Low-order modeling from relay feedback

There have recently been considerable interests in extensions of PID relay autotuning to model-based controllers, and this necessitates transfer function modeling from relay feedback. The methods reported in the literature for this purpose require two tuning tests, and their results are approximate in nature. In this paper, exact expressions for the periods and amplitudes of limit cycles under relay feedback are derived for processes which may be modeled by first-order plus dead-time dynamics. This time-domain information is then combined with frequency response point estimation using Fourier series expansions of the limit cycles so that a first-order plus dead-time model can be identified with a single relay test. Furthermore, no approximation is made in our derivations and the resultant model will be precise if it matches the structure of the process. In the case of the mismatched structure, it is shown through extensive simulations that our procedure yields very accurate results in the sense that the identified model frequency response fits the actual process one well. Both unbiased and biased relays with or without hysteresis are considered.