Adaptive Collocation Methods Using Chebyshev Integration

Abstract

Solutions to differential equations form the bedrock of many guidance, navigation and control operations, such as state propagation and optimal control. While analytical solutions exist for a variety of problems, complex vehicle dynamics often require use of numerical differentiation techniques. The most common are serial, step-wise methods based on a Taylor series expansion about the current state, e.g. the family of Runge-Kutta schemes. Alternatively, global approximation methods–such as Galerkin least squares, collocation, and pseudospectral–approximate the solution and ensure the differential equation is solved through parameter optimization. The focus of this dissertation is to present an innovative Chebyshev collocation technique that leverages exact integral relationships of these polynomials to parameterize both the state and derivatives with a single orthogonal functional basis. This new technique approximates the highest order derivative with a Chebyshev series which is subsequently integrated until the state representation is reached. A general order integration scheme is developed to solve any order differential equation. This technique is applied to a wide set of linear and nonlinear initial value problems and boundary value problems. A novel approach to solving matrix differential equations is presented which offers a new approach to propagating the state transition matrix as well as solving the matrix Riccati equation. Domain subdivision methods are explored to transform one large collocation problem into a set of multiple, manageable problems. This improves computation time and accuracy, especially for highly nonlinear problems.