LAUSR

We study a relay feedback system (RFS) having an ideal relay element and a linear, time-invariant, second-order plant. The relay element is modeled as an ideal on–off switch. And the plant is modeled using a transfer function that as follows: first, is Hurwitz stable, second, is proper, third, has a positive real zero, andfourth,  has a positive dc gain. We analyze this RFS using a state-space description, with closed-form expressions for the state trajectory from one switching time to the next. We prove that the state transformation from one switching time to the next, first, has a Schur stable linearization, and first,  is a contraction mapping. Then using the Banach contraction mapping theorem, we prove that all trajectories of this RFS converge asymptotically to a unique limit cycle. This limit cycle is symmetric, and is unimodal as it has exactly two relay switches per period. This result helps understand the behavior of the relay autotuning method, when applied to second-order plants with no time delay. We also treat the case where the plant either has no finite zero, or has exactly one zero that is negative.