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

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.Comment: 31 pages, 7 figures, 5 table

Advancement and analysis of Gauss pseudospectral transcription for optimal control problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2007.Includes bibliographical references (p. 195-207).As optimal control problems become increasingly complex, innovative numerical methods are needed to solve them. Direct transcription methods, and in particular, methods involving orthogonal collocation have become quite popular in several field areas due to their high accuracy in approximating non-analytic solutions with relatively few discretization points. Several of these methods, known as pseudospectral methods in the aerospace engineering community, have also established costate estimation procedures which can be used to verify the optimality of the resulting solution. This work examines three of these pseudospectral methods in detail, specifically the Legendre, Gauss, and Radau pseudospectral methods, in order to assess their accuracy, efficiency, and applicability to optimal control problems of varying complexity. Emphasis is placed on improving the Gauss pseudospectral method, where advancements to the method include a revised pseudospectral transcription for problems with path constraints and differential dynamic constraints, a new algorithm for the computation of the control at the boundaries, and an analysis of a local versus global implementation of the method. The Gauss pseudospectral method is then applied to solve current problems in the area of tetrahedral spacecraft formation flying. These optimal control problems involve multiple finite-burn maneuvers, nonlinear dynamics, and nonlinear inequality path constraints that depend on both the relative and inertial positions of all four spacecraft. Contributions of this thesis include an improved numerical method for solving optimal control problems, an analysis and numerical comparison of several other competitive direct methods, and a greater understanding of the relative motion of tetrahedral formation flight.by Geoffrey Todd Huntington.Ph.D

A space-time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives

We propose a direct numerical method for the solution of an optimal control problem governed by a two-side space-fractional diffusion equation. The presented method contains two main steps. In the first step, the space variable is discretized by using the Jacobi-Gauss pseudospectral discretization and, in this way, the original problem is transformed into a classical integer-order optimal control problem. The main challenge, which we faced in this step, is to derive the left and right fractional differentiation matrices. In this respect, novel techniques for derivation of these matrices are presented. In the second step, the Legendre-Gauss-Radau pseudospectral method is employed. With these two steps, the original problem is converted into a convex quadratic optimization problem, which can be solved efficiently by available methods. Our approach can be easily implemented and extended to cover fractional optimal control problems with state constraints. Five test examples are provided to demonstrate the efficiency and validity of the presented method. The results show that our method reaches the solutions with good accuracy and a low CPU time.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Vibration and Control', available from [http://journals.sagepub.com/home/jvc]. Submitted 02-June-2018; Revised 03-Sept-2018; Accepted 12-Oct-201

Implementations of the Universal Birkhoff Theory for Fast Trajectory Optimization

This is part II of a two-part paper. Part I presented a universal Birkhoff theory for fast and accurate trajectory optimization. The theory rested on two main hypotheses. In this paper, it is shown that if the computational grid is selected from any one of the Legendre and Chebyshev family of node points, be it Lobatto, Radau or Gauss, then, the resulting collection of trajectory optimization methods satisfy the hypotheses required for the universal Birkhoff theory to hold. All of these grid points can be generated at an $\mathcal{O}(1)$ computational speed. Furthermore, all Birkhoff-generated solutions can be tested for optimality by a joint application of Pontryagin's- and Covector-Mapping Principles, where the latter was developed in Part~I. More importantly, the optimality checks can be performed without resorting to an indirect method or even explicitly producing the full differential-algebraic boundary value problem that results from an application of Pontryagin's Principle. Numerical problems are solved to illustrate all these ideas. The examples are chosen to particularly highlight three practically useful features of Birkhoff methods: (1) bang-bang optimal controls can be produced without suffering any Gibbs phenomenon, (2) discontinuous and even Dirac delta covector trajectories can be well approximated, and (3) extremal solutions over dense grids can be computed in a stable and efficient manner