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

Bandyopadhyay, B.; Kamal, Shyam

doi:10.1007/978-3-319-08621-7

Cited by 119 publications

(73 citation statements)

References 19 publications

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“…By using of the Mittag-Leffler function and the generalized Gronwall inequality, Chen et al [10] proposed a new sufficient condition for the local asymptotic stability and stabilization of a class of fractional-order nonlinear systems with fractional-order α ∈ (1, 2). The authors in [11,12] studied the stabilization and control of fractional systems by a new sliding mode method.…”

Section: Introductionmentioning

confidence: 99%

Dissipativity and contractivity for fractional-order systems

Wang

Xiao

2014

Nonlinear Dyn

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This paper concerns the dissipativity and contractivity of the Caputo fractional initial value problems. We prove that the systems have an absorbing set under the same assumptions as the classic integerorder systems. This directly extends the dissipativity from integer-order systems to the Caputo fractionalorder ones. The fractional dissipativity conditions can be satisfied by many fractional chaotic systems and the systems from the spatial discretization of some timefractional partial differential equations. The fractionalorder systems satisfying the so-called one-sided Lipschitz condition are also considered in a similar way, and the contractivity property of their solutions is proved. Two numerical examples are provided to illustrate the theoretical results.

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Section: Introductionmentioning

confidence: 99%

Dissipativity and contractivity for fractional-order systems

Wang

Xiao

2014

Nonlinear Dyn

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“…This mentioned equation is one of the simplest forms of fractional integro-differential equations, which is important in modelling real world phenomena. Sufficient and necessary conditions and theorems for existence and uniqueness solutions of present differential problem (8) can be found in 31,32 . In general, the particular solution u is not easy to be found, therefore the unknown to be approximated using a suitable alternate method.…”

Section: Implementation Of the Methodsmentioning

confidence: 99%

Using ANNs Approach for Solving Fractional Order Volterra Integro-differential Equations

Jafarian¹,

Rostami²,

Golmankhaneh³

et al. 2017

IJCIS

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Indeed, interesting properties of artificial neural networks approach made this non-parametric model a powerful tool in solving various complicated mathematical problems. The current research attempts to produce an approximate polynomial solution for special type of fractional order Volterra integrodifferential equations. The present technique combines the neural networks approach with the power series method to introduce an efficient iterative technique. To do this, a multi-layer feed-forward neural architecture is depicted for constructing a power series of arbitrary degree. Combining the initial conditions with the resulted series gives us a suitable trial solution. Substituting this solution instead of the unknown function and employing the least mean square rule, converts the origin problem to an approximated unconstrained optimization problem. Subsequently, the resulting nonlinear minimization problem is solved iteratively using the neural networks approach. For this aim, a suitable three-layer feed-forward neural architecture is formed and trained using a back-propagation supervised learning algorithm which is based on the gradient descent rule. In other words, discretizing the differential domain with a classical rule produces some training rules. By importing these to designed architecture as input signals, the indicated learning algorithm can minimize the defined criterion function to achieve the solution series coefficients. Ultimately, the analysis is accompanied by two numerical examples to illustrate the ability of the method. Also, some comparisons are made between the present iterative approach and another traditional technique. The obtained results reveal that our method is very effective, and in these examples leads to the better approximations.

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“…[40] and demonstrated with Example 1 below, the correct way is to define an initial condition over an interval [39], and we refer to the initial condition as initial history. A typical initial history problem is…”

Section: History-dependent Initializations For Equations With the Riementioning

confidence: 99%

Correcting the initialization of models with fractional derivatives via history-dependent conditions

Wang

2015

Acta Mech. Sin.

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Fractional differential equations are more and more used in modeling memory (history-dependent, nonlocal, or hereditary) phenomena. Conventional initial values of fractional differential equations are defined at a point, while recent works define initial conditions over histories. We prove that the conventional initialization of fractional differential equations with a Riemann-Liouville derivative is wrong with a simple counter-example. The initial values were assumed to be arbitrarily given for a typical fractional differential equation, but we find one of these values can only be zero. We show that fractional differential equations are of infinite dimensions, and the initial conditions, initial histories, are defined as functions over intervals. We obtain the equivalent integral equation for Caputo case. With a simple fractional model of materials, we illustrate that the recovery behavior is correct with the initial creep history, but is wrong with initial values at the starting point of the recovery. We demonstrate the application of initial history by solving a forced fractional Lorenz system numerically.

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

Cited by 119 publications

References 19 publications

Dissipativity and contractivity for fractional-order systems

Dissipativity and contractivity for fractional-order systems

Using ANNs Approach for Solving Fractional Order Volterra Integro-differential Equations

Correcting the initialization of models with fractional derivatives via history-dependent conditions

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