Non-Uniform Reducing the Involved Differentiators' Orders and Lyapunov Stability Preservation Problem in Dynamic Systems

Although uniform reduction of the differentiators' orders in the stable state-space integer order models yields in stable fractional order dynamics, stability may not be preserved by non-uniform reducing the orders of the involved differentiators. This fact causes that the powerful methods available for Lyapunov stability analysis of integer order systems cannot be easily applicable for stability analysis of the models obtained by non-uniform reduction of the orders (i.e., incommensurate order models). To overcome such a challenge, this brief aims to facilitate stability analysis of incommensurate order systems via finding whether stability of an incommensurate order system is deduced from the Lyapunov stability of its integer order counterpart. Applicability of this brief achievement is evaluated through stability analysis in some different case studies, such as the special case of linear time invariant systems, time-delay positive linear systems, Lotka-Volterra systems, and biological models.