Estimation of Minimum Conditions for Forced Oscillations in Relay Feedback Systems

Identification of frequency ranges for subharmonic oscillations in a relay feedback system

“…While all these studies were going on, little attention was given to externally forced RFS even though the phenomenon of forced oscillations (FO) have been known since the days of the Huygens' clock. More recently, with the extension of the auto-tuning techniques to multivariable systems, a better understanding of FO was given in [7]. This paper is an extension of the results in [7], with the focus being on another common behaviour, namely that of subharmonic oscillations (SO).…”

“…More recently, with the extension of the auto-tuning techniques to multivariable systems, a better understanding of FO was given in [7]. This paper is an extension of the results in [7], with the focus being on another common behaviour, namely that of subharmonic oscillations (SO). When the external signal, f (t), of frequency ω f induces oscillations of frequency ω f in the RFS, then FO is said to have taken place.…”

“…Other methods by Tsypkin [2] and Atherton [8] attempted to identify the amplitude of the external forcing signal required for FO and SO. The studies in [2], [8] and [7] on the minimum amplitude of f (t) required for FO and SO did not reveal the full extent of the problem on the existence of SO. The prediction of SO is more difficult than FO because they cannot be observed for all f (t) of arbitrary frequencies.…”

“…The problem we address is that of subharmonic oscillations. From our previous work in [7], it was shown that FO can always occur (particularly for delayed system) when the amplitude of f (t) is sufficiently large. However, this is not true in the case of SO.…”

“…In this example, the values of R ν,min are found to be equal to |Cz(0)| as in (15) and they also satisfy (17) and (18). The R 1,min values for forced oscillation are derived from [7]. If T f /2 of the external forcing signal falls within the bounds indicated in Fig.…”