Generalized Mittag-Leffler stability of multi-variables fractional order nonlinear systems

Yu, Jun; Hu, Hua; Zhou, Shangbo; Lin, Xianke

doi:10.1016/j.automatica.2013.02.041

Cited by 89 publications

(40 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [22,23], Mittag-Leffler stability and generalized MittagLeffler stability were introduced to describe fractional Lyapunov stability of fractional-order systems. Generalized Mittag-Leffler stability of multivariables fractional-order systems was investigated by [24] further. Trigeassou et al [6] demonstrated that classical Lyapunov functions are valid for fractional-order systems.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic pseudo-state stabilization of commensurate fractional-order nonlinear systems with additive disturbance

Ding

Peng³

et al. 2015

Nonlinear Dyn

View full text Add to dashboard Cite

The pseudo-state stabilization problem of commensurate fractional-order nonlinear systems is investigated. The concerned fractional-order nonlinear system is of parametric strict-feedback form with both unknown parameters and the additive disturbance. To solve this problem, a new nonlinear adaptive control law is constructed via fractional-order backstepping scheme. The developed fractional-order controller does not require the knowledge about both the interval of uncertain parameters and the upper bound of the additive disturbance. The asymptotic pseudo-state stability of the closed-loop system is proved in terms of fractional Lyapunov stability. Several examples are performed finally, and the efficiency is verified.

show abstract

Section: Introductionmentioning

confidence: 99%

Asymptotic pseudo-state stabilization of commensurate fractional-order nonlinear systems with additive disturbance

Ding

Peng³

et al. 2015

Nonlinear Dyn

View full text Add to dashboard Cite

show abstract

“…In recent years, many mathematical, physical and engineering phenomena have been successfully described using the fractional calculus, the theory of integrals and derivatives of non-integer orders; see [1][2][3][4][5][6][7][8][9]. For examples, fluid mechanics, chemistry [7], biology [8], viscoelasticity [9], engineering, finance and physics [10] fields.…”

Section: Introductionmentioning

confidence: 99%

Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions

2015

View full text Add to dashboard Cite

A spectral shifted Legendre Gauss-Lobatto collocation method is developed and analyzed to solve numerically one-dimensional two-sided space fractional Boussinesq (SFB) equation with non-classical boundary conditions. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomials PL,n(x), x ∈ [0, L] is assumed, for the function and its space-fractional derivatives occurring in the two-sided SFB equation. The Legendre-Gauss-Lobatto quadrature rule is established to treat the non-local conservation conditions, and then the problem with its non-local conservation conditions is reduced to a system of ordinary differential equations (ODEs) in time. Thereby, the expansion coefficients are then determined by reducing the two-sided SFB with its boundary and initial conditions to a system of ODEs for these coefficients. This system may be solved numerically in a step-by-step manner by using implicit Runge-Kutta method of order four. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.Mathematics Subject Classification. 34A08, 65M70, 33C45.

show abstract

“…Then, there was another attempt in [13,14] to derive Mittag-Leffler stability and generalized Mittag-Leffler stability for describing Lyapunovlike stability of FOSs. On the other hand, Yu et al in [31] introduced the generalized Mittag-Leffler stability of multi-variable FOSs.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Mittag-Leffler stability of nonlinear fractional partial differential equations

Hanan¹,

Ahmad²,

Fadhel³

2017

J. Nonlinear Sci. Appl.

View full text Add to dashboard Cite

This paper focuses on the application of fractional backstepping control scheme for nonlinear fractional partial differential equation (FPDE). Two types of fractional derivatives are considered in this paper, Caputo and the Grünwald-Letnikov fractional derivatives. Therefore, obtaining highly accurate approximations for this derivative is of a great importance. Here, the discretized approach for the space variable is used to transform the FPDE into a system of fractional differential equations. The convergence of the closed loop system is guaranteed in the sense of Mittag-Leffler stability. An illustrative example is given to demonstrate the effectiveness of the proposed control scheme.

show abstract

Generalized Mittag-Leffler stability of multi-variables fractional order nonlinear systems

Cited by 89 publications

References 9 publications

Asymptotic pseudo-state stabilization of commensurate fractional-order nonlinear systems with additive disturbance

Asymptotic pseudo-state stabilization of commensurate fractional-order nonlinear systems with additive disturbance

Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions

Nonlinear Mittag-Leffler stability of nonlinear fractional partial differential equations

Contact Info

Product

Resources

About