Electrical and Computer Engineering ETDs

Publication Date



The extended Kalman filter applied to state estimation using nonlinear measurements has demonstrated divergence in many applications involving large initial uncertainties because of "invalid" linearizations performed by the filter on the nonlinear measurement functions. No quantitative results can be found as to how invalid these linearizations must be before divergence occurs. This report attempts to fill this void by deriving the extended Kalman filter as an approximate nonlinear least squares estimator, through the minimization of a measurement squared error function. The convergence of the extended Kalman filter is then determined by examining the squared error function and verifying the usefulness of this function through Monte Carlo simulation.

The results of this analysis are then used to specify parameters for the Gaussian sum approximation. This technique employs a bank of extended Kalman filters processing in parallel to approximate the density function of the state, conditioned on the nonlinear measurement data, by a sum of Gaussian densities. The mean of this density function represents the state estimate determined from the nonlinear measurement data. Simulation results for nonlinear measurements generated by harmonic and Gauss-Markov processes demonstrate that successful state estimation occurs using the Gaussian sum approximation, through proper parameter specification obtained by the convergence analysis. This technique is then applied to estimating the errors in an inertial navigation system through the use of nonlinear radar altimeter measurements and terrain elevation data. The results of applying this technique to terrain-aided navigation demonstrate that the Gaussian sum approximation provides good terrain-aided navigation performance for large initial uncertainties.


Estimation Theory, Kalman Filters

Document Type




Degree Name

Electrical Engineering

Level of Degree


Department Name

Electrical and Computer Engineering

First Committee Member (Chair)

Daniel P. Petersen

Second Committee Member

Clifford R. Qualls

Third Committee Member

Joseph Thomas Cordaro

Third Advisor

Samuel Shaw