Robust absolute stability and nonlinear state feedback stabilization based on polynomial Lur’e functions

This paper provides finite dimensional convex conditions to construct homogeneous polynomially parameter-dependent Lur’e functions which ensure the stability of nonlin-ear systems with state-dependent nonlinearities lying in general sectors and which are af-fected by uncertain parameters belonging to the unit simplex. The proposed conditions are written as linear matrix inequalities parametrized in terms of the degree g of the parameter-dependent solution and in terms of the relaxation level d of the inequality constraints, based on algebraic properties of positive matrix polynomials with parameters in the unit simplex. As g and d increase, progressively less conservative solutions are obtained. The results in the paper include as special cases existing conditions for robust stability and for absolute stability analysis. A convex solution suitable for the design of robust nonlinear state feed-back stabilizing controllers is also provided. Numerical examples illustrate the efficiency of the proposed conditions. ∗Corresponding author. 1 ha l-