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ã«ã«ãã³ãã£ã«ã¿ã¯ãäºæž¬ãšæŽæ°ã®2ã€ã®ã¹ãããã®ããã»ã¹ã§åäœããŸãã
1. äºæž¬ïŒæéæŽæ°ïŒ
äºæž¬ã¹ãããã§ã¯ãã«ã«ãã³ãã£ã«ã¿ã¯ã以åã®ç¶æ æšå®ãšã·ã¹ãã ã¢ãã«ã䜿çšããŠãçŸåšã®ç¶æ ãšãã®é¢é£ããäžç¢ºå®æ§ãäºæž¬ããŸããããã¯ãæ°åŠçã«æ¬¡ã®ããã«è¡šçŸã§ããŸãã
- ç¶æ äºæž¬ïŒ xk- = Fk xk-1 + Bk uk
- å ±åæ£äºæž¬ïŒ Pk- = Fk Pk-1 FkT + Qk
ããã§ã
- xk- ã¯æå» k ã§ã®äºæž¬ç¶æ ã§ã
- xk-1 ã¯æå» k-1 ã§ã®æšå®ç¶æ ã§ã
- Fk ã¯ç¶æ é·ç§»è¡åïŒk-1 ãã k ãžã®ç¶æ ã®é²åãèšè¿°ïŒã§ã
- Bk ã¯å¶åŸ¡å ¥åè¡åã§ã
- uk ã¯å¶åŸ¡å ¥åãã¯ãã«ã§ã
- Pk- ã¯æå» k ã§ã®äºæž¬ç¶æ å ±åæ£è¡åã§ã
- Pk-1 ã¯æå» k-1 ã§ã®æšå®ç¶æ å ±åæ£è¡åã§ã
- Qk ã¯ããã»ã¹ãã€ãºå ±åæ£è¡åïŒã·ã¹ãã ã¢ãã«ã®äžç¢ºå®æ§ã衚ãïŒã§ã
ç¶æ é·ç§»è¡åïŒFkïŒã¯éèŠã§ããããšãã°ãåçŽãªäžå®é床ã¢ãã«ã§ã¯ãFkã¯æ¬¡ã®ããã«ãªããŸãã
F = [[1, dt],
[0, 1]]
ããã§`dt`ã¯æéã¹ãããã§ãããã®è¡åã¯ãåã®äœçœ®ãšé床ã«åºã¥ããŠäœçœ®ãæŽæ°ããé床ãäžå®ã®ãŸãŸã§ãããšä»®å®ããŸãã
ããã»ã¹ãã€ãºå ±åæ£è¡åïŒQkïŒãéèŠã§ããããã¯ãã·ã¹ãã ã¢ãã«ã®äžç¢ºå®æ§ã衚ããŸããã¢ãã«ãéåžžã«æ£ç¢ºãªå ŽåãQkã¯å°ãããªããŸããã¢ãã«ã®ç²ŸåºŠãäœãå ŽåïŒããšãã°ãã¢ãã«åãããŠããªãå€ä¹±ãåå ã§ããå ŽåïŒãQkã¯å€§ãããªããŸãã
2. æŽæ°ïŒæž¬å®æŽæ°ïŒ
æŽæ°ã¹ãããã§ã¯ãã«ã«ãã³ãã£ã«ã¿ã¯ãäºæž¬ãããç¶æ ãææ°ã®æž¬å®å€ãšçµã¿åãããŠãçŸåšã®ç¶æ ã®æŽç·Žãããæšå®ãçæããŸãããã®ã¹ãããã§ã¯ãäºæž¬ãšæž¬å®ã®äž¡æ¹ã®äžç¢ºå®æ§ãèæ ®ãããŸãã
- ã«ã«ãã³ã²ã€ã³ïŒ Kk = Pk- HkT (Hk Pk- HkT + Rk)-1
- ç¶æ æŽæ°ïŒ xk = xk- + Kk (zk - Hk xk-)
- å ±åæ£æŽæ°ïŒ Pk = (I - Kk Hk) Pk-
ããã§ã
- Kk ã¯ã«ã«ãã³ã²ã€ã³è¡åã§ã
- Hk ã¯æž¬å®è¡åïŒç¶æ ãæž¬å®å€ã«é¢é£ä»ããïŒã§ã
- zk ã¯æå» k ã§ã®æž¬å®å€ã§ã
- Rk ã¯æž¬å®ãã€ãºå ±åæ£è¡åïŒæž¬å®ã®äžç¢ºå®æ§ã衚ãïŒã§ã
- I ã¯åäœè¡åã§ã
ã«ã«ãã³ã²ã€ã³ïŒKkïŒã¯ãäºæž¬ã«å¯ŸããŠæž¬å®ã«ã©ãã ãã®éã¿ãäžããããæ±ºå®ããŸããæž¬å®ãéåžžã«æ£ç¢ºãªå ŽåïŒRkãå°ããïŒãã«ã«ãã³ã²ã€ã³ã¯å€§ãããªããæŽæ°ãããç¶æ ã¯æž¬å®å€ã«è¿ããªããŸããäºæž¬ãéåžžã«æ£ç¢ºãªå ŽåïŒPk-ãå°ããïŒãã«ã«ãã³ã²ã€ã³ã¯å°ãããªããæŽæ°ãããç¶æ ã¯äºæž¬ã«è¿ããªããŸãã
ç°¡åãªäŸïŒéè·¯äžã®è»ã®è¿œè·¡
çŽç·éè·¯ã«æ²¿ã£ãŠç§»åããè»ã®åçŽåãããäŸãèããŠã¿ãŸããããäžå®é床ã¢ãã«ãšãè»ã®äœçœ®ã枬å®ããåäžã®ã»ã³ãµãŒã䜿çšããŸãã
ç¶æ ïŒ x = [äœçœ®ãé床]
枬å®ïŒ z = äœçœ®
ã·ã¹ãã ã¢ãã«ïŒ
F = [[1, dt],
[0, 1]] # ç¶æ
é·ç§»è¡å
H = [[1, 0]] # 枬å®è¡å
Q = [[0.1, 0],
[0, 0.01]] # ããã»ã¹ãã€ãºå
񆑜
R = [1] # 枬å®ãã€ãºå
񆑜
ããã§`dt`ã¯æéã¹ãããã§ããè»ã®äœçœ®ãšé床ã®åææšå®ãããã³ç¶æ å ±åæ£è¡åã®åææšå®ã䜿çšããŠãã«ã«ãã³ãã£ã«ã¿ãåæåããŸããæ¬¡ã«ãåã¿ã€ã ã¹ãããã§ãäºæž¬ã¹ããããšæŽæ°ã¹ããããå®è¡ããŸãã
ãã®äŸã¯ãããŸããŸãªããã°ã©ãã³ã°èšèªã§å®è£ ã§ããŸããããšãã°ãNumPyã䜿çšããPythonã§ã¯æ¬¡ã®ããã«ãªããŸãã
import numpy as np
dt = 0.1 # æéã¹ããã
# ã·ã¹ãã ã¢ãã«
F = np.array([[1, dt], [0, 1]])
H = np.array([[1, 0]])
Q = np.array([[0.1, 0], [0, 0.01]])
R = np.array([1])
# åæç¶æ
ãšå
񆑜
x = np.array([[0], [1]]) # åæäœçœ®ãšé床
P = np.array([[1, 0], [0, 1]])
# 枬å®
z = np.array([2]) # äŸã®æž¬å®
# äºæž¬ã¹ããã
x_minus = F @ x
P_minus = F @ P @ F.T + Q
# æŽæ°ã¹ããã
K = P_minus @ H.T @ np.linalg.inv(H @ P_minus @ H.T + R)
x = x_minus + K @ (z - H @ x_minus)
P = (np.eye(2) - K @ H) @ P_minus
print("æšå®ç¶æ
:", x)
print("æšå®å
񆑜:", P)
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