The $\alpha$-Method Direct Transcription in Path Constrained Dynamic Optimization
Nigam Chandra Parida, Soumyendu Raha · SIAM Journal on Scientific Computing · 2009
Numerically discretized dynamic optimization problems having active inequality and equality path constraints that along with the dynamics induce locally high index differential algebraic equations often cause the optimizer to fail in convergence or to produce degraded control solutions. In many applications, regularization of the numerically discretized problem in direct transcription schemes by perturbing the high index path constraints helps the optimizer to converge to useful control solutions. For complex engineering problems with many constraints it is often difficult to find effective nondegenerate perturbations that produce useful solutions in some neighborhood of the correct solution. In this paper we describe a numerical discretization that regularizes the numerically consistent discretized dynamics and does not perturb the path constraints. For all values of the regularization parameter the discretization remains numerically consistent with the dynamics and the path constraints specified in the original problem. The regularization is quantifiable in terms of time step size in the mesh and the regularization parameter. For fully regularized systems the scheme converges linearly in time step size. The method is illustrated with examples.