An asymptotic perturbation method for nonlinear optimal control problems
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A quasianalytical method is presented for solving nonlinear, open-loop, optimal control problems. The approach combines a simple analytical, straightforward expansion from perturbation methods with powerful numerical algorithms (due to Ward and Van Loan) to solve a series of nonhomogeneous, linear, optimal control problems. In the past, the only recourse for solving such nonlinear problems relied almost exclusively on iterative numerical methods, whereas the asymptotic perturbation approach may produce accurate solutions to nonlinear problems without iteration. The nonlinear state and costate equations are derived from the optimal control formulation and expanded in a power series in terms of a small parameter contained either explicitly in the equations or implicitly in the boundary conditions. Each order of the expansion is shown to be governed by a nonhomogeneous, ordinary differential equation. Representing the generally nonintegrable, nonhomogeneous terms by a finite Fourier series, efficient matrix exponential algorithms are then used to solve the system at each order, where the order of the expansion is extended to achieve the appropriate precision. The asymptotic perturbation method is broadly applicable to weakly nonlinear optimal control problems, including the higher-order systems frequently encountered in aerospace vehicle dynamics and control. A number of numerical examples demonstrating the perturbation approach are included. © 1985 American Institute of Aeronautics and Astronautics, Inc.
author list (cited authors)
Junkins, J. L., & Thompson, R. C.