Parameter estimation from relay autotuning with asymmetric limit cycle data

Obtaining the parameters for PID controllers based on limit cycle information for the process in a relay controlled feedback loop has become an accepted practical procedure. Often the only information used is the frequency and amplitude of the limit cycle, which then using the describing function analysis provides estimates for the ultimate gain and frequency of the process. Alternatively, limit cycle measurements may be used to estimate the parameters of an assumed transfer function, usually again using describing function analysis. Both these methods can work well but may lead to problems in cases where the describing function analysis is inaccurate. In this paper, it is shown how the parameters of first and second order plant transfer functions, stable or unstable, with time delay can be found exactly, assuming no measurement errors, from measurements of the parameters of a single asymmetrical limit cycle in a relay controlled feedback loop. Examples are given to compare the approach with results using the describing function method and also to show how good controller design based on a first or second order time delay model can be achieved for plants of higher order.