A phase-plane approach to relay sampled-data feedback systems

The phase plane is used to solve second-order sampled-data feedback systems which contain a relay and zero order hold circuit. The relay quantizes the signal applied to the linear part of the system and each quantization level gives rise to a corresponding set of trajectories in the phase plane. Between sampling instants the output moves along one of these trajectories; however, the error signal which is applied to the relay depends only on the output and input at the end of the sampling interval, i.e., at the sampling instants. The output at the sampling instants is determined in the phase plane with the aid of two difference equations which are obtained from the time solution of the linear part of the system. This method of solution is not restricted to the study of such systems for step function inputs but may be used equally well to study the effects of sine wave inputs on the system as is illustrated in the examples of the paper.