http://link.springer.com/book/10.1007/978-3-642-31549-7

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Stabilization and Control of Fractional Order Systems: A Sliding Mode Approach

Bandyopadhyay, Bijnan, Kamal, Shyam

Book Description

In the last two decades fractional differential equations have been used more frequently in physics, signal processing, fluid mechanics, viscoelasticity, mathematical biology, electro chemistry and many others. It opens a new and more realistic way to capture memory dependent phenomena and irregularities inside the systems by using more sophisticated mathematical analysis. This monograph is based on the authors’ work on stabilization and control design for continuous and discrete fractional order systems. The initial two chapters and some parts of the third chapter are written in tutorial fashion, presenting all the basic concepts of fractional order system and a brief overview of sliding mode control of fractional order systems. The other parts contain deal with robust finite time stability of fractional order systems, integral sliding mode control of fractional order systems, co-operative control of multi-agent systems modeled as fractional differential equation, robust stabilization of discrete fractional order systems, high performance control using soft variable structure control and contraction analysis by integer and fractional order infinitesimal variations. 　

More information on this book can be found by the following link:

http://www.springer.com/engineering/control/book/978-3-319-08620-0

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 Journals

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Nonlinear Dynamics

Volume 77, Issue 4(selected)

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 Paper Highlight
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http://www.sciencedirect.com/science/article/pii/S0021999114005130#

Abstract

The paper makes the first attempt at applyingthe Kansa method, a radial basis function meshless collocation method, to the space-fractional advection–dispersion equations, which have recently been observed to accurately describe solute transport in a variety of field and lab experiments characterized by occasional large jumps with fewer parameters than the classical models of integer-order derivative. However, because of non-local property of integro-differential operator of space-fractional derivative, numerical solution of these novel models is very challenging and little has been reported in literature. It is stressed that local approximation techniques such as the finite element and finite difference methods lose their sparse discretization matrix due to this non-local property. Thus, the global methods appear to have certain advantages in numerical simulation of these non-local models because of their high accuracy and smaller size resultant matrix equation. Compared with the finite difference method, popular in the solution of fractional equations, the Kansa method is a recent meshless global technique and is promising for high-dimensional irregular domain problems. In this study, the resultant matrix of the Kansa method is accurately calculated by the Gauss–Jacobi quadrature rule. Numerical results show that the Kansa method is highly accurate and computationally efficient for space-fractional advection–dispersion problems.

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Chaos in the fractional-order Volta’s system: modeling and simulation

Ivo Petráš

Publication information: Ivo Petráš. Chaos in the fractional-order Volta’s system: modeling and simulation. Nonlinear Dyn. (2009) 57: 157–170. DOI 10.1007/s11071-008-9429-0.

http://link.springer.com/article/10.1007%2Fs11071-008-9429-0

Abstract

This paper deals with a new fractional-order chaotic system. It is based on the concept of Volta’s system, where the mathematical model of Volta’s system contains fractional-order derivatives. This system has simple structure and can display a double-scroll attractor. The behavior of the integer-order and the fractional-order Volta’s system with total order less than three which exhibits chaos is presented as well. Computer simulations are cross-verified by the numerical calculation and the Matlab/Simulink models.

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Anomalous diffusion and ballistic peaks: A quantum perspective

M. Stefancich, P. Allegrini, L. Bonci, P. Grigolini and B. J. West

Publication information: M. Stefancich, P. Allegrini, L. Bonci, P. Grigolini and B. J. West, Anomalous diffusion and ballistic peaks: A quantum perspective, Physical Review E, 1998, 57(6), 6625-6633.

http://journals.aps.org/pre/abstract/10.1103/PhysRevE.57.6625

Abstract

The quantum kicked rotor and the classical kicked rotor are both shown to have truncated Levy distributions in momentum space, when the classical phase space has accelerator modes embedded in a chaotic sea. The survival probability for classical particles at the interface of an accelerator mode and the chaotic sea has an inverse power-law structure, whereas that for quantum particles has a periodically modulated inverse power law, with the period of oscillation being dependent on Planck’s constant. These logarithmic oscillations are a renormalization group property that disappears as h in agreement with the correspondence principle.

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