matlab无迹卡尔曼预测正弦波

你可以使用无迹卡尔曼滤波（Unscented Kalman Filter, UKF）来预测正弦波。以下是使用Matlab实现的示例代码：

% 步骤1：定义系统模型

% 状态转移矩阵
A = [1];

% 系统噪声方差
Q = 0.1;

% 观测矩阵
H = [1];

% 观测噪声方差
R = 0.5;

% 初始状态
x0 = 0;

% 初始状态协方差矩阵
P0 = 1;

% 步骤2：生成真实正弦波数据

% 时间步长
dt = 0.1;

% 时间向量
t = 0:dt:10;

% 真实正弦波的频率和振幅
freq = 0.5;
amplitude = 1;

% 生成真实正弦波数据
x_true = amplitude * sin(2*pi*freq*t);

% 步骤3：实施无迹卡尔曼滤波

% 初始化滤波器
x_est = x0;  % 估计的状态
P_est = P0;  % 估计的状态协方差矩阵

% 存储估计的状态和观测
x_est_history = zeros(size(t));
x_est_history(1) = x_est;

% 实施滤波
for i = 2:length(t)
    % 步骤3.1：预测状态
    
    % 生成sigma点
    sigma_points = sigmaPoints(x_est, P_est);
    
    % sigma点通过状态转移方程预测
    x_pred = predictState(sigma_points, A);
    
    % 预测状态协方差矩阵
    P_pred = predictCovariance(sigma_points, x_pred, Q);
    
    % 步骤3.2：更新状态估计
    
    % 生成sigma点
    sigma_points = sigmaPoints(x_pred, P_pred);
    
    % sigma点通过观测模型预测观测
    z_pred = predictObservation(sigma_points, H);
    
    % 更新状态估计
    [x_est, P_est] = updateState(sigma_points, x_pred, P_pred, z_pred, x_true(i), R);
    
    % 存储估计的状态
    x_est_history(i) = x_est;
end

% 绘制结果
figure;
plot(t, x_true, 'b', 'LineWidth', 2);
hold on;
plot(t, x_est_history, 'r--', 'LineWidth', 1.5);
xlabel('时间');
ylabel('状态');
legend('真实状态', '估计状态');
title('无迹卡尔曼滤波预测正弦波');

% 辅助函数

% 生成sigma点
function sigma_points = sigmaPoints(x, P)
    n = length(x);  % 状态维度
    lambda = 1;  % 参数
    kappa = 0;  % 参数
    
    % sigma点权重
    w_m = zeros(2*n+1, 1);
    w_m(1) = lambda / (n + lambda);
    w_m(2:end) = 1 / (2 * (n + lambda));
    
    w_c = w_m;
    w_c(1) = w_c(1) + (1 - lambda / (n + lambda) + kappa);
    
    % sigma点生成
    sigma_points = zeros(n, 2*n+1);
    sigma_points(:, 1) = x;
    
    sqrt_P = chol((n + lambda) * P, 'lower');
    for i = 1:n
        sigma_points(:, i+1) = x + sqrt(n + lambda) * sqrt_P(:, i);
        sigma_points(:, n+i+1) = x - sqrt(n + lambda) * sqrt_P(:, i);
    end
end

% sigma点通过状态转移方程预测
function x_pred = predictState(sigma_points, A)
    n = size(sigma_points, 1);  % 状态维度
    x_pred = A * sigma_points;
end

% 预测状态协方差矩阵
function P_pred = predictCovariance(sigma_points, x_pred, Q)
    n = size(sigma_points, 1);  % 状态维度
    num_points = size(sigma_points, 2);  % sigma点数量
    
    P_pred = zeros(n);
    for i = 1:num_points
        delta = sigma_points(:, i) - x_pred;
        P_pred = P_pred + delta * delta';
    end
    P_pred = P_pred / num_points + Q;
end

% sigma点通过观测模型预测观测
function z_pred = predictObservation(sigma_points, H)
    z_pred = H * sigma_points;
end

% 更新状态估计
function [x_est, P_est] = updateState(sigma_points, x_pred, P_pred, z_pred, z, R)
    n = size(sigma_points, 1);  % 状态维度
    num_points = size(sigma_points, 2);  % sigma点数量
    
    P_xy = zeros(n, 1);
    P_zz = zeros(1);
    for i = 1:num_points
        delta_x = sigma_points(:, i) - x_pred;
        delta_z = z_pred(:, i) - z;
        
        P_xy = P_xy + delta_x * delta_z';
        P_zz = P_zz + delta_z * delta_z';
    end
    P_xy = P_xy / num_points;
    P_zz = P_zz / num_points + R;
    
    K = P_xy / P_zz;  % 卡尔曼增益
    
    x_est = x_pred + K * (z - z_pred);
    P_est = P_pred - K * P_zz * K';
end

这段代码会生成一个包含真实状态和估计状态的图形。你可以自行调整系统模型参数和正弦波的频率和振幅来进行实验。希望对你有所帮助！