S2.Conventional Kalman filter 0 State space model of the time-varying linear system o Optimal state prediction Relationship between innovation process and state prediction Iterative Predict using innovation process Update of Kalman Gain Update of prediction error Update of prediction error covariance o Suffer from numerical problems,and therefore its importance is more of a theoretical/historic nature 16
16 S2. Conventional Kalman filter State space model of the time-varying linear system Optimal state prediction Relationship between innovation process and state prediction Iterative Predict using innovation process Update of Kalman Gain Update of prediction error Update of prediction error covariance Suffer from numerical problems, and therefore its importance is more of a theoretical/historic nature
(1)State space description 0 State space model of the time-varying linear system h+1 三 Ak"xk十Bg绿十R Vk= Ck·k+Dk'k+隆： The in-and outputs are known. The matrices Ak,Bk,Ck and Dk are the known system matrices at time k. Vk and wk are unknown independent zero-mean white noise sequences -process noise and measurement noise respectively -with known covariances Vk and Wk 17
17 (1) State space description State space model of the time-varying linear system The in- and outputs are known. The matrices Ak , Bk , Ck and Dk are the known system matrices at time k. vk and wk are unknown independent zero-mean white noise sequences –process noise and measurement noise respectively – with known covariances Vk and Wk
The aim of the Kalman filter o Produce an optimal estimate in a least squares sense for all subsequent state vectors by making use of observations of the in-and outputs the available system matrices the known covariances 18
18 The aim of the Kalman filter Produce an optimal estimate in a least squares sense for all subsequent state vectors by making use of observations of the in- and outputs the available system matrices the known covariances
Simple Case o State space description xa+1）=a+l,nxm+y(m） y(n)=c(n)x(n)+v2(n) 19
19 Simple Case State space description xn n nxn v n 1 1 1, yn cnxn v n 2
Problems o Filter 0 at time k an estimate for xk is computed, denoted as "xklk o Smoothing 0 at time k an estimate for xi is computed, denoted as 'xilk(i<k) o Prediction ● at time k an estimate for xk+1 is computed,denoted as xk+1k 20
20 Problems Filter at time k an estimate for xk is computed, denoted as ˆxk|k Smoothing at time k an estimate for xi is computed, denoted as ˆxi|k (i<k) Prediction at time k an estimate for xk+1 is computed, denoted as ˆxk+1|k