Search Results (29)

This paper analyzes the stability of trajectory tracking in fixed-wing UAV swarms subject to time-delayed feedback control. A delay-dependent stability criterion is established using a combination of Routh–Hurwitz analysis and a transcendental characteristic equation method. The study identifies a critical delay threshold beyond which the tracking objective becomes unstable. The influence of delayed feedback on the system dynamics is analyzed, showing how time delays affect the swarm’s ability to maintain formation. Numerical simulations confirm the theoretical predictions and illustrate the loss of stability as the delay increases. The findings underline the importance of accounting for delays when evaluating control performance in UAV swarm coordination. Full article

In this paper, an ancient Babylonian algorithm for calculating the square root of 2 is unveiled, and the potential link between this primitive technique and an ancient Chinese method is explored. The iteration process is a symmetrical property, whereby the approximate root converges to the exact one through harmonious interactions between two approximate roots. Subsequently, the algorithm is extended in an ingenious manner to solve algebraic equations. To demonstrate the effectiveness of the modified algorithm, a transcendental equation that arises in MEMS systems is considered. Furthermore, the established algorithm is adeptly adapted to handle differential equations and fractal-fractional differential equations. Two illustrative examples are presented for consideration: the first is a nonlinear first-order differential equation, and the second is the renowned Duffing equation. The results demonstrate that this age-old Babylonian approach offers a novel and highly effective method for addressing contemporary problems with remarkable ease, presenting a promising solution to a diverse range of modern challenges. Full article

This article presents a buck converter in which the inductor has been replaced by a transmission line. This approach would be practically conceivable if the power supply and load had a greater spatial distance. Alternatively, the model derived in this way could also be regarded as an intermediate model in order to replace a power coil via discretization with a larger number of smaller coils and capacitors. In the time domain, this new converter can be described by a system of coupled partial and ordinary differential equations. In the frequency domain, a transcendental transfer function is obtained. For comparison with an equivalently parameterized conventional converter, Padé approximants are derived. A linear controller is designed for the converter topology under consideration. Full article

This paper aims to investigate the robustness of exponential synchronization in complex dynamic networks (CDNs) with time-varying delays and random disturbances. Via the Gronwall–Bellman lemma and partial inequality methods, by calculating the transcendental equations, the delays limits and maximum disturbance size of the CDNs are estimated. This means that the perturbed system achieves exponential synchronization if the disturbance strength is within our estimation range. The theoretical results are illustrated by several simulations. Full article

The rhythmic and stable operation of trolleybuses and autonomous trolleybuses or urban electric buses, depends to a large extent on the reliability of the equipment installed on the trolleybus. The actual operational reliability of trolleybus electrical equipment (EE) depends on its technical condition. Under the influence of external factors and specific operating modes, the technical condition of the equipment is continuously deteriorating, reliability indicators are decreasing, and the number of failures is increasing. Using the mathematical theory of reliability, probability theory and mathematical statistics, numerical methods of solving nonlinear and transcendental equations, this article defines the conditions of diagnostics depending on the intensity of failures and the given probability of failure-free operation of the equipment. Additionally, the inverse problem of determining the current reliability of electrical engineering systems depends on the terms of diagnostics and the intensity of failures being solved. As a result of the processing of statistical information on failures it is established that for the electrical equipment of a trolleybus, after a number of repair measures, the maximum density of failures occurs at a lower mileage, and the probability of failure-free operation can vary depending on the degree of wear of the equipment, i.e., on the number of previous failures. It is theoretically substantiated and experimentally confirmed that the reliability of trolleybus electrical equipment changes according to the exponential law of distribution of a random variable. It has been established that the real averaged diagnostic terms regulated by instructions are not optimal in most cases and differ several times from those defined in this paper. The dependence of switching equipment run-in on time has been clarified, which served as a prerequisite for specifying the inter-repair period for various types of trolleybus electrical equipment. A method of adjustment of the inter-repair time for the electrical equipment of trolleybuses is proposed. Full article

This study presents a general framework for the control of unknown dynamic systems with unknown input delay. A concise output feedback control system is structured with tuning stabilization/dynamic response by an output feedback low gain, removing steady state error against step reference with a feedforward gain. A series of stability analyses are presented for the designed control systems, (1) a gain/phase margin theorem is proposed for stability analysis by regulating the feedback gain, and (2) a stability theorem based on rational function approximation of the time delay is presented for dealing with the transcendental polynomial characteristic equations, which is equivalent to the analysis from the algebraic polynomial characteristic equation. Both approaches give coherent results for stability analysis by regulating the feedback gain. The approaches are applicable to nonlinear systems, which are linearizable in the neighborhood of the operating points. The low complexity of the controllers does not require hard analytical derivation/numerical calculations to produce an acceptable control performance for the considered systems. Several representative simulation case studies provide demonstrations of computational experiments against those analytically derived and guidance for potential applications. Full article

The well-known power-law fractal element was determined to need several important revisions by the authors of this work. It is now possible to demonstrate that any scaling equation associated with a fractal element is actually K-fold degenerated and includes previously unknown but crucial adjustments. These new discoveries have the potential to significantly alter the preexisting theory and create new connections between it and its experimental support, particularly when it comes to measurements of the impedances of diverse metamaterials. It is now easy to demonstrate that any random curve with a clearly stated tendency in a specific range of scales is self-similar using the method involving reduction to three invariant points (Ymx, Ymn, and Ymin). This useful procedure indicates that the chosen random curve, even after being compressed a certain number of times, still resembles the original curve. Based on this common peculiarity, it is now possible to derive “a universal” fitting function that can be used in a variety of applied sciences, particularly those that deal with complex systems, to parametrize many initial curves when a model fitting function derived from a simple model is not present. This self-similarity principle-derived function demonstrates its effectiveness in data linked to photodiode noise and the smoothed integral curves produced from well-known transcendental numbers E and Pi, which are considered in the paper as an example. Full article

In this study, a new switched-capacitor-based seven-level inverter topology with a photovoltaic system is presented. The proposed topology requires a smaller number of devices and has the ability to self-balance the voltage across the capacitor. The proposed topology configuration is simple and has the ability to extend to higher levels of voltage. This multilevel inverter topology is suitable for low- and medium-voltage applications with photovoltaic (PV) system integration. To improve the PV system efficiency as an input of a DC–DC boost converter, a Fuzzy logic-based maximum power point controller technique is used. A PV system with a DC–DC boost converter integrates with the proposed seven-level inverter topology. The anti-predatory particle swarm optimization (APSO) technique is used to solve the non-linear transduction equations of the seven-level PV switched-capacitor-based multilevel inverter (7L−PV−SCMLI) topology. The proposed APSO is described to minimize the harmonics in the multilevel inverter (MLIs), which is a complex optimization problem involving a non-linear transcendental equation. Furthermore, APSO can be applied in order to solve non-linear transcendental equations for all symmetrical and asymmetrical MLIs that have equal and non-equal DC sources. The APSO-based selected harmonic elimination (SHE) technique obtained the best switching angle value, and the optimized obtained switching angles reduced the total harmonic distortion (THD) of 7L−PV−SCMLI. Full article

The two families of principal stress trajectories can be regarded as an orthogonal curvilinear coordinate system under plane strain and axial symmetry. Under plane strain, the equilibrium equations in conjunction with a yield criterion comprise a statically determinate system. Under axial symmetry, a statically determinate system results from the above equations supplemented with the hypothesis of Haar von Karman. In both cases, the compatibility equations for mapping the principal line coordinate system to a given coordinate system show that the scale factors of the former satisfy a simple algebraic or transcendental equation for many yield criteria. Using this equation, one can develop a method for reducing boundary value problems in plasticity to purely geometric problems. The method is independent of any flow rule that can be chosen to calculate displacement or velocity fields, as well as independent whether elastic strains are included. The present paper summarizes available results related to using principal stress trajectories in plasticity and emphasizes the advantages of the method above. Full article

This study investigates the effect of quantum size and an external magnetic field on the optoelectronic properties of a cylindrical Al${}_x$Ga${}_{1-x}$As/GaAs-based core/shell nanowire. We used the one-band effective mass model to describe the Hamiltonian of an interacting electron-donor impurity system and employed two numerical methods to calculate the ground state energies: the variational and finite element methods. With the finite confinement barrier at the interface between the core and the shell, the cylindrical symmetry of the system revealed proper transcendental equations, leading to the concept of the threshold core radius. Our results show that the optoelectronic properties of the structure strongly depend on core/shell sizes and the strength of the external magnetic field. We found that the maximum probability of finding the electron occurs in either the core or the shell region, depending on the value of the threshold core radius. This threshold radius separates two regions where physical behaviors undergo changes and the applied magnetic field acts as an additional confinement. Full article

The present paper is concerned with the design of the plane-strain bending of wide sheets driven by the sheet’s geometric shape after unloading. The solution is based on the ideal flow theory. It determines the loading path (the dependence between the dimensionless bending moment and the dimensionless tensile force) that produces the desirable shape. The ideal flow theory is valid for the rigid perfectly plastic material model. A method proposed in the literature combines a rigid plastic solution at loading and an elastic solution for springback. The final design solution is practically analytical. A numerical method is only necessary to solve a system of transcendental equations. An illustrative example is provided. This example shows that, in some cases, it is sufficient to use the solution at loading to determine an accurate loading path that produces the desirable sheet’s shape after springback. Full article

In traditional power systems, a dedicated communication channel is utilized to transfer the frequency measurements. With the deregulation and reconstruction of power systems, the information is sent through a shared communication network that makes time delays unavoidable in load frequency control (LFC) systems. With the existence of time delays, the LFC system becomes a standard time delay system that complicates the stability analysis and controller synthesis. In this paper, we present a new approach for analyzing the stability and determining the delay margin of the LFC system. By introducing a new variable, the transcendental equation is converted to nonlinear equations. To find the crossing frequencies, the nonlinear equations are solved, which is simpler than solving a set of linear matrix inequalities. A single-area and a multi-area LFC system are selected as case studies. The new method accurately determines the delay margin of the LFC system with phase and gain margin. The effect of the PI controller gains on the delay margin is also considered. A sensitivity analysis is conducted to discover the effects of the system parameters on the delay margin, and it is found that the primary loop parameters have a powerful influence on the delay margin. The stability region of the LFC system is also clearly identified through the proposed method. The influence of the system parameters on the stability region is studied. Compared to the published methods in the literature, the proposed method has a simpler structure while giving more accurate results. Full article

Rotor unbalance identification plays a critical role in balancing rotors. In this paper, concerned with multi-disc and multi-span rotor-bearing systems, two novel algorithms called the Single Direction Algorithm (SDA) and the Two Orthogonal Direction Algorithm (TODA) are proposed for identifying rotor unbalance from unbalance responses. A matrix method is proposed to solve the problem of the equations being non-linear transcendental, there being too many unknown variables in the equations, and rotor unbalances and bearing coefficients being coupled together. The unbalance responses at all the eccentric discs are necessary for identifying their unbalances. Numerical simulations are conducted to validate the proposed methods. Moreover, an adjustment point is found, and a proper sensor resolution is suggested to achieve high identification accuracy by means of numerical studies. In addition, the identification accuracy of SDA is better than TODA, and SDA is more practical and suitable for medium-speed and high-speed rotors. The proposed algorithms have the flexibility to incorporate any number of bearings and discs and provide a technique for monitoring rotor unbalance without test runs or external exciters. Full article

This article presents the study of the stability of single-input and multiple-input systems with point or distributed state delay and input delay and input saturation. By a linear transformation applied to the initial system with delay, a system is obtained without delay, but which is equivalent from the point of view of stability. Next, using sufficient conditions for the global asymptotic stability of linear systems with bounded control, new sufficient conditions are obtained for global asymptotic stability of the initial system with state delay and input delay and input saturation. In addition, the article presents the results on the instability and estimation of the stability region of the delay and input saturation system. The numerical simulations confirming the results obtained on stability were performed in the MATLAB/Simulink environment. A method is also presented for solving a transcendental matrix equation that results from the process of equating the stability of the systems with and without delay, a method which is based on the computational intelligence, namely, the Particle Swarm Optimization (PSO) method. Full article

A semiclassical approach based on the WKB–Maslov method is developed for the kinetic ionization equation in dense plasma with approximations characteristic of metal vapor active media excited by a contracted discharge. We develop the technique for constructing the leading term of the semiclassical asymptotics of the Cauchy problem solution for the kinetic equation under the supposition of weak diffusion. In terms of the approach developed, the local cubic nonlinear term in the original kinetic equation is considered in a nonlocal form. This allows one to transform the nonlinear nonlocal kinetic equation to an associated linear partial differential equation with a given accuracy of the asymptotic parameter using the dynamical system of moments of the desired solution of the equation. The Cauchy problem solution for the nonlinear nonlocal kinetic equation can be obtained from the solution of the associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation. Within the developed approach, the plasma relaxation in metal vapor active media is studied with asymptotic solutions expressed in terms of higher transcendental functions. The qualitative analysis of such the solutions is given. Full article