The paper presents general solutions for fractional state-space equations. The analysis of the fractional electrical circuit in the transient state is described by the equation of the state and space equations. The results are presented for the voltage of a capacitor and current in a coil, for different alpha values. The Caputo and conformable fractional derivative definitions have been considered. At the end, the results have been obtained.

In this work, the design of current mode Fractional order filter using VDTAs (Voltage differencing trans-conductance amplifier) as an active element with grounded capacitors has been proposed. The approximate transfer functions of low and high pass filters of fractional order on the basis of the integer order transfer has been shown and the form of those functions of filters is also implemented using VDTA as an active building block. In this work, filters of the different sequence have been realized. The frequency domain simulation results of the proposed filters are obtained on Matlab and PSPICE with TSMC CMOS 180 nm technology parameters. Stability and sensitivity is also verified.

An analysis of a given electrical circuit using a fractional derivative. The statespace equation was developed. The dynamics of tensions described by Kirchhoff’s laws equations. The paper used the definition of the integral derivative Caputo and CDF conformable fractional definition. An electrical circuit solution using Caputo and CDF defini- tions for rectangular with zero initial conditions was developed. The results obtained using the Caputo and CDF definitions were compared. The solutions are shown for capacitor voltages, for fractional derivative orders of 0.6, 0.8, 1. The results were compared using graphs.

The contents of total P and its bioavailable forms. as well as of Fe, Al, Mn, Ca, and OM in the bottom sediments of the Solina-Myczkowce (S-E Poland) cascade of reservoirs, are presented. Notwithstanding a relatively low calcium content, it is the apatite fraction that accounts for the largest share of total phosphorus in the shallower parts of the Solina and Myczkowce Reservoirs. In turn, while contents of iron and aluminium (and manganese in the Solina Reservoir) are high, the fraction containing non-apatite inorganic phosphorus accounts for the smallest portion of the total phosphorus in the bottom sediments of both reservoirs. Bottom sediments of the Solina Reservoir are also characterised by significant correlations between total phosphorus content and aluminium content. Otherwise, significant correlations are reported for only some of the stations at each of the reservoirs.

The paper presents the problem of position control of DC motor with rated voltage 24 V loaded by flywheel. The fractional order PD controller implemented in National Instruments NI ELVIS II programmed in LabView is used for controlling. The simple method for determining stability regions in the controller parameters space is given. Knowledge of these regions permits tuning of the controller and ensures required the phase margin of the system.

This study was aimed at evaluating the possibility to use the Friedrich-Braun fractional derivative rheological model to assess the viscoelastic properties of xanthan gum with rice starch and sweet potato starch. The Friedrich-Braun fractional derivative rheological model allows to describe viscoelastic properties comprehensively, starting from the behaviour characteristic of purely viscous fluids to the behaviour corresponding to elastic solids. The Friedrich-Braun fractional derivative rheological model has one more virtue which distinguishes it from other models, it allows to determine the relationship between stress and strain and the impact of each of them on viscoelastic properties on the tested material. An analysis of the data described using the Friedrich-Braun fractional derivative rheological model allows to state that all the tested mixtures of starch with xanthan gum form macromolecular gels exhibiting behaviour typical of viscoelastic quasi-solid bodies. The Friedrich-Braun fractional derivative rheological model and 8 rheological parameters of this model allow to determine changes in the structure of the examined starch - xanthan gum mixtures. Similarly important is the possibility to find out the trend and changes going on in this structure as well as their causes.

This paper adopts a fractional calculus perspective to describe a non-linear electrical inductor. First, the electrical impedance spectroscopy technique is used for measuring the impedance of the device. Second, the experimental data is approximated by means of fractional-order models. The results demonstrate that the proposed approach represents the inductor using a limited number of parameters, while highlighting its most relevant characteristics.

The practical and asymptotic stabilities of delayed fractional discrete-time linear systems described by the model without a time shift in the difference are addressed. The D-decomposition approach is used for stability analysis. New necessary and sufficient stability conditions are established. The conditions in terms of the location of eigenvalues of the system matrix in the complex plane are given.

The Drazin inverse of matrices is applied to analysis of the pointwise completeness and of the pointwise degeneracy of the fractional descriptor linear discrete-time systems. Necessary and sufficient conditions for the pointwise completeness and the pointwise degeneracy of the fractional descriptor linear discrete-time systems are established. It is shown that every fractional descriptor linear discrete-time systems is not pointwise complete and it is pointwise degenerated in one step (for i = 1).

A fractional-order control strategy for a pneumatic position servo-system is presented in this paper. The idea of the fractional calculus application to control theory was introduced in many works, and its advantages were proved. This paper deals with the design of fractional order PIλDµ controllers, in which the orders of the integral and derivative parts, λ and µ, respectively, are fractional. Experiments with fractional-order controller are performed under various conditions, which include position signal with different frequencies and amplitudes or a step position signal. The results show the effectiveness of the proposed schemes and verify their fine control performance for a pneumatic position servo-system.

This paper addresses the nonlinear Cucker–Smale optimal control problem under the interplay of memory effect. The aforementioned effect is included by employing the Caputo fractional derivative in the equation representing the velocity of agents. Sufficient conditions for the existence of solutions to the considered problem are proved and the analysis of some particular problems is illustrated by two numerical examples.

In this paper, we consider an optimal control problem in which a dynamical system is controlled by a nonlinear Caputo fractional state equation. First we get the linearized maximum principle. Further, the concept of a quasi-singular control is introduced and, on this basis, an analogue of the Legendre-Clebsch conditions is obtained. When the analogue of Legendre- Clebsch condition degenerates, a necessary high-order optimality condition is derived. An illustrative example is considered.

The paper presents results of aluminium concentration determination in the samples of surface water and bottom sediments of the Mala Wełna River (West Poland). In the surface water the concentration of aluminium varies in the range from 4.14 to 25.9 ug/dm'. With use of the Mineql+ program the concentration of the aluminium sulphate complexes in the water samples studied has been determined in a model way. In the bottom sediments samples of the river aluminium has been determined in the granulometrie fractions of the grain sizes> 2.0; 2.0-1.0; 1.0-0.5; 0.5-0.25; 0.25-0.1; 0.1-0.063; < 0.063 mm, using the sequential extraction scheme proposed by Tessier el al. The lowest concentration of aluminium has been found in the granulometrie fraction 0.5-0.25 mm, while the highest in the fractions 0.1-0.063 and< 0.063 mm. An elevated concentration of aluminium has been also noted in the fraction> 2.0 mm. Taking into regard the chemical fractions the lowest concentration of aluminium has been found in the exchange fraction and the fraction bounded to carbonates (fractions I and li), whereas the highest concentration of aluminium has been determined in the lithogenic fraction (fraction V). The methods of sample preparation for analysis of aluminium in bottom sediments were compared. Il was observed that higher concentration of aluminium was present in grounded samples without its influence on grain size fractions. The concentration of aluminium in surface water samples has been determined by the GF-AAS, while in bottom sediments by F-AAS.

The use of fractional-order calculus for system modeling is a good alternative to well-known classic integer-order methods, primarily due to the precision with which the modeled object may be mapped. In this study, we created integer and fractional discrete models of a real object – a highspeed brushless micro-motor. The accuracy of the models was verified and compared.

This paper presents a study of the Fourier transform method for parameter identification of a linear dynamic system in the frequency domain using fractional differential equations. Fundamental definitions of fractional differential equations are briefly outlined. The Fourier transform method of identification and their algorithms are generalized so that they include fractional derivatives and integrals.

In the paper a new, state space, fully discrete, fractional model of a heat transfer process in one dimensional body is addressed. The proposed model derives directly from fractional heat transfer equation. It employes the discrete Grünwald-Letnikov operator to express the fractional order differences along both coordinates: time and space. The practical stability and numerical complexity of the model are analysed. Theoretical results are verified using experimental data.

The efficient, stable and reliable operation of the blast furnace secures the proper quality of coke, which is one of the basic components of the blast furnace charge. In modern blast-furnace technology, when using substitute fuels, i.e. coal dust, the role of coke is extremely important. For this reason, the demands placed on its quality increase. Domestic coking plants have a limited base of Polish high quality coking coals at their disposal, therefore the full use of their coking properties is extremely important. The grain composition of the coal blend is one of the basic factors affecting the quality of the produced coke. This influence depends on the quantity and quality of coal components that make up the blend. In the conducted research, 21 coking coals, differing significantly in the degree of rank and origin (Polish and overseas coals), it was shown that the separated grain classes differ in properties, both coking properties and the degree of devolatalization during heating. In analyzing the obtained results, it was observed that the grain volume growth occurs essentially in the temperature range between the beginning and the maximum of fluidity. It has been shown that there is a linear correlation between the temperature corresponding to maximum fluidity and the temperature at which the maximum rate of evolution of volatiles enters. The presented phenomena accompany the emergence of coal expansion pressure during the coking process and they are its primary causes. The presented results can be an important guide for preparing the milling of coal for the coking process.

This study investigates Thomas’ cyclically symmetric attractor dynamics with mathematical and electronic simulations using a proportional fractional derivative to comprehend the dynamics of a given chaotic system. The three-dimensional chaotic flow was examined in detail with Riemann-Liouville derivative for different values of the fractional index to highlight the sensitivity of chaotic systems with initial conditions. Thus, the dynamics of the fractional index system were investigated with Eigenvalues, Kaplan–Yorke dimension, Lyapunov exponent, and NIST testing, and their corresponding trajectories were visualized with phase portraits, 2D density plot, and Poincaré maps. After obtaining the results, we found that the integer index dynamics are more complex than the fractional index dynamics. Furthermore, the chaotic system circuit is simulated with operational amplifiers for different fractional indices to generate analog signals of the symmetric attractor, making it an important aspect of engineering. The qualitative application of our nonlinear chaotic system is then applied to encrypt different data types such as voice, image, and video, to ensure that the developed nonlinear chaotic system can widely applied in the field of cyber security.