Say you have discrete time state space LTI system. Suppose it is uncontrollable, and the uncontrollable modes have eigenvalues which lie on unit circle. Is it stabilizable? or do we need them to be strictly inside unit circle?
2026-03-28 04:42:47.1774672967
I am confused about definition of stabilizability for discrete LTI systems
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Stabilizability means being capable to making the system stable by state-feedback. By stable, I mean here that the eigenvalues of the matrix describing the closed-loop system all lie within the (open) unit disc.
If some modes are uncontrollable, this means that they cannot be changed and, fortiori, cannot be made stable in closed-loop in the case that they would be unstable in open-loop.
Therefore, if a system has at least one unstable mode which is not controllable, then the system is not stabilizable.