A way of reconstructing the states if you have access to a square and invertible matrix, .


If

And C is square and invertible, we can reconstruct the states:

However, if C is not square and invertible, but there are more measurements than states ( has more rows than columns), we can use:

which is a least squares estimations, minimizing: .

Proof

If we find , we get:

\frac{d}{dx}([y-Cx]^T[y-Cx])&=0 \\ -2C^Ty+2C^TCx&= 0\\ x = [C^TC]^{-1}C^Ty \end{align}

It is not very realistic to have more measurements than states.