TABLE OF CONTENTS In the previous chapter, we discussed the least squares approach to parameter estimation. It is the most simple and, perhaps, most highly favoured approach to determine the system characteristics from its input and output time histories. There are several methods that can be used to estimate system parameters. These techniques differ from one another based on the optimal criterion used and the presence of process and measurement noise in the data. The output error concept was described in Chapter 1 (see Fig. 1.1). The maximum likelihood process invokes the probabilistic aspect of random variables (e.g., measurement/errors, etc.) and defines a process by which we obtain estimates of the parameters. These parameters most likely produce the model responses, which closely match the measurements. A likelihood function (akin to probability density function) is defined when measurements are (collected and) used. This likelihood function is maximised to obtain the maximum likelihood estimates of the parameters of the dynamic system. The equation error method is a special case of the maximum likelihood estimator for data containing only process noise and no measurement noise. The output error method is a maximum likelihood estimator for data containing only measurement noise and no process noise. At times, one comes across statements in literature mentioning that maximum likelihood is superior to equation error and output error methods. This falsely gives the impression that equation error and output error methods are not maximum likelihood estimators. The maximum likelihood methods have been extensively studied in the literature [1-5].
