System Identification

There are two basic approaches to obtaining the system models. The first one is based on principles in physics or other sciences. The inverted pendulum in Chap. 1 provides an example to this approach. Its advantage lies in its capability to model nonlinear systems and preservation of the physical parameters. However this approach can be costly and time consuming. The second approach is based on input and output data to extrapolate the underlying system model. This approach treats the system as a black box and is only concerned with its input-output behaviors. While experiments need to be carried out and restrictions on input signals may apply, the second approach overcomes the weakness of the first approach.This chapter examines the input/output approach to modeling of the physical system which is commonly referred to as system identification. In this approach, the mathematical model is first parameterized, and then estimated based on input/output experimental data. Autoregressive moving-average (ARMA) models are often used in feedback control systems due to their ability to capture the system behavior with lower order and fewer parameters than the moving average (MA) models or transversal filters. On the other hand, wireless channels are more suitable to be described by MA models due to the quick die-out of the channel impulse response (CIR). Many identification algorithms exist, and most of them use squared error as the identification criterion. The squared error includes energy or mean power of the model matching error that results in least-squares (LS), or total LS (TLS), or minimum mean-squared error (MMSE) algorithms. These algorithms will be presented and analyzed in two different sections. For ease of the presentation, only real matrices and variable are considered, but the results are readily extendable to the case of complex valued signals and systems. A very important result in estimation is the well known Cramer–Rao lower bound (CRLB).