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.
Calculating the optimal hedge ratio: constant, time varying and the Kalman Filter approach
