Integer-order Ones Research Articles

Frequency Loop-Shaping and IMC-Based Integer-Order Robust PID Controller Design for Fractional-Order Processes with Time-Delay

It is well accepted that fractional-order calculus yields a more accurate description of complex physical processes compared to integer-order one. So, fractional-order modelling is getting a good attention in the process industry. On the other hand, to meet design goals accurately, control designers prefer fractional-order controllers over integer-order ones due to the availability of more parameters. However, the design of such controllers are complex. More often, the design faces the challenge in real-time implementations, and the problem is circumvented by higher order approximations of fractional-order operators. In the presented work, three methods for integer-order Proportional–Integral–Derivative (PID) controller design are proposed based on Frequency Loop-Shaping (FLS) and Internal Model Control (IMC). To prove the efficacy of the presented control law, simulations are conducted and included along with a performance measure based on ISE (Integral of Square Error), ITAE (Integral of Time Absolute Error) and IAE (Integral of Absolute Error).

Numerical Investigation of Fractional-Order Fornberg–Whitham Equations in the Framework of Aboodh Transformation

In this investigation, the fractional Fornberg–Whitham equation (FFWE) is solved and analyzed via the variational iteration method (VIM) and Adomian decomposition method (ADM) with the help of the Aboodh transformation (AT). The FFWE is an important model for describing several nonlinear wave propagations in various fields of science and plasma physics. The AT provides a powerful tool for transforming fractional-order differential equations (DEs) into integer-order ones, making them more amenable to analytical solutions. Accordingly, the main objective of this investigation is to demonstrate the effectiveness and accuracy of ADM and VIM in deriving some approximations for the FFWE. Furthermore, we highlight the advantages and potential applications of these methods in solving other fractional-order nonlinear problems in several scientific fields, especially in plasma physics and some engineering problems.

Fractional-order model and experimental verification of granules-beam coupled vibration

Granular media generally exhibit fluid phase, solid phase, or two coexisting phases, and the mechanism of the phase transition is not clear enough. Therefore, unlike the airflow-structure or fluid–structure coupled vibrations, the granules-structure coupled vibrations have not yet been theoretically modeled, and the interest in such coupled vibrations, which has just started in recent years, is mainly focused on experiments and simulations. In order to develop a theoretical model of the coupled vibrations of granular media and beam structures, in this paper, a novel concept called the inertia-viscous effect is introduced in terms of terminology and mathematical expression, and surprisingly, its constitutive relation can be summarized in a unified framework based on fractional calculus, together with Hooke spring, Newtonian dashpot, and Abel sticky pot. Furthermore, the existence of this effect in the granules-beam coupled vibrations is experimentally confirmed, and more importantly, the effect is more suitable for modeling by fractional-order derivatives rather than the conventional integer-order ones due to the frequency-dependent characteristics of the effect. Based on this, the theoretical model of the granules-beam coupled vibrations and the associated parameters identification procedure are successfully developed, in which a fractional-order model is proposed to describe the additional dynamic load (ADL) of the granular media on the beam. The numerical and experimental results suggest that the system response can be accurately predicted by the developed theoretical model without the computational cost as in the discrete element methods, which is the main contribution of this work. Finally, the effectiveness and robustness of the developed theoretical model are further verified by the results of numerous parametric experiments, and importantly, by combining these results with the analytical results, the mechanism of ADL is revealed, thus providing insights for modeling coupled vibrations between other engineering structures and granular media. From another perspective, this work can also be seen as a novel application of fractional calculus in the dynamics of mechanical systems.

Theoretical analysis and experimental verification of fractional-order RC cobweb circuit network

Recent research has shown that ideal capacitors and inductors do not physically exist, and that the dynamics of real devices can be accurately described by fractional-order (FO) mathematical models. This paper investigates a class of 2×n order RC cobweb FO circuit network with central node. Based on the Kirchhoff’s laws, the impedance magnitude and phase between two points of the network are derived using difference equations and matrix transformations. Three impedance expressions are deduced, and their correctness is verified numerically and by simulations. The influence of various parameters of the electrical network, namely the resistance, capacitance, number of circuit units, frequency and fractional order, on the impedance is studied. Additionally, for the first time, physical experiments are presented to compare the effectiveness of FO and integer-order circuit networks for describing the impedance of actual physical circuits. These experiments confirm that FO circuit network models perform better than the integer-order ones for representing the characteristics of the impedance magnitude and phase.

Fractional Sliding Mode Harmonic Control of an Active Power Filter

A sliding mode control (SMC) strategy with a fractional-order scheme is developed to provide an effective way to compensate for the harmonic current and strengthen the performance of an active power filter (APF). The formulation of a fractional-order sliding mode controller (FOSMC) is incorporated in the current control of an APF control system, obtaining an extra degree of freedom over integer-order ones to improve the dynamic property. The simulation results of the APF utilizing the designed strategy confirm the satisfactory performance and smaller total harmonic distortion (THD) by comparative study.

Stability analysis of quaternion-valued BAM neural networks fractional-order model with impulses and proportional delays

In this paper, a class of quaternion-valued BAM neural networks (QVBAMNNs) fractional-order model with impulses and proportional delays is proposed in discrete-time case. The QVBAMNNs fractional-order model is investigated directly rather than through real decomposition method or the plural one. By employing homeomorphic mapping theorem, Lyapunov stability theory and inequality technology, several criteria for the discrete-time QVBAMNNs fractional-order model are derived to guarantee the existence, uniqueness and global exponential stability (GES) of equilibrium point. The novelty of these results comes from the generality with the fractional-order systems and the integer-order ones. Finally, two examples are given to demonstrate the effectiveness and availability of the derived criteria.

Optimal Control and Spectral Collocation Method for Solving Smoking Models

In this manuscript, we solve the ordinary model of nonlinear smoking mathematically by using the second kind of shifted Chebyshev polynomials. The stability of the equilibrium point is calculated. The schematic of the model illustrates our proposition. We discuss the optimal control of this model, and formularize the optimal control smoking work through the necessary optimality cases. A numerical technique for the simulation of the control problem is adopted. Moreover, a numerical method is presented, and its stability analysis discussed. Numerical simulation then demonstrates our idea. Optimal control for the model is further discussed by clarifying the optimal control through drawing before and after control. Fractional request differential equations (FDEs) are usually used to display frameworks that have memory and exist in a few thermoelasticity models and organic standards. FDEs show the realistic biphasic decline of infection of diseases but at a slower rate. FDEs are more suitable than integer order ones in modeling complex systems, such as biological systems.

Modeling the Virus Infection at the Population Level.

As pointed out by many researchers in the last few decades, differential equations with fractional (non-integer) order differential operators, in comparison with classical integer order ones, have apparent advantages in modeling. A Caputo fractional order system of ordinary differential equations is introduced to model the virus infection at the population level in this chapter. As well known, there are two main methods to study the dynamics of a model: qualitative analysis and numerical modeling. Here the qualitative analysis, including uniqueness, invariant set, and stability, is first presented with intuitive derivation. Then the famous genetic algorithm is introduced to numerically model the dynamics of virus infection, i.e. to adjust the parameters of the Caputo fractional model such that its solution can properly fit real data and predict future.

Modulating functions based model-free fractional order differentiators using a sliding integration window

This paper aims to design model-free fractional order differentiators to non-asymptotically and robustly estimate both the Riemann–Liouville and Caputo fractional derivatives of an unknown signal from its discrete and noisy observation. To achieve this, new fractional integration by parts formulas are first introduced. Then, by applying the generalized modulating functions method to a constructed model, algebraic integral formulas involving the integer order derivatives of the signal are provided. These formulas can be applied to analytically or numerically calculate the fractional derivatives in noise-free case. In order to apply the proposed formulas in discrete noisy cases, a time-delayed integer order differentiator is developed by applying the modulating functions method using a sliding integration window, such that the fractional derivatives can be estimated using the estimations of the integer order ones. This integer order differentiator can cope with non-zero mean noises. Moreover, a recursive algorithm is provided to reduce the computation time for on-line applications. Finally, numerical simulation results are given to illustrate the efficiency and robustness of the proposed method with an application to fractional order PID controller.

A three-dimensional fractional solution for air contaminants dispersal in the planetary boundary layer

In this study, we investigated some closed-form solutions for solving atmospheric dispersion issues under variable atmospherical hypothesis, in a vertically positioned non-homogeneous planetary boundary-layer. In our context, a nonidentical expansion for the solution of the fractional advection-diffusion equation in a non-integer dimensional-space was examined. In a nutshell, a Sturm-Liouville eigenvalue problem with more reliable information concerning the initial value problem is discussed. The performance of the model was estimated by presenting numerical simulations against experimental data. Under these meteorological conditions, fractional-order models performed far better than those of the classical integer-order ones.

Asymptotic and Finite-Time Cluster Synchronization of Coupled Fractional-Order Neural Networks With Time Delay.

This article is devoted to the cluster synchronization issue of coupled fractional-order neural networks. By introducing the stability theory of fractional-order differential systems and the framework of Filippov regularization, some sufficient conditions are derived for ascertaining the asymptotic and finite-time cluster synchronization of coupled fractional-order neural networks, respectively. In addition, the upper bound of the settling time for finite-time cluster synchronization is estimated. Compared with the existing works, the results herein are applicable for fractional-order systems, which could be regarded as an extension of integer-order ones. A numerical example with different cases is presented to illustrate the validity of theoretical results.

A novel image encryption system merging fractional-order edge detection and generalized chaotic maps

This paper presents a novel lossless image encryption algorithm based on edge detection and generalized chaotic maps for key generation. Generalized chaotic maps, including the fractional-order, the delayed, and the Double-Humped logistic maps, are used to design the pseudo-random number key generator. The generalization parameters add extra degrees of freedom to the system and increase the keyspace achieving more secure keys. Fractional order edge detection filters exhibited better noise robustness than the conventional integer-order ones, rendering the system to be suitable for medical imaging security. The proposed system flexibly integrate different edge detectors, as well as various logistic maps for key generation. The sensitivity of the chaotic maps to all parameters guarantees the encryption system key sensitivity. Security analyses aspects assure the efficiency of the proposed algorithm performance, having high pixel correlation coefficients and flat histograms of cipher images reported. A comparison between the proposed scheme with existing cryptosystems is also presented, regarding histogram uniformity, contrast analysis, Shannon entropy measurements. Compared to the state of the art algorithms, the proposed algorithm has higher statistical and cryptanalytic properties.

Global Synchronization of Coupled Fractional-Order Recurrent Neural Networks.

This paper presents new theoretical results on the global synchronization of coupled fractional-order recurrent neural networks. Under the assumptions that the coupled fractional-order recurrent neural networks are sequentially connected in form of a single spanning tree or multiple spanning trees, two sets of sufficient conditions are derived for ascertaining the global synchronization by using the properties of Mittag-Leffler function and stochastic matrices. Compared with existing works, the results herein are applicable for fractional-order systems, which could be viewed as an extension of integer-order ones. Two numerical examples are presented to illustrate the effectiveness and characteristics of the theoretical results.

Improved Internal Model Control-Proportional-Integral- Derivative Fractional-Order Multiloop Controller Design for Non Integer Order Multivariable Systems

This paper is aimed to propose a multiloop control scheme for fractional order multi-input multi-output (FO-MIMO) systems. It is an extension of the FO-multiloop controller design method developed for integer order multivariable systems to FO-MIMO ones. The interactions among the control loops are considered as disturbances and a two degrees-of-freedom (2DOF) paradigm is used to deal with the process outputs performance and the interactions reduction effect, separately. The proposed controller design method is simple, in relation with the desired closed-loop specifications and a tuning parameter. It presents an interest in controlling complex MIMO systems since fractional order models (FO-models) represent some real processes better than integer order ones and high order systems can be approximated by FO-models. Two examples are considered and compared with other existing methods to evaluate the proposed controller.

Stable Physical Human-Robot Interaction Using Fractional Order Admittance Control.

In the near future, humans and robots are expected to perform collaborative tasks involving physical interaction in various environments, such as homes, hospitals, and factories. Robots are good at precision, strength, and repetition, while humans are better at cognitive tasks. The concept, known as physical human-robot interaction (pHRI), takes advantage of these abilities and is highly beneficial by bringing speed, flexibility, and ergonomics to the execution of complex tasks. Current research in pHRI focuses on designing controllers and developing new methods which offer a better tradeoff between robust stability and high interaction performance. In this paper, we propose a new controller, fractional order admittance controller, for pHRI systems. The stability and transparency analyses of the new control system are performed computationally with human-in-the-loop. Impedance matching is proposed to map fractional order control parameters to integer order ones, and then the stability robustness of the system is studied analytically. Furthermore, the interaction performance is investigated experimentally through two human subject studies involving continuous contact with linear and nonlinear viscoelastic environments. The results indicate that the fractional order admittance controller can be made more robust and transparent than the integer order admittance controller and the use of fractional order term can reduce the human effort during tasks involving contact interactions with environment.

Transient and Steady-State Response of a Fractional-Order Dynamic PV Model Under Different Loads

In this paper, a fractional-order dynamic model of the photovoltaic (PV) solar module is introduced. Dynamic modeling of PV solar modules is useful when used in switching circuits and grid-connected situations. The dynamic elements of the proposed model are a fractional-order inductor and capacitor of two independent orders which allow for two extra degrees of freedom over the conventional dynamic model. The step response and transfer function of the load current are investigated for different orders under resistive and supercapacitor loading conditions. Closed-form expressions for the time response of the load current at equal orders of capacitor and inductor are derived. Stability analysis of the load current transfer function is carried out for different orders and loading conditions. The regions for pure real and pure imaginary input admittance scenarios are calculated numerically for both resistive and supercapacitor load cases. It is found that the order of the inductor has a dominant effect on the responses. As a proof of concept, the model is fitted to experimental data to show its flexibility in regenerating the actual response. The fitted fractional-order model response is compared to optimized integer-order ones from literature showing noticeable improvement.

Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection

Fractional order derivative is nonlocal which exhibits a long time memory behavior. With advantage of these, fractional order dynamic system models are more accurate than integer order ones in understanding the dynamic behavior of bioprocesses such as HIV infection. In this paper, we systematically study the existence of positive solutions of an abstract fractional semipositone differential system involving integral boundary conditions arising from the study of HIV infection models. By using the fixed point theorem in cone, some new results are established and an example is given to demonstrate the application of our main results.

The difference between a class of discrete fractional and integer order boundary value problems

In this paper, we point out the differences between a class of fractional difference equations and the integer-order ones. We show that under the same boundary conditions, the problem of the fractional order is nonresonant, while the integer-order one is resonant. Then we analyse the discrete fractional boundary value problem in detail. Then the uniqueness and multiplicity of the solutions for the discrete fractional boundary value problem are obtained by two new tools established in 2012, respectively.

Superiority of fractional order controllers in limit cycle suppression

In this paper, we report an interesting advantage of fractional order controllers over integer order ones. It is found that the three-parameter-fractional-controllers such as PIa and [PI]a outperform integer PID in limit cycle suppression when designed to meet Wang et al. specifications. An elementary servo plant having separable static backlash non-linearity in its model is considered for the tuning purpose. For this plant, designed PIa and [PI]a produce significantly reduced limit cycles in the closed loop non-linear simulation response as compared to PID. The results are further verified with describing function method. Also, the results exhibiting such a ‘fractional superiority’ for a real time application of precision modular servo experimental set-up are included for the better illustration.

Multistage Adomian Decomposition Method for Solving NLP Problems Over a Nonlinear Fractional Dynamical System

This paper deals with implementation of the multistage Adomian decomposition method (MADM) to solve a class of nonlinear programming (NLP) problems, which are reformulated with a nonlinear system of fractional differential equations. The multistage strategy is used to investigate the relation between an equilibrium point of the fractional order dynamical system and an optimal solution of the NLP problem. The preference of the method lies in the fact that the multistage strategy gives this relation in an arbitrary longtime interval, while the Adomian decomposition method (ADM) gives the optimal solution just only in the neighborhood of the initial time. The numerical results taken by the fractional order MADM show that these results are compatible with the solution of NLP problem rather than the ADM. Furthermore, in some cases the fractional order MADM can perform more rapid convergency to the optimal solution of optimization problem than the integer order ones.