In this paper a method is developed for the design of Luenberger-type observers for linear time-invariant control systems whose state equation is of the form Ex = Ax + Bu + Mg where E is a singular matrix and g is an unknown input vector. The method is based on the singular-value decomposition of the matrix E, and on the reduction of the equation Ex=Ax+Bu+Mg to a system consisting of a differential equation of form W1=F1w1+F2w2 +G1u+K1g and an algebraic equation of the form H1w1+H2w2+G2u+K2g = 0. If w2 can be eliminated from the differential equation by the aid of the algebraic equation and original output equation of the system, the method yields a reduced order observer for the generalized state space system.