PI auto-tuning during a single transient

A certain adaptive PI tuner operates by injecting variable-frequency sinewaves at the PI's set-point. Its initialisation requires: (a) a stable preliminary controller, (b) reasonable values (correct within a factor of about 3) of the first probing frequency omega(0) and of the plant's gain \G(p)(j omega(0))\, and (c) an effective choice of the adaptive gain K-a for rapid tuning. The challenge is to prime the parameters of both the PI and the adaptive algorithm during an open-loop starting transient, using data obtained before the output attains its set-point, and then to complete the tuning for a short period while under closed-loop control. A relevant parameter is T-m, the time for the plant's impulse response to attain its maximum (or equivalently the time for a maximum slope of the step-response s(t)), and a simple estimator of T-m is described. One of three generic plant models is adopted according to the relative position of T-m within the primary experimental time [T-1 center dot center dot center dot T-2], and it is shown how initialisation depends on the chosen model. The described approach is shown to be effective for a range of plants.