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The problem of constructing full-order state observers for a class of systems with Lipschitz nonlinearities is addressed. By performing a suitable decomposition of the estimation error dynamics into cascaded systems, conditions have been found that guarantee the asymptotic stability of the estimation error in the absence of disturbances. These conditions can be conveniently expressed by means of linear matrix inequalities (LMIs). In the presence of system and measurement perturbations, when such noises are regarded as unknown deterministic inputs acting on the error dynamics, the estimator can be designed so as to be input-to-state stable (ISS) with respect to the estimation error.