Development of Parallel Indirect Methods for Solving Constrained Optimal Control Problems
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| Publicado en: | ProQuest Dissertations and Theses (2020) |
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| Acceso en línea: | Citation/Abstract Full Text - PDF |
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| Resumen: | Optimal control is a subject where it is desired to determine the inputs to a dynamical system which optimize a specified performance index while satisfying any constraints on the motion of the system at the same time. Because of the complexity of most applications, optimal control problems (OCPs) are most often solved numerically. The indirect method for solving OCPs is based on solving the firs-order necessary conditions for the optimum and these necessary conditions are written as a two-point boundary value problem. This dissertation presents indirect methods for solving OCPs including control variable inequality constraints (CVICs), state variable inequality constraints (SVICs), and parameters. The necessary conditions of the optimum for the OCPs are written as a boundary value problem with differential algebraic equations which are proved to be index-1 (BVP-DAEs). The complementarity conditions in the BVP associated with those inequality constraints are approximated using Kanzow’s smoothed Fisher-Burmeister formula. Two numerical methods for solving the BVP-DAEs are developed. The multiple shooting technique is one of the techniques applied. Except solving the DAE using a single step linearly implicit Runge-Kutta method, a novel implementation based on MATLAB built-in DAE solver ode15s is provided. The other method used is a collocation method where the DAEs are approximated using Lagrange polynomials within each mesh and required to be satisfied at Lobatto points within each interval. Newton’s method is performed to solve the BVP-DAEs systems for both methods and the descent direction is found by solving asparse bordered almost block diagonal (BABD) linear system with a structured orthogonal factorization algorithm. For the MATLAB implementation, the efficiency of the embedded parallel computing toolbox is explored. Moreover, using the graphics processing unit (GPU) to accelerate the numerical algorithm solving process is very promising by using faster hardware. Combining those, this dissertation also presents the GPU based parallel implementation for both numerical methods, which is implemented using Python and CUDA. Numerical examples are presented to illustrate the efficiency of the implementation. The GPU based parallel implementations are shown to be significantly faster than the implementation using Central Processing Unit (CPU) alone or implemented using MATLAB for both methods. Extending the collocation method presented, a study so called the collocation method with ph adaptive mesh refinement is introduced to further improve the efficient and the robustness of the collocation method presented. First, a novel method to estimate the error of the solution from collocation method is presented which serves as a basis of the ph adaptive mesh refinement method. In the original collocation method, the problem is solved based on a global unified number of collocation points used within each mesh interval without the dynamic mesh of the problem. In the ph adaptive method, not only the mesh varies during the solving process but also the collocation points used within each mesh interval keep changing until a desired numerical tolerance is met. The method is called “ph” because the mesh size of each interval (denoted by h) and the number of collocation points which is also the polynomial degrees (denoted by p) within each mesh interval are updated simultaneously. |
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| ISBN: | 9798569995493 |
| Fuente: | ProQuest Dissertations & Theses Global |