Teaching Exploration of Lyapunov’s Second Method

2019 ◽  
Vol 09 (03) ◽  
pp. 343-346
Author(s):  
载涛 梁
Keyword(s):  
Lyapunov’S Second Method

scholarly journals Lyapunov stability theory for dynamic systems on time scales

1992 ◽  
Vol 5 (3) ◽  
pp. 275-281 ◽  
Author(s):  
Billur Kaymakçalan
Keyword(s):  
Time Scales ◽  
Lyapunov Stability ◽  
Comparison Principle ◽  
Dynamic Systems ◽  
Stability Theory ◽  
Lyapunov Stability Theory ◽  
Existence Theory ◽  
Lyapunov’S Second Method ◽  
Stability Results

By use of the necessary calculus and the fundamental existence theory for dynamic systems on time scales, in this paper, we develop Lyapunov's second method in the framework of general comparison principle so that one can cover and include several stability results for both types of equations at the same time.

scholarly journals Stability analysis in terms of two measures for dynamic systems on time scales

1993 ◽  
Vol 6 (4) ◽  
pp. 325-344 ◽  
Author(s):  
Billûr Kaymakçalan
Keyword(s):  
Stability Analysis ◽  
Time Scales ◽  
Dynamic Systems ◽  
Lyapunov’S Second Method ◽  
Two Measures ◽  
Set Up ◽  
Stability Results ◽  
Stability Concepts ◽  
Single Set

Using the theory of Lyapunov's second method developed earlier for time scales, we extend our stability results to two measures which give rise to unification of several stability concepts in a single set up.

Observer-Based Feedback Linearizing Control of an Electromagnetic Suspension

1996 ◽  
Vol 118 (3) ◽  
pp. 615-619 ◽  
Author(s):  
B. C. Fabien
Keyword(s):  
Feedback Controller ◽  
Superior Performance ◽  
Linear Feedback ◽  
Nonlinear Controller ◽  
Electromagnetic Suspension ◽  
Weakly Coupled ◽  
Coupled Riccati Equations ◽  
Linear Feedback Controller ◽  
Lyapunov’S Second Method ◽  
Feedback Linearizable Systems

This paper develops a stabilizing observer-based feedback linearizing controller for a single-axis electromagnetic suspension. The controller uses only the measured output of the system, and is shown to be robust with respect to parameter uncertainty. The controller differs from other robust feedback linearizing controllers that have appeared in recent literature, because it is continuous, and non-adaptive. Lyapunov’s second method is used to prove stability and robustness of the controller. The controller has a simple structure and its gains are determined by solving two weakly coupled Riccati equations. Numerical simulations are performed to compare a linear feedback controller and the observer-based feedback linearizing controller. Results obtained demonstrate that the nonlinear controller yields superior performance when compared with the linear feedback controller. The controller synthesis technique developed in this paper is applicable to other fully feedback linearizable systems, not just electromagnetic suspensions.

scholarly journals Existence and uniqueness of a periodic solution to certain third order nonlinear delay differential equation with multiple deviating arguments

2013 ◽  
Vol 5 (2) ◽  
pp. 113-1 ◽  
Author(s):  
Adeleke Timothy Ademola
Keyword(s):  
Periodic Solution ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Third Order ◽  
Deviating Arguments ◽  
Multiple Deviating Arguments ◽  
Uniform Ultimate Boundedness ◽  
Lyapunov’S Second Method ◽  
Delay Differential ◽  
Nonlinear Delay Differential Equation

Abstract In this paper, we use Lyapunov’s second method, by constructing a complete Lyapunov functional, sufficient conditions which guarantee existence and uniqueness of a periodic solution, uniform asymptotic stability of the trivial solution and uniform ultimate boundedness of solutions of Eq. (2). New results are obtained and proved, an example is given to illustrate the theoretical analysis in the work and to test the effectiveness of the method employed. The results obtained in this investigation extend many existing and exciting results on nonlinear third order delay differential equations.

Sliding Control Without Reaching Phase and Its Application to Bipedal Locomotion

1993 ◽  
Vol 115 (3) ◽  
pp. 447-455 ◽  
Author(s):  
Tai-Heng Chang ◽  
Yildirim Hurmuzlu
Keyword(s):  
Trajectory Planning ◽  
Planning Process ◽  
Control Method ◽  
Variable Structure ◽  
Bipedal Locomotion ◽  
Tracking Errors ◽  
Structure Control ◽  
Control Scheme ◽  
Lyapunov’S Second Method ◽  
Planning Problems

A new variable structure control law based on the Lyapunov’s second method that can be used in trajectory planning problems of robotic systems is developed. A modified approach to the formulation of the sliding domain equations in terms of tracking errors has been presented. This approach possesses three distinct advantages: (i) it eliminates the reaching phase, (ii) it provides means to predict the entire motion and directly control the evolution of tracking errors, (iii) it facilitates the trajectory planning process in the joint and/or cartesian spaces. A planar, five-link bipedal locomotion model has been developed. Five constraint relations that cast the motion of the biped in terms of four parameters are developed. The new control method is applied to regulate the locomotion of the system according to the five constraint relations. Numerical simulation is performed to verify the ability of the controller to achieve steady gait by applying the proposed control scheme. Bifurcation diagrams of the periodic motions of the biped are used to demonstrate the improvements in controller performance that arise from the application of the proposed method.

On Performance Bounds for Control Systems

1967 ◽  
Vol 89 (2) ◽  
pp. 311-314 ◽  
Author(s):  
J. Rissanen ◽  
R. Durbeck
Keyword(s):  
Sensitivity Analysis ◽  
Nonlinear Control ◽  
Control Systems ◽  
Nonlinear Control Systems ◽  
Performance Bounds ◽  
Performance Bound ◽  
Lyapunov’S Second Method ◽  
Lurie Problem ◽  
A Performance

A technique based on Lyapunov’s second method is described for deriving bounds for the performance of nonlinear control systems. As an illustration, a performance bound is derived for systems associated with the so-called Lurie problem. As opposed to conventional sensitivity analysis, performance bounds are also determined for systems designed using grossly inaccurate models.

A Path-Following Feedback Control Law of a Five-Axle, Three-Steering Coupled-Vehicle System

2014 ◽  
Vol 26 (5) ◽  
pp. 551-565 ◽  
Author(s):  
Hiroaki Yamaguchi ◽  
◽  
Ryota Kameyama ◽  
Atsushi Kawakami ◽  
Keyword(s):  
Feedback Control ◽  
Differential Equations ◽  
Path Following ◽  
Control Law ◽  
Operating Environment ◽  
Linear Control Theory ◽  
Single Carrier ◽  
Vehicle System ◽  
Time Differential ◽  
Lyapunov’S Second Method

<div class=""abs_img""><img src=""[disp_template_path]/JRM/abst-image/00260005/03.jpg"" width=""300"" />Experimental coupled-vehicle system</div> This paper presents a new path-following feedback control law of a five-axle, three-steering coupledvehicle system which enables specifying the movements and rotations of its two carriers quantitatively, according to the operating environment. The kinematical equations of the coupled-vehicle system are first converted into time differential equations in a threechain, single-generator chained form. The time differential equations in the chained form are secondly converted into new differential equations with a new variable. The new control law enables the relative orientation between the two carriers to be constant in either a straight-bed carrier configuration or a V-bed carrier configuration, and simultaneously enables the orientations of these carriers functioning as a single carrier relative to the direction of the tangent of the path to be changed quantitatively, according to the locations of obstacles for avoiding collision with them. Asymptotic stability of the new control law is guaranteed by the linear control theory and the Lyapunov’s second method. Especially, the form of the new differential equations facilitates the design of the Lyapunov functions. The validity of the new control law is verified by an experimental five-axle, three-steering coupledvehicle system. </span>

scholarly journals Variational Lyapunov method and stability theory

1998 ◽  
Vol 3 (6) ◽  
pp. 555-571 ◽  
Author(s):  
V. Lakshmikantham ◽  
X. Liu ◽  
S. Leela
Keyword(s):  
Stability Theory ◽  
Lyapunov Method ◽  
Variation Of Parameters ◽  
Lyapunov’S Second Method ◽  
The Stability ◽  
Variational Lyapunov Method ◽  
New Framework

By unifying the method of variation of parameters and Lyapunov's second method, we develop a fruitful technique which we call variational Lyapunov method. We then consider the stability theory in this new framework showing the advantage of this unification.

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