NONLINEAR OBSERVATION SCHEME AND DYNAMIC MODEL (EXTENDED KALMAN FILTER) INTRODUCTION In this section we extend the results for the linear time-invariant and timevariant cases to where the observations are nonlinearly related to the state vector and/or the target dynamics model is a nonlinear relationship [5, pp. 105– 111, 166–171, 298–300]. The approachs involve the use of linearization procedures. This linearization allows us to apply the linear least-squares and minimum-variance theory results obtained so far. When these linearization procedures are used with the Kalman filter, we obtain what is called the extended Kalman filter [7, 122]. NONLINEAR OBSERVATION SCHEME When the. | Tracking and Kalman Filtering Made Easy. Eli Brookner Copyright 1998 John Wiley Sons Inc. ISBNs 0-471-18407-1 Hardback 0-471-22419-7 Electronic 16 NONLINEAR OBSERVATION SCHEME AND DYNAMIC MODEL EXTENDED KALMAN FILTER INTRODUCTION In this section we extend the results for the linear time-invariant and timevariant cases to where the observations are nonlinearly related to the state vector and or the target dynamics model is a nonlinear relationship 5 pp. 105111 166-171 298-300 . The approachs involve the use of linearization procedures. This linearization allows us to apply the linear least-squares and minimum-variance theory results obtained so far. When these linearization procedures are used with the Kalman filter we obtain what is called the extended Kalman filter 7 122 . NONLINEAR OBSERVATION SCHEME When the observation variables are nonlinearly related to the state vector coordinates becomes 5 pp. 166-171 Yn G Xn Nn where G Xn is a vector of nonlinear functions of the state variables. Specifically G Xn g 1 Xn g 2 Xn gn Xn 357 358 NONLINEAR OBSERVATION SCHEME AND DYNAMIC MODEL A common nonlinear observation situation for the radar is where the measurements are obtained in polar coordinates while the target is tracked in cartesian coordinates. Hence the state vector is given by X t X x y z While the observation vector is Y t Y Rs e The nonlinear equation relating Rs d and to x y and z are given by that is gi X g2 X and g3 X are given by respectively to . The inverse equations are given by . The least-squares and minimumvariance estimates developed in Chapters 4 and 9 require a linear observation scheme. It is possible to linearize a nonlinear observation scheme. Such a linearization can be achieved when an approximate estimate of the target trajectory has already been obtained from previous measurements. One important class of applications where the linearization can be applied is when the .
