16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde Continuous:i=Ai+Gu+K(=-Hi Then the continuous Kalman gain K(o=limK P-H'HP-H R() = Im =lmPH(△HPH+R(t =PA R =PH R The distinction between P- and p disappears as we go to the limit of The continuous-discrete update relations for i and P can be analyzed in a similar manner, expanding everything to first order in f and taking the limit a Ar-0. This is done in sufficient detail in the text. The result is the continuous time form of the Kalman filter 文=A+G+K(-压) P= AP+ Pa+BNB-PHR-HP where K= Ph R If this is an approximation to a filter that processes measurements at discrete points in time, with measurement noise variance R,, take R(O=R,At If the app nation is good the discrete and continuous variances will be related as16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 6 of 9 Continuous: ˆˆ ˆ ( ) k x =++ − Ax Gu K z Hx  Then the continuous Kalman gain is ( ) 0 1 0 1 0 1 1 1 ( ) lim 1 () lim lim ( ) k t T T t T T t T T Kt K t R t P H HP H t t P H tHP H R t PH R PH R ∆ → − − − ∆ → − − − ∆ → − − − = ∆ ⎛ ⎞ = + ⎜ ⎟ ∆ ∆ ⎝ ⎠ = ∆+ = = The distinction between P− and P+ disappears as we go to the limit of continuous measurement processing. The continuous-discrete update relations for xˆ and P can be analyzed in a similar manner, expanding everything to first order in ∆t and taking the limit as ∆ →t 0 . This is done in sufficient detail in the text. The result is the continuous time form of the Kalman filter. ( ) 1 ˆˆ ˆ T TT x Ax Gu K z Hx P AP PA BNB PH R HP − =++ − =+ + −   where T 1 K PH R− = . If this is an approximation to a filter that processes measurements at discrete points in time, with measurement noise variance Rk , take ( ) R k t Rt = ∆ If the approximation is good, the discrete and continuous variances will be related as