In principle, general approaches to optimal nonlinear filtering can be described in a unified way from the recursive Bayesian approach. The central idea to this recur- sive Bayesian estimation is to determine the probability density function of the state vector of the nonlinear systems conditioned on the available measurements. However, the optimal exact solution to this Bayesian filtering problem is intractable since it requires an infinite dimensional process. For practical nonlinear filtering applications approximate solutions are required. Recently efficient and accurate approximate non- linear filters as alternatives to the extended Kalman filter are proposed for recursive nonlinear estimation of the states and parameters of dynamical systems. First, as sampling-based nonlinear filters, the sigma point filters, the unscented Kalman fil- ter and the divided difference filter are investigated. Secondly, a direct numerical nonlinear filter is introduced where the state conditional probability density is calcu- lated by applying fast numerical solvers to the Fokker-Planck equation in continuous- discrete system models. As simulation-based nonlinear filters, a universally effective algorithm, called the sequential Monte Carlo filter, that recursively utilizes a set of weighted samples to approximate the distributions of the state variables or param- eters, is investigated for dealing with nonlinear and non-Gaussian systems. Recentparticle filtering algorithms, which are developed independently in various engineer- ing fields, are investigated in a unified way. Furthermore, a new type of particle filter is proposed by integrating the divided difference filter with a particle filtering framework, leading to the divided difference particle filter. Sub-optimality of the ap- proximate nonlinear filters due to unknown system uncertainties can be compensated by using an adaptive filtering method that estimates both the state and system error statistics. For accurate identification of the time-varying parameters of dynamic sys- tems, new adaptive nonlinear filters that integrate the presented nonlinear filtering algorithms with noise estimation algorithms are derived. For qualitative and quantitative performance analysis among the proposed non- linear filters, systematic methods for measuring the nonlinearities, biasness, and op- timality of the proposed nonlinear filters are introduced. The proposed nonlinear optimal and sub-optimal filtering algorithms with applications to spacecraft orbit es- timation and autonomous navigation are investigated. Simulation results indicate that the advantages of the proposed nonlinear filters make these attractive alterna- tives to the extended Kalman filter.

In this thesis, robust set-valued estimation is studied and its application to in-flight alignment of strapdown inertial navigation systems (SINS) with large heading uncertainty is performed. It is known that the performance of the Kalman filter is vulnerable to modeling errors. One of the estimation methods, which are robust against modeling errors, is robust set-valued estimation. In this approach, the filter calculates the set of all possible states, which are consistent with uncertainty inputs satisfying an integral quadratic constraint (IQC) for given measured system outputs. In this thesis, robust set-valued filter with deterministic input is derived. In-flight alignment of SINS with Kalman filtering using external measurements is a widely used technique to eliminate the initial errors. However, if the initial errors are large then the performance of standard Kalman filtering technique is degraded due to modeling error caused by linearization process. To solve this problem, a novel linear norm-bounded uncertain error model is proposed where the remaining second orders terms due to linearization process are considered as norm-bounded uncertainty regarding only the heading error is large. Using the uncertain error model, the robust set-valued filter is applied to in-flight alignment problem. The comparison of the Kalman filter and the robust filter is done on a simulated trajectory and a real-time data. The simulation results show that the modeling errors can be compensated to some extent in Kalman filter by increasing the process noise covariance matrix. However, for very large initial heading errors, the proposed method outperforms the Kalman filter.