10.25394/PGS.7418204.v1
Kshitij Mall
Advancing Optimal Control Theory Using Trigonometry For Solving Complex Aerospace Problems
2019
Purdue University Graduate School
Optimal Control Theory
Bang Bang Control
Singular Control
Human Mars Missions
Pontryagin's Minimum Principle
Direct Methods
Indirect Methods
GPOPS
Path Constraints
Mixed Constraints
Space Shuttle Reentry
Goddard Rocket Problem
Additional Necessary Conditions
Two-Point Boundary Value Problem
Regularization
Trigonometrization
Epsilon-Trig Regularization Method
Smoothing
Hamiltonian
Necessary Conditions of Optimality
Trigonometry
Multi-Point Boundary Value Problem
2019-01-17 13:45:58
article
https://hammer.figshare.com/articles/Advancing_Optimal_Control_Theory_Using_Trigonometry_For_Solving_Complex_Aerospace_Problems/7418204
<div>Optimal control theory (OCT) exists since the 1950s. However, with the advent of modern computers, the design community delegated the task of solving the optimal control problems (OCPs) largely to computationally intensive direct methods instead of methods that use OCT. Some recent work showed that solvers using OCT could leverage parallel computing resources for faster execution. The need for near real-time, high quality solutions for OCPs has therefore renewed interest in OCT in the design community. However, certain challenges still exist that prohibits its use for solving complex practical aerospace problems, such as landing human-class payloads safely on Mars.</div><div><br></div><div>In order to advance OCT, this thesis introduces Epsilon-Trig regularization method to simply and efficiently solve bang-bang and singular control problems. The Epsilon-Trig method resolves the issues pertaining to the traditional smoothing regularization method. Some benchmark problems from the literature including the Van Der Pol oscillator, the boat problem, and the Goddard rocket problem verified and validated the Epsilon-Trig regularization method using GPOPS-II.</div><div><br></div><div>This study also presents and develops the usage of trigonometry for incorporating control bounds and mixed state-control constraints into OCPs and terms it as Trigonometrization. Results from literature and GPOPS-II verified and validated the Trigonometrization technique using certain benchmark OCPs. Unlike traditional OCT, Trigonometrization converts the constrained OCP into a two-point boundary value problem rather than a multi-point boundary value problem, significantly reducing the computational effort required to formulate and solve it. This work uses Trigonometrization to solve some complex aerospace problems including prompt global strike, noise-minimization for general aviation, shuttle re-entry problem, and the g-load constraint problem for an impactor. Future work for this thesis includes the development of the Trigonometrization technique for OCPs with pure state constraints.</div>