## lkf

Run one step of a linear Kalman filter for a discrete linear system described by:

 x_k = F_km1 * x_km1 + F_u_km1 * u_km1 + w_q
z_k = H_k * x_k + w_r

where km1 means k-1, w_q is a zero-mean Gaussian random variable with zero mean and covariance Q_km1. That is, w_q ~ N(0, Q_km1). Similarly, w_r ~ N(0, R_k).

It updates a state estimate and covariance from sample k-1 to sample k.

[x_k, P_k] = lkf(x_km1, P_km1, u_km1, z_k, ...
F_km1, F_u_km1, H_k, Q_km1, R_k);

## Inputs

x_km1 State estimate at sample k-1 State estimate covariance at sample k-1 Input vector at sample k-1 Measurements at sample k State transition matrix at sample k-1 Input-to-state-update matrix for sample k-1 Observation matrix at sample k Process noise at sample k Measurement noise at sample k

Outputs:

x_k Updated state estimate for sample k Updated state covariance for sample k

## Example

We can quickly create a simulation for discrete, dynamic system, generate noisy measurements of the system over time, and pass these to a linear Kalman filter.

First, define the discrete system.

rng(1);
dt    = 0.1;                                  % Time step
F_km1 = expm([0 1; -1 0]*dt);                 % State transition matrix
H_k   = [1 0];                                % Observation matrix
Q_km1 = 0.5^2 * [0.5*dt^2; dt]*[0.5*dt^2 dt]; % Process noise covariance
R_k   = 0.1;                                  % Meas. noise covariance

Now, we'll define the simulation's time step and initial conditions. Note that we define the initial estimate and set the truth as a small error from the estimate (using the covariance).

n       = 100;                     % Number of samples to simulate
x_hat_0 = [1; 0];                  % Initial estimate
P       = diag([0.5 1].^2);        % Initial estimate covariance
x_0     = x_hat_0 + mnddraw(P, 1); % Initial true state

Now we'll just write a loop for the discrete simulation.

% Storage for time histories
x     = [x_0, zeros(2, n-1)];                         % True state
x_hat = [x_hat_0, zeros(2, n-1)];                     % Estimate
z     = [H_k * x_0 + mnddraw(R_k, 1), zeros(1, n-1)]; % Measurement

% Simulate each sample over time.
for k = 2:n

% Propagate the true state.
x(:, k) = F_km1 * x(:, k-1) + mnddraw(Q_km1, 1);

% Create the real measurement at sample k.
z(:, k) = H_k * x(:, k) + mnddraw(R_k, 1);

% Run the Kalman correction.
[x_hat(:, k), P] = lkf(x_hat(:, k-1), P, [], z(:, k), ...
F_km1, [], H_k, Q_km1, R_k);

end

Plot the results.

figure(1);
clf();
t = 0:dt:(n-1)*dt;
h = plot(t, x, ...
t, z, '.', ...
t, x_hat, '--');
legend('True x1', 'True x2', 'Meas.', 'Est. x1', 'Est. x2');
xlabel('Time');