For constant linear systems we are introducing integralreconstructors and generalized proportional-integralcontrollers, which permit to bypass the derivative term in theclassic PID controllers and more generally the usual asymptoticobservers. Our approach, which is mainly of algebraic flavour, isbased on the module-theoretic framework for linear systems and onoperational calculus in Mikusiński's setting. Several examplesare discussed.