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Closed-Loop Nominal and Abort Atmospheric Ascent Guidance for Rocket-Powered Launch VehiclesDukeman, Greg A. 18 January 2005 (has links)
An advanced ascent guidance algorithm for rocket-powered launch vehicles is developed. The ascent guidance function is responsible for commanding attitude, throttle and setting during the powered ascent phase of flight so that the vehicle attains target cutoff conditions in a near-optimal manner while satisfying path constraints such as maximum allowed bending moment and maximum allowed axial acceleration. This algorithm cyclically solves the calculus-of-variations two-point boundary-value problem starting at vertical rise completion through orbit insertion. This is different from traditional ascent guidance algorithms which operate in an open-loop mode until the high dynamic pressure portion of the trajectory is over, at which time there is a switch to a closed loop guidance mode that operates under the assumption of negligible aerodynamic forces.
The main contribution of this research is an algorithm of the predictor-corrector type wherein the state/costate system is propagated with known (navigated) initial state and guessed initial costate to predict the state/costate at engine cutoff. The initial costate guess is corrected, using a multi-dimensional Newtons method, based on errors in the terminal state constraints and the transversality conditions. Path constraints are enforced within the propagation process. A modified multiple shooting method is shown to be a very effective numerical technique for this application. Results for a single stage to orbit launch vehicle are given. In addition, the formulation for the free final time multi-arc trajectory optimization problem is given. Results for a two-stage launch vehicle burn-coast-burn ascent to orbit in a closed-loop guidance mode are shown. An abort to landing site formulation of the algorithm and numerical results are presented. A technique for numerically treating the transversality conditions is discussed that eliminates part of the analytical and coding burden associated with optimal control theory.
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Funções de Green para problemas de valor de contorno com três pontosBarros, André Azevedo Paes de [UNESP] 28 January 2011 (has links) (PDF)
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barros_aap_me_sjrp.pdf: 459604 bytes, checksum: 0a23c4af2e8f9afe3807f0dd603a1237 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo desse trabalho é estudar problemas de valor de contorno com três pontos lineares e não ineares, também conhecidos como problemas não Isto é feito, usando as funções de Green, usadas para resolver problemas de valor de contorno com dois pontos. / The aim of this work is to study boundary value problems with three points also known as non-classical problems. This is done using the Green's functions, which are used to solve two-point boundary value problems.
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Efficient Variable Mesh Techniques to solve Interior Layer ProblemsMbayi, Charles K. January 2020 (has links)
Philosophiae Doctor - PhD / Singularly perturbed problems have been studied extensively over the past
few years from different perspectives. The recent research has focussed on the
problems whose solutions possess interior layers. These interior layers appear
in the interior of the domain, location of which is difficult to determine a-priori
and hence making it difficult to investigate these problems analytically. This
explains the need for approximation methods to gain some insight into the behaviour
of the solution of such problems. Keeping this in mind, in this thesis
we would like to explore a special class of numerical methods, namely, fitted
finite difference methods to determine reliable solutions. As far as the fitted
finite difference methods are concerned, they are grouped into two categories:
fitted mesh finite difference methods (FMFDMs) and the fitted operator finite
difference methods (FOFDMs). The aim of this thesis is to focus on the
former. To this end, we note that FMFDMs have extensively been used for
singularly perturbed two-point boundary value problems (TPBVPs) whose
solutions possess boundary layers. However, they are not fully explored for
problems whose solutions have interior layers. Hence, in this thesis, we intend firstly to design robust FMFDMs for singularly perturbed TPBVPs whose solutions
possess interior layers and to improve accuracy of these approximation
methods via methods like Richardson extrapolation. Then we extend these
two ideas to solve such singularly perturbed TPBVPs with variable diffusion
coefficients. The overall approach is further extended to parabolic singularly
perturbed problems having constant as well as variable diffusion coefficients. / 2023-08-31
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Problemas de valor de contorno não clássicos: uma abordagem usando funções de GreenVerão, Glauce Barbosa [UNESP] 18 February 2007 (has links) (PDF)
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verao_gb_me_sjrp.pdf: 363983 bytes, checksum: c59e477b48d1d71a3199f377018eead3 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo deste trabalho é estudar problemas de valor de contorno do tipo {ÿ + f(t) =0 y(0)=0˙ y(1)= ky(η), (1) onde η ∈ (0, 1), k ∈ R e f ∈C([0, 1],R). Para antingirmos nosso objetivo usamosas funções de Green G(t,s)que nos permitem escrever a solução do problema(1)na seguinte forma: w(t)= ∫ 1 0 G(t,s)f(s)ds. Usando esta solução, investigamos através do ponto fixo de Schauder a solvabilidade do problema não linear { y + f(t,y)=0 y(0)=0˙ y(1)= ky(η). / The main goal of this work is study the following boundary value problems {ÿ + f(t) = 0 =0 y(0)=0˙ y(1)= ky(η), (1), where η ∈ (0, 1), k ∈ R e f ∈C([0, 1],R). To achieve our goal we use the Green's function G(t,s) which allow us to write the solution of the problem (2) in the form: w(t)= ∫ 1 0 G(t,s)f(s)ds. Using this solution and the Schauder point theory, also we study the solvability of a nonlinear problem { y + f(t,y)=0 y(0)=0˙ y(1)= ky(η).
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Problemas de valor de contorno não clássicos : uma abordagem usando funções de Green /Verão, Glauce Barbosa. January 2011 (has links)
Orientador: German Jesus Lozada Cruz / Banca: Luiz Augusto Fernandes de Oliveira / Banca: José Marcio Machado / Resumo: O objetivo deste trabalho é estudar problemas de valor de contorno do tipo {ÿ + f(t) =0 y(0)=0˙ y(1)= ky(η), (1) onde η ∈ (0, 1), k ∈ R e f ∈C([0, 1],R). Para antingirmos nosso objetivo usamosas funções de Green G(t,s)que nos permitem escrever a solução do problema(1)na seguinte forma: w(t)= ∫ 1 0 G(t,s)f(s)ds. Usando esta solução, investigamos através do ponto fixo de Schauder a solvabilidade do problema não linear { y + f(t,y)=0 y(0)=0˙ y(1)= ky(η). / Abstract: The main goal of this work is study the following boundary value problems {ÿ + f(t) = 0 =0 y(0)=0˙ y(1)= ky(η), (1), where η ∈ (0, 1), k ∈ R e f ∈C([0, 1],R). To achieve our goal we use the Green's function G(t,s) which allow us to write the solution of the problem (2) in the form: w(t)= ∫ 1 0 G(t,s)f(s)ds. Using this solution and the Schauder point theory, also we study the solvability of a nonlinear problem { y + f(t,y)=0 y(0)=0˙ y(1)= ky(η). / Mestre
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Funções de Green para problemas de valor de contorno com três pontos /Barros, André Azevedo Paes de. January 2011 (has links)
Orientador: Germán Jesus Lozada Cruz / Banca: Marco Aparecido Queiroz Duarte / Banca: Juliana Conceição Precioso Pereira / Resumo: O objetivo desse trabalho é estudar problemas de valor de contorno com três pontos lineares e não ineares, também conhecidos como problemas não Isto é feito, usando as funções de Green, usadas para resolver problemas de valor de contorno com dois pontos. / Abstract: The aim of this work is to study boundary value problems with three points also known as non-classical problems. This is done using the Green's functions, which are used to solve two-point boundary value problems. / Mestre
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Utility Of Phase Space Behaviour In Solving Two Point Boundary Value ProblemsSai V, V V Sesha 08 1900 (has links) (PDF)
No description available.
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Advancing Optimal Control Theory Using Trigonometry For Solving Complex Aerospace ProblemsKshitij Mall (5930024) 17 January 2019 (has links)
<div>Optimal control theory (OCT) exists since the 1950s. However, with the advent of modern computers, the design community delegated the task of solving the optimal control problems (OCPs) largely to computationally intensive direct methods instead of methods that use OCT. Some recent work showed that solvers using OCT could leverage parallel computing resources for faster execution. The need for near real-time, high quality solutions for OCPs has therefore renewed interest in OCT in the design community. However, certain challenges still exist that prohibits its use for solving complex practical aerospace problems, such as landing human-class payloads safely on Mars.</div><div><br></div><div>In order to advance OCT, this thesis introduces Epsilon-Trig regularization method to simply and efficiently solve bang-bang and singular control problems. The Epsilon-Trig method resolves the issues pertaining to the traditional smoothing regularization method. Some benchmark problems from the literature including the Van Der Pol oscillator, the boat problem, and the Goddard rocket problem verified and validated the Epsilon-Trig regularization method using GPOPS-II.</div><div><br></div><div>This study also presents and develops the usage of trigonometry for incorporating control bounds and mixed state-control constraints into OCPs and terms it as Trigonometrization. Results from literature and GPOPS-II verified and validated the Trigonometrization technique using certain benchmark OCPs. Unlike traditional OCT, Trigonometrization converts the constrained OCP into a two-point boundary value problem rather than a multi-point boundary value problem, significantly reducing the computational effort required to formulate and solve it. This work uses Trigonometrization to solve some complex aerospace problems including prompt global strike, noise-minimization for general aviation, shuttle re-entry problem, and the g-load constraint problem for an impactor. Future work for this thesis includes the development of the Trigonometrization technique for OCPs with pure state constraints.</div>
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Improved Numerical And Numeric-Analytic Schemes In Nonlinear Dynamics And Systems With Finite RotationsGhosh, Susanta 01 1900 (has links)
This thesis deals with different computational techniques related to some classes of nonlinear response regimes of engineering interest. The work is mainly divided into two parts. In the first part different numeric-analytic integration techniques for nonlinear oscillators are developed. In the second part, procedures for handling arbitrarily large rotations are addressed and a few novel developments are reported in the process.
To begin the first part, we have proposed an explicit numeric-analytic technique, based on the Adomian decomposition method, for integrating strongly nonlinear oscillators. Numerical experiments suggest that this method, like most other numerical techniques, is versatile and can accurately solve strongly nonlinear and chaotic systems with relatively larger step-sizes. It is then demonstrated that the procedure may also be effectively employed for solving two-point boundary value problems with the help of a shooting algorithm. This has been followed up with the derivation and numerical exploration of variants of a recently developed numeric-analytic technique, the multi-step transversal linearization (MTrL), in the context of nonlinear oscillators of relevance in engineering dynamics. A considerable generalization and improvement over the original form of a MTrL strategy is achieved in this study. Finally, we have used the concept of MTrL method on the nonlinear variational (rate) equation corresponding to a nonlinear oscillator and thus derive another family of numeric-analytic techniques, presently referred to as the multi-step tangential linearization (MTnL). A comparison of relative errors through the MTrL and MTnL techniques consistently indicate a superior quality of approximation via the MTrL route.
In the second part of the thesis, a scheme for numerical integration of rigid body rotation is proposed using only rudimentary tensor analysis. The equations of motion are rewritten in terms of rotation vectors lying in same tangent spaces, thereby facilitating vector space operations consistent with the underlying geometric structure of rotation. One of the most important findings of this part of the dissertation is that the existing constant-preserving algorithms are not necessarily accurate enough and may not be ideally applicable to cases wherein numerical accuracy is of primary importance. In contrast, the proposed rotation-algorithms, the higher order ones in particular, are significantly more accurate for conservative rotational systems for reasonably long time. Similar accuracy is expected for dissipative rotational systems as well. The operators relating rotation variables corresponding to different tangent spaces are also investigated and this should provide further insight into the understanding of rotation vector parametrization.
A rotation update is next proposed in terms of rotation vectors. This update, employed along with interpolation of relative rotations, gives a strain-objective and path independent finite element implementation of a geometrically exact beam. The method has the computational advantage of requiring considerably less nodal variables due to the use of rotation vector parametrization. We have proposed a new isoparametric interpolation of nodal quaternions for computing the rotation field within an element. This should be a computationally efficient alternative to the interpolation of local rotations. It has been proved that the proposed interpolation of rotation leads to the objectivity of strain measures. Several numerical experiments are conducted to demonstrate the frame invariance, path-independence and other superior aspects of the present approach vis-`a-vis the existing methods based on the rotation vector parametrization. It is emphasized that, in order to develop an objective finite element formulation, the use of relative rotation is not mandatory and an interpolation of total rotation variables conforming with the rotation manifold should suffice.
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High Accuracy Fitted Operator Methods for Solving Interior Layer ProblemsSayi, Mbani T January 2020 (has links)
Philosophiae Doctor - PhD / Fitted operator finite difference methods (FOFDMs) for singularly perturbed
problems have been explored for the last three decades. The construction of
these numerical schemes is based on introducing a fitting factor along with the
diffusion coefficient or by using principles of the non-standard finite difference
methods. The FOFDMs based on the latter idea, are easy to construct and they
are extendible to solve partial differential equations (PDEs) and their systems.
Noting this flexible feature of the FOFDMs, this thesis deals with extension
of these methods to solve interior layer problems, something that was still outstanding.
The idea is then extended to solve singularly perturbed time-dependent
PDEs whose solutions possess interior layers. The second aspect of this work is
to improve accuracy of these approximation methods via methods like Richardson
extrapolation. Having met these three objectives, we then extended our
approach to solve singularly perturbed two-point boundary value problems with
variable diffusion coefficients and analogous time-dependent PDEs. Careful analyses
followed by extensive numerical simulations supporting theoretical findings
are presented where necessary.
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