Necessary conditions for limit cycles in multiloop relay systems

The paper examines the behaviour of multivariable systems under multiloop relay feedback control. The analysis is a generalisation of the Tsypkin method (1984) for the prediction of forced oscillations in single variable systems. It is exact in the sense that a complete harmonic balance is considered in all the loops. The results are valid for systems with characteristic loci with phase lags more than 180°. It is shown that such multivariable systems under multiloop relay feedback may exhibit limit cycle oscillations in three possible modes. The first mode consists of identical relay outputs which are square waves with precisely one fundamental frequency. The second mode is characterised by relay outputs which are square waves of different fundamental frequencies in each loop. In this mode, each loop behaves like a single variable system oscillating at a unique limit cycle frequency. The third mode is one of periodic complex oscillations consisting of multiple relay switches within one fundamental period. The necessary conditions derived show that the modes are related to the strength of the interactions in the respective loops. The authors derive a graphical technique to determine when unique oscillations (with distinct frequencies) at the output of each relay may occur and when single frequency or complex oscillations may exist instead. Simulation results are given to illustrate the possible scenarios