Integer Order Research Articles

Global Stability of Fractional Nonlinear Differential Systems With State-Dependent Delayed Impulses.

The fractional-order (FO) nonlinear differential system with state-dependent (SD) delayed impulses (DI) is considered in this brief. The considered impulses are related to the delayed state of the system and the delays are SD. A novel lemma for the monotonicity of the solution of Caputo's FO derivative equation is given. By means of linear matrix inequality (LMI) and several comparative arguments, criteria of uniform stability, uniform asymptotical stability, and Mittag-Leffler stability are obtained. Compared with other works on integer-order (IO) impulsive delayed systems with SD delays or fixed delays, how to impose constraints on parameters and impulses is explored, without imposing the boundedness on the state delays. Two examples are implemented to examine the practicality and sharpness of our theoretical analysis.

Dynamic Modeling of a Fluidic Soft Actuator: First Results Within a Fractional Approach

Dynamic modeling of soft actuators is a challenging field of research mainly due to the nonlinear properties of the materials they are made of, as well as the infinite degrees of freedom that they exhibit. In this paper, the defection of an eccentric soft fluidic actuator is characterized in terms of the dynamic displacement using both integer and fractional order (FO) models. The actuator, which is cylindrical in shape, is made of a silicone polymer called polydimethylsiloxane (PDMS) from a water-soluble mold and a rod that, when placed eccentrically, allows to create the internal chamber of the actuator. Experimental results are presented to show that the best fitting is obtained with FO models. In fact, the dynamics of the actuator can be fully described with a two-parameter model, namely a fractional integrator of order of about 1.2 with a gain.

Existence, uniqueness and numerical solution of stochastic fractional differential equations with integer and non-integer orders

<abstract><p>The parametrized approach is extended in this study to find solutions to differential equations with fractal, fractional, fractal-fractional, and piecewise derivatives with the inclusion of a stochastic component. The existence and uniqueness of the solution to the stochastic Atangana-Baleanu fractional differential equation are established using Caratheodory's existence theorem. For the solution of differential equations using piecewise differential operators, which take into account combining deterministic and stochastic processes utilizing certain significant mathematical tools such as fractal and fractal-fractional derivatives, the applicability of the parametrized technique is being examined. We discuss the crossover behaviors of the model obtained by including these operators and we present some illustrative examples for some problems with piecewise differential operators.</p></abstract>

Li-ion cell impedance identification in the time domain as an alternative to identification in the frequency domain

Electrochemical Impedance Spectroscopy (EIS) is a widely used tool for selecting a pertinent Equivalent Circuit Model (ECM) of Li-ion cells which is characterized by non integer order operators. The main drawback of EIS is the long time required to scan a whole spectrum down to low frequencies. Thus, chronopotentiometry (CP) is an alternative method which consists in identifying impedance parameters using an excitation current sequence lasting a few seconds and the corresponding induced voltage variations. An impedance fractional model needs to be synthesized in order to allow simulation also in the time domain. The pertinence of CP is demonstrated using experimental results obtained with a Samsung 3.4 Ah Li-ion cell. Due to its inability to take into account the inductive behavior, fractional model used in the time domain exhibit limitations. Nevertherless, parameters identified in the time domain present similar results to the ones identified in the frequency domain under the condition of a restricted frequency range.

A compact fractional-order model for hypnosis in general anesthesia

The research field of clinical practice has witnessed a notable increase in the integration of information technology and control systems engineering tools and the paradigm of drug dosing management for general anesthesia is no exception. Progressing from adequate to optimal drug dosing requires suitable models for closed loop control algorithms. High order, complex parameterized models for hypnosis are available but as one cannot measure real drug concentrations, accurate modelling is not possible. Moreover, ethical limitations impose serious restrictions as to the type of excitatory signals acceptable to patient effect response evaluations. This paper proposes an innovative approach to determine fractional order models to compactly represent the dynamics inherent to the hypnosis response. A simplifying assumption is being made: instead of multi-compartmental models, a single transfer model is proposed consisting of a fractional-order dynamic that directly connects Propofol, the administered drug, and the Bispectral (BIS) index, the measure of hypnosis. The proposed model is validated against clinical data and compared to integer order models to prove its suitability. Results suggest the model may be well used with control algorithms for computerized drug dosing management.

Real-time detection of small objects in transverse electric polarization: Evaluations on synthetic and experimental datasets

<p>It is well-known that if one applies Kirchhoff migration (KM) to identify small objects when their values of magnetic permeabilities differ from those of the background (or transverse electric polarization), their location and outline shape cannot be satisfactorily retrieved because rings of large magnitudes centered at the location of objects appear in the imaging results. Fortunately, it is possible to recognize the existence and approximated location of objects in the 2D Fresnel dataset through the traditional KM, but no theoretical explanation for this phenomenon has been verified. Here we show that the imaging function of KM when tested on the Fresnel dataset can be expressed as squared zero-order and first-order Bessel functions and as an infinite series of Bessel functions of integer order greater than two. We also explain why the existence and approximate location of objects can be identified. This theoretical result is supported by numerical simulations on synthetic and experimental data.</p>

A Deep Neural Network Process for the Influenza Disease Nonlinear Mathematical Model

The current investigations provide a stochastic deep neural network optimization procedure for the numerical solutions of the fractional order influenza disease system, which provide more realistic solutions of the model as compared to integer order derivatives. The mathematical influenza model is categorized into population of birds (susceptible, infected) and humans (susceptible, infected with avian and mutant strains,

A mathematical fractional model of waves on Shallow water surfaces: The Korteweg-de Vries equation

<abstract><p>The homotopy perturbation transform method was examined in the present research to address the nonlinear time-fractional Korteweg-de Vries equations using a nonsingular kernel fractional derivative that Caputo-Fabrizio recently developed. We devoted our research to the nonlinear time-fractional Korteweg-de Vries equation and certain associated phenomena because of some physical applications of this equation. The results are significant and necessary for illuminating a range of physical processes. This paper considered an innovative method and fractional operator in this context to obtain satisfactory approximations to the provided issues. To solve nonlinear time-fractional Korteweg-de Vries equations, we first considered the Yang transform of the Caputo-Fabrizio fractional derivative. In order to confirm the applicability and efficacy of the provided method, we took into consideration two cases of the nonlinear time-fractional Korteweg-de Vries equation. He's polynomials were useful in order to manage nonlinear terms. In this method, the outcome was calculated as a convergent series, and it was demonstrated that the homotopy perturbation transform method solutions converge to the exact solutions. The main benefit of the suggested method was that it offered solutions with a high degree of precision while requiring minimal computation. Graphs were also used to illustrate the series solution for a certain non-integer orders. Finally, a comparison of both examples outcomes were examined using diagrams and numerical data. These graphs showed how the approximated solution's graph and the precise solution's graph eventually converged as the non-integer order gets closer to integer order. When $ \varsigma = 1 $, several numerical comparisons were conducted with the exact solutions. The numerical simulation was offered to illustrate the efficiency and reliability of the proposed approach. In addition, the behavior of the provided solutions was explained using a number of fractional orders. The theoretical analysis matched with the findings obtained using the current technique, and the suggested technique can be extended to tackle many higher-order nonlinear dynamics problems.</p></abstract>

Application of Quasi-Fractional Order PI Controller to IPDT Plant Control

The paper focuses on the design of a fractional order PI controller (FO-PI) and the verification of the properties of the control loop with an integral plus dead time (IPDT) system. The fractional order integrator in the controller is approximated by the Oustaloup method. The multiple dominant pole method (MDPM) is used to tune the controller parameters. The input parameters for controller tuning are obtained by optimization using the algorithm presented in the paper. The properties of the control loop with the FO-PI controller tuned in this way have been verified by simulation and compared with the properties of the control loop using an integer order PI controller.

Novel closed-loop controllers for fractional linear quadratic time-varying systems

A novel method for finding the closed-loop optimal controllers of the finite-time fractional linear quadratic regulator systems is introduced. The necessary optimality conditions for the fractional optimal control problems are obtained. The matrix fractional differential Riccati equation is presented. A new method for solving the fractional differential Riccati equation is given for the first time which can also be applied to the integer order Riccati equation. Several illustrative examples (including a high order system) are presented to show the applicability and capabilities of the method.

Automatic Initialization and Model Selection for Li-ion Battery Impedance Identification in the Frequency Domain

Electrochemical Impedance Spectroscopy (EIS) is a useful tool for selecting a pertinent Equivalent Circuit Model (ECM) of a Li-ion battery. Impedance model is designed to describe low, middle and high frequency electrochemical processes involved. When considering low frequency restricted in the Warburg zone, diffusion impedance is modeled thanks to a Constant Phase Element (CPE) which behaves as a fractional integrator of order n close to 0.5. Phenomena observed in middle frequency are described using specific circuits called Zarc which consist in connecting a CPE in parallel with a resistor. Therefore, the global impedance model is characterized by non integer order operators and parameters can be estimated by a Complex Non linear Least Squares (CNLS) algorithm which requires a proper initialization in order to guarantee the convergence to a global optimum. The paper presents a method to analyze EIS data measurements in order to select automatically the number of middle frequency Zarc circuits required (one or two) and to initialize properly the CNLS algorithm. The method is validated using experimental open source EIS data.

Another formulation of the fractional nonlinear Schrödinger equation

In this letter, we pave the way to establish the formulation of a non-trivial fractional Nonlinear Schrödinger (NLS) equation, which is different from the formulations known so far that consist in directly replacing the integer orders of the derivatives by non-integer ones. Thereafter, we set up some formulations, adapted to some particular physical cases, namely, the cases where the nonlinearity is stronger than the dispersion, in addition to one for which the dispersion strongly dominates the nonlinearity and also the case where the system displays a nonlinearity which is compensated with the dispersion. These formulations highlight the fact that the transition from a formal classical analysis to a fractional one could lead changes in the initial model of a given system. The research for solutions of the equations resulting from this study will undoubtedly reveal new phenomena in the different physical, biological and other systems described by the NLS equation.

Research on the higher-order logic formalization of fractance element

Fractance element reflects the fractional order behavior of circuits, which can show the characteristics of the actual circuits. Higher-order logic theorem proving is based on the rigorous and correct mathematical theories. It becomes more and more important in the verifications of high-reliability systems. Fractance element is formalized using the proof of higher-order logic theorem in this paper. Firstly, the formalized model of fractional calculus which is based on Caputo definition is established in higher-order logic theorem proof tool. Then some properties of fractional calculus are proved, including the zero order property, the fractional differential of a constant and the consistency of fractional calculus and integer order calculus. Finally, fractance element and fractional differential circuit constituted by fractance element are formally analyzed. These formalizations demonstrate the effectiveness of the formal method in the analysis of fractance element.

Enhanced general conformable controller based on Lyapunov technique for DC-DC static converters: Application to a solar system

<abstract> <p>To synthesize the proper control signal while guaranteeing the necessary performance indices (speed, resilience, accuracy, etc.), mathematical models were frequently used to represent physical systems. These descriptions were utilized for control, monitoring, and detection in these kinds of systems. Quality and performance of the process may suffer if the model is inaccurate or incomplete. As a result, conformable systems (CS) may be used to make these mathematical models more near to the real world. However, non-power-electronics experts who need to model and simulate complex systems may find the task of modeling static converters to be rather challenging. Researchers have just recently outlined the properties of the general conformable systems (GCS). This innovative approach built upon the principle of the classical integer order systems, employing the same mathematical foundations for its derivation. With the introduction of this novel description of systems, a fresh array of differential equations emerged, specifically tailored for the realm of direct current to direct current (DC-DC) static converters. GCS has been proved to be more flexible and profitable than the traditional integer-order one for representing DC-DC static converters. This advancement paved the way for more effective control techniques based on the Lyapunov method, with practical applications in photovoltaic (PV) systems and beyond.</p> </abstract>

Gradient-enhanced fractional physics-informed neural networks for solving forward and inverse problems of the multiterm time-fractional Burger-type equation

<p>In this paper, we introduced the gradient-enhanced fractional physics-informed neural networks (gfPINNs) for solving the forward and inverse problems of the multiterm time-fractional Burger-type equation. The gfPINNs leverage gradient information derived from the residual of the fractional partial differential equation and embed the gradient into the loss function. Since the standard chain rule in integer calculus is invalid in fractional calculus, the automatic differentiation of neural networks does not apply to fractional operators. The automatic differentiation for the integer order operators and the finite difference discretization for the fractional operators were used to construct the residual in the loss function. The numerical results demonstrate the effectiveness of gfPINNs in solving the multiterm time-fractional Burger-type equation. By comparing the experimental results of fractional physics-informed neural networks (fPINNs) and gfPINNs, it can be seen that the training performance of gfPINNs is better than fPINNs.</p>

Will Fractional Order Model Based MPC Save Control Energy?

In real-world situations, many complex processes exhibit fractional order dynamic characteristics. In this context, the common integer order modeling methods may not appropriately capture the physics of these processes, leading to a less satisfactory controller design. This paper conducts a comparative analysis between the First order plus time delay (FOPTD) model and the Time Delay with Single Fractional Pole (TDWFP) model. These two modeling methods are employed to characterize a one-dimensional heat propagation process and two model predictive controllers are developed based on the models. The results demonstrate a better fitting performance in both time and frequency response with the TDWFP model. Furthermore, the TDWFP model-based Model Predictive Control exhibits reduced sensitivity to the prediction horizon and requires low control input energy. This indicates the potential of using fractional order model in MPC for improved control energy efficiency in practical applications.

Analysis of fractal-fractional Alzheimer's disease mathematical model in sense of Caputo derivative

<abstract> <p>Alzheimer's disease stands as one of the most widespread neurodegenerative conditions associated with aging, giving rise to dementia and posing significant public health challenges. Mathematical models are considered as valuable tools to gain insights into the mechanisms underlying the onset, progression, and potential therapeutic approaches for AD. In this paper, we introduce a mathematical model for AD that employs the fractal fractional operator in the Caputo sense to characterize the temporal dynamics of key cell populations. This model encompasses essential elements, including amyloid-<italic>β</italic> ($\mathbb{ A_\beta }$), neurons, astroglia and microglia. Using the fractal fractional operator, we have established the existence and uniqueness of solutions for the model under consideration, employing Leray-Schaefer's theorem and the Banach fixed-point methods. Utilizing functional techniques, we have analyzed the proposed model stability under the Ulam-Hyers condition. The suggested model has been numerically simulated by using a fractional Adams-Bashforth approach, which involves a two-step Lagrange polynomial. For numerical simulations, different ranges of fractional order values and fractal dimensions are considered. This new fractal fractional operator in the form of the Caputo derivative was determined to yield better results than an ordinary integer order. Various outcomes are shown graphically by for different fractal dimensions and arbitrary orders.</p> </abstract>

On the application of subspace migration from scattering matrix with constant-valued diagonal elements in microwave imaging

We apply subspace migration (SM) for fast identification of a small object in microwave imaging. Most research in this area is performed under the assumption that the diagonal elements of the scattering matrix can be easily measured if the transmitter and the receiver are in the same location. Unfortunately, it is very difficult to measure such elements in most real-world microwave imaging. To address this issue, several studies have been conducted with the unknown diagonal elements set to zero. In this paper, we generalize the imaging problem by using SM to set the diagonal elements of the scattering matrix to a constant. To demonstrate the applicability of SM and its dependence on the constant, we show that the imaging function of SM can be represented by an infinite series of Bessel functions of integer order, antenna number and arrangement, and the applied constant. This result allows us to discover additional properties, such as the unique determination of the object. We also demonstrated simulation results using synthetic data to back up the theoretical result.

Application of Fractional-order PID controllers in a Greenhouse Climate Control System

A model-based Fractional Order Proportional-Integral-Derivative (FOPID) control design method is applied to a multivariable greenhouse climate control system. A multivariable transfer function model of the greenhouse system, obtained via model Identification experiment, provides plant’s model information for control design. The fractional controller is tuned using the conventional Biggest Log-modulus Tuning (BLT) method associated with integer order PID controllers. However, unlike the conventional PID case, the controller gains are first computed using internal model control design procedure. This produces preliminary values of controller parameters. The FOPID controller gains are further tuned using a single detuning factor (F) until a biggest log modulus of 2N dB is obtained where N is the number of loops in the MIMO system. Simulation studies, using MATLAB and Simulink, show that good compromise between performance and robustness is achievable for both temperature and humidity regulation in a greenhouse control application.

Analysis of the stationary probability density of a generalized and bistable van der Pol system excited by colored noise

The stochastic P-bifurcation behavior of bi-stability in a generalized van der Pol oscillator with the fractional damping under colored noise and thermal excitation is investigated. Firstly, using the principle of minimal mean square error and lin-earization method, the non-linear stiffness terms can be equivalent to a linear stiffness which is a function of the system amplitude, and the original system is simplified to an equivalent integer order van der Pol system. Secondly, the system amplitude stationary probability density function is obtained by the stochastic averaging, and then based on the singularity theory, the critical parametric con-ditions for the system amplitude stochastic P-bifurcation are found. Finally, the types of the stationary probability density function of the system amplitude are qualitatively analyzed in each area divided by the transition set curves. The con-sistency between the analytical results and the numerical results acquired from Monte-Carlo simulation also testifies the theoretical analysis in this paper and the method used in this paper can directly guide the design of the fractional order controller to adjust the response of the system.