Appendix A: Properties of signals, matrices, estimators and estimates

Abstract

A good estimator should possess certain properties in terms of errors in parameter estimation and/or errors in the predicted measurements or responses of the mathematical model thus determined. Since the measured data used in the estimation process are noisy, the parameter estimates can be considered to have some random nature. In fact, the estimates that we would have are the mean of the probability distribution, and hence the estimation error would have some associated covariance matrices. Thus, due to the stochastic nature of the errors, one would want the probability of the estimate being equal to the true value to be 1. We expect an estimator to be unbiased, efficient and consistent - not all of which might be achievable. In this appendix, we collect several properties of signals, matrices, estimators and estimates that would be useful in judging the properties and 'goodness of fit' of the parameter/state estimates and interpreting the results. Many of these definitions, properties and other useful aspects are used or indicated in the various chapters of the book and are compiled in this appendix.