Can the Kalman filter estimate a factor of the model? Product of two state variables

Question

I've written down a discrete state-space model for a simple pendulum, with the state variables angle, angular velocity and angular acceleration.

This can be easily plugged into a simple Kalman filter to estimate the state. However, if the length l of the string was constant but unknown, is there any way of using a Kalman filter to estimate l alongside of the rest of the state?

I can't figure out how to linearise the term angle / l. Would an EKF help here?

g = 9.81;
l = 3;
dt = 30e-3; % 30 ms

A = [   1  dt   0;   % angle_{k+1}   = angle_k   + dt * ang_vel_k
        0   1  dt;   % ang_vel_{k+1} = ang_vel_k + dt * ang_acc_k
      -g/l  0   0 ]; % ang_acc_{k+1} = -g/l * angle_k

B = [ 5/360*2*pi*dt; 0; 0]; 
C = [ 1 0 0 ];
D = 0;
sys = ss(A,B,C,D,dt);

Answer 1

Yes, an extended Kalman (or unscented Kalman, or other extension of the Kalman filter to deal with nonlinear systems) would help here.

Fixing your differential equation for the pendulum motion would help, too -- basically, the angular acceleration isn't a state; treating it as such will just confuse things.

For the EKF or other nonlinear Kalman, let your state vector be $\mathbf x = \begin{bmatrix}\theta, \omega, d\end{bmatrix}^T$, where $\theta$ is the angle, $\omega = \dot \theta$ is the angular rate, and $d$ is length ('l' looks too much like '1'). Note that there's no angular acceleration here, but there is that length. Ignoring noise, your equations of motion are (including the length):

$$\dot {\mathbf x} = f\left( \mathbf x \right) = \begin {bmatrix} \omega \\ -\frac{g}{l} \theta \\ 0 \end{bmatrix} $$

(Note that the above uses $\dot \omega = -\frac{g}{d} \theta$; i.e. it uses the small-angle approximation for the $\sin$ function. You've already crossed the nonlinear-systems Rubicon, so if you wanted to, you could use $\dot \omega = -\frac{g}{l} \sin \theta$, since you're going nonlinear anyway -- doing so depends on whether or not you anticipate the angular excursion to be great enough that the small-angle approximation for $\sin \theta$ holds.

For the EKF, you "simply" need to calculate the vector derivative of $\mathbf x$ to use for your continuous-time $\mathbf A$: $\mathbf A = \frac{\partial}{\partial \mathbf x} f\left( \mathbf x \right)$. Then discretize that, and use it in your EKF.