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This technical note examines attractiveness and minimality of invariant sets for linear systems subject to additive disturbances confined in a state-dependent set. Existence of a minimal attractor is proved under the assumption that the state-dependent set, in which the disturbance is confined, is upper-semi-continuous. In many practical applications, the disturbance may evolve in a compact set while being generated by a dynamic process with a given model and bounded input. Our results for systems subject to general disturbances confined in state-dependent sets are applied to this case to prove the existence of a minimal robust invariant attractor.