Order of Convergence in the Direct Transcription Solution of Optimal Control Problems

Abstract

In the direct transcription approach to the numerical solution of optimal control problems, the optimal control problem is discretized and the resulting nonlinear programming problem is solved numerically. There has been considerable study over the last 10 years on order of convergence of cost, state, multipliers, and control. This paper discusses these questions, and the highly technical results in the literature, in the context of their implications for industrial grade optimal control packages.

Adjoint estimation using direct transcription multipliers: compressed trapezoidal method

“…Computational experiments not detailed here (Engelsone 2006) with HS on (9) show that both of the compressed forms of HS that are implemented in SOCS do not achieve the order of the adjoint estimate proved by Hager for the Butcher array (uncompressed) version of the HS Runge-Kutta method. In fact, the orders we observe for HS are similar to TR with first order at the grid and second order at the midpoint (Engelsone et al 2005).…”

“…This is especially the case when complex constraints are present as is often the case in applications. However, for some problems a good estimate of the continuous adjoint variables from the necessary conditions is useful for sensitivity studies (Alber et al 2002;Borggaard and Vance 2004;Lempio and Maurer 1980;Lucet and Ye 2001;Malanowski and Maurer 1996) or for calculating a better estimate for the control (Engelsone et al 2005;Hager 2000). It is natural to try and use the multipliers from the solution of the discrete NLP problem to estimate the continuous adjoint variables.…”