The Conjugate Gradient Method and Its Application to Aerospace Vehicle Guidance and Control. Part I. Basic Results in the Conjugate Gradient Method

摘要

The effectiveness of the CG Conjugate Gradient) method for the solution of the optimal aerospace vehicle guidance and control problem is demonstrated. The superiority of the convergence rate of the CG method over the steepest descent method is demonstrated. The search of golden section technique is also discussed and shown to be a very powerful technique for one dimensional optimization. As one dimensional search cannot be carried out exactly accumulation of errors due to this source can produce directions of search which increase rather than decrease the objective functional. In this case it is found useful to go back to the local gradient direction and start the process over again. A number of new basic theorems and algorithmic techniques fundamental to the CG method are presented here and the establishment of a monotonic decreasing sequence for the cost functional which results in the determination of the optimal control is shown. A companion paper applies the basic results developed in this paper to the Mars entry guidance and control problem. (Author)