. If it is assumed that the slow and fast subsystem initial conditions are uniformly distributed, optimal gains for the two-time-scale problem provide a second-order approximation to optimal closed-loop performance in the unperturbed system. This is verified with a numerical example.
Several long-standing issues regarding linear estimation and coherent structures are addressed, and some questions that were addressed partially, but never with the benefit of full, three-dimensional information are answered. The objectives were to: determine how well linear estimates approximate the field obtained by true conditional averaging, using events such as those in quadrant analysis; determine the extent to which the three-dimensional linearly estimated fields correspond to coherent structures, and the degree and manner in which they differ; evaluate the type and nature of the structural information gained by employing several different types of events; and learn more about the 3-D structure of important coherent motions that occur in wall turbulence. The results indicate that linear stochastic estimation can be used effectively in the study of numerical data bases consisting of three-dimensional vector fields, both velocity and vorticity. Two-point stochastic estimation yields more structural information and more detail than single-point estimation. The structures observed occurred repeatedly within the flow, but much can not be said about their dominance or the probability of their occurrence without further systematic studies of their frequency.
An abstract approximation framework is developed for the finite and infinite time horizon discrete-time linear-quadratic regulator problem for systems whose state dynamics are described by a linear semigroup of operators on an infinite dimensional Hilbert space. The schemes included the framework yield finite dimensional approximations to the linear state feedback gains which determine the optimal control law. Convergence arguments are given. Examples involving hereditary and parabolic systems and the vibration of a flexible beam are considered. Spline-based finite element schemes for these classes of problems, together with numerical results, are presented and discussed.
An abstract approximation framework is developed for the finite and infinite time horizon discrete-time linear-quadratic regulator problem for systems whose state dynamics are described by a linear semigroup of operators on an infinite dimensional Hilbert space. The schemes included the framework yield finite dimensional approximations to the linear state feedback gains which determine the optimal control law. Convergence arguments are given. Examples involving hereditary and parabolic systems and the vibration of a flexible beam are considered. Spline-based finite element schemes for these classes of problems, together with numerical results, are presented and discussed.
Generalized energetics were studied for nonlinear inviscid symmetric instability (SI). It was found that the linear theory fails to predict the stability in certain cases where the basic state is transitional between stability and instability. The initial growth of the SI perturbations can be fairly well approximated by linear theory, but the long time nonlinear evaluations will be bonded energetically if the SI region is finite. However, a further extension of the energetics to conditional symmetric instability (CSI) shows that the nonlinear evolution of circulation will energetically depend much more on the precipitation in a complicated way. By treating the latent heat as a source which is implicitly related to the motion field, the existence, uniqueness and stability of steady viscous (CSI) circulations are studied. Viscous CSI circulations are proved to be unique and asymptotically stable when the heat sources are weak and less sensitive to the motion perturbations. By considering the fact that moist updrafts are narrow and using eddy viscosity of 0(1,000 m squared/s) the stability criterion suggests that some frontal rainbands were probably dominated by the CSI mechanism even in their mature quasi-steady stage.
A high resolution (approximately 2 km) image of Beta Regio, Venus, obtained with the Arecibo radar system, reveals additional details of structure which seem to confirm past suggestions that this region consists of a rift system and associated volcanism. Numerous long linear features approximately parallel to the north-south trending trough discovered by the Pioneer-Venus Orbiter are interpreted as being indicative of extensive faulting and the detail for the radar reflectivity anomaly coincident with Theia Mons suggests that it is a major volcano situated on what would be the western bounding fault of the rift system.
A high-resolution (approximately 2 km) image of Beta Regio, Venus, obtained with the Arecibo radar system, reveals additional details of its structure which seem to confirm past suggestions that this region consists of a rift system and associated volcanism. Numerous long linear features approximately parallel to the north-south trending trough discovered by the Pioneer-Venus Orbiter are interpreted as being indicative of extensive faulting and the detail for the radar reflectivity anomaly coincident with Theia Mons suggests that it is a major volcano situated on what would be the western bounding fault of the rift system.
Solid-state array imaging has become an indispensable tool for various modem scientific instrumentation and industrial applications. Often in experiments, use of an imaging system with linear response is very desirable. The linearity of commercial products, however, has yet to be accurately determined in order to assess their nonlinearity effects. Linearity is affected both by sensors and electronics. It is also very desirables to employ low-cost products if we can provide the performance comparable of expensive ones through calibration. Here, we present a very simple approach for approximately checking linearity of imaging systems and also a simple calibration method for precisely measuring intensity ratio or relative intensity variation. These methods can be readily implemented with existing equipment. The calibration method works well with an intermediate-grade analog sensor interfaced by a frame grabber. In our initial attempt for calibration, the nonlinearity of the camera can be reduced from about five percent to much less than one percent in standard deviation.
An approximate technique is presented for the prediction of the large-scale turbulent structure sound source in a supersonic jet. A linearized Euler equations code is used to solve for the flow disturbances within and near a jet with a given mean flow. Assuming a normal mode composition for the wave-like disturbances, the linear radial profiles are used in an integration of the Navier-Stokes equations. This results in a set of ordinary differential equations representing the weakly nonlinear self-interactions of the modes along with their interaction with the mean flow. Solutions are then used to correct the amplitude of the disturbances that represent the source of large-scale turbulent structure sound in the jet.
A criterion for linear feature selection is proposed which is based on mean square apporximation of class density functions. It is shown that for the widest possible class of approximants, the criterion reduces to Devijver's Bayesian distance. For linear approximants the criterion is equivalent to well known generalized Fisher criteria.
The paper describes the application of techniques of exact I/O linearization of nonlinear control systems to the flight control of V/STOL aircraft. It is seen that the application of the theory to this example is not straightforward; in particular, the direct application of the theory yielded an undesirable controller. The situation was remedied by neglecting the coupling between the rolling moment input to the aircraft dynamics and the dynamics along the y-axis. An approximate I/O linearization procedure developed for slightly nonminimum phase nonlinear systems is shown to be effective for V/STOL aircraft.
The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.
A nonlinear stationary homogeneous digital filter DIRSIT (derivative information recovery by a selective integration technique) is investigated. The spectrum of a quasi-linear discrete describing function (DDF) to DIRSIT is obtained by a digital measuring scheme. A finite impulse response (FIR) approximation to the quasi-linearization is then obtained. Finally, DIRSIT is compared with its quasi-linear approximation and with a standard digital differentiating technique. Results indicate the effects of DIRSIT on a wide variety of practical signals.