Browsing by Subject "Optimal"
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Item An analytical approach to computing step sizes for finite-difference derivatives(2012-05) Mathur, Ravishankar; Ocampo, Cesar; Hull, David G.; Fowler, Wallace T.; Marchand, Belinda; Senent, JuanFinite-difference methods for computing the derivative of a function with respect to an independent variable require knowledge of the perturbation step size for that variable. Although rules of thumb exist for determining the magnitude of the step size, their effectiveness diminishes for complicated functions or when numerically solving difficult optimization problems. This dissertation investigates the problem of determining the step size that minimizes the total error associated with finite-difference derivative approximations. The total error is defined as the sum of errors from numerical sources (roundoff error) and mathematical approximations (truncation error). Several finite-difference approximations are considered, and expressions are derived for the errors associated with each approximation. Analysis of these errors leads to an algorithm that determines the optimal perturbation step size that minimizes the total error. A benefit of this algorithm is that the computed optimal step size, when used with neighboring values of the independent variable, results in approximately the same magnitude of error in the derivative. This allows the same step size to be used for several successive iterations of the independent variable in an optimization loop. A range of independent variable values for which the optimal step size can safely remain constant is also computed. In addition to roundoff and truncation errors within the finite-difference method, numerical errors within the actual function implementation are also considered. It is shown that the optimal step size can be used to compute an upper bound for these condition errors, without any prior knowledge of the function implementation. Knowledge of a function's condition error is of great assistance during the debugging stages of simulation design. Although the fundamental analysis assumes a scalar function of a scalar independent variable, it is later extended to the general case of a vector function of a vector independent variable. Several numerical examples are shown, ranging from simple polynomial and trigonometric functions to complex trajectory optimization problems. In each example, the step size is computed using the algorithm developed herein, a rule-of-thumb method, and an alternative statistical algorithm, and the resulting finite-difference derivatives are compared to the true derivative where available.Item Analytical approach to the design of optimal satellite constellations for space-based space situational awareness applications(2011-12) Biria, Ashley Darius; Marchand, Belinda G.; Lightsey, E. GlennIn recent years, the accumulation of space debris has become an increasingly pressing issue, and adequately monitoring it is a formidable task for designated ground-based sensors. Supplementing the capabilities of these ground-based networks with orbiting sensing platforms would dramatically enhance the ability of such systems to detect, track, identify, and characterize resident space objects -- the primary goals of modern space situational awareness (SSA). Space-based space situational awareness (SBSSA), then, is concerned with achieving the stated SSA goals through coordinated orbiting sensing platforms. To facilitate the design of satellite constellations that promote SSA goals, an optimization approach is selected, which inherently requires a pre-defined mathematical representation of a cost index or measure of merit. Such representations are often analytically available, but when considering optimal constellation design for SBSSA applications, a closed-form expression for the cost index is only available under certain assumptions. The present study focuses on a subset of cases that admit exact representations. In this case, geometrical arguments are employed to establish an analytical formulation for the coverage area provided as well as for the coverage multiplicity. These analytical results are essential in validating numerical approximations that are able to simulate more complex configurations.Item Continuous canopy temperature as a tool for managing deficit irrigation(2012-12) Young, Andrew W.; Dotray, Peter A.; Mahan, James R.; Payton, Paxton R.Deficit irrigation is becoming a trend in agricultural lands with reduced water. With the declining water resources comes renewed interest in deficit irrigation strategies and enhanced management capabilities to provide water when and where it is needed. However, in the past, plant-monitoring capabilities to assess water status of the plant were very costly and labor intensive. The innovation in infrared thermometry systems has allowed for the technology to become smaller and more cost efficient. This investigation uses the established method of BIOTIC developed by research scientist at USDA/ARS. The BIOTIC method has been patented and licensed by a new technology startup company, Smartfield™, under the moniker Smartcrop™. The research conducted used the Smartcrop™ technology, which consists of wireless infrared sensors and base stations for recording data from sensors. This thesis focused on the 2009 and 2010 cotton growing seasons in the Lubbock area. Water and yield data were discussed and analyzed in detail along with other environmental data relevant to plant growth and yield. Analysis and discussion of large temperature data sets were conducted. Canopy temperature comparisons were made using the BIOTIC method along with air vs. canopy temperature comparisons and treatment temperature comparisons. Vapor pressure deficits were also discussed in detail for selected treatments over the growing seasons. Finally, daytime average canopy temperature comparisons provided accurate estimates of water through the plant as a predictor of yield.Item Optimal Control Strategies for Saccadic Eye Movements in Humans(2010-12) Gaumond, T; Schovanec, Lawrence; Iyer, Ram V.; Ghosh, Bijoy K.Human motor control systems are complicated by issues of nonlinearity, redundancy, and multiple degrees of freedom. A great variety of mathematical and engineering approaches have been applied to the problem of modeling human movement systems: open and closed loop control, dynamic optimization, internal models, and learning. In this dissertation, aspects of these various approaches will be utilized within the context of the human eye system. After establishing the mathematical description of ocular dynamics we shall explore the differences between the linear system and nonlinear system, controllability, parameter sensitivity, and the use of time optimal control as an approach to understanding neural strategies that correspond to eye movement. In particular, optimization theory provides one scenario for the selection process of motor planning. By the choice of appropriate cost functions, we shall examine the cost of movement in terms of measures that correspond to efficiency, smoothness, accuracy and duration. v