Browsing by Author "Doiron, Harold Hughes"
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Item An indirect optimization method with improved convergence characteristics(1970) Doiron, Harold Hughes; Childs, S. Bart; Decell, Henry P., Jr.; Lewallen, Jay M.; Nachlinger, R. Ray.; Hedgcoxe, Pat G.An improved numerical procedure is developed for solving the nonlinear two-point boundary value problem which results from the formulation of optimal control problems for solution by an indirect optimization method. Several modifications and extensions to previously known iterative methods are implemented which dramatically improve the convergence characteristics of the indirect optimization approach. The method of particular solutions for solving linear boundary value problems in ordinary differential equations is extended to solve systems of linear differential equations with boundary values specified in the form of nonlinear functions of the dependent and independent variables. This capability is exploited in the development of a new indirect method for solving trajectory optimization problems where the final state and final time are not specified explicitly. Because the method uses a Perturbation approach for linearizing the nonlinear system of differential equations, and because particular solutions are used to construct the general solution of the linearized system, the method is called the Particular Solution Perturbation Method (PSPM). A power series numerical integration method is adapted for use in solving the nonlinear two-point boundary value problem and is found to have several characteristics which make it uniquely suited for this purpose. The convergence characteristics of the PSPM are compared to those of previous indirect optimization methods for a problem which considers the minimum time orbit transfer of continuous, low-thrust rocket. Conclusions and recommendations for further study are included.Item Numerical integration via power series expansions(1967) Doiron, Harold Hughes; Childs, S. Bart; Dalton, Charles; Hubbard, Martin G.; Motard, Rodolphe L.A method for obtaining numerical solutions to initial value problems by implementing a generalized method of Frobenius for digital computers is presented. Power series which are equivalent to the Taylor's series expansion and which represent the solution of differential equations over finite intervals of the independent variable are constructed with recurrence formulas. A programming approach for overcoming difficulties in obtaining the necessary power series coefficients for nonlinear differential equations is developed. Example applications of the method are made to several second-order nonlinear differential equations containing trigonometric and other transcendental functions. Flow diagram solutions for several examples are included. Evaluation and computer-implemented convergence tests for power series which allow one to obtain numerical solutions of specified accuracy are discussed. The accuracy of the power series expansion method is demonstrated with numerical results, and the method is shown to have some outstanding characteristics when compared to standard integration schemes. Conclusions and recommendations for further study are included.