NECESSARY CONDITIONS FOR LIMIT-CYCLES IN MULTILOOP RELAY SYSTEMS

The paper examines the behaviour of multivariable systems under multil oop relay feedback control. The analysis is a generalisation of the Ts ypkin method for the prediction of forced oscillations in single varia ble systems. It is exact in the sense that a complete harmonic balance is considered in all the loops. The results are valid for systems wit h characteristic loci with phase lags more than 180-degrees. It is sho wn that such multivariable systems under multiloop relay feedback may exhibit limit cycle oscillations in three possible modes. The first mo de consists of identical relay outputs which are square waves with pre cisely one fundamental frequency. The second mode is characterised by relay outputs which are square waves of different fundamental frequenc ies in each loop. In this mode, each loop behaves like a single variab le system oscillating at a unique limit cycle frequency. The third mod e is one of periodic complex oscillations consisting of multiple relay switches within one fundamental period. The necessary conditions deri ved show that the modes are related to the strength of the interaction s 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 o scillations may exist instead. Simulation results are given to illustr ate the possible scenarios.