lec22 (1) - We now use the criterion of optimality to...

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We now use the criterion of optimality to determine Κ k . Since we will assume we know and , we will choose a value for o p(y ) t p(x ) Κ k which minimizes the cost function J (equation 12) for the minimum variance Bayes estimate. Specifically =Ε ν ν aT a kk J[ ( ) k ] = trace ( 1 9 ) Ρ a k [] Evaluating ∂∂ and solving for the so-called “Kalman Gain” matrix we have Κ = J/ 0 Κ k Κ=ΡΗΗΡΗ+ fT 1 k k k [R ] f k ] ( 2 0 ) Substituting (20) into (18) then yields Ρ=Ι−ΚΗΡ a k [ ( 2 1 ) Finally, using the state space equation (7) =+ η oo o x(t) M(t, t )x(t ) (t, t ) we then obtain the estimates of needed in (15) and f k x Ρ f k needed in (21) −− = f 1 k xM x a 1 1 ] ( 2 2 ) −− − Ρ=Μ Ρ Μ + fa T 1 k 1 k 1 k Q (23) where , and =Εη η T k1 k1 k1 Q[ a x and Ρ a are the optimal outputs from the previous iteration of the filter. From our earlier discussion (Section 3),Q could represent random forcing in the system model due to transport model errors. To use the filter we must provide initial (a priori) estimates for x and P. Then from any prior output estimates , we use measurement k information ( ) and model information ( ) together with equations (22), (23), (20), (15), and (21) to provide outputs and Ρ aa (x , ) o y,R H,Q a k x Ρ a k for inputs to the next step. The filter equations are summarized in Table 1. Some intuitive concepts regarding the DKF are useful in understanding its operation. First, from equation (20), the gain matrix Κ →Η 1 , k f k (its “maximum” value) as the measurement error covariance (noise) matrix and (its “minimum” value) as . Since the update in the state vector
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This note was uploaded on 11/27/2011 for the course CHEMICAL E 20.410j taught by Professor Rogerd.kamm during the Spring '03 term at MIT.

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lec22 (1) - We now use the criterion of optimality to...

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