Ill-Conditioning in Matlab Computation of Optimal Control with Time- Delays

Abstract

A direct transcription method transforms an optimal control problem (OCP) into a nonlinear programming problem (NLP). The resulting NLP can be solved by any NLP solver, such as the Matlab's optimization toolbox, the fsqp, etc. On solving optimization problems using the Matlab's optimization toolbox does not obtain an accurate Hessian matrix at the optimal solution due to the fact that the Hessian matrix is not being evaluated directly from the optimal solution. In this paper we compute the condition numbers associated with the optimal control computation, where the classical forth-order Runge-Kutta method is used for the discretization of the state equations. The computations of optimal solutions are done for different numbers of switching points and quadrature points per a switching interval. Test examples show that the condition numbers of the active constraints, projected Hessian and the whole Lagrangian system are more likely to increase with the number of the switching intervals per a delay interval than by the number of the quadrature intervals per a switching interval. Also, the three medium scale optimization algorithm of the Matlab’s optimization toolbox give almost similar condition numbers when used to solve the optimal control problem.