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数学中国总编辑
TA的每日心情 | 衰 2016-11-18 10:46 |
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签到天数: 206 天 [LV.7]常住居民III 超级版主
群组: 2011年第一期数学建模 群组: 第一期sas基础实训课堂 群组: 第二届数模基础实训 群组: 2012第二期MCM/ICM优秀 群组: MCM优秀论文解析专题 |
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发表于 2011-11-28 10:48
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matlab下面的kalman滤波程序
4 c) P# q( ~* Z, sclear N=200; w(1)=0;
X' v4 z5 [8 h7 rw=randn(1,N)
7 E) f! l7 y" A- }2 h" Ox(1)=0;
% b- B% X4 C4 g8 f* H/ s6 Y! Ta=1; 0 a: ?% a; {/ C$ s
for k=2:N; / V3 t2 X1 @8 \8 J
x(k)=a*x(k-1)+w(k-1);
$ v: j4 k2 `! j( {$ t) hend $ m3 A9 z9 D/ B/ L9 R3 ~8 _
V=randn(1,N);
4 C! r/ }% J( c1 @q1=std(V);
9 f; W' I. E2 X0 RRvv=q1.^2; o( w: G' ]8 \; a! R: L
q2=std(x); % J/ |2 o2 O$ T
Rxx=q2.^2;
/ x' n% D7 J0 i% q# A0 Q2 Hq3=std(w);
0 b: R$ _: h7 f# r' t0 i( rRww=q3.^2; ( ]% T3 j* I9 R: M
c=0.2;
- H0 ^( D" D. \5 K$ WY=c*x+V;
8 E5 \ L% o! b/ Zp(1)=0; 2 T) K: |% H8 e; R% g
s(1)=0;. @! {& }9 i' {# s$ P9 t8 b# O
for t=2:N;
" i4 s$ \/ A Gp1(t)=a.^2*p(t-1)+Rww; 8 c4 [' |+ s# Q1 @. C' F
b(t)=c*p1(t)/(c.^2*p1(t)+Rvv); % D: r& C. E, ?" q( x9 K& r. k
s(t)=a*s(t-1)+b(t)*(Y(t)-a*c*s(t-1)); 1 K' j8 r! g* l6 H
p(t)=p1(t)-c*b(t)*p1(t); 9 D2 } I0 r: M: s
end , ~' Y y" f: t+ v
t=1:N; ( v: m2 u6 N( ?& x8 Z& c
plot(t,s,'r',t,Y,'g',t,x,'b');
/ v' \: u( m+ y* c; l/ nfunction [x, V, VV, loglik] = kalman_filter(y, A, C, Q, R, init_x, init_V, varargin)
|6 ]0 x6 V9 {' I) q% Kalman filter.
& N: L( i; U. ~6 K7 M% [x, V, VV, loglik] = kalman_filter(y, A, C, Q, R, init_x, init_V, ...)
`5 K+ ^) K u4 {# @; z* N6 O% ( `, f2 a* g- P g
% INPUTS: 4 g. K9 k- u: T% X4 z, v
% y(:,t) - the observation at time t ( N: x* U* w* d
% A - the system matrix
. ?* p: X4 _) V( G% C - the observation matrix 3 X, l4 |, S! V4 }% g {7 M
% Q - the system covariance $ s$ {5 c4 S: k
% R - the observation covariance ; f1 V4 y0 T9 E$ s& g X
% init_x - the initial state (column) vector 0 x9 @6 K4 ?$ s3 K2 p1 b
% init_V - the initial state covariance
7 v. O* m) T U z% 8 w9 ^5 S- W5 ?; S5 N9 V! q% b6 @
% OPTIONAL INPUTS (string/value pairs [default in brackets])
$ s3 I1 F- ]# A$ x0 L3 H7 T% 'model' - model(t)=m means use params from model m at time t [ones(1,T) ]
% X3 s6 u5 \/ V/ y% In this case, all the above matrices take an additional final dimension, / t5 w- H9 j$ K: l' _, {
% i.e., A(:,:,m), C(:,:,m), Q(:,:,m), R(:,:,m).
) O: k! D6 G! n% However, init_x and init_V are independent of model(1). 1 O1 T- I2 E: b$ H$ k
% 'u' - u(:,t) the control signal at time t [ [] ]
$ ~# \6 B' ], r5 e% 'B' - B(:,:,m) the input regression matrix for model m . m+ R3 d4 ]4 n, T
% ]0 r% P8 h& B5 _5 o" ?1 U# Z
% OUTPUTS (where X is the hidden state being estimated)
+ [% X/ ^% h6 C% x(:,t) = E[X(:,t) | y(:,1:t)]
6 C. H" v. T) a( { _% V(:,:,t) = Cov[X(:,t) | y(:,1:t)]
# R9 y0 N5 O: {" Q% VV(:,:,t) = Cov[X(:,t), X(:,t-1) | y(:,1:t)] t >= 2 / z! K7 i8 a' E$ k
% loglik = sum{t=1}^T log P(y(:,t))
7 h1 {7 o8 I1 a/ m4 ^& l%
* N) Q9 L9 X: } ]! [% If an input signal is specified, we also condition on it:
5 _; ^8 N( k, v2 w% e.g., x(:,t) = E[X(:,t) | y(:,1:t), u(:, 1:t)]
2 u" V$ ^& S: v5 [/ k% If a model sequence is specified, we also condition on it:
, A8 H# D& K5 Q2 p& q' G: @% e.g., x(:,t) = E[X(:,t) | y(:,1:t), u(:, 1:t), m(1:t)]
+ P+ Z7 ~5 K" ^! L[os T] = size(y); 6 k& E4 j( R `6 g' n( k# A
ss = size(A,1); % size of state space % H+ F2 _8 o$ o7 {' w
% set default params % Q z; i3 V, k7 o, b8 L
model = ones(1,T); ; x' T2 c& v b" Q, E c- J- n
u = [];
7 b& y* ^; q {5 e# t4 G4 NB = [];
) `+ |" \. S4 n" ]' A! b; undx = [];
. Q4 h: d% L4 y/ E: Gargs = varargin; - h2 |" q m {% s
nargs = length(args);
2 L! C8 _7 w) [; h' V) c8 ^for i=1:2:nargs 1 V9 j1 E5 e# u# P7 O
switch args c' ]6 i, x/ J5 l; @8 a- m
case 'model', model = args{i+1};
" g1 p6 ]) C) c! L( d T# Ocase 'u', u = args{i+1};
0 Z9 v' i, y U) F4 z/ P4 xcase 'B', B = args{i+1}; - Z! p3 V' S# t! B
case 'ndx', ndx = args{i+1}; 2 j* l9 A2 Z6 |+ Y8 C
otherwise, error(['unrecognized argument ' args])
: V6 O) ~) X. j: W4 K1 Yend 8 |* f0 Y1 F7 M# c5 _
end - W( c4 L* X7 |: q/ q1 ]$ A
x = zeros(ss, T);
& | H Y: j6 KV = zeros(ss, ss, T); " k( }" h; e/ I0 {* O
VV = zeros(ss, ss, T); ( o" x2 w( V" f; X \- V
loglik = 0;
4 R/ c! J( |, d* T6 p" p. S |for t=1:T m = model(t); 7 o/ H4 u0 @4 k* Y
if t==1 %prevx = init_x(:,m);
1 Z. r% O( C4 m%prevV = init_V(:,:,m);
( |7 y3 @. ]) f, j' h: x" _4 iprevx = init_x;
( b G4 I- n5 c- x& |prevV = init_V;
7 F) @/ k+ y( Binitial = 1;
& F4 Y2 O k3 t# a, Jelse prevx = x(:,t-1);
, j, b( T, r1 S( V. gprevV = V(:,:,t-1); 0 p4 N5 w; T" i) s+ J, S" R
initial = 0;
. l/ o0 m" `& w( n; v+ u. mend $ |+ p4 F: k' Q3 c0 M
if isempty(u) * k! p9 ^9 ^. ]7 U: G/ q# r
[x(:,t), V(:,:,t), LL, VV(:,:,t)] = ... ! f6 o- X+ k5 Z
kalman_update(A(:,:,m), C(:,:,m), Q(:,:,m), R(:,:,m), y(:,t), prevx, prevV, 'initial', initial); else % }4 v9 j: h4 e( v6 A+ b8 J
if isempty(ndx) [x(:,t), V(:,:,t), LL, VV(:,:,t)] = ...
! T+ z; ?4 ^: `5 J" @ kalman_update(A(:,:,m), C(:,:,m), Q(:,:,m), R(:,:,m), y(:,t), prevx, prevV, ... 'initial', initial, 'u', u(:,t), 'B', B(:,:,m)); 5 m0 n5 m. J! I; O
else
; @+ V/ d. m4 V' X/ J! li = ndx; - A, s" i4 b5 s T/ o
% copy over all elements; only some will get updated x(:,t) = prevx; ! X( Q" {* b* G/ x* R4 v& v
prevP = inv(prevV); 9 U% u4 C+ {( ]- v- `- U9 ^
prevPsmall = prevP(i,i); " n# B! C$ ^' g* q
prevVsmall = inv(prevPsmall); ) J1 P. [) O/ \0 A3 M0 S, x
[x(i,t), smallV, LL, VV(i,i,t)] = ... kalman_update(A(i,i,m), C(:,i,m), Q(i,i,m), R(:,:,m), y(:,t), prevx(i), prevVsmall, ... 'initial', initial, 'u', u(:,t), 'B', B(i,:,m));
/ a1 N4 s2 w" gsmallP = inv(smallV);
: U" _3 f) U0 C" l" u0 J! b6 _- hprevP(i,i) = smallP; ! T1 b; S* j% Y. F* K# L: L, {% @
V(:,:,t) = inv(prevP);
4 F8 M# r) d) U) v7 g0 }/ kend 5 c" a( A$ v1 V) e- n- e
end / v2 l! A1 x0 r" H
loglik = loglik + LL;
/ K( A+ U, |! w7 wend |
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