Finite-time synchronization of variable-order fractional uncertain coupled systems via adaptive sliding mode control

This chapter aims to investigate the problem of adaptive sliding mode control (SMC) for a class of variable-order fractional (VOF) uncertain coupled systems. First, a novel VOF integral-type sliding surface, composed of nonlinear coupling terms, is designed with the aid of VOF calculus. This surface is crucial for effectively managing the complex dynamics of VOF systems. Second, based on graph theory, novel asymptotical stability criteria are derived for the resulting sliding mode dynamics, ensuring robust stability of the system. Moreover, the finite-time reachability of the predefined VOF integral-type sliding surface is ensured by designing a novel adaptive VOF controller, which adjusts the control parameters in real-time to maintain system performance. Finally, two numerical studies are presented to verify the validity and superiority of the proposed control strategy, demonstrating its effectiveness in handling the uncertainties and complexities inherent in VOF systems. These studies highlight the practical applicability and robustness of the adaptive SMC approach in achieving desired control objectives.