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From a geometrical point of view, looking at the states of each variable of the system to be controlled, every " bad " state of these variables must be controllable and observable to ensure a good behaviour in the closed-loop system.
That is, if one of the eigenvalues of the system is not both controllable and observable, this part of the dynamics will remain untouched in the closed-loop system.
If such an eigenvalue is not stable, the dynamics of this eigenvalue will be present in the closed-loop system which therefore will be unstable.
Unobservable poles are not present in the transfer function realization of a state-space representation, which is why sometimes the latter is preferred in dynamical systems analysis.

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