{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture_7 - 2.160 System Identification Estimation and...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}