Hybrid Gauss Pseudospectral and Generalized Polynomial Chaos Algorithm to Solve Stochastic Optimal Control Problems

A numerical algorithm combining the Gauss Pseudospectral Method (GPM) with a Generalized Polynomial Chaos (gPC) method to solve nonlinear stochastic optimal control problems with constraint uncertainties is presented. The GPM and gPC have been shown to be spectrally accurate numerical methods for solving deterministic optimal control problems and stochastic differential equations, respectively. The gPC uses collocation nodes to sample the random space, which are then inserted into the differential equations and solved using standard solvers to generate a set of deterministic solutions used to characterize the distribution of the solution by constructing a polynomial representation of the output as a function of uncertain parameters. The proposed algorithm investigates using GPM optimization software in place of deterministic differential equation solvers traditionally used in the gPC, providing minimum cost deterministic solutions that meet path, control, and boundary constraints. A trajectory optimization problem is considered where the objectives are to find the path through a two-dimensional space that minimizes the probability a vehicle will be ’killed’ by lethal threats whose locations are uncertain and to characterize the effects those uncertainties have on the solution by estimating the statistical properties

Optimal Control of Fully Routed Air Traffic in the Presence of Uncertainty and Kinodynamic Constraints

A method is presented to extend current graph-based Air Traffic Management optimization frameworks. In general, Air Traffic Management is the process of guiding a finite set of aircraft, each along its pre-determined path within some local airspace, subject to various physical, policy, procedural and operational restrictions. This research addresses several limitations of current graph-based Air Traffic Management optimization methods by incorporating techniques to account for stochastic effects, physical inertia and variable arrival sequencing. In addition, this research provides insight into the performance of multiple methods for approximating non-differentiable air traffic constraints, and incorporates these methods into a generalized weighted-sum representation of the multi-objective Air Traffic Management optimization problem that minimizes the total time of flight, deviation from scheduled arrival time and fuel consumption of all aircraft. The methods developed and tested throughout this dissertation demonstrate the ability of graph-based optimization techniques to model realistic air traffic restrictions and generate viable control strategies

Optimal Collision Avoidance Trajectories for Unmanned/Remotely Piloted Aircraft

The post-911 environment has punctuated the force-multiplying capabilities that Remotely Piloted Aircraft (RPA) provides combatant commanders at all echelons on the battlefield. Not only have unmanned aircraft systems made near-revolutionary impacts on the battlefield, their utility and proliferation in law enforcement, homeland security, humanitarian operations, and commercial applications have likewise increased at a rapid rate. As such, under the Federal Aviation Administration (FAA) Modernization and Reform Act of 2012, the United States Congress tasked the FAA to provide for the safe integration of civil unmanned aircraft systems into the national airspace system (NAS) as soon as practicable, but not later than September 30, 2015. However, a necessary entrance criterion to operate RPAs in the NAS is the ability to Sense and Avoid (SAA) both cooperative and noncooperative air traffic to attain a target level of safety as a traditional manned aircraft platform. The goal of this research effort is twofold: First, develop techniques for calculating optimal avoidance trajectories, and second, develop techniques for estimating an intruder aircraft\u27s trajectory in a stochastic environment. This dissertation describes the optimal control problem associated with SAA and uses a direct orthogonal collocation method to solve this problem and then analyzes these results for different collision avoidance scenarios