Direct Transcription Methods in Optimal Control: Theory and Practice

Abstract

In optimal control as in many other disciplines, individuals developing the theory and those applying it to real life problems do not always see eye to eye. Some results developed by theoreticians have very limited practical value, while other useful results may be unknown to practitioners or incorrectly interpreted. This work aims to bridge the gap between these two groups by presenting theoretical results in a way that will be useful to practitioners. We concentrate specifically on convergence results relating to a class of methods known as direct transcription, where the entire optimal control problem is discretized, in our case using a Runge-Kutta method, to form a nonlinear program.

For unconstrained problems, we present several convergence results, then give an original result that demonstrates that practically designed optimal control software will be unable to attain theoretically possible convergence order in most cases. We present a practical solution to this problem that is currently being implemented in an industrial software package.

In the next chapter, we also prove that many equality constrained problems, including problems unsolvable by other methods, are, for a direct transcription method, equivalent to unconstrained problems, so that convergence results from the previous chapter apply. We provide practical guidelines for regularizing a constrained problem to ensure accurate solution by a direct transcription method.

For inequality constrained problems, we give a detailed overview of different sets of necessary conditions and existing convergence results. We also present a phenomenon we call "virtual boundary arcs", demonstrating the advantage of direct transcription for another class of problems, in this case problems for which a boundary arc is theoretically impossible but the cost structure forces the solution very close to the constraint boundary.