The problem of state estimation for a class of non-linear systems with Lipschitz non-linearities is addressed using sliding-mode estimators. Stability conditions have been found to guarantee asymptotic convergence to zero of the estimation error in the absence of noise and non-divergence if the state perturbations and measurement noise are bounded. A method is proposed to find a suitable solution to the algebraic Riccati equation on which the design of the estimator is based. The performance of the resulting sliding-mode filter minimizes an upper bound on the asymptotic estimation error. Based on such an approach, a sliding-mode estimator may be designed so as to outperform the extended Kalman filter in practical applications with models affected by uncertainty and strong, possibly unknown non-linearities, as shown by means of simulations.