Controllability

A system is controllable if it can be pushed into any desired state by choosing appropriate input values. This depends on how the input affects the states, $B$, and how the states affect each other, $A$.

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Controllability Matrix

The general equation for a controllability matrix given a system on State Space Representation form is: $\mathcal{C}=\left[\begin{array}{c}B|AB|A^{2}B|...|A^{n-1}B\end{array}\right]$ And the system is controllable if $rank(\mathcal{C})=n$, aka we can controll each state.

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Multi-input Controllability

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Controllability Gramian Controllability Index