Integration Of Non-integer Order Research Articles

Nondimensional analysis and experimental assessment of the PII1/2DD1/2 position control of a rotary axis

Fractional Order controllers, based on derivatives and integrals of non-integer order, represent a propitious and promising research area since they can improve the performance of mechatronic systems compared to traditional Integer Order PID controllers without requiring hardware upgrades, but only using their higher tuning flexibility. Nevertheless, Fractional Order controllers are still rarely applied in real automation tasks, since their tuning is more complex and in general requires a model-based optimization for full exploitation of their capabilities. Recently, the PII1/2DD1/2 controller has been proposed, extending the classical PID by introducing a half-integral and a half-derivative term. A Bode plot-based tuning method has been presented for this controller, deriving the PII1/2DD1/2 parameters from a given set of PID gains. This method enables a significant reduction in tracking error for position control of a mechatronic axis without the need for complex model-based optimizations. In this paper, the previous results of this PII1/2DD1/2 tuning methodology have been generalized by employing a nondimensional approach. This approach highlights the conditions under which the benefits of this Fractional Order controller over the PID are most significant. The findings have been validated through experimental tests on a rotary axis, confirming the correctness and usefulness of the method.

Linear and non-linear thermal system identification based on the integral of non-integer order — Application to solve inverse heat conduction linear and non-linear problems

The identification of thermal systems is an approach particularly well suited to solve inverse heat conduction problems (IHCP) in the case of complex systems where many parameters are unknown and where even the sensors must be modeled in order to obtain accurate results. In this paper, we propose significant advances in the formulation of the mathematical model structure by basing it exclusively on the non-integer order integration operator. We show that this approach is consistent with the heat diffusion process and that it allows to get rid of the step of pre-filtering the experimental signals to the sensors which was necessary when using the derivation operator. Demonstrations are made on the simulated configurations through simple analytical solutions and also a complex real configuration. I generalize the use of the non-integer integration operator in the treatment of non-linear heat conduction problems based on the Volterra series decomposition. We show in particular that the use of a generalized non-integer transfer function gives encouraging and generalizable results for a large class of related problems.

Self-Adjoint Sturm-Liouville Dynamic Problem via Proportional Derivative

The concept of a conformable derivative on time scales is a relatively new development in the field of fractional calculus. Traditional fractional calculus deals with derivatives and integrals of non-integer order on continuous time domains. However, time scale calculus extends these concepts to more general time domains that include both continuous and discrete points. The conformable derivative on time scales has several properties that make it advantageous in certain applications. For example, it satisfies a chain rule and has a simple relationship with the conformable integral, which facilitates the development of differential equations involving fractional order dynamics. It also allows for the analysis of systems with both continuous and discrete data points, making it suitable for modeling and control applications in various fields, including physics, engineering, and finance. In this study, the Sturm-Liouville problem and its properties are examined on an arbitrary time scale using the proportional derivative, a more general form of the fractional derivative. Important spectral properties such as self-adjointness, Green formula, Lagrange identity, Abel formula, and orthogonality of eigenfunctions for this problem are expressed in proportional derivatives on an arbitrary time scale.

Fractional Calculus in Signal Processing: The Use of Fractional Calculus in Signal Processing Applications, Such as Image Denoising, Filtering, and Time Series Analysis

Fractional calculus has transformed signal processing by providing a more flexible representation of complex signals. This article explores its applications in image denoising, filtering, and time series analysis, highlighting its potential to revolutionize signal processing methodologies. Fractional calculus, dealing with derivatives and integrals of non-integer orders, captures long-memory and self-similar properties in real-world signals. Image denoising benefits from fractional derivatives and integrals, preserving essential details while removing noise. Filtering signals with fractional calculus allows for dynamic adaptability to changing signal characteristics. Time series analysis benefits from more accurate modelling, enhancing predictive capabilities. The future scope of fractional calculus in signal processing promises further advancements, making it a valuable tool for researchers and practitioners seeking to analyze complex signals across various domains.

Modelling of laminated glass interlayer by fractional viscoelasticity

Fractional calculus, i.e. the theory of derivatives and integrals of non-integer order, can be efficiently used for the theoretical modelling of viscoelastic materials. Our research is focused on the polyvinyl butyral which is used as an interlayer for the laminated glass. Polyvinyl butyral can be classified as a viscoelastic material and the introduction of the fractional viscoelasticity seems to be appropriate tool for its description. This paper briefly introduces the springpot element and its connection into more complex theoretical models. We mainly consider the generalized Maxwell model in its standard and fractional form and show their application by fitting the data obtained by experimental analysis.

Modeling and Control of IPMC-Based Artificial Eukaryotic Flagellum Swimming Robot: Distributed Actuation

Ionic polymer-metal composites (IPMCs) are electrically driven materials that undergo bending deformations in the presence of relatively low external voltages, exhibiting a great potential as actuators in applications in soft robotics, microrobotics, and bioengineering, among others. This paper presents an artificial eukaryotic flagellum (AEF) swimming robot made up of IPMC segments for the study of planar wave generation for robot propulsion by single and distributed actuation, i.e., considering the first flagellum link as an actuator or all of them, respectively. The robot comprises three independent and electrically isolated actuators, manufactured over the same 10 mm long IPMC sheet. For control purposes, a dynamic model of the robot is firstly obtained through its frequency response, acquired by experimentally measuring the flagellum tip deflection thanks to an optical laser meter. In particular, two structures are considered for such a model, consisting of a non-integer order integrator in series with a resonant system of both non-integer and integer order. Secondly, the identified models are analyzed and it is concluded that the tip displacement of each actuator or any IPMC point is characterized by the same dynamics, which remains unchanged through the link with mere variations of the gain for low-frequency applications. Based on these results, a controller robust to gain variations is tuned to control link deflection regardless of link length and enabling the implementation of a distributed actuation with the same controller design. Finally, the deflection of each link is analyzed to determine whether an AEF swimming robot based on IPMC is capable of generating a planar wave motion by distributed actuation.

A different approach for behavior of fractional plant virus model

In the last few decades many authors pointed out that derivatives and integrals of non-integer order are very suitable for the description of properties of various real problems. It has been shown that fractional-order models are more adequate than previously used integer-order models. In this work, we aim to investigate of different features of the plant virus model with its fractional order equivalent. We present an application for reproduction number for these kind of epidemic models with next generation matrix method. Also, existence and uniqueness of solutions have been showed for this fractional order system. Finally we present some figures according to the given numerical scheme.

Using the numerical solution for partial fractional differential equation by ADI numerical method to cryptography in Hill matrix system

Calculus is the subject of evaluating derivatives and integrals of non-integer orders of a given function, fractional differential equations is the subject of studying the solution of differential equations of fractional order is considered in this paper, which contain initial conditions. The general form of a fractional differentia equation is supported by: y (s) = k(c, u), y (st k ) (c 0) = u. Where k = 1, 2, … , n + 1, n < q < n + 1, and n is an integer number. The solution of fractional differential equations has so many difficulties in their analytic solution, therefore numerical methods may be in most cases be the suitable method of solution. This paper introduces and study a numerical solution by ADI methods for solving fractional differential equations. We used the matrix obtained from this solution as a key matrix for cryptography the plain text and transform it to cipher text by using the method of Hill matrix system the quantity demand and investment in research and development, while the other model focuses on a more realistic relationship between the quantity demand and the price.

Integral Equations of Non-Integer Orders and Discrete Maps with Memory

In this paper, we use integral equations of non-integer orders to derive discrete maps with memory. Note that discrete maps with memory were not previously derived from fractional integral equations of non-integer orders. Such a derivation of discrete maps with memory is proposed for the first time in this work. In this paper, we derived discrete maps with nonlocality in time and memory from exact solutions of fractional integral equations with the Riemann–Liouville and Hadamard type fractional integrals of non-integer orders and periodic sequence of kicks that are described by Dirac delta-functions. The suggested discrete maps with nonlocality in time are derived from these fractional integral equations without any approximation and can be considered as exact discrete analogs of these equations. The discrete maps with memory, which are derived from integral equations with the Hadamard type fractional integrals, do not depend on the period of kicks.

A Survey of Fractional Order Calculus Applications of Multiple-Input, Multiple-Output (MIMO) Process Control

Multiple-input multiple-output (MIMO) systems are usually present in process systems engineering. Due to the interaction among the variables and loops in the MIMO system, designing efficient control systems for both servo and regulatory scenarios remains a challenging task. The literature reports the use of several techniques mainly based on classical approaches, such as the proportional-integral-derivative (PID) controller, for single-input single-output (SISO) systems control. Furthermore, control system design approaches based on derivatives and integrals of non-integer order, also known as fractional control or fractional order (FO) control, are frequently used for SISO systems control. A natural consequence, already reported in the literature, is the application of these techniques to MIMO systems to address some inherent issues. Therefore, this work discusses the state-of-the-art of fractional control applied to MIMO systems. It outlines different types of applications, fractional controllers, controller tuning rules, experimental validation, software, and appropriate loop decoupling techniques, leading to literature gaps and research opportunities. The span of publications explored in this survey ranged from the years 1997 to 2019.

Fractional and integer derivatives with continuously distributed lag

We propose new type of derivatives and integrals of non-integer orders. The fractional derivatives and integrals with continuously distributed lag (time delay) are defined as a composition of the translation operator with lag, which is continuously distributed with a weighting function, and the fractional derivatives of non-integer order such as the Riemann–Liouville, Liouville, Caputo, and other fractional derivatives. The weighting function satisfies the condition of non-negativity and the normalization condition. These properties of the weighting function allow us to interpret it as the density of a probability distribution of the lag. This gives a possibility to apply different probability distributions, which are defined on the positive semiaxis, to describe continuously distributions of lags. The properties of the suggested operators with non-integer orders and distributed lag are described. The solution of equations that contain the fractional derivatives with gamma distributed lag and it special cases such as exponential and Erlang distributions of lag are obtained. We use the Laplace method and the method that is based of representation by equations with fractional derivatives without lag. The fractional derivatives and integrals with continuously distributed scaling (dilation) are considered on the basis of the Kober and Erdelyi–Kober fractional operators as a generalization of the approach proposed for continuously distributed lag. The suggested generalization of the fractional integrals and derivatives, which includes the translation and dilation operators, can be used to describe the continuously distributed lag and scale effects in physical, economic, and other processes with memory.

Generalized Memory: Fractional Calculus Approach

The memory means an existence of output (response, endogenous variable) at the present time that depends on the history of the change of the input (impact, exogenous variable) on a finite (or infinite) time interval. The memory can be described by the function that is called the memory function, which is a kernel of the integro-differential operator. The main purpose of the paper is to answer the question of the possibility of using the fractional calculus, when the memory function does not have a power-law form. Using the generalized Taylor series in the Trujillo-Rivero-Bonilla (TRB) form for the memory function, we represent the integro-differential equations with memory functions by fractional integral and differential equations with derivatives and integrals of non-integer orders. This allows us to describe general economic dynamics with memory by the methods of fractional calculus. We prove that equation of the generalized accelerator with the TRB memory function can be represented by as a composition of actions of the accelerator with simplest power-law memory and the multi-parametric power-law multiplier. As an example of application of the suggested approach, we consider a generalization of the Harrod-Domar growth model with continuous time.

Macroeconomic models with long dynamic memory: Fractional calculus approach

This article discusses macroeconomic models, which take into account effects of power-law fading memory. The power-law long memory is described by using the mathematical tool of fractional calculus that includes the fractional derivatives and integrals of non-integer orders. We obtain solutions of the fractional differential equations of these macroeconomic models. Examples of dependence of macroeconomic dynamics on the memory effects are suggested. Asymptotic behaviors of the solutions, which characterize the rate of technological growth with memory, are described. We formulate principles of economic dynamics with one-parametric and multi-parametric memory. It has been shown that the effects of fading long memory can change the economic growth rate and change dominant parameters, which determine growth rates.

Can Cybernetics and Fractional Calculus Be Partners?: Searching for New Ways to Solve Complex Problems

Cybernetics addresses complex problems in space and time, but, surprisingly, it does not often employ fractional calculus. Here, we trace the history of cybernetics and fractional calculus to resolve this apparent paradox. Although the term cybernetics was first used in 1834 by the French physicist and mathematician Andre-Marie Ampere [2] to describe the science of government, its modern definition-control and communication in animals and machines-is traced to Norbert Wiener [3] (Figure 1). Since then, others have coined the words cyberspace and cyborg, and cybernetics is playing an increasing role in the study of human-machine systems and the analysis of the Internet of Things. Often defined as noninteger order integration and differentiation, the idea of fractional calculus first appeared in 1695 in a letter from Guillaume de L'Hopital to Gottfried Wilhelm Leibniz, but more recent interest can be traced to the work of Keith B. Oldham and Jerome Spanier [4] (Figure 2). Today, the techniques of fractional calculus are widely applied in many fields of science and engineering.

Criterion of Existence of Power-Law Memory for Economic Processes.

In this paper, we propose criteria for the existence of memory of power-law type (PLT) memory in economic processes. We give the criterion of existence of power-law long-range dependence in time by using the analogy with the concept of the long-range alpha-interaction. We also suggest the criterion of existence of PLT memory for frequency domain by using the concept of non-integer dimensions. For an economic process, for which it is known that an endogenous variable depends on an exogenous variable, the proposed criteria make it possible to identify the presence of the PLT memory. The suggested criteria are illustrated in various examples. The use of the proposed criteria allows apply the fractional calculus to construct dynamic models of economic processes. These criteria can be also used to identify the linear integro-differential operators that can be considered as fractional derivatives and integrals of non-integer orders.

Accelerator and Multiplier for Macroeconomic Processes with Memory

The paper proposes an approach to the description of macroeconomic phenomena, which takes into account the effects of fading memory. The standard notions of the accelerator and the multiplier are very limited, since the memory of economic agents is neglected. We consider the methods to describe the economic processes with memory, which is characterized by the fading of a power-law type. Using the mathematical tools of derivatives and integrals of non-integer orders, we suggest a generalization of the concept of the accelerator and multiplier. We derive the equations of the accelerator with memory and the multiplier with memory, which take into account the changes of endogenous and exogenous variables on a finite time interval. We prove the duality of the concepts of the multiplier with memory and the accelerator with memory. The proposed generalization includes the standard concepts of the accelerator and the multiplier as special cases. In addition these generalizations provide a range of intermediate characteristics to take into account the memory effects in macroeconomic models.

Electric field in media with power-law spatial dispersion

In this paper, we consider electric fields in media with power-law spatial dispersion (PLSD). Spatial dispersion means that the absolute permittivity of the media depends on the wave vector. Power-law type of this dispersion is described by derivatives and integrals of non-integer orders. We consider electric fields of point charge and dipole in media with PLSD, infinite charged wire, uniformly charged disk, capacitance of spherical capacitor and multipole expansion for PLSD-media.

A NOTE ON LINEAR IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS

Abstract. This paper deals with linear impulsive fractional di er-ential equations involving the Caputo derivative with non-integerorder q. We provide exact solutions of linear impulsive fractionaldi erential equations with constant coecient by mean of the Mittag-Leer functions. Then we apply the exact solutions to improveimpulsive integral inequalities with singularity. 1. IntroductionFractional calculus is a mathematical branch investigating the prop-erties of derivatives and integrals of non-integer orders (called fractionalderivatives and integrals). In particular, this discipline involves the no-tion and methods of solving of fractional di erential equations, i.e., dif-ferential equations involving fractional derivatives of the unknown func-tion. Fractional di erential equations are a generalization of di eren-tial equations through the application of fractional calculus. Recently,fractional di erential equations play a signi cant role in modeling theanomalous dynamics of various processes related to complex systems inmost areas of science and engineering.The exponential function e

Variational principle of stationary action for fractional nonlocal media and fields

Derivatives and integrals of non-integer orders have a wide application to describe complex properties of physical systems and media including nonlocality of power-law type and long-term memory. We suggest an extension of the standard variational principle for fractional nonlocal media with power-law type nonlocality that is described by the Riesz-type derivatives of non-integer orders. As examples of application of the suggested variational principle, we consider an N-dimensional model of waves in anisotropic fractional nonlocal media, and a one-dimensional continuum (string) with power-law spatial dispersion. The main advantage of the suggested fractional variational principle is that it is connected with microstructural lattice approach and the lattice fractional calculus, which is recently proposed.

Fractional thermal diffusion and the heat equation

AbstractFractional calculus is the branch of mathematical analysis that deals with operators interpreted as derivatives and integrals of non-integer order. This mathematical representation is used in the description of non-local behaviors and anomalous complex processes. Fourier’s lawfor the conduction of heat exhibit anomalous behaviors when the order of the derivative is considered as 0 &lt; β,ϒ ≤ 1 for the space-time domain respectively. In this paper we proposed an alternative representation of the fractional Fourier’s law equation, three cases are presented; with fractional spatial derivative, fractional temporal derivative and fractional space-time derivative (both derivatives in simultaneous form). In this analysis we introduce fractional dimensional parameters σ