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

TL;DR

Addresses the challenge of solving state-path constrained optimal control problems where activation times of SVICs are unknown. The paper presents the SPOC method, combining a structure-detection stage with a multiple-domain Legendre-Gauss-Radau collocation formulation and indirect adjoining to enforce high-order tangency on active constraints. Key contributions include an automatic computation of higher-order SVIC derivatives via algorithmic differentiation, the introduction of domain interface variables to optimize activation times, and an adaptive hp-LGR refinement loop with potential re-detection for robustness. The approach yields highly accurate constraint satisfaction and competitive objective values on two aerospace-inspired problems, at the cost of higher computational effort, demonstrating practical applicability for complex SPOCs.

Abstract

A new method is developed for accurately approximating the solution to state-variable inequality path constrained optimal control problems using a multiple-domain adaptive Legendre-Gauss-Radau collocation method. The method consists of the following parts. First, a structure detection method is developed to estimate switch times in the activation and deactivation of state-variable inequality path constraints. Second, using the detected structure, the domain is partitioned into multiple-domains where each domain corresponds to either a constrained or an unconstrained segment. Furthermore, additional decision variables are introduced in the multiple-domain formulation, where these additional decision variables represent the switch times of the detected active state-variable inequality path constraints. Within a constrained domain, the path constraint is differentiated with respect to the independent variable until the control appears explicitly, and this derivative is set to zero along the constrained arc while all preceding derivatives are set to zero at the start of the constrained arc. The time derivatives of the active state-variable inequality path constraints are computed using automatic differentiation and the properties of the chain rule. The method is demonstrated on two problems, the first being a benchmark optimal control problem which has a known analytical solution and the second being a challenging problem from the field of aerospace engineering in which there is no known analytical solution. When compared against previously developed adaptive Legendre-Gauss-Radau methods, the results show that the method developed in this paper is capable of computing accurate solutions to problems whose solution contain active state-variable inequality path constraints.

Figures (7)

• Figure 1: Schematic of process for decomposing the original optimal control problem (with $K=3$ mesh intervals) into $D$ domains where the $D-1$ domain interface variables are included as optimization variables to determine the optimal activation and deactivation in each SVIC.
• Figure 2: Schematic of algorithm for obtaining higher-order time derivatives of any active state-variable inequality path constraint.
• Figure 3: Position, $x(t)$, comparison between the solution obtained on the initial mesh, with the $hp$-LGR method of Ref. Liu2018, and the SPOC method for the Bryson-Denham problem.
• Figure 4: Control, $u(t)$, comparison between the solution obtained on the initial mesh, with the $hp$-LGR method of Ref. Liu2018, and the SPOC method for the Bryson-Denham problem.
• Figure 5: Heating rate profile, $\dot{Q}(t)$, comparison between the solution obtained on the initial mesh, with the $hp$-LGR method of Ref. Liu2018, and the SPOC method for the constrained RLVE problem.