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lecture_7

# lecture_7 - 2.160 System Identification Estimation and...

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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 ) (49) ( Measurement y = Hx + t v ) (50) 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 ) (52) T [ ( t v E ) w ( s ) ] = 0 (53) 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 (57) 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

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Ignoring higher-order small quantities; O ( t 2 ) 0 (58) t t P P = + 1 t t F T P tKH + G tQG T (59) 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 (60) 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 (61) t 0 ± T 1 P = FP + PF T R PH HP + GQG T (62) 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 ) (63) where the Kalman gain is given by 1 K = PH T R (64) This is called the Kalman-Bucy Filter The physical interpretation of the Matrix Riccati Equation ± T 1 P = FP + PF T R PH HP + GQG T (62) ²³ ´ ²µ
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