擴展卡爾曼濾波的實現

擴展卡爾曼濾波（Extended Kalman Filter ）與KF的最大的不同，是允許系統模型和測量模型非線性的存在，它的實現較爲簡單，參照Wikipedia，我把代碼貼出來，方便學習交流。

採用一個簡單的3階非線性模型，仿真結果如下圖：

源代碼：

EKF_Example:

clc; clear all; close all;
%% system and observation equation, with respect of their Jcobian function.
f_model = @( x, u ) [x(2);x(3);0.05*x(1)*(x(2)+x(3))] + u;  % nonlinear state equations
h_model = @( x ) x(1);                               % measurement equation
f_gradient = @ ( x, u ) ( [ 0 1 0; 0 0 1; 0.05 * ( x(2) + x(3)) 0.05 * x(1) 0.05 * x(1) ] );
h_gradient = @ ( x ) ( [ 1 0 0 ] );
%% Specify initial condition and parameters.
StateDim = 3;      %number of state
ObvDim = 1;
q = 0.1;    %std of process 
r = 1.0;    %std of measurement
Q = q^2*eye(StateDim); % covariance of process
R = r^2;        % covariance of measurement  
x0_True = [0;0;1];                                % initial state
x0 = x0_True + q * randn(3,1); %initial state          % initial state with noise
L = 1;P0 = L * eye(StateDim);                               % initial state covraiance
MaxIter = 1000;                                     % total dynamic steps
%% EKF.
[ x_Est P_Est ] = EKF( f_model, h_model, f_gradient, h_gradient, x0, P0, Q, R, ObvDim, MaxIter );

EKF:

%% Extended Kalman Filter, and the equation is exactly same with Wikipedia(http://en.wikipedia.org/wiki/Extended_Kalman_filter)
%% this is the very basic algorithm, if you want get improved or variants EKF, please change corresponding modular in the following code.
function [ x_Est P_Est ] = EKF( f_model, h_model, f_gradient, h_gradient, x0, P0, Q, R, ObvDim, MaxIter )
% EKF   Extended Kalman Filter for nonlinear dynamic systems
% this function returns state estimate, x and state covariance, P 
% for nonlinear dynamic system:
%           x_k = f(x_km1, ukm1) + w_k
%           z_k = h(x_k) + v_k
% where w ~ N(0,Q) meaning w is gaussian noise with covariance Q
%       v ~ N(0,R) meaning v is gaussian noise with covariance R
% Inputs:   f_model: function handle for f( x, u )
%           h_model: function handle for h( x )
%           f_gradient: f'( x, u ), evaluate f_model's gradient at ( x, u )
%           h_gradient: h'( x ), evaluate h_model's gradient at x.
%           x0: "a priori" state estimate
%           P0: "a priori" estimated state covariance
%           Q: process noise covariance of w_k 
%           R: measurement noise covariance of v_k
%           ObvDim: z_k's dimension
%           MaxIter: maximum iteration number.
% Output:   x_Est: "a posteriori" state estimate matrix.
%           P_Est: "a posteriori" state covariance matrix.
%
StateDim = length( x0 );
x_True    = [ x0 ];
x_Noise   = [ x0 ];
x_Predict = [ x0 ];
x_Est     = [ x0 ];
u = zeros( StateDim, MaxIter );
z_True    = [];
z_Noise   = [];
P_Predict = [];
P_Est( :, :, 1 ) = P0;
for k = 2 : 1 : MaxIter
    %% this part is only to generate true data and noise data, in practice, you will only get noise data as input.
    % generate true data.
    x_True( :, k ) = f_model( x_True( :, k - 1 ), u( :, k - 1 ) );
    Z_True( :, k ) = h_model( x_True( :, k ) );
    % Add noise, and these are true measurements value.
    Wk = mvnrnd( zeros( StateDim, 1 ), Q );
    x_Noise( :, k ) = x_True( :, k ) + Wk';
    Vk = mvnrnd( zeros( ObvDim, 1 ), R );
    z_Noise( :, k ) = Z_True( :, k ) + Vk';
    %% the following is EKF.
    % Predict, take the last step's state to predict current state.
    x_Predict( :, k ) = f_model( x_Est( :, k - 1 ), u( :, k - 1 ) ); % predicted state estimate.
    F = f_gradient( x_Est( :, k - 1 ), u( :, k - 1 ) );
    P_Predict( :, :, k ) = F * P_Est( :, :, k - 1 ) * F' + Q;  % predict covariance estimate.
    % Update, besides predicted state calculated above, we incorporate 
    %measurements to get a more general state estimate.
    y(:, k ) = z_Noise( :, k ) - h_model( x_Predict( :, k ) ); % innovation or measurement residual.
    H = h_gradient( x_Predict( :, k - 1 ) );
    S(:, :, k ) = H * P_Predict( :, :, k ) * H' + R; % innovation covariance.
    K( :, :, k ) = P_Predict( :, :, k ) * H' * inv( S( :, :, k ) ); % Near-optimal Kalman gain.
    x_Est(:, k ) = x_Predict( :, k ) + K( :, :, k ) * y( :, k );    % Update state estimate, this is what you need.
    P_Est( :, :, k ) = ( eye(StateDim) - K( :, :, k ) * H ) * P_Predict( :, :, k ); % update covariance estimate.
end
% RMS error metric.
RMS = mean( ( x_Est - x_True ).^2, 2 )
%% Visualize.
figure(2);
plot( x_True( 1, : ), 'b*' );
hold on;plot( x_Est( 1, : ), 'g*' );
hold on; plot( z_Noise( 1, : ), 'r*' );
legend('True State', 'Optimal State', 'Measurement' );
title( ' Extended Kalman Filter ');
xlabel('Time(s)');ylabel( 'Position(m)' );
grid on;
end