Asymptotic Stability of the Sphere's Rolling Equilibrium

2021 ◽  
pp. 1-8
Author(s):  
Tuhin Das
Keyword(s):  
Stability Analysis ◽  
Horizontal Plane ◽  
Stable Equilibrium ◽  
Initial Conditions ◽  
Theoretical Development ◽  
Asymptotically Stable ◽  
Inclined Plane ◽  
Pure Rolling ◽  
Simulation Results ◽  
The Stability

Abstract The classical phenomenon of a sphere transitioning from sliding to rolling during its motion on a horizontal plane is investigated from a novel system theoretic perspective. Specifically, the transition is studied as a problem in stabilization, an approach not reported in the literature. The main contribution of this paper is in proving that pure rolling is an asymptotically stable equilibrium within the state-space of a sliding sphere. It is shown that the stabilization of this equilibrium from arbitrary initial conditions occurs through the natural interplay between the friction force and moment that result from sliding. Simulation results confirm the theoretical development. The stability analysis is extended to motion on an inclined plane.

scholarly journals On the Global Stability Analysis of Corona Virus Disease (COVID-19) Mathematical Model

2021 ◽  
pp. 81-87
Author(s):  
William Atokolo ◽  
Achonu Omale Joseph ◽  
Rose Veronica Paul ◽  
Abdul Sunday ◽  
Thomas Ugbojoide Onoja
Keyword(s):  
Stability Analysis ◽  
Global Stability ◽  
Initial Conditions ◽  
Virus Disease ◽  
Disease Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Global Stability Analysis ◽  
Corona Virus ◽  
The Stability

In this present work, we investigated the Global Stability Analysis of Corona virus disease model formulated by Atokolo et al in [11]. The COVID‑19 pandemic, also known as the coronavirus pandemic, is an ongoing pandemic that is ravaging the whole world. By constructing a Lyapunov function, we investigated the stability of the model Endemic Equilibrium state to be globally asymptotically stable. This results epidemiologically implies that the COVID-19 will invade the population in respective of the initial conditions (population) considered.

An Improved Adaptive Algorithm for INS/GPS System

2013 ◽  
Vol 397-400 ◽  
pp. 1606-1610 ◽  
Author(s):  
Li Dong Wang ◽  
Ying Zhao ◽  
Ni Zhang
Keyword(s):  
Adaptive Filtering ◽  
Adaptive Filter ◽  
Adaptive Algorithm ◽  
Initial Conditions ◽  
Forgetting Factor ◽  
Kalman Filter Algorithm ◽  
Simulation Results ◽  
The Stability ◽  
Gps System

In INS/GPS system, the changing of initial conditions and the quality of the data can affect the convergence of the conventional Kalman filter algorithm. Sage-Husa adaptive filter algorithm is adopted in the INS/GPS system in this paper. The effecting of the forgetting factor to the improved Sage-Husa adaptive filter algorithm is studied and the simulation results show that when the forgetting factor taken near 0.97, the adaptive filtering result is best, the stability of the system is guaranteed and the convergent speed of error can be reduced.

scholarly journals Stability Analysis for Autonomous Dynamical Switched Systems through Nonconventional Lyapunov Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
V. Nosov ◽  
J. A. Meda-Campaña ◽  
J. C. Gomez-Mancilla ◽  
J. O. Escobedo-Alva ◽  
R. G. Hernández-García
Keyword(s):  
Stability Analysis ◽  
Finite Number ◽  
Switched Systems ◽  
Lyapunov Functions ◽  
Switched System ◽  
Linear Functions ◽  
Asymptotically Stable ◽  
Multiple Lyapunov Functions ◽  
Stability Theorems ◽  
The Stability

The stability of autonomous dynamical switched systems is analyzed by means of multiple Lyapunov functions. The stability theorems given in this paper have finite number of conditions to check. It is shown that linear functions can be used as Lyapunov functions. An example of an exponentially asymptotically stable switched system formed by four unstable systems is also given.

Stability analysis and optimal control of an epidemic model with vaccination

2015 ◽  
Vol 08 (03) ◽  
pp. 1550030 ◽  
Author(s):  
Swarnali Sharma ◽  
G. P. Samanta
Keyword(s):  
Optimal Control ◽  
Stability Analysis ◽  
Epidemic Model ◽  
Asymptotically Stable ◽  
Control Pair ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
The Cost ◽  
Objective Functional ◽  
Disease Free

In this paper, we have developed a compartment of epidemic model with vaccination. We have divided the total population into five classes, namely susceptible, exposed, infective, infective in treatment and recovered class. We have discussed about basic properties of the system and found the basic reproduction number (R0) of the system. The stability analysis of the model shows that the system is locally as well as globally asymptotically stable at disease-free equilibrium E0when R0< 1. When R0> 1 endemic equilibrium E1exists and the system becomes locally asymptotically stable at E1under some conditions. We have also discussed the epidemic model with two controls, vaccination control and treatment control. An objective functional is considered which is based on a combination of minimizing the number of exposed and infective individuals and the cost of the vaccines and drugs dose. Then an optimal control pair is obtained which minimizes the objective functional. Our numerical findings are illustrated through computer simulations using MATLAB. Epidemiological implications of our analytical findings are addressed critically.

Stability Analysis of a Class of Nonlinear Controllers

2003 ◽  
Author(s):  
T. Ravichandran ◽  
G. R. Heppler ◽  
D. W. L. Wang
Keyword(s):  
Stability Analysis ◽  
Exponential Stability ◽  
Tracking Control ◽  
Global Exponential Stability ◽  
Feedback Controller ◽  
Nonlinear Gain ◽  
Nonlinear Controllers ◽  
Simulation Results ◽  
Pd Feedback Controller ◽  
The Stability

The stability analysis of a class of nonlinear PD-plus-feedforward controllers is presented for the tracking control of rigid robot manipulators. The controller structure is composed of a nonlinear gain PD feedback controller and a manipulator dynamics feedforward term. The class of representations used for the nonlinear gain PD feedback controller is extended and global exponential stability is proved. Simulation results are included to illustrate the performance of this class of nonlinear controllers.

PERTURBATION AND STABILITY ANALYSIS OF THE MULTI-ANTICIPATIVE INTELLIGENT DRIVER MODEL

2010 ◽  
Vol 21 (05) ◽  
pp. 647-668 ◽  
Author(s):  
XI-QUN CHEN ◽  
WEI-JUN XIE ◽  
JING SHI ◽  
QI-XIN SHI
Keyword(s):  
Stability Analysis ◽  
Traffic Flow ◽  
Autonomous Vehicles ◽  
Driver Model ◽  
Stable Region ◽  
Local Perturbation ◽  
Distance Information ◽  
Car Following ◽  
Simulation Results ◽  
The Stability

This paper discusses three kinds of IDM car-following models that consider both the multi-anticipative behaviors and the reaction delays of drivers. Here, the multi-anticipation comes from two ways: (1) the driver is capable of evaluating the dynamics of several preceding vehicles, and (2) the autonomous vehicles can obtain the velocity and distance information of several preceding vehicles via inter-vehicle communications. In this paper, we study the stability of homogeneous traffic flow. The linear stability analysis indicates that the stable region will generally be enlarged by the multi-anticipative behaviors and reduced by the reaction delays. The temporal amplification and the spatial divergence of velocities for local perturbation are also studied, where the results further prove this conclusion. Simulation results also show that the multi-anticipative behaviors near the bottleneck will lead to a quicker backwards propagation of oscillations.

scholarly journals Computational Stability Analysis of Lotka-Volterra Systems

2016 ◽  
Vol 44 (2) ◽  
pp. 113-120
Author(s):  
Péter Polcz ◽  
Keyword(s):  
Stability Analysis ◽  
Lyapunov Functions ◽  
Stability Region ◽  
Stable Equilibrium ◽  
Linear Matrix ◽  
Stable Equilibrium Point ◽  
Computational Stability ◽  
Volterra Systems ◽  
Selection For ◽  
The Stability

Abstract This paper concerns the computational stability analysis of locally stable Lotka-Volterra (LV) systems by searching for appropriate Lyapunov functions in a general quadratic form composed of higher order monomial terms. The Lyapunov conditions are ensured through the solution of linear matrix inequalities. The stability region is estimated by determining the level set of the Lyapunov function within a suitable convex domain. The paper includes interesting computational results and discussion on the stability regions of higher (3,4) dimensional LV models as well as on the monomial selection for constructing the Lyapunov functions. Finally, the stability region is estimated of an uncertain 2D LV system with an uncertain interior locally stable equilibrium point.

Stability Analysis of a Reach Truck

2009 ◽  
Author(s):  
Jong-Su Bae ◽  
Taewung Kim ◽  
Hyun-Yong Jeong
Keyword(s):  
Stability Analysis ◽  
Variational Method ◽  
Potential Function ◽  
Fe Simulation ◽  
Safety Concern ◽  
Mathematical Tool ◽  
Total Potential ◽  
Pitch Direction ◽  
Simulation Results ◽  
The Stability

There is a need for a higher mast of a reach truck in the market, but a higher mast brings a safety concern. Usually, it is more plausible to fall in the roll direction than in the pitch direction. Since a reach truck with a high mast is a heavy and its center of gravity is high, it is not easy to conduct tests to evaluate its stability. If there is a mathematical tool to evaluate the stability of a reach truck, it is easy to evaluate a design in terms of stability and to modify the design in order to increase its stability. In this study, a variational method using a total potential function was used to make a mathematical means to evaluate the stability of a reach truck. By using the mathematical means the stability of a reach truck was evaluated and compared with FE simulation results.

On the Stability of the Impact Damper

1966 ◽  
Vol 33 (3) ◽  
pp. 586-592 ◽  
Author(s):  
S. F. Masri ◽  
T. K. Caughey
Keyword(s):  
Exact Solution ◽  
Stability Analysis ◽  
Asymptotically Stable ◽  
Impact Damper ◽  
The Stability ◽  
The Impact

The exact solution for the symmetric two-impacts-per-cycle motion of the impact damper is derived analytically, and its asymptotically stable regions are determined. The stability analysis defines the zones where the modulus of all the eigenvalues of a certain matrix relating conditions after each of two consecutive impacts is less than unity.

scholarly journals Approximate analytical solution of the nonlinear system of differential equations having asymptotically stable equilibrium

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2633-2641 ◽  
Author(s):  
Mustafa Turkyilmazoglu
Keyword(s):  
Differential Equations ◽  
Stable Equilibrium ◽  
Initial Conditions ◽  
Analytic Solutions ◽  
Auxiliary Function ◽  
Asymptotically Stable ◽  
Highly Nonlinear ◽  
Homotopy Analysis ◽  
Auxiliary Linear Operator ◽  
Analytical Expressions

The present paper is concerned with the purely analytic solutions of the highly nonlinear systems of differential equations possessing an asymptotically stable equilibrium. A methodology combined with the homotopy analysis method is proposed. The methodology involves proper introduction of an auxiliary linear operator and an auxiliary function during the implementation of the homotopy method so that it can yield uniformly valid solutions, not affected from the existing parameters or initial conditions. The technique is applied to the systems particularly appearing in mathematical biology. The obtained explicit analytical expressions for the solution generate results that compare excellently with the numerically computed ones.

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