Finite-frequency identification of a delay via sliding phase lag

This article proposes a phase sliding improvement of finite-frequency identification algorithm for a linear stable plant with time-delay in presence of unknown-but-bounded external disturbances (with unknown stochastic characteristics). Finite-frequency identification algorithm feeds to the plant's input a testing signal, that consist of a single harmonic or a sum of them. Phase sliding improvement allows to identify two unknown values, such as coefficients of the plant’s transfer function or time-delay value for the successful identification and more than one harmonic if it is needed to increase accuracy or identify a long time-delay value. There are two ideas in the article. Firstly, phase sliding for each harmonic may be opposite for phase deviation, caused by plant's time-delay. Time-delay, phase sliding and harmonic's frequency are analytically related. Secondly, plant's transfer function will be the same for different combinations of harmonics if they are opposite to plant's time-delay. For a definite solution of the delay identification problem the upper bound of the time-delay value is required. The article describes how to select theoretically optimal harmonics, providing time-delay identification without known upper bound. There are three algorithms for the detection equality of phase-slided and time-delayed values, which differ one from each other computational complexity and sensitivity to external disturbances.