16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde K=PHT(HP-HT+R) =红+k(-压) P*=(I-KH)P The filter operates by alternating time propagation and measurement update steps The results were derived here on the basis of preserving zero mean errors and minimizing the error variances. If all errors and noises are assumed normally distributed, so the probability density functions can be manipulated, one can derive the same results using the conditional mean approach: define x at every stage to be the mean of the distribution of x conditioned on all the measurements available up to that stage Example: Conversion of continnous dynamics to discrete time form 2 10 1(0)=x2(0)=0 unit white noise x 010 We could do time propagation by integration(N=1) P=AP+ pa+BB Φ=e=I+At+1A2 00‖0 If this does not work, expand o=4,如0)=1(△M=2) Page 2 of 916.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 2 of 9 ( ) ( ) ( ) 1 ˆˆ ˆ T T K P H HP H R x x K z Hx P I KH P − − − +− − + − = + =+ − = − The filter operates by alternating time propagation and measurement update steps. The results were derived here on the basis of preserving zero mean errors and minimizing the error variances. If all errors and noises are assumed normally distributed, so the probability density functions can be manipulated, one can derive the same results using the conditional mean approach: define x at every stage to be the mean of the distribution of x conditioned on all the measurements available up to that stage. Example: Conversion of continuous dynamics to discrete time form N 1 2 2 10 2 0 10 0 00 2 A B x x x n x x n = = ⎡ ⎤ ⎡⎤ = + ⎢ ⎥ ⎢⎥ ⎣ ⎦ ⎣⎦      We could do time propagation by integration (N = 1): ˆ ˆ T T x Ax P AP PA BB = =+ +   ( )2 1 2 ... 2 A t e I At A t ∆ Φ= = + ∆ + ∆ + 2 0 10 0 10 0 0 0 0 0 0 00 A ⎡ ⎤⎡ ⎤ ⎡ ⎤ = = ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦ If this does not work, expand φ = Aφ  , φ(0) = I (∆t = 2) :