This chapter deals with identification of time-varying systems using Kalman filter approach. Most physical systems exhibit some degree of time-varying behaviour for many reasons. These systems cannot effectively be modelled using time invariant models. A time-varying autoregressive with exogenous input (TVARX) model is good to model these time-varying systems. The Kalman filter approach is a superior way to estimate the system parameters. This approach can track the time-varying parameters and is suitable for recursive estimation. It works well even when there are abrupt changes in the system parameters. Kalman filter is known to be an optimal estimator even when there is significant noise. In the proposed approach, for the purpose of simulation, we employ first order TVARX model and its parameters are estimated using recursive Kalman filter method. The system parameters are varied in continuous and abruptly changing manner to reveal the physical situation. To show the efficacy of the proposed approach, the time-varying parameters are estimated for different noise conditions. The performance is evaluated by calculating error performance measures. The results are found to be satisfactory with reasonable accuracy for noisy conditions even for fast changing parameters. The numerical examples illustrate efficacy of the proposed Kalman filter based approach for identification of time-varying systems.
TopIntroduction
During the recent years, there has been considerable interest in the identification of linear time-varying systems (Niedzwiecki, 2000; Niedzwiecki, 2008; Shahriari, Tarasiewicz, & Adrot, 2008). Most physical systems exhibit some degree of time-varying behaviour. Over short time intervals these systems can be satisfactorily approximated by linear dynamic time invariant models, but over the longer time intervals they reveal time-varying features or characteristics, hence they call for models with time-dependent coefficients. Physical phenomena exhibit time-varying behaviour for many reasons, mainly due to the variation of internal (aging, fatigue) and external (set point changes, time dependent disturbances) operating conditions (Niedzwiecki, 2000; Haddadi, & Zaad, 2008). There have been a large number of really challenging applications of linear time-varying system identification in different areas. For instance, telecommunications, signal processing and automatic control are some of the areas of applications where linear system identification has been extensively used. In telecommunications, models are used for channel equalization, predictive coding and noise cancellation. In signal processing applications, models obtained by system identification are used for spectral estimation, fault detection, pattern recognition, signal reconstruction, outlier elimination, prediction and adaptive filtering. In automatic control, models obtained are used for adaptive control, failure detection etc. The fields like chemical process, socioeconomic system, electric systems, hydrology, aeronautics, and seismology find tremendous applications of systems identification and modelling (Allison, Miller, & Inman, 2009; Budiyono, & Sutarto, 2006; Pan, Westwick, & Nowicki, 2004; Yinfeng, Yingmin, Mingkui, & Ming, 2009; Spiridonakos, & Fassois, 2009). Also there has been very significant progress towards the application of system identification technique to physiological and biomedical problems (MacNeil, Kearney, & Hunter, 1992). Very successful models have been obtained for human performance in man machine environment, control function of beginner and the muscle, metabolism, brain waves, and so on. Thus, the subject of system identification is attracting increasing interest, also from medical and life scientists (Zou, Wang, & Chon, 2003). Similar modelling application can be found in such areas as ecology, transportation, mechanical and structural engineering (Wang, Luo, Qin, Leng, & Wang, 2008; Spiridonakos, & Fassois, 2009; Law, Bu, & Zhu, 2005; Allison, Miller, & Inman, 2008; Poulimenos, & Fassois, 2006).