Stable Control of Multi-link Manipulator Using Collision Phenomena

1991 ◽  
Vol 3 (6) ◽  
pp. 482-490
Author(s):  
Yasumasa Shoji ◽  
◽  
Makoto Inaba ◽  
Toshio Fukuda ◽  
Hidemi Hosokai ◽  
...  
Keyword(s):  
Lyapunov Method ◽  
Direct Method ◽  
Type Model ◽  
Lyapunov Direct Method ◽  
Loss Parameter ◽  
Flexible Wall ◽  
Stabilization Effect ◽  
The Stability ◽  
The Impact ◽  
Stable Control

In this paper, a methodology using the Lyapunov direct method is proposed to analyze the stability of a multi-link manipulator system, which is positioned on a flexible wall, with collision phenomenon. The stability and response of the system are examined by parameter studies of numerical simulation. Because industrial demands for rapid motion of robotics have been increasing in order to achieve higher efficiency, collision has become a problem because every task involves contact when a manipulator interacts with an object. However, few research has been initiated to overcome this problem. In this paper, we employ a Hertz-type model which includes an energy loss parameter to express the impact force between the manipulator and the wall. Using this model, we have verified the stabilization effect of collision by the Lyapunov method. The effect has been confirmed by simulation. As a result, stable positioning of the manipulator on a flexible wall is assured, and the use of collision is sometimes effective to control the manipulator to performs tasks with rapid contact.

scholarly journals Dynamics of the rumor-spreading model with hesitation mechanism in heterogenous networks and bilingual environment

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shuai Yang ◽  
Haijun Jiang ◽  
Cheng Hu ◽  
Juan Yu ◽  
Jiarong Li
Keyword(s):  
Lyapunov Method ◽  
Reproduction Number ◽  
Model Parameters ◽  
Rumor Spreading ◽  
Spreading Model ◽  
Reductio Ad Absurdum ◽  
The Stability ◽  
The Impact ◽  
Theoretical Results ◽  
The Basic Reproduction Number

Abstract In this paper, a novel rumor-spreading model is proposed under bilingual environment and heterogenous networks, which considers that exposures may be converted to spreaders or stiflers at a set rate. Firstly, the nonnegativity and boundedness of the solution for rumor-spreading model are proved by reductio ad absurdum. Secondly, both the basic reproduction number and the stability of the rumor-free equilibrium are systematically discussed. Whereafter, the global stability of rumor-prevailing equilibrium is explored by utilizing Lyapunov method and LaSalle’s invariance principle. Finally, the sensitivity analysis and the numerical simulation are respectively presented to analyze the impact of model parameters and illustrate the validity of theoretical results.

On the stability of a mathematical model for HIV(AIDS) - cancer dynamics

2021 ◽  
Vol 8 (4) ◽  
pp. 783-796
Author(s):  
H. W. Salih ◽  
◽  
A. Nachaoui ◽  
Keyword(s):  
Mathematical Model ◽  
Highly Active Antiretroviral Therapy ◽  
Lyapunov Method ◽  
Equilibrium Points ◽  
Hiv Treatment ◽  
Biochemical Processes ◽  
Highly Active ◽  
Biological Phenomena ◽  
The Stability ◽  
The Impact

In this work, we study an impulsive mathematical model proposed by Chavez et al. [1] to describe the dynamics of cancer growth and HIV infection, when chemotherapy and HIV treatment are combined. To better understand these complex biological phenomena, we study the stability of equilibrium points. To do this, we construct an appropriate Lyapunov function for the first equilibrium point while the indirect Lyapunov method is used for the second one. None of the equilibrium points obtained allow us to study the stability of the chemotherapeutic dynamics, we then propose a bifurcation of the model and make a study of the bifurcated system which contributes to a better understanding of the underlying biochemical processes which govern this highly active antiretroviral therapy. This shows that this mathematical model is sufficiently realistic to formulate the impact of this treatment.

scholarly journals Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Khalid Hattaf
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Direct Method ◽  
Lyapunov Direct Method ◽  
Science And Engineering ◽  
Fractional Differential ◽  
The Stability

This paper aims to study the stability of fractional differential equations involving the new generalized Hattaf fractional derivative which includes the most types of fractional derivatives with nonsingular kernels. The stability analysis is obtained by means of the Lyapunov direct method. First, some fundamental results and lemmas are established in order to achieve the goal of this study. Furthermore, the results related to exponential and Mittag–Leffler stability existing in recent studies are extended and generalized. Finally, illustrative examples are presented to show the applicability of our main results in some areas of science and engineering.

scholarly journals Asymptotic Stability of Distributed-Order Nonlinear Time-Varying Systems with the Prabhakar Fractional Derivatives

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
MohammadHossein Derakhshan ◽  
Azim Aminataei
Keyword(s):  
Asymptotic Stability ◽  
Fractional Derivatives ◽  
Direct Method ◽  
Time Varying ◽  
Lyapunov Direct Method ◽  
Fractional Differential Systems ◽  
Time Varying Systems ◽  
Fractional Differential ◽  
The Stability ◽  
Distributed Order

In this article, we survey the Lyapunov direct method for distributed-order nonlinear time-varying systems with the Prabhakar fractional derivatives. We provide various ways to determine the stability or asymptotic stability for these types of fractional differential systems. Some examples are applied to determine the stability of certain distributed-order systems.

scholarly journals On the Stability of Some Discrete Fractional Nonautonomous Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Fahd Jarad ◽  
Thabet Abdeljawad ◽  
Dumitru Baleanu ◽  
Kübra Biçen
Keyword(s):  
Asymptotic Stability ◽  
Global Stability ◽  
Direct Method ◽  
Uniform Stability ◽  
Lyapunov Direct Method ◽  
Uniform Asymptotic Stability ◽  
Fractional Difference ◽  
Nonautonomous Systems ◽  
The Stability

Using the Lyapunov direct method, the stability of discrete nonautonomous systems within the frame of the Caputo fractional difference is studied. The conditions for uniform stability, uniform asymptotic stability, and uniform global stability are discussed.

scholarly journals Analysis of the most common methods for determining the stability of energy systems

2021 ◽  
pp. 17-26
Author(s):  
Andrii Hnatov ◽  
Shchasiana Arhun ◽  
Ruslan Bagach ◽  
Hanna Hnatova ◽  
Valentina Tarasova ◽  
...  
Keyword(s):  
Power Systems ◽  
Lyapunov Method ◽  
Direct Method ◽  
Energy Systems ◽  
Energy System ◽  
Mathematical Function ◽  
Direct Lyapunov Method ◽  
First Approximation ◽  
Differential Methods ◽  
The Stability

Problem. There are many methods for determining the stability of the energy system. In normal operating condition (normal rated mode), the power system must reliably ensure the consumption of electricity of normalized quality. However, in addition to the normal state, there are emergency and transient states caused by various transients. This is due to the fact that the energy system is constantly changing its parameters. Such changes are determined by variations in the amount of power produced and consumed, as well as the changes in system configuration. Goal. The goal is studying the possibilities of various methods of determining the power systems stability and drawing up the general algorithm of actions for maintenance of their stability. Methodology. When determining the stability of energy systems by the Lyapunov method, two methods can be used: the direct method and the first approximation method. Lyapunov direct method refers to differential methods. To conclude about the stability of the system we do not find a general or particular solution of differential equations, but with their help we find a mathematical function, the complete derivative of which over time allows to obtain a conclusion about the stability of the system. Results. Many methods can be used to determine whether a sustainable energy system is stable or not. The most common are the Lyapunov methods and the Moiseev method. It is determined that the direct Lyapunov method refers to differential methods. The application of the direct Lyapunov method for energy problems is limited. Currently, it can be used only for some individual cases. The method of the first approximation (Lyapunov first method) has received wider application in the solution of power problems. When applying this method, which belongs to the group of methods of full integration, the right-hand sides of the equations are decomposed into power series. Originality. It is determined that one of the perspective directions of increasing the efficiency of the mathematical device work is using the methods of the second order in modeling and optimization of operating modes of electric power systems. This allows you to increase the speed and reliability of the convergence of iterative processes. Practical value. Based on the analysis of various existing methods for solving the problems of stability of energy systems, an algorithm of actions is proposed and developed, which will help to solve the problem of stability in practice.

scholarly journals Stability analysis of mechanical systems with distributed delay via decomposition

2021 ◽  
Vol 17 (1) ◽  
pp. 13-26
Author(s):  
Alexander Yu. Aleksandrov ◽  
◽  
Alexey A. Tikhonov ◽  
Keyword(s):  
Stability Analysis ◽  
Direct Method ◽  
Distributed Delay ◽  
Original System ◽  
Linear Matrix ◽  
Matrix Inequalities ◽  
Lyapunov Direct Method ◽  
First Order ◽  
Linear Mechanical System ◽  
The Stability

The article analyzes a linear mechanical system with a large parameter at the vector of velocity forces and a distributed delay in positional forces. With the aid of the decomposition method, conditions are obtained under which the problem of stability analysis of the original system of the second-order differential equations can be reduced to studying the stability of two auxiliary first-order subsystems. It should be noted that one of the auxiliary subsystems does not contain a delay, whereas for the second subsystem containing a distributed delay, the stability conditions are formulated in terms of the feasibility of systems of linear matrix inequalities. To substantiate this decomposition, the Lyapunov direct method is used. Special constructions of Lyapunov—Krasovskii functionals are proposed. The developed approach is applied to the problem of monoaxial stabilization of a rigid body. The results of a numerical simulation are presented confirming the conclusions obtained analytically.

scholarly journals Lyapunov functions for fractional order h-difference systems

Filomat ◽  
2021 ◽  
Vol 35 (4) ◽  
pp. 1155-1178
Author(s):  
Xiang Liu ◽  
Baoguo Jia ◽  
Lynn Erbe ◽  
Allan Peterson
Keyword(s):  
Fractional Order ◽  
Quadratic Forms ◽  
Lyapunov Functions ◽  
Direct Method ◽  
Difference Operators ◽  
Lyapunov Direct Method ◽  
Polynomial Type ◽  
Difference Systems ◽  
Quadratic Lyapunov Functions ◽  
The Stability

This paper presents some new propositions related to the fractional order h-difference operators, for the case of general quadratic forms and for the polynomial type, which allow proving the stability of fractional order h-difference systems, by means of the discrete fractional Lyapunov direct method, using general quadratic Lyapunov functions, and polynomial Lyapunov functions of any positive integer order, respectively. Some examples are given to illustrate these results.

scholarly journals On the Analysis of Damped Gyroscopic Systems Using Lyapunov Direct Method

2021 ◽  
pp. 63-74
Author(s):  
Ubong D. Akpan
Keyword(s):  
Direct Method ◽  
Stability Conditions ◽  
Lyapunov Direct Method ◽  
Gyroscopic Effect ◽  
Stability Properties ◽  
The Stability ◽  
Gyroscopic Systems

In this work, the stability properties of damped gyroscopic systems have been studied using Lyapunov direct method. These systems are generally stable because of the presence of gyroscopic effect. Conditions for determining the stability of the damped gyroscopic systems have been developed. Solution bounds of amplitude and velocity have been obtained for both homogeneous and inhomogeneous cases. An example is given to show how the stability conditions are applied to systems to determine its stability status.

Lyapunov’s Stability of Nonlinear Misaligned Journal Bearings

1995 ◽  
Vol 117 (3) ◽  
pp. 576-581 ◽  
Author(s):  
P. G. Nikolakopoulos ◽  
C. A. Papadopoulos
Keyword(s):  
Journal Bearing ◽  
Direct Method ◽  
Equations Of Motion ◽  
Orbital Stability ◽  
Element Formulation ◽  
Nonlinear Stiffness ◽  
Lyapunov Direct Method ◽  
Rotor Bearing ◽  
The Stability ◽  
General Method

In this paper the stability of nonlinear misaligned rotor-bearing systems is investigated, using the Lyapunov direct method. A finite element formulation is used to determine the journal bearing pressure distribution. Then the linear and nonlinear stiffness, damping, and hybrid (depending on both displacement and velocity) coefficients are calculated. A general method of analysis based on Lyapunov’s stability criteria is used to investigate the stability of nonlinear misaligned rotor bearing systems. The equations of motion of the rigid rotor on the nonlinear bearings are used to find a Lyapunov function using some of these coefficients, which depend on L/D ratio and the misalignment angles ψx, ψy. The analytical conditions for the stability or instability of some examined cases are given and some examples for the orbital stability are also demonstrated.

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