Is a MIMO (multi-input-multi-output) system $G(s)$ with a pole $s=0$ stable? Or what will happen if the MIMO system has a $s=0$ pole?
2026-03-27 01:44:05.1774575845
Is a MIMO system with a pole $s=0$ stable?
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Stability of MIMO LTI system depends on what kind of definition of stability you are using. For example BIBO (bounded input bounded output) stability is not satisfied if the pole at zero is controllable, because for example a constant input will then be integrated to a ramp.
Asymptotic/exponential stability is also not satisfied, because that requires that all poles have real parts which are strictly negative.
However Lyapunov stability is satisfied iff all poles have a negative real part or have a geometric multiplicity of one when they lie on the imaginary axis. For systems in state space form this comes down to that when writing the system matrix in Jordan form the Jordan blocks associated with eigenvalues on the imaginary axis have at most size one.
So if a MIMO LTI system has one pole at zero and the rest with a strictly negative real part then that will be at least Lyapunov stable. This is because the geometric multiplicity of that pole at zero is then always equal to one.
It can also be noted that a system can be BIBO stable but not Lyapunov stable if it has a uncontrollable unstable pole.