Research on stability and control strategies of fractional-order differential equations in nonlinear dynamic systems

The capacity of fractional-order differential equations (FODEs) to simulate intricate behaviors in nonlinear dynamic systems has drawn significant interest recently. A potent tool for characterizing memory and inherited characteristics present in several physical, biological, and engineering systems, FODEs involve derivatives of arbitrary, non-integer orders, in contrast to conventional integer-order differential equations. The goal of this research is to extensively examine the stability and control techniques of nonlinear FODEs, addressing their vital significance in modeling and managing nonlinear dynamical systems. Lyapunov-based stability is used to theoretically prove a stability theorem, offering a strong basis for fractional-order (FO) system analysis. A linear state feedback controller is designed, specifically to maintain a class of FO nonlinear systems to preserve system stability. The pole placement method from linear FO control theory is used to establish a novel criterion for calculating controller gains. The suggested method makes it possible to systematically select control settings, providing accurate nonlinear dynamics stabilization. Simulation investigations on the FO chaotic Lorenz system are carried out to verify the theoretical framework, and the intended stability and control performance are obtained. The outcomes demonstrate the efficiency of the proposed strategies, offering significant insights for advancing the understanding and function of FODEs in complex nonlinear dynamic systems.