Observability

A system is observable if we can uniquely identify the inital state $x(0)$ given the knowledge of the input $u$ and the output $y$ over the time span of $[0,t_1]$ where $t_1$ is a finite time.

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Observability Matrix

For a system to be observable, the observability matrix needs $rank(\mathcal{O})=n$

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$$\mathcal{O}=\left[\begin{array}{c}C\\ CA\\ C{A}^2\\ ⋮\\ C{A}^{n-1}\end{array}\right]$$Link to original

If and only if the Observability Gramian is nonsingular for any $t>0$, then the state equation is observable

A matrix is nonsingular if it is invertible → The determinant $\ne 0$

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