External stability of Caputo fractional‐order nonlinear control systems

Abstract

SummaryThis paper investigates external stability of Caputo fractional‐order nonlinear control systems. Following the idea of a traditional Lyapunov function method, we point out the problems that would appear when applying it for fractional external stability. These problems are shown to be solvable by employing results on smoothness of solutions, but this method generalized for Caputo fractional‐order nonlinear control systems requires strong conditions to be imposed on vector field functions and inputs. To further explore the fractional external stability, diffusive realizations and Lyapunov‐like functions are taken into consideration. Specifically, a Caputo fractional‐order nonlinear control system with certain assumptions proves to be equivalent to its diffusive realization; a Lyapunov‐like function based on the realization exhibits properties useful to prove the external stability. As expected, this Lyapunov‐like method has weaker requirements. Finally, it is applied to the external stabilization of a Caputo fractional‐order Chua's circuits with inputs.