H[infinity]/H2/Kalman filtering of linear dynamical systems via variational techniques with applications to target tracking

Paul Lawrence Rawicz

doi:10.17918/etd-79

Filtering of linear dynamical systems has historically been accomplished through the application of the Kalman filter. The Kalman filter has a minimum mean squared error type performance for linear systems with stochastic inputs. In practice, the Kalman filter has been adapter for use with systems outside its usual solution set. For example, the extended Kalman filter for nonlinear systems. While the Kalman filter has been rather successful in general, practical applications have found that model uncertainty and non-stochastic inputs are problematic. To resolve same of these issues, filters based on alternative performance criteria have been developed. One of these is the H2 filter which has a minimum error energy type performance for linear systems with unknown deterministic inputs. Another is the H- filter, which places a bound on the energy gain from the deterministic inputs to the filter error. Application of the H2 filter follows the same type of methodology as the Kalman filter. In fact, the structure of the filter and the filter algorithms are quite similar. Application of the H- filter has been sporadic because its methodology is more complicated and more poorly understood. In fact, the structure of the H- filter and filter algorithms also has some similarity with the structure of the Kalman filter. In this thesis, the similarities between the structure of the H-, H2, and Kalman filters are examined. The filters used in this examination have been derived through duality to the full information controller. In addition, a direct variation of parameters derivation of the H- filter is presented for both continuous and discrete time (scaler case). Direct and controller dual derivations using differential games exist in the literature and also employ variational techniques. Using a variational, rather than a differential games, viewpoint has resulted in a simple relationship between the Riccati equations that arise from the derivation and the results of the Bounded Real Lemma. This same relation has previously been found in the literature and used to relate the Riccati inequality for linear systems to the Hamilton Jacobi inequality for nonlinear systems when implementing the H- controller. The H-, H2, and Kalman filters are applied to the two-state target tracking problem. In continuous time, closed form analytic expressions for the trackers and their performance are determined. To evaluate the trackers using a neutral, realistic, criterion, the probability of target escape was developed. That is, the probability that the target position error will be such that the target is outside the radar beam width resulting in a loss of measurement. In discrete time, a numerical example, using the probability of target escape, is presented to illustrate the differences in performance of the trackers.

H[infinity]/H2/Kalman filtering of linear dynamical systems via variational techniques with applications to target tracking

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H[infinity]/H2/Kalman filtering of linear dynamical systems via variational techniques with applications to target tracking

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