扩展卡尔曼滤波的实现

扩展卡尔曼滤波（Extended Kalman Filter ）与KF的最大的不同，是允许系统模型和测量模型非线性的存在，它的实现较为简单，参照Wikipedia，我把代码贴出来，方便学习交流。

采用一个简单的3阶非线性模型，仿真结果如下图：

源代码：

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