117913378 Kalman Filter

L thuyt v cc ng dng ca b lc Kalman Page 1

TRNG I HC BCH KHOA H NI

VIN CNG NGH THNG TIN V TRUYN THNG

----------

BI TP LN

X L TN HIU NGU NHIN

TI:

TM HIU L THUYT V CC NG DNG CA B LC

KALMAN

Nhm sinh vin thc hin: Bi nh Cng 20080355 Nguyn Khnh Hng 20081279

Gio vin hng dn: PGS.TS Nguyn Linh Giang

H NI 8-2012


LI GII THIU ......................................................................................................................... 3

I. L THUYT B LC KALMAN ...................................................................................... 4

1. L thuyt v c lng ..................................................................................................................... 4

1.1. Khi nim .................................................................................................................................. 4

1.2. nh gi cht lng .................................................................................................................. 4

1.3. K vng (Expectation) .............................................................................................................. 5

1.4. Phng sai (Variance) ............................................................................................................... 6

1.5. lch chun............................................................................................................................ 7

1.6. Hip phng sai (Covariance) .................................................................................................. 7

1.7. Ma trn hip phng sai ........................................................................................................... 8

1.8. Phn phi chun (phn phi Gaussian) ..................................................................................... 8

1.9. c lng ca trung bnh v phng sai ................................................................................ 10

1.10. Phng php bnh phng ti thiu .................................................................................... 11

2. B lc Kalman ................................................................................................................................ 12

2.1. Gii thiu chung v b lc Kalman ........................................................................................ 12

2.2. M hnh ton hc .................................................................................................................... 15

2.2.1. H thng v m hnh quan st ......................................................................................... 15

2.2.2. Gi thit ........................................................................................................................... 15

2.2.3. Ngun gc ....................................................................................................................... 16

2.2.4. iu kin khng chch .................................................................................................... 17

2.2.5. Hip phng sai sai s .................................................................................................... 19

2.2.6. li Kalman ................................................................................................................. 20

2.2.7. Tm tt cc phng trnh ca b lc Kalman ................................................................. 21

II. NG DNG CA B LC KALMAN ........................................................................ 24

III. CI T TH NGHIM ............................................................................................... 26

1. Tao nhiu Gaussian ......................................................................................................................... 26

2. Ci t b lc Kalman ..................................................................................................................... 27

2.1. M phng hot ng ca b lc Kalman ................................................................................ 27

2.2. M phng hot ng ca b lc Kalman m rng .................................................................. 31

IV. KT LUN ....................................................................................................................... 32

TI LIU THAM KHO .......................................................................................................... 33


LI GII THIU

Ngy nay, nn cng ngh th gii ang pht trin nhanh chng v hng lot cc gii php cng ngh ra i mi nm. Theo , cc sinh vin ngnh cng ngh ngoi vic

tip thu cc kin thc ging ng cn phi tm hiu nghin cu thm cc cng ngh

tin tin trn th gii c th p ng c yu cu cao ca th trng lao ng. Trong nhng nm gn y cc loi cm bin, thit b o lng c s dng rng ri trong dn

dng cng nh trong cng nghip. Th nhng nhiu loi thit b li rt nhy vi nhiu,

vn lm sao loi nhiu ra khi tn hiu l mt vn thc s khng n gin. Vi nhng u im vt tri, tim nng ng dng ca thut ton Kalman vo

thc t trong vic p dng lc nhiu trong tn hiu l rt kh quan, v vy vic nghin

cu nm r v tin ti lm ch phng php ny l rt cn thit v b ch. Ngoi ra

vi mong mun p dng v lp trnh thut ton Kalman vo thc t nn nhm chng em chn ti: TM HIU L THUYT V CC NG DNG CA B LC

KALMAN.


I. L THUYT B LC KALMAN

Vo nm 1960, R.E Kalman cng b bi bo ni ting v mt gii php truy

hi gii quyt mt bi ton lc thng tin ri rc truyn tnh (discrete data linear

filtering). Tn y ca bi bo l A New Approach to Linear Filtering and Prediction

Problems. T n nay cng vi s pht trin ca tnh ton k thut s, b lc Kalman

tr thnh ch nghin cu si ni v c ng dng trong nhiu ngnh k thut

cng ngh khc nhau: trong t ng ha, trong nh v cng nh trong vin thng v

trong nhiu lnh vc khc.

Mt cch khi qut, b lc Kalman l mt tp hp cc phng trnh ton hc m

t mt phng php tnh ton truy hi hiu qa cho php c on trng thi ca mt qu

trnh sao cho trung bnh phng sai ca l nh nht. B lc Kalman rt hiu qu trong

vic c on cc trng thi trong qu kh, hin ti v tng lai thm ch ngay c khi

tnh chnh xc ca h thng m phng khng c khng nh.

1. L thuyt v c lng

1.1. Khi nim

Trong thng k, mt c lng l mt gi tr c tnh ton t mt mu th v

ngi ta hy vng l gi tr tiu biu cho gi tr cn xc nh trong tp hp. Ngi ta

lun tm mt c lng sao cho l c lng khng chch, hi t, hiu qu v

vng(robust)..

1.2. nh gi cht lng

Mt c lng l mt gi tr x c tnh ton trn mt mu c ly mt cch

ngu nhin, do gi tr ca x l mt bin ngu nhin vi k vng E(x) v phng sai

V(x). Ngha l gi tr x c th dao ng ty theo mu th, n c t c hi c th bng

ng chnh xc gi tr X m n ang c lng. Mc ch y l ta mun c th kim

sot s sai lch gi tr x v gi tr X.

Mt bin ngu nhin lun dao ng xung quanh gi tr k vng ca n. Ta mun l

k vng ca x phi bng X. Khi ta ni c lng l khng chch. Trung bnh tch ly

trong v d v chiu cao trung bnh ca tr 10 tui mt c lng ng, trong khi c


lng v tng s c trong h c tnh nh trong v d l mt c lng khng ng,

l c lng tha: trung bnh tng s c c lng c lun ln hn tng s c c thc

trong h.

Ta cng mun l khi mu th cng rng, th sai lch gia x v X cng nh. Khi

ta ni c lng l hi t. nh ngha theo ngn ng ton hc l nh sau:

(xn) hi t nu lim

(| | > ) = 0 vi mi s thc dng (xc sut sai lch vi

gi tr thc cn c lng ln hn tin v 0 khi kch c ca mu th cng ln).

Bin ngu nhin dao ng quanh gi tr k vng ca n. Nu phng sai V(x) cng

b, th s dao ng cng yu. V vy ta mun phng sai ca c lng l nh nht c

th. Khi ta ni c lng l hiu qu.

Cui cng, trong qu trnh iu tra, c th xut hin mt gi tr bt thng (v d

c tr 10 tui nhng cao 1,80 m). Ta mun gi tr bt thng ny khng nh hng qu

nhiu n gi tr c lng. Khi ta ni c lng l vng. C th thy trung bnh tch

ly trong v d v chiu cao trung bnh tr 10 tui khng phi l mt c lng vng.

1.3. K vng (Expectation)

nh ngha: Gi s l i lng ngu nhin ri rc c th nhn cc gi tr

1, 2, , vi cc xc sut tng ng 1, 2, , .

Khi k vng ca X, k hiu l () hay c xc nh bi cng thc

() =

=1

(1.1)

Nu l i lng ngu nhin lin tc c hm mt xc sut l () th k vng

ca l:

() = ()

+

(1.2)

- Tnh cht

i. () =


ii. (. ) = . (), vi l hng s.

iii. ( + ) = () + ()

iv. Nu X v Y l hai i lng ngu nhin c lp th:

(. ) = (). ()

- ngha: K vng ca mt i lng ngu nhin chnh l gi tr trung bnh

(theo xc sut) ca i lng ngu nhin . N l im trung tm ca phn phi m cc

gi tr c th ca X s tp trung quanh .

1.4. Phng sai (Variance)

nh ngha: Phng sai (trung bnh bnh phng lch) ca i lng ngu

nhin X, k hiu () hay () c xc nh bi cng thc:

() = [( )2] (1.3)

Nu X l i lng ngu nhin ri rc c th nhn cc gi tr 1, 2, , vi xc

xc sut tng ng l 1, 2, , th:

() = [ ]2

=1

(1.4)

Nu X l i lng ngu nhin lin tc c hm mt xc sut l () th:

() = [ ]2()

+

(1.5)

Trong thc t ta thng tnh phng sai bng cng thc:

() = [2] [()]2 (1.6)

Tnh cht:

i. () = 0

ii. (. ) = 2. ();

iii. Nu X, Y l 2 bin ngu nhin c lp th:

( + ) = ( ) = () + () (1.7)

ngha: l lch khi gi tr trung bnh. Do phng sai () gi l

trung bnh bnh phng lch. Nn phng sai phn nh mc phn tn ca cc gi


tr ca i lng ngu nhin quanh gi tr trung bnh hay k vng. i lng ngu nhin

c phng sai cng ln th gi tr cng phn tn v ngc li.

1.5. lch chun

nh ngha: lch chun ca i lng ngu nhin X, k hiu () c xc

nh bi cng thc:

() = () (1.8)

1.6. Hip phng sai (Covariance)

Cho 2 bin ngu nhin X v Y, ta c nh ngha hip phng sai ca X v Y, k

hiu (, ):

(, ) = [( )( )] (1.9)

trong , ln lt l k vng ca X, Y.

Mt cng thc tng ng ca hip phng sai:

(, ) = [] (1.10)

ngha ca hip phng sai l s bin thin cng nhau ca 2 bin ngu nhin:

Nu 2 bin c xu hng thay i cng nhau (ngha l, khi mt bin c gi tr cao hn k

vng th bin kia cng c xu hng cao hn k vng), th hip phng sai ca hai bin

ny c gi tr dng. Mt khc, nu mt bin nm trn gi tr k vng cn bin kia c xu

hng nm di gi tr k vng, th hip phng sai ca hai bin c gi tr m.

Nu 2 bin ngu nhin l c lp th (, ) = 0 tuy nhin iu ngc li

khng ng. Cc bin ngu nhin m c hip phng sai bng 0 c gi l khng tng

quan (uncorrelated), chng c th c lp nhau hoc khng.

Nh vy nu X, Y c lp ta c [] = .

Tnh cht

- (, ) = ()

- (, ) = (, )

- (, ) = (, )


- (1 + 2, 1 + 2) = (1, 1) + (2, 1) + (1, 2) +

(2, 2)

- ( + ) = () + () + 2(, )

1.7. Ma trn hip phng sai

Nh chng ta va trnh by, hip phng sai l i lng tnh ton s tng quan

gia 2 bin ngu nhin.

Vy gi s chng ta c mt vector bin ngu nhin c 3 phn t 1, 2, 3. Nu ta

mun tnh ton s tng quan gia tt c cc cp bin ngu nhin th ta phi tnh tt c 3

hip phng sai (1, 2), (1, 3), (2, 3).

Mt cch tng qut, ma trn hip phng sai ra i cho php ta tnh tt c

cc gia 2 bin ngu nhin trong mt vector bin ngu nhin.

Cho mt vector bin ngu nhin X cha n bin ngu nhin, ma trn hip phng

sai ca X, k hiu l , c nh ngha l:

= [

(1, 1) (1, 2)

(2, 1) (2, 2)

(1, )

(2, )

(, 1) (, 2) (, )

]

Vi = [ 1

]

Quan st trn ng cho ca ma trn hip phng sai (i=j) ta thy ti l cc

phng sai, v ( , ) = ()

1.8. Phn phi chun (phn phi Gaussian)

Trong thc t, ngi ta thng s dng phn phi xc sut c tn l phn phi

chun (normal distribution) hay phn phi Gaussian.

Mt bin ngu nhin X c gi l c phn phi Gaussian khi n c hm mt

l hm Gaussian, k hiu l ~(, ) gi l X c phn phi chun vi tham s , .

Khi hm mt ca X l:

(; , ) =1

2

()2

22


Vi phn phi xc sut nh trn, ngi ta tnh c , ln lt l k vng v

lch chun ca X.

Di y l th ca mt s phn phi chun.

Quan st th ta thy phn phi chun c dng chung. Gi tr k vng ca X l

= l trc i xng. lch chun (hay phng sai 2) cng ln th th cng

bt, ngha l cc gi tr cng phn tn ra xa k vng.

Trong thc t, cc loi nhiu trong cc h thng o lng c th c m phng

mt cch chnh xc bng nhiu trng cng. Hay ni cch khc tp m trng Gaussian l

loi nhiu ph bin nht trong h thng o lng. Loi nhiu ny c mt ph cng

sut ng u trn min tn s v bin tun theo phn b Gaussian. Theo phng thc

tc ng th nhiu Gaussian l nhiu cng. Vy cc h thng o lng ph bin chu tc

ng ca nhiu Gaussian trng cng (AWGN).

Hnh 1.1: th ca mt s phn phi chun


1.9. c lng ca trung bnh v phng sai

Ta chn ngu nhin n c th trong mt dn s gm N c th. Ta quan tm n c

trng nh lng Y ca dn s vi trung bnh v phng sai V(Y). Trong mu , c

trng Y c trung bnh v phng sai o c ln lt l v 2 = 1

( )

2=1 .

Lu l cc gi tr v 2 thay i ty theo mu th, do chng l cc bin ngu nhin

vi trung bnh v phng sai ring khc nhau.

c lng trung bnh ca Y:

Thng thng trung bnh ca Y, tc l c c lng bi: = 1

=1 ,

cn c gi l trung bnh tch ly (hay trung bnh cng). Ta chng minh c y l

c lng khng chch (unbiased), ngha l () =

c lng phng sai ca Y:

2 l mt c lng ca V(Y), nhng l c lng khng ng, ta chng minh

c k vng ca 2 lun nh hn V(Y), tc c lng l thiu.

Cc c lng ng ca V(Y) l:

Hnh 1.2: Nhiu Gaussian


12 (1.11) trong trng hp ly mu c hon li

1

12 (1.12) trong trng hp ly mu khng hon li

Trong trng hp mu ln, php tnh c hon li v php tnh khng hon li l

nh nhau, v

1 xp x bng 1. V vy trong trng hp tng qut c lng ng ca

V(Y) l: 2 = 1

1 ( )

2=1 c gi l phng sai tch ly ca Y.

1.10. Phng php bnh phng ti thiu

Trong ton hc, phng php bnh phng ti thiu, cn gi l bnh phng nh

nht hay bnh phng trung bnh ti thiu, l mt phng php ti u ha la chn

mt ng khp nht cho mt di d liu ng vi cc tr ca tng cc sai s thng k

(error) gia ng khp v d liu.

Phng php ny gi nh cc sai s (error) ca php o c d liu phn phi

ngu nhin. nh l Gauss-Markov chng minh rng kt qu thu c t phng php

bnh phng ti thiu khng thin v v sai s ca vic o c d liu khng nht thit

phi tun theo, v d, phn b Gauss. Mt phng php m rng t phng php ny l

bnh phng ti thiu c trng s.

Phng php bnh phng ti thiu thng c dng trong khp ng cong.

Nhiu bi ton ti u ha cng c quy v vic tm cc tr ca dng bnh phng, v d

nh tm cc tiu ca nng lng hay cc i ca entropy.

Gi s d liu gm cc im (xi, yi) vi i = 1, 2, ..., n. Chng ta cn tm mt hm

s f tha mn:

() (1.13)

Gi s hm f c th thay i hnh dng, ph thuc vo mt s tham s, pj vi j = 1,

2, ..., m.

() ( , ) (1.14)

Ni dung ca phng php l tm gi tr ca cc tham s pj sao cho biu thc sau

t cc tiu:


2 = ( ())2

=1

(1.15)

Ni dung ny gii thch ti sao tn ca phng php l bnh phng ti thiu.

i khi thay v tm gi tr nh nht ca tng bnh phng, ngi ta c th tm gi

tr nh nht ca bnh phng trung bnh:

2 = 1

( ())

2

=1

(1.16)

iu ny dn n tn gi bnh phng trung bnh ti thiu.

Trong hi quy tuyn tnh, ngi ta thay biu thc

()

bng

() = + (1.17)

vi h s nhiu l bin ngu nhin c gi tr k vng bng 0.

Trong biu thc ca hi quy tuyn tnh x c o chnh xc, ch c y chu nhiu lon .

Thm na, hm f tuyn tnh vi cc tham s pj. Nu f khng tuyn tnh vi cc tham s,

ta c hi quy phi tuyn, mt bi ton phc tp hn nhiu hi quy tuyn tnh.

2. B lc Kalman

2.1. Gii thiu chung v b lc Kalman

c xut t nm 1960 bi gio s Kalman thu thp v kt hp linh ng

cc thng tin t cm bin thnh phn. Mt khi phng trnh nh hng v mu thng k

nhiu trn mi cm bin c bit v xc nh, b lc Kalman s cho c lng gi tr

ti u (chnh xc do c loi sai s, nhiu) nh l ang s dng mt tn hiu tinh

khit v c phn b khng i. Trong h thng ny, tn hiu cm bin vo b lc

gm hai tn hiu: t cm bin gc (inclinometer) v cm bin vn tc gc (gyro). Tn hiu

u ra ca b lc l tn hiu ca inclinometer v gyro c loi nhiu nh hai ngun

tn hiu h tr v x l ln nhau trong b lc, thng qua quan h (vn tc gc = o

hm/vi phn ca gi tr gc.


B lc Kalman n gin l thut ton x l d liu hi quy ti u. C nhiu cch

xc nh ti u, ph thuc tiu chun la chn trnh thng s nh gi. N cho thy rng

b lc Kalman ti u i vi chi tit c th trong bt k tiu chun c ngha no. Mt

kha cnh ca s ti u ny l b lc Kalman hp nht tt c thng tin c cung cp ti

n. N x l tt c gi tr sn c, ngoi tr sai s, c lng gi tr hin thi ca

nhng gi tr quan tm, vi cch s dng hiu bit ng hc thit b gi tr v h thng,

m t s liu thng k ca h thng nhiu, gm nhiu n, nhiu o v s khng chc

chn trong m hnh ng hc, v nhng thng tin bt k v iu kin ban u ca gi tr

quan tm.

Hnh 1.3: M hnh o lng c lng ca b lc Kalman


Hnh 1.3 trn m hnh ha hot ng ca mch lc Kalman. Chng ta c tn hiu

o c, chng ta c m hnh ca tn hiu o c (i hi tuyn tnh) v sau l p

dng vo trong h thng phng trnh ca mch lc c lng trng thi quan tm.

Thc ra tn hiu o l khng kh, phng trnh c sn, ci chung ta cn chnh l m

hnh ho h thng. c th ng dng mt cch hiu qu mch lc Kalman th chng ta

phi m hnh ha c mt cch tuyn tnh s thay i ca trng thi cn c lng hoc

d on.

Hnh 1.4: Tn hiu thu trc v sau khi lc qua Kalman


2.2. M hnh ton hc

2.2.1. H thng v m hnh quan st

Chng ta gi s rng c th m hnh ha bi phng trnh chuyn trng thi

+1 = + + (2.1)

Trong l trng thi ti thi im k, l vector iu khin u vo, l h

thng cng hay nhiu qu trnh thng l nhiu Gaussian trng cng (AWGN) , l

ma trn chuyn i u vo v l ma trn chuyn trng thi.

Ngoi ra chng ta gi s rng, kh nng quan st trng thi c thc hin thng

qua mt h thng o lng c th c biu din bi mt phng trnh tuyn tnh nh

sau

= + (2.2)

Trong l thng tin quan st hay o lng thc hin ti thi im , l

trng thi ti thi im , l ma trn quan st v l nhiu cng trong qu trnh o

lng.

2.2.2. Gi thit

Chng ta gi thit nh sau

Nhiu qu trnh v nhiu o lng v l khng tng quan, l nhiu

Gaussian trng cng (AWGN) c gi tr trung bnh bng khng v ma trn

hip phng sai bit.

Hnh 2.1: M hnh khng gian trng thi


Khi ,

[] = {

= 0 ,

(2.3)

[] = {

= 0 ,

(2.4)

[] = 0 , (2.5)

Trong v l cc ma trn i xng na xc nh dng.

Trng thi khi to h thng 0 l mt vector ngu nhin khng tng quan

vi c h thng v nhiu o lng.

Trng thi khi to h thng c gi tr trung bnh v ma trn hip phng

sai bit.

0|0 = [0] 0|0 = [(0|0 0)(0|0 0)] (2.6)

a ra nhng gi nh trn vi mc ch xc nh, a ra tp gi tr quan st

1, , +1, b lc c lng thi im + 1 to ra mt c lng ti u ca trng

thi +1 m chng ta k hiu bi +1, ti thiu ha k vng ca hm tn tht bnh

phng li.

[+1 +12

] = [(+1 +1)

(+1 +1)] (2.7)

2.2.3. Ngun gc

K hiu c lng d on ca trng thi +1 da trn quan st thi im ,

1, , l +1|. c gi l mt bc trc d on hay n gin l d on. By

gi, gii php ti thiu ha phng trnh (2.7) l k vng ca trng thi thi im

+ 1 c c nh da trn quan st thi im . Nh vy,

+1| = [+1|1, , ] = [+1|] (2.8)

Khi trng thi d on c cho bi

+1| = [+1|]

= [ + + |]

= [|] + + [|

]

= | + (2.9)

Khi s dng trong thc t, nhiu qu trnh c gi tr trung bnh l 0 v c

bit chnh xc.


Hip phng sai c lng d on +1| l trung bnh bnh phng sai s trong

c lng +1|.

V vy, bng vic s dng cc s kin m v | l khng tng quan:

+1| = (+1 +1|)

= [(+1 +1|)(+1 +1|)

|]

= (( |) + ) = ( |) + [

]

= | + (2.10)

ang c mt c lng d on +1|, gi s rng chng ta ang c mt gi tr

quan st +1. Lm sao s dng thng tin ny cp nht trng thi d on, tc l

tm +1|+1. Chng ta gi s rng c lng l tng trng s tuyn tnh ca d on v

quan st mi v c th c m t bi phng trnh,

+1|+1 = +1 +1| + +1+1 (2.11)

Trong +1 v +1 l nhng ma trn hiu chnh b hay ma trn li (ca

cc kch thc khc nhau). Vn ca chng ta by gi l tm +1 v +1 ti thiu

ha iu kin c lng trung bnh bnh phng sai s. Sai s d on c cho bi

+1|+1 = +1|+1 +1 (2.12)

2.2.4. iu kin khng chch

b lc khng chch yu cu [+1|+1] = [+1]. Gi s rng | l mt c

lng khng chch. Kt hp phng trnh (2.11) v (2.12) v tnh k vng

[+1|+1] = [+1 +1| + +1+1+1 + +1+1]

= +1 [+1|] + +1+1[+1] + +1[+1] (2.13)

V [+1] = 0, v d on l khng chch:

[+1|] = [| + ]

= [|] +

= [+1] (2.14)


Do kt hp phng trnh (2.13) v (2.14)

[+1|+1] = (+1 + +1+1)[+1]

V iu kin +1|+1 khng chch yu cu

+1 + +1+1 =

+1 = +1+1 (2.15)

c lng khng chch yu cu

+1|+1 = ( +1+1)+1| + +1+1

= +1| + +1[+1 +1+1|] (2.16)

Trong c gi l li ca b lc Kalman (Kalman gain).

+1+1| c th c hiu nh mt quan st hay o lng d on +1|

+1| = [+1|]

= [+1+1 + +1|]

= +1+1| (2.17)

t +1 l lch o lng th hin s sai khc gia gi tr o lng +1 v

c lng ca n +1|, c biu din bi

+1 = +1 +1+1| (2.18)

[+1|] = [+1 +1||

]

= [+1|] +1|

= 0 (2.19)

Kt hp (2.16) v (2.18) ta c

+1|+1 = +1| + +1+1 (2.20)


Hip phng sai lch o lng +1 c cho bi,

+1 = [+1+1 ]

= [(+1 +1+1|)(+1 +1+1|)

]

+1 = +1+1|+1 + +1 (2.21)

2.2.5. Hip phng sai sai s

Chng ta xc nh hip phng sai sai s d on ca phng trnh (2.10). By

gi chng ta tnh ton hip phng sai sai s iu chnh.

+1|+1 = [(+1|+1)(+1|+1)

|]

= [(+1 +1|+1)(+1 +1|+1)

]

= (+1 +1|+1)

= (+1 (+1| + +1(+1 +1+1|))

= (+1 ( +1| + +1(+1+1 + +1 +1+1|)))

= (( +1+1)(+1 +1|) +1+1)

= (( +1+1)(+1 +1|)) + (+1+1)

= ( +1+1)(+1 +1|)( +1+1) + +1(+1)+1

+1|+1 = ( +1+1)+1|( +1+1) + +1+1+1

(2.22)

Ta c th tnh ton +1|+1 theo cch khc nh sau

+1|+1 = (+1 +1|+1)

= (+1 +1| +1+1)

= (+1 +1|) (+1+1 )

= +1| +1+1+1 (2.23)


Trong , +1| = (+1 +1|)

+1 = (+1) = [+1+1 ]

V vy hip phng sai ca c lng iu chnh c biu din qua hip

phng sai d on +1|, nhiu o lng +1 v ma trn li Kalman +1

2.2.6. li Kalman

Mc tiu ca chng ta l lm sao ti thiu ha trung bnh bnh phng sai s

c lng c iu kin vi li Kalman .

= min+1

[(+1|+1)

(+1|+1)|]

= min+1

( [(+1|+1)

(+1|+1)|])

= min+1

(+1|+1) (2.24)

Vi bt k ma trn A v ma trn i xng B ta c

(()) = 2

Kt hp (2.23) v (2.24) v ly vi phn ma trn li v t kt qu bng 0 ta

c

+1= 2( +1+1)+1|+1

+ 2+1+1 = 0

Sp xp li v a ra phng trnh cho ma trn li

+1 = +1|+1 [+1+1|+1

+ +1]1

(2.25)

Kt hp vi (2.21) ta c

+1 = +1|+1 +1

1 (2.26)

Cng vi phng trnh 2.16, nh ngha mt c lng ti u tuyn tnh trung

bnh bnh phng sai s.


T (2.26) ta c

+1+1+1 = +1|+1

+1 (2.27)

Kt hp (2.23) v (2.27)

+1|+1 = +1|( +1 +1

)

= (+1|( +1 +1

) )

= ( +1 +1

)(+1| )

= ( +1+1 )+1| (2.28)

2.2.7. Tm tt cc phng trnh ca b lc Kalman

Trong phn ny chng ta s tm tt cc phng trnh tng qut ca gii thut lc

Kalman. Gii thut bao gm 2 qu trnh: qu trnh c lng v qu trnh iu chnh.

Qu trnh d on

B lc Kalman da vo trng thi c lng iu chnh | - l c lng ca

c lng trng thi +1| l c lng d on ca +1 cho php o +1

Trng thi d on:

+1| = | + (2.29)

Hip phng sai c lng d on:

+1| = | + (2.30)

o lng d on:

+1| = +1+1| (2.31)


Prediction

(1) +1| = | +

(2) +1| = | +

Update

(3) +1 = +1+1|+1 + +1

(4) +1 = +1|+1 +1

1

(5) +1|+1 = +1| + +1(+1 +1+1|)

(6) +1|+1 = ( +1+1 )+1|

Qu trnh iu chnh

lch o lng:

+1 = +1 +1+1| (2.32)

Hip phng sai lch:

+1 = +1+1|+1 + +1 (2.33)

li Kalman:

+1 = +1|+1 +1

1 (2.34)

Trng thi c lng iu chnh:

+1|+1 = +1| + +1+1 (2.35)

Hip phng sai c lng iu chnh:

+1|+1 = ( +1+1 )+1| (2.36)

Hnh 2.2: Tm tt qu trnh khi to ca Kalman

Initial

| and |


Cng vi cc iu kin ban u trong c lng v ma trn hip phng sai li

ca n (phng trnh 2.6) nh nha mt gii thut ri rc ha v thi gian v quy

xc nh hip phng sai c lng tuyn tnh ti thiu c gi l b lc Kalman.


II. NG DNG CA B LC KALMAN

Bi v b lc Kalman gii quyt mt s vn c bn l lc nhiu v ti u cho

cc c lng nn n c ng dng rt rng ri. Ngy nay Kalman c ng dng

nhiu trong cc t t li c kh nng thay th con ngi vn hnh xe, mt chng trnh

my tnh c ci sn b lc Kalman s c nhim v iu khin xe. Nhng chic xe ny

thm ch cn c gii thiu l an ton hn xe li bi con ngi trong mt s trng hp.

Mt ng dng khc, c th chng ta khng thch th lm, l cc tn la khng

i khng (air-to-air missile: AAM). l cc tn la dn hng vic bn t mt my

bay tiu dit my bay khc. Tn la dn hng hot ng theo nguyn l pht hin

mc tiu (thng thng bng rada hoc hng ngoi, i khi cng s dng Lazer hoc

quang hc) sau t ng dn n mc tiu nh qu trnh c lng ca Kalman [3].

Ngoi ra b lc Kalman cn c p dng nhiu vo h thng theo di mc tiu di

ng trong mng cm bin khng dy. Do nhiu o lng trn cc cm bin nn kt qu

thu c thng khng chnh xc, c sai s ln so vi thc t. B lc Kalman c p

dng lc nhiu, d on, c lng trng thi ca mc tiu nh v tr, tc v qu

o. Nh c qu trnh d on v iu chnh ca b lc Kalman gp phn quan trng

vo vic qun l trng thi cc cm bin lm gim thiu nng lng tiu th cng nh

tng cht lng theo di v ko di thi gian sng ca mng [4].

Mt s ng dng c lit k t bi vit Kalman Filter trn Wikipedia [5]:

Li t ng my bay (Autopilot)

c lng trng thi sc ca pin (Battery state of charge (SoC) estimation)

Giao din tng tc vi my tnh bng no (Braincomputer interface)

nh v chuyn ng (Dynamic positioning)

Cc ng dng trong kinh t, c bit l kinh t v m, time series, v econometrics

H thng dn ng qun tnh (Inertial guidance system)

Theo di bng radar (Radar tracker)

H thng nh v v tinh (Satellite navigation systems)

D bo thi tit (Weather forecasting)

H thng nh v (Navigation Systems)

M hnh ha 3 chiu (3D-Modelling)


Vit Nam c mt s ng dng nh:

ng dng lc Kalman trong phn tch bin dng nh cao tng do bc x nhit mt

tri.

Ci thin cht lng truyn ng khng ng b bng cu trc tch knh trc tip

s dng kalman filter quan st t thng. [6]

ng dng Kalman Filter cho d bo nhit 2m t sn phm m hnh HRM.

H thng dn ng qun tnh INS/GPS. [7]

S dng b lc Kalman kt hp vi thut ton bm nh Camshift nhm nng cao

cht lng bm trong cc h thng robot t ng tm kim v bm bt mc tiu.[8]


III. CI T TH NGHIM

1. Tao nhiu Gaussian

S dng chuyn i Box Muller chuyn t phn phi chun trong khong [0; 1]

sang phn phi Gaussian (, 2)

Gi s bin ngu nhin ~(, 2)

= + 2 log() cos(2) (3.1)

Hay

= + 2 log() sin(2) (3.2)

trong , [0; 1]

V d to sinh 1000 im tun theo phn phi Gaussian (0, 1)

Trong hnh v trn, th bn tri l hm mt xc sut v phn b ca 1000

im Gaussian. th bn phi th hin s thay i gia phng sai v k vng. T hnh

v ta thy phng sai cng ln th gi tr cng phn tn.

Hnh 3.1: To sinh 1000 im ~(0, 1)


2. Ci t b lc Kalman

2.1. M phng hot ng ca b lc Kalman

Cc thng s ci t

y chng ta s m hnh ha mt mc tiu chuyn ng c vector trng thi ti

thi im k:

= [() () () ()]

Phng trnh chuyn trng thi ca mc tiu:

+1 = +

Trong :

= [

1 0 1

0 00 0

0 0 0 0

1 0 1

]

= +1 = 1

= []

: l i lng biu din cng ca nhiu qu trnh.

0|0 = [

1 00 000

10

] 0 ; 0|0 = (4, 4)


M hnh o lng:

= +

Trong :

= [10

00

01

00

]

~(0, 1)

= [] = 2 [

1 00 1

]

= 5: l i lng biu din cng ca nhiu o lng.

Kt qu ci t

- Ngn ng s dng C#

- Input: Qu o chuyn ng ca mc tiu c v bng con tr chut c

thm nhiu Gaussian.

- Output: Qu o c c lng.

Giao din


S dng

- Dng con tr chut v mt qu o chuyn ng bt k ca mc tiu v c

biu din bi ng mu en.

- Khi nhn Draw chng trnh s to d liu o lng th hin qu trnh o

lng v tr ca mc tiu c th hin bi ng mu .

- B lc Kalman s cho kt qu l ng mu xanh chnh l d liu c c

lng ti u.

- Chng ta c th thay i cc thng s t kt qu ti u.

Kt qu

Qu o thc ca mc tiu:


Qu o o lng

Qu o sau khi qua lc Kalman


2.2. M phng hot ng ca b lc Kalman m rng

Cc thng s ci t


IV. KT LUN

V l thuyt

Nm r c b lc Kalman

C kh nng pht trin h thng c lng dng b lc Kalman

Hiu c cch m hnh ha h thng trong b lc Kalman

V ng dng

Bng nhng kin thc v b lc Kalman m phng c qu trnh c lng

chuyn ng ca mc tiu.

Hng pht trin

ng dng b lc Kalman vo h thng bt mc tiu trong mng cm bin khng

dy, gip qun l trng thi ca cc cm bin, tng cht lng theo di v ko

di thi gian sng ca mng.


TI LIU THAM KHO

[1] Gio trnh xc sut thng k, Tng nh Qu, NXB Gio dc, H Ni, 1999.

[2] Estimation II, Ian Ried-2001

[3] Tn la khng i khng, http://vi.wikipedia.org/wiki/Tn_la_khng_i_khng

[4] Vinh Tran-Quang, Phat Nguyen Huu, Takumi Miyoshi, A Collaborative Target

Tracking Algorithm Considering Energy Constraint in WSNs 15-17 Sept. 2011.

[5] Kalman filter, http://en.wikipedia.org/wiki/Kalman_filter

[6] Ths. inh Anh Tun, Nguyn Phng Quang, Ci thin cht lng truyn ng khng

ng b bng cu trc tch knh trc tip s dng kalman filter quan st t thng -

20-11-2009

[7] http://www.nchmf.gov.vn/web/vi-VN/71/155/5760/Default.aspx

[8] Ng Mnh Tin, Phan Xun Minh, H Th Kim Duyn, A Method using Kalman

Filter combining with Image Tracking Camshift Algorithm to bring higher tracking

Quality in automatically searching and tracking target Robot System VCCA-2011.

Documents

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