Stability analysis of distributed-order nonlinear dynamic systems

ABSTRACT

The problem of asymptotic stability analysis of equilibrium points in nonlinear distributed-order dynamic systems with non-negative weight functions is considered in this paper. The Lyapunov direct method is extended to be used for this stability analysis. To this end, at first, a discretisation scheme with convergence property is introduced for distributed-order dynamic systems. Then, on the basis of this tool, Lyapunov theorems are proved for asymptotic stability analysis of equilibrium points in distributed-order systems. As the order weight function assumed for the distributed-order systems is general enough, the results are applicable to a wide range of nonlinear distributed-order systems such as fractional-order systems with multiple fractional derivatives. To verify the applicability of the obtained results, these results are applied for the stability analysis of a distributed-order diffusion system and control of a fractional-order Lorenz system with multiple fractional derivatives.

KEYWORDS:

• Distributed-order nonlinear systems
• asymptotic stability analysis
• Lyapunov direct method

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Iran National Science Foundation (INSF).

Notes on contributors

Hamed Taghavian

Hamed Taghavian is an M.Sc. student majoring in control systems in the Department of Electrical Engineering at Sharif University of Technology. His research interests include fractional distributed order calculus and the relevant dynamic systems.

Mohammad Saleh Tavazoei

Mohammad Saleh Tavazoei received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from the Sharif University of Technology, Tehran, Iran, in 2003, 2005, and 2008, respectively. He is currently an Associate Professor with the Department of Electrical Engineering, Sharif University of Technology. His current research interests include dynamical behavior analysis of fractional order systems and applications of these systems in control system design.