While the math behind it can look intimidating, the concept is simple: it’s an algorithm that makes an "educated guess" by combining what it thinks should happen with what it sees happening.
%========================================================================== % MATLAB KALMAN FILTER BEGINNER EXAMPLE 2: MOVING OBJECT (2D STATE) % Description: Tracking position and estimating velocity using matrix math. %========================================================================== clear; clc; close all; %% 1. Simulation Setup dt = 1; % Time step (1 second) num_steps = 50; % Total simulation ticks true_velocity = 2; % True speed (2 meters/second) sensor_noise_std = 3; % GPS noise standard deviation % Generate true positions and noisy measurements true_positions = (0:num_steps-1)' * true_velocity * dt; rng(42); measurements = true_positions + sensor_noise_std * randn(num_steps, 1); %% 2. Matrix Kalman Filter Initialization % State Vector: [Position; Velocity] X = [0; 0]; % Initial guess (Position=0, Velocity=0) % State Transition Matrix (Physics Model: Pos_new = Pos + Vel*dt) A = [1 dt; 0 1]; % Measurement Matrix (We only directly measure position, row 1) H = [1 0]; % Covariance Matrix (Initial Uncertainty) P = [10 0; 0 10]; % Process Noise Covariance (Uncertainty in physics model) Q = [0.1 0; 0 0.1]; % Measurement Noise Covariance (Sensor variance) R = sensor_noise_std^2; % Storage for plotting est_pos = zeros(num_steps, 1); est_vel = zeros(num_steps, 1); %% 3. Matrix Kalman Filter Loop for k = 1:num_steps % --- PREDICT STEP --- X_pred = A * X; P_pred = A * P * A' + Q; % --- UPDATE STEP --- % Measurement Innovation Residual Y = measurements(k) - (H * X_pred); % Innovation Covariance S = H * P_pred * H' + R; % Optimal Kalman Gain K = (P_pred * H') / S; % Update State Estimate X = X_pred + K * Y; % Update Covariance Matrix P = (eye(2) - K * H) * P_pred; % Save data est_pos(k) = X(1); est_vel(k) = X(2); end %% 4. Visualizing Results figure('Color', [1 1 1], 'Position', [100, 100, 800, 500]); % Top Subplot: Position Tracking subplot(2,1,1); plot(true_positions, 'g-', 'LineWidth', 2); hold on; plot(measurements, 'r.', 'MarkerSize', 10); plot(est_pos, 'b--', 'LineWidth', 1.5); grid on; title('Position Estimation'); ylabel('Position (meters)'); legend('True Position', 'GPS Measurements', 'Kalman Estimate', 'Location', 'NW'); % Bottom Subplot: Velocity Estimation subplot(2,1,2); plot(repmat(true_velocity, num_steps, 1), 'g-', 'LineWidth', 2); hold on; plot(est_vel, 'b-', 'LineWidth', 2); grid on; title('Velocity Estimation (Derived from Position)'); xlabel('Time Step'); ylabel('Velocity (m/s)'); legend('True Velocity', 'Kalman Hidden Estimate', 'Location', 'SE'); Use code with caution. Code Explanation:
Classic linear tracking, steady-state steady noise environments. extendedKalmanFilter(stateFcn, measureFcn) kalman filter for beginners with matlab examples download
% Kalman Filter Beginner Example: 1D Tracking clear; clc; % 1. Define the System dt = 1; % Time step (1 second) A = [1 dt; 0 1]; % State transition matrix C = [1 0]; % Measurement matrix Q = [0.01 0; 0 0.01]; % Process noise covariance (trust the model?) R = 5; % Measurement noise covariance (trust the sensor?) % 2. Initialize Variables x = [0; 1]; % Initial state (position=0, velocity=1) P = eye(2); % Initial estimation error covariance true_pos = 0; % Actual position (for simulation) true_vel = 1; % Actual velocity % 3. Simulation Loop steps = 50; history_true = []; history_meas = []; history_est = []; for i = 1:steps % --- Real World Simulation --- true_pos = true_pos + true_vel * dt; measurement = true_pos + sqrt(R) * randn; % Add noise to measurement % --- Kalman Filter Step 1: Predict --- x_pred = A * x; P_pred = A * P * A' + Q; % --- Kalman Filter Step 2: Update --- K = P_pred * C' / (C * P_pred * C' + R); % Kalman Gain x = x_pred + K * (measurement - C * x_pred); P = (eye(2) - K * C) * P_pred; % Store for plotting history_true(i) = true_pos; history_meas(i) = measurement; history_est(i) = x(1); end % 4. Plot Results figure; plot(1:steps, history_meas, 'r.', 'DisplayName', 'Noisy Measurement'); hold on; plot(1:steps, history_true, 'k-', 'LineWidth', 2, 'DisplayName', 'True Path'); plot(1:steps, history_est, 'b--', 'LineWidth', 2, 'DisplayName', 'Kalman Estimate'); xlabel('Time Step'); ylabel('Position'); legend; title('Kalman Filter: Measurement vs. Estimate'); grid on; Use code with caution. Why Use MATLAB for Kalman Filters? MATLAB is the industry standard for this for three reasons:
Kk=Pk∣k−1CT(CPk∣k−1CT+R)-1cap K sub k equals cap P sub k divides k minus 1 end-sub cap C to the cap T-th power open paren cap C cap P sub k divides k minus 1 end-sub cap C to the cap T-th power plus cap R close paren to the negative 1 power While the math behind it can look intimidating,
Your "confidence." High P means you're lost; low P means you're sure.
Let’s track a simple object moving in one dimension (like a car moving at a constant speed) with a noisy position sensor. Step 1: Initialize the Filter Simulation Setup dt = 1; % Time step
These are often used in robotics and navigation applications.