Computational optimal control relies mainly on pseudospectral methods. The use of Chebyshev and Legendre polynomials is ubiquitous in the literature. This family of methods has good accuracy characteristics but constraints the nodal distribution to a certain grid that is denser at the boundaries. In this work, a set of novel Coupled Radial Basis Functions (CRBFs) is introduced as an approximation means for the nonlinear optimal control problem. CRBFs are real-valued Radial Basis Functions (RBFs) augmented with a conical spline. They do not require a specific nodal distribution. A plethora of research articles were published on the optimization of the shape parameter of RBFs. Unlike classic RBFs, CRBFs are insensitive to the shape parameter reducing the computational time needed to find an optimal shape parameter. The method introduced in this dissertation follows an indirect approach of solving optimal control problems. Hence, the method is initiated by deriving the necessary conditions of optimality. Consequently, CRBFs are used to approximate the resulting two-point boundary value problem (TPBVP) into a set of nonlinear algebraic equations (NAEs). The system of NAEs is then solved using a standard nonlinear solver. Numerical experiments of the proposed method are carried out and compared with exact solutions and other computational methods. The method is applied to classical nonlinear optimal control problems: Zermelo's problem, a duffing oscillator with various boundary conditions, and a nonlinear inverted pendulum on a cart. CRBFs-collocation shows superiority of computational speed over other methods and is easy to implement. For future work, this method is suitable for real-time control applications.
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Doctor of Philosophy (Ph.D.)
College of Engineering and Computer Science
Mechanical and Aerospace Engineering
Length of Campus-only Access
Doctoral Dissertation (Open Access)
Seleit, Ahmed Elsadek Ahmed Elshafee, "Shape Parameter & Nodal Distribution Insensitive Radial Basis Functions for Nonlinear Optimal Control Problems" (2022). Electronic Theses and Dissertations, 2020-. 1488.