Search Results (253)

This article investigates modeling issues of fractional-order singular systems. The state estimation can be solved by using the particle filter. An improved Adaptive Moment Estimation (Adam) method—the Amsgrad algorithm can handle the optimization problem caused by parameter estimation. Thus, a hybrid approach that combines the particle filter and Amsgrad is proposed to estimate both parameters and states in fractional-order singular systems. This method leverages the strengths of the particle filter in handling nonlinear and high-dimensional problems, as well as the stability of the Amsgrad algorithm in optimizing parameters for dynamic systems. Then, the identification process is concluded to achieve a more accurate joint estimation. To validate the feasibility of the proposed hybrid algorithm, simulations involving three-order and four-order fractional-order singular systems are conducted. A comparative analysis with other algorithms demonstrates that the proposed method behaves better than the standard particle filter, Amsgrad and Gravitational search algorithm-Kalman filter algorithms. Full article

The fixed-time trajectory tracking control problem of the uncertain nonlinear Euler–Lagrange system is studied. To ensure the fast, high-precision trajectory tracking performance of this system, a non-singular terminal sliding-mode controller based on Gaussian process regression is proposed. The control algorithm proposed in this paper is applicable to periodic motion scenarios, such as spacecraft autonomous orbital rendezvous and repetitive motions of robotic manipulators. Gaussian process regression is employed to establish an offline data-driven model, which is utilized for compensating parametric uncertainties and external disturbances. The non-singular terminal sliding-mode control strategy is used to avoid singularity and ensure fast convergence of tracking errors. In addition, under the Lyapunov framework, the fixed-time convergence stability of the closed-loop system is rigorously demonstrated. The effectiveness of the proposed control scheme is verified through simulations on a spacecraft rendezvous mission and periodic joint trajectory tracking for a robotic manipulator. Full article

Uncertain singular non-causal systems represent a class of singular systems distinguished by the presence of regularity constraints and the involvement of uncertain variables. This paper considers optimal control problems for such systems under the Hurwicz criterion. By integrating dynamic programming with uncertainty theory, a recurrence equation is developed to address these problems. This equation has been shown to be effective in handling the optimal control problems of both linear and nonlinear uncertain singular non-causal systems, thereby enabling the derivation of analytical expressions for their corresponding optimal solutions. Moreover, the transformation of the original system into forward and backward subsystems reveals a fundamental temporal and structural symmetry, which significantly contributes to problem simplification. A detailed example is presented to illustrate the proposed results. Full article

The widespread integration of power electronic devices and renewable energy sources into power systems has significantly exacerbated voltage and current waveform distortion issues, where asymmetric loads—including single-phase nonlinear equipment and unbalanced three-phase power electronic installations—serve as critical harmonic sources whose inherent nonlinear and asymmetric characteristics increasingly compromise power quality. To enhance power quality management, this paper proposes a universal harmonic source modeling and operational state identification methodology integrating physical mechanisms with data-driven algorithms. The approach establishes an RL-series equivalent impedance model as its physical foundation, employing singular value decomposition and Z-score criteria to accurately characterize asymmetric load dynamics; subsequently applies Variational Mode Decomposition (VMD) to extract time-frequency features from equivalent impedance parameters while utilizing Density-Based Spatial Clustering (DBSCAN) for the high-precision identification of operational states in asymmetric loads; and ultimately constructs state-specific harmonic source models by partitioning historical datasets into subsets, substantially improving model generalizability. Simulation and experimental validations demonstrate that the synergistic integration of physical impedance modeling and machine learning methods precisely captures dynamic harmonic characteristics of asymmetric loads, significantly enhancing modeling accuracy, dynamic robustness, and engineering practicality to provide an effective assessment framework for power quality issues caused by harmonic source integration in distribution networks. Full article

The implementation of chaotic behavior and a sensitivity assessment of the newly developed M-fractional Kuralay-II equation are the foremost objectives of the present study. This equation has significant possibilities in control systems, electrical circuits, seismic wave propagation, economic dynamics, groundwater flow, image and signal denoising, complex biological systems, optical fibers, plasma physics, population dynamics, and modern technology. These applications demonstrate the versatility and advantageousness of the stated model for complex systems in various scientific and engineering disciplines. One more essential objective of the present research is to find closed-form wave solutions of the assumed equation based on the $({\displaystyle \frac{G^{\prime }}{G^{\prime }+G+A}})$-expansion approach. The results achieved are in exponential, rational, and trigonometric function forms. Our findings are more novel and also have an exclusive feature in comparison with the existing results. These discoveries substantially expand our understanding of nonlinear wave dynamics in various physical contexts in industry. By simply selecting suitable values of the parameters, three-dimensional (3D), contour, and two-dimensional (2D) illustrations are produced displaying the diagrammatic propagation of the constructed wave solutions that yield the singular periodic, anti-kink, kink, and singular kink-shape solitons. Future improvements to the model may also benefit from what has been obtained as well. The various assortments of solutions are provided by the described procedure. Finally, the framework proposed in this investigation addresses additional fractional nonlinear partial differential equations in mathematical physics and engineering with excellent reliability, quality of effectiveness, and ease of application. Full article

In this study, we investigate the fractional Tzitzéica equation, a nonlinear evolution equation known for modeling complex phenomena in various scientific domains such as solid-state physics, crystal dislocation, electromagnetic waves, chemical kinetics, quantum field theory, and nonlinear optics. Using the (G′/G, 1/G)-expansion approach, we derive different categories of exact solutions, like hyperbolic, trigonometric, and rational functions. The beta fractional derivative is used here to generalize the classical idea of the derivative, which preserves important principles. The derived solutions with broader nonlinear wave structures are periodic waves, breathers, peakons, W-shaped solitons, and singular solitons, which enhance our understanding of nonlinear wave dynamics. In relation to these results, the findings are described by showing the solitons’ physical behaviors, their stabilities, and dispersions under fractional parameters in the form of contour plots and 2D and 3D graphs. Comparisons with earlier studies underscore the originality and consistency of the (G′/G, 1/G)-expansion approach in addressing fractional-order evolution equations. It contributes new solutions to analytical problems of fractional nonlinear integrable systems and helps understand the systems’ dynamic behavior in a wider scope of applications. Full article

This paper establishes the well-posedness of Cauchy-type problems with non-symmetric initial conditions for nonlinear implicit Hilfer fractional differential equations of general fractional orders in weighted function spaces. Using fixed-point techniques, we first prove the existence of solutions via Schaefer’s fixed-point theorem. The uniqueness and Ulam–Hyers stability are then derived using Banach’s contraction principle. By introducing a novel singular-kernel Gronwall inequality, we extend the analysis to Ulam–Hyers–Rassias stability and continuous dependence on initial data. The theoretical framework is unified for general fractional orders and validated through examples, demonstrating its applicability to implicit systems with memory effects. Key contributions include weighted-space analysis and stability criteria for this class of equations. Full article

This paper proposes a multi-party verifiably collaborative system for encrypting the nonlinear and the non-stationary biomedical signals captured by biomedical sensors via the singular spectrum analysis (SSA)-based chaotic networks. In particular, the raw signals are first decomposed into the multiple components by the SSA. Then, these decomposed components are fed into the chaotic filter bank networks for performing the encryption. To perform the multi-party verifiably collaborative encryption, the window length of the SSA and the total number of the layers in the chaotic network are flexibly designed to match the total number of the collaborators. The computer numerical simulation results show that our proposed system achieves a good encryption performance. Full article

This work presents a robust box-constrained nonlinear least-squares algorithm for accurately fitting the Jones–Wilkins–Lee (JWL) equation of state parameters, which describes the isentropic expansion of detonation products from high-energy materials. In the energetic material literature, there are plenty of methods that address this problem, and in some cases, it is not fully clear which method is employed. We provide a fully detailed numerical framework that explicitly enforces Chapman–Jouguet (CJ) constraints and systematically separates the contributions of different terms in the JWL expression. The algorithm leverages a trust-region Gauss–Newton method combined with singular value decomposition to ensure numerical stability and rapid convergence, even in highly overdetermined systems. The methodology is validated through comprehensive comparisons with leading thermochemical codes such as CHEETAH 2.0, ZMWNI, and EXPLO5. The results demonstrate that the proposed approach yields lower residual fitting errors and improved consistency with CJ thermodynamic conditions compared to standard fitting routines. By providing a reproducible and theoretically based methodology, this study advances the state of the art in JWL parameter determination and improves the reliability of energetic material simulations. Full article

This study investigates a nonlinear Schrödinger equation that includes Kudryashov’s quintic power-law refractive index along with dual-form nonlocal nonlinearity. Employing dynamical systems theory, we analyze the model through a traveling-wave transformation, reducing it to a singular yet integrable traveling-wave system. The dynamical behavior of the corresponding regular system is examined, revealing phase trajectories bifurcations under varying parameter conditions. Furthermore, explicit solutions—including periodic, homoclinic, and heteroclinic solutions—are derived for distinct parameter regimes. Full article

This paper concerns accurate spectral collocation solutions, more precisely Chebyshev collocation (ChC), to some third-order nonlinear and singular boundary value problems on unbounded domains. The problems model some draining or coating fluid flows. We use exclusively ChC, in the form of Chebfun, avoid any obsolete shooting-type method, and provide reliable information about the convergence and accuracy of the method, including the order of Newton’s method involved in solving the nonlinear algebraic systems. As a complete novelty, we combine a graphical representation of the convergence of the Newton method with a numerical estimate of its order of convergence for a more realistic value. We treat five challenging examples, some of which have only been solved by approximate methods. The found numerical results are judged in the context of existing ones; at least from a qualitative point of view, they look reasonable. Full article

This study proposes a novel fuzzy disturbance observer (FDO)-augmented adaptive nonsingular terminal sliding mode control (NTSMC) framework for multi-joint robotic manipulators, addressing critical challenges in trajectory tracking precision and disturbance rejection. Unlike conventional disturbance observers requiring prior knowledge of disturbance bounds, the proposed FDO leverages fuzzy logic principles to dynamically estimate composite disturbances—including unmodeled dynamics, parameter perturbations, and external torque variations—without restrictive assumptions about disturbance derivatives. The control architecture achieves rapid finite-time convergence by integrating the FDO with a singularity-free terminal sliding manifold and an adaptive exponential reaching law while significantly suppressing chattering effects. Rigorous Lyapunov stability analysis confirms the uniform ultimate boundedness of tracking errors and disturbance estimation residuals. Comparative simulations on a 2-DOF robotic arm demonstrate a 97.28% reduction in root mean square tracking errors compared to PD-based alternatives and a 73.73% improvement over a nonlinear disturbance observer-enhanced NTSMC. Experimental validation on a physical three-joint manipulator platform reveals that the proposed method reduces torque oscillations by 58% under step-type disturbances while maintaining sub-millimeter tracking accuracy. The framework eliminates reliance on exact system models, offering a generalized solution for industrial manipulators operating under complex dynamic uncertainties. Full article

We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of p-regularity theory, which has been successfully developed over the past four decades. We illustrate the theory with applications to degenerate problems in various areas of mathematics, including optimization and differential equations. In particular, we address the problem of describing the tangent cone to the solution set of nonlinear equations in singular cases. The structure of p-factor operators is used to propose optimality conditions and to construct novel numerical methods for solving degenerate nonlinear equations and optimization problems. The numerical methods presented in this paper represent the first approaches targeting solutions to degenerate problems such as the Van der Pol differential equation, boundary-value problems with small parameters, and partial differential equations where Poincaré’s method of small parameters fails. Additionally, these methods may be extended to nonlinear degenerate dynamical systems and other related problems. Full article

This work presents an alternative method for defining feasible joint-space boundaries and their corresponding geometric workspace in a planar robotic system. Instead of relying on traditional numerical approaches that require extensive sampling and collision detection, the proposed method constructs a continuous boundary by identifying the key intersection points of boundary functions. The feasibility region is further refined through centroid-based scaling, addressing singularity issues and ensuring a well-defined trajectory. Comparative analyses demonstrate that the final robot pose and reachability depend on the selected traversal path, highlighting the nonlinear nature of the workspace. Additionally, an evaluation of traditional numerical methods reveals their limitations in generating continuous boundary trajectories. The proposed approach provides a structured method for defining feasible workspaces, improving trajectory planning in robotic systems. Full article

This paper introduces a novel model-free nonsingular fixed-time sliding mode control (MF-NFxTSMC) strategy for precise trajectory tracking in robot arm systems. Unlike conventional sliding mode control (SMC) approaches that require accurate dynamic models, the proposed method leverages the time delay estimation (TDE) approach to effectively estimate system dynamics and external disturbances in real-time, enabling a fully model-free control solution. This significantly enhances its practicality in real-world scenarios where obtaining precise models is challenging or infeasible. A significant innovation of this work lies in designing a novel fixed-time control framework that achieves faster convergence than traditional fixed-time methods. Building on this, a novel MF-NFxTSMC law is developed, featuring a novel singularity-free fixed-time sliding surface (SF-FxTSS) and a novel fixed-time reaching law (FxTRL). The proposed SF-FxTSS incorporates a dynamic proportional term and an adaptive exponent, ensuring rapid convergence and robust tracking. Notably, its smooth transition between nonlinear and linear dynamics eliminates the singularities often encountered in terminal and fixed-time sliding mode surfaces. Additionally, the designed FxTRL effectively suppresses chattering while guaranteeing fixed-time convergence, leading to smoother control actions and reduced mechanical stress on the robotic hardware. The fixed-time stability of the proposed method is rigorously proven using the Lyapunov theory. Numerical simulations on the SAMSUNG FARA AT2 robotic platform demonstrate the superior performance of the proposed method in terms of tracking accuracy, convergence speed, and control smoothness compared to existing strategies, including conventional SMC, finite-time SMC, approximate fixed-time SMC, and global fixed-time nonsingular terminal SMC (NTSMC). Overall, this approach offers compelling advantages, i.e., model-free implementation, fixed-time convergence, singularity avoidance, and reduced chattering, making it a practical and scalable solution for high-performance control in uncertain robotic systems. Full article