Robust wiener optimal nonlinear estimation for uncertain systems

Abstract

A nonlinear operator based approach to robust estimation is introduced for discrete-time systems. It involves a signal entering a communications channel with nonlinearities, transport delay and uncertainties. The measurements are assumed to be corrupted by coloured noise which is correlated with the signal to be estimated. The signal and noise model parameters are assumed to be subject to perturbations represented by random variables with known means and covariances. The theoretical solution does not involve linearization or approximations. In the limiting case of a linear system the estimator has the form of a Wiener filter in discretetime polynomial matrix system form. I. INTRODUCTIONHE optimal filter for linear systems are well known, like the Wiener ([1]) and the Kalman filter ([2, 3]) that have proved their value in applications. These are based on minimizing a statistical criterion and are optimal in an average sense. Over the last few years a new nonlinear estimation paradigm has emerged which leads to simple filters, smoothers and predictors for classes of nonlinear stochastic systems ([4-7]).There are many techniques for nonlinear estimation and the best known is the Extended Kalman Filter (EKF). The EKF has a similar structure to the Kalman filter but has a nonlinear model within the loop. To accommodate the nonlinearity the model is linearized at each time step to estimate the transition matrix and this is used to update the estimated covariance matrix. The EKF does not include a model for channel dynamics that will be included in the following and the solution involves approximations.In the following a related frequency domain or polynomial system approach to robust nonlinear estimation problems is presented. The system, signal and noise models are assumed to include uncertain elements that can be represented by linear models with probabilistic parameter deviations. The optimal robust filter, smoother, or predictor can be obtained from the results of a frequency weighted estimation problem. The estimation problem involves inferential estimation of a signal which enters a communication channel that contains nonlinearities and S. Inzerillo is with the