Predicting oscillations in relay feedback systems, using fixed points of Poincar\'e maps, and Hopf bifurcations

The relay autotuning method identifies plant parameters, from oscillations of the plant under relay feedback. To predict the presence and nature of such oscillations, we apply the following two approaches: (a) analysis of the switching dynamics, while using an ideal relay, and (b) bifurcation analysis, while using a smooth approximation of the relay. For stable plants with positive DC gains, our analyses predict that: (i) a periodic orbit is guaranteed, for a class of non-minimum phase plants of relative degree one, whose step response starts with an inverse response, and (ii) for a wider class of plants, whose root locus diagrams cross the imaginary axis at complex conjugate values, limit cycles are merely suggested.
Comment: submitted to the IEEE transactions on Automatic Control