Lyapunov Stability of Commensurate Fractional Order Systems: A Physical Interpretation

Abstract

Lyapunov stability of linear commensurate order fractional systems is revisited with the energy balance principle. This methodology is based on the concept of fractional energy stored in inductor and capacitor components, where natural decrease of the stored energy is caused by internal Joule losses. Previous stability results are interpreted, thanks to an equivalent fictitious fractional RLC circuit. Energy balance is used to analyze the usual Lyapunov function and to provide a physical interpretation to the weighting positive matrix. Moreover, the classical linear matrix inequality (LMI) condition is interpreted in terms of internal and external Joule losses.