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.

The study of the generalized $t$-deformation of free convolution from the lens of Cauchy–Stieltjes kernel (CSK) families provides an excellent mathematical framework for understanding noncommutative probability distributions. In this paper, we use the concept of generalized $t$-deformation to demonstrate different elements of the Marchenko–Pastur, free Gamma, and inverse semicircle measures in the CSK families setting. These findings advance our knowledge of generalized $t$-deformation in the non-commutative probability framework.