scholarly journals Finite-Time Lyapunov Functions and Impulsive Control Design

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Huijuan Li ◽  
Qingxia Ma
Keyword(s):  
Finite Time ◽  
Lyapunov Functions ◽  
Lorenz System ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Impulsive System ◽  
Impulsive Systems ◽  
Lyapunov Theorem ◽  
The Lorenz System ◽  
The Stability

In this paper, we introduce finite-time Lyapunov functions for impulsive systems. The relaxed sufficient conditions for asymptotic stability of an equilibrium of an impulsive system are given via finite-time Lyapunov functions. A converse finite-time Lyapunov theorem for controlling the impulsive system is proposed. Three examples are presented to show how to analyze the stability of an equilibrium of the considered impulsive system via finite-time Lyapunov functions. Furthermore, according to the results, we design an impulsive controller for a chaotic system modified from the Lorenz system.

scholarly journals Lyapunov Control of Quantum Systems with Impulsive Control Fields

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Wei Yang ◽  
Jitao Sun
Keyword(s):  
Lyapunov Functions ◽  
Control Method ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Quantum Systems ◽  
Impulsive Systems ◽  
Finite Dimensional ◽  
Lyapunov Control ◽  
Control Of Quantum Systems ◽  
Invariant Principle

We investigate the Lyapunov control of finite-dimensional quantum systems with impulsive control fields, where the studied quantum systems are governed by the Schrödinger equation. By three different Lyapunov functions and the invariant principle of impulsive systems, we study the convergence of quantum systems with impulsive control fields and propose new results for the mentioned quantum systems in the form of sufficient conditions. Two numerical simulations are presented to illustrate the effectiveness of the proposed control method.

scholarly journals Lorenz System Stabilization Using Fuzzy Controllers

2007 ◽  
Vol 2 (3) ◽  
pp. 279 ◽  
Author(s):  
Radu-Emil Precup ◽  
Marius L. Tomescu ◽  
Stefan Preitl
Keyword(s):  
Stability Analysis ◽  
Chaotic Systems ◽  
Fuzzy Logic Controller ◽  
Lorenz System ◽  
Sufficient Conditions ◽  
Fuzzy Rule ◽  
Analysis Method ◽  
The Lorenz System ◽  
The Stability ◽  
Takagi Sugeno

The paper suggests a Takagi Sugeno (TS) fuzzy logic controller (FLC) designed to stabilize the Lorentz chaotic systems. The stability analysis of the fuzzy control system is performed using Barbashin-Krasovskii theorem. This paper proves that if the derivative of Lyapunov function is negative semi-definite for each fuzzy rule then the controlled Lorentz system is asymptotically stable in the sense of Lyapunov. The stability theorem suggested here offers sufficient conditions for the stability of the Lorenz system controlled by TS FLCs. An illustrative example describes the application of the new stability analysis method.

scholarly journals Distributed Impulsive Consensus of the Multiagent System without Velocity Measurement

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Zhi-Wei Liu ◽  
Hong Zhou ◽  
Zhi-Hong Guan ◽  
Wen-Shan Hu ◽  
Li Ding ◽  
...  
Keyword(s):  
Velocity Measurement ◽  
Multiagent System ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Laplacian Matrix ◽  
Impulsive System ◽  
Consensus Algorithm ◽  
Communication Graph ◽  
The Stability ◽  
Necessary And Sufficient

This paper deals with the distributed consensus of the multiagent system. In particular, we consider the case where the velocity (second state) is unmeasurable and the communication among agents occurs at sampling instants. Based on the impulsive control theory, we propose an impulsive consensus algorithm that extends some of our previous work to account for the lack of velocity measurement. By using the stability theory of the impulsive system, some necessary and sufficient conditions are obtained to ensure the consensus of the controlled multiagent system. It is shown that the control gains, the sampled period and the eigenvalues of Laplacian matrix of communication graph play key roles in achieving consensus. Finally, a numerical simulation is provided to illustrate the effectiveness of the proposed algorithm.

IMPULSIVE CONTROL OF CHAOTIC SYSTEM

2002 ◽  
Vol 12 (05) ◽  
pp. 1181-1190 ◽  
Author(s):  
XINZHI LIU ◽  
KOK LAY TEO
Keyword(s):  
Feedback Control ◽  
Control Problem ◽  
Equilibrium Point ◽  
Chaotic System ◽  
Lyapunov Functions ◽  
Lorenz System ◽  
Impulsive Control ◽  
The Lorenz System ◽  
Impulsive Stabilization

This paper studies an impulsive control problem. By utilizing the method of Lyapunov functions, a set of impulsive stabilization criteria are established. These results are then applied to the Lorenz system. It is shown that by using impulsive feedback control, all the solutions of the Lorenz system will converge to an equilibrium point.

Impulsive Control and Synchronization of Memristor-Based Chaotic Circuits

2014 ◽  
Vol 24 (12) ◽  
pp. 1450162 ◽  
Author(s):  
Shiju Yang ◽  
Chuandong Li ◽  
Tingwen Huang
Keyword(s):  
Chaotic System ◽  
Impulsive Control ◽  
Chaotic Behavior ◽  
Memory Function ◽  
Sufficient Conditions ◽  
Impulsive System ◽  
Circuit Element ◽  
Chaotic Circuits ◽  
Control And Synchronization ◽  
Passive Circuit

The memristor is a novel nonlinear passive circuit element which has the memory function, and the circuits based on the memristors might exhibit chaotic behavior. In this paper, we revisit a memristor-based chaotic circuit, and then investigate its stabilization and synchronization via impulsive control. By impulsive system theory, some sufficient conditions for the stabilization and synchronization of the memristor-based chaotic system are established. Moreover, an estimation of the upper bound of the impulse interval is proposed under the condition that the parameters of the chaotic system and the impulsive control law are well defined. To show the effectiveness of the theoretical results, numerical simulations are also presented.

scholarly journals The Stability of Nonlinear Differential Systems with Random Parameters

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Josef Diblík ◽  
Irada Dzhalladova ◽  
Miroslava Růžičková
Keyword(s):  
Trivial Solution ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
New Method ◽  
Random Parameters ◽  
Differential Systems ◽  
The Stability ◽  
Nonlinear Differential Systems

The paper deals with nonlinear differential systems with random parameters in a general form. A new method for construction of the Lyapunov functions is proposed and is used to obtain sufficient conditions forL2-stability of the trivial solution of the considered systems.

Partial stability analysis of nonlinear time-varying impulsive systems

2019 ◽  
Vol 12 (06) ◽  
pp. 1950066
Author(s):  
Boulbaba Ghanmi
Keyword(s):  
Stability Analysis ◽  
Lyapunov Functions ◽  
Time Varying ◽  
Impulsive Systems ◽  
Partial Stability ◽  
Stability Theorems ◽  
Time Varying Systems ◽  
The Stability ◽  
Derivatives Of ◽  
Theoretical Results

This paper investigates the stability analysis with respect to part of the variables of nonlinear time-varying systems with impulse effect. The approach presented is based on the specially introduced piecewise continuous Lyapunov functions. The Lyapunov stability theorems with respect to part of the variables are generalized in the sense that the time derivatives of the Lyapunov functions are allowed to be indefinite. With the help of the notion of stable functions, asymptotic partial stability, exponential partial stability, input-to-state partial stability (ISPS) and integral input-to-state partial stability (iISPS) are considered. Three numerical examples are provided to illustrate the effectiveness of the proposed theoretical results.

scholarly journals Banach fixed-point theorem in semilinear controllability problems – a survey

2016 ◽  
Vol 64 (1) ◽  
pp. 21-35 ◽  
Author(s):  
J. Klamka ◽  
A. Babiarz ◽  
M. Niezabitowski
Keyword(s):  
Differential System ◽  
Functional Equations ◽  
Sufficient Conditions ◽  
Impulsive System ◽  
Banach Fixed Point Theorem ◽  
Complete Controllability ◽  
Impulsive Systems ◽  
System A ◽  
State Of Art ◽  
Controllability Problems

Abstract The main aim of this article is to review the existing state of art concerning the complete controllability of semilinear dynamical systems. The study focus on obtaining the sufficient conditions for the complete controllability for various systems using the Banach fixedpoint theorem. We describe the results for stochastic semilinear functional integro-differential system, stochastic partial differential equations with finite delays, semilinear functional equations, a stochastic semilinear system, a impulsive stochastic integro-differential system, semilinear stochastic impulsive systems, an impulsive neutral functional evolution integro-differential system and a nonlinear stochastic neutral impulsive system. Finally, two examples are presented.

HOMOCLINIC AND HETEROCLINIC ORBITS AND BIFURCATIONS OF A NEW LORENZ-TYPE SYSTEM

2011 ◽  
Vol 21 (09) ◽  
pp. 2695-2712 ◽  
Author(s):  
XIANYI LI ◽  
HAIJUN WANG
Keyword(s):  
Chaotic Attractor ◽  
Lorenz System ◽  
Type System ◽  
Pitchfork Bifurcation ◽  
Heteroclinic Orbits ◽  
Chen System ◽  
Homoclinic And Heteroclinic Orbits ◽  
Dynamical Behaviors ◽  
The Lorenz System ◽  
The Stability

In this paper, a new Lorenz-type system with chaotic attractor is formulated. The structure of the chaotic attractor in this new system is found to be completely different from that in the Lorenz system or the Chen system or the Lü system, etc., which motivates us to further study in detail its complicated dynamical behaviors, such as the number of its equilibrium, the stability of the hyperbolic and nonhyperbolic equilibrium, the degenerate pitchfork bifurcation, the Hopf bifurcation and the local manifold character, etc., when its parameters vary in their space. The existence or nonexistence of homoclinic and heteroclinic orbits of this system is also rigorously proved. Numerical simulation evidences are also presented to examine the corresponding theoretical analytical results.

Lyapunov-Based Sufficient Conditions for Nonlinear Impulsive Systems

2011 ◽  
Vol 219-220 ◽  
pp. 508-512
Author(s):  
Yong Liang Gao ◽  
Xiao Wu Mu
Keyword(s):  
Differential Equation ◽  
Stability Analysis ◽  
Lyapunov Function ◽  
Sufficient Conditions ◽  
Invariant Set ◽  
Positive Definite ◽  
Impulsive Systems ◽  
Stability Theorems ◽  
Sufficient Conditions For Convergence ◽  
The Stability

This paper focuses on the stability analysis and invariant set stability theorems for nonlinear impulsive systems. A set of Lyapunov-based sufficient conditions are established for these convergent properties. These results do not require the Lyapunov function to be positive definite. Inequalities relating the righthandside of the differential equation and the Lyapunov function derivative are involved for these results. These inequalities make it possible to deduce properties of the functions and thus leads to sufficient conditions for convergence and stability.

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