Neighboring Extremal Guidance for Systems with Piecewise Linear Control Using Multiple Optimization

A neighboring extremal guidance law for systems with piecewise linear control using multiple optimization along the suboptimal trajectory is developed. The guidance law computes control gains that relate the difference in the perturbed and nominal trajectory with a control update. Previous research in neighboring extremal control has concentrated on calculating the control gains using a single optimization from the initial conditions to the final constraint manifold. The purpose of this study is to develop a more optimal method of computing the control gains at each node through multiple optimization. Multiple optimization reoptimizes the trajectory from each control node on the suboptimal trajectory. The suboptimal control vectors for each subtrajectory are used to compute a more optimal set of control gains. An example optimization problem, the lunar launch problem, is used to compare the performance of both the single optimization and multiple optimization control laws. Overall, the multiple optimization gains returned results comparable to those achieved with single optimization. For perturbations of + or - 5% in either gravity or thrust acceleration, endpoint conditions for multiple optimization gains never exceeded more than 50 feet in position or 4.3 ft/sec in vertical velocity from the desired boundary conditions. Direct comparisons of performance for specific model perturbations result in mixed conclusions. For thrust perturbations, single optimization gains deliver smaller errors in meeting end point conditions, while for gravity perturbations, multiple gains give better results.