A stochastic analysis of a modified gain extended Kalman filter with applications to estimation with bearings only measurements

Abstract

e d t o t h e t h r e edimensional bearing only measurement problem (BOMP) w h e r e t h e EKF o f t e n shows e r r a t i c b e h a v i o r s . 1. I n t r o d u c t i o n The c o n s t r u c t i o n of i m p l e m e n t a b l e o b s e r v e r s f o r n o n l i n e a r d e t e r m i n i s t i c s y s t e m s a n d n o n l i n e a r f i l t e r s f o r n o n l i n e a r s t o c h a s t i c s y s t e m s h a s had very few succ e s s e s . E x c e p t f o r a f e w n o t a b l e e x c e p t i o n s ( e . g . [ 1 3 ] ) i m p l e m e n t a t i o n i s based upon ad hoc expansions and line a r i z a t i o n t e c h n i q u e s . T h e r e f o r e , i n o r d e r t o o b t a i n a c o m p u t a t i o n a l l y r e a l i z a b l e a n d p r a c t i c a l f i l t e r , t h e EKF i s u s u a l l y s u g g e s t e d f o r m o s t n o n l i n e a r e s t i m a t i o n p r o b l e m s . F o r t u n a t e l y , t h e r e a r e many examples where t h e EKF i s s u c c e s s f u l i n p r o d u c i n g u s a b l e e s t i n a t e s a lthough l i t t l e i s known a b o u t t h e p r o p e r t i e s o f t h e esti m a t e s ( i . e . , s t a b i l i t y , u n b i a s e d n e s s , a n d c o n v e r g e n c e ) t h a t i t o r i t s v a r i a n t s p r o d u c e [ 5 , l l ] . In c o n t r a s t a s p e c i a l c l a s s of n o n l i n e a r e s t i m a t o r s t h a t are similar t o t h a t of t h e EKF, c a l l e d t h e m o d i f i e d g a i n EKF (MGEKF), i s d e s c r i b e d h e r e f o r w h i c h t h e s t a b i l i t y a n al y s i s of j21] i s a p p l i c a b l e . T h i s s p e c i a l c l a s s of nonl i n e a r e s t i m a t o r s was m o t i v a t e d by t h e c l a s s of funct i o n s w h i c h c a n b e m a n i p u l a t e d i n t o a pseudomeasurement f o r m [ 1 , 3 , 1 2 , 1 5 , 2 2 ] . F o r t h e d e t e r m i n i s t i c s y s t e m t h e s e pseudomeasurements a r e l i n e a r f u n c t i o n s o f t h e s t a t e s of t h e s y s t e m a l t h o u g h t h e c o e f f i c i e n t m a t r i x i s a nonl i n e a r f u n c t i o n o f t h e o r i g i n a l m e a s u r e m e n t s . By u s i n g t h e p s e u d o m e a s u r e m e n t s i n a l i n e a r o b s e r v e r s t r u c t u r e [18], g l o b a l s t a b i l i t y c a n b e shown. However, i f t h e pseudomeasurement observer (PMO) i s u s e d i n a n o i s y environment as a pseudomeasurement f i l t e r (PPIF) [ 1 , 3 , 1 2 , 1 5 1 , b i a s e d estimates r e s u l t . In [ 1 8 ] i t i s shown f o r a p a r t i c u l a r e x a m p l e how t h e EKO ( t h e e x t e n d e d K a lman o b s e r v e r ) c a n b e m a n i p u l a t e d i n t o t h e f o r m of t h e PMO. The e s s e n t i a l d i f f e r e n c e l i e s o n l y i n t h e c a l c u l at i o n of t h e o b s e r v e r g a i n s . A m o d i f i c a t i o n o f t h e g a i n s of t h e PMO i s s u g g e s t e d i n [ l a ] w h i c h e n a b l e s t h e EKO t o a c h i e v e p e r f o r m a n c e s i m i l a r t o t h a t o f t h e PNO. T h i s i s c a l l e d t h e m o d i f i e d g a i n...

Sequential Multiple Model Filtering with Interrupted Measurements

“…Notice that the universal linearization approach 11,12) is used for the angle measurement update. Since the occurrence of the range measurement interruption has a stationary distribution with respect to time, we need not calculate the predicted probability of the modal state.…”

“…In this case, the first linear or universal linearization update can provide a more accurate reference point for the second nonlinear measurement up-date. 11,12) Using this algorithm, we can improve filtering performance. As can be seen, the angle measurement for target tracking applications is, in fact, universally linearizable.…”

“…Since universal linearization can give a more satisfactory result in angle-only tracking problems, 11,12) we utilize the universal linearization approach in the angle measurement update. Similarly, applying step-by-step Gaussian approximation, the full sequential multiple model filtering algorithm used for tracking is given in the appendix.…”

“…As can be seen, the angle measurement for target tracking applications is, in fact, universally linearizable. 12) In order to avoid the underflow phenomenon that often happens in mode probability computation, we implement the computation of mode posterior probability in numerical robust forms. Please see relevant work for details on numerical robust implementation of the multiple model filters.…”

State Estimation for Target Tracking Problems with Nonlinear Kalman Filter Algorithms

“…From the sensor either only bearing angle information or both bearing and range information are available. The passive tracking of manoeuvring objects using line of sight (LOS) angle measurements only is an important field of research in the application areas of submarine tracking, aircraft surveillance, Autonomous robotics and mobile systems [1][2][3][4][5]. In 1960, R.E.…”