On the Steady State Motions Control Problem for Mechanical Systems with Relay Controllers

2021 ◽  
Vol 1 (1) ◽  
pp. 48-55
Author(s):  
Aleksandr Andreev ◽  
Olga Peregudova
Keyword(s):  
Differential Equation ◽  
Steady State ◽  
Asymptotic Stability ◽  
Control Problem ◽  
Mechanical System ◽  
Lyapunov Functions ◽  
Robot Manipulator ◽  
Mechanical Systems ◽  
Stabilization Problem ◽  
Stability Of The Solution

The paper presents the solution to the stabilization problem of steady state motions for a holonomic mechanical system by using relay controllers. This solution is achieved by proving new theorems on the asymptotic stability of the solution to a differential equation with a discontinuous right-hand side. The novelty of the theorems is based on the limiting inclusions construction and the use of semidefinite Lyapunov functions. As an example, the stabilization problem of steady-state motion for a five-link robot manipulator is solved by using relay controller.

Fundamental Exercises in Mechanical Systems Engineering Using Robot Manipulator

2011 ◽  
Author(s):  
Hideaki Takanobu
Keyword(s):  
Mechanical System ◽  
Linear Algebra ◽  
Systems Engineering ◽  
Inverse Kinematics ◽  
Degrees Of Freedom ◽  
Engineering Students ◽  
Robot Manipulator ◽  
Mechanical Systems ◽  
Forward Kinematics ◽  
Basic Learning

A five degrees-of-freedom (5-DOF) robot manipulator is used for the basic learning of mechanical system engineering. Students learned the forward kinematics as concrete applications of the mathematics, especially linear algebra. After making a manipulator, baton relay contest was done to understand the inverse kinematics by controlling the manipulator using a manual controller having five levers.

scholarly journals REACTANCES AND SUSCEPTANCES OF MECHANICAL SYSTEMS

2021 ◽  
pp. 64-75
Author(s):  
I.P. Popov ◽  
Keyword(s):  
Steady State ◽  
Mechanical System ◽  
Mechanical Systems ◽  
Harmonic Force ◽  
Parallel Connection ◽  
Electrical Circuits ◽  
Major Interest ◽  
Inert Body ◽  
Complex Mechanical Systems ◽  
External Harmonic Force

The classical solution to the problems associated with calculating the velocities and reactions of elements of complex mechanical systems under harmonic force consists in the compilation and integration of systems of differential equations and is rather cumbersome and time-consuming. In most cases, a steady state is of major interest. The purpose of this study is to develop essentially compact methods for calculating systems under steady-state conditions. The problem is solved by the methods which are typically used to calculate electrical circuits. Representation of harmonic quantities as rotating vectors in a complex plane and the operations with their complex amplitudes can greatly facilitate the calculation of arbitrarily complex mechanical systems under harmonic effects in the steady state. In the proposed method, a key role is played by mechanical reactance, resistance, and impedance for the parallel connection of consumers of mechanical power, as well as susceptance, conductance, and admittance for the serial one. At force resonance, the total reactance of the mechanical system is zero. This means that the system does not exhibit reactive resistance to the external harmonic force. At velocity resonance, the total susceptibility of the mechanical system is zero. This means that the system has infinitely high resistance to the external harmonic force. As a result, the stock of the source of harmonic force is stationary, although the inert body and the elastic element oscillate.

scholarly journals Stability of Generalized Proportional Caputo Fractional Differential Equations by Lyapunov Functions

2022 ◽  
Vol 6 (1) ◽  
pp. 34
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Fractional Derivative ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Nonlinear Fractional Differential Equation ◽  
Comparison Results ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
Nonautonomous Equations

In this paper, nonlinear nonautonomous equations with the generalized proportional Caputo fractional derivative (GPFD) are considered. Some stability properties are studied by the help of the Lyapunov functions and their GPFDs. A scalar nonlinear fractional differential equation with the GPFD is considered as a comparison equation, and some comparison results are proven. Sufficient conditions for stability and asymptotic stability were obtained. Examples illustrating the results and ideas in this paper are also provided.

Mechanical system regulation using matrix quadratic models—a review

1998 ◽  
Vol 212 (2) ◽  
pp. 129-147
Author(s):  
R Whalley
Keyword(s):  
Steady State ◽  
Mechanical System ◽  
Disturbance Rejection ◽  
Closed Loop ◽  
Mechanical Systems ◽  
Control Action ◽  
Set Point ◽  
Quadratic Models ◽  
Transient And Steady State ◽  
Steady State Performance

Matrix quadratic descriptions for series coupled mechanical systems are used in the synthesis of controllers generating prescribed closed-loop pole configurations. Three-term control action is introduced to improve the load disturbance rejection properties of the system while maintaining the specified set-point response. Conditions maximizing the closed-loop model's steady state determinant value, constraining thereby the error to load disturbances, are derived and a study illustrating this feature is presented. The interaction between the transient and steady state performance of the system is commented upon.

Stable Control of Under-Actuated Mechanical System Acrobot

2014 ◽  
Vol 971-973 ◽  
pp. 727-730
Author(s):  
Min Wu ◽  
Hai Pu
Keyword(s):  
Asymptotic Stability ◽  
Mechanical System ◽  
Degrees Of Freedom ◽  
Mechanical Systems ◽  
Vertical Plane ◽  
Drive System ◽  
Control Input ◽  
Underactuated Mechanical Systems ◽  
Two Degrees Of Freedom ◽  
Stable Control

The mechanical system with one control input and two degrees of freedom is one kind of simple under-drive system, As a typical of two-DOF underactuated mechanical systems , Two-link underactuated mechanical arms is a system scholars always research of.This paper mainly take the two link underactuated mechanical arms-Acrobot which motivate in the vertical plane as object to discusses the asymptotic stability of control

Lyapunov functions and a control problem

1967 ◽  
Vol 63 (2) ◽  
pp. 435-438 ◽  
Author(s):  
A. A. Kayande ◽  
D. B. Muley
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Control Problem ◽  
Lyapunov Functions ◽  
Differential Inequality ◽  
Differential Systems ◽  
Systems Of Differential Equations ◽  
Linear Differential Systems ◽  
Lyapunov’S Second Method ◽  
Linear Differential

1. One of the most important techniques in the study of non-linear differential systems is the Lyapunov's second method and its extensions. One of the extensions of the method depends upon the fact that the function satisfying a differential inequality can be majorized by the maximal solution of the corresponding differential equation. This method was used extensively by V. Lakshmikantham and others for obtaining results, in a unified way, on stability and boundedness of systems of differential equations.

scholarly journals Global Asymptotic Stability of Stochastic Nonautonomous Lotka-Volterra Models with Infinite Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Fengying Wei ◽  
Yuhua Cai
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Stochastic Differential Equation ◽  
Global Asymptotic Stability ◽  
Existence And Uniqueness ◽  
Lyapunov Functions ◽  
Infinite Delay ◽  
Volterra Models ◽  
Global Positive Solution ◽  
Time Average

A kind of general stochastic nonautonomous Lotka-Volterra models with infinite delay is investigated in this paper. By constructing several suitable Lyapunov functions, the existence and uniqueness of global positive solution and global asymptotic stability are obtained. Further, the solution asymptotically follows a normal distribution by means of linearizing stochastic differential equation. Moment estimations in time average are derived to improve the approximation distribution. Finally, numerical simulations are given to illustrate our conclusions.

About the Stabilization of Anisotropic Cylindrical Shell With Elastic Filling

2000 ◽  
Author(s):  
Vladimir Sarkisyan ◽  
Sarkis Sarkisyan
Keyword(s):  
Differential Equation ◽  
Cylindrical Shell ◽  
Mechanical System ◽  
Mechanical Systems ◽  
Elastic Base ◽  
Optimal Stabilization ◽  
Convergence Of Series ◽  
Symmetric Kernel ◽  
The Given ◽  
Anisotropic Cylindrical Shell

Abstract The problems about the optimal stabilization of mechanical system of a potency of continuum have the large interest, both theoretical and practical. The solution of such problems is reduced to the nonhomogeneous integro-differential equation with the symmetric kernel. The essential results in the solution of problems of the optimum stabilization for mechanical systems of a potency of continuum are obtained [4], [5]. In work [4,5,6] the convergence of series of solutions and the finiteness of a target functional is proved uniformly. Solved a numerous problems of the optimal stabilization of vibrations of plates and rather slanting shells. In various statements the problem of the optimal stabilization for anisotropic cylindrical shells are solved in [6] etc. The given work is attempt to fill in a gap the problems of the stabilization by the problems about the optimal stabilization of vibrations of shells with filling, where the filling as Winkler’s elastic base, and for filling of Vlasov’s model [3] is considered.

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