Analysis and Synthesis of Gradient Algorithms Based on Fractional-Order System Theory

In this study, a framework for processing gradient algorithms is proposed in accordance with nabla fractional-order system theory. Unlike most of the literature, the gradient algorithm is initially recast into a nabla fractional-order dynamic system. To be specific, the algorithm is designed using control theory and analyzed using the Lyapunov theory, which is a more general and effective way to render high-performance algorithm. Three types of algorithms are built in this study, i.e., the asymptotic convergence case, the finite-time convergence case, and the fixed-time convergence case. Finally, a comprehensive simulation study is conducted verifying the correctness, usefulness, and practicality of the framework.