SPARSE JACOBIAN UPDATES IN THE COLLOCATION METHOD FOR OPTIMAL-CONTROL PROBLEMS

Advanced guidance algorithms for the control of aerospace vehicles can require rapid solution of an optimal control problem. The necessary conditions for the solution of an optimal control problem result in a two-point boundary-value problem. For guidance applications, the boundary-value problem must be solved rapidly in order to reflect real-time navigation input. The collocation method has been proposed by a number of authors as a robust approach to the problem. By introducing piecewise cubic polynomial interpolation of the dynamic variables, the boundary-value problem is reduced to solving a system of nonlinear algebraic equations, the resulting iteration equation involves a large sparse matrix. This paper demonstrates the application of sparse Broyden updates for the iteration matrix that results in significant time savings. The viability of the approach for real-time optimal control applications is illustrated with computational results for maximum downrange and crossrange Shuttle re-entry trajectories and for simpler powered flight ascent trajectories. A substantial reduction in computation cost has been observed for typical cases. © 1990 American Institute of Aeronautics and Astronautics, Inc., All rights reserved.