A Proof of the Non-Singularity of the <i>D</i>Matrix Used in Deriving the Two—Step Butcher’s Hybrid Scheme for the Solution of Initial Value Problems

In this paper, we state and prove the conditions for the non-singularity of the D matrix used in deriving the continuous form of the Two-step Butcher’s hybrid scheme and from it the discrete forms are deduced. We also show that the discrete scheme gives outstanding results for the solution of stiff and non-stiff initial value problems than the 5th order Butcher’s algorithm in predictor-corrector form.

Oscillating gravity,non-singularity and mass quantization from Moffat stochastic gravity arguments

In Moffat stochastic gravity arguments, the spacetime geometry is assumed to be a fluctuating background and the gravitational constant is a control parameter due to the presence of a timedependent Gaussian white noise ξ(t). In such a surrounding, both the singularities of gravitational collapse and the Big Bang have a zero probability of occurring. In this communication, we generalize Moffat’s arguments by adding a random temporal tiny variable for a smoothing purpose and creating a white Gaussian noise process with a short correlation time. The Universe accordingly is found to be non-singular and is dominated by an oscillating gravity. A connection with a quantum oscillator was established and analyzed. Surprisingly, the Hubble mass which emerges in extended supergravity may be quantized.

The Non-Singular Fast Terminal Sliding Mode Control of Interior Permanent Magnet Synchronous Motor Based on Deep Flux Weakening Switching Point Tracking

This paper presents a novel non-singular fast terminal sliding mode control(NFTSMC)based on the deep flux weakening switching point tracking method in order to improve the control performance of permanent interior magnet synchronous motor(IPMSM)drive systems.The mathematical model of flux weakening(FW)control is established,and the deep flux weakening switching point is calculated accurately by analyzing the relationship between the torque curve and voltage decline curve.Next,a second-order NFTSMC is designed for the speed loop controller to ensure that the system converges to the equilibrium state in finite time.Then,an extended sliding mode disturbance observer(ESMDO)is designed to estimate the uncertainty of the system.Finally,compared with both the PI control and sliding mode control(SMC)by simulations and experiments with different working conditions,the method proposed has the merits of accelerating convergence,improving steady-state accuracy,and minimizing the current and torque pulsation.

A Novel Disturbance Observer Based Fixed-Time Sliding Mode Control for Robotic Manipulators With Global Fast Convergence

This paper proposes a new global fixed-time sliding mode control strategy for the trajectory tracking control of uncertain robotic manipulators.First,a fixed-time disturbance observer(FTDO) is designed to deal with the adverse effects of model uncertainties and external disturbances in the manipulator systems.Then an adaptive scheme is used and the adaptive FTDO(AFTDO) is developed,so that the priori knowledge of the lumped disturbance is not required.Further,a new non-singular fast terminal sliding mode(NFTSM) surface is designed by using an arctan function,which helps to overcome the singularity problem and enhance the robustness of the system.Based on the estimation of the lumped disturbance by the AFTDO,a fixed-time non-singular fast terminal sliding mode controller(FTNFTSMC)is developed to guarantee the trajectory tracking errors converge to zero within a fixed time.The settling time is independent of the initial state of the system.In addition,the stability of the AFTDO and FTNFTSMC is strictly proved by using Lyapunov method.Finally,the fixed-time NFESM(FTNFTSM) algorithm is validated on a 2-link manipulator and comparisons with other existing sliding mode controllers(SMCs) are performed.The comparative results confirm that the FTNFTSMC has superior control performance.

Evolutionary Safe Padé Approximation Scheme for Dynamical Study of Nonlinear Cervical Human Papilloma Virus Infection Model

This study proposes a structure-preserving evolutionary framework to find a semi-analytical approximate solution for a nonlinear cervical cancer epidemic(CCE)model.The underlying CCE model lacks a closed-form exact solution.Numerical solutions obtained through traditional finite difference schemes do not ensure the preservation of the model’s necessary properties,such as positivity,boundedness,and feasibility.Therefore,the development of structure-preserving semi-analytical approaches is always necessary.This research introduces an intelligently supervised computational paradigm to solve the underlying CCE model’s physical properties by formulating an equivalent unconstrained optimization problem.Singularity-free safe Padérational functions approximate the mathematical shape of state variables,while the model’s physical requirements are treated as problem constraints.The primary model of the governing differential equations is imposed to minimize the error between approximate solutions.An evolutionary algorithm,the Genetic Algorithm with Multi-Parent Crossover(GA-MPC),executes the optimization task.The resulting method is the Evolutionary Safe PadéApproximation(ESPA)scheme.The proof of unconditional convergence of the ESPA scheme on the CCE model is supported by numerical simulations.The performance of the ESPA scheme on the CCE model is thoroughly investigated by considering various orders of non-singular Padéapproximants.

Inconsistence of Non-Singular Spherically Symmetric Solution and a Satisfactory Energy-Momentum Complex

Two spherically symmetric non-singular black hole solutions in Moiler tetrad theory of gravitation have been obtained. Although the two solutions have the same form of metric （spherically symmetric nonsingular black hole）, their energy contents are different. We use another method given by Gibbons and Hawking to calculate the energy content of these solutions. We also obtained different value of energy. Study the requirements of a satisfactory energymomentum complex given by Moiler we find that the second solution, which behaves as 1/√r, is not transformed as a four-vector under Lorentz transformation.

On non-singular GRADELA crack fields

A brief account is provided on crack-tip solutions that have recently been published in the literature by employing the so-called GRADELA model and its variants. The GRADELA model is a simple gradient elasticity theory involving one internal length in addition to the two Lame＇ constants, in an effort to eliminate elastic singularities and discontinuities and to interpret elastic size effects. The non-singular strains and non-singular （but sometimes singular or even hypersingular） stresses derived this way under different boundary conditions differ from each other and their physical meaning in not clear. This is discussed which focus on the form and physical meaning of non-singular solutions for crack-tip stresses and strains that are possible to obtain within the GRADELA model and its extensions.

Non-Singular Trees, Unicyclic Graphs and Bicyclic Graphs

We called graph G non-singular if adjacency matrix A (G) of G is non-singular. A connected graph with n vertices and n-1, n and n+1 edges are called the tree, the unicyclic graph and the bicyclic graph. Respectively, as we all know, each connected bicyclic graph must contain ∞(a,s,b) or?θ(p,l,q) as the induced subgraph. In this paper, by using three graph transformations which do not change the singularity of the graph, the non-singular trees, unicyclic graphs and bicyclic graphs are obtained.

Creation of a Non-Expanding, Non-Singular Universe

A theory of gravitation in flat space-time is applied to homogeneous, isotropic cosmological models. There are non-singular cosmological models. A natural interpretation is a non-expanding universe. The redshift is an intrinsic effect and not a Doppler effect. The universe contains only energy in the beginning, i.e. no matter exists. In the course of time matter and radiation are created from energy where the whole energy is conserved. Matter increases with time but a certain time after the beginning of the universe the creation of matter is finished and the universe appears like a static one. A modified Hubble law is considered which may explain the high redshifts of objects in the universe without the assumption of dark energy.

A NEW BOUNDARY INTEGRAL FORMULATION WITH NO SINGULARITY

The paper discusses how to reduce higher singularity order of a boundary integral equation. The approach will be discussed in some detail for plane elasticity.Numerical results for the meshes of unequal length boundary elements are reported.Higher precision for both deflection and force is obtained than that obtained with a general boundary element method.

Fixed time integral sliding mode controller and its application to the suppression of chaotic oscillation in power system

Chattering phenomenon and singularity are still the main problems that hinder the practical application of sliding mode control. In this paper, a fixed time integral sliding mode controller is designed based on fixed time stability theory, which ensures precise convergence of the state variables of controlled system, and overcomes the drawback of convergence time growing unboundedly as the initial value increases in finite time controller. It makes the controlled system converge to the control objective within a fixed time bounded by a constant as the initial value grows, and convergence time can be changed by adjusting parameters of controllers properly. Compared with other fixed time controllers, the fixed time integral sliding mode controller proposed in this paper achieves chattering-free control, and integral expression is used to avoid singularity generated by derivation. Finally, the controller is used to stabilize four-order chaotic power system. The results demonstrate that the controller realizes the non-singular chattering-free control of chaotic oscillation in the power system and guarantees the fixed time convergence of state variables, which shows its higher superiority than other finite time controllers.

MHD Maxwell Fluid with Heat Transfer Analysis under Ramp Velocity and Ramp Temperature Subject to Non-Integer Differentiable Operators

The main focus of this study is to investigate the impact of heat generation/absorption with ramp velocity and ramp temperature on magnetohydrodynamic(MHD)time-dependent Maxwell fluid over an unbounded plate embedded in a permeable medium.Non-dimensional parameters along with Laplace transformation and inversion algorithms are used to find the solution of shear stress,energy,and velocity profile.Recently,new fractional differential operators are used to define ramped temperature and ramped velocity.The obtained analytical solutions are plotted for different values of emerging parameters.Fractional time derivatives are used to analyze the impact of fractional parameters(memory effect)on the dynamics of the fluid.While making a comparison,it is observed that the fractional-order model is best to explain the memory effect as compared to classical models.Our results suggest that the velocity profile decrease by increasing the effective Prandtl number.The existence of an effective Prandtl number may reflect the control of the thickness of momentum and enlargement of thermal conductivity.The incremental value of the M is observed for a decrease in the velocity field,which reflects to control resistive force.Further,it is noted that the Atangana-Baleanu derivative in Caputo sense(ABC)is the best to highlight the dynamics of the fluid.The influence of pertinent parameters is analyzed graphically for velocity and energy profile.Expressions for skin friction and Nusselt number are also derived for fractional differential operators.

A Generalization of the Cayley-Hamilton Theorem

It is proposed to generalize the concept of the famous classical Cayley-Hamilton theorem for square matrices wherein for any square matrix A, the det (A-xI) is replaced by det f(x) for arbitrary polynomial matrix f(x).

Active fault-tolerant control scheme of aerial manipulators with actuator faults

In this paper,an active fault-tolerant control(FTC)strategy of aerial manipulators based on non-singular terminal sliding mode(NTSM)and extended state observer(ESO)is proposed.Firstly,back-stepping technology is adopted as the control framework to ensure the global asymptotic stability of the closed-loop system.Next,the NTSM with estimated parameters of actuator faults is used as main robustness controller to deal with actuator faults.Then,the ESO is utilized to estimate and compensate the complex coupling effects and external disturbances.The Lyapunov stability theory can guarantee the asymptotic stability of aerial manipulators system with actuator faults and external disturbances.The proposed FTC scheme considers both actuator fault and modelling errors,combined with the adaptive law of actuator fault,which has better performance than traditional FTC scheme,such as NTSM.Finally,several comparative simulations are conducted to illustrate the effectiveness of the proposed FTC scheme.

T-STRESS IN PIEZOELECTRIC SOLID

The non-singular and bounded terms for stresses near the crack tip were investigated.The crack problem in a transversely isotropic piezoelectric solid for the plane problem was dealt with. The principle of superposition and the Plemelj formulation were introduced. The non-singular terms are given by solving Rieman-Hilbert problem. It is shown that the non-singular terms are influenced by the elastic and electric constants.

UNIQUENESS OF SOLUTION OF FIELD POINT OF SINGULAR SOURCE OUTSIDE-REGION-DISTRIBUTION METHOD

The uniqueness of solution of field point, inside a convex region due to singular source(s) with kernel function decreasing with distance increasing, outside-region-distribution(s) such that the boundary condition expressed by the response of the source(s) is satisfied, is proved by using the condition of kernel function decreasing with distance increasing anal an integral inequality. Examples of part of these singular sources such as Kelvin's point force, Point-Ring-Couple (PRC) etc. are given. The proof of uniqueness of solution of field point in a twisted shaft of revolution due to PRC distribution is given as an example of application.

A novel MPTC sensorless control strategy for ANFTSMC with ESO to control PMSM

Aiming at the problem that the traditional control strategy of permanent magnet synchronous motor(PMSM)for electric vehicles has low control performance,a novel adaptive non-singular fast terminal sliding mode control(ANFTSMC)model predictive torque control(MPTC)strategy is proposed.A new adaptive exponential approach rate is designed,and the traditional switching function sgn()is replaced by the hyperbolic tangent function tanh().A new ANFTSMC with extended state observer(ESO)is constructed as the speed regulator of the system,and ESO can observe disturbances.This improved method weakens chattering and improves the robustness of the system.To realize sensorless control of the speed control system,an ESO speed observer based on tanh(Fal)is constructed.Compared with the traditional ESO based on Fal function,the observation error is smaller,and the observation accuracy is higher.Finally,aiming at the model predictive torque control strategy used,a new objective function construction method is proposed,which avoids the design of weight coefficient,and the traditional voltage vector selection method is improved and optimized,which reduces the calculation amount of the algorithm.

Dual sliding mode coordinated control of manipulator grasping system with visual assistance

The operation efficiency of the manipulator is placed in the primary position in automatic production. This paper proposes a coordinated control strategy for joint servo and visual servo to enable timely transfer and accurate gripping in the working area. Aiming at the issues of chattering and slow convergence of traditional sliding mode controller, a fast variable power reaching rate on the basis of the non-singular fast terminal sliding mode controller is proposed, which can effectively reduce the convergence time and chattering. For the purpose of addressing the problem that the traditional visual servo control method is sensitive to the environment, a visual servo controller based on integral sliding mode is proposed, to ensure the favorable positioning accuracy of the manipulator. Based on the two proposed controllers mentioned above, a coordinated control strategy is used to implement the control of the manipulator. Finally, the upper computer software is developed using the C# programming language to monitor the workstation. The feasibility of the above-mentioned method is verified through multiple simulations and experiments.

Multi-Variable Non-Singular BEM for 2-D Potential Problems

A multi-variable non-singular boundary element method (MNBEM) is presented for 2-D potential problems. This method is based on the coincident collocation of non-singular boundary integral equations (BIEs) of the potential and its derivatives, where the nodal potential derivatives are considered independent of the nodal potential and flux. The system equation is solved to determine the unknown boundary potentials and fluxes, with high accuracy boundary nodal potential derivatives obtained from the solution at the same time. A modified Gaussian elimination algorithm was developed to improve the solution efficiency of the final system equation. Numerical examples verify the validity of the proposed algorithm.

Solutions to the Complex Korteweg-de Vries Equation: Blow-up Solutions and Non-Singular Solutions

In the paper two kinds of solutions are derived for the complex Korteweg-de Vries equation, includ- ing blow-up solutions and non-singular solutions. We derive blow-up solutions from known 1-soliton solution and a double-pole solution. There is a complex Miura transformation between the complex Korteweg-de Vries equation and a modified Kortcweg-de Vries equation. Using the transformation, solitons, breathers and rational solutions to the com- plex Korteweg-de Vries equation are obtained from those of the modified Korteweg-de Vries equation. Dynamics of the obtained solutions are illustrated.