Analysis of Some Methods for Identifying Dynamic Errors-in-variables Systems

Sammanfattning: A system where errors or noises are present on both the inputs and the outputs is called an errors-in-variables (EIV) system. EIV systems appear in industrial and agricultural processes, medical sciences, economical systems, biotechnology, as well as in many other areas. Until now, a considerable number of methods for identifying dynamic errors-in-variables systems have been proposed. This thesis studies the statistic properties of different EIV methods and explores the relationships between some of the existing methods.An EIV approach, based on a bias-compensated least squares scheme, is considered in this thesis. Three promising estimators are in focus, namely, Zheng's bias-eliminated least squares (BELS) methods, Frisch scheme methods and extended compensated least squares (ECLS) methods. A simplified form of the BELS equation is first proposed. The new equation will simplify the computation and the theoretical analysis. Next, an important relationship between the BELS, Frisch and ECLS methods is found. The defining non-linear equations used by these three methods are equivalent, providing that the same extended model is used. This means that despite the use of different techniques to solve these equations, the three methods will have the same asymptotic estimation accuracy. Furthermore, the thesis studies the convergence properties of BELS. An alternative BELS algorithm is proposed, which has less of a divergence problem under low SNR situations as compared to the classic BELS methods.Another important problem which is investigated in the thesis is the asymptotic accuracy of the estimates. For the BELS method and a third-order cumulants based method, explicit expressions for the covariance matrices of the parameter estimates are derived. With such expressions available, one may obtain insight into how different user choices in the algorithms influence the accuracy. By using the expressions for the covariance matrices, comparisons of the estimation accuracies are made between three Frisch methods and between the time-domain maximum likelihood method and the sample maximum likelihood method.Finally, identification of errors-in-variables systems with periodic input signals is considered. How to utilize the periodic data and how to design instrumental variables in order to achieve the optimal estimation accuracy are analyzed as well.