A functional observer and state feedback are proposed for singular systems in the polynomial fraction form that requires no impulsive mode elimination. The order of the compensator is determined by the newly defined generalized observability index that is associated with the McMillan degree of the system. A new generalized Lyapunov equation is also proposed through a realization scheme that can be applied to both ordinary and singular systems. The solution to the equation provides an algebraic approach to the observer of singular systems in the generalized state-space form. >