Boundary Layer Function Research Articles

A novel empirical model for vertical profiles of downburst horizontal wind speed

AbstractThis study proposes an empirical model for preliminary wind‐resist design of downburst flow. Existing empirical models were compared with field data and found to underpredict horizontal wind speed below the height corresponding to the maximum radial velocity, due to the neglect of viscous effects and the evolution of vertical wind profiles along radial direction. To address these deficiencies, semi‐empirical piecewise functions including wall shear effects in the local turbulent boundary layer and interpolation functions were proposed to improve the accuracy of existing models. The wind profile based on Coles' theory was found to agree well with field data, with the parabola interpolation function being the most desirable. Using the proposed method, the vertical profile of horizontal wind speed at different local radial locations can be predicted for wind resist design given the inlet wind speed of the downburst flow. Overall, this model improves upon existing empirical models and allows for more accurate wind‐resist design.

Adaptive iterative learning control for a class of parameterized fractional-order systems subjected to backlash-like hysteresis

In this article, a fractional-order adaptive iterative learning control scheme is proposed for a class of parameterized fractional-order systems with unknown control gain and backlash-like hysteresis nonlinearity under disturbance. Based on the sufficient condition for the stability of linear fractional-order systems, a sliding mode surface of tracking errors is constructed to facilitate the controller design and stability analysis. A new boundary layer function is designed by using Mittag-Leffler function to relax the restriction of the identical initial condition of iterative learning control design. The fractional-order differential-type adaptive laws and difference-type learning law are designed to estimate unknown constant parameters and time-varying parameter, respectively. To deal with the influence of backlash-like hysteresis nonlinearity reminder term and unknown bounded external disturbance, a robust control term is designed by employing the hyperbolic tangent function with a convergent series sequence which can guarantee the learning convergence along iteration axis. By constructing Lyapunov-like composite energy function, the stability analysis is presented to prove the convergence of the system output to a small neighborhood of the desired trajectory and the boundedness of all the closed-loop signals. Finally, a simulation example of second-order nonlinear fractional-order system is presented, which demonstrates the effectiveness of the proposed fractional-order adaptive iterative learning control scheme.

Явление погранслоя в алгебро-дифференциальном уравнении первого порядка

The Cauchy problem for the first-order algebraic differential equation is considered A du/dt=(B+εC+ε^2 D)u(t,ε), u(t_0,ε)=u^0 (ε)∈E_1, where A,B,C,D are closed linear operators acting from a Banach space E_1 to a Banach space E_2 with domains everywhere dense in E_1, u^0 is a holomorphic function at the point ε=0, ε is a small parameter, t∈[t_0; t_max]. Such equations describe, in particular, the processes of filtration and moisture transfer, transverse vibrations of plates, vibrations in DNA molecules, phenomena in electromechanical systems, etc. The operator A is the Fredholm operator with zero index. The aim of the work is to study the boundary layer phenomenon caused by the presence of a small parameter. The necessary information and statements are given. A bifurcation equation is obtained. Two cases are considered: a) boundary layer functions of one type, b) boundary layer functions of two types. Newton’s diagram is used to solve the bifurcation equation. In both, the conditions under which boundary layer phenomenon arises are obtained — these are the conditions for the regularity of degeneracy. Case a) is illustrated by an example of the Cauchy problem with certain operator coefficients acting in the space R^4.

A class of singularly perturbed Robin boundary value problems in critical case

This paper discusses a class of nonlinear singular perturbation problems with Robin boundary values in critical cases. By using the boundary layer function method and successive approximation theory, the corresponding asymptotic expansions of small parameters are constructed, and the existence of uniformly efficient smooth solutions is proved. Meanwhile, we give a concrete example to prove the validity of our results.

Saturated Nonsingular Fast Sliding Mode Control for the Crane-Form Pipeline System.

The crane-form pipeline (CFP) system is a kind of petrochemical mechanical equipment composed of multiple rotating joints and rigid pipelines. It is often used to transport chemical fluid products in the factory to tank trucks. In order to realize the automatic alignment of the CFP and the tank mouth, the trajectory tracking control problem of the CFP must be solved. Therefore, a saturated nonsingular fast terminal sliding mode (SNFTSM) algorithm is proposed in this paper. The new sliding mode manifold is constructed by the nonsingular fast terminal sliding mode (NFTSM) manifold, saturation functions and signum functions. Further, according to the sliding mode control algorithm and the dynamic model of the CFP system, the SNFTSM controller is designed. Owing to the existence of saturation functions in the controller, the stability analysis using the Lyapunov equation needs to be discussed in different cases. The results show that the system states can converge to the equilibrium point in finite time no matter where they are on the state's phase plane. However, due to the existence of signum functions, the control signal will produce chattering. In order to eliminate the chattering problem, the form of the controller is improved by using the boundary layer function. Finally, the control effect of the algorithm is verified by simulation and compared with the NTSM, NFTSM and SNTSM algorithms. From the comparison results, it is obvious that the controller based on the SNFTSM algorithm can effectively reduce the amplitude of the control torque while guaranteeing the fast convergence of the CFP system state error. Specifically, compared with the NFTSM algorithm, the maximum input torque can even be reduced by more than half.

Sliding Mode Controller Based on the Extended State Observer for Plant-Protection Quadrotor Unmanned Aerial Vehicles

Owing to the complex dynamics of quadrotor unmanned aerial vehicles (UAVs) and their susceptibility to unknown interferences in an actual working environment, the flight control accuracy of UAVs is extremely high. Moreover, their anti-interference ability is particularly important. This study designed a sliding-mode controller based on the extended state observer. The position control was obtained through the outer-loop position controller. The attitude control was determined through the inner-loop attitude controller. The input of the UAV system was obtained through the controller. The boundary-layer function was used to weaken the oscillatory response of the system, and the traditional extended state observer was improved to improve the response speed, robustness, and tracking accuracy of the controller. For the entire process, the input and output state information of the system and total internal and external disturbances were estimated in real-time through the extended state observer. A sliding-mode control law was designed to compensate for the estimated disturbance in real-time to realize attitude control. Finally, Lyapunov theory was used to confirm the stability of the system. The simulation results demonstrated the improved anti-interference and tracking ability of the designed controller.

출력 예측 성능을 갖는 제어 입력 변환 터미널 적분 슬라이딩 모드 제어

In this note, the discontinuous and continuous control input transformation integral terminal sliding mode control(TSMC)s by using the integral sliding surface without the reaching phase and with the output prediction performance as the one approach are presented for second order uncertain plants. Theoretically discontinuous and practically continuous control input transformed TSMCs are proposed when Δb ≠ 0, while it is assumed that Δb=0 in most of the TSMCs for easy design. Applying the idea of [32] for the linear sliding mode control(LSMC), the integral sliding surface without the reaching phase is suggested for the TSMCs. The exponent of the power function can be any positive numbers satisfying q＞p＞0 such that 0＜p/q＜1. The ideal sliding dynamics of the integral sliding surface is derived and the real robust output can be predicted, predesigned, predetermined by means of the solution of the ideal sliding dynamics. The transformed control input is suggested for generating the sliding mode for the entire trajectory from any given initial condition to the origin based on defining a new auxiliary nonlinear state. The closed loop exponential stability together with the existence condition of the sliding mode on the predetermined sliding surface is investigated theoretically for the complete formulation of the TSMC design for the output prediction performance. As a remedy of the singularity, a certain limit is imposed on the new auxiliary nonlinear state. For practical applications, a continuous approximation of the discontinuous TSMC is made by means of the modified boundary layer function. In addition, the closed loop bounded stability together with the existence condition of the sliding mode by the continuous TSMC is analyzed. The discontinuity of the control input as the inherent property of the sliding mode control is much improved in view of the practical applications. Through a design example and simulation studies, the effectiveness of the proposed discontinuous and continuous control input transformed TSMCs is verified.

Regularized asymptotics of the solution of systems of parabolic differential equations

The regularization method for singularly perturbed problems of S. A. Lomov is generalized to constructing the asymptotics of the solution of the first boundary value problem for systems of differential equations of parabolic type with a small parameter at all derivatives.It is shown that the asymptotics of the solution of the problem contains n exponential, 2n parabolic and 2n angle boundary layer functions. The exponential boundary layer function describes the boundary layer along t = 0, the boundary layer along x = 0 and x = 1 is described by parabolic boundary layer functions.

ASYMPTOTICS OF THE SOLUTION TO A PIECEWISE-SMOOTH QUASILINEAR SECOND-ORDER DIFFERENTIAL EQUATION

<abstract> We investigate a singularly perturbed boundary value problem for a piecewise-smooth second-order quasilinear differential equation in the case when the discontinuous curve which separates the domain is monotone. Applying the boundary layer function method, the asymptotic expansion of a solution with internal layer appearing in the neighborhoods of some point on the monotone curve and the point itself is constructed. For sufficiently small parameter values, using the matching method, the existence of a smooth solution with an internal transition layer in the neighborhood of a point of the monotone curve is proved. A simple example is given to show the effectiveness of our method. </abstract>

ASYMPTOTICS OF A MULTIZONAL INTERNAL LAYER SOLUTION TO A PIECEWISE-SMOOTH SINGULARLY PERTURBED EQUATION WITH A TRIPLE ROOT OF THE DEGENERATE EQUATION

A singularly perturbed boundary value problem for a stationary equation of reaction-diffusion type in the case when reactive term undergoes discontinuity along some curve that is independent of the small parameter is studied. This is a new class of problems with triple roots of the degenerate equation, which leads to the formation of complex multizonal internal layers in the neighborhood of the discontinuity curve. By the method of asymptotic differential inequalities and matching asymptotic expansion, the existence of a contrast structure solution is proved. Using a different modified boundary layer function method, the asymptotic representation of point itself and this solution are constructed.

On vanishing limits of the shear viscosity and Hall coefficients for the planar compressible Hall‐MHD system

This paper deals with an initial‐boundary value problem of the planar compressible Hall‐magnetohydrodynamic (for short, Hall‐MHD) equations. For the fixed shear viscosity and small Hall coefficients, we obtain the existence of long‐time strong solutions of Hall‐MHD equations and corresponding MHD equations. As both the shear viscosity and the Hall coefficients tend to zero, the convergence rate for the solutions from Hall‐MHD equations to MHD equations is given. The thickness of boundary layer is discussed by spatial weighted estimate and the characteristic of boundary layer is described by constructing a boundary layer function.

A Singularly Perturbed System of Parabolic Equations

The work is devoted to the construction of the asymptotic behavior of the solution of a singularly perturbed system of equations of parabolic type, in the case when the limit equation has a regular singularity as the small parameter tends to zero. The asymptotics of the solution of such problems contains boundary layer functions.

Shear, Stability and Mixing within the Ice-Shelf-Ocean Boundary Current

AbstractWhen the inclined base of an ice shelf melts into the ocean, it induces both a statically-stable stratification and a buoyancy-forced, sheared flow along the interface. Understanding how those competing effects influence the dynamical stability of the boundary current is the key to quantifying the turbulent transfer of heat from far-field ocean to ice. The implications of the close coupling between shear, stability and mixing are explored with the aid of a one-dimensional numerical model that simulates density and current profiles perpendicular to the ice. Diffusivity and viscosity are determined using a mixing length model within the turbulent boundary layer and empirical functions of the gradient Richardson number in the stratified layer below. Starting from rest, the boundary current is initially strongly stratified and dynamically stable, slowly thickening as meltwater diffuses away from the interface. Eventually, the current enters a second phase where dynamical instability generates a relatively well-mixed, turbulent layer adjacent to the ice, while beneath the current maximum, strong stratification suppresses mixing in the region of reverse shear. Under weak buoyancy forcing the timescale for development of the initial dynamical instability can be months or longer, but background flows, which are always present in reality, provide additional current shear that greatly accelerates the process. A third phase can be reached when the ice shelf base is sufficiently steep, with dynamical instability extending beyond the boundary layer into regions of geostrophic flow, generating a marginally-stable pycnocline through which the heat flux is a simple function of ice-ocean interfacial slope.

다입출력 불확실 선형 플랜트를 위한 출력 예측 성능을 갖는 제어 입력 변환 적분 슬라이딩 모드 제어

In this note, the discontinuous and continuous control input transformation integral sliding mode control(SMC)s by using the integral sliding surface without the reaching phase and with the output prediction performance as the another approaches of [34] are presented for multi input multi output(MIMO) uncertain linear plants. Theoretically discontinuous and practically continuous control input transformed variable structure system(VSS)s are proposed. The integral sliding surface with no reaching phase is suggested and its ideal sliding dynamics from a given initial condition to the origin is obtained. By using the solution of the ideal sliding dynamics, the real robust output can be pre-designed, predetermined, and predicted. The transformed control input is suggested for generating the sliding mode for the entire trajectory from any given initial condition to the origin. The closed loop exponential stability together with the MIMO existence condition of the sliding mode on the predetermined sliding surface is investigated theoretically for the complete formulation of the VSS design for the output prediction performance. Another concept of the ideal sliding dynamics is given. For practical applications, a continuous approximation of the discontinuous VSS is made by means of the modified boundary layer function. In addition, the closed loop bounded stability together with the MIMO existence condition of the sliding mode by the continuous VSS is analyzed. The discontinuity of the control input as the inherent property of the VSS is much improved. Through a design example and simulation studies, the usefulness of the proposed discontinuous and continuous control input transformed VSS controller is verified. The projected ideal sliding trajectory is given

Asymptotic Solution of a Singularly Perturbed Cauchy Problem with a Turning Point

Using a modified method of boundary-layer functions, we construct a complete uniform asymptotic solution of the singularly perturbed Cauchy problem for a second-order, linear, inhomogeneous, ordinary differential equation with a multiple turning point on the real axis.

다입출력 불확실 선형 플랜트를 위한 출력 예측 성능을 갖는 슬라이딩 면 변환 적분 슬라이딩 모드 제어

In this note, the discontinuous and continuous sliding surface transformation SMCs by using the transformed integral sliding surface without the reaching phase and with the output prediction performance are presented for MIMO uncertain linear plants. Theoretically discontinuous and practically continuous sliding surface transformed VSSs are proposed. The transformed integral sliding surface with the no reaching phase is suggested and its ideal sliding dynamics from a given initial condition to the origin is obtained. By using the solution of the ideal sliding dynamics, the real robust output can be predetermined and predicted. The closed loop exponential stability together with the MIMO existence condition of the sliding mode on the predetermined sliding surface is investigated theoretically for the complete formulation of the MIMO VSS design for the output prediction performance. Another ideal sliding dynamics is given. For practical applications, a continuous approximation of the discontinuous VSS is made by means of the modified boundary layer function. In addition, the closed loop bounded stability together with the MIMO existence condition of the sliding mode by the continuous VSS is analyzed for the entire trajectory. The discontinuity of the control input as the inherent property of the VSS is much improved in view of the practical aspects. Through a design example and simulation studies, the usefulness of the proposed discontinuous and continuous sliding surface transformed VSS controllers is verified.

ASYMPTOTICS OF CRITICAL LOADS OF A COMPRESSED NARROW ELASTIC PLATE WITH INTERNAL STRESSES

The problem of the asymptotic solution of a modified system of nonlinear Karman equilibrium equations for a longitudinally compressed elongated elastic rectangular plate with internal stresses lying on an elastic base is considered. Internal stresses can be caused by continuously distributed edge dislocations and wedge disclinations, or other sources. The compressive pressure is applied parallel to the long sides of the plate to the two short edges. The boundary conditions are considered: the long edges of the plate are free from loads, and the short edges are freely pinched or movably hinged. A small parameter is introduced, equal to the ratio of the short side of the plate to the long side. The solution of the system – the compressive load, the deflection function, and the stress function – is sought in the form of series expansions over a small parameter. The system of Karman equations with dimensionless variables is reduced to an infinite system of boundary value problems for ordinary differential equations with respect to the coefficients of asymptotic expansions for the critical load, deflection, and stress function. In this case, to meet the boundary conditions, the boundary layer functions are additionally introduced, which are concentrated near the fixed edges and disappear when moving away from them. Boundary value problems for determining the functions of the boundary layer are constructed. It is shown that the main terms of the small parameter expansions for the critical load and deflection are determined from the equilibrium equation of a compressed beam on an elastic base with the boundary conditions of free pinching or movable hinge support of the ends. In this case, the main term of the expansion into a series of the stress function has a fourth order of smallness in the parameter of the relative width of the plate.

Исследование жесткости алгебро-дифференциальной системы первого порядка с возмущением в правой части

The rigidity of a dynamical system described by a first-order differential equationwith an irreversible operator at the highest derivative is investigated. The system is perturbed by an operator addition of the order of the second power of a small parameter. Conditions under which the system is robust with respect to these disturbances are determined as well as conditions under which the influence of disturbances is significant. For this, the bifurcation equation is derived. It is used to set the type of boundary layer functions. As an example, we investigate the initial boundary value problem for a system of partial differential equations with a mixed second partial derivative which occurs in the study of the processes of sorption anddesorption of gases, drying processes, etc.

An Improved Full-Order Sliding-Mode Observer for Rotor Position and Speed Estimation of SPMSM

This paper proposed an improved full-order sliding-mode observer (IFSMO), applied to a sensorless control system of surface-mounted permanent magnet synchronous motor (SPMSM), to obtain a high-precision rotor speed and position. First, the IFSMO method, which combines the new sliding-mode function with the variable boundary layer function, is presented to suppress the chattering and accelerate the convergence speed of the system. Meanwhile, since the observer has the characteristics of a second-order low-pass filter, the high-frequency noise contained in the estimated back EMF signal can be effectively filtered out without an additional low-pass filter. Then, the proposed PLL is designed to obtain rotor position and speed. Finally, simulations and experiments of the SPMSM system based on the IFSMO method under different working conditions are implemented, which demonstrates the effectiveness of the proposed algorithm.

Boundary Layer Phenomenon for a First Order Descriptor Equation with Small Parameter on the Right-Hand Side

We study a first order differential equation with noninvertible operator at the derivative and right-hand side perturbed by a small parameter in a Banach space. We derive a bifurcation equation that can be used to determine the form of the boundary layer function. The result is illustrated by the Leontief dynamic gross output model.