A generalization of Müntz-Legendre polynomials and its implementation in optimal control of nonlinear fractional delay systems

Abstract

A hybrid of Müntz-Legendre polynomials (MLPs) and block-pulse functions (BPFs) is defined and carried out to analyze nonlinear fractional optimal control problems consisting of multiple delays. Instead of using the Caputo fractional derivative, an alternative fractional derivative operator is utilized. The primary optimization problem reduces to an alternative optimization one involving unknown parameters. For this purpose, the fractional Legendre-Gauss quadrature formula is utilized for approximating the associated cost functional and the fractional Legendre-Gaussian nodes are taken as the collocation points. Some new aspects of the proposed basis are demonstrated to verify the effectiveness of the proposed approach based on the Müntz-Legendre basis (MLB) against the classical orthogonal bases. The simulation results certify the feasibility and reliability of the proposed method. Numerous fractional control problems, e.g., bang-bang controls and control systems with any irregularities in the structure of control input can be handled successfully by employing the new fractional basis.