To master Kalman filters, you need to run, break, and modify code. Here are the best sources for MATLAB examples:
stored_x(:, k) = x_est;
Real-world applications usually involve multiple dimensions. For instance, if you are tracking an object moving along an axis, your state vector contains both position ( ) and velocity ( x=[sv]x equals the 2 by 1 column matrix; s, v end-matrix; The state transition matrix incorporates basic kinematics (
end
% --- Storage for analysis --- state_estimates = zeros(n, 1);
% Basic 1D Kalman Filter: Estimating Position from Noisy Measurements dt = 1; % Time step A = [1 dt; 0 1]; % State transition: pos_new = pos + vel*dt H = [1 0]; % Measurement: we only measure position Q = [0.1 0; 0 0.1]; % Process noise covariance R = 5; % Measurement noise covariance (noisy sensor) x = [0; 10]; % Initial state [position; velocity] P = eye(2); % Initial uncertainty for k = 1:50 % 1. Predict x = A * x; P = A * P * A' + Q; % 2. Update (assuming 'z' is your new noisy measurement) z = (10 * k) + randn*sqrt(R); % Simulated noisy measurement K = P * H' / (H * P * H' + R); x = x + K * (z - H * x); P = (eye(2) - K * H) * P; end Use code with caution. Copied to clipboard
) , which dictates whether to trust the prediction or the measurement more. To master Kalman filters, you need to run,
Pk−=APk−1AT+Qcap P sub k raised to the negative power equals cap A cap P sub k minus 1 end-sub cap A to the cap T-th power plus cap Q x̂k−x hat sub k raised to the negative power
% Update K = P_p*H'/(H*P_p*H' + R); xhat = xhat_p + K*(z - H*xhat_p); P = (eye(2) - K*H)*P_p;
%% 1. SIMULATE THE REAL WORLD dt = 0.1; % Time step (seconds) t = 0:dt:10; % Time vector (10 seconds) N = length(t); % Number of time steps Predict x = A * x; P = A * P * A' + Q; % 2
The Kalman filter operates in a loop consisting of two main phases: 1. Predict (Time Update)
Wind, friction, and unpredictable forces cause real-world tracks to deviate from theoretical equations.
Based on how you think the system moves (e.g., "The car should be here based on its last known speed"). Pk−=APk−1AT+Qcap P sub k raised to the negative