Measurement Equation Inversion

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

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If

$$y(t)=Cx(t)$$

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

$$x(t)=C^{-1}y(t)$$

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

$$x(t)=[C^{T}C{]}^{-1}{C}^{T}y(t)$$

which is a least squares estimations, minimizing: $[y-Cx{]}^T[y-Cx]$.

Proof

If we find $\frac{d}{dx}([y-Cx{]}^T[y-Cx])=0$, 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.