Automatic tuning of PID controllers using relay feedback experiments has received attention on and off since it was first proposed and industrially implemented in a control system in the 1980s. While optimal experiment design and modern system identification easily outperform the original automatic tuner, they rely on computational resources that are not always available in industrial control systems. Here we present a combination of experiment and subsequent output-error identification of continuous-time first-order time-delayed (FOTD) system models, that requires very little in terms of computations and memory. The method has been extensively evaluated in simulation, and a prototype has been implemented for the ABB AC 800M controller family.