A kind of generalized proportional-integral(GPI) observer for descriptor linear systems is introduced. We first propose two complete parametric solutions to generalized Sylvester matrix equation corresponding to the left eigenvector matrices in the case of Jordan form. Then a parametric design approach for the observer is presented. The proposed method provides all parametric expression of the gain matrices and the corresponding finite left eigenvector matrix and guarantees the regularity and impulse-freeness of the expanded error system. Two numerical examples are given to explain the design procedure and illustrate the effectiveness of the proposed method.

A complete parametric approach is proposed for the design of the Luenberger type function Kx observers for descriptor linear systems. Based on a complete parametric solution to a class of generalized Sylvester matrix equations, parametric expressions for all the coefficient matrices of the observer are derived. The approach provides all the degrees of design freedom, which can be utilized to achieve some additional design requirements. An illustrative example shows the effect of the proposed approach.