Nonsingular System Research Articles

Adaptive Fuzzy Predefined-Time Control for Third-Order Heterogeneous Vehicular Platoon Systems With Dead Zone

This paper investigates the problem of fuzzy adaptive predefined-time terminal sliding mode (TSM) control for a third-order heterogeneous vehicular platoon system (HVPS) with an unknown dead-zone. For the purpose of approximating unknown nonlinear functions, fuzzy logic systems (FLSs) are utilized. Additionally, the impact of the dead-zone on the performance of the control may be lessened by building the dead-zone compensation. A tracking error based on the modified constant time headway (MCTH) policy is built to get rid of the assumption of zero initial spacing and decrease the distance between vehicles at the same time. A unique nonsingular TSM control system is then built using the predefined-time stability criterion, with the help of Lyapunov functions, it is possible to demonstrate both the individual and the string stability (SS) of the whole heterogeneous vehicle platoon in a predefined time. Finally, a series of simulations are shown to demonstrate the validity of the proposed results.

On forms in prime variables

Let F 1 F_1 , …, F R F_R be homogeneous polynomials of degree d ⩾ 2 d\geqslant 2 with integer coefficients in n n variables, and let F = ( F 1 , … , F R ) \mathbf {F}=(F_1,\ldots ,F_R) . Suppose that F 1 F_1 , …, F R F_R is a non-singular system and n ⩾ 4 d + 2 d 2 R 5 n\geqslant 4^{d+2}d^2R^5 . We prove that there are infinitely many solutions to F ( x ) = 0 \mathbf {F}(\mathbf {x})=\mathbf {0} in prime coordinates if (i) F ( x ) = 0 \mathbf {F}(\mathbf {x})=\mathbf {0} has a non-singular solution over the p p -adic units U p \mathbb {U}_p for all prime numbers p p , and (ii) F ( x ) = 0 \mathbf {F}(\mathbf {x})=\mathbf {0} has a non-singular solution in the open cube ( 0 , 1 ) n (0,1)^n .

THE FRACTIONAL-ORDER SYSTEM OF SINGULAR AND NON-SINGULAR THERMO-ELASTICITY SYSTEM IN THE SENSE OF HOMOTOPY PERTURBATION TRANSFORM METHOD

This paper investigates the mathematical result of the nonlinear fractional scheme of equations representing the singular and non-singular thermoelastic system. The suggested method is a graceful combination of Shehu transformation with homotopy perturbation method and fractional derivative described with the Caputo operator. To show and confirm the effectiveness of the future method, I examined three cases and evaluated in terms of fractional-order the projected model. Furthermore, the physical behavior of the fractional-order solution achieved has been shown in graphs, and the mathematical model is proved to confirm the accuracy. The solutions achieved clarify that the suggested methodology is simple to apply, highly organized, and accurate to study the attitude of nonlinear system differential equations of arbitrary order arising in the related fields of engineering and science.

Quantum oscillations in the black hole horizon

By applying Rosen's quantization approach to the historical Oppenheimer and Snyder gravitational collapse and by setting the constraints for the formation of the Schwarzschild black hole (SBH), in a previous paper [1] two of the Authors (CC and FF) found the gravitational potential, the Schrodinger equation, the solution for the energy levels, the area quantum and the quantum representation of the ground state at the Planck scale of the SBH. Such results are consistent with previous ones in the literature. It was also shown that the traditional classical singularity in the core of the SBH is replaced by a quantum oscillator describing a non-singular two-particle system where the two components, named the "nucleus" and the "electron", strongly interact with each other through a quantum gravitational interaction. In agreement with the de Broglie hypothesis, the "electron" is interpreted in terms of the quantum oscillations of the BH horizon. In other words, the SBH should be the gravitational analogous of the hydrogen atom. In this paper, it is shown that these results allow us to compute the SBH entropy as a function of the BH principal quantum number in terms of Bekenstein-Hawking entropy and three sub-leading corrections. In addition, the coefficient of the formula of Bekenstein-Hawking entropy is reduced to a quarter of the traditional value. Then, it is shown that, by performing a correct rescaling of the energy levels, the semi-classical Bohr-like approach to BH quantum physics, previously developed by one of the Authors (CC), is consistent with the obtained results for large values of the BH principal quantum number. After this, Hawking radiation will be analysed by discussing its connection with the BH quantum structure. Finally, it is shown that the time evolution of the above mentioned system solves the BH information paradox.

Predefined-Time Nonsingular Attitude Control for Vertical-Takeoff Horizontal-Landing Reusable Launch Vehicle

This paper presents a novel predefined-time nonsingular tracking control system for a vertical-takeoff horizontal-landing (VTHL) reusable launch vehicle (RLV) in the face of parameter uncertainties, model couplings and external disturbances. Firstly, this paper proposes a novel predefined-time prescribed performance function (PTPPF) with desired steady-state and transient performance. The convergence time of PTPPF from the transient state to the steady state can be flexibly adjusted by changing one parameter. Moreover, the decreasing rate of PTPPF in the transient phase can also be adjusted by changing one parameter on the premise of not changing the convergence time of PPF to reach steady state. A novel predefined-time terminal sliding mode surface (SMS) is designed to avoid the singularity, and the attitude tracking errors on SMS are predefined-time stable. By utilizing PTPPF and error transformation, this paper designs a novel nonsingular sliding mode controller to guarantee the attitudes of RLV with desired tracking performance. Without using piecewise functions, the phenomenon of singularity can be avoided. The Lyapunov method is used to verify the stability of the controller. Lastly, a numerical simulation is presented to validate the efficiency of the proposed controller.

Robust sliding‐mode observer‐based multiple‐fault diagnosis scheme

AbstractIn this study, we simultaneously evaluate the multiple‐fault diagnosis problem of a class of Lipschitz nonlinear systems with actuator and sensor faults and unknown input disturbances. A nonsingular system transformation is used to transform the original system into two subsystems for multiple‐fault diagnosis: subsystems 1 and 2. At the system level, two robust sliding‐mode observers (RSMOs) are proposed. An RSMO is designed for subsystem 1 to detect actuator faults subjected to unknown input disturbances, and another RSMO is designed for subsystem 2 to detect sensor faults subjected to actuator faults. At the component level, a bank of RSMOs is proposed to detect and isolate actuators (sensors) with faults using a dedicated observer scheme. The reachability of RSMOs is comprehensively investigated in the estimation error space. Accordingly, the proposed observer parameters are designed as an optimization problem and solved using the linear matrix inequality (LMI) optimization technique. The effectiveness of the proposed multiple‐fault diagnosis scheme was validated through simulations of a modified seventh‐order aircraft system.

Adaptive Control on SE(3) for Spacecraft Pose Tracking With Harmonic Disturbance and Input Saturation

This article investigates the coupled attitude and orbit tracking maneuver control problem of a spacecraft system in close space missions with model uncertainty, unknown disturbance, and input saturation. An adaptive compact control scheme with featuring of disturbance rejection and antiwindup is reported. First, the exponential coordinates on Lie group SE(3) are employed to describe the relative pose, which includes the relative position and attitude, and the tracking error system is established by introducing an instrumental error variable, which contains exponential coordinates and velocity tracking error. Then, a dynamic disturbance compensator is constructed based on the idea of internal model design to finely compensate for a class of unknown harmonic disturbances. Furthermore, a nonsingular auxiliary system is proposed to overcome input saturation, in which a time-varying parameter is introduced to improve the performance of the antiwindup auxiliary system. This parameter makes the closed-loop system possess two properties: the timeliness of saturation compensation and the asymptotic convergence of tracking errors. With the developed control law, the stability of the obtained feedback system is strictly proved by means of Lyapunov stability theory. Finally, the numerical simulation is carried out for an on-orbit servicing mission, which validates the effectiveness of the proposed integrated controller in solving the pose tracking control problem.

Wave propagation at the welded interface of an elastic solid and unsaturated poro-thermoelastic solid

PurposeThe purpose of this paper is to investigate propagation characteristics of seismic waves at the welded interface of an elastic solid and unsaturated poro-thermoelastic solid.Design/methodology/approachA theoretical formulation of partially saturated poro-thermoelastic solid is used in this study established by Zhou et al. (2019). The incidence of two primary waves (P and SV) is taken. The incident wave from the elastic solid induces two reflected waves and five refracted waves. Due to viscous pore fluids, partially saturated poro-thermoelastic solid behave dissipative, whereas elastic solid behaves non-dissipative. As a result, both reflected and incident waves are homogeneous. However, all the refracted waves are inhomogeneous. A non-singular system of linear equations is formed by the coefficients of reflection and refraction for a specified incident wave. The energy shares of various reflected and refracted waves are determined by using these reflection and refraction factors. Finally, a sensitivity analysis is performed, and the effect of critical variables on energy partitioning at the interface is observed. The numerical example shows that throughout the process of reflection/refraction, the energy of incidence is conserved at all angles of incidences.FindingsThis study demonstrated two refracted (homogeneous) and five refracted (inhomogeneous) waves due to the incident wave from elastic solid. The reflection and refraction coefficients and partitioning of incident energy are acquired as a part of diverse physical parameters of the partially saturated poro-thermoelastic media. The interference energies between unlike pairs of refracted waves have been discovered due to the dissipative behavior of unsaturated poro-thermoelastic solid.Originality/valueThe sensitivity of different energy shares to various aspects of the considered model is graphically analyzed for a specific numerical model. The energy balance is maintained by combining interaction energy and bulk wave energy shares.

On Reeb components of nonsingular polynomial differential systems on the real plane

Denote by r(n) the maximal number of Reeb components that a nonsingular polynomial differential system of degree n on the real plane can have. It is known that r(0)=r(1)=0, r(2)=r(3)=2 and n−1≤r(n)≤n for n≥4. In this paper we prove that r(n)=n for all n≥4. This completely solves, for nonsingular systems, a problem posed by Chicone and Tian in 1982. We prove our result by explicitly constructing nonsingular Hamiltonian systems of degree n presenting n Reeb components for any n≥4.

Null controllability for a class of stochastic singular parabolic equations with the convection term

<p style='text-indent:20px;'>This paper concerns the null controllability for a class of stochastic singular parabolic equations with the convection term in one dimensional space. Due to the singularity, we first transfer to study an approximate nonsingular system. Next we establish a new Carleman estimate for the backward stochastic singular parabolic equation with convection term and then an observability inequality for the adjoint system of the approximate system. Based on this observability inequality and an approximate argument, we obtain the null controllability result.</p>

Quadcopter nonsingular finite-time adaptive robust saturated command-filtered control system under the presence of uncertainties and input saturation

The nonsingular finite-time adaptive robust saturated command-filtered control problem for quadcopter unmanned aerial vehicles is investigated in this paper. Firstly, an adaptive robust command-filtered control, based on backstepping command-filtered and nonsingular fast terminal sliding mode control, is developed. Secondly, the parametric and nonparametric uncertainties are estimated by using a small number of adaptive laws. Also, a projector function is used to ensure the estimation of quadcopter parameters within an admissible set. Thirdly, error compensating signals are employed to tackle the undesirable effect of command filters. Finally, saturation compensator signals are developed to deal with the adverse effect of saturation in the system. The proposed control strategy can cope with the “explosion of complexity” and “singularity” problems. In addition, it can alleviate the chattering phenomenon and satisfy practical finite-time stability. The numerical simulation results and comparison display the effectiveness of the proposed technique over the other control methods.

Identification of continuous stirred tank reactor based on PCA-interval type-2 fuzzy logic system method

Abstract For unknown model of Continuous Stirred Tank Reactor (CSTR), a model identification based on the combination of principal component analysis (PCA) and a type-2 non-singular interval type-2 fuzzy logic system (Type-2 NSFLS-2) is proposed method. The PCA method can reduce the dimensionality of the high-dimensional input of the model. On this basis, type-2 NSFLS-2 identification model is established, and the backpropagation (BP) algorithm is used to update tthe parameters of antecedent and consequent membership function respectively. The method in this paper is applied to the example of CSTR, and it is compared with other types of FLS method and support vector machine (SVM) under the same conditions. The experimental results show that different types of interval type-2 FLS methods can achieve higher identification accuracy, among which Type-2 NSFLS-2 has higher identification accuracy.

Application of the Method of Non-Classical Approach to Certain Problems of the Theory of Anisotropic Space Plazma in Fluid Description

In this paper, the new well-posed boundary value problem to the system of singular nonlinear differential equations is considered, which describes the stationary radial outflow of anisotropic plasma from the Sun (solar wind). These equations are obtained on the basis of 16-moment MHD (magnetohydrodynamic) transport equations for a collisionless magnetized plasma, which takes into account the temperature anisotropy relative to the direction of the magnetic field and the heat flux carried by the wind. This is a generalization of the classical isotropic Parker model taking into account the effects of anisotropy. In this paper, the equations under study are characterized as a non-autonomous nonlinear system of ordinary differential equations the coefficients in which degenerate and simultaneously have singularities. These equations are related to an unsolved problem in the general theory of ODEs (Ordinary Differential Equations). At first, according to the conditions of the coefficients of the equations, a non-classical boundary value problem is set, and the solvability is established for the same non-autonomous and nonlinear system of equations under consideration. The found analytical solution reconstructs numerical solutions, which are simultaneously automatically established by classical formulation of boundary value problem. Parker’s solutions are also partially included in this obtained class of solutions, which is presented with strictly proves. Further, by means of the methods of “ -regularization” and “fixed point” the theorem of solvability for the considered differential equations is obtained. After constructed nonsingular system equations with well-posed boundary value problem, the analytical solutions are founded. Using the sketch of graph of these solutions their family is established.

Fast algorithms for finding the solution of CUPL-Toeplitz linear system from Markov chain

In this paper, the nonsingular CUPL-Toeplitz linear system from Markov chain is solved. We introduce two fast approaches whose complexity could be considered to be O(nlogn) based on the splitting method of the CUPL-Toeplitz matrix which equals to a Toeplitz matrix minus a rank-one matrix. Finally, we confirm the performance of the new algorithms by three numerical experiments.

Reflection and refraction of elastic waves at the interface of an elastic solid and partially saturated soils

The present problem deals with the reflection and refraction of plane waves at the interface of an elastic solid and partially saturated soils. A mathematical model of partially saturated soils developed by Ghasemzadeh and Abounouri (Soil Dyn Earthq Eng 51:1–8, 2013) is employed. The incidence of two main waves (P and SV) is considered. As a result of the incident wave, two reflected waves (in an elastic solid) and four refracted waves (in partially saturated soils) are generated. The elastic solid behaves non-dissipative, while partially saturated soils behave dissipative due to the presence of viscous pore fluids, as a result of which the incident and reflected waves are homogeneous and all the refracted waves are inhomogeneous. The coefficients of reflection and refraction for a given incident wave result in a non-singular system of linear equations. Further, these reflection and refraction coefficients are used to obtain the energy shares of various reflected and refracted waves. This study has been numerically tested to determine the effects of various properties of partially saturated soils on the various energy shares. The numerical example confirms that during the reflection or refraction process conservation of incident energy is obtained at each angle of incidence.

New lower bounds for the maximal number of inseparable leaves of nonsingular polynomial foliations of the plane

Let F be a nonsingular polynomial differential system of degree n on the real plane and denote by s(n) the maximal number of inseparable leaves that such a system can have. In this paper we prove that s(n) is at least 2n−1 for all n≥4. This improves the known lower bounds for s(n), which are 2n−4 if n≥7 or n=5, and respectively 6 and 9 if n=4 and n=6. Since it is also known that s(n)≤2n for all n≥4 and that s(0)=s(1)=0 and s(2)=s(3)=3, the problem of determining s(n) for all n is now almost solved: any improvement in lower or upper bounds will actually find the exact s(n). Our lower bounds for s(n) are attained in the class of Hamiltonian systems.

Coupled Self-organized Hydrodynamics and Navier–Stokes Models: Local Well-posedness and the Limit from the Self-organized Kinetic-Fluid Models

A coupled system of self-organized hydrodynamics and Navier-Stokes equations (SOH-NS), which models self-propelled particles in a viscous fluid, was recently derived by Degond et al. \cite{DMVY-2017-arXiv}, starting from a micro-macro particle system of Vicsek-Navier-Stokes model, through an intermediate step of a self-organized kinetic-kinetic model by multiple coarse-graining processes. We first transfer SOH-NS into a non-singular system by stereographic projection, then prove the local in time well-posedness of classical solutions by energy method. Furthermore, employing the Hilbert expansion approach, we justify the hydrodynamic limit from the self-organized kinetic-fluid model to macroscopic dynamics. This provides the first analytically rigorous justification of the modeling and asymptotic analysis in \cite{DMVY-2017-arXiv}.

Neural network approach for solving nonsingular multi-linear tensor systems

The main propose of this paper is to develop two neural network models for solving nonsingular multi-linear tensor system. Theoretical analysis shows that each of the neural network models ensures the convergence performance. For possible hardware implementation of the proposed neural network models, based on digital circuits, we adopt the Euler-type difference rule to discretize the corresponding Gradient neural network (GNN) models. The computer simulation results further substantiate that the models can solve a multi-linear system with nonsingular tensors.

Spectral decompositions for the solutions of Lyapunov equations for bilinear dynamical systems

In this paper, novel spectral decompositions are obtained for the solutions of generalized Lyapunov equations, which are observed in the study of controllability and observability of the state vector in deterministic bilinear systems. The same equations are used in the stability analysis and stabilization of stochastic linear control systems. To calculate these spectral decompositions, an iterative algorithm is proposed that uses the residues of the resolvent of the dynamics matrix. This algorithm converges for any initial guess, for a non-singular and stable dynamical system. The practical significance of the obtained results is that they allow one to characterize the contribution of individual eigen-components or their pairwise combinations to the asymptotic dynamics of the perturbation energy in deterministic bilinear and stochastic linear systems. In particular, the norm of the obtained eigen-components increases when frequencies of the corresponding oscillating modes approximate each other. Thus, the proposed decompositions provide a new fundamental approach for quantifying resonant modal interactions in a large and important class of weakly nonlinear systems.

Efficient GMM estimation with singular system of moment conditions

Standard generalised method of moments (GMM) estimation was developed for nonsingular system of moment conditions. However, many important economic models are characterised by singular system of moment conditions. This paper shows that efficient GMM estimation of such models can be achieved by using the reflexive generalised inverses, in particular the Moore–Penrose generalised inverse, of the variance matrix of the sample moment conditions as the weighting matrix. We provide a consistent estimator of the optimal weighting matrix and establish its consistency. Potential issues of using generalised inverse and some remedies are also discussed.