Hierarchical estimation of the region of attraction for systems subject to a state delay and a saturated input

This paper addresses the local stability analysis problem for linear systems subject to input saturation and state delay. Thanks to the construction of a Lyapunov-Krasovskii functional associated to Legendre polynomials and the use of generalized sector conditions, sufficient linear matrix inequalities (LMIs) are derived to guarantee the local stability of the origin for the closed-loop system. In addition, an estimate of the basin of attraction of the origin is provided, which does not include the derivative of the initial condition. A convex optimization problem leans on these conditions to maximize the size of the estimate of the basin of attraction. Optimal LMIs form a hierarchy, which is competitive to improve the size criterium of the estimate of the basin of attraction. An example illustrates the potential of the technique.