Marginal stability

In the theory of dynamical systems and control theory, a linear time-invariant system is marginally stable if it is neither asymptotically stable nor unstable. Roughly speaking, a system is stable if it always returns to and stays near a particular state (called the steady state), and is unstable if it goes farther and farther away from any state, without being bounded. A marginal system, sometimes referred to as having neutral stability, 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. This necessitates the use of appropriately designed control algorithms. 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. A homogeneous continuous linear time-invariant system is marginally stable if and only if the real part of every pole (eigenvalue) in the system's transfer-function is non-positive, one or more poles have zero real part, and all poles with zero real part are simple roots (i.e. the poles on the imaginary axis are all distinct from one another). In contrast, if all the poles have strictly negative real parts, the system is instead asymptotically stable. If the system is neither stable nor marginally stable, it is unstable. If the system is in state space representation, marginal stability can be analyzed by deriving the Jordan normal form: 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.