clear all;
close all;
clc;
%% 上面是调用卡尔曼滤波
% 定义状态维数和初始条件
n = 3;                               % 状态维数
q = 0.2;                             % 过程噪声标准差
r = 0.15;                             % 测量噪声标准差
Q = q * eye(n);                    % 过程噪声协方差矩阵
R = r^2;                             % 测量噪声协方差矩阵
fstate = @(x)[x(2); x(3); 0.5*x(1)*(x(2) + x(3))];  % 状态转移方程
hmeas = @(x)x(1);                                   % 测量方程

% 初始化状态和协方差矩阵
s = [0; 0; 1];                       % 真实初始状态
x = s + q * randn(3,1);              % 加入噪声的初始状态估计
P = eye(n);                          % 初始状态协方差矩阵

% 模拟参数
N = 200;                              % 动态步数
xV = zeros(n, N);                    % 保存状态估计值
sV = zeros(n, N);                    % 保存真实状态值
zV = zeros(1, N);                    % 保存测量值

% 开始模拟
for k = 1:N
    z = hmeas(s) + r * randn;        % 模拟测量值
    sV(:, k) = s;                    % 存储真实状态值
    zV(k) = z;                       % 存储测量值

    [x, P] = ukf(fstate, x, P, hmeas, z, Q, R); % 调用UKF
    xV(:, k) = x;                    % 存储状态估计值

    s = fstate(s) + q * randn(3, 1); % 更新真实状态
end

% 绘制结果
for k = 1:3
    subplot(3, 1, k)
    plot(1:N, sV(k, :), '-', 1:N, xV(k, :), '--')
    title(['State ', num2str(k)])
    legend('True State', 'Estimated State')
end



 function X=sigmas(x,P,c)
    %Sigma points around reference point
    %Inputs:
    %       x: reference point
    %       P: covariance
    %       c: coefficient
    %Output:
    %       X: Sigma points
    
        A = c*chol(P)';
        Y = x(:,ones(1,numel(x)));
        X = [x Y+A Y-A]; 
    end
  function [y,Y,P,Y1]=ut(f,X,Wm,Wc,n,R)
    %Unscented Transformation
    %Input:
    %        f: nonlinear map
    %        X: sigma points
    %       Wm: weights for mean
    %       Wc: weights for covraiance
    %        n: numer of outputs of f
    %        R: additive covariance
    %Output:
    %        y: transformed mean
    %        Y: transformed smapling points
    %        P: transformed covariance
    %       Y1: transformed deviations
    
        L=size(X,2);
        y=zeros(n,1);
        Y=zeros(n,L);
        for k=1:L                   
            Y(:,k)=f(X(:,k));       
            y=y+Wm(k)*Y(:,k);       
        end
        Y1=Y-y(:,ones(1,L));
        P=Y1*diag(Wc)*Y1'+R;          
    end


function [x,P]=ukf(fstate,x,P,hmeas,z,Q,R)
% UKF   Unscented Kalman Filter for nonlinear dynamic systems
% [x, P] = ukf(f,x,P,h,z,Q,R) returns state estimate, x and state covariance, P 
% for nonlinear dynamic system (for simplicity, noises are assumed as additive):
%           x_k+1 = f(x_k) + 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: function handle for f(x)
%           x: "a priori" state estimate
%           P: "a priori" estimated state covariance
%           h: fanction handle for h(x)
%           z: current measurement
%           Q: process noise covariance 
%           R: measurement noise covariance
% Output:   x: "a posteriori" state estimate
%           P: "a posteriori" state covariance
%
% Example:
%{
n=3;      %number of state
q=0.1;    %std of process 
r=0.1;    %std of measurement
Q=q^2*eye(n); % covariance of process
R=r^2;        % covariance of measurement  
f=@(x)[x(2);x(3);0.05*x(1)*(x(2)+x(3))];  % nonlinear state equations
h=@(x)x(1);                               % measurement equation
s=[0;0;1];                                % initial state
x=s+q*randn(3,1); %initial state          % initial state with noise
P = eye(n);                               % initial state covraiance
N=20;                                     % total dynamic steps
xV = zeros(n,N);          %estmate        % allocate memory
sV = zeros(n,N);          %actual
zV = zeros(1,N);
for k=1:N
  z = h(s) + r*randn;                     % measurments
  sV(:,k)= s;                             % save actual state
  zV(k)  = z;                             % save measurment
  [x, P] = ukf(f,x,P,h,z,Q,R);            % ekf 
  xV(:,k) = x;                            % save estimate
  s = f(s) + q*randn(3,1);                % update process 
end
for k=1:3                                 % plot results
  subplot(3,1,k)
  plot(1:N, sV(k,:), '-', 1:N, xV(k,:), '--')
end
%}
% Reference: Julier, SJ. and Uhlmann, J.K., Unscented Filtering and
% Nonlinear Estimation, Proceedings of the IEEE, Vol. 92, No. 3,
% pp.401-422, 2004. 
%
% By Yi Cao at Cranfield University, 04/01/2008
%
    L=numel(x);                                 %numer of states
    m=numel(z);                                 %numer of measurements
    alpha=1e-3;                                 %default, tunable
    ki=0;                                       %default, tunable
    beta=2;                                     %default, tunable
    lambda=alpha^2*(L+ki)-L;                    %scaling factor
    c=L+lambda;                                 %scaling factor
    Wm=[lambda/c 0.5/c+zeros(1,2*L)];           %weights for means
    Wc=Wm;
    Wc(1)=Wc(1)+(1-alpha^2+beta);               %weights for covariance
    c=sqrt(c);
    X=sigmas(x,P,c);                            %sigma points around x
    [x1,X1,P1,X2]=ut(fstate,X,Wm,Wc,L,Q);          %unscented transformation of process
    % X1=sigmas(x1,P1,c);                         %sigma points around x1
    % X2=X1-x1(:,ones(1,size(X1,2)));             %deviation of X1
    [z1,Z1,P2,Z2]=ut(hmeas,X1,Wm,Wc,m,R);       %unscented transformation of measurments
    P12=X2*diag(Wc)*Z2';                        %transformed cross-covariance
    K=P12*inv(P2);
    x=x1+K*(z-z1);                              %state update
    P=P1-K*P12';                                %covariance update
end