The paper considers the problem of observer design for discrete rectangular singular systems with time-delay. First, the observer design for discrete rectangular singular systems with constant time-delay is discussed. The system is described by the following form:where x(k)âˆˆR^{n} is the state vector, u(k)âˆˆR^{p} is the controlled input,y(k)âˆˆR^{q} is the measurable output, z(k)âˆˆR^{l} is the estimated output;E,A,A_{d}âˆˆR^{mÃ—n},BâˆˆR^{mÃ—p},C,C_{d}âˆˆR^{qÃ—n},DâˆˆR^{qÃ—p},LâˆˆR^{lÃ—n}are constant matrices, Eis a singular matrix, rankE =r0, d>0 is a known integer delay.To design an observer for system (2-1) as follows:such that for any admissible initial value it satisfies that At first, it is assumed that rankE = r |