Integral sliding mode control for robust stabilisation of uncertain stochastic time-delay systems driven by fractional Brownian motion

ABSTRACT

In this paper, the stability and controller design for fractional stochastic systems, i.e. stochastic systems driven by fractional Brownian motion (fBm) are investigated. A fractional infinitesimal operator is proposed for stability analysis of this class of stochastic systems and a Lyapunov-based stability criterion is established. Thereafter, the presented stability criterion is utilised to develop the sliding mode control scheme for fractional stochastic systems with state delay and time-varying uncertainties. By applying the proposed fractional infinitesimal operator, the sufficient robust stability conditions are derived in the form of linear matrix inequalities. The proposed method guarantees the reachability of the sliding surface in finite time, and the closed-loop system will be stable in probability for all Hurst indices of the fBm in the range $(\frac{1}{2},1)$. Finally, some simulation examples are given to illustrate the effectiveness of the proposed design method.

KEYWORDS:

• Robust integral sliding mode control
• fractional Brownian motion
• linear matrix inequalities
• fractional infinitesimal operator
• Lyapunov-based stability

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Notes on contributors

Khosro Khandani

Khosro Khandani received his B.Sc. degree in control engineering from Sahand University of Technology, Tabriz, Iran in 2008. Then, he received his MSc degree in control engineering from Iran University of Science and Technology, Tehran, Iran in 2011. He is currently pursuing Ph.D. degree in control engineering at Tarbiat Modares University, Tehran, Iran. He is also a research assistant at intelligent control systems laboratory in Tarbiat Modares University. His research interests include fractional order systems and control, stochastic systems, robust and nonlinear control.

Vahid Johari Majd

Vahid Johari Majd received his B.Sc. degree in 1989 from the Electrical Engineering Department of the University of Tehran, Iran. He then received his M.Sc. and Ph.D. degrees in the area of Control Theory from the Electrical Engineering Department of the University of Pittsburgh, PA, USA in 1991 and 1995, respectively. He is currently an associate professor in the Control System Department of Tarbiat Modares University, Tehran, Iran, and is the director of intelligent control systems laboratory. His areas of interest include: intelligent identification and control, multi-agent learning, fuzzy control, cooperative control, formation control, robust nonlinear control, and fractional order control.

Mahdieh Tahmasebi

Mahdieh Tahmasebi received her B.Sc. degree in applied mathematics from University of Shahid Beheshti, Tehran, Iran in 2002 and M.Sc. and Ph.D. degrees in mathematics from Sharif University of Technology in 2005, 2010, respectively. From 2009 to 2010, she was a research assistant with INRIA Institute, Sophia Antipolis, France. Her research interests include Malliavin calculus, stochastic control, financial mathematics, numerical analysis of SDEs, stochastic analysis, and ordinary and partial stochastic differential equations. Dr Tahmasebi is currently an assistant professor at the Department of Applied Mathematics, School of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran.