Analytical expressions for relay feed back responses

In this work, a systematic approach to derive analytical expressions for relay feed back responses is proposed. It is based on the observation that a relay feedback test consists of a series of step inputs and a stable limit cycle implies a convergent infinite series. The second order plus dead time processes with different damping coefficients are used to illustrate the derivation. Analytical expressions for typical transfer functions up to 5th order are tabulated and a general expression for nth order system is also given. These analytical expressions are useful to identify unknown system parameters and subsequently for autotuning.