Should controls be eliminated while solving optimal control problems via direct methods?

Abstract

Direct methods of solving optimal control problems include techniques based on control discretization, where the control function of time is parameterized, and collocation, where both the control and state functions of time are parameterized. A recently introduced direct approach of solving optimal control problems via differential inclusions parameterizes only the state, and constrains the state rates to lie in a feasible hodograph space. In this method, the controls, which are just artifacts used to parameterize the feasible hodograph space, are completely eliminated from the optimization process. Explicit and implicit schemes of control elimination are discussed. Comparison of the differential inclusions method is made to collocation in terms of number of parameters, number of constraints, CPU time required for solution, and ease of calculation of analytical gradients. A minimum time-to-climb problem for an F-15 aircraft is used as an example for comparison. For a special class of optimal control problems with linearly appearing bounded controls, it is observed that the differential inclusion scheme is better in terms of number of parameters and constraints. Increased robustness of the differential inclusion methodology over collocation for the Goddard problem with singular control as part of the optimal solutions is also observed. Background T HE most precise approach to solve optimal control problems is the variational1 approach based on Pontryagin's minimum principle.2 This is an indirect approach as it involves solving the necessary conditions of optimality associated with the infinite dimensional optimal control problem rather than optimizing the cost of a finite dimensional discretization of the original problem directly. This method requires advanced analytical skills and to generate numerical solutions of the resulting two-point boundary-value problem is highly nontrivial. The controls are eliminated in the indirect method using the minimum principle. Thus, the optimal control is, in general, a nonlinear function of the state and costate variables. The most important application of the indirect method is the generation of benchmark solutions. Usually, good convergence is achieved only with excellent initial guesses for the nonintuitive costates. Additionally, the switching structure has to be guessed correctly in advance. For rapid trajectory prototyping, the safest approaches are the direct methods.3 These methods rely on a finite dimensional discretization of the optimal control problem to a nonlinear programming problem. Even though these methods do not enjoy the high precision and resolution of indirect methods, their convergence robustness makes them the method of choice of most practical applications. Moreover, these methods do not require the advanced mathematical skills necessary to pose and solve the variational problem.