Sliding control for single-degree-of-freedom fractional oscillators

This paper proposes fractional sliding control designs for single-degree-of-freedom fractional oscillators respectively of the Kelvin-Voigt type, the modified Kelvin-Voigt type and D\"{u}ffing type, whose dynamical behaviors are described by second-order differential equations involving fractional derivatives. Firstly, the differential equations of motion are transformed into non-commensurate fractional state equations by introducing state variables with physical significance. Secondly, fractional sliding manifolds are constructed and stability of the corresponding sliding dynamics is addressed via the infinite state approach and Lyapunov stability theory. Thirdly, sliding control laws and adaptive sliding laws are designed for fractional oscillators respectively in cases that the bound of the external exciting force is known or unknown. Finally, numerical simulations are carried out to validate the above control designs.