Sliding Mode Control for Two-Degree-of-Freedom Fractional Zener Oscillator

Abstract

Fractional-order derivatives provide a powerful tool for the characterization of mechanical properties of viscoelastic materials. Fractional oscillators are mechanical models of viscoelastically damped structures, the viscoelastic damping of which is described by fractional-order constitutive equations. This paper proposes sliding mode control for a two-degree-of-freedom fractional Zener oscillator. Firstly, a virtual fractional oscillator is generated by means of a state transformation. Then, the total mechanical energy in the virtual oscillator is determined as the sum of the kinetic energy, the potential energy, and the fractional energy. Furthermore, sliding mode control for the fractional Zener oscillator is designed, in which the Lyapunov function is defined by the total mechanical energy. Finally, numerical simulations are conducted to validate the effectiveness of the proposed controllers.