In this paper, the exact relay response expressions are derived for integrating and unstable 1st or 2nd order processes. Identification algorithms are subsequently developed in terms of the measurable parameters under relay feedback and fitting conditions of the limit cycle. It is demonstrated that the limit cycle can be undoubtedly formed for an integrating process under a biased/unbiased relay feedback test. A tighter limiting condition is given for identification of unstable processes with the standard relay feedback structure. Denoising methods are presented to cope with measurement noise. Illustrative examples are given to demonstrate the effectiveness and merits of the proposed identification algorithms. (C) 2008 Elsevier Ltd. All rights reserved.