Relay Self-Oscillations for Second Order, Stable, Nonminimum Phase Plants

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