Innovative observer design for nonlinear systems using Caputo fractional derivative with respect to another function

This work introduces a novel control framework using the Caputo fractional derivative (CFD) with respect to another function—a derivative that has not been thoroughly treated in control theory. By extending the widely recognized Caputo-Hadamard (CH) fractional-order derivative, we address its utility in nonlinear systems. The core of our contribution is the practical stability for systems governed by this derivative, which ensures convergence toward a bounded region around the origin. Additionally, we extend the Lipschitz condition (LC) to the one-sided Lipschitz (OSL) condition for observer design and observer based-control design in fractional-order systems, ensuring its practical stability. Finally, three numerical examples validate the effectiveness of our proposed framework, providing practical insights for control theory advancements.