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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
, g. w/ [ n) ?" K7 W4 @, M0 \1 wclear N=200; w(1)=0; 0 M/ h$ ^$ \+ L
w=randn(1,N) 3 \1 Y- }% I2 A& m2 D* N
x(1)=0;
% {9 b: J U& Ca=1;
* Q/ l8 k" A& y8 ~2 n8 Jfor k=2:N; 7 I4 B* }; c, |- B
x(k)=a*x(k-1)+w(k-1);
% W( }0 Z$ O6 p, Lend
% G2 h; |( L- S& U3 ?( c. kV=randn(1,N);
# G2 h! o+ L9 c7 X y7 K" qq1=std(V);
" v3 E$ x6 w: I( [. T8 I' {+ p$ K7 oRvv=q1.^2; ( d# x; s! w# h/ ~
q2=std(x);
1 a8 o/ ]5 w9 Y# ~1 ^9 X6 A( eRxx=q2.^2; - L, z3 q1 a8 h9 e
q3=std(w);
" i8 T. n7 c2 T- c* URww=q3.^2;
/ K6 q0 j) }' a+ F8 f9 f) Gc=0.2; . W2 C; W' w2 c9 x3 g/ Y
Y=c*x+V;
1 ?' o" C1 R+ F, T( R0 B" \p(1)=0;
/ l ^! z) P, ^4 h; ws(1)=0;
& m H$ J+ ]. v w5 \% jfor t=2:N; 4 r' [9 w% F* y$ ? o" o
p1(t)=a.^2*p(t-1)+Rww;
2 p6 M% J- z2 J r* G) jb(t)=c*p1(t)/(c.^2*p1(t)+Rvv); - [2 e3 N: h& V2 F
s(t)=a*s(t-1)+b(t)*(Y(t)-a*c*s(t-1)); / a0 [, w `* |2 I! N- j3 w0 L
p(t)=p1(t)-c*b(t)*p1(t);
3 H4 u. D% g* f& B- Zend ' k, L! s$ E% p; L1 Z
t=1:N; 9 R5 X8 _4 z( M* K4 o
plot(t,s,'r',t,Y,'g',t,x,'b');
+ ?. R' l( `+ }5 O( mfunction [x, V, VV, loglik] = kalman_filter(y, A, C, Q, R, init_x, init_V, varargin) / u8 _8 H% V9 v
% Kalman filter.
. s3 I5 L. k; O+ B) F% [x, V, VV, loglik] = kalman_filter(y, A, C, Q, R, init_x, init_V, ...) ( k2 i5 s! e2 f' D
% 6 m& H& u+ y* K- C
% INPUTS: 4 I( s2 y [) F Q! J; M
% y(:,t) - the observation at time t
( t% B& Q) v% R2 S9 { ?* n% A - the system matrix 5 V* N8 T+ k. O/ X/ ]+ v9 {
% C - the observation matrix
% H& J7 e* R( V9 B7 v" B% Q - the system covariance , V( \( }* N& N4 ?! |* n/ ]6 f
% R - the observation covariance
, W" |# i* |* i" p/ h% init_x - the initial state (column) vector
, Y$ p3 L2 v! o; a8 H) W# v2 ^% init_V - the initial state covariance
; A& {1 m& L3 M%
) g: c! Q$ f8 [* X6 Z4 p% OPTIONAL INPUTS (string/value pairs [default in brackets])
9 t8 o' Y5 G) d: x2 \- _" g% 'model' - model(t)=m means use params from model m at time t [ones(1,T) ] 9 o) P x3 Z/ b
% In this case, all the above matrices take an additional final dimension,
/ h, [2 A( X6 g6 {% i.e., A(:,:,m), C(:,:,m), Q(:,:,m), R(:,:,m).
$ } j1 ~2 G$ O% M% However, init_x and init_V are independent of model(1). : D0 { L6 s- A" M; D9 W
% 'u' - u(:,t) the control signal at time t [ [] ] # p3 }7 s0 N5 M- }) D* `
% 'B' - B(:,:,m) the input regression matrix for model m
: N; ]& u+ `- ^% ( u5 B1 R; _ @: b( Z% v2 {3 h
% OUTPUTS (where X is the hidden state being estimated)
. ^& \- |8 N5 K. `% x(:,t) = E[X(:,t) | y(:,1:t)] ( ]) ~3 z5 v" v9 v- h
% V(:,:,t) = Cov[X(:,t) | y(:,1:t)] % k$ M/ |% }, d7 I+ m6 _
% VV(:,:,t) = Cov[X(:,t), X(:,t-1) | y(:,1:t)] t >= 2
1 V. w2 I" K6 D) C8 H. Z% loglik = sum{t=1}^T log P(y(:,t)) # P) Z) {( a+ k" b6 c
%
; r* r- K/ h7 }2 }2 _5 ^8 \% If an input signal is specified, we also condition on it: / K0 I$ j1 g% D4 |
% e.g., x(:,t) = E[X(:,t) | y(:,1:t), u(:, 1:t)]
3 W. K: d! M5 f: B; p3 K, L7 O$ B% If a model sequence is specified, we also condition on it:
, @8 K) H& C N- ?9 K% e.g., x(:,t) = E[X(:,t) | y(:,1:t), u(:, 1:t), m(1:t)] 6 E# N; w$ s% Y) L' {/ e7 l
[os T] = size(y); % y" }/ ~# \8 D2 Y% t2 H
ss = size(A,1); % size of state space
1 c; [6 F* X: m k# ]# X, u% P+ V% set default params & M" S% M. j7 g* n; r. ]
model = ones(1,T); 5 \# w. }# }8 G
u = [];
5 Z7 x; f. }, k/ l9 U% O! P4 s) i# rB = [];
/ l9 j& S4 A( C/ dndx = [];
& ~- e3 _ I; W' vargs = varargin;
" f2 x- |) E: a# mnargs = length(args); ! H# l; t& y4 P8 n
for i=1:2:nargs
% G- l: c9 {8 J5 C, mswitch args
4 y5 W: ^1 O: Q3 m+ m; ]case 'model', model = args{i+1};
* y. D! R* J. f5 R( Wcase 'u', u = args{i+1};
: h% H6 D2 D$ k. z3 vcase 'B', B = args{i+1}; ) O/ Q# z' z0 u7 z2 J' ~; M. g2 T" ^6 v
case 'ndx', ndx = args{i+1}; 1 k5 D5 u' H8 n( e# ~
otherwise, error(['unrecognized argument ' args])
' B/ K; z% a4 T F+ D' d- Zend 7 v" u0 ~9 r D5 B
end
# F+ H: l$ D; g( ]- c1 mx = zeros(ss, T); # A. C# a+ m& e
V = zeros(ss, ss, T); $ R8 b5 i# h) `: X9 c$ W" c' x
VV = zeros(ss, ss, T); # J9 D ?4 D6 E7 C0 h
loglik = 0; ( u- }$ Z( k7 M3 w/ @
for t=1:T m = model(t);
6 a# v w# x3 y, vif t==1 %prevx = init_x(:,m); . K! ]" g# D2 s* ^3 i5 B
%prevV = init_V(:,:,m); $ b0 `2 j# F B
prevx = init_x; ' i/ k& m/ {# D1 w5 D3 l& `' F
prevV = init_V;
9 I7 Q4 |7 t# s4 W, M6 ?+ Kinitial = 1; 4 X8 S9 y' J. g! s6 ~: b
else prevx = x(:,t-1);
. K" b, ^* m5 p1 v( pprevV = V(:,:,t-1); & x) f' p; [6 J
initial = 0;
7 ?+ x# a& g1 _2 l: Kend
; [0 w8 D) I: `# u+ g3 d1 Iif isempty(u) ) f( ^7 \/ k: O4 }& f
[x(:,t), V(:,:,t), LL, VV(:,:,t)] = ...
$ I% V( q+ L5 f5 g" `: ?& kkalman_update(A(:,:,m), C(:,:,m), Q(:,:,m), R(:,:,m), y(:,t), prevx, prevV, 'initial', initial); else W9 _6 {- i0 `* X: R
if isempty(ndx) [x(:,t), V(:,:,t), LL, VV(:,:,t)] = ...
( N2 x/ B! s) v+ o) d( Q' M kalman_update(A(:,:,m), C(:,:,m), Q(:,:,m), R(:,:,m), y(:,t), prevx, prevV, ... 'initial', initial, 'u', u(:,t), 'B', B(:,:,m));
" d5 i" W( B% U w8 c& D: A. Kelse
( c; x. T4 x8 r6 S9 Y* d# C# ~i = ndx; 4 a+ h7 Z( X1 o! W+ @. T
% copy over all elements; only some will get updated x(:,t) = prevx; + \* |' l; C; Q' y# z# f; c
prevP = inv(prevV); : }# }3 p9 a$ \" d4 }6 C
prevPsmall = prevP(i,i); * K+ p8 q5 f$ r2 b& Q- K t
prevVsmall = inv(prevPsmall); 7 [8 D( l2 I' G3 ^
[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)); 7 r3 k; H! X- H
smallP = inv(smallV);
- \, e+ `* ], p+ F: J7 `. _ QprevP(i,i) = smallP;
& o, W' \% O; WV(:,:,t) = inv(prevP);
* n8 g! o8 H1 u; T' Hend
/ n, u. o8 O2 p2 aend 8 r' Q: C8 p: @
loglik = loglik + LL; : j# a# q& m9 H4 R% b5 F
end |
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