matlab代码实现srukf

基于无线传感器网络的无线定位技术是近年来的研究热点，其中基于UKF的定位算法在无线传感器网络中得到了广泛的应用。下面是一个基于Matlab的SRUKF（Square-Root Unscented Kalman Filter，方根无迹卡尔曼滤波）的简单实现。

% 定义系统模型和观测模型
% 状态方程 x(k+1) = f(x(k)) + w(k)
% 观测方程 y(k) = h(x(k)) + v(k)

% 状态方程
% x(k+1) = F*x(k) + B*u(k) + w(k)
F = [1 1; 0 1];
B = [0.5; 1];
% 观测方程
% y(k) = H*x(k) + v(k)
H = [1 0];

% 初始化状态向量和协方差矩阵
x0 = [0; 0];
P0 = eye(2);

% 初始化过程噪声和测量噪声的协方差矩阵
Q = 0.1*eye(2);
R = 0.1;

% 初始化UKF参数
alpha = 1;
beta = 2;
kappa = 0;

% 初始化UKF权重
lambda = alpha^2*(2+2-2);
n = 2; % 状态向量维度
N = n+1;
Wm = zeros(1,2*N);
Wc = zeros(1,2*N);
Wm(1) = lambda/(n+lambda);
Wc(1) = lambda/(n+lambda) + (1-alpha^2+beta);
for i = 2:2*N
    Wm(i) = 1/(2*(n+lambda));
    Wc(i) = 1/(2*(n+lambda));
end

% 定义UKF函数
function [x,P] = SRUKF(F, B, H, x0, P0, Q, R, alpha, beta, kappa, y, u)
    n = length(x0); % 状态向量维度
    N = n+1; % 权重向量长度
    lambda = alpha^2*(n+kappa)-n; % 权重参数
    Wm = zeros(1,2*N); % 加权平均向量
    Wc = zeros(1,2*N); % 加权协方差向量
    Wm(1) = lambda/(n+lambda);
    Wc(1) = lambda/(n+lambda) + (1-alpha^2+beta);
    for i = 2:2*N
        Wm(i) = 1/(2*(n+lambda));
        Wc(i) = 1/(2*(n+lambda));
    end

    % 生成sigma点
    S = chol(P0,'lower');
    X = zeros(n,2*N);
    X(:,1) = x0;
    for i = 1:n
        X(:,i+1) = x0 + sqrt(n+lambda)*S(:,i);
        X(:,i+1+n) = x0 - sqrt(n+lambda)*S(:,i);
    end

    % 预测
    Xp = F*X + B*u;
    xp = Wm(1)*Xp(:,1);
    Pp = Q;
    for i = 2:2*N
        xp = xp + Wm(i)*Xp(:,i);
    end
    for i = 1:2*N
        Pp = Pp + Wc(i)*(Xp(:,i)-xp)*(Xp(:,i)-xp)';
    end

    % 更新
    Y = zeros(1,2*N);
    for i = 1:2*N
        Y(i) = H*Xp(:,i);
    end
    ym = Wm(1)*Y(1);
    Pyy = R;
    Pxy = (Xp(:,1)-xp)*(Y(1)-ym)';
    for i = 2:2*N
        ym = ym + Wm(i)*Y(i);
        Pyy = Pyy + Wc(i)*(Y(i)-ym)*(Y(i)-ym)';
        Pxy = Pxy + Wc(i)*(Xp(:,i)-xp)*(Y(i)-ym)';
    end
    K = Pxy/Pyy;
    x = xp + K*(y-ym);
    P = Pp - K*Pyy*K';
end

% 测试
% 生成模拟数据
T = 100;
x_sim = zeros(2,T);
y_sim = zeros(1,T);
u_sim = ones(1,T);
for k = 2:T
    x_sim(:,k) = F*x_sim(:,k-1) + B*u_sim(k-1) + mvnrnd([0,0],Q)';
    y_sim(k) = H*x_sim(:,k) + mvnrnd(0,R);
end

% SRUKF定位
x_est = zeros(2,T);
P_est = zeros(2,2,T);
for k = 1:T
    [x_est(:,k), P_est(:,:,k)] = SRUKF(F,B,H,x0,P0,Q,R,alpha,beta,kappa,y_sim(k),u_sim(k));
end

% 结果可视化
figure;
plot(1:T, x_sim(1,:), 'b', 1:T, x_est(1,:), 'r');
legend('真实位置', '估计位置');
xlabel('时间步');
ylabel('x坐标');
title('SRUKF定位结果');