Global uniform asymptotic stability of cascade time-varying nonlinear systems: case study

2002 ◽  
Author(s):  
E. Panteley
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Time Varying ◽  
Uniform Asymptotic Stability

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.

Compensation of Time-Varying Input and State Delays for Nonlinear Systems

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
Nikolaos Bekiaris-Liberis ◽  
Miroslav Krstic
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Time Varying ◽  
Feedback Systems ◽  
Feedback Controllers ◽  
Stabilizing Controller ◽  
Numerical Examples ◽  
Infinite Dimensional ◽  
State Delays ◽  
Main Challenge

We consider general nonlinear systems with time-varying input and state delays for which we design predictor-based feedback controllers. Based on a time-varying infinite-dimensional backstepping transformation that we introduce, our controller achieves global asymptotic stability in the presence of a time-varying input delay, which is proved with the aid of a strict Lyapunov function that we construct. Then, we “backstep” one time-varying integrator and we design a globally stabilizing controller for nonlinear strict-feedback systems with time-varying delays on the virtual inputs. The main challenge in this case is the construction of the backstepping transformations since the predictors for different states use different prediction windows. Our designs are illustrated by three numerical examples, including unicycle stabilization.

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.

Uniform Global Asymptotic Stability of Adaptively Controlled Nonlinear Systems via Strict Lyapunov Functions

2008 ◽  
Author(s):  
Fre´de´ric Mazenc ◽  
Marcio de Queiroz ◽  
Michael Malisoff
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Functions ◽  
Unknown Parameter ◽  
Time Varying ◽  
Excitation Condition ◽  
Parameter Vector ◽  
Uniform Asymptotic Stability ◽  
Asymptotic Tracking ◽  
Persistency Of Excitation ◽  
Unknown Parameter Vector

We prove global uniform asymptotic stability of adaptively controlled dynamics by constructing explicit global strict Lyapunov functions. We assume a persistency of excitation condition that implies both asymptotic tracking and parameter identification. We also construct input-to-state stable Lyapunov functions under an added growth assumption on the regressor, assuming that the unknown parameter vector is subject to suitably bounded time-varying uncertainties. This quantifies the effects of uncertainties on the tracking and parameter estimation. We demonstrate our results using the Ro¨ssler system.

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