A novel descriptor observer structure is proposed in the context of discrete-time descriptor linear systems. On the basis of this new structure, the characterization of conditioned-invariant polyhedral sets with the purpose of limiting the estimation error is carried out. Limitation of the estimation error can be achieved by the computation of an as small as possible conditioned-invariant polyhedron, that contains the set of possible initial errors, and by a suitable output injection law. The approach is applied to an experimental tanks platform, for which a descriptor model is identified and validated. The obtained results confirm the merits of the proposed observer for applications in this kind of systems.