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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1 c' p/ i- i. v6 a: J+ I5 s
clear N=200; w(1)=0; 8 v+ M8 E0 d2 H/ L% R
w=randn(1,N)
0 @$ [. q& x6 ]# q; A% a8 p, kx(1)=0;
0 L3 a8 d; a% V9 {0 s0 G1 b5 N) m+ Ta=1; / z1 r) @2 [# Z/ A
for k=2:N;
, S* J$ z/ R% `+ W$ k9 ], b z' Bx(k)=a*x(k-1)+w(k-1); / e( {* q( y- Y# e/ `! E% W" [
end
4 t+ C6 {; ?& tV=randn(1,N);
5 V* V% E$ L3 k$ X' e! x( Eq1=std(V); * J3 I6 d1 r/ h/ \% Z- D
Rvv=q1.^2; " K6 X" t% ~2 L9 _# \7 q
q2=std(x);
& G# u8 ]' ~5 t. `. {Rxx=q2.^2;
7 F" S @: P1 J4 \7 Mq3=std(w);
. z( P7 o2 o" L5 L. u6 zRww=q3.^2;
4 N! r) K% g. [c=0.2;
; ~( n" J' Z4 s$ a1 h6 IY=c*x+V; , G; S' W w1 R( p4 m; v0 c; m7 w' I
p(1)=0; 1 O7 P7 {6 O3 u% S
s(1)=0;' a9 V( f( x2 ~9 u6 p4 t
for t=2:N; 9 q0 }; C8 g w* { v @: `( w
p1(t)=a.^2*p(t-1)+Rww;
" P% E! d; n9 h$ `, \% ^; [b(t)=c*p1(t)/(c.^2*p1(t)+Rvv); 6 q6 e( P: a$ s$ ] I8 ]$ I
s(t)=a*s(t-1)+b(t)*(Y(t)-a*c*s(t-1));
9 c+ o5 r' v2 d6 v9 u- [: b4 |# Q" Fp(t)=p1(t)-c*b(t)*p1(t);
2 b3 o2 @6 `5 M6 o* |end
& w* G ^- G0 f7 D- ^4 Wt=1:N; 5 q! D, y% j, I; P! U
plot(t,s,'r',t,Y,'g',t,x,'b'); 2 X k; q8 A" {
function [x, V, VV, loglik] = kalman_filter(y, A, C, Q, R, init_x, init_V, varargin) ( o8 n0 o" }* q8 {
% Kalman filter. + Q0 L3 G, r9 R- T! l
% [x, V, VV, loglik] = kalman_filter(y, A, C, Q, R, init_x, init_V, ...)
$ S$ V; {, ?4 U: k) Z6 O%
% m+ q; S5 t6 E/ Q: C( w* j; w% INPUTS: & t" z6 |% L5 l3 o7 b
% y(:,t) - the observation at time t
0 y# i% d3 @% V1 Z. V% A - the system matrix
* V$ L/ X: o' h3 R6 v; V* u% C - the observation matrix
5 m7 h* r7 n9 _& T7 X% Q - the system covariance
$ z% F. H2 ^# G% R - the observation covariance
4 ]1 @; P$ m1 L4 ~% F, Y- I$ c3 `: W% init_x - the initial state (column) vector 5 N) ?, u q1 y1 o W& q1 W0 b
% init_V - the initial state covariance : l0 j: O. J( `& i C( N
%
3 X4 E1 k" L1 }. U) R0 o* V0 _' X% OPTIONAL INPUTS (string/value pairs [default in brackets]) / ?/ Y$ u& D0 E: m7 ~; W) j0 E
% 'model' - model(t)=m means use params from model m at time t [ones(1,T) ]
# p' N x7 E2 k K( ]/ J) n* s& P% In this case, all the above matrices take an additional final dimension,
; T9 Y2 Z/ g' n6 p+ L- v, K5 G% i.e., A(:,:,m), C(:,:,m), Q(:,:,m), R(:,:,m).
8 w# O( h* s) S, `( W% However, init_x and init_V are independent of model(1). 6 K' X- |3 D% n. d1 {
% 'u' - u(:,t) the control signal at time t [ [] ]
- r, Q! R. P9 ^3 X; E% 'B' - B(:,:,m) the input regression matrix for model m # J5 D! K7 z4 \
%
" e4 l I2 N: ^$ o+ s% OUTPUTS (where X is the hidden state being estimated)
) R% n4 H& }" k9 M: f% x(:,t) = E[X(:,t) | y(:,1:t)]
3 y3 \4 ~: j! @2 C% V(:,:,t) = Cov[X(:,t) | y(:,1:t)] - s8 S+ l: R: D
% VV(:,:,t) = Cov[X(:,t), X(:,t-1) | y(:,1:t)] t >= 2
( N: [1 ?/ _3 }3 @& b% loglik = sum{t=1}^T log P(y(:,t))
/ O$ q0 T1 e9 d2 i2 D, i, N0 M% 4 q0 r2 D; g2 t. N3 I4 J
% If an input signal is specified, we also condition on it:
- h. a- M1 o4 p% Y; M3 o% e.g., x(:,t) = E[X(:,t) | y(:,1:t), u(:, 1:t)]
# h2 _ z4 s/ c/ J2 [% If a model sequence is specified, we also condition on it: 5 c" N" `8 Z" }3 E
% e.g., x(:,t) = E[X(:,t) | y(:,1:t), u(:, 1:t), m(1:t)] ( R0 {# ?+ J7 ^; M
[os T] = size(y);
) r# ^0 t! x$ G [2 @7 qss = size(A,1); % size of state space
. g ]: P5 I3 T" X- N6 q) a3 p% set default params
# y- |+ d6 a# ^; u h2 U* U" \model = ones(1,T);
/ @% M* }8 R# y; }# \u = [];
f' t3 f; a/ ?7 d2 B) |1 HB = []; ! e/ l2 a- J q, r) M t
ndx = []; * x0 E* b1 \% p, v4 H7 w' u/ s
args = varargin; & O' B/ K2 S0 C4 i7 k i! ~$ T
nargs = length(args); 5 Q7 B" ?6 ^- I4 S' C) T+ Q( K
for i=1:2:nargs
7 \6 E' Q& `& e9 v4 ~* o, y4 Xswitch args 1 a1 B! N0 Y0 L3 o
case 'model', model = args{i+1};
' Z0 y* v& Y& Q/ B( Hcase 'u', u = args{i+1}; 1 z: B3 l- E7 Y+ p- J$ A
case 'B', B = args{i+1}; 4 x# s4 K' ~' G5 F
case 'ndx', ndx = args{i+1};
( M5 u9 j3 [& \otherwise, error(['unrecognized argument ' args]) , c5 Q- O% Z) _6 t
end + {$ y9 A: H6 f8 y8 x! \% Q
end 7 o$ b o; M O# b
x = zeros(ss, T);
7 l, H: |8 J% D7 x* _V = zeros(ss, ss, T);
7 @& l( E: S- ~& ?/ ^$ rVV = zeros(ss, ss, T);
n1 ~& \ ~! M8 ~loglik = 0;
) r3 I# V6 r1 ~; T5 G" e! vfor t=1:T m = model(t); 4 h7 W: m9 s3 X, N7 N
if t==1 %prevx = init_x(:,m);
* J" m9 k) R3 H1 H0 c" S%prevV = init_V(:,:,m);
$ u# U: u5 ]! T8 gprevx = init_x;
( M `( l {* x, [4 _% UprevV = init_V; " Z; ? [$ ^" t
initial = 1;
+ J. I8 I, e3 |* R8 ` nelse prevx = x(:,t-1);
. D+ f! h& V3 f- XprevV = V(:,:,t-1);
1 i3 _8 j7 L P$ \2 [" {$ M7 Sinitial = 0;
0 C/ v6 ?: Z7 S6 vend
" M" V' L; ]; Y* k' uif isempty(u) [' `9 s9 _1 @, U# o9 F. ?
[x(:,t), V(:,:,t), LL, VV(:,:,t)] = ... 6 F% A H* t |$ R/ l+ ~
kalman_update(A(:,:,m), C(:,:,m), Q(:,:,m), R(:,:,m), y(:,t), prevx, prevV, 'initial', initial); else
: [4 }- L% e; y if isempty(ndx) [x(:,t), V(:,:,t), LL, VV(:,:,t)] = ... $ p, I+ D. \0 w" @$ O) n
kalman_update(A(:,:,m), C(:,:,m), Q(:,:,m), R(:,:,m), y(:,t), prevx, prevV, ... 'initial', initial, 'u', u(:,t), 'B', B(:,:,m)); 9 B. N' D1 A' ~( z+ e$ I) R
else ( [$ T' p7 [: r6 n. C
i = ndx;
1 |$ T: X' S0 O0 e% copy over all elements; only some will get updated x(:,t) = prevx;
: O! u v& G5 ]) B$ a! A3 c- L& U6 KprevP = inv(prevV); " y' r$ x* x5 X. P6 n
prevPsmall = prevP(i,i);
% C" ~) a' p1 g" y( S( B" TprevVsmall = inv(prevPsmall); ) w& S8 Z" ]8 P& v* a0 v
[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));
: a! J8 @7 G& h, S' fsmallP = inv(smallV); 0 c3 D2 _/ ~; Y; M( j
prevP(i,i) = smallP; - O3 s, V: ] [9 q3 l2 \
V(:,:,t) = inv(prevP);
+ ?4 N Q3 O2 k/ s+ nend
1 g$ d. v; i1 Y9 W7 H W/ k; jend 4 y: k" p! t7 L9 _. y" v9 [; u- J
loglik = loglik + LL;
% R( H, c' }/ O" A* S1 Pend |
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