| dc.description.abstract | Solving nonlinear optimal control problems can be a difficult and time intensive process. This is especially true when the state dynamics of the problem are computationally expensive to solve. Using the Theory of Functional Connections, linear combinations of orthogonal manifolds are used to approximate computationally expensive terms within the state dynamics. Doing so can yield significantly faster solution time with minimal error compared to directly calculating these terms.
To test this method, a longitudinal dynamic model of a hypersonic vehicle was created. The aerodynamic lift and drag forces, as well as the pitching moment, were calculated using a panel method along with Pradtl-Meyer expansion and compression wave equations. The calculated forces and moment were then verified by comparing to CFD solutions generated by SOLIDWORKS Flow Analysis Tool at various angles of attack and Mach numbers.
This panel method still took up a bulk of the computation time needed to solve optimal control problems as it has to run for every timestep in each trajectory iteration created in the solution process. A new method to approximate the aerodynamic forces and moment was created by using this panel method alongside a least squares algorithm to solve for weights of recursively generated orthogonal manifolds. Lift, drag, and pitching moment could then be approximated by evaluating each orthogonal manifold at locations corresponding to states and controls. Compared to the original panel method, which took 117.30 seconds to calculate 10,000 different flight conditions, this least squares method calculated the same conditions in 0.13 seconds, almost 1000 times faster.
The new least squares method was then used alongside a nonlinear optimal control problem algorithm known as Dynamic Programming with Interior Points. Using the hypersonic model with the aerodynamic forces and moment approximated using orthogonal manifolds, solutions to several optimal control problems were found in a less than a minute for problems which would take over 2 hours otherwise. | en |
