Method for Solving State-Path Constrained Optimal Control Problems Using Adaptive Radau Collocation

A new method is developed for accurately approximating the solution tostate-variable inequality path constrained optimal control problems using amultiple-domain adaptive Legendre-Gauss-Radau collocation method. The methodconsists of the following parts. First, a structure detection method isdeveloped to estimate switch times in the activation and deactivation ofstate-variable inequality path constraints. Second, using the detectedstructure, the domain is partitioned into multiple-domains where each domaincorresponds to either a constrained or an unconstrained segment. Furthermore,additional decision variables are introduced in the multiple-domainformulation, where these additional decision variables represent the switchtimes of the detected active state-variable inequality path constraints. Withina constrained domain, the path constraint is differentiated with respect to theindependent variable until the control appears explicitly, and this derivativeis set to zero along the constrained arc while all preceding derivatives areset to zero at the start of the constrained arc. The time derivatives of theactive state-variable inequality path constraints are computed using automaticdifferentiation and the properties of the chain rule. The method isdemonstrated on two problems, the first being a benchmark optimal controlproblem which has a known analytical solution and the second being achallenging problem from the field of aerospace engineering in which there isno known analytical solution. When compared against previously developedadaptive Legendre-Gauss-Radau methods, the results show that the methoddeveloped in this paper is capable of computing accurate solutions to problemswhose solution contain active state-variable inequality path constraints.