scholarly journals Rational Lyapunov functions for estimating and controlling the robust domain of attraction

Automatica ◽  
2013 ◽  
Vol 49 (4) ◽  
pp. 1051-1057 ◽  
Author(s):  
Graziano Chesi
Keyword(s):  
Lyapunov Functions ◽  
Domain Of Attraction

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.

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.

scholarly journals Robust State Feedback Stabilization of Uncertain Discrete-Time Switched Linear Systems Subject to Actuator Saturation

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Xinquan Zhang ◽  
Guoliang Wang ◽  
Jun Zhao
Keyword(s):  
Linear Systems ◽  
Discrete Time ◽  
Lyapunov Functions ◽  
State Feedback ◽  
Robust Stabilization ◽  
Domain Of Attraction ◽  
Feedback Stabilization ◽  
Multiple Lyapunov Functions ◽  
Switched Linear Systems ◽  
Lyapunov Functions Method

The robust stabilization problem is investigated for a class of discrete-time switched linear systems with time-varying norm-bounded uncertainties and saturating actuators by using the multiple Lyapunov functions method. A switching law and a state feedback law are designed to asymptotically stabilize the system with a large domain of attraction. Based on the multiple Lyapunov functions method, sufficient conditions are obtained for robust stabilization. Furthermore, when some parameters are given in advance, the state feedback controllers and the estimation of domain of attraction are presented by solving a convex optimization problem subject to a set of linear matrix inequalities (LMI) constraints. A numerical example is given to show the effectiveness of the proposed technique.

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