A marginal system, sometimes referred to as having neutral stability,[1] is between these two types: when displaced, it does not return to near a common steady state, nor does it go away from where it started without limit.
Marginal stability, like instability, is a feature that control theory seeks to avoid; we wish that, when perturbed by some external force, a system will return to a desired state.
In econometrics, the presence of a unit root in observed time series, rendering them marginally stable, can lead to invalid regression results regarding effects of the independent variables upon a dependent variable, unless appropriate techniques are used to convert the system to a stable system.
In contrast, if all the poles have strictly negative real parts, the system is instead asymptotically stable.
If the system is in state space representation, marginal stability can be analyzed by deriving the Jordan normal form:[2] if and only if the Jordan blocks corresponding to poles with zero real part are scalar is the system marginally stable.
A homogeneous discrete time linear time-invariant system is marginally stable if and only if the greatest magnitude of any of the poles (eigenvalues) of the transfer function is 1, and the poles with magnitude equal to 1 are all distinct.
A simple example involves a single first-order linear difference equation: Suppose a state variable x evolves according to with parameter a > 0.
This explains why for a system to be BIBO stable, the real parts of the poles have to be strictly negative (and not just non-positive).
A system with a pole at the origin is also marginally stable but in this case there will be no oscillation in the response as the imaginary part is also zero (jw = 0 means w = 0 rad/sec).
The mass will come to rest due to friction however, and the sidewards movement will remain bounded.
Since the locations of the marginal poles must be exactly on the imaginary axis or unit circle (for continuous time and discrete time systems respectively) for a system to be marginally stable, this situation is unlikely to occur in practice unless marginal stability is an inherent theoretical feature of the system.
Marginal stability is also an important concept in the context of stochastic dynamics.
Marginally stable Markov processes are those that possess null recurrent classes.