Necessary conditions for limit cycles in multiloop relay systems

Abstract

The paper examines the behaviour of multivariable systems under multiloop relay feedback control. The analysis is a generalisation of the Tsypkin method 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 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.