Comments on direct transcription solution of DAE constrained optimal control problems with two discretization approaches

Abstract

There have been a number of results in the literature showing that a direct transcription numerical approach to optimal control problems could have a number of surprising and desirable behaviors that were not always predicted by the existing theory especially when differential algebraic equations are involved. Most of these results were developed with an implicit Runge-Kutta (IRK) discretization being used. It is important to know which of these observations hold for direct transcription software using different types of discretizations and which are discretization specific. This paper reexamines several of these questions but using an hp-pseudospectral code. It is seen that while philosophically the results are often similar to the IRK results, there are some differences that should be understood by users solving constrained optimal control problems. This paper also discusses a type of regularization for higher index problems that does not reduce the index.

Semianalytical minimum‐time solution for the optimal control of a vehicle subject to limited acceleration

“…The solution of the OCP can be obtained via general optimal control software, as described,() where various methods for OCP are presented. However, they cannot take advantage of the analytic solution and hence, although having a more general purpose, they are slower than the present method when applied to this specific problem.…”

“…All the boundary conditions are fixed: without loss of generality, the initial position is zero, the final position is L (the length of the curve), and the initial and final velocities are v i and v f , respectively, both assumed nonnegative. The solution of the OCP can be obtained via general optimal control software, as described, 2,13,14,[29][30][31][32] where various methods for OCP are presented. However, they cannot take advantage of the analytic solution and hence, although having a more general purpose, they are slower than the present method when applied to this specific problem.…”

A symplectic indirect approach for a class of nonlinear optimal control problems of differential‐algebraic systems

“…Differential indexes of DAEs also have effects on the properties of optimal control problems subject to DAEs. For problems that are subject to higher‐index DAEs, the solution is more challenging 8‐9 . Consistent initial values need to be considered.…”

“…At present, there are two categories of numerical methods: direct methods and indirect methods. In a direct method, the original optimal control problem is approximated as finite‐dimensional nonlinear programming (NLP) 1,8‐9 using a specified transcription method and discretization scheme. By solving the NLP, numerical solutions of state variables and control inputs are obtained.…”

“…For problems that are subject to higher-index DAEs, the solution is more challenging. [8][9] Consistent initial values need to be considered. It often requires that initial state variables, algebraic variables, and control inputs satisfy not only algebraic equations but also hidden algebraic equations.…”