Let's say I would like to use the discrete version of the Kalman filter in a role of a state observer of a linear dynamic system. The observed continuous time domain dynamic system can be described with following state space model:
$$ \begin{eqnarray} \dot{\mathbf{x}} &=& {\mathbf{A}}\cdot{\mathbf{x}} + {\mathbf{B}}\cdot{\mathbf{u}} \\ {\mathbf{y}} &=& {\mathbf{C}}\cdot{\mathbf{x}} \end{eqnarray} $$
where the system matrix ${\mathbf{A}}$ is $[4\times 4]$, the input matrix ${\mathbf{B}}$ is $[4\times 2]$ and the output matrix ${\mathbf{C}}$ is $[2\times 4]$.
My question is how I should set the initial values of:
- the elements of the covariance matrix of the model noise ${\mathbf{Q}}$ $[4\times 4]$
- the elements of the covariance matrix of the measurement noise ${\mathbf{R}}$ $[2\times 2]$
- the elements of the covariance matrix of the estimate error ${\mathbf{P}}$ $[4\times 4]$
It is worthwhile to say that I have no statistical information neither about the model noise nor the measurement noise. Is it ever possible to come up with some reasonable initial values in such circumstances? Thanks in advance.