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Title: Nonlinear Filtering-Extensions and Application to Target Tracking Problems
Authors: Radhakrishnan, R.
Keywords: Electrical Engineering
Issue Date: 2018
Publisher: IIT Patna
Series/Report no.: TH-116;
Abstract: This thesis concentrates on developing new nonlinear filters and their application to real-life target tracking problems. The filters are developed for discrete and continuous-discrete systems such that they possess comparable or better filtering accuracy with moderate computational cost, with respect to existing solutions. To start with, a detailed literature survey is conducted which helped in identifying the scope of possible contributions. From literature survey, it has been found that for nonlinear systems, in general, there exists no closed form solution for the filtering problem. However, several suboptimal solutions are present, such as the extended Kalman filter (EKF), unscented Kalman filter (UKF), cubature Kalman filter (CKF), cubature quadrature Kalman filter (CQKF), Gauss-Hermite filter (GHF), sparse-grid Gauss-Hermite filter (SGHF), multiple quadrature Kalman filter (MQKF), particle filter (PF) to name a few. It has also been realized that there is a scope of improvement. Literature survey also helped in identifying test problems, on which the proposed methods are implemented and their performance in terms of accuracy and computational burden is compared with the existing techniques. Moreover, literature survey on aerospace and underwater military target tracking problems has also been carried out with an emphasis to search for solutions which are developed exclusively for certain tracking problems. One such solution is the shifted Rayleigh filter (SRF), developed for solving bearings-only target tracking problems. In the present thesis, two new nonlinear filters, which assume the conditional probability density function of states as Gaussian, are proposed. Among them, one filter concentrates on bringing down the computational time while maintaining accuracy levels for higher dimensional systems, in the context of GHF. The second one, named as new unscented Kalman filter (NUKF), defines a new set of sigma points and weights for approximating the conditional densities with more accuracy. Filtering in continuous-discrete systems has proved to be more realistic as the process is modeled in the continuous time domain and measurements in discrete time due to the finite sampling interval. Hence various existing filters and the proposed NUKF have been modified to deal with continuous-discrete processes. This is achieved by making use of an efficient discretization method and incorporating it into discrete filter framework. It is a fact that after nonlinear transformation, Gaussian density will not remain Gaussian in nature. Hence, the assumption of deterministic sample point filters (which approximate the prior and posterior pdfs as Gaussian) stands as a source of error. A more realistic approach can be achieved by adopting Gaussian sum framework of filtering where the conditional densities are represented using a weighted sum of many Gaussian densities, where the individual densities are realized using Gaussian filters. To further enhance the accuracy, a weight adaptation scheme is also incorporated. In this thesis, several accurate filters including the one proposed, is incorporated as proposals of Gaussian sum filter and their results compared with existing techniques. The developed filters have been applied to two military tracking problems namely (i) underwater passive bearings-only target tracking and (ii) ballistic target tracking and interception on reentry. The contributions of this thesis can be summarized as follows: (1) Developed a nonlinear filtering algorithm, whose accuracy level is as high as GHF but with much less computational cost. It is termed as multiple sparse-grid Gauss-Hermite filter (MSGHF). This filter can perform with similar accuracy levels as compared to the multiple (2) Formulated a new unscented Kalman filter (NUKF) which defines a new set of sigma points and weights to approximate the probability densities. To tackle the problem of preserving the positive definite nature of covariance matrix, a square root version of the algorithm is also proposed. (3) NUKF, SGHF and SRF are modified to deal with continuous-discrete (CD) state-space models. These algorithms are named as CD-NUKF, CD-SGHF and CD-SRF. (4) NUKF, SGHF and SRF are used to update and predict individual Gaussian components of a Gaussian sum filter. Such incorporation improves the filtering accuracy. To further reduce the estimation error, weight adaptation method is incorporated with all the above methods. (5) All the proposed filters mentioned above are implemented for solving an underwater bearings-only target tracking problem for two popular observer manoeuvring cases. They are compared with the existing methods in terms of i) track-loss ii) RMSE iii) computational time and iv) robustness. It has been found that the proposed Gaussian sum shifted Rayleigh filter (GS-SRF) and continuous-discrete shifted Rayleigh filter (CD-SRF) are the best among all the filters. (6) Tracking and interception of a ballistic target using seeker measurements are solved where estimator and guidance law are considered in closed-loop, that is the output of estimator is fed to guidance block (realized using PNG law) for generating the guidance commands. The performance of the interceptor in closed-loop was evaluated by calculating the RMSE and final miss-distance. It was found that adaptive Gaussian sum filters performed with superior accuracy with a miss-distance of only 13 meters.
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