Studies on identification of time-variant and time-varying nonlinear systems

Abstract: This thesis addresses same aspects of nonlinear systemidentification. In par ticular, it is assumed that theconsidered systems can be described by the Wiener model whichconsists of a dynamit linear part followed by a nonbnear partor the Haxnmerstein model in which the order of these twoblocks is reversed. The parameters of the linear part ofrespective model may be time-varying, while the nonlinear blockis described on either a polynomial or a state space form withhxed parametrization.First, an Extended Kalman Filter, EKF is derived. There isno guaran tee that this algorithm Will be stable. To ensurestability a new constrained algorithm is derived based on areformulation of the EKF in terms of a nonlin ear minimizationproblem with a quadratic inequality constraint. It is demonstrated that this latter algorithm, named the CEKF, yieldsbetter performance in terms of convergence rate and sensitivityto initialization than does the EKF. Furthermore, the problemof estimating the measurement noise variance and the covariancematrix which defines the parameter variations is addressed.Second, a Recursive Prediction Error Method, RPEM combined witha method for on-line adjustment of the forgetting factor isproposed. This algorithm is computationally less burdensomethan the Kalman filter type algorithms. The performance of theproposed algorithms is compared to time-varying Cramér Raolower bounds. In addition,it is shown how the proposedalgorithms can be applied to the areas of blind channelequalization and nonlinear time-varying acoustic echocancelation.Next, the problem of identifying Wiener- and Hammersteintype systems which incorporate an unknown, possiblytime-varying, delay is addressed. A RPEM algorithm is derivedfor identification of the system-parameters, includ ing thedelay which may assume non-integer values of the sampleinterval.Subsequently, the topic of quantifying errors due to modelmismatch is considered. In particular, it is assumed that apurely linear model is used in order to identify Wiener andHammerstein systems with nonlinearities on a polyno mial form.An algorithm which provides an estimate of the resultingamplitude error bound of the estimated transfer function isdevised.For identification of Wiener systems with a knownnonlinearity the RLS algorithm based on a linear model may beconsidered, e. g. due to faster (parameter) convergence ascompared to the RPEM. However, this approach implies amplification of output noise and would also be impossible in casethe nonlinearity is noninvertible. To avoid these problems amodified form of the RLS algorithm is proposed.Finally, Cramér-Rao bounds are derived fortimeinvariant Wiener and Hammerstein systems, where the linearpart consists of a FIR filter and the nonlin earity is given ona polynomial form. Numerical examples, which indicate that theIndirect Prediction Error Method almost exactly attains thegiven bounds, are presented.Keywords:: Nonlinear system identification, Wiener,Hammerstein, time-varying, constrained EKF, RPEM, delayestimation, error estimation, modeling errors, modified RLS,Cramér-Rao bounds.