Existing tuning formulas for PID based on relay feedback experiments are derived from continuous-time systems theory, even though most such systems are implemented digitally. These formulas rely on the fact that, according to this continuous time theory, the relay feedback experiment allows the identification of the ultimate point of the plant's frequency response - the point at which its Nyquist plot crosses the negative real axis. We have shown in a recent paper (Bazanella and Parraga (2016)) that a sampled relay feedback experiment may exhibit various limit-cycles at possibly quite different frequencies, even for quite reasonable sampling rates - that is, well within the ranges recommended by sampling theory and control textbooks. In this paper we show the deleterious effect of this reality on the tuning of PID controllers and propose an improvement to the tuning formulas to overcome this limitation.
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