scholarly journals New approach to asymptotic stability: time-varying nonlinear systems

1997 ◽  
Vol 20 (2) ◽  
pp. 347-366 ◽  
Author(s):  
L. T. Grujić
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Sufficient Conditions ◽  
Stability Domain ◽  
Time Varying ◽  
Uniform Asymptotic Stability ◽  
Stability Time ◽  
Necessary And Sufficient

The results of the paper concern a broad family of time-varying nonlinear systems with differentiable motions. The solutions are established in a form of the necessary and sufficient conditions for: 1) uniform asymptotic stability of the zero state, 2) for an exact single construction of a system Lyapunov function and 3) for an accurate single determination of the (uniform) asymptotic stability domain. They permit arbitrary selection of a functionp(⋅)from a defined functional family to determine a Lyapunov functionv(⋅),[v(⋅)], by solvingv′(⋅)=−p(⋅){or equivalently,v′(⋅)=−p(⋅)[1−v(⋅)]}, respectively. Illstrative examples are worked out.

scholarly journals Solutions to Lyapunov stability problems of sets: nonlinear systems with differentiable motions

1994 ◽  
Vol 17 (1) ◽  
pp. 103-112 ◽  
Author(s):  
Ljubomir T. Grujic
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Stability Domain ◽  
Invariant Set ◽  
Necessary And Sufficient ◽  
Time Invariant ◽  
Selection Of

Time-invariant nonlinear systems with differentiable motions are considered. The algorithmic necessary and sufficient conditions are established in various forms for one-shot construction of a Lyapunov function, for asymptotic stability of a compact invariant set and for the exact determination of the asymptotic stability domain of the invariant set.The classical conditions are expressed in terms of existence of a system Lyapunov functions. The conditions of theorems presented herein are expressed via properties of the solution?to??=-p, or of the solution?to??=-(1-?)p, for arbitrarily selectedp?P(S;f)orp?P1(S;f), where familiesP(S;f)andP1(S;f)are well defined. The equation??=-p, or its equivalent??=-(1-?)p, should be solved only for one selection of the functionp.

scholarly journals Global Stability of Positive Discrete-time Time-varying Nonlinear Feedback Systems

2021 ◽  
Vol 3 (1) ◽  
pp. 17-20
Author(s):  
Tadeusz Kaczorek ◽  
Łukasz Sajewski
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Global Stability ◽  
Linear Systems ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Time Varying ◽  
Nonlinear Feedback ◽  
Feedback Systems ◽  
Positive Time

The global stability of positive  discrete-time time-varying nonlinear systems with time-varying scalar feedbacks is investigated. Sufficient conditions for the asymptotic stability of discrete-time positive time-varying linear systems are given. The new conditions are applied to discrete-time positive time-varying nonlinear systems with time-varying feedbacks. Sufficient conditions are established for the global stability of the discrete-time positive time-varying nonlinear systems with feedbacks.

5.—The Uniform Asymptotic Stability of Certain Linear Differential-difference Equations

1976 ◽  
Vol 74 ◽  
pp. 71-79
Author(s):  
R. Datko
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Difference Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Asymptotic Stability ◽  
Linear Ordinary Differential Equations ◽  
Linear Differential ◽  
Necessary And Sufficient

SynopsisA necessary and sufficient condition is developed for determination of the uniform stability of a class of non-autonomous linear differential-difference equations. This condition is the analogue of the Liapunov criterion for linear ordinary differential equations.

scholarly journals Global Uniform Asymptotic Stability Criteria for Linear Uncertain Switched Positive Time-Varying Delay Systems with All Unstable Subsystems

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2118
Author(s):  
Thaned Rojsiraphisal ◽  
Piyapong Niamsup ◽  
Suriyon Yimnet
Keyword(s):  
Asymptotic Stability ◽  
Robust Stability ◽  
Sufficient Conditions ◽  
Delay Systems ◽  
Time Varying ◽  
Uniform Asymptotic Stability ◽  
Time Varying Delay ◽  
Unstable Subsystems ◽  
Positive Time ◽  
Varying Delay

In this paper, the problem of robust stability for a class of linear switched positive time-varying delay systems with all unstable subsystems and interval uncertainties is investigated. By establishing suitable time-scheduled multiple copositive Lyapunov-Krasovskii functionals (MCLKF) and adopting a mode-dependent dwell time (MDDT) switching strategy, new delay-dependent sufficient conditions guaranteeing global uniform asymptotic stability of the considered systems are formulated. Apart from past studies that studied switched systems with at least one stable subsystem, in the present study, the MDDT switching technique has been applied to ensure robust stability of the considered systems with all unstable subsystems. Compared with the existing results, our results are more general and less conservative than some of the previous studies. Two numerical examples are provided to illustrate the effectiveness of the proposed methods.

scholarly journals Analysis of positivity and stability of fractional discrete-time nonlinear systems

2016 ◽  
Vol 64 (3) ◽  
pp. 491-494 ◽  
Author(s):  
T. Kaczorek
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Discrete Time ◽  
Lyapunov Method ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stability Tests ◽  
Necessary And Sufficient

Abstract The positivity and asymptotic stability of the fractional discrete-time nonlinear systems are addressed. Necessary and sufficient conditions for the positivity and sufficient conditions for the asymptotic stability of the fractional nonlinear systems are established. The proposed stability tests are based on an extension of the Lyapunov method to the positive fractional nonlinear systems. The effectiveness of tests is demonstrated on examples.

scholarly journals Solutions to Lyapunov stability problems: nonlinear systems with continuous motions

1994 ◽  
Vol 17 (3) ◽  
pp. 587-596 ◽  
Author(s):  
Ljubomir T. Grujic
Keyword(s):  
Lyapunov Function ◽  
Nonlinear Dynamical System ◽  
Sufficient Conditions ◽  
Stability Domain ◽  
Necessary And Sufficient Conditions ◽  
Nonlinear Dynamical ◽  
Bounded Set ◽  
Necessary And Sufficient ◽  
Bounded Sets ◽  
Selection Of

The necessary and sufficient conditions for accurate construction of a Lyapunov function and the necessary and sufficient conditions for a set to be the asymptotic stability domain are algorithmically solved for a nonlinear dynamical system with continuous motions. The conditions are established by utilizing properties of o-uniquely bounded sets, which are explained in the paper. They allow arbitrary selection of an o-uniquely bounded set to generate a Lyapunov function.Simple examples illustrate the theory and its applications.

Sign in / Sign up

Export Citation Format

Share Document