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%========================================================================== % 基于 Huber 函数和最大相关熵的抗差滤波算法实现 GNSS 导航定位粗差处理 % 参考文献：https://matlabzhushou.blog.csdn.net/article/details/ %==========================================================================

% 设置非交互式图形后端 graphics_toolkit("gnuplot");

clear all; close all; clc;

%% 参数设置 n = 4; % 状态变量维度 (位置 x,y,z 和钟差) m = 6; % 观测值维度 (卫星数量) T = 100; % 仿真时间步长

% 系统噪声协方差 Q = diag([0.1, 0.1, 0.1, 0.01]);

% 观测噪声协方差 R = diag([1, 1, 1, 1, 1, 1]);

% 初始状态 x0 = [100; 50; 10; 0.5];

% 初始协方差 P0 = eye(n) * 10;

% 观测矩阵 H = [1 0 0 1;

 0 1 0 1; 0 0 1 1; 1 1 0 1; 0 1 1 1; 1 0 1 1]; 

%% 生成仿真数据 rng(0); % 固定随机种子 X_true = zeros(n, T); Y = zeros(m, T);

% 生成真实轨迹 for t = 1:T

if t == 1 X_true(:, t) = x0; else X_true(:, t) = X_true(:, t-1) + normrnd(0, sqrt(Q)); end 

end

% 生成观测值 for t = 1:T

Y(:, t) = H * X_true(:, t) + mvnrnd(zeros(m,1), R)'; % 在特定时刻注入粗差 if t == 30 || t == 60 || t == 80 Y(:, t) = Y(:, t) + [10; -15; 20; -10; 15; -20]; % 粗差 end 

end

%% 卡尔曼滤波 X_kf = zeros(n, T); P_kf = zeros(n, n, T); X_kf(:, 1) = x0; P_kf(:, :, 1) = P0;

for t = 2:T

% 预测 xbar = X_kf(:, t-1); pbar = P_kf(:, :, t-1) + Q; % 更新 ybar = H * xbar; K = pbar * H' * inv(H * pbar * H' + R); X_kf(:, t) = xbar + K * (Y(:, t) - ybar); P_kf(:, :, t) = (eye(n) - K * H) * pbar; 

end

%% 基于 Huber 函数和最大相关熵的抗差滤波 X_robust = zeros(n, T); P_robust = zeros(n, n, T); X_robust(:, 1) = x0; P_robust(:, :, 1) = P0;

% Huber 函数参数 k = 1.345; % 通常取值 1.345

% 最大相关熵参数 sigma = 1.0; % 核宽度

for t = 2:T

% 预测 xbar = X_robust(:, t-1); pbar = P_robust(:, :, t-1) + Q; % 计算残差 ybar = H * xbar; residual = Y(:, t) - ybar; % Huber 函数处理 w_huber = huber_weight(residual, k); % 最大相关熵权重 w_mcc = mcc_weight(residual, sigma); % 组合权重 W = diag(w_huber .* w_mcc); % 加权更新 S = H * pbar * H' + R; S_w = H * pbar * H' + W * R * W; K = pbar * H' * inv(S_w); X_robust(:, t) = xbar + K * W * residual; P_robust(:, :, t) = (eye(n) - K * H) * pbar; 

end

%% 结果可视化并保存图片 figure(‘Position’, [100, 100, 1200, 800]);

% 位置估计对比 subplot(2, 2, 1); plot(1:T, X_true(1, :), ‘k-’, ‘LineWidth’, 2); hold on; plot(1:T, X_kf(1, :), ‘b–’, ‘LineWidth’, 1.5); plot(1:T, X_robust(1, :), ‘r-.’, ‘LineWidth’, 1.5); xlabel(‘Time Step’); ylabel(‘X Position’); legend(‘True’, ‘Kalman Filter’, ‘Robust Filter’); title(‘X Position Estimation’); grid on; print(‘-dpng’, ‘-r150’, ‘position_estimation.png’);

figure(); set(gcf, ‘PaperPositionMode’, ‘auto’); subplot(2, 2, 1); plot(1:T, X_true(1, :), ‘k-’, ‘LineWidth’, 2); hold on; plot(1:T, X_kf(1, :), ‘b–’, ‘LineWidth’, 1.5); plot(1:T, X_robust(1, :), ‘r-.’, ‘LineWidth’, 1.5); xlabel(‘Time Step’); ylabel(‘X Position’); legend(‘True’, ‘Kalman Filter’, ‘Robust Filter’); title(‘X Position Estimation’); grid on;

subplot(2, 2, 2); plot(1:T, X_true(2, :), ‘k-’, ‘LineWidth’, 2); hold on; plot(1:T, X_kf(2, :), ‘b–’, ‘LineWidth’, 1.5); plot(1:T, X_robust(2, :), ‘r-.’, ‘LineWidth’, 1.5); xlabel(‘Time Step’); ylabel(‘Y Position’); legend(‘True’, ‘Kalman Filter’, ‘Robust Filter’); title(‘Y Position Estimation’); grid on;

subplot(2, 2, 3); plot(1:T, X_true(3, :), ‘k-’, ‘LineWidth’, 2); hold on; plot(1:T, X_kf(3, :), ‘b–’, ‘LineWidth’, 1.5); plot(1:T, X_robust(3, :), ‘r-.’, ‘LineWidth’, 1.5); xlabel(‘Time Step’); ylabel(‘Z Position’); legend(‘True’, ‘Kalman Filter’, ‘Robust Filter’); title(‘Z Position Estimation’); grid on;

subplot(2, 2, 4); plot(1:T, X_true(4, :), ‘k-’, ‘LineWidth’, 2); hold on; plot(1:T, X_kf(4, :), ‘b–’, ‘LineWidth’, 1.5); plot(1:T, X_robust(4, :), ‘r-.’, ‘LineWidth’, 1.5); xlabel(‘Time Step’); ylabel(‘Clock Bias’); legend(‘True’, ‘Kalman Filter’, ‘Robust Filter’); title(‘Clock Bias Estimation’); grid on;

% 保存位置估计图 print(‘-dpng’, ‘position_estimation.png’);

% 误差对比图 figure(‘Position’, [100, 100, 1200, 400]);

error_kf = sqrt(sum((X_kf - X_true).^2, 1)); error_robust = sqrt(sum((X_robust - X_true).^2, 1));

subplot(1, 2, 1); plot(1:T, error_kf, ‘b-’, ‘LineWidth’, 1.5); hold on; plot(1:T, error_robust, ‘r-’, ‘LineWidth’, 1.5); xlabel(‘Time Step’); ylabel(‘RMSE’); legend(‘Kalman Filter’, ‘Robust Filter’); title(‘Root Mean Square Error Comparison’); grid on;

% 粗差时刻标记 xline(30, ‘k:’, ‘Outlier 1’); xline(60, ‘k:’, ‘Outlier 2’); xline(80, ‘k:’, ‘Outlier 3’);

subplot(1, 2, 2); bar([mean(error_kf), mean(error_robust)]); set(gca, ‘XTickLabel’, {‘Kalman Filter’, ‘Robust Filter’}); ylabel(‘Mean RMSE’); title(‘Average RMSE Comparison’); grid on;

% 保存误差对比图 print(‘-dpng’, ‘-r150’, ‘error_comparison.png’);

% 打印统计结果 fprintf(‘=== 算法性能对比 === ‘); fprintf(‘卡尔曼滤波平均 RMSE: %.4f ‘, mean(error_kf)); fprintf(‘抗差滤波平均 RMSE: %.4f ‘, mean(error_robust)); fprintf(‘性能提升：%.2f%% ‘, (mean(error_kf) - mean(error_robust)) / mean(error_kf) * 100);

%% Huber 权重函数 function w = huber_weight(r, k)

abs_r = abs(r); w = ones(size(r)); idx = abs_r > k; w(idx) = k ./ abs_r(idx); 

end

%% 最大相关熵权重函数 function w = mcc_weight(r, sigma)

w = exp(-r.^2 / (2 * sigma^2)); 

end