The Tiger Optimization Software - A Pseudospectral Optimal Control Package
Type of DegreeMaster's Thesis
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The fields of trajectory optimization and optimal control are closely connected. Practical trajectory optimization techniques are built on the indirect and direct methods for solving optimal control problems. Indirect methods seek to satisfy the first-order necessary conditions of optimality, while direct methods discretize the problem and transcribe it into a large-scale nonlinear programming (NLP) problem. This study provides a basic overview of the direct and indirect methods. The indirect necessary conditions and Pontryagin's Minimum Principle are reviewed, and the approaches taken by direct methods are presented. The focus of this study then shifts to pseudospectral direct methods, which belong to the class of collocation methods. The theoretical groundwork of pseudospectral methods is laid, leading to a pseudospectral formulation and software for solving general optimal control problems. Several types of pseudospectral methods are presented, including the Legendre-Gauss and Chebyshev-Gauss methods. Special attention is given to the Legendre-Gauss-Radau (LGR) method, which is the primary transcription employed by the MATLAB-based Tiger Optimization Software (TOPS). TOPS is a general-purpose pseudospectral optimal control software developed by the author as part of their research in the Aero-Astro Computational and Experimental (ACE) Lab. A multi-interval pseudospectral method is presented, and the discrete form of the optimality conditions are derived. The study shifts focus from theory to application, and the practical aspects of pseudospectral optimal control methods are discussed. Another objective of this research is to compile the author's knowledge with regard to implementing pseudospectral techniques, thus enabling the reader to easily implement their own pseudospectral method. Several mesh refinement methods are compared and their merits are compared. In addition, several techniques that improve the efficiency of an NLP solver are presented, including efficient and exact derivative formulas and several scaling techniques. Finally, several example optimal control problems are solved using TOPS. The solutions obtained from TOPS are compared to the solutions obtained using indirect techniques to verify their accuracy and demonstrate the capabilities of TOPS.