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Unformatted text preview: 2.160 System Identification, Estimation, and Learning Lecture Notes No. 7 March 1, 2006 4.7. Continuous Kalman Filter Converting the Discrete Filter to a Continuous Filter Continuous process x ¡ = Fx + Gw ( t ) ( 4 9 ) ( Measurement y = Hx + t v ) ( 5 ) Assumptions ( T E [ t w ) w ( s ) ]= Q δ ( t − s ) δ ( t − s ) = Dirac delta function (51) [ ( ) v t v E T ( s ) ]= R δ ( t − s ) ( 5 2 ) T [ ( t v E ) w ( s ) ]= 0 ( 5 3 ) Converting R t and Q in the discrete Kalman filter to Q and R of the above equations, t (see Brown and Hwang, Section 7.1 for detail) R Q t = Q ∆ t R t = ∆ t = sampling interval (54) ∆ t From (4) T T − 1 K t = P t t − 1 H t ( H P t t − 1 H t + R t ) R t tP ∆ = T − 1 ∆ t t t − 1 H t T ( ⋅ ∆ H t P t t − 1 H t + R ) (55) t T − 1 ∆ ≅ tP H t R for ∆ t << 1 t t − 1 ∆ H t T R − 1 Define K = P t t − 1 (56) From (8) P = A P A t T + G Q G T ∆ tK t t − 1 t t t t t ( 5 7 ) T = A ( I − K t H t ) P t t − 1 A t + G Q G T t t t t A t = I ∆ + tF 1 Ignoring higherorder small quantities; O (∆ t 2 ) ≅ 0 ( 5 8 ) t t P P = + 1 t t F T ∆ − P tKH + G ∆ tQG T ( 5 9 ) − 1 ∆ + tFP t t − 1 ∆ + tP t t − 1 t t t t P t + 1 t − P t t − 1 R H − 1 t t t t ( 6 ) T FP = t t − 1 + P t t − 1 F T − P t t − 1 t P H + QG G T ∆ t ∆ t → 0 lim P t t − 1 = P t − 1 ( 6 1 ) ∆ t → 0 ¡ T − 1 P = FP + PF T − R PH HP + GQG T ( 6 2 ) This is called the Matrix Riccati Equation. Similarly, we can reduce the discrete time form of state estimation correction to the one of continuous time: ( x ¡ ˆ = x F ˆ + y K − x H ˆ ) ( 6 3 ) where the Kalman gain is given by − 1 K = PH T R (64) This is called the KalmanBucy Filter The physical interpretation of the Matrix Riccati Equation...
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.160 taught by Professor Harryasada during the Spring '06 term at MIT.
 Spring '06
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