48 research outputs found
Adaptive Backstepping Control for Fractional-Order Nonlinear Systems with External Disturbance and Uncertain Parameters Using Smooth Control
In this paper, we consider controlling a class of single-input-single-output
(SISO) commensurate fractional-order nonlinear systems with parametric
uncertainty and external disturbance. Based on backstepping approach, an
adaptive controller is proposed with adaptive laws that are used to estimate
the unknown system parameters and the bound of unknown disturbance. Instead of
using discontinuous functions such as the function, an
auxiliary function is employed to obtain a smooth control input that is still
able to achieve perfect tracking in the presence of bounded disturbances.
Indeed, global boundedness of all closed-loop signals and asymptotic perfect
tracking of fractional-order system output to a given reference trajectory are
proved by using fractional directed Lyapunov method. To verify the
effectiveness of the proposed control method, simulation examples are
presented.Comment: Accepted by the IEEE Transactions on Systems, Man and Cybernetics:
Systems with Minor Revision
Stability of fractional order systems
The theory and applications of fractional calculus (FC) had a considerable progress during the last years. Dynamical systems and control are one of the most active areas, and several authors focused on the stability of fractional order systems. Nevertheless, due to the multitude of efforts in a short period of time, contributions are scattered along the literature, and it becomes difficult for researchers to have a complete and systematic picture of the present day knowledge. This paper is an attempt to overcome this situation by reviewing the state of the art and putting this topic in a systematic form. While the problem is formulated with rigour, from the mathematical point of view, the exposition intends to be easy to read by the applied researchers. Different types of systems are considered, namely, linear/nonlinear, positive, with delay, distributed, and continuous/discrete. Several possible routes of future progress that emerge are also tackled
Recent Advances and Applications of Fractional-Order Neural Networks
This paper focuses on the growth, development, and future of various forms of fractional-order neural networks. Multiple advances in structure, learning algorithms, and methods have been critically investigated and summarized. This also includes the recent trends in the dynamics of various fractional-order neural networks. The multiple forms of fractional-order neural networks considered in this study are Hopfield, cellular, memristive, complex, and quaternion-valued based networks. Further, the application of fractional-order neural networks in various computational fields such as system identification, control, optimization, and stability have been critically analyzed and discussed
Fractional-Order Sliding Mode Synchronization for Fractional-Order Chaotic Systems
Some sufficient conditions, which are valid for stability check of fractional-order nonlinear systems, are given in this paper. Based on these results, the synchronization of two fractional-order chaotic systems is investigated. A novel fractional-order sliding surface, which is composed of a synchronization error and its fractional-order integral, is introduced. The asymptotical stability of the synchronization error dynamical system can be guaranteed by the proposed fractional-order sliding mode controller. Finally, two numerical examples are given to show the feasibility of the proposed methods
Optimal routh-hurwitz conditions and picard’s successive approximation method for system of fractional differential equations
Fractional calculusisabranchofmathematicalanalysisinvestigatingthederivatives
and integralsofarbitraryorder.Fractionalcalculushasawideapplicationsincemany
realistic phenomenaaredefinedinfractionalorderderivativeandintegral.Moreover,
fractional differentialequationsprovideanexcellentframeworkfordiscussingthe
possibility ofunlimitedmemoryandhereditaryproperties,consideringmoredegrees
of freedom.Inthisthesis,thestabilitycriteriaofthefractionalShimizu-Morioka
system andfractionaloceancirculationmodelinthesenseofCaputoderivative
are developedanalyticallyusingoptimalRouth-Hurwitzconditions.Hence,Routh-
Hurwitz conditionsforcubicandquadraticpolynomialsarepresented.Theadvantage
of Routh-Hurwitzconditionsisthattheyallowonetoobtainstabilityconditions
without solvingthefractionaldifferentialequations.Inthiscase,wefindthecritical
range foradjustablecontrolparameterandfractionalorder �, whichconcludesthat
the equilibriaofsystemsarelocallyasymptoticallystable.Aftermath,thenumerical
results arepresentedtosupportourtheoreticalconclusionsusingtheAdams-type
predictor-correctormethod.Ontheotherhand,wederivetheanalyticalsolutionfor
the inhomogeneoussystemofdifferentialequationswithincommensuratefractional
order 1 < �;�< 2, wherethefractionalorders � and � are uniqueandindependent
of eachother.ThesystemsarefirstwritteninVolterraintegralequationsofthesecond
kind. Further,Picard’ssuccessiveapproximationmethodisperformed,whichisan
explicitanalyticalmethodthatconvergesveryclosetoexactsolutions,andthesolution
is derivedinmultipleseriesandsomespecialfunctionexpressions,suchasGamma
function, Mittag-Lefflerfunctionsandhypergeometricfunctions.Somespecialcases
are discussedwhereallthesolutionsareverifiedusingsubstitution
Pseudo-State Sliding Mode Control of Fractional SISO Nonlinear Systems
This paper deals with the problem of pseudo-state sliding mode
control of fractional SISO nonlinear systems with model inaccuracies. Firstly,
a stable fractional sliding mode surface is constructed based on the Routh-Hurwitz conditions for fractional differential equations. Secondly, a sliding
mode control law is designed using the theory of Mittag-Leffler stability. Further,
we utilize the control methodology to synchronize two fractional chaotic
systems, which serves as an example of verifying the viability and effectiveness
of the proposed technique
Stabilization of the Fractional-Order Chua Chaotic Circuit via the Caputo Derivative of a Single Input
A modified fractional-order Chua chaotic circuit is proposed in this paper, and the chaotic attractor is obtained for q=0.98. Based on the Mittag-Leffler function in two parameters and Gronwall’s Lemma, two control schemes are proposed to stabilize the modified fractional-order Chua chaotic system via the Caputo derivative of a single input. The numerical simulation shows the validity and feasibility of the control scheme