Simple formulae are derived to tune/design the PI and PID controllers to meet user-specified gain margin and phase margin. These formulae are particularly useful in the context of adaptive control and auto-tuning, where the controller parameters have to be calculated on-line. The results in this paper can be used to predict the achievable rise time of the closed-loop system, which is useful for self-diagnosis-a desirable feature of ’intelligent’ controllers. New insights into the internal model control design for the PID controller are also given.