An LMI Technique for the Global Stabilization of Nonlinear Polynomial Systems

Abstract

This paper deals with the global asymptotic stabilization of nonlinear polynomial systems within the framework of Linear Matrix Inequalities (LMIs). By employing the well-known Lyapunov stability direct method and the Kronecker product properties, we develop a technique of designing a state feedback control law which stabilizes quadratically the studied systems. Our main goal is to derive sufficient LMI stabilization conditions which resolution yields a stabilizing control law of polynomial systems.

Robust non-fragile control of a class of nonlinear uncertain systems

“…Using a quadratic Lyapunov function V (X) = X T P X sufficient condition of the global asymptotic stabilization of the polynomial system is given as [10]: The nonlinear polynomial system defined by the equation (5) is globally stabilized by the control law (8), if there exist…”

Design of stabilizing control for synchronous machines via polynomial modelling and linear matrix inequalities approach

“…In fact, control of synchronous machines is usually leaded from linearized models, using a reduction order method or a decoupling of slow and fast dynamics by singular perturbations method, yielding reduced order or composite controllers [1] which were used after for optimal control of synchronous machines. With an ambition to achieve a better performance, we propose a new design for synthesis of a nonlinear polynomial control law using recent results on stabilization of polynomial systems associated to the LMI technique [21].…”

“…(ii) Provides satisfactory performance over a wide range of agressive state perturbations. Lyapunov's direct method is concerned in this work for assessing the stability analysis and synthesis of the power dynamic system described by a set of nonlinear equations of the form (20)(21). For this goal, we consider a quadratic Lyapunov function [28,29]:…”

“…The 7th order state space model of the synchronous machine is developed around the operating point on a third order polynomial system to improve the performances of the considered power system controlling, and particularly in the sense of the widening of the stability domain around the operating point. Furthermore, in this present work, we make use of recent results on nonlinear polynomial system stabilization [21], to derive a polynomial stabilizer of a synchronous machine connected to an infinite busbar. This approach has the advantage to lead to a closed loop system, which stability is guaranteed in a large domain around the operating point.…”

Enlarging the Domain of Attraction in Nonlinear Polynomial Systems

“…The proposed controller is polynomial in the measurable output; it exploits relaxations based on the sum of squares of polynomials in order to prove that the lower bound of the maximum achievable largest estimated domain of attraction and a corresponding controller can be computed via a generalized eigenvalue problem. The main advantage of the methodology is that the problem is formulated as a quasi-convex Linear Matrix Inequalities (LMI) ( [1], [3], [10]). …”