Lyapunov Stability of Noncommensurate Fractional Order Systems: An Energy Balance Approach

Lyapunov stability of linear noncommensurate order fractional systems is treated in this paper. The proposed 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. The Lyapunov function is expressed as the sum of the different reversible fractional energies, whereas its derivative is interpreted in terms of internal and external Joule losses. Stability conditions are derived from the energy balance principle, adapted to the fractional case. Examples are taken from electrical systems, but this methodology applies also directly to mechanical and electromechanical systems.