16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Lecture 25 K2=PH(HPH+R) If ==x+y H=l K2=P(P-+R Then if p-≤R, Alternatively,ifR≤P, K,→I The effect is that 式=+K(-所) The quantity K, is known as the Kalman gain. It is the optimum gain in the mean squared error sense. Substitute it into the expression for P P*=(I-KH)P-(/-KH)+KRK (I-KH)P--(I-KHP-HK+KRK (I-KH)P--P-H'K+KHP-H'K+KRK (-KH)P--PHK+K(HPH +R)KT (-KH)P--P-H'K+ P-H(HP-HT+)(HP-HT+R)K (I-KH)P--P-H'K+P-H'K (I-KH)P- The form at the top is true for any choice of K. The last form is true only for the alman gain. The first form is better behaved numerically if you process a measurement which is very accurate relative to your prior information. So the measurement update step is Page 1 of 916.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 1 of 9 Lecture 25 Last time: ( ) 1 2 T T K P H HP H R − − − = + If k k z xv = + , ( ) 1 2 H I K PP R − − − = = + Then if P R −  , 1 K PR 2 → − − Alternatively, if R P−  , K I 2 → The effect is that xˆˆ ˆ x K z Hx ( ) +− − =+ − The quantity K2 is known as the Kalman gain. It is the optimum gain in the mean squared error sense. Substitute it into the expression for P+ . ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) 1 T T TT T TT TT T TT T T TT T T T T TT TT P I KH P I KH KRK I KH P I KH P H K KRK I KH P P H K KHP H K KRK I KH P P H K K HP H R K I KH P P H K P H HP H R HP H R K I KH P P H K P H K I KH P + − − − −− − −− − − −− − − − −− − − =− − + =− −− + =− − + + =− − + + =− − + + + =− − + = − The form at the top is true for any choice of K . The last form is true only for the Kalman gain. The first form is better behaved numerically if you process a measurement which is very accurate relative to your prior information. So the measurement update step is: