Differintegral Operator Research Articles

A novel continuous fractional sliding mode control

ABSTRACTA new fractional-order controller is proposed, whose novelty is twofold: (i) it withstands a class of continuous but not necessarily differentiable disturbances as well as uncertainties and unmodelled dynamics, and (ii) based on a principle of dynamic memory resetting of the differintegral operator, it is enforced an invariant sliding mode in finite time. Both (i) and (ii) account for exponential convergence of tracking errors, where such principle is instrumental to demonstrate the closed-loop stability, robustness and a sustained sliding motion, as well as that high frequencies are filtered out from the control signal. The proposed methodology is illustrated with a representative simulation study.

On a problem for the loaded degenerating mixed type equation involving integral-differential operators

On a problem for the loaded degenerating mixed type equation involving integral-differential operators

Some properties of certain extended fractional differintegral operator

In this paper, we derive some inequality properties, convolution properties, subordination and superordination properties, and sandwich-type results involving certain extended fractional differintegral operator.

GENERALIZED FRACTIONAL DIFFERINTEGRAL OPERATORS OF THE K-SERIES

In the present paper, we further study the generalized fractional differintegral (integral and differential) operators involving Appell's function <TEX>$F_3$</TEX> introduced by Saigo-Maeda [9], and are applied to the K-Series defined by Gehlot and Ram [3]. On account of the general nature of our main results, a large number of results obtained earlier by several authors such as Ram et al. [7], Saxena et al. [14], Saxena and Saigo [15] and many more follow as special cases.

A ground motion prediction equation for novel peak ground fractional order response intensity measures

Peak ground fractional order responses (PGRα) as a generalization of conventional seismic intensity measures (IMs) such as peak ground acceleration (PGA) and peak ground velocity (PGV) can better predict the seismic performance of structural systems. This paper proposes the first ground motion prediction equation (GMPE) for PGRα for active shallow crustal regions using a subset of the PEER NGA-West2 ground motion database. The model development database consists of 4491 accelerograms from 82 different earthquakes in California with the magnitude and rupture distance ranges of Mw 4.0–7.9 and RRUP 0–300 km, respectively. PGRα intensity measures are computed from the modified Oustaloup’s recursive approximation to Caputo’s definition of differintegral operator. The main functional form of the predictive model is decided by implementing statistical ground motion data-driven testing methods such as the likelihood approach and Euclidean distance concept. The final functional form of the predictive model accounts for magnitude, distance, style-of-faulting, linear and nonlinear site, hanging wall, basin response, and anelastic distance attenuation effects, and models the aleatory variability with respect to Mw and VS30. The final predictive model produces PGA (α = 0), PGV (α = −1), and peak ground fractional order responses at 19 different α values ranging from −0.05 to −0.95 for the average horizontal component. The proposed predictive model draws estimates of ground motion amplitudes that are consistent with those from the NGA-West2 models for PGA and PGV for sample earthquake scenarios. Moreover, it can offer a basis for predictive modeling of peak ground fractional order response quantities for performance assessment of structures and infrastructures across a region.

Inclusion results for certain classes of analytic functions associated with a new fractional differintegral operator

By using a new fractional differintegral operator which was studied in a recent work (Mediterr J Math 13:1535–1553 2016), we first introduce various classes of analytic functions in terms of the subordination and then prove several inclusion results for each of these classes of analytic functions. We also consider some useful special cases and consequences of the results which are presented in this paper.

Design of fractional-order variants of complex LMS and NLMS algorithms for adaptive channel equalization

Equalization filtering is an effective technique applied to minimize the inter-symbol interference (ISI) in multipath fading channels; the problem gets worse for higher-order constellations which are required for high data rates in today’s communication systems. The least mean square (LMS) filter is a computationally efficient and easily implementable algorithm but suffers from slow convergence; highly complex filters are required to nullify the effects of ISI. In this paper, we develop complex modified fractional-order (FO) nonlinear variants of the LMS and the NLMS algorithms and apply in adaptive channel equalization, in both feed-forward and decision feedback configurations. In addition to the standard first-order derivative, the update in the modified LMS also depends on the FO derivative of the mean square error, the final update is formed using a combination of conventional update term and a nonlinear term obtained through Riemann–Liouville fractional derivative. The step size of the FNLMS scheme in fractional part is not only a function of the input energy but also the FO. The differintegral operator working as differentiator helps improve the convergence rate because the algorithm becomes nonlinear; the fractional algorithms provide more parameters to control the rate of convergence and have simple implementation with almost similar complexity. The performances of the schemes are validated through extensive simulation results for block fading channels (frequency flat and selective) to evaluate the symbol error rate for higher-order quadrature amplitude modulation schemes, mean square error and combined channel and equalizer responses to show the improved inverse modeling of the channel. Simulation experiments confirm the superiority of the proposed algorithm over the traditional counterparts.

Majorization Properties for Subclasses of Analytic P-Valent Functions Defined by Generalized Differintegral Operator

Majorization Properties for Subclasses of Analytic P-Valent Functions Defined by Generalized Differintegral Operator

Fractional integral sliding modes for robust tracking of nonlinear systems

Memory and heritage of differintegral operators require knowledge of the error manifold derivative at the initial time to sustain a sliding motion for any initial condition. Moreover, when the system is subject to (unknown) disturbances, such initial condition is unknown; thus, the enforcement of an integral sliding motion has been elusive with a chatter-less controller. In this paper, a novel fractional-order integral sliding mode (FISM) is proposed to maintain an invariant sliding mode due to an exact estimation of disturbances at first step. Our scheme is continuous after initial condition, avoiding chattering effects thanks to the topological properties of differintegral operators. In contrast to other FISM approaches, the proposed scheme induces a fractional-order reaching dynamics of order \((1+\nu )\in (1,2)\) to enforce an integral sliding mode for any initial condition, even in the presence of Holder (continuous but not necessarily differentiable) disturbances and model uncertainties. Simulations show the reliability of the proposed scheme.

Continuous Fractional‐Order Sliding PI Control for Nonlinear Systems Subject to Non‐Differentiable Disturbances

AbstractAiming at designing a robust controller to withstand a class of continuous, but not necessarily differentiable, disturbances, such as Hölder type, a continuous and chattering‐free sliding mode control is proposed. The key idea is a judicious synthesis of a resetting memory principle for the differintegral operators to show that a sliding mode is induced, and sustained, in finite‐time, to guarantee asymptotic tracking. The closed‐loop system achieves exact rejection of Hölder disturbances even in case of uncertain flow, but assuming certain knowledge of the input matrix. Furthermore, it is shown that our methodology generalizes continuous high‐oder sliding mode schemes by using an integral action of fractional‐order. A representative simulation study is discussed to show the feasibility of the proposal.

Some Geometric Properties of Analytic Functions Involving a New Fractional Operator

In this paper, we introduce and study a new fractional operator and its implications in terms of the Ruscheweyh derivative operator, the Sălăgean operator and a certain fractional differintegral operator. Some geometric properties of the analytic functions involving this operator are derived. We also consider some applications and derive certain corollaries of our main results. Some useful consequences and relationship of certain results with known results are also pointed out.

On certain subclasses of multivalent functions defined by a generalized fractional differintegral operator

The purpose of this paper is to introduce subclasses of multivalent functions by using linear operator defined by generalized fractional differintegral operator. We investigate various properties for functions of these subclasses.

Some Characterizations of Integral Operators Associated with Certain Classes of p-Valent Functions Defined by the Srivastava–Saigo–Owa Fractional Differintegral Operator

The purpose of this paper is to introduce new integral operators associated with Srivastava–Saigo–Owa fractional differintegral operator. We investigate some properties for the integral operators \({\mathcal {F}}_{p,\eta ,\mu }^{\lambda ,\delta }(z)\) and \({\mathcal {G}}_{p,\eta ,\mu }^{\lambda ,\delta }(z)\) to be in the classes \({\mathcal {R}}_{k}^{\zeta }\left( p,\rho \right) \) and \({\mathcal {V}}_{k}^{\zeta }\left( p,\rho \right) \).

Attitude control of quadrotors based on fractional sliding modes: theory and experiments

Light weight quadrotors are highly manoeuvrable autonomous aerial robots (UAV) that outperform most UAVs; however, quadrotors are prone to a wide class of aerodynamic disturbances, vibrations, and uncertainties that demand not only robust but fast attitude control structures. These arguments are compelling enough to motivate the design of a novel fractional-order control scheme that enforces finite-time convergence of the sliding manifold to guarantee local exponential tracking. The controller does not exhibit chattering and does not require the knowledge of the dynamic model. The stability analysis is based on a proposed resetting memory principle to deal with memory effects of the differintegral operator, which allows the controller to tackle also robustness against continuous disturbances that are not necessarily differentiable in the conventional sense such as some aerodynamic phenomena. Experimental evidence is presented and discussed to expose the enforcement of the sliding manifold based on a fractional-order reaching phase that demonstrates the feasibility of the proposed control scheme.

Uniformly continuous differintegral sliding mode control of nonlinear systems subject to Hölder disturbances

An integral sliding mode controller based on fractional order differintegral operators is proposed. This controller generalizes the classical discontinuous (integer order) integral sliding mode scheme. By using differintegral operators, their topological properties lead to a uniformly continuous controller that enforces an integral sliding mode for any initial condition. In addition, it is demonstrated that the proposed scheme is robust against matched Hölder continuous, but not necessarily differentiable, disturbances and uncertainties. Also, asymptotic convergence of tracking errors is assured for any initial condition by means of an ideal controller, even in presence of anomalous but Hölder continuous disturbances. It is worth to mention that the salient properties of our proposal (i.e. invariance at any initial condition, uniform continuity, and robustness to non-differentiable disturbances) are not provided by any existing integer order sliding mode based controller of the literature. The viability of the proposed scheme is shown in a representative simulation study.

The Small Ball Asymptotics in Hilbert Norm for the Kac--Kiefer--Wolfowitz Processes

We consider the problem of a small ball behavior in $L_2$-norm for some Gaussian processes of statistical interest. The problem is reduced to the spectral asymptotics for some integral-differential operators. To find these asymptotics we construct complete asymptotic expansions of some oscillation integrals with slowly varying amplitudes.

ВЫРОЖДЕННЫЕ УРАВНЕНИЯ ВОЛЬТЕРРА ТИПА СВЕРТКИ В БАНАХОВЫХ ПРОСТРАНСТВАХ И ИХ ПРИЛОЖЕНИЯ

The article is devoted to the problem of unique solvability of linear integral and integral-differential Volterra equations in Banach spaces with irreversible operator in the main part. Operator-valued kernel has a special form, K ( t , s ) = g ( t – s ) A , where g = g ( t ) is a numeric function, and A is a linear operator. Abstract equations of this kind are very typical for applications. For the study of such equations it is possible to use structural stack theory of two linear operators, which has been developed by Professor G.A. Sviridyuk and his students. Another peculiarity of the studied problems is multiple zero of function g = g ( t ) at the point t = 0. Fundamental operator-functions of considered integral and integral-differential operators in Banach spaces are constructed under the assumption of relative spectrally boundness of operator A with respect to degenerated main part of equations. On this basis, theorems of unique existence of solutions in the class of distributions with left-bounded support are proved. The dependence between the order of singularity of generalized solutions and multiplicity of zero of integral kernel at the initial point is ascertained. Also we have obtained conditions under which generalized solutions are equal to the classical solutions. Theorems formulated for abstract equations are applied to the study of significant initial boundary value problems arising in plasma physics and mathematical theory of elasticity

Some Subclasses of p-Valent Functions Defined by Generalised Fractional Differintegral Operator - I

In this paper, by applying a generalized extended fractional differintegral operator S λ;μ;η 0;z (z 2 4 ; p 2 N; μ;η 2 R; μ < p+ 1;−∞ < λ < η + p+ 1) we define a new class convex functions C V λ;μ;η p (α; A;B) and several sharp inclusion relationships and other interesting properties were discussed by using the te chniques of differential subordination.

Impact of colored noise in pulse amplitude measurements: A time-domain approach using differintegrals

In particle detectors, pulse shaping is the process of changing the waveform of the pulses in order to maximize the signal to noise ratio. This shaping usually only takes into account white, pink (flicker) and red (Brownian) noise. In this paper, a generalization of noise indexes as a function to an arbitrary fβ noise type, where β is a real number, is presented. This generalization has been created using the differintegral operator, defined in Fractional Calculus. These formulas are used to calculate the Equivalent Noise Change (ENC) in detector particle systems.

Subordination and superordination properties of p-valent functions defined by a generalized fractional differintegral operator

For a generalized fractional differintegral operator associated with p- valent functions, we study different properties of differential subordination and superordination related to this operator.