In this article, we propose a dynamic augmentation scheme for the asymptotic solution of the nonlinear algebraic loops arising in well-known input saturated feedbacks typically designed by solving linear matrix inequalities. We prove that the existing approach based on dynamic augmentation, which replaces the static loop by a dynamic one through the introduction of a sufficiently small time constant, works under some restrictive sufficient well-posedness conditions, requiring the existence of a diagonal Lyapunov matrix. However it can fail in general, even when the algebraic loop is well-posed. Then, we propose a novel approach whose effectiveness is guaranteed whenever well-posedness holds. We also show how this augmentation allows preserving the guaranteed region of attraction with Lyapunov-based designs, as long as a gain parameter is sufficiently large. We finally propose an adaptive version of the scheme where this parameter is adjusted online. Simulation results show the effectiveness of the proposed solutions.

Starting from a symmetric state-feedback solution ensuring α-exponential convergence in an ellipsoidal sublevel set, with asymmetric saturation and single-input linear plants, we propose a novel asymmetric scheduled extension preserving the original symmetric solution in that sublevel set and extending the guaranteed stability region to the union of all possible contractive ellipsoids centered at a shifted equilibrium. Our design being based on the solution of a parametric optimization problem, we prove Lipschitz properties of the ensuing feedback law and we compute its explicit state-feedback expression.