For Beginners With Matlab Examples Best | --- Kalman Filter

%% Initialize Kalman Filter % State vector: [position; velocity] x_est = [0; 10]; % Initial guess (position, velocity) P = [1 0; 0 1]; % Initial uncertainty covariance

%% Kalman Filter for 1D Position Tracking clear; clc; close all; % Simulation parameters dt = 0.1; % Time step (seconds) T = 10; % Total time (seconds) t = 0:dt:T; % Time vector N = length(t); % Number of steps

figure; subplot(2,1,1); plot(1:50, K_history, 'b-', 'LineWidth', 2); xlabel('Time Step'); ylabel('Kalman Gain (Position)'); title('Kalman Gain Convergence'); grid on;

%% Run Kalman Filter for k = 1:N % --- PREDICT STEP --- x_pred = F * x_est; P_pred = F * P * F' + Q; --- Kalman Filter For Beginners With MATLAB Examples BEST

%% Plot results figure('Position', [100 100 800 600]);

subplot(2,1,2); plot(t, true_vel, 'g-', 'LineWidth', 2); hold on; plot(t, est_vel, 'b-', 'LineWidth', 1.5); xlabel('Time (s)'); ylabel('Velocity (m/s)'); title('Velocity Estimate'); legend('True', 'Kalman Estimate'); grid on;

K_history(k) = K(1); P_history(k) = P(1,1); end %% Initialize Kalman Filter % State vector: [position;

% --- UPDATE STEP (using measurement)--- z = measurements(k); y = z - H * x_pred; % Innovation (residual) S = H * P_pred * H' + R; % Innovation covariance K = P_pred * H' / S; % Kalman Gain

% Update (using a dummy measurement) S = H * P_pred * H' + R; K = P_pred * H' / S; P = (eye(2) - K * H) * P_pred;

The filter starts with an initial guess (0 m position, 10 m/s velocity). As each noisy GPS reading arrives, the Kalman filter computes the optimal blend between the model prediction and the measurement. Notice how the position estimate (blue line) is much smoother than the noisy measurements (red dots), and the velocity converges to the true value (10 m/s). Example 2: Visualizing the Kalman Gain This example shows how the filter becomes more confident over time. Example 2: Visualizing the Kalman Gain This example

subplot(2,1,1); plot(t, true_pos, 'g-', 'LineWidth', 2); hold on; plot(t, measurements, 'r.', 'MarkerSize', 8); plot(t, est_pos, 'b-', 'LineWidth', 1.5); xlabel('Time (s)'); ylabel('Position (m)'); title('Kalman Filter: Position Tracking'); legend('True', 'Noisy Measurements', 'Kalman Estimate'); grid on;

% Measurement noise covariance R R = measurement_noise^2;

% True system: constant velocity of 10 m/s true_pos = 0:dt 10:T 10; % Starting at 0, moving at 10 m/s true_vel = 10 * ones(size(t));

x_est = [0; 0]; P = [100 0; 0 100]; % High initial uncertainty