DERIVATION AND INTEGRAL SLIDING MODE VARIABLE STRUCTURE CONTROL OF HYDRAULIC VELOCITY TRACKING SYSTEM

The velocity tracking control of a hydraulic servo system is studied. Sincethe dynamics of the system are highly nonlinear and have large extent of model uncertainties, suchas big changes in load and parameters, a derivation and integral sliding mode variable structurecontrol scheme (DI-SVSC) is proposed. An integral controller is introduced to avoid the assumptionthat the derivative of desired signal must be known in conventional sliding mode variable structurecontrol, a nonlinear derivation controller is used to weaken the chattering of system. The designmethod of switching function in integral sliding mode control, nonlinear derivation coefficient andcontrollers of DI-SVSC is presented respectively. Simulation shows that the control approach is ofnice robustness and improves velocity tracking accuracy considerably.

Numerical analysis for viscoelastic fluid flow with distributed/variable order time fractional Maxwell constitutive models

Fractional calculus has been widely used to study the flow of viscoelastic fluids recently,and fractional differential equations have attracted a lot of attention.However,the research has shown that the fractional equation with constant order operators has certain limitations in characterizing some physical phenomena.In this paper,the viscoelastic fluid flow of generalized Maxwell fluids in an infinite straight pipe driven by a periodic pressure gradient is investigated systematically.Consider the complexity of the material structure and multi-scale effects in the viscoelastic fluid flow.The modified time fractional Maxwell models and the corresponding governing equations with distributed/variable order time fractional derivatives are proposed.Based on the L1-approximation formula of Caputo fractional derivatives,the implicit finite difference schemes for the distributed/variable order time fractional governing equations are presented,and the numerical solutions are derived.In order to test the correctness and availability of numerical schemes,two numerical examples are established to give the exact solutions.The comparisons between the numerical solutions and the exact solutions have been made,and their high consistency indicates that the present numerical methods are effective.Then,this paper analyzes the velocity distributions of the distributed/variable order fractional Maxwell governing equations under specific conditions,and discusses the effects of the weight coefficient(α)in distributed order time fractional derivatives,the orderα(r,t)in variable fractional order derivatives,the relaxation timeλ,and the frequencyωof the periodic pressure gradient on the fluid flow velocity.Finally,the flow rates of the distributed/variable order fractional Maxwell governing equations are also studied.

Pfaff-Birkhoff Variational Problem and Noether Symmetry in Terms of RiemannLiouville Fractional Derivatives of Variable Order

The Pfaff-Birkhoff variational problem and its Noether symmetry are studied in terms of Riemann-Liouville fractional derivatives of variable order. Based on the combination of variational principle and fractional calculus of variable order,the Pfaff-Birkhoff variational principle with Riemann-Liouville fractional derivatives of variable order is proposed, and the fractional Birkhoff's equations of variable order are derived. Then,the Noether 's theorem for the fractional Birkhoffian system of variable order is given. At last,an example is expressed to illustrate the application of the results.

Moving Least Squares Interpolation Based A-Posteriori Error Technique in Finite Element Elastic Analysis

The performance of a-posteriori error methodology based on moving least squares(MLS)interpolation is explored in this paper by varying the finite element error recovery parameters,namely recovery points and field variable derivatives recovery.The MLS interpolation based recovery technique uses the weighted least squares method on top of the finite element method’s field variable derivatives solution to build a continuous field variable derivatives approximation.The boundary of the node support(mesh free patch of influenced nodes within a determined distance)is taken as circular,i.e.,circular support domain constructed using radial weights is considered.The field variable derivatives(stress and strains)are recovered at two kinds of points in the support domain,i.e.,Gauss points(super-convergent stress locations)and nodal points.The errors are computed as the difference between the stress from the finite element results and projected stress from the post-processed energy norm at both elemental and global levels.The benchmark numerical tests using quadrilateral and triangular meshes measure the finite element errors in strain and stress fields.The numerical examples showed the support domain-based recovery technique’s capabilities for effective and efficient error estimation in the finite element analysis of elastic problems.The MLS interpolation based recovery technique performs better for stress extraction at Gauss points with the quadrilateral discretization of the problem domain.It is also shown that the behavior of the MLS interpolation based a-posteriori error technique in stress extraction is comparable to classical Zienkiewicz-Zhu(ZZ)a-posteriori error technique.

Seasonal variability of cirrus depolarization properties derived from CALIPSO lidar measurements over Beijing in China

The seasonal variability of cirrus depolarization ratio and its altitude at the region of Beijing （39.93°N, 116.43°E, the capital of China） are presented. From the results obtained from the cloud aerosol lidar and infrared pathfinder satellite observations lidar measurements, it appears that the values of depolarization ratio and altitude of cirrus are generally higher in autumn and summer than those in spring and winter, and the cirrus altitude is modulated by the height of tropopause. Additionally, the depolarization ratio tends to linearly vary with the increase of altitude and the decrease of temperature.

Analysis on Krasnoselskii’s fixed point theorem of fuzzy variable fractional differential equation for a novel coronavirus(COVID-19)model with singular operator

The fuzzy variable fractional differential equations(FVFDEs)play a very important role in mathematical modeling of COVID-19.The scientists are studying and developing several aspects of these COVID-19 models.The existence and uniqueness of the solution,stability analysis are the most common and important study aspects.There is no study in the literature to establish the existence,uniqueness,and UH stability for fuzzy variable fractional(FVF)order COVID-19 models.Due to high demand of this study,we investigate results for the existence,uniqueness,and UH stability for the considered COVID-19 model based on FVFDEs using a fixed point theory approach with the singular operator.Additionally,discuss the maximal/minimal solution for the FVFDE of the COVID-19 model.

The Numerical Simulation of Space-Time Variable Fractional Order Diffusion Equation

Many physical processes appear to exhibit fractional order behavior that may vary with time or space.The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable.Numerical methods and analysis of stability and convergence of numerical scheme for the variable fractional order partial differential equations are quite limited and difficult to derive.This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the space-time variable fractional order diffusion equation on a finite domain.It is worth mentioning that here we use the Coimbradefinition variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems.An implicit Euler approximation is proposed and then the stability and convergence of the numerical scheme are investigated.Finally,numerical examples are provided to show that the implicit Euler approximation is computationally efficient.