Guaranteed cost-based feedback control design for fractional-order neutral systems with input-delayed and nonlinear perturbations.

Time delay in actuators is mainly caused by electrical and mechanical components. The effect is visible in the system response particularly when changing in the input command. Therefore, input delay is a problem in the control system design that must be taken into account. Besides, ignoring uncertainty in the dynamic models may compromise the controller design. Thus, how to mitigate the effect of this issue on the system stability and performance is a challenging topic. This article deals with the stabilization of fractional neutral systems considering input-delayed and nonlinear perturbations using the guaranteed cost-based feedback control technique. The main focus is to design the state- and output-feedback controllers to achieve a good performance. The stability criteria are formulated in the Lyapunov sense, which are described in terms of matrix inequalities. The proposed idea is validated using simulations.

Modified SIQR model for the COVID-19 outbreak in several countries.

In this paper, we propose a modified Susceptible-Infected-Quarantine-Recovered (mSIQR) model, for the COVID-19 pandemic. We start by proving the well-posedness of the model and then compute its reproduction number and the corresponding sensitivity indices. We discuss the values of these indices for epidemiological relevant parameters, namely, the contact rate, the proportion of unknown infectious, and the recovering rate. The mSIQR model is simulated, and the outputs are fit to COVID-19 pandemic data from several countries, including France, US, UK, and Portugal. We discuss the epidemiological relevance of the results and provide insights on future patterns, subjected to health policies.

Optimal Solution of a Fractional HIV/AIDS Epidemic Mathematical Model.

This article presents a fractional mathematical model of the human immunodeficiency virus (HIV)/AIDS spread with a fractional derivative of the Caputo type. The model includes five compartments corresponding to the variables describing the susceptible patients, HIV-infected patients, people with AIDS but not receiving antiretroviral treatment, patients being treated, and individuals who are immune to HIV infection by sexual contact. Moreover, it is assumed that the total population is constant. We construct an optimization technique supported by a class of basis functions, consisting of the generalized shifted Jacobi polynomials (GSJPs). The solution of the fractional HIV/AIDS epidemic model is approximated by means of GSJPs with coefficients and parameters in the matrix form. After calculating and combining the operational matrices with the Lagrange multipliers, we obtain the optimization method. The theorems on the existence, unique, and convergence results of the method are proved. Several illustrative examples show the performance of the proposed method. Mathematics Subject Classification: 97M60; 41A58; 92C42.

A local stabilized approach for approximating the modified time-fractional diffusion problem arising in heat and mass transfer.

INTRODUCTION: During the last years the modeling of dynamical phenomena has been advanced by including concepts borrowed from fractional order differential equations. The diffusion process plays an important role not only in heat transfer and fluid flow problems, but also in the modelling of pattern formation that arises in porous media. The modified time-fractional diffusion equation provides a deeper understanding of several dynamic phenomena. OBJECTIVES: The purpose of the paper is to develop an efficient meshless technique for approximating the modified time-fractional diffusion problem formulated in the Riemann-Liouville sense. METHODS: The temporal discretization is performed by integrating both sides of the modified time-fractional diffusion model. The unconditional stability of the time discretization scheme and the optimal convergence rate are obtained. Then, the spatial derivatives are discretized through a local hybridization of the cubic and Gaussian radial basis function. This hybrid kernel improves the condition of the system matrix. Therefore, the solution of the linear system can be obtained using direct solvers that reduce significantly computational cost. The main idea of the method is to consider the distribution of data points over the local support domain where the number of points is almost constant. RESULTS: Three examples show that the numerical procedure has good accuracy and applicable over complex domains with various node distributions. Numerical results on regular and irregular domains illustrate the accuracy, efficiency and validity of the technique. CONCLUSION: This paper adopts a local hybrid kernel meshless approach to solve the modified time-fractional diffusion problem. The main results of the research is the numerical technique with non-uniform distribution in irregular grids.

Fractional and fractal processes applied to cryptocurrencies price series.

Introduction: Cryptocurrencies have been attracting the attention from media, investors, regulators and academia during the last years. In spite of some scepticism in the financial area, cryptocurrencies are a relevant subject of academic research. Objectives: In this paper, several tools are adopted as an instrument that can help market agents and investors to more clearly assess the cryptocurrencies price dynamics and, thus, guide investment decisions more assertively while mitigating risks. Methods: We consider three methods, namely the Auto-Regressive Integrated Moving Average (ARIMA), Auto-Regressive Fractionally Integrated Moving Average (ARFIMA) and Detrended Fluctuation Analysis, and three indices given by the Hurst and Lyapunov exponents or the Fractal Dimension. This information allows assessing the behaviour of the time series, such as their persistence, randomness, predictability and chaoticity. Results: The results suggest that, except for the Bitcoin, the other cryptocurrencies exhibit the characteristic of mean reverting, showing a lower predictability when compared to the Bitcoin. The results for the Bitcoin also indicate a persistent behavior that is related to the long memory effect. Conclusions: The ARFIMA reveals better predictive performance than the ARIMA for all cryptocurrencies. Indeed, the obtained residual values for the ARFIMA are smaller for the auto and partial auto correlations functions, as well as for confidence intervals.

Advances in the computational analysis of SARS-COV2 genome.

Given a data-set of Ribonucleic acid (RNA) sequences we can infer the phylogenetics of the samples and tackle the information for scientific purposes. Based on current data and knowledge, the SARS-CoV-2 seemingly mutates much more slowly than the influenza virus that causes seasonal flu. However, very recent evolution poses some doubts about such conjecture and shadows the out-coming light of people vaccination. This paper adopts mathematical and computational tools for handling the challenge of analyzing the data-set of different clades of the severe acute respiratory syndrome virus-2 (SARS-CoV-2). On one hand, based on the mathematical paraphernalia of tools, the concept of distance associated with the Kolmogorov complexity and Shannon information theories, as well as with the Hamming scheme, are considered. On the other, advanced data processing computational techniques, such as, data compression, clustering and visualization, are borrowed for tackling the problem. The results of the synergistic approach reveal the complex time dynamics of the evolutionary process and may help to clarify future directions of the SARS-CoV-2 evolution.

Optimal solution of the fractional order breast cancer competition model.

In this article, a fractional order breast cancer competition model (F-BCCM) under the Caputo fractional derivative is analyzed. A new set of basis functions, namely the generalized shifted Legendre polynomials, is proposed to deal with the solutions of F-BCCM. The F-BCCM describes the dynamics involving a variety of cancer factors, such as the stem, tumor and healthy cells, as well as the effects of excess estrogen and the body's natural immune response on the cell populations. After combining the operational matrices with the Lagrange multipliers technique we obtain an optimization method for solving the F-BCCM whose convergence is investigated. Several examples show that a few number of basis functions lead to the satisfactory results. In fact, numerical experiments not only confirm the accuracy but also the practicability and computational efficiency of the devised technique.

Entropy analysis of human death uncertainty.

Uncertainty about the time of death is part of one's life, and plays an important role in demographic and actuarial sciences. Entropy is a measure useful for characterizing complex systems. This paper analyses death uncertainty through the concept of entropy. For that purpose, the Shannon and the cumulative residual entropies are adopted. The first may be interpreted as an average information. The second was proposed more recently and is related to reliability measures such as the mean residual lifetime. Data collected from the Human Mortality Database and describing the evolution of 40 countries during several decades are studied using entropy measures. The emerging country and inter-country entropy patterns are used to characterize the dynamics of mortality. The locus of the two entropies gives a deeper insight into the dynamical evolution of the human mortality data series.

Design of fractional evolutionary processing for reactive power planning with FACTS devices.

Reactive power dispatch is a vital problem in the operation, planning and control of power system for obtaining a fixed economic load expedition. An optimal dispatch reduces the grid congestion through the minimization of the active power loss. This strategy involves adjusting the transformer tap settings, generator voltages and reactive power sources, such as flexible alternating current transmission systems (FACTS). The optimal dispatch improves the system security, voltage profile, power transfer capability and overall network efficiency. In the present work, a fractional evolutionary approach achieves the desired objectives of reactive power planning by incorporating FACTS devices. Two compensation arrangements are possible: the shunt type compensation, through Static Var compensator (SVC) and the series compensation through the Thyristor controlled series compensator (TCSC). The fractional order Darwinian Particle Swarm Optimization (FO-DPSO) is implemented on the standard IEEE 30, IEEE 57 and IEEE 118 bus test systems. The power flow analysis is used for determining the location of TCSC, while the voltage collapse proximity indication (VCPI) method identifies the location of the SVC. The superiority of the FO-DPSO is demonstrated by comparing the results with those obtained by other techniques in terms of measure of central tendency, variation indices and time complexity.

Robust stability of uncertain fractional order systems of neutral type with distributed delays and control input saturation.

Time delay occurs naturally due to the limited bandwidth of any real-world system. However, this problem can deteriorate the system performance and can even result in system instability. Input saturation is also an essential issue due to the energy constraint in real actuators that makes the control design procedure more difficult. This article concerns with the stability of uncertain fractional order (FO) delay systems of neutral type including structured uncertainties, distributed delays and actuator saturation. A Lyapunov-Krasovskii functional allows the formulation of the conditions to insure the asymptotic robust stability of such systems via the linear matrix inequalities (LMI) and to compute the gain of a state feedback controller. In addition, by using the cone complementarity linearization method, we obtain the controller gains that extend the domain of attraction. Several simulations validate the theoretical analysis.

Commensurate and Non-Commensurate Fractional-Order Discrete Models of an Electric Individual-Wheel Drive on an Autonomous Platform.

This paper presents integer and linear time-invariant fractional order (FO) models of a closed-loop electric individual-wheel drive implemented on an autonomous platform. Two discrete-time FO models are tested: non-commensurate and commensurate. A classical model described by the second-order linear difference equation is used as the reference. According to the sum of the squared error criterion (SSE), we compare a two-parameter integer order model with four-parameter non-commensurate and three-parameter commensurate FO descriptions. The computer simulation results are compared with the measured velocity of a real autonomous platform powered by a closed-loop electric individual-wheel drive.

New discrete-time fractional derivatives based on the bilinear transformation: Definitions and properties.

In this paper we introduce new discrete-time derivative concepts based on the bilinear (Tustin) transformation. From the new formulation, we obtain derivatives that exhibit a high degree of similarity with the continuous-time Grünwald-Letnikov derivatives. Their properties are described highlighting one important feature, namely that such derivatives have always long memory.

Analysis and implementation of fractional-order chaotic system with standard components.

This paper is devoted to the problem of uncertainty in fractional-order Chaotic systems implemented by means of standard electronic components. The fractional order element (FOE) is typically substituted by one complex impedance network containing a huge number of discrete resistors and capacitors. In order to balance the complexity and accuracy of the circuit, a sparse optimization based parameter selection method is proposed. The random error and the uncertainty of system implementation are analyzed through numerical simulations. The effectiveness of the method is verified by numerical and circuit simulations, tested experimentally with electronic circuit implementations. The simulations and experiments show that the proposed method reduces the order of circuit systems and finds a minimum number for the combination of commercially available standard components.

Multidimensional scaling locus of memristor and fractional order elements.

This paper combines the synergies of three mathematical and computational generalizations. The concepts of fractional calculus, memristor and information visualization extend the classical ideas of integro-differential calculus, electrical elements and data representation, respectively. The study embeds these notions in a common framework, with the objective of organizing and describing the "continuum" of fractional order elements (FOE). Each FOE is characterized by its behavior, either in the time or in the frequency domains, and the differences between the FOE are captured by a variety of distinct indices, such as the Arccosine, Canberra, Jaccard and Sørensen distances. The dissimilarity information is processed by the multidimensional scaling (MDS) computational algorithm to unravel possible clusters and to allow a direct pattern visualization. The MDS yields 3-dimensional loci organized according to the FOE characteristics both for linear and nonlinear elements. The new representation generalizes the standard Cartesian 2-dimensional periodic table of elements.

Numerical evaluation of fractional Tricomi-type model arising from physical problems of gas dynamics.

This paper deals with approximating the time fractional Tricomi-type model in the sense of the Caputo derivative. The model is often adopted for describing the anomalous process of nearly sonic speed gas dynamics. The temporal semi-discretization is computed via a finite difference algorithm, while the spatial discretization is obtained using the local radial basis function in a finite difference mode. The local collocation method approximates the differential operators using a weighted sum of the function values over a local collection of nodes (named stencil) through a radial basis function expansion. This technique considers merely the discretization nodes of each subdomain around the collocation node. This leads to sparse systems and tackles the ill-conditioning produced of global collocation. The theoretical convergence and stability analyses of the proposed time semi-discrete scheme are proved by means of the discrete energy method. Numerical results confirm the accuracy and efficiency of the new approach.

Computational analysis of the SARS-CoV-2 and other viruses based on the Kolmogorov's complexity and Shannon's information theories.

This paper tackles the information of 133 RNA viruses available in public databases under the light of several mathematical and computational tools. First, the formal concepts of distance metrics, Kolmogorov complexity and Shannon information are recalled. Second, the computational tools available presently for tackling and visualizing patterns embedded in datasets, such as the hierarchical clustering and the multidimensional scaling, are discussed. The synergies of the common application of the mathematical and computational resources are then used for exploring the RNA data, cross-evaluating the normalized compression distance, entropy and Jensen-Shannon divergence, versus representations in two and three dimensions. The results of these different perspectives give extra light in what concerns the relations between the distinct RNA viruses.

Fractional-order modelling of epoxy resin.

This paper describes epoxy resins by means of electrical impedance spectroscopy (EIS) and the mathematical tool of fractional calculus (FC). Two stages are considered: first, the EIS is used for testing the samples and, second, the measured data are approximated using integer and fractional order models. The FC-based modelling describes the epoxy resins using a small number of parameters that reflect their main characteristics. The EIS data gathered for the epoxies samples are compared with those of different adhesives and sealants by means of a hierarchical clustering algorithm that unravels the relationships between the distinct materials. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

Rare and extreme events: the case of COVID-19 pandemic.

Complex systems have characteristics that give rise to the emergence of rare and extreme events. This paper addresses an example of such type of crisis, namely the spread of the new Coronavirus disease 2019 (COVID-19). The study deals with the statistical comparison and visualization of country-based real-data for the period December 31, 2019, up to April 12, 2020, and does not intend to address the medical treatment of the disease. Two distinct approaches are considered, the description of the number of infected people across time by means of heuristic models fitting the real-world data, and the comparison of countries based on hierarchical clustering and multidimensional scaling. The computational and mathematical modeling lead to the emergence of patterns, highlighting similarities and differences between the countries, pointing toward the main characteristics of the complex dynamics.

Electrochemical impedance spectroscopy characterization of beverages.

This paper compares the results of standard chemical analytical processes and electrochemical impedance spectroscopy (EIS) in the characterization of different beverages, namely ground coffee, soluble coffee, coffee substitutes, barley, cow milk, vegetable drinks, tea, plant infusions and plant mixtures. For the two approaches, the similarities between the experimental data are assessed by means of the Euclidean and Canberra distances. The resulting information is processed by means of the multidimensional scaling (MDS) clustering and visualization algorithm. The results of the chemical analytical processes and EIS reveal identical clusters for the two adopted distances. Furthermore, the robustness of the experimental and computational scheme are assessed by means of the Procrustes technique. The results confirm the effectiveness of combining the EIS and MDS.