Thesis (M.S. in in Aeronautical Engineering)--Air Force Institute of Technology, March 2008. / "Presented to the Faculty, Department of Aeronautics and Astronautics Graduate School of Engineering and Management, Air Force Institute of Technology Air University, Air Education and Training Command in partial fulfillment of the requirements for the Degree of Master of Science in Aeronautical Engineering, March 2008."--P. [ii]. Thesis advisor: Lt. Col. Chris Shearer. "March 2008." "AFIT/GAE/ENY/08-M13." Includes bibliographical references. Also available online in PDF from the DTIC Online Web site.

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2005. / Includes bibliographical references (leaves 81-86). Also available in electronic version.

Fractional calculus enables the possibility of using real number powers or complex number powers of the differentiation operator. The fundamental connection between fractional calculus and subordination processes is explored and affords a physical interpretation for a fractional trajectory, that being an average over an ensemble of stochastic trajectories. With an ensemble average perspective, the explanation of the behavior of fractional chaotic systems changes dramatically. Before now what has been interpreted as intrinsic friction is actually a form of non-Markovian dissipation that automatically arises from adopting the fractional calculus, is shown to be a manifestation of decorrelations between trajectories. Nonlinear Langevin equation describes the mean field of a finite size complex network at criticality. Critical phenomena and temporal complexity are two very important issues of modern nonlinear dynamics and the link between them found by the author can significantly improve the understanding behavior of dynamical systems at criticality. The subject of temporal complexity addresses the challenging and especially helpful in addressing fundamental physical science issues beyond the limits of reductionism.

Trajectory optimization is an important part of launch vehicle design and operation. With the high costs of launching payload into orbit, every pound that can be saved increases affordability. One way to save weight in launch vehicle design and operation is by optimizing the ascent trajectory. Launch vehicle trajectory optimization is a field that has been studied since the 1950’s. Originally, analytic solutions were sought because computers were slow and inefficient. With the advent of computers, however, different algorithms were developed for the purpose of trajectory optimization. Computer resources were still limited, and as such the algorithms were limited to local optimization methods, which can get stuck in specific regions of the design space. Local methods for trajectory optimization have been well studied and developed. Computer technology continues to advance, and in recent years global optimization has become available for application to a wide variety of problems, including trajectory optimization. The aim of this thesis is to create a methodology that applies global optimization to the trajectory optimization problem. Using information from a global search, the optimization design space can be reduced and a much smaller design space can be analyzed using already existing local methods. This allows for areas of interest in the design space to be identified and further studied and helps overcome the fact that many local methods can get stuck in local optima. The design space included in trajectory optimization is also considered in this thesis. The typical optimization variables are initial conditions and flight control variables. For direct optimization methods, the trajectory phase structure is currently chosen a priori. Including trajectory phase structure variables in the optimization process can yield better solutions. The methodology and phase structure optimization is demonstrated using an earth-to-orbit trajectory of a Delta IV Medium launch vehicle. Different methods of performing the global search and reducing the design space are compared. Local optimization is performed using the industry standard trajectory optimization tool POST. Finally, methods for varying the trajectory phase structure are presented and the results are compared.

This dissertation progressively addresses research problems related to the trajectory optimization for fuel cell powered UAVs, from propulsion system model development, to optimal trajectory analyses and optimal trajectory applications. A dynamic model of a fuel cell powered UAV propulsion system is derived by combining a fuel cell system dynamic model, an electric motor dynamic model, and a propeller performance model. The influence of the fuel cell system dynamics on the optimal trajectories of a fuel cell powered UAV is investigated in two phases. In the first phase, the optimal trajectories of a fuel cell powered configuration and that of a conventional gas powered configuration are compared for point-to-point trajectory optimization problems with different performance index functions. In the second phase, the influence of the fuel cell system parameters on the optimal fuel consumption cost of the minimum fuel point-to-point optimal trajectories is investigated. This dissertation also presents two applications for the minimum fuel point-to-point optimal trajectories of a fuel cell powered UAV: three-dimensional minimum fuel route planning and path generation, and fuel cell system size optimization with respect to a UAV mission.

The escape of trajectories is a ubiquitous phenomenon in open dynamical systems and stochastic processes. If escape occurs repetitively for a statistical ensemble of trajectories, the population of remaining trajectories often undergoes an exponential decay characterised by the so-called escape rate. Its inverse defines the lifetime of the decaying state, which represents an intrinsic property of the system. This paradigm is fundamental to nucleation theory and reaction-rate theory in chemistry, physics, and biology.<p><p>In many circumstances, escape is activated by the presence of noise, which may be of internal or external origin. This is the case for thermally activated escape over a potential energy barrier and, more generally, for noise-induced escape in continuous-time or discrete-time dynamics. <p><p>In the weak-noise limit, the escape rate is often observed to decrease exponentially with the inverse of the noise amplitude, a behaviour which is given by the van't Hoff-Arrhenius law of chemical kinetics. In particular, the two important quantities to determine in this case are the exponential dependence (the ``activation energy') and its prefactor.<p><p>The purpose of the present thesis is to develop an analytical method to determine these two quantities. We consider in particular one-dimensional continuous and discrete-time systems perturbed by Gaussian white noise and we focus on the escape from the basin of attraction of an attracting fixed point.<p><p>In both classes of systems, using path-integral methods, a formula is deduced for the noise-induced escape rate from the attracting fixed point across an unstable fixed point, which forms the boundary of the basin of attraction. The calculation starts from the trace formula for the eigenvalues of the operator ruling the time evolution of the probability density in noisy maps. The escape rate is determined by the loop formed by two heteroclinic orbits connecting back and forth the two fixed points in a two-dimensional auxiliary deterministic dynamical system. The escape rate is obtained, including the expression of the prefactor to van't Hoff-Arrhenius exponential factor./L'échappement des trajectoires est un phénomène omniprésent dans les systèmes dynamiques ouverts et les processus stochastiques. Si l'échappement se produit de façon répétitive pour un ensemble statistique de trajectoires, la population des trajectoires restantes subit souvent une décroissance exponentielle caractérisée par le taux d'échappement. L'inverse du taux d'échappement définit alors la durée de vie de l'état transitoire associé, ce qui représente une propriété intrinsèque du système. Ce paradigme est fondamental pour la théorie de la nucléation et, de manière générale, pour la théorie des taux de transitions en chimie, en physique et en biologie.<p><p>Dans de nombreux cas, l'échappement est induit par la présence de bruit, qui peut être d'origine interne ou externe. Ceci concerne en particulier l'échappement activé thermiquement à travers une barrière d'énergie potentielle, et plus généralement, l'échappement dû au bruit dans les systèmes dynamiques à temps continu ou à temps discret.<p><p>Dans la limite de faible bruit, on observe souvent une décroissance exponentielle du taux d'échappement en fonction de l'inverse de l'amplitude du bruit, un comportement qui est régi par la loi de van't Hoff-Arrhenius de la cinétique chimique. En particulier, les deux quantités importantes de cette loi sont le coefficient de la dépendance exponentielle (c'est-à-dire ``l'énergie d'activation') et son préfacteur.<p><p>L'objectif de cette thèse est de développer une théorie analytique pour déterminer ces deux quantités. La théorie que nous présentons concerne les systèmes unidimensionnels à temps continu ou discret perturbés par un bruit blanc gaussien et nous considérons le problème de l'échappement du bassin d'attraction d'un point fixe attractif. Pour s'échapper, les trajectoires du système bruité initialement contenues dans ce bassin d'attraction doivent alors traverser un point fixe instable qui forme la limite du bassin.<p><p>Dans le présent travail, et pour les deux types de systèmes, une formule est dérivée pour le taux d'échappement du point fixe attractif en utilisant des méthodes d'intégrales de chemin. Le calcul utilise la formule de trace pour les valeurs propres de l'opérateur gouvernant l'évolution temporelle de la densité de probabilité dans le système bruité. Le taux d'échappement est déterminé en considérant la boucle formée par deux orbites hétéroclines liant dans les deux sens les deux points fixes dans un système dynamique auxiliaire symplectique et bidimensionnel. On obtient alors le taux d'échappement, comprenant l'expression du préfacteur de l'exponentielle de la loi de van't Hoff-Arrhenius. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished