Transfer Function Modelling

Abstract

The preceding two chapters indicate that relay identification usually yields information on process frequency response. However, most modern control designs make use of parametric models. Transfer functions might be the most welcome parametric model. Fitting parametric models to frequency data is thus important (Ninness, 1996). Most existing methods for transfer function identification are for delay-free processes (Ljung, 1985; Sagara and Zhao, 1990; Pin-telon et al., 1994; Tugnait and Tontiruttananon, 1998) or assume that the delay is known a priori. It has long been recognized that the inclusion of a time-delay term in a transfer function can drastically reduce the model order and facilitate cheap and more efficient implementation of both conventional and model-based controllers (Taiwo, 1999). A frequently used method for dealing with unknown delays has been to use a shift operator model with an expanded numerator polynomial (Kurz and Goedecke, 1981). Another popular approach is based on the approximation of the dead time by a rational transfer function such as the polynomial approximation (Gawthrop and Nihtila, 1985), Padé approximation (Souza et al., 1992) or Laguerre expansion (Malti et al., 1998). Such approaches require estimation of more parameters because of the increased model order, and could result in large unacceptable approximation errors, especially when the system has a large delay. These methods have been developed for discrete systems while continuous systems are more familiar to practising control engineers.