Phase-locked loops for plant tuning and monitoring

A loop's phase-margin phi(m) provides helpful information: often it predictably affects damping, and the corresponding frequency omega(m) is related to the closed-loop bandwidth. However, its use in controller design is frequently hampered by lack of knowledge of the plant's transfer-function G(s). Hence, as with the popular autotuning methods, the authors consider a simple algorithm that automatically determines the frequency omega(d) and gain \G(jomega(d))\ corresponding to a prescribed plant phase-lag phi(d) as this information can be used simply to tune a PI regulator This adaptive approach is shown to have the structure of a phased-locked loop (PLL). The key ingredient in a PLL is its phase-sensitive detector (PSD), as phase cannot be estimated instantaneously and the deduced value is affected by harmonics and noise. Four common PSDs are compared and it is found that the Hilbert-transform PSD has uniformly effective behaviour. Predictions of the convergence and noise-rejection properties of the algorithm are presented and confirmed via simulation. The same algorithm can be used for loop monitoring, in which plant variations are tracked via on-line estimation of omega(d)(t). A benefit of the algorithm is that it enables active probing of a plant by low-amplitude sinewaves, as good tracking is possible even with signal-to-noise ratios much less than one.