Optimal guidance of dynamic systems

Abstract

The solution of optimal control problems is usually done in two succeeding stages: before the dynamic process starts, a so-called nominal solution is precalculated with great accuracy by sophisticated methods. Since the real process is still to begin, there is plenty of time for these computations. But when the process has been started, there will be not enough time to use these elaborate methods in order to correct deviations from the precalculated nominal path. Therefore, during the process, feedback-schemes have to be applied to control the dynamic system with minimal amount of computation. This paper develops a linear guidance scheme by linearizing the necessary conditions for the disturbed trajectory along the precalculated reference path. For linear problems or problems with inequality constraints, this deduction is valid as long as the nominal and the actual solution have the same switching structure. But the linearization of the conditions at the switching points needs special care in order to get suitable conditions for the development of a fast algorithm. The resulting feedback-scheme requires only one matrix-times-vector operation per correction step, but a succession of correction maneuvres is required in order to dampen the influence of the linearization error. Finally the feedback algorithm is applied to the re-entry of Space-Shuttle vehicle. The associated controllability tubes demonstrate the performance of the method.