Keywords: Riemann-Liouville integral, diffusive representation, infinite state representation, Laplace transform, admissible transformation, numerical approximation
Abstract: The goal of this talk is to provide a theoretical foundation for the construction of very efficient numerical algorithms for computing fractional integrals and for solving fractional differential equations. In this context, efficiency relates to three aspects: (i) small run time, (ii) low memory requirements, and (iii) simple integrability into general computational frameworks like finite element software packages. While the first aspect has been discussed intensively for many years, the two others have attracted interest only much more recently. To achieve our goal, we recall the concept of diffusive representations (or infinite state representations) of fractional integral operators. We demonstrate how such representations naturally lead to numerical methods that automatically satisfy all three requirements. A few approaches of this type have been developed in the past, but often rather unsystematically. By looking at the issue from a slightly more abstract perspective, we attempt to provide some general structural information and to initiate a systematic analysis of such methods.

Keywords: Automatic Control & Stability, History of Fractional-Order Calculus, System identification & Modeling
Abstract: 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.

Keywords: Mathematical methods, Automatic Control & Stability, Electrochemistry
Abstract: Fractional equivalent circuit models have been used in numerous publications to represent long-term memory phenomena of the charging/discharging behavior of Lithium-ion batteries. However, despite the fact that fractional models allow for capturing (infinite horizon) memory properties, the same feature also complicates the numerical evaluation. This is especially critical in cases, in which state estimation procedures are implemented that need to be executed in real time. For this type of applications, it is necessary that the execution time in each time step is bounded a-priori and that it does not grow over time. For that purpose, classical simulation approaches exploit, for example, Grünwald-Letnikov schemes with a finite length of previous state information. However, truncating the memory window without considering the arising errors may lead to wrong state estimates. This is especially critical if interval observers are designed which are meant to include the sets of reachable states with certainty.

Keywords: System identification & Modeling, Automatic Control & Stability
Abstract: In this paper, an analytical procedure to identify a fractional first-order plus dead-time (FFOPDT) model is presented. This procedure is applied to processes with overdamped or underdamped step response, i.e., fractional order values in the range 0.5 ≤ α ≤ 2.0 can be estimated. The required process information is obtained through a simple open-loop step experiment and is based on fitting three arbitrary points (x1-x2-x3%) on the process reaction curve. Some numerical examples are proposed to show the effectiveness and applicability of this procedure for the identification of fractional-order models. The main feature of this approach is that it estimates the FFOPDT model of overdamped and underdamped processes.

Keywords: System identification & Modeling, Signal Processing
Abstract: This paper aims to design a non-asymptotic and robust state estimator applicable both for Caputo fractional linear and nonlinear systems with noisy outputs. For this purpose, a fractional observable canonical form is considered to show the idea of the proposed method. First, the considered form is transformed by taking a fractional derivative operator. Second, based on the new form, the modulating functions method with the additive index law of fractional derivatives is used to get exact formulas for the sought variables in continuous noise-free case, which only contain the integrals involving the output and the fractional derivative of the input, without producing any source of errors. Moreover, the integral forms can reduce noisy effect as low-pass filters. Hence, fast convergent and robust estimation can be obtained using discrete noisy outputs. After constructing the required modulating functions, two numerical examples are given to demonstrate the advantages of the proposed estimator by comparing with the fractional order Luenberger-like observer.

Keywords: System identification & Modeling, System Analysis & Dynamics, Electrical Engineering & Electromagnetism
Abstract: An accurate component modeling of power electronics converters is crucial to understanding their real performance. Thus, a fractional-order representation of energy storage components like inductors and capacitors could be helpful to reflect the hidden dynamics of a power converter in voltage regulation tasks. This paper presents the fractional-order non-ideal modelling and behavioral matching of an isolation transformer used on a flyback converter. A multiphysics simulation model is built to replicate the flyback converter non-ideal dynamics. It is optimized to precisely reflect the real converter response by testing various transformer configurations, including ideal, non-ideal, integer and fractional-order. The transformer parameters are adjusted using raw data from the physical power converter. Likewise, the fractional-order energy storage elements are represented using discrete circuit elements for its realization. The results obtained reveal that using a non-ideal fractional-order transformer on the flyback converter with a non-integer inductance provides the maximum fit for the power converter operation.

Keywords: System identification & Modeling, Thermal Engineering
Abstract: The European Space agency is highly interested in a full atmospheric reentry from low Earth orbit conditions using an automatic lifting body. In this respect, a numerical method is performed to estimate the heat transfer conditions at the rear surface of a stainless steel material through solving the heat equation. The proposed models should be easily invertible in order to reconstruct the heat flux. The present study presents the analytical modeling of a 2D thermal diffusion which will be compared to different approximations: Taylor approximation, Padé approximation and finite difference method.

Keywords: Mathematical methods
Abstract: In this paper, we deal with the link between fractional calculus, Riemann zeta functions and prime numbers. In particular, we compute the logarithmic fractional derivative of the Riemann zeta function. In addition, we study the possibility to express zeta^{(alpha)}(s) as an Euler product. This leads us to generalize the Euler's product formula for Dirichlet series where the hypothesis of multiplicative function is replaced by a homomorphism.

Keywords: Mathematical methods, Special Functions, Singularities Analysis and Integral Representations
Abstract: The method of Mikusi'{n}ski operational calculus relies on an abstract algebra understanding of integral and derivative operators, which can be used for solving differential equations. This has been used for fractional operators of many types, including with Sonine kernels. Here, we use the general fractional integrals with Sonine kernels to construct a different algebraic setting in which the general fractional derivatives and general fractional integrals can be understood in a different way.

Keywords: Mathematical methods, Special Functions, Singularities Analysis and Integral Representations
Abstract: Mikusinski's operational calculus provides a way of applying the machinery of abstract algebra to the spaces and operators of calculus, thus allowing integro-differential equations to be solved by reducing them to algebraic equations. We summarise the application of this method to several operators of fractional calculus, defined by various convolution kernel functions at different levels of generality, and how the corresponding fractional differential equations can be solved.

Keywords: Mathematical methods, Special Functions, Singularities Analysis and Integral Representations
Abstract: Fractional calculus operators can be defined using many different convolution kernels, which often take the form of analytic functions. We extended this idea to a bivariate setting, in three different ways. Firstly, we use a general univariate analytic function with a fractional power substitution to define a bivariate fractional integral operator. Secondly, we use a general bivariate analytic function with fractional power substitutions to define a univariate fractional integral operator. Finally, we use a general bivariate analytic function with fractional power substitutions to define a bivariate fractional integral operator. All of these general operators can be expressed as (single or double) infinite sums of classical Riemann--Liouville operators, using analyticity, and this allows many fundamental properties to be proved in a natural way using the existing theory of fractional calculus. To motivate the proposed models, we consider some illustrative examples which have already found applications in the literature.

Keywords: Mathematical methods, Special Functions, System Analysis & Dynamics
Abstract: The paper investigates linear fractional differential equations involving the Liouville derivative. Solution to these equations under a non-local boundary condition are derived in explicit form, their uniqueness is established using integral transforms technique.

Keywords: Singularities Analysis and Integral Representations, Mathematical methods, Special Functions
Abstract: This work deals with the computation of the Mittag-Leffler function of regularly accretive operators by means of the sinc rule applied to the Stieltjes integral representation after a logarithmic change of variable. The error analysis, together with the selection strategy for the parameters of the sinc formula, is considerate. Several numerical examples are presented.

Keywords: Automatic Control & Stability, Mathematical methods
Abstract: This paper is devoted to studying three-dimensional non-commensurate fractional order differential equation systems with Caputo derivatives. Necessary and sufficient conditions for the asymptotic stability of such systems are obtained.

Keywords: Automatic Control & Stability, Mathematical methods, System Analysis & Dynamics
Abstract: This paper focuses on boundary disturbance rejection control (BDRC) for fractional-order multi-agent systems (FMASs) with reaction diffusion terms. Firstly, based on the output information, the corresponding observer is designed to obtain information about all the agents and the unknown external perturbations. Then, a novel distributed boundary control protocol is designed based on the observed values. Thus, the convergence analysis of the closed-loop system is accomplished through the Mittag-Leffler stability theory and the linear matrix inequality condition. Finally, a simulation example is used to verify the correctness of the results.

Keywords: Automatic Control & Stability, System Analysis & Dynamics
Abstract: When a linear controller is replaced by a reset controller, it is possible to keep the gain behaviour essentially the same, while improving the phase behaviour. However, because reset control is nonlinear, higher order harmonics appear, which may deteriorate the results. In this paper, reset controllers are combined with fractional derivatives, decreasing the magnitude of higher order harmonics. In this way, better results are found in simulations of a Clegg integrator, of a FORE (first order reset element), and of a CgLp (constant in gain lead in phase) controller. The reason why better results are found is explained.

Keywords: Automatic Control & Stability, System Analysis & Dynamics, Mathematical methods
Abstract: This paper investigates the hybrid impulsive control synchronization problem of multi-term fractional-order neural networks (MFNNs) with switching parameters. Firstly, a novel MFNN with switching parameters model is introduced by incorporating multi-term Caputo fractional-order derivative to extend the existing framework for fractional-order cases. Then, the relationship between multi-term fractional-order derivative and distributed-order derivative is analyzed, and a synchronization criterion for a class of multi-term fractional-order impulsive switched systems is derived by utilizing the properties of the distributed-order derivative weight function. Furthermore, a hybrid impulsive controller is designed to obtain sufficient conditions for synchronization of MFNNs with switching parameters. To validate the effectiveness of the obtained conclusions, a numerical example is presented to demonstrate the validity of the proposed MFNN model and synchronization criterion.

Keywords: Automatic Control & Stability, Mathematical methods
Abstract: This work introduces a new realization method of fractional-order integrators (FOI) and differentiators (FOD) of complex orders. The corresponding Laplace variables of the form s^(+-alpha +-jbeta); 0

Keywords: System Analysis & Dynamics, Automatic Control & Stability, System identification & Modeling
Abstract: This paper presents an alternative way to do control performance assessment (CPA). The task to measure controller quality has to meet contradictory goals. Generally, the tuning of any controller means reaching a compromise between the accuracy and speed. The demanded ratio between them depends on the process requirements, technological limitations and just the engineering skills. Two fundamental indexes, like the overshoot and settling time fit perfectly into the picture. Considered investigations follow this research path, but with the use of modern measures: L-moments, tail index and ARFIMA filter fractional order estimator. The assessment uses two dimensional Index Ratio Diagrams (IRD), which allow to present and compare contradictory measures in a single diagram. They also allow to define new multi-criteria index able to compare different loops. The validation is compared against commonly used integral measures: means square and absolute errors.

Keywords: System Analysis & Dynamics, Mathematical methods, Automatic Control & Stability
Abstract: Since the first fractional integration operator dedicated to simulation 25 years ago, the Infinite State representation has evolved into a general methodology for the analysis of fractional order system dynamics. Based on the frequency-distributed model of the fractional integrator, a fractional system is decomposed into an infinite dimensional set of ODEs, whose state variables are the frequency-distributed state variables of the fractional system. Thus, the Infinite State approach provides a conventional integerorder solution to the initialization problem, a new interpretation of the initial conditions of fractional derivatives, and a definition of fractional energy, preliminary to the analysis of system stability based on the Lyapunov approach. Its finite dimensional approximation enables the study of fractional transients thanks to numerical simulation and the discovery of hidden properties of fractional systems. Moreover, it is a helpful and necessary technique for the analysis of nonlinear and chaotic systems and represents a powerful characterization tool.

Keywords: Automatic Control & Stability, Mathematical methods
Abstract: Since the first fractional integration operator dedicated to simulation 25 years ago, the Infinite State representation has evolved into a general methodology for the analysis of fractional order system dynamics. Based on the frequency-distributed model of the fractional integrator, a fractional system is decomposed into an infinite dimensional set of ODEs, whose state variables are the frequency-distributed state variables of the fractional system. Thus, the Infinite State approach provides a conventional integerorder solution to the initialization problem, a new interpretation of the initial conditions of fractional derivatives, and a definition of fractional energy, preliminary to the analysis of system stability based on the Lyapunov approach. Its finite dimensional approximation enables the study of fractional transients thanks to numerical simulation and the discovery of hidden properties of fractional systems. Moreover, it is a helpful and necessary technique for the analysis of nonlinear and chaotic systems and represents a powerful characterization tool.

Keywords: Mathematical methods, Others
Abstract: Unique solvability conditions for general fractional functional integro-differential equations with nonlocal initial conditions are established using fixed-point theory. Ulam-Hyers and generalized Ulam-Hyers stability conditions for a solution of these problems are formulated. An example of the nonlinear fractional functional integro-differential equation related to wheel shimmy models is considered as well.

Keywords: System Analysis & Dynamics, Mathematical methods
Abstract: This paper aims to establish some novel conditions for revealing the stability of nonlinear incommensurate fractional-order systems formulated using Caputo differential operator. This will be accomplished through introducing satisfactory proofs for these conditions with the help of using Lyapunov function method. All theoretical findings will be verified numerically by observing the wished stability is occurring on the systems at hand.

Keywords: Mathematical methods
Abstract: This study is concerned with the existence and stability of solutions of a class of initial value problem for fractional differential equations with a variable order generalized proportional Caputo fractional derivative with respect to another function. The statement of the problem is discussed and compared with the existing in the literature. Existence results are studied and the Hyers-Ulam stability is defined and established. The approach is more broad-based and the same methodology can be used for a number of additional issues

Keywords: Mathematical methods
Abstract: Using Banach’s contractive principle and Leray–Schauder’s fixed point theorem, we study the uniqueness and existence for a nonlinear fractional differential equation with functional boundary condition based on the two-parameter Mittag-Leffler function and an implicit integral equation. Several examples are also presented to demonstrate applications of our key theorems. The methods used can also deal with other types of differential equations with various initial or boundary conditions.

Keywords: Mathematical methods, History of Fractional-Order Calculus, Special Functions
Abstract: In this survey paper, the general fractional integrals and derivatives with the Sonin kernels, the regularized general fractional derivatives, the sequential generalized fractional derivatives as well as some of their applications and the fractional differential equations with these derivatives are discussed. We start with a short summary of different kinds of the general Fractional Calculus operators with the Sonin kernels and then proceed with a presentation of some recent results including the fundamental theorems of Fractional Calculus for the general fractional derivatives of arbitrary order, for the regularized general fractional derivatives of arbitrary order, and for the sequential general fractional derivatives. As an application of these results, the generalized convolution Taylor formulas and the generalized convolution Taylor series with the coefficients and remainders in terms of the general fractional derivatives and the regularized general fractional derivatives are presented. Finally, we discuss a method for construction of solutions to the fractional differential equations with the general fractional derivatives in terms of the so called convolution series.

Keywords: Mathematical methods, Singularities Analysis and Integral Representations, History of Fractional-Order Calculus
Abstract: Numerical simulation of fractional-order partial differential equations is a challenging task and the majority of computing environments does not provide support for these problems. In this paper we describe how to exploit some of the Matlab features (a programming language not supporting fractional calculus in a naive way) to solve partial differential equations with the spectral fractional Laplacian. For shortness we focus on fractional Poisson equations but the proposed approach can be extended, with just some technical difficulties, to more involved problems. This approach cannot be considered as a highly efficient and accurate way to solve fractional partial differential equations, but as an easy-to-use tool for non specialists in numerical computation to obtain solutions without having to produce sophisticated numerical codes.

Keywords: Mathematical methods, System identification & Modeling, History of Fractional-Order Calculus
Abstract: Computationally efficient Exponential Time Differencing (ETD) methods for Riesz space fractional reaction-diffusion models are presented. The Riesz space-fractional derivatives are approximated using the fourth order compact fractional finite differences scheme. Single real pole restricted Pad'e rational approximation is used to avoid complex arithmetic. A splitting technique is applied to enhance computational efficiency of the new methods. Algorithms based on the methods are developed to easily implement the methods on computer. The superiority of new methods in terms of accuracy, reliability, and computational efficiency is demonstrated through numerical experiments. Time evolution graphs are plotted to show the reliability of the methods. Order of convergences are computed, and CPU times are recorded to show the accuracy and computational efficiency of the methods.

Keywords: Others, History of Fractional-Order Calculus
Abstract: Two different formulae have been proposed for fractional d-bar operators in complex analysis, using infinite series in the style of the fractional binomial theorem. In the context of the fractional Leibniz rule, it is known that a similar pair of infinite series formulae can be unified into a single, more general, double series formula. We investigate whether the same idea can be applied to fractional d-bar operators or not.

Keywords: Automatic Control & Stability, Others, System Analysis & Dynamics
Abstract: This article deals with the Fault Tolerant Control (FTC) and Fault Diagnosis (FD) for trajectory tracking in a class of nonlinear Fractional Order (FO) systems. The proposed control scheme consists of a set of FO differential equations describing a FO dynamic controller, the estimation of the unknown pseudo-states of the system and faults, the estimators that are designed have different structures, therefore, this scheme can be considered as a hybrid type. In addition, the existence and uniqueness of the solution for the controller is proved using the Banach fixed point theorem. On the other hand, it is shown that the dynamic controller, estimators and the tracking error are globally Mittag-Leffler bounded ensuring the desired trajectory tracking, in consequence a separation principle for nonlinear FO systems is presented. This FO control scheme is illustrated by considering a FO chaotic system as an example.

Keywords: Automatic Control & Stability, System Analysis & Dynamics
Abstract: This paper deals with the implementation of a gain scheduling technique for fractional-order-proportional-integral-derivative (FOPID) controllers. The presence of fractional-order integer and derivative parts requires special considerations when the controller parameters are changed during the control task due to the initialization issue of the fractional operators when the fractional order changes. Simulation results show the effectiveness of the proposed technique.

Keywords: Automatic Control & Stability, System Analysis & Dynamics
Abstract: The paper deals with stability of distributed order systems. A recently proposed method for stability analysis of a wide class of linear stationary systems is shown to be applicable to distributed order systems as well. In addition, we demonstrate that the well-known distributed order PID control algorithm should be used with utmost care, since the corresponding transfer functions introduce dipoles in the open right-half complex plane. The analysis is illustrated by a numerical example.

Keywords: Automatic Control & Stability, System Analysis & Dynamics
Abstract: Data-driven controller synthesis methods allow finding controller parameters based only on the system input-output response. Among these methods, Virtual Reference Feedback Tuning (VRFT) is accepted thanks to its reference-model based direct parameters synthesis, including robustness and noise rejection considerations for integer and fractional-order controller design. However, for fractional-order VRFT controller synthesis, a direct solution is harder to reach involving a complex optimization problem to solve. This paper presents a recursive VRFT method for synthesizing arbitrary degree fractional-order controllers with feedforward based on a integer or fractional-order reference model. This methodology leverages existing optimization tools for fractional-order systems identification to obtain integer and fractional-order controller gains when integer and fractional-order reference models are used for the virtual error computation. The proposed method synthesizes a set of feedforward VRFT controllers implemented to control a uniform temperature control system. The results show that combining a fractional-order model or compensator improves the energy consumption of the controller and the transient temperature response in tracking and regulation tasks.

Keywords: Automatic Control & Stability
Abstract: This study investigates the control of discrete fractional order Hammerstein systems using first-order sliding mode control (FOSMC) and upgrades its performance by incorporating second-order sliding mode control (2SMC). Two simulation examples are presented to evaluate the effectiveness of these methods. Results indicate that 2SMC outperforms FOSMC, showcasing its superiority in governing these complex systems, offering promising advancements in control strategies.

Keywords: Automatic Control & Stability
Abstract: Fractional-order controllers are modeled using irrational continuous-time transfer functions. Implementing these controllers in analogue form can be challenging, particularly if their parameters need to be adjusted in real-time. This paper presents an analogue implementation of a fractional PID controller that can be adjusted and configured in real-time. The controller can be configured as PIαDf , PIDβ, PIα, PDβ, Iα, Dβ. The signal within the proposed controller’s circuit stays analog and is not converted into discrete values. A prototype of the controller is tested. The results of the frequency and time domain validations indicated similarity between the theoretical, simulated and experimental measurements.

Keywords: Mechanics & Viscoelasticity, Mathematical methods
Abstract: The rheological properties of polymer solutions are extremely complex and it leads to a specific power-law behavior in response to creep experiments, which can not be better characterized by the classical viscoelastic constitutive models. The renowned viscoelastic fluid models with fractional derivatives have shown astonishing advantages in characterizing the broad relaxation process of the multiple materials with power-law response behavior. In this paper, we use the fractional Maxwell constitutive model to simulate the power-law responses for several polymer solutions through experimental measuring, and better fitting results than classical models can be found.

Keywords: Mechanics & Viscoelasticity, Mathematical methods
Abstract: We analyze, spatially one dimensional, non-local elastic body with deformation measure given in the form of General fractional derivative of Riesz type with truncated power-law kernel. It is assumed that stress at an arbitrary point x and time t depends on the strain at point y at time t-(x-y)/c where c is a constant having dimension of time. We formulate spatially one-dimensional wave equation for such a body and show the influence of the parameter c on the solution.

Keywords: Mechanics & Viscoelasticity, Mathematical methods, Earth Science
Abstract: Piles are commonly used in Civil and Mechanical Engineering to provide support to superstructures and to transmit loads to deeper ground layers. During earthquakes, pile instability can result in the collapse of the entire structure. In this paper, the pile is modeled as a column surrounded by Winkler soil foundations with fractional damping. Investigation of the axially loaded pile leads to a fractional Mathieu differential equation of motion. The Bolotin method, employing harmonic balance, is proposed to obtain the approximate instability boundaries of the pile in the stability diagrams. A practical example is presented to conduct a parametric study on pile instability concerning the fractional order. The study observes parametric resonance in the first order, representing the principal instability region, which requires special attention to maintain stability. Furthermore, increasing the fractional damping order leads to a higher critical dynamic load and a slight reduction in the critical frequency ratio in each instability region. Higher-order instability regions exhibit greater sensitivity to changes in the fractional order. These results provided insights into the stability of piles under earthquake conditions.

Keywords: Mechanics & Viscoelasticity, Physics
Abstract: We study the effects of aging properties of subordinated fractional Brownian motion (FBM) with drift and in harmonic confinement, when the measurement of the stochastic process starts a time after its original initiation. Specifically, we consider the aged versions of the ensemble mean squared displacement (MSD) and the time averaged MSD (TAMSD), along with the aging factor. Our results are favorably compared with simulations results. The aging subordinated FBM exhibits a disparity between MSD and TAMSD and is thus weakly nonergodic, while strong aging is shown to effect a convergence of the MSD and TAMSD. The information on the aging factor with respect to the lag time exhibits an identical form to the aging behavior of subdiffusive continuous time random walks (CTRW). The statistical properties of the MSD and TAMSD for the confined subordinated FBM are also derived. At long times, the MSD in the harmonic potential has a stationary value, that depends on the Hurst index of the parental (non-equilibrium) FBM. The TAMSD of confined subordinated FBM does not relax to a stationary value but increases sublinearly with lag time, analogously to confined CTRW. Interestingly, for the confined subordinated FBM, small aging time has no effect on the aged MSD, while at large aging time the aged MSD has monotonic increase ultimately with a power-law and is identical to the aged TAMSD.

Keywords: Mechanics & Viscoelasticity, Physics, Others
Abstract: The stationary behavior of freely moving spherical particles under harmonic flow forcing from a Newtonian fluid ranging from low to high particle Reynolds (Re_{!p}) numbers and from low to high Strouhal (Sl) numbers is studied numerically. This study extends the classical studies on harmonic Stokes flows by exploring particle dynamics in regimes where the convective contributions can no longer be neglected. High-order finite-element numerical simulations determine the order of the derivative that satisfies the long term (stationary) solution for the particle velocities. We propose a new history drag expression that correlates well (R^2 > 0.995) with the numerical results for (radius-based) particle Reynolds numbers up to 10, and for dimensionless frequency (S,=,Sl Re_p) values up to 10.

Keywords: Mechanics & Viscoelasticity
Abstract: This paper mainly investigates the convective diffusion of non-Newtonian fluids in fractal main channels. The spatial fractional derivative constitutive equations of non-Newtonian fluids are coupled with the flow equations to derive the governing equations for fluid flow and heat transfer. The shifted Grünwald–Letnikov formula is used to obtain numerical solutions for the model. The effects of fractional derivatives, Reynolds number, and Prandtl number on fluid flow and heat transfer are analyzed.

Keywords: Mathematical methods
Abstract: Fractional differential equations have become central tools for the accurate modeling of real-world phenomena in various fields. This work focuses on the discretization of the space-time fractional diffusion problem with Caputo derivative in time and Riesz-Caputo derivative in space. We introduce a collocation method based on a B-spline representation of the solution. This approach strategically exploits the symmetry properties of both the spline basis functions and the Riesz-Caputo operator, resulting in an efficient method for solving the given fractional differential problem. Preliminary numerical tests are presented to validate the proposed collocation method.

Keywords: Mathematical methods, Others, System Analysis & Dynamics
Abstract: The article investigates a mathematical model of the Duffing oscillator with a fractional variable-order difference operator of the Caputo and convolution type. The use of new dynamic in oscillator type systems provide a new aspects and parameters in a system. The chaotic behaviour in a fractional variable order modified discrete-time Duffing system are studied numerically by phase portraits. The relationship between the order of the fractional derivative and the quality factor of the oscillatory system is established.

Keywords: Mathematical methods, Special Functions, Others
Abstract: Existence of solutions for a class of boundary value problems associated to a Hilfer generalized proportional fractional integro-differential inclusion is studied. A new result is obtained when the right hand side has nonconvex values.

Keywords: Mathematical methods, Special Functions, Others
Abstract: This article establishes sufficient conditions for the existence of positive solutions to initial value problems associated with the discrete fractional oscillation equation using the well-known conical shell fixed point theorem. Two numerical examples illustrate the applicability of the established results.

Keywords: Singularities Analysis and Integral Representations, Mathematical methods
Abstract: In this paper, we revisit the diffusive representations of fractional integrals established in Diethelm (2023a) to explore novel variants of such representations which provide highly efficient numerical algorithms for the approximate numerical evaluation of fractional integrals.

Keywords: Singularities Analysis and Integral Representations, Mathematical methods
Abstract: This study reexamines diffusive representations for fractional integrals with the goal of pioneering new variants of such representations. These variants aim to offer highly efficient numerical algorithms for the approximate computation of fractional integrals. The approach seamlessly aligns with established techniques used in addressing problems involving integer-order operators, contributing to a unified framework for numerical solutions.

Keywords: Automatic Control & Stability, Robotics
Abstract: A mobile tracking antenna requires a precise control system that combines good trajectory tracking and disturbance rejection performances. The system has several operating scenarios, requiring robust control design that ensures good performances and stability degree for a set of uncertain models. This paper presents a comparative study between CRONE and H_∞ robust controllers to assess the feasibility of applying these methodologies. A practical validation in antenna tracking demonstrates that both controllers yield equivalent results both in time and frequency domains, thus ensuring excellent stability and outstanding performances. The choice of controller ultimately depends on the designer's expertise, software and/or hardware requirements, and the complexity of the system model.

Keywords: Automatic Control & Stability, System Analysis & Dynamics, Mechatronics
Abstract: The objective of this paper is to present a new design method for an unknown input estimator with a closed-loop structure. A frequency domain approach is carried out using a model-based design method and a loop-shaping technique. The loop-shaping approach used to the estimator design is usually applied in the context of control system design. More precisely, it is the second generation CRONE system design methodology that is used. This method is applied to the estimation of a speed bump by using one of the sensors initially intended for suspension control. Note that the noise measurement used in the simulation is recorded from on-board sensors used to control the piloted suspension of a DS7 Crossback. The time responses of the estimator, obtained with a non-linear quarter-car simulator, illustrate the very good results obtained.

Keywords: Robotics, System identification & Modeling
Abstract: Off-road autonomous vehicles such as tractors and agricultural machinery require an important power traction to cope with mobility issues on soft soils and rough terrains. This paper presents a simplified version of tyre modeling as an adapted Burckhardt tire model based on an analytical Bekker-Wong wheel-soil interaction model. The Bekker-Wong model is a fractional model in space and takes into account the plasticity theory and soil mechanics, however such a model is too complex to be usable in real-time. An adapted version of the Burckhardt model helps having a more compact model that can be usable in real-time. In this paper, an adapted Burckhardt tire model has been optimized for the E-Tract simulator and experiments have been carried out to validate the tractor simulator.

Keywords: System identification & Modeling, Robotics
Abstract: Dynamic modeling of soft actuators is a challenging research field mainly due to the non-linear properties of the materials they are made of, as well as the infinite degrees of freedom that they exhibit. This paper characterizes the deflection of an eccentric soft fluidic actuator in terms of the dynamic displacement using both integer and fractional order (FO) models. The actuator, cylindrical in shape, is made of a silicone polymer called polydimethylsiloxane (PDMS) from a water-soluble mold and a rod that, eccentrically placed, allows to create the internal chamber of the actuator. Experimental results are given to show that the best fitting is obtained with FO models. In fact, the dynamics of the actuator can be fully described with only two parameter model by means of a fractional integrator of order of about 1.2.

Keywords: System identification & Modeling, Robotics, System Analysis & Dynamics
Abstract: This paper develops a mechanism to detect the instant at which an underwater haptic flexible antenna detects an object. This sensor is implemented as an aid to navigation in mobile underwater robots. It is used to localize obstacles by performing an active sensing strategy in which the servomotor system moves the beam back and forth until it hits an object. This impact instant is estimated by our mechanism which subsequently triggers an estimator of the contact point of the antenna with the obstacle and switches the rover control system from free to constrained navigation modes. Our detection mechanism is based on detecting if the difference between a measured force/torque signal and the value predicted by a dynamic model surpasses a threshold. This threshold and, then, the delay in the estimation of the impact instant, depend on the accuracy of this model. Several models have been fitted to experimentally recorded data showing that a fractional-order model of complexity equivalent to that of the standard integer model outperforms by more than five times the accuracy of this last one. This results in a much more efficient impact detection mechanism that has a much lower delay time.

Keywords: System identification & Modeling
Abstract: This paper presents a hybrid method for estimating state of charge of Li-ion batteries from frequency domain input-output data using electrochemical impedance spectroscopy (EIS). The hybrid method combines impedance modeling and a fuzzy-logic estimator. Based on an equivalent electrical circuit, a fractional impedance model is identified by minimizing a quadratic norm of the output error. Then, a fuzzy logic estimator is developed on the basis of the estimated parameters

Keywords: Automatic Control & Stability, System Analysis & Dynamics, Mechanics & Viscoelasticity
Abstract: In the field of passenger vehicle suspension, the primary objective of this article is to introduce a new strategy entirely focused on vibrational comfort, known as the Generalized CRONE Sky Hook strategy. The generalization of the nominal CRONE Sky Hook strategy is achieved by introducing a non-integer integration order v within the interval [0; 2]. Three steps are taken to evaluate the influence of this non-integer order v on performance, comparing it with the nominal CRONE Sky Hook suspension where v; is equal to the unit. The first step involves a comparison of performance at iso-rapidity conditions in case where the integer value of v takes on the values 0 (inertial behavior), 1 (viscous or Sky Hook behavior), and 2 (elastic behavior). The second step compares, always under constant speed conditions, the performance in the case where v between 0 and 1 (visco-inertial behavior), and the third step deals with the case where v between 0 and 2 (visco-elastic behavior).