Estimation of the Domain of Attraction for Uncertain Nonlinear Autonomous Systems via rational Lyapunov functions: A Bezoutian Approach

2020 ◽  
Author(s):  
Thomas Pursche ◽  
Roland Claus ◽  
Bern Tibken
Keyword(s):  
Lyapunov Functions ◽  
Autonomous Systems ◽  
Domain Of Attraction

scholarly journals Stability of Nonlinear Autonomous Quadratic Discrete Systems in the Critical Case

2010 ◽  
Vol 2010 ◽  
pp. 1-23 ◽  
Author(s):  
Josef Diblík ◽  
Denys Ya. Khusainov ◽  
Irina V. Grytsay ◽  
Zdenĕk Šmarda
Keyword(s):  
Lyapunov Functions ◽  
Critical Case ◽  
Autonomous Systems ◽  
Discrete Systems ◽  
Simple Eigenvalue ◽  
Discrete Equations ◽  
The Matrix ◽  
The Stability ◽  
Important Characterization

Many processes are mathematically simulated by systems of discrete equations with quadratic right-hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalueλ=1of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.

Recursive Lyapunov Functions

1989 ◽  
Vol 111 (4) ◽  
pp. 641-645 ◽  
Author(s):  
Andrzej Olas
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Exponential Stability ◽  
Performance Measures ◽  
Lyapunov Functions ◽  
Stability Problem ◽  
Autonomous Systems ◽  
Sequence Of Functions

The paper presents the concept of recursive Lyapunov function. The concept is applied to investigation of asymptotic stability problem of autonomous systems. The sequence of functions {Uα(i)} and corresponding performance measures λ(i) are introduced. It is proven that λ(i+1) ≤ λ(i) and in most cases the inequality is a strong one. This fact leads to a concept of a recursive Lyapunov function. For the very important applications case of exponential stability the procedure is effective under very weak conditions imposed on the function V = U(0). The procedure may be particularly applicable for the systems dependent on parameters, when the Lyapunov function determined from one set of parameters may be employed at the first step of the procedure.

scholarly journals Lyapunov-based Methods for Maximizing the Domain of Attraction

2020 ◽  
Vol 15 (5) ◽  
Author(s):  
Houssem Mahmoud JERBI ◽  
Faiçal HAMIDI ◽  
Sondess BEN AOUN ◽  
Severus Constantin OLTEANU ◽  
Dumitru POPESCU
Keyword(s):  
Lyapunov Functions ◽  
Nonlinear Models ◽  
Domain Of Attraction ◽  
Attraction Domain ◽  
Linear Matrix ◽  
Van Der Pol ◽  
Lmi Optimization ◽  
Quadratic Lyapunov Functions ◽  
Van Der Pol Model ◽  
High Degree

This paper investigates Lyapunov approaches to expand the domain of attraction (DA) of nonlinear autonomous models. These techniques had been examined for creating generic numerical procedures centred on the search of rational and quadratic Lyapunov functions. The outcomes are derived from all investigated methods: the method of estimation via Threshold Accepted Algorithm (TAA), the method of estimation via a Zubov technique and the method of estimation via a linear matrix inequality (LMI) optimization and genetic algorithms (GA). These methods are effective for a large group of nonlinear models, they have a significant ability of improvement of the attraction domain area and they are distinguished by an apparent propriety of direct application for compact and nonlinear models of high degree. The validity and the effectiveness of the examined techniques are established based on a simulation case analysis. The effectiveness of the presented methods is evaluated and discussed through the study of the renowned Van der Pol model.

Robust stabilization of non-linear non-autonomous control systems with periodic linear approximation

2020 ◽  
Author(s):  
V I Slyn’ko ◽  
Cemil Tunç ◽  
V O Bivziuk
Keyword(s):  
Autonomous Systems ◽  
Robust Stabilization ◽  
Domain Of Attraction ◽  
Linear Feedback ◽  
Partial Sums ◽  
Controlled System ◽  
Periodic Linear ◽  
Non Linear ◽  
Lyapunov Matrix ◽  
Matrix Differential

Abstract The paper deals with the problem of stabilizing the equilibrium states of a family of non-linear non-autonomous systems. It is assumed that the nominal system is a linear controlled system with periodic coefficients. For the nominal controlled system, a new method for constructing a Lyapunov function in the quadratic form with a variable matrix is proposed. This matrix is defined as an approximate solution of the Lyapunov matrix differential equation in the form of a piecewise exponential function based on partial sums of a W. Magnus series. A stabilizing control in the form of a linear feedback with a piecewise constant periodic matrix is constructed. This control simultaneously stabilizes the considered family of systems. The estimates of the domain of attraction of an asymptotically stable equilibrium state of a closed-loop system that are common for all systems are obtained. A numerical example is given.

Control of Polynomial Nonlinear Systems Using Higher Degree Lyapunov Functions

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
Shuowei Yang ◽  
Fen Wu
Keyword(s):  
Nonlinear Systems ◽  
Lyapunov Functions ◽  
Optimization Problems ◽  
Gain Control ◽  
Domain Of Attraction ◽  
State Variables ◽  
Control Approach ◽  
Nonlinear Controllers ◽  
Quadratic Lyapunov Functions ◽  
L2 Gain

In this paper, we propose a new control design approach for polynomial nonlinear systems based on higher degree Lyapunov functions. To derive higher degree Lyapunov functions and polynomial nonlinear controllers effectively, the original nonlinear systems are augmented under the rule of power transformation. The augmented systems have more state variables and the additional variables represent higher order combinations of the original ones. As a result, the stabilization and L2 gain control problems with higher degree Lyapunov functions can be recast to the search of quadratic Lyapunov functions for augmented nonlinear systems. The sum-of-squares (SOS) programming is then used to solve the quadratic Lyapunov function of augmented state variables (higher degree in terms of original states) and its associated nonlinear controllers through convex optimization problems. The proposed control approach has also been extended to polynomial nonlinear systems subject to actuator saturations for better performance including domain of attraction (DOA) expansion and regional L2 gain minimization. Several examples are used to illustrate the advantages and benefits of the proposed approach for unsaturated and saturated polynomial nonlinear systems.

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