Covariance Matrix of the Estimation Error

Resources: Covariance Matrix $\dot{P}=(A-LC)P+P(A-LC{)}^T+G{Q}_w{G}^T+L{R}_v{L}^T$

The same way ${\mathcal{C}}_x$ worked (see link above), we now produce a matrix $P$ which is defined as

$$P(t)\triangleq E[e(t)e(t{)}^T]$$

$P(t)$ is symmetric, with the elements on the diagonal being the variance of each state.

IMPORTANT: We want the trace to be as low as possible. Aka minimize this.

$$\frac{\partial tr(P)}{\partial L}$$