how to code a kalman filter? in matlab

main.m
% Define the model
A = [1 1; 0 1]; % state transition matrix
C = [1 0]; % measurement matrix
Q = [0.01 0; 0 0.01]; % process noise covariance
R = 1; % measurement noise covariance

% Initialize variables
xhat = [0; 0]; % initial state estimate
P = Q; % initial error covariance
K = P*C'/(C*P*C' + R); % Kalman gain

% Simulate measurements
T = 100;
y = zeros(T,1);
x = zeros(T,2);
for t = 1:T
    % Simulate state
    x(t,:) = A*xhat + chol(Q)*randn(2,1);
    
    % Simulate measurement
    y(t) = C*x(t,:)' + sqrt(R)*randn;
    
    % Perform Kalman filter update
    xhat = A*xhat + K*(y(t) - C*A*xhat);
    P = (eye(2) - K*C)*P;
end

% Plot results
t = 1:T;
figure
subplot(211)
plot(t,x(:,1),'b-',t,xhat(1)*ones(T,1),'r--')
ylabel('Position')
legend('True position','Estimated position')
subplot(212)
plot(t,x(:,2),'b-',t,xhat(2)*ones(T,1),'r--')
ylabel('Velocity')
xlabel('Time')
legend('True velocity','Estimated velocity')
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