The CARMA model generally assumed in self-tuning control frequently leads to an offset between the measurement and the desired value when regulating a plant subjected to constant load disturbances about a nonzero set point. The normalad hocinsertion of an integrator solves the problem but can introduce others (such as convergence difficulties). The solution explored in this paper is to use a CARIMA plant model in which the noise term is nonstationary, and it is shown that good adaptive behaviour is then possible both with generalised minimum-variance and pole-placement self-tuning controllers. The approach produces integral control terms in a natural way, and, for simplified plant models, leads directly to a general PID algorithm.