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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( Q% [* E" }1 h9 P, C
clear N=200; w(1)=0;
+ X. s$ J$ Y2 x8 `; Qw=randn(1,N)
% p0 [9 o6 ]" \5 M7 I, bx(1)=0; 8 t: p2 F7 |6 B4 X
a=1;
" `2 {$ m8 q) t: z. |for k=2:N;
1 j2 N G2 W6 {* ]* fx(k)=a*x(k-1)+w(k-1); ! T; a/ D: i; O5 K+ W& ]/ l
end ' d5 D6 D" B6 ^& V6 q7 P
V=randn(1,N);
, X/ T$ `4 ~+ F0 i& Yq1=std(V); 4 Q* Z& B4 [6 p* ]
Rvv=q1.^2;
2 z0 b7 q2 U0 Z: p3 ~0 Fq2=std(x); ' @ v, m( r/ T0 G7 l! N
Rxx=q2.^2; 4 o% A0 P! n) D+ g- X! X
q3=std(w);
+ \! U; t5 C; E, z* dRww=q3.^2; 6 d0 `7 }! {; s0 O, R- H
c=0.2;
% |0 O' ^; N; N+ jY=c*x+V;
: Z! k/ ]" S( a8 t' y3 g$ v! N; wp(1)=0;
/ ]) Z8 y2 l9 I @! [& }$ R. j9 {s(1)=0;) C+ y! c. V# }
for t=2:N; * u; A6 T7 Y! z! K
p1(t)=a.^2*p(t-1)+Rww; ' T* W' E3 o0 x8 S" k- `1 p
b(t)=c*p1(t)/(c.^2*p1(t)+Rvv); . T$ m1 a$ [9 Q6 _( |2 |$ c; b! @
s(t)=a*s(t-1)+b(t)*(Y(t)-a*c*s(t-1));
7 p5 Y+ z7 q8 `p(t)=p1(t)-c*b(t)*p1(t);
& M3 [, k7 Y$ b- S- a$ Kend 8 \( _( T8 L9 D8 z( J1 f: b
t=1:N; |; H6 D0 V8 F+ L8 H
plot(t,s,'r',t,Y,'g',t,x,'b');
4 G+ R! I3 c4 e( U& }0 Cfunction [x, V, VV, loglik] = kalman_filter(y, A, C, Q, R, init_x, init_V, varargin) ) X$ J' M" @) S- t a# C4 m% U& V
% Kalman filter. 2 g/ U" r7 ?) A; E
% [x, V, VV, loglik] = kalman_filter(y, A, C, Q, R, init_x, init_V, ...)
; z! e) a( n0 z& w; a5 U- {% ) L4 O' h9 H, H9 V
% INPUTS:
# B$ ]! m0 \" r: w& {' Q6 X# ?% y(:,t) - the observation at time t ; u7 P( l! ~& @; f F% g
% A - the system matrix
1 c& Y! N2 F% a$ L4 _* z% C - the observation matrix 9 l' |1 h! a4 N, t `& U
% Q - the system covariance * u) W: Z) y5 T/ ~
% R - the observation covariance
4 a8 U) z4 _4 s! N" I. L$ l- k% init_x - the initial state (column) vector & y F/ ^7 ]) v9 B8 f
% init_V - the initial state covariance
# ^3 w) V/ I& ^& `6 f- a; T9 @% ( I6 X# R7 f$ B# q! O
% OPTIONAL INPUTS (string/value pairs [default in brackets])
- Z( U. v3 @9 [ B0 M! H$ x8 b% 'model' - model(t)=m means use params from model m at time t [ones(1,T) ]
0 W* ^/ N& @, U z1 D- u' b% q% In this case, all the above matrices take an additional final dimension,
E$ R# C" W& }( {% i.e., A(:,:,m), C(:,:,m), Q(:,:,m), R(:,:,m).
- M) h$ Q+ d. j5 ^& X$ Z: m |2 J% However, init_x and init_V are independent of model(1).
" V/ l9 `- Q; O. S# \" g+ J% 'u' - u(:,t) the control signal at time t [ [] ] + l. |& Z$ L- P& T+ i: ~
% 'B' - B(:,:,m) the input regression matrix for model m / J! W/ p5 o" @ ~9 L4 Z. |! k
%
7 E" s& D- Q0 q) f# Y4 W2 `/ M! h% OUTPUTS (where X is the hidden state being estimated) 4 L7 z. y8 w+ l- e
% x(:,t) = E[X(:,t) | y(:,1:t)]
$ R9 J) t3 M9 y* h9 z7 K2 E3 h% V(:,:,t) = Cov[X(:,t) | y(:,1:t)] ; O7 Y1 C. X3 K0 K6 c2 V3 m
% VV(:,:,t) = Cov[X(:,t), X(:,t-1) | y(:,1:t)] t >= 2
( o1 U5 V1 ~" ?0 D8 y# ?7 q; e+ y) t+ I% loglik = sum{t=1}^T log P(y(:,t))
/ G$ z, w- Y8 G1 H; l" c. Z% H; Q+ M! A8 K! o9 e
% If an input signal is specified, we also condition on it: - g* g0 m' m4 b4 A3 v7 f+ d8 H
% e.g., x(:,t) = E[X(:,t) | y(:,1:t), u(:, 1:t)]
# a# _: {% w- @# N" R% If a model sequence is specified, we also condition on it: $ ]: y& z6 H$ g- z. a: g! q+ @7 d
% e.g., x(:,t) = E[X(:,t) | y(:,1:t), u(:, 1:t), m(1:t)]
2 ?& i v* v: u# u7 P. k[os T] = size(y); & L, j/ a* J" Q* b8 }- @# |
ss = size(A,1); % size of state space
( o* @. ^0 x6 b% set default params ) I, P" p) Y3 U2 u
model = ones(1,T);
- n9 e, l7 X/ ~' L9 `7 ]u = [];
; x. B% t7 H0 j: P5 T' L2 A7 _5 hB = []; ( W9 \- Q1 ]: F
ndx = []; % n0 G/ E7 V; I1 ?
args = varargin;
( @4 y/ N. A0 k! j, unargs = length(args); * Q3 R; @3 S- p0 U D
for i=1:2:nargs $ ^& Z( \ K J3 U; X @
switch args ! z' ]1 @% F+ z3 Z9 p
case 'model', model = args{i+1}; 8 [8 y$ O( ]' C
case 'u', u = args{i+1}; 8 c* v0 u3 r# z) |6 e1 S1 s
case 'B', B = args{i+1}; ! p) T. g8 w# U) h# D& L
case 'ndx', ndx = args{i+1};
! n5 w, y/ Q$ G' G Eotherwise, error(['unrecognized argument ' args]) ) D% V" O6 R' v3 w
end 4 s! G9 \# @# a' j* F, o
end
: g% f! D% S2 `/ j/ j0 R* qx = zeros(ss, T);
# I5 e/ }5 ]8 ~' g% F" nV = zeros(ss, ss, T); + l0 k- Q1 N! {2 I6 U" M* }, S
VV = zeros(ss, ss, T);
' d; O1 C m! D/ {. u' wloglik = 0; - { {( e) u+ U3 s! x
for t=1:T m = model(t);
z/ |1 }' O/ R3 b+ ^' _' A6 @if t==1 %prevx = init_x(:,m);
5 N- V" W" H9 [% }! _6 q6 C%prevV = init_V(:,:,m); 2 U# S) w( R- }- W
prevx = init_x;
* o$ D# j2 ]. E5 PprevV = init_V;
2 u9 N# C1 O; @* ?initial = 1; . J3 r0 P5 O7 s# m+ J% B' a6 v
else prevx = x(:,t-1);
- o( ~. k5 T }+ y7 y/ V* `, rprevV = V(:,:,t-1); $ E5 @5 N. Z5 I
initial = 0;
7 d0 C, w+ I5 X& S4 B6 [end % G2 [: N( S4 d" |3 R# T9 g
if isempty(u) 7 k- s8 ?3 e, |: K% \+ O
[x(:,t), V(:,:,t), LL, VV(:,:,t)] = ...
* y% _. F9 J9 Nkalman_update(A(:,:,m), C(:,:,m), Q(:,:,m), R(:,:,m), y(:,t), prevx, prevV, 'initial', initial); else
9 O% K; B1 M4 \! O) z8 O( N" H4 i" ] if isempty(ndx) [x(:,t), V(:,:,t), LL, VV(:,:,t)] = ... . V2 M; b X; W; h0 X
kalman_update(A(:,:,m), C(:,:,m), Q(:,:,m), R(:,:,m), y(:,t), prevx, prevV, ... 'initial', initial, 'u', u(:,t), 'B', B(:,:,m)); ) `: v9 P6 e1 h0 g8 a7 g
else
r7 _* ~7 r- m, }9 g2 ni = ndx; . u+ p3 n9 W5 x- g4 X( M
% copy over all elements; only some will get updated x(:,t) = prevx;
/ o# B% z; O! t& K3 ]3 k! gprevP = inv(prevV); 3 o- Q/ I# N I- a+ Z. {
prevPsmall = prevP(i,i); 2 `" Q0 u# o, P
prevVsmall = inv(prevPsmall); " I8 E# F G+ ~
[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));
1 G4 b" ?( }* @* a0 csmallP = inv(smallV);
- v3 E" h% _2 Z# g6 n( a% C6 n) lprevP(i,i) = smallP; 9 S$ i/ T+ s+ G/ O4 [4 L% i
V(:,:,t) = inv(prevP); V' n! g, N! T1 `4 m/ G1 ?
end 2 Q* ]7 q1 Q# F! r6 A9 t H
end
& i& v1 }7 Q) z: Y/ ~loglik = loglik + LL;
, d+ U, g) | X! {( pend |
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